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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
209952.kc.1.a1 $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ ✓ ✓ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $C_1$ $1$ ✓
209952.kc.2.A $C_3^6:(S_3\times \SL(2,3))$ $2^{4} \cdot 3^{8}$ ✓ ✓ ✓ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $C_2$ $2$ ✓
209952.kc.2.B $C_3^6:(C_3\times \GL(2,3))$ $2^{4} \cdot 3^{8}$ ✓ ✓ ✓ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $C_2$ $2$ ✓
209952.kc.2.C $C_3^6:(C_3:\GL(2,3))$ $2^{4} \cdot 3^{8}$ ✓ ✓ ✓ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $C_2$ $2$ ✓
209952.kc.3.A $C_3^6.Q_8.D_6$ $2^{5} \cdot 3^{7}$ ✓ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $3$ $-$
209952.kc.3.B $C_3^6.C_{12}.C_2^3$ $2^{5} \cdot 3^{7}$ ✓ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $3$ $-$
209952.kc.4.A $C_3^6.Q_8.C_3^2$ $2^{3} \cdot 3^{8}$ ✓ ✓ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $C_2^2$ $2^{2}$ ✓
209952.kc.4.B $C_3^4.C_3^4.C_2^3$ $2^{3} \cdot 3^{8}$ ✓ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{2}$ $-$
209952.kc.6.A $C_3^6.(S_3\times Q_8)$ $2^{4} \cdot 3^{7}$ ✓ ✓ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $S_3$ $2 \cdot 3$
209952.kc.6.B $C_3^6.Q_8.S_3$ $2^{4} \cdot 3^{7}$ ✓ ✓ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $S_3$ $2 \cdot 3$
209952.kc.6.C $C_3^6.Q_8.C_6$ $2^{4} \cdot 3^{7}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2 \cdot 3$ $-$
209952.kc.6.D $C_3^4.S_3^3.C_2$ $2^{4} \cdot 3^{7}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2 \cdot 3$ $-$
209952.kc.6.E $C_3^4:(S_3\times F_9)$ $2^{4} \cdot 3^{7}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2 \cdot 3$ $-$
209952.kc.6.F $C_3^6.C_6.D_4$ $2^{4} \cdot 3^{7}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2 \cdot 3$ $-$
209952.kc.6.G $C_3^6.(C_3\times \SD_{16})$ $2^{4} \cdot 3^{7}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2 \cdot 3$ $-$
209952.kc.6.H $C_3^5:S_3.D_{12}$ $2^{4} \cdot 3^{7}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2 \cdot 3$ $-$
209952.kc.6.I $C_3^5.F_9:C_2$ $2^{4} \cdot 3^{7}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2 \cdot 3$ $-$
209952.kc.6.J $C_3^6.Q_8.S_3$ $2^{4} \cdot 3^{7}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2 \cdot 3$ $-$
209952.kc.6.K $C_3^6.Q_8.S_3$ $2^{4} \cdot 3^{7}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2 \cdot 3$ $-$
209952.kc.8.A $C_3^6.(C_6\times S_3)$ $2^{2} \cdot 3^{8}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{3}$ $-$
209952.kc.8.B $C_3^4.C_3^4.C_2^2$ $2^{2} \cdot 3^{8}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{3}$ $-$
209952.kc.8.C $C_3^6.S_3^2$ $2^{2} \cdot 3^{8}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{3}$ $-$
209952.kc.8.D $C_3^5.C_3^3.C_2^2$ $2^{2} \cdot 3^{8}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{3}$ $-$
209952.kc.8.E $C_3^6.S_3^2$ $2^{2} \cdot 3^{8}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{3}$ $-$
209952.kc.8.F $C_3^6.(C_6\times S_3)$ $2^{2} \cdot 3^{8}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{3}$ $-$
209952.kc.8.G $C_3^4.C_3^3.D_6$ $2^{2} \cdot 3^{8}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{3}$ $-$
209952.kc.9.A $S_3\times C_3^4:\GL(2,3)$ $2^{5} \cdot 3^{6}$ ✓ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $3^{2}$ $-$
209952.kc.9.B $C_3^6.\SD_{16}.C_2$ $2^{5} \cdot 3^{6}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $3^{2}$ $-$
209952.kc.9.C $C_3^4.Q_8.S_3^2$ $2^{5} \cdot 3^{6}$ ✓ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $3^{2}$ $-$
209952.kc.12.A $C_3^6.C_{12}.C_2$ $2^{3} \cdot 3^{7}$ ✓ ✓ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $D_6$ $2^{2} \cdot 3$
209952.kc.12.B $C_3^6.Q_8.C_3$ $2^{3} \cdot 3^{7}$ ✓ ✓ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $D_6$ $2^{2} \cdot 3$
209952.kc.12.C $C_3^4.C_3^3.C_2^3$ $2^{3} \cdot 3^{7}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{2} \cdot 3$ $-$
209952.kc.12.D $C_3\wr C_3.S_3^3$ $2^{3} \cdot 3^{7}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{2} \cdot 3$ $-$
209952.kc.12.E $C_3.C_3^5.D_6.C_2$ $2^{3} \cdot 3^{7}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{2} \cdot 3$ $-$
209952.kc.12.F $(C_3\times \He_3).S_3^3$ $2^{3} \cdot 3^{7}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{2} \cdot 3$ $-$
209952.kc.12.G $C_3^5.C_3:S_3.C_2^2$ $2^{3} \cdot 3^{7}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{2} \cdot 3$ $-$
209952.kc.12.H $C_3^6.C_3.D_4$ $2^{3} \cdot 3^{7}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{2} \cdot 3$ $-$
209952.kc.12.I $C_3^6.C_6.C_2^2$ $2^{3} \cdot 3^{7}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{2} \cdot 3$ $-$
209952.kc.12.J $C_3^6.D_{12}$ $2^{3} \cdot 3^{7}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{2} \cdot 3$ $-$
209952.kc.12.K $C_3^6.C_3.Q_8$ $2^{3} \cdot 3^{7}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{2} \cdot 3$ $-$
209952.kc.12.L $C_3^6.C_{24}$ $2^{3} \cdot 3^{7}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{2} \cdot 3$ $-$
209952.kc.12.M $C_3^4:C_3.F_9$ $2^{3} \cdot 3^{7}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{2} \cdot 3$ $-$
209952.kc.12.N $C_3^6.\SL(2,3)$ $2^{3} \cdot 3^{7}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{2} \cdot 3$ $-$
209952.kc.16.A $C_3^5.C_3^3.C_2$ $2 \cdot 3^{8}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{4}$ $-$
209952.kc.16.B $C_3^5.C_3^3.C_2$ $2 \cdot 3^{8}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{4}$ $-$
209952.kc.16.C $C_3^5.C_3^3.C_2$ $2 \cdot 3^{8}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{4}$ $-$
209952.kc.16.D $C_3^4.\He_3.C_6$ $2 \cdot 3^{8}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{4}$ $-$
209952.kc.16.E $C_3^6.C_3.C_6$ $2 \cdot 3^{8}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{4}$ $-$
209952.kc.16.F $C_3^4.C_3^4.C_2$ $2 \cdot 3^{8}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{4}$ $-$
209952.kc.16.G $C_3^6.C_3.S_3$ $2 \cdot 3^{8}$ $C_3^6.(S_3\times \GL(2,3))$ $2^{5} \cdot 3^{8}$ $2^{4}$ $-$
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