Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
660 conjugacy classes of subgroups; only the 237 normal subgroups are shown.
This group has 660 conjugacy classes of subgroups — too many to draw. Only its 237 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#120 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#385 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #2905 | C2 | semidirect |
| 5. #1868 | C2 | semidirect |
| 6. C2×#344 | C2 | semidirect |
| 18. #424 | C22 | semidirect |
| 26. #344 | C22 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#568 | 2 |
| #2905 | 2 |
| C2×#410 | 2 |
| #2879 | 2 |
| #1868 | 2 |
| C2×#344 | 2 |