Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
453 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 453 conjugacy classes of subgroups — too many to draw. Only its 135 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 | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊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 |
| 3. #2452 | C2 | semidirect |
| 4. #2065 | C2 | semidirect |
| 5. #2023 | C2 | semidirect |
| 8. #1944 | C2 | semidirect |
| 9. #1943 | C2 | semidirect |
| 10. #1864 | C2 | semidirect |
| 11. #1771 | C2 | semidirect |
| 46. C2×D8 | Q8 | semidirect |
| 182. C8 | C22⋊Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#498 | 2 |
| C2×#442 | 2 |
| #2452 | 2 |
| #2065 | 2 |
| #2023 | 2 |
| #1972 | 2 |
| #1970 | 2 |
| #1944 | 2 |
| #1943 | 2 |
| #1864 | 2 |
| #1771 | 2 |