Finite Group DatabasebetaFinite Group DatabasebetaOrder 8192.
488194502149 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #72274291 | C2 | 1 | 2080 | yes | maximal, complemented |
| 2 | C2×2−1+10 | C2 | 1 | 2015 | yes | maximal, complemented |
| 3 | C2×#72258201 | C22 | 1 | 1063920 | yes | complemented |
| 4 | C22×2−1+8 | C22 | 1 | 332475 | yes | complemented |
| 5 | C22×#63103 | C23 | 1 | 45216600 | yes | complemented |
| 6 | C23×2−1+6 | C23 | 1 | 5652075 | yes | complemented |
| 7 | 2−1+10 | C22 | 1 | 1031680 | yes | complemented |
| 8 | H(11,𝔽2) | C22 | 1 | 366080 | yes | complemented |
| 9 | #72258201 | C23 | 1 | 181575680 | yes | complemented |
| 10 | C2×2−1+8 | C23 | 1 | 127670400 | yes | complemented |
| 11 | C2×H(9,𝔽2) | C23 | 1 | 48231040 | yes | complemented |
| 12 | 2−1+8 | C24 | 1 | 3631513600 | yes | complemented |
| 13 | H(9,𝔽2) | C24 | 1 | 2469429248 | yes | complemented |
| 14 | C2×#63103 | C24 | 1 | 5787724800 | yes | complemented |
| 15 | C22×2−1+6 | C24 | 1 | 1266064800 | yes | complemented |
| 16 | C22×H(7,𝔽2) | C24 | 1 | 542599200 | yes | complemented |
| 17 | C23×#585 | C24 | 1 | 203474700 | yes | complemented |
| 18 | C24×2−1+4 | C24 | 1 | 10173735 | yes | complemented |
| 19 | #63103 | C25 | 1 | 49388584960 | yes | complemented |
| 20 | C2×2−1+6 | C25 | 1 | 27009382400 | yes | complemented |
| 21 | C2×H(7,𝔽2) | C25 | 1 | 20835809280 | yes | complemented |
| 22 | C22×#585 | C25 | 1 | 15192777600 | yes | complemented |
| 23 | C23×2−1+4 | C25 | 1 | 1220848200 | yes | complemented |
| 24 | C23×H(5,𝔽2) | C25 | 1 | 678249000 | yes | complemented |
| 25 | C24×#41 | C25 | 1 | 101737350 | yes | complemented |
| 26 | C25×Q8 | C25 | 1 | 1640925 | yes | complemented |
| 27 | 2−1+6 | C26 | 1 | 49388584960 | yes | — |
| 28 | H(7,𝔽2) | C26 | 1 | 49388584960 | yes | — |
| 29 | C2×#585 | C26 | 1 | 97233776640 | yes | — |
| 30 | C22×2−1+4 | C26 | 1 | 15192777600 | yes | — |
| 31 | C22×H(5,𝔽2) | C26 | 1 | 15192777600 | yes | — |
| 32 | C23×#41 | C26 | 1 | 4069494000 | yes | — |
| 33 | C24×Q8 | C26 | 1 | 101737350 | yes | — |
| 34 | C24×D4 | C26 | 1 | 101737350 | yes | — |
| 35 | C25×C4 | C26 | 1 | 4922775 | yes | — |
| 36 | C26 | C27 | 1 | 1640925 | yes | — |
| 37 | #585 | C27 | 1 | 49388584960 | yes | — |
| 38 | C2×2−1+4 | C27 | 1 | 20835809280 | yes | — |
| 39 | C2×H(5,𝔽2) | C27 | 1 | 27009382400 | yes | — |
| 40 | C22×#41 | C27 | 1 | 15192777600 | yes | — |
| 41 | C23×Q8 | C27 | 1 | 678249000 | yes | — |
| 42 | C23×D4 | C27 | 1 | 1220848200 | yes | — |
| 43 | C24×C4 | C27 | 1 | 101737350 | yes | — |
| 44 | C25 | — | 32 | 1640925 | no | — |
| 45 | C25 | C28 | 1 | 10173735 | yes | — |
| 46 | 2−1+4 | C28 | 1 | 2469429248 | yes | — |
| 47 | H(5,𝔽2) | C28 | 1 | 3631513600 | yes | — |
| 48 | C2×#41 | C28 | 1 | 5787724800 | yes | — |
| 49 | C22×Q8 | C28 | 1 | 542599200 | yes | — |
| 50 | C22×D4 | C28 | 1 | 1266064800 | yes | — |
| 51 | C23×C4 | C28 | 1 | 203474700 | yes | — |
| 52 | C24 | — | 16 | 10173735 | no | — |
| 53 | C24 | C29 | 1 | 5652075 | yes | — |
| 54 | #41 | C29 | 1 | 181575680 | yes | — |
| 55 | C2×Q8 | C29 | 1 | 48231040 | yes | — |
| 56 | C2×D4 | C29 | 1 | 127670400 | yes | — |
| 57 | C22×C4 | C29 | 1 | 45216600 | yes | — |
| 58 | C23 | — | 8 | 5652075 | no | — |
| 59 | C23 | C210 | 1 | 332475 | yes | — |
| 60 | Q8 | C210 | 1 | 366080 | yes | — |
| 61 | D4 | C210 | 1 | 1031680 | yes | — |
| 62 | C2×C4 | C210 | 1 | 1063920 | yes | — |
| 63 | C22 | — | 4 | 332475 | no | — |
| 64 | C22 | C211 | 1 | 2015 | yes | — |
| 65 | C4 | C211 | 1 | 2080 | yes | — |
| 66 | C2 | — | 2 | 2015 | no | — |
| 67 | C2 | C212 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C2 | C212 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ G | |
| LECS | G ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #72274291 | C2 | semidirect |
| 2. C2×2−1+10 | C2 | semidirect |
| 3. C2×#72258201 | C22 | semidirect |
| 4. C22×2−1+8 | C22 | semidirect |
| 5. C22×#63103 | C23 | semidirect |
| 6. C23×2−1+6 | C23 | semidirect |
| 7. 2−1+10 | C22 | semidirect |
| 8. H(11,𝔽2) | C22 | semidirect |
| 9. #72258201 | C23 | semidirect |
| 10. C2×2−1+8 | C23 | semidirect |
| 11. C2×H(9,𝔽2) | C23 | semidirect |
| 12. 2−1+8 | C24 | semidirect |
| 13. H(9,𝔽2) | C24 | semidirect |
| 14. C2×#63103 | C24 | semidirect |
| 15. C22×2−1+6 | C24 | semidirect |
| 16. C22×H(7,𝔽2) | C24 | semidirect |
| 17. C23×#585 | C24 | semidirect |
| 18. C24×2−1+4 | C24 | semidirect |
| 19. #63103 | C25 | semidirect |
| 20. C2×2−1+6 | C25 | semidirect |
| 21. C2×H(7,𝔽2) | C25 | semidirect |
| 22. C22×#585 | C25 | semidirect |
| 23. C23×2−1+4 | C25 | semidirect |
| 24. C23×H(5,𝔽2) | C25 | semidirect |
| 25. C24×#41 | C25 | semidirect |
| 26. C25×Q8 | C25 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #72274291 | 2 |
| C2×2−1+10 | 2 |