Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
429 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 429 conjugacy classes of subgroups — too many to draw. Only its 168 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2177 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| prime radical | C2×#2263 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2108 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #33465 | C2 | semidirect |
| 3. #33456 | C2 | semidirect |
| 4. #13658 | C2 | semidirect |
| 5. #13638 | C2 | semidirect |
| 6. C2×#1419 | C2 | semidirect |
| 7. #13622 | C2 | semidirect |
| 8. #13551 | C2 | semidirect |
| 13. #2263 | C4 | semidirect |
| 15. C2×SD64 | C4 | semidirect |
| 16. C2×D32 | C4 | semidirect |
| 19. #2179 | C22 | semidirect |
| 21. #2177 | C22 | semidirect |
| 28. #1426 | C22 | semidirect |
| 29. #1419 | C22 | semidirect |
| 30. #1418 | C22 | semidirect |
| 31. #1398 | C22 | semidirect |
| 38. #508 | C2×C4 | semidirect |
| 40. C2×Q32 | C2×C4 | semidirect |
| 47. C2×D16 | C2×C4 | semidirect |
| 60. SD64 | C2×C4 | semidirect |
| 61. D32 | C2×C4 | semidirect |
| 63. M64 | C2×C4 | semidirect |
| 67. C2×C32 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2263 | 2 |
| #33465 | 2 |
| #33456 | 2 |
| #13658 | 2 |
| #13638 | 2 |
| C2×#1419 | 2 |
| #13622 | 2 |
| #13551 | 2 |