Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
751 conjugacy classes of subgroups; only the 399 normal subgroups are shown.
This group has 751 conjugacy classes of subgroups — too many to draw. Only its 399 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2745 | C2 | semidirect |
| 3. #2527 | C2 | semidirect |
| 7. #401 | C22 | semidirect |
| 8. #401 | C22 | semidirect |
| 9. #400 | C22 | semidirect |
| 11. #395 | C22 | semidirect |
| 13. #391 | C22 | semidirect |
| 16. #387 | C22 | semidirect |
| 17. #387 | C22 | semidirect |
| 26. #376 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2840 | 2 |
| #2745 | 2 |
| #2527 | 2 |
| #2483 | 2 |
| #2312 | 2 |