Abstract group 31104.fx: $(C_2^2\times C_6^2).S_3^3$

• Label: 31104.fx
• Order: \( 2^{7} \cdot 3^{5} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{8} \cdot 3^{5} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $26$
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$(C_2^2\times C_6^2).S_3^3$
Order:	\(31104\)\(\medspace = 2^{7} \cdot 3^{5} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_3^2:S_4^2:D_6$, of order \(62208\)\(\medspace = 2^{8} \cdot 3^{5} \)
Composition factors:	$C_2$ x 7, $C_3$ x 5
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	1167	2186	2160	18030	7560	31104
Conjugacy classes	1	11	13	8	40	17	90
Divisions	1	11	13	8	39	17	89
Autjugacy classes	1	9	9	6	27	11	63

Dimension	1	2	4	6	8	9	12	18	24	36	48	72
Irr. complex chars.	8	12	6	12	1	8	12	12	6	12	0	1	90
Irr. rational chars.	8	12	6	12	1	8	12	12	4	12	1	1	89

Minimal presentations

Permutation degree:	$26$
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	$111041280$

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	12
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid a^{2}=b^{6}=c^{6}=d^{2}=e^{6}=f^{6}=g^{6}= \!\cdots\! \rangle}$
Permutation group:	Degree $26$ $\langle(1,2,5,11,16,18)(3,7,13,17,9,6)(4,10,8,15,12,14)(19,20,22,26)(21,24,25,23) \!\cdots\! \rangle$
Transitive group:	36T13453	36T13454	36T13455	36T13456	more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(A_4^2:S_3^2)$ $\,\rtimes\,$ $S_3$	$(C_3\times C_6^2:S_4)$ $\,\rtimes\,$ $D_6$	$(C_6^3:C_2^2)$ $\,\rtimes\,$ $S_3^2$	$C_2^4$ $\,\rtimes\,$ $(C_3^2.S_3^3)$	all 21
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_6^2:S_4)$ . $S_3^2$	$(C_2^2\times C_6^2)$ . $S_3^3$	$C_3^2$ . $(C_2^4:S_3^3)$	$(C_6^2:A_4)$ . $(S_3\times D_6)$	all 6

Elements of the group are displayed as words in the presentation generators from the presentation above.

Homology

Abelianization:	$C_{2}^{3} $
Schur multiplier:	$C_{2}^{4}$
Commutator length:	$1$

Subgroups

There are 616782 subgroups in 5307 conjugacy classes, 51 normal (19 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $(C_2^2\times C_6^2).S_3^3$
Commutator:	$G' \simeq$ $C_2^4.C_3^4.C_3$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_3^2$	$G/\Phi \simeq$ $C_2^4:S_3^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_2\times C_6^3$	$G/\operatorname{Fit} \simeq$ $S_3\times D_6$
Radical:	$R \simeq$ $(C_2^2\times C_6^2).S_3^3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6^2$	$G/\operatorname{soc} \simeq$ $S_3^3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4:D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4:C_3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: subgroups only stored up to automorphism

Classes of subgroups up to automorphism

No diagram available

Normal subgroups

No diagram available: subgroups only stored up to automorphism

Normal subgroups up to automorphism

No diagram available

Classes of subgroups up to conjugation

Order 31104: $(C_2^2\times C_6^2).S_3^3$

Order 15552: $(C_2\times C_6^3):S_3^2$ x 2, $(C_3^2\times A_4^2):D_6$ x 2, $C_2^4.C_3^4.D_6$, $(C_2\times C_6^3):S_3^2$, $(C_2\times C_6^3):S_3^2$

Order 10368: $C_6^2.(D_6\times S_4)$ x 2, $A_4^2:(S_3\times D_6)$

Order 7776: $(C_3^2\times A_4^2):S_3$ x 2, $(C_3^2\times A_4^2):C_6$ x 2, $C_2^4.C_3^4.S_3$, $C_2^4.C_3^4.C_6$, $C_3^3.A_4^2.C_2$

Order 5184: $C_2^4.C_3^3:S_3.C_2$ x 2, $(C_2\times C_6^3):D_6$ x 2, $(S_3\times C_6^2):S_4$ x 2, $C_6^2:(S_3\times S_4)$ x 2, $(C_2\times C_6^3):D_6$ x 2, $C_6^2.(A_4\times D_6)$ x 2, $(S_3\times C_6^2):S_4$ x 2, $C_3^2.\POPlus(4,3)$ x 2, $A_4^2:S_3^2$ x 2, $A_4^2:S_3^2$ x 2, $C_3\times A_4^2:D_6$ x 2, $C_6^2.(S_3\times C_3:D_4)$, $(A_4\times C_6^2):D_6$

Order 3888: $C_2^4.C_3^4.C_3$, $C_6^2.C_3^3.C_2^2$

Order 3456: $C_6^2.(C_2^2\times S_4)$ x 2, $C_2^4:S_3^3$ x 2, $D_6^2:S_4$ x 2, $C_2^4:S_3^3$

Order 2592: $C_3^2.A_4\wr C_2$ x 4, $(C_3\times A_4)\wr C_2$ x 4, $C_3\times C_2^4.C_3^2:S_3$ x 3, $C_3\times C_2^4:\He_3.C_2$ x 2, $C_2^4.C_3^3:S_3$ x 2, $(C_2\times C_6^3).C_6$ x 2, $S_3\times C_6^2:D_6$ x 2, $(C_3\times C_6^2):S_4$ x 2, $C_3\times C_6^2:S_4$ x 2, $(C_2\times C_6^3):C_6$ x 2, $(C_2\times C_6^3):C_6$ x 2, $C_3\times C_6^2:C_3.D_4$, $C_2\times C_6^2.S_3^2$, $C_6^2:C_3.C_3:D_4$, $C_6^2.D_6:S_3$, $C_6^2.C_3^2:C_4.C_2$, $C_2\times C_3.C_6^2:C_6.C_2$, $(C_3\times C_6\times A_4).C_6.C_2$, $(A_4\times C_6^2):S_3$, $C_3:S_3\times A_4^2$, $C_3^2\times \PSOPlus(4,3)$

Order 1944: $C_3^3:(C_6\times A_4)$, $(A_4\times C_3^3):S_3$, $(A_4\times C_3^3):C_6$, $C_3^2.S_3^3$

Order 1728: $A_4^2:D_6$ x 6, $C_3\times C_2^4:S_3^2$ x 6, $(C_2^2\times C_6^2):D_6$ x 6, $A_4^2:D_6$ x 4, $C_6^2:(C_2\times S_4)$ x 4, $(C_6\times D_6):S_4$ x 4, $A_4^2:D_6$ x 3, $C_6^3.C_2^3$ x 3, $C_6^3.C_2^3$ x 3, $C_2^3.S_3^3$ x 3, $C_2^3.S_3^3$ x 3, $C_6^2:(C_2\times S_4)$ x 2, $C_6^2:(C_2\times S_4)$ x 2, $A_4^2:D_6$ x 2, $C_3\times \POPlus(4,3)$ x 2, $C_3.\POPlus(4,3)$ x 2, $D_6^2:A_4$ x 2, $C_6^2.(C_2^2\times A_4)$ x 2, $(C_2\times C_6^2):S_4$ x 2, $(C_2\times C_6^2):S_4$ x 2, $C_2^3:S_3^3$, $C_3:S_3\times \GL(2,\mathbb{Z}/4)$, $C_6^2:(C_2\times S_4)$, $C_6^3.C_2^3$, $C_2^3.S_3^3$

Order 1296: $C_6^2:S_3^2$ x 6, $C_3^2.A_4^2$ x 3, $C_3\times C_6^2:A_4$ x 2, $C_6^3:C_6$ x 2, $C_6^2:C_6\times S_3$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2.S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2:S_3^2$ x 2, $C_6^2.S_3^2$ x 2, $C_3^2\times A_4^2$, $C_6^2:S_3^2$, $C_6^3:C_6$, $C_6^3:S_3$, $C_6^3:S_3$, $C_6^2:S_3^2$, $C_6^3.S_3$, $C_6^3.S_3$

Order 1152: $S_3\times C_2^3:S_4$ x 4, $D_6^2:D_4$ x 3, $A_4^2:C_2^3$

Order 972: $C_3^3:S_3^2$ x 2, $C_3^4:D_6$ x 2, $C_6^2:\He_3$, $C_3^3:S_3^2$, $C_3^4:D_6$, $C_3^3:S_3^2$

Order 864: $C_6^2:S_4$ x 6, $A_4^2:S_3$ x 6, $A_4^2:S_3$ x 6, $A_4^2:C_6$ x 6, $C_6^2:S_4$ x 6, $C_6^2:S_4$ x 6, $C_3.A_4\wr C_2$ x 6, $(C_6\times D_6):A_4$ x 6, $C_6^2.D_{12}$ x 6, $(C_2^2\times C_6^2):C_6$ x 4, $C_6^2:S_4$ x 4, $C_6^3:C_2^2$ x 4, $C_6^2:D_{12}$ x 4, $C_6^2:D_{12}$ x 4, $C_6^3.C_2^2$ x 4, $S_3\times A_4^2$ x 3, $A_4^2:C_6$ x 3, $C_6^3:C_2^2$ x 3, $C_2^2:S_3^3$ x 3, $C_6.D_6^2$ x 3, $C_6.D_6^2$ x 3, $C_6.D_6^2$ x 3, $C_2\times C_6^2:D_6$ x 3, $(C_3\times A_4):S_4$ x 3, $C_6^3.C_2^2$ x 3, $C_6^3.C_2^2$ x 3, $C_6^2.D_{12}$ x 3, $C_6^2.D_{12}$ x 3, $C_6^2:(C_2\times A_4)$ x 2, $C_6^2:S_4$ x 2, $S_4\times S_3^2$ x 2, $C_3^2\times \GL(2,\mathbb{Z}/4)$ x 2, $C_6^2:(C_2\times A_4)$ x 2, $C_6^2:S_4$ x 2, $(C_2\times C_6^2):A_4$ x 2, $C_6^3:C_2^2$, $C_6:D_6\times A_4$, $C_6:S_3\times S_4$, $C_6^3:C_2^2$, $C_6^2:S_4$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_6.(S_3\times S_4)$, $C_2^2\times C_6^2:C_6$, $C_6^2:S_4$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_6.(S_3\times S_4)$

Order 648: $C_3^3:S_4$ x 9, $(C_3\times C_6^2):C_6$ x 6, $C_3^3:S_4$ x 5, $S_3\times C_3^2:A_4$ x 2, $C_3\times C_6^2:C_6$ x 2, $C_3^3:S_4$ x 2, $C_3^3:S_4$ x 2, $C_3.S_3^3$ x 2, $C_3\times C_6^2:C_6$, $C_3^3:S_4$, $C_3^3:S_4$, $C_3:S_3^3$, $C_2\times C_3^3:D_6$, $C_6^3:C_3$, $C_2\times C_3^2:S_3^2$, $C_3\times C_6^2:S_3$, $(C_3\times C_6^2):C_6$, $C_3^3:D_{12}$, $\He_3:D_{12}$, $C_3^3:(C_4\times S_3)$

Order 576: $C_2^4:S_3^2$ x 12, $C_2^4:S_3^2$ x 10, $\POPlus(4,3)$ x 9, $C_2^4:S_3^2$ x 7, $C_3\times C_2^3:S_4$ x 6, $C_3\times C_2^4:D_6$ x 6, $C_2^4:S_3^2$ x 6, $(C_2\times C_{12}):D_{12}$ x 6, $(C_2\times D_6):D_{12}$ x 6, $D_6^2:C_4$ x 6, $S_3\times \GL(2,\mathbb{Z}/4)$ x 4, $S_3\times C_2^3:A_4$ x 4, $(C_2^2\times C_6):S_4$ x 4, $D_6^2:C_2^2$ x 3, $C_6^2.C_2^4$ x 3, $(C_6\times D_4).D_6$ x 3, $A_4^2:C_2^2$ x 2, $A_4^2:C_2^2$

Order 486: $C_3^4:S_3$ x 2, $C_3^4:C_6$ x 2, $C_3^4:S_3$, $C_3^4:C_6$, $C_3^4:S_3$

Order 432: $C_6^2:D_6$ x 16, $C_6^2:D_6$ x 16, $A_4:S_3^2$ x 8, $C_3\times A_4^2$ x 6, $C_6^2:D_6$ x 6, $C_6^2:D_6$ x 6, $C_6^2:D_6$ x 6, $C_6^2.D_6$ x 6, $C_2.S_3^3$ x 6, $D_6:S_3^2$ x 6, $C_2\times C_6^2:C_6$ x 6, $C_3.A_4^2$ x 6, $C_6^2.D_6$ x 5, $C_6^3:C_2$ x 4, $C_6^2:A_4$ x 4, $C_6^3:C_2$ x 4, $D_6:C_6^2$ x 4, $C_6^2.D_6$ x 4, $C_6^2.D_6$ x 4, $C_6^2.D_6$ x 4, $C_6^2:A_4$ x 4, $C_6^2:D_6$ x 4, $C_6^2:C_{12}$ x 4, $C_6^2:C_{12}$ x 4, $C_3:D_6^2$ x 3, $C_6^2:D_6$ x 3, $C_6^2.D_6$ x 3, $C_6^2:D_6$ x 3, $C_6^2.D_6$ x 3, $C_2.S_3^3$ x 3, $C_2.S_3^3$ x 3, $D_6:S_3^2$ x 3, $D_6:S_3^2$ x 3, $D_6:S_3^2$ x 3, $C_2.S_3^3$ x 3, $C_6^2:D_6$ x 3, $A_4\times C_6^2$ x 2, $C_6:S_3\times A_4$ x 2, $A_4:C_6^2$ x 2, $A_4\times S_3^2$ x 2, $A_4:S_3^2$ x 2, $C_3\times A_4:C_{12}$ x 2, $C_6^2:A_4$ x 2, $(C_3\times A_4):C_{12}$ x 2, $C_2\times C_6^3$, $C_6^2:D_6$, $C_2\times S_3^3$, $C_6^2:D_6$, $(C_3^2\times A_4):C_4$, $C_2.S_3^3$, $C_6^2:D_6$, $(C_3\times C_6).S_4$

Order 384: $C_2\wr C_2^2\times S_3$ x 3, $C_2^4:S_4$ x 2

Order 324: $C_3^3:A_4$ x 9, $C_3\wr C_2^2$ x 4, $C_3^2:S_3^2$ x 4, $C_3^2:S_3^2$ x 4, $C_3^3:D_6$ x 4, $C_3^2:S_3^2$ x 4, $C_3^2:S_3^2$ x 3, $C_3^3:D_6$ x 3, $C_3^2:S_3^2$ x 2, $\He_3:C_{12}$, $C_3^3:C_{12}$, $A_4\times C_3^3$, $C_3^2:C_6^2$, $C_3^3:D_6$

Order 288: $C_6^2:D_4$ x 26, $C_3\times C_2^2:S_4$ x 20, $D_6^2:C_2$ x 16, $C_6^2:D_4$ x 14, $(C_2\times C_6):S_4$ x 14, $A_4\wr C_2$ x 14, $C_2^3.S_3^2$ x 13, $D_{12}:D_6$ x 12, $C_6^2.D_4$ x 12, $C_6^2.D_4$ x 12, $\PSOPlus(4,3)$ x 11, $C_2^3:S_3^2$ x 9, $C_3\times \GL(2,\mathbb{Z}/4)$ x 8, $(C_2\times D_6):A_4$ x 8, $D_6\times S_4$ x 8, $C_6^2.C_2^3$ x 6, $D_4:S_3^2$ x 6, $D_4\times S_3^2$ x 6, $C_3\times C_2^3:A_4$ x 6, $C_6^2.D_4$ x 6, $C_6^2.D_4$ x 6, $S_3\times C_2^2:A_4$ x 6, $C_3:\GL(2,\mathbb{Z}/4)$ x 5, $D_6:S_4$ x 4, $D_6:S_4$ x 4, $D_6.S_4$ x 4, $C_6^2:D_4$ x 4, $C_6^2:D_4$ x 4, $C_6^2.C_2^3$ x 4, $C_6^2:C_2^3$ x 4, $C_2\times A_4\times D_6$ x 4, $C_6^2:C_2^3$ x 4, $D_{12}:D_6$ x 3, $D_{12}:D_6$ x 3, $C_6^2:D_4$ x 3, $C_6^2:D_4$ x 3, $C_2\times A_4^2$

Order 243: $C_3^4:C_3$

Order 216: $C_3^2:S_4$ x 17, $C_3^2:S_4$ x 14, $C_6^2:C_6$ x 13, $C_3^2:S_4$ x 13, $C_6^2:S_3$ x 12, $C_6^2:C_6$ x 10, $C_3^2:S_4$ x 10, $C_3^2\times S_4$ x 10, $C_6^2:C_6$ x 10, $C_6:S_3^2$ x 9, $C_3^2:D_{12}$ x 9, $C_6^2:C_6$ x 8, $C_3^2:D_{12}$ x 8, $C_6.S_3^2$ x 8, $C_6^2:C_6$ x 6, $S_3^3$ x 6, $C_3^2:D_{12}$ x 6, $C_6.S_3^2$ x 6, $C_6^2:C_6$ x 6, $C_6^2:C_6$ x 4, $C_6^2.C_6$ x 4, $C_6.C_6^2$ x 4, $S_3\times C_6^2$ x 3, $C_6\times S_3^2$ x 3, $C_3^2:D_{12}$ x 3, $C_6.S_3^2$ x 3, $C_3^3:Q_8$ x 3, $C_3^3:D_4$ x 3, $C_6.S_3^2$ x 3, $C_6.S_3^2$ x 3, $C_6^2:C_6$ x 2, $C_6^3$ x 2, $C_6^2:C_6$ x 2, $C_6^2:S_3$, $C_3^2:D_{12}$, $\He_3:D_4$, $C_6.S_3^2$, $C_6:S_3^2$, $C_6^2:S_3$, $C_3^3:Q_8$, $C_3^3:D_4$, $C_6.S_3^2$, $C_6^2:C_6$

Order 192: $C_2^3:S_4$ x 12, $C_2^4:D_6$ x 10, $C_2^4:D_6$ x 9, $C_2^3:C_4\times S_3$ x 9, $C_2^3:D_{12}$ x 9, $C_2^3:S_4$ x 4, $C_3\times C_2\wr C_2^2$ x 3, $(C_2\times C_{12}):D_4$ x 3, $D_{12}:C_2^3$ x 3, $C_2^4:A_4$ x 2, $C_2\times \GL(2,\mathbb{Z}/4)$

Order 162: $C_3^3:C_6$ x 8, $C_3^3:C_6$ x 7, $C_3^2\wr C_2$ x 6, $C_3^3:S_3$ x 6, $C_3^3:S_3$ x 2, $S_3\times \He_3$ x 2, $C_3^3:S_3$, $C_6\times \He_3$

Order 144: $D_6:D_6$ x 70, $S_3\times S_4$ x 38, $C_6^2:C_2^2$ x 32, $C_6:D_{12}$ x 32, $C_6^2:C_2^2$ x 29, $D_6:D_6$ x 28, $C_6^2:C_4$ x 26, $D_6.D_6$ x 24, $C_6.D_{12}$ x 20, $C_{12}:D_6$ x 16, $C_2^2.S_3^2$ x 16, $C_2^4:C_3^2$ x 14, $C_6\times S_4$ x 14, $A_4\times D_6$ x 12, $C_{12}.D_6$ x 12, $C_{12}:D_6$ x 12, $S_3\times D_{12}$ x 12, $C_{12}.D_6$ x 12, $D_6^2$ x 10, $C_6:S_4$ x 10, $A_4^2$ x 10, $C_2^2:C_6^2$ x 9, $C_6^2:C_2^2$ x 8, $A_4:C_{12}$ x 8, $D_{12}:C_6$ x 6, $D_6.D_6$ x 6, $C_4\times S_3^2$ x 6, $C_{12}.D_6$ x 6, $D_{12}:S_3$ x 6, $C_6.D_{12}$ x 5, $C_6.S_4$ x 5, $C_2^2\times C_6^2$ x 4, $C_4:C_6^2$ x 4, $C_6:D_{12}$ x 4, $C_{12}:D_6$ x 4, $C_6^2:C_4$ x 4, $C_6^2:C_4$ x 4, $C_{12}:D_6$ x 3, $C_{12}.D_6$ x 3

Order 128: $C_2^4:D_4$

Order 108: $C_3:S_3^2$ x 24, $C_3^2:A_4$ x 22, $C_3^2\times A_4$ x 18, $C_3\times S_3^2$ x 17, $C_3^2:D_6$ x 14, $C_3^2:D_6$ x 10, $C_3^2:D_6$ x 10, $C_3\times C_6^2$ x 7, $C_3^2:D_6$ x 6, $C_3^2\times D_6$ x 6, $C_3^2:C_{12}$ x 6, $C_3:S_3^2$ x 5, $C_3^2:C_{12}$ x 4, $C_3^2:D_6$ x 3, $C_3^2:C_{12}$ x 2, $C_2^2\times \He_3$ x 2, $C_3^3:C_4$, $\He_3:C_4$

Order 96: $D_4\times D_6$ x 26, $D_4:D_6$ x 18, $C_2^2.D_{12}$ x 18, $C_2^2:D_{12}$ x 17, $D_6.D_4$ x 17, $D_6:D_4$ x 17, $C_2^2:S_4$ x 16, $C_2^4:C_6$ x 10, $C_2^4:S_3$ x 10, $C_2^3:C_{12}$ x 9, $C_2^3.D_6$ x 9, $D_4:D_6$ x 9, $D_4:D_6$ x 9, $C_2^3:A_4$ x 8, $\GL(2,\mathbb{Z}/4)$ x 6, $C_2^3\times D_6$ x 4, $C_2^2\times S_4$ x 3, $C_6.C_2^4$ x 3, $C_2^3:A_4$ x 2, $C_2^3\times A_4$, $C_2^2.S_4$

Order 81: $C_3\times \He_3$ x 6, $C_3^4$

Order 72: $C_6:D_6$ x 100, $C_6\wr C_2$ x 72, $C_3:D_{12}$ x 68, $S_3\times D_6$ x 47, $C_3\times S_4$ x 46, $C_3:S_4$ x 44, $C_6:C_{12}$ x 32, $C_6.D_6$ x 32, $S_3\times A_4$ x 30, $C_6\times D_6$ x 24, $C_6^2:C_2$ x 21, $C_6\times A_4$ x 17, $C_2\times C_6^2$ x 16, $C_6.D_6$ x 16, $S_3\times C_{12}$ x 12, $D_6:S_3$ x 10, $D_4\times C_3^2$ x 8, $C_3:D_{12}$ x 8, $C_{12}:S_3$ x 8, $C_3\times D_{12}$ x 6, $C_3^2:Q_8$ x 6, $C_3^2:Q_8$ x 6, $C_6\times C_{12}$ x 4, $C_6.D_6$ x 4

Order 64: $C_2\wr C_2^2$ x 8, $C_2^4:C_4$ x 3, $C_2^3:D_4$ x 3, $D_4:C_2^3$

Order 54: $S_3\times C_3^2$ x 24, $C_3^2:C_6$ x 23, $C_3^2:C_6$ x 22, $C_3^2:S_3$ x 12, $C_3^2:S_3$ x 10, $C_2\times \He_3$ x 6, $C_3^2\times C_6$ x 4

Order 48: $S_3\times D_4$ x 113, $C_6:D_4$ x 53, $C_2^2\times D_6$ x 39, $D_4:S_3$ x 27, $D_{12}:C_2$ x 27, $C_6\times D_4$ x 26, $C_2\times D_{12}$ x 26, $C_4\times D_6$ x 26, $D_6:C_4$ x 25, $C_2\times S_4$ x 24, $C_2^2:C_{12}$ x 17, $C_6.D_4$ x 17, $D_4:C_6$ x 9, $D_{12}:C_2$ x 9, $C_2^2\times A_4$ x 7, $C_2^2:A_4$ x 6, $C_2^3\times C_6$ x 4, $A_4:C_4$ x 4, $S_3\times Q_8$ x 3

Order 36: $C_6:S_3$ x 92, $S_3^2$ x 65, $C_6\times S_3$ x 60, $C_3\times A_4$ x 57, $C_6^2$ x 44, $C_3:C_{12}$ x 34, $C_3^2:C_4$ x 9, $C_3\times C_{12}$ x 4

Order 32: $C_2^2\wr C_2$ x 14, $C_2^3:C_4$ x 12, $D_4:C_2^2$ x 10, $C_2^2\times D_4$ x 7, $C_2^3:C_4$ x 4, $D_4:C_2^2$ x 3, $C_2^5$

Order 27: $C_3^3$ x 13, $\He_3$ x 12

Order 24: $C_2\times D_6$ x 151, $C_3:D_4$ x 92, $C_4\times S_3$ x 52, $D_{12}$ x 43, $C_3\times D_4$ x 43, $S_4$ x 30, $C_6:C_4$ x 27, $C_2^2\times C_6$ x 27, $C_2\times C_{12}$ x 26, $C_2\times A_4$ x 18, $C_3:Q_8$ x 9, $C_3\times Q_8$ x 3

Order 18: $C_3\times S_3$ x 78, $C_3:S_3$ x 50, $C_3\times C_6$ x 33

Order 16: $C_2\times D_4$ x 59, $D_4:C_2$ x 24, $C_2^2:C_4$ x 13, $C_2^4$ x 9, $C_2^2\times C_4$ x 7, $C_2\times Q_8$

Order 12: $D_6$ x 146, $C_2\times C_6$ x 58, $C_3:C_4$ x 21, $A_4$ x 17, $C_{12}$ x 17

Order 9: $C_3^2$ x 37

Order 8: $D_4$ x 44, $C_2^3$ x 34, $C_2\times C_4$ x 28, $Q_8$ x 4

Order 6: $S_3$ x 47, $C_6$ x 39

Order 4: $C_2^2$ x 33, $C_4$ x 8

Order 3: $C_3$ x 13

Order 2: $C_2$ x 11

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 31104: $(C_2^2\times C_6^2).S_3^3$

Order 15552: $C_2^4.C_3^4.D_6$, $(C_2\times C_6^3):S_3^2$, $(C_2\times C_6^3):S_3^2$, $(C_3^2\times A_4^2):D_6$, $(C_2\times C_6^3):S_3^2$

Order 10368: $C_6^2.(D_6\times S_4)$, $A_4^2:(S_3\times D_6)$

Order 7776: $C_2^4.C_3^4.S_3$, $C_2^4.C_3^4.C_6$, $C_3^3.A_4^2.C_2$, $(C_3^2\times A_4^2):S_3$, $(C_3^2\times A_4^2):C_6$

Order 5184: $A_4^2:S_3^2$ x 2, $C_6^2.(S_3\times C_3:D_4)$, $C_2^4.C_3^3:S_3.C_2$, $(A_4\times C_6^2):D_6$, $(C_2\times C_6^3):D_6$, $(S_3\times C_6^2):S_4$, $C_6^2:(S_3\times S_4)$, $(C_2\times C_6^3):D_6$, $C_6^2.(A_4\times D_6)$, $(S_3\times C_6^2):S_4$, $C_3^2.\POPlus(4,3)$, $A_4^2:S_3^2$, $C_3\times A_4^2:D_6$

Order 3888: $C_2^4.C_3^4.C_3$, $C_6^2.C_3^3.C_2^2$

Order 3456: $C_6^2.(C_2^2\times S_4)$, $C_2^4:S_3^3$, $D_6^2:S_4$, $C_2^4:S_3^3$

Order 2592: $C_3\times C_2^4.C_3^2:S_3$ x 2, $C_3^2.A_4\wr C_2$ x 2, $(C_3\times A_4)\wr C_2$ x 2, $C_3\times C_6^2:C_3.D_4$, $C_2\times C_6^2.S_3^2$, $C_6^2:C_3.C_3:D_4$, $C_6^2.D_6:S_3$, $C_6^2.C_3^2:C_4.C_2$, $C_2\times C_3.C_6^2:C_6.C_2$, $C_3\times C_2^4:\He_3.C_2$, $(C_3\times C_6\times A_4).C_6.C_2$, $C_2^4.C_3^3:S_3$, $(C_2\times C_6^3).C_6$, $(A_4\times C_6^2):S_3$, $C_3:S_3\times A_4^2$, $C_3^2\times \PSOPlus(4,3)$, $S_3\times C_6^2:D_6$, $(C_3\times C_6^2):S_4$, $C_3\times C_6^2:S_4$, $(C_2\times C_6^3):C_6$, $(C_2\times C_6^3):C_6$

Order 1944: $C_3^3:(C_6\times A_4)$, $(A_4\times C_3^3):S_3$, $(A_4\times C_3^3):C_6$, $C_3^2.S_3^3$

Order 1728: $A_4^2:D_6$ x 3, $C_3\times C_2^4:S_3^2$ x 3, $(C_2^2\times C_6^2):D_6$ x 3, $A_4^2:D_6$ x 2, $A_4^2:D_6$ x 2, $C_6^2:(C_2\times S_4)$ x 2, $C_6^3.C_2^3$ x 2, $(C_6\times D_6):S_4$ x 2, $C_6^3.C_2^3$ x 2, $C_2^3.S_3^3$ x 2, $C_2^3.S_3^3$ x 2, $C_6^2:(C_2\times S_4)$, $C_6^2:(C_2\times S_4)$, $A_4^2:D_6$, $C_3\times \POPlus(4,3)$, $C_2^3:S_3^3$, $C_3:S_3\times \GL(2,\mathbb{Z}/4)$, $C_3.\POPlus(4,3)$, $D_6^2:A_4$, $C_6^2:(C_2\times S_4)$, $C_6^3.C_2^3$, $C_2^3.S_3^3$, $C_6^2.(C_2^2\times A_4)$, $(C_2\times C_6^2):S_4$, $(C_2\times C_6^2):S_4$

Order 1296: $C_6^2:S_3^2$ x 4, $C_6^3:C_6$ x 2, $C_3^2.A_4^2$ x 2, $C_3^2\times A_4^2$, $C_6^2:S_3^2$, $C_3\times C_6^2:A_4$, $C_6^3:C_6$, $C_6^3:S_3$, $C_6^3:S_3$, $C_6^2:C_6\times S_3$, $C_6^2:S_3^2$, $C_6^2.S_3^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2.S_3^2$, $C_6^2:S_3^2$, $C_6^3.S_3$, $C_6^3.S_3$

Order 1152: $D_6^2:D_4$ x 2, $S_3\times C_2^3:S_4$ x 2, $A_4^2:C_2^3$

Order 972: $C_6^2:\He_3$, $C_3^3:S_3^2$, $C_3^4:D_6$, $C_3^3:S_3^2$, $C_3^4:D_6$, $C_3^3:S_3^2$

Order 864: $A_4^2:S_3$ x 4, $C_6^2:S_4$ x 3, $A_4^2:S_3$ x 3, $A_4^2:C_6$ x 3, $C_6^3:C_2^2$ x 3, $C_6^2:S_4$ x 3, $C_6^2:S_4$ x 3, $C_3.A_4\wr C_2$ x 3, $C_6^2:D_{12}$ x 3, $C_6^2:D_{12}$ x 3, $C_6^3.C_2^2$ x 3, $(C_6\times D_6):A_4$ x 3, $C_6^2.D_{12}$ x 3, $(C_2^2\times C_6^2):C_6$ x 2, $C_6^2:S_4$ x 2, $S_3\times A_4^2$ x 2, $A_4^2:C_6$ x 2, $C_6^3:C_2^2$ x 2, $C_3^2\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^2:S_3^3$ x 2, $C_6.D_6^2$ x 2, $C_6.D_6^2$ x 2, $C_6.D_6^2$ x 2, $C_2\times C_6^2:D_6$ x 2, $(C_3\times A_4):S_4$ x 2, $C_6^3.C_2^2$ x 2, $C_6^3.C_2^2$ x 2, $C_6^2:S_4$ x 2, $C_6^2.D_{12}$ x 2, $C_6^2.D_{12}$ x 2, $C_6^3:C_2^2$, $C_6^2:(C_2\times A_4)$, $C_6:D_6\times A_4$, $C_6^2:S_4$, $C_6:S_3\times S_4$, $S_4\times S_3^2$, $C_6^3:C_2^2$, $C_6^2:S_4$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_6.(S_3\times S_4)$, $C_6^2:(C_2\times A_4)$, $C_2^2\times C_6^2:C_6$, $C_6^2:S_4$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_6.(S_3\times S_4)$, $(C_2\times C_6^2):A_4$

Order 648: $C_3^3:S_4$ x 6, $(C_3\times C_6^2):C_6$ x 4, $C_3^3:S_4$ x 3, $C_3\times C_6^2:C_6$, $C_3^3:S_4$, $C_3^3:S_4$, $C_3:S_3^3$, $C_2\times C_3^3:D_6$, $C_6^3:C_3$, $C_2\times C_3^2:S_3^2$, $S_3\times C_3^2:A_4$, $C_3\times C_6^2:C_6$, $C_3^3:S_4$, $C_3^3:S_4$, $C_3.S_3^3$, $C_3\times C_6^2:S_3$, $(C_3\times C_6^2):C_6$, $C_3^3:D_{12}$, $\He_3:D_{12}$, $C_3^3:(C_4\times S_3)$

Order 576: $\POPlus(4,3)$ x 7, $C_2^4:S_3^2$ x 6, $C_2^4:S_3^2$ x 5, $C_2^4:S_3^2$ x 4, $C_2^4:S_3^2$ x 4, $(C_2\times D_6):D_{12}$ x 4, $S_3\times \GL(2,\mathbb{Z}/4)$ x 3, $C_3\times C_2^3:S_4$ x 3, $C_3\times C_2^4:D_6$ x 3, $(C_2\times C_{12}):D_{12}$ x 3, $D_6^2:C_4$ x 3, $D_6^2:C_2^2$ x 2, $C_6^2.C_2^4$ x 2, $S_3\times C_2^3:A_4$ x 2, $(C_2^2\times C_6):S_4$ x 2, $(C_6\times D_4).D_6$ x 2, $A_4^2:C_2^2$, $A_4^2:C_2^2$

Order 486: $C_3^4:S_3$, $C_3^4:C_6$, $C_3^4:S_3$, $C_3^4:S_3$, $C_3^4:C_6$

Order 432: $C_6^2:D_6$ x 10, $C_6^2:D_6$ x 9, $A_4:S_3^2$ x 5, $C_2\times C_6^2:C_6$ x 5, $C_6^3:C_2$ x 4, $C_3\times A_4^2$ x 4, $C_6^2:D_6$ x 4, $C_3.A_4^2$ x 4, $C_6^2.D_6$ x 4, $C_6^2:D_6$ x 3, $C_6^3:C_2$ x 3, $D_6:C_6^2$ x 3, $C_6^2.D_6$ x 3, $C_6^2.D_6$ x 3, $C_6^2.D_6$ x 3, $C_6^2:D_6$ x 3, $C_6^2.D_6$ x 3, $C_2.S_3^3$ x 3, $D_6:S_3^2$ x 3, $C_6^2:D_6$ x 3, $C_6^2:C_{12}$ x 3, $C_6^2:C_{12}$ x 3, $C_6^2:A_4$ x 2, $A_4\times C_6^2$ x 2, $C_3:D_6^2$ x 2, $C_6:S_3\times A_4$ x 2, $A_4:C_6^2$ x 2, $C_6^2:D_6$ x 2, $C_6^2.D_6$ x 2, $C_6^2:D_6$ x 2, $C_6^2.D_6$ x 2, $C_3\times A_4:C_{12}$ x 2, $C_2.S_3^3$ x 2, $C_2.S_3^3$ x 2, $D_6:S_3^2$ x 2, $D_6:S_3^2$ x 2, $D_6:S_3^2$ x 2, $C_2.S_3^3$ x 2, $C_6^2:A_4$ x 2, $C_6^2:A_4$ x 2, $C_6^2:D_6$ x 2, $(C_3\times A_4):C_{12}$ x 2, $C_2\times C_6^3$, $C_6^2:D_6$, $C_2\times S_3^3$, $A_4\times S_3^2$, $A_4:S_3^2$, $C_6^2:D_6$, $(C_3^2\times A_4):C_4$, $C_2.S_3^3$, $C_6^2:D_6$, $(C_3\times C_6).S_4$

Order 384: $C_2\wr C_2^2\times S_3$ x 2, $C_2^4:S_4$

Order 324: $C_3^3:A_4$ x 6, $C_3^2:S_3^2$ x 3, $C_3^3:D_6$ x 3, $C_3^2:S_3^2$ x 3, $C_3\wr C_2^2$ x 2, $C_3^2:S_3^2$ x 2, $C_3^3:D_6$ x 2, $C_3^2:S_3^2$ x 2, $\He_3:C_{12}$, $C_3^3:C_{12}$, $A_4\times C_3^3$, $C_3^2:C_6^2$, $C_3^3:D_6$, $C_3^2:S_3^2$

Order 288: $C_6^2:D_4$ x 16, $D_6^2:C_2$ x 10, $C_3\times C_2^2:S_4$ x 10, $\PSOPlus(4,3)$ x 9, $C_6^2:D_4$ x 8, $C_2^3.S_3^2$ x 8, $(C_2\times C_6):S_4$ x 7, $A_4\wr C_2$ x 7, $C_2^3:S_3^2$ x 6, $D_{12}:D_6$ x 6, $C_3\times \GL(2,\mathbb{Z}/4)$ x 6, $C_6^2.D_4$ x 6, $C_6^2.D_4$ x 6, $D_6\times S_4$ x 5, $C_3:\GL(2,\mathbb{Z}/4)$ x 4, $(C_2\times D_6):A_4$ x 4, $C_6^2.D_4$ x 4, $C_6^2.C_2^3$ x 3, $D_4:S_3^2$ x 3, $D_4\times S_3^2$ x 3, $D_6:S_4$ x 3, $D_6:S_4$ x 3, $D_6.S_4$ x 3, $C_6^2:D_4$ x 3, $C_6^2:D_4$ x 3, $C_6^2.C_2^3$ x 3, $C_3\times C_2^3:A_4$ x 3, $C_6^2.D_4$ x 3, $C_6^2:C_2^3$ x 3, $S_3\times C_2^2:A_4$ x 3, $C_2\times A_4\times D_6$ x 3, $C_6^2:C_2^3$ x 3, $D_{12}:D_6$ x 2, $D_{12}:D_6$ x 2, $C_6^2:D_4$ x 2, $C_6^2:D_4$ x 2, $C_2\times A_4^2$

Order 243: $C_3^4:C_3$

Order 216: $C_3^2:S_4$ x 10, $C_6^2:S_3$ x 10, $C_3^2:S_4$ x 9, $C_6^2:C_6$ x 8, $C_3^2:S_4$ x 8, $C_6:S_3^2$ x 7, $C_6^2:C_6$ x 7, $C_6^2:C_6$ x 6, $C_3^2:S_4$ x 6, $C_3^2\times S_4$ x 6, $C_3^2:D_{12}$ x 6, $C_6.S_3^2$ x 6, $C_6^2:C_6$ x 5, $C_3^2:D_{12}$ x 5, $C_6^2:C_6$ x 5, $C_6^2:C_6$ x 4, $S_3^3$ x 4, $C_6^2:C_6$ x 3, $C_6^2.C_6$ x 3, $C_6.C_6^2$ x 3, $C_3^2:D_{12}$ x 3, $C_6.S_3^2$ x 3, $C_6^2:C_6$ x 2, $C_6^3$ x 2, $S_3\times C_6^2$ x 2, $C_6^2:C_6$ x 2, $C_6\times S_3^2$ x 2, $C_3^2:D_{12}$ x 2, $C_6.S_3^2$ x 2, $C_3^3:Q_8$ x 2, $C_3^3:D_4$ x 2, $C_6.S_3^2$ x 2, $C_6.S_3^2$ x 2, $C_6^2:S_3$, $C_3^2:D_{12}$, $\He_3:D_4$, $C_6.S_3^2$, $C_6:S_3^2$, $C_6^2:S_3$, $C_3^3:Q_8$, $C_3^3:D_4$, $C_6.S_3^2$, $C_6^2:C_6$

Order 192: $C_2^3:S_4$ x 6, $C_2^4:D_6$ x 6, $C_2^4:D_6$ x 5, $C_2^3:C_4\times S_3$ x 5, $C_2^3:D_{12}$ x 5, $C_3\times C_2\wr C_2^2$ x 2, $(C_2\times C_{12}):D_4$ x 2, $C_2^3:S_4$ x 2, $D_{12}:C_2^3$ x 2, $C_2\times \GL(2,\mathbb{Z}/4)$, $C_2^4:A_4$

Order 162: $C_3^3:C_6$ x 5, $C_3^2\wr C_2$ x 4, $C_3^3:S_3$ x 4, $C_3^3:C_6$ x 4, $C_3^3:S_3$, $C_6\times \He_3$, $C_3^3:S_3$, $S_3\times \He_3$

Order 144: $D_6:D_6$ x 41, $C_6^2:C_2^2$ x 22, $S_3\times S_4$ x 21, $C_6^2:C_2^2$ x 19, $C_6:D_{12}$ x 19, $C_6^2:C_4$ x 16, $D_6:D_6$ x 15, $C_6.D_{12}$ x 13, $D_6.D_6$ x 12, $C_{12}:D_6$ x 10, $C_2^2.S_3^2$ x 10, $C_6\times S_4$ x 9, $A_4\times D_6$ x 8, $A_4^2$ x 8, $C_2^4:C_3^2$ x 7, $C_2^2:C_6^2$ x 7, $C_6:S_4$ x 7, $D_6^2$ x 6, $C_6^2:C_2^2$ x 6, $C_{12}.D_6$ x 6, $C_{12}:D_6$ x 6, $S_3\times D_{12}$ x 6, $C_{12}.D_6$ x 6, $A_4:C_{12}$ x 6, $C_6.D_{12}$ x 4, $D_6.D_6$ x 4, $C_{12}.D_6$ x 4, $C_6.S_4$ x 4, $C_2^2\times C_6^2$ x 3, $C_4:C_6^2$ x 3, $C_6:D_{12}$ x 3, $C_{12}:D_6$ x 3, $D_{12}:C_6$ x 3, $C_4\times S_3^2$ x 3, $D_{12}:S_3$ x 3, $C_6^2:C_4$ x 3, $C_6^2:C_4$ x 3, $C_{12}:D_6$ x 2, $C_{12}.D_6$ x 2

Order 128: $C_2^4:D_4$

Order 108: $C_3:S_3^2$ x 15, $C_3^2:A_4$ x 14, $C_3^2\times A_4$ x 12, $C_3^2:D_6$ x 9, $C_3\times S_3^2$ x 9, $C_3^2:D_6$ x 8, $C_3^2:D_6$ x 8, $C_3\times C_6^2$ x 6, $C_3^2:D_6$ x 5, $C_3^2\times D_6$ x 5, $C_3^2:C_{12}$ x 5, $C_3:S_3^2$ x 3, $C_3^2:C_{12}$ x 3, $C_3^2:C_{12}$ x 2, $C_2^2\times \He_3$ x 2, $C_3^2:D_6$ x 2, $C_3^3:C_4$, $\He_3:C_4$

Order 96: $D_4\times D_6$ x 16, $C_2^2:D_{12}$ x 11, $D_6.D_4$ x 11, $D_6:D_4$ x 11, $D_4:D_6$ x 10, $C_2^2.D_{12}$ x 9, $C_2^2:S_4$ x 8, $\GL(2,\mathbb{Z}/4)$ x 6, $C_2^4:C_6$ x 6, $C_2^4:S_3$ x 6, $C_2^3:C_{12}$ x 5, $C_2^3.D_6$ x 5, $D_4:D_6$ x 5, $D_4:D_6$ x 5, $C_2^3:A_4$ x 4, $C_2^3\times D_6$ x 3, $C_2^2\times S_4$ x 2, $C_6.C_2^4$ x 2, $C_2^3:A_4$, $C_2^3\times A_4$, $C_2^2.S_4$

Order 81: $C_3\times \He_3$ x 4, $C_3^4$

Order 72: $C_6:D_6$ x 69, $C_6\wr C_2$ x 42, $C_3:D_{12}$ x 39, $S_3\times D_6$ x 30, $C_3:S_4$ x 27, $C_3\times S_4$ x 25, $C_6.D_6$ x 20, $C_6:C_{12}$ x 19, $S_3\times A_4$ x 17, $C_6\times D_6$ x 14, $C_6^2:C_2$ x 14, $C_2\times C_6^2$ x 12, $C_6\times A_4$ x 12, $C_6.D_6$ x 9, $D_6:S_3$ x 7, $C_{12}:S_3$ x 6, $S_3\times C_{12}$ x 6, $D_4\times C_3^2$ x 5, $C_3:D_{12}$ x 5, $C_3^2:Q_8$ x 4, $C_6\times C_{12}$ x 3, $C_6.D_6$ x 3, $C_3\times D_{12}$ x 3, $C_3^2:Q_8$ x 3

Order 64: $C_2\wr C_2^2$ x 6, $C_2^4:C_4$ x 2, $C_2^3:D_4$ x 2, $D_4:C_2^3$

Order 54: $C_3^2:C_6$ x 14, $S_3\times C_3^2$ x 14, $C_3^2:C_6$ x 13, $C_3^2:S_3$ x 9, $C_3^2:S_3$ x 6, $C_2\times \He_3$ x 5, $C_3^2\times C_6$ x 4

Order 48: $S_3\times D_4$ x 63, $C_6:D_4$ x 33, $C_2^2\times D_6$ x 28, $D_6:C_4$ x 17, $C_6\times D_4$ x 16, $C_2\times D_{12}$ x 16, $C_4\times D_6$ x 16, $C_2\times S_4$ x 15, $D_4:S_3$ x 14, $D_{12}:C_2$ x 14, $C_2^2:C_{12}$ x 11, $C_6.D_4$ x 11, $C_2^2\times A_4$ x 6, $D_4:C_6$ x 5, $D_{12}:C_2$ x 5, $A_4:C_4$ x 4, $C_2^3\times C_6$ x 3, $C_2^2:A_4$ x 3, $S_3\times Q_8$ x 2

Order 36: $C_6:S_3$ x 65, $S_3^2$ x 40, $C_6\times S_3$ x 37, $C_3\times A_4$ x 34, $C_6^2$ x 31, $C_3:C_{12}$ x 22, $C_3^2:C_4$ x 7, $C_3\times C_{12}$ x 3

Order 32: $C_2^2\wr C_2$ x 10, $C_2^3:C_4$ x 7, $D_4:C_2^2$ x 7, $C_2^2\times D_4$ x 5, $C_2^3:C_4$ x 3, $D_4:C_2^2$ x 2, $C_2^5$

Order 27: $C_3^3$ x 9, $\He_3$ x 8

Order 24: $C_2\times D_6$ x 100, $C_3:D_4$ x 55, $C_4\times S_3$ x 31, $D_{12}$ x 25, $C_3\times D_4$ x 25, $C_2^2\times C_6$ x 19, $C_6:C_4$ x 17, $S_4$ x 17, $C_2\times C_{12}$ x 16, $C_2\times A_4$ x 12, $C_3:Q_8$ x 5, $C_3\times Q_8$ x 2

Order 18: $C_3\times S_3$ x 44, $C_3:S_3$ x 35, $C_3\times C_6$ x 24

Order 16: $C_2\times D_4$ x 38, $D_4:C_2$ x 14, $C_2^2:C_4$ x 10, $C_2^4$ x 8, $C_2^2\times C_4$ x 5, $C_2\times Q_8$

Order 12: $D_6$ x 97, $C_2\times C_6$ x 40, $C_3:C_4$ x 15, $C_{12}$ x 11, $A_4$ x 10

Order 9: $C_3^2$ x 25

Order 8: $C_2^3$ x 27, $D_4$ x 27, $C_2\times C_4$ x 19, $Q_8$ x 3

Order 6: $S_3$ x 31, $C_6$ x 27

Order 4: $C_2^2$ x 26, $C_4$ x 6

Order 3: $C_3$ x 9

Order 2: $C_2$ x 9

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 31104: $(C_2^2\times C_6^2).S_3^3$ ($C_1$)

Order 15552: $(C_2\times C_6^3):S_3^2$ ($C_2$) x 2, $(C_3^2\times A_4^2):D_6$ ($C_2$) x 2, $C_2^4.C_3^4.D_6$ ($C_2$), $(C_2\times C_6^3):S_3^2$ ($C_2$), $(C_2\times C_6^3):S_3^2$ ($C_2$)

Order 7776: $(C_3^2\times A_4^2):S_3$ ($C_2^2$) x 2, $(C_3^2\times A_4^2):C_6$ ($C_2^2$) x 2, $C_2^4.C_3^4.S_3$ ($C_2^2$), $C_2^4.C_3^4.C_6$ ($C_2^2$), $C_3^3.A_4^2.C_2$ ($C_2^2$)

Order 5184: $C_6^2:(S_3\times S_4)$ ($S_3$) x 2, $A_4^2:S_3^2$ ($S_3$)

Order 3888: $C_2^4.C_3^4.C_3$ ($C_2^3$)

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

Order 1296: $C_3\times C_6^2:A_4$ ($C_2\times D_6$) x 2, $C_3^2\times A_4^2$ ($C_2\times D_6$)

Order 864: $C_6^2:S_4$ ($S_3^2$) x 2, $C_6^3:C_2^2$ ($S_3^2$)

Order 432: $C_6^2:A_4$ ($S_3\times D_6$) x 2, $C_2\times C_6^3$ ($S_3\times D_6$)

Order 288: $(C_2\times C_6):S_4$ ($C_3^2:D_6$) x 2

Order 144: $C_2^4:C_3^2$ ($C_6.S_3^2$) x 2, $C_2^2\times C_6^2$ ($S_3^3$)

Order 54: $C_3^2:S_3$ ($\POPlus(4,3)$)

Order 48: $C_2^3\times C_6$ ($C_3.S_3^3$) x 2

Order 27: $C_3^3$ ($A_4^2:C_2^3$)

Order 16: $C_2^4$ ($C_3^2.S_3^3$)

Order 9: $C_3^2$ ($C_2^4:S_3^3$)

Order 3: $C_3$ ($(C_2^3\times C_6).S_3^3$) x 2

Order 1: $C_1$ ($(C_2^2\times C_6^2).S_3^3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 31104: $(C_2^2\times C_6^2).S_3^3$ ($C_1$)

Order 15552: $C_2^4.C_3^4.D_6$ ($C_2$), $(C_2\times C_6^3):S_3^2$ ($C_2$), $(C_2\times C_6^3):S_3^2$ ($C_2$), $(C_3^2\times A_4^2):D_6$ ($C_2$), $(C_2\times C_6^3):S_3^2$ ($C_2$)

Order 7776: $C_2^4.C_3^4.S_3$ ($C_2^2$), $C_2^4.C_3^4.C_6$ ($C_2^2$), $C_3^3.A_4^2.C_2$ ($C_2^2$), $(C_3^2\times A_4^2):S_3$ ($C_2^2$), $(C_3^2\times A_4^2):C_6$ ($C_2^2$)

Order 5184: $C_6^2:(S_3\times S_4)$ ($S_3$), $A_4^2:S_3^2$ ($S_3$)

Order 3888: $C_2^4.C_3^4.C_3$ ($C_2^3$)

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

Order 1296: $C_3^2\times A_4^2$ ($C_2\times D_6$), $C_3\times C_6^2:A_4$ ($C_2\times D_6$)

Order 864: $C_6^3:C_2^2$ ($S_3^2$), $C_6^2:S_4$ ($S_3^2$)

Order 432: $C_2\times C_6^3$ ($S_3\times D_6$), $C_6^2:A_4$ ($S_3\times D_6$)

Order 288: $(C_2\times C_6):S_4$ ($C_3^2:D_6$)

Order 144: $C_2^2\times C_6^2$ ($S_3^3$), $C_2^4:C_3^2$ ($C_6.S_3^2$)

Order 54: $C_3^2:S_3$ ($\POPlus(4,3)$)

Order 48: $C_2^3\times C_6$ ($C_3.S_3^3$)

Order 27: $C_3^3$ ($A_4^2:C_2^3$)

Order 16: $C_2^4$ ($C_3^2.S_3^3$)

Order 9: $C_3^2$ ($C_2^4:S_3^3$)

Order 3: $C_3$ ($(C_2^3\times C_6).S_3^3$)

Order 1: $C_1$ ($(C_2^2\times C_6^2).S_3^3$)

Series

Derived series

$(C_2^2\times C_6^2).S_3^3$

$\rhd$

$(C_2^2\times C_6^2).S_3^3$

$\rhd$

$C_2^4.C_3^4.C_3$

$\rhd$

$C_2^4.C_3^4.C_3$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_1$

$\rhd$

$C_1$

Chief series

$(C_2^2\times C_6^2).S_3^3$

$\rhd$

$(C_2^2\times C_6^2).S_3^3$

$\rhd$

$(C_2\times C_6^3):S_3^2$

$\rhd$

$(C_2\times C_6^3):S_3^2$

$\rhd$

$C_2^4.C_3^4.C_6$

$\rhd$

$C_2^4.C_3^4.C_6$

$\rhd$

$C_2^4.C_3^4.C_3$

$\rhd$

$C_2^4.C_3^4.C_3$

$\rhd$

$C_3\times C_6^2:A_4$

$\rhd$

$C_3^2\times A_4^2$

$\rhd$

$C_3^2\times A_4^2$

$\rhd$

$C_6^2:A_4$

$\rhd$

$C_6^2:A_4$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_2^2\times C_6^2$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2^3\times C_6$

$\rhd$

$C_2^4$

$\rhd$

$C_2^4$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$(C_2^2\times C_6^2).S_3^3$

$\rhd$

$(C_2^2\times C_6^2).S_3^3$

$\rhd$

$C_2^4.C_3^4.C_3$

$\rhd$

$C_2^4.C_3^4.C_3$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

This group is a maximal quotient of 10 larger groups in the database.

Character theory

Complex character table

See the $90 \times 90$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $89 \times 89$ rational character table.