Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
147 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 147 conjugacy classes of subgroups — too many to draw. Only its 37 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2184 |
| commutator, Frattini, cosocle | C2×C16 | C23 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C22×C8 ◁ C2×M32 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3414 | C2 | semidirect |
| 2. #2263 | C2 | semidirect |
| 3. #2218 | C2 | semidirect |
| 4. #1418 | C2 | semidirect |
| 5. #1400 | C2 | semidirect |
| 11. C2×D16 | C22 | semidirect |
| 16. M64 | C22 | semidirect |
| 17. #367 | C22 | semidirect |
| 31. Q32 | D4 | semidirect |
| 36. D16 | D4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3414 | 2 |
| #2263 | 2 |
| #2218 | 2 |
| #1418 | 2 |
| #1400 | 2 |