Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
842 conjugacy classes of subgroups; only the 266 normal subgroups are shown.
This group has 842 conjugacy classes of subgroups — too many to draw. Only its 266 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 | C25 | D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #12920 | C2 | semidirect |
| 4. #12919 | C2 | semidirect |
| 5. #12836 | C2 | semidirect |
| 11. C2×#367 | C22 | semidirect |
| 12. C2×#366 | C22 | semidirect |
| 13. #2112 | C4 | semidirect |
| 20. #1380 | C22 | semidirect |
| 35. C2×M32 | C2×C4 | semidirect |
| 50. C22⋊C16 | C2×C4 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2112 | 2 |
| C2×#1808 | 2 |
| #12920 | 2 |
| #12919 | 2 |
| #12836 | 2 |