Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
20746 conjugacy classes of subgroups; only the 4258 normal subgroups are shown.
This group has 20746 conjugacy classes of subgroups — too many to draw. Only its 4258 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, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×#127 | C2 | semidirect |
| 2. C23×#124 | C2 | semidirect |
| 3. C2×#2657 | C2 | direct (N × H) |
| 4. C22×#392 | C2 | semidirect |
| 5. C22×#376 | C2 | semidirect |
| 10. C22×#127 | C22 | semidirect |
| 12. C22×#124 | C22 | semidirect |
| 15. #2657 | C22 | direct (N × H) |
| 16. C2×#392 | C22 | semidirect |
| 17. C2×#376 | C22 | semidirect |
| 32. C2×#127 | C23 | semidirect |
| 34. C2×#124 | C23 | semidirect |
| 37. C22×C42 | D4 | semidirect |
| 39. #392 | C23 | semidirect |
| 40. #376 | C23 | semidirect |
| 69. C2×C42 | C2×D4 | semidirect |
| 101. C42 | C22×D4 | semidirect |
| 128. C22 | #2657 | direct (N × H) |
| 137. C2 | C2×#2657 | 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 |
| C23×#124 | 2 |
| C2×#2657 | 2 |
| C22×#392 | 2 |
| C22×#376 | 2 |