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 | #3414 | 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. #2161 | C2 | semidirect |
| 6. C2×#359 | C2 | semidirect |
| 7. #2108 | C2 | semidirect |
| 10. #510 | C4 | semidirect |
| 11. #509 | C4 | semidirect |
| 16. #443 | C22 | semidirect |
| 20. #368 | C22 | semidirect |
| 21. #359 | C22 | semidirect |
| 22. #359 | C22 | semidirect |
| 23. #346 | C22 | semidirect |
| 24. C4×C16 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3414 | 2 |
| #2957 | 2 |
| #2179 | 2 |
| #2178 | 2 |
| #2161 | 2 |
| C2×#359 | 2 |
| #2108 | 2 |