Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
6135 conjugacy classes of subgroups; only the 3183 normal subgroups are shown.
This group has 6135 conjugacy classes of subgroups — too many to draw. Only its 3183 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#41 | C2 | semidirect |
| 2. C2×#537 | C2 | direct (N × H) |
| 3. C22×C22⋊Q8 | C2 | semidirect |
| 4. C22×#121 | C2 | semidirect |
| 5. C22×#121 | C2 | semidirect |
| 6. C23×C4⋊C4 | C2 | semidirect |
| 8. C22×#41 | C22 | semidirect |
| 16. #537 | C22 | direct (N × H) |
| 17. C2×C22⋊Q8 | C22 | semidirect |
| 18. C2×#121 | C22 | semidirect |
| 19. C2×#121 | C22 | semidirect |
| 22. C22×C4⋊C4 | C22 | semidirect |
| 27. C2×#41 | C23 | semidirect |
| 38. C23×C4 | D4 | semidirect |
| 45. C22⋊Q8 | C23 | semidirect |
| 46. #121 | C23 | semidirect |
| 47. #121 | C23 | semidirect |
| 50. C2×C4⋊C4 | C23 | semidirect |
| 75. C22×C4 | C2×D4 | semidirect |
| 108. C2×C4 | C22×D4 | semidirect |
| 119. C22 | #537 | direct (N × H) |
| 130. C2 | C2×#537 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#41 | 2 |
| C2×#537 | 2 |
| C22×C22⋊Q8 | 2 |
| C22×#121 | 2 |
| C23×C4⋊C4 | 2 |