Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1043 conjugacy classes of subgroups; only the 287 normal subgroups are shown.
This group has 1043 conjugacy classes of subgroups — too many to draw. Only its 287 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C24 |
| commutator | C23 | C23×C4 |
| prime radical | C25 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 4. #1446 | C2 | semidirect |
| 5. #1445 | C2 | semidirect |
| 6. #1441 | C2 | semidirect |
| 13. C2×C22⋊Q8 | C4 | semidirect |
| 30. C22×Q8 | C2×C4 | semidirect |
| 39. C23×C4 | Q8 | semidirect |
| 53. C2×C4⋊C4 | C2×C4 | semidirect |
| 54. C2×C4⋊C4 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C22⋊Q8 | 2 |
| C2×#380 | 2 |
| C22×#95 | 2 |
| #1446 | 2 |
| #1445 | 2 |
| #1441 | 2 |
| C22⋊#95 | 2 |