Abstract group 2064384.p: $C_2^8.(S_3\times C_2^3:\GL(3,2))$

• Label: 2064384.p
• Order: \( 2^{15} \cdot 3^{2} \cdot 7 \)
• Exponent: \( 2^{3} \cdot 3 \cdot 7 \)
• Nilpotent: no
• Solvable: no
• $\card{G^{\mathrm{ab}}}$: \( 2 \)
• $\card{Z(G)}$: \( 2^{2} \)
• $\card{\Aut(G)}$: \( 2^{16} \cdot 3^{3} \cdot 7 \)
• $\card{\mathrm{Out}(G)}$: \( 2^{3} \cdot 3 \)
• Perm deg.: $35$
• Trans deg.: $96$
• Rank: $2$

Group information

Description:	$C_2^8.(S_3\times C_2^3:\GL(3,2))$
Order:	\(2064384\)\(\medspace = 2^{15} \cdot 3^{2} \cdot 7 \)
Exponent:	\(168\)\(\medspace = 2^{3} \cdot 3 \cdot 7 \)
Automorphism group:	$C_2^8.C_3^2.C_2^5.\PSL(2,7)$, of order \(12386304\)\(\medspace = 2^{16} \cdot 3^{3} \cdot 7 \)
Composition factors:	$C_2$ x 12, $C_3$, $\PSL(2,7)$
Derived length:	$2$

This group is nonabelian and nonsolvable.

Group statistics

Order	1	2	3	4	6	7	8	12	14	21	24	42
Elements	1	16255	10754	379008	427454	24576	129024	447552	368640	49152	64512	147456	2064384
Conjugacy classes	1	53	3	90	57	2	6	54	14	2	3	6	291
Divisions	1	53	3	90	57	1	6	54	7	1	3	3	279
Autjugacy classes	1	21	3	31	23	2	2	20	4	2	1	2	112

Minimal presentations

Permutation degree:	$35$
Transitive degree:	$96$
Rank:	$2$
Inequivalent generating pairs:	not computed

Minimal degrees of faithful linear representations

	Over $\mathbb{C}$	Over $\mathbb{R}$	Over $\mathbb{Q}$
Irreducible	none	not computed	none
Arbitrary	not computed	not computed	not computed

Constructions

Permutation group:	Degree $35$ $\langle(18,19)(20,21)(22,23)(24,25)(26,28)(27,29)(30,32)(31,33)(34,35), (1,2)(3,6) \!\cdots\! \rangle$
Direct product:	not computed
Semidirect product:	not computed
Trans. wreath product:	not isomorphic to a non-trivial transitive wreath product
Possibly split product:	$C_2^8$ . $(S_3\times C_2^3:\GL(3,2))$	$C_2^4$ . $(C_6.C_2^7:\GL(3,2))$	$(C_2^8.C_2^3.\PSL(2,7))$ . $S_3$	$S_3$ . $(C_2^5.C_2^6.\PSL(2,7))$	all 27
Aut. group:	$\Aut(C_{12}:C_2^4)$	$\Aut(D_6:C_2^4)$

Elements of the group are displayed as permutations of degree 35.

Homology

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

Subgroups

There are 63 normal subgroups (17 characteristic).

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

Special subgroups

Center:	$Z \simeq$ $C_2^2$	$G/Z \simeq$ $(C_2^3\times S_3).C_2^6.\PSL(2,7)$
Commutator:	$G' \simeq$ $(C_2^4\times C_6).C_2^6.\PSL(2,7)$	$G/G' \simeq$ $C_2$
Frattini:	$\Phi \simeq$ $C_2^2$	$G/\Phi \simeq$ $(C_2^3\times S_3).C_2^6.\PSL(2,7)$
Fitting:	$\operatorname{Fit} \simeq$ $C_3:(C_2^5.C_2^6)$	$G/\operatorname{Fit} \simeq$ $C_2\times \GL(3,2)$
Radical:	$R \simeq$ $C_3:(C_2^5.C_2^6.C_2)$	$G/R \simeq$ $\PSL(2,7)$
Socle:	$\operatorname{soc} \simeq$ $C_2\times C_6$	$G/\operatorname{soc} \simeq$ $C_2^{10}.\PSL(2,7)$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^5.C_2^6.C_2^4$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3^2$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7$

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 96 (order at least 21504) are shown, as well as all normal subgroups of any index.

Order 2064384: $C_2^8.(S_3\times C_2^3:\GL(3,2))$

Order 1032192: $(C_2^4\times C_6).C_2^6.\PSL(2,7)$

Order 688128: $C_2^6.C_2^6.\PSL(2,7)$

Order 344064: $C_2^8.C_2^3.\PSL(2,7)$

Order 294912: $C_2^5.C_2^6.D_6^2$, $C_2^8.C_2^5:S_3^2$

Order 258048: $(C_2^5\times S_3).C_2^3.\PSL(2,7)$ x 3, $D_6.C_2^7:\GL(3,2)$ x 3, $C_2^5.C_2^6.C_{21}.C_6$, $(C_2^7\times C_6).C_2.\PSL(2,7)$, $(C_2^6\times C_6).C_2^2.\PSL(2,7)$

Order 147456: $C_2^5.C_2^6.S_3^2.C_2$ x 3, $C_2^5.C_2^6.C_3.D_6.C_2$ x 3, $C_2^5.C_2^6.C_3^2.C_2^3$ x 3, $C_2^9.(D_6\times S_4)$ x 3, $C_2^8.C_2^4.S_3^2$, $C_2^8.C_2^4.S_3^2$, $C_2^5.C_2^6.C_3.D_6.C_2$, $C_2^6.C_2^6.S_3^2$, $C_2^6.C_2^6.C_3.D_6$, $C_2^6.C_2^6.S_3^2$, $C_2^8.C_2^4.(C_6\times S_3)$, $(C_2^2\times C_6).C_2^6.A_4.C_2^3$, $(C_2^5\times C_6).C_2^6.C_6.C_2$, $C_2^5.C_2^6.C_3^2.C_2^3$

Order 129024: $(C_2^5\times C_6).C_2^2.\PSL(2,7)$ x 6, $C_6.C_2^7:\GL(3,2)$ x 6, $(C_2\times C_6).C_2^6.\PSL(2,7)$ x 3, $C_6.C_2^7:\GL(3,2)$ x 3, $C_2^5.C_2^6.C_{21}.C_3$, $(C_2^7\times C_6).\PSL(2,7)$, $(C_2^7\times C_6).\PSL(2,7)$

Order 98304: $C_2^5.C_2^6.C_6.C_2^3$, $C_3:(C_2^7.C_2^5.C_2^3)$, $C_2^8.C_2^2\wr S_3$

Order 86016: $C_2^5.C_2^4.\PSL(2,7)$ x 3, $C_2^2.C_2^7:\GL(3,2)$ x 3, $C_2^9.\PSL(2,7)$, $(C_2\times C_6).C_2^6.F_8.C_2$, $C_2^8.F_8.C_6$, $C_2^9.\PSL(2,7)$

Order 73728: $C_2^5.C_2^6.S_3^2$ x 6, $C_2^5.C_2^4.D_6^2$ x 3, $C_2^5.C_2^6.S_3^2$ x 3, $(C_2^4\times C_6).C_2^6.C_6.C_2$ x 3, $C_2^5.C_2^6.S_3^2$ x 3, $C_2^4.C_2^4.D_6^2.C_2$ x 3, $C_2^5.C_2^6.S_3^2$ x 3, $C_2^5.C_2^6.(C_6\times S_3)$ x 3, $C_2^5.C_2^6.C_3.D_6$ x 3, $(C_2^4\times C_6).C_2^6.C_6.C_2$ x 3, $C_2^5.C_2^6.(C_6\times S_3)$ x 3, $C_2^5.C_2^6.C_3.D_6$ x 3, $(C_2^5\times C_6).C_2^6.C_6$, $C_2^4.C_2^4.D_6^2.C_2$, $C_2^6.C_2^6.C_3.C_6$, $C_2^5.C_2^6.C_6^2$, $C_2^6.C_2^6.C_3.C_6$, $C_2^8.C_2^4.C_3.C_6$, $C_2^6.C_2^6.C_3.S_3$, $C_2^6.C_2^6.C_3.S_3$, $C_2^6.A_4.D_6.C_2^3$, $(C_2^5\times C_6).C_2^6.C_6$, $C_2^5.C_2^6.S_3^2$, $C_2^6.C_6^2.C_2^5$

Order 64512: $(C_2^3\times S_3).C_2^3.\PSL(2,7)$ x 4, $(C_2^6\times C_6).\PSL(2,7)$ x 3, $C_6.C_2^6.\PSL(2,7)$ x 3

Order 49152: $C_2^5.C_2^6.D_6.C_2$ x 5, $C_2^5.C_2^6:S_4$ x 5, $C_3:(C_2^5.C_2^6.C_2^3)$ x 3, $C_3:(C_2^5.C_2^6.C_2^3)$ x 3, $C_3:(C_2^6.C_2^5.C_2^3)$ x 3, $C_3:(C_2^7.C_2^4.C_2^3)$ x 3, $C_3:(C_2^7.C_2.C_2^6)$ x 3, $C_3:(C_2^7.D_4^2.C_2)$ x 3, $C_3:(C_2^5.C_2^4.C_2^5)$ x 3, $C_2^5.C_2^6.D_6.C_2$ x 3, $C_3:(C_2^5.C_2^4.C_2^5)$ x 3, $C_3:(C_2^5.C_2^6.C_2^3)$ x 3, $C_3:(C_2^7.C_2^4.C_2^3)$ x 3, $C_2^5.C_2^6.D_6.C_2$ x 3, $C_3:(C_2^7.C_2^5.C_2^2)$ x 3, $C_3:(C_2^6.C_2^5.C_2^3)$ x 3, $C_2^9.A_4.C_2^3$ x 3, $C_3:(C_2^5.C_2^4.C_2^5)$ x 3, $C_3:(C_2^5.C_2^5.C_2^4)$ x 3, $C_3:(C_2^5.C_2^4.C_2^5)$ x 3, $C_3:(C_2^5.C_2^6.C_2^3)$ x 3, $C_3:(C_2^8.C_2.C_2^5)$, $C_3:(C_2^5.C_2^6.C_2^3)$, $C_2^6.C_2^6.D_6$, $C_3:(C_2^5.C_2^6.C_2^3)$, $C_2^5.C_2^6.C_6.C_2^2$, $C_3:(C_2^8.C_2^6)$, $C_3:(C_2^5.C_2^5.C_2^4)$, $C_2^6.C_2^6.C_6.C_2$, $C_3:(C_2^5.C_2^6.C_2^3)$, $C_3:(C_2^6.C_2^6.C_2^2)$, $C_3:(C_2^7.C_2^6.C_2)$, $C_3:(C_2^6.C_2^4.C_2^4)$, $C_3:(C_2^5.C_2^6.C_2^3)$, $C_3:(C_2^5.C_2^6.C_2^3)$, $C_2^5.C_2^6.C_6.C_2^2$, $C_2^6.C_2^6.D_6$, $C_3:(C_2^5.C_2^6.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^4)$, $C_3:(C_2^5.C_2^5.C_2^4)$, $C_3:(C_2^5.C_2^5.C_2^4)$

Order 43008: $C_2.C_2^7:\GL(3,2)$ x 9, $C_2^8.\GL(3,2)$ x 7, $C_2^5.C_2^3.\PSL(2,7)$ x 3, $C_2^5.C_2^6.C_7.C_3$ x 2, $C_2^8.F_8.C_3$, $C_2^8:\PSL(2,7)$

Order 36864: $C_2^7.(D_6\times S_4)$ x 9, $C_2.C_2^6.A_4.D_6.C_2$ x 6, $C_2^4.C_2^4.D_6^2$ x 5, $C_2^6.D_6^2.C_2^2$ x 3, $C_2^4.C_2^4.D_6^2$ x 3, $C_2^5.C_2^4.S_3^2.C_2$ x 3, $C_2^5.C_2^4.C_3.D_6.C_2$ x 3, $C_2^6.D_6^2.C_2^2$ x 3, $C_2^4.C_2^4.D_6^2$ x 3, $C_2^6.A_4.D_{12}.C_2$ x 3, $C_2.C_2^6.A_4.D_6.C_2$ x 3, $C_2^5.C_2^4.C_6^2.C_2$ x 3, $C_2^4.C_2^4.C_6^2.C_2^2$ x 3, $C_2^6.A_4.D_6.C_2^2$ x 3, $C_2^4.C_2^4.D_6^2$ x 3, $C_2^5.C_2^4.C_3.D_6.C_2$ x 3, $C_2.C_2^6.(C_2\times S_3\times S_4)$ x 3, $C_2^6.C_6^2.C_2^4$ x 3, $(C_2^6\times C_6).A_4.C_2^3$ x 3, $C_2.C_2^6.A_4.D_6.C_2$ x 3, $C_2^6.C_2^4.S_3^2$ x 3, $C_2^6.C_6^2.C_2^4$ x 3, $C_2^6.A_4.D_6.C_2^2$ x 3, $C_2^5.C_2^6.C_3.C_6$ x 3, $C_2^6.C_6^2.C_2^4$ x 3, $C_2^4.C_2^4.C_3.D_6.C_2^2$ x 3, $(C_2^4\times C_6).C_2^4.C_6.C_2^2$ x 3, $C_2^6.C_2^2.D_6^2$ x 3, $C_2.C_2^6.(C_2\times S_3\times S_4)$ x 3, $C_2^5.C_2^4.C_6^2.C_2$ x 3, $C_2^5.C_2^4.C_3.D_6.C_2$ x 3, $C_2^6.C_2^2.D_6^2$ x 3, $(C_2^4\times C_6).C_2^6.C_6$ x 3, $C_2^4.C_2^4.D_6^2$ x 3, $C_2^6.C_6^2.C_2^4$ x 3, $C_2^5.C_2^4.C_3^2.C_2^3$ x 3, $C_2.C_2^6.A_4.D_6.C_2$ x 3, $C_2^4.C_2^4.D_6^2$ x 3, $C_2^4.C_2^4.D_6^2$ x 3, $C_2^5.C_2^6.C_3.S_3$ x 3, $(C_2^5\times C_6).C_2^4.C_6.C_2$ x 3, $C_2^5.C_2^6.C_3.C_6$ x 3, $C_2^4.C_2^4.D_6^2$ x 3, $C_2^4.C_2^4.D_6^2$ x 3, $C_2^4.C_2^4.D_6^2$ x 3, $C_2^5.C_2^4.C_3^2.C_2^3$ x 3, $C_2^6.(C_6\times A_4).C_2^3$ x 3, $C_2^6.C_6^2.C_2^4$ x 3, $C_2^6.A_4.D_{12}.C_2$ x 3, $C_2^6.C_2^2.D_6^2$ x 3, $C_2^6.D_6^2.C_2^2$ x 3, $(C_2^5\times C_6).C_2^4.C_6.C_2$ x 3, $C_2^6.C_2^2.D_6^2$ x 3, $C_2^6.(C_6\times A_4).C_2^3$ x 3, $C_2.C_2^6.A_4.D_6.C_2$ x 3, $C_2^5.C_2^4.C_3.D_6.C_2$ x 3, $C_2^6.C_2^2.D_6^2$ x 3, $C_2^6.C_2^2.D_6^2$ x 3, $C_2^6.A_4.C_6.C_2^3$ x 3, $C_2^6.A_4.C_6.C_2^3$ x 3, $C_2^6.A_4.C_6.C_2^3$ x 3, $C_2^6.A_4.D_6.C_2^2$ x 3, $C_2^7.(D_6\times S_4)$ x 3, $C_2^6.C_2^6.C_3^2$, $C_2^6.C_2^2.D_6^2$, $(C_2^6\times C_6).A_4.C_2^3$, $C_2^6.A_4.C_6.C_2^3$, $C_2^4.C_2^4.C_6^2.C_2^2$, $C_2^6.D_6^2.C_2^2$, $(C_2^4\times C_6).C_2^6.C_6$, $C_2^4.C_2^4.D_6^2$, $C_2^5.C_2^6.C_3.C_6$, $(C_2\times C_6).C_2^6.C_6.C_2^3$, $(C_2^7\times C_6).C_6.C_2^3$, $C_2^4.C_2^4.C_6^2.C_2^2$, $C_2^4.C_2^4.D_6^2$, $(C_2^7\times C_6).A_4.C_2^2$, $C_2^6.C_2^2.D_6^2$, $C_2^6.A_4.D_6.C_2^2$, $C_2^6.C_6^2.C_2^4$, $(C_2^5\times C_6).A_4.C_2^4$, $C_2^6.C_2^2.D_6^2$, $C_2^6.C_2^2.D_6^2$, $C_2^4.C_2^4.D_6^2$, $C_2^6.C_6^2.C_2^4$, $C_2^6.C_6^2.C_2^4$, $C_2^5.C_2^6.C_3.S_3$, $C_2^6.C_6^2.C_2^4$, $C_2^6.C_6^2.C_2^4$, $C_2^6.C_2^4.S_3^2$

Order 32768: $C_2^5.C_2^6.C_2^4$

Order 32256: $(D_6\times C_2^4).\GL(3,2)$ x 7, $(C_2^5\times S_3).\PSL(2,7)$ x 6, $C_2^4.F_8.(C_6\times S_3)$ x 3, $(C_2^4\times S_3).\SL(2,7)$ x 3, $C_2^6.C_{21}.C_6.C_2^2$, $C_3\times C_2^6.\PSL(2,7)$

Order 28672: $C_2^5.C_2^6.C_{14}$

Order 24576: $C_2^5.C_2^6.D_6$ x 12, $C_3:(C_2^6.C_2^4.C_2^3)$ x 12, $C_2^5.C_2^6.D_6$ x 12, $C_3:(C_2^6.C_2^6.C_2)$ x 6, $C_3:(C_2^4.C_2^6.C_2^3)$ x 6, $C_3:(C_2^7.C_2.C_2^5)$ x 6, $C_3:(C_2^5.C_2^6.C_2^2)$ x 6, $C_3:(C_2^5.C_2^5.C_2^3)$ x 6, $C_3:(C_2^7.C_2.C_2^5)$ x 6, $C_3:(C_2^5.C_2^4.C_2^4)$ x 6, $C_3:(C_2^7.D_4^2)$ x 6, $C_3:(C_2^5.C_2^6.C_2^2)$ x 6, $C_3:(C_2^5.C_2^4.C_2^4)$ x 6, $C_3:(C_2^6.D_4^2.C_2)$ x 6, $C_3:(C_2^6.C_2^4.C_2^3)$ x 6, $C_3:(C_2^5.C_2^4.C_2^4)$ x 6, $C_3:(C_2^5.C_2^6.C_2^2)$ x 6, $C_3:(C_2^6.C_2^3.C_2^4)$ x 6, $C_3:(C_2^5.C_2^4.C_2^4)$ x 6, $C_3:(C_2^5.C_2^4.C_2^4)$ x 6, $C_3:(C_2^6.C_2^6.C_2)$ x 6, $C_3:(C_2^8.C_2^5)$ x 6, $C_3:(C_2^8.C_2^5)$ x 6, $C_3:(C_2^5.C_2^4.C_2^4)$ x 6, $C_3:(C_2^5.C_2^6.C_2^2)$ x 6, $C_3:(C_2^8.C_2^5)$ x 6, $C_3:(C_2^5.C_2^4.C_2^4)$ x 6, $C_3:(C_2^7.C_2^3.C_2^3)$ x 6, $C_3:(C_2^5.C_2^5.C_2^3)$ x 6, $C_3:(C_2^6.C_2^6.C_2)$ x 6, $C_3:(C_2^6.C_2^4.C_2^3)$ x 6, $C_3:(C_2^6.C_2^6.C_2)$ x 6, $C_3:(C_2^6.C_2^3.C_2^4)$ x 6, $C_3:(C_2^6.C_2^4.C_2^3)$ x 6, $C_3:(C_2^6.C_2^6.C_2)$ x 6, $C_3:(C_2^6.C_2^4.C_2^3)$ x 6, $C_3:(C_2^4.C_2^6.C_2^3)$ x 6, $C_3:(C_2^5.C_2^5.C_2^3)$ x 6, $C_3:(C_2^5.C_2^4.C_2^4)$ x 6, $C_3:(C_2^5.C_2^6.C_2^2)$ x 6, $C_3:(C_2^5.C_2^4.C_2^4)$ x 6, $C_3:(C_2^7.C_2.C_2^5)$ x 6, $C_3:(C_2^7.D_4^2)$ x 6, $C_3:(C_2^5.C_2^5.C_2^3)$ x 6, $C_3:(C_2^6.C_2^6.C_2)$ x 6, $C_3:(C_2^5.C_2^4.C_2^4)$ x 6, $C_3:(C_2^5.C_2^5.C_2^3)$ x 6, $C_3:(C_2^5.C_2^4.C_2^4)$ x 6, $C_3:(C_2^5.C_2^5.C_2^3)$ x 6, $C_3:(C_2^6.C_2^6.C_2)$ x 6, $C_3:(C_2^5.C_2^5.C_2^3)$ x 6, $C_3:(C_2^5.C_2^4.C_2^4)$ x 6, $C_3:(C_2^7.C_2.C_2^5)$ x 6, $C_3:(C_2^5.C_2^5.C_2^3)$ x 6, $C_3:(C_2^5.C_2^6.C_2^2)$ x 6, $C_3:(C_2^7.D_4^2)$ x 6, $C_3:(C_2^5.C_2^4.C_2^4)$ x 6, $C_3:(C_2^6.C_2^6.C_2)$ x 6, $C_3:(C_2^5.C_2^5.C_2^3)$ x 6, $C_3:(C_2^8.C_2^5)$ x 5, $C_3:(C_2^8.C_2^5)$ x 5, $C_3:(C_2^8.C_2^5)$ x 5, $C_2^5.C_2^6.C_6.C_2$ x 5, $C_3:(C_2^7.C_2^6)$ x 4, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^6.C_2^5.C_2^2)$ x 3, $C_3:(C_2^7.C_2^4.C_2^2)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2.C_2^6.C_2^6)$ x 3, $C_3:(C_2^6.C_2^5.C_2^2)$ x 3, $C_3:(C_2^8.C_2^5)$ x 3, $C_3:(C_2^7.C_2^6)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^7.C_2^4.C_2^2)$ x 3, $C_3:(C_2^5.C_2^6.C_2^2)$ x 3, $C_3:(C_2^5.C_2^4.C_2^4)$ x 3, $C_3:(C_2^5.C_2^6.C_2^2)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^7.C_2.C_2^5)$ x 3, $C_3:(C_2^5.C_2^6.C_2^2)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^6.C_2.C_2^6)$ x 3, $C_3:(C_2.C_2^6.C_2^6)$ x 3, $C_3:(C_2^6.D_4^2.C_2)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_2^7.C_2^4.D_6$ x 3, $C_3:(C_2^7.C_2^4.C_2^2)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^8.C_2.C_2^4)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^7.C_2^6)$ x 3, $C_3:(C_2^7.C_2^6)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^7.C_4.C_2^4)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^6.C_2.C_2^6)$ x 3, $C_3:(C_2^6.C_2^5.C_2^2)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^7.C_2^6)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^7.C_2^6)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^5.C_2^6.C_2^2)$ x 3, $C_3:(C_2^7.C_2.C_2^5)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_2^5.C_2^6.C_6.C_2$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^7.C_2^6)$ x 3, $C_3:(C_2^5.C_2^6.C_2^2)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^8.C_2.C_2^4)$ x 3, $C_2^5.C_2^6.D_6$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^8.C_2.C_2^4)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_2^6.A_4.C_2.C_2^4$ x 3, $C_3:(C_2^6.C_2.C_2^6)$ x 3, $C_3:(C_2^4.D_4.C_2^6)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^6.C_2^5.C_2^2)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^5.C_2^6.C_2^2)$ x 3, $C_3:(C_2^5.C_2^6.C_2^2)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^5.C_2^6.C_2^2)$ x 3, $C_2^5.C_2^4.C_6.C_2^3$ x 3, $C_2^6.C_2^6.S_3$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^7.C_2^6)$ x 3, $C_3:(C_2^5.C_2^6.C_2^2)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^7.C_2^6)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^7.C_2^3.C_2^3)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^7.C_2^6)$ x 3, $C_3:(C_2^5.C_2^6.C_2^2)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^5.C_2^6.C_2^2)$ x 3, $C_2^7.C_2^4.D_6$ x 3, $C_3:(C_2^5.C_2^6.C_2^2)$ x 3, $C_3:(C_2^7.C_2^3.C_2^3)$ x 3, $C_3:(C_2^5.C_2^6.C_2^2)$ x 3, $C_3:(C_2^7.C_2^6)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^5.C_2^6.C_2^2)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^6.C_2^5.C_2^2)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^5.C_2^6.C_2^2)$ x 3, $C_3:(C_2^7.C_2^3.C_2^3)$ x 3, $C_3:(C_2^7.C_2.C_2^5)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^6.C_2^5.C_2^2)$ x 3, $C_3:(C_2^7.C_2.C_2^5)$ x 3, $C_3:(C_2^6.C_2^3.C_2^4)$ x 3, $C_3:(C_2^5.C_2^6.C_2^2)$ x 3, $C_3:(C_2^5.C_2^6.C_2^2)$ x 3, $C_3:(C_2^6.C_2^3.C_2^4)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^6.C_2^2.C_2^5)$ x 3, $C_3:(C_2^6.C_2^6.C_2)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^5.C_2^4.C_2^4)$ x 3, $C_2^6.C_2^6.C_6$ x 3, $C_3:(C_2^7.C_2^4.C_2^2)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^6.C_2^5.C_2^2)$ x 3, $C_3:(C_2^6.C_2^2.C_2^5)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^5.C_2^6.C_2^2)$ x 3, $C_3:(C_2^6.D_4^2.C_2)$ x 3, $C_3:(C_2^5.C_2^6.C_2^2)$ x 3, $C_3:(C_2.C_2^6.C_2^6)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_3:(C_2^6.C_2^5.C_2^2)$ x 3, $C_3:(C_2^8.C_2.C_2^4)$ x 3, $C_3:(C_2^6.C_2^4.C_2^3)$ x 3, $C_3:(C_2^7.C_2^4.C_2^2)$ x 3, $C_3:(C_2^5.C_2^5.C_2^3)$ x 3, $C_2^6.C_2^6.S_3$ x 2, $C_3:(C_2^6.C_2^6.C_2)$ x 2, $C_3:(C_2^6.C_2^6.C_2)$ x 2, $C_3:(C_2^6.C_2.C_2^6)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^8.C_2^5)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_2^6.C_2^6.C_6$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_2^7.C_2^4.D_6$, $C_3:(C_2^8.C_2.C_2^4)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_2^6.A_4.C_2^5$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^8.C_2^5)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_2^4.C_2^6.S_4$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^8.C_2^5)$, $C_3:(C_2^8.C_2.C_2^4)$, $C_2^4.C_2^4.C_6.C_2^4$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^8.C_2.C_2^4)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^8.C_2^4.C_2)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_2^7.(D_4\times S_4)$

Order 21504: $C_2^7:\GL(3,2)$ x 7, $C_2^4:C_2^3:\GL(3,2)$ x 3

Order 12288: $C_3:(C_2^5.C_2^6.C_2)$

Order 6144: $C_3:(C_2^5.C_2^6)$

Order 2048: $C_2^5.C_2^6$

Order 1536: $C_2\times D_4.C_2^4\times S_3$ x 3, $C_2^8\times S_3$

Order 768: $C_6\times C_2.C_2^6$ x 3, $C_2^7\times C_6$

Order 256: $C_2^4:C_2^4$ x 3, $C_2^8$

Order 192: $D_6\times C_2^4$ x 4

Order 96: $C_2^3\times D_6$ x 6, $C_2^4\times C_6$ x 4

Order 48: $C_2^3\times C_6$ x 3

Order 32: $C_2^5$ x 4

Order 24: $C_2\times D_6$

Order 16: $C_2^4$ x 3

Order 12: $D_6$ x 6, $C_2\times C_6$

Order 9: $C_3^2$

Order 7: $C_7$

Order 6: $S_3$ x 4, $C_6$ x 3

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$ x 3

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 96 (order at least 21504) are shown, as well as all normal subgroups of any index.

Order 2064384: $C_2^8.(S_3\times C_2^3:\GL(3,2))$

Order 1032192: $(C_2^4\times C_6).C_2^6.\PSL(2,7)$

Order 688128: $C_2^6.C_2^6.\PSL(2,7)$

Order 344064: $C_2^8.C_2^3.\PSL(2,7)$

Order 294912: $C_2^5.C_2^6.D_6^2$, $C_2^8.C_2^5:S_3^2$

Order 258048: $(C_2^5\times S_3).C_2^3.\PSL(2,7)$, $C_2^5.C_2^6.C_{21}.C_6$, $(C_2^7\times C_6).C_2.\PSL(2,7)$, $(C_2^6\times C_6).C_2^2.\PSL(2,7)$, $D_6.C_2^7:\GL(3,2)$

Order 147456: $C_2^5.C_2^6.S_3^2.C_2$, $C_2^5.C_2^6.C_3.D_6.C_2$, $C_2^5.C_2^6.C_3^2.C_2^3$, $C_2^8.C_2^4.S_3^2$, $C_2^8.C_2^4.S_3^2$, $C_2^5.C_2^6.C_3.D_6.C_2$, $C_2^6.C_2^6.S_3^2$, $C_2^6.C_2^6.C_3.D_6$, $C_2^6.C_2^6.S_3^2$, $C_2^8.C_2^4.(C_6\times S_3)$, $(C_2^2\times C_6).C_2^6.A_4.C_2^3$, $(C_2^5\times C_6).C_2^6.C_6.C_2$, $C_2^5.C_2^6.C_3^2.C_2^3$, $C_2^9.(D_6\times S_4)$

Order 129024: $C_2^5.C_2^6.C_{21}.C_3$, $(C_2^5\times C_6).C_2^2.\PSL(2,7)$, $(C_2^7\times C_6).\PSL(2,7)$, $(C_2^7\times C_6).\PSL(2,7)$, $(C_2\times C_6).C_2^6.\PSL(2,7)$, $C_6.C_2^7:\GL(3,2)$, $C_6.C_2^7:\GL(3,2)$

Order 98304: $C_2^5.C_2^6.C_6.C_2^3$, $C_3:(C_2^7.C_2^5.C_2^3)$, $C_2^8.C_2^2\wr S_3$

Order 86016: $C_2^9.\PSL(2,7)$, $C_2^5.C_2^4.\PSL(2,7)$, $(C_2\times C_6).C_2^6.F_8.C_2$, $C_2^8.F_8.C_6$, $C_2^9.\PSL(2,7)$, $C_2^2.C_2^7:\GL(3,2)$

Order 73728: $C_2^5.C_2^4.D_6^2$, $(C_2^5\times C_6).C_2^6.C_6$, $C_2^4.C_2^4.D_6^2.C_2$, $C_2^6.C_2^6.C_3.C_6$, $C_2^5.C_2^6.C_6^2$, $C_2^5.C_2^6.S_3^2$, $C_2^6.C_2^6.C_3.C_6$, $C_2^8.C_2^4.C_3.C_6$, $C_2^6.C_2^6.C_3.S_3$, $C_2^5.C_2^6.S_3^2$, $(C_2^4\times C_6).C_2^6.C_6.C_2$, $C_2^6.C_2^6.C_3.S_3$, $C_2^5.C_2^6.S_3^2$, $C_2^4.C_2^4.D_6^2.C_2$, $C_2^5.C_2^6.S_3^2$, $C_2^6.A_4.D_6.C_2^3$, $C_2^5.C_2^6.(C_6\times S_3)$, $C_2^5.C_2^6.C_3.D_6$, $(C_2^5\times C_6).C_2^6.C_6$, $C_2^5.C_2^6.S_3^2$, $(C_2^4\times C_6).C_2^6.C_6.C_2$, $C_2^5.C_2^6.(C_6\times S_3)$, $C_2^5.C_2^6.C_3.D_6$, $C_2^6.C_6^2.C_2^5$

Order 64512: $(C_2^6\times C_6).\PSL(2,7)$, $C_6.C_2^6.\PSL(2,7)$, $(C_2^3\times S_3).C_2^3.\PSL(2,7)$

Order 49152: $C_2^5.C_2^6:S_4$ x 5, $C_2^5.C_2^6.D_6.C_2$ x 3, $C_3:(C_2^5.C_2^6.C_2^3)$, $C_3:(C_2^8.C_2.C_2^5)$, $C_3:(C_2^5.C_2^6.C_2^3)$, $C_3:(C_2^6.C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^6.C_2^3)$, $C_2^6.C_2^6.D_6$, $C_3:(C_2^7.C_2^4.C_2^3)$, $C_3:(C_2^7.C_2.C_2^6)$, $C_3:(C_2^5.C_2^6.C_2^3)$, $C_3:(C_2^7.D_4^2.C_2)$, $C_3:(C_2^5.C_2^4.C_2^5)$, $C_2^5.C_2^6.D_6.C_2$, $C_2^5.C_2^6.C_6.C_2^2$, $C_3:(C_2^8.C_2^6)$, $C_3:(C_2^5.C_2^4.C_2^5)$, $C_3:(C_2^5.C_2^5.C_2^4)$, $C_3:(C_2^5.C_2^6.C_2^3)$, $C_2^6.C_2^6.C_6.C_2$, $C_3:(C_2^5.C_2^6.C_2^3)$, $C_3:(C_2^7.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2^6.C_2^2)$, $C_3:(C_2^7.C_2^6.C_2)$, $C_3:(C_2^6.C_2^4.C_2^4)$, $C_3:(C_2^5.C_2^6.C_2^3)$, $C_2^5.C_2^6.D_6.C_2$, $C_3:(C_2^7.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^6.C_2^3)$, $C_2^9.A_4.C_2^3$, $C_2^5.C_2^6.C_6.C_2^2$, $C_3:(C_2^5.C_2^4.C_2^5)$, $C_2^6.C_2^6.D_6$, $C_3:(C_2^5.C_2^6.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^4)$, $C_3:(C_2^5.C_2^5.C_2^4)$, $C_3:(C_2^5.C_2^5.C_2^4)$, $C_3:(C_2^5.C_2^5.C_2^4)$, $C_3:(C_2^5.C_2^4.C_2^5)$, $C_3:(C_2^5.C_2^6.C_2^3)$

Order 43008: $C_2^5.C_2^6.C_7.C_3$ x 2, $C_2.C_2^7:\GL(3,2)$ x 2, $C_2^8.\GL(3,2)$ x 2, $C_2^8.F_8.C_3$, $C_2^5.C_2^3.\PSL(2,7)$, $C_2^8:\PSL(2,7)$

Order 36864: $C_2^4.C_2^4.D_6^2$ x 3, $C_2^7.(D_6\times S_4)$ x 3, $C_2.C_2^6.A_4.D_6.C_2$ x 2, $C_2^6.D_6^2.C_2^2$, $C_2^4.C_2^4.D_6^2$, $C_2^6.C_2^6.C_3^2$, $C_2^5.C_2^4.S_3^2.C_2$, $C_2^5.C_2^4.C_3.D_6.C_2$, $C_2^6.C_2^2.D_6^2$, $C_2^6.D_6^2.C_2^2$, $(C_2^6\times C_6).A_4.C_2^3$, $C_2^4.C_2^4.D_6^2$, $C_2^6.A_4.D_{12}.C_2$, $C_2.C_2^6.A_4.D_6.C_2$, $C_2^5.C_2^4.C_6^2.C_2$, $C_2^6.A_4.C_6.C_2^3$, $C_2^4.C_2^4.C_6^2.C_2^2$, $C_2^4.C_2^4.C_6^2.C_2^2$, $C_2^6.D_6^2.C_2^2$, $C_2^6.A_4.D_6.C_2^2$, $(C_2^4\times C_6).C_2^6.C_6$, $C_2^4.C_2^4.D_6^2$, $C_2^5.C_2^4.C_3.D_6.C_2$, $C_2^4.C_2^4.D_6^2$, $C_2.C_2^6.(C_2\times S_3\times S_4)$, $C_2^5.C_2^6.C_3.C_6$, $(C_2\times C_6).C_2^6.C_6.C_2^3$, $C_2^6.C_6^2.C_2^4$, $(C_2^7\times C_6).C_6.C_2^3$, $(C_2^6\times C_6).A_4.C_2^3$, $C_2.C_2^6.A_4.D_6.C_2$, $C_2^6.C_2^4.S_3^2$, $C_2^6.C_6^2.C_2^4$, $C_2^6.A_4.D_6.C_2^2$, $C_2^5.C_2^6.C_3.C_6$, $C_2^6.C_6^2.C_2^4$, $C_2^4.C_2^4.C_3.D_6.C_2^2$, $(C_2^4\times C_6).C_2^4.C_6.C_2^2$, $C_2^4.C_2^4.C_6^2.C_2^2$, $C_2^4.C_2^4.D_6^2$, $C_2^6.C_2^2.D_6^2$, $(C_2^7\times C_6).A_4.C_2^2$, $C_2.C_2^6.(C_2\times S_3\times S_4)$, $C_2^6.C_2^2.D_6^2$, $C_2^6.A_4.D_6.C_2^2$, $C_2^5.C_2^4.C_6^2.C_2$, $C_2^5.C_2^4.C_3.D_6.C_2$, $C_2^6.C_6^2.C_2^4$, $C_2^6.C_2^2.D_6^2$, $(C_2^5\times C_6).A_4.C_2^4$, $C_2^6.C_2^2.D_6^2$, $(C_2^4\times C_6).C_2^6.C_6$, $C_2^6.C_2^2.D_6^2$, $C_2^4.C_2^4.D_6^2$, $C_2^6.C_6^2.C_2^4$, $C_2^5.C_2^4.C_3^2.C_2^3$, $C_2.C_2^6.A_4.D_6.C_2$, $C_2^4.C_2^4.D_6^2$, $C_2^4.C_2^4.D_6^2$, $C_2^5.C_2^6.C_3.S_3$, $(C_2^5\times C_6).C_2^4.C_6.C_2$, $C_2^5.C_2^6.C_3.C_6$, $C_2^4.C_2^4.D_6^2$, $C_2^4.C_2^4.D_6^2$, $C_2^4.C_2^4.D_6^2$, $C_2^4.C_2^4.D_6^2$, $C_2^5.C_2^4.C_3^2.C_2^3$, $C_2^6.(C_6\times A_4).C_2^3$, $C_2^6.C_6^2.C_2^4$, $C_2^6.A_4.D_{12}.C_2$, $C_2^6.C_6^2.C_2^4$, $C_2^6.C_6^2.C_2^4$, $C_2^6.C_2^2.D_6^2$, $C_2^6.D_6^2.C_2^2$, $(C_2^5\times C_6).C_2^4.C_6.C_2$, $C_2^6.C_2^2.D_6^2$, $C_2^6.(C_6\times A_4).C_2^3$, $C_2.C_2^6.A_4.D_6.C_2$, $C_2^5.C_2^4.C_3.D_6.C_2$, $C_2^5.C_2^6.C_3.S_3$, $C_2^6.C_6^2.C_2^4$, $C_2^6.C_2^2.D_6^2$, $C_2^6.C_2^2.D_6^2$, $C_2^6.A_4.C_6.C_2^3$, $C_2^6.A_4.C_6.C_2^3$, $C_2^6.C_6^2.C_2^4$, $C_2^6.A_4.C_6.C_2^3$, $C_2^6.C_2^4.S_3^2$, $C_2^6.A_4.D_6.C_2^2$, $C_2^7.(D_6\times S_4)$

Order 32768: $C_2^5.C_2^6.C_2^4$

Order 32256: $(D_6\times C_2^4).\GL(3,2)$ x 3, $(C_2^5\times S_3).\PSL(2,7)$ x 2, $C_2^6.C_{21}.C_6.C_2^2$, $C_2^4.F_8.(C_6\times S_3)$, $C_3\times C_2^6.\PSL(2,7)$, $(C_2^4\times S_3).\SL(2,7)$

Order 28672: $C_2^5.C_2^6.C_{14}$

Order 24576: $C_2^5.C_2^6.D_6$ x 4, $C_3:(C_2^8.C_2^5)$ x 3, $C_3:(C_2^8.C_2^5)$ x 3, $C_3:(C_2^8.C_2^5)$ x 3, $C_2^6.C_2^6.S_3$ x 3, $C_2^5.C_2^6.D_6$ x 3, $C_2^6.C_2^6.C_6$ x 3, $C_2^5.C_2^6.C_6.C_2$ x 3, $C_3:(C_2^6.C_2^6.C_2)$ x 2, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2^6.C_2^3.C_2^4)$ x 2, $C_3:(C_2^6.C_2^6.C_2)$ x 2, $C_3:(C_2^8.C_2^5)$ x 2, $C_3:(C_2^8.C_2^5)$ x 2, $C_3:(C_2^8.C_2^5)$ x 2, $C_3:(C_2^6.C_2^6.C_2)$ x 2, $C_3:(C_2^6.C_2^6.C_2)$ x 2, $C_3:(C_2^6.C_2^4.C_2^3)$ x 2, $C_3:(C_2^6.C_2^4.C_2^3)$ x 2, $C_3:(C_2^6.C_2^6.C_2)$ x 2, $C_3:(C_2^5.C_2^4.C_2^4)$ x 2, $C_2^6.C_2^6.S_3$ x 2, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2^6.C_2^6.C_2)$ x 2, $C_3:(C_2^6.C_2^6.C_2)$ x 2, $C_3:(C_2^7.C_2^6)$ x 2, $C_3:(C_2^6.C_2^6.C_2)$ x 2, $C_3:(C_2^5.C_2^5.C_2^3)$ x 2, $C_3:(C_2^6.C_2^6.C_2)$ x 2, $C_3:(C_2^7.C_2.C_2^5)$ x 2, $C_3:(C_2^6.C_2^6.C_2)$ x 2, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^7.C_2^4.C_2^2)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^4.C_2^6.C_2^3)$, $C_3:(C_2.C_2^6.C_2^6)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^8.C_2^5)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^7.C_2^4.C_2^2)$, $C_3:(C_2^6.C_2.C_2^6)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^5.C_2^4.C_2^4)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^6.C_2.C_2^6)$, $C_3:(C_2.C_2^6.C_2^6)$, $C_3:(C_2^6.D_4^2.C_2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_2^7.C_2^4.D_6$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^8.C_2^5)$, $C_3:(C_2^7.C_2^4.C_2^2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_2^6.C_2^6.C_6$, $C_3:(C_2^5.C_2^4.C_2^4)$, $C_3:(C_2^8.C_2.C_2^4)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^7.D_4^2)$, $C_3:(C_2^7.C_2^6)$, $C_2^7.C_2^4.D_6$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^7.C_4.C_2^4)$, $C_3:(C_2^8.C_2.C_2^4)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^5.C_2^4.C_2^4)$, $C_3:(C_2^6.C_2.C_2^6)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_2^6.A_4.C_2^5$, $C_3:(C_2^6.D_4^2.C_2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^4.C_2^4)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^5.C_2^4.C_2^4)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_2^5.C_2^6.C_6.C_2$, $C_3:(C_2^5.C_2^4.C_2^4)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^8.C_2^5)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^4.C_2^4)$, $C_3:(C_2^8.C_2.C_2^4)$, $C_2^5.C_2^6.D_6$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^8.C_2.C_2^4)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_2^6.A_4.C_2.C_2^4$, $C_3:(C_2^6.C_2.C_2^6)$, $C_3:(C_2^4.D_4.C_2^6)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^4.C_2^4)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^7.C_2^3.C_2^3)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_2^5.C_2^4.C_6.C_2^3$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^6.C_2^3.C_2^4)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^4.C_2^6.C_2^3)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^7.C_2^3.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_2^7.C_2^4.D_6$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^7.C_2^3.C_2^3)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_2^4.C_2^6.S_4$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^5.C_2^4.C_2^4)$, $C_3:(C_2^7.D_4^2)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^4.C_2^4)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^8.C_2^5)$, $C_3:(C_2^8.C_2.C_2^4)$, $C_3:(C_2^5.C_2^4.C_2^4)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_2^4.C_2^4.C_6.C_2^4$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^7.C_2^3.C_2^3)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^7.C_2.C_2^5)$, $C_3:(C_2^6.C_2^3.C_2^4)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^8.C_2.C_2^4)$, $C_3:(C_2^5.C_2^4.C_2^4)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^6.C_2^3.C_2^4)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^6.C_2^2.C_2^5)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^4.C_2^4)$, $C_3:(C_2^7.C_2^6)$, $C_3:(C_2^7.C_2^4.C_2^2)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^6.C_2^2.C_2^5)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^6.D_4^2.C_2)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2.C_2^6.C_2^6)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^7.D_4^2)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^5.C_2^6.C_2^2)$, $C_3:(C_2^6.C_2^5.C_2^2)$, $C_3:(C_2^8.C_2.C_2^4)$, $C_3:(C_2^5.C_2^4.C_2^4)$, $C_3:(C_2^6.C_2^4.C_2^3)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_3:(C_2^8.C_2^4.C_2)$, $C_3:(C_2^7.C_2^4.C_2^2)$, $C_3:(C_2^6.C_2^6.C_2)$, $C_3:(C_2^5.C_2^5.C_2^3)$, $C_2^7.(D_4\times S_4)$

Order 21504: $C_2^7:\GL(3,2)$ x 2, $C_2^4:C_2^3:\GL(3,2)$

Order 12288: $C_3:(C_2^5.C_2^6.C_2)$

Order 6144: $C_3:(C_2^5.C_2^6)$

Order 2048: $C_2^5.C_2^6$

Order 1536: $C_2^8\times S_3$, $C_2\times D_4.C_2^4\times S_3$

Order 768: $C_2^7\times C_6$, $C_6\times C_2.C_2^6$

Order 256: $C_2^8$, $C_2^4:C_2^4$

Order 192: $D_6\times C_2^4$ x 2

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

Order 48: $C_2^3\times C_6$

Order 32: $C_2^5$ x 2

Order 24: $C_2\times D_6$

Order 16: $C_2^4$

Order 12: $C_2\times C_6$, $D_6$

Order 9: $C_3^2$

Order 7: $C_7$

Order 6: $C_6$, $S_3$

Order 4: $C_2^2$

Order 3: $C_3$

Order 2: $C_2$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 2064384: $C_2^8.(S_3\times C_2^3:\GL(3,2))$ ($C_1$)

Order 1032192: $(C_2^4\times C_6).C_2^6.\PSL(2,7)$ ($C_2$)

Order 344064: $C_2^8.C_2^3.\PSL(2,7)$ ($S_3$)

Order 12288: $C_3:(C_2^5.C_2^6.C_2)$ ($\PSL(2,7)$)

Order 6144: $C_3:(C_2^5.C_2^6)$ ($C_2\times \GL(3,2)$)

Order 2048: $C_2^5.C_2^6$ ($S_3\times \GL(3,2)$)

Order 1536: $C_2\times D_4.C_2^4\times S_3$ ($C_2^3:\GL(3,2)$) x 3, $C_2^8\times S_3$ ($C_2^3:\GL(3,2)$)

Order 768: $C_6\times C_2.C_2^6$ ($C_2^4:\GL(3,2)$) x 3, $C_2^7\times C_6$ ($C_2^4:\GL(3,2)$)

Order 256: $C_2^4:C_2^4$ ($(C_2^3\times S_3).\PSL(2,7)$) x 3, $C_2^8$ ($S_3\times C_2^3:\GL(3,2)$)

Order 192: $D_6\times C_2^4$ ($C_2^6:\GL(3,2)$) x 3, $D_6\times C_2^4$ ($C_2^6:\GL(3,2)$)

Order 96: $C_2^3\times D_6$ ($C_2.C_2^6.\PSL(2,7)$) x 6, $C_2^4\times C_6$ ($C_2^7:\PSL(2,7)$) x 3, $C_2^4\times C_6$ ($C_2^7:\GL(3,2)$)

Order 48: $C_2^3\times C_6$ ($C_2.C_2^7:\GL(3,2)$) x 3

Order 32: $C_2^5$ ($(C_2^5\times C_6).C_2.\PSL(2,7)$) x 3, $C_2^5$ ($S_3.C_2^6.\PSL(2,7)$)

Order 24: $C_2\times D_6$ ($C_2^9.\PSL(2,7)$)

Order 16: $C_2^4$ ($C_6.C_2^7:\GL(3,2)$) x 3

Order 12: $D_6$ ($C_2^7.C_2^3.\PSL(2,7)$) x 6, $C_2\times C_6$ ($C_2^{10}.\PSL(2,7)$)

Order 6: $S_3$ ($C_2^5.C_2^6.\PSL(2,7)$) x 4, $C_6$ ($C_2^7.C_2^4.\PSL(2,7)$) x 3

Order 4: $C_2^2$ ($(C_2^3\times S_3).C_2^6.\PSL(2,7)$)

Order 3: $C_3$ ($C_2^6.C_2^6.\PSL(2,7)$)

Order 2: $C_2$ ($(C_2^4\times S_3).C_2^6.\PSL(2,7)$) x 3

Order 1: $C_1$ ($C_2^8.(S_3\times C_2^3:\GL(3,2))$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 2064384: $C_2^8.(S_3\times C_2^3:\GL(3,2))$ ($C_1$)

Order 1032192: $(C_2^4\times C_6).C_2^6.\PSL(2,7)$ ($C_2$)

Order 344064: $C_2^8.C_2^3.\PSL(2,7)$ ($S_3$)

Order 12288: $C_3:(C_2^5.C_2^6.C_2)$ ($\PSL(2,7)$)

Order 6144: $C_3:(C_2^5.C_2^6)$ ($C_2\times \GL(3,2)$)

Order 2048: $C_2^5.C_2^6$ ($S_3\times \GL(3,2)$)

Order 1536: $C_2^8\times S_3$ ($C_2^3:\GL(3,2)$), $C_2\times D_4.C_2^4\times S_3$ ($C_2^3:\GL(3,2)$)

Order 768: $C_2^7\times C_6$ ($C_2^4:\GL(3,2)$), $C_6\times C_2.C_2^6$ ($C_2^4:\GL(3,2)$)

Order 256: $C_2^8$ ($S_3\times C_2^3:\GL(3,2)$), $C_2^4:C_2^4$ ($(C_2^3\times S_3).\PSL(2,7)$)

Order 192: $D_6\times C_2^4$ ($C_2^6:\GL(3,2)$), $D_6\times C_2^4$ ($C_2^6:\GL(3,2)$)

Order 96: $C_2^4\times C_6$ ($C_2^7:\PSL(2,7)$), $C_2^4\times C_6$ ($C_2^7:\GL(3,2)$), $C_2^3\times D_6$ ($C_2.C_2^6.\PSL(2,7)$)

Order 48: $C_2^3\times C_6$ ($C_2.C_2^7:\GL(3,2)$)

Order 32: $C_2^5$ ($(C_2^5\times C_6).C_2.\PSL(2,7)$), $C_2^5$ ($S_3.C_2^6.\PSL(2,7)$)

Order 24: $C_2\times D_6$ ($C_2^9.\PSL(2,7)$)

Order 16: $C_2^4$ ($C_6.C_2^7:\GL(3,2)$)

Order 12: $C_2\times C_6$ ($C_2^{10}.\PSL(2,7)$), $D_6$ ($C_2^7.C_2^3.\PSL(2,7)$)

Order 6: $C_6$ ($C_2^7.C_2^4.\PSL(2,7)$), $S_3$ ($C_2^5.C_2^6.\PSL(2,7)$)

Order 4: $C_2^2$ ($(C_2^3\times S_3).C_2^6.\PSL(2,7)$)

Order 3: $C_3$ ($C_2^6.C_2^6.\PSL(2,7)$)

Order 2: $C_2$ ($(C_2^4\times S_3).C_2^6.\PSL(2,7)$)

Order 1: $C_1$ ($C_2^8.(S_3\times C_2^3:\GL(3,2))$)

Series

Derived series

$C_2^8.(S_3\times C_2^3:\GL(3,2))$

$\rhd$

$(C_2^4\times C_6).C_2^6.\PSL(2,7)$

$\rhd$

$C_2^8.C_2^3.\PSL(2,7)$

Chief series

$C_2^8.(S_3\times C_2^3:\GL(3,2))$

$\rhd$

$C_3:(C_2^5.C_2^6.C_2)$

$\rhd$

$C_3:(C_2^5.C_2^6)$

$\rhd$

$C_6\times C_2.C_2^6$

$\rhd$

$C_2^4\times C_6$

$\rhd$

$C_2\times C_6$

$\rhd$

$C_6$

$\rhd$

$C_3$

$\rhd$

$C_1$

Lower central series

$C_2^8.(S_3\times C_2^3:\GL(3,2))$

$\rhd$

$(C_2^4\times C_6).C_2^6.\PSL(2,7)$

Upper central series

$C_1$

$\lhd$

$C_2^2$

Supergroups

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

This group is a maximal quotient of 0 larger groups in the database.

Character theory

Complex character table

The $291 \times 291$ character table is not available for this group.

Rational character table

The $279 \times 279$ rational character table is not available for this group.