Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
341 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 341 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×#493 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3160 | C2 | semidirect |
| 3. #2244 | C2 | semidirect |
| 4. #2240 | C2 | semidirect |
| 5. #2140 | C2 | semidirect |
| 6. C4×M32 | C2 | semidirect |
| 12. #500 | C22 | semidirect |
| 14. #499 | C22 | semidirect |
| 23. #357 | C22 | semidirect |
| 24. #346 | C22 | semidirect |
| 25. C4×C16 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3160 | 2 |
| C2×#493 | 2 |
| #2244 | 2 |
| #2240 | 2 |
| #2140 | 2 |
| C4×M32 | 2 |