Abstract group 559872.l: $(C_3:S_3)^3.\GL(2,\mathbb{Z}/4)$

• Label: 559872.l
• Order: \( 2^{8} \cdot 3^{7} \)
• Exponent: \( 2^{3} \cdot 3^{2} \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{9} \cdot 3^{7} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $27$
• Trans deg.: $27$
• Rank: $2$

Group information

Description:	$(C_3:S_3)^3.\GL(2,\mathbb{Z}/4)$
Order:	\(559872\)\(\medspace = 2^{8} \cdot 3^{7} \)
Exponent:	\(72\)\(\medspace = 2^{3} \cdot 3^{2} \)
Automorphism group:	$(C_3:S_3)^3.(D_4\times S_4)$, of order \(1119744\)\(\medspace = 2^{9} \cdot 3^{7} \)
Composition factors:	$C_2$ x 8, $C_3$ x 7
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	9	12	18	24
Elements	1	2727	3320	59832	103464	178848	20736	97632	62208	31104	559872
Conjugacy classes	1	5	7	11	19	10	2	12	2	4	73
Divisions	1	5	7	11	18	5	1	12	1	2	63
Autjugacy classes	1	5	6	11	14	9	1	8	1	2	58

Dimension	1	2	3	4	6	12	24	48	96	128	192	256	384
Irr. complex chars.	4	7	4	1	15	1	8	14	10	6	1	0	2	73
Irr. rational chars.	4	3	4	3	7	5	8	14	6	2	3	2	2	63

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:	${\langle a, b, c, d, e, f, g, h, i \mid b^{6}=c^{4}=d^{3}=e^{12}=f^{12}= \!\cdots\! \rangle}$
Permutation group:	Degree $27$ $\langle(1,7,6,4,8,2,9,5)(10,21,12,19,18,22,16,24)(11,23)(13,27,17,26,15,25,14,20) \!\cdots\! \rangle$
Transitive group:	27T1664	36T31617			more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$(C_3^5:D_6.A_4)$ $\,\rtimes\,$ $\SD_{16}$	$(C_3^6:(C_4^3.S_3))$ $\,\rtimes\,$ $C_2$	$(C_3^6:(C_4^3.S_3))$ $\,\rtimes\,$ $C_2$	$C_3^6$ $\,\rtimes\,$ $(C_2^3.\GL(2,\mathbb{Z}/4))$	more information
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Non-split product:	$(C_3^6.C_4^3)$ . $D_6$	$(C_3^6.(C_2^2\times D_4))$ . $S_4$	$(C_3^6.C_2^3.C_2^4)$ . $S_3$	$(C_3^6:C_4\wr C_3)$ . $C_2^2$	all 12

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

Homology

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

Subgroups

There are 8580210 subgroups in 5518 conjugacy classes, 19 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $(C_3:S_3)^3.\GL(2,\mathbb{Z}/4)$
Commutator:	$G' \simeq$ $C_3^6:C_4\wr C_3$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $(C_3:S_3)^3.\GL(2,\mathbb{Z}/4)$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^6$	$G/\operatorname{Fit} \simeq$ $C_2^3.\GL(2,\mathbb{Z}/4)$
Radical:	$R \simeq$ $(C_3:S_3)^3.\GL(2,\mathbb{Z}/4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_3^6$	$G/\operatorname{soc} \simeq$ $C_2^3.\GL(2,\mathbb{Z}/4)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:\SD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2\wr C_3$

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 559872: $(C_3:S_3)^3.\GL(2,\mathbb{Z}/4)$

Order 279936: $C_3^6:(C_4^3:C_6)$, $C_3^6:(C_4^3.S_3)$, $C_3^6:(C_4^3.S_3)$

Order 186624: $C_3^6.C_4^3.C_2^2$

Order 139968: $C_3^6:C_4\wr C_3$, $C_3^5:S_3.\GL(2,\mathbb{Z}/4)$, $(C_3:S_3)^3.S_4$, $C_3^6:C_2^4.A_4$

Order 93312: $C_3^6.C_4^3.C_2$, $C_3^6.C_2^2.C_2^4.C_2$, $C_3^6.C_4.C_4^2.C_2$, $C_3^6.C_4^2.C_2^3$, $C_3^6.C_2^2.C_2^4.C_2$, $C_3^6.C_2^3.C_2^4$, $C_3^6.C_4^2.C_2^3$

Order 69984: $C_3^6:(C_4^2:C_6)$, $C_3^6:(C_4^2:C_6)$, $(C_3:S_3)^3.D_6$, $C_3^6:(A_4:C_8)$, $C_3^6:(D_4\times A_4)$

Order 46656: $C_3^6.(C_4.D_8)$ x 2, $C_3^6.C_2^3.C_2^3$ x 2, $C_3^6.C_2.C_2^4.C_2$ x 2, $C_3^6.Q_8:D_4$ x 2, $C_3^5:D_6.\SD_{16}$ x 2, $C_3^6.(C_4.D_8)$, $C_3^6.C_4:\OD_{16}$, $C_3^6.C_4:D_4.C_2$, $C_3^6.(C_4.D_8)$, $C_3^6.C_4.C_4^2$, $C_3^6.(C_4.\OD_{16})$, $C_3^6.C_4:D_4.C_2$, $C_3^6.C_4:D_4.C_2$, $C_3^6.C_4:D_4.C_2$, $C_3^6.C_4:\SD_{16}$, $C_3^6.C_4:\SD_{16}$, $C_3^6.C_4^3$, $C_3^6.C_4:D_4.C_2$, $C_3^6.C_2^3.C_2^3$, $C_3^6.C_2^3.C_2^3$, $C_3^6.(C_4.\OD_{16})$, $C_3^4:(C_{12}^2:C_4)$

Order 34992: $C_3^5:S_3.S_4$, $C_3^6.(C_4\times A_4)$, $C_3^6.(C_2^2\times A_4)$, $C_3^2\wr C_3:\SD_{16}$, $C_3^5:D_6.A_4$

Order 23328: $C_3^6.C_4:D_4$ x 4, $C_3^5:C_{12}.D_4$ x 4, $C_3^5:C_6.D_8$ x 4, $C_3^6:(C_2\times \SD_{16})$ x 4, $C_3^6.C_4.C_2^3$ x 3, $C_3^6.C_4:D_4$ x 2, $C_3^6.C_4:D_4$ x 2, $C_3^6.D_4:C_4$ x 2, $C_3^6.C_4.C_2^3$ x 2, $C_3^6.C_2.C_2^4$ x 2, $C_3^6.D_4:C_4$ x 2, $C_3^6.C_4.C_2^3$ x 2, $C_3^5:C_{12}:C_8$ x 2, $C_3^6.(C_2^2\times D_4)$ x 2, $C_3^6.C_2^2:Q_8$, $C_3^6.C_4^2.C_2$, $C_3^6.C_4:C_8$, $C_3^6.C_4:C_8$, $C_3^6.C_4:Q_8$, $C_3^6.C_4^2.C_2$, $C_3^6.C_4:C_8$, $C_3^6.C_4^2.C_2$, $C_3^4.C_{12}^2.C_2$, $C_3^6.C_4:D_4$, $C_3^6.C_2^2:C_8$, $C_3^6.C_2^3.C_2^2$, $C_3^6.C_4:Q_8$, $C_3^6.C_2^3.C_2^2$, $C_3^6.C_4^2.C_2$, $C_3^6.C_4:C_8$, $C_3^6.C_4^2.C_2$, $C_3^6:(C_4\times Q_8)$, $C_3^6:(C_4^2:C_2)$, $C_3^6:(C_2\times \OD_{16})$, $C_3^2:S_3^2:F_9$, $C_3^4.C_6^2:Q_8$

Order 20736: $C_3^4.C_4^3.C_2^2$

Order 17496: $C_3^2\wr C_3:C_8$, $C_3^2\wr C_3:Q_8$, $C_3^6:(C_2\times A_4)$, $C_3^6.(C_2\times A_4)$, $C_3^2\wr C_3:D_4$

Order 15552: $C_3^4:C_4^2:D_6$

Order 11664: $C_3^6.(C_2\times D_4)$ x 6, $C_3^6.C_2^3.C_2$ x 4, $C_3^5.C_6.C_2^3$ x 4, $C_3^6:(C_2\times D_4)$ x 4, $C_3^6.\SD_{16}$ x 4, $C_3^6:(C_2\times D_4)$ x 3, $C_3^6:C_4^2$ x 3, $C_3^5.C_4.C_6.C_2$ x 2, $C_3^6.C_4.C_2^2$ x 2, $C_3^6.C_4^2$ x 2, $C_3^5.C_6.C_2^3$ x 2, $C_3^6.\SD_{16}$ x 2, $C_3^6.D_4.C_2$ x 2, $C_3^6.C_4.C_2^2$ x 2, $C_3^6.C_4^2$ x 2, $C_3^6:C_2^4$ x 2, $C_3^4.(C_2\times F_9)$ x 2, $C_3^6.C_4:C_4$ x 2, $C_3^6.C_2^3.C_2$, $C_3^6.C_2.C_2^3$, $C_3^6.D_4.C_2$, $C_3^5.D_6.C_2^2$, $C_3^6.C_4.C_2^2$, $C_3^6.C_4.C_2^2$, $C_3^6.C_2.C_2^3$, $C_3^6.C_4.C_2^2$, $C_3^6.C_4^2$, $C_3^6.C_4^2$, $C_3^6.C_4^2$, $C_3^6.C_4:C_4$, $C_3^6.C_4:C_4$, $C_3^5:C_6.D_4$, $C_3^5:C_{12}.C_4$, $C_3^6:(C_2^2\times C_4)$, $C_3^6:(C_2\times C_8)$, $C_3^5:C_{12}:C_4$, $C_3^6.(C_2\times Q_8)$, $C_3^6:\OD_{16}$, $C_3^6:\OD_{16}$

Order 10368: $C_3^4.C_4^2.C_2^3$, $C_3^4.(C_2\times C_4^2).C_2^2$, $C_3^4.C_4^2.C_2^3$, $C_3^4.(C_2\times C_4).C_2^4$, $C_3^4.C_2^3.C_2^4$, $C_3^4.(C_2\times C_4).C_2^4$, $C_3^4.C_4^3.C_2$

Order 8748: $C_3^6.A_4$, $C_3^2\wr C_3:C_4$, $C_3^2\wr C_3:C_2^2$, $C_3^6.C_{12}$

Order 7776: $C_3^4:C_{12}:D_4$ x 4, $C_3^4:(D_4\times D_6)$ x 4, $C_3^5.C_4.C_2^3$ x 2, $C_3^4:C_{12}:D_4$ x 2, $C_3^4:(D_4\times D_6)$ x 2, $S_3\times C_3^4:C_4^2$, $C_3^4:C_{12}:D_4$, $C_3^3:C_{12}:D_{12}$

Order 5832: $C_3^6:D_4$ x 5, $C_3^6:C_2^3$ x 4, $C_3^5:D_{12}$ x 4, $C_3^5:D_{12}$ x 4, $C_3^6:D_4$ x 4, $C_3^3:S_3^3$ x 4, $C_3^6.D_4$ x 3, $C_3^4.(C_3^2\times D_4)$ x 3, $C_3^6:(C_2\times C_4)$ x 3, $C_3^6.C_4.C_2$ x 2, $C_3^6:(C_2\times C_4)$ x 2, $C_3^6:Q_8$ x 2, $C_3^6:C_8$ x 2, $C_3^6:(C_2\times C_4)$ x 2, $C_3^6:C_2^3$ x 2, $C_3^5.D_6.C_2$, $C_3^6.C_4.C_2$, $C_3^4.(C_3^2\times Q_8)$, $C_3^6.C_2^2.C_2$, $C_3^6.Q_8$, $C_3^6.C_2^2.C_2$, $C_3^6.C_4.C_2$, $C_3^6.C_8$, $C_3^6.C_8$, $(C_3:S_3)^3$, $C_3^6:C_8$

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

Order 4374: $C_3^6.C_6$, $C_3\wr C_6$

Order 3888: $C_3\wr C_4:D_6$ x 8, $C_3^4:(S_3\times D_4)$ x 7, $C_3^4:(C_6\times D_4)$ x 4, $S_3\times C_3^4:D_4$ x 4, $C_3^3:(S_3\times D_{12})$ x 4, $S_3\times C_3^3:D_{12}$ x 4, $C_3\wr C_4:D_6$ x 4, $S_3\times C_3^4:D_4$ x 4, $C_3:S_3^4$ x 4, $C_3^4:(S_3\times D_4)$ x 4, $C_3^5:(C_2\times D_4)$ x 4, $C_3^4.C_{12}.C_2^2$ x 2, $C_3^4.C_{12}.C_2^2$ x 2, $C_3^5.C_4.C_2^2$ x 2, $C_3^4.C_{12}.C_2^2$ x 2, $C_3^5.\SD_{16}$ x 2, $C_3^4.D_6.C_2^2$ x 2, $C_3^5:C_4^2$ x 2, $C_3\wr C_4:D_6$ x 2, $C_3^4:(C_2\times D_{12})$ x 2, $C_3^4:(C_6\times D_4)$ x 2, $C_3^5.C_4^2$, $C_3^5:C_4^2$, $C_3^3:F_9:C_2$, $C_3^4:(C_4\times D_6)$

Order 2916: $C_3^4:C_6^2$ x 4, $C_3^5:D_6$ x 4, $C_3^6.C_4$ x 3, $C_3^5.D_6$ x 2, $C_3^5:C_{12}$ x 2, $C_3^6:C_4$ x 2, $C_3^5:C_{12}$ x 2, $C_3^5.D_6$, $C_3^6.C_4$, $C_3^6.C_4$, $C_3^5:D_6$, $C_3^4:C_6^2$, $C_3^5:(C_2\times C_6)$

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

Order 2304: $C_3^2.C_4^3.C_2^2$

Order 2187: $C_3^2\wr C_3$

Order 1944: $C_3^2:S_3^3$ x 12, $C_3^5:D_4$ x 12, $C_3^2:S_3^3$ x 8, $C_3^2:S_3^3$ x 8, $C_3^5:D_4$ x 8, $C_3^4:D_{12}$ x 8, $C_3^4:D_{12}$ x 7, $C_3^4:D_{12}$ x 7, $C_3^2:S_3^3$ x 5, $C_3^4:(C_4\times S_3)$ x 5, $C_3^2:S_3^3$ x 4, $C_3^4:D_{12}$ x 4, $C_3^5:D_4$ x 4, $C_3^4:D_{12}$ x 4, $C_3^5:D_4$ x 4, $C_3\wr C_4:S_3$ x 4, $C_3^5:Q_8$ x 2, $C_3^5:Q_8$ x 2, $C_3^5:Q_8$ x 2, $C_3^4:(C_4\times S_3)$ x 2, $C_3^4:C_4\times S_3$ x 2, $C_3\wr C_4:S_3$ x 2, $C_3^4:C_4\times S_3$ x 2, $C_3^3:(S_3\times C_{12})$ x 2, $S_3\times C_3^3:C_{12}$ x 2, $C_3^4:C_{24}$ x 2, $C_3^4:C_{24}$ x 2, $C_3^5:(C_2\times C_4)$, $C_3^4:(C_2\times C_{12})$, $C_3^3:\PSU(3,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:F_9$

Order 1728: $(C_2\times S_3^3).C_2^2$ x 2

Order 1458: $C_3^3\wr C_2$ x 2, $C_3^5:C_6$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5.C_6$, $C_3^5:C_6$, $C_3^5:C_6$, $C_3^5:C_6$

Order 1296: $C_3^3:(S_3\times D_4)$ x 49, $C_3^4:(C_2\times D_4)$ x 22, $C_3^4:C_4^2$ x 15, $C_3^4:\SD_{16}$ x 12, $C_2\times C_3^4:D_4$ x 10, $C_2\times C_3^4:D_4$ x 8, $C_6\times C_3^2:D_{12}$ x 8, $S_3^3:S_3$ x 8, $C_3^4:\SD_{16}$ x 8, $C_3^2\wr C_2.Q_8$ x 6, $C_3^3:(S_3\times Q_8)$ x 5, $C_3^2:(C_2\times F_9)$ x 5, $C_3:(S_3\times F_9)$ x 5, $C_6:S_3^3$ x 4, $S_3^4$ x 4, $S_3\times C_3^2:D_{12}$ x 4, $S_3^2:S_3^2$ x 4, $S_3^3:C_6$ x 4, $C_3^4:\SD_{16}$ x 4, $C_3\times C_{12}:S_3^2$ x 4, $C_6.S_3^3$ x 4, $C_2\times C_3^2:F_9$ x 3, $(C_3\times S_3^2):D_6$ x 3, $C_3^2:C_4\times S_3^2$ x 3, $C_3^2\wr C_2.D_4$ x 3, $(C_3\times C_{12}):S_3^2$ x 3, $C_6.S_3^3$ x 3, $(C_3^3\times C_6).D_4$ x 3, $C_3^4:C_4^2$ x 3, $C_3^4:(C_2^2\times C_4)$ x 2, $C_3^3:(C_4\times D_6)$ x 2, $C_3^4:(C_2\times Q_8)$ 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, $C_3^4:\SD_{16}$ x 2, $C_3^4:\OD_{16}$ x 2, $C_3^2:(C_2\times F_9)$ x 2, $(C_3^2\times S_3^2):C_4$ x 2, $C_3^4:C_4^2$ x 2, $C_3^4:\SD_{16}$ x 2, $C_3^4:C_4^2$ x 2, $C_3^4:\SD_{16}$ x 2, $C_2\times C_3^4:Q_8$, $C_3^3:(S_3\times Q_8)$, $C_3:S_3\times F_9$, $C_3^2\wr C_2.D_4$, $S_3^3:C_6$, $C_4\times C_3^2:S_3^2$, $C_3^3:(S_3\times Q_8)$, $(C_3^3\times C_{12}):C_4$, $C_3^4:(C_2\times C_8)$, $C_3^4:C_4^2$, $C_3^2:C_{12}^2$, $(C_3^3\times C_6).Q_8$, $(C_3^3\times C_6).D_4$, $C_3^3:C_{12}:C_4$, $C_3^3:(S_3\times C_8)$

Order 1152: $C_3:S_3.C_4:D_4.C_2$ x 2, $C_4:F_9.C_2^2$ x 2, $C_{12}^2:D_4$, $(C_4\times C_3:S_3).C_4^2$, $C_3:S_3:\OD_{16}.C_2^2$

Order 972: $C_3^3:S_3^2$ x 24, $C_3^3:S_3^2$ x 15, $C_3^3:C_6^2$ x 12, $C_3^3:C_6^2$ x 6, $C_3^3:S_3^2$ x 5, $C_3^3:S_3^2$ x 5, $C_3^5:C_4$ x 5, $C_3^4:C_{12}$ x 5, $C_3^4:C_{12}$ x 4, $C_3^4:C_{12}$ x 4, $C_3^5:C_4$ x 2, $C_3^4:C_{12}$ x 2, $C_3^3:S_3^2$, $\He_3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:C_3:C_4$, $C_3^4:C_{12}$

Order 864: $S_3^3:C_2^2$ x 8, $(S_3\times D_6).D_6$ x 8, $C_4:S_3^3$ x 4, $C_{12}:\SOPlus(4,2)$ x 4, $F_9:D_6$ x 2, $C_{12}:\SOPlus(4,2)$ x 2, $C_{12}:\SOPlus(4,2)$ x 2, $S_3\times C_3^2:C_4^2$ x 2, $C_4.S_3^3$ x 2

Order 768: $C_2^3.\GL(2,\mathbb{Z}/4)$

Order 729: $C_3^6$, $C_3^5.C_3$, $C_3^5:C_3$

Order 648: $C_3^4:D_4$ x 40, $C_3^3:D_{12}$ x 38, $C_3:S_3^3$ x 36, $C_3^4:D_4$ x 33, $C_3^3:(C_4\times S_3)$ x 26, $C_3:S_3^3$ x 19, $C_3^4:(C_2\times C_4)$ x 15, $C_3:S_3^3$ x 12, $C_3\times S_3^3$ x 12, $C_3^4:Q_8$ x 10, $C_3^3:(C_4\times S_3)$ x 9, $C_3^2:F_9$ x 9, $C_3^4:C_2^3$ x 8, $C_3^3:D_{12}$ x 8, $C_3\wr D_4$ x 8, $C_3^4:Q_8$ x 6, $C_3^4:Q_8$ x 6, $C_3^2:F_9$ x 6, $C_3^3:C_{24}$ x 6, $C_3^4:C_2^3$ x 5, $C_2\times C_3^4:C_4$ x 5, $C_3^4:D_4$ x 4, $C_3^3:C_4\times S_3$ x 4, $S_3\times C_3^2:C_{12}$ x 4, $C_3^3:D_{12}$ x 4, $C_3^3:D_{12}$ x 3, $C_3^4:D_4$ x 3, $C_3^3:D_{12}$ x 3, $C_3^4:D_4$ x 3, $C_2\times C_3^4:C_4$ x 2, $C_6\times C_3^2:C_{12}$ x 2, $C_3^4:Q_8$ x 2, $C_3^2:F_9$ x 2, $S_3\wr C_3$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3^4:Q_8$, $C_3^2:F_9$, $C_3^2\times F_9$, $C_2\times C_3^3:A_4$, $C_4\times C_3^3:S_3$, $C_3^2\wr C_2\times C_4$, $C_3^4:Q_8$, $C_3^4:Q_8$, $C_3^3:(C_4\times S_3)$, $C_3^4:C_8$, $C_3^3:C_{24}$

Order 576: $D_6^2.C_2^2$ x 7, $(C_3\times C_{12}).\SD_{16}$ x 5, $C_6^2:\SD_{16}$ x 4, $\PSU(3,2):D_4$ x 4, $F_9:D_4$ x 3, $(C_6\times C_{12}):C_8$ x 2, $(C_4\times F_9):C_2$ x 2, $S_3^2:\SD_{16}$ x 2, $(C_6\times C_{12}):Q_8$, $C_4^2:S_3^2$, $C_{12}^2:C_4$, $(C_6\times C_{12}).Q_8$, $Q_8:\SOPlus(4,2)$, $C_8:\SOPlus(4,2)$, $C_{12}^2:C_4$, $(C_6\times C_{12}).Q_8$

Order 486: $C_3^4:C_6$ x 15, $C_3^4:C_6$ x 15, $C_3^4:S_3$ x 6, $C_3^4:C_6$ x 2, $C_3^4:C_6$, $S_3\times C_3^4$, $C_3^3:C_{18}$, $C_3^4:C_6$, $C_3^4:C_6$, $C_3^4:C_6$, $C_3^4.C_6$, $C_3^4:C_6$

Order 432: $S_3^3:C_2$ x 39, $S_3^2:D_6$ x 16, $C_2\times S_3^3$ x 10, $C_2\times C_3^2:D_{12}$ x 8, $C_6\times \SOPlus(4,2)$ x 8, $D_6:S_3^2$ x 8, $C_{12}:S_3^2$ x 7, $C_3^2:C_4\times D_6$ x 4, $C_{12}:S_3^2$ x 4, $C_{12}:S_3^2$ x 4, $C_{12}:S_3^2$ x 4, $D_6:S_3^2$ x 4, $D_6:S_3^2$ x 4, $C_3^3:C_4^2$ x 4, $C_3^3:\SD_{16}$ x 3, $S_3\times \PSU(3,2)$ x 2, $F_9:S_3$ x 2, $C_3^3:\SD_{16}$ x 2, $F_9:C_6$ x 2, $S_3\times F_9$ x 2, $C_{12}:S_3^2$ x 2, $C_{12}.S_3^2$ x 2, $C_3^3:C_4^2$ x 2, $C_3^3:C_4^2$ x 2, $C_{12}.S_3^2$ x 2, $C_{12}.S_3^2$ x 2, $C_{12}.S_3^2$ x 2, $C_{12}.S_3^2$ x 2, $C_{12}.S_3^2$ x 2

Order 384: $C_4^3:C_6$, $C_4^3.S_3$, $C_4^3.S_3$

Order 324: $C_3^2:S_3^2$ x 70, $C_3\wr C_2^2$ x 68, $C_3^4:C_4$ x 36, $C_3^2:S_3^2$ x 32, $C_3^4:C_4$ x 22, $C_3^3:C_{12}$ x 22, $C_3^2:S_3^2$ x 19, $C_3^2\times S_3^2$ x 19, $C_3^2:S_3^2$ x 18, $C_3\wr C_4$ x 12, $C_3^2:C_6^2$ x 5, $C_3^3:C_{12}$ x 2, $\He_3:D_6$, $C_3^3:D_6$, $C_3^3:A_4$, $C_3^3\times C_{12}$, $C_3^4:C_4$, $C_3^2:S_3^2$

Order 288: $C_4:\SOPlus(4,2)$ x 32, $F_9:C_2^2$ x 12, $C_6^2:D_4$ x 10, $\SOPlus(4,2):C_4$ x 8, $D_{12}:D_6$ x 6, $C_4:F_9$ x 5, $C_2\times C_3^2:C_4^2$ x 4, $C_4.\SOPlus(4,2)$ x 4, $C_4.\SOPlus(4,2)$ x 4, $D_4\times S_3^2$ x 3, $C_4:\PSU(3,2)$ x 3, $C_4\times F_9$ x 3, $C_6^2.C_2^3$ x 3, $C_4.\PSU(3,2)$ x 3, $C_6^2:Q_8$ x 2, $C_4\times \PSU(3,2)$ x 2, $S_3^2:Q_8$ x 2, $C_6^2:C_8$ x 2, $Q_8:S_3^2$ x 2, $D_4.S_3^2$ x 2, $C_{24}:D_6$ x 2, $(D_4\times C_3^2):C_4$ x 2, $S_3^2:C_8$ x 2, $C_3^2:C_4\times Q_8$, $(C_6\times C_{12}):C_4$, $(C_6\times C_{12}).C_4$, $C_4.\PSU(3,2)$, $C_4.\SOPlus(4,2)$, $C_{12}^2:C_2$, $C_{12}:D_{12}$, $C_3^2:C_4\times C_8$, $C_4.\SOPlus(4,2)$

Order 256: $C_4^2:\SD_{16}$

Order 243: $C_3^5$ x 6, $C_3^4:C_3$ x 2, $C_3^3:C_9$ x 2, $C_3^4.C_3$, $C_3^4:C_3$

Order 216: $S_3^3$ x 64, $S_3^2:S_3$ x 44, $C_3^2:C_4\times S_3$ x 25, $S_3^2:C_6$ x 24, $C_3^2:D_{12}$ x 23, $C_6\times S_3^2$ x 20, $C_6:S_3^2$ x 18, $C_3^3:D_4$ x 12, $C_6:S_3^2$ x 10, $C_3^2:D_{12}$ x 8, $C_3^2:D_{12}$ x 7, $C_3^2\times D_{12}$ x 7, $C_3^3:Q_8$ x 5, $C_4\times C_3^2:S_3$ x 5, $C_2\times C_3^3:C_4$ x 4, $C_2\times C_3^2:C_{12}$ x 4, $C_3^2:D_{12}$ x 4, $C_6.S_3^2$ x 4, $C_3^3:D_4$ x 4, $C_3:F_9$ x 3, $C_3^2:C_{24}$ x 2, $C_3^2:C_{24}$ x 2, $C_3^3:Q_8$ x 2, $C_3\times F_9$ x 2, $C_3:S_3\times C_{12}$ x 2, $C_3^3:Q_8$ x 2, $C_6.C_6^2$ x 2, $C_3^3:Q_8$ x 2, $C_3^3:Q_8$ x 2, $C_6.S_3^2$ x 2, $C_6.S_3^2$ x 2

Order 192: $A_4:\SD_{16}$, $C_4\wr C_3$, $C_2^3.S_4$, $C_4^2:D_6$, $C_2^4.A_4$

Order 162: $C_3^2\wr C_2$ x 110, $C_3^3:S_3$ x 64, $C_3^3:C_6$ x 18, $S_3\times C_3^3$ x 18, $C_3^3:C_6$ x 6, $C_3^3\times C_6$, $C_3^3:C_6$, $C_3^2:C_{18}$, $C_3^3.C_6$, $S_3\times \He_3$, $C_3^3:C_6$, $C_3^3:C_6$

Order 144: $S_3^2:C_2^2$ x 62, $C_{12}:D_6$ x 37, $C_3^2:C_4^2$ x 21, $F_9:C_2$ x 20, $D_6^2$ x 8, $D_6:D_6$ x 8, $C_3^2:\SD_{16}$ x 8, $C_2\times F_9$ x 7, $(C_3\times C_{12}):C_4$ x 7, $C_6\times D_{12}$ x 6, $D_6:D_6$ x 6, $C_{12}.D_6$ x 5, $C_{12}.D_6$ x 5, $C_3^2:(C_2\times C_8)$ x 5, $C_2.\PSU(3,2)$ x 5, $C_2\times \PSU(3,2)$ x 4, $C_{12}:D_6$ x 4, $D_6:D_6$ x 4, $S_3\times D_{12}$ x 4, $C_2.\SOPlus(4,2)$ x 4, $C_6^2:C_4$ x 3, $C_{12}:D_6$ x 3, $C_4\times S_3^2$ x 3, $C_3^2:C_4^2$ x 2, $C_{12}.D_6$ x 2, $C_{12}.D_6$ x 2, $C_{24}:C_6$ x 2, $C_6.D_{12}$ x 2, $C_6.D_{12}$ x 2, $C_{12}.D_6$ x 2, $C_{12}:D_6$ x 2, $S_3^2:C_4$ x 2, $C_{24}:S_3$, $C_{12}.D_6$, $C_6^2:C_4$, $C_3^2:\OD_{16}$, $C_3^2:\OD_{16}$, $C_{12}^2$

Order 128: $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2:C_8$, $C_4^2:Q_8$, $C_4^3:C_2$

Order 108: $C_3:S_3^2$ x 131, $C_3\times S_3^2$ x 97, $C_3:S_3^2$ x 42, $C_3^2:D_6$ x 18, $C_3^2\times D_6$ x 18, $C_3^3:C_4$ x 18, $C_3^2:C_{12}$ x 18, $C_3^2:D_6$ x 7, $C_3^2\times C_{12}$ x 5, $C_3^3:C_4$ x 5, $C_3^2:C_{12}$ x 4, $C_3^2:C_{12}$ x 4, $C_3^2:D_6$

Order 96: $D_4\times D_6$ x 6, $C_{12}:D_4$ x 6, $S_3\times \SD_{16}$ x 2, $C_4:D_{12}$, $S_3\times C_4^2$, $C_4^2:C_6$, $C_4^2:C_6$, $A_4:C_8$, $D_4\times A_4$, $A_4:Q_8$, $C_{12}:D_4$

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

Order 72: $S_3\times D_6$ x 75, $\SOPlus(4,2)$ x 59, $D_6:S_3$ x 32, $C_3\times D_{12}$ x 30, $C_2\times C_3^2:C_4$ x 29, $C_{12}:S_3$ x 22, $F_9$ x 13, $C_6\times D_6$ x 12, $\PSU(3,2)$ x 12, $C_6.D_6$ x 9, $C_6\wr C_2$ x 8, $C_3:D_{12}$ x 8, $C_3^2:Q_8$ x 6, $C_3^2:Q_8$ x 6, $C_3^2:C_8$ x 6, $C_3:C_{24}$ x 6, $C_6:D_6$ x 5, $C_6^2:C_2$ x 4, $S_3\times C_{12}$ x 4, $C_6.D_6$ x 4, $D_4\times C_3^2$ x 3, $C_3:D_{12}$ x 3, $C_6\times C_{12}$ x 2, $C_6.D_6$ x 2, $C_3^2:Q_8$ x 2, $Q_8\times C_3^2$, $C_3\times C_{24}$, $C_3^2:C_8$

Order 64: $C_4^2:C_2^2$ x 4, $Q_8:D_4$ x 4, $C_2^2:\SD_{16}$ x 4, $C_4.D_8$ x 4, $C_4^2:C_4$ x 2, $C_8:D_4$ x 2, $C_4.D_8$ x 2, $C_4^2.C_4$ x 2, $C_4^3$, $C_4^2.C_2^2$, $C_4:\OD_{16}$

Order 54: $C_3^2:S_3$ x 137, $C_3^2:C_6$ x 108, $S_3\times C_3^2$ x 108, $C_3^2\times C_6$ x 7, $C_3^2:C_6$ x 3, $S_3\times C_9$, $C_9:C_6$, $C_2\times \He_3$

Order 48: $S_3\times D_4$ x 27, $C_6:D_4$ x 12, $C_6\times D_4$ x 6, $C_2\times D_{12}$ x 6, $C_2^2\times D_6$ x 4, $C_4\times D_6$ x 3, $C_3:\SD_{16}$ x 3, $C_{12}:C_4$ x 3, $C_{24}:C_2$ x 2, $S_3\times Q_8$ x 2, $S_3\times C_8$ x 2, $C_3\times \SD_{16}$ x 2, $Q_8:S_3$ x 2, $C_2^2\times A_4$, $C_4\times A_4$, $A_4:C_4$, $C_4^2:C_3$, $C_4\times C_{12}$

Order 36: $S_3^2$ x 120, $C_6\times S_3$ x 73, $C_3^2:C_4$ x 48, $C_6:S_3$ x 35, $C_3\times C_{12}$ x 18, $C_3^2:C_4$ x 18, $C_3:C_{12}$ x 12, $C_6^2$ x 5

Order 32: $C_4:D_4$ x 14, $D_4:C_4$ x 8, $C_2\times \SD_{16}$ x 8, $C_2^2\times D_4$ x 7, $C_4:C_8$ x 6, $C_2\times C_4^2$ x 3, $C_2^2:C_8$ x 2, $C_4:Q_8$ x 2, $C_4\times C_8$ x 2, $C_2^2:Q_8$ x 2, $C_4\times Q_8$ x 2, $C_2\times \OD_{16}$, $C_4.Q_8$, $C_4^2:C_2$

Order 27: $C_3^3$ x 131, $C_9:C_3$ x 2, $\He_3$ x 2, $C_3\times C_9$

Order 24: $C_2\times D_6$ x 33, $C_3:D_4$ x 28, $C_4\times S_3$ x 17, $C_3\times D_4$ x 16, $D_{12}$ x 15, $C_3:Q_8$ x 5, $C_2^2\times C_6$ x 4, $C_2\times C_{12}$ x 3, $C_6:C_4$ x 3, $C_3:C_8$ x 3, $C_{24}$ x 2, $C_2\times A_4$ x 2, $C_3\times Q_8$ x 2

Order 18: $C_3\times S_3$ x 100, $C_3:S_3$ x 88, $C_3\times C_6$ x 26, $C_{18}$

Order 16: $C_2\times D_4$ x 32, $C_4^2$ x 10, $\SD_{16}$ x 8, $C_4:C_4$ x 6, $C_2\times C_8$ x 5, $C_2^2\times C_4$ x 3, $\OD_{16}$ x 2, $C_2^4$ x 2, $C_2\times Q_8$ x 2, $C_2^2:C_4$

Order 12: $D_6$ x 51, $C_2\times C_6$ x 18, $C_{12}$ x 12, $C_3:C_4$ x 12, $A_4$

Order 9: $C_3^2$ x 64, $C_9$

Order 8: $D_4$ x 23, $C_2\times C_4$ x 14, $C_2^3$ x 11, $C_8$ x 5, $Q_8$ x 4

Order 6: $S_3$ x 22, $C_6$ x 18

Order 4: $C_2^2$ x 13, $C_4$ x 11

Order 3: $C_3$ x 7

Order 2: $C_2$ x 5

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 559872: $(C_3:S_3)^3.\GL(2,\mathbb{Z}/4)$

Order 279936: $C_3^6:(C_4^3:C_6)$, $C_3^6:(C_4^3.S_3)$, $C_3^6:(C_4^3.S_3)$

Order 186624: $C_3^6.C_4^3.C_2^2$

Order 139968: $C_3^6:C_4\wr C_3$, $C_3^5:S_3.\GL(2,\mathbb{Z}/4)$, $(C_3:S_3)^3.S_4$, $C_3^6:C_2^4.A_4$

Order 93312: $C_3^6.C_4^3.C_2$, $C_3^6.C_2^2.C_2^4.C_2$, $C_3^6.C_4.C_4^2.C_2$, $C_3^6.C_4^2.C_2^3$, $C_3^6.C_2^2.C_2^4.C_2$, $C_3^6.C_2^3.C_2^4$, $C_3^6.C_4^2.C_2^3$

Order 69984: $C_3^6:(C_4^2:C_6)$, $C_3^6:(C_4^2:C_6)$, $(C_3:S_3)^3.D_6$, $C_3^6:(A_4:C_8)$, $C_3^6:(D_4\times A_4)$

Order 46656: $C_3^6.(C_4.D_8)$ x 2, $C_3^6.C_2^3.C_2^3$ x 2, $C_3^6.Q_8:D_4$ x 2, $C_3^5:D_6.\SD_{16}$ x 2, $C_3^6.(C_4.D_8)$, $C_3^6.C_4:\OD_{16}$, $C_3^6.C_4:D_4.C_2$, $C_3^6.(C_4.D_8)$, $C_3^6.C_4.C_4^2$, $C_3^6.(C_4.\OD_{16})$, $C_3^6.C_4:D_4.C_2$, $C_3^6.C_4:D_4.C_2$, $C_3^6.C_4:D_4.C_2$, $C_3^6.C_4:\SD_{16}$, $C_3^6.C_4:\SD_{16}$, $C_3^6.C_4^3$, $C_3^6.C_2.C_2^4.C_2$, $C_3^6.C_4:D_4.C_2$, $C_3^6.C_2^3.C_2^3$, $C_3^6.C_2^3.C_2^3$, $C_3^6.(C_4.\OD_{16})$, $C_3^4:(C_{12}^2:C_4)$

Order 34992: $C_3^5:S_3.S_4$, $C_3^6.(C_4\times A_4)$, $C_3^6.(C_2^2\times A_4)$, $C_3^2\wr C_3:\SD_{16}$, $C_3^5:D_6.A_4$

Order 23328: $C_3^6.C_4:D_4$ x 4, $C_3^5:C_{12}.D_4$ x 4, $C_3^5:C_6.D_8$ x 4, $C_3^6:(C_2\times \SD_{16})$ x 4, $C_3^6.C_4:D_4$ x 2, $C_3^6.C_4.C_2^3$ x 2, $C_3^6.C_2.C_2^4$ x 2, $C_3^5:C_{12}:C_8$ x 2, $C_3^6.(C_2^2\times D_4)$ x 2, $C_3^6.C_2^2:Q_8$, $C_3^6.C_4^2.C_2$, $C_3^6.C_4:D_4$, $C_3^6.C_4:C_8$, $C_3^6.C_4:C_8$, $C_3^6.C_4:Q_8$, $C_3^6.C_4^2.C_2$, $C_3^6.D_4:C_4$, $C_3^6.C_4:C_8$, $C_3^6.C_4.C_2^3$, $C_3^6.C_4^2.C_2$, $C_3^4.C_{12}^2.C_2$, $C_3^6.C_4:D_4$, $C_3^6.C_2^2:C_8$, $C_3^6.C_2^3.C_2^2$, $C_3^6.C_4:Q_8$, $C_3^6.D_4:C_4$, $C_3^6.C_4.C_2^3$, $C_3^6.C_2^3.C_2^2$, $C_3^6.C_4^2.C_2$, $C_3^6.C_4:C_8$, $C_3^6.C_4^2.C_2$, $C_3^6:(C_4\times Q_8)$, $C_3^6:(C_4^2:C_2)$, $C_3^6:(C_2\times \OD_{16})$, $C_3^2:S_3^2:F_9$, $C_3^4.C_6^2:Q_8$

Order 20736: $C_3^4.C_4^3.C_2^2$

Order 17496: $C_3^2\wr C_3:C_8$, $C_3^2\wr C_3:Q_8$, $C_3^6:(C_2\times A_4)$, $C_3^6.(C_2\times A_4)$, $C_3^2\wr C_3:D_4$

Order 15552: $C_3^4:C_4^2:D_6$

Order 11664: $C_3^6.(C_2\times D_4)$ x 6, $C_3^6:(C_2\times D_4)$ x 4, $C_3^6.\SD_{16}$ x 4, $C_3^6:C_4^2$ x 3, $C_3^6.C_4^2$ x 2, $C_3^5.C_6.C_2^3$ x 2, $C_3^6.C_2^3.C_2$ x 2, $C_3^6.D_4.C_2$ x 2, $C_3^6.C_4.C_2^2$ x 2, $C_3^5.C_6.C_2^3$ x 2, $C_3^6.C_4^2$ x 2, $C_3^6:(C_2\times D_4)$ x 2, $C_3^6:C_2^4$ x 2, $C_3^4.(C_2\times F_9)$ x 2, $C_3^6.C_4:C_4$ x 2, $C_3^6.C_2^3.C_2$, $C_3^6.C_2.C_2^3$, $C_3^5.C_4.C_6.C_2$, $C_3^6.D_4.C_2$, $C_3^6.C_4.C_2^2$, $C_3^5.D_6.C_2^2$, $C_3^6.C_4.C_2^2$, $C_3^6.C_4.C_2^2$, $C_3^6.\SD_{16}$, $C_3^6.C_2.C_2^3$, $C_3^6.C_4.C_2^2$, $C_3^6.C_4^2$, $C_3^6.C_4^2$, $C_3^6.C_4^2$, $C_3^6.C_4:C_4$, $C_3^6.C_4:C_4$, $C_3^5:C_6.D_4$, $C_3^5:C_{12}.C_4$, $C_3^6:(C_2^2\times C_4)$, $C_3^6:(C_2\times C_8)$, $C_3^5:C_{12}:C_4$, $C_3^6.(C_2\times Q_8)$, $C_3^6:\OD_{16}$, $C_3^6:\OD_{16}$

Order 10368: $C_3^4.C_4^2.C_2^3$, $C_3^4.(C_2\times C_4^2).C_2^2$, $C_3^4.C_4^2.C_2^3$, $C_3^4.(C_2\times C_4).C_2^4$, $C_3^4.C_2^3.C_2^4$, $C_3^4.(C_2\times C_4).C_2^4$, $C_3^4.C_4^3.C_2$

Order 8748: $C_3^6.A_4$, $C_3^2\wr C_3:C_4$, $C_3^2\wr C_3:C_2^2$, $C_3^6.C_{12}$

Order 7776: $C_3^4:C_{12}:D_4$ x 2, $C_3^4:C_{12}:D_4$ x 2, $C_3^4:(D_4\times D_6)$ x 2, $C_3^4:(D_4\times D_6)$ x 2, $C_3^5.C_4.C_2^3$, $S_3\times C_3^4:C_4^2$, $C_3^4:C_{12}:D_4$, $C_3^3:C_{12}:D_{12}$

Order 5832: $C_3^6:D_4$ x 5, $C_3^5:D_{12}$ x 4, $C_3^6:D_4$ x 4, $C_3^6:(C_2\times C_4)$ x 3, $C_3^6:C_2^3$ x 3, $C_3^3:S_3^3$ x 3, $C_3^6.C_4.C_2$ x 2, $C_3^6.D_4$ x 2, $C_3^4.(C_3^2\times D_4)$ x 2, $C_3^6:(C_2\times C_4)$ x 2, $C_3^6:Q_8$ x 2, $C_3^6:C_8$ x 2, $C_3^6:(C_2\times C_4)$ x 2, $C_3^5:D_{12}$ x 2, $C_3^6:C_2^3$ x 2, $C_3^5.D_6.C_2$, $C_3^6.C_4.C_2$, $C_3^4.(C_3^2\times Q_8)$, $C_3^6.C_2^2.C_2$, $C_3^6.Q_8$, $C_3^6.C_2^2.C_2$, $C_3^6.C_4.C_2$, $C_3^6.C_8$, $C_3^6.C_8$, $(C_3:S_3)^3$, $C_3^6:C_8$

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

Order 4374: $C_3^6.C_6$, $C_3\wr C_6$

Order 3888: $C_3^4:(S_3\times D_4)$ x 4, $C_3\wr C_4:D_6$ x 4, $S_3\times C_3^4:D_4$ x 4, $C_3^4:(S_3\times D_4)$ x 4, $C_3:S_3^4$ x 3, $C_3^5:(C_2\times D_4)$ x 3, $C_3^4:(C_6\times D_4)$ x 2, $S_3\times C_3^4:D_4$ x 2, $C_3^3:(S_3\times D_{12})$ x 2, $S_3\times C_3^3:D_{12}$ x 2, $C_3\wr C_4:D_6$ x 2, $C_3^4:(C_2\times D_{12})$ x 2, $C_3^4:(C_6\times D_4)$ x 2, $C_3^4.C_{12}.C_2^2$, $C_3^4.C_{12}.C_2^2$, $C_3^5.C_4.C_2^2$, $C_3^4.C_{12}.C_2^2$, $C_3^5.\SD_{16}$, $C_3^4.D_6.C_2^2$, $C_3^5:C_4^2$, $C_3^5.C_4^2$, $C_3^5:C_4^2$, $C_3\wr C_4:D_6$, $C_3^3:F_9:C_2$, $C_3^4:(C_4\times D_6)$

Order 2916: $C_3^6.C_4$ x 3, $C_3^4:C_6^2$ x 3, $C_3^5:D_6$ x 3, $C_3^5.D_6$ x 2, $C_3^5:C_{12}$ x 2, $C_3^6:C_4$ x 2, $C_3^5:C_{12}$ x 2, $C_3^5.D_6$, $C_3^6.C_4$, $C_3^6.C_4$, $C_3^5:D_6$, $C_3^4:C_6^2$, $C_3^5:(C_2\times C_6)$

Order 2592: $C_3^3:C_{12}:D_4$ x 18, $C_3^4:(C_2^2\times D_4)$ x 5, $C_3^4.C_4:D_4$ x 4, $C_3^3:(D_4\times D_6)$ x 4, $(C_3^3\times C_6).D_8$ x 4, $C_2\times C_3^4:\SD_{16}$ x 4, $C_3^3:C_{12}:D_4$ x 4, $C_3^4:(C_2\times \SD_{16})$ x 4, $C_2\times C_3^4:C_4^2$ x 3, $(C_3^3\times C_6).\OD_{16}$ x 3, $(C_3^3\times C_6).C_2^4$ x 2, $C_3^4.C_4:C_8$ x 2, $C_3^4.C_4:D_4$ x 2, $S_3^4:C_2$ x 2, $C_3^2\wr C_2.\SD_{16}$ x 2, $C_3^4:(C_2\times \SD_{16})$ x 2, $C_3^4.C_4^2.C_2$, $C_3^4.C_4:Q_8$, $C_3^4.C_2^2:Q_8$, $C_3^4.C_4:C_8$, $C_3^4.C_4:D_4$, $C_3^4.D_4:C_4$, $C_3^4.C_4:Q_8$, $C_3^4.(C_2\times \SD_{16})$, $C_3^4.D_4:C_4$, $C_3^2.C_{12}^2.C_2$, $C_3^4.C_2^3.C_2^2$, $C_3^4.C_4^2.C_2$, $C_3^4.(C_4.Q_8)$, $C_3^4.C_4:C_8$, $C_3^4.C_4:C_8$, $C_3^4.C_2^2:C_8$, $C_3^4.C_4^2.C_2$, $C_3^4.(C_2\times \SD_{16})$, $C_3^4.C_4^2.C_2$, $C_3^3:C_{12}:C_8$, $C_3^3:C_{12}:Q_8$, $C_3^2:C_4\times F_9$, $C_3^4:(C_4\times Q_8)$, $F_9:S_3^2$, $\PSU(3,2):S_3^2$, $C_3^3:C_{12}:Q_8$, $C_3^2:(C_4\times F_9)$, $C_3^4:(C_4\times Q_8)$, $S_3^2:\PSU(3,2)$, $S_3^2:F_9$, $\SOPlus(4,2).S_3^2$, $C_3^2\wr C_2.\SD_{16}$, $C_2\times C_3^4:\OD_{16}$, $(C_3^3\times C_6).\OD_{16}$, $C_3^2:S_3^2:Q_8$, $C_3^4:C_4^2:C_2$

Order 2304: $C_3^2.C_4^3.C_2^2$

Order 2187: $C_3^2\wr C_3$

Order 1944: $C_3^2:S_3^3$ x 7, $C_3^5:D_4$ x 7, $C_3^2:S_3^3$ x 6, $C_3^5:D_4$ x 6, $C_3^2:S_3^3$ x 4, $C_3^2:S_3^3$ x 4, $C_3^4:D_{12}$ x 4, $C_3^5:D_4$ x 4, $C_3^4:D_{12}$ x 4, $C_3^4:D_{12}$ x 4, $C_3^4:D_{12}$ x 4, $C_3^2:S_3^3$ x 3, $C_3^4:(C_4\times S_3)$ x 3, $C_3^4:D_{12}$ x 2, $C_3^5:D_4$ x 2, $C_3^4:(C_4\times S_3)$ x 2, $C_3^4:C_4\times S_3$ x 2, $C_3\wr C_4:S_3$ x 2, $C_3^5:Q_8$, $C_3^5:Q_8$, $C_3^5:Q_8$, $C_3^5:(C_2\times C_4)$, $C_3^4:(C_2\times C_{12})$, $C_3\wr C_4:S_3$, $C_3^4:C_4\times S_3$, $C_3^3:(S_3\times C_{12})$, $S_3\times C_3^3:C_{12}$, $C_3^4:C_{24}$, $C_3^4:C_{24}$, $C_3^3:\PSU(3,2)$, $C_3^3:\SOPlus(4,2)$, $C_3^3:F_9$

Order 1728: $(C_2\times S_3^3).C_2^2$

Order 1458: $C_3^3\wr C_2$ x 2, $C_3^5:C_6$, $C_3^5:S_3$, $C_3^5:C_6$, $C_3^5.C_6$, $C_3^5:C_6$, $C_3^5:C_6$, $C_3^5:C_6$

Order 1296: $C_3^3:(S_3\times D_4)$ x 28, $C_3^4:(C_2\times D_4)$ x 22, $C_3^4:C_4^2$ x 15, $C_3^4:\SD_{16}$ x 12, $C_2\times C_3^4:D_4$ x 10, $C_3^2\wr C_2.Q_8$ x 5, $C_2\times C_3^4:D_4$ x 4, $C_6\times C_3^2:D_{12}$ x 4, $S_3^3:S_3$ x 4, $C_3^4:\SD_{16}$ x 4, $C_3^4:\SD_{16}$ x 4, $C_3^2:(C_2\times F_9)$ x 4, $C_6:S_3^3$ x 3, $C_2\times C_3^2:F_9$ x 3, $S_3^4$ x 3, $C_3^3:(S_3\times Q_8)$ x 3, $S_3^2:S_3^2$ x 3, $C_3:(S_3\times F_9)$ x 3, $(C_3^3\times C_6).D_4$ x 3, $C_3^4:C_4^2$ x 3, $C_3^4:(C_2^2\times C_4)$ x 2, $C_3^3:(C_4\times D_6)$ x 2, $C_3^4:(C_2\times Q_8)$ x 2, $S_3\times C_3^2:D_{12}$ x 2, $(C_3\times S_3^2):D_6$ x 2, $S_3^3:C_6$ x 2, $C_3^2:C_4\times S_3^2$ x 2, $C_3^4:\OD_{16}$ x 2, $C_3^2:(C_2\times F_9)$ x 2, $C_3^2\wr C_2.D_4$ x 2, $(C_3\times C_{12}):S_3^2$ x 2, $C_3\times C_{12}:S_3^2$ x 2, $C_3^4:C_4^2$ x 2, $C_6.S_3^3$ x 2, $C_6.S_3^3$ x 2, $C_3^4:C_4^2$ x 2, $C_2\times C_3^4:Q_8$, $C_3^3:(S_3\times Q_8)$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3^4:\SD_{16}$, $C_3:S_3\times F_9$, $C_3^2\wr C_2.D_4$, $(C_3^2\times S_3^2):C_4$, $S_3^3:C_6$, $C_4\times C_3^2:S_3^2$, $C_3^3:(S_3\times Q_8)$, $(C_3^3\times C_{12}):C_4$, $C_3^4:(C_2\times C_8)$, $C_3^4:C_4^2$, $C_3^2:C_{12}^2$, $(C_3^3\times C_6).Q_8$, $C_3^4:\SD_{16}$, $(C_3^3\times C_6).D_4$, $C_3^3:C_{12}:C_4$, $C_3^3:(S_3\times C_8)$, $C_3^4:\SD_{16}$

Order 1152: $C_3:S_3.C_4:D_4.C_2$ x 2, $C_4:F_9.C_2^2$ x 2, $C_{12}^2:D_4$, $(C_4\times C_3:S_3).C_4^2$, $C_3:S_3:\OD_{16}.C_2^2$

Order 972: $C_3^3:S_3^2$ x 14, $C_3^3:S_3^2$ x 11, $C_3^3:C_6^2$ x 7, $C_3^3:S_3^2$ x 4, $C_3^3:S_3^2$ x 4, $C_3^3:C_6^2$ x 4, $C_3^5:C_4$ x 3, $C_3^4:C_{12}$ x 3, $C_3^5:C_4$ x 2, $C_3^4:C_{12}$ x 2, $C_3^4:C_{12}$ x 2, $C_3^4:C_{12}$ x 2, $C_3^3:S_3^2$, $\He_3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:C_3:C_4$, $C_3^4:C_{12}$

Order 864: $S_3^3:C_2^2$ x 4, $(S_3\times D_6).D_6$ x 4, $C_4:S_3^3$ x 2, $C_{12}:\SOPlus(4,2)$ x 2, $F_9:D_6$, $C_{12}:\SOPlus(4,2)$, $C_{12}:\SOPlus(4,2)$, $S_3\times C_3^2:C_4^2$, $C_4.S_3^3$

Order 768: $C_2^3.\GL(2,\mathbb{Z}/4)$

Order 729: $C_3^6$, $C_3^5.C_3$, $C_3^5:C_3$

Order 648: $C_3^4:D_4$ x 33, $C_3:S_3^3$ x 23, $C_3^4:D_4$ x 23, $C_3^3:(C_4\times S_3)$ x 23, $C_3^3:D_{12}$ x 22, $C_3^4:(C_2\times C_4)$ x 14, $C_3:S_3^3$ x 12, $C_3^4:Q_8$ x 10, $C_3^2:F_9$ x 9, $C_3:S_3^3$ x 7, $C_3\times S_3^3$ x 7, $C_3^3:(C_4\times S_3)$ x 6, $C_2\times C_3^4:C_4$ x 5, $C_3^2:F_9$ x 5, $C_3^4:C_2^3$ x 4, $C_3^4:C_2^3$ x 4, $C_3^3:D_{12}$ x 4, $C_3\wr D_4$ x 4, $C_3^4:Q_8$ x 3, $C_3^4:Q_8$ x 3, $C_3^4:D_4$ x 3, $C_3^3:C_{24}$ x 3, $C_2\times C_3^4:C_4$ x 2, $C_6\times C_3^2:C_{12}$ x 2, $C_3^4:Q_8$ x 2, $C_3^3:D_{12}$ x 2, $C_3^4:D_4$ x 2, $C_3^3:C_4\times S_3$ x 2, $S_3\times C_3^2:C_{12}$ x 2, $C_3^2:F_9$ x 2, $S_3\wr C_3$ x 2, $C_3^3:D_{12}$ x 2, $C_3^4:D_4$ x 2, $C_3^3:(C_4\times S_3)$ x 2, $C_3^3:D_{12}$ x 2, $C_3^4:Q_8$, $C_3^2:F_9$, $C_3^2\times F_9$, $C_2\times C_3^3:A_4$, $C_4\times C_3^3:S_3$, $C_3^2\wr C_2\times C_4$, $C_3^4:Q_8$, $C_3^4:Q_8$, $C_3^3:(C_4\times S_3)$, $C_3^4:C_8$, $C_3^3:C_{24}$

Order 576: $D_6^2.C_2^2$ x 6, $(C_3\times C_{12}).\SD_{16}$ x 5, $\PSU(3,2):D_4$ x 4, $C_6^2:\SD_{16}$ x 3, $F_9:D_4$ x 3, $(C_6\times C_{12}):C_8$ x 2, $(C_4\times F_9):C_2$ x 2, $(C_6\times C_{12}):Q_8$, $C_4^2:S_3^2$, $C_{12}^2:C_4$, $(C_6\times C_{12}).Q_8$, $Q_8:\SOPlus(4,2)$, $S_3^2:\SD_{16}$, $C_8:\SOPlus(4,2)$, $C_{12}^2:C_4$, $(C_6\times C_{12}).Q_8$

Order 486: $C_3^4:C_6$ x 11, $C_3^4:C_6$ x 11, $C_3^4:S_3$ x 5, $C_3^4:C_6$ x 2, $C_3^4:C_6$, $S_3\times C_3^4$, $C_3^3:C_{18}$, $C_3^4:C_6$, $C_3^4:C_6$, $C_3^4:C_6$, $C_3^4.C_6$, $C_3^4:C_6$

Order 432: $S_3^3:C_2$ x 22, $S_3^2:D_6$ x 8, $C_2\times S_3^3$ x 6, $C_2\times C_3^2:D_{12}$ x 4, $C_6\times \SOPlus(4,2)$ x 4, $C_{12}:S_3^2$ x 4, $D_6:S_3^2$ x 4, $C_3^2:C_4\times D_6$ x 2, $C_3^3:\SD_{16}$ x 2, $C_{12}:S_3^2$ x 2, $C_{12}:S_3^2$ x 2, $C_{12}:S_3^2$ x 2, $D_6:S_3^2$ x 2, $D_6:S_3^2$ x 2, $C_3^3:C_4^2$ x 2, $S_3\times \PSU(3,2)$, $F_9:S_3$, $C_3^3:\SD_{16}$, $F_9:C_6$, $S_3\times F_9$, $C_{12}:S_3^2$, $C_{12}.S_3^2$, $C_3^3:C_4^2$, $C_3^3:C_4^2$, $C_{12}.S_3^2$, $C_{12}.S_3^2$, $C_{12}.S_3^2$, $C_{12}.S_3^2$, $C_{12}.S_3^2$

Order 384: $C_4^3:C_6$, $C_4^3.S_3$, $C_4^3.S_3$

Order 324: $C_3^2:S_3^2$ x 52, $C_3\wr C_2^2$ x 39, $C_3^4:C_4$ x 33, $C_3^2:S_3^2$ x 19, $C_3^4:C_4$ x 19, $C_3^3:C_{12}$ x 19, $C_3^2:S_3^2$ x 13, $C_3^2:S_3^2$ x 12, $C_3^2\times S_3^2$ x 12, $C_3\wr C_4$ x 6, $C_3^2:C_6^2$ x 4, $C_3^3:C_{12}$ x 2, $\He_3:D_6$, $C_3^3:D_6$, $C_3^3:A_4$, $C_3^3\times C_{12}$, $C_3^4:C_4$, $C_3^2:S_3^2$

Order 288: $C_4:\SOPlus(4,2)$ x 27, $F_9:C_2^2$ x 10, $C_6^2:D_4$ x 7, $\SOPlus(4,2):C_4$ x 6, $C_4:F_9$ x 5, $D_{12}:D_6$ x 4, $C_2\times C_3^2:C_4^2$ x 4, $C_4:\PSU(3,2)$ x 3, $C_4\times F_9$ x 3, $C_4.\PSU(3,2)$ x 3, $D_4\times S_3^2$ x 2, $C_6^2:Q_8$ x 2, $C_4\times \PSU(3,2)$ x 2, $C_4.\SOPlus(4,2)$ x 2, $C_6^2:C_8$ x 2, $C_6^2.C_2^3$ x 2, $C_4.\SOPlus(4,2)$ x 2, $C_3^2:C_4\times Q_8$, $(C_6\times C_{12}):C_4$, $(C_6\times C_{12}).C_4$, $C_4.\PSU(3,2)$, $C_4.\SOPlus(4,2)$, $S_3^2:Q_8$, $C_{12}^2:C_2$, $C_{12}:D_{12}$, $Q_8:S_3^2$, $D_4.S_3^2$, $C_{24}:D_6$, $(D_4\times C_3^2):C_4$, $C_3^2:C_4\times C_8$, $C_4.\SOPlus(4,2)$, $S_3^2:C_8$

Order 256: $C_4^2:\SD_{16}$

Order 243: $C_3^5$ x 5, $C_3^4:C_3$ x 2, $C_3^3:C_9$ x 2, $C_3^4.C_3$, $C_3^4:C_3$

Order 216: $S_3^3$ x 39, $S_3^2:S_3$ x 25, $S_3^2:C_6$ x 15, $C_3^2:C_4\times S_3$ x 15, $C_3^2:D_{12}$ x 14, $C_6:S_3^2$ x 11, $C_6\times S_3^2$ x 11, $C_3^3:D_4$ x 7, $C_6:S_3^2$ x 6, $C_3^2:D_{12}$ x 4, $C_3^2\times D_{12}$ x 4, $C_3^2:D_{12}$ x 4, $C_3^3:Q_8$ x 3, $C_4\times C_3^2:S_3$ x 3, $C_2\times C_3^3:C_4$ x 2, $C_2\times C_3^2:C_{12}$ x 2, $C_3:F_9$ x 2, $C_3^2:D_{12}$ x 2, $C_6.S_3^2$ x 2, $C_3^3:D_4$ x 2, $C_3^2:C_{24}$, $C_3^2:C_{24}$, $C_3^3:Q_8$, $C_3\times F_9$, $C_3:S_3\times C_{12}$, $C_3^3:Q_8$, $C_6.C_6^2$, $C_3^3:Q_8$, $C_3^3:Q_8$, $C_6.S_3^2$, $C_6.S_3^2$

Order 192: $A_4:\SD_{16}$, $C_4\wr C_3$, $C_2^3.S_4$, $C_4^2:D_6$, $C_2^4.A_4$

Order 162: $C_3^2\wr C_2$ x 76, $C_3^3:S_3$ x 51, $C_3^3:C_6$ x 13, $S_3\times C_3^3$ x 13, $C_3^3:C_6$ x 5, $C_3^3\times C_6$, $C_3^3:C_6$, $C_3^2:C_{18}$, $C_3^3.C_6$, $S_3\times \He_3$, $C_3^3:C_6$, $C_3^3:C_6$

Order 144: $S_3^2:C_2^2$ x 44, $C_{12}:D_6$ x 24, $C_3^2:C_4^2$ x 21, $F_9:C_2$ x 18, $C_2\times F_9$ x 7, $D_6^2$ x 6, $(C_3\times C_{12}):C_4$ x 6, $C_2.\PSU(3,2)$ x 5, $C_2\times \PSU(3,2)$ x 4, $C_6\times D_{12}$ x 4, $D_6:D_6$ x 4, $D_6:D_6$ x 4, $C_3^2:(C_2\times C_8)$ x 4, $C_3^2:\SD_{16}$ x 4, $C_{12}.D_6$ x 3, $C_6^2:C_4$ x 3, $D_6:D_6$ x 3, $C_{12}.D_6$ x 3, $C_2.\SOPlus(4,2)$ x 3, $C_3^2:C_4^2$ x 2, $C_{12}:D_6$ x 2, $C_{12}:D_6$ x 2, $C_{12}:D_6$ x 2, $S_3\times D_{12}$ x 2, $C_4\times S_3^2$ x 2, $C_{24}:S_3$, $C_{12}.D_6$, $C_{12}.D_6$, $C_{24}:C_6$, $C_6.D_{12}$, $C_6.D_{12}$, $C_{12}.D_6$, $C_{12}.D_6$, $C_6^2:C_4$, $C_3^2:\OD_{16}$, $C_3^2:\OD_{16}$, $S_3^2:C_4$, $C_{12}^2$

Order 128: $C_4^2.D_4$ x 2, $C_4^2.D_4$ x 2, $C_4^2:C_8$, $C_4^2:Q_8$, $C_4^3:C_2$

Order 108: $C_3:S_3^2$ x 86, $C_3\times S_3^2$ x 57, $C_3:S_3^2$ x 26, $C_3^3:C_4$ x 12, $C_3^2:C_{12}$ x 12, $C_3^2:D_6$ x 11, $C_3^2\times D_6$ x 11, $C_3^2:D_6$ x 5, $C_3^2\times C_{12}$ x 3, $C_3^3:C_4$ x 3, $C_3^2:C_{12}$ x 2, $C_3^2:C_{12}$ x 2, $C_3^2:D_6$

Order 96: $D_4\times D_6$ x 4, $C_{12}:D_4$ x 4, $C_4:D_{12}$, $S_3\times C_4^2$, $C_4^2:C_6$, $C_4^2:C_6$, $A_4:C_8$, $D_4\times A_4$, $A_4:Q_8$, $C_{12}:D_4$, $S_3\times \SD_{16}$

Order 81: $C_3^4$ x 50, $C_3\wr C_3$ x 4, $C_3^2:C_9$ x 2, $C_9:C_3^2$, $C_3\times \He_3$

Order 72: $\SOPlus(4,2)$ x 47, $S_3\times D_6$ x 46, $C_2\times C_3^2:C_4$ x 28, $C_{12}:S_3$ x 19, $D_6:S_3$ x 19, $C_3\times D_{12}$ x 18, $F_9$ x 13, $\PSU(3,2)$ x 12, $C_6\times D_6$ x 7, $C_6.D_6$ x 6, $C_3^2:C_8$ x 5, $C_6:D_6$ x 4, $C_6\wr C_2$ x 4, $C_3:D_{12}$ x 4, $C_6^2:C_2$ x 3, $C_3^2:Q_8$ x 3, $C_3^2:Q_8$ x 3, $C_3:C_{24}$ x 3, $D_4\times C_3^2$ x 2, $C_6\times C_{12}$ x 2, $C_6.D_6$ x 2, $C_3:D_{12}$ x 2, $C_3^2:Q_8$ x 2, $S_3\times C_{12}$ x 2, $C_6.D_6$ x 2, $Q_8\times C_3^2$, $C_3\times C_{24}$, $C_3^2:C_8$

Order 64: $C_4^2:C_2^2$ x 4, $Q_8:D_4$ x 4, $C_4.D_8$ x 4, $C_2^2:\SD_{16}$ x 3, $C_4^2:C_4$ x 2, $C_8:D_4$ x 2, $C_4.D_8$ x 2, $C_4^2.C_4$ x 2, $C_4^3$, $C_4^2.C_2^2$, $C_4:\OD_{16}$

Order 54: $C_3^2:S_3$ x 92, $C_3^2:C_6$ x 72, $S_3\times C_3^2$ x 72, $C_3^2\times C_6$ x 5, $C_3^2:C_6$ x 3, $S_3\times C_9$, $C_9:C_6$, $C_2\times \He_3$

Order 48: $S_3\times D_4$ x 18, $C_6:D_4$ x 7, $C_6\times D_4$ x 4, $C_2\times D_{12}$ x 4, $C_2^2\times D_6$ x 3, $C_4\times D_6$ x 2, $C_3:\SD_{16}$ x 2, $C_{12}:C_4$ x 2, $C_{24}:C_2$, $C_2^2\times A_4$, $S_3\times Q_8$, $S_3\times C_8$, $C_4\times A_4$, $A_4:C_4$, $C_4^2:C_3$, $C_3\times \SD_{16}$, $C_4\times C_{12}$, $Q_8:S_3$

Order 36: $S_3^2$ x 79, $C_3^2:C_4$ x 45, $C_6\times S_3$ x 42, $C_6:S_3$ x 28, $C_3\times C_{12}$ x 15, $C_3^2:C_4$ x 15, $C_3:C_{12}$ x 6, $C_6^2$ x 4

Order 32: $C_4:D_4$ x 13, $D_4:C_4$ x 6, $C_2^2\times D_4$ x 6, $C_2\times \SD_{16}$ x 6, $C_4:C_8$ x 6, $C_2\times C_4^2$ x 3, $C_2^2:C_8$ x 2, $C_4:Q_8$ x 2, $C_4\times C_8$ x 2, $C_2^2:Q_8$ x 2, $C_4\times Q_8$ x 2, $C_2\times \OD_{16}$, $C_4.Q_8$, $C_4^2:C_2$

Order 27: $C_3^3$ x 88, $C_9:C_3$ x 2, $\He_3$ x 2, $C_3\times C_9$

Order 24: $C_2\times D_6$ x 21, $C_3:D_4$ x 17, $C_4\times S_3$ x 11, $C_3\times D_4$ x 11, $D_{12}$ x 10, $C_3:Q_8$ x 3, $C_2^2\times C_6$ x 3, $C_2\times C_{12}$ x 2, $C_6:C_4$ x 2, $C_2\times A_4$ x 2, $C_3:C_8$ x 2, $C_{24}$, $C_3\times Q_8$

Order 18: $C_3:S_3$ x 70, $C_3\times S_3$ x 66, $C_3\times C_6$ x 21, $C_{18}$

Order 16: $C_2\times D_4$ x 26, $C_4^2$ x 10, $\SD_{16}$ x 6, $C_4:C_4$ x 6, $C_2\times C_8$ x 5, $C_2^2\times C_4$ x 3, $\OD_{16}$ x 2, $C_2^4$ x 2, $C_2\times Q_8$ x 2, $C_2^2:C_4$

Order 12: $D_6$ x 34, $C_2\times C_6$ x 12, $C_{12}$ x 8, $C_3:C_4$ x 8, $A_4$

Order 9: $C_3^2$ x 51, $C_9$

Order 8: $D_4$ x 19, $C_2\times C_4$ x 14, $C_2^3$ x 9, $C_8$ x 5, $Q_8$ x 4

Order 6: $S_3$ x 17, $C_6$ x 14

Order 4: $C_2^2$ x 11, $C_4$ x 11

Order 3: $C_3$ x 6

Order 2: $C_2$ x 5

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 559872: $(C_3:S_3)^3.\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 279936: $C_3^6:(C_4^3:C_6)$ ($C_2$), $C_3^6:(C_4^3.S_3)$ ($C_2$), $C_3^6:(C_4^3.S_3)$ ($C_2$)

Order 139968: $C_3^6:C_4\wr C_3$ ($C_2^2$)

Order 93312: $C_3^6.C_2^3.C_2^4$ ($S_3$)

Order 69984: $C_3^6:(C_4^2:C_6)$ ($D_4$)

Order 46656: $C_3^6.C_4^3$ ($D_6$)

Order 34992: $C_3^5:D_6.A_4$ ($\SD_{16}$)

Order 23328: $C_3^6.C_2^3.C_2^2$ ($C_3:D_4$), $C_3^6.(C_2^2\times D_4)$ ($S_4$)

Order 11664: $C_3^6.C_4^2$ ($C_3:\SD_{16}$), $C_3^6:(C_2^2\times C_4)$ ($C_2\times S_4$)

Order 5832: $(C_3:S_3)^3$ ($\GL(2,\mathbb{Z}/4)$)

Order 2916: $C_3^6.C_4$ ($C_4^2:D_6$), $C_3^5:D_6$ ($A_4:\SD_{16}$)

Order 1458: $C_3^5:S_3$ ($C_2^2.\GL(2,\mathbb{Z}/4)$)

Order 729: $C_3^6$ ($C_2^3.\GL(2,\mathbb{Z}/4)$)

Order 1: $C_1$ ($(C_3:S_3)^3.\GL(2,\mathbb{Z}/4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 559872: $(C_3:S_3)^3.\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 279936: $C_3^6:(C_4^3:C_6)$ ($C_2$), $C_3^6:(C_4^3.S_3)$ ($C_2$), $C_3^6:(C_4^3.S_3)$ ($C_2$)

Order 139968: $C_3^6:C_4\wr C_3$ ($C_2^2$)

Order 93312: $C_3^6.C_2^3.C_2^4$ ($S_3$)

Order 69984: $C_3^6:(C_4^2:C_6)$ ($D_4$)

Order 46656: $C_3^6.C_4^3$ ($D_6$)

Order 34992: $C_3^5:D_6.A_4$ ($\SD_{16}$)

Order 23328: $C_3^6.C_2^3.C_2^2$ ($C_3:D_4$), $C_3^6.(C_2^2\times D_4)$ ($S_4$)

Order 11664: $C_3^6.C_4^2$ ($C_3:\SD_{16}$), $C_3^6:(C_2^2\times C_4)$ ($C_2\times S_4$)

Order 5832: $(C_3:S_3)^3$ ($\GL(2,\mathbb{Z}/4)$)

Order 2916: $C_3^6.C_4$ ($C_4^2:D_6$), $C_3^5:D_6$ ($A_4:\SD_{16}$)

Order 1458: $C_3^5:S_3$ ($C_2^2.\GL(2,\mathbb{Z}/4)$)

Order 729: $C_3^6$ ($C_2^3.\GL(2,\mathbb{Z}/4)$)

Order 1: $C_1$ ($(C_3:S_3)^3.\GL(2,\mathbb{Z}/4)$)

Series

Derived series

$(C_3:S_3)^3.\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_3^6:C_4\wr C_3$

$\rhd$

$C_3^6.C_4^2$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Chief series

$(C_3:S_3)^3.\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_3^6:(C_4^3:C_6)$

$\rhd$

$C_3^6:C_4\wr C_3$

$\rhd$

$C_3^6:(C_4^2:C_6)$

$\rhd$

$C_3^5:D_6.A_4$

$\rhd$

$C_3^6.C_4^2$

$\rhd$

$C_3^5:D_6$

$\rhd$

$C_3^6$

$\rhd$

$C_1$

Lower central series

$(C_3:S_3)^3.\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_3^6:C_4\wr C_3$

$\rhd$

$C_3^6:(C_4^2:C_6)$

$\rhd$

$C_3^5:D_6.A_4$

Upper central series

$C_1$

Supergroups

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

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

Character theory

Complex character table

See the $73 \times 73$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $63 \times 63$ rational character table.