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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.hr.1.a1 $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ ✓ ✓ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $C_1$ $1$ ✓
11664.hr.2.a1 $\He_3^2:C_2^3$ $2^{3} \cdot 3^{6}$ ✓ ✓ ✓ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $C_2$ $2$ ✓
11664.hr.2.b1 $C_3^4.C_3^2.C_2^3$ $2^{3} \cdot 3^{6}$ ✓ ✓ ✓ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $C_2$ $2$ ✓
11664.hr.2.c1 $C_2\times \He_3^2:C_4$ $2^{3} \cdot 3^{6}$ ✓ ✓ ✓ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $C_2$ $2$ ✓
11664.hr.2.d1 $\He_3^2:D_4$ $2^{3} \cdot 3^{6}$ ✓ ✓ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $C_2$ $2$ ✓
11664.hr.4.a1 $C_3^4.C_3:S_3.C_2$ $2^{2} \cdot 3^{6}$ ✓ ✓ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $C_2^2$ $2^{2}$ ✓
11664.hr.4.b1 $C_3^4:S_3^2$ $2^{2} \cdot 3^{6}$ ✓ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $C_2^2$ $2^{2}$ ✓
11664.hr.4.c1 $C_3^4.C_3:S_3.C_2$ $2^{2} \cdot 3^{6}$ ✓ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $C_2^2$ $2^{2}$ ✓
11664.hr.4.d1 $\He_3^2:C_4$ $2^{2} \cdot 3^{6}$ ✓ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $C_2^2$ $2^{2}$ ✓
11664.hr.4.e1 $\He_3^2:C_2^2$ $2^{2} \cdot 3^{6}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2}$ $-$
11664.hr.4.f1 $C_3^4.C_6^2$ $2^{2} \cdot 3^{6}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2}$ $-$
11664.hr.4.g1 $C_3^4:S_3^2$ $2^{2} \cdot 3^{6}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2}$ $-$
11664.hr.4.h1 $(C_3^2\times \He_3).D_6$ $2^{2} \cdot 3^{6}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2}$ $-$
11664.hr.6.a1 $C_2\times C_3^3:S_3^2$ $2^{3} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2 \cdot 3$ $-$
11664.hr.6.b1 $C_2\times C_3^3:S_3^2$ $2^{3} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2 \cdot 3$ $-$
11664.hr.6.c1 $C_2\times C_3^4:D_6$ $2^{3} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2 \cdot 3$ $-$
11664.hr.6.d1 $C_2\times C_3^3:S_3^2$ $2^{3} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2 \cdot 3$ $-$
11664.hr.8.a1 $C_2\times \He_3^2$ $2 \cdot 3^{6}$ ✓ ✓ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $D_4$ $2^{3}$
11664.hr.8.b1 $\He_3^2:C_2$ $2 \cdot 3^{6}$ ✓ ✓ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $C_2^3$ $2^{3}$ ✓
11664.hr.8.c1 $\He_3^2:C_2$ $2 \cdot 3^{6}$ ✓ ✓ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $D_4$ $2^{3}$
11664.hr.8.d1 $\He_3\wr C_2$ $2 \cdot 3^{6}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{3}$ $-$
11664.hr.8.e1 $\He_3^2:C_2$ $2 \cdot 3^{6}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{3}$ $-$
11664.hr.9.a1 $C_2\times C_3^3:D_{12}$ $2^{4} \cdot 3^{4}$ ✓ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $3^{2}$ $-$
11664.hr.9.b1 $(C_6\times \He_3):D_4$ $2^{4} \cdot 3^{4}$ ✓ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $3^{2}$ $-$
11664.hr.12.a1 $C_3^3.C_6^2$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.b1 $C_3^3.C_6^2$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.c1 $C_3^4:D_6$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.d1 $C_3^4:D_6$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.e1 $C_3^4:D_6$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.f1 $C_3^4:D_6$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.g1 $C_3^4:D_6$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.h1 $C_3^4:D_6$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.i1 $C_3^4:D_6$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.j1 $C_3^4:D_6$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.k1 $C_3^4:D_6$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.l1 $C_3^3:C_6^2$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.m1 $C_3^3:C_6^2$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.n1 $C_2\times C_3^4:C_6$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.o1 $C_3^4:D_6$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.p1 $C_3^3:S_3^2$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.q1 $C_3^3:S_3^2$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.r1 $C_3^3:S_3^2$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.s1 $C_3^3:S_3^2$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.t1 $C_3^4:D_6$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.u1 $C_3^3:S_3^2$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.v1 $C_3^4:D_6$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.12.w1 $C_3^3:S_3^2$ $2^{2} \cdot 3^{5}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2^{2} \cdot 3$ $-$
11664.hr.16.a1 $\He_3^2$ $3^{6}$ $3$ ✓ ✓ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $C_2\times D_4$ $2^{4}$
11664.hr.18.a1 $C_3^4:C_2^3$ $2^{3} \cdot 3^{4}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2 \cdot 3^{2}$ $-$
11664.hr.18.b1 $C_3^4:C_2^3$ $2^{3} \cdot 3^{4}$ $C_2\times \He_3^2:D_4$ $2^{4} \cdot 3^{6}$ $2 \cdot 3^{2}$ $-$
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