Finite Group DatabasebetaFinite Group DatabasebetaOrder 256.
297 conjugacy classes of subgroups; only the 99 normal subgroups are shown.
This group has 297 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 | #1625 | 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. #3394 | C2 | semidirect |
| 2. #3361 | C2 | semidirect |
| 3. #3353 | C2 | semidirect |
| 4. #3331 | C2 | semidirect |
| 5. #1719 | C2 | semidirect |
| 6. #1697 | C2 | semidirect |
| 7. #1685 | C2 | semidirect |
| 8. #1684 | C2 | semidirect |
| 9. #1674 | C2 | semidirect |
| 10. #1644 | C2 | semidirect |
| 11. #1640 | C2 | semidirect |
| 12. #1638 | C2 | semidirect |
| 14. #1588 | C2 | semidirect |
| 15. #1588 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3394 | 2 |
| #3361 | 2 |
| #3353 | 2 |
| #3331 | 2 |
| #1719 | 2 |
| #1697 | 2 |
| #1685 | 2 |
| #1684 | 2 |
| #1674 | 2 |
| #1644 | 2 |
| #1640 | 2 |
| #1638 | 2 |
| #1625 | 2 |
| #1588 | 2 |