Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
401 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 401 conjugacy classes of subgroups — too many to draw. Only its 119 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 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #2538 | C2 | semidirect |
| 2. #2094 | C2 | semidirect |
| 3. #2084 | C2 | semidirect |
| 4. #2065 | C2 | semidirect |
| 5. #1985 | C2 | semidirect |
| 7. #1944 | C2 | semidirect |
| 9. #1862 | C2 | semidirect |
| 40. C2×D8 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2538 | 2 |
| #2094 | 2 |
| #2084 | 2 |
| #2065 | 2 |
| #1985 | 2 |
| #1972 | 2 |
| #1944 | 2 |
| #1879 | 2 |
| #1862 | 2 |