Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
227 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 227 conjugacy classes of subgroups — too many to draw. Only its 99 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 | #466 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #117 ◁ #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. #3164 | C2 | semidirect |
| 3. #2255 | C2 | semidirect |
| 4. #2254 | C2 | semidirect |
| 5. #2243 | C2 | semidirect |
| 6. #2242 | C2 | semidirect |
| 7. #2237 | C2 | semidirect |
| 8. #2235 | C2 | semidirect |
| 11. #2229 | C2 | semidirect |
| 12. #2228 | C2 | semidirect |
| 13. #2226 | C2 | semidirect |
| 14. #2225 | C2 | semidirect |
| 15. #2164 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3175 | 2 |
| #3164 | 2 |
| #2255 | 2 |
| #2254 | 2 |
| #2243 | 2 |
| #2242 | 2 |
| #2237 | 2 |
| #2235 | 2 |
| #2234 | 2 |
| #2233 | 2 |
| #2229 | 2 |
| #2228 | 2 |
| #2226 | 2 |
| #2225 | 2 |
| #2164 | 2 |