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Subgroup label Subgroup order Cyclic Abelian Solvable
Normal Characteristic Perfect Maximal Central Non-trivial proper
Ambient label Ambient order Direct product Semidirect product
Quotient label Subgroup index Cyclic quotient Abelian quotient Solvable quotient Maximal quotient
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Results (1-50 of 275379753 matches)

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Label Subgroup Ambient Quotient
Name Order norm char cyc ab solv max perf cent Name Order Name Size cyc ab solv max
1.1.1.a1.a1 $C_1$ $1$ ✓ ✓ ✓ ✓ ✓ ✓ ✓ $C_1$ $1$ $C_1$ $1$ ✓ ✓ ✓
2.1.1.a1.a1 $C_2$ $2$ ✓ ✓ ✓ ✓ ✓ ✓ $C_2$ $2$ $C_1$ $1$ ✓ ✓ ✓ ✓
2.1.2.a1.a1 $C_1$ $1$ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ $C_2$ $2$ $C_2$ $2$ ✓ ✓ ✓
3.1.1.a1.a1 $C_3$ $3$ ✓ ✓ ✓ ✓ ✓ ✓ $C_3$ $3$ $C_1$ $1$ ✓ ✓ ✓ ✓
3.1.3.a1.a1 $C_1$ $1$ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ $C_3$ $3$ $C_3$ $3$ ✓ ✓ ✓
4.1.1.a1.a1 $C_4$ $2^{2}$ ✓ ✓ ✓ ✓ ✓ ✓ $C_4$ $2^{2}$ $C_1$ $1$ ✓ ✓ ✓
4.1.2.a1.a1 $C_2$ $2$ ✓ ✓ ✓ ✓ ✓ ✓ ✓ $C_4$ $2^{2}$ $C_2$ $2$ ✓ ✓ ✓ ✓
4.1.4.a1.a1 $C_1$ $1$ ✓ ✓ ✓ ✓ ✓ ✓ ✓ $C_4$ $2^{2}$ $C_4$ $2^{2}$ ✓ ✓ ✓
4.2.1.a1.a1 $C_2^2$ $2^{2}$ ✓ ✓ ✓ ✓ ✓ $C_2^2$ $2^{2}$ $C_1$ $1$ ✓ ✓ ✓
4.2.2.a1.c1 $C_2$ $2$ ✓ ✓ ✓ ✓ ✓ ✓ $C_2^2$ $2^{2}$ $C_2$ $2$ ✓ ✓ ✓ ✓
4.2.2.a1.a1 $C_2$ $2$ ✓ ✓ ✓ ✓ ✓ ✓ $C_2^2$ $2^{2}$ $C_2$ $2$ ✓ ✓ ✓ ✓
4.2.2.a1.b1 $C_2$ $2$ ✓ ✓ ✓ ✓ ✓ ✓ $C_2^2$ $2^{2}$ $C_2$ $2$ ✓ ✓ ✓ ✓
4.2.4.a1.a1 $C_1$ $1$ ✓ ✓ ✓ ✓ ✓ ✓ ✓ $C_2^2$ $2^{2}$ $C_2^2$ $2^{2}$ ✓ ✓
5.1.1.a1.a1 $C_5$ $5$ ✓ ✓ ✓ ✓ ✓ ✓ $C_5$ $5$ $C_1$ $1$ ✓ ✓ ✓ ✓
5.1.5.a1.a1 $C_1$ $1$ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ $C_5$ $5$ $C_5$ $5$ ✓ ✓ ✓
6.1.1.a1.a1 $S_3$ $2 \cdot 3$ ✓ ✓ ✓ $S_3$ $2 \cdot 3$ $C_1$ $1$ ✓ ✓ ✓
6.1.2.a1.a1 $C_3$ $3$ ✓ ✓ ✓ ✓ ✓ ✓ $S_3$ $2 \cdot 3$ $C_2$ $2$ ✓ ✓ ✓ ✓
6.1.3.a1.a1 $C_2$ $2$ ✓ ✓ ✓ ✓ $S_3$ $2 \cdot 3$ $3$ $-$ $-$ $-$
6.1.6.a1.a1 $C_1$ $1$ ✓ ✓ ✓ ✓ ✓ ✓ ✓ $S_3$ $2 \cdot 3$ $S_3$ $2 \cdot 3$ ✓
6.2.1.a1.a1 $C_6$ $2 \cdot 3$ ✓ ✓ ✓ ✓ ✓ ✓ $C_6$ $2 \cdot 3$ $C_1$ $1$ ✓ ✓ ✓
6.2.2.a1.a1 $C_3$ $3$ ✓ ✓ ✓ ✓ ✓ ✓ ✓ $C_6$ $2 \cdot 3$ $C_2$ $2$ ✓ ✓ ✓ ✓
6.2.3.a1.a1 $C_2$ $2$ ✓ ✓ ✓ ✓ ✓ ✓ ✓ $C_6$ $2 \cdot 3$ $C_3$ $3$ ✓ ✓ ✓ ✓
6.2.6.a1.a1 $C_1$ $1$ ✓ ✓ ✓ ✓ ✓ ✓ ✓ $C_6$ $2 \cdot 3$ $C_6$ $2 \cdot 3$ ✓ ✓ ✓
7.1.1.a1.a1 $C_7$ $7$ ✓ ✓ ✓ ✓ ✓ ✓ $C_7$ $7$ $C_1$ $1$ ✓ ✓ ✓ ✓
7.1.7.a1.a1 $C_1$ $1$ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ $C_7$ $7$ $C_7$ $7$ ✓ ✓ ✓
8.1.1.a1.a1 $C_8$ $2^{3}$ ✓ ✓ ✓ ✓ ✓ ✓ $C_8$ $2^{3}$ $C_1$ $1$ ✓ ✓ ✓
8.1.2.a1.a1 $C_4$ $2^{2}$ ✓ ✓ ✓ ✓ ✓ ✓ ✓ $C_8$ $2^{3}$ $C_2$ $2$ ✓ ✓ ✓
8.1.4.a1.a1 $C_2$ $2$ ✓ ✓ ✓ ✓ ✓ ✓ $C_8$ $2^{3}$ $C_4$ $2^{2}$ ✓ ✓ ✓ ✓
8.1.8.a1.a1 $C_1$ $1$ ✓ ✓ ✓ ✓ ✓ ✓ ✓ $C_8$ $2^{3}$ $C_8$ $2^{3}$ ✓ ✓ ✓
8.2.1.a1.a1 $C_2\times C_4$ $2^{3}$ ✓ ✓ ✓ ✓ ✓ $C_2\times C_4$ $2^{3}$ $C_1$ $1$ ✓ ✓ ✓
8.2.2.b1.a1 $C_4$ $2^{2}$ ✓ ✓ ✓ ✓ ✓ ✓ $C_2\times C_4$ $2^{3}$ $C_2$ $2$ ✓ ✓ ✓
8.2.2.b1.b1 $C_4$ $2^{2}$ ✓ ✓ ✓ ✓ ✓ ✓ $C_2\times C_4$ $2^{3}$ $C_2$ $2$ ✓ ✓ ✓
8.2.2.a1.a1 $C_2^2$ $2^{2}$ ✓ ✓ ✓ ✓ ✓ ✓ $C_2\times C_4$ $2^{3}$ $C_2$ $2$ ✓ ✓ ✓
8.2.4.a1.b1 $C_2$ $2$ ✓ ✓ ✓ ✓ ✓ $C_2\times C_4$ $2^{3}$ $C_4$ $2^{2}$ ✓ ✓ ✓ ✓
8.2.4.a1.a1 $C_2$ $2$ ✓ ✓ ✓ ✓ ✓ $C_2\times C_4$ $2^{3}$ $C_4$ $2^{2}$ ✓ ✓ ✓ ✓
8.2.4.b1.a1 $C_2$ $2$ ✓ ✓ ✓ ✓ ✓ ✓ $C_2\times C_4$ $2^{3}$ $C_2^2$ $2^{2}$ ✓ ✓ ✓
8.2.8.a1.a1 $C_1$ $1$ ✓ ✓ ✓ ✓ ✓ ✓ ✓ $C_2\times C_4$ $2^{3}$ $C_2\times C_4$ $2^{3}$ ✓ ✓
8.3.1.a1.a1 $D_4$ $2^{3}$ ✓ ✓ ✓ $D_4$ $2^{3}$ $C_1$ $1$ ✓ ✓ ✓
8.3.2.b1.a1 $C_4$ $2^{2}$ ✓ ✓ ✓ ✓ ✓ ✓ $D_4$ $2^{3}$ $C_2$ $2$ ✓ ✓ ✓
8.3.2.a1.a1 $C_2^2$ $2^{2}$ ✓ ✓ ✓ ✓ $D_4$ $2^{3}$ $C_2$ $2$ ✓ ✓ ✓
8.3.2.a1.b1 $C_2^2$ $2^{2}$ ✓ ✓ ✓ ✓ $D_4$ $2^{3}$ $C_2$ $2$ ✓ ✓ ✓
8.3.4.b1.b1 $C_2$ $2$ ✓ ✓ ✓ $D_4$ $2^{3}$ $2^{2}$ $-$ $-$ $-$
8.3.4.b1.a1 $C_2$ $2$ ✓ ✓ ✓ $D_4$ $2^{3}$ $2^{2}$ $-$ $-$ $-$
8.3.4.a1.a1 $C_2$ $2$ ✓ ✓ ✓ ✓ ✓ ✓ $D_4$ $2^{3}$ $C_2^2$ $2^{2}$ ✓ ✓ ✓
8.3.8.a1.a1 $C_1$ $1$ ✓ ✓ ✓ ✓ ✓ ✓ ✓ $D_4$ $2^{3}$ $D_4$ $2^{3}$ ✓
8.4.1.a1.a1 $Q_8$ $2^{3}$ ✓ ✓ ✓ $Q_8$ $2^{3}$ $C_1$ $1$ ✓ ✓ ✓
8.4.2.a1.c1 $C_4$ $2^{2}$ ✓ ✓ ✓ ✓ ✓ $Q_8$ $2^{3}$ $C_2$ $2$ ✓ ✓ ✓
8.4.2.a1.a1 $C_4$ $2^{2}$ ✓ ✓ ✓ ✓ ✓ $Q_8$ $2^{3}$ $C_2$ $2$ ✓ ✓ ✓
8.4.2.a1.b1 $C_4$ $2^{2}$ ✓ ✓ ✓ ✓ ✓ $Q_8$ $2^{3}$ $C_2$ $2$ ✓ ✓ ✓
8.4.4.a1.a1 $C_2$ $2$ ✓ ✓ ✓ ✓ ✓ ✓ $Q_8$ $2^{3}$ $C_2^2$ $2^{2}$ ✓ ✓ ✓
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