Abstract group 82944.fs: $S_3^4:C_2^3.D_4$

• Label: 82944.fs
• Order: \( 2^{10} \cdot 3^{4} \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{12} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \)
• Perm deg.: $20$
• Trans deg.: $36$
• Rank: $4$

Group information

Description:	$S_3^4:C_2^3.D_4$
Order:	\(82944\)\(\medspace = 2^{10} \cdot 3^{4} \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	$C_3^4.C_4:\SD_{16}.C_2^6$, of order \(331776\)\(\medspace = 2^{12} \cdot 3^{4} \)
Composition factors:	$C_2$ x 10, $C_3$ x 4
Derived length:	$4$

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

Group statistics

Order	1	2	3	4	6	8	12	16	24
Elements	1	1255	80	16632	7088	29088	13824	10368	4608	82944
Conjugacy classes	1	13	2	22	18	30	10	4	8	108
Divisions	1	13	2	22	18	18	10	2	4	90
Autjugacy classes	1	10	2	15	13	18	6	2	4	71

Dimension	1	2	4	8	16	32	64	128
Irr. complex chars.	16	12	36	6	16	12	8	2	108
Irr. rational chars.	16	12	12	14	18	4	12	2	90

Minimal presentations

Permutation degree:	$20$
Transitive degree:	$36$
Rank:	$4$
Inequivalent generating quadruples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	16	16	16
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid c^{8}=d^{24}=e^{6}=f^{3}=g^{3}=[d,g]=[e,f]= \!\cdots\! \rangle}$
Permutation group:	Degree $20$ $\langle(1,2,4,11)(3,7,5,13)(6,15,12,18)(10,17,16,14)(19,20), (1,3,8,17,15,9,11,13) \!\cdots\! \rangle$
Transitive group:	36T19404	36T19405	36T19407	36T19410	more information
Direct product:	$C_2$ $\, \times\, $ $(\SOPlus(4,2)^2.D_4)$
Semidirect product:	$(S_3^4:C_2^3.C_4)$ $\,\rtimes\,$ $C_2$	$(F_9\wr C_2:C_2^2)$ $\,\rtimes\,$ $C_2$	$(F_9\wr C_2:C_2)$ $\,\rtimes\,$ $C_2^2$	$(C_3^4:(Q_8^2.C_2^3))$ $\,\rtimes\,$ $C_2$	all 12
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(S_3^4:C_2^3)$ . $D_4$	$(S_3\wr D_4:C_2)$ . $C_2^2$	$(C_3^4:D_8:D_4)$ . $C_2^3$	$(C_3^2:F_9.D_8)$ . $C_2^3$	all 36
Aut. group:	$\Aut(C_3^4:(C_2\times C_{12}))$	$\Aut(C_3^5.C_4^2)$	$\Aut(C_3^4:C_{12}:D_4)$	$\Aut(C_3^5:D_4^2)$

Elements of the group are displayed as permutations of degree 20.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{6}$
Commutator length:	$1$

Subgroups

There are 1401854 subgroups in 7979 conjugacy classes, 119 normal (51 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $\SOPlus(4,2)^2.D_4$
Commutator:	$G' \simeq$ $C_3^2:F_9:D_4$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $S_3^4:C_2^3.D_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^3\times C_6$	$G/\operatorname{Fit} \simeq$ $\SD_{16}\wr C_2$
Radical:	$R \simeq$ $S_3^4:C_2^3.D_4$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^3\times C_6$	$G/\operatorname{soc} \simeq$ $\SD_{16}\wr C_2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2.C_2^5.C_2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

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Classes of subgroups up to automorphism

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Normal subgroups

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Normal subgroups up to automorphism

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Subgroup information

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Classes of subgroups up to conjugation

All subgroups of index up to 96 (order at least 864) are shown, as well as all normal subgroups of any index.

Order 82944: $S_3^4:C_2^3.D_4$

Order 41472: $\SOPlus(4,2)^2.D_4$ x 8, $C_2\times C_3^4.D_4^2.C_2^2$, $S_3^4:C_2^3.C_4$, $C_3^4.D_4^2.C_2^3$, $C_2\times C_3^4:(Q_8^2:C_4)$, $F_9\wr C_2:C_2^2$, $C_3^4:(Q_8^2.C_2^3)$, $C_2\times C_3^4:(C_8^2:C_4)$

Order 20736: $C_3^4.D_4^2.C_2^2$ x 10, $S_3\wr D_4:C_2$ x 6, $F_9\wr C_2:C_2$ x 6, $C_3^4:(Q_8^2:C_2^2)$ x 6, $C_3^4:(Q_8^2:C_4)$ x 4, $C_3^4:(C_8^2:C_4)$ x 4, $\SOPlus(4,2)^2.C_4$ x 4, $C_2\times C_3^4.D_4^2.C_2$, $C_2\times C_3^4.C_4^2.C_2^3$, $C_2\times C_3^4.C_4^2.C_2^3$, $C_2\times C_3^4.C_4^2.C_2^3$, $C_2\times C_3^4.(C_2\times D_4).C_2^3$, $C_2\times C_3^4.(C_4\times \SD_{16}).C_2$, $C_2\times C_3^4.C_4^2.C_2^3$, $C_2\times C_3^4.C_4^2.C_2^3$, $C_2\times C_3^4.C_4^2.C_2^3$, $C_2\times C_3^4:C_4^2.D_4$, $C_3^4:C_4^2.(C_2^2\times C_4)$, $C_3^4:\OD_{32}.C_2^3$, $C_3^4:(Q_8^2:C_2^2)$, $C_2\times C_3^2:F_9.D_8$, $C_2\times C_3^4:C_4^2.D_4$, $C_3^4:C_4^2.(C_2^2\times C_4)$, $C_3^2:S_3^2.D_4^2$, $C_2\times F_9\wr C_2$, $C_2\times S_3\wr D_4$

Order 10368: $\SOPlus(4,2)^2.C_2$ x 12, $C_3^4.C_4^2.C_2^3$ x 8, $C_3^4.(C_4\times \SD_{16}).C_2$ x 8, $C_3^4.C_4^2.C_2^3$ x 8, $C_3^4.C_4^2.C_2^3$ x 8, $C_3^4.C_4^2.C_2^3$ x 8, $C_3^4:D_8:D_4$ x 7, $C_3^4.C_4^2.C_2^3$ x 6, $C_3^4.D_4^2.C_2$ x 6, $C_3^4.C_4^2.C_2^3$ x 6, $C_3^4:D_8:Q_8$ x 6, $\PSU(3,2)\wr C_2$ x 6, $S_3\wr D_4$ x 6, $C_3^4.(D_4\times Q_8).C_2$ x 4, $C_3^4:C_4^2.D_4$ x 4, $C_3^4:C_4^2.D_4$ x 4, $F_9\wr C_2$ x 4, $C_3^2:(C_4\times F_9):C_4$ x 3, $C_3^2:F_9.D_8$ x 3, $C_3^2:(C_4\times F_9).C_4$ x 3, $C_2\times C_3^4.D_4:C_4.C_2$ x 2, $C_2\times C_3^4.(C_4\times \SD_{16})$ x 2, $C_2\times C_3^4.Q_8:Q_8$ x 2, $C_2\times C_3^4.C_8:Q_8$ x 2, $C_3^4:C_4^2.C_2^3$ x 2, $C_2\times C_3^4:C_4^2.C_4$ x 2, $C_2\times C_3^4.C_4:\SD_{16}$, $C_2\times C_3^4.C_2^2:\SD_{16}$, $C_2\times C_3^4.C_4^2.C_2^2$, $C_2\times C_3^4.(C_4.D_8)$, $C_2\times C_3^4.C_8:Q_8$, $C_2\times C_3^4.C_4:C_8.C_2$, $C_2\times C_3^4.(C_8\times D_4)$, $C_2\times C_3^4.(C_2\times \SD_{16}).C_2$, $C_2\times C_3^4.Q_8^2$, $C_2\times C_3^4.(D_4\times Q_8)$, $C_2\times C_3^4.C_8^2$, $C_2\times C_3^4.(C_8\times Q_8)$, $C_2\times C_3^4.(C_4:D_4.C_2)$, $C_2\times C_3^4.C_4:\SD_{16}$, $C_2\times C_3^4.Q_8:C_8$, $C_2\times C_3^4.D_4.C_2^3$, $C_2\times C_3^4.D_4:C_8$, $C_2\times C_3^4.C_4:Q_8.C_2$, $C_2\times C_3^4.(C_2\times \SD_{16}).C_2$, $C_2\times C_3^4.(C_2^2\times D_4).C_2$, $S_3^4:C_2^3$, $C_3^4:\OD_{16}.C_2^3$, $C_3^4:\OD_{32}:C_4$, $C_3^4:\OD_{32}:C_2^2$, $C_3^4:D_8:C_2^3$, $C_3^4:C_2^3.\OD_{16}$, $C_3^4:C_4^2.C_2^3$, $C_6:S_3^3.D_4$, $C_6:S_3^3.D_4$, $C_2\times S_3\wr C_4$, $S_3^4:C_2^3$

Order 5184: $C_3^4:(C_4\times \SD_{16})$ x 16, $C_3^4.D_4:C_4.C_2$ x 14, $C_3^4.Q_8:Q_8$ x 14, $C_3^4.C_8:Q_8$ x 14, $S_3\times F_9:D_6$ x 12, $C_3^4:D_4.D_4$ x 12, $C_3^4:D_8:C_2^2$ x 12, $C_3^4:(D_4\times Q_8)$ x 9, $F_9:\SOPlus(4,2)$ x 8, $(C_3^2\times S_3^2):\SD_{16}$ x 8, $(C_3^2\times S_3^2):\SD_{16}$ x 8, $C_3^2:F_9.D_4$ x 8, $C_3^4:C_4^2.C_4$ x 7, $C_3^4.(C_8\times Q_8)$ x 6, $C_3^4.Q_8^2$ x 6, $F_9\times \SOPlus(4,2)$ x 6, $F_9:\SOPlus(4,2)$ x 6, $C_3^3:C_{12}:\SD_{16}$ x 6, $C_3^3:C_{12}:\SD_{16}$ x 6, $\SOPlus(4,2)^2$ x 6, $C_3^4:D_8:C_4$ x 6, $C_3^4:D_4:D_4$ x 6, $S_3\wr C_2^2$ x 6, $C_3^2:F_9:D_4$ x 5, $C_3^4.C_4^2.C_2^2$ x 4, $C_3^4.D_4:C_8$ x 4, $C_3^4.C_4:Q_8.C_2$ x 4, $C_3^4.C_8:Q_8$ x 4, $C_3^4.Q_8:C_8$ x 4, $C_3^4:C_{16}:C_4$ x 4, $C_3^4:D_8:C_4$ x 4, $S_3\wr C_4$ x 4, $C_3^4.C_4:C_8.C_2$ x 3, $C_3^4.C_8^2$ x 3, $C_2\times C_3^4.C_4:Q_8$ x 3, $C_2\times C_3^4.C_8:C_4$ x 3, $C_3^4.(C_4.D_8)$ x 3, $C_3^4:Q_{16}:C_2^2$ x 3, $C_3^2:F_9.C_8$ x 3, $C_3^4:C_4^2:C_4$ x 3, $C_2\times C_3^4.C_2^2:Q_8$ x 2, $C_2\times C_3^4.Q_8:C_4$ x 2, $C_2\times C_3^4.(C_4\times Q_8)$ x 2, $C_2\times C_3^4.(C_4\times C_8)$ x 2, $C_2\times C_3^4.D_4:C_4$ x 2, $C_2\times C_3^4.C_8:C_4$ x 2, $C_2\times C_3^4.C_4:C_8$ x 2, $C_2\times C_3^4.C_4:Q_8$ x 2, $S_3^2:C_6^2:C_4$ x 2, $C_3^4:(C_2^2\times \SD_{16})$ x 2, $C_3^4:\OD_{16}.C_2^2$ x 2, $C_3^4:D_8:C_2^2$ x 2, $C_3^4:C_4^2:C_2^2$ x 2, $S_3^4:C_2^2$ x 2, $C_6:S_3^3:C_4$ x 2, $C_3^2:F_9:C_2^3$ x 2, $C_2\times C_3^4.\SD_{16}.C_2$, $C_2\times C_3.(S_3\times F_9).C_2$, $C_2\times C_3^4.C_8:C_4$, $C_2\times C_3^4.\SD_{16}.C_2$, $C_2\times C_3^4.D_4.C_2^2$, $C_2\times C_3^4.\SD_{16}.C_2$, $C_2\times C_3^4.(C_2^2\times Q_8)$, $C_2\times C_3^4.Q_8:C_4$, $C_2\times C_3^4.C_4:C_8$, $C_2\times C_3^4.D_4.C_2^2$, $C_2\times C_3^4.\SD_{16}.C_2$, $C_2\times C_3^4.C_2^2:C_8$, $C_2\times C_3^4.C_4^2.C_2$, $C_2\times C_3^4.\SD_{16}.C_2$, $C_2\times C_3^4.D_4:C_4$, $C_2\times C_3^3.(C_6\times \SD_{16})$, $C_2\times C_3^4.\SD_{16}.C_2$, $C_3^4:C_4^2:C_2^2$, $C_3:S_3^3:D_4$, $S_3^2:D_6^2$, $C_3^4:C_2^3.Q_8$, $C_3^4:\OD_{16}:C_2^2$, $C_3^4:C_4.C_2^4$, $C_2\times C_3^4:\OD_{32}$, $C_6:S_3^3.C_4$, $C_6:S_3^3:C_2^2$

Order 4608: $C_2\times C_3:S_3.D_4^2.C_2$

Order 3456: $C_2\times (C_6\times S_3\wr C_2).C_2^2$

Order 2592: $C_3^4:Q_{16}:C_2$ x 20, $C_3^3:C_{12}:Q_8$ x 16, $S_3^2:\PSU(3,2)$ x 14, $C_3^4:(C_2\times \SD_{16})$ x 14, $C_3^4:D_8:C_2$ x 14, $C_3^4:D_8:C_2$ x 14, $C_3^4.C_8:C_4$ x 12, $C_3^4:(C_4\times Q_8)$ x 12, $C_3^4:(C_4\times D_4)$ x 12, $S_3^2\wr C_2$ x 12, $C_3^3:C_{12}:Q_8$ x 11, $S_3^2\times \PSU(3,2)$ x 9, $S_3^4:C_2$ x 9, $(C_3^2\times F_9):C_4$ x 8, $C_3^2:C_4\times F_9$ x 8, $(C_3\times F_9):D_6$ x 8, $C_3^2\wr C_2.\SD_{16}$ x 8, $F_9:S_3^2$ x 8, $\PSU(3,2):S_3^2$ x 8, $S_3\times C_3^3:\SD_{16}$ x 8, $F_9:C_6\times S_3$ x 8, $C_3^2\wr C_2.\SD_{16}$ x 8, $C_3^4:C_4\wr C_2$ x 8, $C_3^4.C_4:C_8$ x 7, $C_3:S_3^3:C_4$ x 7, $F_9:S_3^2$ x 6, $S_3^2\times F_9$ x 6, $\PSU(3,2):S_3^2$ x 6, $S_3^2:\SOPlus(4,2)$ x 6, $\SOPlus(4,2).S_3^2$ x 6, $C_3^4:C_4.D_4$ x 6, $C_3^4:\OD_{16}:C_2$ x 6, $C_3^4:C_4.C_2^3$ x 6, $C_3:S_3^3:C_2^2$ x 6, $C_3^4:(C_2\times \SD_{16})$ x 5, $C_2\times C_3^4.C_4:C_4$ x 4, $C_3^2\wr C_2.\SD_{16}$ x 4, $C_3^3:C_{12}:D_4$ x 4, $C_3^2\wr C_2.\SD_{16}$ x 4, $S_3^2:F_9$ x 4, $C_3^4:D_4:C_2^2$ x 4, $C_2\times C_3^4:\SD_{16}$ x 4, $C_3^3:C_{12}:C_8$ x 3, $C_2\times C_3^4:C_4^2$ x 3, $C_3^2\wr C_2.\SD_{16}$ x 3, $C_3^2:(C_4\times F_9)$ x 3, $C_2\times C_3^4:Q_{16}$ x 3, $C_3^4:(C_2^2\times Q_8)$ x 3, $C_2\times C_3^4:\OD_{16}$ x 3, $C_3^4:(C_2^2\times D_4)$ x 3, $C_3:S_3^3:C_4$ x 3, $C_3^2:F_9.C_4$ x 3, $C_3^4:\OD_{32}$ x 3, $C_3:S_3^3.C_4$ x 3, $C_2\times C_3^4.(C_2\times C_8)$ x 2, $C_2\times C_3^4.C_4:C_4$ x 2, $C_3^2:C_6^2.D_4$ x 2, $C_3\wr C_4:C_2^3$ x 2, $C_2\times C_3^4:\SD_{16}$ x 2, $C_3^2:(C_2^2\times F_9)$ x 2, $C_2\times S_3^2:S_3^2$ x 2, $C_2\times C_3^4:D_8$ x 2, $C_3:S_3^3:C_4$ x 2, $C_2\times C_3^4.\SD_{16}$, $C_2\times C_3.(C_6\times F_9)$, $C_2\times C_3^3.(C_3\times \SD_{16})$, $C_2\times C_3^4.C_8.C_2$, $C_2\times C_3^4.\SD_{16}$, $C_2\times C_3^3.(C_6\times Q_8)$, $C_2\times C_3:S_3.C_6^2.C_2$, $C_2\times C_3^4.Q_8.C_2$, $C_2\times C_3^4.\SD_{16}$, $C_2\times C_3^4.\SD_{16}$, $C_2\times C_3^4.C_8.C_2$, $C_2\times C_3^4.\SD_{16}$, $C_2\times C_3^4.Q_8.C_2$, $C_2\times C_3^4.(C_2\times Q_8)$, $C_2\times C_3^4.\SD_{16}$, $C_2\times C_3^3.(C_3\times \SD_{16})$, $C_2\times C_3^4.C_8.C_2$, $C_2\times C_3^3.(C_3\times \SD_{16})$, $C_2\times C_3^4.Q_8.C_2$, $C_2\times C_3^4.(C_2\times Q_8)$, $C_2\times C_3^4.\SD_{16}$, $C_3^3:(D_4\times D_6)$, $C_2\times S_3^3:C_6$, $(C_6\times S_3^2):D_6$, $C_2\times S_3^4$, $C_3^2:D_{12}\times D_6$, $S_3^3:D_6$, $C_2\times C_3^4:C_{16}$

Order 2304: $C_3:S_3.D_4^2.C_2$ x 16, $C_2\times C_3:S_3.C_2^2:D_4.C_2$, $C_2\times C_3:S_3.(D_4\times Q_8)$, $(C_4\times C_3:S_3).C_2^4.C_2$, $C_2\times C_3^2.D_4^2.C_2$, $(C_4\times C_3:S_3).C_2^4.C_2$, $C_2\times C_3:S_3.(C_8\times D_4)$, $(C_4\times C_3:S_3).C_2^4.C_2$, $(C_4\times C_3:S_3).C_2^5$, $C_2\times C_3:S_3.C_4:\SD_{16}$, $C_2\times C_3:S_3.C_4:D_4.C_2$, $C_2\times C_4:S_3\wr C_2.C_2^2$, $(C_4\times C_3:S_3).C_2^5$, $C_2\times C_3^2.(C_8\times D_4).C_2$, $(C_4\times C_3:S_3).C_2^4.C_2$, $D_4\times F_9:C_2^2$

Order 1728: $C_2\times F_9:D_6$ x 24, $S_3\times C_6^2:Q_8$, $D_6^2:D_6$, $C_2\times F_9:D_6$, $C_2^2\times C_3^3:\SD_{16}$, $C_2^2\times C_3^3:\SD_{16}$, $C_2^2\times F_9:C_6$, $C_2\times D_6\times F_9$

Order 1296: $C_3^4:D_4:C_2$ x 24, $C_3^4:(C_2\times D_4)$ x 16, $C_3^4:\SD_{16}$ x 16, $C_3^4:(C_2\times Q_8)$ x 14, $S_3\times C_3^3:Q_8$ x 14, $C_3^2\wr C_2.Q_8$ x 14, $S_3^2:S_3^2$ x 12, $C_3^2:C_4\times S_3^2$ x 12, $(C_3^2\times S_3^2):C_4$ x 12, $C_3^3:(S_3\times Q_8)$ x 11, $C_3^4:Q_{16}$ x 11, $C_3^4:\OD_{16}$ x 10, $C_3^4:C_4^2$ x 10, $S_3^3:S_3$ x 8, $C_3^4:\SD_{16}$ x 8, $C_3^2:(C_2\times F_9)$ x 8, $C_3^4:D_8$ x 7, $C_3^2\wr C_2.D_4$ x 7, $C_3^4:(C_2^2\times C_4)$ x 6, $S_3^4$ x 6, $C_3^3:(S_3\times Q_8)$ x 6, $S_3\times C_3^3:Q_8$ x 6, $S_3\times C_3^2:D_{12}$ x 6, $(C_3\times S_3^2):D_6$ x 6, $S_3^3:C_6$ x 6, $C_3^2:(C_2\times F_9)$ x 6, $C_3^2\wr C_2.D_4$ x 6, $C_3^2\wr C_2.D_4$ x 6, $C_2\times C_3^4:D_4$ x 5, $C_3^3:(S_3\times D_4)$ x 5, $C_2\times C_3^4:Q_8$ x 4, $C_3^4:\SD_{16}$ x 4, $C_3^4:\SD_{16}$ x 4, $C_3^4:\SD_{16}$ x 4, $C_3^4:\SD_{16}$ x 4, $C_3^4:\SD_{16}$ x 4, $C_3^4:\SD_{16}$ x 4, $C_3^4:\SD_{16}$ x 4, $C_3^4:\SD_{16}$ x 4, $C_3^4:\SD_{16}$ x 4, $C_3^4:\SD_{16}$ x 4, $C_3^4:\SD_{16}$ x 4, $C_3:S_3\times F_9$ x 4, $S_3\times C_3:F_9$ x 4, $C_3\times S_3\times F_9$ x 4, $C_3^3:(C_4\times D_6)$ x 3, $C_2\times C_3^2:F_9$ x 3, $C_3:(S_3\times F_9)$ x 3, $C_6:S_3^3$ x 2, $C_6:S_3^3$ x 2, $C_2\times C_3^4:Q_8$ x 2, $C_2\times C_3^4:Q_8$ x 2, $C_6\times C_3^3:Q_8$ x 2, $C_2\times C_3^3:D_{12}$ x 2, $C_2\times C_3\wr D_4$ x 2, $C_3^3:(C_4\times D_6)$ x 2, $C_3^3:C_4\times D_6$ x 2, $C_3^2:C_{12}\times D_6$ x 2, $C_2\times C_3^2:F_9$ x 2, $C_3^4:C_{16}$ x 2, $C_6:S_3^3$, $C_6\times S_3^3$, $C_6\times C_3^3:Q_8$, $C_2\times C_3^3:D_{12}$, $C_2\times C_3^4:D_4$, $C_2\times C_3^4:D_4$, $C_6\times C_3^2:D_{12}$, $S_3^2:C_6^2$, $C_2\times C_3^2:F_9$, $C_2\times C_3^2:F_9$, $C_3\times C_6:F_9$, $C_3\times C_6\times F_9$

Order 1152: $C_3:S_3.(D_4\times Q_8)$ x 12, $(C_4\times C_3:S_3).C_2^4$ x 12, $D_4\times F_9:C_2$ x 12, $\SD_{16}\times \SOPlus(4,2)$ x 12, $C_4:S_3\wr C_2.C_2^2$ x 8, $C_3^2.C_8:Q_8.C_2$ x 8, $C_3^2.C_8:Q_8.C_2$ x 8, $C_3:S_3.C_4:D_4.C_2$ x 8, $C_4:S_3\wr C_2.C_2^2$ x 8, $C_3^2.C_8:Q_8.C_2$ x 8, $C_3^2.C_8:Q_8.C_2$ x 8, $C_3:S_3.D_4:D_4$ x 8, $C_3:S_3.D_4:D_4$ x 8, $C_3^2.C_8^2.C_2$ x 8, $C_4:F_9.C_2^2$ x 8, $C_2\times C_3^2.C_8:Q_8$ x 3, $C_4\times F_9:C_2^2$ x 2, $C_2\times C_3:S_3.\SD_{16}.C_2$ x 2, $C_2\times C_3^2.C_8:Q_8$ x 2, $C_2\times F_9:D_4$ x 2, $C_2\times C_3^2.Q_8:Q_8$ x 2, $C_2\times C_3^2.Q_8:Q_8$ x 2, $C_4:F_9.C_2^2$ x 2, $C_2\times C_4:\PSU(3,2).C_2$ x 2, $(C_6\times C_{12}):\SD_{16}$, $C_2\times C_4:S_3\wr C_2.C_2$, $(C_6\times C_{12}):\SD_{16}$, $C_2\times C_2.S_3\wr C_2.C_2.C_2$, $C_2\times C_6^2:\SD_{16}$, $C_2\times C_4:S_3^2.C_2^2$, $C_2\times C_6^2:\SD_{16}$, $C_2\times C_3:S_3:\SD_{16}.C_2$, $C_2\times C_3^2.(C_8\times Q_8)$, $C_2\times C_4:F_9.C_2$, $C_2\times S_3^2:C_8.C_2$, $C_2\times D_4\times F_9$, $C_2\times C_3^2.Q_8:C_8$, $C_2\times C_4:F_9.C_2$, $C_2\times C_3^2.D_4:C_8$, $C_2\times C_4:F_9.C_2$, $C_2\times C_8\times F_9$, $F_9:C_2^4$, $C_2\times \SD_{16}\times S_3^2$, $C_2\times C_3^2.Q_8^2$, $C_6^2:(C_2^2\times Q_8)$, $C_2\times S_3^2:Q_8.C_2$, $D_6^2:C_2^3$, $C_2\times C_4:S_3^2.C_2^2$, $(C_4\times F_9).C_2^2$, $C_2\times C_4:F_9.C_2$, $C_2\times C_3:S_3.Q_8:C_4$, $C_2\times C_3:S_3.D_4:C_4$, $C_2\times C_4:F_9.C_2$, $C_2\times C_3:S_3.D_4:C_4$, $C_2\times F_9:D_4$, $C_2\times C_4:S_3^2.C_2^2$

Order 1024: $C_4^2.C_2^5.C_2$

Order 864: $F_9:D_6$ x 64, $S_3^3:C_2^2$ x 19, $D_6\times \PSU(3,2)$ x 18, $F_9:D_6$ x 12, $\PSU(3,2):D_6$ x 12, $C_2\times C_3^3:\SD_{16}$ x 12, $C_2\times F_9:C_6$ x 12, $D_6\times F_9$ x 12, $C_2^2\times C_3^3:Q_8$ x 2, $C_6^2:C_4\times S_3$ x 2, $S_3\times D_6^2$, $C_3\times C_6^2:Q_8$, $C_6^2:D_{12}$, $D_6^2:S_3$, $D_6^2:C_6$, $C_2\times C_6:F_9$, $C_6^2:C_{24}$

Order 648: $C_3^4:(C_2\times C_4)$ x 2, $C_3^4:C_2^3$

Order 512: $\SD_{16}\wr C_2$ x 8

Order 324: $C_3^2:S_3^2$ x 2, $C_3^3:D_6$

Order 162: $C_3^3:S_3$ x 2, $C_3^3\times C_6$

Order 81: $C_3^4$

Order 16: $C_2\times D_4$ x 4

Order 8: $D_4$ x 8

Order 4: $C_2^2$ x 15

Order 2: $C_2$ x 10

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 96 (order at least 864) are shown, as well as all normal subgroups of any index.

Order 82944: $S_3^4:C_2^3.D_4$

Order 41472: $C_2\times C_3^4.D_4^2.C_2^2$, $\SOPlus(4,2)^2.D_4$, $S_3^4:C_2^3.C_4$, $C_3^4.D_4^2.C_2^3$, $C_2\times C_3^4:(Q_8^2:C_4)$, $F_9\wr C_2:C_2^2$, $C_3^4:(Q_8^2.C_2^3)$, $C_2\times C_3^4:(C_8^2:C_4)$

Order 20736: $C_3^4.D_4^2.C_2^2$ x 4, $S_3\wr D_4:C_2$ x 2, $F_9\wr C_2:C_2$ x 2, $C_3^4:(Q_8^2:C_2^2)$ x 2, $C_2\times C_3^4.D_4^2.C_2$, $C_2\times C_3^4.C_4^2.C_2^3$, $C_2\times C_3^4.C_4^2.C_2^3$, $C_2\times C_3^4.C_4^2.C_2^3$, $C_2\times C_3^4.(C_2\times D_4).C_2^3$, $C_2\times C_3^4.(C_4\times \SD_{16}).C_2$, $C_2\times C_3^4.C_4^2.C_2^3$, $C_2\times C_3^4.C_4^2.C_2^3$, $C_2\times C_3^4.C_4^2.C_2^3$, $C_3^4:(Q_8^2:C_4)$, $C_3^4:(C_8^2:C_4)$, $C_2\times C_3^4:C_4^2.D_4$, $C_3^4:C_4^2.(C_2^2\times C_4)$, $C_3^4:\OD_{32}.C_2^3$, $C_3^4:(Q_8^2:C_2^2)$, $C_2\times C_3^2:F_9.D_8$, $C_2\times C_3^4:C_4^2.D_4$, $C_3^4:C_4^2.(C_2^2\times C_4)$, $C_3^2:S_3^2.D_4^2$, $C_2\times F_9\wr C_2$, $C_2\times S_3\wr D_4$, $\SOPlus(4,2)^2.C_4$

Order 10368: $C_3^4:D_8:D_4$ x 5, $C_3^4.C_4^2.C_2^3$ x 4, $C_3^4.D_4^2.C_2$ x 4, $C_3^4.C_4^2.C_2^3$ x 4, $\SOPlus(4,2)^2.C_2$ x 3, $C_2\times C_3^4.D_4:C_4.C_2$ x 2, $C_3^4.C_4^2.C_2^3$ x 2, $C_2\times C_3^4.(C_4\times \SD_{16})$ x 2, $C_3^4.(C_4\times \SD_{16}).C_2$ x 2, $C_2\times C_3^4.Q_8:Q_8$ x 2, $C_3^4.C_4^2.C_2^3$ x 2, $C_3^4.C_4^2.C_2^3$ x 2, $C_2\times C_3^4.C_8:Q_8$ x 2, $C_3^4.C_4^2.C_2^3$ x 2, $C_3^4:C_4^2.C_2^3$ x 2, $C_2\times C_3^4:C_4^2.C_4$ x 2, $C_3^2:(C_4\times F_9):C_4$ x 2, $C_3^4:D_8:Q_8$ x 2, $\PSU(3,2)\wr C_2$ x 2, $C_3^2:F_9.D_8$ x 2, $C_3^2:(C_4\times F_9).C_4$ x 2, $S_3\wr D_4$ x 2, $C_2\times C_3^4.C_4:\SD_{16}$, $C_2\times C_3^4.C_2^2:\SD_{16}$, $C_2\times C_3^4.C_4^2.C_2^2$, $C_2\times C_3^4.(C_4.D_8)$, $C_2\times C_3^4.C_8:Q_8$, $C_2\times C_3^4.C_4:C_8.C_2$, $C_2\times C_3^4.(C_8\times D_4)$, $C_2\times C_3^4.(C_2\times \SD_{16}).C_2$, $C_2\times C_3^4.Q_8^2$, $C_2\times C_3^4.(D_4\times Q_8)$, $C_2\times C_3^4.C_8^2$, $C_2\times C_3^4.(C_8\times Q_8)$, $C_2\times C_3^4.(C_4:D_4.C_2)$, $C_2\times C_3^4.C_4:\SD_{16}$, $C_2\times C_3^4.Q_8:C_8$, $C_3^4.(D_4\times Q_8).C_2$, $C_2\times C_3^4.D_4.C_2^3$, $C_2\times C_3^4.D_4:C_8$, $C_2\times C_3^4.C_4:Q_8.C_2$, $C_2\times C_3^4.(C_2\times \SD_{16}).C_2$, $C_2\times C_3^4.(C_2^2\times D_4).C_2$, $S_3^4:C_2^3$, $C_3^4:\OD_{16}.C_2^3$, $C_3^4:\OD_{32}:C_4$, $C_3^4:\OD_{32}:C_2^2$, $C_3^4:D_8:C_2^3$, $C_3^4:C_2^3.\OD_{16}$, $C_3^4:C_4^2.C_2^3$, $C_6:S_3^3.D_4$, $C_6:S_3^3.D_4$, $C_2\times S_3\wr C_4$, $S_3^4:C_2^3$, $C_3^4:C_4^2.D_4$, $C_3^4:C_4^2.D_4$, $F_9\wr C_2$

Order 5184: $C_3^4:D_4.D_4$ x 6, $C_3^4:D_8:C_2^2$ x 6, $C_3^4.D_4:C_4.C_2$ x 5, $C_3^4.C_8:Q_8$ x 5, $C_3^2:F_9:D_4$ x 5, $C_3^4.Q_8^2$ x 4, $C_3^4.Q_8:Q_8$ x 4, $C_3^4:(C_4\times \SD_{16})$ x 4, $C_3^4:(D_4\times Q_8)$ x 4, $\SOPlus(4,2)^2$ x 4, $C_3^2:F_9.D_4$ x 4, $C_3^4:D_8:C_4$ x 4, $C_3^4:D_4:D_4$ x 4, $C_2\times C_3^4.C_4:Q_8$ x 3, $C_2\times C_3^4.C_8:C_4$ x 3, $C_3^4.C_8:Q_8$ x 3, $F_9:\SOPlus(4,2)$ x 3, $C_3^3:C_{12}:\SD_{16}$ x 3, $S_3\times F_9:D_6$ x 3, $C_3^4:Q_{16}:C_2^2$ x 3, $C_3^4:C_4^2.C_4$ x 3, $C_2\times C_3^4.C_2^2:Q_8$ x 2, $C_3^4.(C_8\times Q_8)$ x 2, $C_2\times C_3^4.Q_8:C_4$ x 2, $C_3^4.C_4:C_8.C_2$ x 2, $C_2\times C_3^4.(C_4\times Q_8)$ x 2, $C_3^4.C_8^2$ x 2, $C_3^4.D_4:C_8$ x 2, $C_2\times C_3^4.(C_4\times C_8)$ x 2, $C_3^4.(C_4.D_8)$ x 2, $C_2\times C_3^4.D_4:C_4$ x 2, $C_2\times C_3^4.C_8:C_4$ x 2, $C_2\times C_3^4.C_4:C_8$ x 2, $C_3^4.Q_8:C_8$ x 2, $C_2\times C_3^4.C_4:Q_8$ x 2, $F_9\times \SOPlus(4,2)$ x 2, $F_9:\SOPlus(4,2)$ x 2, $C_3^3:C_{12}:\SD_{16}$ x 2, $S_3^2:C_6^2:C_4$ x 2, $(C_3^2\times S_3^2):\SD_{16}$ x 2, $C_3^4:(C_2^2\times \SD_{16})$ x 2, $C_3^4:\OD_{16}.C_2^2$ x 2, $C_3^4:D_8:C_2^2$ x 2, $(C_3^2\times S_3^2):\SD_{16}$ x 2, $C_3^4:C_4^2:C_2^2$ x 2, $S_3^4:C_2^2$ x 2, $C_6:S_3^3:C_4$ x 2, $C_3^2:F_9:C_2^3$ x 2, $C_3^2:F_9.C_8$ x 2, $C_3^4:C_4^2:C_4$ x 2, $S_3\wr C_2^2$ x 2, $C_2\times C_3^4.\SD_{16}.C_2$, $C_2\times C_3.(S_3\times F_9).C_2$, $C_3^4.C_4^2.C_2^2$, $C_2\times C_3^4.C_8:C_4$, $C_2\times C_3^4.\SD_{16}.C_2$, $C_2\times C_3^4.D_4.C_2^2$, $C_2\times C_3^4.\SD_{16}.C_2$, $C_2\times C_3^4.(C_2^2\times Q_8)$, $C_2\times C_3^4.Q_8:C_4$, $C_2\times C_3^4.C_4:C_8$, $C_2\times C_3^4.D_4.C_2^2$, $C_2\times C_3^4.\SD_{16}.C_2$, $C_3^4.C_4:Q_8.C_2$, $C_2\times C_3^4.C_2^2:C_8$, $C_2\times C_3^4.C_4^2.C_2$, $C_2\times C_3^4.\SD_{16}.C_2$, $C_2\times C_3^4.D_4:C_4$, $C_2\times C_3^3.(C_6\times \SD_{16})$, $C_2\times C_3^4.\SD_{16}.C_2$, $C_3^4:C_4^2:C_2^2$, $C_3:S_3^3:D_4$, $S_3^2:D_6^2$, $C_3^4:C_2^3.Q_8$, $C_3^4:\OD_{16}:C_2^2$, $C_3^4:C_4.C_2^4$, $C_2\times C_3^4:\OD_{32}$, $C_6:S_3^3.C_4$, $C_6:S_3^3:C_2^2$, $C_3^4:C_{16}:C_4$, $C_3^4:D_8:C_4$, $S_3\wr C_4$

Order 4608: $C_2\times C_3:S_3.D_4^2.C_2$

Order 3456: $C_2\times (C_6\times S_3\wr C_2).C_2^2$

Order 2592: $C_3^3:C_{12}:Q_8$ x 11, $C_3^4:(C_2\times \SD_{16})$ x 10, $C_3^4:Q_{16}:C_2$ x 9, $C_3^4:D_8:C_2$ x 8, $C_3^4:(C_4\times Q_8)$ x 7, $C_3^4:D_8:C_2$ x 7, $C_3^3:C_{12}:Q_8$ x 6, $C_3^4:(C_4\times D_4)$ x 6, $S_3^2\wr C_2$ x 6, $C_3^4.C_8:C_4$ x 5, $S_3^4:C_2$ x 5, $C_2\times C_3^4.C_4:C_4$ x 4, $S_3^2\times \PSU(3,2)$ x 4, $C_3^3:C_{12}:D_4$ x 4, $S_3^2:\PSU(3,2)$ x 4, $C_3^4:D_4:C_2^2$ x 4, $C_2\times C_3^4:\SD_{16}$ x 4, $C_3^4:C_4.D_4$ x 4, $C_3^4:\OD_{16}:C_2$ x 4, $C_3^4:C_4.C_2^3$ x 4, $C_3:S_3^3:C_2^2$ x 4, $C_3^4.C_4:C_8$ x 3, $(C_3^2\times F_9):C_4$ x 3, $C_3^2:C_4\times F_9$ x 3, $C_2\times C_3^4:C_4^2$ x 3, $F_9:S_3^2$ x 3, $\PSU(3,2):S_3^2$ x 3, $C_3^2:(C_4\times F_9)$ x 3, $C_2\times C_3^4:Q_{16}$ x 3, $C_3^4:(C_2^2\times Q_8)$ x 3, $S_3^2:\SOPlus(4,2)$ x 3, $C_3^4:(C_2\times \SD_{16})$ x 3, $C_3^2\wr C_2.\SD_{16}$ x 3, $C_2\times C_3^4:\OD_{16}$ x 3, $C_3^4:(C_2^2\times D_4)$ x 3, $C_3:S_3^3:C_4$ x 3, $C_3^4:C_4\wr C_2$ x 3, $C_3:S_3^3:C_4$ x 3, $C_2\times C_3^4.(C_2\times C_8)$ x 2, $C_2\times C_3^4.C_4:C_4$ x 2, $C_3^3:C_{12}:C_8$ x 2, $(C_3\times F_9):D_6$ x 2, $C_3^2\wr C_2.\SD_{16}$ x 2, $C_3^2:C_6^2.D_4$ x 2, $F_9:S_3^2$ x 2, $S_3^2\times F_9$ x 2, $\PSU(3,2):S_3^2$ x 2, $S_3\times C_3^3:\SD_{16}$ x 2, $F_9:C_6\times S_3$ x 2, $C_3\wr C_4:C_2^3$ x 2, $C_3^2\wr C_2.\SD_{16}$ x 2, $C_2\times C_3^4:\SD_{16}$ x 2, $C_3^2:(C_2^2\times F_9)$ x 2, $C_3^2\wr C_2.\SD_{16}$ x 2, $\SOPlus(4,2).S_3^2$ x 2, $C_2\times S_3^2:S_3^2$ x 2, $C_2\times C_3^4:D_8$ x 2, $C_3:S_3^3:C_4$ x 2, $C_3^2:F_9.C_4$ x 2, $C_3^4:\OD_{32}$ x 2, $C_3:S_3^3.C_4$ x 2, $C_2\times C_3^4.\SD_{16}$, $C_2\times C_3.(C_6\times F_9)$, $C_2\times C_3^3.(C_3\times \SD_{16})$, $C_2\times C_3^4.C_8.C_2$, $C_2\times C_3^4.\SD_{16}$, $C_2\times C_3^3.(C_6\times Q_8)$, $C_2\times C_3:S_3.C_6^2.C_2$, $C_2\times C_3^4.Q_8.C_2$, $C_2\times C_3^4.\SD_{16}$, $C_2\times C_3^4.\SD_{16}$, $C_2\times C_3^4.C_8.C_2$, $C_2\times C_3^4.\SD_{16}$, $C_2\times C_3^4.Q_8.C_2$, $C_2\times C_3^4.(C_2\times Q_8)$, $C_2\times C_3^4.\SD_{16}$, $C_2\times C_3^3.(C_3\times \SD_{16})$, $C_2\times C_3^4.C_8.C_2$, $C_2\times C_3^3.(C_3\times \SD_{16})$, $C_2\times C_3^4.Q_8.C_2$, $C_2\times C_3^4.(C_2\times Q_8)$, $C_2\times C_3^4.\SD_{16}$, $C_3^2\wr C_2.\SD_{16}$, $C_3^3:(D_4\times D_6)$, $C_2\times S_3^3:C_6$, $(C_6\times S_3^2):D_6$, $C_2\times S_3^4$, $C_3^2:D_{12}\times D_6$, $S_3^3:D_6$, $C_2\times C_3^4:C_{16}$, $S_3^2:F_9$

Order 2304: $C_3:S_3.D_4^2.C_2$ x 4, $C_2\times C_3:S_3.C_2^2:D_4.C_2$, $C_2\times C_3:S_3.(D_4\times Q_8)$, $(C_4\times C_3:S_3).C_2^4.C_2$, $C_2\times C_3^2.D_4^2.C_2$, $(C_4\times C_3:S_3).C_2^4.C_2$, $C_2\times C_3:S_3.(C_8\times D_4)$, $(C_4\times C_3:S_3).C_2^4.C_2$, $(C_4\times C_3:S_3).C_2^5$, $C_2\times C_3:S_3.C_4:\SD_{16}$, $C_2\times C_3:S_3.C_4:D_4.C_2$, $C_2\times C_4:S_3\wr C_2.C_2^2$, $(C_4\times C_3:S_3).C_2^5$, $C_2\times C_3^2.(C_8\times D_4).C_2$, $(C_4\times C_3:S_3).C_2^4.C_2$, $D_4\times F_9:C_2^2$

Order 1728: $C_2\times F_9:D_6$ x 12, $S_3\times C_6^2:Q_8$, $D_6^2:D_6$, $C_2\times F_9:D_6$, $C_2^2\times C_3^3:\SD_{16}$, $C_2^2\times C_3^3:\SD_{16}$, $C_2^2\times F_9:C_6$, $C_2\times D_6\times F_9$

Order 1296: $C_3^4:D_4:C_2$ x 14, $C_3^4:(C_2\times D_4)$ x 11, $C_3^4:(C_2\times Q_8)$ x 10, $C_3^4:\SD_{16}$ x 9, $C_3^2\wr C_2.Q_8$ x 9, $C_3^4:C_4^2$ x 7, $C_3^4:(C_2^2\times C_4)$ x 6, $C_3^3:(S_3\times Q_8)$ x 6, $C_3^2:C_4\times S_3^2$ x 6, $C_3^4:\SD_{16}$ x 6, $C_3^4:\OD_{16}$ x 6, $C_3^2:(C_2\times F_9)$ x 6, $C_2\times C_3^4:D_4$ x 5, $C_3^3:(S_3\times D_4)$ x 5, $S_3^2:S_3^2$ x 5, $C_3^4:Q_{16}$ x 5, $C_2\times C_3^4:Q_8$ x 4, $S_3^4$ x 4, $C_3^3:(S_3\times Q_8)$ x 4, $S_3\times C_3^3:Q_8$ x 4, $(C_3\times S_3^2):D_6$ x 4, $S_3^3:S_3$ x 4, $C_3^4:D_8$ x 4, $C_3^2:(C_2\times F_9)$ x 4, $C_3^2\wr C_2.D_4$ x 4, $(C_3^2\times S_3^2):C_4$ x 4, $C_3^2\wr C_2.D_4$ x 4, $C_3^3:(C_4\times D_6)$ x 3, $C_2\times C_3^2:F_9$ x 3, $S_3\times C_3^2:D_{12}$ x 3, $S_3^3:C_6$ x 3, $C_3^2\wr C_2.D_4$ x 3, $C_6:S_3^3$ x 2, $C_6:S_3^3$ x 2, $C_2\times C_3^4:Q_8$ x 2, $C_2\times C_3^4:Q_8$ x 2, $C_6\times C_3^3:Q_8$ x 2, $C_2\times C_3^3:D_{12}$ x 2, $C_2\times C_3\wr D_4$ x 2, $C_3^3:(C_4\times D_6)$ x 2, $C_3^3:C_4\times D_6$ x 2, $C_3^2:C_{12}\times D_6$ x 2, $C_2\times C_3^2:F_9$ x 2, $S_3\times C_3^3:Q_8$ x 2, $C_3^4:\SD_{16}$ x 2, $C_3^4:\SD_{16}$ x 2, $C_3^4:\SD_{16}$ x 2, $C_3^4:\SD_{16}$ x 2, $C_3^4:\SD_{16}$ x 2, $C_3:(S_3\times F_9)$ x 2, $C_3:S_3\times F_9$ x 2, $C_6:S_3^3$, $C_6\times S_3^3$, $C_6\times C_3^3:Q_8$, $C_2\times C_3^3:D_{12}$, $C_2\times C_3^4:D_4$, $C_2\times C_3^4:D_4$, $C_6\times C_3^2:D_{12}$, $S_3^2:C_6^2$, $C_2\times C_3^2:F_9$, $C_2\times C_3^2:F_9$, $C_3\times C_6:F_9$, $C_3\times C_6\times F_9$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $S_3\times C_3:F_9$, $C_3\times S_3\times F_9$, $C_3^4:C_{16}$

Order 1152: $C_4:S_3\wr C_2.C_2^2$ x 4, $C_3:S_3.D_4:D_4$ x 4, $C_3:S_3.D_4:D_4$ x 4, $C_3:S_3.(D_4\times Q_8)$ x 3, $(C_4\times C_3:S_3).C_2^4$ x 3, $D_4\times F_9:C_2$ x 3, $\SD_{16}\times \SOPlus(4,2)$ x 3, $C_2\times C_3^2.C_8:Q_8$ x 3, $C_4\times F_9:C_2^2$ x 2, $C_2\times C_3:S_3.\SD_{16}.C_2$ x 2, $C_3^2.C_8:Q_8.C_2$ x 2, $C_3^2.C_8:Q_8.C_2$ x 2, $C_3:S_3.C_4:D_4.C_2$ x 2, $C_4:S_3\wr C_2.C_2^2$ x 2, $C_3^2.C_8:Q_8.C_2$ x 2, $C_3^2.C_8:Q_8.C_2$ x 2, $C_3^2.C_8^2.C_2$ x 2, $C_4:F_9.C_2^2$ x 2, $C_2\times C_3^2.C_8:Q_8$ x 2, $C_2\times F_9:D_4$ x 2, $C_2\times C_3^2.Q_8:Q_8$ x 2, $C_2\times C_3^2.Q_8:Q_8$ x 2, $C_4:F_9.C_2^2$ x 2, $C_2\times C_4:\PSU(3,2).C_2$ x 2, $(C_6\times C_{12}):\SD_{16}$, $C_2\times C_4:S_3\wr C_2.C_2$, $(C_6\times C_{12}):\SD_{16}$, $C_2\times C_2.S_3\wr C_2.C_2.C_2$, $C_2\times C_6^2:\SD_{16}$, $C_2\times C_4:S_3^2.C_2^2$, $C_2\times C_6^2:\SD_{16}$, $C_2\times C_3:S_3:\SD_{16}.C_2$, $C_2\times C_3^2.(C_8\times Q_8)$, $C_2\times C_4:F_9.C_2$, $C_2\times S_3^2:C_8.C_2$, $C_2\times D_4\times F_9$, $C_2\times C_3^2.Q_8:C_8$, $C_2\times C_4:F_9.C_2$, $C_2\times C_3^2.D_4:C_8$, $C_2\times C_4:F_9.C_2$, $C_2\times C_8\times F_9$, $F_9:C_2^4$, $C_2\times \SD_{16}\times S_3^2$, $C_2\times C_3^2.Q_8^2$, $C_6^2:(C_2^2\times Q_8)$, $C_2\times S_3^2:Q_8.C_2$, $D_6^2:C_2^3$, $C_2\times C_4:S_3^2.C_2^2$, $(C_4\times F_9).C_2^2$, $C_2\times C_4:F_9.C_2$, $C_2\times C_3:S_3.Q_8:C_4$, $C_2\times C_3:S_3.D_4:C_4$, $C_2\times C_4:F_9.C_2$, $C_2\times C_3:S_3.D_4:C_4$, $C_2\times F_9:D_4$, $C_2\times C_4:S_3^2.C_2^2$

Order 1024: $C_4^2.C_2^5.C_2$

Order 864: $F_9:D_6$ x 20, $S_3^3:C_2^2$ x 13, $D_6\times \PSU(3,2)$ x 10, $F_9:D_6$ x 6, $\PSU(3,2):D_6$ x 6, $C_2\times C_3^3:\SD_{16}$ x 6, $C_2\times F_9:C_6$ x 6, $D_6\times F_9$ x 6, $C_2^2\times C_3^3:Q_8$ x 2, $C_6^2:C_4\times S_3$ x 2, $S_3\times D_6^2$, $C_3\times C_6^2:Q_8$, $C_6^2:D_{12}$, $D_6^2:S_3$, $D_6^2:C_6$, $C_2\times C_6:F_9$, $C_6^2:C_{24}$

Order 648: $C_3^4:(C_2\times C_4)$ x 2, $C_3^4:C_2^3$

Order 512: $\SD_{16}\wr C_2$

Order 324: $C_3^2:S_3^2$ x 2, $C_3^3:D_6$

Order 162: $C_3^3:S_3$ x 2, $C_3^3\times C_6$

Order 81: $C_3^4$

Order 16: $C_2\times D_4$ x 3

Order 8: $D_4$ x 2

Order 4: $C_2^2$ x 12

Order 2: $C_2$ x 7

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 82944: $S_3^4:C_2^3.D_4$ ($C_1$)

Order 41472: $\SOPlus(4,2)^2.D_4$ ($C_2$) x 8, $C_2\times C_3^4.D_4^2.C_2^2$ ($C_2$), $S_3^4:C_2^3.C_4$ ($C_2$), $C_3^4.D_4^2.C_2^3$ ($C_2$), $C_2\times C_3^4:(Q_8^2:C_4)$ ($C_2$), $F_9\wr C_2:C_2^2$ ($C_2$), $C_3^4:(Q_8^2.C_2^3)$ ($C_2$), $C_2\times C_3^4:(C_8^2:C_4)$ ($C_2$)

Order 20736: $C_3^4.D_4^2.C_2^2$ ($C_2^2$) x 4, $S_3\wr D_4:C_2$ ($C_2^2$) x 4, $C_3^4:(Q_8^2:C_4)$ ($C_2^2$) x 4, $F_9\wr C_2:C_2$ ($C_2^2$) x 4, $C_3^4:(Q_8^2:C_2^2)$ ($C_2^2$) x 4, $C_3^4:(C_8^2:C_4)$ ($C_2^2$) x 4, $\SOPlus(4,2)^2.C_4$ ($C_2^2$) x 4, $C_2\times C_3^4.D_4^2.C_2$ ($C_2^2$), $C_2\times C_3^4.C_4^2.C_2^3$ ($C_2^2$), $C_2\times C_3^4.C_4^2.C_2^3$ ($C_2^2$), $C_3^4:C_4^2.(C_2^2\times C_4)$ ($C_2^2$), $C_2\times C_3^2:F_9.D_8$ ($C_2^2$), $C_3^4:C_4^2.(C_2^2\times C_4)$ ($C_2^2$), $C_3^2:S_3^2.D_4^2$ ($C_2^2$)

Order 10368: $C_3^4.C_4^2.C_2^3$ ($C_2^3$) x 2, $C_3^4.C_4^2.C_2^3$ ($D_4$) x 2, $C_3^4.D_4^2.C_2$ ($C_2^3$) x 2, $C_3^4.D_4^2.C_2$ ($D_4$) x 2, $C_3^4.C_4^2.C_2^3$ ($C_2^3$) x 2, $C_3^4.C_4^2.C_2^3$ ($D_4$) x 2, $C_3^2:(C_4\times F_9):C_4$ ($C_2^3$) x 2, $C_3^2:F_9.D_8$ ($C_2^3$) x 2, $C_3^2:(C_4\times F_9).C_4$ ($C_2^3$) x 2, $C_3^4:D_8:D_4$ ($C_2^3$) x 2, $C_2\times C_3^4.C_4:\SD_{16}$ ($C_2^3$), $C_2\times C_3^4.(C_4.D_8)$ ($D_4$), $C_2\times C_3^4.C_8:Q_8$ ($D_4$), $C_2\times C_3^4.C_4:C_8.C_2$ ($D_4$), $C_2\times C_3^4.Q_8^2$ ($D_4$), $C_2\times C_3^4.C_8^2$ ($D_4$), $S_3^4:C_2^3$ ($D_4$)

Order 5184: $C_3^2:F_9:D_4$ ($C_2\times D_4$) x 3, $C_3^4.C_4:C_8.C_2$ ($C_2\times D_4$) x 2, $C_3^4.C_8^2$ ($C_2\times D_4$) x 2, $C_3^4.(C_4.D_8)$ ($C_2\times D_4$) x 2, $C_3^4.Q_8^2$ ($C_2\times D_4$) x 2, $C_3^4.C_8:Q_8$ ($C_2\times D_4$) x 2, $\SOPlus(4,2)^2$ ($C_2\times D_4$) x 2, $C_2\times C_3^4.C_4:Q_8$ ($C_2\times D_4$), $C_2\times C_3^4.C_4^2.C_2$ ($C_2\times D_4$), $C_3^4:C_4^2:C_2^2$ ($C_2\times D_4$), $C_3^2:F_9:D_4$ ($C_2^4$)

Order 2592: $C_2\times C_3^4:C_4^2$ ($C_2^2\wr C_2$), $C_3^3:C_{12}:Q_8$ ($C_2^2\times D_4$), $C_3^3:C_{12}:Q_8$ ($C_2^2\wr C_2$), $C_3^2:(C_4\times F_9)$ ($C_2^2\times D_4$), $C_3^2:(C_4\times F_9)$ ($C_2^2\wr C_2$), $C_3^3:C_{12}:D_4$ ($C_2^2\times D_4$), $C_3^3:C_{12}:D_4$ ($C_2^2\wr C_2$)

Order 1296: $C_3^4:(C_2^2\times C_4)$ ($C_2\wr C_2^2$), $C_3^4:C_4^2$ ($C_2^3:D_4$), $C_3^4:C_4^2$ ($C_2\wr C_2^2$)

Order 648: $C_3^4:C_2^3$ ($C_2\wr D_4$), $C_3^4:(C_2\times C_4)$ ($C_2\wr D_4$), $C_3^4:(C_2\times C_4)$ ($C_2^4:D_4$)

Order 324: $C_3^3:D_6$ ($D_4^2.C_2^2$), $C_3^2:S_3^2$ ($D_4^2.C_2^2$), $C_3^2:S_3^2$ ($D_4^2:C_2^2$)

Order 162: $C_3^3\times C_6$ ($\SD_{16}\wr C_2$), $C_3^3:S_3$ ($\SD_{16}\wr C_2$), $C_3^3:S_3$ ($D_4^2.C_2^3$)

Order 81: $C_3^4$ ($C_4^2.C_2^5.C_2$)

Order 2: $C_2$ ($\SOPlus(4,2)^2.D_4$)

Order 1: $C_1$ ($S_3^4:C_2^3.D_4$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 82944: $S_3^4:C_2^3.D_4$ ($C_1$)

Order 41472: $C_2\times C_3^4.D_4^2.C_2^2$ ($C_2$), $\SOPlus(4,2)^2.D_4$ ($C_2$), $S_3^4:C_2^3.C_4$ ($C_2$), $C_3^4.D_4^2.C_2^3$ ($C_2$), $C_2\times C_3^4:(Q_8^2:C_4)$ ($C_2$), $F_9\wr C_2:C_2^2$ ($C_2$), $C_3^4:(Q_8^2.C_2^3)$ ($C_2$), $C_2\times C_3^4:(C_8^2:C_4)$ ($C_2$)

Order 20736: $C_2\times C_3^4.D_4^2.C_2$ ($C_2^2$), $C_2\times C_3^4.C_4^2.C_2^3$ ($C_2^2$), $C_3^4.D_4^2.C_2^2$ ($C_2^2$), $C_2\times C_3^4.C_4^2.C_2^3$ ($C_2^2$), $S_3\wr D_4:C_2$ ($C_2^2$), $C_3^4:(Q_8^2:C_4)$ ($C_2^2$), $F_9\wr C_2:C_2$ ($C_2^2$), $C_3^4:(Q_8^2:C_2^2)$ ($C_2^2$), $C_3^4:(C_8^2:C_4)$ ($C_2^2$), $C_3^4:C_4^2.(C_2^2\times C_4)$ ($C_2^2$), $C_2\times C_3^2:F_9.D_8$ ($C_2^2$), $C_3^4:C_4^2.(C_2^2\times C_4)$ ($C_2^2$), $C_3^2:S_3^2.D_4^2$ ($C_2^2$), $\SOPlus(4,2)^2.C_4$ ($C_2^2$)

Order 10368: $C_2\times C_3^4.C_4:\SD_{16}$ ($C_2^3$), $C_3^4.C_4^2.C_2^3$ ($C_2^3$), $C_3^4.C_4^2.C_2^3$ ($D_4$), $C_2\times C_3^4.(C_4.D_8)$ ($D_4$), $C_2\times C_3^4.C_8:Q_8$ ($D_4$), $C_2\times C_3^4.C_4:C_8.C_2$ ($D_4$), $C_2\times C_3^4.Q_8^2$ ($D_4$), $C_2\times C_3^4.C_8^2$ ($D_4$), $C_3^4.D_4^2.C_2$ ($C_2^3$), $C_3^4.D_4^2.C_2$ ($D_4$), $C_3^4.C_4^2.C_2^3$ ($C_2^3$), $C_3^4.C_4^2.C_2^3$ ($D_4$), $S_3^4:C_2^3$ ($D_4$), $C_3^2:(C_4\times F_9):C_4$ ($C_2^3$), $C_3^2:F_9.D_8$ ($C_2^3$), $C_3^2:(C_4\times F_9).C_4$ ($C_2^3$), $C_3^4:D_8:D_4$ ($C_2^3$)

Order 5184: $C_3^2:F_9:D_4$ ($C_2\times D_4$) x 3, $C_3^4.C_4:C_8.C_2$ ($C_2\times D_4$), $C_3^4.C_8^2$ ($C_2\times D_4$), $C_2\times C_3^4.C_4:Q_8$ ($C_2\times D_4$), $C_3^4.(C_4.D_8)$ ($C_2\times D_4$), $C_3^4.Q_8^2$ ($C_2\times D_4$), $C_3^4.C_8:Q_8$ ($C_2\times D_4$), $C_2\times C_3^4.C_4^2.C_2$ ($C_2\times D_4$), $C_3^4:C_4^2:C_2^2$ ($C_2\times D_4$), $C_3^2:F_9:D_4$ ($C_2^4$), $\SOPlus(4,2)^2$ ($C_2\times D_4$)

Order 2592: $C_2\times C_3^4:C_4^2$ ($C_2^2\wr C_2$), $C_3^3:C_{12}:Q_8$ ($C_2^2\times D_4$), $C_3^3:C_{12}:Q_8$ ($C_2^2\wr C_2$), $C_3^2:(C_4\times F_9)$ ($C_2^2\times D_4$), $C_3^2:(C_4\times F_9)$ ($C_2^2\wr C_2$), $C_3^3:C_{12}:D_4$ ($C_2^2\times D_4$), $C_3^3:C_{12}:D_4$ ($C_2^2\wr C_2$)

Order 1296: $C_3^4:(C_2^2\times C_4)$ ($C_2\wr C_2^2$), $C_3^4:C_4^2$ ($C_2^3:D_4$), $C_3^4:C_4^2$ ($C_2\wr C_2^2$)

Order 648: $C_3^4:C_2^3$ ($C_2\wr D_4$), $C_3^4:(C_2\times C_4)$ ($C_2\wr D_4$), $C_3^4:(C_2\times C_4)$ ($C_2^4:D_4$)

Order 324: $C_3^3:D_6$ ($D_4^2.C_2^2$), $C_3^2:S_3^2$ ($D_4^2.C_2^2$), $C_3^2:S_3^2$ ($D_4^2:C_2^2$)

Order 162: $C_3^3\times C_6$ ($\SD_{16}\wr C_2$), $C_3^3:S_3$ ($\SD_{16}\wr C_2$), $C_3^3:S_3$ ($D_4^2.C_2^3$)

Order 81: $C_3^4$ ($C_4^2.C_2^5.C_2$)

Order 2: $C_2$ ($\SOPlus(4,2)^2.D_4$)

Order 1: $C_1$ ($S_3^4:C_2^3.D_4$)

Series

Derived series

$S_3^4:C_2^3.D_4$

$\rhd$

$S_3^4:C_2^3.D_4$

$\rhd$

$C_3^2:F_9:D_4$

$\rhd$

$C_3^2:F_9:D_4$

$\rhd$

$C_3^4:(C_2\times C_4)$

$\rhd$

$C_3^4:(C_2\times C_4)$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$S_3^4:C_2^3.D_4$

$\rhd$

$S_3^4:C_2^3.D_4$

$\rhd$

$C_2\times C_3^4.D_4^2.C_2^2$

$\rhd$

$C_2\times C_3^4.D_4^2.C_2^2$

$\rhd$

$C_2\times C_3^4.D_4^2.C_2$

$\rhd$

$C_2\times C_3^4.D_4^2.C_2$

$\rhd$

$C_2\times C_3^4.C_4:\SD_{16}$

$\rhd$

$C_2\times C_3^4.C_4:\SD_{16}$

$\rhd$

$C_3^2:F_9:D_4$

$\rhd$

$C_3^2:F_9:D_4$

$\rhd$

$C_3^3:C_{12}:D_4$

$\rhd$

$C_3^3:C_{12}:D_4$

$\rhd$

$C_3^4:C_4^2$

$\rhd$

$C_3^4:C_4^2$

$\rhd$

$C_3^4:(C_2\times C_4)$

$\rhd$

$C_3^4:(C_2\times C_4)$

$\rhd$

$C_3^2:S_3^2$

$\rhd$

$C_3^2:S_3^2$

$\rhd$

$C_3^3:S_3$

$\rhd$

$C_3^3:S_3$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$S_3^4:C_2^3.D_4$

$\rhd$

$S_3^4:C_2^3.D_4$

$\rhd$

$C_3^2:F_9:D_4$

$\rhd$

$C_3^2:F_9:D_4$

$\rhd$

$C_3^4:C_4^2$

$\rhd$

$C_3^4:C_4^2$

$\rhd$

$C_3^4:(C_2\times C_4)$

$\rhd$

$C_3^4:(C_2\times C_4)$

$\rhd$

$C_3^2:S_3^2$

$\rhd$

$C_3^2:S_3^2$

$\rhd$

$C_3^3:S_3$

$\rhd$

$C_3^3:S_3$

$\rhd$

$C_3^4$

$\rhd$

$C_3^4$

Upper central series

$C_1$

$\lhd$

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

This group is a maximal quotient of 4 larger groups in the database.

Character theory

Complex character table

See the $108 \times 108$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $90 \times 90$ rational character table.