Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
360 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 123 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 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3163 | C2 | semidirect |
| 2. #3081 | C2 | semidirect |
| 4. #2063 | C2 | semidirect |
| 5. #2027 | C2 | semidirect |
| 8. #1951 | C2 | semidirect |
| 9. #1946 | C2 | semidirect |
| 10. #1856 | C2 | semidirect |
| 11. #1773 | C2 | semidirect |
| 49. C2×D8 | Q8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3163 | 2 |
| #3081 | 2 |
| #2948 | 2 |
| #2063 | 2 |
| #2027 | 2 |
| #1978 | 2 |
| #1972 | 2 |
| #1951 | 2 |
| #1946 | 2 |
| #1856 | 2 |
| #1773 | 2 |