Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
296 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 296 conjugacy classes of subgroups — too many to draw. Only its 99 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3305 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #3372 | C2 | semidirect |
| 2. #3368 | C2 | semidirect |
| 3. #3357 | C2 | semidirect |
| 5. #1719 | C2 | semidirect |
| 6. #1701 | C2 | semidirect |
| 7. #1698 | C2 | semidirect |
| 8. #1696 | C2 | semidirect |
| 9. #1683 | C2 | semidirect |
| 10. #1646 | C2 | semidirect |
| 11. #1636 | C2 | semidirect |
| 12. #1631 | C2 | semidirect |
| 13. #1622 | C2 | semidirect |
| 14. #1592 | C2 | semidirect |
| 15. #1582 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3372 | 2 |
| #3368 | 2 |
| #3357 | 2 |
| #3305 | 2 |
| #1719 | 2 |
| #1701 | 2 |
| #1698 | 2 |
| #1696 | 2 |
| #1683 | 2 |
| #1646 | 2 |
| #1636 | 2 |
| #1631 | 2 |
| #1622 | 2 |
| #1592 | 2 |
| #1582 | 2 |