Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
14802 conjugacy classes of subgroups; only the 5634 normal subgroups are shown.
This group has 14802 conjugacy classes of subgroups — too many to draw. Only its 5634 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator | C22 | C25×C4 |
| Frattini, cosocle | C23 | C26 |
| prime radical | C23×#127 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#2380 | C2 | direct (N × H) |
| 3. C22×#386 | C2 | semidirect |
| 4. C22×#383 | C2 | semidirect |
| 7. C22×#127 | C4 | semidirect |
| 11. #2380 | C22 | direct (N × H) |
| 12. C2×#386 | C22 | semidirect |
| 13. C2×#383 | C22 | semidirect |
| 18. C23×D4 | C2×C4 | semidirect |
| 24. C2×#127 | C2×C4 | semidirect |
| 28. C22×C42 | C2×C4 | semidirect |
| 30. #386 | C23 | semidirect |
| 31. #383 | C23 | semidirect |
| 41. C22×D4 | C22×C4 | semidirect |
| 51. #127 | C22×C4 | semidirect |
| 54. C2×C42 | C22×C4 | semidirect |
| 70. C2×D4 | C23×C4 | semidirect |
| 81. C42 | C23×C4 | semidirect |
| 105. C22 | #2380 | direct (N × H) |
| 113. C2 | C2×#2380 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#127 | 2 |
| C2×#2380 | 2 |
| C22×#386 | 2 |
| C22×#383 | 2 |