Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1223 conjugacy classes of subgroups; only the 743 normal subgroups are shown.
This group has 1223 conjugacy classes of subgroups — too many to draw. Only its 743 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C25 |
| commutator | C22 | C24×C4 |
| prime radical | C25 | C23 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2343 | C2 | semidirect |
| 4. #2313 | C2 | semidirect |
| 5. #2283 | C2 | semidirect |
| 9. C2×#117 | C4 | semidirect |
| 16. #387 | C22 | semidirect |
| 18. #383 | C22 | semidirect |
| 19. #382 | C22 | semidirect |
| 24. C4×C4⋊C4 | C22 | semidirect |
| 27. #376 | C22 | semidirect |
| 38. #117 | C2×C4 | semidirect |
| 39. C2×C4⋊C4 | C2×C4 | semidirect |
| 49. C2×C42 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#117 | 2 |
| #2343 | 2 |
| #2315 | 2 |
| #2313 | 2 |
| #2283 | 2 |
| C2×#381 | 2 |
| #2273 | 2 |