Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1273 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 1273 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 | C24 | C24 |
| commutator, Frattini, cosocle | C23 | C25 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #2643 | C2 | direct (N × H) |
| 2. C2×#401 | C2 | semidirect |
| 3. C2×#400 | C2 | semidirect |
| 4. C2×#397 | C2 | semidirect |
| 5. C2×#396 | C2 | semidirect |
| 7. C2×#385 | C2 | semidirect |
| 8. C22×#95 | C2 | semidirect |
| 16. #401 | C22 | semidirect |
| 17. #400 | C22 | semidirect |
| 18. #397 | C22 | semidirect |
| 19. #396 | C22 | semidirect |
| 21. #385 | C22 | semidirect |
| 28. C2×#95 | C22 | semidirect |
| 153. C2 | #2643 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2643 | 2 |
| C2×#401 | 2 |
| C2×#400 | 2 |
| C2×#397 | 2 |
| C2×#396 | 2 |
| C2×#386 | 2 |
| C2×#385 | 2 |
| C22×#95 | 2 |