Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
25274 conjugacy classes of subgroups; only the 6658 normal subgroups are shown.
This group has 25274 conjugacy classes of subgroups — too many to draw. Only its 6658 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C26 | C23 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C27 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C26 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×C4⋊C4 | C2 | semidirect |
| 3. C22×#399 | C2 | direct (N × H) |
| 4. C23×#95 | C2 | semidirect |
| 9. C23×C4⋊C4 | C22 | semidirect |
| 12. C2×#399 | C22 | direct (N × H) |
| 13. C22×#95 | C22 | semidirect |
| 26. C22×C4⋊C4 | C23 | semidirect |
| 29. #399 | C23 | direct (N × H) |
| 30. C2×#95 | C23 | semidirect |
| 51. C2×C4⋊C4 | C24 | semidirect |
| 54. #95 | C24 | semidirect |
| 86. C23 | #399 | direct (N × H) |
| 105. C22 | C2×#399 | direct (N × H) |
| 115. C2 | C22×#399 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×C4⋊C4 | 2 |
| C24×C22⋊C4 | 2 |
| C22×#399 | 2 |
| C23×#95 | 2 |