Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
703 conjugacy classes of subgroups; only the 287 normal subgroups are shown.
This group has 703 conjugacy classes of subgroups — too many to draw. Only its 287 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C23 |
| commutator | C22 | C23×C8 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C23 | C22×C8 |
| prime radical | C25 | C8 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C22×M32 | C2 | semidirect |
| 3. #2112 | C2 | direct (N × H) |
| 4. C2×C22⋊C16 | C2 | semidirect |
| 14. C2×M32 | C22 | semidirect |
| 15. C22×C16 | C22 | semidirect |
| 16. C22⋊C16 | C22 | semidirect |
| 43. C2×C16 | C23 | semidirect |
| 88. C23 | M32 | semidirect |
| 121. C22 | C2×M32 | semidirect |
| 130. C2 | #2112 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C8 | 2 |
| C22×M32 | 2 |
| #2112 | 2 |
| C2×C22⋊C16 | 2 |