Abstract group $C_2^6.C_2\wr C_4$

• Label: 4096.bqf
• Order: \( 2^{12} \)
• Exponent: \( 2^{3} \)
• Nilpotent: yes
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\mathrm{Aut}(G)}$: \( 2^{16} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $16$
• Trans deg.: $16$
• Rank: $3$

Group information

Description:	$C_2^6.C_2\wr C_4$
Order:	\(4096\)\(\medspace = 2^{12} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_5^2:C_{10}^2:C_8$, of order \(65536\)\(\medspace = 2^{16} \) (generators)
Outer automorphisms:	$C_2^2\times D_4$, of order \(32\)\(\medspace = 2^{5} \)
Composition factors:	$C_2$ x 12
Nilpotency class:	$8$
Derived length:	$3$

This group is nonabelian and a $p$-group (hence nilpotent, solvable, supersolvable, monomial, elementary, and hyperelementary).

Group statistics

Order	1	2	4	8
Elements	1	335	1712	2048	4096
Conjugacy classes	1	19	27	14	61
Divisions	1	19	25	10	55
Autjugacy classes	1	14	17	4	36

Dimension	1	2	4	8	16	32
Irr. complex chars.	16	4	14	16	11	0	61
Irr. rational chars.	8	8	12	17	9	1	55

Minimal Presentations

Permutation degree:	$16$
Transitive degree:	$16$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	8	8	8
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid b^{4}=c^{2}=d^{2}=e^{4}=f^{2}=g^{8}=[a,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,13,2,14)(3,16,7,11,4,15,8,12)(5,9,6,10), (1,9,6,13)(2,10,5,14)(3,12)(4,11)(7,16,8,15), (1,9,8,12,6,14,3,16)(2,10,7,11,5,13,4,15)\rangle$
Transitive group:	16T1622	16T1628			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2^6.C_2^4)$ $\,\rtimes\,$ $C_4$	$((C_2\times C_4^3).D_8)$ $\,\rtimes\,$ $C_2$	$(C_2^5.C_2^2{\rm wrC}_2)$ $\,\rtimes\,$ $C_4$	$((C_2\times C_4^3).Q_8)$ $\,\rtimes\,$ $C_2^2$	all 8
Trans. wreath product:	not computed
Non-split product:	$C_2^6$ . $(C_2\wr C_4)$	$(C_2^6.C_2^3)$ . $D_4$	$(C_2^5.C_2^4)$ . $D_4$	$C_2^5$ . $(C_2^4.D_4)$	all 36

Elements of the group are displayed as permutations of degree 16.

Homology

Abelianization:	$C_{2}^{2} \times C_{4} $
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 155563 subgroups in 9277 conjugacy classes, 57 normal (43 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $(C_{10}\times D_5^2):C_4$
Commutator:	$G' \simeq$ $C_2^5.C_2^3$	$G/G' \simeq$ $C_2^2\times C_4$
Frattini:	$\Phi \simeq$ $C_2^4.C_2^4.C_2$	$G/\Phi \simeq$ $C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^6.C_2\wr C_4$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $C_2^6.C_2\wr C_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $(C_{10}\times D_5^2):C_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^6.C_2\wr C_4$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Classes of subgroups up to conjugation

Order 4096: $C_2^6.C_2\wr C_4$

Order 2048: $(C_2\times C_4^3).D_8$ x 2, $(C_2\times C_4^3).\SD_{16}$ x 2, $C_2^5.C_2^2.C_2^3.C_2$, $C_4^3:C_2^2.C_2^3$, $C_2^5.C_2^4.C_2^2$

Order 1024: $C_2^4.C_2^3:C_8$ x 2, $C_2^5.C_2^2{\rm wrC}_2$ x 2, $C_2^5.C_2^2{\rm wrC}_2$ x 2, $C_2^5.C_2^4.C_2$, $C_2^5.C_2^4.C_2$, $C_2^4.C_2^3:C_8$, $(C_2\times C_4^3).Q_8$, $(C_2^2\times C_4^2).C_4^2$, $C_2^4.C_2^5.C_2$, $C_2^6:\OD_{16}$, $(C_2\times C_4^3).D_4$, $C_2.C_2^6.C_2^3$, $C_2^5.C_2^3.C_2^2$, $C_2^6.C_2^4$, $C_2^5.C_2^2{\rm wrC}_2$, $C_2^5.C_2^2{\rm wrC}_2$

Order 512: $C_2^5.C_2^3.C_2$ x 4, $C_2^4.C_2^4.C_2$ x 3, $C_2^5.C_4^2$ x 3, $C_2^5.C_2^4$ x 3, $C_2^5.C_2^3.C_2$ x 2, $C_2^4.C_2^4.C_2$ x 2, $C_2^4.C_2^4.C_2$ x 2, $C_2^5.C_2^3.C_2$ x 2, $C_2^3.D_4^2$ x 2, $C_2^3.D_4^2$ x 2, $C_2^3.D_4^2$ x 2, $C_2^3.D_4^2$ x 2, $C_2^4.C_2^2:C_8$ x 2, $C_2^5.C_2^4$ x 2, $C_2^6.C_2^3$, $C_2^5.C_2^4$, $C_2^6.C_2^3$, $C_2^2.C_2^5.C_2^2$, $C_2^6.C_2^3$, $C_2^3.C_2^4.C_2^2$, $(C_2^2\times D_4).C_2^4$, $C_2^6.C_2^3$, $C_2^5.C_2^3.C_2$, $C_2^4.C_2^4.C_2$, $C_2^4.C_2^4.C_2$, $C_2^3.C_2^5.C_2$, $C_2^3.C_2^4.C_2^2$, $C_2^3.C_2^4.C_2^2$, $(C_2^4\times C_4).C_2^3$, $(C_2^3\times D_4).C_2^3$, $C_2.C_2^6.C_2^2$, $C_2^4.C_2^4.C_2$, $C_2^5.C_4.C_2^2$, $C_2^5.C_2^4$, $C_2^6.C_2^3$, $C_2^6.C_2^3$

Order 256: $C_2^5:D_4$ x 12, $C_2^5.D_4$ x 10, $C_2^5.D_4$ x 8, $C_4.D_4^2$ x 8, $C_2^5.C_2^3$ x 5, $C_2^5.D_4$ x 4, $C_4^2:C_2^4$ x 4, $(C_2\times C_4^2).D_4$ x 4, $C_2^5.D_4$ x 4, $C_2^5:D_4$ x 4, $C_2^3.C_2^5$ x 4, $C_2^5.D_4$ x 4, $C_2^5.D_4$ x 4, $C_2^5.D_4$ x 4, $C_2^5.D_4$ x 4, $C_2^6:C_4$ x 4, $C_2^2.D_4^2$ x 3, $C_2^6:C_4$ x 3, $C_2^5:D_4$ x 3, $C_2^2.D_4^2$ x 2, $C_2^5.D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^5:D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^5.D_4$ x 2, $(C_2\times C_4^2).D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^3.(C_4\times D_4)$ x 2, $(C_2\times C_4^2).D_4$ x 2, $C_2^5.D_4$ x 2, $(C_2\times \OD_{16}):D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^5.D_4$ x 2, $(C_2\times \OD_{16}):D_4$ x 2, $(C_2\times C_4^3):C_2$ x 2, $C_2^5:C_8$ x 2, $C_2^5.D_4$ x 2, $C_2^3.C_2^5$ x 2, $C_2^4.C_4^2$ x 2, $C_2^2:D_4^2$ x 2, $C_2^3.C_2^5$ x 2, $C_2^3.C_2^5$ x 2, $C_2^3.C_2^5$ x 2, $C_2^5:C_2^3$ x 2, $C_2^5:D_4$ x 2, $C_2^2.D_4^2$ x 2, $C_4.D_4^2$ x 2, $C_4.D_4^2$ x 2, $C_2^5.D_4$ x 2, $C_2^5.D_4$ x 2, $(C_2\times \OD_{16}):D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^5.C_2^3$ x 2, $C_2^6:C_4$ x 2, $C_2^5.D_4$ x 2, $(C_2\times C_4^2).D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^6.C_4$ x 2, $C_4^2:\OD_{16}$, $C_2^2.D_4^2$, $C_2^4.C_2^4$, $C_2^5.D_4$, $D_4:C_2^5$, $C_4^2:C_2^4$, $C_2^2\times D_4^2$, $C_2^4.C_4^2$, $\OD_{16}:C_2^4$, $C_2^4:\OD_{16}$, $C_2^4.C_4^2$, $C_2^5.D_4$, $C_2^4.\OD_{16}$, $C_4^3:C_2^2$, $C_2^3.C_2^5$, $C_2^2.D_4^2$, $C_2^3.C_2^5$, $C_2^3.C_2^5$, $C_2^3.C_2^5$, $C_2^6:C_4$, $C_4.D_4^2$, $C_4.D_4^2$, $C_4.D_4^2$, $C_4.D_4^2$, $(C_4\times \OD_{16}):C_2^2$, $C_2\times \OD_{16}:D_4$, $(C_2\times D_4):\OD_{16}$, $C_2^5.D_4$, $C_2^4.C_4^2$, $C_4^2.C_2^4$

Order 128: $C_2^4:D_4$ x 44, $C_4^2:C_2^3$ x 34, $C_2^4:D_4$ x 32, $C_2^5:C_4$ x 30, $D_4:C_2^4$ x 23, $C_2^4.D_4$ x 20, $C_2.D_4^2$ x 17, $C_2^5:C_4$ x 16, $C_2^4.D_4$ x 16, $C_2^4:D_4$ x 16, $C_2^4:D_4$ x 14, $\OD_{16}:C_2^3$ x 14, $C_4^2:D_4$ x 12, $\OD_{16}:D_4$ x 12, $C_2^4.D_4$ x 12, $C_2^3.C_2^4$ x 11, $C_2^4:D_4$ x 11, $D_4^2:C_2$ x 10, $C_2\times D_4^2$ x 9, $\OD_{16}:D_4$ x 8, $\OD_{16}:D_4$ x 8, $C_2^4.D_4$ x 8, $C_2^5.C_4$ x 8, $C_2^4.D_4$ x 8, $C_4^2:C_2^3$ x 8, $C_2^5:C_4$ x 8, $C_2^3.C_4^2$ x 8, $C_2^3.C_2^4$ x 8, $C_2^3.C_2^4$ x 8, $C_2.D_4^2$ x 8, $C_2^4.D_4$ x 8, $C_4^2:C_2^3$ x 7, $C_2^4.D_4$ x 6, $C_2^3.C_2^4$ x 6, $C_2.D_4^2$ x 6, $D_4\times C_2^4$ x 5, $C_2^4.D_4$ x 5, $C_2^4.D_4$ x 5, $C_2^4.D_4$ x 4, $C_2^4.D_4$ x 4, $C_2^3.D_8$ x 4, $(C_2\times C_8):D_4$ x 4, $(C_2\times C_4):D_8$ x 4, $C_2^3:\SD_{16}$ x 4, $C_2^3:D_8$ x 4, $C_2^4.D_4$ x 4, $C_2^3.C_4^2$ x 4, $C_8:C_2^4$ x 4, $C_4^2:C_2^3$ x 4, $C_2^4:D_4$ x 4, $C_2^4.D_4$ x 4, $C_2^4.D_4$ x 4, $C_2^4.D_4$ x 4, $C_2^4:D_4$ x 4, $C_4^2:D_4$ x 4, $C_2^3.C_2^4$ x 4, $C_2^4.D_4$ x 4, $C_2^3.C_2^4$ x 4, $C_2^5:C_4$ x 4, $C_2^4:C_8$ x 3, $C_2^3:\OD_{16}$ x 3, $\OD_{16}:D_4$ x 2, $C_4.(C_4\times D_4)$ x 2, $C_4.(C_4\times D_4)$ x 2, $C_4.(C_4\times D_4)$ x 2, $(C_2\times C_4).D_8$ x 2, $\OD_{16}:D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^3:\OD_{16}$ x 2, $C_2^3.\OD_{16}$ x 2, $C_2^4.D_4$ x 2, $C_2^3.C_4^2$ x 2, $C_2^4.D_4$ x 2, $C_2^3.C_4^2$ x 2, $C_4^2.D_4$ x 2, $D_4:C_2^4$ x 2, $C_4^2:C_2^3$ x 2, $C_2^4:D_4$ x 2, $C_4^2:C_2^3$ x 2, $C_2^4.D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^3:D_8$ x 2, $C_2^3:\SD_{16}$ x 2, $C_2^3:\SD_{16}$ x 2, $C_2^3:D_8$ x 2, $C_2^3:C_4^2$ x 2, $C_2^4.D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^4:D_4$ x 2, $C_2^3\wr C_2$ x 2, $C_2^3.C_2^4$ x 2, $C_2^3.C_2^4$ x 2, $C_2^3.C_2^4$ x 2, $C_2^3.C_2^4$ x 2, $C_2^3.C_2^4$ x 2, $C_2^3.C_2^4$ x 2, $C_2^3.C_2^4$ x 2, $C_2^3:C_4^2$ x 2, $C_2\times C_4^3$, $C_4^2:C_8$, $C_4^2.D_4$, $C_2^3\times \OD_{16}$, $C_4^2:C_2^3$, $C_4^2:C_2^3$, $C_2^3:\OD_{16}$, $C_2^4.C_2^3$, $C_4^2.C_2^3$, $C_4^3:C_2$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$

Order 64: $D_4:C_2^3$ x 252, $C_2^3:D_4$ x 135, $D_4\times C_2^3$ x 124, $C_2^3.D_4$ x 82, $C_2^4:C_4$ x 70, $C_2^3:D_4$ x 70, $C_2\wr C_2^2$ x 64, $D_4:D_4$ x 64, $D_4:C_2^3$ x 54, $\OD_{16}:C_2^2$ x 50, $D_4^2$ x 48, $C_2^3:D_4$ x 45, $C_4^2:C_2^2$ x 44, $C_4^2:C_2^2$ x 40, $C_2^4:C_4$ x 35, $C_8:C_2^3$ x 34, $D_4:D_4$ x 32, $C_2^4:C_4$ x 31, $C_2^3.C_2^3$ x 24, $C_2\wr C_4$ x 20, $C_2^3.D_4$ x 18, $C_2^3:D_4$ x 16, $D_4:D_4$ x 16, $C_4^2:C_2^2$ x 15, $C_2^3.D_4$ x 14, $C_2^2\times \OD_{16}$ x 13, $C_2^2.D_8$ x 12, $C_2^2:C_4^2$ x 12, $C_4^2:C_2^2$ x 12, $C_4^2:C_2^2$ x 12, $C_4^2:C_4$ x 12, $C_2^3.D_4$ x 10, $C_2^2:\OD_{16}$ x 10, $C_2^3.D_4$ x 10, $C_2^3.D_4$ x 8, $C_2^2.C_4^2$ x 8, $C_4^2:C_4$ x 8, $C_8:D_4$ x 8, $C_8:D_4$ x 8, $C_8:D_4$ x 8, $C_8:D_4$ x 8, $Q_8:D_4$ x 8, $D_4:D_4$ x 8, $C_2^3:C_8$ x 6, $C_2^3:Q_8$ x 6, $C_2^3.C_2^3$ x 6, $C_2^3.D_4$ x 6, $C_4^2:C_2^2$ x 6, $\OD_{16}:C_4$ x 6, $C_2^3.D_4$ x 5, $C_2^6$ x 5, $C_2^4\times C_4$ x 5, $C_2^2.Q_{16}$ x 4, $C_2^2\times \SD_{16}$ x 4, $C_2^2\times D_8$ x 4, $C_4.C_4^2$ x 4, $C_2^3:Q_8$ x 4, $C_2^2\times C_4^2$ x 4, $C_2^2.C_4^2$ x 4, $C_4.D_8$ x 4, $C_2^3:C_8$ x 3, $C_2^3.C_2^3$ x 2, $C_4:C_4^2$ x 2, $\OD_{16}:C_4$ x 2, $C_2^2.C_4^2$ x 2, $C_4^2:C_2^2$ x 2, $C_4^2:C_2^2$ x 2, $C_2^3.D_4$ x 2, $C_4^2.C_4$ x 2, $C_4^2.C_4$ x 2, $C_2^2.D_8$ x 2, $C_2^3.D_4$ x 2, $C_4:\OD_{16}$ x 2, $C_4\times \OD_{16}$, $C_4^3$, $Q_8\times C_2^3$, $C_2^3\times C_8$

Order 32: $C_2^2\times D_4$ x 839, $D_4:C_2^2$ x 544, $D_4:C_2^2$ x 364, $C_2^2\wr C_2$ x 222, $C_2^3:C_4$ x 195, $C_4:D_4$ x 100, $C_2^3:C_4$ x 74, $C_2^3\times C_4$ x 67, $C_4:D_4$ x 64, $C_2^5$ x 62, $D_8:C_2$ x 48, $C_2\times \OD_{16}$ x 44, $C_4\times D_4$ x 40, $\OD_{16}:C_2$ x 36, $C_4\wr C_2$ x 32, $C_2.C_4^2$ x 29, $D_4:C_4$ x 24, $C_2^2.D_4$ x 24, $C_2\times C_4^2$ x 22, $C_2^2.D_4$ x 19, $C_2\times \SD_{16}$ x 18, $C_2\times D_8$ x 18, $C_2^2:C_8$ x 15, $C_2^2:Q_8$ x 12, $Q_8:C_4$ x 12, $C_2^2\times Q_8$ x 11, $C_2^2\times C_8$ x 9, $C_4^2:C_2$ x 8, $C_4^2:C_2$ x 8, $C_8:C_4$ x 4, $C_8:C_4$ x 4, $C_4:C_8$ x 4, $C_8:C_4$, $C_4\times C_8$

Order 16: $C_2\times D_4$ x 1234, $D_4:C_2$ x 400, $C_2^4$ x 296, $C_2^2\times C_4$ x 249, $C_2^2:C_4$ x 170, $C_2\times Q_8$ x 36, $\OD_{16}$ x 30, $C_4:C_4$ x 24, $C_4^2$ x 23, $C_2\times C_8$ x 22, $\SD_{16}$ x 16, $D_8$ x 16

Order 8: $C_2^3$ x 439, $D_4$ x 260, $C_2\times C_4$ x 194, $Q_8$ x 20, $C_8$ x 10

Order 4: $C_2^2$ x 174, $C_4$ x 25

Order 2: $C_2$ x 19

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 4096: $C_2^6.C_2\wr C_4$

Order 2048: $C_2^5.C_2^2.C_2^3.C_2$, $(C_2\times C_4^3).D_8$, $(C_2\times C_4^3).\SD_{16}$, $C_4^3:C_2^2.C_2^3$, $C_2^5.C_2^4.C_2^2$

Order 1024: $C_2^5.C_2^4.C_2$, $C_2^4.C_2^3:C_8$, $C_2^5.C_2^4.C_2$, $C_2^4.C_2^3:C_8$, $(C_2\times C_4^3).Q_8$, $(C_2^2\times C_4^2).C_4^2$, $C_2^4.C_2^5.C_2$, $C_2^6:\OD_{16}$, $(C_2\times C_4^3).D_4$, $C_2.C_2^6.C_2^3$, $C_2^5.C_2^3.C_2^2$, $C_2^6.C_2^4$, $C_2^5.C_2^2{\rm wrC}_2$, $C_2^5.C_2^2{\rm wrC}_2$, $C_2^5.C_2^2{\rm wrC}_2$, $C_2^5.C_2^2{\rm wrC}_2$

Order 512: $C_2^4.C_2^4.C_2$ x 3, $C_2^5.C_2^3.C_2$ x 2, $C_2^4.C_2^4.C_2$ x 2, $C_2^4.C_2^4.C_2$ x 2, $C_2^5.C_2^3.C_2$ x 2, $C_2^5.C_4^2$ x 2, $C_2^5.C_2^4$ x 2, $C_2^6.C_2^3$, $C_2^5.C_2^4$, $C_2^6.C_2^3$, $C_2^2.C_2^5.C_2^2$, $C_2^6.C_2^3$, $C_2^3.C_2^4.C_2^2$, $(C_2^2\times D_4).C_2^4$, $C_2^6.C_2^3$, $C_2^5.C_2^3.C_2$, $C_2^5.C_2^3.C_2$, $C_2^4.C_2^4.C_2$, $C_2^4.C_2^4.C_2$, $C_2^3.D_4^2$, $C_2^3.D_4^2$, $C_2^3.D_4^2$, $C_2^3.D_4^2$, $C_2^3.C_2^5.C_2$, $C_2^3.C_2^4.C_2^2$, $C_2^3.C_2^4.C_2^2$, $(C_2^4\times C_4).C_2^3$, $(C_2^3\times D_4).C_2^3$, $C_2.C_2^6.C_2^2$, $C_2^4.C_2^4.C_2$, $C_2^4.C_2^2:C_8$, $C_2^5.C_4.C_2^2$, $C_2^5.C_2^4$, $C_2^5.C_2^4$, $C_2^6.C_2^3$, $C_2^6.C_2^3$

Order 256: $C_2^5.D_4$ x 5, $C_2^5:D_4$ x 5, $C_2^5.C_2^3$ x 5, $C_2^5.D_4$ x 4, $C_2^2.D_4^2$ x 3, $C_2^6:C_4$ x 3, $C_4^2:C_2^4$ x 3, $C_2^5:D_4$ x 3, $C_2^5.D_4$ x 3, $C_2^6:C_4$ x 3, $C_2^2.D_4^2$ x 2, $C_2^5:D_4$ x 2, $(C_2\times C_4^2).D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^5:D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^3.C_2^5$ x 2, $C_2^4.C_4^2$ x 2, $C_2^2:D_4^2$ x 2, $C_2^3.C_2^5$ x 2, $C_2^3.C_2^5$ x 2, $C_2^5:C_2^3$ x 2, $C_2^5:D_4$ x 2, $C_2^2.D_4^2$ x 2, $C_2^5.D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^5.C_2^3$ x 2, $C_2^6:C_4$ x 2, $C_2^5.D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^6.C_4$ x 2, $C_4^2:\OD_{16}$, $C_2^2.D_4^2$, $C_2^4.C_2^4$, $C_2^5.D_4$, $C_2^5.D_4$, $D_4:C_2^5$, $C_4^2:C_2^4$, $C_2^2\times D_4^2$, $C_2^4.C_4^2$, $C_2^5.D_4$, $C_2^5.D_4$, $\OD_{16}:C_2^4$, $C_2^4:\OD_{16}$, $C_2^5.D_4$, $(C_2\times C_4^2).D_4$, $C_2^5.D_4$, $C_2^3.(C_4\times D_4)$, $(C_2\times C_4^2).D_4$, $C_2^5.D_4$, $(C_2\times \OD_{16}):D_4$, $C_2^5.D_4$, $C_2^5.D_4$, $(C_2\times \OD_{16}):D_4$, $(C_2\times C_4^3):C_2$, $C_2^4.C_4^2$, $C_2^5.D_4$, $C_2^4.\OD_{16}$, $C_2^5:C_8$, $C_4^3:C_2^2$, $C_2^3.C_2^5$, $C_2^3.C_2^5$, $C_2^3.C_2^5$, $C_2^2.D_4^2$, $C_2^3.C_2^5$, $C_2^3.C_2^5$, $C_2^3.C_2^5$, $C_2^6:C_4$, $C_2^5.D_4$, $C_2^5.D_4$, $C_4.D_4^2$, $C_4.D_4^2$, $C_4.D_4^2$, $C_4.D_4^2$, $C_4.D_4^2$, $C_4.D_4^2$, $C_4.D_4^2$, $C_2^5.D_4$, $C_2^5.D_4$, $(C_2\times \OD_{16}):D_4$, $C_2^5.D_4$, $(C_4\times \OD_{16}):C_2^2$, $(C_2\times C_4^2).D_4$, $C_2\times \OD_{16}:D_4$, $C_2^5.D_4$, $(C_2\times D_4):\OD_{16}$, $C_2^5.D_4$, $C_2^5.D_4$, $C_2^4.C_4^2$, $C_4^2.C_2^4$

Order 128: $C_2^4:D_4$ x 25, $C_2^5:C_4$ x 20, $D_4:C_2^4$ x 13, $C_2^4:D_4$ x 13, $C_4^2:C_2^3$ x 11, $C_2^4:D_4$ x 11, $C_2^4:D_4$ x 10, $C_2^4:D_4$ x 9, $\OD_{16}:C_2^3$ x 8, $C_2^5:C_4$ x 7, $C_2\times D_4^2$ x 7, $C_4^2:C_2^3$ x 7, $C_2.D_4^2$ x 7, $C_2^3.C_2^4$ x 7, $C_2^4.D_4$ x 7, $C_2^4.D_4$ x 6, $C_2^5:C_4$ x 6, $C_2^5.C_4$ x 5, $D_4\times C_2^4$ x 5, $C_2^4.D_4$ x 5, $C_2.D_4^2$ x 5, $C_2^4.D_4$ x 4, $C_2^4.D_4$ x 4, $C_2^4:D_4$ x 4, $C_4^2:C_2^3$ x 4, $C_4^2:D_4$ x 4, $C_2^3.C_4^2$ x 4, $C_2^3.C_2^4$ x 4, $C_2.D_4^2$ x 4, $C_2^3.C_2^4$ x 4, $C_2^4.D_4$ x 4, $C_2^3.C_2^4$ x 4, $C_2^5:C_4$ x 4, $C_2^4.D_4$ x 3, $D_4^2:C_2$ x 3, $C_2^3.C_2^4$ x 3, $C_2^4.D_4$ x 3, $C_2^4.D_4$ x 2, $C_2^3.D_8$ x 2, $(C_2\times C_8):D_4$ x 2, $(C_2\times C_4):D_8$ x 2, $C_4^2:D_4$ x 2, $C_2^3:\SD_{16}$ x 2, $C_2^3:D_8$ x 2, $C_2^4.D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^3:\OD_{16}$ x 2, $C_2^3.\OD_{16}$ x 2, $C_2^4.D_4$ x 2, $C_2^4:C_8$ x 2, $C_2^3.C_4^2$ x 2, $D_4:C_2^4$ x 2, $C_8:C_2^4$ x 2, $C_4^2:C_2^3$ x 2, $C_2^4:D_4$ x 2, $\OD_{16}:D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^3:C_4^2$ x 2, $C_2^4.D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^3:\OD_{16}$ x 2, $C_2^4.D_4$ x 2, $C_2^4:D_4$ x 2, $C_2^3\wr C_2$ x 2, $C_2^4:D_4$ x 2, $C_2^3.C_2^4$ x 2, $C_2^3.C_2^4$ x 2, $C_2^3.C_2^4$ x 2, $C_2^3.C_2^4$ x 2, $C_2^3:C_4^2$ x 2, $C_2\times C_4^3$, $C_4^2:C_8$, $\OD_{16}:D_4$, $\OD_{16}:D_4$, $C_4.(C_4\times D_4)$, $C_4.(C_4\times D_4)$, $C_4.(C_4\times D_4)$, $(C_2\times C_4).D_8$, $\OD_{16}:D_4$, $\OD_{16}:D_4$, $C_2^4.D_4$, $C_2^4.D_4$, $C_2^3.C_4^2$, $C_2^4.D_4$, $C_2^3.C_4^2$, $C_4^2.D_4$, $C_4^2.D_4$, $C_2^3\times \OD_{16}$, $C_4^2:C_2^3$, $C_4^2:C_2^3$, $C_4^2:C_2^3$, $C_4^2:C_2^3$, $C_2^3:\OD_{16}$, $C_2^4.D_4$, $C_2^4.D_4$, $C_2^3:D_8$, $C_2^3:\SD_{16}$, $C_2^3:\SD_{16}$, $C_2^3:D_8$, $C_2^4.C_2^3$, $C_4^2.C_2^3$, $C_2^4.D_4$, $C_4^3:C_2$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$

Order 64: $D_4\times C_2^3$ x 75, $C_2^3:D_4$ x 61, $D_4:C_2^3$ x 54, $C_2^3.D_4$ x 41, $C_2^4:C_4$ x 34, $C_2^3:D_4$ x 34, $C_2^4:C_4$ x 33, $C_4^2:C_2^2$ x 31, $C_2^3:D_4$ x 28, $\OD_{16}:C_2^2$ x 22, $C_2^4:C_4$ x 22, $D_4:C_2^3$ x 22, $D_4^2$ x 14, $C_2^3.C_2^3$ x 13, $C_8:C_2^3$ x 12, $C_4^2:C_2^2$ x 12, $C_4^2:C_2^2$ x 11, $C_2\wr C_2^2$ x 10, $C_2^3.D_4$ x 8, $C_2^2:C_4^2$ x 8, $C_2^2.C_4^2$ x 8, $C_4^2:C_4$ x 8, $C_2^2\times \OD_{16}$ x 7, $C_2^2.D_8$ x 6, $C_2^2:\OD_{16}$ x 6, $C_2^3.D_4$ x 6, $C_2\wr C_4$ x 6, $D_4:D_4$ x 6, $C_2^3.D_4$ x 5, $C_2^6$ x 5, $C_2^4\times C_4$ x 5, $C_2^3.D_4$ x 4, $C_2^3.D_4$ x 4, $C_2^3:C_8$ x 4, $C_2^3:Q_8$ x 4, $D_4:D_4$ x 4, $C_4^2:C_2^2$ x 4, $C_2^3:D_4$ x 4, $C_4^2:C_2^2$ x 4, $C_4^2:C_4$ x 4, $C_2^2\times C_4^2$ x 4, $C_2^3.C_2^3$ x 3, $C_2^3.D_4$ x 3, $C_2^3.D_4$ x 3, $C_4^2:C_2^2$ x 3, $C_2^2.Q_{16}$ x 2, $C_2^3.C_2^3$ x 2, $C_4:C_4^2$ x 2, $C_2^3:C_8$ x 2, $C_2^2\times \SD_{16}$ x 2, $C_2^2\times D_8$ x 2, $C_2^2.C_4^2$ x 2, $C_4.C_4^2$ x 2, $C_2^3:Q_8$ x 2, $C_2^3.D_4$ x 2, $C_2^2.C_4^2$ x 2, $D_4:D_4$ x 2, $C_4.D_8$ x 2, $C_4^2.C_4$ x 2, $C_4^2.C_4$ x 2, $\OD_{16}:C_4$ x 2, $C_4\times \OD_{16}$, $C_4^3$, $Q_8\times C_2^3$, $C_2^3\times C_8$, $\OD_{16}:C_4$, $C_4^2:C_2^2$, $C_4^2:C_2^2$, $C_8:D_4$, $C_8:D_4$, $C_8:D_4$, $C_8:D_4$, $Q_8:D_4$, $D_4:D_4$, $C_2^2.D_8$, $C_2^3.D_4$, $C_4:\OD_{16}$

Order 32: $C_2^2\times D_4$ x 285, $C_2^3:C_4$ x 108, $D_4:C_2^2$ x 68, $D_4:C_2^2$ x 66, $C_2^2\wr C_2$ x 60, $C_2^5$ x 45, $C_2^3\times C_4$ x 45, $C_4:D_4$ x 32, $C_2^3:C_4$ x 30, $C_4:D_4$ x 28, $C_2\times \OD_{16}$ x 19, $C_2\times C_4^2$ x 18, $C_4\times D_4$ x 16, $C_2.C_4^2$ x 16, $\OD_{16}:C_2$ x 13, $C_2^2.D_4$ x 12, $C_2^2:C_8$ x 8, $D_8:C_2$ x 8, $D_4:C_4$ x 6, $C_2^2\times Q_8$ x 6, $C_2\times \SD_{16}$ x 6, $C_2\times D_8$ x 6, $C_2^2\times C_8$ x 4, $C_2^2.D_4$ x 4, $C_2^2:Q_8$ x 4, $C_4^2:C_2$ x 4, $C_4\wr C_2$ x 4, $C_4^2:C_2$ x 2, $C_4:C_8$ x 2, $Q_8:C_4$ x 2, $C_8:C_4$, $C_4\times C_8$, $C_8:C_4$, $C_8:C_4$

Order 16: $C_2\times D_4$ x 313, $C_2^4$ x 162, $C_2^2\times C_4$ x 121, $C_2^2:C_4$ x 74, $D_4:C_2$ x 50, $C_4^2$ x 17, $\OD_{16}$ x 12, $C_2\times Q_8$ x 11, $C_4:C_4$ x 10, $C_2\times C_8$ x 9, $\SD_{16}$ x 2, $D_8$ x 2

Order 8: $C_2^3$ x 211, $C_2\times C_4$ x 94, $D_4$ x 65, $Q_8$ x 5, $C_8$ x 4

Order 4: $C_2^2$ x 93, $C_4$ x 17

Order 2: $C_2$ x 14

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 4096: $C_2^6.C_2\wr C_4$ ($C_1$)

Order 2048: $(C_2\times C_4^3).D_8$ ($C_2$) x 2, $(C_2\times C_4^3).\SD_{16}$ ($C_2$) x 2, $C_2^5.C_2^2.C_2^3.C_2$ ($C_2$), $C_4^3:C_2^2.C_2^3$ ($C_2$), $C_2^5.C_2^4.C_2^2$ ($C_2$)

Order 1024: $C_2^4.C_2^3:C_8$ ($C_2^2$) x 2, $C_2^5.C_2^2{\rm wrC}_2$ ($C_4$) x 2, $C_2^5.C_2^2{\rm wrC}_2$ ($C_2^2$) x 2, $(C_2\times C_4^3).Q_8$ ($C_2^2$), $C_2^4.C_2^5.C_2$ ($C_2^2$), $(C_2\times C_4^3).D_4$ ($C_2^2$), $C_2^5.C_2^3.C_2^2$ ($C_4$), $C_2^6.C_2^4$ ($C_4$)

Order 512: $C_2^5.C_4^2$ ($C_2\times C_4$) x 2, $C_2^5.C_2^4$ ($C_2\times C_4$) x 2, $C_2^6.C_2^3$ ($C_2\times C_4$), $C_2^5.C_2^4$ ($D_4$), $C_2^6.C_2^3$ ($D_4$), $C_2^4.C_2^4.C_2$ ($D_4$), $C_2^4.C_2^4.C_2$ ($C_2\times C_4$), $C_2^4.C_2^4.C_2$ ($C_2^3$), $C_2^4.C_2^4.C_2$ ($D_4$)

Order 256: $C_2^5.C_2^3$ ($C_2^2:C_4$) x 2, $C_2^2.D_4^2$ ($C_2^2:C_4$), $C_2^2\times D_4^2$ ($C_2^2:C_4$), $C_4^3:C_2^2$ ($C_2\times D_4$), $C_2^5.C_2^3$ ($C_2\times D_4$), $C_2^5.C_2^3$ ($C_2^2\times C_4$)

Order 128: $C_2^4:D_4$ ($C_2^3:C_4$) x 2, $C_2\times C_4^3$ ($D_8:C_2$), $C_4^2:C_2^3$ ($D_8:C_2$), $C_4^2:C_2^3$ ($C_2^3:C_4$)

Order 64: $C_2^6$ ($C_4^2:C_4$), $C_2^6$ ($C_2\wr C_4$), $D_4\times C_2^3$ ($C_2^4:C_4$), $D_4\times C_2^3$ ($C_4^2:C_2^2$), $C_2^4:C_4$ ($C_4^2:C_4$), $C_2^4:C_4$ ($C_2\wr C_4$), $C_2^2\times C_4^2$ ($C_2^3.D_4$)

Order 32: $C_2^5$ ($C_2^4.D_4$), $C_2^5$ ($C_2^5:C_4$), $C_2^3\times C_4$ ($C_2^4.D_4$)

Order 16: $C_2^4$ ($C_2^5.D_4$)

Order 8: $C_2^3$ ($C_2^5.C_2.C_2^3$)

Order 4: $C_2^2$ ($C_2^4.C_2\wr C_4$)

Order 2: $C_2$ ($C_2^5.C_2^4.C_2^2$)

Order 1: $C_1$ ($C_2^6.C_2\wr C_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 4096: $C_2^6.C_2\wr C_4$ ($C_1$)

Order 2048: $C_2^5.C_2^2.C_2^3.C_2$ ($C_2$), $(C_2\times C_4^3).D_8$ ($C_2$), $(C_2\times C_4^3).\SD_{16}$ ($C_2$), $C_4^3:C_2^2.C_2^3$ ($C_2$), $C_2^5.C_2^4.C_2^2$ ($C_2$)

Order 1024: $C_2^4.C_2^3:C_8$ ($C_2^2$), $(C_2\times C_4^3).Q_8$ ($C_2^2$), $C_2^4.C_2^5.C_2$ ($C_2^2$), $(C_2\times C_4^3).D_4$ ($C_2^2$), $C_2^5.C_2^3.C_2^2$ ($C_4$), $C_2^6.C_2^4$ ($C_4$), $C_2^5.C_2^2{\rm wrC}_2$ ($C_4$), $C_2^5.C_2^2{\rm wrC}_2$ ($C_2^2$)

Order 512: $C_2^6.C_2^3$ ($C_2\times C_4$), $C_2^5.C_2^4$ ($D_4$), $C_2^6.C_2^3$ ($D_4$), $C_2^4.C_2^4.C_2$ ($D_4$), $C_2^4.C_2^4.C_2$ ($C_2\times C_4$), $C_2^4.C_2^4.C_2$ ($C_2^3$), $C_2^4.C_2^4.C_2$ ($D_4$), $C_2^5.C_4^2$ ($C_2\times C_4$), $C_2^5.C_2^4$ ($C_2\times C_4$)

Order 256: $C_2^5.C_2^3$ ($C_2^2:C_4$) x 2, $C_2^2.D_4^2$ ($C_2^2:C_4$), $C_2^2\times D_4^2$ ($C_2^2:C_4$), $C_4^3:C_2^2$ ($C_2\times D_4$), $C_2^5.C_2^3$ ($C_2\times D_4$), $C_2^5.C_2^3$ ($C_2^2\times C_4$)

Order 128: $C_2^4:D_4$ ($C_2^3:C_4$) x 2, $C_2\times C_4^3$ ($D_8:C_2$), $C_4^2:C_2^3$ ($D_8:C_2$), $C_4^2:C_2^3$ ($C_2^3:C_4$)

Order 64: $C_2^6$ ($C_4^2:C_4$), $C_2^6$ ($C_2\wr C_4$), $D_4\times C_2^3$ ($C_2^4:C_4$), $D_4\times C_2^3$ ($C_4^2:C_2^2$), $C_2^4:C_4$ ($C_4^2:C_4$), $C_2^4:C_4$ ($C_2\wr C_4$), $C_2^2\times C_4^2$ ($C_2^3.D_4$)

Order 32: $C_2^5$ ($C_2^4.D_4$), $C_2^5$ ($C_2^5:C_4$), $C_2^3\times C_4$ ($C_2^4.D_4$)

Order 16: $C_2^4$ ($C_2^5.D_4$)

Order 8: $C_2^3$ ($C_2^5.C_2.C_2^3$)

Order 4: $C_2^2$ ($C_2^4.C_2\wr C_4$)

Order 2: $C_2$ ($C_2^5.C_2^4.C_2^2$)

Order 1: $C_1$ ($C_2^6.C_2\wr C_4$)

Series

Derived series

$C_2^6.C_2\wr C_4$

$\rhd$

$C_2^5.C_2^3$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

Chief series

$C_2^6.C_2\wr C_4$

$\rhd$

$C_2^5.C_2^4.C_2^2$

$\rhd$

$C_2^4.C_2^5.C_2$

$\rhd$

$C_2^4.C_2^4.C_2$

$\rhd$

$C_2^5.C_2^3$

$\rhd$

$C_4^2:C_2^3$

$\rhd$

$D_4\times C_2^3$

$\rhd$

$C_2^5$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^6.C_2\wr C_4$

$\rhd$

$C_2^5.C_2^3$

$\rhd$

$C_4^2:C_2^3$

$\rhd$

$C_2^3\times C_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

$\lhd$

$C_2^3$

$\lhd$

$C_2^4$

$\lhd$

$D_4\times C_2^3$

$\lhd$

$C_2^2\times D_4^2$

$\lhd$

$C_2^4.C_2^5.C_2$

$\lhd$

$C_2^6.C_2\wr C_4$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

See the $61 \times 61$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $55 \times 55$ rational character table.