Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
8266 conjugacy classes of subgroups; only the 1602 normal subgroups are shown.
This group has 8266 conjugacy classes of subgroups — too many to draw. Only its 1602 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 | C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#121 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×C22⋊C8 | C2 | semidirect |
| 3. C23×#95 | C2 | semidirect |
| 4. C22×#327 | C2 | direct (N × H) |
| 6. C24×D4 | C4 | semidirect |
| 12. C22×#121 | C4 | semidirect |
| 14. C23×C4⋊C4 | C4 | semidirect |
| 17. C22×C22⋊C8 | C22 | semidirect |
| 19. C22×#95 | C22 | semidirect |
| 20. C2×#327 | C22 | direct (N × H) |
| 29. C23×D4 | C2×C4 | semidirect |
| 47. C2×#121 | C2×C4 | semidirect |
| 50. C22×C4⋊C4 | C2×C4 | semidirect |
| 54. C2×C22⋊C8 | C23 | semidirect |
| 56. C2×#95 | C23 | semidirect |
| 57. #327 | C23 | direct (N × H) |
| 77. C22×D4 | C22×C4 | semidirect |
| 105. #121 | C22×C4 | semidirect |
| 108. C2×C4⋊C4 | C22×C4 | semidirect |
| 112. C22⋊C8 | C24 | semidirect |
| 114. #95 | C24 | semidirect |
| 143. C2×D4 | C23×C4 | semidirect |
| 172. C4⋊C4 | C23×C4 | semidirect |
| 187. C23 | #327 | direct (N × H) |
| 233. C22 | C2×#327 | direct (N × H) |
| 251. C2 | C22×#327 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×#121 | 2 |
| C23×C22⋊C8 | 2 |
| C23×#95 | 2 |
| C22×#327 | 2 |