Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
4355 conjugacy classes of subgroups; only the 2967 normal subgroups are shown.
This group has 4355 conjugacy classes of subgroups — too many to draw. Only its 2967 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C24 |
| commutator, Frattini, cosocle | C22 | C26 |
| prime radical | C22×#127 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#559 | C2 | direct (N × H) |
| 3. C22×#124 | C2 | semidirect |
| 4. C22×#123 | C2 | semidirect |
| 5. C22×C4×Q8 | C2 | semidirect |
| 6. C22×C4×D4 | C2 | semidirect |
| 11. #559 | C22 | direct (N × H) |
| 13. C2×#124 | C22 | semidirect |
| 14. C2×#123 | C22 | semidirect |
| 15. C2×C4×Q8 | C22 | semidirect |
| 16. C2×C4×D4 | C22 | semidirect |
| 37. #124 | C23 | semidirect |
| 38. #123 | C23 | semidirect |
| 39. C4×Q8 | C23 | semidirect |
| 40. C4×D4 | C23 | semidirect |
| 95. C22 | #559 | direct (N × H) |
| 104. C2 | C2×#559 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#559 | 2 |
| C22×#127 | 2 |
| C22×#124 | 2 |
| C22×#123 | 2 |
| C22×C4×Q8 | 2 |
| C22×C4×D4 | 2 |