Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
317 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 317 conjugacy classes of subgroups — too many to draw. Only its 127 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#500 | C2 | semidirect |
| 3. #2243 | C2 | direct (N × H) |
| 4. C2×#358 | C2 | semidirect |
| 6. C2×#346 | C2 | semidirect |
| 14. #500 | C22 | semidirect |
| 20. #358 | C22 | semidirect |
| 22. #346 | C22 | semidirect |
| 129. C2 | #2243 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#500 | 2 |
| C2×#495 | 2 |
| #2243 | 2 |
| C2×#358 | 2 |
| C2×#357 | 2 |
| C2×#346 | 2 |