Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
3619 conjugacy classes of subgroups; only the 2887 normal subgroups are shown.
This group has 3619 conjugacy classes of subgroups — too many to draw. Only its 2887 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 | C23×D4 | C22 |
| 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×#563 | C2 | direct (N × H) |
| 2. C22×C4⋊Q8 | C2 | semidirect |
| 3. C22×#126 | C2 | semidirect |
| 4. C22×#125 | C2 | semidirect |
| 7. C22×C22⋊Q8 | C2 | semidirect |
| 12. #563 | C22 | direct (N × H) |
| 13. C2×C4⋊Q8 | C22 | semidirect |
| 14. C2×#126 | C22 | semidirect |
| 15. C2×#125 | C22 | semidirect |
| 18. C2×C22⋊Q8 | C22 | semidirect |
| 43. C4⋊Q8 | C23 | semidirect |
| 44. #126 | C23 | semidirect |
| 45. #125 | C23 | semidirect |
| 48. C22⋊Q8 | C23 | semidirect |
| 115. C22 | #563 | direct (N × H) |
| 125. C2 | C2×#563 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#563 | 2 |
| C22×C4⋊Q8 | 2 |
| C22×#126 | 2 |
| C22×#125 | 2 |
| C22×#124 | 2 |
| C22×#123 | 2 |
| C22×C22⋊Q8 | 2 |