Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
419 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 419 conjugacy classes of subgroups — too many to draw. Only its 117 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3038 | C2 | semidirect |
| 2. C2×#459 | C2 | semidirect |
| 3. C4×AGL(1,ℤ/8) | C2 | semidirect |
| 4. #2925 | C2 | semidirect |
| 5. #2051 | C2 | semidirect |
| 6. #2018 | C2 | semidirect |
| 7. #1978 | C2 | semidirect |
| 8. #1965 | C2 | semidirect |
| 9. #1950 | C2 | semidirect |
| 10. #1931 | C2 | semidirect |
| 11. #1800 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3038 | 2 |
| C2×#459 | 2 |
| C4×AGL(1,ℤ/8) | 2 |
| #2925 | 2 |
| #2051 | 2 |
| #2018 | 2 |
| #1978 | 2 |
| #1965 | 2 |
| #1950 | 2 |
| #1931 | 2 |
| #1800 | 2 |