Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
363 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 363 conjugacy classes of subgroups — too many to draw. Only its 111 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 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | D4×D8 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3380 | C2 | semidirect |
| 2. #3295 | C2 | semidirect |
| 4. #1706 | C2 | semidirect |
| 5. #1677 | C2 | semidirect |
| 6. #1675 | C2 | semidirect |
| 7. #1668 | C2 | semidirect |
| 8. #1629 | C2 | semidirect |
| 9. #1622 | C2 | semidirect |
| 10. C8×SD16 | C2 | semidirect |
| 11. C8×D8 | C2 | semidirect |
| 13. #554 | C22 | semidirect |
| 15. #500 | C22 | semidirect |
| 18. #492 | C22 | semidirect |
| 19. #486 | C22 | semidirect |
| 23. #476 | C22 | semidirect |
| 26. #460 | C22 | semidirect |
| 30. C8×Q8 | C22 | semidirect |
| 35. C8×D4 | C22 | semidirect |
| 37. #335 | C22 | semidirect |
| 38. #334 | C22 | semidirect |
| 39. #333 | C22 | semidirect |
| 40. #331 | C22 | semidirect |
| 41. Q8⋊C8 | C22 | semidirect |
| 43. D4⋊C8 | C22 | semidirect |
| 44. C82 | C22 | semidirect |
| 118. D8 | SD16 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3380 | 2 |
| #3295 | 2 |
| D4×D8 | 2 |
| #1706 | 2 |
| #1677 | 2 |
| #1675 | 2 |
| #1668 | 2 |
| #1629 | 2 |
| #1622 | 2 |
| C8×SD16 | 2 |
| C8×D8 | 2 |