Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
339 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 339 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #95 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #1288 | C2 | direct (N × H) |
| 9. #432 | C4 | semidirect |
| 15. C4×M16 | C4 | semidirect |
| 18. C2×#97 | C4 | semidirect |
| 19. C2×C4×C8 | C4 | semidirect |
| 55. #97 | C2×C4 | semidirect |
| 57. C4×C8 | C2×C4 | semidirect |
| 180. C2 | #1288 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#432 | 2 |
| C2×C4×M16 | 2 |
| C2×#383 | 2 |
| #1288 | 2 |