Abstract group 419904.fx: $C_3^6:(A_4\times \GL(2,3))$

• Label: 419904.fx
• Order: \( 2^{6} \cdot 3^{8} \)
• Exponent: \( 2^{3} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{8} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $27$
• Trans deg.: $27$
• Rank: $2$

Group information

Description:	$C_3^6:(A_4\times \GL(2,3))$
Order:	\(419904\)\(\medspace = 2^{6} \cdot 3^{8} \)
Exponent:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Automorphism group:	$C_3^6.(S_4\times \GL(2,3))$, of order \(839808\)\(\medspace = 2^{7} \cdot 3^{8} \)
Composition factors:	$C_2$ x 6, $C_3$ x 8
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	9	12	18	24
Elements	1	2295	12392	17496	154440	34992	46656	34992	46656	69984	419904
Conjugacy classes	1	5	14	2	28	4	6	2	2	4	68
Divisions	1	5	12	2	25	2	3	1	1	1	53
Autjugacy classes	1	5	12	2	21	4	3	1	1	2	52

Dimension	1	2	3	4	6	8	9	12	24	32	48	64	96	128	144	192	288	384
Irr. complex chars.	6	9	8	3	3	0	2	1	8	6	8	3	2	0	4	3	2	0	68
Irr. rational chars.	2	3	4	3	3	2	2	2	8	2	8	3	2	1	4	1	2	1	53

Minimal presentations

Permutation degree:	$27$
Transitive degree:	$27$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j \mid b^{6}=d^{4}=e^{6}=f^{3}=g^{3}= \!\cdots\! \rangle}$
Permutation group:	Degree $27$ $\langle(1,7,8,4,6,9,2,3)(10,15,11,14,13,17,12,18)(20,23,21,25,27,24,26,22), (1,24,14,4,23,18,3,26,17,8,25,11,9,27,10,6,22,12,7,19,16,5,20,13)(2,21,15)\rangle$
Transitive group:	27T1614	36T28795	36T29451		more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^6.A_4)$ $\,\rtimes\,$ $\GL(2,3)$	$C_3^6$ $\,\rtimes\,$ $(A_4\times \GL(2,3))$	$(C_3^6:(Q_8\times A_4))$ $\,\rtimes\,$ $S_3$	$(C_3^6:Q_8)$ $\,\rtimes\,$ $(S_3\times A_4)$	all 9
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^5:S_3)$ . $(A_4\times S_4)$	$(C_3^6.(C_2\times A_4))$ . $S_4$	$((C_3:S_3)^3)$ . $(C_3\times S_4)$		more information

Elements of the group are displayed as permutations of degree 27.

Homology

Abelianization:	$C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier:	$C_{2}$
Commutator length:	$1$

Subgroups

There are 6520834 subgroups in 4766 conjugacy classes, 16 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^6:(A_4\times \GL(2,3))$
Commutator:	$G' \simeq$ $C_3^6:(C_2^2\times \SL(2,3))$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3^6:(A_4\times \GL(2,3))$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^6$	$G/\operatorname{Fit} \simeq$ $A_4\times \GL(2,3)$
Radical:	$R \simeq$ $C_3^6:(A_4\times \GL(2,3))$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^6$	$G/\operatorname{soc} \simeq$ $A_4\times \GL(2,3)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times \SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^6:C_3^2$

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 419904: $C_3^6:(A_4\times \GL(2,3))$

Order 209952: $C_3^6:(A_4\times \SL(2,3))$

Order 139968: $C_3^6:(C_2^2\times \GL(2,3))$, $C_3^6:(A_4\times \SD_{16})$

Order 104976: $C_3^6.(A_4\times D_6)$, $C_3^6:(C_3\times \GL(2,3))$

Order 69984: $C_3^6.Q_8.D_6$, $C_3^6:(C_2\times \GL(2,3))$, $C_3^6:(C_2^2\times \SL(2,3))$, $C_3^6:(Q_8\times A_4)$, $C_3^6:Q_8:A_4$, $C_3^6:(C_8\times A_4)$, $C_3^6:(D_4\times A_4)$

Order 52488: $C_3^6:(C_6\times A_4)$, $C_3^6:(S_3\times A_4)$, $C_3^6:(S_3\times A_4)$, $C_3^6:(C_3\times \SL(2,3))$

Order 46656: $C_3^6:(C_2^2\times \SD_{16})$

Order 34992: $C_3^6.Q_8.S_3$, $C_3^6.Q_8.C_6$, $C_3^6.Q_8.S_3$, $C_3^6.(C_4\times A_4)$, $C_3^6.(C_2^2\times A_4)$, $C_3^3.S_3^4$, $C_3^6:(C_3\times \SD_{16})$

Order 26244: $C_3^6.(C_3\times A_4)$, $C_3^6:(C_6\times S_3)$

Order 23328: $C_3^6.(C_2\times \SD_{16})$ x 2, $C_3^6:(C_2\times \SD_{16})$ x 2, $C_3^6:(C_2^2\times C_8)$, $C_3^6.(C_2^2\times Q_8)$, $C_3^6.(C_2^2\times D_4)$

Order 17496: $C_3^6.\SL(2,3)$ x 2, $C_3^6.(C_2\times A_4)$ x 2, $C_3^4.S_3^3$, $C_3^6.Q_8.C_3$, $C_3^4.S_3^3$, $C_3^4.S_3^3$, $C_3^4:S_3^3$, $C_3^4:S_3^3$, $C_3^4:S_3^3$, $C_3^4.S_3^3$, $C_3^2\wr C_3:Q_8$, $C_3^6.C_{24}$, $C_3^6:(C_2\times A_4)$, $C_3^2\wr C_3:D_4$

Order 15552: $C_3^4:(C_2^2\times \GL(2,3))$

Order 13122: $C_3^6:(C_3\times S_3)$, $C_3^6:(C_3\times S_3)$, $C_3^6.(C_3\times C_6)$

Order 11664: $C_3^6.\SD_{16}$ x 4, $C_3^6.(C_2\times D_4)$ x 3, $C_3^6.\SD_{16}$ x 2, $C_3^6.(C_2\times Q_8)$ x 2, $C_3^6.C_4.C_2^2$, $C_3^4:D_6^2$, $C_3^4:(C_3\times \GL(2,3))$, $C_3^6:(C_2^2\times C_4)$, $C_3^6:C_2^4$, $C_3^4.(C_2\times F_9)$

Order 8748: $C_3^6:D_6$ x 2, $C_3^6.A_4$ x 2, $C_3^5.(C_6\times S_3)$, $C_3^4.C_3^2.D_6$, $C_3^6.C_6.C_2$, $C_3^4.C_3^2.D_6$, $C_3^5.C_3.D_6$, $C_3^4.C_3^2.D_6$, $C_3^4.C_3^3.C_2^2$, $C_3^6:D_6$, $C_3^6:D_6$, $C_3^5.C_6^2$, $C_3^5.(C_6\times S_3)$, $C_3^5:(C_6\times S_3)$, $C_3^2\wr C_3:C_2^2$, $C_3^6.C_{12}$

Order 7776: $C_3^4:(C_2\times \GL(2,3))$ x 5, $C_2\times C_3^4.Q_8.S_3$, $C_2\times C_3^4:\GL(2,3)$, $C_3^4:(C_2^2\times \SL(2,3))$

Order 6561: $C_3^6:C_3^2$

Order 5832: $C_3^6:D_4$ x 4, $C_3^3.S_3^3$ x 3, $C_3^6:Q_8$ x 3, $C_3^6:(C_2\times C_4)$ x 2, $C_3^6:C_2^3$ x 2, $C_3^3.S_3^3$ x 2, $C_3^3:S_3^3$ x 2, $C_3^3.S_3^3$ x 2, $C_3^3.S_3^3$ x 2, $C_3^6.C_8$, $C_3^4.C_3^2.C_2^3$, $(C_3:S_3)^3$, $C_3^6:C_8$, $C_3^3:S_3^3$, $C_3^3:S_3^3$, $C_3^3.S_3^3$, $C_3^3:S_3^3$, $C_3^3:S_3^3$, $C_3^4:(C_3\times \SL(2,3))$, $C_3^3:S_3^3$, $C_3^6:C_2^3$

Order 5184: $C_3^4:(C_2^2\times \SD_{16})$

Order 4374: $C_3^6.C_6$ x 2, $C_3^5.C_3.C_6$, $C_3^3.\He_3.C_6$, $C_3^4.C_3^3.C_2$, $C_3^6.C_6$, $C_3^4.C_3^3.C_2$, $C_3^4.C_3^3.C_2$, $C_3^4.C_3^3.C_2$, $C_3^5:(C_3\times C_6)$, $C_3^5:(C_3\times S_3)$, $C_3^5.(C_3\times S_3)$, $C_3^5.(C_3\times S_3)$, $C_3^5:(C_3\times S_3)$, $C_3\wr C_6$

Order 3888: $C_3^4:\GL(2,3)$ x 4, $C_3^4:\GL(2,3)$ x 3, $C_3^4:(C_2\times \SL(2,3))$ x 3, $C_2\times C_3^4.Q_8.C_3$, $C_3^3:D_6^2$, $C_3^3:D_6^2$, $C_3^4:(C_3\times \SD_{16})$, $C_3:S_3^4$, $S_3^4:C_3$

Order 2916: $C_3^4:C_3.C_6.C_2$ x 6, $C_3^5:D_6$ x 4, $C_3^5.C_6.C_2$ x 3, $C_3^4.(C_6\times S_3)$ x 3, $C_3^5.D_6$ x 2, $(C_3^2\times \He_3).D_6$ x 2, $C_3^6.C_4$ x 2, $C_3^4.C_3:S_3.C_2$ x 2, $C_3^4:C_3.C_6.C_2$ x 2, $(C_3^2\times \He_3).D_6$ x 2, $C_3^3.C_3^3.C_2^2$ x 2, $C_3^4:C_6^2$ x 2, $C_3^5:D_6$ x 2, $C_3^5:D_6$ x 2, $C_3^5:D_6$ x 2, $C_3^5:D_6$ x 2, $C_3^5:D_6$ x 2, $C_3^5:D_6$ x 2, $C_3^4.C_6^2$ x 2, $C_3^4:C_6^2$ x 2, $C_3^5.D_6$, $C_3^4.C_3:S_3.C_2$, $C_3^4.C_3:S_3.C_2$, $C_3^4.S_3^2$, $(C_3^2\times \He_3).D_6$, $C_3^4.C_6^2$, $C_3^4.S_3^2$, $C_3^4.(C_6\times S_3)$, $C_3^5.C_6.C_2$, $C_3^4.C_3:S_3.C_2$, $C_3^5:D_6$, $C_3^4:C_6^2$, $C_3^5:D_6$, $C_3^4:S_3^2$, $C_3^4:(C_6\times S_3)$, $C_3^4:(C_6\times S_3)$, $C_3^5:(C_2\times C_6)$

Order 2592: $C_3^4:(C_2\times \SD_{16})$ x 10, $C_2\times C_3^4:\SD_{16}$ x 2, $C_2\times C_3^4:\SD_{16}$ x 2, $C_3^4:(C_2^2\times Q_8)$, $C_3^2:(C_2^2\times F_9)$, $C_3^4:(C_2^2\times D_4)$

Order 2187: $C_3^4.C_3^3$ x 2, $C_3^2\wr C_3$ x 2, $C_3^5:C_3^2$, $C_3^6.C_3$, $C_3^4.C_3^3$

Order 1944: $C_3^2.S_3^3$ x 12, $C_3^2:S_3^3$ x 6, $C_3^2:S_3^3$ x 4, $C_3^2:S_3^3$ x 4, $C_3^2.S_3^3$ x 4, $C_3^2:S_3^3$ x 3, $C_3^2:S_3^3$ x 3, $C_3^4:\SL(2,3)$ x 3, $C_3^2:S_3^3$ x 3, $C_3^2:S_3^3$ x 2, $C_3^2:S_3^3$ x 2, $C_2\times C_3^3:S_3^2$ x 2, $C_2\times C_3^4:D_6$ x 2, $C_3^4:\SL(2,3)$ x 2, $C_3^2:S_3^3$, $S_3\times C_3^3:A_4$, $C_3^3:(S_3\times A_4)$, $S_3^3:C_3^2$, $C_3^2:S_3^3$, $C_3^2:S_3^3$, $C_3^2.S_3^3$, $C_2\times C_3^3:S_3^2$, $C_2\times C_3^4:D_6$, $C_2\times C_3^3:S_3^2$, $C_3^3:\PSU(3,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^4:C_{24}$

Order 1728: $C_6^2:\GL(2,3)$

Order 1458: $C_3^5:S_3$ x 6, $C_3^5:S_3$ x 6, $C_3^5:C_6$ x 6, $C_3^5:S_3$ x 3, $C_3^5:S_3$ x 3, $C_3^5:C_6$ x 3, $C_3^5:C_6$ x 3, $C_3^5:C_6$ x 3, $C_3^4.(C_3\times S_3)$ x 2, $C_3^3\wr C_2$ x 2, $C_3^4:(C_3\times C_6)$ x 2, $C_3^5:S_3$ x 2, $C_3^5:C_6$ x 2, $C_3^4:(C_3\times C_6)$ x 2, $C_3^4:C_{18}$, $C_3^4.(C_3\times S_3)$, $C_3^5:C_6$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^4:(C_3\times S_3)$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5.C_6$, $(C_3^2\times \He_3):C_6$, $C_3^4.(C_3\times S_3)$, $C_3^4:(C_3\times S_3)$, $C_3^4:(C_3\times S_3)$, $C_3^5:C_6$, $C_3^5:C_6$

Order 1296: $C_3^4:\SD_{16}$ x 14, $C_3^4:\SD_{16}$ x 12, $C_3^4:(C_2\times D_4)$ x 9, $C_3^2:(C_2\times F_9)$ x 7, $C_3^4:(C_2\times Q_8)$ x 6, $C_2\times C_3^4:D_4$ x 3, $S_3^4$ x 3, $C_2\times C_3^4:Q_8$ x 2, $C_3^2:D_6^2$ x 2, $C_6:S_3^3$, $C_3^4:(C_2^2\times C_4)$, $C_2\times C_3^2:F_9$, $C_2\times C_3^2:F_9$, $S_3^3:C_6$, $C_3^3:\GL(2,3)$

Order 972: $C_3^3:S_3^2$ x 18, $C_3^4:D_6$ x 18, $C_3^3:S_3^2$ x 17, $C_3^3:S_3^2$ x 14, $C_3^4:D_6$ x 10, $C_3^3:S_3^2$ x 10, $C_3^3:S_3^2$ x 9, $C_3^3:S_3^2$ x 8, $C_3^3:C_6^2$ x 8, $C_3^3:C_6^2$ x 6, $C_3^3:C_6^2$ x 6, $C_3^3:S_3^2$ x 6, $C_3^3:S_3^2$ x 6, $C_3^3:S_3^2$ x 6, $C_3^3:S_3^2$ x 6, $C_3^4:D_6$ x 4, $C_3^3:S_3^2$ x 3, $C_3^3:S_3^2$ x 3, $C_3^3:S_3^2$ x 3, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^4:D_6$ x 2, $C_3^3.C_6^2$ x 2, $\He_3:S_3^2$ x 2, $C_3^3:S_3^2$, $C_3\wr A_4$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $\He_3\times S_3^2$, $C_3^4:D_6$, $C_2\times C_3^4:C_6$, $C_3^3:C_6^2$, $C_3^4:C_{12}$

Order 864: $C_2\times C_3^2:\GL(2,3)$ x 7, $C_6^2:\SL(2,3)$

Order 729: $C_3^5:C_3$ x 6, $C_3^4.C_3^2$ x 4, $C_3^5.C_3$ x 3, $C_3^5:C_3$ x 3, $C_3^4.C_3^2$ x 2, $C_3^3.C_3^3$ x 2, $C_3^4:C_3^2$ x 2, $C_3^4.C_3^2$, $C_3^4:C_9$, $C_3^6$, $C_3^4:C_3^2$, $C_3^3.C_3^3$

Order 648: $C_3^4:D_4$ x 24, $C_3.S_3^3$ x 22, $C_3:S_3^3$ x 18, $C_3:S_3^3$ x 17, $C_3^4:Q_8$ x 13, $C_3\times S_3^3$ x 11, $C_3:S_3^3$ x 10, $C_3^4:(C_2\times C_4)$ x 9, $C_3^2:F_9$ x 9, $C_3^2:F_9$ x 5, $S_3\wr C_3$ x 4, $C_2\times C_3^2:S_3^2$ x 4, $C_6\times C_3^2:D_6$ x 4, $S_3\times C_3^2:D_6$ x 4, $C_3^4:C_2^3$ x 3, $C_2\times C_3^2:S_3^2$ x 3, $C_3^4:C_2^3$ x 2, $C_2\times C_3^4:C_4$ x 2, $S_3\times C_3^2:D_6$ x 2, $C_2\times C_3^3:A_4$, $C_3^3:\SL(2,3)$

Order 576: $F_9:C_2^3$, $A_4\times \GL(2,3)$

Order 486: $C_3^4:C_6$ x 19, $C_3^4:S_3$ x 18, $C_3^4:C_6$ x 15, $C_3^4:C_6$ x 14, $C_3^4:C_6$ x 14, $C_3^4:C_6$ x 12, $C_3^4:S_3$ x 11, $C_3^4:S_3$ x 11, $C_3^4:S_3$ x 10, $C_3^4:C_6$ x 9, $C_3^4:C_6$ x 8, $C_3^4:S_3$ x 6, $C_3^4:S_3$ x 6, $C_3^4:C_6$ x 6, $C_3^4:C_6$ x 6, $C_3^4:C_6$ x 3, $C_3^3:(C_3\times C_6)$ x 2, $C_3^3:(C_3\times C_6)$ x 2, $C_3^4:C_6$ x 2, $C_3^4:C_6$, $S_3\times C_3^4$, $C_3^4:S_3$, $C_3^3.(C_3\times S_3)$, $C_3^3:(C_3\times S_3)$, $C_3^4:C_6$, $C_3^4:C_6$, $C_3^3:C_{18}$, $C_3^3.(C_3\times S_3)$, $C_3^3.(C_3\times S_3)$, $C_3\wr C_3:C_6$, $C_3^4.C_6$

Order 432: $C_3^2:\GL(2,3)$ x 7, $C_2\times S_3^3$ x 5, $C_2\times \PU(3,2)$ x 4, $F_9:C_6$, $C_6^2:D_6$

Order 324: $C_3^2:S_3^2$ x 68, $C_3\wr C_2^2$ x 43, $C_3^2:S_3^2$ x 39, $C_3^2:S_3^2$ x 33, $C_3^3:D_6$ x 32, $C_3^2:S_3^2$ x 29, $C_3^4:C_4$ x 24, $C_3^2:S_3^2$ x 23, $C_3^2\times S_3^2$ x 22, $C_3^2:S_3^2$ x 21, $C_3^2:S_3^2$ x 18, $C_3^2:S_3^2$ x 15, $C_3^2:C_6^2$ x 8, $C_3^3:D_6$ x 6, $C_3^3:D_6$ x 6, $C_3^2:C_6^2$ x 4, $C_3^3:A_4$ x 3, $C_3^3:D_6$ x 2, $D_6\times \He_3$ x 2, $\He_3:D_6$, $C_3^3:D_6$

Order 288: $F_9:C_2^2$ x 14, $A_4\times \SL(2,3)$, $C_6^2:Q_8$, $C_6^2:D_4$, $C_2^2\times F_9$

Order 243: $C_3^4:C_3$ x 24, $C_3^2\times \He_3$ x 11, $C_3^4:C_3$ x 10, $C_3^3:C_9$ x 8, $C_3^5$ x 6, $C_3^3:C_3^2$ x 6, $C_3^4.C_3$ x 3, $\He_3:C_3^2$ x 2, $C_3^2.C_3^3$ x 2, $C_9.C_3^3$, $C_3^3:C_3^2$

Order 216: $S_3^3$ x 53, $C_6.S_3^2$ x 17, $C_6:S_3^2$ x 15, $C_6\times S_3^2$ x 15, $C_6:S_3^2$ x 5, $\PU(3,2)$ x 5, $C_6^2:C_6$ x 2, $C_3^3:Q_8$, $S_3^2:C_6$, $C_3\times F_9$, $C_6^2:S_3$

Order 192: $C_2^2\times \GL(2,3)$, $A_4\times \SD_{16}$

Order 162: $C_3^2\wr C_2$ x 114, $C_3^3:S_3$ x 75, $C_3^3:C_6$ x 62, $C_3^3:S_3$ x 47, $C_3^3:C_6$ x 46, $S_3\times C_3^3$ x 27, $C_3^3:C_6$ x 22, $C_3^3:C_6$ x 19, $S_3\times \He_3$ x 18, $C_3^3:S_3$ x 12, $C_6\times \He_3$ x 6, $C_3^3.C_6$ x 4, $C_3^3\times C_6$, $\He_3.C_6$, $C_3^2:C_{18}$, $C_3^3:C_6$, $\He_3.C_6$, $\He_3.C_6$, $C_3\wr S_3$

Order 144: $F_9:C_2$ x 26, $S_3^2:C_2^2$ x 12, $C_2\times F_9$ x 9, $C_2\times \PSU(3,2)$ x 8, $D_6^2$ x 4, $C_6^2:C_4$, $A_4\times D_6$, $C_3\times \GL(2,3)$

Order 108: $C_3:S_3^2$ x 153, $C_3\times S_3^2$ x 118, $C_3^2:D_6$ x 43, $C_3:S_3^2$ x 35, $C_3^2:D_6$ x 26, $C_3^2\times D_6$ x 20, $C_3^2:D_6$ x 17, $C_3^2:D_6$ x 12, $C_3^2:D_6$ x 7, $C_3^2:C_{12}$, $C_2^2\times \He_3$

Order 96: $C_2\times \GL(2,3)$ x 2, $C_8\times A_4$, $Q_8:A_4$, $Q_8\times A_4$, $C_2^2\times \SL(2,3)$, $D_4\times A_4$

Order 81: $C_3^4$ x 77, $C_3\times \He_3$ x 75, $C_3\wr C_3$ x 24, $C_9:C_3^2$ x 6, $C_3^2:C_9$ x 5, $\He_3:C_3$ x 2, $C_3.\He_3$ x 2, $C_9.C_3^2$ x 2

Order 72: $S_3\times D_6$ x 60, $\SOPlus(4,2)$ x 24, $F_9$ x 14, $\PSU(3,2)$ x 13, $C_2\times C_3^2:C_4$ x 11, $C_6\times D_6$ x 8, $C_6:D_6$ x 4, $S_3\times A_4$ x 2, $C_6\times A_4$, $C_3\times \SL(2,3)$

Order 64: $C_2^2\times \SD_{16}$

Order 54: $C_3^2:S_3$ x 180, $S_3\times C_3^2$ x 161, $C_3^2:C_6$ x 137, $C_3^2:C_6$ x 69, $C_3^2:S_3$ x 27, $C_2\times \He_3$ x 14, $C_3^2\times C_6$ x 10, $S_3\times C_9$ x 4, $C_9:C_6$

Order 48: $C_2^2\times D_6$ x 2, $\GL(2,3)$ x 2, $C_2^2\times A_4$, $C_2\times \SL(2,3)$, $C_4\times A_4$, $C_3\times \SD_{16}$

Order 36: $S_3^2$ x 140, $C_6\times S_3$ x 82, $C_6:S_3$ x 42, $C_3^2:C_4$ x 24, $C_6^2$ x 5, $C_3\times A_4$

Order 32: $C_2\times \SD_{16}$ x 4, $C_2^2\times Q_8$, $C_2^2\times D_4$, $C_2^2\times C_8$

Order 27: $C_3^3$ x 197, $\He_3$ x 55, $C_9:C_3$ x 11, $C_3\times C_9$ x 6

Order 24: $C_2\times D_6$ x 23, $\SL(2,3)$ x 3, $C_2\times A_4$ x 3, $C_2^2\times C_6$ x 2, $C_{24}$, $C_3\times Q_8$, $C_3\times D_4$

Order 18: $C_3\times S_3$ x 159, $C_3:S_3$ x 110, $C_3\times C_6$ x 41, $C_{18}$

Order 16: $\SD_{16}$ x 6, $C_2\times D_4$ x 3, $C_2\times C_8$ x 2, $C_2\times Q_8$ x 2, $C_2^4$, $C_2^2\times C_4$

Order 12: $D_6$ x 59, $C_2\times C_6$ x 14, $A_4$ x 2, $C_{12}$

Order 9: $C_3^2$ x 102, $C_9$ x 3

Order 8: $C_2^3$ x 5, $D_4$ x 4, $Q_8$ x 3, $C_2\times C_4$ x 2, $C_8$ x 2

Order 6: $S_3$ x 32, $C_6$ x 25

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 3: $C_3$ x 12

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 419904: $C_3^6:(A_4\times \GL(2,3))$

Order 209952: $C_3^6:(A_4\times \SL(2,3))$

Order 139968: $C_3^6:(C_2^2\times \GL(2,3))$, $C_3^6:(A_4\times \SD_{16})$

Order 104976: $C_3^6.(A_4\times D_6)$, $C_3^6:(C_3\times \GL(2,3))$

Order 69984: $C_3^6.Q_8.D_6$, $C_3^6:(C_2\times \GL(2,3))$, $C_3^6:(C_2^2\times \SL(2,3))$, $C_3^6:(Q_8\times A_4)$, $C_3^6:Q_8:A_4$, $C_3^6:(C_8\times A_4)$, $C_3^6:(D_4\times A_4)$

Order 52488: $C_3^6:(C_6\times A_4)$, $C_3^6:(S_3\times A_4)$, $C_3^6:(S_3\times A_4)$, $C_3^6:(C_3\times \SL(2,3))$

Order 46656: $C_3^6:(C_2^2\times \SD_{16})$

Order 34992: $C_3^6.Q_8.S_3$, $C_3^6.Q_8.C_6$, $C_3^6.Q_8.S_3$, $C_3^6.(C_4\times A_4)$, $C_3^6.(C_2^2\times A_4)$, $C_3^3.S_3^4$, $C_3^6:(C_3\times \SD_{16})$

Order 26244: $C_3^6.(C_3\times A_4)$, $C_3^6:(C_6\times S_3)$

Order 23328: $C_3^6.(C_2\times \SD_{16})$ x 2, $C_3^6:(C_2\times \SD_{16})$ x 2, $C_3^6:(C_2^2\times C_8)$, $C_3^6.(C_2^2\times Q_8)$, $C_3^6.(C_2^2\times D_4)$

Order 17496: $C_3^6.\SL(2,3)$ x 2, $C_3^6.(C_2\times A_4)$ x 2, $C_3^4.S_3^3$, $C_3^6.Q_8.C_3$, $C_3^4.S_3^3$, $C_3^4.S_3^3$, $C_3^4:S_3^3$, $C_3^4:S_3^3$, $C_3^4:S_3^3$, $C_3^4.S_3^3$, $C_3^2\wr C_3:Q_8$, $C_3^6.C_{24}$, $C_3^6:(C_2\times A_4)$, $C_3^2\wr C_3:D_4$

Order 15552: $C_3^4:(C_2^2\times \GL(2,3))$

Order 13122: $C_3^6:(C_3\times S_3)$, $C_3^6:(C_3\times S_3)$, $C_3^6.(C_3\times C_6)$

Order 11664: $C_3^6.(C_2\times D_4)$ x 3, $C_3^6.\SD_{16}$ x 3, $C_3^6.\SD_{16}$ x 2, $C_3^6.(C_2\times Q_8)$ x 2, $C_3^6.C_4.C_2^2$, $C_3^4:D_6^2$, $C_3^4:(C_3\times \GL(2,3))$, $C_3^6:(C_2^2\times C_4)$, $C_3^6:C_2^4$, $C_3^4.(C_2\times F_9)$

Order 8748: $C_3^6.A_4$ x 2, $C_3^5.(C_6\times S_3)$, $C_3^4.C_3^2.D_6$, $C_3^6.C_6.C_2$, $C_3^4.C_3^2.D_6$, $C_3^5.C_3.D_6$, $C_3^4.C_3^2.D_6$, $C_3^4.C_3^3.C_2^2$, $C_3^6:D_6$, $C_3^6:D_6$, $C_3^6:D_6$, $C_3^5.C_6^2$, $C_3^5.(C_6\times S_3)$, $C_3^5:(C_6\times S_3)$, $C_3^2\wr C_3:C_2^2$, $C_3^6.C_{12}$

Order 7776: $C_3^4:(C_2\times \GL(2,3))$ x 3, $C_2\times C_3^4.Q_8.S_3$, $C_2\times C_3^4:\GL(2,3)$, $C_3^4:(C_2^2\times \SL(2,3))$

Order 6561: $C_3^6:C_3^2$

Order 5832: $C_3^6:D_4$ x 4, $C_3^6:Q_8$ x 3, $C_3^3.S_3^3$ x 2, $C_3^6:(C_2\times C_4)$ x 2, $C_3^6:C_2^3$ x 2, $C_3^3.S_3^3$ x 2, $C_3^6.C_8$, $C_3^4.C_3^2.C_2^3$, $(C_3:S_3)^3$, $C_3^6:C_8$, $C_3^3.S_3^3$, $C_3^3:S_3^3$, $C_3^3:S_3^3$, $C_3^3:S_3^3$, $C_3^3.S_3^3$, $C_3^3:S_3^3$, $C_3^3.S_3^3$, $C_3^3:S_3^3$, $C_3^4:(C_3\times \SL(2,3))$, $C_3^3:S_3^3$, $C_3^6:C_2^3$

Order 5184: $C_3^4:(C_2^2\times \SD_{16})$

Order 4374: $C_3^6.C_6$ x 2, $C_3^5.C_3.C_6$, $C_3^3.\He_3.C_6$, $C_3^4.C_3^3.C_2$, $C_3^6.C_6$, $C_3^4.C_3^3.C_2$, $C_3^4.C_3^3.C_2$, $C_3^4.C_3^3.C_2$, $C_3^5:(C_3\times C_6)$, $C_3^5:(C_3\times S_3)$, $C_3^5.(C_3\times S_3)$, $C_3^5.(C_3\times S_3)$, $C_3^5:(C_3\times S_3)$, $C_3\wr C_6$

Order 3888: $C_3^4:\GL(2,3)$ x 4, $C_3^4:\GL(2,3)$ x 2, $C_3^4:(C_2\times \SL(2,3))$ x 2, $C_2\times C_3^4.Q_8.C_3$, $C_3^3:D_6^2$, $C_3^3:D_6^2$, $C_3^4:(C_3\times \SD_{16})$, $C_3:S_3^4$, $S_3^4:C_3$

Order 2916: $C_3^4:C_3.C_6.C_2$ x 4, $C_3^5.D_6$ x 2, $(C_3^2\times \He_3).D_6$ x 2, $C_3^6.C_4$ x 2, $C_3^5.C_6.C_2$ x 2, $C_3^4:C_3.C_6.C_2$ x 2, $C_3^4.(C_6\times S_3)$ x 2, $C_3^4:C_6^2$ x 2, $C_3^5:D_6$ x 2, $C_3^5:D_6$ x 2, $C_3^5:D_6$ x 2, $C_3^4:C_6^2$ x 2, $C_3^5.D_6$, $C_3^4.C_3:S_3.C_2$, $C_3^4.C_3:S_3.C_2$, $C_3^4.S_3^2$, $C_3^4.C_3:S_3.C_2$, $(C_3^2\times \He_3).D_6$, $C_3^4.C_6^2$, $C_3^4.S_3^2$, $(C_3^2\times \He_3).D_6$, $C_3^4.(C_6\times S_3)$, $C_3^3.C_3^3.C_2^2$, $C_3^5.C_6.C_2$, $C_3^4.C_3:S_3.C_2$, $C_3^5:D_6$, $C_3^4:C_6^2$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^5:D_6$, $C_3^4.C_6^2$, $C_3^4:S_3^2$, $C_3^4:(C_6\times S_3)$, $C_3^4:(C_6\times S_3)$, $C_3^5:(C_2\times C_6)$

Order 2592: $C_3^4:(C_2\times \SD_{16})$ x 6, $C_2\times C_3^4:\SD_{16}$ x 2, $C_2\times C_3^4:\SD_{16}$ x 2, $C_3^4:(C_2^2\times Q_8)$, $C_3^2:(C_2^2\times F_9)$, $C_3^4:(C_2^2\times D_4)$

Order 2187: $C_3^4.C_3^3$ x 2, $C_3^2\wr C_3$ x 2, $C_3^5:C_3^2$, $C_3^6.C_3$, $C_3^4.C_3^3$

Order 1944: $C_3^2.S_3^3$ x 9, $C_3^2:S_3^3$ x 4, $C_3^2:S_3^3$ x 3, $C_3^2:S_3^3$ x 3, $C_3^4:\SL(2,3)$ x 3, $C_3^2:S_3^3$ x 2, $C_3^2:S_3^3$ x 2, $C_3^2:S_3^3$ x 2, $C_3^2:S_3^3$ x 2, $C_3^2.S_3^3$ x 2, $C_3^2:S_3^3$ x 2, $C_3^4:\SL(2,3)$ x 2, $C_3^2:S_3^3$, $S_3\times C_3^3:A_4$, $C_3^3:(S_3\times A_4)$, $S_3^3:C_3^2$, $C_3^2:S_3^3$, $C_3^2:S_3^3$, $C_3^2.S_3^3$, $C_2\times C_3^3:S_3^2$, $C_2\times C_3^4:D_6$, $C_2\times C_3^3:S_3^2$, $C_2\times C_3^4:D_6$, $C_2\times C_3^3:S_3^2$, $C_3^3:\PSU(3,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^4:C_{24}$

Order 1728: $C_6^2:\GL(2,3)$

Order 1458: $C_3^5:S_3$ x 6, $C_3^5:S_3$ x 4, $C_3^5:C_6$ x 4, $C_3^5:S_3$ x 3, $C_3^5:C_6$ x 3, $C_3^5:C_6$ x 3, $C_3^3\wr C_2$ x 2, $C_3^4:(C_3\times C_6)$ x 2, $C_3^5:S_3$ x 2, $C_3^5:S_3$ x 2, $C_3^5:C_6$ x 2, $C_3^5:C_6$ x 2, $C_3^4:(C_3\times C_6)$ x 2, $C_3^4:C_{18}$, $C_3^4.(C_3\times S_3)$, $C_3^4.(C_3\times S_3)$, $C_3^5:C_6$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^4:(C_3\times S_3)$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5.C_6$, $(C_3^2\times \He_3):C_6$, $C_3^4.(C_3\times S_3)$, $C_3^4:(C_3\times S_3)$, $C_3^4:(C_3\times S_3)$, $C_3^5:C_6$, $C_3^5:C_6$

Order 1296: $C_3^4:\SD_{16}$ x 11, $C_3^4:\SD_{16}$ x 7, $C_3^4:(C_2\times D_4)$ x 6, $C_3^2:(C_2\times F_9)$ x 5, $C_3^4:(C_2\times Q_8)$ x 4, $C_2\times C_3^4:D_4$ x 3, $S_3^4$ x 3, $C_2\times C_3^4:Q_8$ x 2, $C_6:S_3^3$, $C_3^4:(C_2^2\times C_4)$, $C_2\times C_3^2:F_9$, $C_2\times C_3^2:F_9$, $S_3^3:C_6$, $C_3^3:\GL(2,3)$, $C_3^2:D_6^2$

Order 972: $C_3^3:S_3^2$ x 11, $C_3^3:S_3^2$ x 11, $C_3^3:S_3^2$ x 10, $C_3^4:D_6$ x 10, $C_3^4:D_6$ x 9, $C_3^3:S_3^2$ x 8, $C_3^3:S_3^2$ x 7, $C_3^3:C_6^2$ x 4, $C_3^3:C_6^2$ x 4, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 4, $C_3^3:C_6^2$ x 4, $C_3^3:S_3^2$ x 3, $C_3^3:S_3^2$ x 3, $C_3^3:S_3^2$ x 3, $C_3^4:D_6$ x 3, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $C_3^3:S_3^2$ x 2, $\He_3:S_3^2$ x 2, $C_3^3:S_3^2$, $C_3\wr A_4$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $\He_3\times S_3^2$, $C_3^4:D_6$, $C_2\times C_3^4:C_6$, $C_3^4:D_6$, $C_3^3:C_6^2$, $C_3^3.C_6^2$, $C_3^4:C_{12}$

Order 864: $C_2\times C_3^2:\GL(2,3)$ x 5, $C_6^2:\SL(2,3)$

Order 729: $C_3^5:C_3$ x 6, $C_3^5.C_3$ x 3, $C_3^5:C_3$ x 3, $C_3^4.C_3^2$ x 2, $C_3^4.C_3^2$ x 2, $C_3^3.C_3^3$ x 2, $C_3^4:C_3^2$ x 2, $C_3^4.C_3^2$, $C_3^4:C_9$, $C_3^6$, $C_3^4:C_3^2$, $C_3^3.C_3^3$

Order 648: $C_3^4:D_4$ x 18, $C_3:S_3^3$ x 13, $C_3:S_3^3$ x 13, $C_3.S_3^3$ x 12, $C_3^4:Q_8$ x 11, $C_3^2:F_9$ x 9, $C_3\times S_3^3$ x 8, $C_3:S_3^3$ x 7, $C_3^4:(C_2\times C_4)$ x 7, $C_3^2:F_9$ x 4, $S_3\wr C_3$ x 4, $C_3^4:C_2^3$ x 3, $C_2\times C_3^4:C_4$ x 2, $C_2\times C_3^2:S_3^2$ x 2, $C_2\times C_3^2:S_3^2$ x 2, $C_6\times C_3^2:D_6$ x 2, $S_3\times C_3^2:D_6$ x 2, $C_3^4:C_2^3$, $C_2\times C_3^3:A_4$, $C_3^3:\SL(2,3)$, $S_3\times C_3^2:D_6$

Order 576: $F_9:C_2^3$, $A_4\times \GL(2,3)$

Order 486: $C_3^4:C_6$ x 14, $C_3^4:C_6$ x 13, $C_3^4:S_3$ x 12, $C_3^4:C_6$ x 11, $C_3^4:C_6$ x 11, $C_3^4:S_3$ x 10, $C_3^4:S_3$ x 9, $C_3^4:C_6$ x 9, $C_3^4:S_3$ x 9, $C_3^4:C_6$ x 8, $C_3^4:S_3$ x 6, $C_3^4:C_6$ x 6, $C_3^4:C_6$ x 6, $C_3^4:S_3$ x 4, $C_3^4:C_6$ x 4, $C_3^4:C_6$ x 3, $C_3^3:(C_3\times C_6)$ x 2, $C_3^3:(C_3\times C_6)$ x 2, $C_3^4:C_6$ x 2, $C_3^4:C_6$, $S_3\times C_3^4$, $C_3^4:S_3$, $C_3^3.(C_3\times S_3)$, $C_3^3:(C_3\times S_3)$, $C_3^4:C_6$, $C_3^4:C_6$, $C_3^3:C_{18}$, $C_3^3.(C_3\times S_3)$, $C_3^3.(C_3\times S_3)$, $C_3\wr C_3:C_6$, $C_3^4.C_6$

Order 432: $C_3^2:\GL(2,3)$ x 6, $C_2\times S_3^3$ x 4, $C_2\times \PU(3,2)$ x 3, $F_9:C_6$, $C_6^2:D_6$

Order 324: $C_3^2:S_3^2$ x 51, $C_3\wr C_2^2$ x 28, $C_3^2:S_3^2$ x 24, $C_3^4:C_4$ x 20, $C_3^2:S_3^2$ x 20, $C_3^3:D_6$ x 19, $C_3^2:S_3^2$ x 17, $C_3^2:S_3^2$ x 17, $C_3^2:S_3^2$ x 16, $C_3^2\times S_3^2$ x 16, $C_3^2:S_3^2$ x 13, $C_3^2:S_3^2$ x 9, $C_3^3:D_6$ x 4, $C_3^3:D_6$ x 4, $C_3^2:C_6^2$ x 4, $C_3^2:C_6^2$ x 3, $C_3^3:A_4$ x 3, $\He_3:D_6$, $C_3^3:D_6$, $C_3^3:D_6$, $D_6\times \He_3$

Order 288: $F_9:C_2^2$ x 10, $A_4\times \SL(2,3)$, $C_6^2:Q_8$, $C_6^2:D_4$, $C_2^2\times F_9$

Order 243: $C_3^4:C_3$ x 22, $C_3^2\times \He_3$ x 10, $C_3^4:C_3$ x 10, $C_3^3:C_9$ x 8, $C_3^5$ x 6, $C_3^3:C_3^2$ x 6, $C_3^4.C_3$ x 3, $\He_3:C_3^2$ x 2, $C_3^2.C_3^3$ x 2, $C_9.C_3^3$, $C_3^3:C_3^2$

Order 216: $S_3^3$ x 37, $C_6.S_3^2$ x 11, $C_6:S_3^2$ x 10, $C_6\times S_3^2$ x 9, $\PU(3,2)$ x 5, $C_6:S_3^2$ x 4, $C_6^2:C_6$ x 2, $C_3^3:Q_8$, $S_3^2:C_6$, $C_3\times F_9$, $C_6^2:S_3$

Order 192: $C_2^2\times \GL(2,3)$, $A_4\times \SD_{16}$

Order 162: $C_3^2\wr C_2$ x 79, $C_3^3:S_3$ x 61, $C_3^3:C_6$ x 41, $C_3^3:S_3$ x 37, $C_3^3:C_6$ x 37, $S_3\times C_3^3$ x 21, $C_3^3:C_6$ x 17, $C_3^3:C_6$ x 15, $S_3\times \He_3$ x 13, $C_3^3:S_3$ x 8, $C_6\times \He_3$ x 4, $C_3^3.C_6$ x 3, $C_3^3\times C_6$, $\He_3.C_6$, $C_3^2:C_{18}$, $C_3^3:C_6$, $\He_3.C_6$, $\He_3.C_6$, $C_3\wr S_3$

Order 144: $F_9:C_2$ x 18, $S_3^2:C_2^2$ x 9, $C_2\times F_9$ x 7, $C_2\times \PSU(3,2)$ x 6, $D_6^2$ x 4, $C_6^2:C_4$, $A_4\times D_6$, $C_3\times \GL(2,3)$

Order 108: $C_3:S_3^2$ x 103, $C_3\times S_3^2$ x 73, $C_3^2:D_6$ x 28, $C_3:S_3^2$ x 25, $C_3^2:D_6$ x 17, $C_3^2\times D_6$ x 13, $C_3^2:D_6$ x 11, $C_3^2:D_6$ x 8, $C_3^2:D_6$ x 6, $C_3^2:C_{12}$, $C_2^2\times \He_3$

Order 96: $C_2\times \GL(2,3)$ x 2, $C_8\times A_4$, $Q_8:A_4$, $Q_8\times A_4$, $C_2^2\times \SL(2,3)$, $D_4\times A_4$

Order 81: $C_3^4$ x 63, $C_3\times \He_3$ x 63, $C_3\wr C_3$ x 19, $C_9:C_3^2$ x 5, $C_3^2:C_9$ x 4, $\He_3:C_3$ x 2, $C_3.\He_3$ x 2, $C_9.C_3^2$ x 2

Order 72: $S_3\times D_6$ x 41, $\SOPlus(4,2)$ x 18, $F_9$ x 13, $\PSU(3,2)$ x 11, $C_2\times C_3^2:C_4$ x 9, $C_6\times D_6$ x 6, $C_6:D_6$ x 4, $S_3\times A_4$ x 2, $C_6\times A_4$, $C_3\times \SL(2,3)$

Order 64: $C_2^2\times \SD_{16}$

Order 54: $C_3^2:S_3$ x 126, $S_3\times C_3^2$ x 113, $C_3^2:C_6$ x 94, $C_3^2:C_6$ x 53, $C_3^2:S_3$ x 21, $C_2\times \He_3$ x 10, $C_3^2\times C_6$ x 8, $S_3\times C_9$ x 3, $C_9:C_6$

Order 48: $C_2^2\times D_6$ x 2, $\GL(2,3)$ x 2, $C_2^2\times A_4$, $C_2\times \SL(2,3)$, $C_4\times A_4$, $C_3\times \SD_{16}$

Order 36: $S_3^2$ x 96, $C_6\times S_3$ x 52, $C_6:S_3$ x 32, $C_3^2:C_4$ x 20, $C_6^2$ x 5, $C_3\times A_4$

Order 32: $C_2\times \SD_{16}$ x 4, $C_2^2\times Q_8$, $C_2^2\times D_4$, $C_2^2\times C_8$

Order 27: $C_3^3$ x 140, $\He_3$ x 48, $C_9:C_3$ x 10, $C_3\times C_9$ x 5

Order 24: $C_2\times D_6$ x 19, $\SL(2,3)$ x 3, $C_2\times A_4$ x 3, $C_2^2\times C_6$ x 2, $C_{24}$, $C_3\times Q_8$, $C_3\times D_4$

Order 18: $C_3\times S_3$ x 113, $C_3:S_3$ x 88, $C_3\times C_6$ x 31, $C_{18}$

Order 16: $\SD_{16}$ x 5, $C_2\times D_4$ x 3, $C_2\times C_8$ x 2, $C_2\times Q_8$ x 2, $C_2^4$, $C_2^2\times C_4$

Order 12: $D_6$ x 44, $C_2\times C_6$ x 12, $A_4$ x 2, $C_{12}$

Order 9: $C_3^2$ x 84, $C_9$ x 3

Order 8: $C_2^3$ x 5, $D_4$ x 4, $Q_8$ x 3, $C_2\times C_4$ x 2, $C_8$ x 2

Order 6: $S_3$ x 28, $C_6$ x 21

Order 4: $C_2^2$ x 9, $C_4$ x 2

Order 3: $C_3$ x 12

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 419904: $C_3^6:(A_4\times \GL(2,3))$ ($C_1$)

Order 209952: $C_3^6:(A_4\times \SL(2,3))$ ($C_2$)

Order 139968: $C_3^6:(C_2^2\times \GL(2,3))$ ($C_3$)

Order 69984: $C_3^6:(C_2^2\times \SL(2,3))$ ($C_6$), $C_3^6:(Q_8\times A_4)$ ($S_3$)

Order 34992: $C_3^6.Q_8.S_3$ ($A_4$)

Order 23328: $C_3^6.(C_2^2\times Q_8)$ ($C_3\times S_3$)

Order 17496: $C_3^6.Q_8.C_3$ ($C_2\times A_4$), $C_3^6.(C_2\times A_4)$ ($S_4$)

Order 8748: $C_3^6.A_4$ ($\GL(2,3)$)

Order 5832: $(C_3:S_3)^3$ ($C_3\times S_4$), $C_3^6:Q_8$ ($S_3\times A_4$)

Order 2916: $C_3^5:D_6$ ($C_3\times \GL(2,3)$)

Order 1458: $C_3^5:S_3$ ($A_4\times S_4$)

Order 729: $C_3^6$ ($A_4\times \GL(2,3)$)

Order 1: $C_1$ ($C_3^6:(A_4\times \GL(2,3))$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 419904: $C_3^6:(A_4\times \GL(2,3))$ ($C_1$)

Order 209952: $C_3^6:(A_4\times \SL(2,3))$ ($C_2$)

Order 139968: $C_3^6:(C_2^2\times \GL(2,3))$ ($C_3$)

Order 69984: $C_3^6:(C_2^2\times \SL(2,3))$ ($C_6$), $C_3^6:(Q_8\times A_4)$ ($S_3$)

Order 34992: $C_3^6.Q_8.S_3$ ($A_4$)

Order 23328: $C_3^6.(C_2^2\times Q_8)$ ($C_3\times S_3$)

Order 17496: $C_3^6.Q_8.C_3$ ($C_2\times A_4$), $C_3^6.(C_2\times A_4)$ ($S_4$)

Order 8748: $C_3^6.A_4$ ($\GL(2,3)$)

Order 5832: $(C_3:S_3)^3$ ($C_3\times S_4$), $C_3^6:Q_8$ ($S_3\times A_4$)

Order 2916: $C_3^5:D_6$ ($C_3\times \GL(2,3)$)

Order 1458: $C_3^5:S_3$ ($A_4\times S_4$)

Order 729: $C_3^6$ ($A_4\times \GL(2,3)$)

Order 1: $C_1$ ($C_3^6:(A_4\times \GL(2,3))$)

Series

Derived series

$C_3^6:(A_4\times \GL(2,3))$

$\rhd$

$C_3^6:(C_2^2\times \SL(2,3))$

$\rhd$

$C_3^6:Q_8$

$\rhd$

$C_3^5:S_3$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Chief series

$C_3^6:(A_4\times \GL(2,3))$

$\rhd$

$C_3^6:(A_4\times \SL(2,3))$

$\rhd$

$C_3^6:(C_2^2\times \SL(2,3))$

$\rhd$

$C_3^6.Q_8.C_3$

$\rhd$

$C_3^6:Q_8$

$\rhd$

$C_3^5:S_3$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Lower central series

$C_3^6:(A_4\times \GL(2,3))$

$\rhd$

$C_3^6:(C_2^2\times \SL(2,3))$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 3 larger groups in the database.

This group is a maximal quotient of 2 larger groups in the database.

Character theory

Complex character table

See the $68 \times 68$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $53 \times 53$ rational character table.