Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
269 conjugacy classes of subgroups; only the 89 normal subgroups are shown.
This group has 269 conjugacy classes of subgroups — too many to draw. Only its 89 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2749 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ 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. #2102 | C2 | semidirect |
| 2. #2081 | C2 | semidirect |
| 3. #2081 | C2 | semidirect |
| 4. #2070 | C2 | semidirect |
| 7. #1967 | C2 | semidirect |
| 8. #1932 | C2 | semidirect |
| 9. #1915 | C2 | semidirect |
| 10. #1912 | C2 | semidirect |
| 12. #1888 | C2 | semidirect |
| 13. #1861 | C2 | semidirect |
| 14. #1841 | C2 | semidirect |
| 15. #1830 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2102 | 2 |
| #2081 | 2 |
| #2070 | 2 |
| #2048 | 2 |
| #2003 | 2 |
| #1967 | 2 |
| #1932 | 2 |
| #1915 | 2 |
| #1912 | 2 |
| #1903 | 2 |
| #1888 | 2 |
| #1861 | 2 |
| #1841 | 2 |
| #1830 | 2 |