Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
452 conjugacy classes of subgroups; only the 120 normal subgroups are shown.
This group has 452 conjugacy classes of subgroups — too many to draw. Only its 120 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ D4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#2263 | C2 | semidirect |
| 2. #33846 | C2 | semidirect |
| 3. #13728 | C2 | semidirect |
| 4. #13699 | C2 | semidirect |
| 5. #13689 | C2 | semidirect |
| 6. #13666 | C2 | semidirect |
| 7. #13661 | C2 | semidirect |
| 8. C4×D32 | C2 | semidirect |
| 9. #13646 | C2 | semidirect |
| 11. C22×D16 | C22 | semidirect |
| 14. C2×Q64 | C22 | semidirect |
| 15. C2×SD64 | C22 | semidirect |
| 16. C2×SD64 | C22 | semidirect |
| 20. C2×M64 | C22 | semidirect |
| 21. #2249 | C22 | semidirect |
| 22. #2216 | C22 | semidirect |
| 27. C4×D16 | C22 | semidirect |
| 29. #1423 | C22 | semidirect |
| 30. C4⋊C32 | C22 | semidirect |
| 31. #1416 | C22 | semidirect |
| 32. #1415 | C22 | semidirect |
| 33. #1415 | C22 | semidirect |
| 34. C22⋊C32 | C22 | semidirect |
| 35. C4×C32 | C22 | semidirect |
| 63. D32 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2263 | 2 |
| #33846 | 2 |
| #13728 | 2 |
| #13699 | 2 |
| #13689 | 2 |
| #13666 | 2 |
| #13661 | 2 |
| C4×D32 | 2 |
| #13646 | 2 |