Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
463 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 463 conjugacy classes of subgroups — too many to draw. Only its 231 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | D4 |
| commutator | C4 | C43 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C23 | C4×D4 |
| prime radical | C24 | #41 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 12. #405 | C4 | semidirect |
| 16. C2×C4×C8 | C4 | semidirect |
| 17. #341 | C4 | semidirect |
| 18. #340 | C4 | semidirect |
| 19. #339 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#405 | 2 |
| C22×C4×C8 | 2 |
| C2×#341 | 2 |
| #1735 | 2 |
| C2×#339 | 2 |