Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
421205 conjugacy classes of subgroups.
| # | subgroup | G/H | class size | multiplicity | normal? | properties |
|---|---|---|---|---|---|---|
| 1 | #63103 | C2 | 1 | 120 | yes | maximal, complemented |
| 2 | C2×H(7,𝔽2) | C2 | 1 | 135 | yes | maximal, complemented |
| 3 | 2−1+6 | C22 | 1 | 1120 | yes | complemented |
| 4 | H(7,𝔽2) | C22 | 1 | 4320 | yes | complemented |
| 5 | C2×#585 | C22 | 1 | 3780 | yes | complemented |
| 6 | C22×H(5,𝔽2) | C22 | 1 | 1575 | yes | complemented |
| 7 | #585 | C23 | 1 | 40320 | yes | complemented |
| 8 | C2×2−1+4 | C23 | 1 | 7560 | yes | complemented |
| 9 | C2×H(5,𝔽2) | C23 | 1 | 37800 | yes | complemented |
| 10 | C22×#41 | C23 | 1 | 9450 | yes | complemented |
| 11 | C23×D4 | C23 | 1 | 2025 | yes | complemented |
| 12 | C25 | C24 | 1 | 270 | yes | complemented |
| 13 | 2−1+4 | C24 | 1 | 24192 | yes | complemented |
| 14 | H(5,𝔽2) | C24 | 1 | 67200 | yes | complemented |
| 15 | C2×#41 | C24 | 1 | 75600 | yes | complemented |
| 16 | C22×Q8 | C24 | 1 | 3150 | yes | complemented |
| 17 | C22×D4 | C24 | 1 | 28350 | yes | complemented |
| 18 | C23×C4 | C24 | 1 | 2025 | yes | complemented |
| 19 | C24 | — | 16 | 270 | no | — |
| 20 | C24 | C25 | 1 | 2025 | yes | — |
| 21 | #41 | C25 | 1 | 40320 | yes | — |
| 22 | C2×Q8 | C25 | 1 | 7560 | yes | — |
| 23 | C2×D4 | C25 | 1 | 37800 | yes | — |
| 24 | C22×C4 | C25 | 1 | 9450 | yes | — |
| 25 | C23 | — | 8 | 2025 | no | — |
| 26 | C23 | C26 | 1 | 1575 | yes | — |
| 27 | Q8 | C26 | 1 | 1120 | yes | — |
| 28 | D4 | C26 | 1 | 4320 | yes | — |
| 29 | C2×C4 | C26 | 1 | 3780 | yes | — |
| 30 | C22 | — | 4 | 1575 | no | — |
| 31 | C22 | C27 | 1 | 135 | yes | — |
| 32 | C4 | C27 | 1 | 120 | yes | — |
| 33 | C2 | — | 2 | 135 | no | — |
| 34 | C2 | C28 | 1 | 1 | yes | characteristic |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C2 | C28 |
| 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. #63103 | C2 | semidirect |
| 2. C2×H(7,𝔽2) | C2 | semidirect |
| 3. 2−1+6 | C22 | semidirect |
| 4. H(7,𝔽2) | C22 | semidirect |
| 5. C2×#585 | C22 | semidirect |
| 6. C22×H(5,𝔽2) | C22 | semidirect |
| 7. #585 | C23 | semidirect |
| 8. C2×2−1+4 | C23 | semidirect |
| 9. C2×H(5,𝔽2) | C23 | semidirect |
| 10. C22×#41 | C23 | semidirect |
| 11. C23×D4 | C23 | semidirect |
| 12. C25 | C24 | semidirect |
| 13. 2−1+4 | C24 | semidirect |
| 14. H(5,𝔽2) | C24 | semidirect |
| 15. C2×#41 | C24 | semidirect |
| 16. C22×Q8 | C24 | semidirect |
| 17. C22×D4 | C24 | semidirect |
| 18. C23×C4 | C24 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #63103 | 2 |
| C2×H(7,𝔽2) | 2 |