Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
321 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 321 conjugacy classes of subgroups — too many to draw. Only its 157 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×D8 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C2×C4×D8 |
| prime radical | C2×#508 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×M16 ◁ #405 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2957 | C2 | semidirect |
| 3. #2179 | C2 | semidirect |
| 4. #2178 | C2 | semidirect |
| 5. C2×#368 | C2 | semidirect |
| 6. #2143 | C2 | semidirect |
| 7. #2108 | C2 | semidirect |
| 10. #508 | C4 | semidirect |
| 11. #508 | C4 | semidirect |
| 14. C2×Q32 | C4 | semidirect |
| 15. C2×SD32 | C4 | semidirect |
| 16. C2×D16 | C4 | semidirect |
| 20. #443 | C22 | semidirect |
| 24. #368 | C22 | semidirect |
| 25. #368 | C22 | semidirect |
| 26. #359 | C22 | semidirect |
| 27. #346 | C22 | semidirect |
| 28. C4×C16 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#508 | 2 |
| #2957 | 2 |
| #2179 | 2 |
| #2178 | 2 |
| C2×#368 | 2 |
| #2143 | 2 |
| #2108 | 2 |