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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
314928.qb.1.a1 $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ ✓ ✓ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $C_1$ $1$ ✓
314928.qb.2.A $C_9^4.C_6.C_2^2$ $2^{3} \cdot 3^{9}$ ✓ ✓ ✓ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $C_2$ $2$ ✓
314928.qb.2.B $C_3^4.C_3^5.C_4.C_2$ $2^{3} \cdot 3^{9}$ ✓ ✓ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $C_2$ $2$ ✓
314928.qb.3.a1 $C_9^4.C_2.D_4$ $2^{4} \cdot 3^{8}$ ✓ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $3$ $-$
314928.qb.4.A $C_9^4.C_6.C_2$ $2^{2} \cdot 3^{9}$ ✓ ✓ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $C_2^2$ $2^{2}$ ✓
314928.qb.4.B $C_9^4.C_6.C_2$ $2^{2} \cdot 3^{9}$ ✓ ✓ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $C_4$ $2^{2}$ ✓
314928.qb.4.C $C_9^4.C_6.C_2$ $2^{2} \cdot 3^{9}$ ✓ ✓ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $C_4$ $2^{2}$ ✓
314928.qb.4.D $(C_3\times C_9^3).C_6^2$ $2^{2} \cdot 3^{9}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{2}$ $-$
314928.qb.4.E $C_3^5.C_3^3.C_{12}$ $2^{2} \cdot 3^{9}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{2}$ $-$
314928.qb.6.A $C_9^4.C_2^3$ $2^{3} \cdot 3^{8}$ ✓ ✓ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $S_3$ $2 \cdot 3$
314928.qb.6.B $C_9^3.C_3^2.C_2^3$ $2^{3} \cdot 3^{8}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2 \cdot 3$ $-$
314928.qb.6.C $C_3^4.C_3^4.C_4.C_2$ $2^{3} \cdot 3^{8}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2 \cdot 3$ $-$
314928.qb.6.D $C_9^4.C_4.C_2$ $2^{3} \cdot 3^{8}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2 \cdot 3$ $-$
314928.qb.8.A $C_9^2.C_3^3.C_3.C_6$ $2 \cdot 3^{9}$ ✓ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $D_4$ $2^{3}$
314928.qb.8.B $C_9^4.C_6$ $2 \cdot 3^{9}$ ✓ ✓ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $C_2\times C_4$ $2^{3}$ ✓
314928.qb.8.C $C_9^2.C_3^3.C_3.C_6$ $2 \cdot 3^{9}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{3}$ $-$
314928.qb.8.D $C_9^2.C_3^3.C_3.C_6$ $2 \cdot 3^{9}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{3}$ $-$
314928.qb.9.a1 $C_3^4.C_3^3.C_2.D_4$ $2^{4} \cdot 3^{7}$ ✓ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $3^{2}$ $-$
314928.qb.12.A $C_9^4.C_2^2$ $2^{2} \cdot 3^{8}$ ✓ ✓ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $D_6$ $2^{2} \cdot 3$
314928.qb.12.B $C_9^4.C_2^2$ $2^{2} \cdot 3^{8}$ ✓ ✓ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $C_3:C_4$ $2^{2} \cdot 3$
314928.qb.12.C $C_9^4.C_2^2$ $2^{2} \cdot 3^{8}$ ✓ ✓ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $C_3:C_4$ $2^{2} \cdot 3$
314928.qb.12.D $C_9^2.C_3^3.C_6.C_2$ $2^{2} \cdot 3^{8}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{2} \cdot 3$ $-$
314928.qb.12.E $C_3^3.C_3^4.C_6.C_2$ $2^{2} \cdot 3^{8}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{2} \cdot 3$ $-$
314928.qb.12.F $C_9^2.C_3^3.C_6.C_2$ $2^{2} \cdot 3^{8}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{2} \cdot 3$ $-$
314928.qb.12.G $C_9^3.C_6^2$ $2^{2} \cdot 3^{8}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{2} \cdot 3$ $-$
314928.qb.12.H $C_9^3.C_{18}.C_2$ $2^{2} \cdot 3^{8}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{2} \cdot 3$ $-$
314928.qb.12.I $C_9^4.C_2^2$ $2^{2} \cdot 3^{8}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{2} \cdot 3$ $-$
314928.qb.12.J $C_9^2.C_3^3.C_{12}$ $2^{2} \cdot 3^{8}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{2} \cdot 3$ $-$
314928.qb.12.K $C_3^5.C_3^3.C_4$ $2^{2} \cdot 3^{8}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{2} \cdot 3$ $-$
314928.qb.12.L $C_3^3.C_3^4.C_{12}$ $2^{2} \cdot 3^{8}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{2} \cdot 3$ $-$
314928.qb.12.M $C_9^3.C_{36}$ $2^{2} \cdot 3^{8}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{2} \cdot 3$ $-$
314928.qb.12.N $C_9^3.(C_6\times S_3)$ $2^{2} \cdot 3^{8}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{2} \cdot 3$ $-$
314928.qb.12.O $(C_3\times C_9^3).C_6.C_2$ $2^{2} \cdot 3^{8}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{2} \cdot 3$ $-$
314928.qb.12.P $C_9^3.(C_6\times S_3)$ $2^{2} \cdot 3^{8}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{2} \cdot 3$ $-$
314928.qb.12.Q $(C_3\times C_9^3).C_6.C_2$ $2^{2} \cdot 3^{8}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{2} \cdot 3$ $-$
314928.qb.12.R $C_3^3.C_3^5.C_2^2$ $2^{2} \cdot 3^{8}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{2} \cdot 3$ $-$
314928.qb.12.S $C_9^3.(C_6\times S_3)$ $2^{2} \cdot 3^{8}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{2} \cdot 3$ $-$
314928.qb.16.a1 $C_9^4.C_3$ $3^{9}$ $3$ ✓ ✓ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $C_2^2:C_4$ $2^{4}$
314928.qb.18.A $C_3^2.C_3^5.C_2^3$ $2^{3} \cdot 3^{7}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2 \cdot 3^{2}$ $-$
314928.qb.18.B $C_3^3.C_3^4.C_2^3$ $2^{3} \cdot 3^{7}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2 \cdot 3^{2}$ $-$
314928.qb.18.C $C_9^2.C_3^3.C_2^3$ $2^{3} \cdot 3^{7}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2 \cdot 3^{2}$ $-$
314928.qb.18.D $C_9^3.C_6.C_2^2$ $2^{3} \cdot 3^{7}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2 \cdot 3^{2}$ $-$
314928.qb.18.E $C_3^2.C_3^5.C_4.C_2$ $2^{3} \cdot 3^{7}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2 \cdot 3^{2}$ $-$
314928.qb.18.F $C_3^3.C_3^4.C_4.C_2$ $2^{3} \cdot 3^{7}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2 \cdot 3^{2}$ $-$
314928.qb.18.G $C_9^3.D_6.C_2$ $2^{3} \cdot 3^{7}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2 \cdot 3^{2}$ $-$
314928.qb.18.H $(C_3\times C_9^3).C_4.C_2$ $2^{3} \cdot 3^{7}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2 \cdot 3^{2}$ $-$
314928.qb.24.A $C_9^4.C_2$ $2 \cdot 3^{8}$ ✓ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $C_3:D_4$ $2^{3} \cdot 3$
314928.qb.24.B $C_9^4.C_2$ $2 \cdot 3^{8}$ ✓ ✓ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $C_6:C_4$ $2^{3} \cdot 3$
314928.qb.24.C $C_9.C_3^4.C_3.C_6$ $2 \cdot 3^{8}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{3} \cdot 3$ $-$
314928.qb.24.D $C_3^2.C_3^4.C_3.C_6$ $2 \cdot 3^{8}$ $C_9^4.C_6.D_4$ $2^{4} \cdot 3^{9}$ $2^{3} \cdot 3$ $-$
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