Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
300 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 300 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 | #3397 | 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 |
|---|---|---|
| 2. #3360 | C2 | semidirect |
| 3. #3356 | C2 | semidirect |
| 4. #3304 | C2 | semidirect |
| 5. #1714 | C2 | semidirect |
| 6. #1704 | C2 | semidirect |
| 7. #1701 | C2 | semidirect |
| 8. #1671 | C2 | semidirect |
| 9. #1666 | C2 | semidirect |
| 10. #1650 | C2 | semidirect |
| 11. #1650 | C2 | semidirect |
| 12. #1631 | C2 | semidirect |
| 13. #1622 | C2 | semidirect |
| 14. #1587 | C2 | semidirect |
| 15. #1586 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3397 | 2 |
| #3360 | 2 |
| #3356 | 2 |
| #3304 | 2 |
| #1714 | 2 |
| #1704 | 2 |
| #1701 | 2 |
| #1671 | 2 |
| #1666 | 2 |
| #1650 | 2 |
| #1631 | 2 |
| #1622 | 2 |
| #1587 | 2 |
| #1586 | 2 |