Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
360 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 360 conjugacy classes of subgroups — too many to draw. Only its 115 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 | #3060 | C2 |
| 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. #3098 | C2 | semidirect |
| 3. #2952 | C2 | semidirect |
| 4. #2078 | C2 | semidirect |
| 5. #2040 | C2 | semidirect |
| 6. #1974 | C2 | semidirect |
| 7. #1951 | C2 | semidirect |
| 8. #1860 | C2 | semidirect |
| 9. #1846 | C2 | semidirect |
| 10. #1815 | C2 | semidirect |
| 11. #1764 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3098 | 2 |
| #3060 | 2 |
| #2952 | 2 |
| #2078 | 2 |
| #2040 | 2 |
| #1974 | 2 |
| #1951 | 2 |
| #1860 | 2 |
| #1846 | 2 |
| #1815 | 2 |
| #1764 | 2 |