Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
293 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 293 conjugacy classes of subgroups — too many to draw. Only its 159 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #106 |
| commutator | C8 | C2×C42 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | #340 |
| prime radical | C24 | D8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #1384 | C2 | direct (N × H) |
| 8. C2×M32 | C4 | semidirect |
| 10. C22×C16 | C4 | semidirect |
| 13. #427 | C4 | semidirect |
| 16. C2×#106 | C4 | semidirect |
| 39. C2×C16 | C2×C4 | semidirect |
| 43. #106 | C2×C4 | semidirect |
| 134. C2 | #1384 | 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×#427 | 2 |
| #1384 | 2 |