Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
347 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 347 conjugacy classes of subgroups — too many to draw. Only its 119 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 | C2×#508 | C2 |
| 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 |
|---|---|---|
| 1. C2×#510 | C2 | semidirect |
| 3. #3156 | C2 | semidirect |
| 4. #2951 | C2 | semidirect |
| 5. #2211 | C2 | semidirect |
| 6. #2210 | C2 | semidirect |
| 7. #2209 | C2 | semidirect |
| 8. #2208 | C2 | semidirect |
| 9. #2151 | C2 | semidirect |
| 10. #2139 | C2 | semidirect |
| 11. C2×#358 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#510 | 2 |
| C2×#508 | 2 |
| #3156 | 2 |
| #2951 | 2 |
| #2211 | 2 |
| #2210 | 2 |
| #2209 | 2 |
| #2208 | 2 |
| #2151 | 2 |
| #2139 | 2 |
| C2×#358 | 2 |