Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
703 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 703 conjugacy classes of subgroups — too many to draw. Only its 231 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 | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| prime radical | C23×D8 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×M32 | C2 | semidirect |
| 3. C2×#427 | C2 | semidirect |
| 4. #2140 | C2 | direct (N × H) |
| 5. C2×#357 | C2 | semidirect |
| 10. C22×D8 | C4 | semidirect |
| 16. C2×M32 | C22 | semidirect |
| 17. C22×C16 | C22 | semidirect |
| 19. #427 | C22 | semidirect |
| 20. C2×#107 | C22 | semidirect |
| 22. #357 | C22 | semidirect |
| 38. C2×D8 | C2×C4 | semidirect |
| 53. C2×C16 | C23 | semidirect |
| 55. #107 | C23 | semidirect |
| 142. C2 | #2140 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×D8 | 2 |
| C22×M32 | 2 |
| C2×#427 | 2 |
| #2140 | 2 |
| C2×#357 | 2 |