Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
4158 conjugacy classes of subgroups; only the 1610 normal subgroups are shown.
This group has 4158 conjugacy classes of subgroups — too many to draw. Only its 1610 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #13832 | C2 | semidirect |
| 4. #13830 | C2 | semidirect |
| 5. C4×#383 | C2 | semidirect |
| 6. C4×#386 | C2 | semidirect |
| 10. #2380 | C4 | direct (N × H) |
| 11. C4×#127 | C4 | semidirect |
| 30. C2×C4×D4 | C2×C4 | semidirect |
| 40. #390 | C2×C4 | semidirect |
| 41. #386 | C2×C4 | semidirect |
| 42. #384 | C2×C4 | semidirect |
| 43. #383 | C2×C4 | semidirect |
| 48. C43 | C2×C4 | semidirect |
| 66. #127 | C42 | semidirect |
| 99. C2×D4 | C2×C42 | semidirect |
| 128. C42 | C2×C42 | semidirect |
| 175. C4 | #2380 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2380 | 2 |
| C2×C4×#127 | 2 |
| #13832 | 2 |
| #13830 | 2 |
| C4×#383 | 2 |
| C4×#386 | 2 |