Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
3815 conjugacy classes of subgroups; only the 3183 normal subgroups are shown.
This group has 3815 conjugacy classes of subgroups — too many to draw. Only its 3183 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 | C25 | C23 |
| 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×#544 | C2 | direct (N × H) |
| 2. C22×C4⋊Q8 | C2 | semidirect |
| 3. C22×#125 | C2 | semidirect |
| 10. #544 | C22 | direct (N × H) |
| 11. C2×C4⋊Q8 | C22 | semidirect |
| 12. C2×#125 | C22 | semidirect |
| 27. C23×C4 | Q8 | semidirect |
| 34. C4⋊Q8 | C23 | semidirect |
| 35. #125 | C23 | semidirect |
| 58. C22×C4 | C2×Q8 | semidirect |
| 87. C2×C4 | C22×Q8 | semidirect |
| 95. C22 | #544 | direct (N × H) |
| 106. C2 | C2×#544 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#544 | 2 |
| C22×C4⋊Q8 | 2 |
| C22×#125 | 2 |
| C22×C22⋊Q8 | 2 |
| C22×#117 | 2 |
| C23×C4⋊C4 | 2 |