Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
587 conjugacy classes of subgroups; only the 271 normal subgroups are shown.
This group has 587 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×C8 | D4 |
| commutator | C4 | C23×C8 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C23 | C22⋊C8 |
| prime radical | C22×#41 | C4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ 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. C23×C16 | C2 | semidirect |
| 4. C2×#350 | C2 | direct (N × H) |
| 15. C2×M32 | C22 | semidirect |
| 19. C22×C16 | C22 | semidirect |
| 20. #350 | C22 | direct (N × H) |
| 49. M32 | C23 | semidirect |
| 53. C2×C16 | C23 | semidirect |
| 127. C22 | #350 | direct (N × H) |
| 139. C2 | C2×#350 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#131 | 2 |
| C22×M32 | 2 |
| C23×C16 | 2 |
| C2×#350 | 2 |