Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
403 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 123 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 ◁ C2×C42 ◁ C2×C4×C8 ◁ 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×#509 | C2 | semidirect |
| 2. C2×#493 | C2 | semidirect |
| 3. C4×#135 | C2 | semidirect |
| 4. #2207 | C2 | semidirect |
| 5. #2206 | C2 | semidirect |
| 6. #2150 | C2 | semidirect |
| 7. #2140 | C2 | semidirect |
| 14. C2×SD32 | C22 | semidirect |
| 15. C2×D16 | C22 | semidirect |
| 19. C4×Q16 | C22 | semidirect |
| 21. C4×D8 | C22 | semidirect |
| 25. C4⋊C16 | C22 | semidirect |
| 26. #357 | C22 | semidirect |
| 31. #135 | D4 | semidirect |
| 81. Q16 | C2×D4 | semidirect |
| 88. D8 | C2×D4 | semidirect |
| 147. C4 | #509 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#509 | 2 |
| C2×#493 | 2 |
| C4×#135 | 2 |
| #2207 | 2 |
| #2206 | 2 |
| #2150 | 2 |
| #2140 | 2 |