Abstract group 9216.lt: $(D_4\times C_2^3).D_6^2$

• Label: 9216.lt
• Order: \( 2^{10} \cdot 3^{2} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{4} \)
• $\card{Z(G)}$: 2
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3^{2} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{5} \)
• Perm deg.: $15$
• Trans deg.: not computed
• Rank: $4$

Group information

Description:	$(D_4\times C_2^3).D_6^2$
Order:	\(9216\)\(\medspace = 2^{10} \cdot 3^{2} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$(C_6\times A_4).C_2^5.C_2^6$, of order \(147456\)\(\medspace = 2^{14} \cdot 3^{2} \)
Composition factors:	$C_2$ x 10, $C_3$ x 2
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	8	12	24
Elements	1	903	26	2552	1446	640	3136	512	9216
Conjugacy classes	1	32	3	43	34	4	34	5	156
Divisions	1	32	3	43	34	4	34	5	156
Autjugacy classes	1	19	3	26	23	4	19	4	99

Dimension	1	2	3	4	6	8	12	16	24
Irr. complex chars.	16	28	16	32	20	18	18	2	6	156
Irr. rational chars.	16	28	16	32	20	18	18	2	6	156

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	24	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g \mid a^{2}=b^{2}=c^{12}=d^{12}=e^{2}=f^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,3,2)(4,7,6)(8,15)(12,14), (1,2)(5,7), (1,3,2)(4,7,5,6)(8,9,15,10)(11,14) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(A_4:D_4^2)$ . $D_6$	$A_4$ . $(D_4^2:D_6)$	$(C_2^5.D_{12})$ . $D_6$ (2)	$(C_2^5.D_{12})$ . $D_6$ (2)	all 160

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

Homology

Abelianization:	$C_{2}^{4} $
Schur multiplier:	$C_{2}^{8}$
Commutator length:	$1$

Subgroups

There are 221 normal subgroups (109 characteristic).

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

Special subgroups

Center:	a subgroup isomorphic to $C_2$
Commutator:	a subgroup isomorphic to $C_2^4:C_6^2$
Frattini:	a subgroup isomorphic to $C_2\times D_4$
Fitting:	not computed
Radical:	not computed
Socle:	not computed
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:D_4^2$

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 512) are shown, as well as all normal subgroups of any index.

Order 9216: $(D_4\times C_2^3).D_6^2$

Order 4608: $(C_2^2\times D_6):\GL(2,\mathbb{Z}/4)$ x 2, $(C_2^3\times S_4):D_{12}$ x 2, $C_2^5.(C_4\times S_3^2)$ x 2, $C_2^5.D_6^2$ x 2, $C_2^2.D_6^2.C_2^3$, $C_6:\GL(2,\mathbb{Z}/4).D_4$, $(C_4\times C_{12}):\GL(2,\mathbb{Z}/4)$, $C_3\times C_4^2:\GL(2,\mathbb{Z}/4)$, $C_2^2:C_{12}^2:D_4$, $C_{12}:C_2^4:D_{12}$, $C_{12}:\GL(2,\mathbb{Z}/4):C_4$

Order 3072: $C_4^2.(D_4\times S_4)$, $(C_4\times C_{12}).D_4^2$

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

Order 1536: $C_2\wr C_2^2\times S_4$ x 2, $D_4:D_4\times S_4$ x 2, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 2, $C_2\wr C_4\times S_4$ x 2, $C_2^6:D_{12}$ x 2, $(C_2\times C_{12}):D_4^2$ x 2, $(C_2\times C_{12}).D_4^2$ x 2, $(D_4\times C_2^3):D_{12}$ x 2, $(C_2^3\times D_{12}):D_4$ x 2, $(C_2\times C_{12}).D_4^2$ x 2, $(D_6\times C_2^4):D_4$ x 2, $(C_2^3\times D_{12}):D_4$ x 2, $(C_2\times C_{12}).D_4^2$ x 2, $S_4\times D_4^2$, $C_4^2:\GL(2,\mathbb{Z}/4)$, $C_4^2:\GL(2,\mathbb{Z}/4)$, $C_4^2:C_4\times S_4$, $(A_4\times C_4^2):D_4$, $A_4\times C_2\wr D_4$, $(C_6\times D_4^2):C_2^2$, $C_3:D_4^3$, $(D_6\times C_2^4):D_4$, $C_2\wr D_4\times D_6$, $(C_3\times D_4):D_4^2$, $C_2^4.(D_4\times D_6)$, $(C_2^2\times C_4^2):D_{12}$, $(C_3\times D_4^2):D_4$, $(C_2\times D_4^2).D_6$, $(D_6\times C_4^2):D_4$, $(C_2\times D_4^2).D_6$, $(C_2\times D_4^2).D_6$, $(D_6\times C_4^2):D_4$, $(C_2^2\times C_6).D_4^2$, $C_2^4.(D_4\times D_6)$

Order 1152: $\GL(2,\mathbb{Z}/4):D_6$ x 16, $C_6:S_4\times D_4$ x 7, $\GL(2,\mathbb{Z}/4):D_6$ x 6, $D_6:\GL(2,\mathbb{Z}/4)$ x 6, $(C_2\times S_4):D_{12}$ x 6, $C_2^5.S_3^2$ x 6, $A_4\times D_6:D_4$ x 5, $C_3\times C_2^2:\GL(2,\mathbb{Z}/4)$ x 5, $D_6:\GL(2,\mathbb{Z}/4)$ x 5, $C_2^5.S_3^2$ x 5, $C_2^5.S_3^2$ x 5, $(C_6\times D_4):S_4$ x 4, $C_2^5:S_3^2$ x 4, $D_6\times \GL(2,\mathbb{Z}/4)$ x 4, $C_{12}:\GL(2,\mathbb{Z}/4)$ x 4, $D_6:\GL(2,\mathbb{Z}/4)$ x 4, $C_2^5.S_3^2$ x 4, $D_6:(C_4\times S_4)$ x 4, $(C_2^3\times C_{12}):C_{12}$ x 4, $C_3\times C_2^2.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^3:D_6^2$ x 3, $C_2^3.D_6^2$ x 2, $(C_2^2\times D_6):D_{12}$ x 2, $C_{12}^2:D_4$ x 2, $C_3^2:C_2\wr D_4$ x 2, $C_{12}^2:D_4$ x 2, $(C_2\times D_6^2):C_4$ x 2, $D_4\times A_4\times D_6$ x 2, $(D_4\times A_4):D_6$ x 2, $(D_4\times A_4):D_6$ x 2, $C_6\times D_4\times S_4$ x 2, $C_2^3.D_6^2$ x 2, $C_2^3.D_6^2$ x 2, $C_2^3.D_6^2$ x 2, $A_4\times C_{12}:D_4$ x 2, $C_{12}:\GL(2,\mathbb{Z}/4)$ x 2, $C_{12}:\GL(2,\mathbb{Z}/4)$ x 2, $D_6:(C_4\times S_4)$ x 2, $C_{12}:\GL(2,\mathbb{Z}/4)$ x 2, $(C_2\times S_4):D_{12}$ x 2, $C_{12}^2:C_2^3$, $C_{12}^2:(C_2\times C_4)$, $C_2^3.D_6^2$, $C_4\times A_4\times D_{12}$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{24}:(C_2\times S_4)$, $C_{24}:(C_2\times S_4)$, $C_{24}:(C_2\times S_4)$, $D_8\times C_3:S_4$, $C_{24}:(C_2\times S_4)$, $C_{24}:(C_2\times S_4)$, $(C_4\times C_{12}):S_4$, $C_4\times C_{12}:S_4$, $C_4\times C_{12}:S_4$, $(A_4\times D_{12}):C_4$, $C_{12}:(C_4\times S_4)$, $C_{12}:(C_4\times S_4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $(C_4\times C_{12}):S_4$, $C_2^5:C_6^2$, $A_4\times D_8:C_6$, $C_2^5.C_6^2$, $D_4:C_{12}\times A_4$, $A_4\times \OD_{16}:C_6$, $D_{12}^2:C_2$

Order 1024: $C_4^2:D_4^2$

Order 768: $D_4^2:D_6$ x 9, $(C_2^3\times D_6):D_4$ x 8, $D_6:D_4^2$ x 5, $C_2^6:D_6$ x 5, $(C_2^3\times C_{12}):D_4$ x 4, $(C_2^3\times D_6):D_4$ x 4, $(C_2^3\times C_{12}):D_4$ x 4, $(C_2\times C_4^2).D_{12}$ x 4, $(C_2^3\times C_{12}):D_4$ x 4, $C_2\times C_4^2:D_{12}$ x 4, $(C_2^2\times D_{12}):D_4$ x 4, $(C_2\times C_6).D_4^2$ x 4, $(C_2^2\times D_{12}):D_4$ x 4, $(C_2^3\times C_{12}):D_4$ x 4, $(C_2\times C_6).D_4^2$ x 4, $C_2^5:D_{12}$ x 4, $(C_2^2\times D_{12}):D_4$ x 4, $C_2^5:(C_4\times S_3)$ x 4, $C_2^5:D_{12}$ x 4, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 4, $C_2^3:C_4\times S_4$ x 4, $D_8:C_2\times S_4$ x 3, $A_4:D_4^2$ x 3, $C_4^2:(C_2\times S_4)$ x 3, $D_4:\GL(2,\mathbb{Z}/4)$ x 3, $C_4\wr C_2\times S_4$ x 3, $D_6\times D_4^2$ x 3, $D_6:D_4^2$ x 3, $D_6:D_4^2$ x 3, $D_6:D_4^2$ x 3, $(C_2^3\times C_{12}).D_4$ x 2, $(C_2^3\times C_{12}).D_4$ x 2, $C_2^5.D_{12}$ x 2, $(C_2^3\times C_{12}):D_4$ x 2, $(C_2^3\times C_{12}):D_4$ x 2, $C_2^5:D_{12}$ x 2, $C_2^4.(S_3\times D_4)$ x 2, $C_2^4.(C_4\times D_6)$ x 2, $C_2^4.(C_4\times D_6)$ x 2, $C_2^5.D_{12}$ x 2, $C_2^4.(S_3\times D_4)$ x 2, $(C_2^3\times C_{12}).D_4$ x 2, $(C_2^3\times D_{12}):C_4$ x 2, $(C_2^3\times D_{12}):C_4$ x 2, $(C_2^3\times C_{12}):D_4$ x 2, $C_2^3.(C_4\times D_{12})$ x 2, $C_2^4.(C_4\times D_6)$ x 2, $C_2^5.D_{12}$ x 2, $(D_6\times C_2^4):C_4$ x 2, $(D_6\times C_2^4):C_4$ x 2, $(C_2^2\times D_6):C_4^2$ x 2, $(D_4\times C_2^3):C_{12}$ x 2, $C_2^3.(D_4\times C_{12})$ x 2, $D_6:D_4^2$ x 2, $C_{12}:D_4^2$ x 2, $D_6:D_4^2$ x 2, $D_6:D_4^2$ x 2, $C_2^5:D_{12}$ x 2, $C_2^5.D_{12}$ x 2, $C_2\wr C_4\times D_6$ x 2, $C_2^3.(D_4\times D_6)$ x 2, $(C_2^2\times D_{12}):D_4$ x 2, $C_2\wr D_4\times C_6$ x 2, $C_3\times C_2^5:D_4$ x 2, $C_2^3.(D_4\times D_6)$ x 2, $\GL(2,\mathbb{Z}/4):C_2^3$ x 2, $\GL(2,\mathbb{Z}/4):C_2^3$ x 2, $A_4:D_4^2$ x 2, $C_4:D_4\times S_4$ x 2, $C_4:D_4\times S_4$ x 2, $D_4\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 2, $D_4:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5.D_{12}$ x 2, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 2, $S_4\times \OD_{16}:C_2$ x 2, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5.D_{12}$ x 2, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 2, $A_4\times C_2\wr C_2^2$ x 2, $A_4\times D_4:D_4$ x 2, $C_2^6:C_{12}$ x 2, $(C_2^3\times C_{12}):D_4$ x 2, $C_2\wr C_2^2\times D_6$ x 2, $C_{12}:D_4^2$ x 2, $C_{12}:D_4^2$ x 2, $C_{12}:D_4^2$ x 2, $(C_2^3\times C_{12}).D_4$, $(C_2^3\times C_{12}).D_4$, $C_2^4.(S_3\times D_4)$, $(C_3\times D_4^2):C_4$, $(D_6\times C_4^2):C_4$, $(C_4\times C_{12}):C_4^2$, $(C_2^2\times C_4^2):C_{12}$, $C_3\times C_4^2:C_2^4$, $C_{12}:D_4^2$, $C_{12}:D_4^2$, $C_{12}:D_4^2$, $C_{12}:D_4^2$, $C_{12}:D_4^2$, $D_{12}:(C_4\times D_4)$, $C_{12}:D_4^2$, $D_4^2:D_6$, $(C_6\times D_4).C_2^4$, $C_4^2:C_4\times D_6$, $C_2^3.(D_4\times D_6)$, $(C_2^2\times D_{12}):C_2^3$, $(D_4\times D_6):D_4$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_2^3.(D_4\times D_6)$, $A_4:D_4^2$, $A_4:D_4^2$, $C_4\times D_4\times S_4$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $A_4\times D_4^2$, $A_4\times C_4^2:C_4$, $(D_4\times C_2^3):D_6$, $C_2\times C_6:D_4^2$, $C_{12}.D_4^2$, $D_4:D_4\times D_6$, $C_{12}:D_4^2$, $C_{12}:D_4^2$

Order 576: $C_2^2:D_6^2$ x 28, $D_4\times C_3:S_4$ x 24, $\GL(2,\mathbb{Z}/4):S_3$ x 24, $C_2^4:S_3^2$ x 16, $S_3\times \GL(2,\mathbb{Z}/4)$ x 16, $C_6:\GL(2,\mathbb{Z}/4)$ x 13, $S_3\times D_4\times A_4$ x 8, $\GL(2,\mathbb{Z}/4):C_6$ x 8, $D_{12}:S_4$ x 8, $S_4\times D_{12}$ x 8, $S_3\times C_4:S_4$ x 8, $C_4\times S_3\times S_4$ x 8, $(C_2\times C_{12}):D_{12}$ x 8, $C_2^4:C_6^2$ x 7, $C_2\times C_{12}:S_4$ x 7, $C_6:D_4\times A_4$ x 6, $C_6\times \GL(2,\mathbb{Z}/4)$ x 6, $C_6:\GL(2,\mathbb{Z}/4)$ x 6, $C_2^4.S_3^2$ x 6, $C_6.\GL(2,\mathbb{Z}/4)$ x 6, $C_2^2\times C_6:S_4$ x 5, $(C_3\times D_4):S_4$ x 5, $(C_2\times C_{12}):S_4$ x 5, $C_2\times C_{12}:S_4$ x 5, $D_6^2:C_2^2$ x 5, $A_4\times C_6.D_4$ x 5, $C_6.\GL(2,\mathbb{Z}/4)$ x 5, $C_2^4.S_3^2$ x 4, $D_6^2:C_2^2$ x 4, $A_4\times D_6:C_4$ x 4, $C_6.\GL(2,\mathbb{Z}/4)$ x 4, $C_6.\GL(2,\mathbb{Z}/4)$ x 4, $C_6.\GL(2,\mathbb{Z}/4)$ x 4, $(C_6\times C_{12}):D_4$ x 4, $(C_6\times D_4).D_6$ x 4, $(C_2^2\times D_6):C_{12}$ x 4, $(C_2\times C_{12}).D_{12}$ x 4, $C_2^2\times A_4\times D_6$ x 3, $C_2^2\times C_6\times S_4$ x 3, $C_3^2:D_4^2$ x 3, $C_2^4.C_6^2$ x 2, $C_4:D_6^2$ x 2, $C_4\times A_4\times D_6$ x 2, $(C_3\times Q_8):S_4$ x 2, $C_2\times C_{12}\times S_4$ x 2, $C_6:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4.S_3^2$ x 2, $C_2^4.S_3^2$ x 2, $C_{12}^2:C_2^2$ x 2, $C_2^2.D_6^2$ x 2, $C_6.\GL(2,\mathbb{Z}/4)$ x 2, $C_4\times C_6.S_4$ x 2, $C_6.\GL(2,\mathbb{Z}/4)$ x 2, $C_{12}^2:C_4$ x 2, $(C_6\times D_4).D_6$ x 2, $C_6^2.C_2^4$, $C_2\times A_4\times D_{12}$, $Q_8\times C_3:S_4$, $C_2\times C_{12}:S_4$, $C_{24}:(C_2\times D_6)$, $C_{12}^2:C_2^2$, $C_3\times A_4\times \SD_{16}$, $C_3\times A_4\times D_8$, $C_3\times A_4\times \OD_{16}$, $C_2^2:C_{12}^2$, $D_{12}^2$, $C_4^2:S_3^2$, $A_4\times C_{12}:C_4$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_{12}.(C_2\times S_4)$, $C_{24}:S_4$, $C_{24}:S_4$, $C_{24}:S_4$, $C_{24}:S_4$, $(C_4\times A_4):C_{12}$, $C_4^2:C_6^2$, $(D_4\times C_3^2):D_4$, $(C_6\times D_4):D_6$, $C_{12}^2:C_2^2$, $C_{12}^2:C_2^2$, $C_{12}^2:C_2^2$, $C_{12}.(S_3\times D_4)$, $C_{12}.(S_3\times D_4)$

Order 512: $C_2^6:D_4$ x 4, $C_2^6:D_4$ x 4, $D_4^2:C_2^3$ x 2, $C_2^3:D_4^2$ x 2, $D_4^2:D_4$ x 2, $C_2^3.D_4^2$ x 2, $C_2^3.D_4^2$ x 2, $C_2^3.D_4^2$ x 2, $C_2^3.D_4^2$ x 2, $C_2^3.D_4^2$ x 2, $D_4^3$, $D_4:D_4^2$, $C_4^3:D_4$, $C_4^3:D_4$, $C_4^3:D_4$, $C_2^3.D_4^2$, $C_4^3:D_4$

Order 384: $C_2^2.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5:D_6$ x 2, $C_2^2:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5.D_6$ x 2, $C_2^5.C_{12}$, $(C_4\times C_{12}):D_4$, $C_{12}.C_2^5$, $A_4\times D_4:C_2^2$, $C_2^2\times C_{12}:D_4$, $A_4\times C_4:D_4$, $C_2^5.C_{12}$

Order 288: $C_2\times C_6:S_4$ x 3, $C_2^3:C_6^2$ x 2, $C_2^3.C_6^2$, $C_2\times C_6.S_4$

Order 256: $C_4^2:C_2^4$

Order 192: $C_2.\GL(2,\mathbb{Z}/4)$ x 2, $(C_2\times C_{12}):D_4$ x 2, $C_2^3.D_{12}$ x 2, $D_6\times C_2^4$ x 2, $C_2^3\times S_4$ x 2, $C_{12}:C_2^4$ x 2, $C_2^5:C_6$ x 2, $C_2^4.D_6$ x 2, $A_4\times C_4^2$, $C_3\times D_4:D_4$, $C_2^2\times C_4\times C_{12}$, $C_3:D_4^2$, $(C_4\times C_{12}):C_4$

Order 144: $C_6:S_4$ x 2, $C_2^2:C_6^2$

Order 128: $D_4:C_2^4$, $C_4^2:C_2^3$, $\OD_{16}:C_2^3$

Order 96: $C_2^3.D_6$ x 2, $C_2^4\times C_6$ x 2, $C_2^3\times A_4$ x 2, $D_6:D_4$ x 2, $\OD_{16}:C_6$, $C_6.C_2^4$, $C_2^3\times C_{12}$, $C_2^3:C_{12}$, $C_{12}:D_4$

Order 72: $C_3:S_4$ x 2, $C_6\times A_4$

Order 64: $D_4\times C_2^3$ x 2, $C_2^2\times C_4^2$, $D_4:D_4$

Order 48: $C_2^2\times D_6$ x 2, $C_6\times D_4$ x 2, $C_6.D_4$ x 2, $C_2^3\times C_6$, $C_2^2\times A_4$, $C_4\times C_{12}$

Order 36: $C_3\times A_4$

Order 32: $C_2^5$ x 2, $\OD_{16}:C_2$, $D_4:C_2^2$, $C_2^3\times C_4$, $C_4:D_4$

Order 24: $C_2^2\times C_6$ x 3, $C_2\times C_{12}$, $C_2\times A_4$

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

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

Order 8: $C_2^3$ x 3, $C_2\times C_4$

Order 6: $C_6$

Order 4: $C_2^2$ x 2

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

not computed

Normal subgroups (quotient in parentheses)

Order 9216: $(D_4\times C_2^3).D_6^2$ ($C_1$)

Order 4608: $(C_2^2\times D_6):\GL(2,\mathbb{Z}/4)$ ($C_2$) x 2, $(C_2^3\times S_4):D_{12}$ ($C_2$) x 2, $C_2^5.(C_4\times S_3^2)$ ($C_2$) x 2, $C_2^5.D_6^2$ ($C_2$) x 2, $C_2^2.D_6^2.C_2^3$ ($C_2$), $C_6:\GL(2,\mathbb{Z}/4).D_4$ ($C_2$), $(C_4\times C_{12}):\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_3\times C_4^2:\GL(2,\mathbb{Z}/4)$ ($C_2$), $C_2^2:C_{12}^2:D_4$ ($C_2$), $C_{12}:C_2^4:D_{12}$ ($C_2$), $C_{12}:\GL(2,\mathbb{Z}/4):C_4$ ($C_2$)

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

Order 1536: $C_4^2:\GL(2,\mathbb{Z}/4)$ ($S_3$), $(D_6\times C_2^4):D_4$ ($S_3$)

Order 1152: $C_2^3:D_6^2$ ($D_4$) x 2, $A_4\times D_6:D_4$ ($C_2^3$) x 2, $C_3\times C_2^2:\GL(2,\mathbb{Z}/4)$ ($C_2^3$) x 2, $D_6:\GL(2,\mathbb{Z}/4)$ ($D_4$) x 2, $C_2^5.S_3^2$ ($D_4$) x 2, $C_2^5.S_3^2$ ($D_4$) x 2, $(C_2^3\times C_{12}):C_{12}$ ($C_2^3$) x 2, $C_3\times C_2^2.\GL(2,\mathbb{Z}/4)$ ($C_2^3$) x 2, $(C_6\times D_4):S_4$ ($C_2^3$), $C_6:S_4\times D_4$ ($C_2^3$), $C_6:S_4\times D_4$ ($D_4$), $C_{12}:\GL(2,\mathbb{Z}/4)$ ($C_2^3$), $C_{12}:\GL(2,\mathbb{Z}/4)$ ($D_4$), $(C_4\times C_{12}):S_4$ ($D_4$), $C_4\times C_{12}:S_4$ ($D_4$), $C_{12}.\GL(2,\mathbb{Z}/4)$ ($C_2^3$), $C_2^5:C_6^2$ ($C_2^3$), $C_2^5.C_6^2$ ($C_2^3$), $A_4\times \OD_{16}:C_6$ ($C_2^3$)

Order 768: $C_2^5.D_{12}$ ($D_6$) x 2, $C_2^3:\GL(2,\mathbb{Z}/4)$ ($D_6$) x 2, $C_2^5.D_{12}$ ($D_6$) x 2, $(C_2^3\times C_{12}):D_4$ ($D_6$) x 2, $C_3\times C_4^2:C_2^4$ ($D_6$), $(C_6\times D_4).C_2^4$ ($D_6$), $A_4:D_4^2$ ($D_6$), $C_2^3.\GL(2,\mathbb{Z}/4)$ ($D_6$), $A_4\times D_4:D_4$ ($D_6$), $C_2\times C_6:D_4^2$ ($D_6$)

Order 576: $C_2^2\times A_4\times D_6$ ($C_2\times D_4$) x 2, $C_2^2\times C_6:S_4$ ($C_2\times D_4$) x 2, $C_2^2\times C_6\times S_4$ ($C_2\times D_4$) x 2, $C_6:\GL(2,\mathbb{Z}/4)$ ($C_2\times D_4$) x 2, $C_2\times C_{12}:S_4$ ($C_2\times D_4$) x 2, $A_4\times C_6.D_4$ ($C_2\times D_4$) x 2, $C_6.\GL(2,\mathbb{Z}/4)$ ($C_2\times D_4$) x 2, $C_2^4:C_6^2$ ($C_2^4$), $C_2^4:C_6^2$ ($C_2\times D_4$), $C_2\times C_{12}:S_4$ ($C_2\times D_4$), $C_2^2:C_{12}^2$ ($C_2\times D_4$), $C_4\times C_6.S_4$ ($C_2\times D_4$)

Order 384: $C_2^2.\GL(2,\mathbb{Z}/4)$ ($C_2\times D_6$) x 2, $C_2^5:D_6$ ($C_2\times D_6$) x 2, $C_2^2:\GL(2,\mathbb{Z}/4)$ ($C_2\times D_6$) x 2, $C_2^5.D_6$ ($C_2\times D_6$) x 2, $C_2^5.C_{12}$ ($C_2\times D_6$), $(C_4\times C_{12}):D_4$ ($S_4$), $C_{12}.C_2^5$ ($C_2\times D_6$), $A_4\times D_4:C_2^2$ ($C_2\times D_6$), $C_2^2\times C_{12}:D_4$ ($C_2\times D_6$), $A_4\times C_4:D_4$ ($C_2\times D_6$), $C_2^5.C_{12}$ ($C_2\times D_6$)

Order 288: $C_2\times C_6:S_4$ ($C_2^2\wr C_2$) x 3, $C_2^3:C_6^2$ ($C_2^2\times D_4$) x 2, $C_2^3.C_6^2$ ($C_2^2\times D_4$), $C_2\times C_6.S_4$ ($C_2^2\wr C_2$)

Order 256: $C_4^2:C_2^4$ ($S_3^2$)

Order 192: $C_2.\GL(2,\mathbb{Z}/4)$ ($S_3\times D_4$) x 2, $(C_2\times C_{12}):D_4$ ($C_2\times S_4$) x 2, $C_2^3.D_{12}$ ($C_2\times S_4$) x 2, $D_6\times C_2^4$ ($S_3\times D_4$) x 2, $C_2^3\times S_4$ ($S_3\times D_4$) x 2, $C_2^4.D_6$ ($S_3\times D_4$) x 2, $A_4\times C_4^2$ ($S_3\times D_4$), $C_3\times D_4:D_4$ ($C_2\times S_4$), $C_{12}:C_2^4$ ($C_2^2\times D_6$), $C_{12}:C_2^4$ ($S_3\times D_4$), $C_2^5:C_6$ ($C_2^2\times D_6$), $C_2^5:C_6$ ($S_3\times D_4$), $C_2^2\times C_4\times C_{12}$ ($S_3\times D_4$), $C_3:D_4^2$ ($C_2\times S_4$), $(C_4\times C_{12}):C_4$ ($C_2\times S_4$)

Order 144: $C_6:S_4$ ($C_2\wr C_2^2$) x 2, $C_2^2:C_6^2$ ($C_2^3:D_4$)

Order 128: $D_4:C_2^4$ ($S_3\times D_6$), $C_4^2:C_2^3$ ($S_3\times D_6$), $\OD_{16}:C_2^3$ ($S_3\times D_6$)

Order 96: $C_2^3.D_6$ ($C_2^2\times S_4$) x 2, $C_2^4\times C_6$ ($D_4\times D_6$) x 2, $C_2^3\times A_4$ ($D_4\times D_6$) x 2, $D_6:D_4$ ($C_2^2\times S_4$) x 2, $\OD_{16}:C_6$ ($C_2^2\times S_4$), $C_6.C_2^4$ ($C_2^2\times S_4$), $C_2^3\times C_{12}$ ($D_4\times D_6$), $C_2^3:C_{12}$ ($D_4\times D_6$), $C_{12}:D_4$ ($C_2^2\times S_4$)

Order 72: $C_3:S_4$ ($C_2\wr D_4$) x 2, $C_6\times A_4$ ($C_2^4:D_4$)

Order 64: $D_4\times C_2^3$ ($D_6^2$), $D_4\times C_2^3$ ($D_6:D_6$), $C_2^2\times C_4^2$ ($D_6:D_6$), $D_4:D_4$ ($S_3\times S_4$)

Order 48: $C_2^2\times D_6$ ($D_4\times S_4$) x 2, $C_6.D_4$ ($D_4\times S_4$) x 2, $C_2^3\times C_6$ ($C_2^4:D_6$), $C_2^2\times A_4$ ($C_2^4:D_6$), $C_6\times D_4$ ($C_2^3\times S_4$), $C_6\times D_4$ ($D_4\times S_4$), $C_4\times C_{12}$ ($D_4\times S_4$)

Order 36: $C_3\times A_4$ ($D_4^2:C_2^2$)

Order 32: $C_2^5$ ($D_4\times S_3^2$) x 2, $\OD_{16}:C_2$ ($D_6\times S_4$), $D_4:C_2^2$ ($D_6\times S_4$), $C_2^3\times C_4$ ($D_6^2:C_2$), $C_4:D_4$ ($D_6\times S_4$)

Order 24: $C_2^2\times C_6$ ($\GL(2,\mathbb{Z}/4):C_2^2$) x 2, $C_2\times C_{12}$ ($\GL(2,\mathbb{Z}/4):C_2^2$), $C_2^2\times C_6$ ($C_2\wr C_2^2\times S_3$), $C_2\times A_4$ ($C_2\wr C_2^2\times S_3$)

Order 16: $C_4^2$ ($\GL(2,\mathbb{Z}/4):S_3$), $C_2^4$ ($D_6^2:C_2^2$), $C_2\times D_4$ ($C_2^2:D_6^2$), $C_2\times D_4$ ($\GL(2,\mathbb{Z}/4):S_3$)

Order 12: $C_2\times C_6$ ($D_4^2:D_6$), $C_2\times C_6$ ($C_2^6:D_6$), $A_4$ ($D_4^2:D_6$)

Order 8: $C_2^3$ ($\GL(2,\mathbb{Z}/4):D_6$) x 2, $C_2^3$ ($D_6^2:D_4$), $C_2\times C_4$ ($\GL(2,\mathbb{Z}/4):D_6$)

Order 6: $C_6$ ($C_2\wr C_2^2\times S_4$)

Order 4: $C_2^2$ ($C_2^4.D_6^2$), $C_2^2$ ($C_{12}^2:(C_2\times D_4)$)

Order 3: $C_3$ ($C_4^2.(D_4\times S_4)$)

Order 2: $C_2$ ($C_2^5.D_6^2$)

Order 1: $C_1$ ($(D_4\times C_2^3).D_6^2$)

Normal subgroups up to automorphism (quotient in parentheses)

not computed

Series

Derived series	not computed
Chief series	not computed
Lower central series	not computed
Upper central series	not computed

Supergroups

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

This group is a maximal quotient of 0 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 $156 \times 156$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.