Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
413 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 413 conjugacy classes of subgroups — too many to draw. Only its 135 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 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#510 | C2 | semidirect |
| 2. C2×#509 | C2 | semidirect |
| 3. C2×#508 | C2 | semidirect |
| 4. C22×SD32 | C2 | semidirect |
| 5. #3156 | C2 | semidirect |
| 6. #3155 | C2 | semidirect |
| 7. #2251 | C2 | semidirect |
| 8. #2250 | C2 | semidirect |
| 9. #2249 | C2 | semidirect |
| 10. #2248 | C2 | semidirect |
| 11. #2108 | C2 | semidirect |
| 23. C2×Q32 | C22 | semidirect |
| 24. C2×Q32 | C22 | semidirect |
| 29. C2×SD32 | C22 | semidirect |
| 30. C2×SD32 | C22 | semidirect |
| 31. C2×SD32 | C22 | semidirect |
| 33. C2×D16 | C22 | semidirect |
| 39. #495 | C22 | semidirect |
| 40. #494 | C22 | semidirect |
| 43. #346 | C22 | semidirect |
| 44. C4×C16 | C22 | semidirect |
| 94. M32 | D4 | semidirect |
| 95. C2×C16 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#510 | 2 |
| C2×#509 | 2 |
| C2×#508 | 2 |
| C22×SD32 | 2 |
| #3156 | 2 |
| #3155 | 2 |
| #2251 | 2 |
| #2250 | 2 |
| #2249 | 2 |
| #2248 | 2 |
| #2108 | 2 |