Abstract group 393216.es: $C_2^8.C_2^4:\GL(2,\mathbb{Z}/4)$

• Label: 393216.es
• Order: \( 2^{17} \cdot 3 \)
• Exponent: \( 2^{3} \cdot 3 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2^{2} \)
• $\card{Z(G)}$: \( 2 \)
• $\card{\Aut(G)}$: \( 2^{23} \cdot 3 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{7} \)
• Perm deg.: $24$
• Trans deg.: $24$
• Rank: $2$

Group information

Description:	$C_2^8.C_2^4:\GL(2,\mathbb{Z}/4)$
Order:	\(393216\)\(\medspace = 2^{17} \cdot 3 \)
Exponent:	\(24\)\(\medspace = 2^{3} \cdot 3 \)
Automorphism group:	Group of order \(25165824\)\(\medspace = 2^{23} \cdot 3 \)
Composition factors:	$C_2$ x 17, $C_3$
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
Elements	1	5695	8192	133568	57344	122880	65536	393216
Conjugacy classes	1	73	1	214	4	52	5	350
Divisions	1	73	1	184	3	36	3	301
Autjugacy classes	1	37	1	72	3	14	3	131

Minimal presentations

Permutation degree:	$24$
Transitive degree:	$24$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

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

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k, l, m \mid b^{6}=c^{2}=e^{2}=f^{4}= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,21)(2,22)(3,19,4,20)(5,18,6,17)(7,14,9,15)(8,13,10,16)(11,24,12,23), (1,24,10,19,6,16)(2,23,9,20,5,15)(3,13,11,22,7,17)(4,14,12,21,8,18)\rangle$
Transitive group:	24T19764				more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$(C_2^8.C_2^6)$ . $S_4$ (2)	$(C_2^9.C_2^6)$ . $D_6$	$C_2^{10}$ . $(C_2^4:S_4)$	$(C_2^8.C_2^6:A_4)$ . $C_2$	all 29

Elements of the group are displayed as permutations of degree 24.

Homology

Abelianization:	$C_{2}^{2} $
Schur multiplier:	$C_{2}^{7}$
Commutator length:	$2$

Subgroups

There are 51 normal subgroups (33 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2$	$G/Z \simeq$ $C_2^9.C_2^4.C_3.D_4$
Commutator:	$G' \simeq$ $C_2^8.C_2^6.C_6$	$G/G' \simeq$ $C_2^2$
Frattini:	$\Phi \simeq$ $C_2^8.C_2^3$	$G/\Phi \simeq$ $C_2^3:S_4$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^8.C_2^6.C_2^2$	$G/\operatorname{Fit} \simeq$ $S_3$
Radical:	$R \simeq$ $C_2^8.C_2^4:\GL(2,\mathbb{Z}/4)$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^3$	$G/\operatorname{soc} \simeq$ $C_2^7.C_2\wr S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^9.C_2^6.C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3$

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 16 (order at least 24576) are shown, as well as all normal subgroups of any index.

Order 393216: $C_2^8.C_2^4:\GL(2,\mathbb{Z}/4)$

Order 196608: $C_2^8.C_2^6.D_6$, $C_2^6.(C_2^7.S_4)$, $C_2^8.C_2^6:A_4$

Order 131072: $C_2^9.C_2^6.C_2^2$

Order 98304: $C_2^{10}.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^8.C_2^6.S_3$, $C_2^8.C_2^6.C_6$, $C_4^3.C_2\wr S_4$, $C_2^6.C_2^8:C_6$

Order 65536: $C_2^8.C_2^6.C_2^2$ x 2, $C_2^9.C_2^4.C_2^3$ x 2, $C_2^9.C_2^5.C_2^2$ x 2, $C_2^9.C_2^5.C_2^2$ x 2, $C_2^8.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^3$, $C_2^8.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^3$, $C_2^7.C_2^6.C_2^3$, $C_2^6.C_2^6.C_2^4$, $C_2^7.C_2^6.C_2^3$

Order 49152: $C_2^{10}.A_4:C_4$ x 4, $C_2^3.(C_2^8.S_4)$ x 3, $C_2^3.C_2^6.(C_4\times S_4)$ x 2, $C_2^6.C_2^6.D_6$ x 2, $C_2^8.C_2^4.D_6$ x 2, $C_4^3.C_2^5:S_4$ x 2, $C_2^6.C_2\wr A_4$ x 2, $C_2^8.C_2^6.C_3$, $C_2^6.C_2^6.D_6$, $C_2^8.C_2^4.D_6$, $C_2^7.C_2^4.C_{12}.C_2$, $C_4^3.C_2\wr A_4$, $C_2^6.C_2^6:A_4$

Order 32768: $C_2^9.C_2^4.C_2^2$ x 4, $C_2^8.C_2^5.C_2^2$ x 4, $C_2^7.C_2^6.C_2^2$ x 4, $C_2^7.C_2^6.C_2^2$ x 4, $C_2^7.C_2^6.C_2^2$ x 4, $C_2^7.C_2^6.C_2^2$ x 3, $C_2^6.C_2^6.C_2^3$ x 3, $C_2^7.C_2^6.C_2^2$ x 3, $C_2^7.C_2^6.C_2^2$ x 3, $C_2^6.C_2^6.C_2^3$ x 3, $C_2^7.C_2^6.C_2^2$ x 3, $C_2^8.C_2^5.C_2^2$ x 2, $C_2^7.C_2^6.C_2^2$ x 2, $C_2^8.C_2^4.C_2^3$ x 2, $C_2^6.C_2^6.C_2^3$ x 2, $C_2^6.C_2^6.C_2^3$ x 2, $C_2^9.C_2^4.C_2^2$ x 2, $C_2^9.C_2^4.C_2^2$ x 2, $C_2^8.C_2^6.C_2$ x 2, $C_2^9.C_2^4.C_2^2$ x 2, $C_2^4.C_2^6.C_2^4.C_2$ x 2, $C_2^8.C_2^6.C_2$ x 2, $C_2^7.C_2^6.C_2^2$ x 2, $C_2^8.C_2^6.C_2$ x 2, $C_2^7.C_2^6.C_2^2$ x 2, $C_2^7.C_2^6.C_2^2$ x 2, $C_2^7.C_2^6.C_2^2$ x 2, $C_2^9.C_2^4.C_2^2$ x 2, $C_2^9.C_2^4.C_2^2$ x 2, $C_2^6.C_2^6.C_2^3$ x 2, $C_2^9.C_2^4.C_2^2$ x 2, $C_2^9.C_2^4.C_2^2$ x 2, $C_2^5.C_2^6.C_2^4$ x 2, $C_2^7.C_2^6.C_2^2$ x 2, $C_2^6.C_2^6.C_2^3$ x 2, $C_2^9.C_2^4.C_2^2$ x 2, $C_2^7.C_2^6.C_2^2$ x 2, $C_2^7.C_2^6.C_2^2$ x 2, $C_2^9.C_2^4.C_2^2$ x 2, $C_2^7.C_2^5.C_2^3$ x 2, $C_2^7.C_2^6.C_2^2$ x 2, $C_2^9.C_2^4.C_2^2$ x 2, $C_2^7.C_2^6.C_2^2$ x 2, $C_2^7.C_2^6.C_2^2$ x 2, $C_2^9.C_2^5.C_2$ x 2, $C_2^6.C_2^6.C_2^3$ x 2, $C_2^6.C_2^6.C_2^3$ x 2, $C_2^8.C_2^6.C_2$ x 2, $C_2^7.C_2^6.C_2^2$ x 2, $C_2^9.C_2^6$, $C_2^6.C_2^6.C_2^3$, $C_2^6.C_2^6.C_2^3$, $C_2^5.C_2^6.C_2^4$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^6.C_2^6.C_2^3$, $C_2^7.C_2^6.C_2^2$, $C_2^6.C_2^6.C_2^3$, $C_2^5.C_2^6.C_2^4$, $C_2^6.C_2^6.C_2^3$, $C_2^8.C_2^6.C_2$, $C_2^6.C_2^6.C_2^3$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^5.C_2^6.C_2^4$, $C_2^7.C_2^6.C_2^2$, $C_2^8.C_2^6.C_2$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^6.C_2^6.C_2^3$, $C_2^6.C_2^6.C_2^3$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$

Order 24576: $C_2^8.C_2^4.C_6$ x 6, $C_2^8.C_2^4.C_6$ x 5, $C_2^8.C_2^4.C_6$ x 4, $C_2^{10}.S_4$ x 4, $C_2^8.C_2^4.C_6$ x 3, $C_2^8.C_2^4.C_6$ x 3, $C_2^8.C_2^4.S_3$ x 2, $C_2^6.C_2^6.S_3$ x 2, $C_2^5.C_2^6.C_6.C_2$ x 2, $C_2^9.A_4.C_2^2$ x 2, $C_2^{10}.A_4.C_2$ x 2, $C_2^8.C_2^4.C_6$ x 2, $C_2^5.C_2^6.C_{12}$ x 2, $C_2^8.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^3.C_2^8:A_4$ x 2, $C_2^8.\GL(2,\mathbb{Z}/4)$ x 2, $C_2^5.C_2\wr A_4$ x 2, $C_2^7.C_2^4.C_6.C_2$, $C_2^5.C_2^6.D_6$, $C_2^8.C_2^4.C_6$, $C_2^8.C_2^4.S_3$, $C_2^7.C_2^4.C_{12}$, $C_2^5.C_2\wr A_4$, $(C_2^2\times C_4^3).\GL(2,\mathbb{Z}/4)$, $(C_2^2\times C_4^3).\GL(2,\mathbb{Z}/4)$, $C_2^5.C_2^6:A_4$, $C_2^8.\GL(2,\mathbb{Z}/4)$

Order 16384: $C_2^8.C_2^6$ x 2, $C_2^9.C_2^5$, $C_2^6.C_2^6.C_2^2$

Order 8192: $C_2^7.C_2^6$ x 2, $C_2^7.C_2^6$

Order 4096: $C_2^8.C_2^4$ x 2, $C_2^8.C_2^4$, $C_2^8.C_2^4$

Order 2048: $C_2^6.C_2^5$ x 2, $C_2^7.C_2^4$ x 2, $C_2^8.C_2^3$ x 2, $C_2\times C_4^5$

Order 1024: $C_2^6.C_2^4$ x 2, $C_2^8.C_2^2$ x 2, $C_2^{10}$ x 2, $C_2^2\times C_4^4$

Order 512: $C_2^8.C_2$ x 2, $C_2^3\times C_4^3$

Order 256: $C_2^8$ x 2, $C_2^4\times C_4^2$

Order 128: $C_2^5\times C_4$

Order 64: $C_2^4\times C_4$ x 2, $C_2^6$

Order 32: $C_2^5$

Order 16: $C_2^4$ x 2

Order 8: $C_2^3$

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

All subgroups of index up to 16 (order at least 24576) are shown, as well as all normal subgroups of any index.

Order 393216: $C_2^8.C_2^4:\GL(2,\mathbb{Z}/4)$

Order 196608: $C_2^8.C_2^6.D_6$, $C_2^6.(C_2^7.S_4)$, $C_2^8.C_2^6:A_4$

Order 131072: $C_2^9.C_2^6.C_2^2$

Order 98304: $C_2^8.C_2^6.S_3$, $C_2^8.C_2^6.C_6$, $C_4^3.C_2\wr S_4$, $C_2^6.C_2^8:C_6$, $C_2^{10}.\GL(2,\mathbb{Z}/4)$

Order 65536: $C_2^8.C_2^6.C_2^2$, $C_2^8.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^3$, $C_2^8.C_2^6.C_2^2$, $C_2^9.C_2^4.C_2^3$, $C_2^7.C_2^6.C_2^3$, $C_2^9.C_2^5.C_2^2$, $C_2^7.C_2^6.C_2^3$, $C_2^6.C_2^6.C_2^4$, $C_2^7.C_2^6.C_2^3$, $C_2^9.C_2^5.C_2^2$

Order 49152: $C_2^3.(C_2^8.S_4)$ x 2, $C_4^3.C_2^5:S_4$ x 2, $C_2^{10}.A_4:C_4$ x 2, $C_2^8.C_2^6.C_3$, $C_2^6.C_2^6.D_6$, $C_2^3.C_2^6.(C_4\times S_4)$, $C_2^8.C_2^4.D_6$, $C_2^6.C_2^6.D_6$, $C_2^8.C_2^4.D_6$, $C_2^7.C_2^4.C_{12}.C_2$, $C_4^3.C_2\wr A_4$, $C_2^6.C_2^6:A_4$, $C_2^6.C_2\wr A_4$

Order 32768: $C_2^7.C_2^6.C_2^2$ x 3, $C_2^7.C_2^6.C_2^2$ x 2, $C_2^7.C_2^6.C_2^2$ x 2, $C_2^6.C_2^6.C_2^3$ x 2, $C_2^7.C_2^6.C_2^2$ x 2, $C_2^7.C_2^6.C_2^2$ x 2, $C_2^7.C_2^6.C_2^2$ x 2, $C_2^6.C_2^6.C_2^3$ x 2, $C_2^7.C_2^6.C_2^2$ x 2, $C_2^7.C_2^6.C_2^2$ x 2, $C_2^7.C_2^6.C_2^2$ x 2, $C_2^8.C_2^5.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^9.C_2^6$, $C_2^6.C_2^6.C_2^3$, $C_2^8.C_2^4.C_2^3$, $C_2^6.C_2^6.C_2^3$, $C_2^6.C_2^6.C_2^3$, $C_2^6.C_2^6.C_2^3$, $C_2^9.C_2^4.C_2^2$, $C_2^5.C_2^6.C_2^4$, $C_2^9.C_2^4.C_2^2$, $C_2^8.C_2^6.C_2$, $C_2^9.C_2^4.C_2^2$, $C_2^4.C_2^6.C_2^4.C_2$, $C_2^8.C_2^6.C_2$, $C_2^7.C_2^6.C_2^2$, $C_2^8.C_2^6.C_2$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^6.C_2^6.C_2^3$, $C_2^9.C_2^4.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^6.C_2^6.C_2^3$, $C_2^5.C_2^6.C_2^4$, $C_2^9.C_2^4.C_2^2$, $C_2^9.C_2^4.C_2^2$, $C_2^6.C_2^6.C_2^3$, $C_2^6.C_2^6.C_2^3$, $C_2^9.C_2^4.C_2^2$, $C_2^9.C_2^4.C_2^2$, $C_2^5.C_2^6.C_2^4$, $C_2^8.C_2^5.C_2^2$, $C_2^8.C_2^6.C_2$, $C_2^6.C_2^6.C_2^3$, $C_2^6.C_2^6.C_2^3$, $C_2^9.C_2^4.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^9.C_2^4.C_2^2$, $C_2^7.C_2^5.C_2^3$, $C_2^5.C_2^6.C_2^4$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^9.C_2^4.C_2^2$, $C_2^8.C_2^6.C_2$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^6.C_2^6.C_2^3$, $C_2^7.C_2^6.C_2^2$, $C_2^9.C_2^5.C_2$, $C_2^6.C_2^6.C_2^3$, $C_2^6.C_2^6.C_2^3$, $C_2^6.C_2^6.C_2^3$, $C_2^8.C_2^6.C_2$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$, $C_2^7.C_2^6.C_2^2$

Order 24576: $C_2^8.C_2^4.C_6$ x 4, $C_2^8.C_2^4.C_6$ x 3, $C_2^8.C_2^4.C_6$ x 2, $C_2^8.C_2^4.C_6$ x 2, $C_2^8.C_2^4.C_6$ x 2, $C_2^8.C_2^4.C_6$ x 2, $C_2^{10}.S_4$ x 2, $C_2^8.C_2^4.S_3$, $C_2^6.C_2^6.S_3$, $C_2^5.C_2^6.C_6.C_2$, $C_2^9.A_4.C_2^2$, $C_2^{10}.A_4.C_2$, $C_2^7.C_2^4.C_6.C_2$, $C_2^5.C_2^6.D_6$, $C_2^8.C_2^4.C_6$, $C_2^5.C_2^6.C_{12}$, $C_2^8.C_2^4.S_3$, $C_2^7.C_2^4.C_{12}$, $C_2^5.C_2\wr A_4$, $(C_2^2\times C_4^3).\GL(2,\mathbb{Z}/4)$, $(C_2^2\times C_4^3).\GL(2,\mathbb{Z}/4)$, $C_2^8.\GL(2,\mathbb{Z}/4)$, $C_2^5.C_2^6:A_4$, $C_2^3.C_2^8:A_4$, $C_2^8.\GL(2,\mathbb{Z}/4)$, $C_2^8.\GL(2,\mathbb{Z}/4)$, $C_2^5.C_2\wr A_4$

Order 16384: $C_2^9.C_2^5$, $C_2^6.C_2^6.C_2^2$, $C_2^8.C_2^6$

Order 8192: $C_2^7.C_2^6$, $C_2^7.C_2^6$

Order 4096: $C_2^8.C_2^4$, $C_2^8.C_2^4$, $C_2^8.C_2^4$

Order 2048: $C_2^8.C_2^3$ x 2, $C_2^6.C_2^5$, $C_2^7.C_2^4$, $C_2\times C_4^5$

Order 1024: $C_2^8.C_2^2$ x 2, $C_2^6.C_2^4$, $C_2^2\times C_4^4$, $C_2^{10}$

Order 512: $C_2^8.C_2$, $C_2^3\times C_4^3$

Order 256: $C_2^8$, $C_2^4\times C_4^2$

Order 128: $C_2^5\times C_4$

Order 64: $C_2^4\times C_4$ x 2, $C_2^6$

Order 32: $C_2^5$

Order 16: $C_2^4$ x 2

Order 8: $C_2^3$

Order 4: $C_2^2$ x 2

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 393216: $C_2^8.C_2^4:\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 196608: $C_2^8.C_2^6.D_6$ ($C_2$), $C_2^6.(C_2^7.S_4)$ ($C_2$), $C_2^8.C_2^6:A_4$ ($C_2$)

Order 98304: $C_2^8.C_2^6.C_6$ ($C_2^2$)

Order 65536: $C_2^8.C_2^6.C_2^2$ ($S_3$)

Order 49152: $C_2^8.C_2^6.C_3$ ($D_4$)

Order 32768: $C_2^9.C_2^6$ ($D_6$)

Order 16384: $C_2^8.C_2^6$ ($S_4$) x 2, $C_2^9.C_2^5$ ($C_3:D_4$), $C_2^6.C_2^6.C_2^2$ ($S_4$)

Order 8192: $C_2^7.C_2^6$ ($C_2\times S_4$) x 2, $C_2^7.C_2^6$ ($C_2\times S_4$)

Order 4096: $C_2^8.C_2^4$ ($\GL(2,\mathbb{Z}/4)$) x 2, $C_2^8.C_2^4$ ($\GL(2,\mathbb{Z}/4)$), $C_2^8.C_2^4$ ($C_2^2:S_4$)

Order 2048: $C_2^6.C_2^5$ ($C_2^3:S_4$) x 2, $C_2^7.C_2^4$ ($C_2^3:S_4$) x 2, $C_2\times C_4^5$ ($C_2^3:S_4$), $C_2^8.C_2^3$ ($C_2^3:S_4$), $C_2^8.C_2^3$ ($C_2^3:S_4$)

Order 1024: $C_2^6.C_2^4$ ($C_2^4:S_4$) x 2, $C_2^{10}$ ($C_2^4:S_4$) x 2, $C_2^8.C_2^2$ ($C_2^4:S_4$), $C_2^8.C_2^2$ ($C_2\wr S_3$), $C_2^2\times C_4^4$ ($C_2^4:S_4$)

Order 512: $C_2^8.C_2$ ($C_2^5:S_4$) x 2, $C_2^3\times C_4^3$ ($C_2^5:S_4$)

Order 256: $C_2^8$ ($C_2^4:\GL(2,\mathbb{Z}/4)$) x 2, $C_2^4\times C_4^2$ ($C_2^4:\GL(2,\mathbb{Z}/4)$)

Order 128: $C_2^5\times C_4$ ($C_2^8:D_6$)

Order 64: $C_2^6$ ($C_2^6.\GL(2,\mathbb{Z}/4)$), $C_2^4\times C_4$ ($C_2^5.C_2^4.D_6$), $C_2^4\times C_4$ ($C_2^5.C_2^4.D_6$)

Order 32: $C_2^5$ ($C_2.C_2^6.A_4.D_4$)

Order 16: $C_2^4$ ($C_2^6.C_2^4.C_3.D_4$), $C_2^4$ ($C_2^4.C_2^6.C_3.D_4$)

Order 8: $C_2^3$ ($C_2^7.C_2\wr S_3$)

Order 4: $C_2^2$ ($C_4^4.C_2^4.C_3.D_4$), $C_2^2$ ($C_2^8.C_2^4.C_3.D_4$)

Order 2: $C_2$ ($C_2^9.C_2^4.C_3.D_4$)

Order 1: $C_1$ ($C_2^8.C_2^4:\GL(2,\mathbb{Z}/4)$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 393216: $C_2^8.C_2^4:\GL(2,\mathbb{Z}/4)$ ($C_1$)

Order 196608: $C_2^8.C_2^6.D_6$ ($C_2$), $C_2^6.(C_2^7.S_4)$ ($C_2$), $C_2^8.C_2^6:A_4$ ($C_2$)

Order 98304: $C_2^8.C_2^6.C_6$ ($C_2^2$)

Order 65536: $C_2^8.C_2^6.C_2^2$ ($S_3$)

Order 49152: $C_2^8.C_2^6.C_3$ ($D_4$)

Order 32768: $C_2^9.C_2^6$ ($D_6$)

Order 16384: $C_2^9.C_2^5$ ($C_3:D_4$), $C_2^6.C_2^6.C_2^2$ ($S_4$), $C_2^8.C_2^6$ ($S_4$)

Order 8192: $C_2^7.C_2^6$ ($C_2\times S_4$), $C_2^7.C_2^6$ ($C_2\times S_4$)

Order 4096: $C_2^8.C_2^4$ ($\GL(2,\mathbb{Z}/4)$), $C_2^8.C_2^4$ ($C_2^2:S_4$), $C_2^8.C_2^4$ ($\GL(2,\mathbb{Z}/4)$)

Order 2048: $C_2^6.C_2^5$ ($C_2^3:S_4$), $C_2^7.C_2^4$ ($C_2^3:S_4$), $C_2\times C_4^5$ ($C_2^3:S_4$), $C_2^8.C_2^3$ ($C_2^3:S_4$), $C_2^8.C_2^3$ ($C_2^3:S_4$)

Order 1024: $C_2^6.C_2^4$ ($C_2^4:S_4$), $C_2^8.C_2^2$ ($C_2^4:S_4$), $C_2^8.C_2^2$ ($C_2\wr S_3$), $C_2^2\times C_4^4$ ($C_2^4:S_4$), $C_2^{10}$ ($C_2^4:S_4$)

Order 512: $C_2^8.C_2$ ($C_2^5:S_4$), $C_2^3\times C_4^3$ ($C_2^5:S_4$)

Order 256: $C_2^8$ ($C_2^4:\GL(2,\mathbb{Z}/4)$), $C_2^4\times C_4^2$ ($C_2^4:\GL(2,\mathbb{Z}/4)$)

Order 128: $C_2^5\times C_4$ ($C_2^8:D_6$)

Order 64: $C_2^6$ ($C_2^6.\GL(2,\mathbb{Z}/4)$), $C_2^4\times C_4$ ($C_2^5.C_2^4.D_6$), $C_2^4\times C_4$ ($C_2^5.C_2^4.D_6$)

Order 32: $C_2^5$ ($C_2.C_2^6.A_4.D_4$)

Order 16: $C_2^4$ ($C_2^6.C_2^4.C_3.D_4$), $C_2^4$ ($C_2^4.C_2^6.C_3.D_4$)

Order 8: $C_2^3$ ($C_2^7.C_2\wr S_3$)

Order 4: $C_2^2$ ($C_4^4.C_2^4.C_3.D_4$), $C_2^2$ ($C_2^8.C_2^4.C_3.D_4$)

Order 2: $C_2$ ($C_2^9.C_2^4.C_3.D_4$)

Order 1: $C_1$ ($C_2^8.C_2^4:\GL(2,\mathbb{Z}/4)$)

Series

Derived series

$C_2^8.C_2^4:\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_2^8.C_2^6.C_6$

$\rhd$

$C_2^9.C_2^5$

$\rhd$

$C_2^6$

$\rhd$

$C_1$

Chief series

$C_2^8.C_2^4:\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_2^8.C_2^6.D_6$

$\rhd$

$C_2^8.C_2^6.C_6$

$\rhd$

$C_2^8.C_2^6.C_3$

$\rhd$

$C_2^9.C_2^5$

$\rhd$

$C_2^8.C_2^4$

$\rhd$

$C_2^{10}$

$\rhd$

$C_2^8$

$\rhd$

$C_2^6$

$\rhd$

$C_2^4$

$\rhd$

$C_2^2$

$\rhd$

$C_2$

$\rhd$

$C_1$

Lower central series

$C_2^8.C_2^4:\GL(2,\mathbb{Z}/4)$

$\rhd$

$C_2^8.C_2^6.C_6$

$\rhd$

$C_2^8.C_2^6.C_3$

Upper central series

$C_1$

$\lhd$

$C_2$

$\lhd$

$C_2^2$

Supergroups

This group is a maximal subgroup of 5 larger groups in the database.

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

Character theory

Complex character table

The $350 \times 350$ character table is not available for this group.

Rational character table

The $301 \times 301$ rational character table is not available for this group.