Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
525 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 525 conjugacy classes of subgroups — too many to draw. Only its 134 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 | #2504 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #2619 | C22 |
| LCS | G ▷ C2×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. #20807 | C2 | semidirect |
| 3. #10633 | C2 | semidirect |
| 4. #10631 | C2 | semidirect |
| 5. #9774 | C2 | semidirect |
| 8. #9193 | C2 | semidirect |
| 9. #9191 | C2 | semidirect |
| 10. #8794 | C2 | semidirect |
| 11. #8157 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33045 | 2 |
| #20807 | 2 |
| #10633 | 2 |
| #10631 | 2 |
| #9774 | 2 |
| #9197 | 2 |
| #9193 | 2 |
| #9191 | 2 |
| #8794 | 2 |
| #8157 | 2 |