Abstract group $(C_2^2\times D_4^2):\SD_{16}$

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

Group information

Description:	$(C_2^2\times D_4^2):\SD_{16}$
Order:	\(4096\)\(\medspace = 2^{12} \)
Exponent:	\(8\)\(\medspace = 2^{3} \)
Automorphism group:	$C_{100}\times D_{25}$, 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	16	27	14	58
Divisions	1	16	27	10	54
Autjugacy classes	1	11	15	4	31

Dimension	1	2	4	8	16	32
Irr. complex chars.	8	10	13	16	11	0	58
Irr. rational chars.	8	6	13	17	9	1	54

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:	${\langle a, b, c, d, e, f, g \mid b^{8}=c^{2}=d^{4}=e^{4}=f^{2}=g^{4}=[a,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,10,13,7)(2,9,14,8)(3,11,15,5,4,12,16,6), (1,14,2,13)(3,15)(4,16)(5,11,7,9)(6,12,8,10), (1,11)(2,12)(3,9,4,10)(5,14,6,13)(7,15)(8,16)\rangle$
Transitive group:	16T1640				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_2^2\times D_4^2)$ $\,\rtimes\,$ $\SD_{16}$	$(C_2^2.D_4^2)$ $\,\rtimes\,$ $\SD_{16}$	$((C_2\times C_4^3).D_8)$ $\,\rtimes\,$ $C_2$	$((C_2^2\times D_4^2):Q_8)$ $\,\rtimes\,$ $C_2$	all 7
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$C_2^5$ . $(C_2\wr D_4)$	$(C_2^5.D_4^2)$ . $C_2$	$(C_2^5.C_2^4)$ . $D_4$	$(C_2^5.C_4^2)$ . $D_4$	all 22

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

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 114227 subgroups in 6485 conjugacy classes, 41 normal (25 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_5^4:(C_2^2\times \OD_{16})$
Commutator:	$G' \simeq$ $C_2^4.C_2^4.C_2$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2^4.C_2^4.C_2$	$G/\Phi \simeq$ $C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $(C_2^2\times D_4^2):\SD_{16}$	$G/\operatorname{Fit} \simeq$ $C_1$
Radical:	$R \simeq$ $(C_2^2\times D_4^2):\SD_{16}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2$	$G/\operatorname{soc} \simeq$ $C_5^4:(C_2^2\times \OD_{16})$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $(C_2^2\times D_4^2):\SD_{16}$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 4096: $(C_2^2\times D_4^2):\SD_{16}$

Order 2048: $(C_2\times C_4^3).D_8$ x 2, $(C_2\times C_4^3):\SD_{16}$ x 2, $C_2^4.C_2^4.C_2^2.C_2$, $C_2^5.D_4^2$, $(C_2^2\times D_4^2):Q_8$

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

Order 512: $C_2^4.C_2^4.C_2$ x 3, $C_2^5.C_4^2$ x 3, $C_2^5.C_2^4$ x 3, $C_2^3.C_2^4.C_2^2$ x 2, $C_2^5.D_4.C_2$ x 2, $C_2^5.D_4:C_2$ x 2, $C_2^5.D_4.C_2$ x 2, $C_2^5.C_2^3.C_2$ x 2, $C_2^4.C_2^4.C_2$ x 2, $C_2^4.C_2^4.C_2$ x 2, $C_2^4.C_2^4.C_2$ x 2, $C_2^4.C_2^4.C_2$ x 2, $C_2^4.C_2^4.C_2$ x 2, $C_2^5.C_4.C_2^2$ 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, $C_2^3.C_2^5.C_2$ x 2, $C_2^3.D_4^2$ x 2, $C_2^3.C_2{\rm wrC}_4$ x 2, $C_2^4.C_2^2:C_8$ x 2, $C_2^5.C_2^4$, $C_2.C_2^6.C_2^2$, $C_2^5.C_2^4$, $C_2^3.C_2^4.C_2^2$, $C_2^4.C_2^4.C_2$, $(C_2^4\times C_4).C_2^3$, $C_2^3.C_2^4.C_2^2$, $C_2^4.C_2^4.C_2$

Order 256: $C_4.D_4^2$ x 16, $C_2^5.D_4$ x 8, $C_2^5.D_4$ x 6, $C_2^5.D_4$ x 6, $\OD_{16}:(C_2^2\times C_4)$ x 6, $C_2^5.C_2^3$ x 5, $C_2^5.D_4$ x 4, $C_2^5.D_4$ x 4, $C_4^2:C_2^4$ x 4, $C_2^5.D_4$ x 4, $C_2^4.C_4^2$ x 4, $C_2^5:D_4$ x 4, $C_2^5.D_4$ x 4, $C_4.D_4^2$ x 4, $C_2^4.(C_2\times D_4)$ x 3, $C_2^5:D_4$ x 3, $C_2\times \OD_{16}:D_4$ x 3, $C_2^2.D_4^2$ x 2, $C_2^2.D_4^2$ x 2, $(C_2\times C_4^2).D_4$ 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, $C_2^6:C_4$ x 2, $D_4^2:C_2^2$ 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, $C_2^4:\SD_{16}$ x 2, $C_2^5.D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^4.Q_{16}$ 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, $(C_2\times \OD_{16}):D_4$ x 2, $(C_2\times C_4^3):C_2$ x 2, $C_4.D_4^2$ x 2, $C_2^3.(C_4\times D_4)$ x 2, $C_2^2.D_4^2$ x 2, $C_2^5:C_8$ x 2, $C_2^4.C_4^2$ x 2, $C_2^3.C_2^5$ x 2, $C_2^3.C_2^5$ x 2, $C_2^2.D_4^2$ x 2, $D_4^2:C_2^2$ x 2, $C_2^5:Q_8$ x 2, $C_4.D_4^2$ x 2, $C_4.D_4^2$ x 2, $(C_2\times \OD_{16}):D_4$ x 2, $C_2^5.D_4$ x 2, $C_2\times \OD_{16}:D_4$ x 2, $(C_2\times C_4^2).D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^5.D_4$ x 2, $C_2^5.D_4$ x 2, $C_4^2:\OD_{16}$, $C_2^2.D_4^2$, $C_2^4.C_2^4$, $C_8:C_2^5$, $C_2^2\times D_4^2$, $C_2^5.D_4$, $C_4.D_4^2$, $C_4:D_4^2$, $(C_2\times C_4^2).D_4$, $(C_2\times C_4^2):D_4$, $C_4.D_4^2$, $(C_2\times C_4^2).D_4$, $C_4.D_4^2$, $C_4^3:C_2^2$, $C_2^2.D_4^2$, $C_2^3.C_2^5$, $C_2^3.C_2^5$, $(C_4\times \OD_{16}):C_2^2$, $C_2^5.D_4$, $(C_2\times D_4):\OD_{16}$, $C_2^5.D_4$, $C_4^2.C_2^4$

Order 128: $C_4^2:C_2^3$ x 28, $C_8:C_2^4$ x 24, $C_4^2:D_4$ x 20, $\OD_{16}:(C_2\times C_4)$ x 20, $D_4^2:C_2$ x 20, $C_2.D_4^2$ x 20, $\OD_{16}:D_4$ x 16, $C_2^5:C_4$ x 15, $C_2^4:D_4$ x 14, $C_2\times D_4^2$ x 13, $C_2^5:C_4$ x 12, $C_2^4.D_4$ x 12, $C_2.D_4^2$ x 10, $C_2\wr D_4$ x 8, $C_2^4.D_4$ x 8, $\OD_{16}:D_4$ x 8, $\OD_{16}:D_4$ x 8, $C_2^4.D_4$ x 8, $C_2^4.D_4$ x 8, $C_2^4.D_4$ x 8, $C_4^2:C_2^3$ x 8, $C_2^4:D_4$ x 8, $C_4^2:C_2^3$ x 8, $C_2^3.C_4^2$ x 8, $C_2.D_4^2$ x 8, $C_2.D_4^2$ x 8, $C_4^2:C_2^3$ x 6, $C_2^4.D_4$ x 6, $\OD_{16}:C_2^3$ x 6, $C_4^2:D_4$ x 6, $C_2.D_4^2$ x 6, $C_2^4:D_4$ x 6, $C_2^4:D_4$ x 5, $C_2^3:D_8$ x 5, $C_2^3.C_2^4$ x 5, $(C_2\times C_8):D_4$ 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, $(C_2\times C_4):D_8$ x 4, $(C_2\times C_4):\SD_{16}$ x 4, $C_2^4.D_4$ x 4, $C_2^3.D_8$ x 4, $C_2^3.C_4^2$ x 4, $C_4^2:C_2^3$ x 4, $C_4^2:C_2^3$ x 4, $C_2^4:D_4$ x 4, $C_2^4.D_4$ x 4, $C_2^3:\SD_{16}$ x 4, $C_2^4.D_4$ x 4, $C_2.D_4^2$ x 4, $C_2.D_4^2$ x 4, $C_2^3.C_2^4$ x 4, $C_2^4.D_4$ x 4, $C_2.D_4^2$ x 4, $C_2.D_4^2$ x 4, $C_2^3.C_2^4$ 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, $C_2.D_4^2$ x 4, $C_2^3.C_4^2$ x 3, $C_2^2.C_2^5$ x 3, $C_2^3:\OD_{16}$ x 3, $C_2^4:Q_8$ x 2, $C_2^4:Q_8$ x 2, $C_2^3.D_8$ x 2, $\OD_{16}:D_4$ x 2, $C_2^4.D_4$ x 2, $C_2^3.D_8$ x 2, $C_2^3:\OD_{16}$ x 2, $C_2^3.\OD_{16}$ x 2, $C_2^5.C_4$ x 2, $C_2^4:C_8$ x 2, $C_2^4.D_4$ x 2, $C_2^3.C_4^2$ x 2, $C_2^3:\SD_{16}$ x 2, $C_4^2.D_4$ x 2, $D_4:C_2^4$ x 2, $C_4^2.C_2^3$ x 2, $C_4^2.C_2^3$ x 2, $C_4^2.C_2^3$ x 2, $C_2^4.D_4$ x 2, $C_2^4.D_4$ x 2, $C_2.D_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2^3.C_2^4$ x 2, $C_2^3.C_2^4$ x 2, $C_2.D_4^2$ x 2, $C_2^3.C_2^4$ x 2, $C_2^3.C_2^4$ x 2, $C_2.D_4^2$ 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, $C_4^2:D_4$ 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, $C_2\times C_4^3$, $C_4^2:C_8$, $C_4^2.D_4$, $D_4:C_2^4$, $D_4\times C_2^4$, $C_2^3\times \SD_{16}$, $C_2^3\times D_8$, $C_2^3\times \OD_{16}$, $C_4^2:C_2^3$, $C_2^3:\SD_{16}$, $C_2^3:C_4^2$, $C_2^4.C_2^3$, $C_4^2.C_2^3$, $C_4^3:C_2$, $C_2^3.C_2^4$, $C_2^4.D_4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$

Order 64: $C_8:C_2^3$ x 150, $C_2^3:D_4$ x 86, $D_4^2$ x 70, $C_2^3:D_4$ x 64, $C_2^3:D_4$ x 57, $D_4:D_4$ x 52, $D_4:D_4$ x 48, $C_4^2:C_2^2$ x 48, $D_4\times C_2^3$ x 47, $C_4^2:C_2^2$ x 40, $C_2^3.C_2^3$ x 38, $C_2^3.D_4$ x 38, $C_4^2:C_2^2$ x 28, $C_4^2:C_2^2$ x 28, $C_2^4:C_4$ x 27, $C_4^2:C_2^2$ x 27, $C_2^2.C_2^4$ x 26, $\OD_{16}:C_2^2$ x 24, $D_4:C_2^3$ x 24, $C_2\wr C_2^2$ x 24, $C_2^3.D_4$ x 22, $C_2\wr C_4$ x 20, $D_4:D_4$ x 20, $D_4:C_2^3$ x 18, $C_2^2\times \SD_{16}$ x 18, $C_2^2\times D_8$ x 18, $C_2^2:\SD_{16}$ x 18, $C_2^3.D_4$ x 16, $C_2^3.C_2^3$ x 16, $C_2^2:C_4^2$ x 16, $Q_8:D_4$ x 16, $Q_8:D_4$ x 16, $C_2^4:C_4$ x 15, $C_2^2.D_8$ x 14, $C_4^2:C_4$ x 14, $C_2^2\times \OD_{16}$ x 13, $C_2^4:C_4$ x 12, $C_4^2.C_2^2$ x 12, $C_4^2:C_4$ x 12, $C_2^3.D_4$ x 10, $\OD_{16}:C_4$ x 10, $C_2^3:Q_8$ x 10, $C_2^2:\OD_{16}$ x 9, $C_2^3.Q_8$ x 8, $D_4:D_4$ x 8, $C_2^3.D_4$ x 6, $C_2^3:C_8$ 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, $C_2^3.D_4$ x 6, $C_2^2\times C_4^2$ x 5, $C_2^3.C_2^3$ x 4, $C_2^3.C_2^3$ x 4, $C_4:C_4^2$ x 4, $C_2^2.C_4^2$ x 4, $C_4.C_4^2$ x 4, $C_4^2:C_2^2$ x 4, $C_2^3.D_4$ x 4, $C_2^2.C_4^2$ x 4, $Q_8:D_4$ x 4, $C_4.D_8$ x 4, $C_2^2.C_4^2$ x 3, $C_2^2.Q_{16}$ x 2, $C_2^2.Q_{16}$ x 2, $C_2^3.C_2^3$ x 2, $C_2^3:Q_8$ x 2, $C_2^3:C_8$ x 2, $C_2^4\times C_4$ x 2, $C_2^3:Q_8$ x 2, $C_4^2.C_2^2$ x 2, $C_4^2.C_4$ x 2, $C_4^2.C_4$ x 2, $C_4:\OD_{16}$ x 2, $C_4\times \OD_{16}$, $C_2^3.D_4$, $C_4^3$, $C_2^6$, $Q_8\times C_2^3$, $C_2^3\times C_8$

Order 32: $C_2^2\times D_4$ x 323, $C_4:D_4$ x 204, $D_8:C_2$ x 168, $C_2^2\wr C_2$ x 166, $C_2^3:C_4$ x 137, $D_4:C_2^2$ x 124, $C_4\times D_4$ x 84, $C_4:D_4$ x 78, $C_2\times \SD_{16}$ x 72, $C_2\times D_8$ x 70, $C_2\times \OD_{16}$ x 50, $D_4:C_2^2$ x 48, $C_2^2:Q_8$ x 48, $C_2^3:C_4$ x 46, $C_2^3\times C_4$ x 45, $C_2^2.D_4$ x 39, $C_2\times C_4^2$ x 28, $C_4\wr C_2$ x 28, $D_4:C_4$ x 26, $C_2.C_4^2$ x 25, $\OD_{16}:C_2$ x 24, $C_4^2:C_2$ x 24, $C_2^2.D_4$ x 24, $C_2^5$ x 22, $C_2^2:C_8$ x 16, $C_4^2:C_2$ x 16, $C_2^2\times Q_8$ x 11, $C_2^2\times C_8$ x 10, $C_4\times Q_8$ x 8, $C_4:C_8$ x 4, $Q_8:C_4$ x 4, $C_8:C_4$, $C_4\times C_8$

Order 16: $C_2\times D_4$ x 590, $C_2^2:C_4$ x 166, $C_2^2\times C_4$ x 164, $C_2^4$ x 133, $D_4:C_2$ x 132, $C_4:C_4$ x 51, $\SD_{16}$ x 44, $D_8$ x 40, $C_4^2$ x 33, $\OD_{16}$ x 32, $C_2\times Q_8$ x 31, $C_2\times C_8$ x 26

Order 8: $C_2^3$ x 241, $D_4$ x 192, $C_2\times C_4$ x 151, $Q_8$ x 18, $C_8$ x 10

Order 4: $C_2^2$ x 120, $C_4$ x 27

Order 2: $C_2$ x 16

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 4096: $(C_2^2\times D_4^2):\SD_{16}$

Order 2048: $C_2^4.C_2^4.C_2^2.C_2$, $C_2^5.D_4^2$, $(C_2\times C_4^3).D_8$, $(C_2^2\times D_4^2):Q_8$, $(C_2\times C_4^3):\SD_{16}$

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

Order 512: $C_2^4.C_2^4.C_2$ x 3, $C_2^5.C_4^2$ x 2, $C_2^5.C_2^4$ x 2, $C_2^5.C_2^4$, $C_2.C_2^6.C_2^2$, $C_2^3.C_2^4.C_2^2$, $C_2^5.D_4.C_2$, $C_2^5.C_2^4$, $C_2^5.D_4:C_2$, $C_2^5.D_4.C_2$, $C_2^5.C_2^3.C_2$, $C_2^4.C_2^4.C_2$, $C_2^4.C_2^4.C_2$, $C_2^4.C_2^4.C_2$, $C_2^4.C_2^4.C_2$, $C_2^4.C_2^4.C_2$, $C_2^5.C_4.C_2^2$, $C_2^5.C_2^3.C_2$, $C_2^3.D_4^2$, $C_2^3.D_4^2$, $C_2^3.D_4^2$, $C_2^3.C_2^5.C_2$, $C_2^3.C_2^4.C_2^2$, $C_2^3.D_4^2$, $C_2^4.C_2^4.C_2$, $(C_2^4\times C_4).C_2^3$, $C_2^3.C_2^4.C_2^2$, $C_2^4.C_2^4.C_2$, $C_2^3.C_2{\rm wrC}_4$, $C_2^4.C_2^2:C_8$

Order 256: $C_2^5.D_4$ x 5, $C_2^5.D_4$ x 4, $C_2^5.D_4$ x 3, $C_2^5.C_2^3$ x 3, $\OD_{16}:(C_2^2\times C_4)$ x 3, $C_2^2.D_4^2$ x 2, $C_2^5.D_4$ x 2, $C_2^5.D_4$ x 2, $C_4^2:C_2^4$ x 2, $C_2^4.(C_2\times D_4)$ x 2, $C_2^5.D_4$ x 2, $C_2^2.D_4^2$ x 2, $C_2^5:D_4$ x 2, $C_2^4.C_4^2$ x 2, $C_2^5:D_4$ x 2, $C_2^5.D_4$ x 2, $C_4.D_4^2$ x 2, $C_4.D_4^2$ x 2, $C_4.D_4^2$ x 2, $C_2\times \OD_{16}:D_4$ x 2, $C_4^2:\OD_{16}$, $C_2^2.D_4^2$, $C_2^2.D_4^2$, $C_2^4.C_2^4$, $(C_2\times C_4^2).D_4$, $C_2^5:D_4$, $(C_2\times C_4^2).D_4$, $C_2^5.D_4$, $C_2^6:C_4$, $C_8:C_2^5$, $D_4^2:C_2^2$, $C_2^2\times D_4^2$, $C_2^5.D_4$, $C_2^5.D_4$, $C_2^2.D_4^2$, $C_4:D_4^2$, $C_4.D_4^2$, $C_4.D_4^2$, $C_4:D_4^2$, $C_2^4:\SD_{16}$, $(C_2\times C_4^2).D_4$, $C_2^5.D_4$, $C_2^5.D_4$, $C_2^4.Q_{16}$, $(C_2\times C_4^2).D_4$, $(C_2\times C_4^2):D_4$, $C_2^5.D_4$, $(C_2\times \OD_{16}):D_4$, $C_2^5.D_4$, $(C_2\times \OD_{16}):D_4$, $(C_2\times C_4^3):C_2$, $C_4.D_4^2$, $(C_2\times C_4^2).D_4$, $C_4.D_4^2$, $C_4.D_4^2$, $C_2^3.(C_4\times D_4)$, $C_2^5:C_8$, $C_4^3:C_2^2$, $C_2^4.C_4^2$, $C_2^3.C_2^5$, $C_2^3.C_2^5$, $C_2^2.D_4^2$, $C_2^3.C_2^5$, $C_2^2.D_4^2$, $C_2^3.C_2^5$, $D_4^2:C_2^2$, $C_2^5:Q_8$, $C_4.D_4^2$, $(C_2\times \OD_{16}):D_4$, $C_2^5.D_4$, $(C_4\times \OD_{16}):C_2^2$, $C_2\times \OD_{16}:D_4$, $C_2^5.D_4$, $(C_2\times C_4^2).D_4$, $C_2^5.D_4$, $(C_2\times D_4):\OD_{16}$, $C_2^5.D_4$, $C_2^5.D_4$, $C_2^5.D_4$, $C_4^2.C_2^4$

Order 128: $C_8:C_2^4$ x 9, $C_2\times D_4^2$ x 9, $C_2^4:D_4$ x 9, $C_2^5:C_4$ x 8, $C_2^4.D_4$ x 6, $C_4^2:C_2^3$ x 6, $C_4^2:C_2^3$ x 6, $C_2.D_4^2$ x 6, $C_2^5:C_4$ x 5, $C_2.D_4^2$ x 5, $C_4^2:D_4$ x 5, $C_2^4.D_4$ x 4, $C_2^4.D_4$ x 4, $D_4^2:C_2$ x 4, $C_2^4:D_4$ x 4, $C_4^2:C_2^3$ x 4, $C_2^3.C_4^2$ x 4, $C_2.D_4^2$ x 4, $C_2.D_4^2$ x 4, $C_2.D_4^2$ x 4, $C_2^4:D_4$ x 4, $C_4^2:D_4$ x 3, $\OD_{16}:(C_2\times C_4)$ x 3, $C_2^3.D_8$ x 3, $C_2^3.C_4^2$ x 3, $C_4^2:C_2^3$ x 3, $C_2^4:D_4$ x 3, $C_2^4:D_4$ x 3, $C_2^4.D_4$ x 3, $C_2^3:D_8$ x 3, $C_2^4.D_4$ x 3, $C_2^4.D_4$ x 3, $\OD_{16}:C_2^3$ x 3, $C_2^3.C_2^4$ x 3, $(C_2\times C_8):D_4$ x 2, $(C_2\times C_4):D_8$ x 2, $\OD_{16}:D_4$ x 2, $C_2^3:\SD_{16}$ x 2, $C_2^3:D_8$ x 2, $C_2^3.D_8$ x 2, $(C_2\times C_4):D_8$ x 2, $(C_2\times C_4):\SD_{16}$ 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, $C_2^2.C_2^5$ x 2, $C_4^2:C_2^3$ x 2, $C_4^2:C_2^3$ x 2, $C_2^4.D_4$ x 2, $C_2^3:\SD_{16}$ x 2, $C_2^3:\OD_{16}$ x 2, $C_2.D_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2^3.C_2^4$ x 2, $C_2^4.D_4$ x 2, $C_2.D_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2.D_4^2$ x 2, $C_2^3.C_2^4$ x 2, $C_2^3.C_2^4$ x 2, $C_2.D_4^2$ x 2, $C_2^3.C_2^4$ 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, $C_2.D_4^2$ x 2, $C_2^3.C_2^4$ x 2, $C_2\times C_4^3$, $C_2\wr D_4$, $C_4^2:C_8$, $C_2^4:Q_8$, $C_2^4:Q_8$, $C_2^3.D_8$, $\OD_{16}:D_4$, $\OD_{16}:D_4$, $C_2^4.D_4$, $\OD_{16}:D_4$, $C_2^4.D_4$, $C_2^5.C_4$, $C_2^4:C_8$, $C_2^4.D_4$, $C_2^3.C_4^2$, $C_2^3:\SD_{16}$, $C_4^2.D_4$, $C_4^2.D_4$, $D_4:C_2^4$, $D_4:C_2^4$, $D_4\times C_2^4$, $C_2^3\times \SD_{16}$, $C_2^3\times D_8$, $C_2^3\times \OD_{16}$, $C_4^2.C_2^3$, $C_4^2.C_2^3$, $C_4^2.C_2^3$, $C_4^2:C_2^3$, $C_2^4.D_4$, $C_2^3:\SD_{16}$, $C_2^3:C_4^2$, $C_2^4.C_2^3$, $C_4^2.C_2^3$, $C_2^4.D_4$, $C_4^3:C_2$, $C_2.D_4^2$, $C_2.D_4^2$, $C_2.D_4^2$, $C_2.D_4^2$, $C_2^3.C_2^4$, $C_2^4.D_4$, $C_2^3.C_2^4$, $C_2.D_4^2$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2.D_4^2$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_4^2:D_4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3.C_2^4$, $C_2^3:C_4^2$

Order 64: $C_2^3:D_4$ x 39, $D_4\times C_2^3$ x 31, $C_2^3:D_4$ x 31, $C_2^3:D_4$ x 28, $C_8:C_2^3$ x 28, $C_4^2:C_2^2$ x 28, $C_2^3.D_4$ x 20, $D_4^2$ x 18, $C_4^2:C_2^2$ x 17, $C_2^3.C_2^3$ x 16, $C_2^4:C_4$ x 14, $D_4:C_2^3$ x 13, $C_2^3.D_4$ x 12, $C_4^2:C_2^2$ x 12, $C_4^2:C_4$ x 10, $\OD_{16}:C_2^2$ x 9, $C_2^4:C_4$ x 9, $C_2^3.C_2^3$ x 8, $C_2^4:C_4$ x 8, $C_2^2:C_4^2$ x 8, $C_2\wr C_2^2$ x 8, $C_2^2.D_8$ x 7, $D_4:C_2^3$ x 7, $C_2^2\times \OD_{16}$ x 7, $C_2^3.D_4$ x 6, $C_2^3.D_4$ x 6, $C_2^2\times \SD_{16}$ x 6, $C_2^2\times D_8$ x 6, $D_4:D_4$ x 6, $C_4^2:C_2^2$ x 6, $C_4^2:C_2^2$ x 6, $D_4:D_4$ x 6, $C_2^2:\OD_{16}$ x 5, $C_2\wr C_4$ x 5, $C_2^2.C_2^4$ x 5, $C_2^3:Q_8$ x 5, $C_2^3:C_8$ x 4, $C_2^3.Q_8$ 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, $D_4:D_4$ x 4, $C_2^3.D_4$ x 3, $C_2^3.C_2^3$ x 3, $C_4:C_4^2$ x 3, $\OD_{16}:C_4$ x 3, $C_4^2:C_2^2$ x 3, $C_2^3.D_4$ x 3, $C_2^3.D_4$ x 3, $C_2^2:\SD_{16}$ x 3, $C_2^3.C_2^3$ x 2, $C_2^3:Q_8$ x 2, $C_2^3.C_2^3$ x 2, $C_2^3.C_2^3$ x 2, $C_2^2.C_4^2$ x 2, $C_2^4\times C_4$ x 2, $Q_8:D_4$ x 2, $C_2^2.C_4^2$ x 2, $Q_8:D_4$ x 2, $C_4.C_4^2$ x 2, $C_4^2:C_2^2$ x 2, $C_2^2.C_4^2$ x 2, $D_4:D_4$ x 2, $Q_8:D_4$ x 2, $C_4.D_8$ x 2, $C_4^2.C_4$ x 2, $C_4^2.C_4$ x 2, $C_2^2.Q_{16}$, $C_2^2.Q_{16}$, $C_4\times \OD_{16}$, $C_2^3.D_4$, $C_4^3$, $C_2^3:C_8$, $C_2^6$, $Q_8\times C_2^3$, $C_2^3\times C_8$, $C_2^3:Q_8$, $C_4^2.C_2^2$, $C_4:\OD_{16}$

Order 32: $C_2^2\times D_4$ x 145, $C_2^3:C_4$ x 61, $C_2^2\wr C_2$ x 47, $C_4:D_4$ x 46, $D_4:C_2^2$ x 42, $C_4:D_4$ x 31, $C_2^3\times C_4$ x 28, $C_4\times D_4$ x 24, $C_2^2.D_4$ x 20, $C_2^3:C_4$ x 18, $C_2\times \OD_{16}$ x 18, $C_2\times C_4^2$ x 18, $D_4:C_2^2$ x 16, $D_8:C_2$ x 16, $C_2^5$ x 14, $C_2.C_4^2$ x 14, $C_2\times \SD_{16}$ x 13, $C_2\times D_8$ x 12, $C_2^2:Q_8$ x 10, $D_4:C_4$ x 9, $\OD_{16}:C_2$ x 7, $C_2^2:C_8$ x 7, $C_2^2\times Q_8$ x 6, $C_4^2:C_2$ x 6, $C_2^2\times C_8$ x 4, $C_4^2:C_2$ x 4, $C_2^2.D_4$ x 4, $C_4\wr C_2$ x 4, $C_4\times Q_8$ x 2, $C_4:C_8$ x 2, $Q_8:C_4$ x 2, $C_8:C_4$, $C_4\times C_8$

Order 16: $C_2\times D_4$ x 203, $C_2^2\times C_4$ x 83, $C_2^4$ x 64, $C_2^2:C_4$ x 58, $D_4:C_2$ x 34, $C_4:C_4$ x 18, $C_4^2$ x 18, $C_2\times Q_8$ x 12, $\OD_{16}$ x 11, $C_2\times C_8$ x 9, $\SD_{16}$ x 5, $D_8$ x 4

Order 8: $C_2^3$ x 101, $C_2\times C_4$ x 74, $D_4$ x 55, $Q_8$ x 6, $C_8$ x 4

Order 4: $C_2^2$ x 56, $C_4$ x 15

Order 2: $C_2$ x 11

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 4096: $(C_2^2\times D_4^2):\SD_{16}$ ($C_1$)

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, $C_2^4.C_2^4.C_2^2.C_2$ ($C_2$), $C_2^5.D_4^2$ ($C_2$), $(C_2^2\times D_4^2):Q_8$ ($C_2$)

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

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

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

Order 128: $C_2\times C_4^3$ ($D_8:C_2$), $C_4^2:C_2^3$ ($C_2\times \SD_{16}$), $C_4^2:C_2^3$ ($C_2^2\wr C_2$)

Order 64: $D_4\times C_2^3$ ($C_2\wr C_2^2$), $D_4\times C_2^3$ ($C_4^2:C_2^2$), $C_2^2\times C_4^2$ ($C_2^2:\SD_{16}$)

Order 32: $C_2^5$ ($C_2\wr D_4$) x 2, $C_2^3\times C_4$ ($C_2^3:\SD_{16}$)

Order 16: $C_2^4$ ($C_2^4:\SD_{16}$)

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

Order 4: $C_2^2$ ($C_2^6:\SD_{16}$)

Order 2: $C_2$ ($C_2^5.C_2{\rm wrC}_2^2$)

Order 1: $C_1$ ($(C_2^2\times D_4^2):\SD_{16}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 4096: $(C_2^2\times D_4^2):\SD_{16}$ ($C_1$)

Order 2048: $C_2^4.C_2^4.C_2^2.C_2$ ($C_2$), $C_2^5.D_4^2$ ($C_2$), $(C_2\times C_4^3).D_8$ ($C_2$), $(C_2^2\times D_4^2):Q_8$ ($C_2$), $(C_2\times C_4^3):\SD_{16}$ ($C_2$)

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

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

Order 256: $C_2^2.D_4^2$ ($\SD_{16}$), $C_2^2\times D_4^2$ ($\SD_{16}$), $C_4^3:C_2^2$ ($C_2\times D_4$), $C_2^5.C_2^3$ ($C_2\times D_4$)

Order 128: $C_2\times C_4^3$ ($D_8:C_2$), $C_4^2:C_2^3$ ($C_2\times \SD_{16}$), $C_4^2:C_2^3$ ($C_2^2\wr C_2$)

Order 64: $D_4\times C_2^3$ ($C_2\wr C_2^2$), $D_4\times C_2^3$ ($C_4^2:C_2^2$), $C_2^2\times C_4^2$ ($C_2^2:\SD_{16}$)

Order 32: $C_2^5$ ($C_2\wr D_4$), $C_2^3\times C_4$ ($C_2^3:\SD_{16}$)

Order 16: $C_2^4$ ($C_2^4:\SD_{16}$)

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

Order 4: $C_2^2$ ($C_2^6:\SD_{16}$)

Order 2: $C_2$ ($C_2^5.C_2{\rm wrC}_2^2$)

Order 1: $C_1$ ($(C_2^2\times D_4^2):\SD_{16}$)

Series

Derived series

$(C_2^2\times D_4^2):\SD_{16}$

$\rhd$

$C_2^4.C_2^4.C_2$

$\rhd$

$C_2^3\times C_4$

$\rhd$

$C_1$

Chief series

$(C_2^2\times D_4^2):\SD_{16}$

$\rhd$

$C_2^5.D_4^2$

$\rhd$

$C_2^4.C_2^5.C_2$

$\rhd$

$C_2^4.C_2^4.C_2$

$\rhd$

$C_4^3:C_2^2$

$\rhd$

$C_4^2:C_2^3$

$\rhd$

$D_4\times C_2^3$

$\rhd$

$C_2^3\times C_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$(C_2^2\times D_4^2):\SD_{16}$

$\rhd$

$C_2^4.C_2^4.C_2$

$\rhd$

$C_4^2:C_2^3$

$\rhd$

$C_2^3\times C_4$

$\rhd$

$C_2^4$

$\rhd$

$C_2^3$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

$\lhd$

$C_2^3$

$\lhd$

$C_2^4$

$\lhd$

$D_4\times C_2^3$

$\lhd$

$C_4^2:C_2^3$

$\lhd$

$C_2^4.C_2^4.C_2$

$\lhd$

$(C_2^2\times D_4^2):\SD_{16}$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 3 larger groups in the database.

Character theory

Complex character table

See the $58 \times 58$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $54 \times 54$ rational character table.