Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
491 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 491 conjugacy classes of subgroups — too many to draw. Only its 135 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 | C2×#469 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C4×M16 | C2 | semidirect |
| 3. #2854 | C2 | semidirect |
| 4. #1972 | C2 | semidirect |
| 5. #1970 | C2 | semidirect |
| 6. #1943 | C2 | semidirect |
| 7. #1941 | C2 | semidirect |
| 8. #1936 | C2 | semidirect |
| 9. #1934 | C2 | semidirect |
| 10. #1895 | C2 | semidirect |
| 11. #1809 | C2 | semidirect |
| 201. C8 | #123 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#469 | 2 |
| C2×C4×M16 | 2 |
| #2854 | 2 |
| #1972 | 2 |
| #1970 | 2 |
| #1943 | 2 |
| #1941 | 2 |
| #1936 | 2 |
| #1934 | 2 |
| #1895 | 2 |
| #1809 | 2 |