Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
2815 conjugacy classes of subgroups; only the 647 normal subgroups are shown.
This group has 2815 conjugacy classes of subgroups — too many to draw. Only its 647 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 | C23 | C25 |
| prime radical | C22×#121 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×Q8 | C2 | semidirect |
| 2. C22×#124 | C2 | semidirect |
| 4. #2436 | C2 | direct (N × H) |
| 5. C2×#393 | C2 | semidirect |
| 6. C2×#391 | C2 | semidirect |
| 7. C2×#386 | C2 | semidirect |
| 8. C2×C4×C22⋊C4 | C2 | semidirect |
| 13. C23×Q8 | C22 | semidirect |
| 19. C2×#124 | C22 | semidirect |
| 27. #393 | C22 | semidirect |
| 28. #391 | C22 | semidirect |
| 29. #386 | C22 | semidirect |
| 30. C4×C22⋊C4 | C22 | semidirect |
| 42. C22×Q8 | D4 | semidirect |
| 94. C2×Q8 | C2×D4 | semidirect |
| 142. C23 | #124 | semidirect |
| 184. C22 | C2×#124 | semidirect |
| 198. C2 | #2436 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×Q8 | 2 |
| C22×#124 | 2 |
| C22×#121 | 2 |
| #2436 | 2 |
| C2×#393 | 2 |
| C2×#391 | 2 |
| C2×#386 | 2 |
| C2×C4×C22⋊C4 | 2 |