Abstract group 82944.gr: $C_6^2:C_2^2.S_4^2$

• Label: 82944.gr
• Order: \( 2^{10} \cdot 3^{4} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $17$
• Trans deg.: $72$
• Rank: $2$

Group information

Description:	$C_6^2:C_2^2.S_4^2$
Order:	\(82944\)\(\medspace = 2^{10} \cdot 3^{4} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^9.A_4\wr C_3$, of order \(995328\)\(\medspace = 2^{12} \cdot 3^{5} \)
Composition factors:	$C_2$ x 10, $C_3$ x 4
Derived length:	$5$

This group is nonabelian and solvable. Whether it is monomial has not been computed.

Group statistics

Order	1	2	3	4	6	8	12	24
Elements	1	2023	2672	10776	24464	13824	22272	6912	82944
Conjugacy classes	1	20	7	23	44	22	22	4	143
Divisions	1	20	7	23	44	11	22	2	130
Autjugacy classes	1	10	7	12	26	10	12	2	80

Dimension	1	2	3	4	6	8	9	12	16	18	24	32	48	64	96	128
Irr. complex chars.	4	8	16	9	22	13	12	13	9	2	15	6	8	4	1	1	143
Irr. rational chars.	4	4	16	9	10	10	12	17	9	2	16	7	8	4	1	1	130

Minimal presentations

Permutation degree:	$17$
Transitive degree:	$72$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	32	32	32
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid a^{2}=b^{6}=d^{12}=e^{2}=f^{6}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $17$ $\langle(10,11)(12,15)(13,17)(14,16), (13,17)(14,16), (1,2,4)(3,6,8)(5,7,9), (2,4) \!\cdots\! \rangle$
Direct product:	$(C_2^3:S_4)$ $\, \times\, $ $(C_3^2:\GL(2,3))$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_6^2:C_2^2)$ . $S_4^2$	$(\GL(2,3).C_2^6)$ . $S_4$	$\PU(3,2)$ . $(C_2^4:S_4)$	$(C_6^2:C_2^2:S_4)$ . $S_4$	all 35
Aut. group:	$\Aut(C_6^2:C_4)$	$\Aut(C_6^2.D_6)$	$\Aut(C_6^2:C_{12})$

Elements of the group are displayed as permutations of degree 17.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 4049448 subgroups in 24986 conjugacy classes, 52 normal (32 characteristic).

Characteristic subgroups are shown in this color. Normal (but not characteristic) subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^2:S_4\times \AGL(2,3)$
Commutator:	$G' \simeq$ $C_3^2.(C_2^3\times Q_8).A_4.C_3$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2$	$G/\Phi \simeq$ $C_2^2:S_4\times \AGL(2,3)$
Fitting:	$\operatorname{Fit} \simeq$ $D_4:C_6^2$	$G/\operatorname{Fit} \simeq$ $S_3\times \GL(2,3)$
Radical:	$R \simeq$ $C_6^2:C_2^2.S_4^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2:S_4\times \GL(2,3)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4.C_2^5.C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\times \He_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 128 (order at least 648) are shown, as well as all normal subgroups of any index.

Order 82944: $C_6^2:C_2^2.S_4^2$

Order 41472: $C_3:S_3.(C_2^3\times A_4).A_4.C_2$, $(C_2\times C_6^2).(Q_8\times A_4).C_6$, $(C_2^3\times C_3:S_3).A_4^2.C_2$

Order 27648: $(C_2\times C_6^2).(D_4\times S_4).C_2$, $C_2\times \SL(2,3).C_2^6$

Order 20736: $C_6^2:(D_6\times \GL(2,3))$ x 3, $C_3^2.(C_2^3\times Q_8).A_4.C_3$, $(C_3\times Q_8:C_2^2).S_3^3$

Order 13824: $\ASL(2,3).C_2^3.C_2^3$ x 3, $\ASL(2,3).C_2\wr C_2^2$ x 3, $\ASL(2,3).C_2\wr C_2^2$ x 3, $C_6^2:(D_4\times \GL(2,3))$ x 3, $\ASL(2,3).C_2^3.C_2^3$, $(C_2^3\times C_3:S_3).(D_4\times A_4)$, $(C_2\times C_6^2).A_4.\SD_{16}$, $(C_2\times C_6^2).A_4.\SD_{16}$, $(C_2\times C_6^2).(Q_8\times S_4)$, $(C_2\times C_6^2).A_4.\SD_{16}$, $(C_3\times C_6).C_2^5.S_4$, $Q_8:C_2^2.(S_3\times F_9)$, $\ASL(2,3).C_2\wr C_2^2$, $C_3:S_3.C_2^5.C_6.C_2^2$, $D_6^2:C_2^2:S_4$

Order 10368: $S_4\times C_3^2:\GL(2,3)$ x 12, $C_6^2:C_6:\GL(2,3)$ x 3, $C_6^2:(C_6\times \GL(2,3))$ x 3, $C_6^2:(D_6\times \SL(2,3))$ x 3, $Q_8:C_2^2.C_3^3.C_6.C_2$, $C_2^3:A_4.\He_3.C_2^2$, $C_3:S_3.(Q_8\times A_4).C_6$, $(C_3\times C_2^3:A_4).S_3^2$, $C_6.(C_3\times C_2^4:C_3).D_6$, $(C_3\times C_2^3:A_4).S_3^2$, $(C_2^3\times \He_3).A_4.C_2^2$, $(C_3\times C_2^3:A_4).C_3.D_6$

Order 9216: $C_3:S_3.C_2^5.C_2^4$, $C_2^3:S_4\times \GL(2,3)$

Order 6912: $\PSU(3,2).C_2^3:D_6$ x 6, $\ASL(2,3).C_2^2\wr C_2$ x 6, $\ASL(2,3).C_2^3.C_2^2$ x 6, $C_2\times C_2^4.\SL(3,3)$ x 6, $\ASL(2,3).C_2^4.C_2$ x 3, $\ASL(2,3).C_2^3.C_2^2$ x 3, $C_3:S_3.(C_2^3\times A_4).C_2^2$ x 3, $C_3:S_3.(C_2^4\times A_4).C_2$ x 3, $\ASL(2,3).C_2^3.C_2^2$ x 3, $\ASL(2,3).C_2^2.C_2^3$ x 3, $\ASL(2,3).Q_8:C_2^2$ x 3, $F_9:C_2^2\times S_4$ x 3, $C_2^3:S_4\times S_3^2$ x 2, $(C_2\times C_6^2).A_4.C_8$, $C_2^3:A_4.F_9$, $\ASL(2,3).D_4.C_2^2$, $(C_2\times C_6^2).A_4.Q_8$, $(C_2\times C_6^2).(Q_8\times A_4)$, $(C_3\times C_6).C_2^5.A_4$, $(C_2\times \ASL(2,3)).C_2^4$, $(C_2^4\times \He_3).C_2^4$, $\ASL(2,3).D_4.C_2^2$, $\GL(2,3).C_2^6$, $C_6^2.C_2^4.D_6$, $D_4:C_6^2:D_{12}$, $D_6^2:C_2^2:A_4$, $(C_2\times C_6^2):\GL(2,\mathbb{Z}/4)$

Order 5184: $(C_3^2\times A_4):\GL(2,3)$ x 6, $A_4\times C_3^2:\GL(2,3)$ x 6, $S_4\times \PU(3,2)$ x 6, $C_6^2:(C_6\times \SL(2,3))$ x 3, $C_6^2:D_6^2$ x 3, $Q_8:C_2^2.C_3^3.C_6$, $C_2^3:A_4.\He_3.C_2$, $C_2.(C_2^4\times \He_3).C_6$, $(C_2\times C_6^2).A_4.C_6$, $Q_8:C_2^2.C_3^3.C_6$, $(C_2^3\times \He_3).A_4.C_2$, $(C_2^2\times C_6).C_3^2:S_4$, $C_3^2.Q_8^2.C_3^2$, $C_3^2:\GL(2,3)\times D_6$

Order 4608: $(C_2^3\times C_3:S_3).C_2^4.C_2$ x 3, $C_3:S_3.C_2^5.C_2^3$ x 3, $C_3:S_3.C_2^5.C_2^3$ x 3, $C_3:S_3.C_2^5.C_2^3$ x 3, $C_3:S_3.C_2^5.C_2^3$ x 3, $(C_2\times C_3:S_3).C_2^4.C_2^3$ x 3, $(C_2^2\times C_3:S_3).D_4^2$ x 3, $(C_2^2\times C_3:S_3).D_4^2$ x 3, $C_6^2.(C_2\times C_4\times D_4).C_2$, $C_2^4.A_4^2.C_2$, $(C_2\times C_3:S_3).C_2^6.C_2$, $C_2^4.A_4^2.C_2$, $C_6^2.C_2^4.C_2^3$, $(C_2^3\times C_3:S_3).C_2^4.C_2$, $(C_2^3\times C_3:S_3).C_2^4.C_2$, $C_6^2.C_2^4.C_2^3$, $(C_2\times C_3:S_3).C_2^6.C_2$, $Q_8.(C_3\times C_2^3:A_4).C_2$

Order 3456: $F_9:C_2\times S_4$ x 24, $C_2^4.\SL(3,3)$ x 21, $\GL(2,3).C_2^6$ x 15, $C_2\times C_6^2:\GL(2,3)$ x 12, $\PSU(3,2).D_6:C_4$ x 9, $\He_3.C_2\wr C_2^2.C_2$ x 6, $\He_3.C_2\wr C_2^2.C_2$ x 6, $C_3:S_3.(C_2^5.S_3)$ x 6, $(C_6\times C_{12}):\GL(2,3)$ x 6, $(C_6\times C_{12}):\GL(2,3)$ x 6, $C_6^2:(C_2^2\times \SL(2,3))$ x 6, $C_6^2:C_2^2:S_4$ x 4, $\ASL(2,3).D_4:C_2$ x 3, $(C_2^3\times \He_3).C_2^4$ x 3, $(C_2^4\times \He_3).C_2^3$ x 3, $C_3:S_3.(C_2^3\times A_4).C_2$ x 3, $C_3:S_3.(A_4\times C_2^2:C_4)$ x 3, $(C_2^4\times \He_3).C_2^3$ x 3, $\ASL(2,3).D_4.C_2$ x 3, $(C_2^3\times \He_3).C_2^4$ x 3, $A_4\times F_9:C_2^2$ x 3, $C_2\times S_4\times F_9$ x 3, $C_2\times F_9:S_4$ x 3, $A_4:F_9:C_2^2$ x 3, $A_4:F_9:C_2^2$ x 3, $C_2\times C_6^2:\GL(2,3)$ x 3, $C_6^2:(Q_8\times D_6)$ x 3, $Q_8:A_4:S_3^2$ x 3, $C_2\times D_6^2:D_6$ x 3, $(C_2\times C_6^2):\GL(2,3)$ x 3, $C_6^2.(C_6\times D_4).C_2$ x 2, $(C_3\times C_2^3:A_4).D_6$ x 2, $\He_3.C_2\wr C_2^2.C_2$ x 2, $C_6^2:C_2^2:S_4$ x 2, $Q_8:A_4\times S_3^2$ x 2, $Q_8:A_4:S_3^2$ x 2, $C_3:S_3.(Q_8\times S_4)$, $(C_3\times C_6).(A_4\times \SD_{16})$, $(C_2^3\times \He_3).C_2^4$, $\He_3.C_2\wr C_2^2.C_2$, $C_3:S_3.(D_4\times A_4).C_2$, $C_2^4\times \ASL(2,3)$, $(C_2\times \He_3).C_2^6$, $C_3:S_3.Q_8:S_4$, $D_4:C_6^2:C_{12}$, $C_6^2:C_2^2.S_4$

Order 3072: $C_2^3.(D_4\times S_4).C_2$, $C_2\wr C_2^2\times \GL(2,3)$

Order 2592: $C_3^2:D_6\times S_4$ x 24, $C_2\times C_6^2:S_3^2$ x 6, $C_3^2:D_6\times S_4$ x 6, $S_3\times C_3^2:\GL(2,3)$ x 4, $C_2\times C_6^2:S_3^2$ x 3, $C_3^2:D_6\times S_4$ x 3, $C_6^2:(C_6\times D_6)$ x 3, $A_4\times \PU(3,2)$ x 3, $(C_3\times C_2^3:A_4).C_3^2$, $C_6.(A_4\times S_3^2)$, $C_2\times C_3^3:\GL(2,3)$, $D_6\times \PU(3,2)$, $C_6\times C_3^2:\GL(2,3)$

Order 2304: $(C_2^3\times C_3:S_3).C_2^3.C_2$ x 6, $(C_2\times C_3:S_3).D_4^2$ x 6, $(C_2^3\times C_3:S_3.C_2).C_2^3$ x 6, $(C_2^3\times C_3:S_3).C_2^3.C_2$ x 6, $(C_2^3\times C_3:S_3.C_2).C_2^3$ x 6, $(C_2^3\times C_3:S_3.C_2).C_2^3$ x 6, $(C_2\times C_3:S_3).D_4^2$ x 6, $(C_2^2\times C_3:S_3).C_2^4.C_2$ x 6, $C_3:S_3.C_2^5.C_2^2$ x 6, $(C_2^3\times C_3:S_3).C_2^3.C_2$ x 6, $(C_2^3\times C_3:S_3.C_2).C_2^3$ x 6, $(C_2\times C_3:S_3).D_4^2$ x 6, $D_4\times F_9:C_2^2$ x 6, $C_6^2.(C_8\times D_4)$ x 3, $C_3:S_3.C_2^5.C_2^2$ x 3, $(C_2\times C_3:S_3).C_2^4.C_2^2$ x 3, $C_6^2.(C_2^3\times C_4).C_2$ x 3, $(C_2\times C_3:S_3).C_2^4.C_2^2$ x 3, $C_6^2.(C_2\times C_4\times Q_8)$ x 3, $C_3:S_3.C_2^6.C_2$ x 3, $(C_2^2\times C_3:S_3).C_2^4.C_2$ x 3, $(C_2^2\times C_3:S_3).C_2^4.C_2$ x 3, $(C_2\times C_3:S_3).D_4^2$ x 3, $C_3:S_3.C_2^5.C_2^2$ x 3, $C_6^2.C_4^2.C_2^2$ x 3, $(C_2\times C_3:S_3).C_2^5.C_2$ x 3, $(C_2\times C_3:S_3).C_2^5.C_2$ x 3, $C_6^2.(D_4\times Q_8)$ x 3, $(C_2\times C_3:S_3).C_2^5.C_2$ x 3, $(C_2\times C_3:S_3).C_2^4.C_2^2$ x 3, $C_6^2.C_2^4.C_2^2$ x 3, $C_6^2.D_4^2$ x 3, $(C_2\times C_3:S_3).C_2^4.C_2^2$ x 3, $(C_2^2\times C_3:S_3).C_2^4.C_2$ x 3, $(C_2\times C_3:S_3).C_2^5.C_2$ x 3, $(C_2\times C_3:S_3).C_2^5.C_2$ x 3, $(C_2\times C_3:S_3).C_2^5.C_2$ x 3, $C_6^2.(D_4\times Q_8)$ x 3, $(C_2^2\times C_3:S_3).C_2^4.C_2$ x 3, $(C_2\times C_3:S_3).D_4^2$ x 3, $C_6^2.(C_2^3\times C_8)$ x 3, $C_6^2:D_4^2$ x 3, $C_6^2.D_4^2$ x 3, $C_6^2.D_4^2$ x 3, $C_6^2.D_4^2$ x 3, $C_6^2.D_4^2$ x 3, $C_2^2.S_4^2$ x 3, $C_2^4:D_6^2$ x 2, $S_3\times C_2^4:S_4$ x 2, $(C_2\times C_3:S_3).C_2^5.C_2$, $(C_2\times C_3:S_3).C_2^5.C_2$, $(C_2\times C_3:S_3).C_2^5.C_2$, $C_2^2.C_2^6.C_3^2$, $(C_2\times C_3:S_3).C_2^5.C_2$, $C_6^2.(C_2^3\times C_4).C_2$, $(C_2\times C_3:S_3).C_2^6$, $C_2^4\times \PSU(3,2):C_2$, $(C_2\times C_3:S_3).C_2^5.C_2$, $C_6^2.C_2^4.C_2^2$, $D_6^2.C_2^4$, $(C_2\times D_6^2):D_4$, $(C_2\times D_6^2).C_2^3$, $C_2^4:C_6^2:C_4$

Order 1728: $C_6^2:\GL(2,3)$ x 56, $D_6^2:D_6$ x 18, $C_6^2:\GL(2,3)$ x 18, $S_4\times \PSU(3,2)$ x 12, $A_4\times F_9:C_2$ x 12, $\PSU(3,2):S_4$ x 12, $\SOPlus(4,2).S_4$ x 12, $F_9:S_4$ x 12, $S_4\times F_9$ x 12, $C_6^2.(C_2\times D_{12})$ x 12, $C_6^2.(C_2\times D_{12})$ x 12, $C_6^2:(S_3\times D_4)$ x 12, $C_6^2:(C_2\times D_{12})$ x 12, $C_4\times C_3^2:\GL(2,3)$ x 10, $C_2^3\times \PU(3,2)$ x 9, $D_4\times \PU(3,2)$ x 9, $\PSU(3,2):D_{12}$ x 9, $(C_2^2\times C_6^2):D_6$ x 6, $C_6^2.(C_4\times D_6)$ x 6, $(C_6\times C_{12}):D_{12}$ x 6, $(C_2^2\times C_6^2):D_6$ x 6, $(C_6\times C_{12}):D_{12}$ x 6, $(C_6\times C_{12}):D_{12}$ x 6, $C_6^2.(S_3\times D_4)$ x 6, $C_6^2.(C_4\times D_6)$ x 6, $A_4:D_6^2$ x 6, $C_6^2:\GL(2,3)$ x 6, $(C_6\times C_{12}):\SL(2,3)$ x 6, $C_6^2.\GL(2,3)$ x 6, $C_6^2.(C_4\times D_6)$ x 6, $C_6^2.(C_2\times D_{12})$ x 6, $C_{12}.D_6^2$ x 6, $C_{12}.D_6^2$ x 6, $C_6^2:(C_6\times D_4)$ x 6, $C_6^2:(S_3\times D_4)$ x 6, $C_6^2:(S_3\times D_4)$ x 6, $C_6^2:(S_3\times D_4)$ x 6, $C_6^2:(C_4\times D_6)$ x 6, $D_4:C_6^2:C_6$ x 4, $(C_2\times C_6^2):S_4$ x 4, $C_6^2.(C_4\times D_6)$ x 3, $(C_6\times C_{12}):D_{12}$ x 3, $C_6^2.(C_4\times D_6)$ x 3, $C_6^2:(C_6\times Q_8)$ x 3, $C_2\times D_6^2:C_6$ x 3, $D_6^2:D_6$ x 3, $C_2\times C_6^2:D_{12}$ x 3, $C_6^2:(C_4\times D_6)$ x 3, $C_6^2:C_6:Q_8$ x 3, $C_2\times C_6^2:\SL(2,3)$ x 3, $C_2\times A_4\times F_9$ x 3, $C_2\times A_4:F_9$ x 3, $C_3^2\times C_2^3:S_4$ x 3, $(C_2\times C_6^2):\SL(2,3)$ x 3, $C_{12}.D_6^2$ x 3, $(C_2\times C_6^2):S_4$ x 3, $C_6^2:(S_3\times D_4)$ x 3, $C_6^2:(S_3\times D_4)$ x 3, $(C_2^2\times C_6^2):D_6$ x 3, $C_6^2.(C_6\times D_4)$ x 2, $D_4:C_6^2:C_6$ x 2, $(C_2\times C_6^2):S_4$ x 2, $\PSU(3,2):\SL(2,3)$ x 2, $(D_4\times \He_3):C_2^3$ x 2, $C_{12}.D_6^2$ x 2, $D_4:C_6^2:C_6$ x 2, $(C_2\times C_6^2):S_4$ x 2, $C_2\times F_9:D_6$, $\SL(2,3)\times \PSU(3,2)$, $\SL(2,3)\times \SOPlus(4,2)$, $Q_8\times \PU(3,2)$, $F_9\times \SL(2,3)$, $(C_3\times C_{12}).\GL(2,3)$, $\He_3:C_2^6$, $(D_4\times \He_3):C_2^3$, $C_{12}.D_6^2$, $C_{12}.D_6^2$, $C_2\wr C_2^2\times \He_3$, $(C_2^3\times \He_3):D_4$

Order 1536: $C_2^2\wr C_2\times \GL(2,3)$ x 3, $C_2^4.C_2^4.S_3$ x 3, $\GL(2,3)\times C_2^2.D_4$ x 3, $C_2^4.C_2^4.S_3$ x 3, $C_2^4.C_2^4.S_3$, $C_2^3:S_4\times Q_8$, $D_4\times C_2^3:S_4$, $\GL(2,3)\times Q_8:C_2^2$, $C_4.Q_8:C_2^2.D_6$, $C_4.Q_8:C_2^2.D_6$, $C_4.Q_8:C_2^2.D_6$, $C_8\times C_2^3:S_4$, $C_2^4.C_2^4.S_3$, $C_2^2.C_2^6.C_6$, $C_2^4.C_2^5.C_3$

Order 1296: $C_6^2:S_3^2$ x 24, $C_6^2:S_3^2$ x 24, $A_4\times C_3^2:D_6$ x 12, $C_6^2:S_3^2$ x 12, $C_6^2:S_3^2$ x 12, $C_6^2:C_6^2$ x 6, $C_2\times \He_3:S_4$ x 6, $C_3^2:D_6\times A_4$ x 3, $C_2\times \He_3:S_4$ x 3, $C_2\times S_4\times \He_3$ x 3, $S_3\times \PU(3,2)$ x 2, $C_3^3:\GL(2,3)$ x 2, $C_3^3:\GL(2,3)$ x 2, $C_3^2:C_6\times \SL(2,3)$ x 2, $C_6\times \PU(3,2)$, $C_3^2:D_6^2$, $C_3^2:S_3\times \SL(2,3)$

Order 1152: $D_4\times F_9:C_2$ x 51, $F_9:C_2^4$ x 27, $C_2\times C_6^2:\SD_{16}$ x 18, $C_2\times C_6^2:\SD_{16}$ x 18, $C_3:S_3.(C_2^5.C_2)$ x 12, $C_3:S_3.(C_2^5.C_2)$ x 12, $S_3\times C_2^3:S_4$ x 12, $S_4\times \GL(2,3)$ x 12, $C_2\times F_9:D_4$ x 12, $D_6^2:D_4$ x 9, $D_6^2:D_4$ x 9, $C_3:S_3.C_2.C_2^4.C_2$ x 9, $C_3:S_3.C_2.C_2^4.C_2$ x 9, $C_3:S_3.C_2.C_2^4.C_2$ x 9, $C_3:S_3.C_2.C_2^4.C_2$ x 9, $C_3:S_3.C_2.(C_4\times D_4)$ x 9, $(C_2^2\times F_9).C_2^2$ x 9, $(C_2^2\times F_9).C_2^2$ x 9, $(C_2\times S_3\wr C_2).C_2^3$ x 9, $C_2.\PSU(3,2).C_2^3$ x 9, $C_3:S_3.C_2^4:C_2^2$ x 9, $(C_6\times C_{12}):\SD_{16}$ x 6, $(C_6\times C_{12}):\SD_{16}$ x 6, $D_6^2:D_4$ x 6, $D_6^2:D_4$ x 6, $C_3:S_3.C_4\wr C_2.C_2$ x 6, $C_3:S_3.C_4\wr C_2.C_2$ x 6, $C_4\times F_9:C_2^2$ x 6, $C_2\times D_4\times F_9$ x 6, $D_6^2.D_4$ x 6, $(C_2\times C_3:S_3.C_2).C_4^2$ x 6, $(C_2^2\times C_3:S_3).C_4^2$ x 6, $C_3:S_3.C_2^2\wr C_2.C_2$ x 6, $D_6^2.C_2^3$ x 6, $D_6^2:D_4$ x 6, $C_3:S_3.C_2^2\wr C_2.C_2$ x 6, $D_6^2.C_2^3$ x 6, $D_6^2:D_4$ x 6, $(C_2^2\times C_3:S_3).C_2^4$ x 6, $D_6^2.C_2^3$ x 6, $D_6^2:D_4$ x 6, $C_6^2.C_4^2.C_2$ x 6, $C_6^2.(C_4\times D_4)$ x 6, $D_6^2.D_4$ x 6, $C_3:S_3.(C_2^3.D_4)$ x 6, $C_3:S_3.(C_2^3.D_4)$ x 6, $(C_2^2\times F_9).C_2^2$ x 6, $(C_2^2\times F_9).C_2^2$ x 6, $C_3^2.(C_2\times C_8:C_4).C_2$ x 6, $(C_2^2\times F_9).C_2^2$ x 6, $C_6^2:(C_2^2\times Q_8)$ x 6, $D_6^2:C_2^3$ x 6, $D_6^2:D_4$ x 6, $C_2\times F_9:D_4$ x 6, $(C_6\times D_{12}):D_4$ x 4, $D_6^2:D_4$ x 3, $C_2^3.D_6^2$ x 3, $C_2^2:F_9.C_2^2$ x 3, $(C_2^2\times C_3:S_3).C_2^4$ x 3, $D_6^2.C_2^3$ x 3, $D_6^2:D_4$ x 3, $C_2^2:F_9.C_2^2$ x 3, $C_6^2:(C_4\times D_4)$ x 3, $C_3^2.C_2^3:Q_8.C_2$ x 3, $D_6^2.D_4$ x 3, $D_6^2:D_4$ x 3, $D_6^2:(C_2\times C_4)$ x 3, $C_6^2.(C_4\times D_4)$ x 3, $(C_2^3\times C_3:S_3).C_2^3$ x 3, $C_6^2.(C_4\times D_4)$ x 3, $(C_2^3\times C_3:S_3).C_2^3$ x 3, $C_6:D_{12}.Q_8$ x 3, $D_6^2.D_4$ x 3, $D_6^2.D_4$ x 3, $(C_2^2\times F_9).C_2^2$ x 3, $(C_6\times C_{12}):C_4^2$ x 3, $C_2\times A_4:\GL(2,3)$ x 3, $C_2\times A_4\times \GL(2,3)$ x 3, $C_2\times S_4\times \SL(2,3)$ x 3, $C_3:S_3.D_4.C_2^3$ x 3, $D_6^2.C_2^3$ x 3, $D_6^2:C_2^3$ x 3, $C_3:S_3.Q_8:D_4$ x 3, $(C_4\times F_9).C_2^2$ x 3, $C_3:S_3.C_2\wr C_2^2$ x 3, $C_2^2:F_9.C_2^2$ x 3, $C_3:S_3.C_2\wr C_2^2$ x 3, $(C_2\times C_6^2):\SD_{16}$ x 3, $C_3:S_3.C_2^2:D_4.C_2$ x 3, $C_3:S_3.C_4:D_4.C_2$ x 3, $C_3:S_3.C_4:D_4.C_2$ x 3, $C_3:S_3.C_2^2:D_4.C_2$ x 3, $C_3:S_3.C_4^2:C_2^2$ x 3, $C_3:S_3.C_4^2:C_2^2$ x 3, $C_2^3.D_6^2$ x 2, $(C_2^2\times C_6^2):C_2^3$ x 2, $Q_8:A_4\times D_6$ x 2, $(C_2^3\times C_6):S_4$ x 2, $C_6\times C_2^3:S_4$ x 2, $D_4:D_6^2$ x 2, $C_2^2:F_9.C_2^2$, $C_2^2:F_9.C_2^2$, $C_2^4:\PSU(3,2)$, $D_4:C_6^2:C_4$, $\GL(2,3)\times \SL(2,3)$, $C_2^4\times \PSU(3,2)$, $D_6^2:C_2^3$, $C_2^4\times F_9$, $C_3:S_3.D_4.C_2^3$, $D_6^2.C_2^3$, $D_6^2.C_2^3$, $C_3:S_3.D_4.C_2^3$, $D_6^2.C_2^3$, $C_6^2.(C_2^3\times C_4)$, $C_3:S_3.(D_4\times Q_8)$, $C_4:S_3\wr C_2.C_2^2$, $C_3:S_3.C_2^2:D_4.C_2$

Order 1024: $C_2^4.C_2^5.C_2$

Order 864: $C_2\times C_3^2:\GL(2,3)$ x 46, $C_6.D_6^2$ x 45, $S_4\times S_3^2$ x 42, $C_6.D_6^2$ x 27, $C_6.D_6^2$ x 24, $(C_6\times C_{12}):D_6$ x 18, $(C_6\times C_{12}):D_6$ x 18, $C_6^2:\SL(2,3)$ x 17, $C_6\times S_3\times S_4$ x 12, $S_3^2:S_4$ x 12, $C_3^2:C_4\times S_4$ x 12, $C_6^2.D_{12}$ x 12, $C_6^2.D_{12}$ x 12, $C_2\times C_6^2:D_6$ x 12, $C_6.D_6^2$ x 12, $C_6.D_6^2$ x 12, $C_6.D_6^2$ x 12, $(C_6\times C_{12}):D_6$ x 12, $C_6^2:D_{12}$ x 12, $(C_6\times C_{12}):D_6$ x 12, $C_6^2:D_{12}$ x 12, $C_6^2:D_{12}$ x 12, $(C_2\times C_6^2).D_6$ x 12, $C_6^2:(C_4\times S_3)$ x 12, $S_3\times C_6:S_4$ x 9, $D_6^2:C_6$ x 9, $C_6^2:D_{12}$ x 9, $A_4:\PSU(3,2)$ x 9, $C_6.D_6^2$ x 9, $(C_2\times C_6^2):D_6$ x 9, $(C_6\times C_{12}):D_6$ x 9, $(C_6\times C_{12}):D_6$ x 9, $F_9:D_6$ x 8, $C_6^2.D_{12}$ x 6, $D_6^2:C_6$ x 6, $C_2\times A_4:S_3^2$ x 6, $C_6:S_3\times S_4$ x 6, $A_4\times \PSU(3,2)$ x 6, $A_4\times F_9$ x 6, $A_4:F_9$ x 6, $(C_6\times C_{12}):C_{12}$ x 6, $C_2\times C_6^2:D_6$ x 6, $C_6^2.D_{12}$ x 6, $C_2\times C_6^2:D_6$ x 6, $(C_6\times C_{12}):D_6$ x 6, $C_6.D_6^2$ x 6, $C_6.D_6^2$ x 6, $C_6.D_6^2$ x 6, $C_6.D_6^2$ x 6, $C_6.D_6^2$ x 6, $C_6.D_6^2$ x 6, $(C_6\times C_{12}):D_6$ x 6, $C_6^2:D_{12}$ x 6, $(C_6\times C_{12}):D_6$ x 6, $C_6^2:(C_3\times D_4)$ x 6, $C_6^2:(C_4\times S_3)$ x 6, $C_4\times \PU(3,2)$ x 4, $(C_3\times C_6).\GL(2,3)$ x 4, $(C_6\times C_{12}):C_{12}$ x 3, $(C_2^3\times \He_3):C_4$ x 3, $C_2\times C_6^2:C_{12}$ x 3, $C_6^2:C_6:C_4$ x 3, $C_6^2:\SL(2,3)$ x 3, $C_3^2\times Q_8:A_4$ x 3, $(C_6\times C_{12}):D_6$ x 3, $(C_2\times C_6^2):A_4$ x 3, $C_6.D_6^2$ x 3, $C_6.D_6^2$ x 3, $C_6.D_6^2$ x 3, $C_2^2\wr C_2\times \He_3$ x 3, $(C_2^2\times C_6^2):S_3$ x 3, $S_3^2\times \SL(2,3)$ x 2, $\He_3:C_2^5$ x 2, $(C_6\times C_{12}):D_6$ x 2, $(C_6\times C_{12}):D_6$ x 2, $C_6.D_6^2$ x 2, $C_6.D_6^2$ x 2, $D_6\times \PSU(3,2)$, $S_3^3:C_2^2$, $F_9:D_6$, $\PSU(3,2):D_6$, $C_2\times C_3^3:\SD_{16}$, $C_2\times F_9:C_6$, $D_6\times F_9$, $C_3^2:C_4\times \SL(2,3)$, $\He_3:C_2^5$, $D_4:C_2^2\times \He_3$, $(C_6\times C_{12}):D_6$, $(C_6\times C_{12}):D_6$, $C_6.D_6^2$, $C_6.D_6^2$

Order 768: $C_2\times D_4\times \GL(2,3)$ x 6, $(C_2\times \GL(2,3)):D_4$ x 6, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 6, $C_2^2:C_4\times \GL(2,3)$ x 6, $Q_8.(C_2^2\times S_3):C_4$ x 6, $Q_8.D_{12}:C_2^2$ x 3, $C_2\times \SD_{16}\times S_4$ x 3, $C_2^4:\GL(2,3)$ x 3, $Q_8.(C_6.D_4).C_2$ x 3, $C_2^2\wr C_2\times \SL(2,3)$ x 3, $C_2^4.C_2^4.C_3$ x 3, $C_2\wr C_2^2\times D_6$ x 2, $C_2^5:S_4$, $C_2^4\times \GL(2,3)$, $C_2^3:A_4.Q_8$, $(C_2^3\times C_4):S_4$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $C_4\times C_2^3:S_4$, $Q_8.(S_3\times D_4).C_2$, $Q_8^2:D_6$, $C_2^3:A_4.C_8$, $C_2^4.C_2^4.C_3$, $C_2^4.C_2^4.C_3$, $C_2^4.C_2^4.C_3$, $D_4\times Q_8:A_4$, $(C_2^3\times C_8).A_4$

Order 648: $A_4\times C_3^2:C_6$ x 12, $\He_3:S_4$ x 12, $C_3.S_3^3$ x 8, $A_4\times C_3^2:S_3$ x 6, $\He_3:S_4$ x 6, $S_4\times \He_3$ x 6, $C_2\times A_4\times \He_3$ x 3, $C_2\times C_3^2:S_3^2$ x 2, $S_3\times C_3^2:D_6$ x 2, $C_3^3:\SL(2,3)$, $C_2\times C_3^2:S_3^2$, $S_3\times C_3^2:D_6$, $C_6\times C_3^2:D_6$, $\He_3\times \SL(2,3)$

Order 576: $C_2^3\times \PSU(3,2)$ x 3, $C_6^2.C_2^4$

Order 432: $C_3^2:\GL(2,3)$ x 2, $C_2\times \PU(3,2)$

Order 288: $D_4:C_6^2$

Order 216: $\PU(3,2)$

Order 192: $C_2^3:S_4$

Order 144: $C_6^2:C_2^2$ x 3, $C_2\times \PSU(3,2)$

Order 96: $Q_8:A_4$

Order 81: $C_3\times \He_3$

Order 72: $C_2\times C_6^2$ x 3, $\PSU(3,2)$

Order 36: $C_6:S_3$

Order 32: $D_4:C_2^2$

Order 18: $C_3:S_3$ x 2, $C_3\times C_6$

Order 9: $C_3^2$

Order 8: $C_2^3$ x 3

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 128 (order at least 648) are shown, as well as all normal subgroups of any index.

Order 82944: $C_6^2:C_2^2.S_4^2$

Order 41472: $C_3:S_3.(C_2^3\times A_4).A_4.C_2$, $(C_2\times C_6^2).(Q_8\times A_4).C_6$, $(C_2^3\times C_3:S_3).A_4^2.C_2$

Order 27648: $(C_2\times C_6^2).(D_4\times S_4).C_2$, $C_2\times \SL(2,3).C_2^6$

Order 20736: $C_3^2.(C_2^3\times Q_8).A_4.C_3$, $(C_3\times Q_8:C_2^2).S_3^3$, $C_6^2:(D_6\times \GL(2,3))$

Order 13824: $\ASL(2,3).C_2^3.C_2^3$, $(C_2^3\times C_3:S_3).(D_4\times A_4)$, $(C_2\times C_6^2).A_4.\SD_{16}$, $(C_2\times C_6^2).A_4.\SD_{16}$, $(C_2\times C_6^2).(Q_8\times S_4)$, $(C_2\times C_6^2).A_4.\SD_{16}$, $\ASL(2,3).C_2^3.C_2^3$, $\ASL(2,3).C_2\wr C_2^2$, $(C_3\times C_6).C_2^5.S_4$, $Q_8:C_2^2.(S_3\times F_9)$, $\ASL(2,3).C_2\wr C_2^2$, $\ASL(2,3).C_2\wr C_2^2$, $C_3:S_3.C_2^5.C_6.C_2^2$, $C_6^2:(D_4\times \GL(2,3))$, $D_6^2:C_2^2:S_4$

Order 10368: $Q_8:C_2^2.C_3^3.C_6.C_2$, $C_2^3:A_4.\He_3.C_2^2$, $C_3:S_3.(Q_8\times A_4).C_6$, $(C_3\times C_2^3:A_4).S_3^2$, $C_6.(C_3\times C_2^4:C_3).D_6$, $(C_3\times C_2^3:A_4).S_3^2$, $(C_2^3\times \He_3).A_4.C_2^2$, $(C_3\times C_2^3:A_4).C_3.D_6$, $C_6^2:C_6:\GL(2,3)$, $C_6^2:(C_6\times \GL(2,3))$, $C_6^2:(D_6\times \SL(2,3))$, $S_4\times C_3^2:\GL(2,3)$

Order 9216: $C_3:S_3.C_2^5.C_2^4$, $C_2^3:S_4\times \GL(2,3)$

Order 6912: $\PSU(3,2).C_2^3:D_6$ x 2, $\ASL(2,3).C_2^2\wr C_2$ x 2, $C_2\times C_2^4.\SL(3,3)$ x 2, $C_2^3:S_4\times S_3^2$ x 2, $\ASL(2,3).C_2^4.C_2$, $(C_2\times C_6^2).A_4.C_8$, $C_2^3:A_4.F_9$, $\ASL(2,3).C_2^3.C_2^2$, $\ASL(2,3).D_4.C_2^2$, $C_3:S_3.(C_2^3\times A_4).C_2^2$, $C_3:S_3.(C_2^4\times A_4).C_2$, $(C_2\times C_6^2).A_4.Q_8$, $\ASL(2,3).C_2^3.C_2^2$, $\ASL(2,3).C_2^3.C_2^2$, $\ASL(2,3).C_2^2.C_2^3$, $(C_2\times C_6^2).(Q_8\times A_4)$, $(C_3\times C_6).C_2^5.A_4$, $(C_2\times \ASL(2,3)).C_2^4$, $(C_2^4\times \He_3).C_2^4$, $\ASL(2,3).D_4.C_2^2$, $\ASL(2,3).Q_8:C_2^2$, $\GL(2,3).C_2^6$, $F_9:C_2^2\times S_4$, $C_6^2.C_2^4.D_6$, $D_4:C_6^2:D_{12}$, $D_6^2:C_2^2:A_4$, $(C_2\times C_6^2):\GL(2,\mathbb{Z}/4)$

Order 5184: $Q_8:C_2^2.C_3^3.C_6$, $C_2^3:A_4.\He_3.C_2$, $C_2.(C_2^4\times \He_3).C_6$, $(C_2\times C_6^2).A_4.C_6$, $Q_8:C_2^2.C_3^3.C_6$, $(C_2^3\times \He_3).A_4.C_2$, $(C_2^2\times C_6).C_3^2:S_4$, $C_3^2.Q_8^2.C_3^2$, $C_3^2:\GL(2,3)\times D_6$, $C_6^2:(C_6\times \SL(2,3))$, $C_6^2:D_6^2$, $(C_3^2\times A_4):\GL(2,3)$, $A_4\times C_3^2:\GL(2,3)$, $S_4\times \PU(3,2)$

Order 4608: $(C_2^3\times C_3:S_3).C_2^4.C_2$, $C_3:S_3.C_2^5.C_2^3$, $C_6^2.(C_2\times C_4\times D_4).C_2$, $C_3:S_3.C_2^5.C_2^3$, $C_3:S_3.C_2^5.C_2^3$, $C_2^4.A_4^2.C_2$, $(C_2\times C_3:S_3).C_2^6.C_2$, $C_2^4.A_4^2.C_2$, $C_3:S_3.C_2^5.C_2^3$, $(C_2\times C_3:S_3).C_2^4.C_2^3$, $C_6^2.C_2^4.C_2^3$, $(C_2^3\times C_3:S_3).C_2^4.C_2$, $(C_2^2\times C_3:S_3).D_4^2$, $(C_2^3\times C_3:S_3).C_2^4.C_2$, $(C_2^2\times C_3:S_3).D_4^2$, $C_6^2.C_2^4.C_2^3$, $(C_2\times C_3:S_3).C_2^6.C_2$, $Q_8.(C_3\times C_2^3:A_4).C_2$

Order 3456: $\GL(2,3).C_2^6$ x 6, $C_2\times C_6^2:\GL(2,3)$ x 4, $C_6^2:C_2^2:S_4$ x 4, $\PSU(3,2).D_6:C_4$ x 3, $Q_8:A_4:S_3^2$ x 3, $C_2^4.\SL(3,3)$ x 3, $C_6^2.(C_6\times D_4).C_2$ x 2, $(C_3\times C_2^3:A_4).D_6$ x 2, $\He_3.C_2\wr C_2^2.C_2$ x 2, $\He_3.C_2\wr C_2^2.C_2$ x 2, $\He_3.C_2\wr C_2^2.C_2$ x 2, $C_3:S_3.(C_2^5.S_3)$ x 2, $(C_6\times C_{12}):\GL(2,3)$ x 2, $C_6^2:C_2^2:S_4$ x 2, $(C_6\times C_{12}):\GL(2,3)$ x 2, $C_6^2:(C_2^2\times \SL(2,3))$ x 2, $Q_8:A_4\times S_3^2$ x 2, $Q_8:A_4:S_3^2$ x 2, $F_9:C_2\times S_4$ x 2, $\ASL(2,3).D_4:C_2$, $(C_2^3\times \He_3).C_2^4$, $(C_2^4\times \He_3).C_2^3$, $C_3:S_3.(C_2^3\times A_4).C_2$, $C_3:S_3.(A_4\times C_2^2:C_4)$, $C_3:S_3.(Q_8\times S_4)$, $(C_2^4\times \He_3).C_2^3$, $(C_3\times C_6).(A_4\times \SD_{16})$, $(C_2^3\times \He_3).C_2^4$, $\He_3.C_2\wr C_2^2.C_2$, $C_3:S_3.(D_4\times A_4).C_2$, $\ASL(2,3).D_4.C_2$, $C_2^4\times \ASL(2,3)$, $(C_2\times \He_3).C_2^6$, $(C_2^3\times \He_3).C_2^4$, $C_3:S_3.Q_8:S_4$, $A_4\times F_9:C_2^2$, $C_2\times S_4\times F_9$, $C_2\times F_9:S_4$, $A_4:F_9:C_2^2$, $A_4:F_9:C_2^2$, $C_2\times C_6^2:\GL(2,3)$, $C_6^2:(Q_8\times D_6)$, $D_4:C_6^2:C_{12}$, $C_6^2:C_2^2.S_4$, $C_2\times D_6^2:D_6$, $(C_2\times C_6^2):\GL(2,3)$

Order 3072: $C_2^3.(D_4\times S_4).C_2$, $C_2\wr C_2^2\times \GL(2,3)$

Order 2592: $C_2\times C_6^2:S_3^2$ x 2, $C_3^2:D_6\times S_4$ x 2, $C_3^2:D_6\times S_4$ x 2, $(C_3\times C_2^3:A_4).C_3^2$, $C_6.(A_4\times S_3^2)$, $C_2\times C_6^2:S_3^2$, $C_3^2:D_6\times S_4$, $C_2\times C_3^3:\GL(2,3)$, $D_6\times \PU(3,2)$, $C_6\times C_3^2:\GL(2,3)$, $C_6^2:(C_6\times D_6)$, $A_4\times \PU(3,2)$, $S_3\times C_3^2:\GL(2,3)$

Order 2304: $(C_2\times C_3:S_3).D_4^2$ x 2, $(C_2\times C_3:S_3).D_4^2$ x 2, $(C_2^2\times C_3:S_3).C_2^4.C_2$ x 2, $(C_2^3\times C_3:S_3.C_2).C_2^3$ x 2, $(C_2\times C_3:S_3).D_4^2$ x 2, $C_2^4:D_6^2$ x 2, $D_4\times F_9:C_2^2$ x 2, $S_3\times C_2^4:S_4$ x 2, $C_6^2.(C_8\times D_4)$, $C_3:S_3.C_2^5.C_2^2$, $(C_2\times C_3:S_3).C_2^5.C_2$, $(C_2\times C_3:S_3).C_2^5.C_2$, $(C_2^3\times C_3:S_3).C_2^3.C_2$, $(C_2\times C_3:S_3).C_2^4.C_2^2$, $C_6^2.(C_2^3\times C_4).C_2$, $(C_2\times C_3:S_3).C_2^5.C_2$, $(C_2\times C_3:S_3).C_2^4.C_2^2$, $C_6^2.(C_2\times C_4\times Q_8)$, $C_2^2.C_2^6.C_3^2$, $(C_2^3\times C_3:S_3.C_2).C_2^3$, $(C_2\times C_3:S_3).C_2^5.C_2$, $(C_2^3\times C_3:S_3).C_2^3.C_2$, $(C_2^3\times C_3:S_3.C_2).C_2^3$, $C_3:S_3.C_2^6.C_2$, $(C_2^2\times C_3:S_3).C_2^4.C_2$, $(C_2^2\times C_3:S_3).C_2^4.C_2$, $(C_2\times C_3:S_3).D_4^2$, $C_3:S_3.C_2^5.C_2^2$, $(C_2^3\times C_3:S_3.C_2).C_2^3$, $C_6^2.C_4^2.C_2^2$, $(C_2\times C_3:S_3).C_2^5.C_2$, $(C_2\times C_3:S_3).C_2^5.C_2$, $C_6^2.(C_2^3\times C_4).C_2$, $C_6^2.(D_4\times Q_8)$, $(C_2\times C_3:S_3).C_2^5.C_2$, $(C_2\times C_3:S_3).C_2^4.C_2^2$, $C_3:S_3.C_2^5.C_2^2$, $C_6^2.C_2^4.C_2^2$, $C_6^2.D_4^2$, $(C_2^3\times C_3:S_3).C_2^3.C_2$, $(C_2\times C_3:S_3).C_2^4.C_2^2$, $(C_2\times C_3:S_3).C_2^6$, $C_2^4\times \PSU(3,2):C_2$, $(C_2^2\times C_3:S_3).C_2^4.C_2$, $(C_2\times C_3:S_3).C_2^5.C_2$, $(C_2\times C_3:S_3).C_2^5.C_2$, $(C_2\times C_3:S_3).C_2^5.C_2$, $C_6^2.C_2^4.C_2^2$, $(C_2\times C_3:S_3).C_2^5.C_2$, $C_6^2.(D_4\times Q_8)$, $(C_2^2\times C_3:S_3).C_2^4.C_2$, $(C_2\times C_3:S_3).D_4^2$, $C_6^2.(C_2^3\times C_8)$, $D_6^2.C_2^4$, $C_6^2:D_4^2$, $C_6^2.D_4^2$, $(C_2\times D_6^2):D_4$, $C_6^2.D_4^2$, $C_6^2.D_4^2$, $(C_2\times D_6^2).C_2^3$, $C_2^4:C_6^2:C_4$, $C_6^2.D_4^2$, $C_2^2.S_4^2$

Order 1728: $C_6^2:\GL(2,3)$ x 13, $C_2^3\times \PU(3,2)$ x 4, $D_4:C_6^2:C_6$ x 4, $(C_2\times C_6^2):S_4$ x 4, $C_6^2:(S_3\times D_4)$ x 4, $C_6^2:(C_2\times D_{12})$ x 4, $C_3^2\times C_2^3:S_4$ x 3, $C_6^2:\GL(2,3)$ x 3, $C_4\times C_3^2:\GL(2,3)$ x 3, $(C_2\times C_6^2):S_4$ x 3, $C_6^2.(C_6\times D_4)$ x 2, $(C_2^2\times C_6^2):D_6$ x 2, $C_6^2.(C_4\times D_6)$ x 2, $(C_6\times C_{12}):D_{12}$ x 2, $(C_6\times C_{12}):D_{12}$ x 2, $(C_6\times C_{12}):D_{12}$ x 2, $C_6^2.(S_3\times D_4)$ x 2, $C_6^2.(C_4\times D_6)$ x 2, $A_4:D_6^2$ x 2, $S_4\times \PSU(3,2)$ x 2, $D_6^2:D_6$ x 2, $A_4\times F_9:C_2$ x 2, $\PSU(3,2):S_4$ x 2, $D_4:C_6^2:C_6$ x 2, $(C_2\times C_6^2):S_4$ x 2, $\PSU(3,2):\SL(2,3)$ x 2, $D_4\times \PU(3,2)$ x 2, $(C_6\times C_{12}):\SL(2,3)$ x 2, $C_6^2.\GL(2,3)$ x 2, $\PSU(3,2):D_{12}$ x 2, $C_6^2.(C_2\times D_{12})$ x 2, $C_6^2.(C_2\times D_{12})$ x 2, $(D_4\times \He_3):C_2^3$ x 2, $C_{12}.D_6^2$ x 2, $C_{12}.D_6^2$ x 2, $C_{12}.D_6^2$ x 2, $D_4:C_6^2:C_6$ x 2, $(C_2\times C_6^2):S_4$ x 2, $C_6^2:(C_6\times D_4)$ x 2, $C_6^2:(S_3\times D_4)$ x 2, $C_6^2:(S_3\times D_4)$ x 2, $C_6^2:(S_3\times D_4)$ x 2, $C_6^2:(C_4\times D_6)$ x 2, $C_6^2.(C_4\times D_6)$, $(C_6\times C_{12}):D_{12}$, $(C_2^2\times C_6^2):D_6$, $C_6^2.(C_4\times D_6)$, $C_6^2:(C_6\times Q_8)$, $C_2\times D_6^2:C_6$, $D_6^2:D_6$, $C_2\times C_6^2:D_{12}$, $C_6^2:(C_4\times D_6)$, $C_6^2:C_6:Q_8$, $C_2\times C_6^2:\SL(2,3)$, $C_2\times A_4\times F_9$, $C_2\times A_4:F_9$, $C_2\times F_9:D_6$, $C_6^2:\GL(2,3)$, $\SOPlus(4,2).S_4$, $F_9:S_4$, $S_4\times F_9$, $\SL(2,3)\times \PSU(3,2)$, $\SL(2,3)\times \SOPlus(4,2)$, $(C_2\times C_6^2):\SL(2,3)$, $Q_8\times \PU(3,2)$, $F_9\times \SL(2,3)$, $(C_3\times C_{12}).\GL(2,3)$, $\He_3:C_2^6$, $C_6^2.(C_4\times D_6)$, $C_6^2.(C_2\times D_{12})$, $(D_4\times \He_3):C_2^3$, $C_{12}.D_6^2$, $C_{12}.D_6^2$, $C_{12}.D_6^2$, $C_6^2:(S_3\times D_4)$, $C_6^2:(S_3\times D_4)$, $C_2\wr C_2^2\times \He_3$, $(C_2^3\times \He_3):D_4$, $(C_2^2\times C_6^2):D_6$

Order 1536: $C_2^4.C_2^4.S_3$, $C_2^3:S_4\times Q_8$, $D_4\times C_2^3:S_4$, $\GL(2,3)\times Q_8:C_2^2$, $C_4.Q_8:C_2^2.D_6$, $C_4.Q_8:C_2^2.D_6$, $C_4.Q_8:C_2^2.D_6$, $C_8\times C_2^3:S_4$, $C_2^2\wr C_2\times \GL(2,3)$, $C_2^4.C_2^4.S_3$, $C_2^4.C_2^4.S_3$, $\GL(2,3)\times C_2^2.D_4$, $C_2^4.C_2^4.S_3$, $C_2^2.C_2^6.C_6$, $C_2^4.C_2^5.C_3$

Order 1296: $C_6^2:S_3^2$ x 3, $C_6^2:S_3^2$ x 3, $C_6^2:C_6^2$ x 2, $C_2\times \He_3:S_4$ x 2, $A_4\times C_3^2:D_6$ x 2, $C_3^2:C_6\times \SL(2,3)$ x 2, $C_6\times \PU(3,2)$, $S_3\times \PU(3,2)$, $C_3^3:\GL(2,3)$, $C_3^3:\GL(2,3)$, $C_3^2:D_6\times A_4$, $C_2\times \He_3:S_4$, $C_2\times S_4\times \He_3$, $C_3^2:D_6^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_3^2:S_3\times \SL(2,3)$

Order 1152: $S_3\times C_2^3:S_4$ x 12, $F_9:C_2^4$ x 11, $D_4\times F_9:C_2$ x 7, $C_2\times C_6^2:\SD_{16}$ x 6, $C_2\times C_6^2:\SD_{16}$ x 6, $(C_6\times D_{12}):D_4$ x 4, $C_3:S_3.(C_2^5.C_2)$ x 4, $C_3:S_3.(C_2^5.C_2)$ x 4, $C_2\times F_9:D_4$ x 4, $C_2^3.D_6^2$ x 3, $D_6^2:D_4$ x 3, $D_6^2:D_4$ x 3, $C_3:S_3.C_2.C_2^4.C_2$ x 3, $C_3:S_3.C_2.C_2^4.C_2$ x 3, $C_3:S_3.C_2.C_2^4.C_2$ x 3, $C_3:S_3.C_2.C_2^4.C_2$ x 3, $C_3:S_3.C_2.(C_4\times D_4)$ x 3, $(C_2^2\times F_9).C_2^2$ x 3, $(C_2^2\times F_9).C_2^2$ x 3, $(C_6\times C_{12}):\SD_{16}$ x 2, $(C_6\times C_{12}):\SD_{16}$ x 2, $C_2^3.D_6^2$ x 2, $D_6^2:D_4$ x 2, $D_6^2:D_4$ x 2, $(C_2^2\times C_6^2):C_2^3$ x 2, $C_4\times F_9:C_2^2$ x 2, $C_2\times D_4\times F_9$ x 2, $D_6^2.D_4$ x 2, $C_3:S_3.C_2^2\wr C_2.C_2$ x 2, $D_6^2.C_2^3$ x 2, $D_6^2:D_4$ x 2, $C_3:S_3.C_2^2\wr C_2.C_2$ x 2, $D_6^2.C_2^3$ x 2, $D_6^2:D_4$ x 2, $(C_2^2\times C_3:S_3).C_2^4$ x 2, $D_6^2.C_2^3$ x 2, $C_3:S_3.(C_2^3.D_4)$ x 2, $C_3:S_3.(C_2^3.D_4)$ x 2, $(C_2^2\times F_9).C_2^2$ x 2, $(C_2^2\times F_9).C_2^2$ x 2, $C_3^2.(C_2\times C_8:C_4).C_2$ x 2, $(C_2^2\times F_9).C_2^2$ x 2, $Q_8:A_4\times D_6$ x 2, $(C_2^3\times C_6):S_4$ x 2, $C_6\times C_2^3:S_4$ x 2, $D_4:D_6^2$ x 2, $C_6^2:(C_2^2\times Q_8)$ x 2, $D_6^2:C_2^3$ x 2, $D_6^2:D_4$ x 2, $C_3:S_3.Q_8:D_4$ x 2, $(C_2\times S_3\wr C_2).C_2^3$ x 2, $C_2.\PSU(3,2).C_2^3$ x 2, $C_3:S_3.C_2^4:C_2^2$ x 2, $C_2\times F_9:D_4$ x 2, $D_6^2:D_4$, $C_2^2:F_9.C_2^2$, $C_3:S_3.C_4\wr C_2.C_2$, $C_2^2:F_9.C_2^2$, $C_3:S_3.C_4\wr C_2.C_2$, $C_2^2:F_9.C_2^2$, $(C_2^2\times C_3:S_3).C_2^4$, $D_6^2.C_2^3$, $D_6^2:D_4$, $C_2^2:F_9.C_2^2$, $(C_2\times C_3:S_3.C_2).C_4^2$, $(C_2^2\times C_3:S_3).C_4^2$, $C_6^2:(C_4\times D_4)$, $C_3^2.C_2^3:Q_8.C_2$, $D_6^2.D_4$, $C_2^4:\PSU(3,2)$, $D_6^2:D_4$, $D_6^2:D_4$, $C_6^2.C_4^2.C_2$, $C_6^2.(C_4\times D_4)$, $D_6^2.D_4$, $D_6^2:(C_2\times C_4)$, $C_6^2.(C_4\times D_4)$, $(C_2^3\times C_3:S_3).C_2^3$, $C_6^2.(C_4\times D_4)$, $(C_2^3\times C_3:S_3).C_2^3$, $C_6:D_{12}.Q_8$, $D_6^2.D_4$, $D_4:C_6^2:C_4$, $D_6^2.D_4$, $(C_2^2\times F_9).C_2^2$, $(C_6\times C_{12}):C_4^2$, $C_2\times A_4:\GL(2,3)$, $C_2\times A_4\times \GL(2,3)$, $C_2\times S_4\times \SL(2,3)$, $S_4\times \GL(2,3)$, $\GL(2,3)\times \SL(2,3)$, $C_2^4\times \PSU(3,2)$, $D_6^2:C_2^3$, $C_2^4\times F_9$, $C_3:S_3.D_4.C_2^3$, $C_3:S_3.D_4.C_2^3$, $D_6^2.C_2^3$, $D_6^2.C_2^3$, $D_6^2.C_2^3$, $D_6^2:C_2^3$, $C_3:S_3.D_4.C_2^3$, $D_6^2.C_2^3$, $C_6^2.(C_2^3\times C_4)$, $C_3:S_3.(D_4\times Q_8)$, $C_4:S_3\wr C_2.C_2^2$, $C_3:S_3.C_2^2:D_4.C_2$, $(C_4\times F_9).C_2^2$, $C_3:S_3.C_2\wr C_2^2$, $C_2^2:F_9.C_2^2$, $C_3:S_3.C_2\wr C_2^2$, $(C_2\times C_6^2):\SD_{16}$, $C_3:S_3.C_2^2:D_4.C_2$, $C_3:S_3.C_4:D_4.C_2$, $C_3:S_3.C_4:D_4.C_2$, $C_3:S_3.C_2^2:D_4.C_2$, $C_3:S_3.C_4^2:C_2^2$, $C_3:S_3.C_4^2:C_2^2$

Order 1024: $C_2^4.C_2^5.C_2$

Order 864: $C_2\times C_3^2:\GL(2,3)$ x 12, $C_6.D_6^2$ x 11, $C_6.D_6^2$ x 8, $C_6^2:\SL(2,3)$ x 6, $C_6.D_6^2$ x 6, $(C_6\times C_{12}):D_6$ x 6, $(C_6\times C_{12}):D_6$ x 6, $C_6\times S_3\times S_4$ x 4, $S_4\times S_3^2$ x 4, $C_2\times C_6^2:D_6$ x 4, $C_6.D_6^2$ x 4, $C_6.D_6^2$ x 4, $(C_6\times C_{12}):D_6$ x 4, $C_6^2:D_{12}$ x 4, $(C_6\times C_{12}):D_6$ x 4, $C_6^2:D_{12}$ x 4, $C_6^2:D_{12}$ x 4, $(C_2\times C_6^2).D_6$ x 4, $C_6^2:(C_4\times S_3)$ x 4, $S_3\times C_6:S_4$ x 3, $C_3^2\times Q_8:A_4$ x 3, $(C_2\times C_6^2):A_4$ x 3, $(C_2\times C_6^2):D_6$ x 3, $(C_6\times C_{12}):D_6$ x 3, $(C_6\times C_{12}):D_6$ x 3, $D_6^2:C_6$ x 2, $C_2\times A_4:S_3^2$ x 2, $C_6:S_3\times S_4$ x 2, $A_4\times \PSU(3,2)$ x 2, $D_6^2:C_6$ x 2, $C_6^2:D_{12}$ x 2, $C_3^2:C_4\times S_4$ x 2, $A_4:\PSU(3,2)$ x 2, $F_9:D_6$ x 2, $(C_6\times C_{12}):C_{12}$ x 2, $S_3^2\times \SL(2,3)$ x 2, $C_4\times \PU(3,2)$ x 2, $(C_3\times C_6).\GL(2,3)$ x 2, $\He_3:C_2^5$ x 2, $C_2\times C_6^2:D_6$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $C_2\times C_6^2:D_6$ x 2, $(C_6\times C_{12}):D_6$ x 2, $(C_6\times C_{12}):D_6$ x 2, $(C_6\times C_{12}):D_6$ x 2, $C_6.D_6^2$ x 2, $C_6.D_6^2$ x 2, $C_6.D_6^2$ x 2, $C_6.D_6^2$ x 2, $C_6.D_6^2$ x 2, $C_6.D_6^2$ x 2, $C_6.D_6^2$ x 2, $C_6.D_6^2$ x 2, $(C_6\times C_{12}):D_6$ x 2, $C_6^2:D_{12}$ x 2, $(C_6\times C_{12}):D_6$ x 2, $C_6^2:(C_3\times D_4)$ x 2, $C_6^2:(C_4\times S_3)$ x 2, $(C_6\times C_{12}):C_{12}$, $(C_2^3\times \He_3):C_4$, $C_6^2.D_{12}$, $C_2\times C_6^2:C_{12}$, $C_6^2:C_6:C_4$, $D_6\times \PSU(3,2)$, $S_3^3:C_2^2$, $F_9:D_6$, $\PSU(3,2):D_6$, $C_2\times C_3^3:\SD_{16}$, $C_2\times F_9:C_6$, $D_6\times F_9$, $S_3^2:S_4$, $C_6^2:\SL(2,3)$, $A_4\times F_9$, $A_4:F_9$, $C_3^2:C_4\times \SL(2,3)$, $\He_3:C_2^5$, $C_6^2.D_{12}$, $D_4:C_2^2\times \He_3$, $(C_6\times C_{12}):D_6$, $(C_6\times C_{12}):D_6$, $(C_6\times C_{12}):D_6$, $C_6.D_6^2$, $C_6.D_6^2$, $C_6.D_6^2$, $C_6.D_6^2$, $C_6.D_6^2$, $C_6.D_6^2$, $C_6.D_6^2$, $C_2^2\wr C_2\times \He_3$, $(C_2^2\times C_6^2):S_3$

Order 768: $C_2\times D_4\times \GL(2,3)$ x 2, $(C_2\times \GL(2,3)):D_4$ x 2, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^2:C_4\times \GL(2,3)$ x 2, $C_2\wr C_2^2\times D_6$ x 2, $C_2^5:S_4$, $C_2^4\times \GL(2,3)$, $C_2^3:A_4.Q_8$, $(C_2^3\times C_4):S_4$, $C_2^3:\GL(2,\mathbb{Z}/4)$, $C_4\times C_2^3:S_4$, $Q_8.(S_3\times D_4).C_2$, $Q_8.D_{12}:C_2^2$, $Q_8^2:D_6$, $C_2\times \SD_{16}\times S_4$, $C_2^3:A_4.C_8$, $C_2^4:\GL(2,3)$, $Q_8.(C_2^2\times S_3):C_4$, $Q_8.(C_6.D_4).C_2$, $C_2^4.C_2^4.C_3$, $C_2^4.C_2^4.C_3$, $C_2^4.C_2^4.C_3$, $D_4\times Q_8:A_4$, $(C_2^3\times C_8).A_4$, $C_2^2\wr C_2\times \SL(2,3)$, $C_2^4.C_2^4.C_3$

Order 648: $A_4\times C_3^2:C_6$ x 3, $C_2\times C_3^2:S_3^2$ x 2, $S_3\times C_3^2:D_6$ x 2, $\He_3:S_4$ x 2, $C_3.S_3^3$ x 2, $C_3^3:\SL(2,3)$, $C_2\times A_4\times \He_3$, $C_2\times C_3^2:S_3^2$, $S_3\times C_3^2:D_6$, $C_6\times C_3^2:D_6$, $A_4\times C_3^2:S_3$, $\He_3:S_4$, $S_4\times \He_3$, $\He_3\times \SL(2,3)$

Order 576: $C_2^3\times \PSU(3,2)$, $C_6^2.C_2^4$

Order 432: $C_2\times \PU(3,2)$, $C_3^2:\GL(2,3)$

Order 288: $D_4:C_6^2$

Order 216: $\PU(3,2)$

Order 192: $C_2^3:S_4$

Order 144: $C_6^2:C_2^2$, $C_2\times \PSU(3,2)$

Order 96: $Q_8:A_4$

Order 81: $C_3\times \He_3$

Order 72: $C_2\times C_6^2$, $\PSU(3,2)$

Order 36: $C_6:S_3$

Order 32: $D_4:C_2^2$

Order 18: $C_3:S_3$ x 2, $C_3\times C_6$

Order 9: $C_3^2$

Order 8: $C_2^3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 82944: $C_6^2:C_2^2.S_4^2$ ($C_1$)

Order 41472: $C_3:S_3.(C_2^3\times A_4).A_4.C_2$ ($C_2$), $(C_2\times C_6^2).(Q_8\times A_4).C_6$ ($C_2$), $(C_2^3\times C_3:S_3).A_4^2.C_2$ ($C_2$)

Order 20736: $C_3^2.(C_2^3\times Q_8).A_4.C_3$ ($C_2^2$)

Order 13824: $\ASL(2,3).C_2^3.C_2^3$ ($S_3$), $(C_2\times C_6^2).(Q_8\times S_4)$ ($S_3$)

Order 6912: $(C_2\times C_6^2).(Q_8\times A_4)$ ($D_6$), $(C_2\times \ASL(2,3)).C_2^4$ ($D_6$)

Order 3456: $\GL(2,3).C_2^6$ ($S_4$) x 3, $C_6^2:C_2^2:S_4$ ($S_4$)

Order 2304: $(C_2\times C_3:S_3).C_2^6$ ($S_3^2$)

Order 1728: $C_2^3\times \PU(3,2)$ ($C_2\times S_4$) x 3, $D_4:C_6^2:C_6$ ($C_2\times S_4$), $(C_2\times C_6^2):S_4$ ($\GL(2,3)$), $C_3^2\times C_2^3:S_4$ ($\GL(2,3)$)

Order 864: $C_2\times C_3^2:\GL(2,3)$ ($C_2^2:S_4$), $C_3^2\times Q_8:A_4$ ($C_2\times \GL(2,3)$)

Order 576: $C_2^3\times \PSU(3,2)$ ($S_3\times S_4$) x 3, $C_6^2.C_2^4$ ($S_3\times S_4$)

Order 432: $C_3^2:\GL(2,3)$ ($C_2^3:S_4$) x 2, $C_2\times \PU(3,2)$ ($C_2^3:S_4$)

Order 288: $D_4:C_6^2$ ($S_3\times \GL(2,3)$)

Order 216: $\PU(3,2)$ ($C_2^4:S_4$)

Order 192: $C_2^3:S_4$ ($C_3^2:\GL(2,3)$)

Order 144: $C_6^2:C_2^2$ ($S_4^2$) x 3, $C_2\times \PSU(3,2)$ ($C_2^4:S_3^2$)

Order 96: $Q_8:A_4$ ($C_2\times C_3^2:\GL(2,3)$)

Order 72: $C_2\times C_6^2$ ($S_4\times \GL(2,3)$) x 3, $\PSU(3,2)$ ($S_3\times C_2^3:S_4$)

Order 36: $C_6:S_3$ ($C_2^2:S_4^2$)

Order 32: $D_4:C_2^2$ ($S_3\times C_3^2:\GL(2,3)$)

Order 18: $C_3\times C_6$ ($C_2^2:S_4\times \GL(2,3)$), $C_3:S_3$ ($Q_8.C_2^4.S_3^2$), $C_3:S_3$ ($C_2^3:S_4^2$)

Order 9: $C_3^2$ ($C_2^3:S_4\times \GL(2,3)$)

Order 8: $C_2^3$ ($S_4\times C_3^2:\GL(2,3)$) x 3

Order 2: $C_2$ ($C_2^2:S_4\times \AGL(2,3)$)

Order 1: $C_1$ ($C_6^2:C_2^2.S_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 82944: $C_6^2:C_2^2.S_4^2$ ($C_1$)

Order 41472: $C_3:S_3.(C_2^3\times A_4).A_4.C_2$ ($C_2$), $(C_2\times C_6^2).(Q_8\times A_4).C_6$ ($C_2$), $(C_2^3\times C_3:S_3).A_4^2.C_2$ ($C_2$)

Order 20736: $C_3^2.(C_2^3\times Q_8).A_4.C_3$ ($C_2^2$)

Order 13824: $\ASL(2,3).C_2^3.C_2^3$ ($S_3$), $(C_2\times C_6^2).(Q_8\times S_4)$ ($S_3$)

Order 6912: $(C_2\times C_6^2).(Q_8\times A_4)$ ($D_6$), $(C_2\times \ASL(2,3)).C_2^4$ ($D_6$)

Order 3456: $C_6^2:C_2^2:S_4$ ($S_4$), $\GL(2,3).C_2^6$ ($S_4$)

Order 2304: $(C_2\times C_3:S_3).C_2^6$ ($S_3^2$)

Order 1728: $C_2^3\times \PU(3,2)$ ($C_2\times S_4$), $D_4:C_6^2:C_6$ ($C_2\times S_4$), $(C_2\times C_6^2):S_4$ ($\GL(2,3)$), $C_3^2\times C_2^3:S_4$ ($\GL(2,3)$)

Order 864: $C_2\times C_3^2:\GL(2,3)$ ($C_2^2:S_4$), $C_3^2\times Q_8:A_4$ ($C_2\times \GL(2,3)$)

Order 576: $C_2^3\times \PSU(3,2)$ ($S_3\times S_4$), $C_6^2.C_2^4$ ($S_3\times S_4$)

Order 432: $C_2\times \PU(3,2)$ ($C_2^3:S_4$), $C_3^2:\GL(2,3)$ ($C_2^3:S_4$)

Order 288: $D_4:C_6^2$ ($S_3\times \GL(2,3)$)

Order 216: $\PU(3,2)$ ($C_2^4:S_4$)

Order 192: $C_2^3:S_4$ ($C_3^2:\GL(2,3)$)

Order 144: $C_6^2:C_2^2$ ($S_4^2$), $C_2\times \PSU(3,2)$ ($C_2^4:S_3^2$)

Order 96: $Q_8:A_4$ ($C_2\times C_3^2:\GL(2,3)$)

Order 72: $C_2\times C_6^2$ ($S_4\times \GL(2,3)$), $\PSU(3,2)$ ($S_3\times C_2^3:S_4$)

Order 36: $C_6:S_3$ ($C_2^2:S_4^2$)

Order 32: $D_4:C_2^2$ ($S_3\times C_3^2:\GL(2,3)$)

Order 18: $C_3\times C_6$ ($C_2^2:S_4\times \GL(2,3)$), $C_3:S_3$ ($Q_8.C_2^4.S_3^2$), $C_3:S_3$ ($C_2^3:S_4^2$)

Order 9: $C_3^2$ ($C_2^3:S_4\times \GL(2,3)$)

Order 8: $C_2^3$ ($S_4\times C_3^2:\GL(2,3)$)

Order 2: $C_2$ ($C_2^2:S_4\times \AGL(2,3)$)

Order 1: $C_1$ ($C_6^2:C_2^2.S_4^2$)

Series

Derived series

$C_6^2:C_2^2.S_4^2$

$\rhd$

$C_3^2.(C_2^3\times Q_8).A_4.C_3$

$\rhd$

$(C_2\times C_3:S_3).C_2^6$

$\rhd$

$C_6:S_3$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Chief series

$C_6^2:C_2^2.S_4^2$

$\rhd$

$(C_2\times C_6^2).(Q_8\times A_4).C_6$

$\rhd$

$C_3^2.(C_2^3\times Q_8).A_4.C_3$

$\rhd$

$(C_2\times C_6^2).(Q_8\times A_4)$

$\rhd$

$(C_2\times C_3:S_3).C_2^6$

$\rhd$

$C_2^3\times \PSU(3,2)$

$\rhd$

$C_2\times \PSU(3,2)$

$\rhd$

$C_6:S_3$

$\rhd$

$C_3\times C_6$

$\rhd$

$C_3^2$

$\rhd$

$C_1$

Lower central series

$C_6^2:C_2^2.S_4^2$

$\rhd$

$C_3^2.(C_2^3\times Q_8).A_4.C_3$

Upper central series

$C_1$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 1 larger groups in the database.

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

See the $143 \times 143$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $130 \times 130$ rational character table (warning: may be slow to load).