Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
1499 conjugacy classes of subgroups; only the 487 normal subgroups are shown.
This group has 1499 conjugacy classes of subgroups — too many to draw. Only its 487 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 | C23 | C25 |
| prime radical | C22×#121 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C22⋊Q8 | C2 | semidirect |
| 3. #2816 | C2 | direct (N × H) |
| 4. C2×#399 | C2 | semidirect |
| 5. C2×#398 | C2 | semidirect |
| 6. C2×#394 | C2 | semidirect |
| 7. C2×#394 | C2 | semidirect |
| 8. C2×#393 | C2 | semidirect |
| 9. C2×#381 | C2 | semidirect |
| 15. C2×C22⋊Q8 | C22 | semidirect |
| 27. #399 | C22 | semidirect |
| 28. #398 | C22 | semidirect |
| 29. #394 | C22 | semidirect |
| 30. #394 | C22 | semidirect |
| 31. #393 | C22 | semidirect |
| 32. #381 | C22 | semidirect |
| 202. C2 | #2816 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C22⋊Q8 | 2 |
| C22×#121 | 2 |
| #2816 | 2 |
| C2×#399 | 2 |
| C2×#398 | 2 |
| C2×#394 | 2 |
| C2×#393 | 2 |
| C2×#381 | 2 |