Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1143 conjugacy classes of subgroups; only the 463 normal subgroups are shown.
This group has 1143 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. #2785 | C2 | direct (N × H) |
| 2. C2×#395 | C2 | semidirect |
| 4. C2×#393 | C2 | semidirect |
| 5. C2×#388 | C2 | semidirect |
| 6. C2×#382 | C2 | semidirect |
| 14. #395 | C22 | semidirect |
| 16. #393 | C22 | semidirect |
| 17. #388 | C22 | semidirect |
| 18. #382 | C22 | semidirect |
| 111. C2 | #2785 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2785 | 2 |
| C2×#395 | 2 |
| C2×#394 | 2 |
| C2×#393 | 2 |
| C2×#388 | 2 |
| C2×#382 | 2 |