Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
321 conjugacy classes of subgroups; only the 107 normal subgroups are shown.
This group has 321 conjugacy classes of subgroups — too many to draw. Only its 107 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 |
| prime radical | C2×#466 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×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×#467 | C2 | semidirect |
| 3. #2235 | C2 | semidirect |
| 4. #2233 | C2 | semidirect |
| 5. #2217 | C2 | semidirect |
| 6. #2187 | C2 | semidirect |
| 7. #2185 | C2 | semidirect |
| 8. C2×C22⋊C16 | C2 | semidirect |
| 9. C22×Q16 | C22 | semidirect |
| 14. C2×SD32 | C22 | semidirect |
| 15. C22×C16 | C22 | semidirect |
| 19. #467 | C22 | semidirect |
| 25. C2×#107 | C22 | semidirect |
| 28. #367 | C22 | semidirect |
| 29. #358 | C22 | semidirect |
| 30. #357 | C22 | semidirect |
| 32. C22⋊C16 | C22 | semidirect |
| 129. C23 | SD32 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#467 | 2 |
| C2×#466 | 2 |
| #2235 | 2 |
| #2233 | 2 |
| #2217 | 2 |
| #2187 | 2 |
| #2185 | 2 |
| C2×C22⋊C16 | 2 |