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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
41472.br.1.a1.a1 $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ ✓ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_1$ $1$ ✓
41472.br.2._.A $C_6.S_3^3.C_2^4$ $2^{8} \cdot 3^{4}$ ✓ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2$ $2$ ✓
41472.br.2._.B $C_6.S_3^3.C_2^4$ $2^{8} \cdot 3^{4}$ ✓ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2$ $2$ ✓
41472.br.2._.C $(C_6^2.D_6^2):C_4$ $2^{8} \cdot 3^{4}$ ✓ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2$ $2$ ✓
41472.br.2._.D $C_3^4:(C_2^6.C_4)$ $2^{8} \cdot 3^{4}$ ✓ ✓ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2$ $2$ ✓
41472.br.2._.E $C_2^2.S_3\wr C_2^2$ $2^{8} \cdot 3^{4}$ ✓ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2$ $2$ ✓
41472.br.2._.F $C_2^2.S_3\wr C_2^2$ $2^{8} \cdot 3^{4}$ ✓ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2$ $2$ ✓
41472.br.2._.G $C_3^3:C_{12}.D_4^2$ $2^{8} \cdot 3^{4}$ ✓ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2$ $2$ ✓
41472.br.2._.H $C_3^4.C_2^6.C_2^2$ $2^{8} \cdot 3^{4}$ ✓ ✓ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2$ $2$ ✓
41472.br.2._.I $C_6.S_3^3.C_2^4$ $2^{8} \cdot 3^{4}$ ✓ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2$ $2$ ✓
41472.br.2._.J $C_6.S_3^3.C_2^4$ $2^{8} \cdot 3^{4}$ ✓ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2$ $2$ ✓
41472.br.2._.K $(C_6^2.D_6^2):C_4$ $2^{8} \cdot 3^{4}$ ✓ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2$ $2$ ✓
41472.br.2._.L $C_3^4.C_2^6.C_2^2$ $2^{8} \cdot 3^{4}$ ✓ ✓ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2$ $2$ ✓
41472.br.2._.M $C_2^2.S_3\wr C_2^2$ $2^{8} \cdot 3^{4}$ ✓ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2$ $2$ ✓
41472.br.2._.N $C_2^2.S_3\wr C_2^2$ $2^{8} \cdot 3^{4}$ ✓ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2$ $2$ ✓
41472.br.2._.O $C_3^3:C_{12}.D_4^2$ $2^{8} \cdot 3^{4}$ ✓ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2$ $2$ ✓
41472.br.4._.A $C_2\times C_3^4.C_2^5.C_2$ $2^{7} \cdot 3^{4}$ ✓ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.B $C_3^4.C_2^5.C_2^2$ $2^{7} \cdot 3^{4}$ ✓ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.C $C_6^3:C_6:D_4$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.D $C_6^3:C_6.D_4$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.E $C_2\times C_3^4.C_2^4.C_2^2$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.F $C_3^4.C_2^5.C_2^2$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.G $C_6^2.D_6\wr C_2$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.H $C_6^2.D_6\wr C_2$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.I $C_3^4:C_4^2:C_2^3$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.J $C_6^2:S_3^2.D_4$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.K $C_3^4:C_4^2:C_2^3$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.L $C_6^2:S_3^2.D_4$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.M $C_3:S_3^3:\OD_{16}$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.N $C_6^2:S_3^2.D_4$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.O $C_6^2:S_3^2.D_4$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.P $C_3:S_3^3:\OD_{16}$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.Q $C_3^4.C_2^5.C_2^2$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.R $C_3^4.C_2^5.C_2^2$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.S $C_6^2.D_6\wr C_2$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.T $C_6^2.D_6\wr C_2$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.U $C_3:S_3^3:\OD_{16}$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.V $C_3:S_3^3:\OD_{16}$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.W $C_6^2:S_3^2.D_4$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.X $C_6^2:S_3^2.D_4$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.Y $C_3^4:C_4^2:C_2^3$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.Z $C_3^4:C_4^2:C_2^3$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.BA $C_6^2:S_3^2.D_4$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.BB $C_6^2:S_3^2.D_4$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.BC $C_3^4.C_2^5.C_2^2$ $2^{7} \cdot 3^{4}$ ✓ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.BD $C_6^2.D_6\wr C_2$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.BE $C_6^3:C_6.D_4$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.BF $C_6^2.D_6\wr C_2$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.BG $C_6^3:C_6:D_4$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
41472.br.4._.BH $C_6^2.D_6\wr C_2$ $2^{7} \cdot 3^{4}$ ✓ $C_2^3.S_3\wr C_2^2$ $2^{9} \cdot 3^{4}$ $C_2^2$ $2^{2}$ ✓
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