Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
447 conjugacy classes of subgroups; only the 167 normal subgroups are shown.
This group has 447 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#498 | C2 | semidirect |
| 4. #2084 | C2 | semidirect |
| 5. #2058 | C2 | semidirect |
| 6. #1941 | C2 | semidirect |
| 7. #1941 | C2 | semidirect |
| 9. #1844 | C2 | semidirect |
| 46. C22×C8 | Q8 | semidirect |
| 77. C22⋊C8 | Q8 | semidirect |
| 180. C8 | C22⋊Q8 | semidirect |
| 185. C22 | #498 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#498 | 2 |
| C22×#106 | 2 |
| #2440 | 2 |
| #2084 | 2 |
| #2058 | 2 |
| #1941 | 2 |
| #1853 | 2 |
| #1844 | 2 |
| #1807 | 2 |
| C8×C22⋊C4 | 2 |