Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
369 conjugacy classes of subgroups; only the 303 normal subgroups are shown.
This group has 369 conjugacy classes of subgroups — too many to draw. Only its 303 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C42 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C22 | C2×#97 |
| prime radical | C24 | M16 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #2110 | C2 | direct (N × H) |
| 4. C2×#347 | C2 | semidirect |
| 8. C2×M32 | C4 | semidirect |
| 9. C2×M32 | C4 | semidirect |
| 12. C22×C16 | C4 | semidirect |
| 17. #347 | C22 | semidirect |
| 36. M32 | C2×C4 | semidirect |
| 39. C2×C16 | C2×C4 | semidirect |
| 116. C2 | #2110 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×M32 | 2 |
| C2×#405 | 2 |
| #2110 | 2 |
| C2×#347 | 2 |