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Subgroup label Subgroup order Cyclic Abelian Solvable
Normal Characteristic Perfect Maximal Central
Ambient label Ambient order Direct product Semidirect product
Quotient label Subgroup index Cyclic quotient Abelian quotient Solvable quotient
Maximal quotient Non-trivial proper Hall subgroup Sylow subgroup
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Label Subgroup Ambient Quotient
Name Order Sylow norm char max cent ab Name Order Name Size max ab
11664.kv.1.a1 $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ ✓ ✓ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $C_1$ $1$ ✓
11664.kv.2.A $C_3^3.S_3^3$ $2^{3} \cdot 3^{6}$ ✓ ✓ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $C_2$ $2$ ✓
11664.kv.2.B $C_3^2.C_3^4.C_4.C_2$ $2^{3} \cdot 3^{6}$ ✓ ✓ ✓ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $C_2$ $2$ ✓
11664.kv.2.C $C_3^4.S_3\wr C_2$ $2^{3} \cdot 3^{6}$ ✓ ✓ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $C_2$ $2$ ✓
11664.kv.2.D $C_3^2.C_3^4.D_4$ $2^{3} \cdot 3^{6}$ ✓ ✓ ✓ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $C_2$ $2$ ✓
11664.kv.2.E $C_3^2.C_3^4.D_4$ $2^{3} \cdot 3^{6}$ ✓ ✓ ✓ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $C_2$ $2$ ✓
11664.kv.4.A $C_3^4:C_3.D_6$ $2^{2} \cdot 3^{6}$ ✓ ✓ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $C_2^2$ $2^{2}$ ✓
11664.kv.4.B $C_3^4:C_3.D_6$ $2^{2} \cdot 3^{6}$ ✓ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $C_2^2$ $2^{2}$ ✓
11664.kv.4.C $(C_3^2\times \He_3).D_6$ $2^{2} \cdot 3^{6}$ ✓ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $C_2^2$ $2^{2}$ ✓
11664.kv.4.D $C_3^2.C_3^4.C_4$ $2^{2} \cdot 3^{6}$ ✓ ✓ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $C_2^2$ $2^{2}$ ✓
11664.kv.4.E $C_3^4.C_3^2.C_4$ $2^{2} \cdot 3^{6}$ ✓ ✓ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $C_2^2$ $2^{2}$ ✓
11664.kv.4.F $C_3^4.(C_6\times S_3)$ $2^{2} \cdot 3^{6}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{2}$ $-$
11664.kv.6.a1 $C_3^2.S_3^3$ $2^{3} \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2 \cdot 3$ $-$
11664.kv.8.A $\He_3\wr C_2$ $2 \cdot 3^{6}$ ✓ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $D_4$ $2^{3}$
11664.kv.8.B $\He_3^2:C_2$ $2 \cdot 3^{6}$ ✓ ✓ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $C_2^3$ $2^{3}$ ✓
11664.kv.8.C $\He_3\wr C_2$ $2 \cdot 3^{6}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{3}$ $-$
11664.kv.8.D $\He_3^2:C_2$ $2 \cdot 3^{6}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{3}$ $-$
11664.kv.9.a1 $S_3\times \He_3:D_4$ $2^{4} \cdot 3^{4}$ ✓ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $3^{2}$ $-$
11664.kv.12.A $C_3^3:S_3^2$ $2^{2} \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.kv.12.B $C_3^4:D_6$ $2^{2} \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.kv.12.C $C_3^4:D_6$ $2^{2} \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.kv.12.D $C_3^3:S_3^2$ $2^{2} \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.kv.12.E $C_3^3:S_3^2$ $2^{2} \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.kv.12.F $C_3^3:S_3^2$ $2^{2} \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.kv.12.G $C_3^3:S_3^2$ $2^{2} \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.kv.12.H $C_3^4:D_6$ $2^{2} \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.kv.12.I $C_3^3:S_3^2$ $2^{2} \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.kv.16.a1 $\He_3^2$ $3^{6}$ $3$ ✓ ✓ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $C_2\times D_4$ $2^{4}$
11664.kv.18.A $C_3:S_3^3$ $2^{3} \cdot 3^{4}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2 \cdot 3^{2}$ $-$
11664.kv.18.B $C_3.S_3^3$ $2^{3} \cdot 3^{4}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2 \cdot 3^{2}$ $-$
11664.kv.18.C $C_3.S_3^3$ $2^{3} \cdot 3^{4}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2 \cdot 3^{2}$ $-$
11664.kv.18.D $S_3\times \He_3:C_4$ $2^{3} \cdot 3^{4}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2 \cdot 3^{2}$ $-$
11664.kv.18.E $C_3^3:D_{12}$ $2^{3} \cdot 3^{4}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2 \cdot 3^{2}$ $-$
11664.kv.18.F $C_3^2.\SOPlus(4,2)$ $2^{3} \cdot 3^{4}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2 \cdot 3^{2}$ $-$
11664.kv.18.G $C_3\times \He_3:D_4$ $2^{3} \cdot 3^{4}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2 \cdot 3^{2}$ $-$
11664.kv.18.H $C_3\times \He_3:D_4$ $2^{3} \cdot 3^{4}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2 \cdot 3^{2}$ $-$
11664.kv.18.I $\He_3:D_{12}$ $2^{3} \cdot 3^{4}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2 \cdot 3^{2}$ $-$
11664.kv.18.J $\He_3:D_{12}$ $2^{3} \cdot 3^{4}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2 \cdot 3^{2}$ $-$
11664.kv.24.A $C_3^4:C_6$ $2 \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{3} \cdot 3$ $-$
11664.kv.24.B $C_3^4:S_3$ $2 \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{3} \cdot 3$ $-$
11664.kv.24.C $C_3^4:C_6$ $2 \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{3} \cdot 3$ $-$
11664.kv.24.D $C_3^4:C_6$ $2 \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{3} \cdot 3$ $-$
11664.kv.24.E $C_3^4:S_3$ $2 \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{3} \cdot 3$ $-$
11664.kv.24.F $C_3^4:S_3$ $2 \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{3} \cdot 3$ $-$
11664.kv.24.G $C_3^4:S_3$ $2 \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{3} \cdot 3$ $-$
11664.kv.24.H $C_3^4:S_3$ $2 \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{3} \cdot 3$ $-$
11664.kv.24.I $C_3^4:S_3$ $2 \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{3} \cdot 3$ $-$
11664.kv.24.J $C_3^4:S_3$ $2 \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{3} \cdot 3$ $-$
11664.kv.24.K $C_3^4:S_3$ $2 \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{3} \cdot 3$ $-$
11664.kv.24.L $C_3^4:C_6$ $2 \cdot 3^{5}$ $\He_3^2.(C_2\times D_4)$ $2^{4} \cdot 3^{6}$ $2^{3} \cdot 3$ $-$
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