Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
615 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 615 conjugacy classes of subgroups — too many to draw. Only its 171 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 | #2856 | 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 |
|---|---|---|
| 1. C2×#492 | C2 | semidirect |
| 2. C2×#468 | C2 | semidirect |
| 3. C2×C4×M16 | C2 | semidirect |
| 4. #2857 | C2 | semidirect |
| 6. #1973 | C2 | semidirect |
| 7. #1957 | C2 | semidirect |
| 8. #1956 | C2 | semidirect |
| 9. #1845 | C2 | semidirect |
| 45. C2×M16 | D4 | semidirect |
| 74. C4×C8 | D4 | semidirect |
| 168. C8 | #121 | semidirect |
| 189. C4 | #468 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#492 | 2 |
| C2×#468 | 2 |
| C2×C4×M16 | 2 |
| #2857 | 2 |
| #2856 | 2 |
| #1973 | 2 |
| #1957 | 2 |
| #1956 | 2 |
| #1845 | 2 |