Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1135 conjugacy classes of subgroups; only the 567 normal subgroups are shown.
This group has 1135 conjugacy classes of subgroups — too many to draw. Only its 567 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 | C2×D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C22×D4 |
| prime radical | C25 | D4 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #1736 | C2 | semidirect |
| 4. #1735 | C2 | semidirect |
| 7. C22×M16 | C4 | semidirect |
| 13. C2×#117 | C4 | semidirect |
| 19. #340 | C22 | semidirect |
| 33. C2×M16 | C2×C4 | semidirect |
| 35. C22×C8 | C2×C4 | semidirect |
| 42. #117 | C2×C4 | semidirect |
| 45. C2×C4⋊C4 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×M16 | 2 |
| C22×#117 | 2 |
| #1736 | 2 |
| #1735 | 2 |