Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
511 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 511 conjugacy classes of subgroups — too many to draw. Only its 158 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 | #2328 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #19667 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #17970 | C2 | semidirect |
| 3. #9762 | C2 | semidirect |
| 4. #9283 | C2 | semidirect |
| 5. #9281 | C2 | semidirect |
| 6. #8796 | C2 | semidirect |
| 7. #8138 | C2 | semidirect |
| 12. #1717 | C4 | semidirect |
| 13. #1712 | C4 | semidirect |
| 14. #1710 | C4 | semidirect |
| 15. #1706 | C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #19667 | 2 |
| #17970 | 2 |
| #9762 | 2 |
| #9283 | 2 |
| #9281 | 2 |
| #8796 | 2 |
| #8138 | 2 |