Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
302 conjugacy classes of subgroups; only the 114 normal subgroups are shown.
This group has 302 conjugacy classes of subgroups — too many to draw. Only its 114 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 | C2×D32 |
| commutator, Frattini, cosocle | C32 | C24 |
| LCS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ C22×C32 ◁ G | |
| LECS | G ▷ C32 ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#2262 | C2 | semidirect |
| 2. C22×D32 | C2 | semidirect |
| 3. #13742 | C2 | direct (N × H) |
| 4. C2×SD128 | C2 | semidirect |
| 5. C2×D64 | C2 | semidirect |
| 6. C2×M128 | C2 | semidirect |
| 10. #2262 | C22 | semidirect |
| 11. C2×Q64 | C22 | semidirect |
| 16. C2×D32 | C22 | semidirect |
| 17. C2×D32 | C22 | semidirect |
| 19. SD128 | C22 | semidirect |
| 20. D64 | C22 | semidirect |
| 21. M128 | C22 | semidirect |
| 22. C2×C64 | C22 | semidirect |
| 36. Q64 | C23 | semidirect |
| 42. D32 | C23 | semidirect |
| 47. C64 | C23 | semidirect |
| 139. C2 | #13742 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2262 | 2 |
| C22×D32 | 2 |
| #13742 | 2 |
| C2×SD128 | 2 |
| C2×D64 | 2 |
| C2×M128 | 2 |