Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
539 conjugacy classes of subgroups; only the 327 normal subgroups are shown.
This group has 539 conjugacy classes of subgroups — too many to draw. Only its 327 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C23 |
| commutator | C22 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #1746 | C2 | semidirect |
| 5. #1744 | C2 | semidirect |
| 11. C4×M16 | C4 | semidirect |
| 13. C2×C4×C8 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×M16 | 2 |
| C22×C4×C8 | 2 |
| C4×#117 | 2 |
| #1746 | 2 |
| #1744 | 2 |
| #1729 | 2 |