Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
500 conjugacy classes of subgroups; only the 138 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 138 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 | #2563 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #3147 | 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 |
|---|---|---|
| 1. #33257 | C2 | semidirect |
| 3. #10368 | C2 | semidirect |
| 4. #10366 | C2 | semidirect |
| 6. #8794 | C2 | semidirect |
| 7. #8674 | C2 | semidirect |
| 8. #8674 | C2 | semidirect |
| 9. #8267 | C2 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #33257 | 2 |
| #21705 | 2 |
| #10368 | 2 |
| #10366 | 2 |
| #9760 | 2 |
| #8794 | 2 |
| #8674 | 2 |
| #8267 | 2 |