Abstract group 77760.t: $C_3^3:(S_4\times S_5)$

• Label: 77760.t
• Order: \( 2^{6} \cdot 3^{5} \cdot 5 \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 5 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{7} \cdot 3^{5} \cdot 5 \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $14$
• Trans deg.: $45$
• Rank: $2$

Group information

Description:	$C_3^3:(S_4\times S_5)$
Order:	\(77760\)\(\medspace = 2^{6} \cdot 3^{5} \cdot 5 \)
Exponent:	\(180\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5 \)
Automorphism group:	$S_3\wr S_3\times S_5$, of order \(155520\)\(\medspace = 2^{7} \cdot 3^{5} \cdot 5 \)
Composition factors:	$C_2$ x 4, $C_3$ x 4, $A_5$
Derived length:	$4$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	5	6	9	10	12	15	18	20	30	45
Elements	1	1635	2078	7452	24	23178	3024	1080	22680	2352	2160	3888	4752	3456	77760
Conjugacy classes	1	8	9	7	1	29	2	2	9	5	1	1	6	2	83
Divisions	1	8	9	7	1	29	2	2	9	4	1	1	4	1	79
Autjugacy classes	1	8	9	7	1	29	2	2	9	4	1	1	4	1	79

Dimension	1	2	3	4	5	6	8	10	12	15	16	18	24	30	32	36	40	48	60	64	72	80	96
Irr. complex chars.	4	2	4	4	4	6	4	2	10	4	1	2	8	4	2	4	2	7	5	1	2	1	0	83
Irr. rational chars.	4	2	4	4	4	6	4	2	10	4	1	2	4	4	2	2	2	7	5	1	3	1	1	79

Minimal presentations

Permutation degree:	$14$
Transitive degree:	$45$
Rank:	$2$
Inequivalent generating pairs:	$4446$

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

Permutation group:	Degree $14$ $\langle(1,3,2,4,5,8)(6,9,7)(10,12,14,13), (1,2,4,6)(3,5,7)(8,9)(10,11,13)(12,14)\rangle$
Transitive group:	45T933				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	$A_5$ $\,\rtimes\,$ $(S_3\wr S_3)$	$(A_5:S_3^3)$ $\,\rtimes\,$ $S_3$	$(C_3^3:S_5)$ $\,\rtimes\,$ $S_4$	$(C_3^3:S_4)$ $\,\rtimes\,$ $S_5$	all 13
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product

Elements of the group are displayed as permutations of degree 14.

Homology

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

Subgroups

There are 944448 subgroups in 2826 conjugacy classes, 15 normal, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_3^3:(S_4\times S_5)$
Commutator:	$G' \simeq$ $A_5\times C_3^3:A_4$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_3^3:(S_4\times S_5)$
Fitting:	$\operatorname{Fit} \simeq$ $C_3^3$	$G/\operatorname{Fit} \simeq$ $S_4\times S_5$
Radical:	$R \simeq$ $C_3^3:S_4$	$G/R \simeq$ $S_5$
Socle:	$\operatorname{soc} \simeq$ $C_3^3\times A_5$	$G/\operatorname{soc} \simeq$ $C_2\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $D_4^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4:C_3$
5-Sylow subgroup:	$P_{ 5 } \simeq$ $C_5$

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 77760: $C_3^3:(S_4\times S_5)$

Order 38880: $C_3^3:(A_4\times S_5)$, $A_5\times C_3^3:S_4$, $(C_3^3\times A_5):S_4$

Order 25920: $S_3^2:(S_3\times S_5)$

Order 19440: $C_3^3:(S_3\times S_5)$, $A_5\times C_3^3:A_4$

Order 15552: $C_3^3:S_4^2$

Order 12960: $C_3^3:(C_4\times S_5)$, $C_3\times S_3^2:S_5$, $A_5\times S_3^2:S_3$, $\GL(2,4):\SOPlus(4,2)$, $A_5:S_3^3$, $\GL(2,4):\SOPlus(4,2)$, $A_5:S_3^3$, $C_3^3:(F_5\times S_4)$

Order 9720: $C_3^3:(C_3\times S_5)$, $A_5\times C_3\wr S_3$, $C_3\wr C_3:S_5$

Order 8640: $S_5\times \SOPlus(4,2)$

Order 7776: $C_3^3:(A_4\times S_4)$, $(A_4\times C_3^3):S_4$, $A_4\times C_3^3:S_4$, $S_3^4:S_3$

Order 6480: $(C_3\times \GL(2,4)):C_{12}$, $\GL(2,4):S_3^2$, $\GL(2,4):S_3^2$, $S_3^2\times \GL(2,4)$, $A_5\times C_3^3:C_4$, $\GL(2,4):S_3^2$, $\GL(2,4):S_3^2$, $\GL(2,4):S_3^2$, $\GL(2,4):S_3^2$, $\GL(2,4).S_3^2$, $C_3^3:(A_4\times F_5)$, $D_5\times C_3^3:S_4$, $(D_5\times C_3^3).S_4$

Order 5184: $D_6^2:S_3^2$, $C_3^3:(D_4\times S_4)$

Order 4860: $C_3^3:\GL(2,4)$

Order 4320: $S_3^2\times S_5$ x 2, $S_3^2:S_5$ x 2, $S_3^2:(S_3\times F_5)$, $S_3\times C_6:S_5$, $A_5:\SOPlus(4,2)$, $C_3^2:C_4\times S_5$, $A_5\times \SOPlus(4,2)$

Order 3888: $C_3^3:(S_3\times S_4)$, $C_3^3:A_4^2$, $S_3\times C_3^3:S_4$, $S_3\times C_3^3:S_4$, $C_3^3:(S_3\times S_4)$, $(C_3\times S_3^3):S_3$, $S_3^4:C_3$, $C_3\times S_3\wr S_3$, $C_3^3:(S_3\times S_4)$

Order 3240: $\He_3:S_5$ x 2, $C_3^3:S_5$, $C_3:S_3\times \GL(2,4)$, $C_3\times S_3\times \GL(2,4)$, $C_3^3:S_5$, $C_3^3:S_5$, $C_3^2:S_3\times A_5$, $C_9:(C_3\times S_5)$, $C_3^3:(S_3\times F_5)$, $C_5\times C_3^3:S_4$, $D_5\times C_3^3:A_4$, $(C_3^2\times C_{15}):S_4$

Order 2880: $C_3:(D_4\times S_5)$, $S_4\times S_5$

Order 2592: $C_3^3:(C_4\times S_4)$, $(C_3\times C_6^2):D_{12}$, $C_3^3:\GL(2,\mathbb{Z}/4)$, $C_3^3:\GL(2,\mathbb{Z}/4)$, $A_4:S_3^3$, $A_4\times S_3^2:S_3$, $A_4:S_3^3$, $C_3^3:(C_4\times S_4)$, $C_3^3:\GL(2,\mathbb{Z}/4)$, $C_2^2\times C_3^3:S_4$, $C_2.S_3\wr S_3$, $C_3^3:\GL(2,\mathbb{Z}/4)$, $(C_2\times S_3^3):C_6$, $S_3^4:C_2$, $S_3^3:D_6$, $(C_3\times C_6^2):S_4$

Order 2160: $A_5:S_3^2$ x 8, $S_3^2\times A_5$ x 2, $A_5:S_3^2$ x 2, $\GL(2,4):D_6$ x 2, $(C_3^2\times C_{15}):C_4^2$, $C_3\times S_3^2:F_5$, $(C_3^2\times D_5).D_{12}$, $D_5.S_3^3$, $D_5.S_3^3$, $\GL(2,4):D_6$, $\GL(2,4):D_6$, $D_6\times \GL(2,4)$, $D_5\times S_3^2:S_3$, $(D_5\times C_3^3).D_4$, $\GL(2,4):D_6$, $C_3^2:C_4\times A_5$, $(C_3\times \GL(2,4)):C_4$

Order 1944: $S_3\times C_3^3:A_4$, $C_3^3:(S_3\times A_4)$, $S_3^3:C_3^2$, $C_3^4:S_4$, $C_3^4:S_4$, $C_3^4:S_4$, $C_3\wr S_4$, $A_4\times C_3\wr S_3$, $C_3\wr C_3:S_4$, $C_3^3:(C_3\times S_4)$, $C_3^2.S_3^3$

Order 1728: $D_6^2:D_6$, $D_6^2:D_6$

Order 1620: $C_3\wr C_3:F_5$, $(C_3^2\times C_{15}):C_{12}$, $C_5\times C_3^3:A_4$, $C_3^3\times A_5$, $C_9:\GL(2,4)$, $\He_3\times A_5$, $C_3\wr S_3\times D_5$

Order 1440: $D_6\times S_5$ x 2, $F_5\times \SOPlus(4,2)$, $\GL(2,4):C_2^3$, $S_4\times A_5$, $A_4:S_5$, $A_4\times S_5$, $\GL(2,4):D_4$, $\GL(2,4):D_4$, $\GL(2,4):D_4$, $D_6:S_5$, $C_3:(C_4\times S_5)$

Order 1296: $S_3^3:S_3$ x 4, $S_3\wr S_3$ x 3, $S_3^4$ x 2, $S_3\times C_3^2:D_{12}$ x 2, $S_3^2:S_3^2$ x 2, $S_3^3:C_6$ x 2, $C_2\times C_3^3:S_4$ x 2, $C_6^2.S_3^2$ x 2, $C_6^2.S_3^2$, $A_4\times C_3:S_3^2$, $C_6^2:C_6^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $(C_3\times C_6^2):A_4$, $C_2^2\times C_3^3:A_4$, $A_4\times C_3^3:C_4$, $(C_3^2\times A_4):C_{12}$, $C_3^3:(S_3\times D_4)$, $(C_3\times S_3^2):D_6$, $C_3^2:C_4\times S_3^2$, $S_3^3:C_6$, $C_2\times C_3^3:S_4$, $C_3^3:(C_4\times A_4)$, $(C_3^2\times C_6).S_4$, $C_6^2:S_3^2$

Order 1080: $S_3\times \GL(2,4)$ x 6, $C_3^2:S_5$ x 6, $C_3^2:S_5$ x 5, $D_5\times C_3:S_3^2$, $C_3\times D_5\times S_3^2$, $C_3^3:D_{20}$, $C_{15}:\SOPlus(4,2)$, $S_3^2:D_{15}$, $C_5\times S_3^2:S_3$, $C_3^3:C_4\times D_5$, $(C_3^2\times D_5):C_{12}$, $C_3^2:(S_3\times F_5)$, $C_3^2:(S_3\times F_5)$, $S_3\times C_3^2:F_5$, $C_{15}:(S_3\times C_{12})$, $S_3\times C_{15}:C_{12}$, $C_6\times \GL(2,4)$, $\GL(2,4):S_3$, $C_3^2\times S_5$, $C_3^2:(S_3\times F_5)$, $C_9:S_5$

Order 972: $C_3\wr A_4$, $C_6^2.C_3^3$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:D_6$, $\He_3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:D_6$

Order 960: $D_4\times S_5$

Order 864: $S_4\times S_3^2$ x 3, $S_3^3:C_2^2$ x 2, $S_3^2:S_4$ x 2, $D_6^2:S_3$, $S_3\times C_6:S_4$, $D_6^2:C_6$, $C_6^2:D_{12}$, $C_3^2:C_4\times S_4$, $C_2^2:S_3^3$, $D_6^2:S_3$, $C_6^2:D_{12}$, $(C_3\times C_6^2):D_4$, $D_6^2:C_6$, $C_6^2:(C_4\times S_3)$, $(S_3\times D_6).D_6$, $(S_3\times D_6).D_6$, $(C_3\times S_3^2):D_4$, $D_6:\SOPlus(4,2)$, $S_3^2:D_{12}$, $D_6:\SOPlus(4,2)$, $S_3^2:(C_4\times S_3)$

Order 810: $C_3^3:D_{15}$, $\He_3:C_{30}$, $C_3\wr C_3\times D_5$

Order 720: $S_3\times S_5$ x 7, $C_6:S_5$ x 6, $F_5\times S_3^2$ x 2, $S_3^2:F_5$ x 2, $D_6\times A_5$ x 2, $C_6\times S_5$ x 2, $D_{10}.S_3^2$, $S_3^2:D_{10}$, $D_5.\SOPlus(4,2)$, $C_3^2:C_4\times F_5$, $C_2^2\times \GL(2,4)$, $A_4\times A_5$, $\GL(2,4):C_4$, $A_5:C_{12}$

Order 648: $C_3:S_3^3$ x 5, $C_3:S_3^3$ x 5, $C_3\times S_3^3$ x 5, $C_3^3:D_{12}$ x 4, $C_3\wr D_4$ x 4, $C_3^3:S_4$ x 3, $C_3^3:S_4$ x 3, $C_3:S_3^3$ x 2, $C_3^4:D_4$ x 2, $C_3^4:D_4$ x 2, $C_3^3:D_{12}$ x 2, $C_3^3:C_4\times S_3$ x 2, $S_3\times C_3^2:C_{12}$ x 2, $S_3\wr C_3$ x 2, $\He_3:S_4$ x 2, $(C_3^2\times A_4):S_3$ x 2, $(C_3^2\times A_4):S_3$ x 2, $\He_3:S_4$ x 2, $\He_3:S_4$ x 2, $C_3\times C_6^2:C_6$, $S_3\times C_3^2\times A_4$, $C_3^3:S_4$, $C_3^3:S_4$, $C_3^3:S_4$, $C_3^3:D_{12}$, $C_3^4:D_4$, $C_3^3:(C_4\times S_3)$, $C_3^3:(C_4\times S_3)$, $C_2\times C_3^3:A_4$, $C_3\wr C_3:D_4$, $C_3^3:D_{12}$, $C_3^3:(C_4\times S_3)$, $A_4\times C_3^2:S_3$, $C_9:(C_3\times S_4)$, $C_3.S_3^3$, $C_2^2\times C_3\wr S_3$, $C_2\times C_3^3:D_6$, $C_3^3:S_4$, $(C_3\times C_6^2):S_3$, $(C_3\times C_6^2):C_6$

Order 576: $S_4^2$, $C_3^2:D_4^2$, $\GL(2,\mathbb{Z}/4):S_3$

Order 540: $C_3\times \GL(2,4)$ x 4, $\He_3:F_5$ x 2, $C_{15}:C_6^2$, $(C_3^2\times C_{15}):C_4$, $C_3^3:C_{20}$, $C_3^3:F_5$, $C_3^3:F_5$, $C_3^3:F_5$, $C_3^3:F_5$, $\He_3:D_{10}$, $C_{45}:C_{12}$, $C_9\times A_5$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}\times S_3^2$, $C_3^3:D_{10}$

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

Order 480: $C_2^2:S_5$ x 2, $C_2^2\times S_5$ x 2, $D_4\times A_5$, $C_4:S_5$, $C_4\times S_5$, $F_5\times S_4$, $C_{15}:(C_4\times D_4)$

Order 432: $A_4:S_3^2$ x 13, $S_3^3:C_2$ x 13, $S_3^2:D_6$ x 7, $C_2\times S_3^3$ x 5, $C_6^2:D_6$ x 4, $C_2\times C_3^2:D_{12}$ x 3, $C_6\times \SOPlus(4,2)$ x 3, $A_4\times S_3^2$ x 3, $A_4:S_3^2$ x 3, $C_3^2:C_4\times D_6$ x 2, $C_6^2:D_6$ x 2, $D_6:S_3^2$ x 2, $C_6.\SOPlus(4,2)$ x 2, $S_3^2:C_{12}$ x 2, $C_3:D_6^2$, $C_3\times D_6^2$, $C_2^2\times C_3^3:C_4$, $C_6\times S_3\times A_4$, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2:C_{12}$, $(C_3^2\times A_4):C_4$, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2:C_{12}$, $D_6:S_3^2$, $D_6:S_3^2$, $D_6:S_3^2$, $D_6:S_3^2$, $C_2.S_3^3$, $C_2.S_3^3$, $C_6.\SOPlus(4,2)$, $(C_3\times S_3^2):C_4$, $C_6.\SOPlus(4,2)$, $C_3^2:C_{12}:C_4$, $(C_3^2\times D_6):C_4$, $C_3^3:C_4^2$, $C_6^2:D_6$

Order 405: $C_3^3:C_{15}$

Order 360: $D_5.S_3^2$ x 8, $C_3:S_5$ x 8, $C_6\times A_5$ x 4, $S_3\times A_5$ x 4, $C_3\times S_5$ x 4, $D_5\times S_3^2$ x 2, $S_3^2:D_5$ x 2, $D_5.S_3^2$ x 2, $D_{15}:C_{12}$ x 2, $C_{30}:D_6$, $C_2\times C_3^2:F_5$, $C_{30}:C_{12}$, $C_3^2:D_{20}$, $S_3^2:C_{10}$, $(C_3^2\times D_5):C_4$, $C_3^2:C_4\times D_5$, $C_3:S_3\times F_5$

Order 324: $C_3^2:S_3^2$ x 8, $C_3^2:S_3^2$ x 5, $C_3\wr C_2^2$ x 5, $C_3^2\times S_3^2$ x 5, $C_3^3:D_6$ x 4, $C_3^2:S_3^2$ x 3, $C_3^3:A_4$ x 3, $C_3^3:D_6$ x 2, $C_6^2.C_3^2$ x 2, $\He_3:A_4$ x 2, $C_3^2:S_3^2$ x 2, $C_3\wr C_4$ x 2, $C_3^2:S_3^2$ x 2, $C_3^2:S_3^2$ x 2, $C_3^2.C_6^2$, $C_3^3:D_6$, $\He_3:D_6$, $C_3^3:A_4$, $C_3\wr C_3:C_4$, $A_4\times C_3^3$, $C_3^4:C_4$, $C_3^3:C_{12}$, $C_3^3:C_{12}$, $C_6^2.C_3^2$, $A_4\times \He_3$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2.S_3^2$, $C_3^3:D_6$

Order 288: $D_6\times S_4$ x 5, $D_6\wr C_2$ x 4, $D_4\times S_3^2$ x 3, $C_6^2:D_4$ x 2, $S_3^2:D_4$ x 2, $C_6^2:D_4$ x 2, $A_4\times S_4$ x 2, $D_6^2:C_2$, $C_3^2:C_4\times D_4$, $A_4\times C_3:D_4$, $C_3\times \GL(2,\mathbb{Z}/4)$, $C_4:\SOPlus(4,2)$, $C_4\times \SOPlus(4,2)$, $D_6:S_4$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $C_2\times C_6:S_4$, $\PSOPlus(4,3)$

Order 270: $C_{45}:C_6$, $D_5\times \He_3$, $C_3^2:D_{15}$, $C_3^2:C_{30}$, $C_3^2\times D_{15}$, $C_3^2:C_{30}$, $D_5\times C_3^3$, $C_3^2:D_{15}$, $\He_3:C_{10}$

Order 243: $C_3^4:C_3$

Order 240: $C_2\times S_5$ x 5, $D_6\times F_5$ x 2, $C_2^2\times A_5$ x 2, $D_{10}.D_6$, $D_{30}:C_4$, $C_{15}:C_4^2$, $C_4\times A_5$, $A_5:C_4$, $D_{10}.D_6$, $D_5\times S_4$, $A_4\times F_5$, $A_4:F_5$, $D_6:D_{10}$, $D_{10}:C_{12}$

Order 216: $S_3^3$ x 30, $S_3^2:S_3$ x 17, $C_6\times S_3^2$ x 12, $C_6^2:C_6$ x 11, $C_3^2:S_4$ x 11, $C_3^2:S_4$ x 9, $C_6:S_3^2$ x 8, $C_3^2:D_{12}$ x 8, $S_3^2:C_6$ x 8, $C_3^2:C_4\times S_3$ x 7, $C_6:S_3^2$ x 6, $C_2\times C_3^3:C_4$ x 3, $C_2\times C_3^2:C_{12}$ x 3, $C_3^2:S_4$ x 2, $C_3^2.S_4$ x 2, $C_6^2:C_6$ x 2, $C_3^2:D_{12}$ x 2, $C_6^2:C_6$ x 2, $C_3^2\times S_4$ x 2, $C_3^3:D_4$ x 2, $C_3^2:D_{12}$ x 2, $C_9:S_4$, $D_{18}:C_6$, $C_6.S_3^2$, $C_6^2:C_6$, $S_3\times C_6^2$, $C_6^2:C_6$, $C_6^2:S_3$, $C_6^2:C_6$, $C_6^2:C_6$, $C_3^2:D_{12}$, $C_6.S_3^2$, $C_3^2:D_{12}$, $C_3^2:D_{12}$, $C_6.S_3^2$, $C_6.S_3^2$, $C_3^3:D_4$, $C_6.S_3^2$, $C_6.S_3^2$, $C_6^2:S_3$, $C_6.S_3^2$

Order 192: $D_4\times S_4$ x 2, $C_2^4:D_6$

Order 180: $C_{15}:D_6$ x 8, $C_{15}:C_{12}$ x 6, $C_3^2:F_5$ x 5, $\GL(2,4)$ x 4, $C_{15}:D_6$ x 2, $S_3\times D_{15}$ x 2, $C_5\times S_3^2$ x 2, $C_{45}:C_4$, $C_3\times D_{30}$, $S_3\times C_{30}$, $C_3^2\times D_{10}$, $C_{15}:D_6$, $C_3^2:F_5$, $(C_3\times C_{15}):C_4$, $C_3^2:C_{20}$, $C_3^2\times F_5$

Order 162: $C_3^2\wr C_2$ x 4, $C_3^3:S_3$ x 4, $C_3\wr S_3$ x 4, $C_3^3:C_6$ x 3, $S_3\times C_3^3$ x 3, $C_3^3:C_6$ x 3, $C_3^3:C_6$ x 2, $C_3^3:C_6$ x 2, $C_3^3:S_3$, $C_3^3:S_3$, $C_3^3.S_3$, $C_3^3:S_3$, $C_3^3.C_6$, $D_9:C_3^2$, $S_3\times \He_3$, $C_3^3:C_6$

Order 160: $D_4\times F_5$

Order 144: $S_3\times S_4$ x 25, $D_6:D_6$ x 18, $S_3^2:C_2^2$ x 13, $C_6:S_4$ x 12, $D_6^2$ x 8, $D_6:D_6$ x 6, $S_3^2:C_4$ x 6, $A_4\times D_6$ x 5, $C_6\times S_4$ x 5, $C_{12}:D_6$ x 4, $S_3\times D_{12}$ x 4, $C_{12}:D_6$ x 3, $C_4\times S_3^2$ x 3, $C_6^2:C_4$ x 2, $C_{12}:D_6$ x 2, $C_6^2:C_4$ x 2, $C_6^2:C_2^2$, $C_2^2:C_6^2$, $A_4^2$, $C_6^2:C_2^2$, $C_6^2:C_2^2$, $C_6:D_{12}$, $D_6:D_6$, $D_6.D_6$, $(C_3\times C_{12}):C_4$, $C_3^2:C_4^2$, $A_4\times C_3:C_4$, $A_4:C_{12}$, $C_2.\SOPlus(4,2)$

Order 135: $C_3^2\times C_{15}$, $C_9:C_{15}$, $C_5\times \He_3$

Order 120: $S_3\times F_5$ x 7, $C_{30}:C_4$ x 6, $C_6\times F_5$ x 3, $S_5$ x 3, $S_3\times D_{10}$ x 2, $C_2\times A_5$ x 2, $C_6.D_{10}$, $C_6\times D_{10}$, $D_5\times A_4$, $C_5:S_4$, $C_5\times S_4$, $C_{15}:D_4$, $C_{15}:D_4$, $C_3:D_{20}$, $C_{15}:D_4$

Order 108: $C_3\times S_3^2$ x 47, $C_3:S_3^2$ x 41, $C_3:S_3^2$ x 20, $C_3^2:D_6$ x 9, $C_3^2\times D_6$ x 9, $C_3^2\times A_4$ x 8, $C_3^3:C_4$ x 5, $C_3^2:C_{12}$ x 5, $C_3^2:D_6$ x 4, $C_3^2:D_6$ x 3, $C_3^2:C_{12}$ x 2, $C_3^2:D_6$ x 2, $C_3^2:D_6$ x 2, $C_3^2:A_4$ x 2, $C_3^2.A_4$ x 2, $C_9:C_{12}$, $C_3\times C_6^2$, $C_3^3:C_4$, $C_3^2:C_{12}$, $C_3^2:C_{12}$, $C_{18}:C_6$, $C_2^2\times \He_3$, $C_{18}:C_6$, $C_9:A_4$, $C_9\times A_4$, $S_3\times D_9$

Order 96: $C_2^2\times S_4$ x 5, $D_4\times D_6$ x 4, $\GL(2,\mathbb{Z}/4)$ x 4, $D_6:D_4$ x 2, $C_2^2:D_{12}$ x 2, $C_2^3.D_6$ x 2, $C_2^3:D_6$ x 2, $D_4\times A_4$ x 2, $C_4:S_4$ x 2, $C_4\times S_4$ x 2, $C_2^3.D_6$ x 2, $C_2^2:S_4$, $C_2^4:C_6$, $C_{12}:D_4$, $D_6:D_4$

Order 90: $C_3\times D_{15}$ x 6, $S_3\times C_{15}$ x 6, $C_3^2\times D_5$ x 5, $C_3:D_{15}$, $C_{15}:S_3$, $C_9\times D_5$, $C_3\times C_{30}$

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

Order 80: $C_2^2\times F_5$ x 2, $D_{10}:C_4$ x 2, $D_4\times D_5$, $C_{20}:C_4$, $C_4\times F_5$

Order 72: $S_3\times D_6$ x 59, $C_3:S_4$ x 28, $C_6\times D_6$ x 19, $S_3\times A_4$ x 16, $C_3\times S_4$ x 16, $C_6\wr C_2$ x 14, $C_3:D_{12}$ x 14, $\SOPlus(4,2)$ x 13, $C_6\times A_4$ x 10, $C_6^2:C_2$ x 10, $D_6:S_3$ x 10, $C_6.D_6$ x 10, $C_6:D_6$ x 6, $C_2\times C_3^2:C_4$ x 6, $C_3\times D_{12}$ x 4, $S_3\times C_{12}$ x 4, $C_6.D_6$ x 3, $D_4\times C_3^2$ x 2, $C_3:D_{12}$ x 2, $C_{12}:S_3$ x 2, $C_2^2:D_9$ x 2, $C_9:D_4$, $C_2\times C_6^2$, $C_6.D_6$, $C_6:C_{12}$

Order 64: $D_4^2$

Order 60: $C_{15}:C_4$ x 8, $S_3\times D_5$ x 7, $C_3\times D_{10}$ x 6, $C_3\times F_5$ x 5, $D_{30}$ x 2, $S_3\times C_{10}$ x 2, $C_5\times A_4$, $A_5$, $C_{15}:C_4$, $C_2\times C_{30}$, $C_3:C_{20}$

Order 54: $C_3^2:C_6$ x 28, $S_3\times C_3^2$ x 28, $C_3^2:S_3$ x 10, $C_3^2:C_6$ x 6, $C_3^2:S_3$ x 4, $C_9:C_6$ x 3, $C_3^2\times C_6$ x 3, $C_3:D_9$, $S_3\times C_9$, $C_3\times D_9$, $C_9:C_6$, $C_2\times \He_3$

Order 48: $S_3\times D_4$ x 26, $C_2\times S_4$ x 20, $C_6:D_4$ x 19, $C_2^2\times D_6$ x 7, $C_2^2\times A_4$ x 5, $C_6\times D_4$ x 5, $C_2\times D_{12}$ x 5, $C_4\times D_6$ x 4, $D_6:C_4$ x 4, $C_2^2:C_{12}$ x 3, $C_6.C_2^3$ x 2, $C_4\times A_4$ x 2, $A_4:C_4$ x 2, $C_6.D_4$ x 2, $C_2^3\times C_6$, $C_2^2:A_4$, $C_6.D_4$, $C_{12}:C_4$

Order 45: $C_3\times C_{15}$ x 4, $C_{45}$

Order 40: $C_2\times F_5$ x 6, $C_5:D_4$ x 2, $C_2\times D_{10}$ x 2, $D_{20}$, $C_4\times D_5$, $C_5\times D_4$

Order 36: $S_3^2$ x 88, $C_6\times S_3$ x 67, $C_6:S_3$ x 31, $C_3\times A_4$ x 18, $C_6^2$ x 10, $C_3:C_{12}$ x 8, $C_3^2:C_4$ x 6, $C_3^2:C_4$ x 4, $C_3\times C_{12}$ x 2, $C_2^2:C_9$ x 2, $C_2\times C_{18}$, $D_{18}$, $C_9:C_4$

Order 32: $C_2^2\times D_4$ x 4, $C_4:D_4$ x 4, $C_2^2\wr C_2$ x 4, $C_4\times D_4$ x 2, $C_4:D_4$

Order 30: $C_3\times D_5$ x 8, $C_{30}$ x 4, $D_{15}$ x 4, $C_5\times S_3$ x 4

Order 27: $C_3^3$ x 8, $C_9:C_3$ x 3, $\He_3$ x 3, $C_3\times C_9$

Order 24: $C_2\times D_6$ x 52, $C_3:D_4$ x 43, $S_4$ x 20, $D_{12}$ x 15, $C_3\times D_4$ x 15, $C_4\times S_3$ x 14, $C_2^2\times C_6$ x 11, $C_2\times A_4$ x 11, $C_6:C_4$ x 9, $C_2\times C_{12}$ x 5

Order 20: $D_{10}$ x 5, $F_5$ x 4, $C_2\times C_{10}$ x 2, $C_{20}$, $C_5:C_4$

Order 18: $C_3\times S_3$ x 66, $C_3:S_3$ x 33, $C_3\times C_6$ x 20, $D_9$ x 2, $C_{18}$

Order 16: $C_2\times D_4$ x 24, $C_2^2:C_4$ x 8, $C_2^4$ x 4, $C_2^2\times C_4$ x 4, $C_4:C_4$ x 2, $C_4^2$

Order 15: $C_{15}$ x 4

Order 12: $D_6$ x 87, $C_2\times C_6$ x 34, $C_3:C_4$ x 13, $A_4$ x 9, $C_{12}$ x 9

Order 10: $D_5$ x 3, $C_{10}$ x 2

Order 9: $C_3^2$ x 17, $C_9$ x 2

Order 8: $D_4$ x 24, $C_2^3$ x 17, $C_2\times C_4$ x 11

Order 6: $S_3$ x 34, $C_6$ x 29

Order 5: $C_5$

Order 4: $C_2^2$ x 25, $C_4$ x 7

Order 3: $C_3$ x 9

Order 2: $C_2$ x 8

Order 1: $C_1$

Classes of subgroups up to automorphism

Order 77760: $C_3^3:(S_4\times S_5)$

Order 38880: $C_3^3:(A_4\times S_5)$, $A_5\times C_3^3:S_4$, $(C_3^3\times A_5):S_4$

Order 25920: $S_3^2:(S_3\times S_5)$

Order 19440: $C_3^3:(S_3\times S_5)$, $A_5\times C_3^3:A_4$

Order 15552: $C_3^3:S_4^2$

Order 12960: $C_3^3:(C_4\times S_5)$, $C_3\times S_3^2:S_5$, $A_5\times S_3^2:S_3$, $\GL(2,4):\SOPlus(4,2)$, $A_5:S_3^3$, $\GL(2,4):\SOPlus(4,2)$, $A_5:S_3^3$, $C_3^3:(F_5\times S_4)$

Order 9720: $C_3^3:(C_3\times S_5)$, $A_5\times C_3\wr S_3$, $C_3\wr C_3:S_5$

Order 8640: $S_5\times \SOPlus(4,2)$

Order 7776: $C_3^3:(A_4\times S_4)$, $(A_4\times C_3^3):S_4$, $A_4\times C_3^3:S_4$, $S_3^4:S_3$

Order 6480: $(C_3\times \GL(2,4)):C_{12}$, $\GL(2,4):S_3^2$, $\GL(2,4):S_3^2$, $S_3^2\times \GL(2,4)$, $A_5\times C_3^3:C_4$, $\GL(2,4):S_3^2$, $\GL(2,4):S_3^2$, $\GL(2,4):S_3^2$, $\GL(2,4):S_3^2$, $\GL(2,4).S_3^2$, $C_3^3:(A_4\times F_5)$, $D_5\times C_3^3:S_4$, $(D_5\times C_3^3).S_4$

Order 5184: $D_6^2:S_3^2$, $C_3^3:(D_4\times S_4)$

Order 4860: $C_3^3:\GL(2,4)$

Order 4320: $S_3^2\times S_5$ x 2, $S_3^2:S_5$ x 2, $S_3^2:(S_3\times F_5)$, $S_3\times C_6:S_5$, $A_5:\SOPlus(4,2)$, $C_3^2:C_4\times S_5$, $A_5\times \SOPlus(4,2)$

Order 3888: $C_3^3:(S_3\times S_4)$, $C_3^3:A_4^2$, $S_3\times C_3^3:S_4$, $S_3\times C_3^3:S_4$, $C_3^3:(S_3\times S_4)$, $(C_3\times S_3^3):S_3$, $S_3^4:C_3$, $C_3\times S_3\wr S_3$, $C_3^3:(S_3\times S_4)$

Order 3240: $\He_3:S_5$ x 2, $C_3^3:S_5$, $C_3:S_3\times \GL(2,4)$, $C_3\times S_3\times \GL(2,4)$, $C_3^3:S_5$, $C_3^3:S_5$, $C_3^2:S_3\times A_5$, $C_9:(C_3\times S_5)$, $C_3^3:(S_3\times F_5)$, $C_5\times C_3^3:S_4$, $D_5\times C_3^3:A_4$, $(C_3^2\times C_{15}):S_4$

Order 2880: $C_3:(D_4\times S_5)$, $S_4\times S_5$

Order 2592: $C_3^3:(C_4\times S_4)$, $(C_3\times C_6^2):D_{12}$, $C_3^3:\GL(2,\mathbb{Z}/4)$, $C_3^3:\GL(2,\mathbb{Z}/4)$, $A_4:S_3^3$, $A_4\times S_3^2:S_3$, $A_4:S_3^3$, $C_3^3:(C_4\times S_4)$, $C_3^3:\GL(2,\mathbb{Z}/4)$, $C_2^2\times C_3^3:S_4$, $C_2.S_3\wr S_3$, $C_3^3:\GL(2,\mathbb{Z}/4)$, $(C_2\times S_3^3):C_6$, $S_3^4:C_2$, $S_3^3:D_6$, $(C_3\times C_6^2):S_4$

Order 2160: $A_5:S_3^2$ x 8, $S_3^2\times A_5$ x 2, $A_5:S_3^2$ x 2, $\GL(2,4):D_6$ x 2, $(C_3^2\times C_{15}):C_4^2$, $C_3\times S_3^2:F_5$, $(C_3^2\times D_5).D_{12}$, $D_5.S_3^3$, $D_5.S_3^3$, $\GL(2,4):D_6$, $\GL(2,4):D_6$, $D_6\times \GL(2,4)$, $D_5\times S_3^2:S_3$, $(D_5\times C_3^3).D_4$, $\GL(2,4):D_6$, $C_3^2:C_4\times A_5$, $(C_3\times \GL(2,4)):C_4$

Order 1944: $S_3\times C_3^3:A_4$, $C_3^3:(S_3\times A_4)$, $S_3^3:C_3^2$, $C_3^4:S_4$, $C_3^4:S_4$, $C_3^4:S_4$, $C_3\wr S_4$, $A_4\times C_3\wr S_3$, $C_3\wr C_3:S_4$, $C_3^3:(C_3\times S_4)$, $C_3^2.S_3^3$

Order 1728: $D_6^2:D_6$, $D_6^2:D_6$

Order 1620: $C_3\wr C_3:F_5$, $(C_3^2\times C_{15}):C_{12}$, $C_5\times C_3^3:A_4$, $C_3^3\times A_5$, $C_9:\GL(2,4)$, $\He_3\times A_5$, $C_3\wr S_3\times D_5$

Order 1440: $D_6\times S_5$ x 2, $F_5\times \SOPlus(4,2)$, $\GL(2,4):C_2^3$, $S_4\times A_5$, $A_4:S_5$, $A_4\times S_5$, $\GL(2,4):D_4$, $\GL(2,4):D_4$, $\GL(2,4):D_4$, $D_6:S_5$, $C_3:(C_4\times S_5)$

Order 1296: $S_3^3:S_3$ x 4, $S_3\wr S_3$ x 3, $S_3^4$ x 2, $S_3\times C_3^2:D_{12}$ x 2, $S_3^2:S_3^2$ x 2, $S_3^3:C_6$ x 2, $C_2\times C_3^3:S_4$ x 2, $C_6^2.S_3^2$, $A_4\times C_3:S_3^2$, $C_6^2:C_6^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $C_6^2:S_3^2$, $(C_3\times C_6^2):A_4$, $C_2^2\times C_3^3:A_4$, $A_4\times C_3^3:C_4$, $(C_3^2\times A_4):C_{12}$, $C_3^3:(S_3\times D_4)$, $(C_3\times S_3^2):D_6$, $C_3^2:C_4\times S_3^2$, $S_3^3:C_6$, $C_2\times C_3^3:S_4$, $C_3^3:(C_4\times A_4)$, $(C_3^2\times C_6).S_4$, $C_6^2:S_3^2$, $C_6^2.S_3^2$

Order 1080: $S_3\times \GL(2,4)$ x 6, $C_3^2:S_5$ x 6, $C_3^2:S_5$ x 5, $D_5\times C_3:S_3^2$, $C_3\times D_5\times S_3^2$, $C_3^3:D_{20}$, $C_{15}:\SOPlus(4,2)$, $S_3^2:D_{15}$, $C_5\times S_3^2:S_3$, $C_3^3:C_4\times D_5$, $(C_3^2\times D_5):C_{12}$, $C_3^2:(S_3\times F_5)$, $C_3^2:(S_3\times F_5)$, $S_3\times C_3^2:F_5$, $C_{15}:(S_3\times C_{12})$, $S_3\times C_{15}:C_{12}$, $C_6\times \GL(2,4)$, $\GL(2,4):S_3$, $C_3^2\times S_5$, $C_3^2:(S_3\times F_5)$, $C_9:S_5$

Order 972: $C_3\wr A_4$, $C_6^2.C_3^3$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:D_6$, $\He_3:S_3^2$, $C_3^3:S_3^2$, $C_3^4:D_6$

Order 960: $D_4\times S_5$

Order 864: $S_4\times S_3^2$ x 3, $S_3^3:C_2^2$ x 2, $S_3^2:S_4$ x 2, $D_6^2:S_3$, $S_3\times C_6:S_4$, $D_6^2:C_6$, $C_6^2:D_{12}$, $C_3^2:C_4\times S_4$, $C_2^2:S_3^3$, $D_6^2:S_3$, $C_6^2:D_{12}$, $(C_3\times C_6^2):D_4$, $D_6^2:C_6$, $C_6^2:(C_4\times S_3)$, $(S_3\times D_6).D_6$, $(S_3\times D_6).D_6$, $(C_3\times S_3^2):D_4$, $D_6:\SOPlus(4,2)$, $S_3^2:D_{12}$, $D_6:\SOPlus(4,2)$, $S_3^2:(C_4\times S_3)$

Order 810: $C_3^3:D_{15}$, $\He_3:C_{30}$, $C_3\wr C_3\times D_5$

Order 720: $S_3\times S_5$ x 7, $C_6:S_5$ x 6, $F_5\times S_3^2$ x 2, $S_3^2:F_5$ x 2, $D_6\times A_5$ x 2, $C_6\times S_5$ x 2, $D_{10}.S_3^2$, $S_3^2:D_{10}$, $D_5.\SOPlus(4,2)$, $C_3^2:C_4\times F_5$, $C_2^2\times \GL(2,4)$, $A_4\times A_5$, $\GL(2,4):C_4$, $A_5:C_{12}$

Order 648: $C_3:S_3^3$ x 5, $C_3:S_3^3$ x 5, $C_3\times S_3^3$ x 5, $C_3^3:D_{12}$ x 4, $C_3\wr D_4$ x 4, $C_3^3:S_4$ x 3, $C_3^3:S_4$ x 3, $C_3:S_3^3$ x 2, $C_3^4:D_4$ x 2, $C_3^4:D_4$ x 2, $C_3^3:D_{12}$ x 2, $C_3^3:C_4\times S_3$ x 2, $S_3\times C_3^2:C_{12}$ x 2, $S_3\wr C_3$ x 2, $\He_3:S_4$ x 2, $C_3\times C_6^2:C_6$, $S_3\times C_3^2\times A_4$, $C_3^3:S_4$, $C_3^3:S_4$, $C_3^3:S_4$, $C_3^3:D_{12}$, $C_3^4:D_4$, $C_3^3:(C_4\times S_3)$, $C_3^3:(C_4\times S_3)$, $C_2\times C_3^3:A_4$, $C_3\wr C_3:D_4$, $C_3^3:D_{12}$, $C_3^3:(C_4\times S_3)$, $A_4\times C_3^2:S_3$, $C_9:(C_3\times S_4)$, $C_3.S_3^3$, $C_2^2\times C_3\wr S_3$, $C_2\times C_3^3:D_6$, $(C_3^2\times A_4):S_3$, $C_3^3:S_4$, $(C_3^2\times A_4):S_3$, $\He_3:S_4$, $\He_3:S_4$, $(C_3\times C_6^2):S_3$, $(C_3\times C_6^2):C_6$

Order 576: $S_4^2$, $C_3^2:D_4^2$, $\GL(2,\mathbb{Z}/4):S_3$

Order 540: $C_3\times \GL(2,4)$ x 4, $\He_3:F_5$ x 2, $C_{15}:C_6^2$, $(C_3^2\times C_{15}):C_4$, $C_3^3:C_{20}$, $C_3^3:F_5$, $C_3^3:F_5$, $C_3^3:F_5$, $C_3^3:F_5$, $\He_3:D_{10}$, $C_{45}:C_{12}$, $C_9\times A_5$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}:S_3^2$, $C_{15}\times S_3^2$, $C_3^3:D_{10}$

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

Order 480: $C_2^2:S_5$ x 2, $C_2^2\times S_5$ x 2, $D_4\times A_5$, $C_4:S_5$, $C_4\times S_5$, $F_5\times S_4$, $C_{15}:(C_4\times D_4)$

Order 432: $S_3^3:C_2$ x 13, $A_4:S_3^2$ x 11, $S_3^2:D_6$ x 7, $C_2\times S_3^3$ x 5, $C_2\times C_3^2:D_{12}$ x 3, $C_6\times \SOPlus(4,2)$ x 3, $A_4\times S_3^2$ x 3, $A_4:S_3^2$ x 3, $C_6^2:D_6$ x 3, $C_3^2:C_4\times D_6$ x 2, $C_6^2:D_6$ x 2, $D_6:S_3^2$ x 2, $C_6.\SOPlus(4,2)$ x 2, $S_3^2:C_{12}$ x 2, $C_3:D_6^2$, $C_3\times D_6^2$, $C_2^2\times C_3^3:C_4$, $C_6\times S_3\times A_4$, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2:C_{12}$, $(C_3^2\times A_4):C_4$, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2:D_6$, $C_6^2:C_{12}$, $D_6:S_3^2$, $D_6:S_3^2$, $D_6:S_3^2$, $D_6:S_3^2$, $C_2.S_3^3$, $C_2.S_3^3$, $C_6.\SOPlus(4,2)$, $(C_3\times S_3^2):C_4$, $C_6.\SOPlus(4,2)$, $C_3^2:C_{12}:C_4$, $(C_3^2\times D_6):C_4$, $C_3^3:C_4^2$, $C_6^2:D_6$

Order 405: $C_3^3:C_{15}$

Order 360: $D_5.S_3^2$ x 8, $C_3:S_5$ x 8, $C_6\times A_5$ x 4, $S_3\times A_5$ x 4, $C_3\times S_5$ x 4, $D_5\times S_3^2$ x 2, $S_3^2:D_5$ x 2, $D_5.S_3^2$ x 2, $D_{15}:C_{12}$ x 2, $C_{30}:D_6$, $C_2\times C_3^2:F_5$, $C_{30}:C_{12}$, $C_3^2:D_{20}$, $S_3^2:C_{10}$, $(C_3^2\times D_5):C_4$, $C_3^2:C_4\times D_5$, $C_3:S_3\times F_5$

Order 324: $C_3^2:S_3^2$ x 8, $C_3^2:S_3^2$ x 5, $C_3\wr C_2^2$ x 5, $C_3^2\times S_3^2$ x 5, $C_3^3:D_6$ x 4, $C_3^2:S_3^2$ x 3, $C_3^3:A_4$ x 3, $C_3^3:D_6$ x 2, $C_3^2:S_3^2$ x 2, $C_3\wr C_4$ x 2, $C_3^2:S_3^2$ x 2, $C_3^2:S_3^2$ x 2, $C_3^2.C_6^2$, $C_3^3:D_6$, $\He_3:D_6$, $C_3^3:A_4$, $C_6^2.C_3^2$, $\He_3:A_4$, $C_3\wr C_3:C_4$, $A_4\times C_3^3$, $C_3^4:C_4$, $C_3^3:C_{12}$, $C_3^3:C_{12}$, $C_6^2.C_3^2$, $A_4\times \He_3$, $C_3^2:S_3^2$, $C_3^2:S_3^2$, $C_3^2.S_3^2$, $C_3^3:D_6$

Order 288: $D_6\wr C_2$ x 4, $D_6\times S_4$ x 4, $D_4\times S_3^2$ x 3, $C_6^2:D_4$ x 2, $S_3^2:D_4$ x 2, $C_6^2:D_4$ x 2, $A_4\times S_4$ x 2, $D_6^2:C_2$, $C_3^2:C_4\times D_4$, $A_4\times C_3:D_4$, $C_3\times \GL(2,\mathbb{Z}/4)$, $C_4:\SOPlus(4,2)$, $C_4\times \SOPlus(4,2)$, $D_6:S_4$, $C_2^3.S_3^2$, $C_2^3.S_3^2$, $C_2\times C_6:S_4$, $\PSOPlus(4,3)$

Order 270: $C_{45}:C_6$, $D_5\times \He_3$, $C_3^2:D_{15}$, $C_3^2:C_{30}$, $C_3^2\times D_{15}$, $C_3^2:C_{30}$, $D_5\times C_3^3$, $C_3^2:D_{15}$, $\He_3:C_{10}$

Order 243: $C_3^4:C_3$

Order 240: $C_2\times S_5$ x 5, $D_6\times F_5$ x 2, $C_2^2\times A_5$ x 2, $D_{10}.D_6$, $D_{30}:C_4$, $C_{15}:C_4^2$, $C_4\times A_5$, $A_5:C_4$, $D_{10}.D_6$, $D_5\times S_4$, $A_4\times F_5$, $A_4:F_5$, $D_6:D_{10}$, $D_{10}:C_{12}$

Order 216: $S_3^3$ x 30, $S_3^2:S_3$ x 17, $C_6\times S_3^2$ x 12, $C_6^2:C_6$ x 9, $C_3^2:S_4$ x 9, $C_6:S_3^2$ x 8, $C_3^2:S_4$ x 8, $C_3^2:D_{12}$ x 8, $S_3^2:C_6$ x 8, $C_3^2:C_4\times S_3$ x 7, $C_6:S_3^2$ x 6, $C_2\times C_3^3:C_4$ x 3, $C_2\times C_3^2:C_{12}$ x 3, $C_3^2:S_4$ x 2, $C_6^2:C_6$ x 2, $C_3^2:D_{12}$ x 2, $C_6^2:C_6$ x 2, $C_3^2\times S_4$ x 2, $C_3^3:D_4$ x 2, $C_3^2:D_{12}$ x 2, $C_9:S_4$, $C_3^2.S_4$, $D_{18}:C_6$, $C_6.S_3^2$, $C_6^2:C_6$, $S_3\times C_6^2$, $C_6^2:C_6$, $C_6^2:S_3$, $C_6^2:C_6$, $C_6^2:C_6$, $C_3^2:D_{12}$, $C_6.S_3^2$, $C_3^2:D_{12}$, $C_3^2:D_{12}$, $C_6.S_3^2$, $C_6.S_3^2$, $C_3^3:D_4$, $C_6.S_3^2$, $C_6.S_3^2$, $C_6^2:S_3$, $C_6.S_3^2$

Order 192: $D_4\times S_4$ x 2, $C_2^4:D_6$

Order 180: $C_{15}:D_6$ x 8, $C_{15}:C_{12}$ x 6, $C_3^2:F_5$ x 5, $\GL(2,4)$ x 4, $C_{15}:D_6$ x 2, $S_3\times D_{15}$ x 2, $C_5\times S_3^2$ x 2, $C_{45}:C_4$, $C_3\times D_{30}$, $S_3\times C_{30}$, $C_3^2\times D_{10}$, $C_{15}:D_6$, $C_3^2:F_5$, $(C_3\times C_{15}):C_4$, $C_3^2:C_{20}$, $C_3^2\times F_5$

Order 162: $C_3^2\wr C_2$ x 4, $C_3^3:S_3$ x 4, $C_3\wr S_3$ x 4, $C_3^3:C_6$ x 3, $S_3\times C_3^3$ x 3, $C_3^3:C_6$ x 3, $C_3^3:C_6$ x 2, $C_3^3:C_6$ x 2, $C_3^3:S_3$, $C_3^3:S_3$, $C_3^3.S_3$, $C_3^3:S_3$, $C_3^3.C_6$, $D_9:C_3^2$, $S_3\times \He_3$, $C_3^3:C_6$

Order 160: $D_4\times F_5$

Order 144: $S_3\times S_4$ x 19, $D_6:D_6$ x 18, $S_3^2:C_2^2$ x 13, $C_6:S_4$ x 10, $D_6^2$ x 8, $D_6:D_6$ x 6, $S_3^2:C_4$ x 6, $A_4\times D_6$ x 4, $C_6\times S_4$ x 4, $C_{12}:D_6$ x 4, $S_3\times D_{12}$ x 4, $C_{12}:D_6$ x 3, $C_4\times S_3^2$ x 3, $C_6^2:C_4$ x 2, $C_{12}:D_6$ x 2, $C_6^2:C_4$ x 2, $C_6^2:C_2^2$, $C_2^2:C_6^2$, $A_4^2$, $C_6^2:C_2^2$, $C_6^2:C_2^2$, $C_6:D_{12}$, $D_6:D_6$, $D_6.D_6$, $(C_3\times C_{12}):C_4$, $C_3^2:C_4^2$, $A_4\times C_3:C_4$, $A_4:C_{12}$, $C_2.\SOPlus(4,2)$

Order 135: $C_3^2\times C_{15}$, $C_9:C_{15}$, $C_5\times \He_3$

Order 120: $S_3\times F_5$ x 7, $C_{30}:C_4$ x 6, $C_6\times F_5$ x 3, $S_5$ x 3, $S_3\times D_{10}$ x 2, $C_2\times A_5$ x 2, $C_6.D_{10}$, $C_6\times D_{10}$, $D_5\times A_4$, $C_5:S_4$, $C_5\times S_4$, $C_{15}:D_4$, $C_{15}:D_4$, $C_3:D_{20}$, $C_{15}:D_4$

Order 108: $C_3\times S_3^2$ x 47, $C_3:S_3^2$ x 41, $C_3:S_3^2$ x 20, $C_3^2:D_6$ x 9, $C_3^2\times D_6$ x 9, $C_3^2\times A_4$ x 7, $C_3^3:C_4$ x 5, $C_3^2:C_{12}$ x 5, $C_3^2:D_6$ x 4, $C_3^2:D_6$ x 3, $C_3^2:C_{12}$ x 2, $C_3^2:D_6$ x 2, $C_3^2:D_6$ x 2, $C_3^2:A_4$ x 2, $C_9:C_{12}$, $C_3\times C_6^2$, $C_3^3:C_4$, $C_3^2:C_{12}$, $C_3^2:C_{12}$, $C_{18}:C_6$, $C_2^2\times \He_3$, $C_{18}:C_6$, $C_3^2.A_4$, $C_9:A_4$, $C_9\times A_4$, $S_3\times D_9$

Order 96: $C_2^2\times S_4$ x 5, $D_4\times D_6$ x 4, $\GL(2,\mathbb{Z}/4)$ x 4, $D_6:D_4$ x 2, $C_2^2:D_{12}$ x 2, $C_2^3.D_6$ x 2, $C_2^3:D_6$ x 2, $D_4\times A_4$ x 2, $C_4:S_4$ x 2, $C_4\times S_4$ x 2, $C_2^3.D_6$ x 2, $C_2^2:S_4$, $C_2^4:C_6$, $C_{12}:D_4$, $D_6:D_4$

Order 90: $C_3\times D_{15}$ x 6, $S_3\times C_{15}$ x 6, $C_3^2\times D_5$ x 5, $C_3:D_{15}$, $C_{15}:S_3$, $C_9\times D_5$, $C_3\times C_{30}$

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

Order 80: $C_2^2\times F_5$ x 2, $D_{10}:C_4$ x 2, $D_4\times D_5$, $C_{20}:C_4$, $C_4\times F_5$

Order 72: $S_3\times D_6$ x 59, $C_3:S_4$ x 22, $C_6\times D_6$ x 19, $C_6\wr C_2$ x 14, $C_3:D_{12}$ x 14, $\SOPlus(4,2)$ x 13, $S_3\times A_4$ x 12, $C_3\times S_4$ x 12, $C_6^2:C_2$ x 10, $D_6:S_3$ x 10, $C_6.D_6$ x 10, $C_6\times A_4$ x 8, $C_6:D_6$ x 6, $C_2\times C_3^2:C_4$ x 6, $C_3\times D_{12}$ x 4, $S_3\times C_{12}$ x 4, $C_6.D_6$ x 3, $D_4\times C_3^2$ x 2, $C_3:D_{12}$ x 2, $C_{12}:S_3$ x 2, $C_9:D_4$, $C_2\times C_6^2$, $C_6.D_6$, $C_6:C_{12}$, $C_2^2:D_9$

Order 64: $D_4^2$

Order 60: $C_{15}:C_4$ x 8, $S_3\times D_5$ x 7, $C_3\times D_{10}$ x 6, $C_3\times F_5$ x 5, $D_{30}$ x 2, $S_3\times C_{10}$ x 2, $C_5\times A_4$, $A_5$, $C_{15}:C_4$, $C_2\times C_{30}$, $C_3:C_{20}$

Order 54: $C_3^2:C_6$ x 28, $S_3\times C_3^2$ x 28, $C_3^2:S_3$ x 10, $C_3^2:C_6$ x 6, $C_3^2:S_3$ x 4, $C_9:C_6$ x 3, $C_3^2\times C_6$ x 3, $C_3:D_9$, $S_3\times C_9$, $C_3\times D_9$, $C_9:C_6$, $C_2\times \He_3$

Order 48: $S_3\times D_4$ x 26, $C_6:D_4$ x 19, $C_2\times S_4$ x 17, $C_2^2\times D_6$ x 7, $C_2^2\times A_4$ x 5, $C_6\times D_4$ x 5, $C_2\times D_{12}$ x 5, $C_4\times D_6$ x 4, $D_6:C_4$ x 4, $C_2^2:C_{12}$ x 3, $C_6.C_2^3$ x 2, $C_4\times A_4$ x 2, $A_4:C_4$ x 2, $C_6.D_4$ x 2, $C_2^3\times C_6$, $C_2^2:A_4$, $C_6.D_4$, $C_{12}:C_4$

Order 45: $C_3\times C_{15}$ x 4, $C_{45}$

Order 40: $C_2\times F_5$ x 6, $C_5:D_4$ x 2, $C_2\times D_{10}$ x 2, $D_{20}$, $C_4\times D_5$, $C_5\times D_4$

Order 36: $S_3^2$ x 88, $C_6\times S_3$ x 67, $C_6:S_3$ x 31, $C_3\times A_4$ x 14, $C_6^2$ x 10, $C_3:C_{12}$ x 8, $C_3^2:C_4$ x 6, $C_3^2:C_4$ x 4, $C_3\times C_{12}$ x 2, $C_2\times C_{18}$, $D_{18}$, $C_2^2:C_9$, $C_9:C_4$

Order 32: $C_2^2\times D_4$ x 4, $C_4:D_4$ x 4, $C_2^2\wr C_2$ x 4, $C_4\times D_4$ x 2, $C_4:D_4$

Order 30: $C_3\times D_5$ x 8, $C_{30}$ x 4, $D_{15}$ x 4, $C_5\times S_3$ x 4

Order 27: $C_3^3$ x 8, $C_9:C_3$ x 3, $\He_3$ x 3, $C_3\times C_9$

Order 24: $C_2\times D_6$ x 52, $C_3:D_4$ x 43, $S_4$ x 17, $D_{12}$ x 15, $C_3\times D_4$ x 15, $C_4\times S_3$ x 14, $C_2^2\times C_6$ x 11, $C_6:C_4$ x 9, $C_2\times A_4$ x 9, $C_2\times C_{12}$ x 5

Order 20: $D_{10}$ x 5, $F_5$ x 4, $C_2\times C_{10}$ x 2, $C_{20}$, $C_5:C_4$

Order 18: $C_3\times S_3$ x 66, $C_3:S_3$ x 33, $C_3\times C_6$ x 20, $D_9$ x 2, $C_{18}$

Order 16: $C_2\times D_4$ x 24, $C_2^2:C_4$ x 8, $C_2^4$ x 4, $C_2^2\times C_4$ x 4, $C_4:C_4$ x 2, $C_4^2$

Order 15: $C_{15}$ x 4

Order 12: $D_6$ x 87, $C_2\times C_6$ x 34, $C_3:C_4$ x 13, $C_{12}$ x 9, $A_4$ x 8

Order 10: $D_5$ x 3, $C_{10}$ x 2

Order 9: $C_3^2$ x 17, $C_9$ x 2

Order 8: $D_4$ x 24, $C_2^3$ x 17, $C_2\times C_4$ x 11

Order 6: $S_3$ x 34, $C_6$ x 29

Order 5: $C_5$

Order 4: $C_2^2$ x 25, $C_4$ x 7

Order 3: $C_3$ x 9

Order 2: $C_2$ x 8

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 77760: $C_3^3:(S_4\times S_5)$ ($C_1$)

Order 38880: $C_3^3:(A_4\times S_5)$ ($C_2$), $A_5\times C_3^3:S_4$ ($C_2$), $(C_3^3\times A_5):S_4$ ($C_2$)

Order 19440: $A_5\times C_3^3:A_4$ ($C_2^2$)

Order 12960: $A_5:S_3^3$ ($S_3$)

Order 6480: $\GL(2,4):S_3^2$ ($D_6$)

Order 3240: $C_3^3:S_5$ ($S_4$)

Order 1620: $C_3^3\times A_5$ ($C_2\times S_4$)

Order 648: $C_3^3:S_4$ ($S_5$)

Order 324: $C_3^3:A_4$ ($C_2\times S_5$)

Order 108: $C_3:S_3^2$ ($S_3\times S_5$)

Order 60: $A_5$ ($S_3\wr S_3$)

Order 27: $C_3^3$ ($S_4\times S_5$)

Order 1: $C_1$ ($C_3^3:(S_4\times S_5)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 77760: $C_3^3:(S_4\times S_5)$ ($C_1$)

Order 38880: $C_3^3:(A_4\times S_5)$ ($C_2$), $A_5\times C_3^3:S_4$ ($C_2$), $(C_3^3\times A_5):S_4$ ($C_2$)

Order 19440: $A_5\times C_3^3:A_4$ ($C_2^2$)

Order 12960: $A_5:S_3^3$ ($S_3$)

Order 6480: $\GL(2,4):S_3^2$ ($D_6$)

Order 3240: $C_3^3:S_5$ ($S_4$)

Order 1620: $C_3^3\times A_5$ ($C_2\times S_4$)

Order 648: $C_3^3:S_4$ ($S_5$)

Order 324: $C_3^3:A_4$ ($C_2\times S_5$)

Order 108: $C_3:S_3^2$ ($S_3\times S_5$)

Order 60: $A_5$ ($S_3\wr S_3$)

Order 27: $C_3^3$ ($S_4\times S_5$)

Order 1: $C_1$ ($C_3^3:(S_4\times S_5)$)

Series

Derived series

$C_3^3:(S_4\times S_5)$

$\rhd$

$C_3^3:(S_4\times S_5)$

$\rhd$

$A_5\times C_3^3:A_4$

$\rhd$

$A_5\times C_3^3:A_4$

$\rhd$

$\GL(2,4):S_3^2$

$\rhd$

$\GL(2,4):S_3^2$

$\rhd$

$C_3^3\times A_5$

$\rhd$

$C_3^3\times A_5$

$\rhd$

$A_5$

$\rhd$

$A_5$

Chief series

$C_3^3:(S_4\times S_5)$

$\rhd$

$C_3^3:(S_4\times S_5)$

$\rhd$

$A_5\times C_3^3:S_4$

$\rhd$

$A_5\times C_3^3:S_4$

$\rhd$

$C_3^3:S_4$

$\rhd$

$C_3^3:S_4$

$\rhd$

$C_3^3:A_4$

$\rhd$

$C_3^3:A_4$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3:S_3^2$

$\rhd$

$C_3^3$

$\rhd$

$C_3^3$

$\rhd$

$C_1$

$\rhd$

$C_1$

Lower central series

$C_3^3:(S_4\times S_5)$

$\rhd$

$C_3^3:(S_4\times S_5)$

$\rhd$

$A_5\times C_3^3:A_4$

$\rhd$

$A_5\times C_3^3:A_4$

Upper central series

$C_1$

$\lhd$

$C_1$

Supergroups

This group is a maximal subgroup of 3 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 $83 \times 83$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $79 \times 79$ rational character table.