Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
397 conjugacy classes of subgroups; only the 147 normal subgroups are shown.
This group has 397 conjugacy classes of subgroups — too many to draw. Only its 147 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. #2085 | C2 | semidirect |
| 5. #2057 | C2 | semidirect |
| 6. #1941 | C2 | semidirect |
| 7. #1941 | C2 | semidirect |
| 9. #1846 | C2 | semidirect |
| 45. C22×C8 | Q8 | semidirect |
| 74. C22⋊C8 | 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×#427 | 2 |
| #2439 | 2 |
| #2085 | 2 |
| #2057 | 2 |
| #1941 | 2 |
| #1854 | 2 |
| #1846 | 2 |
| #1807 | 2 |
| C8×C22⋊C4 | 2 |