Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
5559 conjugacy classes of subgroups; only the 5287 normal subgroups are shown.
This group has 5559 conjugacy classes of subgroups — too many to draw. Only its 5287 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator | C2 | C25×C4 |
| Frattini, cosocle | C4 | C26 |
| socle | C23 | C23×C4 |
| prime radical | C23×#41 | C2 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#568 | C2 | direct (N × H) |
| 3. C22×#131 | C2 | semidirect |
| 4. C23×M16 | C2 | semidirect |
| 12. #568 | C22 | direct (N × H) |
| 13. C2×#131 | C22 | semidirect |
| 15. C22×M16 | C22 | semidirect |
| 16. C22×M16 | C22 | semidirect |
| 17. C23×C8 | C22 | semidirect |
| 34. #131 | C23 | semidirect |
| 36. C2×M16 | C23 | semidirect |
| 37. C2×M16 | C23 | semidirect |
| 39. C22×C8 | C23 | semidirect |
| 63. M16 | C24 | semidirect |
| 65. C2×C8 | C24 | semidirect |
| 89. C22 | #568 | direct (N × H) |
| 99. C2 | C2×#568 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#41 | 2 |
| C2×#568 | 2 |
| C22×#131 | 2 |
| C23×M16 | 2 |