Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
710 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 710 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #21698 | C2 | semidirect |
| 4. #13207 | C2 | semidirect |
| 6. #13014 | C2 | semidirect |
| 7. #13013 | C2 | semidirect |
| 8. #12889 | C2 | semidirect |
| 9. #12592 | C2 | semidirect |
| 19. #2248 | C4 | semidirect |
| 46. C2×SD32 | C2×C4 | semidirect |
| 50. #494 | C2×C4 | semidirect |
| 81. C4×C16 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2248 | 2 |
| #21698 | 2 |
| #21697 | 2 |
| #13207 | 2 |
| #13206 | 2 |
| #13014 | 2 |
| #13013 | 2 |
| #12889 | 2 |
| #12592 | 2 |