Abstract group 27648.ec: $(C_2^2\times D_6).S_4^2$

• Label: 27648.ec
• Order: \( 2^{10} \cdot 3^{3} \)
• Exponent: \( 2^{2} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{11} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2 \)
• Perm deg.: $15$
• Trans deg.: not computed
• Rank: $4$

Group information

Description:	$(C_2^2\times D_6).S_4^2$
Order:	\(27648\)\(\medspace = 2^{10} \cdot 3^{3} \)
Exponent:	\(12\)\(\medspace = 2^{2} \cdot 3 \)
Automorphism group:	$C_2^6.(S_3\times C_3:D_4)\times S_3$, of order \(55296\)\(\medspace = 2^{11} \cdot 3^{3} \)
Composition factors:	$C_2$ x 10, $C_3$ x 3
Derived length:	$3$

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

Group statistics

Order	1	2	3	4	6	12
Elements	1	1759	890	6432	9014	9552	27648
Conjugacy classes	1	41	7	54	59	48	210
Divisions	1	41	7	54	59	48	210
Autjugacy classes	1	29	7	34	47	32	150

Dimension	1	2	3	4	6	8	9	12	18	24	36
Irr. complex chars.	16	24	32	12	48	2	16	24	24	4	8	210
Irr. rational chars.	16	24	32	12	48	2	16	24	24	4	8	210

Minimal presentations

Permutation degree:	$15$
Transitive degree:	not computed
Rank:	$4$
Inequivalent generating quadruples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	36	36	36
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i \mid c^{6}=d^{6}=e^{4}=f^{2}=g^{2}=h^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(4,5,7), (1,2), (4,7)(5,6), (1,3,2)(4,6,7,5)(8,12)(9,15)(10,14)(11,13), (2,3) \!\cdots\! \rangle$
Direct product:	$S_3$ $\, \times\, $ $S_4$ $\, \times\, $ $(C_2^3:S_4)$
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2^3:S_4^2)$ . $S_3$	$(C_2^5:S_4)$ . $S_3^2$	$C_2^5$ . $(S_4\times S_3^2)$	$C_2^3$ . $(S_3\times S_4^2)$	all 158
Aut. group:	$\Aut(C_2^4.C_6^2)$	$\Aut(C_6\times \SL(2,3):C_2^2)$	$\Aut((C_2\times C_6).S_4^2)$	$\Aut((A_4\times C_2^4):S_3^2)$	all 8

Elements of the group are displayed as permutations of degree 15.

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{9}$
Commutator length:	$1$

Subgroups

There are 208 normal subgroups (140 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $(C_2^2\times D_6).S_4^2$
Commutator:	$G' \simeq$ $C_2^6:C_3^3$	$G/G' \simeq$ $C_2^4$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $D_6\times S_4^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^6:C_6$	$G/\operatorname{Fit} \simeq$ $S_3\times D_6$
Radical:	$R \simeq$ $(C_2^2\times D_6).S_4^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^2:D_6^2$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^4:D_4^2$
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 18 (order at least 1536) are shown, as well as all normal subgroups of any index.

Order 27648: $(C_2^2\times D_6).S_4^2$

Order 13824: $C_2^6.S_3^3$ x 2, $C_2^6.S_3^3$ x 2, $C_2^6.S_3^3$ x 2, $C_2^6.S_3^3$ x 2, $C_2^5:(A_4\times S_3^2)$, $C_2^3.A_4^2:D_6$, $C_2^6:(C_6\times S_3^2)$, $(C_2^2\times C_6).S_4^2$, $(C_2^2\times C_6).S_4^2$, $(C_2^2\times C_6).S_4^2$, $(C_2^2\times C_6).S_4^2$

Order 9216: $C_2^6.D_6^2$, $C_2^6.D_6^2$, $C_2^4.S_4^2$

Order 6912: $C_2^6.C_3:S_3\times S_3$ x 2, $A_4\times C_2^4:S_3^2$ x 2, $(A_4\times C_2^4):S_3^2$ x 2, $A_4\times C_2^4:S_3^2$ x 2, $(C_2\times C_6):S_4^2$ x 2, $(C_2\times C_6):S_4^2$ x 2, $(C_2\times C_6).S_4^2$ x 2, $(C_2\times C_6).S_4^2$ x 2, $C_3\times C_2^2:S_4^2$ x 2, $C_3\times C_2^2.S_4^2$ x 2, $(C_2\times C_6):S_4^2$ x 2, $(C_2\times C_6).S_4^2$ x 2, $C_2^6:(C_3\times S_3^2)$, $C_2^2:A_4^2:D_6$, $S_3\times C_2^3:A_4^2$, $C_2^6:(C_3\times S_3^2)$, $(A_4\times C_2^4):C_6^2$, $C_2^6:(C_3\times S_3^2)$, $C_2^4.(A_4\times S_3^2)$, $C_2^3:S_4\times S_3^2$, $C_2^5:(C_3^2\times S_4)$, $C_2^3:C_6^2:S_4$, $C_2^3:C_6^2:S_4$, $C_2^3:C_6^2:S_4$, $D_6\times S_4^2$

Order 4608: $C_2^3:S_4^2$ x 10, $C_2^5.D_6^2$ x 4, $C_2^5.D_6^2$ x 4, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 3, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5:D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^3:S_4^2$ x 2, $C_2^3.S_4^2$ x 2, $(C_6\times A_4).C_2^6$, $C_2^5.D_6^2$, $C_2^5.D_6^2$, $(C_6\times \GL(2,\mathbb{Z}/4)):D_4$, $C_2^5.D_6^2$, $S_3\times C_2^5:S_4$, $C_2^5.D_6^2$, $C_6:\GL(2,\mathbb{Z}/4):D_4$, $C_2^5.D_6^2$, $C_2^5.D_6^2$, $C_2\wr C_6.D_6$, $C_2^6:(C_6\times D_6)$, $C_2\wr C_6:D_6$, $C_2^5.D_6^2$, $C_2^5:(C_6\times S_4)$, $(A_4\times C_2^4):S_4$, $C_2^4.(A_4\times S_4)$, $A_4\times C_2^4:S_4$

Order 3456: $S_3\times S_4^2$ x 10, $D_6^2:S_4$ x 4, $C_2^6.C_3^3.C_2$ x 2, $C_6^2.(C_2^2\times S_4)$ x 2, $C_2^4:(C_6\times S_3^2)$ x 2, $D_6^2:S_4$ x 2, $C_2^4:(C_3^2\times S_4)$ x 2, $C_2^2:C_6^2:S_4$ x 2, $C_2^2:C_6^2:S_4$ x 2, $C_2\times A_4^2.D_6$ x 2, $A_4\times D_6\times S_4$ x 2, $D_6^2:S_4$ x 2, $(C_2^4\times A_4).C_3.C_6$, $C_2^6.C_3^2\times S_3$, $(C_2\times D_6):A_4^2$, $C_3\times C_2^3:A_4^2$, $C_2^2:A_4^2:S_3$, $C_6^2.(C_2^2\times S_4)$, $C_6^2.(C_2^2\times S_4)$, $C_2^2:A_4^2:S_3$, $C_2^4:C_3^2\times S_4$, $D_6\times \PSOPlus(4,3)$, $C_2^3:A_4\times S_3^2$, $C_6\times S_4^2$, $C_6:S_4^2$

Order 3072: $C_3:(C_2^5.C_2^5)$, $C_2^4:D_4\times S_4$, $D_4\times C_2^4:S_4$

Order 2304: $C_2^4:D_6^2$ x 24, $C_2^2:S_4^2$ x 12, $C_2^2.S_4^2$ x 10, $C_2^4:D_6^2$ x 9, $(C_2^3\times A_4):S_4$ x 7, $S_4\times C_2^2.D_{12}$ x 6, $(C_2^3\times S_4):D_6$ x 6, $(C_2^2\times C_6\times A_4).C_2^3$ x 6, $C_2^4.D_6^2$ x 6, $(C_2\times \GL(2,\mathbb{Z}/4)).D_6$ x 6, $C_2^4.D_6^2$ x 6, $C_2^4.D_6^2$ x 6, $C_2^3:A_4\times S_4$ x 6, $A_4\times C_2^3:S_4$ x 6, $(C_2\times D_6):\GL(2,\mathbb{Z}/4)$ x 6, $(C_2^2\times S_4):D_{12}$ x 6, $C_2^4.D_6^2$ x 5, $C_2^4.D_6^2$ x 5, $C_2^4.D_6^2$ x 5, $C_2^4.D_6^2$ x 5, $C_2^4.D_6^2$ x 5, $C_2^4.D_6^2$ x 5, $C_2^4.D_6^2$ x 5, $C_2^3.(D_6\times S_4)$ x 4, $(C_2\times D_6):\GL(2,\mathbb{Z}/4)$ x 4, $C_2^6:S_3^2$ x 4, $C_2^6:S_3^2$ x 4, $C_2^4:D_6^2$ x 4, $C_2^3.(D_6\times S_4)$ x 4, $C_2^6:S_3^2$ x 4, $C_2^6:S_3^2$ x 4, $C_2^4:D_6^2$ x 4, $C_2^4:(S_3\times S_4)$ x 4, $C_2^4:(S_3\times S_4)$ x 4, $(C_2^2\times D_{12}):S_4$ x 4, $S_3\times C_2^4:S_4$ x 4, $(C_2^2\times D_{12}):S_4$ x 4, $C_2\wr C_6:S_3$ x 4, $C_2^6:S_3^2$ x 4, $C_2^6:S_3^2$ x 4, $C_3\times C_2\wr D_6$ x 4, $C_2^3:C_{12}\times S_4$ x 3, $(C_2^3\times C_{12}):D_{12}$ x 3, $C_3\times C_2^5:D_{12}$ x 3, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$ x 3, $C_2^6:S_3^2$ x 3, $C_2^4:C_6\times S_4$ x 3, $S_3\times C_2^2.\GL(2,\mathbb{Z}/4)$ x 3, $C_2^4.D_6^2$ x 3, $(A_4\times C_2^4):D_6$ x 3, $C_2^4.D_6^2$ x 3, $A_4\times C_2^4:D_6$ x 3, $(C_2^3\times D_{12}).C_{12}$ x 3, $C_2^4.D_6^2$ x 3, $C_2^6:S_3^2$ x 3, $A_4\times C_2^3:D_{12}$ x 3, $A_4\times C_2^4:D_6$ x 3, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$ x 3, $C_2^4.D_6^2$ x 3, $C_2^3.(D_6\times S_4)$ x 3, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$ x 3, $(C_2^3\times C_{12}):D_{12}$ x 3, $(C_6\times \GL(2,\mathbb{Z}/4)):C_4$ x 3, $C_3\times C_2^3:\GL(2,\mathbb{Z}/4)$ x 3, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$ x 3, $(C_2^3\times C_{12}):D_{12}$ x 3, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$ x 3, $C_6:\GL(2,\mathbb{Z}/4):C_4$ x 3, $C_2^4:C_3.D_4\times S_3$ x 2, $(C_2\times C_6\times A_4).D_4.C_2$ x 2, $C_2^6.S_3^2$ x 2, $C_2\times C_2^6.C_3:S_3$ x 2, $S_3\times C_2^4:D_{12}$ x 2, $C_2^4:(S_3\times D_{12})$ x 2, $C_2^2:S_4\times D_{12}$ x 2, $(C_2^2\times D_{12}):S_4$ x 2, $(C_2^4\times C_{12}):D_6$ x 2, $(C_2^2\times C_6):\GL(2,\mathbb{Z}/4)$ x 2, $(C_2^2\times C_6):\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.(D_6\times S_4)$ x 2, $C_2^4:(S_3\times D_{12})$ x 2, $C_3\times C_2^3:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^6:S_3^2$ x 2, $C_2^6:S_3^2$ x 2, $C_2^4:A_4\times D_6$ x 2, $C_4\times C_2^4:S_3^2$ x 2, $C_4\times C_2^4:S_3^2$ x 2, $(C_2^2\times D_{12}):S_4$ x 2, $(C_2^2\times D_{12}):S_4$ x 2, $(C_2^4\times C_{12}):D_6$ x 2, $C_2^4:(C_4\times S_3^2)$ x 2, $(C_2^4\times C_6):S_4$ x 2, $C_2^5:(S_3\times A_4)$ x 2, $C_2\wr C_6:S_3$ x 2, $(C_2^2\times C_6):\GL(2,\mathbb{Z}/4)$ x 2, $C_6\times C_2^4:S_4$ x 2, $C_2^6:S_3^2$ x 2, $(C_2^4\times C_{12}):D_6$ x 2, $C_3\times C_2^6:D_6$ x 2, $C_2^4:(C_4\times S_3^2)$ x 2, $S_3\times C_2^4:D_{12}$ x 2, $C_2^6:S_3^2$ x 2, $C_2\wr C_6\times S_3$ x 2, $A_4\times C_2^3:S_4$ x 2, $(C_6\times A_4).C_2^4.C_2$, $(C_2^3\times C_{12}):S_4$, $(C_2^3\times C_{12}):S_4$, $C_2^3:S_4\times C_{12}$, $S_3\times C_2^4.S_4$, $C_2^3.(D_6\times S_4)$, $C_3\times C_2^3:\GL(2,\mathbb{Z}/4)$, $C_3\times C_2^5:S_4$, $A_4\times D_{12}:C_2^3$, $C_{12}.(C_2^3\times S_4)$, $S_3\times C_2^5:A_4$, $(C_2^3\times C_{12}):S_4$, $C_2^3.(D_6\times S_4)$, $C_2^6:(C_6\times S_3)$, $C_2^4:(C_{12}\times D_6)$, $C_2^4.(A_4\times D_6)$, $(C_2^4\times D_{12}):C_6$, $C_2^3:A_4\times D_{12}$, $(C_2^4\times C_6):S_4$, $C_2^6:C_6\times S_3$, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$, $C_2^5.(C_6\times D_6)$, $C_2^6:C_6^2$, $C_3\times \GL(2,\mathbb{Z}/4):C_2^3$, $C_2^4:D_6^2$, $C_2^4:D_6^2$, $C_2^6:C_6^2$, $C_2^3:A_4\times S_4$, $C_2^4:(C_6\times S_4)$, $C_2^4:A_4^2$, $C_2^2\times S_4^2$

Order 1728: $A_4^2:D_6$ x 12, $C_3:S_4^2$ x 12, $A_4^2:D_6$ x 6, $C_3:S_4^2$ x 6, $C_3\times S_4^2$ x 6, $(C_6\times D_6):S_4$ x 4, $C_3\times C_2^4:S_3^2$ x 4, $(C_6\times D_6):S_4$ x 4, $C_3\times C_2^4:S_3^2$ x 4, $C_6^2:(C_2\times S_4)$ x 2, $C_6^2:(C_2\times S_4)$ x 2, $A_4^2:D_6$ x 2, $C_6\times A_4\times S_4$ x 2, $A_4:D_6^2$ x 2, $(C_2^3\times C_6):C_6^2$ x 2, $(C_2\times C_6^2):S_4$ x 2, $D_6^2:A_4$ x 2, $(C_2^2\times C_6^2):D_6$ x 2, $(C_2^2\times C_6^2):D_6$ x 2, $C_2^6:C_3^3$, $D_6^2:A_4$, $D_6\times A_4^2$, $A_4^2:D_6$, $C_6\times \PSOPlus(4,3)$, $C_6^2.(C_2^2\times A_4)$, $(C_2\times C_6^2):S_4$, $C_3^2\times C_2^3:S_4$, $D_6^2:A_4$

Order 1536: $C_2\wr C_2^2\times S_4$ x 17, $D_4\times C_2^3:S_4$ x 17, $(C_2\times C_{12}):D_4^2$ x 6, $(C_2\times C_{12}):D_4^2$ x 6, $S_3\times C_2^5:D_4$ x 6, $S_3\times C_2^5:D_4$ x 6, $C_2^7:D_6$ x 4, $C_2^7:D_6$ x 3, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 3, $C_2^4:C_4\times S_4$ x 3, $C_2^6:D_{12}$ x 3, $C_2^2:D_4^2\times S_3$ x 3, $D_4\times C_2^3:D_{12}$ x 3, $(C_2\times C_{12}):D_4^2$ x 3, $D_4\times C_2^4:D_6$ x 3, $(C_2\times C_{12}):D_4^2$ x 3, $(D_6\times C_4^2):D_4$ x 3, $(D_6\times C_4^2):D_4$ x 3, $C_2^5:(C_4\times D_6)$ x 3, $C_2^7:D_6$ x 2, $C_2^6:S_4$ x 2, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 2, $(D_4\times C_2^3):S_4$ x 2, $C_2^4:D_4\times D_6$ x 2, $S_3\times C_2^5:D_4$ x 2, $(C_2\times C_{12}):D_4^2$ x 2, $(C_2\times C_{12}):D_4^2$ x 2, $C_3\times C_2^3:D_4^2$, $\GL(2,\mathbb{Z}/4):C_2^4$, $C_2^6:S_4$, $C_2^4:\GL(2,\mathbb{Z}/4)$, $(C_2^4\times C_4):S_4$, $C_4\times C_2^4:S_4$, $A_4\times C_2^4:D_4$, $D_4\times C_2^4:A_4$, $D_4\times D_{12}:C_2^3$, $(D_6\times C_4^2):D_4$, $(C_2\times C_{12}):D_4^2$, $(C_2\times C_{12}):D_4^2$, $C_2\wr C_2^2\times D_{12}$, $(D_6\times C_4^2):D_4$

Order 1152: $(C_2^3\times C_6):S_4$ x 2, $C_3\times C_2^2\wr S_3$ x 2, $(C_2^2\times A_4):S_4$ x 2, $A_4\times C_2^2:S_4$ x 2, $C_2^2\wr C_3\times S_3$, $C_2^3\times A_4\times D_6$, $C_2^3\times C_6:S_4$, $C_6\times C_2^3\times S_4$, $C_2^3:D_6^2$, $C_2^2:A_4\times S_4$, $A_4\times C_2^4:S_3$, $C_2^4:(C_6\times D_6)$, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$, $C_3\times C_2^2:\GL(2,\mathbb{Z}/4)$, $C_2^3:A_4^2$, $S_3\times C_2^3:S_4$, $(C_2^2\times S_4):A_4$, $C_2^5:C_6^2$, $C_2^5:C_6^2$

Order 1024: $C_2^4:D_4^2$

Order 768: $C_2^5:S_4$, $C_2^6:D_6$, $C_2^6:D_6$

Order 576: $C_2^2:D_6^2$ x 4, $C_2^4:S_3^2$ x 2, $C_2^4:S_3^2$ x 2, $C_2^4:C_6^2$, $C_2^4:C_6^2$, $C_2^2\times A_4\times D_6$, $C_2^2\times C_6:S_4$, $C_2^2\times C_6\times S_4$, $C_2^2:A_4^2$, $S_3\times C_2^3:A_4$, $(C_2^2\times C_6):S_4$, $C_3\times C_2^3:S_4$

Order 384: $C_2^2\wr S_3$ x 2, $D_6\times C_2^5$, $C_2^4\times S_4$, $C_2^6:C_6$, $C_2^5:D_6$, $C_2^6:C_6$, $C_2^5:A_4$, $C_2^2:\GL(2,\mathbb{Z}/4)$

Order 288: $C_2\times A_4\times D_6$ x 2, $(C_2\times C_6):S_4$ x 2, $C_3\times C_2^2:S_4$ x 2, $C_2\times C_6:S_4$ x 2, $C_2\times C_6\times S_4$ x 2, $C_3\times C_2^3:A_4$, $(C_2\times D_6):A_4$, $C_2^3:C_6^2$, $S_3\times C_2^2:A_4$

Order 192: $C_2^3:S_4$, $C_2^5\times C_6$, $D_6\times C_2^4$, $C_2^2\wr C_3$, $A_4\times C_2^4$, $C_2^3\times S_4$, $C_2^4:D_6$

Order 144: $C_2^4:C_3^2$, $C_2^2:C_6^2$, $S_3\times S_4$

Order 128: $C_2^4:D_4$

Order 96: $C_2^3\times D_6$ x 3, $C_2^2:S_4$ x 2, $C_2^2\times S_4$ x 2, $C_2^3:A_4$, $C_2^4\times C_6$, $C_2^3\times A_4$, $C_2^4:C_6$, $C_2^4:S_3$

Order 72: $S_3\times A_4$, $C_3:S_4$, $C_3\times S_4$

Order 64: $C_2^6$

Order 48: $C_2^3\times C_6$ x 2, $C_2^2\times D_6$, $C_2^2:A_4$, $C_2^2\times A_4$

Order 36: $C_3\times A_4$

Order 32: $C_2^5$, $C_2^2\wr C_2$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 3, $C_2^2\times C_6$, $S_4$

Order 16: $C_2^4$ x 2

Order 12: $C_2\times C_6$ x 2, $A_4$

Order 8: $C_2^3$

Order 6: $S_3$

Order 4: $C_2^2$ x 2

Order 3: $C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 18 (order at least 1536) are shown, as well as all normal subgroups of any index.

Order 27648: $(C_2^2\times D_6).S_4^2$

Order 13824: $C_2^5:(A_4\times S_3^2)$, $C_2^3.A_4^2:D_6$, $C_2^6:(C_6\times S_3^2)$, $C_2^6.S_3^3$, $(C_2^2\times C_6).S_4^2$, $(C_2^2\times C_6).S_4^2$, $C_2^6.S_3^3$, $C_2^6.S_3^3$, $(C_2^2\times C_6).S_4^2$, $(C_2^2\times C_6).S_4^2$, $C_2^6.S_3^3$

Order 9216: $C_2^6.D_6^2$, $C_2^6.D_6^2$, $C_2^4.S_4^2$

Order 6912: $C_2^6.C_3:S_3\times S_3$, $C_2^6:(C_3\times S_3^2)$, $C_2^2:A_4^2:D_6$, $S_3\times C_2^3:A_4^2$, $C_2^6:(C_3\times S_3^2)$, $(A_4\times C_2^4):C_6^2$, $A_4\times C_2^4:S_3^2$, $(A_4\times C_2^4):S_3^2$, $C_2^6:(C_3\times S_3^2)$, $C_2^4.(A_4\times S_3^2)$, $A_4\times C_2^4:S_3^2$, $C_2^3:S_4\times S_3^2$, $(C_2\times C_6):S_4^2$, $(C_2\times C_6):S_4^2$, $(C_2\times C_6).S_4^2$, $C_2^5:(C_3^2\times S_4)$, $(C_2\times C_6).S_4^2$, $C_2^3:C_6^2:S_4$, $C_2^3:C_6^2:S_4$, $C_3\times C_2^2:S_4^2$, $C_3\times C_2^2.S_4^2$, $(C_2\times C_6):S_4^2$, $C_2^3:C_6^2:S_4$, $(C_2\times C_6).S_4^2$, $D_6\times S_4^2$

Order 4608: $C_2^3:S_4^2$ x 8, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5:D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $C_2^5.D_6^2$ x 2, $(C_6\times A_4).C_2^6$, $C_2^5.D_6^2$, $C_2^5.D_6^2$, $(C_6\times \GL(2,\mathbb{Z}/4)):D_4$, $C_2^5.D_6^2$, $S_3\times C_2^5:S_4$, $C_2^5.D_6^2$, $C_6:\GL(2,\mathbb{Z}/4):D_4$, $C_2^5.D_6^2$, $C_2^5.D_6^2$, $C_2^5.D_6^2$, $C_2^5.D_6^2$, $C_2^5.D_6^2$, $C_2\wr C_6.D_6$, $C_2^6:(C_6\times D_6)$, $C_2\wr C_6:D_6$, $C_2^5.D_6^2$, $C_2^5.D_6^2$, $C_2^5:(C_6\times S_4)$, $(A_4\times C_2^4):S_4$, $C_2^3:S_4^2$, $C_2^3.S_4^2$, $C_2^4.(A_4\times S_4)$, $A_4\times C_2^4:S_4$

Order 3456: $S_3\times S_4^2$ x 5, $D_6^2:S_4$ x 2, $C_6^2.(C_2^2\times S_4)$ x 2, $C_2^4:(C_6\times S_3^2)$ x 2, $C_2\times A_4^2.D_6$ x 2, $A_4\times D_6\times S_4$ x 2, $(C_2^4\times A_4).C_3.C_6$, $C_2^6.C_3^3.C_2$, $C_2^6.C_3^2\times S_3$, $(C_2\times D_6):A_4^2$, $C_3\times C_2^3:A_4^2$, $C_2^2:A_4^2:S_3$, $D_6^2:S_4$, $C_6^2.(C_2^2\times S_4)$, $C_6^2.(C_2^2\times S_4)$, $C_2^2:A_4^2:S_3$, $C_2^4:C_3^2\times S_4$, $C_2^4:(C_3^2\times S_4)$, $C_2^2:C_6^2:S_4$, $C_2^2:C_6^2:S_4$, $D_6\times \PSOPlus(4,3)$, $C_2^3:A_4\times S_3^2$, $C_6\times S_4^2$, $C_6:S_4^2$, $D_6^2:S_4$

Order 3072: $C_3:(C_2^5.C_2^5)$, $C_2^4:D_4\times S_4$, $D_4\times C_2^4:S_4$

Order 2304: $C_2^4:D_6^2$ x 18, $C_2^4:D_6^2$ x 6, $C_2^3:A_4\times S_4$ x 6, $C_2^2:S_4^2$ x 6, $(C_2^3\times A_4):S_4$ x 6, $A_4\times C_2^3:S_4$ x 5, $C_2^2.S_4^2$ x 5, $C_2^4.D_6^2$ x 4, $C_2^4.D_6^2$ x 4, $C_2^4.D_6^2$ x 4, $S_4\times C_2^2.D_{12}$ x 3, $(C_2^3\times S_4):D_6$ x 3, $(C_2^2\times C_6\times A_4).C_2^3$ x 3, $C_2^4.D_6^2$ x 3, $C_2^4.D_6^2$ x 3, $C_2^4.D_6^2$ x 3, $C_2^4.D_6^2$ x 3, $(C_2\times \GL(2,\mathbb{Z}/4)).D_6$ x 3, $C_2^4.D_6^2$ x 3, $C_2^4.D_6^2$ x 3, $C_2^4.D_6^2$ x 3, $(C_2\times D_6):\GL(2,\mathbb{Z}/4)$ x 3, $(C_2^2\times S_4):D_{12}$ x 3, $C_2^3:C_{12}\times S_4$ x 2, $(C_2^2\times C_6):\GL(2,\mathbb{Z}/4)$ x 2, $(C_2^2\times C_6):\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.(D_6\times S_4)$ x 2, $C_2^3.(D_6\times S_4)$ x 2, $(C_2\times D_6):\GL(2,\mathbb{Z}/4)$ x 2, $(C_2^3\times C_{12}):D_{12}$ x 2, $C_3\times C_2^5:D_{12}$ x 2, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$ x 2, $C_2^6:S_3^2$ x 2, $C_2^4:C_6\times S_4$ x 2, $C_3\times C_2^3:\GL(2,\mathbb{Z}/4)$ x 2, $S_3\times C_2^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^6:S_3^2$ x 2, $C_2^6:S_3^2$ x 2, $C_2^4:D_6^2$ x 2, $C_2^4.D_6^2$ x 2, $(A_4\times C_2^4):D_6$ x 2, $C_2^4.D_6^2$ x 2, $C_2^4:A_4\times D_6$ x 2, $C_2^3.(D_6\times S_4)$ x 2, $(C_2^4\times C_6):S_4$ x 2, $C_2^5:(S_3\times A_4)$ x 2, $C_2\wr C_6:S_3$ x 2, $A_4\times C_2^4:D_6$ x 2, $(C_2^3\times D_{12}).C_{12}$ x 2, $C_2^4.D_6^2$ x 2, $C_2^6:S_3^2$ x 2, $(C_2^2\times C_6):\GL(2,\mathbb{Z}/4)$ x 2, $C_6\times C_2^4:S_4$ x 2, $C_2^6:S_3^2$ x 2, $C_2^6:S_3^2$ x 2, $A_4\times C_2^3:D_{12}$ x 2, $A_4\times C_2^4:D_6$ x 2, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4.D_6^2$ x 2, $C_2^3.(D_6\times S_4)$ x 2, $C_2^4:D_6^2$ x 2, $C_2^4:(S_3\times S_4)$ x 2, $C_2^4:(S_3\times S_4)$ x 2, $(C_2^2\times D_{12}):S_4$ x 2, $S_3\times C_2^4:S_4$ x 2, $(C_2^2\times D_{12}):S_4$ x 2, $C_2\wr C_6:S_3$ x 2, $C_2^6:S_3^2$ x 2, $C_2^6:S_3^2$ x 2, $C_2\wr C_6\times S_3$ x 2, $C_3\times C_2\wr D_6$ x 2, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$ x 2, $(C_2^3\times C_{12}):D_{12}$ x 2, $(C_6\times \GL(2,\mathbb{Z}/4)):C_4$ x 2, $C_3\times C_2^3:\GL(2,\mathbb{Z}/4)$ x 2, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$ x 2, $(C_2^3\times C_{12}):D_{12}$ x 2, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$ x 2, $C_6:\GL(2,\mathbb{Z}/4):C_4$ x 2, $C_2^4:C_3.D_4\times S_3$, $(C_2\times C_6\times A_4).D_4.C_2$, $C_2^6.S_3^2$, $C_2\times C_2^6.C_3:S_3$, $(C_6\times A_4).C_2^4.C_2$, $S_3\times C_2^4:D_{12}$, $C_2^4:(S_3\times D_{12})$, $(C_2^3\times C_{12}):S_4$, $C_2^2:S_4\times D_{12}$, $(C_2^2\times D_{12}):S_4$, $(C_2^4\times C_{12}):D_6$, $(C_2^3\times C_{12}):S_4$, $C_2^3:S_4\times C_{12}$, $S_3\times C_2^4.S_4$, $C_2^3.(D_6\times S_4)$, $C_2^4:(S_3\times D_{12})$, $C_3\times C_2^3:\GL(2,\mathbb{Z}/4)$, $C_3\times C_2^5:S_4$, $C_2^6:S_3^2$, $A_4\times D_{12}:C_2^3$, $C_2^6:S_3^2$, $C_{12}.(C_2^3\times S_4)$, $S_3\times C_2^5:A_4$, $C_4\times C_2^4:S_3^2$, $C_4\times C_2^4:S_3^2$, $(C_2^2\times D_{12}):S_4$, $(C_2^2\times D_{12}):S_4$, $(C_2^4\times C_{12}):D_6$, $(C_2^3\times C_{12}):S_4$, $C_2^3.(D_6\times S_4)$, $C_2^4:(C_4\times S_3^2)$, $C_2^6:(C_6\times S_3)$, $C_2^4:(C_{12}\times D_6)$, $C_2^4.(A_4\times D_6)$, $(C_2^4\times D_{12}):C_6$, $C_2^3:A_4\times D_{12}$, $(C_2^4\times C_6):S_4$, $C_2^6:S_3^2$, $C_2^6:C_6\times S_3$, $(C_2^4\times C_{12}):D_6$, $C_3\times C_2^6:D_6$, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$, $C_2^5.(C_6\times D_6)$, $C_2^6:C_6^2$, $C_3\times \GL(2,\mathbb{Z}/4):C_2^3$, $C_2^4:D_6^2$, $C_2^4:D_6^2$, $C_2^6:C_6^2$, $C_2^3:A_4\times S_4$, $C_2^4:(C_4\times S_3^2)$, $S_3\times C_2^4:D_{12}$, $C_2^6:S_3^2$, $C_2^4:(C_6\times S_4)$, $A_4\times C_2^3:S_4$, $C_2^4:A_4^2$, $C_2^2\times S_4^2$

Order 1728: $A_4^2:D_6$ x 8, $C_3:S_4^2$ x 6, $A_4^2:D_6$ x 3, $C_3:S_4^2$ x 3, $C_3\times S_4^2$ x 3, $(C_6\times D_6):S_4$ x 2, $C_3\times C_2^4:S_3^2$ x 2, $A_4^2:D_6$ x 2, $C_6\times A_4\times S_4$ x 2, $A_4:D_6^2$ x 2, $(C_2^3\times C_6):C_6^2$ x 2, $(C_2\times C_6^2):S_4$ x 2, $D_6^2:A_4$ x 2, $(C_6\times D_6):S_4$ x 2, $C_3\times C_2^4:S_3^2$ x 2, $C_2^6:C_3^3$, $D_6^2:A_4$, $C_6^2:(C_2\times S_4)$, $C_6^2:(C_2\times S_4)$, $D_6\times A_4^2$, $A_4^2:D_6$, $C_6\times \PSOPlus(4,3)$, $C_6^2.(C_2^2\times A_4)$, $(C_2\times C_6^2):S_4$, $C_3^2\times C_2^3:S_4$, $D_6^2:A_4$, $(C_2^2\times C_6^2):D_6$, $(C_2^2\times C_6^2):D_6$

Order 1536: $C_2\wr C_2^2\times S_4$ x 13, $D_4\times C_2^3:S_4$ x 13, $(C_2\times C_{12}):D_4^2$ x 4, $(C_2\times C_{12}):D_4^2$ x 4, $S_3\times C_2^5:D_4$ x 4, $S_3\times C_2^5:D_4$ x 4, $C_2^7:D_6$ x 2, $C_2^6:S_4$ x 2, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4:C_4\times S_4$ x 2, $C_2^6:D_{12}$ x 2, $C_2^7:D_6$ x 2, $C_2^4:D_4\times D_6$ x 2, $C_2^2:D_4^2\times S_3$ x 2, $S_3\times C_2^5:D_4$ x 2, $(C_2\times C_{12}):D_4^2$ x 2, $D_4\times C_2^3:D_{12}$ x 2, $(C_2\times C_{12}):D_4^2$ x 2, $(C_2\times C_{12}):D_4^2$ x 2, $D_4\times C_2^4:D_6$ x 2, $(C_2\times C_{12}):D_4^2$ x 2, $(D_6\times C_4^2):D_4$ x 2, $(D_6\times C_4^2):D_4$ x 2, $C_2^5:(C_4\times D_6)$ x 2, $C_3\times C_2^3:D_4^2$, $C_2^7:D_6$, $\GL(2,\mathbb{Z}/4):C_2^4$, $C_2^6:S_4$, $C_2^4:\GL(2,\mathbb{Z}/4)$, $(D_4\times C_2^3):S_4$, $(C_2^4\times C_4):S_4$, $C_4\times C_2^4:S_4$, $A_4\times C_2^4:D_4$, $D_4\times C_2^4:A_4$, $D_4\times D_{12}:C_2^3$, $(D_6\times C_4^2):D_4$, $(C_2\times C_{12}):D_4^2$, $(C_2\times C_{12}):D_4^2$, $C_2\wr C_2^2\times D_{12}$, $(D_6\times C_4^2):D_4$

Order 1152: $C_2^2\wr C_3\times S_3$, $C_2^3\times A_4\times D_6$, $(C_2^3\times C_6):S_4$, $C_3\times C_2^2\wr S_3$, $C_2^3\times C_6:S_4$, $C_6\times C_2^3\times S_4$, $C_2^3:D_6^2$, $(C_2^2\times A_4):S_4$, $C_2^2:A_4\times S_4$, $A_4\times C_2^2:S_4$, $A_4\times C_2^4:S_3$, $C_2^4:(C_6\times D_6)$, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$, $C_3\times C_2^2:\GL(2,\mathbb{Z}/4)$, $C_2^3:A_4^2$, $S_3\times C_2^3:S_4$, $(C_2^2\times S_4):A_4$, $C_2^5:C_6^2$, $C_2^5:C_6^2$

Order 1024: $C_2^4:D_4^2$

Order 768: $C_2^5:S_4$, $C_2^6:D_6$, $C_2^6:D_6$

Order 576: $C_2^2:D_6^2$ x 4, $C_2^4:C_6^2$, $C_2^4:C_6^2$, $C_2^2\times A_4\times D_6$, $C_2^2\times C_6:S_4$, $C_2^2\times C_6\times S_4$, $C_2^2:A_4^2$, $C_2^4:S_3^2$, $S_3\times C_2^3:A_4$, $(C_2^2\times C_6):S_4$, $C_3\times C_2^3:S_4$, $C_2^4:S_3^2$

Order 384: $D_6\times C_2^5$, $C_2^2\wr S_3$, $C_2^4\times S_4$, $C_2^6:C_6$, $C_2^5:D_6$, $C_2^6:C_6$, $C_2^5:A_4$, $C_2^2:\GL(2,\mathbb{Z}/4)$

Order 288: $C_2\times A_4\times D_6$ x 2, $C_2\times C_6:S_4$ x 2, $C_2\times C_6\times S_4$ x 2, $C_3\times C_2^3:A_4$, $(C_2\times D_6):A_4$, $C_2^3:C_6^2$, $S_3\times C_2^2:A_4$, $(C_2\times C_6):S_4$, $C_3\times C_2^2:S_4$

Order 192: $C_2^3:S_4$, $C_2^5\times C_6$, $D_6\times C_2^4$, $C_2^2\wr C_3$, $A_4\times C_2^4$, $C_2^3\times S_4$, $C_2^4:D_6$

Order 144: $C_2^4:C_3^2$, $C_2^2:C_6^2$, $S_3\times S_4$

Order 128: $C_2^4:D_4$

Order 96: $C_2^3\times D_6$ x 3, $C_2^2\times S_4$ x 2, $C_2^3:A_4$, $C_2^4\times C_6$, $C_2^3\times A_4$, $C_2^2:S_4$, $C_2^4:C_6$, $C_2^4:S_3$

Order 72: $S_3\times A_4$, $C_3:S_4$, $C_3\times S_4$

Order 64: $C_2^6$

Order 48: $C_2^3\times C_6$ x 2, $C_2^2\times D_6$, $C_2^2:A_4$, $C_2^2\times A_4$

Order 36: $C_3\times A_4$

Order 32: $C_2^5$, $C_2^2\wr C_2$

Order 27: $C_3^3$

Order 24: $C_2\times D_6$ x 3, $C_2^2\times C_6$, $S_4$

Order 16: $C_2^4$ x 2

Order 12: $C_2\times C_6$ x 2, $A_4$

Order 8: $C_2^3$

Order 6: $S_3$

Order 4: $C_2^2$ x 2

Order 3: $C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 27648: $(C_2^2\times D_6).S_4^2$ ($C_1$)

Order 13824: $C_2^6.S_3^3$ ($C_2$) x 2, $C_2^6.S_3^3$ ($C_2$) x 2, $C_2^6.S_3^3$ ($C_2$) x 2, $C_2^6.S_3^3$ ($C_2$) x 2, $C_2^5:(A_4\times S_3^2)$ ($C_2$), $C_2^3.A_4^2:D_6$ ($C_2$), $C_2^6:(C_6\times S_3^2)$ ($C_2$), $(C_2^2\times C_6).S_4^2$ ($C_2$), $(C_2^2\times C_6).S_4^2$ ($C_2$), $(C_2^2\times C_6).S_4^2$ ($C_2$), $(C_2^2\times C_6).S_4^2$ ($C_2$)

Order 6912: $C_2^6.C_3:S_3\times S_3$ ($C_2^2$) x 2, $A_4\times C_2^4:S_3^2$ ($C_2^2$) x 2, $(A_4\times C_2^4):S_3^2$ ($C_2^2$) x 2, $A_4\times C_2^4:S_3^2$ ($C_2^2$) x 2, $(C_2\times C_6):S_4^2$ ($C_2^2$) x 2, $(C_2\times C_6):S_4^2$ ($C_2^2$) x 2, $(C_2\times C_6).S_4^2$ ($C_2^2$) x 2, $(C_2\times C_6).S_4^2$ ($C_2^2$) x 2, $C_3\times C_2^2:S_4^2$ ($C_2^2$) x 2, $C_3\times C_2^2.S_4^2$ ($C_2^2$) x 2, $(C_2\times C_6):S_4^2$ ($C_2^2$) x 2, $(C_2\times C_6).S_4^2$ ($C_2^2$) x 2, $C_2^6:(C_3\times S_3^2)$ ($C_2^2$), $C_2^2:A_4^2:D_6$ ($C_2^2$), $S_3\times C_2^3:A_4^2$ ($C_2^2$), $C_2^6:(C_3\times S_3^2)$ ($C_2^2$), $(A_4\times C_2^4):C_6^2$ ($C_2^2$), $C_2^6:(C_3\times S_3^2)$ ($C_2^2$), $C_2^4.(A_4\times S_3^2)$ ($C_2^2$), $C_2^5:(C_3^2\times S_4)$ ($C_2^2$), $C_2^3:C_6^2:S_4$ ($C_2^2$), $C_2^3:C_6^2:S_4$ ($C_2^2$), $C_2^3:C_6^2:S_4$ ($C_2^2$)

Order 4608: $C_2^5:D_6^2$ ($S_3$), $C_2^5.D_6^2$ ($S_3$), $C_2^3:S_4^2$ ($S_3$)

Order 3456: $C_2^6.C_3^3.C_2$ ($C_2^3$) x 2, $C_2^4:(C_3^2\times S_4)$ ($C_2^3$) x 2, $C_2^2:C_6^2:S_4$ ($C_2^3$) x 2, $C_2^2:C_6^2:S_4$ ($C_2^3$) x 2, $(C_2^4\times A_4).C_3.C_6$ ($C_2^3$), $C_2^6.C_3^2\times S_3$ ($C_2^3$), $(C_2\times D_6):A_4^2$ ($C_2^3$), $C_3\times C_2^3:A_4^2$ ($C_2^3$), $C_2^2:A_4^2:S_3$ ($C_2^3$), $C_2^2:A_4^2:S_3$ ($C_2^3$), $C_2^4:C_3^2\times S_4$ ($C_2^3$)

Order 2304: $C_2^4:D_6^2$ ($D_6$) x 2, $C_2^4:D_6^2$ ($D_6$) x 2, $C_2^2:S_4^2$ ($D_6$) x 2, $C_2^2.S_4^2$ ($D_6$) x 2, $(C_6\times A_4).C_2^4.C_2$ ($D_6$), $C_2^6:S_3^2$ ($D_6$), $C_2^4:C_6\times S_4$ ($D_6$), $(A_4\times C_2^4):D_6$ ($D_6$), $C_2^4:A_4\times D_6$ ($D_6$), $(C_2^4\times C_6):S_4$ ($D_6$), $A_4\times C_2^4:D_6$ ($D_6$), $C_2^6:S_3^2$ ($D_6$), $C_6\times C_2^4:S_4$ ($D_6$), $C_2^4:D_6^2$ ($D_6$), $C_2^3:A_4\times S_4$ ($D_6$), $A_4\times C_2^3:S_4$ ($D_6$), $(C_2^3\times A_4):S_4$ ($D_6$)

Order 1728: $C_2^6:C_3^3$ ($C_2^4$)

Order 1152: $(C_2^3\times C_6):S_4$ ($C_2\times D_6$) x 2, $C_3\times C_2^2\wr S_3$ ($C_2\times D_6$) x 2, $(C_2^2\times A_4):S_4$ ($C_2\times D_6$) x 2, $A_4\times C_2^2:S_4$ ($C_2\times D_6$) x 2, $C_2^2\wr C_3\times S_3$ ($C_2\times D_6$), $C_2^3\times A_4\times D_6$ ($C_2\times D_6$), $C_2^3\times C_6:S_4$ ($C_2\times D_6$), $C_6\times C_2^3\times S_4$ ($C_2\times D_6$), $C_2^3:D_6^2$ ($S_4$), $C_2^2:A_4\times S_4$ ($C_2\times D_6$), $A_4\times C_2^4:S_3$ ($C_2\times D_6$), $C_2^4:(C_6\times D_6)$ ($C_2\times D_6$), $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$ ($C_2\times D_6$), $C_3\times C_2^2:\GL(2,\mathbb{Z}/4)$ ($C_2\times D_6$), $C_2^3:A_4^2$ ($C_2\times D_6$), $S_3\times C_2^3:S_4$ ($S_4$), $(C_2^2\times S_4):A_4$ ($C_2\times D_6$), $C_2^5:C_6^2$ ($C_2\times D_6$), $C_2^5:C_6^2$ ($C_2\times D_6$)

Order 768: $C_2^5:S_4$ ($S_3^2$), $C_2^6:D_6$ ($S_3^2$), $C_2^6:D_6$ ($S_3^2$)

Order 576: $C_2^2:D_6^2$ ($C_2\times S_4$) x 4, $C_2^4:S_3^2$ ($C_2\times S_4$) x 2, $C_2^4:S_3^2$ ($C_2\times S_4$) x 2, $C_2^4:C_6^2$ ($C_2^2\times D_6$), $C_2^4:C_6^2$ ($C_2^2\times D_6$), $C_2^2\times A_4\times D_6$ ($C_2\times S_4$), $C_2^2\times C_6:S_4$ ($C_2\times S_4$), $C_2^2\times C_6\times S_4$ ($C_2\times S_4$), $C_2^2:A_4^2$ ($C_2^2\times D_6$), $S_3\times C_2^3:A_4$ ($C_2\times S_4$), $(C_2^2\times C_6):S_4$ ($C_2\times S_4$), $C_3\times C_2^3:S_4$ ($C_2\times S_4$)

Order 384: $C_2^2\wr S_3$ ($S_3\times D_6$) x 2, $D_6\times C_2^5$ ($S_3\times D_6$), $C_2^4\times S_4$ ($S_3\times D_6$), $C_2^6:C_6$ ($S_3\times D_6$), $C_2^5:D_6$ ($S_3\times D_6$), $C_2^6:C_6$ ($S_3\times D_6$), $C_2^5:A_4$ ($S_3\times D_6$), $C_2^2:\GL(2,\mathbb{Z}/4)$ ($S_3\times D_6$)

Order 288: $C_2\times A_4\times D_6$ ($C_2^2\times S_4$) x 2, $(C_2\times C_6):S_4$ ($C_2^2\times S_4$) x 2, $C_3\times C_2^2:S_4$ ($C_2^2\times S_4$) x 2, $C_2\times C_6:S_4$ ($C_2^2\times S_4$) x 2, $C_2\times C_6\times S_4$ ($C_2^2\times S_4$) x 2, $C_3\times C_2^3:A_4$ ($C_2^2\times S_4$), $(C_2\times D_6):A_4$ ($C_2^2\times S_4$), $C_2^3:C_6^2$ ($C_2^2\times S_4$), $S_3\times C_2^2:A_4$ ($C_2^2\times S_4$)

Order 192: $C_2^3:S_4$ ($S_3\times S_4$), $C_2^5\times C_6$ ($D_6^2$), $D_6\times C_2^4$ ($S_3\times S_4$), $C_2^2\wr C_3$ ($D_6^2$), $A_4\times C_2^4$ ($D_6^2$), $C_2^3\times S_4$ ($S_3\times S_4$), $C_2^4:D_6$ ($S_3\times S_4$)

Order 144: $C_2^4:C_3^2$ ($C_2^3\times S_4$), $C_2^2:C_6^2$ ($C_2^3\times S_4$), $S_3\times S_4$ ($C_2^3:S_4$)

Order 128: $C_2^4:D_4$ ($S_3^3$)

Order 96: $C_2^3\times D_6$ ($D_6\times S_4$) x 3, $C_2^2:S_4$ ($D_6\times S_4$) x 2, $C_2^2\times S_4$ ($D_6\times S_4$) x 2, $C_2^3:A_4$ ($D_6\times S_4$), $C_2^4\times C_6$ ($D_6\times S_4$), $C_2^3\times A_4$ ($D_6\times S_4$), $C_2^4:C_6$ ($D_6\times S_4$), $C_2^4:S_3$ ($D_6\times S_4$)

Order 72: $S_3\times A_4$ ($C_2^4:S_4$), $C_3:S_4$ ($C_2^4:S_4$), $C_3\times S_4$ ($C_2^4:S_4$)

Order 64: $C_2^6$ ($C_2\times S_3^3$)

Order 48: $C_2^3\times C_6$ ($C_2^2:D_6^2$) x 2, $C_2^2\times D_6$ ($S_4^2$), $C_2^2:A_4$ ($C_2^2:D_6^2$), $C_2^2\times A_4$ ($C_2^2:D_6^2$)

Order 36: $C_3\times A_4$ ($C_2^5:S_4$)

Order 32: $C_2^5$ ($S_4\times S_3^2$), $C_2^2\wr C_2$ ($S_4\times S_3^2$)

Order 24: $C_2\times D_6$ ($C_2\times S_4^2$) x 2, $C_2^2\times C_6$ ($C_2\times S_4^2$), $C_2\times D_6$ ($S_3\times C_2^3:S_4$), $S_4$ ($S_3\times C_2^3:S_4$)

Order 16: $C_2^4$ ($A_4:D_6^2$) x 2

Order 12: $C_2\times C_6$ ($C_2^4:D_6^2$), $C_2\times C_6$ ($C_2^2\times S_4^2$), $A_4$ ($C_2^4:D_6^2$)

Order 8: $C_2^3$ ($S_3\times S_4^2$)

Order 6: $S_3$ ($C_2^3:S_4^2$)

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

Order 3: $C_3$ ($C_2^4.S_4^2$)

Order 1: $C_1$ ($(C_2^2\times D_6).S_4^2$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 27648: $(C_2^2\times D_6).S_4^2$ ($C_1$)

Order 13824: $C_2^5:(A_4\times S_3^2)$ ($C_2$), $C_2^3.A_4^2:D_6$ ($C_2$), $C_2^6:(C_6\times S_3^2)$ ($C_2$), $C_2^6.S_3^3$ ($C_2$), $(C_2^2\times C_6).S_4^2$ ($C_2$), $(C_2^2\times C_6).S_4^2$ ($C_2$), $C_2^6.S_3^3$ ($C_2$), $C_2^6.S_3^3$ ($C_2$), $(C_2^2\times C_6).S_4^2$ ($C_2$), $(C_2^2\times C_6).S_4^2$ ($C_2$), $C_2^6.S_3^3$ ($C_2$)

Order 6912: $C_2^6.C_3:S_3\times S_3$ ($C_2^2$), $C_2^6:(C_3\times S_3^2)$ ($C_2^2$), $C_2^2:A_4^2:D_6$ ($C_2^2$), $S_3\times C_2^3:A_4^2$ ($C_2^2$), $C_2^6:(C_3\times S_3^2)$ ($C_2^2$), $(A_4\times C_2^4):C_6^2$ ($C_2^2$), $A_4\times C_2^4:S_3^2$ ($C_2^2$), $(A_4\times C_2^4):S_3^2$ ($C_2^2$), $C_2^6:(C_3\times S_3^2)$ ($C_2^2$), $C_2^4.(A_4\times S_3^2)$ ($C_2^2$), $A_4\times C_2^4:S_3^2$ ($C_2^2$), $(C_2\times C_6):S_4^2$ ($C_2^2$), $(C_2\times C_6):S_4^2$ ($C_2^2$), $(C_2\times C_6).S_4^2$ ($C_2^2$), $C_2^5:(C_3^2\times S_4)$ ($C_2^2$), $(C_2\times C_6).S_4^2$ ($C_2^2$), $C_2^3:C_6^2:S_4$ ($C_2^2$), $C_2^3:C_6^2:S_4$ ($C_2^2$), $C_3\times C_2^2:S_4^2$ ($C_2^2$), $C_3\times C_2^2.S_4^2$ ($C_2^2$), $(C_2\times C_6):S_4^2$ ($C_2^2$), $C_2^3:C_6^2:S_4$ ($C_2^2$), $(C_2\times C_6).S_4^2$ ($C_2^2$)

Order 4608: $C_2^5:D_6^2$ ($S_3$), $C_2^5.D_6^2$ ($S_3$), $C_2^3:S_4^2$ ($S_3$)

Order 3456: $(C_2^4\times A_4).C_3.C_6$ ($C_2^3$), $C_2^6.C_3^3.C_2$ ($C_2^3$), $C_2^6.C_3^2\times S_3$ ($C_2^3$), $(C_2\times D_6):A_4^2$ ($C_2^3$), $C_3\times C_2^3:A_4^2$ ($C_2^3$), $C_2^2:A_4^2:S_3$ ($C_2^3$), $C_2^2:A_4^2:S_3$ ($C_2^3$), $C_2^4:C_3^2\times S_4$ ($C_2^3$), $C_2^4:(C_3^2\times S_4)$ ($C_2^3$), $C_2^2:C_6^2:S_4$ ($C_2^3$), $C_2^2:C_6^2:S_4$ ($C_2^3$)

Order 2304: $(C_6\times A_4).C_2^4.C_2$ ($D_6$), $C_2^6:S_3^2$ ($D_6$), $C_2^4:C_6\times S_4$ ($D_6$), $C_2^4:D_6^2$ ($D_6$), $(A_4\times C_2^4):D_6$ ($D_6$), $C_2^4:A_4\times D_6$ ($D_6$), $(C_2^4\times C_6):S_4$ ($D_6$), $A_4\times C_2^4:D_6$ ($D_6$), $C_2^6:S_3^2$ ($D_6$), $C_6\times C_2^4:S_4$ ($D_6$), $C_2^4:D_6^2$ ($D_6$), $C_2^4:D_6^2$ ($D_6$), $C_2^3:A_4\times S_4$ ($D_6$), $C_2^2:S_4^2$ ($D_6$), $A_4\times C_2^3:S_4$ ($D_6$), $(C_2^3\times A_4):S_4$ ($D_6$), $C_2^2.S_4^2$ ($D_6$)

Order 1728: $C_2^6:C_3^3$ ($C_2^4$)

Order 1152: $C_2^2\wr C_3\times S_3$ ($C_2\times D_6$), $C_2^3\times A_4\times D_6$ ($C_2\times D_6$), $(C_2^3\times C_6):S_4$ ($C_2\times D_6$), $C_3\times C_2^2\wr S_3$ ($C_2\times D_6$), $C_2^3\times C_6:S_4$ ($C_2\times D_6$), $C_6\times C_2^3\times S_4$ ($C_2\times D_6$), $C_2^3:D_6^2$ ($S_4$), $(C_2^2\times A_4):S_4$ ($C_2\times D_6$), $C_2^2:A_4\times S_4$ ($C_2\times D_6$), $A_4\times C_2^2:S_4$ ($C_2\times D_6$), $A_4\times C_2^4:S_3$ ($C_2\times D_6$), $C_2^4:(C_6\times D_6)$ ($C_2\times D_6$), $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$ ($C_2\times D_6$), $C_3\times C_2^2:\GL(2,\mathbb{Z}/4)$ ($C_2\times D_6$), $C_2^3:A_4^2$ ($C_2\times D_6$), $S_3\times C_2^3:S_4$ ($S_4$), $(C_2^2\times S_4):A_4$ ($C_2\times D_6$), $C_2^5:C_6^2$ ($C_2\times D_6$), $C_2^5:C_6^2$ ($C_2\times D_6$)

Order 768: $C_2^5:S_4$ ($S_3^2$), $C_2^6:D_6$ ($S_3^2$), $C_2^6:D_6$ ($S_3^2$)

Order 576: $C_2^2:D_6^2$ ($C_2\times S_4$) x 4, $C_2^4:C_6^2$ ($C_2^2\times D_6$), $C_2^4:C_6^2$ ($C_2^2\times D_6$), $C_2^2\times A_4\times D_6$ ($C_2\times S_4$), $C_2^2\times C_6:S_4$ ($C_2\times S_4$), $C_2^2\times C_6\times S_4$ ($C_2\times S_4$), $C_2^2:A_4^2$ ($C_2^2\times D_6$), $C_2^4:S_3^2$ ($C_2\times S_4$), $S_3\times C_2^3:A_4$ ($C_2\times S_4$), $(C_2^2\times C_6):S_4$ ($C_2\times S_4$), $C_3\times C_2^3:S_4$ ($C_2\times S_4$), $C_2^4:S_3^2$ ($C_2\times S_4$)

Order 384: $D_6\times C_2^5$ ($S_3\times D_6$), $C_2^2\wr S_3$ ($S_3\times D_6$), $C_2^4\times S_4$ ($S_3\times D_6$), $C_2^6:C_6$ ($S_3\times D_6$), $C_2^5:D_6$ ($S_3\times D_6$), $C_2^6:C_6$ ($S_3\times D_6$), $C_2^5:A_4$ ($S_3\times D_6$), $C_2^2:\GL(2,\mathbb{Z}/4)$ ($S_3\times D_6$)

Order 288: $C_2\times A_4\times D_6$ ($C_2^2\times S_4$) x 2, $C_2\times C_6:S_4$ ($C_2^2\times S_4$) x 2, $C_2\times C_6\times S_4$ ($C_2^2\times S_4$) x 2, $C_3\times C_2^3:A_4$ ($C_2^2\times S_4$), $(C_2\times D_6):A_4$ ($C_2^2\times S_4$), $C_2^3:C_6^2$ ($C_2^2\times S_4$), $S_3\times C_2^2:A_4$ ($C_2^2\times S_4$), $(C_2\times C_6):S_4$ ($C_2^2\times S_4$), $C_3\times C_2^2:S_4$ ($C_2^2\times S_4$)

Order 192: $C_2^3:S_4$ ($S_3\times S_4$), $C_2^5\times C_6$ ($D_6^2$), $D_6\times C_2^4$ ($S_3\times S_4$), $C_2^2\wr C_3$ ($D_6^2$), $A_4\times C_2^4$ ($D_6^2$), $C_2^3\times S_4$ ($S_3\times S_4$), $C_2^4:D_6$ ($S_3\times S_4$)

Order 144: $C_2^4:C_3^2$ ($C_2^3\times S_4$), $C_2^2:C_6^2$ ($C_2^3\times S_4$), $S_3\times S_4$ ($C_2^3:S_4$)

Order 128: $C_2^4:D_4$ ($S_3^3$)

Order 96: $C_2^3\times D_6$ ($D_6\times S_4$) x 3, $C_2^2\times S_4$ ($D_6\times S_4$) x 2, $C_2^3:A_4$ ($D_6\times S_4$), $C_2^4\times C_6$ ($D_6\times S_4$), $C_2^3\times A_4$ ($D_6\times S_4$), $C_2^2:S_4$ ($D_6\times S_4$), $C_2^4:C_6$ ($D_6\times S_4$), $C_2^4:S_3$ ($D_6\times S_4$)

Order 72: $S_3\times A_4$ ($C_2^4:S_4$), $C_3:S_4$ ($C_2^4:S_4$), $C_3\times S_4$ ($C_2^4:S_4$)

Order 64: $C_2^6$ ($C_2\times S_3^3$)

Order 48: $C_2^3\times C_6$ ($C_2^2:D_6^2$) x 2, $C_2^2\times D_6$ ($S_4^2$), $C_2^2:A_4$ ($C_2^2:D_6^2$), $C_2^2\times A_4$ ($C_2^2:D_6^2$)

Order 36: $C_3\times A_4$ ($C_2^5:S_4$)

Order 32: $C_2^5$ ($S_4\times S_3^2$), $C_2^2\wr C_2$ ($S_4\times S_3^2$)

Order 24: $C_2\times D_6$ ($C_2\times S_4^2$) x 2, $C_2^2\times C_6$ ($C_2\times S_4^2$), $C_2\times D_6$ ($S_3\times C_2^3:S_4$), $S_4$ ($S_3\times C_2^3:S_4$)

Order 16: $C_2^4$ ($A_4:D_6^2$) x 2

Order 12: $C_2\times C_6$ ($C_2^4:D_6^2$), $C_2\times C_6$ ($C_2^2\times S_4^2$), $A_4$ ($C_2^4:D_6^2$)

Order 8: $C_2^3$ ($S_3\times S_4^2$)

Order 6: $S_3$ ($C_2^3:S_4^2$)

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

Order 3: $C_3$ ($C_2^4.S_4^2$)

Order 1: $C_1$ ($(C_2^2\times D_6).S_4^2$)

Series

Derived series

$(C_2^2\times D_6).S_4^2$

$\rhd$

$C_2^6:C_3^3$

$\rhd$

$C_2^6$

$\rhd$

$C_1$

Chief series

$(C_2^2\times D_6).S_4^2$

$\rhd$

$C_2^6.S_3^3$

$\rhd$

$C_3\times C_2^2:S_4^2$

$\rhd$

$C_2^4:(C_3^2\times S_4)$

$\rhd$

$C_2^6:C_3^3$

$\rhd$

$C_2^2:A_4^2$

$\rhd$

$C_2^2\wr C_3$

$\rhd$

$C_2^6$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_1$

Lower central series

$(C_2^2\times D_6).S_4^2$

$\rhd$

$C_2^6:C_3^3$

Upper central series

$C_1$

Supergroups

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

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

Character theory

Complex character table

Every character has rational values, so the complex character table is the same as the rational character table below.

Rational character table

See the $210 \times 210$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.