Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
529 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 529 conjugacy classes of subgroups — too many to draw. Only its 139 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×#459 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#475 | C2 | semidirect |
| 2. C2×#474 | C2 | semidirect |
| 3. C2×#460 | C2 | semidirect |
| 5. #2858 | C2 | semidirect |
| 6. C4×#124 | C2 | semidirect |
| 7. #2092 | C2 | semidirect |
| 8. #2012 | C2 | semidirect |
| 9. #1973 | C2 | semidirect |
| 10. #1972 | C2 | semidirect |
| 11. #1767 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#475 | 2 |
| C2×#474 | 2 |
| C2×#460 | 2 |
| C2×#459 | 2 |
| #2858 | 2 |
| C4×#124 | 2 |
| #2092 | 2 |
| #2012 | 2 |
| #1973 | 2 |
| #1972 | 2 |
| #1767 | 2 |