Abstract group 110592.cy: $A_4^2.C_2\wr D_6$

• Label: 110592.cy
• Order: \( 2^{12} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \cdot 3 \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{15} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: not computed
• Trans deg.: $36$
• Rank: $3$

Group information

Description:	$A_4^2.C_2\wr D_6$
Order:	\(110592\)\(\medspace = 2^{12} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2\times C_2^8.C_3^3.C_2^6$, of order \(884736\)\(\medspace = 2^{15} \cdot 3^{3} \)
Composition factors:	$C_2$ x 12, $C_3$ x 3
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	2175	2672	14208	40080	51456	110592
Conjugacy classes	1	57	11	44	184	95	392
Divisions	1	57	7	44	101	50	260
Autjugacy classes	1	43	7	30	74	28	183

Minimal presentations

Permutation degree:	not computed
Transitive degree:	$36$
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	18	not computed	not computed
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k, l \mid c^{6}=d^{2}=e^{2}=f^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $36$ $\langle(1,12)(2,11)(3,9)(4,10)(5,8)(6,7)(13,24,14,23)(15,21,16,22)(17,20,18,19) \!\cdots\! \rangle$
Transitive group:	36T20490				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2^6.S_4^2)$ . $C_3$	$(C_2^5.S_4^2)$ . $C_6$	$(C_2^5.S_4^2)$ . $C_6$	$(C_2^5.S_4^2)$ . $C_6$	all 99

Elements of the group are displayed as permutations of degree 36.

Homology

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

Subgroups

There are 119 normal subgroups (109 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^8.(C_6\times S_3^2)$
Commutator:	$G' \simeq$ $C_2^9:C_3^2$	$G/G' \simeq$ $C_2^2\times C_6$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $A_4^3.C_2^3$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^{10}$	$G/\operatorname{Fit} \simeq$ $C_3\times S_3^2$
Radical:	$R \simeq$ $A_4^2.C_2\wr D_6$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^7$	$G/\operatorname{soc} \simeq$ $C_6\times S_3\times S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^9.D_4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^3$

Hi

Subgroup diagram and profile

Classes of subgroups up to conjugation

No diagram available: inclusions not computed

Classes of subgroups up to automorphism

No diagram available: inclusions not computed

Normal subgroups

No diagram available: inclusions not computed

Normal subgroups up to automorphism

No diagram available: inclusions not computed

Classes of subgroups up to conjugation

All subgroups of index up to 36 (order at least 3072) are shown, as well as all normal subgroups of any index.

Order 110592: $A_4^2.C_2\wr D_6$

Order 55296: $C_2^4:C_3.A_4^2.C_2^3$, $C_2^9.(C_3\times S_3^2)$, $C_2^8.(C_6\times S_3^2)$, $C_2^9.(C_3\times S_3^2)$, $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$, $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$, $A_4^2.C_2\wr C_6$

Order 36864: $C_2^4:C_3.C_2^4.C_6.C_2^3$, $C_2^6.S_4^2$, $C_2^8.(C_6\times S_4)$

Order 27648: $C_2^4:C_3.A_4^2.C_2^2$ x 4, $C_2^8.(C_3\times S_3^2)$ x 2, $C_2^8.C_3.C_6^2$, $S_4\times A_4^2:D_4$, $(C_2^3\times A_4^2).S_4$, $(C_2^3\times A_4^2).S_4$, $C_2^8.(C_3^2\times D_6)$, $C_2^9.(S_3\times C_3^2)$

Order 18432: $C_2^4:C_3.C_2^4.C_6.C_2^2$ x 4, $C_2^4:C_3.C_2^3.C_6.C_2^3$ x 2, $C_2^4.(C_2\times C_6^2).C_2^4$ x 2, $C_2^4.(C_2\times C_6^2).C_2^4$ x 2, $C_2^4:C_3.C_2^3.C_6.C_2^3$ x 2, $C_2^6.(A_4\times S_4)$ x 2, $C_2^4.(A_4\times \GL(2,\mathbb{Z}/4))$ x 2, $C_2^6.C_6^2.C_2^3$, $C_2^4.C_6^2.C_2^5$, $C_2\times C_2^2.A_4^2.C_2^4$, $C_2^4:C_3.C_2^3.C_6.C_2^3$, $C_2^5.S_4^2$, $C_2^5.S_4^2$, $C_2^5.S_4^2$, $C_2^5.S_4^2$, $C_2^4.(A_4\times \GL(2,\mathbb{Z}/4))$, $C_2^6.(A_4\times S_4)$, $C_2^7.(C_6\times S_4)$, $C_2^6.(C_{12}\times S_4)$, $C_2^6.\PSOPlus(4,3)$, $C_2^7.(A_4\times D_6)$, $C_2^7.(C_6\times S_4)$, $C_2^9.C_6^2$

Order 13824: $C_2^4:C_3.A_4^2.C_2$ x 3, $A_4.A_4^2.C_2^3$ x 2, $C_2^8.C_3^3.C_2$ x 2, $A_4^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4.(C_6^2\times A_4).C_2$, $A_4^2:C_4\times S_4$, $A_4^2.\GL(2,\mathbb{Z}/4)$, $A_4^3:D_4$, $A_4^3.C_2^3$, $A_4^3:D_4$, $(C_2^2\times A_4^2).S_4$, $C_2^8:(S_3\times C_3^2)$

Order 12288: $C_2^8.C_6.C_2^3$, $C_2^4:C_3.C_2^6.C_2^2$, $C_2^4:C_3.C_2^6.C_2^2$

Order 9216: $C_2^6.C_6^2.C_2^2$ x 14, $C_2\times C_2^2.A_4^2.C_2^3$ x 10, $C_2^4.C_6^2.C_2^4$ x 9, $C_2^4.C_6^2.C_2^3.C_2$ x 8, $C_2^5.(A_4\times S_4)$ x 8, $(C_2^3\times A_4^2):D_4$ x 8, $C_2^4.C_6^2.C_2^4$ x 4, $C_2^4.C_6^2.C_4^2$ x 4, $C_2^2.A_4^2.C_2^4$ x 4, $C_2^4.C_6^2.C_2.C_2^3$ x 4, $C_2^4.C_6^2.C_2^4$ x 4, $C_2^4.C_6^2.C_2^4$ x 3, $C_2^4.C_6^2.C_2^3.C_2$ x 3, $C_2^5.C_6^2.C_2^3$ x 2, $C_2^5.C_6^2.C_2^3$ x 2, $C_2^{10}:C_3^2$ x 2, $C_2^{10}:C_3^2$ x 2, $C_2^5.(A_4\times S_4)$ x 2, $C_2^5.(A_4\times S_4)$ x 2, $C_2^5.(A_4\times S_4)$ x 2, $C_2^4.S_4^2$ x 2, $C_2^4.C_6^2.C_2^4$, $C_2^4.C_6^2.C_2^4$, $C_2^4.C_6^2.C_2.C_2^3$, $C_2^4.C_6^2.C_2^4$, $C_2\times C_2^6.C_6^2.C_2$, $C_2^4.C_6^2.C_2^4$, $C_2^8.C_6^2$, $A_4^2.C_2^6$, $C_2^4.S_4^2$, $C_2^4.S_4^2$, $S_4\times C_2\wr C_6$, $C_2^6.(C_6\times S_4)$, $C_2^5.\PSOPlus(4,3)$, $C_2^5.\PSOPlus(4,3)$, $C_2^7.(S_3\times A_4)$, $C_2^6.(A_4\times D_6)$, $C_2^6.(C_6\times S_4)$, $C_2^7.(C_3\times S_4)$, $C_2^6.A_4^2$, $C_2^6.(C_{12}\times A_4)$

Order 6912: $A_4.A_4^2.C_2^2$ x 10, $C_2^2.A_4^2.C_6.C_2$ x 6, $C_2^6.C_3^3.C_2^2$ x 3, $A_4^3:C_4$ x 2, $A_4^3:C_2^2$ x 2, $A_4^3:C_2^2$ x 2, $C_2^6.C_3^3.C_2^2$, $C_2^8.C_3^3$, $S_3\times A_4^2:D_4$, $C_3\times C_2^6:S_3^2$, $A_4^3:C_2^2$, $A_4^3:C_4$

Order 6144: $C_2^4:C_3.C_2^5.C_2^2$ x 4, $C_2^8.C_6.C_2^2$ x 4, $C_2^7.C_6.C_2^3$ x 3, $C_2^4:C_3.C_2^6.C_2$ x 3, $C_2^4:C_3.C_2^6.C_2$ x 3, $C_2^4:C_3.C_2^6.C_2$ x 3, $C_2^4:C_3.C_2^6.C_2$ x 3, $C_2^9.D_6$ x 3, $C_2^7.C_6.C_2^3$ x 2, $C_2^4:C_3.C_2^5.C_2^2$ x 2, $C_2^6.C_6.C_2^4$ x 2, $C_2^4:C_3.C_2^5.C_2^2$ x 2, $C_2^4:C_3.C_2^5.C_2^2$ x 2, $C_2^7.C_6.C_2^3$ x 2, $C_2^7.C_6.C_2^3$ x 2, $(C_2^4\times A_4).C_2^5$, $C_2^7.C_6.C_2^3$, $C_2^6.C_6.C_2^4$, $C_2^4:C_3.C_2^5.C_2^2$, $C_2^4:C_3.C_2^5.C_2^2$, $C_2^{10}.C_6$, $C_2^6.\GL(2,\mathbb{Z}/4)$, $C_2^6\times C_2^2:S_4$

Order 4608: $C_2^4.(A_4\times S_4)$ x 54, $C_2^6.C_6^2.C_2$ x 36, $C_2^5.C_6^2.C_2^2$ x 34, $(C_2^2\times A_4)\wr C_2$ x 33, $C_2^4.C_6^2.C_2^3$ x 22, $C_2^5.C_6^2.C_2^2$ x 19, $A_4^2.C_2^5$ x 13, $C_2^4.C_6^2.C_2^3$ x 11, $C_2^4.C_6^2.C_2^3$ x 8, $C_2^2\wr S_3:C_{12}$ x 8, $C_2^6.C_6^2.C_2$ x 7, $C_2^4:(A_4\times S_4)$ x 6, $(C_2^3\times A_4^2):C_4$ x 6, $(D_4\times A_4^2):C_2^2$ x 6, $C_2^4.C_6^2.C_2^3$ x 4, $C_2^4.C_6^2.C_2^3$ x 4, $C_2^4.C_6^2.C_2^3$ x 4, $C_2^4.C_6^2.C_2^3$ x 4, $C_2^4.C_6^2.C_2^3$ x 4, $C_2^9:C_3^2$ x 4, $C_2^4.(A_4\times S_4)$ x 4, $C_2^9:C_3^2$ x 4, $C_2^8.C_3.C_6$ x 3, $A_4^2.C_2^5$ x 3, $(C_2^2\times A_4):\GL(2,\mathbb{Z}/4)$ x 3, $(C_2^2\times \GL(2,\mathbb{Z}/4)):A_4$ x 3, $C_2^6.C_6^2.C_2$ x 3, $C_2^4.C_6^2.C_2^3$ x 2, $C_2^4.C_6^2.C_2^3$ x 2, $C_2^4.C_6^2.C_2^3$ x 2, $C_2^4.C_6^2.C_2^3$ x 2, $C_2^4.C_6^2.C_2^3$ x 2, $C_2^4.C_6^2.C_2^3$ x 2, $A_4^2.C_2^4.C_2$ x 2, $A_4^2:C_2^2\times D_4$ x 2, $A_4\times C_2\wr S_3$ x 2, $C_2^7.C_6^2$ x 2, $C_2^6:(C_3\times S_4)$ x 2, $C_2^4.C_6^2.C_2^3$, $C_2^4.C_6^2.C_2^3$, $C_2^4.C_6^2.C_2^3$, $C_2^4.C_6^2.C_2^3$, $C_2^3.S_4^2$, $C_2^3.S_4^2$, $C_2^3.S_4^2$, $C_2^3:S_4^2$, $C_2^3.S_4^2$, $C_2^3.S_4^2$, $C_2^3.S_4^2$, $C_2^3.S_4^2$, $C_2^4:C_{12}\times S_4$, $C_2^4.(A_4\times S_4)$, $(S_4\times C_2^5):C_6$, $C_2^4:(A_4\times S_4)$, $A_4\times C_2^4:S_4$, $C_2^6:(S_3\times C_{12})$, $A_4\times C_2^4:S_4$, $A_4\times C_2^4:D_{12}$, $C_2\wr S_3:A_4$, $C_2^4:\PSOPlus(4,3)$, $C_2^5:A_4^2$, $C_2^6.(C_3\times S_4)$, $C_2^6.(S_3\times A_4)$

Order 4096: $C_2^9.D_4$

Order 3456: $C_2\times A_4^3$ x 7, $S_4\times A_4^2$ x 7, $C_2^4:(C_3^2\times S_4)$ x 4, $C_2^6.C_3^3.C_2$ x 3, $C_2^2:C_6^2\times S_4$ x 3, $A_4^2:S_4$ x 2, $C_2^4.(C_6^2\times S_3)$, $C_2^4.(C_6\times S_3^2)$, $C_2\times D_6\times A_4^2$, $A_4\wr C_2\times D_6$, $A_4^2:D_{12}$, $(C_3\times A_4^2):D_4$, $A_4^2:C_4\times S_3$, $C_6^2.\GL(2,\mathbb{Z}/4)$, $C_6^2.(C_4\times S_4)$, $(C_2^2\times C_6^2):D_{12}$, $(C_3\times A_4^2):D_4$, $C_6^2:\GL(2,\mathbb{Z}/4)$, $(C_2^4\times C_6^2):C_6$, $C_3\times A_4^2:D_4$, $A_4^3:C_2$

Order 3072: $(C_2^4\times A_4).C_2^4$ x 56, $C_2^6.C_6.C_2^3$ x 28, $C_2^8:D_6$ x 20, $C_2^7.S_4$ x 18, $C_2^4:C_3.C_2^5.C_2$ x 14, $C_2^4:C_3.C_2^5.C_2$ x 12, $C_2^6.C_6.C_2^3$ x 12, $C_2^4:C_3.C_2^5.C_2$ x 11, $C_2^6.C_6.C_2^3$ x 11, $C_2^4.C_2^4:S_3.C_2$ x 8, $C_2^7.C_6.C_2^2$ x 8, $C_2^4:C_3.C_2^5.C_2$ x 7, $C_2^4:C_3.C_2^5.C_2$ x 7, $C_2^4:C_3.C_2^5.C_2$ x 7, $C_2^4:C_3.C_2^4.C_2^2$ x 4, $C_2^4:C_3.C_2^4.C_2^2$ x 4, $C_2^6.C_6.C_2^3$ x 4, $C_2^5.(C_2^4\times C_6)$ x 4, $C_2^4:C_3.C_2^4.C_2^2$ x 4, $C_2^6.C_6.C_4.C_2$ x 4, $C_2^4:C_3.D_4^2$ x 4, $C_2^6.C_6.C_2^3$ x 4, $C_2^4:C_3.C_2^4.C_2^2$ x 4, $C_2^5.(C_2^3\times C_{12})$ x 4, $C_2^6.C_6.C_2^3$ x 4, $C_2^6.C_6.C_2^3$ x 4, $C_2^6.C_6.C_2^3$ x 3, $(C_2^4\times A_4).C_2^4$ x 3, $C_2^4:C_3.C_2^4.C_2^2$ x 3, $(C_2^4\times A_4).C_2^4$ x 3, $C_2^8.D_6$ x 3, $C_2^8.C_6.C_2$ x 2, $C_2^4:C_3.C_2^6$ x 2, $C_2^4:C_3.C_2^4.C_2^2$, $C_2^4\times C_2^2\times C_2^4:C_3$, $A_4.C_2^6.C_2^2$, $C_2^6.C_6.C_2^3$, $A_4.C_2^6.C_2^2$, $C_2^6.C_6.C_2^3$, $A_4.C_2^6.C_2^2$, $C_2^5.(C_2^4\times C_6)$, $C_2^4.C_2^4:S_3:C_2$, $C_2^5.(C_2^3\times C_{12})$, $C_2^4:C_3.C_2^4.C_2^2$, $C_2^6.C_6.C_2^3$, $C_2^4:C_3.C_2^4.C_2^2$, $C_2^8.C_{12}$, $C_2^7.S_4$, $C_2^9.C_6$, $C_2^7.S_4$, $C_2^{10}:C_3$

Order 2304: $A_4^2.C_2^4$ x 2, $C_2^4:A_4^2$ x 2, $A_4^2.C_2^3.C_2$, $A_4^2:C_2^4$

Order 1536: $C_2^9.C_3$ x 3, $C_2^6:S_4$ x 2, $C_2^2\times C_2^4:C_3.D_4$

Order 1152: $C_2^3\times A_4^2$

Order 1024: $C_2^{10}$

Order 768: $C_2^8.C_3$ x 5, $C_2^5:S_4$, $C_2^2\times C_2^2:(A_4:C_4)$

Order 576: $C_2^2\times A_4^2$ x 2

Order 512: $C_2^9$

Order 384: $C_2^5:A_4$ x 2, $C_2^2\wr S_3$

Order 288: $C_2\times A_4^2$

Order 256: $C_2^8$ x 2

Order 192: $C_2^2\wr C_3$ x 5, $C_2^3:S_4$ x 2

Order 144: $A_4^2$

Order 128: $C_2^7$

Order 96: $C_2^3:A_4$ x 3, $C_2^2:S_4$ x 2

Order 64: $C_2^6$ x 3

Order 48: $C_2^2:A_4$ x 3

Order 32: $C_2^5$ x 2

Order 27: $C_3^3$

Order 16: $C_2^4$ x 3

Order 8: $C_2^3$

Order 4: $C_2^2$ x 2

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 36 (order at least 3072) are shown, as well as all normal subgroups of any index.

Order 110592: $A_4^2.C_2\wr D_6$

Order 55296: $C_2^4:C_3.A_4^2.C_2^3$, $C_2^9.(C_3\times S_3^2)$, $C_2^8.(C_6\times S_3^2)$, $C_2^9.(C_3\times S_3^2)$, $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$, $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$, $A_4^2.C_2\wr C_6$

Order 36864: $C_2^4:C_3.C_2^4.C_6.C_2^3$, $C_2^6.S_4^2$, $C_2^8.(C_6\times S_4)$

Order 27648: $C_2^4:C_3.A_4^2.C_2^2$ x 3, $C_2^8.C_3.C_6^2$, $C_2^8.(C_3\times S_3^2)$, $S_4\times A_4^2:D_4$, $(C_2^3\times A_4^2).S_4$, $(C_2^3\times A_4^2).S_4$, $C_2^8.(C_3^2\times D_6)$, $C_2^9.(S_3\times C_3^2)$

Order 18432: $C_2^4:C_3.C_2^4.C_6.C_2^2$ x 4, $C_2^4:C_3.C_2^3.C_6.C_2^3$ x 2, $C_2^4.(C_2\times C_6^2).C_2^4$ x 2, $C_2^4.(C_2\times C_6^2).C_2^4$ x 2, $C_2^4:C_3.C_2^3.C_6.C_2^3$ x 2, $C_2^6.(A_4\times S_4)$ x 2, $C_2^4.(A_4\times \GL(2,\mathbb{Z}/4))$ x 2, $C_2^6.C_6^2.C_2^3$, $C_2^4.C_6^2.C_2^5$, $C_2\times C_2^2.A_4^2.C_2^4$, $C_2^4:C_3.C_2^3.C_6.C_2^3$, $C_2^5.S_4^2$, $C_2^5.S_4^2$, $C_2^5.S_4^2$, $C_2^5.S_4^2$, $C_2^4.(A_4\times \GL(2,\mathbb{Z}/4))$, $C_2^6.(A_4\times S_4)$, $C_2^7.(C_6\times S_4)$, $C_2^6.(C_{12}\times S_4)$, $C_2^6.\PSOPlus(4,3)$, $C_2^7.(A_4\times D_6)$, $C_2^7.(C_6\times S_4)$, $C_2^9.C_6^2$

Order 13824: $A_4.A_4^2.C_2^3$ x 2, $C_2^8.C_3^3.C_2$ x 2, $C_2^4:C_3.A_4^2.C_2$ x 2, $A_4^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4.(C_6^2\times A_4).C_2$, $A_4^2:C_4\times S_4$, $A_4^2.\GL(2,\mathbb{Z}/4)$, $A_4^3:D_4$, $A_4^3.C_2^3$, $A_4^3:D_4$, $(C_2^2\times A_4^2).S_4$, $C_2^8:(S_3\times C_3^2)$

Order 12288: $C_2^8.C_6.C_2^3$, $C_2^4:C_3.C_2^6.C_2^2$, $C_2^4:C_3.C_2^6.C_2^2$

Order 9216: $C_2^4.C_6^2.C_2^4$ x 9, $C_2^6.C_6^2.C_2^2$ x 9, $C_2^4.C_6^2.C_2^3.C_2$ x 8, $C_2^5.(A_4\times S_4)$ x 6, $C_2^4.C_6^2.C_2^4$ x 4, $C_2^4.C_6^2.C_4^2$ x 4, $C_2^2.A_4^2.C_2^4$ x 4, $C_2\times C_2^2.A_4^2.C_2^3$ x 4, $C_2^4.C_6^2.C_2.C_2^3$ x 4, $C_2^4.C_6^2.C_2^4$ x 4, $C_2^4.C_6^2.C_2^4$ x 3, $C_2^4.C_6^2.C_2^3.C_2$ x 3, $(C_2^3\times A_4^2):D_4$ x 3, $C_2^5.C_6^2.C_2^3$ x 2, $C_2^5.C_6^2.C_2^3$ x 2, $C_2^{10}:C_3^2$ x 2, $C_2^{10}:C_3^2$ x 2, $C_2^5.(A_4\times S_4)$ x 2, $C_2^5.(A_4\times S_4)$ x 2, $C_2^4.C_6^2.C_2^4$, $C_2^4.C_6^2.C_2^4$, $C_2^4.C_6^2.C_2.C_2^3$, $C_2^4.C_6^2.C_2^4$, $C_2\times C_2^6.C_6^2.C_2$, $C_2^4.C_6^2.C_2^4$, $C_2^8.C_6^2$, $A_4^2.C_2^6$, $C_2^4.S_4^2$, $C_2^4.S_4^2$, $S_4\times C_2\wr C_6$, $C_2^6.(C_6\times S_4)$, $C_2^5.\PSOPlus(4,3)$, $C_2^5.\PSOPlus(4,3)$, $C_2^7.(S_3\times A_4)$, $C_2^6.(A_4\times D_6)$, $C_2^6.(C_6\times S_4)$, $C_2^7.(C_3\times S_4)$, $C_2^6.A_4^2$, $C_2^6.(C_{12}\times A_4)$, $C_2^5.(A_4\times S_4)$, $C_2^4.S_4^2$

Order 6912: $A_4.A_4^2.C_2^2$ x 6, $C_2^6.C_3^3.C_2^2$ x 3, $C_2^2.A_4^2.C_6.C_2$ x 3, $A_4^3:C_4$ x 2, $A_4^3:C_2^2$ x 2, $C_2^6.C_3^3.C_2^2$, $C_2^8.C_3^3$, $S_3\times A_4^2:D_4$, $C_3\times C_2^6:S_3^2$, $A_4^3:C_2^2$, $A_4^3:C_4$, $A_4^3:C_2^2$

Order 6144: $C_2^4:C_3.C_2^5.C_2^2$ x 4, $C_2^8.C_6.C_2^2$ x 4, $C_2^7.C_6.C_2^3$ x 3, $C_2^4:C_3.C_2^6.C_2$ x 3, $C_2^4:C_3.C_2^6.C_2$ x 3, $C_2^4:C_3.C_2^6.C_2$ x 3, $C_2^4:C_3.C_2^6.C_2$ x 3, $C_2^9.D_6$ x 3, $C_2^7.C_6.C_2^3$ x 2, $C_2^4:C_3.C_2^5.C_2^2$ x 2, $C_2^6.C_6.C_2^4$ x 2, $C_2^4:C_3.C_2^5.C_2^2$ x 2, $C_2^4:C_3.C_2^5.C_2^2$ x 2, $C_2^7.C_6.C_2^3$ x 2, $C_2^7.C_6.C_2^3$ x 2, $(C_2^4\times A_4).C_2^5$, $C_2^7.C_6.C_2^3$, $C_2^6.C_6.C_2^4$, $C_2^4:C_3.C_2^5.C_2^2$, $C_2^4:C_3.C_2^5.C_2^2$, $C_2^{10}.C_6$, $C_2^6.\GL(2,\mathbb{Z}/4)$, $C_2^6\times C_2^2:S_4$

Order 4608: $(C_2^2\times A_4)\wr C_2$ x 29, $C_2^4.(A_4\times S_4)$ x 25, $C_2^5.C_6^2.C_2^2$ x 24, $C_2^4.C_6^2.C_2^3$ x 22, $C_2^6.C_6^2.C_2$ x 14, $A_4^2.C_2^5$ x 12, $C_2^5.C_6^2.C_2^2$ x 11, $C_2^4.C_6^2.C_2^3$ x 11, $C_2^4.C_6^2.C_2^3$ x 8, $C_2^6.C_6^2.C_2$ x 5, $C_2^4.C_6^2.C_2^3$ x 4, $C_2^4.C_6^2.C_2^3$ x 4, $C_2^4.C_6^2.C_2^3$ x 4, $C_2^4.C_6^2.C_2^3$ x 4, $C_2^4.C_6^2.C_2^3$ x 4, $C_2^4:(A_4\times S_4)$ x 4, $C_2^9:C_3^2$ x 4, $C_2^4.(A_4\times S_4)$ x 4, $C_2^9:C_3^2$ x 4, $C_2^2\wr S_3:C_{12}$ x 4, $(C_2^3\times A_4^2):C_4$ x 4, $(D_4\times A_4^2):C_2^2$ x 4, $(C_2^2\times A_4):\GL(2,\mathbb{Z}/4)$ x 3, $(C_2^2\times \GL(2,\mathbb{Z}/4)):A_4$ x 3, $C_2^4.C_6^2.C_2^3$ x 2, $C_2^8.C_3.C_6$ x 2, $A_4^2.C_2^5$ x 2, $C_2^4.C_6^2.C_2^3$ x 2, $C_2^4.C_6^2.C_2^3$ x 2, $C_2^4.C_6^2.C_2^3$ x 2, $C_2^4.C_6^2.C_2^3$ x 2, $C_2^4.C_6^2.C_2^3$ x 2, $A_4^2.C_2^4.C_2$ x 2, $A_4^2:C_2^2\times D_4$ x 2, $A_4\times C_2\wr S_3$ x 2, $C_2^6.C_6^2.C_2$ x 2, $C_2^7.C_6^2$ x 2, $C_2^6:(C_3\times S_4)$ x 2, $C_2^4.C_6^2.C_2^3$, $C_2^4.C_6^2.C_2^3$, $C_2^4.C_6^2.C_2^3$, $C_2^4.C_6^2.C_2^3$, $C_2^3.S_4^2$, $C_2^3.S_4^2$, $C_2^3.S_4^2$, $C_2^3:S_4^2$, $C_2^3.S_4^2$, $C_2^3.S_4^2$, $C_2^3.S_4^2$, $C_2^3.S_4^2$, $C_2^4:C_{12}\times S_4$, $C_2^4.(A_4\times S_4)$, $(S_4\times C_2^5):C_6$, $C_2^4:(A_4\times S_4)$, $A_4\times C_2^4:S_4$, $C_2^6:(S_3\times C_{12})$, $A_4\times C_2^4:S_4$, $A_4\times C_2^4:D_{12}$, $C_2\wr S_3:A_4$, $C_2^4:\PSOPlus(4,3)$, $C_2^5:A_4^2$, $C_2^6.(C_3\times S_4)$, $C_2^6.(S_3\times A_4)$

Order 4096: $C_2^9.D_4$

Order 3456: $C_2\times A_4^3$ x 6, $S_4\times A_4^2$ x 4, $C_2^6.C_3^3.C_2$ x 2, $C_2^4:(C_3^2\times S_4)$ x 2, $C_2^2:C_6^2\times S_4$ x 2, $A_4^2:S_4$ x 2, $C_2^4.(C_6^2\times S_3)$, $C_2^4.(C_6\times S_3^2)$, $C_2\times D_6\times A_4^2$, $A_4\wr C_2\times D_6$, $A_4^2:D_{12}$, $(C_3\times A_4^2):D_4$, $A_4^2:C_4\times S_3$, $C_6^2.\GL(2,\mathbb{Z}/4)$, $C_6^2.(C_4\times S_4)$, $(C_2^2\times C_6^2):D_{12}$, $(C_3\times A_4^2):D_4$, $C_6^2:\GL(2,\mathbb{Z}/4)$, $(C_2^4\times C_6^2):C_6$, $C_3\times A_4^2:D_4$, $A_4^3:C_2$

Order 3072: $(C_2^4\times A_4).C_2^4$ x 24, $C_2^6.C_6.C_2^3$ x 18, $C_2^7.S_4$ x 13, $C_2^4:C_3.C_2^5.C_2$ x 12, $C_2^6.C_6.C_2^3$ x 12, $C_2^8:D_6$ x 12, $C_2^4:C_3.C_2^5.C_2$ x 11, $C_2^6.C_6.C_2^3$ x 11, $C_2^4:C_3.C_2^5.C_2$ x 10, $C_2^4:C_3.C_2^5.C_2$ x 6, $C_2^4:C_3.C_2^5.C_2$ x 6, $C_2^4:C_3.C_2^5.C_2$ x 6, $C_2^4:C_3.C_2^4.C_2^2$ x 4, $C_2^4:C_3.C_2^4.C_2^2$ x 4, $C_2^6.C_6.C_2^3$ x 4, $C_2^5.(C_2^4\times C_6)$ x 4, $C_2^4:C_3.C_2^4.C_2^2$ x 4, $C_2^6.C_6.C_4.C_2$ x 4, $C_2^4:C_3.D_4^2$ x 4, $C_2^6.C_6.C_2^3$ x 4, $C_2^4:C_3.C_2^4.C_2^2$ x 4, $C_2^5.(C_2^3\times C_{12})$ x 4, $C_2^6.C_6.C_2^3$ x 4, $C_2^6.C_6.C_2^3$ x 4, $C_2^6.C_6.C_2^3$ x 3, $C_2^4:C_3.C_2^4.C_2^2$ x 3, $C_2^4.C_2^4:S_3.C_2$ x 3, $C_2^7.C_6.C_2^2$ x 3, $C_2^8.D_6$ x 3, $C_2^8.C_6.C_2$ x 2, $C_2^4:C_3.C_2^6$ x 2, $(C_2^4\times A_4).C_2^4$ x 2, $(C_2^4\times A_4).C_2^4$ x 2, $C_2^4:C_3.C_2^4.C_2^2$, $C_2^4\times C_2^2\times C_2^4:C_3$, $A_4.C_2^6.C_2^2$, $C_2^6.C_6.C_2^3$, $A_4.C_2^6.C_2^2$, $C_2^6.C_6.C_2^3$, $A_4.C_2^6.C_2^2$, $C_2^5.(C_2^4\times C_6)$, $C_2^4.C_2^4:S_3:C_2$, $C_2^5.(C_2^3\times C_{12})$, $C_2^4:C_3.C_2^4.C_2^2$, $C_2^6.C_6.C_2^3$, $C_2^4:C_3.C_2^4.C_2^2$, $C_2^8.C_{12}$, $C_2^7.S_4$, $C_2^9.C_6$, $C_2^7.S_4$, $C_2^{10}:C_3$

Order 2304: $A_4^2.C_2^4$ x 2, $C_2^4:A_4^2$ x 2, $A_4^2.C_2^3.C_2$, $A_4^2:C_2^4$

Order 1536: $C_2^9.C_3$ x 3, $C_2^6:S_4$, $C_2^2\times C_2^4:C_3.D_4$

Order 1152: $C_2^3\times A_4^2$

Order 1024: $C_2^{10}$

Order 768: $C_2^8.C_3$ x 5, $C_2^5:S_4$, $C_2^2\times C_2^2:(A_4:C_4)$

Order 576: $C_2^2\times A_4^2$ x 2

Order 512: $C_2^9$

Order 384: $C_2^5:A_4$ x 2, $C_2^2\wr S_3$

Order 288: $C_2\times A_4^2$

Order 256: $C_2^8$ x 2

Order 192: $C_2^2\wr C_3$ x 5, $C_2^3:S_4$ x 2

Order 144: $A_4^2$

Order 128: $C_2^7$

Order 96: $C_2^3:A_4$ x 3, $C_2^2:S_4$

Order 64: $C_2^6$ x 3

Order 48: $C_2^2:A_4$ x 3

Order 32: $C_2^5$ x 2

Order 27: $C_3^3$

Order 16: $C_2^4$ x 3

Order 8: $C_2^3$

Order 4: $C_2^2$ x 2

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 110592: $A_4^2.C_2\wr D_6$ ($C_1$)

Order 55296: $C_2^4:C_3.A_4^2.C_2^3$ ($C_2$), $C_2^9.(C_3\times S_3^2)$ ($C_2$), $C_2^8.(C_6\times S_3^2)$ ($C_2$), $C_2^9.(C_3\times S_3^2)$ ($C_2$), $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$ ($C_2$), $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$ ($C_2$), $A_4^2.C_2\wr C_6$ ($C_2$)

Order 36864: $C_2^6.S_4^2$ ($C_3$)

Order 27648: $C_2^4:C_3.A_4^2.C_2^2$ ($C_2^2$) x 2, $C_2^8.C_3.C_6^2$ ($C_2^2$), $(C_2^3\times A_4^2).S_4$ ($C_2^2$), $(C_2^3\times A_4^2).S_4$ ($C_2^2$), $C_2^8.(C_3^2\times D_6)$ ($C_2^2$), $C_2^9.(S_3\times C_3^2)$ ($C_2^2$)

Order 18432: $C_2^4.(C_2\times C_6^2).C_2^4$ ($S_3$), $C_2^6.(A_4\times S_4)$ ($C_6$), $C_2^6.(A_4\times S_4)$ ($S_3$), $C_2^5.S_4^2$ ($C_6$), $C_2^5.S_4^2$ ($C_6$), $C_2^5.S_4^2$ ($C_6$), $C_2^5.S_4^2$ ($C_6$), $C_2^6.\PSOPlus(4,3)$ ($C_6$), $C_2^4.(A_4\times \GL(2,\mathbb{Z}/4))$ ($C_6$)

Order 13824: $C_2^4:C_3.A_4^2.C_2$ ($D_4$) x 2, $C_2^8.C_3^3.C_2$ ($C_2^3$)

Order 9216: $C_2^5.(A_4\times S_4)$ ($C_2\times C_6$) x 2, $C_2^5.(A_4\times S_4)$ ($D_6$) x 2, $C_2^4.C_6^2.C_2^3.C_2$ ($D_6$), $A_4^2.C_2^6$ ($D_6$), $C_2^4.C_6^2.C_2^4$ ($D_6$), $C_2^{10}:C_3^2$ ($C_2\times C_6$), $C_2^{10}:C_3^2$ ($D_6$), $C_2^5.\PSOPlus(4,3)$ ($C_2\times C_6$), $C_2^5.\PSOPlus(4,3)$ ($C_2\times C_6$), $C_2^5.(A_4\times S_4)$ ($C_2\times C_6$), $C_2^5.(A_4\times S_4)$ ($C_2\times C_6$)

Order 6912: $C_2^8.C_3^3$ ($C_2\times D_4$)

Order 6144: $C_2^4:C_3.C_2^6.C_2$ ($C_3\times S_3$), $C_2^6\times C_2^2:S_4$ ($C_3\times S_3$)

Order 4608: $C_2^4:(A_4\times S_4)$ ($C_3:D_4$) x 2, $C_2^4:(A_4\times S_4)$ ($C_3\times D_4$) x 2, $A_4^2.C_2^5$ ($C_2\times D_6$), $C_2^4.C_6^2.C_2^3$ ($S_4$), $C_2^9:C_3^2$ ($C_2^2\times C_6$), $C_2^9:C_3^2$ ($C_2\times D_6$)

Order 3072: $C_2^7.S_4$ ($C_6\times S_3$) x 2, $C_2^4:C_3.C_2^5.C_2$ ($C_6\times S_3$), $C_2^4\times C_2^2\times C_2^4:C_3$ ($C_6\times S_3$), $C_2^4:C_3.C_2^6$ ($C_6\times S_3$), $C_2^4:C_3.C_2^6$ ($S_3^2$), $C_2^4:C_3.C_2^5.C_2$ ($C_6\times S_3$)

Order 2304: $A_4^2.C_2^4$ ($C_2\times S_4$), $A_4^2.C_2^4$ ($S_3\times D_4$), $A_4^2.C_2^3.C_2$ ($C_2\times S_4$), $A_4^2:C_2^4$ ($C_2\times S_4$), $C_2^4:A_4^2$ ($C_6\times D_4$), $C_2^4:A_4^2$ ($C_6:D_4$)

Order 1536: $C_2^6:S_4$ ($C_6\wr C_2$) x 2, $C_2^9.C_3$ ($C_6\times D_6$) x 2, $C_2^2\times C_2^4:C_3.D_4$ ($C_3\times S_4$), $C_2^9.C_3$ ($S_3\times D_6$)

Order 1152: $C_2^3\times A_4^2$ ($C_2^2\times S_4$)

Order 1024: $C_2^{10}$ ($C_3\times S_3^2$)

Order 768: $C_2^5:S_4$ ($C_6\times S_4$), $C_2^2\times C_2^2:(A_4:C_4)$ ($C_6\times S_4$), $C_2^8.C_3$ ($C_6\times S_4$), $C_2^8.C_3$ ($S_3\times S_4$), $C_2^8.C_3$ ($C_6^2:C_2^2$), $C_2^8.C_3$ ($C_{12}:D_6$), $C_2^8.C_3$ ($D_6:D_6$)

Order 576: $C_2^2\times A_4^2$ ($C_2^3:S_4$), $C_2^2\times A_4^2$ ($D_4\times S_4$)

Order 512: $C_2^9$ ($C_6\times S_3^2$)

Order 384: $C_2^5:A_4$ ($C_2\times C_6\times S_4$), $C_2^5:A_4$ ($D_6\times S_4$), $C_2^2\wr S_3$ ($A_4\wr C_2$)

Order 288: $C_2\times A_4^2$ ($C_2^4:S_4$)

Order 256: $C_2^8$ ($C_6^2:D_6$), $C_2^8$ ($C_6^2:D_6$)

Order 192: $C_2^3:S_4$ ($A_4^2:C_2^2$) x 2, $C_2^2\wr C_3$ ($A_4^2:C_2^2$), $C_2^2\wr C_3$ ($\GL(2,\mathbb{Z}/4):C_6$), $C_2^2\wr C_3$ ($C_2^4:S_3^2$), $C_2^2\wr C_3$ ($C_3\times C_2^3:S_4$), $C_2^2\wr C_3$ ($C_2^4:S_3^2$)

Order 144: $A_4^2$ ($C_2\wr D_6$)

Order 128: $C_2^7$ ($C_6\times S_3\times S_4$)

Order 96: $C_2^2:S_4$ ($A_4^2:D_4$) x 2, $C_2^3:A_4$ ($A_4^2:C_2^3$), $C_2^3:A_4$ ($C_6\times C_2^3:S_4$), $C_2^3:A_4$ ($C_2^5:S_3^2$)

Order 64: $C_2^6$ ($A_4^2:D_6$), $C_2^6$ ($C_6\wr C_2\times S_4$), $C_2^6$ ($C_3\times C_2^4:S_3^2$)

Order 48: $C_2^2:A_4$ ($C_2^6:S_3^2$), $C_2^2:A_4$ ($C_3\times C_2\wr D_6$), $C_2^2:A_4$ ($C_2\times A_4^2:D_4$)

Order 32: $C_2^5$ ($A_4\wr C_2\times D_6$), $C_2^5$ ($C_6\times C_2^4:S_3^2$)

Order 16: $C_2^4$ ($S_3\times A_4^2:D_4$), $C_2^4$ ($C_3\times C_2^6:S_3^2$), $C_2^4$ ($A_4^3:C_2^2$)

Order 8: $C_2^3$ ($A_4^3.C_2^3$)

Order 4: $C_2^2$ ($C_2^8.(C_3\times S_3^2)$), $C_2^2$ ($S_4\times A_4^2:D_4$)

Order 2: $C_2$ ($C_2^8.(C_6\times S_3^2)$)

Order 1: $C_1$ ($A_4^2.C_2\wr D_6$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 110592: $A_4^2.C_2\wr D_6$ ($C_1$)

Order 55296: $C_2^4:C_3.A_4^2.C_2^3$ ($C_2$), $C_2^9.(C_3\times S_3^2)$ ($C_2$), $C_2^8.(C_6\times S_3^2)$ ($C_2$), $C_2^9.(C_3\times S_3^2)$ ($C_2$), $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$ ($C_2$), $(C_2^2\times A_4^2).\GL(2,\mathbb{Z}/4)$ ($C_2$), $A_4^2.C_2\wr C_6$ ($C_2$)

Order 36864: $C_2^6.S_4^2$ ($C_3$)

Order 27648: $C_2^4:C_3.A_4^2.C_2^2$ ($C_2^2$) x 2, $C_2^8.C_3.C_6^2$ ($C_2^2$), $(C_2^3\times A_4^2).S_4$ ($C_2^2$), $(C_2^3\times A_4^2).S_4$ ($C_2^2$), $C_2^8.(C_3^2\times D_6)$ ($C_2^2$), $C_2^9.(S_3\times C_3^2)$ ($C_2^2$)

Order 18432: $C_2^4.(C_2\times C_6^2).C_2^4$ ($S_3$), $C_2^6.(A_4\times S_4)$ ($C_6$), $C_2^6.(A_4\times S_4)$ ($S_3$), $C_2^5.S_4^2$ ($C_6$), $C_2^5.S_4^2$ ($C_6$), $C_2^5.S_4^2$ ($C_6$), $C_2^5.S_4^2$ ($C_6$), $C_2^6.\PSOPlus(4,3)$ ($C_6$), $C_2^4.(A_4\times \GL(2,\mathbb{Z}/4))$ ($C_6$)

Order 13824: $C_2^8.C_3^3.C_2$ ($C_2^3$), $C_2^4:C_3.A_4^2.C_2$ ($D_4$)

Order 9216: $C_2^5.(A_4\times S_4)$ ($C_2\times C_6$) x 2, $C_2^5.(A_4\times S_4)$ ($D_6$) x 2, $C_2^4.C_6^2.C_2^3.C_2$ ($D_6$), $A_4^2.C_2^6$ ($D_6$), $C_2^4.C_6^2.C_2^4$ ($D_6$), $C_2^{10}:C_3^2$ ($C_2\times C_6$), $C_2^{10}:C_3^2$ ($D_6$), $C_2^5.\PSOPlus(4,3)$ ($C_2\times C_6$), $C_2^5.\PSOPlus(4,3)$ ($C_2\times C_6$), $C_2^5.(A_4\times S_4)$ ($C_2\times C_6$), $C_2^5.(A_4\times S_4)$ ($C_2\times C_6$)

Order 6912: $C_2^8.C_3^3$ ($C_2\times D_4$)

Order 6144: $C_2^4:C_3.C_2^6.C_2$ ($C_3\times S_3$), $C_2^6\times C_2^2:S_4$ ($C_3\times S_3$)

Order 4608: $A_4^2.C_2^5$ ($C_2\times D_6$), $C_2^4.C_6^2.C_2^3$ ($S_4$), $C_2^4:(A_4\times S_4)$ ($C_3:D_4$), $C_2^4:(A_4\times S_4)$ ($C_3\times D_4$), $C_2^9:C_3^2$ ($C_2^2\times C_6$), $C_2^9:C_3^2$ ($C_2\times D_6$)

Order 3072: $C_2^7.S_4$ ($C_6\times S_3$) x 2, $C_2^4:C_3.C_2^5.C_2$ ($C_6\times S_3$), $C_2^4\times C_2^2\times C_2^4:C_3$ ($C_6\times S_3$), $C_2^4:C_3.C_2^6$ ($C_6\times S_3$), $C_2^4:C_3.C_2^6$ ($S_3^2$), $C_2^4:C_3.C_2^5.C_2$ ($C_6\times S_3$)

Order 2304: $A_4^2.C_2^4$ ($C_2\times S_4$), $A_4^2.C_2^4$ ($S_3\times D_4$), $A_4^2.C_2^3.C_2$ ($C_2\times S_4$), $A_4^2:C_2^4$ ($C_2\times S_4$), $C_2^4:A_4^2$ ($C_6\times D_4$), $C_2^4:A_4^2$ ($C_6:D_4$)

Order 1536: $C_2^9.C_3$ ($C_6\times D_6$) x 2, $C_2^6:S_4$ ($C_6\wr C_2$), $C_2^2\times C_2^4:C_3.D_4$ ($C_3\times S_4$), $C_2^9.C_3$ ($S_3\times D_6$)

Order 1152: $C_2^3\times A_4^2$ ($C_2^2\times S_4$)

Order 1024: $C_2^{10}$ ($C_3\times S_3^2$)

Order 768: $C_2^5:S_4$ ($C_6\times S_4$), $C_2^2\times C_2^2:(A_4:C_4)$ ($C_6\times S_4$), $C_2^8.C_3$ ($C_6\times S_4$), $C_2^8.C_3$ ($S_3\times S_4$), $C_2^8.C_3$ ($C_6^2:C_2^2$), $C_2^8.C_3$ ($C_{12}:D_6$), $C_2^8.C_3$ ($D_6:D_6$)

Order 576: $C_2^2\times A_4^2$ ($C_2^3:S_4$), $C_2^2\times A_4^2$ ($D_4\times S_4$)

Order 512: $C_2^9$ ($C_6\times S_3^2$)

Order 384: $C_2^5:A_4$ ($C_2\times C_6\times S_4$), $C_2^5:A_4$ ($D_6\times S_4$), $C_2^2\wr S_3$ ($A_4\wr C_2$)

Order 288: $C_2\times A_4^2$ ($C_2^4:S_4$)

Order 256: $C_2^8$ ($C_6^2:D_6$), $C_2^8$ ($C_6^2:D_6$)

Order 192: $C_2^3:S_4$ ($A_4^2:C_2^2$) x 2, $C_2^2\wr C_3$ ($A_4^2:C_2^2$), $C_2^2\wr C_3$ ($\GL(2,\mathbb{Z}/4):C_6$), $C_2^2\wr C_3$ ($C_2^4:S_3^2$), $C_2^2\wr C_3$ ($C_3\times C_2^3:S_4$), $C_2^2\wr C_3$ ($C_2^4:S_3^2$)

Order 144: $A_4^2$ ($C_2\wr D_6$)

Order 128: $C_2^7$ ($C_6\times S_3\times S_4$)

Order 96: $C_2^3:A_4$ ($A_4^2:C_2^3$), $C_2^3:A_4$ ($C_6\times C_2^3:S_4$), $C_2^3:A_4$ ($C_2^5:S_3^2$), $C_2^2:S_4$ ($A_4^2:D_4$)

Order 64: $C_2^6$ ($A_4^2:D_6$), $C_2^6$ ($C_6\wr C_2\times S_4$), $C_2^6$ ($C_3\times C_2^4:S_3^2$)

Order 48: $C_2^2:A_4$ ($C_2^6:S_3^2$), $C_2^2:A_4$ ($C_3\times C_2\wr D_6$), $C_2^2:A_4$ ($C_2\times A_4^2:D_4$)

Order 32: $C_2^5$ ($A_4\wr C_2\times D_6$), $C_2^5$ ($C_6\times C_2^4:S_3^2$)

Order 16: $C_2^4$ ($S_3\times A_4^2:D_4$), $C_2^4$ ($C_3\times C_2^6:S_3^2$), $C_2^4$ ($A_4^3:C_2^2$)

Order 8: $C_2^3$ ($A_4^3.C_2^3$)

Order 4: $C_2^2$ ($C_2^8.(C_3\times S_3^2)$), $C_2^2$ ($S_4\times A_4^2:D_4$)

Order 2: $C_2$ ($C_2^8.(C_6\times S_3^2)$)

Order 1: $C_1$ ($A_4^2.C_2\wr D_6$)

Series

Derived series

$A_4^2.C_2\wr D_6$

$\rhd$

$C_2^9:C_3^2$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Chief series

$A_4^2.C_2\wr D_6$

$\rhd$

$C_2^4:C_3.A_4^2.C_2^3$

$\rhd$

$C_2^8.C_3.C_6^2$

$\rhd$

$C_2^8.C_3^3.C_2$

$\rhd$

$C_2^9:C_3^2$

$\rhd$

$C_2^4:A_4^2$

$\rhd$

$C_2^8.C_3$

$\rhd$

$C_2^8$

$\rhd$

$C_2^6$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$A_4^2.C_2\wr D_6$

$\rhd$

$C_2^9:C_3^2$

$\rhd$

$C_2^4:A_4^2$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

Supergroups

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

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

Character theory

Complex character table

The $392 \times 392$ character table is not available for this group.

Rational character table

The $260 \times 260$ rational character table is not available for this group.