Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
384 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 110 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #331 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C43 | C23 |
| socle | C2 | #7569 |
| prime radical | #3147 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C43 ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #8266 | C2 | semidirect |
| 3. #7106 | C2 | semidirect |
| 18. #494 | C8 | semidirect |
| 19. #493 | C8 | semidirect |
| 20. #492 | C8 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #21705 | 2 |
| #8266 | 2 |
| #7106 | 2 |