Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
403 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#127 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3230 | C2 | semidirect |
| 3. #3170 | C2 | semidirect |
| 4. #3165 | C2 | semidirect |
| 7. #1722 | C2 | semidirect |
| 8. #1719 | C2 | semidirect |
| 9. #1714 | C2 | semidirect |
| 10. #1714 | C2 | semidirect |
| 11. #1451 | C2 | semidirect |
| 18. #500 | C22 | semidirect |
| 19. #500 | C22 | semidirect |
| 20. #500 | C22 | semidirect |
| 21. #498 | C22 | semidirect |
| 22. #494 | C22 | semidirect |
| 28. #487 | C22 | semidirect |
| 29. #487 | C22 | semidirect |
| 30. #487 | C22 | semidirect |
| 45. #322 | C22 | semidirect |
| 46. #322 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3230 | 2 |
| #3208 | 2 |
| #3170 | 2 |
| #3165 | 2 |
| C2×#492 | 2 |
| #3138 | 2 |
| #1722 | 2 |
| #1719 | 2 |
| #1714 | 2 |
| #1451 | 2 |