Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1111 conjugacy classes of subgroups; only the 463 normal subgroups are shown.
This group has 1111 conjugacy classes of subgroups — too many to draw. Only its 463 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. #2801 | C2 | direct (N × H) |
| 2. C2×#401 | C2 | semidirect |
| 3. C2×#399 | C2 | semidirect |
| 4. C2×#396 | C2 | semidirect |
| 6. C2×#389 | C2 | semidirect |
| 7. C2×#388 | C2 | semidirect |
| 8. C2×#388 | C2 | semidirect |
| 18. #401 | C22 | semidirect |
| 19. #399 | C22 | semidirect |
| 20. #396 | C22 | semidirect |
| 22. #389 | C22 | semidirect |
| 23. #388 | C22 | semidirect |
| 24. #388 | C22 | semidirect |
| 151. C2 | #2801 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2801 | 2 |
| C2×#401 | 2 |
| C2×#399 | 2 |
| C2×#396 | 2 |
| C2×#394 | 2 |
| C2×#389 | 2 |
| C2×#388 | 2 |