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

• Label: 27648.fm
• Order: \( 2^{10} \cdot 3^{3} \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{13} \cdot 3^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{4} \)
• Perm deg.: $15$
• Trans deg.: not computed
• Rank: $3$

Group information

Description:	$(C_2^2\times D_6).S_4^2$
Order:	\(27648\)\(\medspace = 2^{10} \cdot 3^{3} \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	$C_2^3\times C_2^6.C_3^3.C_2^4$, of order \(221184\)\(\medspace = 2^{13} \cdot 3^{3} \)
Composition factors:	$C_2$ x 10, $C_3$ x 3
Derived length:	$4$

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	1543	890	5496	8582	1152	7680	2304	27648
Conjugacy classes	1	26	7	33	48	4	31	3	153
Divisions	1	26	7	33	48	4	31	3	153
Autjugacy classes	1	19	7	26	35	4	23	3	118

Dimension	1	2	3	4	6	8	9	12	16	18	24	32	36	48
Irr. complex chars.	8	12	16	14	24	13	8	24	6	12	9	1	4	2	153
Irr. rational chars.	8	12	16	14	24	13	8	24	6	12	9	1	4	2	153

Minimal presentations

Permutation degree:	$15$
Transitive degree:	not computed
Rank:	$3$
Inequivalent generating triples:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	24	24	24
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h \mid b^{6}=c^{6}=d^{4}=e^{12}=f^{2}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $15$ $\langle(1,3)(4,7)(8,10)(9,15)(11,12)(13,14), (1,3)(4,7,5,6)(11,12)(13,14), (1,2) \!\cdots\! \rangle$
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:	$(C_2^3.S_4^2)$ . $S_3$	$(C_2^5:S_4)$ . $S_3^2$	$C_2^5$ . $(S_4\times S_3^2)$	$C_2^3$ . $(S_3\times S_4^2)$	all 75

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

Homology

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

Subgroups

There are 2060382 subgroups in 28233 conjugacy classes, 76 normal (74 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^6.S_3^3$
Commutator:	$G' \simeq$ $C_3\times Q_8:A_4^2$	$G/G' \simeq$ $C_2^3$
Frattini:	$\Phi \simeq$ $C_2^3$	$G/\Phi \simeq$ $S_3\times S_4^2$
Fitting:	$\operatorname{Fit} \simeq$ $C_{12}.C_2^5$	$G/\operatorname{Fit} \simeq$ $S_3\times D_6$
Radical:	$R \simeq$ $(C_2^2\times D_6).S_4^2$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^2\times C_6$	$G/\operatorname{soc} \simeq$ $S_3\times C_2^3:S_4$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_4^2:D_4^2$
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 36 (order at least 768) are shown, as well as all normal subgroups of any index.

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

Order 13824: $Q_8:A_4^2:D_6$, $(C_2^2\times C_6).S_4^2$, $(C_3\times Q_8).S_4^2$, $C_2^5.(A_4\times S_3^2)$, $Q_8:A_4^2:D_6$, $C_3\times C_2^3.S_4^2$, $(C_2^2\times C_6):S_4^2$

Order 9216: $C_2^5.(D_6\times S_4)$, $(D_4\times C_2^3).D_6^2$, $C_2^4.S_4^2$

Order 6912: $Q_8:A_4^2:S_3$, $C_2^5:(C_3^2\times S_4)$, $(C_2^2\times D_6):A_4^2$, $Q_8:A_4^2:C_6$, $Q_8:A_4^2:C_6$, $Q_8:A_4^2:S_3$, $Q_8:A_4^2:S_3$, $(C_2\times D_6^2):S_4$, $D_6\times S_4^2$

Order 4608: $C_2^3:S_4^2$ x 2, $C_2^3.S_4^2$ x 2, $C_2^4.(D_6\times S_4)$, $C_2^4.(D_6\times S_4)$, $(D_6\times C_2^4):S_4$, $(C_2^2\times D_6):\GL(2,\mathbb{Z}/4)$, $C_2^4.(D_6\times S_4)$, $C_2^4.(D_6\times S_4)$, $(C_2^2\times D_6):\GL(2,\mathbb{Z}/4)$, $C_6.C_2\wr D_6$, $(C_2^3\times D_{12}):S_4$, $(C_2\times D_{12}):\GL(2,\mathbb{Z}/4)$, $(C_2\times D_{12}):\GL(2,\mathbb{Z}/4)$, $(C_2^2\times D_6):\GL(2,\mathbb{Z}/4)$, $(C_2^5\times C_6):D_{12}$, $(D_6\times C_2^4):D_{12}$, $C_2^6:C_6\wr C_2$, $(C_2^3\times S_4):D_{12}$, $C_{12}:\GL(2,\mathbb{Z}/4):C_4$, $(C_2^3\times C_4).D_6^2$, $C_6:\GL(2,\mathbb{Z}/4).D_4$, $C_2^5.(C_4\times S_3^2)$, $(D_6\times C_2^4):S_4$, $C_2^5:(C_6\times S_4)$, $C_2^5.D_6^2$, $C_6:\GL(2,\mathbb{Z}/4):D_4$, $C_2^5.D_6^2$, $C_2^5.(A_4\times D_6)$, $C_2^4.(D_6\times S_4)$, $C_3\times C_2^4:\GL(2,\mathbb{Z}/4)$, $C_2^4.(D_6\times S_4)$, $(C_2^3\times D_{12}):S_4$, $C_2^3.S_4^2$, $Q_8:S_4^2$, $C_2\wr A_4:S_4$, $C_2\wr A_4\times S_4$, $A_4\times C_2\wr S_4$, $C_2\wr S_4\times D_6$

Order 3456: $S_3\times S_4^2$ x 8, $C_2\times A_4^2.D_6$ x 3, $C_3^2:C_2\wr S_4$ x 2, $C_3^2:C_2\wr S_4$ x 2, $A_4\times D_6\times S_4$ x 2, $C_3\times Q_8:A_4^2$, $C_6.S_4^2$, $C_6^2:C_2^2:S_4$, $D_6\times \PSOPlus(4,3)$, $(C_2\times D_6^2):A_4$, $Q_8:A_4:S_3^2$, $C_6\times S_4^2$, $C_6:S_4^2$

Order 3072: $C_4^2.(D_4\times S_4)$, $D_4\times C_2\wr S_4$, $(C_2\times D_{12}).D_4^2$

Order 2304: $S_3\times C_2\wr S_4$ x 5, $(C_2^3\times A_4):S_4$ x 4, $(C_2^3\times D_6):S_4$ x 4, $(A_4\times C_2^4):D_6$ x 2, $C_2^4:D_6^2$ x 2, $A_4\times C_2^3:S_4$ x 2, $A_4\times Q_8:S_4$ x 2, $(C_2^3\times S_4):A_4$ x 2, $Q_8:A_4\times S_4$ 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, $(C_2^4\times C_6):S_4$, $(C_2^3\times D_6):S_4$, $C_3\times C_2^5.S_4$, $(D_6\times C_2^4):A_4$, $(C_2^3\times C_{12}):S_4$, $(C_2^3\times C_{12}):D_{12}$, $C_3\times C_2^3:\GL(2,\mathbb{Z}/4)$, $C_2^4.D_6^2$, $C_2^4.D_6^2$, $C_2^4.D_6^2$, $C_2^4.D_6^2$, $(C_2^2\times C_6):\GL(2,\mathbb{Z}/4)$, $C_{12}.(C_2^3\times S_4)$, $C_2^4.D_6^2$, $(C_2^3\times D_{12}):A_4$, $(C_3\times Q_8):\GL(2,\mathbb{Z}/4)$, $C_2^4.(S_3\times S_4)$, $C_2^4.(S_3\times S_4)$, $(C_2^2\times C_6).\GL(2,\mathbb{Z}/4)$, $C_3\times D_4:\GL(2,\mathbb{Z}/4)$, $(C_2^3\times C_{12}):S_4$, $C_3\times C_2^5:S_4$, $(C_2^3\times C_{12}):S_4$, $C_3\times A_4:D_4^2$, $(D_6\times C_2^4):A_4$, $C_2^5:(C_6\times A_4)$, $(C_2^4\times C_6).S_4$, $C_3\times Q_8:\GL(2,\mathbb{Z}/4)$, $C_2^4:D_6^2$, $C_2^4.(S_3\times S_4)$, $C_2^4.(S_3\times S_4)$, $C_2^5.C_6\wr C_2$, $(C_2^3\times D_{12}):C_{12}$, $C_2^4.D_6^2$, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$, $C_2^5.(C_6\times D_6)$, $C_2^6:C_6^2$, $C_2^6:S_3^2$, $C_2^4.D_6^2$, $C_2^4.D_6^2$, $A_4\times C_2^4:D_6$, $C_2^4.D_6^2$, $C_2^4.D_6^2$, $\GL(2,\mathbb{Z}/4):D_{12}$, $C_2^4.D_6^2$, $C_2^4.D_6^2$, $C_2^3.(D_6\times S_4)$, $C_6\times C_2^4:S_4$, $(C_2^2\times C_6):\GL(2,\mathbb{Z}/4)$, $C_2^5:(S_3\times A_4)$, $(C_2\times D_6^2):D_4$, $(Q_8\times A_4):S_4$, $C_2^4:A_4^2$, $C_2\wr A_4\times D_6$, $D_6\times Q_8:S_4$, $C_2\wr S_4\times C_6$, $S_3\times C_2^4:S_4$, $C_2^2\times S_4^2$, $C_6:C_2\wr S_4$, $(C_2^3\times D_6):S_4$, $(C_2^2\times C_6).\GL(2,\mathbb{Z}/4)$, $C_2^4.(S_3\times S_4)$, $(C_2^2\times C_6).\GL(2,\mathbb{Z}/4)$, $D_4:(D_6\times S_4)$, $C_{12}.(D_4\times S_4)$, $C_2^4.(A_4\times D_6)$, $C_2^4.(A_4\times D_6)$, $(C_2^3\times C_{12}).S_4$, $(C_2^2\times C_6).\GL(2,\mathbb{Z}/4)$, $C_{12}.(D_4\times S_4)$, $C_{12}.(D_4\times S_4)$, $C_3\times C_2^3.\GL(2,\mathbb{Z}/4)$, $(C_2\times C_{12}).\GL(2,\mathbb{Z}/4)$, $C_{12}.(D_4\times S_4)$, $C_3\times C_2^3.\GL(2,\mathbb{Z}/4)$, $(C_2\times C_{12}).\GL(2,\mathbb{Z}/4)$, $(C_2\times C_{12}).\GL(2,\mathbb{Z}/4)$, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$, $(C_2^3\times S_4):C_{12}$, $(D_6\times C_2^4):C_{12}$, $(C_2^2\times C_6).\GL(2,\mathbb{Z}/4)$, $C_{12}.(D_4\times S_4)$, $(C_3\times D_4):\GL(2,\mathbb{Z}/4)$, $(C_2^5\times C_6):C_{12}$

Order 1728: $C_3:S_4^2$ x 12, $A_4^2:D_6$ x 8, $A_4^2:D_6$ x 4, $C_3:S_4^2$ x 4, $C_3\times S_4^2$ x 4, $A_4^2:D_6$ x 3, $C_6\times A_4\times S_4$ x 3, $A_4^2:D_6$ x 2, $A_4:D_6^2$ x 2, $(Q_8\times C_3^2):S_4$ x 2, $C_3^2:C_2\wr A_4$ x 2, $D_6\times A_4^2$, $C_6\times \PSOPlus(4,3)$, $D_4:C_6^2:C_6$, $(C_2\times C_6^2):S_4$, $C_3^2\times C_2^3:S_4$, $C_3\times A_4:\GL(2,3)$, $A_4\times C_3:\GL(2,3)$, $C_3:S_4\times \SL(2,3)$

Order 1536: $C_2\wr C_2^2\times S_4$ x 3, $(D_4\times C_2^3):S_4$ x 3, $C_2^6:S_4$ x 2, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 2, $C_2\wr C_4\times S_4$ x 2, $C_2^6:D_{12}$ x 2, $C_2^5:(C_2\times S_4)$ x 2, $D_4\times Q_8:S_4$, $D_4\times C_2^3:S_4$, $S_4\times D_4^2$, $C_2^6:S_4$, $D_4:D_4\times S_4$, $C_4^2:\GL(2,\mathbb{Z}/4)$, $C_4^2:\GL(2,\mathbb{Z}/4)$, $C_4^2:C_4\times S_4$, $(A_4\times C_4^2):D_4$, $(D_4\times C_2^3):S_4$, $(D_4\times C_2^3):S_4$, $C_4:C_2\wr S_4$, $C_4\times C_2\wr S_4$, $A_4\times C_2\wr D_4$, $D_4\times C_2\wr A_4$, $C_3\times C_2^6:D_4$, $C_3:D_4^3$, $(C_2^3\times D_{12}):D_4$, $C_2\wr D_4\times D_6$, $(C_2\times C_{12}):D_4^2$, $(C_2\times C_{12}):D_4^2$, $(C_3\times D_4):D_4^2$, $C_2^4.(D_4\times D_6)$, $C_2^6:D_{12}$, $(C_3\times D_4^2):D_4$, $(C_2^2\times C_6).D_4^2$, $(C_2\times C_{12}).D_4^2$, $(C_2\times C_{12}).D_4^2$, $(C_2^3\times D_{12}):D_4$, $(C_2\times C_{12}).D_4^2$, $(D_6\times C_2^4):D_4$, $(D_6\times C_2^4):D_4$, $(C_2^3\times D_{12}):D_4$, $(D_6\times C_4^2):D_4$, $(C_2^2\times C_6).D_4^2$, $(C_2\times C_{12}).D_4^2$, $(C_2\times C_{12}).D_4^2$, $(C_2^3\times D_{12}):D_4$, $(D_6\times C_4^2):D_4$, $(D_6\times C_2^4):D_4$, $(C_2^3\times D_{12}):D_4$, $C_2^5:(S_3\times D_4)$, $(C_2^2\times C_6).D_4^2$, $(C_2\times C_{12}).D_4^2$, $C_2^6:(C_4\times S_3)$, $C_2^4.(D_4\times D_6)$

Order 1152: $\GL(2,\mathbb{Z}/4):D_6$ x 24, $\GL(2,\mathbb{Z}/4):D_6$ x 19, $C_2\times S_4^2$ x 17, $(C_2^2\times D_6):S_4$ x 6, $(C_2^2\times D_6):S_4$ x 6, $C_6:S_4\times D_4$ x 5, $C_2^5:S_3^2$ x 5, $C_2^3:D_6^2$ x 4, $Q_8:A_4^2$ x 4, $S_3\times C_2^3:S_4$ x 4, $D_6\times \GL(2,\mathbb{Z}/4)$ x 4, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$ x 4, $(C_2^2\times A_4):D_{12}$ x 4, $(C_2^3\times C_{12}):D_6$ x 4, $S_3\times Q_8:S_4$ x 3, $C_2^3.D_6^2$ x 3, $S_3\times C_2\wr A_4$ x 3, $(C_2\times C_6).\GL(2,\mathbb{Z}/4)$ x 3, $(C_2^3\times C_6):S_4$ x 3, $C_3\times C_2\wr S_4$ x 3, $(C_2\times D_{12}):D_{12}$ x 2, $(C_6\times D_{12}):D_4$ x 2, $(C_6\times D_{12}):D_4$ x 2, $(C_6\times D_{12}):D_4$ x 2, $A_4^2:C_2^3$ x 2, $D_4\times A_4\times D_6$ x 2, $(C_6\times Q_8):S_4$ x 2, $C_6\times C_2^3:S_4$ x 2, $(C_6\times D_4):S_4$ 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, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5.S_3^2$ x 2, $(C_6\times S_4):D_4$ x 2, $D_6:\GL(2,\mathbb{Z}/4)$ x 2, $D_6:(C_4\times S_4)$ x 2, $(C_2^3\times D_6):A_4$ x 2, $(C_2\times C_6).\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5:C_6^2$ x 2, $C_2^5.C_6^2$ x 2, $C_2^3.D_6^2$, $(C_2^2\times C_6^2):C_2^3$, $(C_6\times D_{12}):C_2^3$, $C_{12}:D_{12}:C_4$, $C_2^2\wr C_2.S_3^2$, $(C_2\times D_6^2):C_4$, $C_2^3\times C_6:S_4$, $C_2^5:S_3^2$, $A_4^2:C_2^3$, $Q_8:A_4\times D_6$, $C_2^3:D_6\times A_4$, $(C_2^3\times C_6):S_4$, $C_6\times Q_8:S_4$, $(D_4\times A_4):D_6$, $(D_4\times A_4):D_6$, $C_2\times C_6\times \GL(2,\mathbb{Z}/4)$, $C_2^3.D_6^2$, $C_2^3.D_6^2$, $C_2^3.D_6^2$, $S_4\times \GL(2,3)$, $C_2^4.(S_3\times A_4)$, $A_4\times D_4:D_6$, $A_4\times D_6:D_4$, $A_4\times C_{12}:D_4$, $A_4\times C_2^3.D_6$, $A_4\times D_6:D_4$, $A_4\times C_2^2:D_{12}$, $C_2^4:(S_3\times C_{12})$, $(C_2^3\times C_6).S_4$, $C_3\times C_2^4.S_4$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_3\times C_2^2:\GL(2,\mathbb{Z}/4)$, $C_{12}:\GL(2,\mathbb{Z}/4)$, $C_3\times C_2^2:\GL(2,\mathbb{Z}/4)$, $(C_6\times S_4):D_4$, $C_3\times C_2^4:D_{12}$, $\GL(2,\mathbb{Z}/4):C_{12}$, $D_6:\GL(2,\mathbb{Z}/4)$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $D_6:\GL(2,\mathbb{Z}/4)$, $D_6:\GL(2,\mathbb{Z}/4)$, $D_6:\GL(2,\mathbb{Z}/4)$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_3:(D_4\times \GL(2,3))$, $Q_8:(S_3\times S_4)$, $Q_8:(S_3\times S_4)$, $D_4:(S_3\times S_4)$, $C_3:(D_8\times S_4)$, $(C_2\times S_4):D_{12}$, $C_2.(S_4\times D_{12})$, $C_{12}:\GL(2,\mathbb{Z}/4)$, $(C_2\times S_4):D_{12}$, $D_6:\GL(2,\mathbb{Z}/4)$, $D_6:(C_4\times S_4)$, $D_6:(C_4\times S_4)$, $(C_{12}\times A_4):D_4$, $(C_4\times D_6):S_4$, $C_6.(D_4\times S_4)$, $C_2^2.(D_6\times S_4)$, $C_6.(D_4\times S_4)$, $(C_6\times S_4):D_4$, $C_6.(D_4\times S_4)$, $C_{12}:(C_4\times S_4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_4.(D_6\times S_4)$, $C_3:(\OD_{16}\times S_4)$, $A_4\times C_{12}.D_4$, $A_4\times C_{12}.D_4$, $(C_2^3\times C_{12}):C_{12}$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_3\times C_2^2.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_2^5.S_3^2$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.(C_4\times S_4)$, $C_3\times C_2^5:A_4$, $C_6\times C_2^4:A_4$, $C_2^5:C_6^2$, $C_2\wr A_4\times C_6$, $(C_2^3\times C_{12}):A_4$, $D_6^2:C_2^3$

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

Order 864: $S_4\times S_3^2$ x 16, $A_4^2:S_3$ x 6, $A_4^2:C_6$ x 6, $S_3\times C_6:S_4$ x 4, $C_6\times S_3\times S_4$ x 4, $A_4^2:S_3$ x 4, $C_6:S_3\times S_4$ x 3, $C_6\times A_4^2$ x 2, $D_6^2:C_6$ x 2, $C_2\times A_4:S_3^2$ x 2, $S_3\times A_4^2$ x 2, $A_4^2:C_6$ x 2, $C_3^2\times Q_8:A_4$, $C_6.A_4^2$, $\SL(2,3):S_3^2$

Order 768: $C_2^5:S_4$ x 12, $(C_2^2\times D_{12}):D_4$ x 10, $D_4^2:D_6$ x 9, $C_2^6:D_6$ x 7, $C_2^5:D_{12}$ x 6, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 6, $C_2^3:C_4\times S_4$ x 6, $C_2^5.D_{12}$ x 5, $C_2^5:D_{12}$ x 5, $D_6:D_4^2$ x 3, $(C_2\times C_6):D_4^2$ x 3, $D_6:D_4^2$ x 3, $C_2^6:D_6$ x 3, $(C_2\times C_6).D_4^2$ x 3, $(C_2^2\times D_{12}):D_4$ x 3, $C_2^5.S_4$ x 3, $\GL(2,\mathbb{Z}/4):C_2^3$ x 3, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 3, $A_4:D_4^2$ x 3, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 3, $(C_2^3\times C_4):S_4$ x 3, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 3, $A_4\times C_2\wr C_2^2$ x 3, $(D_4\times C_2^3):A_4$ x 3, $D_6:D_4^2$ x 3, $(C_2^3\times C_{12}):D_4$ x 2, $(C_2^2\times D_{12}):D_4$ x 2, $C_2^4.(S_3\times D_4)$ x 2, $(C_2^3\times D_6):D_4$ 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, $C_2^4.(S_3\times D_4)$ x 2, $(C_2^3\times C_{12}).D_4$ x 2, $C_2^4.(S_3\times D_4)$ x 2, $(D_6\times C_2^4):C_4$ x 2, $(C_2^3\times C_{12}):D_4$ x 2, $C_3\times C_2^5:D_4$ x 2, $(C_2^3\times C_{12}):D_4$ x 2, $C_{12}:D_4^2$ x 2, $D_6:D_4^2$ x 2, $C_3:(C_4\times D_4^2)$ x 2, $(C_2^3\times D_6):D_4$ x 2, $C_2\wr C_4\times D_6$ x 2, $C_2^3.(D_4\times D_6)$ x 2, $C_2\wr D_4\times C_6$ x 2, $(C_2^2\times D_{12}):D_4$ x 2, $C_2^5:D_{12}$ x 2, $(C_2\times C_6).D_4^2$ x 2, $C_2^6:D_6$ x 2, $C_2^5:(C_4\times S_3)$ x 2, $C_2^5:S_4$ x 2, $C_2^5:S_4$ x 2, $\GL(2,\mathbb{Z}/4):C_2^3$ x 2, $Q_8:\GL(2,\mathbb{Z}/4)$ x 2, $A_4:D_4^2$ x 2, $C_4:D_4\times S_4$ x 2, $D_4\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^5:S_4$ 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, $C_2^3.(C_4\times S_4)$ x 2, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^6:A_4$ x 2, $C_2^6:C_{12}$ x 2, $C_2^6:D_6$ x 2, $D_6\times D_4^2$ x 2, $C_2\wr C_2^2\times D_6$ x 2, $(C_2\times C_6):D_4^2$ x 2, $D_6:D_4^2$ x 2, $D_6:D_4^2$ x 2, $(C_2\times C_6):D_4^2$ x 2, $D_6:D_4^2$ x 2, $D_6:D_4^2$ x 2, $(C_2^3\times C_{12}).D_4$, $C_2^5.D_{12}$, $(C_2\times C_4^2).D_{12}$, $(C_2^3\times C_{12}).D_4$, $(C_2\times C_4^2).D_{12}$, $C_6.C_2\wr D_4$, $(C_2^3\times C_{12}):D_4$, $C_2^5.D_{12}$, $(C_2\times C_4^2):D_{12}$, $(C_2^3\times C_{12}):D_4$, $(C_2\times C_4^2):D_{12}$, $(C_2^4\times C_6):D_4$, $(D_4\times C_2^3).D_6$, $(C_2^3\times D_{12}):C_4$, $C_2^5.D_{12}$, $C_2^5:(C_4\times S_3)$, $(C_2^3\times C_{12}).D_4$, $(C_2^3\times D_{12}):C_4$, $(C_2^3\times C_{12}).D_4$, $D_6:C_2\wr C_4$, $C_2^3.(C_4\times D_{12})$, $(C_2^3\times C_{12}):D_4$, $(C_2\times C_4^2):D_{12}$, $C_2^3.(C_4\times D_{12})$, $(C_2\times C_4^2).D_{12}$, $(C_2^3\times C_{12}).D_4$, $C_2^4.(S_3\times D_4)$, $C_2^4.(C_4\times D_6)$, $C_2^4.(C_4\times D_6)$, $(C_3\times D_4^2):C_4$, $(C_2\times C_4^2).D_{12}$, $C_2^5.D_{12}$, $(D_6\times C_4^2):C_4$, $C_2^5.D_{12}$, $(C_2\times D_{12}):C_4^2$, $(C_2^3\times C_6):C_4^2$, $(C_2^2\times D_6):C_4^2$, $C_3\times C_2^5.D_4$, $(D_4\times C_2^3):C_{12}$, $C_2^6:C_{12}$, $C_6\times C_2^4:D_4$, $C_3\times C_2^2:D_4^2$, $C_2\times C_4^2:D_{12}$, $D_4^2:D_6$, $C_2^5.D_{12}$, $C_4^2:C_4\times D_6$, $(C_2^5\times C_6):C_4$, $(D_6\times C_2^4):C_4$, $C_2^3.(D_4\times D_6)$, $(C_2^2\times D_{12}):D_4$, $(C_2^3\times C_{12}):D_4$, $(C_2^2\times D_{12}):D_4$, $(D_4\times D_6):D_4$, $C_3\times C_2^5:D_4$, $C_3\times C_2^5.D_4$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $(C_2^2\times D_{12}):D_4$, $(C_2^3\times C_{12}):D_4$, $(C_2\times C_6).D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_2^5:D_{12}$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_2^5:(C_4\times S_3)$, $C_2^3.(D_4\times D_6)$, $C_2^3.(D_4\times D_6)$, $C_2^3.(D_4\times D_6)$, $C_2^5.S_4$, $(C_2^3\times C_4):S_4$, $(C_4\times Q_8):S_4$, $C_4\times C_2^3:S_4$, $C_4\times Q_8:S_4$, $D_8:C_2\times S_4$, $A_4:D_4^2$, $C_4:D_4\times S_4$, $A_4:D_4^2$, $C_4\times D_4\times S_4$, $D_4:\GL(2,\mathbb{Z}/4)$, $C_4^2:(C_2\times S_4)$, $C_2^5.D_{12}$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $D_4:\GL(2,\mathbb{Z}/4)$, $C_4\wr C_2\times S_4$, $S_4\times \OD_{16}:C_2$, $C_2^5.S_4$, $(C_2^3\times C_4):S_4$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $D_4\times Q_8:A_4$, $A_4\times D_4^2$, $C_2^6:A_4$, $A_4\times D_4:D_4$, $A_4\times C_4^2:C_4$, $(D_4\times C_2^3):A_4$, $C_4\times C_2\wr A_4$, $(D_4\times C_2^3):D_6$, $C_{12}.D_4^2$, $(C_2^3\times C_{12}):D_4$, $(C_2\times D_{12}):C_2^4$, $D_4:D_4\times D_6$, $C_6.(C_2\times D_4^2)$, $D_6:D_4^2$

Order 576: $C_2^2\times A_4\times D_6$, $C_2^2\times C_6:S_4$, $C_2^2\times C_6\times S_4$, $(C_3\times Q_8):S_4$, $C_3\times C_2^3:S_4$, $(C_2^2\times D_6):A_4$

Order 384: $C_{12}.C_2^5$, $C_2^5:A_4$, $A_4\times D_4:C_2^2$

Order 288: $C_3\times Q_8:A_4$, $C_2^3:C_6^2$

Order 192: $(C_2\times C_{12}):D_4$, $D_6\times C_2^4$, $C_2^3\times S_4$, $C_2^3:S_4$

Order 144: $C_6:S_4$

Order 128: $D_4:C_2^4$

Order 96: $C_2^4\times C_6$, $C_2^3\times A_4$, $C_6.C_2^4$, $Q_8:A_4$

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

Order 48: $C_2^2\times D_6$

Order 36: $C_3\times A_4$

Order 32: $C_2^5$, $D_4:C_2^2$

Order 27: $C_3^3$

Order 24: $C_2^2\times C_6$ x 2, $C_2\times A_4$

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

Order 8: $C_2^3$ x 2

Order 6: $C_6$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 36 (order at least 768) are shown, as well as all normal subgroups of any index.

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

Order 13824: $Q_8:A_4^2:D_6$, $(C_2^2\times C_6).S_4^2$, $(C_3\times Q_8).S_4^2$, $C_2^5.(A_4\times S_3^2)$, $Q_8:A_4^2:D_6$, $C_3\times C_2^3.S_4^2$, $(C_2^2\times C_6):S_4^2$

Order 9216: $C_2^5.(D_6\times S_4)$, $(D_4\times C_2^3).D_6^2$, $C_2^4.S_4^2$

Order 6912: $Q_8:A_4^2:S_3$, $C_2^5:(C_3^2\times S_4)$, $(C_2^2\times D_6):A_4^2$, $Q_8:A_4^2:C_6$, $Q_8:A_4^2:C_6$, $Q_8:A_4^2:S_3$, $Q_8:A_4^2:S_3$, $(C_2\times D_6^2):S_4$, $D_6\times S_4^2$

Order 4608: $C_2^3:S_4^2$ x 2, $C_2^3.S_4^2$ x 2, $C_2^4.(D_6\times S_4)$, $C_2^4.(D_6\times S_4)$, $(D_6\times C_2^4):S_4$, $(C_2^2\times D_6):\GL(2,\mathbb{Z}/4)$, $C_2^4.(D_6\times S_4)$, $C_2^4.(D_6\times S_4)$, $(C_2^2\times D_6):\GL(2,\mathbb{Z}/4)$, $C_6.C_2\wr D_6$, $(C_2^3\times D_{12}):S_4$, $(C_2\times D_{12}):\GL(2,\mathbb{Z}/4)$, $(C_2\times D_{12}):\GL(2,\mathbb{Z}/4)$, $(C_2^2\times D_6):\GL(2,\mathbb{Z}/4)$, $(C_2^5\times C_6):D_{12}$, $(D_6\times C_2^4):D_{12}$, $C_2^6:C_6\wr C_2$, $(C_2^3\times S_4):D_{12}$, $C_{12}:\GL(2,\mathbb{Z}/4):C_4$, $(C_2^3\times C_4).D_6^2$, $C_6:\GL(2,\mathbb{Z}/4).D_4$, $C_2^5.(C_4\times S_3^2)$, $(D_6\times C_2^4):S_4$, $C_2^5:(C_6\times S_4)$, $C_2^5.D_6^2$, $C_6:\GL(2,\mathbb{Z}/4):D_4$, $C_2^5.D_6^2$, $C_2^5.(A_4\times D_6)$, $C_2^4.(D_6\times S_4)$, $C_3\times C_2^4:\GL(2,\mathbb{Z}/4)$, $C_2^4.(D_6\times S_4)$, $(C_2^3\times D_{12}):S_4$, $C_2^3.S_4^2$, $Q_8:S_4^2$, $C_2\wr A_4:S_4$, $C_2\wr A_4\times S_4$, $A_4\times C_2\wr S_4$, $C_2\wr S_4\times D_6$

Order 3456: $C_2\times A_4^2.D_6$ x 3, $C_3^2:C_2\wr S_4$ x 2, $C_3^2:C_2\wr S_4$ x 2, $A_4\times D_6\times S_4$ x 2, $C_3\times Q_8:A_4^2$, $C_6.S_4^2$, $C_6^2:C_2^2:S_4$, $D_6\times \PSOPlus(4,3)$, $(C_2\times D_6^2):A_4$, $Q_8:A_4:S_3^2$, $S_3\times S_4^2$, $C_6\times S_4^2$, $C_6:S_4^2$

Order 3072: $C_4^2.(D_4\times S_4)$, $D_4\times C_2\wr S_4$, $(C_2\times D_{12}).D_4^2$

Order 2304: $S_3\times C_2\wr S_4$ x 5, $(C_2^3\times A_4):S_4$ x 4, $(C_2^3\times D_6):S_4$ x 4, $(A_4\times C_2^4):D_6$ x 2, $C_2^4:D_6^2$ x 2, $A_4\times C_2^3:S_4$ x 2, $A_4\times Q_8:S_4$ x 2, $(C_2^3\times S_4):A_4$ x 2, $Q_8:A_4\times S_4$ 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, $(C_2^4\times C_6):S_4$, $(C_2^3\times D_6):S_4$, $C_3\times C_2^5.S_4$, $(D_6\times C_2^4):A_4$, $(C_2^3\times C_{12}):S_4$, $(C_2^3\times C_{12}):D_{12}$, $C_3\times C_2^3:\GL(2,\mathbb{Z}/4)$, $C_2^4.D_6^2$, $C_2^4.D_6^2$, $C_2^4.D_6^2$, $C_2^4.D_6^2$, $(C_2^2\times C_6):\GL(2,\mathbb{Z}/4)$, $C_{12}.(C_2^3\times S_4)$, $C_2^4.D_6^2$, $(C_2^3\times D_{12}):A_4$, $(C_3\times Q_8):\GL(2,\mathbb{Z}/4)$, $C_2^4.(S_3\times S_4)$, $C_2^4.(S_3\times S_4)$, $(C_2^2\times C_6).\GL(2,\mathbb{Z}/4)$, $C_3\times D_4:\GL(2,\mathbb{Z}/4)$, $(C_2^3\times C_{12}):S_4$, $C_3\times C_2^5:S_4$, $(C_2^3\times C_{12}):S_4$, $C_3\times A_4:D_4^2$, $(D_6\times C_2^4):A_4$, $C_2^5:(C_6\times A_4)$, $(C_2^4\times C_6).S_4$, $C_3\times Q_8:\GL(2,\mathbb{Z}/4)$, $C_2^4:D_6^2$, $C_2^4.(S_3\times S_4)$, $C_2^4.(S_3\times S_4)$, $C_2^5.C_6\wr C_2$, $(C_2^3\times D_{12}):C_{12}$, $C_2^4.D_6^2$, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$, $C_2^5.(C_6\times D_6)$, $C_2^6:C_6^2$, $C_2^6:S_3^2$, $C_2^4.D_6^2$, $C_2^4.D_6^2$, $A_4\times C_2^4:D_6$, $C_2^4.D_6^2$, $C_2^4.D_6^2$, $\GL(2,\mathbb{Z}/4):D_{12}$, $C_2^4.D_6^2$, $C_2^4.D_6^2$, $C_2^3.(D_6\times S_4)$, $C_6\times C_2^4:S_4$, $(C_2^2\times C_6):\GL(2,\mathbb{Z}/4)$, $C_2^5:(S_3\times A_4)$, $(C_2\times D_6^2):D_4$, $(Q_8\times A_4):S_4$, $C_2^4:A_4^2$, $C_2\wr A_4\times D_6$, $D_6\times Q_8:S_4$, $C_2\wr S_4\times C_6$, $S_3\times C_2^4:S_4$, $C_2^2\times S_4^2$, $C_6:C_2\wr S_4$, $(C_2^3\times D_6):S_4$, $(C_2^2\times C_6).\GL(2,\mathbb{Z}/4)$, $C_2^4.(S_3\times S_4)$, $(C_2^2\times C_6).\GL(2,\mathbb{Z}/4)$, $D_4:(D_6\times S_4)$, $C_{12}.(D_4\times S_4)$, $C_2^4.(A_4\times D_6)$, $C_2^4.(A_4\times D_6)$, $(C_2^3\times C_{12}).S_4$, $(C_2^2\times C_6).\GL(2,\mathbb{Z}/4)$, $C_{12}.(D_4\times S_4)$, $C_{12}.(D_4\times S_4)$, $C_3\times C_2^3.\GL(2,\mathbb{Z}/4)$, $(C_2\times C_{12}).\GL(2,\mathbb{Z}/4)$, $C_{12}.(D_4\times S_4)$, $C_3\times C_2^3.\GL(2,\mathbb{Z}/4)$, $(C_2\times C_{12}).\GL(2,\mathbb{Z}/4)$, $(C_2\times C_{12}).\GL(2,\mathbb{Z}/4)$, $(C_2\times C_{12}):\GL(2,\mathbb{Z}/4)$, $(C_2^3\times S_4):C_{12}$, $(D_6\times C_2^4):C_{12}$, $(C_2^2\times C_6).\GL(2,\mathbb{Z}/4)$, $C_{12}.(D_4\times S_4)$, $(C_3\times D_4):\GL(2,\mathbb{Z}/4)$, $(C_2^5\times C_6):C_{12}$

Order 1728: $A_4^2:D_6$ x 3, $C_6\times A_4\times S_4$ x 3, $C_3:S_4^2$ x 3, $A_4^2:D_6$ x 2, $A_4:D_6^2$ x 2, $A_4^2:D_6$ x 2, $(Q_8\times C_3^2):S_4$ x 2, $C_3^2:C_2\wr A_4$ x 2, $D_6\times A_4^2$, $C_6\times \PSOPlus(4,3)$, $A_4^2:D_6$, $C_3:S_4^2$, $C_3\times S_4^2$, $D_4:C_6^2:C_6$, $(C_2\times C_6^2):S_4$, $C_3^2\times C_2^3:S_4$, $C_3\times A_4:\GL(2,3)$, $A_4\times C_3:\GL(2,3)$, $C_3:S_4\times \SL(2,3)$

Order 1536: $C_2\wr C_2^2\times S_4$ x 3, $(D_4\times C_2^3):S_4$ x 3, $C_2^6:S_4$ x 2, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 2, $C_2\wr C_4\times S_4$ x 2, $C_2^6:D_{12}$ x 2, $C_2^5:(C_2\times S_4)$ x 2, $D_4\times Q_8:S_4$, $D_4\times C_2^3:S_4$, $S_4\times D_4^2$, $C_2^6:S_4$, $D_4:D_4\times S_4$, $C_4^2:\GL(2,\mathbb{Z}/4)$, $C_4^2:\GL(2,\mathbb{Z}/4)$, $C_4^2:C_4\times S_4$, $(A_4\times C_4^2):D_4$, $(D_4\times C_2^3):S_4$, $(D_4\times C_2^3):S_4$, $C_4:C_2\wr S_4$, $C_4\times C_2\wr S_4$, $A_4\times C_2\wr D_4$, $D_4\times C_2\wr A_4$, $C_3\times C_2^6:D_4$, $C_3:D_4^3$, $(C_2^3\times D_{12}):D_4$, $C_2\wr D_4\times D_6$, $(C_2\times C_{12}):D_4^2$, $(C_2\times C_{12}):D_4^2$, $(C_3\times D_4):D_4^2$, $C_2^4.(D_4\times D_6)$, $C_2^6:D_{12}$, $(C_3\times D_4^2):D_4$, $(C_2^2\times C_6).D_4^2$, $(C_2\times C_{12}).D_4^2$, $(C_2\times C_{12}).D_4^2$, $(C_2^3\times D_{12}):D_4$, $(C_2\times C_{12}).D_4^2$, $(D_6\times C_2^4):D_4$, $(D_6\times C_2^4):D_4$, $(C_2^3\times D_{12}):D_4$, $(D_6\times C_4^2):D_4$, $(C_2^2\times C_6).D_4^2$, $(C_2\times C_{12}).D_4^2$, $(C_2\times C_{12}).D_4^2$, $(C_2^3\times D_{12}):D_4$, $(D_6\times C_4^2):D_4$, $(D_6\times C_2^4):D_4$, $(C_2^3\times D_{12}):D_4$, $C_2^5:(S_3\times D_4)$, $(C_2^2\times C_6).D_4^2$, $(C_2\times C_{12}).D_4^2$, $C_2^6:(C_4\times S_3)$, $C_2^4.(D_4\times D_6)$

Order 1152: $\GL(2,\mathbb{Z}/4):D_6$ x 12, $C_2\times S_4^2$ x 9, $(C_2^2\times D_6):S_4$ x 6, $(C_2^2\times D_6):S_4$ x 6, $C_6:S_4\times D_4$ x 5, $C_2^5:S_3^2$ x 5, $C_2^3:D_6^2$ x 4, $Q_8:A_4^2$ x 4, $S_3\times C_2^3:S_4$ x 4, $D_6\times \GL(2,\mathbb{Z}/4)$ x 4, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$ x 4, $(C_2^2\times A_4):D_{12}$ x 4, $(C_2^3\times C_{12}):D_6$ x 4, $S_3\times Q_8:S_4$ x 3, $\GL(2,\mathbb{Z}/4):D_6$ x 3, $C_2^3.D_6^2$ x 3, $S_3\times C_2\wr A_4$ x 3, $(C_2\times C_6).\GL(2,\mathbb{Z}/4)$ x 3, $(C_2^3\times C_6):S_4$ x 3, $C_3\times C_2\wr S_4$ x 3, $(C_2\times D_{12}):D_{12}$ x 2, $(C_6\times D_{12}):D_4$ x 2, $(C_6\times D_{12}):D_4$ x 2, $(C_6\times D_{12}):D_4$ x 2, $A_4^2:C_2^3$ x 2, $D_4\times A_4\times D_6$ x 2, $(C_6\times Q_8):S_4$ x 2, $C_6\times C_2^3:S_4$ x 2, $(C_6\times D_4):S_4$ 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, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5.S_3^2$ x 2, $(C_6\times S_4):D_4$ x 2, $D_6:\GL(2,\mathbb{Z}/4)$ x 2, $D_6:(C_4\times S_4)$ x 2, $(C_2^3\times D_6):A_4$ x 2, $(C_2\times C_6).\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5:C_6^2$ x 2, $C_2^5.C_6^2$ x 2, $C_2^3.D_6^2$, $(C_2^2\times C_6^2):C_2^3$, $(C_6\times D_{12}):C_2^3$, $C_{12}:D_{12}:C_4$, $C_2^2\wr C_2.S_3^2$, $(C_2\times D_6^2):C_4$, $C_2^3\times C_6:S_4$, $C_2^5:S_3^2$, $A_4^2:C_2^3$, $Q_8:A_4\times D_6$, $C_2^3:D_6\times A_4$, $(C_2^3\times C_6):S_4$, $C_6\times Q_8:S_4$, $(D_4\times A_4):D_6$, $(D_4\times A_4):D_6$, $C_2\times C_6\times \GL(2,\mathbb{Z}/4)$, $C_2^3.D_6^2$, $C_2^3.D_6^2$, $C_2^3.D_6^2$, $S_4\times \GL(2,3)$, $C_2^4.(S_3\times A_4)$, $A_4\times D_4:D_6$, $A_4\times D_6:D_4$, $A_4\times C_{12}:D_4$, $A_4\times C_2^3.D_6$, $A_4\times D_6:D_4$, $A_4\times C_2^2:D_{12}$, $C_2^4:(S_3\times C_{12})$, $(C_2^3\times C_6).S_4$, $C_3\times C_2^4.S_4$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_3\times C_2^2:\GL(2,\mathbb{Z}/4)$, $C_{12}:\GL(2,\mathbb{Z}/4)$, $C_3\times C_2^2:\GL(2,\mathbb{Z}/4)$, $(C_6\times S_4):D_4$, $C_3\times C_2^4:D_{12}$, $\GL(2,\mathbb{Z}/4):C_{12}$, $D_6:\GL(2,\mathbb{Z}/4)$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $D_6:\GL(2,\mathbb{Z}/4)$, $D_6:\GL(2,\mathbb{Z}/4)$, $D_6:\GL(2,\mathbb{Z}/4)$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_3:(D_4\times \GL(2,3))$, $Q_8:(S_3\times S_4)$, $Q_8:(S_3\times S_4)$, $D_4:(S_3\times S_4)$, $C_3:(D_8\times S_4)$, $(C_2\times S_4):D_{12}$, $C_2.(S_4\times D_{12})$, $C_{12}:\GL(2,\mathbb{Z}/4)$, $(C_2\times S_4):D_{12}$, $D_6:\GL(2,\mathbb{Z}/4)$, $D_6:(C_4\times S_4)$, $D_6:(C_4\times S_4)$, $(C_{12}\times A_4):D_4$, $(C_4\times D_6):S_4$, $C_6.(D_4\times S_4)$, $C_2^2.(D_6\times S_4)$, $C_6.(D_4\times S_4)$, $(C_6\times S_4):D_4$, $C_6.(D_4\times S_4)$, $C_{12}:(C_4\times S_4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_4.(D_6\times S_4)$, $C_3:(\OD_{16}\times S_4)$, $A_4\times C_{12}.D_4$, $A_4\times C_{12}.D_4$, $(C_2^3\times C_{12}):C_{12}$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_3\times C_2^2.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_2^5.S_3^2$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.\GL(2,\mathbb{Z}/4)$, $C_{12}.(C_4\times S_4)$, $C_3\times C_2^5:A_4$, $C_6\times C_2^4:A_4$, $C_2^5:C_6^2$, $C_2\wr A_4\times C_6$, $(C_2^3\times C_{12}):A_4$, $D_6^2:C_2^3$

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

Order 864: $S_3\times C_6:S_4$ x 4, $C_6\times S_3\times S_4$ x 4, $C_6:S_3\times S_4$ x 3, $A_4^2:S_3$ x 3, $A_4^2:C_6$ x 3, $C_6\times A_4^2$ x 2, $D_6^2:C_6$ x 2, $C_2\times A_4:S_3^2$ x 2, $A_4^2:S_3$ x 2, $S_4\times S_3^2$ x 2, $S_3\times A_4^2$, $A_4^2:C_6$, $C_3^2\times Q_8:A_4$, $C_6.A_4^2$, $\SL(2,3):S_3^2$

Order 768: $C_2^5:S_4$ x 12, $(C_2^2\times D_{12}):D_4$ x 10, $D_4^2:D_6$ x 9, $C_2^6:D_6$ x 6, $C_2^5.D_{12}$ x 5, $C_2^5:D_{12}$ x 5, $C_2^5:D_{12}$ x 5, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 5, $C_2^3:C_4\times S_4$ x 5, $D_6:D_4^2$ x 3, $(C_2\times C_6):D_4^2$ x 3, $D_6:D_4^2$ x 3, $C_2^6:D_6$ x 3, $(C_2\times C_6).D_4^2$ x 3, $(C_2^2\times D_{12}):D_4$ x 3, $C_2^5.S_4$ x 3, $\GL(2,\mathbb{Z}/4):C_2^3$ x 3, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 3, $A_4:D_4^2$ x 3, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 3, $(C_2^3\times C_4):S_4$ x 3, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 3, $A_4\times C_2\wr C_2^2$ x 3, $(D_4\times C_2^3):A_4$ x 3, $D_6:D_4^2$ x 3, $(C_2^3\times C_{12}):D_4$ x 2, $(C_2^2\times D_{12}):D_4$ x 2, $C_2^4.(S_3\times D_4)$ x 2, $(C_2^3\times D_6):D_4$ 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, $C_2^4.(S_3\times D_4)$ x 2, $(C_2^3\times C_{12}).D_4$ x 2, $C_2^4.(S_3\times D_4)$ x 2, $(D_6\times C_2^4):C_4$ x 2, $(C_2^3\times C_{12}):D_4$ x 2, $C_3\times C_2^5:D_4$ x 2, $(C_2^3\times C_{12}):D_4$ x 2, $C_{12}:D_4^2$ x 2, $D_6:D_4^2$ x 2, $C_3:(C_4\times D_4^2)$ x 2, $(C_2^3\times D_6):D_4$ x 2, $C_2\wr C_4\times D_6$ x 2, $C_2^3.(D_4\times D_6)$ x 2, $C_2\wr D_4\times C_6$ x 2, $(C_2^2\times D_{12}):D_4$ x 2, $C_2^5:D_{12}$ x 2, $(C_2\times C_6).D_4^2$ x 2, $C_2^6:D_6$ x 2, $C_2^5:(C_4\times S_3)$ x 2, $C_2^5:S_4$ x 2, $C_2^5:S_4$ x 2, $\GL(2,\mathbb{Z}/4):C_2^3$ x 2, $Q_8:\GL(2,\mathbb{Z}/4)$ x 2, $A_4:D_4^2$ x 2, $C_4:D_4\times S_4$ x 2, $D_4\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^5:S_4$ 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, $C_2^3.(C_4\times S_4)$ x 2, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^6:A_4$ x 2, $C_2^6:C_{12}$ x 2, $C_2^6:D_6$ x 2, $D_6\times D_4^2$ x 2, $C_2\wr C_2^2\times D_6$ x 2, $(C_2\times C_6):D_4^2$ x 2, $D_6:D_4^2$ x 2, $D_6:D_4^2$ x 2, $(C_2\times C_6):D_4^2$ x 2, $D_6:D_4^2$ x 2, $D_6:D_4^2$ x 2, $(C_2^3\times C_{12}).D_4$, $C_2^5.D_{12}$, $(C_2\times C_4^2).D_{12}$, $(C_2^3\times C_{12}).D_4$, $(C_2\times C_4^2).D_{12}$, $C_6.C_2\wr D_4$, $(C_2^3\times C_{12}):D_4$, $C_2^5.D_{12}$, $(C_2\times C_4^2):D_{12}$, $(C_2^3\times C_{12}):D_4$, $(C_2\times C_4^2):D_{12}$, $(C_2^4\times C_6):D_4$, $(D_4\times C_2^3).D_6$, $(C_2^3\times D_{12}):C_4$, $C_2^5.D_{12}$, $C_2^5:(C_4\times S_3)$, $(C_2^3\times C_{12}).D_4$, $(C_2^3\times D_{12}):C_4$, $(C_2^3\times C_{12}).D_4$, $D_6:C_2\wr C_4$, $C_2^3.(C_4\times D_{12})$, $(C_2^3\times C_{12}):D_4$, $(C_2\times C_4^2):D_{12}$, $C_2^3.(C_4\times D_{12})$, $(C_2\times C_4^2).D_{12}$, $(C_2^3\times C_{12}).D_4$, $C_2^4.(S_3\times D_4)$, $C_2^4.(C_4\times D_6)$, $C_2^4.(C_4\times D_6)$, $(C_3\times D_4^2):C_4$, $(C_2\times C_4^2).D_{12}$, $C_2^5.D_{12}$, $(D_6\times C_4^2):C_4$, $C_2^5.D_{12}$, $(C_2\times D_{12}):C_4^2$, $(C_2^3\times C_6):C_4^2$, $(C_2^2\times D_6):C_4^2$, $C_3\times C_2^5.D_4$, $(D_4\times C_2^3):C_{12}$, $C_2^6:C_{12}$, $C_6\times C_2^4:D_4$, $C_3\times C_2^2:D_4^2$, $C_2\times C_4^2:D_{12}$, $D_4^2:D_6$, $C_2^5.D_{12}$, $C_4^2:C_4\times D_6$, $(C_2^5\times C_6):C_4$, $(D_6\times C_2^4):C_4$, $C_2^3.(D_4\times D_6)$, $(C_2^2\times D_{12}):D_4$, $(C_2^3\times C_{12}):D_4$, $(C_2^2\times D_{12}):D_4$, $(D_4\times D_6):D_4$, $C_3\times C_2^5:D_4$, $C_3\times C_2^5.D_4$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $(C_2^2\times D_{12}):D_4$, $(C_2^3\times C_{12}):D_4$, $(C_2\times C_6).D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_2^5:D_{12}$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_{12}.D_4^2$, $C_2^5:(C_4\times S_3)$, $C_2^3.(D_4\times D_6)$, $C_2^3.(D_4\times D_6)$, $C_2^3.(D_4\times D_6)$, $C_2^5.S_4$, $(C_2^3\times C_4):S_4$, $(C_4\times Q_8):S_4$, $C_4\times C_2^3:S_4$, $C_4\times Q_8:S_4$, $D_8:C_2\times S_4$, $A_4:D_4^2$, $C_4:D_4\times S_4$, $A_4:D_4^2$, $C_4\times D_4\times S_4$, $D_4:\GL(2,\mathbb{Z}/4)$, $C_4^2:(C_2\times S_4)$, $C_2^5.D_{12}$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $D_4:\GL(2,\mathbb{Z}/4)$, $C_4\wr C_2\times S_4$, $S_4\times \OD_{16}:C_2$, $C_2^5.S_4$, $(C_2^3\times C_4):S_4$, $C_2^3.\GL(2,\mathbb{Z}/4)$, $D_4\times Q_8:A_4$, $A_4\times D_4^2$, $C_2^6:A_4$, $A_4\times D_4:D_4$, $A_4\times C_4^2:C_4$, $(D_4\times C_2^3):A_4$, $C_4\times C_2\wr A_4$, $(D_4\times C_2^3):D_6$, $C_{12}.D_4^2$, $(C_2^3\times C_{12}):D_4$, $(C_2\times D_{12}):C_2^4$, $D_4:D_4\times D_6$, $C_6.(C_2\times D_4^2)$, $D_6:D_4^2$

Order 576: $C_2^2\times A_4\times D_6$, $C_2^2\times C_6:S_4$, $C_2^2\times C_6\times S_4$, $(C_3\times Q_8):S_4$, $C_3\times C_2^3:S_4$, $(C_2^2\times D_6):A_4$

Order 384: $C_{12}.C_2^5$, $C_2^5:A_4$, $A_4\times D_4:C_2^2$

Order 288: $C_3\times Q_8:A_4$, $C_2^3:C_6^2$

Order 192: $(C_2\times C_{12}):D_4$, $D_6\times C_2^4$, $C_2^3\times S_4$, $C_2^3:S_4$

Order 144: $C_6:S_4$

Order 128: $D_4:C_2^4$

Order 96: $C_2^4\times C_6$, $C_2^3\times A_4$, $C_6.C_2^4$, $Q_8:A_4$

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

Order 48: $C_2^2\times D_6$

Order 36: $C_3\times A_4$

Order 32: $C_2^5$, $D_4:C_2^2$

Order 27: $C_3^3$

Order 24: $C_2^2\times C_6$ x 2, $C_2\times A_4$

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

Order 8: $C_2^3$ x 2

Order 6: $C_6$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

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

Order 13824: $Q_8:A_4^2:D_6$ ($C_2$), $(C_2^2\times C_6).S_4^2$ ($C_2$), $(C_3\times Q_8).S_4^2$ ($C_2$), $C_2^5.(A_4\times S_3^2)$ ($C_2$), $Q_8:A_4^2:D_6$ ($C_2$), $C_3\times C_2^3.S_4^2$ ($C_2$), $(C_2^2\times C_6):S_4^2$ ($C_2$)

Order 6912: $Q_8:A_4^2:S_3$ ($C_2^2$), $C_2^5:(C_3^2\times S_4)$ ($C_2^2$), $(C_2^2\times D_6):A_4^2$ ($C_2^2$), $Q_8:A_4^2:C_6$ ($C_2^2$), $Q_8:A_4^2:C_6$ ($C_2^2$), $Q_8:A_4^2:S_3$ ($C_2^2$), $Q_8:A_4^2:S_3$ ($C_2^2$)

Order 4608: $(D_6\times C_2^4):S_4$ ($S_3$), $C_2^5.D_6^2$ ($S_3$), $C_2^3.S_4^2$ ($S_3$)

Order 3456: $C_3\times Q_8:A_4^2$ ($C_2^3$)

Order 2304: $C_3\times C_2^3:\GL(2,\mathbb{Z}/4)$ ($D_6$), $C_{12}.(C_2^3\times S_4)$ ($D_6$), $(D_6\times C_2^4):A_4$ ($D_6$), $(C_2^4\times C_6).S_4$ ($D_6$), $C_2^5.(C_6\times D_6)$ ($D_6$), $C_6\times C_2^4:S_4$ ($D_6$), $(Q_8\times A_4):S_4$ ($D_6$), $A_4\times C_2^3:S_4$ ($D_6$), $(C_2^3\times S_4):A_4$ ($D_6$)

Order 1152: $C_2^3:D_6^2$ ($S_4$), $Q_8:A_4^2$ ($C_2\times D_6$), $(C_2^2\times D_6):S_4$ ($S_4$), $C_6\times C_2^4:A_4$ ($C_2\times D_6$), $C_2^5:C_6^2$ ($C_2\times D_6$)

Order 768: $C_2^5:S_4$ ($S_3^2$), $C_2^3:\GL(2,\mathbb{Z}/4)$ ($S_3^2$), $(C_2^3\times C_{12}):D_4$ ($S_3^2$)

Order 576: $C_2^2\times A_4\times D_6$ ($C_2\times S_4$), $C_2^2\times C_6:S_4$ ($C_2\times S_4$), $C_2^2\times C_6\times S_4$ ($C_2\times S_4$), $(C_3\times Q_8):S_4$ ($C_2\times S_4$), $C_3\times C_2^3:S_4$ ($C_2\times S_4$), $(C_2^2\times D_6):A_4$ ($C_2\times S_4$)

Order 384: $C_{12}.C_2^5$ ($S_3\times D_6$), $C_2^5:A_4$ ($S_3\times D_6$), $A_4\times D_4:C_2^2$ ($S_3\times D_6$)

Order 288: $C_3\times Q_8:A_4$ ($C_2^2\times S_4$), $C_2^3:C_6^2$ ($C_2^2\times S_4$)

Order 192: $(C_2\times C_{12}):D_4$ ($S_3\times S_4$), $D_6\times C_2^4$ ($S_3\times S_4$), $C_2^3\times S_4$ ($S_3\times S_4$), $C_2^3:S_4$ ($S_3\times S_4$)

Order 144: $C_6:S_4$ ($C_2^3:S_4$)

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

Order 96: $C_2^4\times C_6$ ($D_6\times S_4$), $C_2^3\times A_4$ ($D_6\times S_4$), $C_6.C_2^4$ ($D_6\times S_4$), $Q_8:A_4$ ($D_6\times S_4$)

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

Order 48: $C_2^2\times D_6$ ($S_4^2$)

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

Order 32: $C_2^5$ ($S_4\times S_3^2$), $D_4:C_2^2$ ($S_4\times S_3^2$)

Order 24: $C_2^2\times C_6$ ($C_2\times S_4^2$), $C_2^2\times C_6$ ($S_3\times C_2^3:S_4$), $C_2\times A_4$ ($S_3\times C_2^3:S_4$)

Order 12: $C_2\times C_6$ ($C_2^3.(C_6\times S_4).C_2$), $A_4$ ($C_2^3:A_4.D_6.C_2$)

Order 8: $C_2^3$ ($S_3\times S_4^2$), $C_2^3$ ($D_6^2:S_4$)

Order 6: $C_6$ ($C_2^3:S_4^2$)

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

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

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

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: $Q_8:A_4^2:D_6$ ($C_2$), $(C_2^2\times C_6).S_4^2$ ($C_2$), $(C_3\times Q_8).S_4^2$ ($C_2$), $C_2^5.(A_4\times S_3^2)$ ($C_2$), $Q_8:A_4^2:D_6$ ($C_2$), $C_3\times C_2^3.S_4^2$ ($C_2$), $(C_2^2\times C_6):S_4^2$ ($C_2$)

Order 6912: $Q_8:A_4^2:S_3$ ($C_2^2$), $C_2^5:(C_3^2\times S_4)$ ($C_2^2$), $(C_2^2\times D_6):A_4^2$ ($C_2^2$), $Q_8:A_4^2:C_6$ ($C_2^2$), $Q_8:A_4^2:C_6$ ($C_2^2$), $Q_8:A_4^2:S_3$ ($C_2^2$), $Q_8:A_4^2:S_3$ ($C_2^2$)

Order 4608: $(D_6\times C_2^4):S_4$ ($S_3$), $C_2^5.D_6^2$ ($S_3$), $C_2^3.S_4^2$ ($S_3$)

Order 3456: $C_3\times Q_8:A_4^2$ ($C_2^3$)

Order 2304: $C_3\times C_2^3:\GL(2,\mathbb{Z}/4)$ ($D_6$), $C_{12}.(C_2^3\times S_4)$ ($D_6$), $(D_6\times C_2^4):A_4$ ($D_6$), $(C_2^4\times C_6).S_4$ ($D_6$), $C_2^5.(C_6\times D_6)$ ($D_6$), $C_6\times C_2^4:S_4$ ($D_6$), $(Q_8\times A_4):S_4$ ($D_6$), $A_4\times C_2^3:S_4$ ($D_6$), $(C_2^3\times S_4):A_4$ ($D_6$)

Order 1152: $C_2^3:D_6^2$ ($S_4$), $Q_8:A_4^2$ ($C_2\times D_6$), $(C_2^2\times D_6):S_4$ ($S_4$), $C_6\times C_2^4:A_4$ ($C_2\times D_6$), $C_2^5:C_6^2$ ($C_2\times D_6$)

Order 768: $C_2^5:S_4$ ($S_3^2$), $C_2^3:\GL(2,\mathbb{Z}/4)$ ($S_3^2$), $(C_2^3\times C_{12}):D_4$ ($S_3^2$)

Order 576: $C_2^2\times A_4\times D_6$ ($C_2\times S_4$), $C_2^2\times C_6:S_4$ ($C_2\times S_4$), $C_2^2\times C_6\times S_4$ ($C_2\times S_4$), $(C_3\times Q_8):S_4$ ($C_2\times S_4$), $C_3\times C_2^3:S_4$ ($C_2\times S_4$), $(C_2^2\times D_6):A_4$ ($C_2\times S_4$)

Order 384: $C_{12}.C_2^5$ ($S_3\times D_6$), $C_2^5:A_4$ ($S_3\times D_6$), $A_4\times D_4:C_2^2$ ($S_3\times D_6$)

Order 288: $C_3\times Q_8:A_4$ ($C_2^2\times S_4$), $C_2^3:C_6^2$ ($C_2^2\times S_4$)

Order 192: $(C_2\times C_{12}):D_4$ ($S_3\times S_4$), $D_6\times C_2^4$ ($S_3\times S_4$), $C_2^3\times S_4$ ($S_3\times S_4$), $C_2^3:S_4$ ($S_3\times S_4$)

Order 144: $C_6:S_4$ ($C_2^3:S_4$)

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

Order 96: $C_2^4\times C_6$ ($D_6\times S_4$), $C_2^3\times A_4$ ($D_6\times S_4$), $C_6.C_2^4$ ($D_6\times S_4$), $Q_8:A_4$ ($D_6\times S_4$)

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

Order 48: $C_2^2\times D_6$ ($S_4^2$)

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

Order 32: $C_2^5$ ($S_4\times S_3^2$), $D_4:C_2^2$ ($S_4\times S_3^2$)

Order 24: $C_2^2\times C_6$ ($C_2\times S_4^2$), $C_2^2\times C_6$ ($S_3\times C_2^3:S_4$), $C_2\times A_4$ ($S_3\times C_2^3:S_4$)

Order 12: $C_2\times C_6$ ($C_2^3.(C_6\times S_4).C_2$), $A_4$ ($C_2^3:A_4.D_6.C_2$)

Order 8: $C_2^3$ ($S_3\times S_4^2$), $C_2^3$ ($D_6^2:S_4$)

Order 6: $C_6$ ($C_2^3:S_4^2$)

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

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

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

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

Series

Derived series

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

$\rhd$

$C_3\times Q_8:A_4^2$

$\rhd$

$D_4:C_2^4$

$\rhd$

$C_2$

$\rhd$

$C_1$

Chief series

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

$\rhd$

$C_2^5.(A_4\times S_3^2)$

$\rhd$

$Q_8:A_4^2:S_3$

$\rhd$

$C_3\times Q_8:A_4^2$

$\rhd$

$C_2^5:C_6^2$

$\rhd$

$C_{12}.C_2^5$

$\rhd$

$D_4:C_2^4$

$\rhd$

$C_2^5$

$\rhd$

$C_2^3$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

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

$\rhd$

$C_3\times Q_8:A_4^2$

Upper central series

$C_1$

$\lhd$

$C_2$

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 $153 \times 153$ rational character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.