Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
144 conjugacy classes of subgroups; only the 37 normal subgroups are shown.
This group has 144 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 ◁ #131 ◁ #504 ◁ 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. #3415 | C2 | semidirect |
| 2. #2263 | C2 | semidirect |
| 3. C2×D32 | C2 | semidirect |
| 4. #2221 | C2 | semidirect |
| 5. #1419 | C2 | semidirect |
| 6. #1417 | C2 | semidirect |
| 7. #1401 | C2 | semidirect |
| 15. C2×D16 | C22 | semidirect |
| 21. M64 | C22 | semidirect |
| 22. C2×C32 | C22 | semidirect |
| 23. #368 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3415 | 2 |
| #2263 | 2 |
| C2×D32 | 2 |
| #2221 | 2 |
| #1419 | 2 |
| #1417 | 2 |
| #1401 | 2 |