Abstract group 331776.a: $C_2^8.C_3^4:\OD_{16}$

• Label: 331776.a
• Order: \( 2^{12} \cdot 3^{4} \)
• Exponent: \( 2^{4} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{3} \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{14} \cdot 3^{4} \)
• $\card{\mathrm{Out}(G)}$: \( 2^{2} \)
• Perm deg.: $16$
• Trans deg.: $16$
• Rank: $2$

Group information

Description:	$C_2^8.C_3^4:\OD_{16}$
Order:	\(331776\)\(\medspace = 2^{12} \cdot 3^{4} \)
Exponent:	\(48\)\(\medspace = 2^{4} \cdot 3 \)
Automorphism group:	$A_4^2\wr C_2.C_2^2.D_4$, of order \(1327104\)\(\medspace = 2^{14} \cdot 3^{4} \)
Composition factors:	$C_2$ x 12, $C_3$ x 4
Derived length:	$4$

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

Group statistics

Order	1	2	3	4	6	8	12	16
Elements	1	2703	6560	43632	23136	145152	27648	82944	331776
Conjugacy classes	1	9	6	15	18	11	9	4	73
Divisions	1	9	6	14	18	6	9	2	65
Autjugacy classes	1	9	5	13	13	5	6	1	53

Dimension	1	2	4	8	12	16	18	24	36	48	72	81	108	144	162	216	324
Irr. complex chars.	8	2	0	4	4	4	8	10	4	4	8	8	4	1	2	2	0	73
Irr. rational chars.	4	2	1	4	4	4	4	10	6	4	8	4	4	1	2	2	1	65

Minimal presentations

Permutation degree:	$16$
Transitive degree:	$16$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	12	12	12
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k \mid c^{3}=d^{6}=e^{6}=f^{6}=g^{2}= \!\cdots\! \rangle}$
Permutation group:	Degree $16$ $\langle(1,16,5,10,4,13,7,12,3,15,8,9,2,14,6,11), (1,14,8,9,3,15,5,12,4,16,7,11,2,13,6,10)\rangle$
Transitive group:	16T1907	24T19596	24T19597	24T19598	all 9
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:	$(C_2^8.C_3^4)$ . $\OD_{16}$	$C_2^8$ . $(C_3^4:\OD_{16})$	$(C_2^8.C_3^3.D_6)$ . $C_4$	$(C_2^8.C_3^2:F_9)$ . $C_2$	all 8

Elements of the group are displayed as permutations of degree 16.

Homology

Abelianization:	$C_{2} \times C_{4} $
Schur multiplier:	$C_{6}$
Commutator length:	$1$

Subgroups

There are 12673402 subgroups in 15861 conjugacy classes, 11 normal (9 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_2^8.C_3^4:\OD_{16}$
Commutator:	$G' \simeq$ $C_2^8.C_3^4.C_2$	$G/G' \simeq$ $C_2\times C_4$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^8.C_3^4:\OD_{16}$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^8$	$G/\operatorname{Fit} \simeq$ $C_3^4:\OD_{16}$
Radical:	$R \simeq$ $C_2^8.C_3^4:\OD_{16}$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^8$	$G/\operatorname{soc} \simeq$ $C_3^4:\OD_{16}$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2\wr \OD_{16}$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^4$

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 648 (order at least 512) are shown, as well as all normal subgroups of any index.

Order 331776: $C_2^8.C_3^4:\OD_{16}$

Order 165888: $C_2^8.C_3^2:F_9$ x 2, $A_4^2.(C_2\times A_4^2:C_4)$

Order 82944: $A_4^2:(A_4^2:C_4)$ x 2, $C_2^8.C_3^3.D_6$

Order 41472: $A_4^2.A_4^2.C_2$, $C_2^8.C_3^4.C_2$

Order 27648: $C_2^8.C_3^2.D_6$, $C_2^6.C_3^3.C_2^3.C_2$

Order 20736: $C_2^8.C_3^4$, $A_4^3.D_6$

Order 18432: $C_2^8.F_9$ x 2, $A_4^2.C_2^5:C_4$

Order 13824: $C_2^8.C_3^3.C_2$ x 2, $C_2^2:A_4^2.S_4$ x 2, $C_2^6.C_3^3.D_4$, $C_2^6.C_3^3.D_4$, $A_4^2.C_2^4.C_6$, $C_2^6.C_3^2.D_6.C_2$, $A_4^2.A_4.C_2^3$, $C_2^4:C_3.A_4^2.C_2$, $A_4.C_2^5.C_6^2$, $C_2^6.C_3^3.D_4$, $A_4^3.C_2^3$, $C_2^2.A_4^2:S_4$

Order 10368: $A_4^3:S_3$, $A_4^3:S_3$, $A_4^3:C_6$, $(C_3^2\times A_4^2):(C_2\times C_4)$

Order 9216: $C_2^4.A_4^2:C_4$ x 3, $C_2^6.(S_3\times C_3:D_4)$ x 2, $C_2^4.A_4^2:C_4$ x 2, $A_4^2:C_2\wr C_4$ x 2, $A_4^2:C_2\wr C_4$ x 2, $A_4^2.C_2^3.C_2^3$, $A_4^2.D_4^2$, $(C_2^2:S_4)^2$, $A_4^2:\OD_{16}:C_2^2$, $A_4^2:C_2^4:C_4$

Order 6912: $A_4:S_4^2$ x 5, $C_2^8.C_3^3$ x 2, $C_3:S_4\times \GL(2,\mathbb{Z}/4)$ x 2, $A_4^3:C_2^2$ x 2, $(C_2\times C_6):S_4^2$ x 2, $C_2^2.A_4^3$ x 2, $A_4.A_4^2.C_4$, $C_2^6.C_3^3.C_2^2$, $C_2^6.C_3^3.C_4$, $A_4^3:C_2^2$, $(C_2^2\times A_4^2):D_6$, $A_4.A_4^2.C_2^2$, $C_2^2.A_4^3$

Order 5184: $(C_3^2\times A_4^2):C_4$ x 2, $C_3\times A_4^3$, $C_3^2:S_4^2$, $(A_4\times C_6^2):D_6$

Order 4608: $C_2^4.\PSOPlus(4,3)$ x 6, $(C_2^2\times A_4):\GL(2,\mathbb{Z}/4)$ x 4, $C_2^4:\PSOPlus(4,3)$ x 3, $(C_2^3\times A_4^2).C_4$ x 3, $C_2^4.\PSOPlus(4,3)$ x 3, $A_4.C_2^5.C_6.C_2$ x 2, $A_4^2.C_2^2\wr C_2$ x 2, $(C_2^4\times A_4).D_6.C_2$ x 2, $A_4^2.C_2^2:D_4$ x 2, $A_4^2.C_2^2:D_4$ x 2, $(C_2\times A_4^2).C_2^4$ x 2, $A_4^2.(C_4\times D_4)$ x 2, $C_2^4.D_6:S_4$ x 2, $C_2^8.C_3.S_3$ x 2, $A_4^2.C_2^2\wr C_2$ x 2, $A_4^2.C_2^4.C_2$ x 2, $A_4^2.C_2^2\wr C_2$ x 2, $C_2^3:S_4^2$ x 2, $A_4^2:C_2^2\times D_4$ x 2, $C_2^4:(A_4\times S_4)$ x 2, $C_2^4.(A_4\times S_4)$ x 2, $C_2^3:S_4^2$ x 2, $C_2^3.S_4^2$ x 2, $C_2^4:\PSOPlus(4,3)$ x 2, $(C_2^3\times A_4^2):C_4$ x 2, $A_4^2.C_4:D_4$, $A_4^2.C_2^4.C_2$, $C_2^6.C_6^2.C_2$, $A_4^2.C_2^5$, $A_4^2.C_2^4.C_2$, $A_4^2.C_2^2.C_2^3$, $C_2^3:(A_4^2:C_4)$, $A_4^2:(C_2\times \OD_{16})$, $A_4^2:C_2^3:C_4$

Order 4096: $C_2\wr \OD_{16}$

Order 3456: $A_4^2:S_4$ x 17, $A_4^2:S_4$ x 5, $C_2^2:C_6^2:S_4$ x 4, $S_4\times A_4^2$ x 4, $(C_3\times A_4):\GL(2,\mathbb{Z}/4)$ x 3, $C_6.S_4^2$ x 2, $C_6.S_4^2$ x 2, $C_2^2:A_4^2:S_3$ x 2, $C_2^4:(C_3^2\times S_4)$ x 2, $A_4^2:D_{12}$ x 2, $C_3\times A_4\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2\times A_4^2:D_6$ x 2, $C_2\times A_4^2.D_6$ x 2, $C_2^2:A_4^2:S_3$ x 2, $C_3:D_4\times A_4^2$, $(C_3\times A_4):\GL(2,\mathbb{Z}/4)$, $C_2\times A_4^3$, $A_4^2:C_2^2\times C_6$, $(C_3\times A_4^2):D_4$, $(C_2\times A_4^2).D_6$, $D_6\times \PSOPlus(4,3)$

Order 3072: $A_4.C_2^5.C_2^3$, $C_2^8.D_6$

Order 2592: $(A_4\times C_6^2):S_3$ x 2, $A_4\times C_3^2:S_4$ x 2, $C_3:S_3\times A_4^2$, $C_3^2\times \PSOPlus(4,3)$

Order 2304: $S_4\times \GL(2,\mathbb{Z}/4)$ x 14, $C_2^2:S_4^2$ x 10, $C_2\times A_4\times \GL(2,\mathbb{Z}/4)$ x 8, $C_2\times A_4:\GL(2,\mathbb{Z}/4)$ x 7, $D_4\times \PSOPlus(4,3)$ x 6, $C_2^4:A_4^2$ x 6, $A_4^2:C_2^4$ x 5, $C_2^4:A_4^2$ x 4, $(C_2^3\times A_4).S_4$ x 4, $(C_2^3\times A_4):S_4$ x 4, $A_4\times C_2^3:S_4$ x 4, $A_4^2.C_2^3.C_2$ x 3, $C_2^2.S_4^2$ x 3, $C_2^4:A_4^2$ x 3, $C_2^4:A_4^2$ x 3, $(C_2^2\times A_4^2):C_4$ x 3, $A_4^2.C_2^4$ x 2, $A_4.C_2^4:C_3.C_4$ x 2, $A_4^2.C_4:C_4$ x 2, $A_4^2.C_2^2:C_4$ x 2, $C_2^8.C_3^2$ x 2, $A_4^2.C_4.C_2^2$ x 2, $A_4^2.C_2^2:C_4$ x 2, $A_4^2.C_2^3.C_2$ x 2, $A_4^2.C_2^2:C_4$ x 2, $C_2^2.S_4^2$ x 2, $C_2^2.S_4^2$ x 2, $C_2\times A_4^2:D_4$ x 2, $C_4\times A_4^2:C_2^2$ x 2, $C_2^6:S_3^2$ x 2, $C_2^6:S_3^2$ x 2, $A_4^2:C_2^4$ x 2, $C_2^6:C_6^2$ x 2, $C_2^3:A_4\times S_4$ x 2, $C_2^3.\PSOPlus(4,3)$ x 2, $C_2^4:A_4^2$ x 2, $A_4^2:\OD_{16}$ x 2, $C_2^3:\PSOPlus(4,3)$ x 2, $A_4^2.C_2^3.C_2$, $A_4^2.C_2^3.C_2$, $C_2^5.C_6^2.C_2$, $C_2^5.(C_6\times C_{12})$, $A_4^2.C_4^2$, $(A_4\times C_2^4):D_6$, $A_4^2:C_4\times C_2^2$, $A_4^2:\OD_{16}$, $A_4^2:(C_2\times C_8)$, $C_2^2\times S_4^2$

Order 2048: $C_2^6.C_2^2:C_8$ x 2, $C_2\wr C_8$ x 2, $C_2^6.C_2^4.C_2$, $C_2^7.\OD_{16}$, $C_2^7:\OD_{16}$

Order 1728: $A_4^3$ x 17, $C_3:S_4^2$ x 10, $C_2^6:C_3^3$ x 4, $A_4^2:D_6$ x 4, $C_2^6:C_3^3$ x 3, $A_4^2:D_6$ x 3, $A_4^2:D_6$ x 3, $(C_3\times A_4^2):C_4$ x 3, $C_2^6:C_3^3$ x 2, $C_6\times \PSOPlus(4,3)$ x 2, $C_6\times A_4\times S_4$ x 2, $C_6^2:(C_2\times S_4)$ x 2, $A_4^2:C_{12}$ x 2, $C_6^2:(C_2\times S_4)$, $D_6\times A_4^2$, $C_3:S_3\times \GL(2,\mathbb{Z}/4)$, $C_3:C_4\times A_4^2$

Order 1536: $C_2^7:D_6$ x 3, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 3, $C_2^6:D_{12}$ x 3, $C_2^7.D_6$ x 3, $C_2^7:D_6$ x 3, $C_2^6.C_2.D_6$ x 3, $C_2^6.C_2.D_6$ x 3, $C_2^6.C_2.D_6$ x 3, $C_2^6.C_2.D_6$ x 3, $C_2^6.C_2.D_6$ x 3, $C_2^5.C_2^3.S_3$ x 3, $\GL(2,\mathbb{Z}/4)\times C_2^2:C_4$ x 3, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^6:S_4$, $C_2^6:S_4$, $C_2^4\times \GL(2,\mathbb{Z}/4)$, $C_2^7:D_6$, $C_2^8.S_3$, $C_2^6.C_2.D_6$, $C_2^6.C_2.D_6$, $A_4:C_4\times C_2^2\wr C_2$, $C_2^8:C_6$, $C_2^2\times C_2^3.(C_6\times D_4)$

Order 1296: $C_3^2\times A_4^2$ x 2, $C_6^2:S_3^2$, $C_3^4:\OD_{16}$

Order 1152: $(C_2^2\times A_4):S_4$ x 42, $A_4:\GL(2,\mathbb{Z}/4)$ x 29, $C_2^2:\PSOPlus(4,3)$ x 24, $A_4^2:C_2^3$ x 16, $A_4\times \GL(2,\mathbb{Z}/4)$ x 16, $A_4^2:C_2^3$ x 14, $C_2\times S_4^2$ x 11, $A_4:\GL(2,\mathbb{Z}/4)$ x 11, $A_4\times C_2^2:S_4$ x 10, $A_4:\GL(2,\mathbb{Z}/4)$ x 10, $C_2.S_4^2$ x 10, $C_2^2:A_4\times S_4$ x 8, $C_2^3\times A_4^2$ x 6, $C_2\times A_4^2:C_4$ x 5, $C_2^3:A_4^2$ x 4, $C_2\times A_4^2:C_4$ x 4, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$ x 4, $C_4\times \PSOPlus(4,3)$ x 4, $C_2\times A_4^2:C_4$ x 4, $C_6:S_4\times D_4$ x 3, $D_4\times A_4^2$ x 3, $A_4^2:D_4$ x 3, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$ x 3, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$ x 3, $(C_2^2\times A_4):D_{12}$ x 3, $(C_2^3\times C_{12}):D_6$ x 3, $(C_2^3\times C_6):S_4$ x 2, $C_3\times C_2^2\wr S_3$ x 2, $C_2^3:A_4^2$ x 2, $C_2^5:S_3^2$ x 2, $C_2^4:(S_3\times A_4)$ x 2, $C_2^3:D_6\times A_4$ x 2, $C_2\times C_6\times \GL(2,\mathbb{Z}/4)$ x 2, $C_2^5.C_6^2$ x 2, $C_2^5.S_3^2$ x 2, $C_2^5.S_3^2$ x 2, $C_2^5.S_3^2$ x 2, $C_2^5:S_3^2$ x 2, $D_6\times \GL(2,\mathbb{Z}/4)$ x 2, $A_4^2:C_8$ x 2, $D_6:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5.S_3^2$ x 2, $C_2^5.S_3^2$ x 2, $D_6:\GL(2,\mathbb{Z}/4)$ x 2, $D_6:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5.S_3^2$ x 2, $C_2^5.S_3^2$ x 2, $C_2^5.S_3^2$ x 2, $C_2^5.S_3^2$ x 2, $C_2^5.S_3^2$ x 2, $C_6^2:D_4:C_4$, $C_2^3\times C_6:S_4$, $C_2^5:C_6^2$

Order 1024: $C_2^6.C_2^3.C_2$ x 2, $C_2^6.C_2^3.C_2$ x 2, $C_2^6.C_4.C_2^2$ x 2, $C_2^2\wr C_4$ x 2, $C_2^7.C_8$ x 2, $C_2^5.C_2^4.C_2$, $C_2^5.C_2^4.C_2$, $C_2^5.C_2^4.C_2$, $C_2^5.C_2^4.C_2$, $C_2^5.C_2^4.C_2$, $(C_2^2\wr C_2)^2$, $C_2^6.\OD_{16}$, $C_2^6.\OD_{16}$, $C_2^6.\OD_{16}$

Order 864: $A_4^2:S_3$ x 35, $A_4^2:S_3$ x 10, $A_4^2:C_6$ x 8, $C_6\times A_4^2$ x 3, $A_4^2:C_6$ x 3, $C_6^2:S_4$ x 3, $C_6^2:S_4$ x 2, $C_6^2:S_4$ x 2, $S_3\times C_6:S_4$ x 2, $S_3\times A_4^2$ x 2, $C_6\wr C_2\times A_4$ x 2, $C_3^2:\GL(2,\mathbb{Z}/4)$ x 2, $C_6^2:D_{12}$ x 2, $C_3^2:(C_4\times S_4)$ x 2, $C_6^2:(C_2\times A_4)$, $C_6:D_6\times A_4$, $C_6^2:S_4$, $C_6:S_3\times S_4$, $C_3^2\times \GL(2,\mathbb{Z}/4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_6.(S_3\times S_4)$

Order 768: $D_4\times \GL(2,\mathbb{Z}/4)$ x 24, $C_2^3\times \GL(2,\mathbb{Z}/4)$ x 13, $C_2^6:D_6$ x 12, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 12, $A_4.(C_2.C_4:D_4)$ x 12, $C_2^6:D_6$ x 11, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 10, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 9, $C_2^7:C_6$ x 8, $C_2^5:S_4$ x 7, $A_4.(C_2^5.C_2)$ x 7, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 6, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 6, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 6, $(C_2^3\times S_4).C_2^2$ x 6, $A_4.(C_2.C_4:D_4)$ x 6, $A_4.(C_2^3.D_4)$ x 6, $(C_2^3\times A_4).C_2^3$ x 6, $A_4.(C_2^3.D_4)$ x 6, $A_4.(C_2^3.D_4)$ x 6, $A_4.C_2^5.C_2$ x 6, $C_2^5:D_{12}$ x 3, $C_2^6.D_6$ x 3, $\GL(2,\mathbb{Z}/4):C_2^3$ x 3, $C_2^5:D_{12}$ x 3, $C_2^6.D_6$ x 3, $C_2^6.D_6$ x 3, $C_2^5:D_{12}$ x 3, $C_2^6.D_6$ x 3, $C_2^6.D_6$ x 3, $C_2^6.D_6$ x 3, $A_4.(C_2^3.D_4)$ x 3, $(C_2^3\times A_4).C_2^3$ x 3, $(C_2^3\times A_4).C_2^3$ x 3, $A_4:C_4\times C_2^2:C_4$ x 3, $C_2^7:C_6$ x 3, $C_2^7:C_6$ x 3, $C_2^6:C_{12}$ x 3, $C_2^7.C_6$ x 3, $C_2^5:S_4$ x 2, $C_2^5.S_4$ x 2, $C_2^8.C_3$ x 2, $C_2^6:A_4$ x 2, $C_2^6:D_6$, $S_4\times C_2^5$, $C_2^4\times A_4:C_4$, $C_2^8.C_3$, $C_2^6:A_4$

Order 648: $C_3^2:F_9$ x 2, $C_3\times C_6^2:C_6$, $C_3^3:S_4$, $C_3^3:S_4$, $C_3^4:(C_2\times C_4)$

Order 576: $C_2^2:A_4^2$ x 42, $A_4^2:C_2^2$ x 34, $C_2^2\times A_4^2$ x 30, $A_4^2:C_2^2$ x 30, $S_4^2$ x 25, $C_2^2:A_4^2$ x 24, $A_4^2:C_4$ x 23, $C_2^4:S_3^2$ x 14, $A_4^2:C_4$ x 12, $C_6:\GL(2,\mathbb{Z}/4)$ x 12, $D_4\times C_3:S_4$ x 12, $S_3\times \GL(2,\mathbb{Z}/4)$ x 10, $A_4^2:C_4$ x 10, $C_6.\GL(2,\mathbb{Z}/4)$ x 9, $C_6:D_4\times A_4$ x 8, $C_6\times \GL(2,\mathbb{Z}/4)$ x 8, $C_2^2\times C_6:S_4$ x 7, $C_2^4:S_3^2$ x 6, $C_6:\GL(2,\mathbb{Z}/4)$ x 6, $C_6.\GL(2,\mathbb{Z}/4)$ x 6, $C_2^4:C_6^2$ x 4, $(C_2^2\times C_6):S_4$ x 4, $C_2^2:S_4\times C_6$ x 4, $(C_2^2\times C_6).S_4$ x 4, $C_6.\GL(2,\mathbb{Z}/4)$ x 4, $C_6.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^4:C_6^2$ x 3, $C_2^4:C_6^2$ x 3, $C_2\times C_{12}:S_4$ x 3, $C_2\times C_{12}:S_4$ x 3, $C_2^4.C_6^2$ x 3, $C_2^6:C_3^2$ x 2, $C_2^2:A_4\times D_6$ x 2, $C_2^2\times A_4\times D_6$ x 2, $C_2^2\times C_6\times S_4$ x 2, $C_2^2:D_6^2$ x 2, $(C_2^3\times C_6):C_{12}$ x 2, $C_6.C_2^3\times A_4$ x 2, $C_2^2\times A_4:C_{12}$ x 2, $C_4\times A_4^2$ 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, $C_2^4.S_3^2$ x 2, $C_6.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4.S_3^2$ x 2, $C_6.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4.S_3^2$ x 2, $(C_2^2\times C_6^2):C_4$ x 2, $C_3^2:C_2\wr C_4$ x 2, $C_6^2.C_2^4$, $C_2^4:S_3^2$, $C_6^2.(C_2^2\times C_4)$, $C_6^2.(C_2^2\times C_4)$

Order 512: $C_2^7.C_2^2$ x 5, $C_2^7.C_4$ x 5, $C_2^7:C_4$ x 4, $C_2^7.C_2^2$ x 3, $C_2^4.C_2^5$ x 3, $C_2^3.C_2^5.C_2$ x 3, $C_2^6.C_8$ x 3, $C_2^3:D_4^2$ x 2, $C_2^6.C_2^3$ x 2, $C_2^6:D_4$ x 2, $C_2^3.C_2^4.C_2^2$ x 2, $(C_2^3\times D_4).C_2^3$ x 2, $C_2^4.C_2^5$ x 2, $C_2^5.C_2^3.C_2$ x 2, $C_2^6:C_8$ x 2, $C_2^2.C_2^6.C_2$, $C_2^2.C_2^6.C_2$, $C_2^6.C_2^3$, $C_2^4\wr C_2$, $C_2^5.\OD_{16}$, $C_2^5:\OD_{16}$, $C_2^5.(C_2\times D_4)$, $C_2^6.D_4$, $C_2^3.(C_2^3\times C_4).C_2$, $C_2^5.C_2^3.C_2$, $C_2^4.C_2^4.C_2$, $(C_2^2\times D_4).C_2^4$, $C_2^7:C_4$, $C_2^3.C_2^5.C_2$, $C_2^3.C_2^5.C_2$, $C_2^5.\OD_{16}$, $C_2^6:D_4$

Order 256: $C_2^8$

Order 81: $C_3^4$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 648 (order at least 512) are shown, as well as all normal subgroups of any index.

Order 331776: $C_2^8.C_3^4:\OD_{16}$

Order 165888: $C_2^8.C_3^2:F_9$, $A_4^2.(C_2\times A_4^2:C_4)$

Order 82944: $A_4^2:(A_4^2:C_4)$ x 2, $C_2^8.C_3^3.D_6$

Order 41472: $A_4^2.A_4^2.C_2$, $C_2^8.C_3^4.C_2$

Order 27648: $C_2^8.C_3^2.D_6$, $C_2^6.C_3^3.C_2^3.C_2$

Order 20736: $C_2^8.C_3^4$, $A_4^3.D_6$

Order 18432: $C_2^8.F_9$, $A_4^2.C_2^5:C_4$

Order 13824: $C_2^8.C_3^3.C_2$ x 2, $C_2^6.C_3^3.D_4$, $C_2^6.C_3^3.D_4$, $A_4^2.C_2^4.C_6$, $C_2^6.C_3^2.D_6.C_2$, $A_4^2.A_4.C_2^3$, $C_2^4:C_3.A_4^2.C_2$, $A_4.C_2^5.C_6^2$, $C_2^6.C_3^3.D_4$, $A_4^3.C_2^3$, $C_2^2.A_4^2:S_4$, $C_2^2:A_4^2.S_4$

Order 10368: $A_4^3:S_3$, $A_4^3:S_3$, $A_4^3:C_6$, $(C_3^2\times A_4^2):(C_2\times C_4)$

Order 9216: $C_2^4.A_4^2:C_4$ x 2, $C_2^4.A_4^2:C_4$ x 2, $A_4^2:C_2\wr C_4$ x 2, $A_4^2:C_2\wr C_4$ x 2, $C_2^6.(S_3\times C_3:D_4)$, $A_4^2.C_2^3.C_2^3$, $A_4^2.D_4^2$, $(C_2^2:S_4)^2$, $A_4^2:\OD_{16}:C_2^2$, $A_4^2:C_2^4:C_4$

Order 6912: $A_4:S_4^2$ x 4, $C_2^8.C_3^3$ x 2, $A_4^3:C_2^2$ x 2, $A_4.A_4^2.C_4$, $C_2^6.C_3^3.C_2^2$, $C_2^6.C_3^3.C_4$, $C_3:S_4\times \GL(2,\mathbb{Z}/4)$, $A_4^3:C_2^2$, $(C_2\times C_6):S_4^2$, $(C_2^2\times A_4^2):D_6$, $A_4.A_4^2.C_2^2$, $C_2^2.A_4^3$, $C_2^2.A_4^3$

Order 5184: $(C_3^2\times A_4^2):C_4$ x 2, $C_3\times A_4^3$, $C_3^2:S_4^2$, $(A_4\times C_6^2):D_6$

Order 4608: $C_2^4.\PSOPlus(4,3)$ x 3, $C_2^4:\PSOPlus(4,3)$ x 3, $(C_2^3\times A_4^2).C_4$ x 3, $A_4^2.C_2^2\wr C_2$ x 2, $A_4^2.C_2^2\wr C_2$ x 2, $A_4^2.C_2^2\wr C_2$ x 2, $A_4^2:C_2^2\times D_4$ x 2, $(C_2^2\times A_4):\GL(2,\mathbb{Z}/4)$ x 2, $(C_2^3\times A_4^2):C_4$ x 2, $C_2^4.\PSOPlus(4,3)$ x 2, $A_4.C_2^5.C_6.C_2$, $(C_2^4\times A_4).D_6.C_2$, $A_4^2.C_4:D_4$, $A_4^2.C_2^4.C_2$, $A_4^2.C_2^2:D_4$, $C_2^6.C_6^2.C_2$, $A_4^2.C_2^5$, $A_4^2.C_2^4.C_2$, $A_4^2.C_2^2.C_2^3$, $A_4^2.C_2^2:D_4$, $(C_2\times A_4^2).C_2^4$, $A_4^2.(C_4\times D_4)$, $C_2^4.D_6:S_4$, $C_2^8.C_3.S_3$, $A_4^2.C_2^4.C_2$, $C_2^3:S_4^2$, $C_2^4:(A_4\times S_4)$, $C_2^4.(A_4\times S_4)$, $C_2^3:S_4^2$, $C_2^3.S_4^2$, $C_2^4:\PSOPlus(4,3)$, $C_2^3:(A_4^2:C_4)$, $A_4^2:(C_2\times \OD_{16})$, $A_4^2:C_2^3:C_4$

Order 4096: $C_2\wr \OD_{16}$

Order 3456: $A_4^2:S_4$ x 8, $A_4^2:S_4$ x 4, $S_4\times A_4^2$ x 3, $C_2^2:C_6^2:S_4$ x 2, $(C_3\times A_4):\GL(2,\mathbb{Z}/4)$ x 2, $C_3:D_4\times A_4^2$, $C_6.S_4^2$, $C_6.S_4^2$, $(C_3\times A_4):\GL(2,\mathbb{Z}/4)$, $C_2\times A_4^3$, $C_2^2:A_4^2:S_3$, $A_4^2:C_2^2\times C_6$, $C_2^4:(C_3^2\times S_4)$, $(C_3\times A_4^2):D_4$, $(C_2\times A_4^2).D_6$, $A_4^2:D_{12}$, $C_3\times A_4\times \GL(2,\mathbb{Z}/4)$, $D_6\times \PSOPlus(4,3)$, $C_2\times A_4^2:D_6$, $C_2\times A_4^2.D_6$, $C_2^2:A_4^2:S_3$

Order 3072: $A_4.C_2^5.C_2^3$, $C_2^8.D_6$

Order 2592: $(A_4\times C_6^2):S_3$ x 2, $A_4\times C_3^2:S_4$, $C_3:S_3\times A_4^2$, $C_3^2\times \PSOPlus(4,3)$

Order 2304: $C_2\times A_4:\GL(2,\mathbb{Z}/4)$ x 6, $D_4\times \PSOPlus(4,3)$ x 6, $S_4\times \GL(2,\mathbb{Z}/4)$ x 6, $A_4^2:C_2^4$ x 5, $C_2\times A_4\times \GL(2,\mathbb{Z}/4)$ x 4, $C_2^2:S_4^2$ x 4, $C_2^4:A_4^2$ x 3, $C_2^4:A_4^2$ x 3, $(C_2^2\times A_4^2):C_4$ x 3, $A_4^2.C_2^4$ x 2, $A_4^2.C_2^2:C_4$ x 2, $A_4^2.C_2^3.C_2$ x 2, $A_4^2.C_2^2:C_4$ x 2, $C_2\times A_4^2:D_4$ x 2, $C_4\times A_4^2:C_2^2$ x 2, $C_2^2.S_4^2$ x 2, $C_2^6:C_6^2$ x 2, $C_2^4:A_4^2$ x 2, $(C_2^3\times A_4).S_4$ x 2, $(C_2^3\times A_4):S_4$ x 2, $A_4\times C_2^3:S_4$ x 2, $A_4^2:\OD_{16}$ x 2, $C_2^4:A_4^2$ x 2, $A_4^2.C_2^3.C_2$, $A_4.C_2^4:C_3.C_4$, $A_4^2.C_4:C_4$, $A_4^2.C_2^3.C_2$, $C_2^8.C_3^2$, $C_2^5.C_6^2.C_2$, $C_2^5.(C_6\times C_{12})$, $A_4^2.C_4^2$, $A_4^2.C_4.C_2^2$, $A_4^2.C_2^3.C_2$, $A_4^2.C_2^2:C_4$, $(A_4\times C_2^4):D_6$, $C_2^2.S_4^2$, $C_2^2.S_4^2$, $C_2^6:S_3^2$, $C_2^6:S_3^2$, $A_4^2:C_2^4$, $C_2^3:A_4\times S_4$, $C_2^3.\PSOPlus(4,3)$, $C_2^4:A_4^2$, $A_4^2:C_4\times C_2^2$, $A_4^2:\OD_{16}$, $A_4^2:(C_2\times C_8)$, $C_2^3:\PSOPlus(4,3)$, $C_2^2\times S_4^2$

Order 2048: $C_2^6.C_2^4.C_2$, $C_2^6.C_2^2:C_8$, $C_2^7.\OD_{16}$, $C_2^7:\OD_{16}$, $C_2\wr C_8$

Order 1728: $A_4^3$ x 8, $C_3:S_4^2$ x 4, $A_4^2:D_6$ x 3, $C_2^6:C_3^3$ x 2, $C_2^6:C_3^3$ x 2, $A_4^2:D_6$ x 2, $C_6\times \PSOPlus(4,3)$ x 2, $A_4^2:D_6$ x 2, $(C_3\times A_4^2):C_4$ x 2, $C_2^6:C_3^3$, $C_6^2:(C_2\times S_4)$, $D_6\times A_4^2$, $C_6\times A_4\times S_4$, $C_6^2:(C_2\times S_4)$, $C_3:S_3\times \GL(2,\mathbb{Z}/4)$, $C_3:C_4\times A_4^2$, $A_4^2:C_{12}$

Order 1536: $C_2^7:D_6$ x 2, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^4:\GL(2,\mathbb{Z}/4)$ x 2, $C_2^6:D_{12}$ x 2, $C_2^7.D_6$ x 2, $C_2^7:D_6$ x 2, $C_2^6.C_2.D_6$ x 2, $C_2^6.C_2.D_6$ x 2, $C_2^6.C_2.D_6$ x 2, $C_2^6.C_2.D_6$ x 2, $C_2^6.C_2.D_6$ x 2, $C_2^5.C_2^3.S_3$ x 2, $\GL(2,\mathbb{Z}/4)\times C_2^2:C_4$ x 2, $C_2^6:S_4$, $C_2^6:S_4$, $C_2^4:\GL(2,\mathbb{Z}/4)$, $C_2^4\times \GL(2,\mathbb{Z}/4)$, $C_2^7:D_6$, $C_2^8.S_3$, $C_2^6.C_2.D_6$, $C_2^6.C_2.D_6$, $A_4:C_4\times C_2^2\wr C_2$, $C_2^8:C_6$, $C_2^2\times C_2^3.(C_6\times D_4)$

Order 1296: $C_3^2\times A_4^2$ x 2, $C_6^2:S_3^2$, $C_3^4:\OD_{16}$

Order 1152: $(C_2^2\times A_4):S_4$ x 16, $A_4^2:C_2^3$ x 16, $A_4:\GL(2,\mathbb{Z}/4)$ x 16, $C_2^2:\PSOPlus(4,3)$ x 9, $A_4\times \GL(2,\mathbb{Z}/4)$ x 7, $A_4^2:C_2^3$ x 6, $C_2^3\times A_4^2$ x 5, $C_2\times S_4^2$ x 5, $A_4:\GL(2,\mathbb{Z}/4)$ x 5, $A_4\times C_2^2:S_4$ x 4, $C_2\times A_4^2:C_4$ x 4, $C_2\times A_4^2:C_4$ x 4, $C_4\times \PSOPlus(4,3)$ x 4, $A_4:\GL(2,\mathbb{Z}/4)$ x 4, $C_2.S_4^2$ x 4, $C_2^2:A_4\times S_4$ x 3, $D_4\times A_4^2$ x 3, $A_4^2:D_4$ x 3, $C_2^3:A_4^2$ x 2, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$ x 2, $C_6:S_4\times D_4$ x 2, $C_2^5.C_6^2$ x 2, $C_2\times A_4^2:C_4$ x 2, $A_4^2:C_8$ x 2, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$ x 2, $(C_2\times C_6):\GL(2,\mathbb{Z}/4)$ x 2, $(C_2^2\times A_4):D_{12}$ x 2, $(C_2^3\times C_{12}):D_6$ x 2, $C_6^2:D_4:C_4$, $(C_2^3\times C_6):S_4$, $C_3\times C_2^2\wr S_3$, $C_2^3\times C_6:S_4$, $C_2^3:A_4^2$, $C_2^5:S_3^2$, $C_2^4:(S_3\times A_4)$, $C_2^3:D_6\times A_4$, $C_2\times C_6\times \GL(2,\mathbb{Z}/4)$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^5:S_3^2$, $D_6\times \GL(2,\mathbb{Z}/4)$, $D_6:\GL(2,\mathbb{Z}/4)$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $D_6:\GL(2,\mathbb{Z}/4)$, $D_6:\GL(2,\mathbb{Z}/4)$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^5.S_3^2$, $C_2^5:C_6^2$

Order 1024: $C_2^6.C_2^3.C_2$ x 2, $C_2^6.C_2^3.C_2$ x 2, $C_2^2\wr C_4$ x 2, $C_2^5.C_2^4.C_2$, $C_2^5.C_2^4.C_2$, $C_2^5.C_2^4.C_2$, $C_2^5.C_2^4.C_2$, $C_2^5.C_2^4.C_2$, $C_2^6.C_4.C_2^2$, $(C_2^2\wr C_2)^2$, $C_2^6.\OD_{16}$, $C_2^7.C_8$, $C_2^6.\OD_{16}$, $C_2^6.\OD_{16}$

Order 864: $A_4^2:S_3$ x 16, $A_4^2:S_3$ x 4, $A_4^2:C_6$ x 3, $A_4^2:C_6$ x 3, $C_6^2:S_4$ x 2, $C_6\times A_4^2$ x 2, $S_3\times A_4^2$ x 2, $C_6^2:S_4$ x 2, $C_6^2:(C_2\times A_4)$, $C_6:D_6\times A_4$, $C_6^2:S_4$, $C_6^2:S_4$, $S_3\times C_6:S_4$, $C_6:S_3\times S_4$, $C_6\wr C_2\times A_4$, $C_3^2\times \GL(2,\mathbb{Z}/4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_6^2:D_{12}$, $C_3^2:(C_4\times S_4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_3^2:\GL(2,\mathbb{Z}/4)$, $C_6.(S_3\times S_4)$

Order 768: $D_4\times \GL(2,\mathbb{Z}/4)$ x 12, $C_2^3\times \GL(2,\mathbb{Z}/4)$ x 9, $C_2^6:D_6$ x 8, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 8, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 7, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 6, $C_2^6:D_6$ x 6, $A_4.(C_2.C_4:D_4)$ x 6, $C_2^7:C_6$ x 6, $C_2^5:S_4$ x 5, $A_4.(C_2^5.C_2)$ x 5, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 4, $C_2^3:\GL(2,\mathbb{Z}/4)$ x 4, $A_4.(C_2^3.D_4)$ x 4, $A_4.C_2^5.C_2$ x 4, $C_2^3.\GL(2,\mathbb{Z}/4)$ x 3, $(C_2^3\times S_4).C_2^2$ x 3, $A_4.(C_2.C_4:D_4)$ x 3, $A_4.(C_2^3.D_4)$ x 3, $(C_2^3\times A_4).C_2^3$ x 3, $A_4.(C_2^3.D_4)$ x 3, $C_2^5:D_{12}$ x 2, $C_2^6.D_6$ x 2, $\GL(2,\mathbb{Z}/4):C_2^3$ x 2, $C_2^5:D_{12}$ x 2, $C_2^6.D_6$ x 2, $C_2^6.D_6$ x 2, $C_2^5:D_{12}$ x 2, $C_2^6.D_6$ x 2, $C_2^6.D_6$ x 2, $C_2^6.D_6$ x 2, $A_4.(C_2^3.D_4)$ x 2, $(C_2^3\times A_4).C_2^3$ x 2, $(C_2^3\times A_4).C_2^3$ x 2, $A_4:C_4\times C_2^2:C_4$ x 2, $C_2^8.C_3$ x 2, $C_2^7:C_6$ x 2, $C_2^7:C_6$ x 2, $C_2^6:C_{12}$ x 2, $C_2^7.C_6$ x 2, $C_2^6:D_6$, $C_2^5:S_4$, $S_4\times C_2^5$, $C_2^5.S_4$, $C_2^4\times A_4:C_4$, $C_2^8.C_3$, $C_2^6:A_4$, $C_2^6:A_4$

Order 648: $C_3\times C_6^2:C_6$, $C_3^3:S_4$, $C_3^3:S_4$, $C_3^4:(C_2\times C_4)$, $C_3^2:F_9$

Order 576: $A_4^2:C_2^2$ x 22, $C_2^2\times A_4^2$ x 17, $C_2^2:A_4^2$ x 16, $A_4^2:C_2^2$ x 12, $S_4^2$ x 11, $A_4^2:C_4$ x 11, $C_2^2:A_4^2$ x 9, $A_4^2:C_4$ x 8, $C_6:\GL(2,\mathbb{Z}/4)$ x 7, $D_4\times C_3:S_4$ x 6, $C_2^4:S_3^2$ x 6, $C_2^2\times C_6:S_4$ x 5, $S_3\times \GL(2,\mathbb{Z}/4)$ x 5, $C_6.\GL(2,\mathbb{Z}/4)$ x 5, $C_6:D_4\times A_4$ x 4, $C_6\times \GL(2,\mathbb{Z}/4)$ x 4, $A_4^2:C_4$ x 4, $C_2^4:S_3^2$ x 3, $C_6:\GL(2,\mathbb{Z}/4)$ x 3, $C_6.\GL(2,\mathbb{Z}/4)$ x 3, $C_2^4:C_6^2$ x 2, $C_2^4:C_6^2$ x 2, $(C_2^2\times C_6):S_4$ x 2, $C_2^2:S_4\times C_6$ x 2, $C_2^4:C_6^2$ x 2, $(C_2^2\times C_6).S_4$ x 2, $C_2\times C_{12}:S_4$ x 2, $C_2\times C_{12}:S_4$ x 2, $C_4\times A_4^2$ x 2, $C_2^4.C_6^2$ x 2, $C_6.\GL(2,\mathbb{Z}/4)$ x 2, $C_6.\GL(2,\mathbb{Z}/4)$ x 2, $(C_2^2\times C_6^2):C_4$ x 2, $C_3^2:C_2\wr C_4$ x 2, $C_2^6:C_3^2$, $C_2^2:A_4\times D_6$, $C_2^2\times A_4\times D_6$, $C_2^2\times C_6\times S_4$, $C_2^2:D_6^2$, $(C_2^3\times C_6):C_{12}$, $C_6.C_2^3\times A_4$, $C_2^2\times A_4:C_{12}$, $C_6:\GL(2,\mathbb{Z}/4)$, $C_2^4.S_3^2$, $C_2^4.S_3^2$, $C_2^4.S_3^2$, $C_6^2.C_2^4$, $C_2^4:S_3^2$, $C_6^2.(C_2^2\times C_4)$, $C_6^2.(C_2^2\times C_4)$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_2^4.S_3^2$, $C_6.\GL(2,\mathbb{Z}/4)$, $C_2^4.S_3^2$

Order 512: $C_2^7.C_4$ x 5, $C_2^7.C_2^2$ x 4, $C_2^7:C_4$ x 4, $C_2^7.C_2^2$ x 3, $C_2^4.C_2^5$ x 3, $C_2^3.C_2^5.C_2$ x 3, $C_2^3:D_4^2$ x 2, $C_2^6.C_2^3$ x 2, $C_2^6:D_4$ x 2, $(C_2^3\times D_4).C_2^3$ x 2, $C_2^4.C_2^5$ x 2, $C_2^5.C_2^3.C_2$ x 2, $C_2^6.C_8$ x 2, $C_2^2.C_2^6.C_2$, $C_2^2.C_2^6.C_2$, $C_2^6.C_2^3$, $C_2^4\wr C_2$, $C_2^5.\OD_{16}$, $C_2^5:\OD_{16}$, $C_2^3.C_2^4.C_2^2$, $C_2^5.(C_2\times D_4)$, $C_2^6.D_4$, $C_2^3.(C_2^3\times C_4).C_2$, $C_2^5.C_2^3.C_2$, $C_2^4.C_2^4.C_2$, $(C_2^2\times D_4).C_2^4$, $C_2^7:C_4$, $C_2^3.C_2^5.C_2$, $C_2^3.C_2^5.C_2$, $C_2^5.\OD_{16}$, $C_2^6:C_8$, $C_2^6:D_4$

Order 256: $C_2^8$

Order 81: $C_3^4$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 331776: $C_2^8.C_3^4:\OD_{16}$ ($C_1$)

Order 165888: $C_2^8.C_3^2:F_9$ ($C_2$) x 2, $A_4^2.(C_2\times A_4^2:C_4)$ ($C_2$)

Order 82944: $C_2^8.C_3^3.D_6$ ($C_4$), $A_4^2:(A_4^2:C_4)$ ($C_2^2$), $A_4^2:(A_4^2:C_4)$ ($C_4$)

Order 41472: $C_2^8.C_3^4.C_2$ ($C_2\times C_4$)

Order 20736: $C_2^8.C_3^4$ ($\OD_{16}$)

Order 256: $C_2^8$ ($C_3^4:\OD_{16}$)

Order 1: $C_1$ ($C_2^8.C_3^4:\OD_{16}$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 331776: $C_2^8.C_3^4:\OD_{16}$ ($C_1$)

Order 165888: $C_2^8.C_3^2:F_9$ ($C_2$), $A_4^2.(C_2\times A_4^2:C_4)$ ($C_2$)

Order 82944: $C_2^8.C_3^3.D_6$ ($C_4$), $A_4^2:(A_4^2:C_4)$ ($C_2^2$), $A_4^2:(A_4^2:C_4)$ ($C_4$)

Order 41472: $C_2^8.C_3^4.C_2$ ($C_2\times C_4$)

Order 20736: $C_2^8.C_3^4$ ($\OD_{16}$)

Order 256: $C_2^8$ ($C_3^4:\OD_{16}$)

Order 1: $C_1$ ($C_2^8.C_3^4:\OD_{16}$)

Series

Derived series

$C_2^8.C_3^4:\OD_{16}$

$\rhd$

$C_2^8.C_3^4.C_2$

$\rhd$

$C_2^8.C_3^4$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Chief series

$C_2^8.C_3^4:\OD_{16}$

$\rhd$

$A_4^2.(C_2\times A_4^2:C_4)$

$\rhd$

$A_4^2:(A_4^2:C_4)$

$\rhd$

$C_2^8.C_3^4.C_2$

$\rhd$

$C_2^8.C_3^4$

$\rhd$

$C_2^8$

$\rhd$

$C_1$

Lower central series

$C_2^8.C_3^4:\OD_{16}$

$\rhd$

$C_2^8.C_3^4.C_2$

$\rhd$

$C_2^8.C_3^4$

Upper central series

$C_1$

Supergroups

This group is a maximal subgroup of 7 larger groups in the database.

This group is a maximal quotient of 7 larger groups in the database.

Character theory

Complex character table

See the $73 \times 73$ character table. Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $65 \times 65$ rational character table.