Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
373 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 373 conjugacy classes of subgroups — too many to draw. Only its 111 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×#127 | 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. #2567 | C2 | semidirect |
| 3. #2067 | C2 | semidirect |
| 5. #2023 | C2 | semidirect |
| 6. #1985 | C2 | semidirect |
| 7. #1936 | C2 | semidirect |
| 8. #1925 | C2 | semidirect |
| 9. #1921 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #2567 | 2 |
| #2548 | 2 |
| #2067 | 2 |
| #2041 | 2 |
| #2023 | 2 |
| #1985 | 2 |
| #1936 | 2 |
| #1925 | 2 |
| #1921 | 2 |