Abstract group 28449792.b: $C_2^9.C_7^3:C_3\wr S_3$

• Label: 28449792.b
• Order: \( 2^{10} \cdot 3^{4} \cdot 7^{3} \)
• Exponent: \( 2^{2} \cdot 3^{2} \cdot 7 \)
• Nilpotent: no
• Solvable: yes
• $\card{G^{\mathrm{ab}}}$: \( 2 \cdot 3 \)
• $\card{Z(G)}$: \( 1 \)
• $\card{\Aut(G)}$: \( 2^{10} \cdot 3^{4} \cdot 7^{3} \)
• $\card{\mathrm{Out}(G)}$: \( 1 \)
• Perm deg.: $24$
• Trans deg.: $24$
• Rank: $2$

Group information

Description:	$C_2^9.C_7^3:C_3\wr S_3$
Order:	\(28449792\)\(\medspace = 2^{10} \cdot 3^{4} \cdot 7^{3} \)
Exponent:	\(252\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 7 \)
Automorphism group:	$C_2^9.C_7^3:C_3\wr S_3$, of order \(28449792\)\(\medspace = 2^{10} \cdot 3^{4} \cdot 7^{3} \)
Composition factors:	$C_2$ x 10, $C_3$ x 4, $C_7$ x 3
Derived length:	$5$

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

Group statistics

Order	1	2	3	4	6	7	9	12	14	18	21	28	42	84
Elements	1	4543	241640	28224	5380760	117648	3161088	3782016	1436400	3161088	3564288	169344	6048000	1354752	28449792
Conjugacy classes	1	5	10	2	60	9	2	16	17	2	18	2	44	4	192
Divisions	1	5	6	2	32	5	1	8	9	1	6	1	12	1	90
Autjugacy classes	1	5	10	2	60	9	2	16	17	2	18	2	44	4	192

Dimension	1	2	3	4	6	9	18	21	27	36	42	54	72	81	84	108	126	147	162	189	252	294	343	378	441	504	588	686	756	882	1029	1372	1764	2058	3528
Irr. complex chars.	6	3	12	0	1	12	6	18	16	0	9	5	0	4	0	0	18	18	0	12	0	9	6	3	12	0	0	3	0	6	12	0	0	1	0	192
Irr. rational chars.	2	3	0	1	7	0	2	2	0	3	9	5	1	0	4	4	0	2	2	0	1	9	2	3	0	4	4	3	3	2	0	1	3	7	1	90

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	21	21	21
Arbitrary	not computed	not computed	not computed

Constructions

Presentation:	${\langle a, b, c, d, e, f, g, h, i, j, k, l, m, n \mid b^{3}=d^{21}=e^{7}= \!\cdots\! \rangle}$
Permutation group:	Degree $24$ $\langle(1,20,9,8,19,14,6,24,12)(2,17,15)(3,22,16,4,18,13,5,21,10)(7,23,11), (1,22,6,24,4,23)(2,21,8,18,5,19)(3,17)(7,20)(9,12,14,10,11,16,15)\rangle$
Transitive group:	24T24313	42T3074	42T3075		more information
Direct product:	not isomorphic to a non-trivial direct product
Semidirect product:	not computed
Trans. wreath product:	not computed
Possibly split product:	$C_2^9$ . $(C_7^3:C_3\wr S_3)$	$(C_2^9.C_7^3.C_3^3)$ . $S_3$	$(C_2^9:(C_7^3:\He_3))$ . $C_6$	$(C_2^9.C_7^3:C_3^2:S_3)$ . $C_3$	all 8
Aut. group:	$\Aut(C_2^9:(C_7^3:\He_3))$	$\Aut(C_2^9.C_7^3:C_3^2:S_3)$	$\Aut(C_2^{10}:(C_7^3:\He_3))$	$\Aut(C_2^9:(C_7^3:C_3\wr C_3))$	all 5

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

Homology

Abelianization:	$C_{6} \simeq C_{2} \times C_{3}$
Schur multiplier:	$C_1$
Commutator length:	$1$

Subgroups

There are 10 normal subgroups, and all normal subgroups are characteristic.

Characteristic subgroups are shown in this color.

Special subgroups

Center:	$Z \simeq$ $C_1$	$G/Z \simeq$ $C_2^9.C_7^3:C_3\wr S_3$
Commutator:	$G' \simeq$ $C_2^9:(C_7^3:\He_3)$	$G/G' \simeq$ $C_6$
Frattini:	$\Phi \simeq$ $C_1$	$G/\Phi \simeq$ $C_2^9.C_7^3:C_3\wr S_3$
Fitting:	$\operatorname{Fit} \simeq$ $C_2^9$	$G/\operatorname{Fit} \simeq$ $C_7^3:C_3\wr S_3$
Radical:	$R \simeq$ $C_2^9.C_7^3:C_3\wr S_3$	$G/R \simeq$ $C_1$
Socle:	$\operatorname{soc} \simeq$ $C_2^9$	$G/\operatorname{soc} \simeq$ $C_7^3:C_3\wr S_3$
2-Sylow subgroup:	$P_{ 2 } \simeq$ $C_2^2\times C_2^6.C_2^2$
3-Sylow subgroup:	$P_{ 3 } \simeq$ $C_3\wr C_3$
7-Sylow subgroup:	$P_{ 7 } \simeq$ $C_7^3$

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 10368 (order at least 2744) are shown, as well as all normal subgroups of any index.

Order 28449792: $C_2^9.C_7^3:C_3\wr S_3$

Order 14224896: $C_2^9:(C_7^3:C_3\wr C_3)$

Order 9483264: $C_2^6.C_7^2.F_8.C_3^3.C_2$, $C_2^9.C_7^3:C_3^2:S_3$

Order 4741632: $C_2^9.C_7^3.C_3^3$, $C_2^9:(C_7^3:C_9:C_3)$, $C_2^9:(C_7^3:\He_3)$

Order 3161088: $C_2^6.C_7^2.F_8.C_3.C_6$ x 2, $C_2^6.C_7^2.C_3.C_6\times F_8$, $C_2^6.C_7^2.F_8.C_3.C_6$, $C_2^9.C_7^3.C_3.C_6$, $C_2^9.C_7^3:(C_3\times S_3)$

Order 1580544: $C_2^9.C_7^3.C_3^2$ x 2, $C_2^6.C_7^2.F_8.C_3^2$, $C_2^9.C_7^3.C_3^2$, $C_2^9.C_7^3.C_3^2$, $C_2^9:(C_7^3:C_9)$, $C_2^9.C_7^3:C_3^2$

Order 1354752: $C_2^6.C_7^2.(C_6\times S_3\times A_4)$

Order 1185408: $C_2^6.C_7^3.C_3^3.C_2$

Order 1053696: $C_2^6:C_7.F_7\times F_8$, $C_2^9.C_7^3.C_6$, $C_2^6.C_7^2.S_3\times F_8$, $C_2^9.C_7^3.C_6$, $C_2^9.C_7^3.C_6$, $C_2^9.C_7\wr S_3$

Order 677376: $C_2^6.C_7^2.C_3.A_4.C_6$ x 2, $C_2^6.C_{14}^2.C_3^3.C_2$

Order 592704: $C_2^6.C_7^3.C_3^3$

Order 526848: $C_2^9.C_7^3.C_3$, $F_8.C_2^6.C_7.C_{21}$, $C_2^6.C_7^2.F_8.C_3$, $C_2^9.C_7^3.C_3$, $C_2^6.C_7^2.F_8.C_3$, $C_2^9.C_7\wr C_3$

Order 451584: $C_2^6.C_7^2.(C_6\times A_4).C_2$ x 2, $C_2^6.C_7^2.C_6^2.C_2^2$, $C_2^9.C_7^2.C_3.C_6$, $C_2^6.C_{14}^2.C_6^2$, $C_2^6.C_7^2.(C_6\times A_4).C_2$, $C_2^9.C_7.C_{21}.C_6$, $C_2^9.C_7^2:(C_3\times S_3)$

Order 395136: $C_2^6.C_7^3.C_3.C_6$ x 2, $C_2^6.C_7^3.C_3.C_6$, $C_2^6.C_7^3.C_3.C_6$, $C_7\times C_2^6.C_7^2.C_3.C_6$

Order 351232: $F_8\times C_2^3:F_8.C_{14}$

Order 338688: $C_2\times C_2^6.C_7^2.C_3^3.C_2$, $C_2^6.C_{14}^2.C_3^3$

Order 225792: $C_2^6.C_7^2.(S_3\times A_4)$ x 4, $C_2^2\times C_2^6.C_7^2.C_3.C_6$ x 2, $C_2^6.C_7^2.A_4.C_6$ x 2, $C_2^6.C_7^2.A_4.C_6$ x 2, $C_2^9.C_7.C_{21}.C_3$, $C_2\times A_4\times C_2^6.C_7^2.C_3$, $C_2^6.C_{14}^2.C_3.C_6$, $C_2^6.C_7^2.A_4.C_6$, $C_2^9.C_7^2.C_3^2$, $C_2^3\times C_2^6.C_7^2.C_3^2$, $F_8.C_2^5.C_{21}.C_6$, $C_2^6.C_7^2.S_3\times A_4$, $C_2^6.C_7^2.(S_3\times A_4)$, $C_2^6.C_7^2.A_4.C_6$, $C_2^6.C_{14}^2.C_3.C_6$, $C_2^9:(C_7^2:C_3^2)$

Order 197568: $C_2^6.C_7^3.C_3^2$ x 3, $F_8:C_3\times C_2^3:(C_7^2:C_3)$, $F_8\times C_2^3:(C_7^2:C_3:C_3)$, $C_2^6.C_7^3.C_3^2$, $C_7\times C_2^6.C_7^2.C_3^2$, $C_2^6.C_7^3.C_3^2$, $C_2^6.C_7^3.C_3^2$

Order 193536: $C_2^7.C_{42}.C_6^2$

Order 175616: $C_2^6.C_7^2\times F_8$

Order 169344: $C_2^6.C_7^2.C_3^3.C_2$ x 2, $C_2^6.C_7^2.C_3^3.C_2$

Order 150528: $C_2\times C_2^6.C_{14}^2.C_6$ x 2, $C_2^9.C_7.F_7$ x 2, $C_2^6.F_7\times F_8$, $C_2^3\times C_2^6:C_7.F_7$, $C_2^6:C_7.C_6\times F_8$, $C_2^9.C_7^2.C_6$, $C_2\times C_2^6.C_{14}^2.C_6$, $C_2^3\times C_2^6.C_7^2.S_3$, $F_8.C_2^6.C_{42}$, $C_2^9.C_7^2.C_6$, $C_2^9.C_7^2.C_6$, $C_2^9.C_7:F_7$, $C_2^9.C_7.F_7$, $C_2^9.C_7^2:S_3$

Order 148176: $C_7^2.F_8.C_3^3.C_2$

Order 131712: $C_7\times C_2^6.C_7^2.C_6$, $C_2^6.C_7^3.C_6$, $C_2^6.C_7^3.C_6$, $C_7\times C_2^6.C_7^2.S_3$, $C_2^6.C_7^3.C_6$

Order 112896: $(F_8:C_3)\wr C_2\times C_2$ x 4, $C_2^6.C_7^2.A_4.C_3$ x 3, $C_2^6.C_7^2.A_4.C_3$, $C_2^6.C_{14}^2.C_3^2$, $C_2^6.C_7^2.A_4.C_3$, $C_2^6.C_7^2.(C_6\times S_3)$, $C_2^6.C_7^2.C_6^2$, $C_2^6.C_7^2.A_4.C_3$, $F_8.C_2^5.C_{21}.C_3$, $C_2^6.C_7^2.C_6^2$

Order 96768: $F_8.A_4^2.C_6.C_2$, $C_2^7.C_{21}.C_6^2$, $C_2^7.C_7.C_3^3.C_4$

Order 84672: $C_2^6.C_7^2.C_3^3$, $(C_7\times F_8).A_4.C_3.C_6$

Order 82944: $C_2\times A_4^3.S_4$

Order 75264: $C_2^6.C_{14}^2.C_6$ x 4, $C_2^6.C_{14}^2.C_6$ x 2, $C_2\times C_2^6.C_7^2.C_6.C_2$ x 2, $C_2^6.C_{14}^2.C_6$ x 2, $C_2^9.C_7^2.C_3$ x 2, $F_8.C_2^5.C_{42}$ x 2, $C_2\times C_2^6.C_7^2.D_6$ x 2, $C_2^9.C_7^2.C_3$, $C_2^6.C_{14}^2.C_6$, $F_8.C_2^6.C_{21}$, $C_2^9.C_7^2.C_3$, $C_2^6.C_7^2.C_6.C_2^2$, $C_2^8.C_7.C_{42}$, $C_2^9.C_7^2.C_3$, $F_8.F_8.C_6.C_2^2$, $C_2^6:C_7.F_8.C_3$, $C_2^3:F_8.F_8.C_3$, $C_2^6.C_7^2.C_6.C_2^2$, $F_8.C_2^6.C_{21}$, $C_2^9.C_7^2.C_3$, $C_2^9:(C_7:C_{21})$, $C_2^9:(C_7^2:C_3)$

Order 74088: $F_8.C_7^2.C_3^3$

Order 65856: $(C_7\times F_8).F_8.C_3$, $F_8.F_8.C_{21}$, $(C_7\times F_8).F_8.C_3$, $C_2^6.C_7^3.C_3$, $C_2^6.C_7^3.C_3$, $C_2^6.C_7^3.C_3$, $C_7.(C_2^3\times F_8).C_{21}$, $C_2^6.C_7^3.C_3$, $C_2^6.C_7^3.C_3$

Order 64512: $C_2^7.C_{42}.C_6.C_2$ x 2, $C_2^6:C_7.A_4.C_6.C_2$, $C_2^4.C_6^2.C_2\times F_8$, $F_8.C_2^5.C_6^2$, $C_2^7.C_{42}.C_6.C_2$, $A_4.C_2\wr F_7$, $C_2^9.C_{21}:C_6$

Order 56448: $C_2^6.C_7^2.C_3.C_6$ x 7, $(F_8:C_3)\wr C_2$ x 6, $C_2^6.C_7^2.C_3.C_6$ x 2, $C_2^6:C_7.C_{21}.C_6$, $C_2^3:F_8.C_{21}.C_6$, $C_2^6.C_7^2.C_3.C_6$, $C_2^6.C_7^2.C_3.C_6$, $C_2^6.C_7^2.C_3.C_6$, $C_2^6.C_7^2.C_3.C_6$, $F_8.F_8.C_3.C_6$

Order 55566: $C_7^3:C_3\wr S_3$

Order 50176: $C_2^3:F_8.F_8.C_2$ x 2, $C_2^6.D_7\times F_8$, $C_2^6.C_{14}\times F_8$, $C_2^3\times C_2^3:F_8.C_{14}$

Order 49392: $C_7^2.F_8.C_3.C_6$ x 2, $C_7^2.F_8.C_3.C_6$, $C_2^3.C_7^3.C_3.C_6$, $C_7^2.(C_3\times F_8).C_6$

Order 48384: $F_8.A_4^2.C_6$ x 2, $C_2^7.C_{21}.C_3.C_6$

Order 43904: $C_7\times C_2^3:F_8.C_{14}$

Order 42336: $(C_7\times F_8).A_4.C_3^2$

Order 41472: $A_4^3.S_4$ x 2, $C_2^6.C_6^2.C_3.C_6$

Order 37632: $F_8\wr C_2:C_6$ x 4, $C_2^6.C_7^2.A_4$ x 2, $F_8.C_2^5.C_{21}$ x 2, $C_2^6:(C_7^2:D_6)$ x 2, $F_8.F_8.C_6.C_2$, $C_2^6.C_7^2.A_4$, $C_6\times C_2^3:F_8.C_{14}$, $C_2^6.C_7^2\times A_4$, $C_2^6.C_7^2.C_6.C_2$, $C_2^6.C_7^2.C_6.C_2$

Order 32256: $F_8.A_4^2.C_2^2$ x 3, $F_8.A_4^2.C_2^2$ x 2, $C_2^7.C_{21}.C_{12}$ x 2, $C_2^7.C_{21}.C_6.C_2$ x 2, $F_8.C_2^4.C_6^2$ x 2, $C_2^6:C_7.A_4.C_6$ x 2, $A_4\times C_2^6:F_7$ x 2, $C_2^7.C_{21}.C_6.C_2$, $F_8.C_2^4.C_3.C_{12}$, $C_2^6:C_7.A_4.C_6$, $C_2^7.C_{21}.C_{12}$, $C_2^7.C_7.C_6^2$, $F_8.C_2^4.C_6^2$, $C_2^3\times F_8.A_4.C_6$, $C_2^4:C_3.F_8.C_6.C_2$, $C_2^4:C_3.F_8.C_{12}$, $C_2^8.C_{21}:C_6$, $C_2^9:(C_{21}:C_3)$

Order 28224: $C_2^6.C_7^2.C_3^2$ x 9, $(C_7\times F_8).A_4.C_6$ x 4, $C_2^3\times C_2^3:(C_7^2:C_3:C_3)$, $C_3\times C_2^6.C_7^2.C_3$, $(C_7\times F_8).A_4.C_6$, $C_2^6.C_7^2.C_3^2$, $F_8.(C_{42}\times A_4)$, $F_8.C_7:A_4.C_6$, $(C_7\times F_8).(C_6\times A_4)$, $(C_7\times F_8).A_4.C_6$, $(C_7\times F_8).(C_6\times A_4)$, $C_2^6.C_7^2.C_3^2$, $C_2^6.C_7^2.C_3^2$, $(C_7\times F_8).A_4.C_6$, $(C_2^4\times C_{14}).C_{21}.C_6$, $C_3\times F_8:C_3\times F_8$

Order 27783: $C_7^3:C_3\wr C_3$

Order 27648: $C_2^4.(C_6^2\times A_4).C_2^2$, $C_2^3:A_4^2.S_4$

Order 25088: $C_2^9.C_7^2$ x 2, $C_2^2\times C_2^3:F_8.C_{14}$ x 2, $C_2^6:C_7\times F_8$, $C_2^3\times C_2^6.C_7^2$, $F_8\times C_2^3:F_8$

Order 24696: $C_2^3.C_7^3.C_3^2$ x 2, $C_7\times C_2^3:(C_7^2:C_3:C_3)$, $(C_7\times F_8).C_{21}.C_3$, $F_8\times C_7^2:C_3:C_3$, $F_8.C_7^2.C_3^2$, $C_2^3.C_7^3.C_3^2$, $C_2^3.C_7^3.C_3^2$, $(C_7\times F_8).C_{21}.C_3$

Order 24192: $C_2^4.C_{42}.C_6^2$, $F_8.A_4^2.C_3$, $F_8.A_4.C_6^2$

Order 21952: $F_8\times C_7\times F_8$

Order 21504: $C_2^6:C_7.A_4.C_2^2$ x 2, $F_8.C_2^5.C_6.C_2$ x 2, $C_2^9.F_7$ x 2, $C_2^4:C_3.D_4\times F_8$, $C_2^5.(C_2\times C_6)\times F_8$, $C_2^3\times C_2^6:C_7.C_6$, $C_2^8.C_{42}.C_2$, $C_2^9.C_7.C_6$, $C_2^3\times C_2^6.F_7$, $F_8.(C_2^5\times C_6).C_2$, $A_4.C_2\wr D_7$, $C_2^6.F_8:C_6$, $C_2^3\wr C_3.C_{14}$

Order 21168: $C_7^2.(C_6\times S_3\times A_4)$, $(C_7\times F_8).C_3^3.C_2$

Order 20736: $A_4^3.A_4$, $A_4^3:D_6$

Order 18816: $F_8\wr C_2:C_3$ x 8, $C_2\times C_2^6.C_7^2.C_3$ x 3, $C_2\times F_8:C_3\times F_8$ x 2, $C_2\times C_2^6.C_7^2.C_3$ x 2, $C_2^3:F_8.C_{42}$ x 2, $C_2^6:(C_7^2:S_3)$ x 2, $C_7\times C_2^6.F_7$, $F_8\times C_2\times F_8:C_3$, $F_8.F_8.C_6$, $C_2\times F_8.F_8.C_3$, $C_7.C_2^6:C_7.C_6$, $C_2^6.C_7^2.C_6$, $C_2^6.C_7^2.C_6$, $C_2^6.C_7^2.C_6$, $C_2^6.C_7^2.C_6$, $C_7\times C_2^6:C_7.C_6$, $C_2^6.C_7^2.C_6$, $C_2^6.C_7:F_7$, $C_2^6.C_7^2.C_6$, $C_2\times C_2^6.C_7^2.C_3$

Order 18522: $C_7^3:(S_3\times C_3^2)$, $C_7^3:C_3^2:S_3$

Order 16464: $C_7^2.(S_3\times F_8)$, $C_2^3.C_7^3.C_6$, $F_8.C_7^2.C_6$, $C_7^2.F_8.C_6$, $(C_7\times F_8).F_7$

Order 16128: $F_8.A_4^2.C_2$ x 7, $F_8.A_4^2.C_2$ x 6, $F_8.C_2^4.C_3.C_6$ x 4, $C_2\times F_8.A_4.C_6.C_2$ x 2, $C_2^7.C_{21}.C_6$ x 2, $C_2^7.C_{21}.C_6$ x 2, $A_4\wr C_2\times F_8$, $F_8.A_4.C_6.C_2^2$, $F_8.C_2^4.C_3.C_6$, $C_2^7.C_{21}.C_6$, $F_8.(C_3\times D_4\times A_4)$, $C_2^6:C_7.A_4.C_3$, $C_2\times C_2^6.C_{21}.C_6$, $C_2^5.C_{42}.C_6.C_2$, $C_2\wr F_7\times C_3$, $C_2^8.C_{21}:C_3$

Order 14112: $(C_7\times F_8).A_4.C_3$ x 2, $(C_7\times F_8).A_4.C_3$ x 2, $C_2^3.C_{14}^2.C_3^2$ x 2, $(C_7\times F_8).A_4.C_3$ x 2, $(C_2\times C_{14}).F_8.C_3^2$, $(C_7\times F_8).A_4.C_3$, $C_2^2\times C_2^3:(C_7^2:C_3:C_3)$, $F_8.(C_{21}\times A_4)$, $(C_7\times F_8).A_4.C_3$

Order 13824: $A_4^3:D_4$ x 4, $C_2^2:A_4^2.S_4$ x 2, $C_2^4.(C_6^2\times A_4).C_2$, $C_2\times C_2^6.C_3^3.C_2^2$, $C_2^4.(C_6^2\times A_4).C_2$, $C_2\wr C_9.C_3$, $C_2^9:\He_3$

Order 12544: $C_2\times F_8\wr C_2$ x 2, $C_2^6.C_{14}^2$

Order 12096: $C_6\times F_8.A_4.C_3$ x 2, $C_7.A_4^2.C_6.C_2$, $F_8.(C_3\times A_4).C_6$, $(C_2^3\times C_{14}).C_3^3.C_4$, $(C_7\times C_2^4:C_3).C_6^2$, $(C_6\times F_8).C_6^2$

Order 10752: $C_2^6:C_7.A_4.C_2$ x 4, $C_2^8.F_7$ x 4, $F_8.C_2^4.C_6.C_2$ x 3, $C_2^7.C_{42}.C_2$ x 2, $F_8.A_4.C_2^4$ x 2, $C_2^8.C_{42}$ x 2, $F_8.C_2^4.C_6.C_2$ x 2, $C_2\times C_2^6:C_7.C_6.C_2$ x 2, $F_8.C_2^4.C_{12}$ x 2, $C_2^6.F_8:C_3$ x 2, $A_4\times C_2^6:D_7$ x 2, $C_2\wr C_7.A_4$ x 2, $C_2^4.F_8:A_4$ x 2, $C_2^8.F_7$ x 2, $C_2\times C_2^8.C_{21}$, $C_2^4:C_3.F_8.C_4$, $C_2^4.(S_3\times F_8).C_2$, $F_8:C_3.C_2^6$, $C_2\times C_2^5.C_{42}.C_2^2$, $C_2^3:F_8.C_6.C_2^2$, $(C_2\times F_8).C_6.C_2^4$, $C_2^6:C_7.C_6.C_2^2$, $C_2^7.C_{84}$, $C_2^7.C_{42}.C_2$, $F_8.(C_2^4\times C_6).C_2$, $C_2^9:C_{21}$, $C_2^9:C_{21}$, $C_2^9:C_7:C_3$

Order 10584: $C_7^2.(C_3\times A_4).C_6$ x 2, $(C_7\times F_8).C_3^3$, $C_{14}^2.C_3^3.C_2$

Order 10368: $A_4\wr S_3$ x 2, $C_2^6.C_3^3.C_6$

Order 9408: $F_8.F_8.C_3$ x 9, $C_2^6.C_7^2.C_3$ x 9, $C_2^6.C_7^2.C_3$ x 5, $(C_7\times F_8).A_4.C_2$ x 2, $C_7\times F_8.A_4.C_2$ x 2, $(C_2\times C_{14}).F_8.C_6$ x 2, $C_2\times (C_7\times F_8).C_6.C_2$, $F_8.C_{42}.C_2^2$, $C_2^6.C_7^2.C_3$, $C_2^3\times C_2^3:(C_7^2:C_3)$, $C_2^6.C_7^2.C_3$, $C_7\times C_2^3:F_8.C_3$, $C_2^3\times C_2^3:(C_7:(C_7:C_3))$, $C_2^6.C_7^2.C_3$, $C_2^6.C_7^2.C_3$, $F_8.C_7:A_4.C_2$, $(C_7\times F_8).A_4.C_2$, $C_2^5.C_7^2.C_6$, $C_2^6.C_7^2.C_3$, $C_2^6:C_7\times C_7:C_3$, $C_2^3:F_8\times C_7:C_3$, $C_7\times C_2^6:C_7.C_3$

Order 9261: $C_7^3:C_3^3$, $C_7^3:C_9:C_3$, $C_7^3:\He_3$

Order 9216: $C_2^4.(C_6\times A_4).C_2^3$ x 2, $C_2^4.(C_6\times A_4).C_2^3$, $C_2^4.C_6^2.C_2^4$, $C_2^7.C_6^2.C_2$, $C_2^5.(A_4\times S_4)$

Order 8232: $C_2^3:(C_7\times C_7^2:C_3)$, $C_2^3:(C_7^2\times C_7:C_3)$, $C_2^3:(C_7\times C_7^2:C_3)$, $C_7\times C_2^3:(C_7\times C_7:C_3)$, $C_2^3:(C_7\times C_7:(C_7:C_3))$, $C_2^3:(C_7^3:C_3)$, $C_2^3:(C_7^3:C_3)$, $C_2^3:(C_7^3:C_3)$, $C_2^3:(C_7\times C_7:(C_7:C_3))$

Order 8064: $F_8.A_4.C_6.C_2$ x 7, $F_8.A_4^2$ x 3, $F_8.(C_6\times A_4).C_2$ x 3, $C_2\times F_8.A_4.C_6$ x 3, $C_2^6.C_{21}.C_6$ x 2, $F_8.A_4^2$ x 2, $C_2^4.C_{42}.C_6.C_2$ x 2, $F_8.C_2^4.C_3^2$ x 2, $C_3\times C_2^6:F_7$ x 2, $C_2^6:C_7.C_3.C_6$, $F_8:C_3\times C_4\times A_4$, $F_8:C_3\times A_4:C_4$, $C_2^4.C_{42}.C_6.C_2$, $C_7\times C_2^4.C_6^2.C_2$, $F_8:C_3\times C_2^2\times A_4$, $C_2\times A_4\times C_2\times F_8:C_3$, $C_2^3\times C_6\times F_8:C_3$, $C_2^5.(C_6\times C_{42})$, $F_8.(C_6\times A_4).C_2$, $C_2\times F_8.A_4.C_6$, $(C_2^4\times C_{14}).C_6^2$, $C_2^7.C_{21}.C_3$, $F_8:C_6\times S_4$, $C_2\wr C_7:C_3^2$, $C_2^6:C_{21}:C_6$

Order 7168: $C_2^9.C_{14}$, $C_2^3\times C_2^6.C_{14}$, $F_8\times C_2^6.C_2$, $C_2^3\times C_2^6.D_7$, $C_2^9.C_{14}$

Order 7056: $C_2\times C_2^3:(C_7^2:C_3:C_3)$ x 4, $(C_7\times F_8).C_3.C_6$ x 3, $F_8.C_{21}.C_6$ x 2, $F_8.C_7:C_3.C_6$ x 2, $C_7^2.(C_6\times A_4).C_2$ x 2, $(C_7\times F_8).C_3.C_6$, $(C_7\times F_8).C_3.C_6$, $(C_7\times F_8).C_3.C_6$, $C_{14}^2.C_6^2$, $F_8.C_{21}.C_6$, $C_7^2.(C_6\times A_4).C_2$, $C_2^3\times C_7^2:(C_3\times S_3)$, $F_7\times F_8:C_3$

Order 6912: $A_4^3:C_2^2$ x 4, $C_2\times C_2^6.C_3^3.C_2$ x 3, $A_4^3:C_4$ x 2, $C_2^4.C_6^3.C_2$, $C_2^6.C_6^2.C_3$, $C_2\times C_2^6.\He_3.C_2$, $C_2^2:A_4^2.A_4$

Order 6272: $F_8\wr C_2$ x 4, $C_2^6.C_7^2.C_2$ x 2, $C_7\times C_2^6.C_{14}$, $C_7\times C_2^6.D_7$

Order 6174: $C_7^3:(C_3\times S_3)$ x 2, $C_7^3:(C_3\times S_3)$, $C_7^3:(C_3\times S_3)$, $C_7^2:(S_3\times C_{21})$, $C_7^3:(C_3\times C_6)$

Order 6048: $C_7.A_4^2.C_6$ x 2, $F_8.C_3^3.C_2^2$, $F_8.C_3^3.C_4$, $F_8.C_3^3.C_2^2$, $(C_7\times A_4^2):C_6$, $A_4\times F_8:C_3^2$

Order 5488: $(C_7\times F_8).C_{14}$

Order 5376: $F_8.C_2^4.C_6$ x 6, $F_8.A_4.C_2^3$ x 5, $C_2^4:F_8:C_6$ x 4, $C_2\wr F_7$ x 4, $C_2\times C_2^7.C_{21}$ x 3, $F_8:C_3.C_2^5$ x 2, $C_2^5.C_{42}.C_2^2$ x 2, $F_8.C_2^4.C_6$ x 2, $F_8.(D_4\times A_4)$ x 2, $C_2^3.F_8:A_4$ x 2, $C_2^3.F_8:A_4$ x 2, $C_2\times F_8.C_2^4.C_3$, $C_2\times F_8.A_4.C_2^2$, $C_2\times C_2^5.C_{42}.C_2$, $F_8\times C_2^2:S_4$, $(C_2\times F_8).C_6.C_2^3$, $F_8\times \GL(2,\mathbb{Z}/4)$, $F_8\times C_2^2\wr C_2:C_3$, $C_2^7.C_{42}$, $C_2\times C_2^6.C_{42}$, $D_4\times A_4\times F_8$, $C_2^8.C_{21}$, $F_8.(C_2^3\times C_6).C_2$, $(C_2\times F_8).C_6.C_2^3$, $F_8.(C_6\times D_4).C_2$, $C_2\wr D_7\times C_3$, $C_2^8:C_{21}$, $C_2\wr C_7.C_6$, $C_2^5:F_8:C_3$

Order 5292: $C_7^2:(C_3^2\times A_4)$, $C_7^2:(C_3^2\times D_6)$

Order 5184: $A_4\wr C_3$

Order 4704: $(C_7\times F_8).A_4$ x 2, $F_8.C_7:A_4$ x 2, $F_8.(C_7\times A_4)$ x 2, $(C_2^4\times C_{14}).C_{21}$, $(C_2\times C_{14}).F_8.C_3$, $(C_7\times F_8).C_6.C_2$, $F_8.C_{42}.C_2$, $C_2^5.C_7^2.C_3$, $(C_7\times F_8).C_6.C_2$, $(C_7\times F_8).C_6.C_2$

Order 4608: $C_2\wr S_3:A_4$ x 8, $C_2^4.C_6^2.C_2^3$ x 4, $A_4\times C_2\wr S_3$ x 4, $C_2^7.C_6^2$ x 4, $C_2\times C_2^6.C_6^2$ x 2, $C_2\times C_2^6.C_6^2$ x 2, $C_2^4.(C_6\times A_4).C_2^2$ x 2, $C_2^5.(A_4\times D_6)$ x 2, $C_2^2\times A_4^2.C_2^3$, $C_2^6.(S_3\times A_4)$, $C_2^2\times A_4^2.C_2^3$, $C_2\times C_2^6.C_6^2$, $C_2^2.A_4^2.C_2^2.C_2$, $A_4^2.C_2^4.C_2$, $C_2^6.C_6^2.C_2$, $C_2^4.(A_4\times S_4)$, $C_2^4:(A_4\times S_4)$, $C_2\wr C_9$, $C_2^9:C_3^2$

Order 4032: $A_4\times F_8:C_6$ x 37, $F_8.(C_6\times A_4)$ x 4, $C_7.A_4^2.C_2^2$ x 3, $F_8.C_6^2.C_2$ x 3, $(D_6\times F_8):C_6$ x 3, $C_7.A_4^2.C_2^2$ x 2, $(C_7\times C_2^4:C_3).C_6.C_2$ x 2, $C_3\times C_2^5.C_{42}$ x 2, $(C_2^3\times C_{14}).C_6^2$ x 2, $(C_7\times C_2^4:C_3).C_{12}$ x 2, $(C_2^3\times C_{14}).C_6^2$, $(C_6\times F_8).C_6.C_2$, $C_2^4.(C_6\times C_{42})$, $C_2\times C_2^2:S_4\times C_7:C_3$, $C_2^4:C_3.C_{84}$, $C_3\times A_4\times C_2\times F_8$, $(C_2^4\times C_7:C_3).C_{12}$, $F_8.A_4.C_6$, $C_2\times F_8.(C_3\times A_4)$, $C_2^6.C_{21}.C_3$, $C_2\times (A_4\times C_7:C_3).C_2^3$, $C_2^2:(A_4:C_4)\times C_7:C_3$, $C_2^4.(C_{42}\times S_3)$, $C_2^6:C_7.C_3^2$, $D_4\times F_8:C_3^2$, $C_2^6:C_{21}:C_3$, $F_8:C_3\times S_4$

Order 3584: $C_2^2\times C_2^6.C_{14}$ x 2, $C_2^7:D_{14}$ x 2, $C_2^3\times C_2^3:F_8$, $(C_2^3\times F_8).C_2^3$, $C_2^3\times C_2^6:C_7$, $F_8.C_2^6$, $(C_2^3\times F_8).C_2^3$, $C_2^9.C_7$, $C_2^9:C_7$

Order 3528: $(C_7\times F_8):C_3^2$ x 14, $C_7^2:(S_3\times A_4)$ x 4, $C_7^2:(C_6\times A_4)$ x 2, $C_7^2:(C_6\times A_4)$ x 2, $C_7^2:(C_6\times A_4)$ x 2, $C_7^2:(S_3\times A_4)$ x 2, $C_2^2\times C_7^2:(C_3\times S_3)$ x 2, $C_7^2:(C_6\times A_4)$, $C_{21}\times F_8:C_3$, $C_3\times C_2^3:(C_7^2:C_3)$, $C_7^2:(C_6\times A_4)$, $C_2^3\times C_7^2:C_3:C_3$, $C_2^3:(C_3\times C_7:(C_7:C_3))$, $C_3\times F_8\times C_7:C_3$, $C_7^2:(C_6\times A_4)$, $C_2\times C_{14}^2:C_3^2$

Order 3456: $C_3\times A_4^2:D_4$ x 4, $C_2\times A_4^3$ x 3, $A_4^3:C_2$ x 2, $C_2^2:A_4^2:S_3$ x 2, $C_2^6.\He_3.C_2$, $C_2^4:C_3.C_6^2.C_2$, $C_2^4.C_6^3$, $A_4^2:C_2^2\times C_6$, $C_2^6:C_9:C_6$

Order 3136: $C_2^6.C_7^2$ x 7, $F_8.C_{14}.C_2^2$, $C_2^6.C_7^2$, $C_2^6:C_7^2$

Order 3087: $C_7^3:C_3^2$ x 2, $C_7^3:C_3^2$, $C_7^3:C_3^2$, $C_7^3:C_3^2$, $C_7^3:C_3^2$, $C_7^3:C_9$

Order 3072: $C_2^7.C_6.C_2^2$ x 2, $(C_2^4\times A_4).C_2^4$, $C_2^4:C_3.C_2^5.C_2$, $C_2^5.(C_2^4\times C_6)$, $C_2^7.S_4$

Order 3024: $F_8.C_3^3.C_2$, $C_3\times C_6\times F_8:C_3$, $C_7.(C_6^2\times A_4)$, $C_7:C_3\times A_4^2$, $S_3\times F_8:C_3^2$

Order 2744: $C_7^2\times F_8$

Order 1024: $C_2^2\times C_2^6.C_2^2$

Order 512: $C_2^9$

Order 343: $C_7^3$

Order 81: $C_3\wr C_3$

Order 1: $C_1$

Classes of subgroups up to automorphism

All subgroups of index up to 10368 (order at least 2744) are shown, as well as all normal subgroups of any index.

Order 28449792: $C_2^9.C_7^3:C_3\wr S_3$

Order 14224896: $C_2^9:(C_7^3:C_3\wr C_3)$

Order 9483264: $C_2^6.C_7^2.F_8.C_3^3.C_2$, $C_2^9.C_7^3:C_3^2:S_3$

Order 4741632: $C_2^9.C_7^3.C_3^3$, $C_2^9:(C_7^3:C_9:C_3)$, $C_2^9:(C_7^3:\He_3)$

Order 3161088: $C_2^6.C_7^2.F_8.C_3.C_6$ x 2, $C_2^6.C_7^2.C_3.C_6\times F_8$, $C_2^6.C_7^2.F_8.C_3.C_6$, $C_2^9.C_7^3.C_3.C_6$, $C_2^9.C_7^3:(C_3\times S_3)$

Order 1580544: $C_2^9.C_7^3.C_3^2$ x 2, $C_2^6.C_7^2.F_8.C_3^2$, $C_2^9.C_7^3.C_3^2$, $C_2^9.C_7^3.C_3^2$, $C_2^9:(C_7^3:C_9)$, $C_2^9.C_7^3:C_3^2$

Order 1354752: $C_2^6.C_7^2.(C_6\times S_3\times A_4)$

Order 1185408: $C_2^6.C_7^3.C_3^3.C_2$

Order 1053696: $C_2^6:C_7.F_7\times F_8$, $C_2^9.C_7^3.C_6$, $C_2^6.C_7^2.S_3\times F_8$, $C_2^9.C_7^3.C_6$, $C_2^9.C_7^3.C_6$, $C_2^9.C_7\wr S_3$

Order 677376: $C_2^6.C_7^2.C_3.A_4.C_6$ x 2, $C_2^6.C_{14}^2.C_3^3.C_2$

Order 592704: $C_2^6.C_7^3.C_3^3$

Order 526848: $C_2^9.C_7^3.C_3$, $F_8.C_2^6.C_7.C_{21}$, $C_2^6.C_7^2.F_8.C_3$, $C_2^9.C_7^3.C_3$, $C_2^6.C_7^2.F_8.C_3$, $C_2^9.C_7\wr C_3$

Order 451584: $C_2^6.C_7^2.(C_6\times A_4).C_2$ x 2, $C_2^6.C_7^2.C_6^2.C_2^2$, $C_2^9.C_7^2.C_3.C_6$, $C_2^6.C_{14}^2.C_6^2$, $C_2^6.C_7^2.(C_6\times A_4).C_2$, $C_2^9.C_7.C_{21}.C_6$, $C_2^9.C_7^2:(C_3\times S_3)$

Order 395136: $C_2^6.C_7^3.C_3.C_6$ x 2, $C_2^6.C_7^3.C_3.C_6$, $C_2^6.C_7^3.C_3.C_6$, $C_7\times C_2^6.C_7^2.C_3.C_6$

Order 351232: $F_8\times C_2^3:F_8.C_{14}$

Order 338688: $C_2\times C_2^6.C_7^2.C_3^3.C_2$, $C_2^6.C_{14}^2.C_3^3$

Order 225792: $C_2^6.C_7^2.(S_3\times A_4)$ x 4, $C_2^2\times C_2^6.C_7^2.C_3.C_6$ x 2, $C_2^6.C_7^2.A_4.C_6$ x 2, $C_2^6.C_7^2.A_4.C_6$ x 2, $C_2^9.C_7.C_{21}.C_3$, $C_2\times A_4\times C_2^6.C_7^2.C_3$, $C_2^6.C_{14}^2.C_3.C_6$, $C_2^6.C_7^2.A_4.C_6$, $C_2^9.C_7^2.C_3^2$, $C_2^3\times C_2^6.C_7^2.C_3^2$, $F_8.C_2^5.C_{21}.C_6$, $C_2^6.C_7^2.S_3\times A_4$, $C_2^6.C_7^2.(S_3\times A_4)$, $C_2^6.C_7^2.A_4.C_6$, $C_2^6.C_{14}^2.C_3.C_6$, $C_2^9:(C_7^2:C_3^2)$

Order 197568: $C_2^6.C_7^3.C_3^2$ x 3, $F_8:C_3\times C_2^3:(C_7^2:C_3)$, $F_8\times C_2^3:(C_7^2:C_3:C_3)$, $C_2^6.C_7^3.C_3^2$, $C_7\times C_2^6.C_7^2.C_3^2$, $C_2^6.C_7^3.C_3^2$, $C_2^6.C_7^3.C_3^2$

Order 193536: $C_2^7.C_{42}.C_6^2$

Order 175616: $C_2^6.C_7^2\times F_8$

Order 169344: $C_2^6.C_7^2.C_3^3.C_2$ x 2, $C_2^6.C_7^2.C_3^3.C_2$

Order 150528: $C_2\times C_2^6.C_{14}^2.C_6$ x 2, $C_2^9.C_7.F_7$ x 2, $C_2^6.F_7\times F_8$, $C_2^3\times C_2^6:C_7.F_7$, $C_2^6:C_7.C_6\times F_8$, $C_2^9.C_7^2.C_6$, $C_2\times C_2^6.C_{14}^2.C_6$, $C_2^3\times C_2^6.C_7^2.S_3$, $F_8.C_2^6.C_{42}$, $C_2^9.C_7^2.C_6$, $C_2^9.C_7^2.C_6$, $C_2^9.C_7:F_7$, $C_2^9.C_7.F_7$, $C_2^9.C_7^2:S_3$

Order 148176: $C_7^2.F_8.C_3^3.C_2$

Order 131712: $C_7\times C_2^6.C_7^2.C_6$, $C_2^6.C_7^3.C_6$, $C_2^6.C_7^3.C_6$, $C_7\times C_2^6.C_7^2.S_3$, $C_2^6.C_7^3.C_6$

Order 112896: $(F_8:C_3)\wr C_2\times C_2$ x 4, $C_2^6.C_7^2.A_4.C_3$ x 3, $C_2^6.C_7^2.A_4.C_3$, $C_2^6.C_{14}^2.C_3^2$, $C_2^6.C_7^2.A_4.C_3$, $C_2^6.C_7^2.(C_6\times S_3)$, $C_2^6.C_7^2.C_6^2$, $C_2^6.C_7^2.A_4.C_3$, $F_8.C_2^5.C_{21}.C_3$, $C_2^6.C_7^2.C_6^2$

Order 96768: $F_8.A_4^2.C_6.C_2$, $C_2^7.C_{21}.C_6^2$, $C_2^7.C_7.C_3^3.C_4$

Order 84672: $C_2^6.C_7^2.C_3^3$, $(C_7\times F_8).A_4.C_3.C_6$

Order 82944: $C_2\times A_4^3.S_4$

Order 75264: $C_2^6.C_{14}^2.C_6$ x 4, $C_2^6.C_{14}^2.C_6$ x 2, $C_2\times C_2^6.C_7^2.C_6.C_2$ x 2, $C_2^6.C_{14}^2.C_6$ x 2, $C_2^9.C_7^2.C_3$ x 2, $F_8.C_2^5.C_{42}$ x 2, $C_2\times C_2^6.C_7^2.D_6$ x 2, $C_2^9.C_7^2.C_3$, $C_2^6.C_{14}^2.C_6$, $F_8.C_2^6.C_{21}$, $C_2^9.C_7^2.C_3$, $C_2^6.C_7^2.C_6.C_2^2$, $C_2^8.C_7.C_{42}$, $C_2^9.C_7^2.C_3$, $F_8.F_8.C_6.C_2^2$, $C_2^6:C_7.F_8.C_3$, $C_2^3:F_8.F_8.C_3$, $C_2^6.C_7^2.C_6.C_2^2$, $F_8.C_2^6.C_{21}$, $C_2^9.C_7^2.C_3$, $C_2^9:(C_7:C_{21})$, $C_2^9:(C_7^2:C_3)$

Order 74088: $F_8.C_7^2.C_3^3$

Order 65856: $(C_7\times F_8).F_8.C_3$, $F_8.F_8.C_{21}$, $(C_7\times F_8).F_8.C_3$, $C_2^6.C_7^3.C_3$, $C_2^6.C_7^3.C_3$, $C_2^6.C_7^3.C_3$, $C_7.(C_2^3\times F_8).C_{21}$, $C_2^6.C_7^3.C_3$, $C_2^6.C_7^3.C_3$

Order 64512: $C_2^7.C_{42}.C_6.C_2$ x 2, $C_2^6:C_7.A_4.C_6.C_2$, $C_2^4.C_6^2.C_2\times F_8$, $F_8.C_2^5.C_6^2$, $C_2^7.C_{42}.C_6.C_2$, $A_4.C_2\wr F_7$, $C_2^9.C_{21}:C_6$

Order 56448: $C_2^6.C_7^2.C_3.C_6$ x 7, $(F_8:C_3)\wr C_2$ x 6, $C_2^6.C_7^2.C_3.C_6$ x 2, $C_2^6:C_7.C_{21}.C_6$, $C_2^3:F_8.C_{21}.C_6$, $C_2^6.C_7^2.C_3.C_6$, $C_2^6.C_7^2.C_3.C_6$, $C_2^6.C_7^2.C_3.C_6$, $C_2^6.C_7^2.C_3.C_6$, $F_8.F_8.C_3.C_6$

Order 55566: $C_7^3:C_3\wr S_3$

Order 50176: $C_2^3:F_8.F_8.C_2$ x 2, $C_2^6.D_7\times F_8$, $C_2^6.C_{14}\times F_8$, $C_2^3\times C_2^3:F_8.C_{14}$

Order 49392: $C_7^2.F_8.C_3.C_6$ x 2, $C_7^2.F_8.C_3.C_6$, $C_2^3.C_7^3.C_3.C_6$, $C_7^2.(C_3\times F_8).C_6$

Order 48384: $F_8.A_4^2.C_6$ x 2, $C_2^7.C_{21}.C_3.C_6$

Order 43904: $C_7\times C_2^3:F_8.C_{14}$

Order 42336: $(C_7\times F_8).A_4.C_3^2$

Order 41472: $A_4^3.S_4$ x 2, $C_2^6.C_6^2.C_3.C_6$

Order 37632: $F_8\wr C_2:C_6$ x 4, $C_2^6.C_7^2.A_4$ x 2, $F_8.C_2^5.C_{21}$ x 2, $C_2^6:(C_7^2:D_6)$ x 2, $F_8.F_8.C_6.C_2$, $C_2^6.C_7^2.A_4$, $C_6\times C_2^3:F_8.C_{14}$, $C_2^6.C_7^2\times A_4$, $C_2^6.C_7^2.C_6.C_2$, $C_2^6.C_7^2.C_6.C_2$

Order 32256: $F_8.A_4^2.C_2^2$ x 3, $F_8.A_4^2.C_2^2$ x 2, $C_2^7.C_{21}.C_{12}$ x 2, $C_2^7.C_{21}.C_6.C_2$ x 2, $F_8.C_2^4.C_6^2$ x 2, $C_2^6:C_7.A_4.C_6$ x 2, $A_4\times C_2^6:F_7$ x 2, $C_2^7.C_{21}.C_6.C_2$, $F_8.C_2^4.C_3.C_{12}$, $C_2^6:C_7.A_4.C_6$, $C_2^7.C_{21}.C_{12}$, $C_2^7.C_7.C_6^2$, $F_8.C_2^4.C_6^2$, $C_2^3\times F_8.A_4.C_6$, $C_2^4:C_3.F_8.C_6.C_2$, $C_2^4:C_3.F_8.C_{12}$, $C_2^8.C_{21}:C_6$, $C_2^9:(C_{21}:C_3)$

Order 28224: $C_2^6.C_7^2.C_3^2$ x 9, $(C_7\times F_8).A_4.C_6$ x 4, $C_2^3\times C_2^3:(C_7^2:C_3:C_3)$, $C_3\times C_2^6.C_7^2.C_3$, $(C_7\times F_8).A_4.C_6$, $C_2^6.C_7^2.C_3^2$, $F_8.(C_{42}\times A_4)$, $F_8.C_7:A_4.C_6$, $(C_7\times F_8).(C_6\times A_4)$, $(C_7\times F_8).A_4.C_6$, $(C_7\times F_8).(C_6\times A_4)$, $C_2^6.C_7^2.C_3^2$, $C_2^6.C_7^2.C_3^2$, $(C_7\times F_8).A_4.C_6$, $(C_2^4\times C_{14}).C_{21}.C_6$, $C_3\times F_8:C_3\times F_8$

Order 27783: $C_7^3:C_3\wr C_3$

Order 27648: $C_2^4.(C_6^2\times A_4).C_2^2$, $C_2^3:A_4^2.S_4$

Order 25088: $C_2^9.C_7^2$ x 2, $C_2^2\times C_2^3:F_8.C_{14}$ x 2, $C_2^6:C_7\times F_8$, $C_2^3\times C_2^6.C_7^2$, $F_8\times C_2^3:F_8$

Order 24696: $C_2^3.C_7^3.C_3^2$ x 2, $C_7\times C_2^3:(C_7^2:C_3:C_3)$, $(C_7\times F_8).C_{21}.C_3$, $F_8\times C_7^2:C_3:C_3$, $F_8.C_7^2.C_3^2$, $C_2^3.C_7^3.C_3^2$, $C_2^3.C_7^3.C_3^2$, $(C_7\times F_8).C_{21}.C_3$

Order 24192: $C_2^4.C_{42}.C_6^2$, $F_8.A_4^2.C_3$, $F_8.A_4.C_6^2$

Order 21952: $F_8\times C_7\times F_8$

Order 21504: $C_2^6:C_7.A_4.C_2^2$ x 2, $F_8.C_2^5.C_6.C_2$ x 2, $C_2^9.F_7$ x 2, $C_2^4:C_3.D_4\times F_8$, $C_2^5.(C_2\times C_6)\times F_8$, $C_2^3\times C_2^6:C_7.C_6$, $C_2^8.C_{42}.C_2$, $C_2^9.C_7.C_6$, $C_2^3\times C_2^6.F_7$, $F_8.(C_2^5\times C_6).C_2$, $A_4.C_2\wr D_7$, $C_2^6.F_8:C_6$, $C_2^3\wr C_3.C_{14}$

Order 21168: $C_7^2.(C_6\times S_3\times A_4)$, $(C_7\times F_8).C_3^3.C_2$

Order 20736: $A_4^3.A_4$, $A_4^3:D_6$

Order 18816: $F_8\wr C_2:C_3$ x 8, $C_2\times C_2^6.C_7^2.C_3$ x 3, $C_2\times F_8:C_3\times F_8$ x 2, $C_2\times C_2^6.C_7^2.C_3$ x 2, $C_2^3:F_8.C_{42}$ x 2, $C_2^6:(C_7^2:S_3)$ x 2, $C_7\times C_2^6.F_7$, $F_8\times C_2\times F_8:C_3$, $F_8.F_8.C_6$, $C_2\times F_8.F_8.C_3$, $C_7.C_2^6:C_7.C_6$, $C_2^6.C_7^2.C_6$, $C_2^6.C_7^2.C_6$, $C_2^6.C_7^2.C_6$, $C_2^6.C_7^2.C_6$, $C_7\times C_2^6:C_7.C_6$, $C_2^6.C_7^2.C_6$, $C_2^6.C_7:F_7$, $C_2^6.C_7^2.C_6$, $C_2\times C_2^6.C_7^2.C_3$

Order 18522: $C_7^3:(S_3\times C_3^2)$, $C_7^3:C_3^2:S_3$

Order 16464: $C_7^2.(S_3\times F_8)$, $C_2^3.C_7^3.C_6$, $F_8.C_7^2.C_6$, $C_7^2.F_8.C_6$, $(C_7\times F_8).F_7$

Order 16128: $F_8.A_4^2.C_2$ x 7, $F_8.A_4^2.C_2$ x 6, $F_8.C_2^4.C_3.C_6$ x 4, $C_2\times F_8.A_4.C_6.C_2$ x 2, $C_2^7.C_{21}.C_6$ x 2, $C_2^7.C_{21}.C_6$ x 2, $A_4\wr C_2\times F_8$, $F_8.A_4.C_6.C_2^2$, $F_8.C_2^4.C_3.C_6$, $C_2^7.C_{21}.C_6$, $F_8.(C_3\times D_4\times A_4)$, $C_2^6:C_7.A_4.C_3$, $C_2\times C_2^6.C_{21}.C_6$, $C_2^5.C_{42}.C_6.C_2$, $C_2\wr F_7\times C_3$, $C_2^8.C_{21}:C_3$

Order 14112: $(C_7\times F_8).A_4.C_3$ x 2, $(C_7\times F_8).A_4.C_3$ x 2, $C_2^3.C_{14}^2.C_3^2$ x 2, $(C_7\times F_8).A_4.C_3$ x 2, $(C_2\times C_{14}).F_8.C_3^2$, $(C_7\times F_8).A_4.C_3$, $C_2^2\times C_2^3:(C_7^2:C_3:C_3)$, $F_8.(C_{21}\times A_4)$, $(C_7\times F_8).A_4.C_3$

Order 13824: $A_4^3:D_4$ x 4, $C_2^2:A_4^2.S_4$ x 2, $C_2^4.(C_6^2\times A_4).C_2$, $C_2\times C_2^6.C_3^3.C_2^2$, $C_2^4.(C_6^2\times A_4).C_2$, $C_2\wr C_9.C_3$, $C_2^9:\He_3$

Order 12544: $C_2\times F_8\wr C_2$ x 2, $C_2^6.C_{14}^2$

Order 12096: $C_6\times F_8.A_4.C_3$ x 2, $C_7.A_4^2.C_6.C_2$, $F_8.(C_3\times A_4).C_6$, $(C_2^3\times C_{14}).C_3^3.C_4$, $(C_7\times C_2^4:C_3).C_6^2$, $(C_6\times F_8).C_6^2$

Order 10752: $C_2^6:C_7.A_4.C_2$ x 4, $C_2^8.F_7$ x 4, $F_8.C_2^4.C_6.C_2$ x 3, $C_2^7.C_{42}.C_2$ x 2, $F_8.A_4.C_2^4$ x 2, $C_2^8.C_{42}$ x 2, $F_8.C_2^4.C_6.C_2$ x 2, $C_2\times C_2^6:C_7.C_6.C_2$ x 2, $F_8.C_2^4.C_{12}$ x 2, $C_2^6.F_8:C_3$ x 2, $A_4\times C_2^6:D_7$ x 2, $C_2\wr C_7.A_4$ x 2, $C_2^4.F_8:A_4$ x 2, $C_2^8.F_7$ x 2, $C_2\times C_2^8.C_{21}$, $C_2^4:C_3.F_8.C_4$, $C_2^4.(S_3\times F_8).C_2$, $F_8:C_3.C_2^6$, $C_2\times C_2^5.C_{42}.C_2^2$, $C_2^3:F_8.C_6.C_2^2$, $(C_2\times F_8).C_6.C_2^4$, $C_2^6:C_7.C_6.C_2^2$, $C_2^7.C_{84}$, $C_2^7.C_{42}.C_2$, $F_8.(C_2^4\times C_6).C_2$, $C_2^9:C_{21}$, $C_2^9:C_{21}$, $C_2^9:C_7:C_3$

Order 10584: $C_7^2.(C_3\times A_4).C_6$ x 2, $(C_7\times F_8).C_3^3$, $C_{14}^2.C_3^3.C_2$

Order 10368: $A_4\wr S_3$ x 2, $C_2^6.C_3^3.C_6$

Order 9408: $F_8.F_8.C_3$ x 9, $C_2^6.C_7^2.C_3$ x 9, $C_2^6.C_7^2.C_3$ x 5, $(C_7\times F_8).A_4.C_2$ x 2, $C_7\times F_8.A_4.C_2$ x 2, $(C_2\times C_{14}).F_8.C_6$ x 2, $C_2\times (C_7\times F_8).C_6.C_2$, $F_8.C_{42}.C_2^2$, $C_2^6.C_7^2.C_3$, $C_2^3\times C_2^3:(C_7^2:C_3)$, $C_2^6.C_7^2.C_3$, $C_7\times C_2^3:F_8.C_3$, $C_2^3\times C_2^3:(C_7:(C_7:C_3))$, $C_2^6.C_7^2.C_3$, $C_2^6.C_7^2.C_3$, $F_8.C_7:A_4.C_2$, $(C_7\times F_8).A_4.C_2$, $C_2^5.C_7^2.C_6$, $C_2^6.C_7^2.C_3$, $C_2^6:C_7\times C_7:C_3$, $C_2^3:F_8\times C_7:C_3$, $C_7\times C_2^6:C_7.C_3$

Order 9261: $C_7^3:C_3^3$, $C_7^3:C_9:C_3$, $C_7^3:\He_3$

Order 9216: $C_2^4.(C_6\times A_4).C_2^3$ x 2, $C_2^4.(C_6\times A_4).C_2^3$, $C_2^4.C_6^2.C_2^4$, $C_2^7.C_6^2.C_2$, $C_2^5.(A_4\times S_4)$

Order 8232: $C_2^3:(C_7\times C_7^2:C_3)$, $C_2^3:(C_7^2\times C_7:C_3)$, $C_2^3:(C_7\times C_7^2:C_3)$, $C_7\times C_2^3:(C_7\times C_7:C_3)$, $C_2^3:(C_7\times C_7:(C_7:C_3))$, $C_2^3:(C_7^3:C_3)$, $C_2^3:(C_7^3:C_3)$, $C_2^3:(C_7^3:C_3)$, $C_2^3:(C_7\times C_7:(C_7:C_3))$

Order 8064: $F_8.A_4.C_6.C_2$ x 7, $F_8.A_4^2$ x 3, $F_8.(C_6\times A_4).C_2$ x 3, $C_2\times F_8.A_4.C_6$ x 3, $C_2^6.C_{21}.C_6$ x 2, $F_8.A_4^2$ x 2, $C_2^4.C_{42}.C_6.C_2$ x 2, $F_8.C_2^4.C_3^2$ x 2, $C_3\times C_2^6:F_7$ x 2, $C_2^6:C_7.C_3.C_6$, $F_8:C_3\times C_4\times A_4$, $F_8:C_3\times A_4:C_4$, $C_2^4.C_{42}.C_6.C_2$, $C_7\times C_2^4.C_6^2.C_2$, $F_8:C_3\times C_2^2\times A_4$, $C_2\times A_4\times C_2\times F_8:C_3$, $C_2^3\times C_6\times F_8:C_3$, $C_2^5.(C_6\times C_{42})$, $F_8.(C_6\times A_4).C_2$, $C_2\times F_8.A_4.C_6$, $(C_2^4\times C_{14}).C_6^2$, $C_2^7.C_{21}.C_3$, $F_8:C_6\times S_4$, $C_2\wr C_7:C_3^2$, $C_2^6:C_{21}:C_6$

Order 7168: $C_2^9.C_{14}$, $C_2^3\times C_2^6.C_{14}$, $F_8\times C_2^6.C_2$, $C_2^3\times C_2^6.D_7$, $C_2^9.C_{14}$

Order 7056: $C_2\times C_2^3:(C_7^2:C_3:C_3)$ x 4, $(C_7\times F_8).C_3.C_6$ x 3, $F_8.C_{21}.C_6$ x 2, $F_8.C_7:C_3.C_6$ x 2, $C_7^2.(C_6\times A_4).C_2$ x 2, $(C_7\times F_8).C_3.C_6$, $(C_7\times F_8).C_3.C_6$, $(C_7\times F_8).C_3.C_6$, $C_{14}^2.C_6^2$, $F_8.C_{21}.C_6$, $C_7^2.(C_6\times A_4).C_2$, $C_2^3\times C_7^2:(C_3\times S_3)$, $F_7\times F_8:C_3$

Order 6912: $A_4^3:C_2^2$ x 4, $C_2\times C_2^6.C_3^3.C_2$ x 3, $A_4^3:C_4$ x 2, $C_2^4.C_6^3.C_2$, $C_2^6.C_6^2.C_3$, $C_2\times C_2^6.\He_3.C_2$, $C_2^2:A_4^2.A_4$

Order 6272: $F_8\wr C_2$ x 4, $C_2^6.C_7^2.C_2$ x 2, $C_7\times C_2^6.C_{14}$, $C_7\times C_2^6.D_7$

Order 6174: $C_7^3:(C_3\times S_3)$ x 2, $C_7^3:(C_3\times S_3)$, $C_7^3:(C_3\times S_3)$, $C_7^2:(S_3\times C_{21})$, $C_7^3:(C_3\times C_6)$

Order 6048: $C_7.A_4^2.C_6$ x 2, $F_8.C_3^3.C_2^2$, $F_8.C_3^3.C_4$, $F_8.C_3^3.C_2^2$, $(C_7\times A_4^2):C_6$, $A_4\times F_8:C_3^2$

Order 5488: $(C_7\times F_8).C_{14}$

Order 5376: $F_8.C_2^4.C_6$ x 6, $F_8.A_4.C_2^3$ x 5, $C_2^4:F_8:C_6$ x 4, $C_2\wr F_7$ x 4, $C_2\times C_2^7.C_{21}$ x 3, $F_8:C_3.C_2^5$ x 2, $C_2^5.C_{42}.C_2^2$ x 2, $F_8.C_2^4.C_6$ x 2, $F_8.(D_4\times A_4)$ x 2, $C_2^3.F_8:A_4$ x 2, $C_2^3.F_8:A_4$ x 2, $C_2\times F_8.C_2^4.C_3$, $C_2\times F_8.A_4.C_2^2$, $C_2\times C_2^5.C_{42}.C_2$, $F_8\times C_2^2:S_4$, $(C_2\times F_8).C_6.C_2^3$, $F_8\times \GL(2,\mathbb{Z}/4)$, $F_8\times C_2^2\wr C_2:C_3$, $C_2^7.C_{42}$, $C_2\times C_2^6.C_{42}$, $D_4\times A_4\times F_8$, $C_2^8.C_{21}$, $F_8.(C_2^3\times C_6).C_2$, $(C_2\times F_8).C_6.C_2^3$, $F_8.(C_6\times D_4).C_2$, $C_2\wr D_7\times C_3$, $C_2^8:C_{21}$, $C_2\wr C_7.C_6$, $C_2^5:F_8:C_3$

Order 5292: $C_7^2:(C_3^2\times A_4)$, $C_7^2:(C_3^2\times D_6)$

Order 5184: $A_4\wr C_3$

Order 4704: $(C_7\times F_8).A_4$ x 2, $F_8.C_7:A_4$ x 2, $F_8.(C_7\times A_4)$ x 2, $(C_2^4\times C_{14}).C_{21}$, $(C_2\times C_{14}).F_8.C_3$, $(C_7\times F_8).C_6.C_2$, $F_8.C_{42}.C_2$, $C_2^5.C_7^2.C_3$, $(C_7\times F_8).C_6.C_2$, $(C_7\times F_8).C_6.C_2$

Order 4608: $C_2\wr S_3:A_4$ x 8, $C_2^4.C_6^2.C_2^3$ x 4, $A_4\times C_2\wr S_3$ x 4, $C_2^7.C_6^2$ x 4, $C_2\times C_2^6.C_6^2$ x 2, $C_2\times C_2^6.C_6^2$ x 2, $C_2^4.(C_6\times A_4).C_2^2$ x 2, $C_2^5.(A_4\times D_6)$ x 2, $C_2^2\times A_4^2.C_2^3$, $C_2^6.(S_3\times A_4)$, $C_2^2\times A_4^2.C_2^3$, $C_2\times C_2^6.C_6^2$, $C_2^2.A_4^2.C_2^2.C_2$, $A_4^2.C_2^4.C_2$, $C_2^6.C_6^2.C_2$, $C_2^4.(A_4\times S_4)$, $C_2^4:(A_4\times S_4)$, $C_2\wr C_9$, $C_2^9:C_3^2$

Order 4032: $A_4\times F_8:C_6$ x 37, $F_8.(C_6\times A_4)$ x 4, $C_7.A_4^2.C_2^2$ x 3, $F_8.C_6^2.C_2$ x 3, $(D_6\times F_8):C_6$ x 3, $C_7.A_4^2.C_2^2$ x 2, $(C_7\times C_2^4:C_3).C_6.C_2$ x 2, $C_3\times C_2^5.C_{42}$ x 2, $(C_2^3\times C_{14}).C_6^2$ x 2, $(C_7\times C_2^4:C_3).C_{12}$ x 2, $(C_2^3\times C_{14}).C_6^2$, $(C_6\times F_8).C_6.C_2$, $C_2^4.(C_6\times C_{42})$, $C_2\times C_2^2:S_4\times C_7:C_3$, $C_2^4:C_3.C_{84}$, $C_3\times A_4\times C_2\times F_8$, $(C_2^4\times C_7:C_3).C_{12}$, $F_8.A_4.C_6$, $C_2\times F_8.(C_3\times A_4)$, $C_2^6.C_{21}.C_3$, $C_2\times (A_4\times C_7:C_3).C_2^3$, $C_2^2:(A_4:C_4)\times C_7:C_3$, $C_2^4.(C_{42}\times S_3)$, $C_2^6:C_7.C_3^2$, $D_4\times F_8:C_3^2$, $C_2^6:C_{21}:C_3$, $F_8:C_3\times S_4$

Order 3584: $C_2^2\times C_2^6.C_{14}$ x 2, $C_2^7:D_{14}$ x 2, $C_2^3\times C_2^3:F_8$, $(C_2^3\times F_8).C_2^3$, $C_2^3\times C_2^6:C_7$, $F_8.C_2^6$, $(C_2^3\times F_8).C_2^3$, $C_2^9.C_7$, $C_2^9:C_7$

Order 3528: $(C_7\times F_8):C_3^2$ x 14, $C_7^2:(S_3\times A_4)$ x 4, $C_7^2:(C_6\times A_4)$ x 2, $C_7^2:(C_6\times A_4)$ x 2, $C_7^2:(C_6\times A_4)$ x 2, $C_7^2:(S_3\times A_4)$ x 2, $C_2^2\times C_7^2:(C_3\times S_3)$ x 2, $C_7^2:(C_6\times A_4)$, $C_{21}\times F_8:C_3$, $C_3\times C_2^3:(C_7^2:C_3)$, $C_7^2:(C_6\times A_4)$, $C_2^3\times C_7^2:C_3:C_3$, $C_2^3:(C_3\times C_7:(C_7:C_3))$, $C_3\times F_8\times C_7:C_3$, $C_7^2:(C_6\times A_4)$, $C_2\times C_{14}^2:C_3^2$

Order 3456: $C_3\times A_4^2:D_4$ x 4, $C_2\times A_4^3$ x 3, $A_4^3:C_2$ x 2, $C_2^2:A_4^2:S_3$ x 2, $C_2^6.\He_3.C_2$, $C_2^4:C_3.C_6^2.C_2$, $C_2^4.C_6^3$, $A_4^2:C_2^2\times C_6$, $C_2^6:C_9:C_6$

Order 3136: $C_2^6.C_7^2$ x 7, $F_8.C_{14}.C_2^2$, $C_2^6.C_7^2$, $C_2^6:C_7^2$

Order 3087: $C_7^3:C_3^2$ x 2, $C_7^3:C_3^2$, $C_7^3:C_3^2$, $C_7^3:C_3^2$, $C_7^3:C_3^2$, $C_7^3:C_9$

Order 3072: $C_2^7.C_6.C_2^2$ x 2, $(C_2^4\times A_4).C_2^4$, $C_2^4:C_3.C_2^5.C_2$, $C_2^5.(C_2^4\times C_6)$, $C_2^7.S_4$

Order 3024: $F_8.C_3^3.C_2$, $C_3\times C_6\times F_8:C_3$, $C_7.(C_6^2\times A_4)$, $C_7:C_3\times A_4^2$, $S_3\times F_8:C_3^2$

Order 2744: $C_7^2\times F_8$

Order 1024: $C_2^2\times C_2^6.C_2^2$

Order 512: $C_2^9$

Order 343: $C_7^3$

Order 81: $C_3\wr C_3$

Order 1: $C_1$

Normal subgroups (quotient in parentheses)

Order 28449792: $C_2^9.C_7^3:C_3\wr S_3$ ($C_1$)

Order 14224896: $C_2^9:(C_7^3:C_3\wr C_3)$ ($C_2$)

Order 9483264: $C_2^9.C_7^3:C_3^2:S_3$ ($C_3$)

Order 4741632: $C_2^9.C_7^3.C_3^3$ ($S_3$), $C_2^9:(C_7^3:\He_3)$ ($C_6$)

Order 1580544: $C_2^9.C_7^3.C_3^2$ ($C_3\times S_3$)

Order 526848: $C_2^9.C_7^3.C_3$ ($C_3^2:C_6$)

Order 175616: $C_2^6.C_7^2\times F_8$ ($C_3\wr S_3$)

Order 512: $C_2^9$ ($C_7^3:C_3\wr S_3$)

Order 1: $C_1$ ($C_2^9.C_7^3:C_3\wr S_3$)

Normal subgroups up to automorphism (quotient in parentheses)

Order 28449792: $C_2^9.C_7^3:C_3\wr S_3$ ($C_1$)

Order 14224896: $C_2^9:(C_7^3:C_3\wr C_3)$ ($C_2$)

Order 9483264: $C_2^9.C_7^3:C_3^2:S_3$ ($C_3$)

Order 4741632: $C_2^9.C_7^3.C_3^3$ ($S_3$), $C_2^9:(C_7^3:\He_3)$ ($C_6$)

Order 1580544: $C_2^9.C_7^3.C_3^2$ ($C_3\times S_3$)

Order 526848: $C_2^9.C_7^3.C_3$ ($C_3^2:C_6$)

Order 175616: $C_2^6.C_7^2\times F_8$ ($C_3\wr S_3$)

Order 512: $C_2^9$ ($C_7^3:C_3\wr S_3$)

Order 1: $C_1$ ($C_2^9.C_7^3:C_3\wr S_3$)

Series

Derived series

$C_2^9.C_7^3:C_3\wr S_3$

$\rhd$

$C_2^9:(C_7^3:\He_3)$

$\rhd$

$C_2^9.C_7^3.C_3$

$\rhd$

$C_2^6.C_7^2\times F_8$

$\rhd$

$C_2^9$

$\rhd$

$C_1$

Chief series

$C_2^9.C_7^3:C_3\wr S_3$

$\rhd$

$C_2^9:(C_7^3:C_3\wr C_3)$

$\rhd$

$C_2^9:(C_7^3:\He_3)$

$\rhd$

$C_2^9.C_7^3.C_3^2$

$\rhd$

$C_2^9.C_7^3.C_3$

$\rhd$

$C_2^6.C_7^2\times F_8$

$\rhd$

$C_2^9$

$\rhd$

$C_1$

Lower central series

$C_2^9.C_7^3:C_3\wr S_3$

$\rhd$

$C_2^9:(C_7^3:\He_3)$

Upper central series

$C_1$

Supergroups

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

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

Character theory

Complex character table

See the $192 \times 192$ character table (warning: may be slow to load). Alternatively, you may search for characters of this group with desired properties.

Rational character table

See the $90 \times 90$ rational character table.