Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
631 conjugacy classes of subgroups; only the 271 normal subgroups are shown.
This group has 631 conjugacy classes of subgroups — too many to draw. Only its 271 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22⋊C4 |
| commutator | C22 | C23×C8 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C23 | C22⋊C8 |
| prime radical | C25 | C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×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. C2×#349 | C2 | direct (N × H) |
| 12. C2×M32 | C22 | semidirect |
| 14. #349 | C22 | direct (N × H) |
| 38. M32 | C23 | semidirect |
| 110. C22 | #349 | direct (N × H) |
| 122. C2 | C2×#349 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×M16 | 2 |
| C22×M32 | 2 |
| C2×#349 | 2 |