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 | #1624 | 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. #3373 | C2 | semidirect |
| 3. #3340 | C2 | semidirect |
| 4. #3329 | C2 | semidirect |
| 5. #1714 | C2 | semidirect |
| 6. #1692 | C2 | semidirect |
| 7. #1685 | C2 | semidirect |
| 8. #1684 | C2 | semidirect |
| 9. #1681 | C2 | semidirect |
| 10. #1650 | C2 | semidirect |
| 11. #1649 | C2 | semidirect |
| 12. #1630 | C2 | semidirect |
| 14. #1588 | C2 | semidirect |
| 15. #1585 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #3394 | 2 |
| #3373 | 2 |
| #3340 | 2 |
| #3329 | 2 |
| #1714 | 2 |
| #1692 | 2 |
| #1685 | 2 |
| #1684 | 2 |
| #1681 | 2 |
| #1650 | 2 |
| #1649 | 2 |
| #1630 | 2 |
| #1624 | 2 |
| #1588 | 2 |
| #1585 | 2 |