Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
251 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 251 conjugacy classes of subgroups — too many to draw. Only its 103 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×D16 |
| commutator, Frattini, cosocle | C16 | C24 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C22×C16 ◁ G | |
| LECS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#508 | C2 | semidirect |
| 2. C22×D16 | C2 | semidirect |
| 3. #2263 | C2 | direct (N × H) |
| 4. C2×SD64 | C2 | semidirect |
| 5. C2×D32 | C2 | semidirect |
| 6. C2×M64 | C2 | semidirect |
| 10. #508 | C22 | semidirect |
| 11. C2×Q32 | C22 | semidirect |
| 16. C2×D16 | C22 | semidirect |
| 17. C2×D16 | C22 | semidirect |
| 19. SD64 | C22 | semidirect |
| 20. D32 | C22 | semidirect |
| 21. M64 | C22 | semidirect |
| 22. C2×C32 | C22 | semidirect |
| 36. Q32 | C23 | semidirect |
| 42. D16 | C23 | semidirect |
| 47. C32 | C23 | semidirect |
| 112. C2 | #2263 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#508 | 2 |
| C22×D16 | 2 |
| #2263 | 2 |
| C2×SD64 | 2 |
| C2×D32 | 2 |
| C2×M64 | 2 |