Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
587 conjugacy classes of subgroups; only the 231 normal subgroups are shown.
This group has 587 conjugacy classes of subgroups — too many to draw. Only its 231 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 | D8 |
| commutator | C8 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C23 | #102 |
| prime radical | C22×#135 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×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. C23×C16 | C2 | semidirect |
| 3. C22×#108 | C2 | semidirect |
| 4. C2×#359 | C2 | direct (N × H) |
| 17. C22×C16 | C22 | semidirect |
| 19. C2×#108 | C22 | semidirect |
| 20. #359 | C22 | direct (N × H) |
| 52. C2×C16 | C23 | semidirect |
| 54. #108 | C23 | semidirect |
| 129. C22 | #359 | direct (N × H) |
| 141. C2 | C2×#359 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#135 | 2 |
| C23×C16 | 2 |
| C22×#108 | 2 |
| C2×#359 | 2 |