Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
343 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 343 conjugacy classes of subgroups — too many to draw. Only its 109 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×#121 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3038 | C2 |
| 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. #3208 | C2 | semidirect |
| 2. #3107 | C2 | semidirect |
| 4. #2991 | C2 | semidirect |
| 5. #1701 | C2 | semidirect |
| 6. #1685 | C2 | semidirect |
| 7. #1682 | C2 | semidirect |
| 8. #1666 | C2 | semidirect |
| 9. #1574 | C2 | semidirect |
| 10. #1561 | C2 | semidirect |
| 11. #1545 | C2 | semidirect |
| 19. #475 | C22 | semidirect |
| 20. #474 | C22 | semidirect |
| 21. #462 | C22 | semidirect |
| 22. #461 | C22 | semidirect |
| 31. C8×Q8 | C22 | semidirect |
| 35. #436 | C22 | semidirect |
| 43. #335 | C22 | semidirect |
| 44. #334 | C22 | semidirect |
| 45. #332 | C22 | semidirect |
| 46. #332 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3208 | 2 |
| #3107 | 2 |
| #3038 | 2 |
| #2991 | 2 |
| #1701 | 2 |
| #1685 | 2 |
| #1682 | 2 |
| #1666 | 2 |
| #1574 | 2 |
| #1561 | 2 |
| #1545 | 2 |