Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1167 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 1167 conjugacy classes of subgroups — too many to draw. Only its 487 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C8 | C25 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#135 | C2 | semidirect |
| 2. C23×D8 | C2 | semidirect |
| 3. C2×#509 | C2 | direct (N × H) |
| 4. C22×SD32 | C2 | semidirect |
| 5. C22×D16 | C2 | semidirect |
| 6. C22×M32 | C2 | semidirect |
| 10. C2×#135 | C22 | semidirect |
| 11. C22×Q16 | C22 | semidirect |
| 16. C22×D8 | C22 | semidirect |
| 17. C22×D8 | C22 | semidirect |
| 19. #509 | C22 | direct (N × H) |
| 20. C2×SD32 | C22 | semidirect |
| 21. C2×D16 | C22 | semidirect |
| 22. C2×M32 | C22 | semidirect |
| 23. C22×C16 | C22 | semidirect |
| 37. #135 | C23 | semidirect |
| 39. C2×Q16 | C23 | semidirect |
| 46. C2×D8 | C23 | semidirect |
| 47. C2×D8 | C23 | semidirect |
| 52. SD32 | C23 | semidirect |
| 53. D16 | C23 | semidirect |
| 54. M32 | C23 | semidirect |
| 55. C2×C16 | C23 | semidirect |
| 80. Q16 | C24 | semidirect |
| 86. D8 | C24 | semidirect |
| 92. C16 | C24 | semidirect |
| 126. C22 | #509 | direct (N × H) |
| 136. C2 | C2×#509 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#135 | 2 |
| C23×D8 | 2 |
| C2×#509 | 2 |
| C22×SD32 | 2 |
| C22×D16 | 2 |
| C22×M32 | 2 |