Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2427 conjugacy classes of subgroups; only the 557 normal subgroups are shown.
This group has 2427 conjugacy classes of subgroups — too many to draw. Only its 557 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#120 |
| commutator, Frattini, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×D4 ◁ G | |
| LECS | G ▷ C23 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. H(7,𝔽2) | C2 | semidirect |
| 2. #3485 | C2 | semidirect |
| 3. #3028 | C2 | semidirect |
| 4. #3026 | C2 | semidirect |
| 5. #3025 | C2 | semidirect |
| 6. C2×#458 | C2 | semidirect |
| 7. #2883 | C2 | semidirect |
| 13. C22×#41 | C22 | semidirect |
| 16. #536 | C22 | semidirect |
| 17. #534 | C22 | semidirect |
| 18. #525 | C22 | semidirect |
| 19. C2×#123 | C22 | semidirect |
| 20. #518 | C22 | semidirect |
| 21. #458 | C22 | semidirect |
| 22. #458 | C22 | semidirect |
| 23. B(4,𝔽2) | C22 | semidirect |
| 24. #410 | C22 | semidirect |
| 25. C2×#99 | C22 | semidirect |
| 37. C2×#41 | D4 | semidirect |
| 39. C22×Q8 | C23 | semidirect |
| 42. C22×D4 | D4 | semidirect |
| 45. C22×D4 | C23 | semidirect |
| 46. C23×C4 | C23 | semidirect |
| 47. C23×C4 | C23 | semidirect |
| 52. #123 | C23 | semidirect |
| 53. #123 | C23 | semidirect |
| 59. #120 | C23 | semidirect |
| 62. #117 | C23 | semidirect |
| 65. C2×C22⋊C4 | C23 | semidirect |
| 66. #99 | C23 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| H(7,𝔽2) | 2 |
| #3485 | 2 |
| #3028 | 2 |
| #3026 | 2 |
| #3025 | 2 |
| C2×#458 | 2 |
| #2883 | 2 |