Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
7170 conjugacy classes of subgroups; only the 4138 normal subgroups are shown.
This group has 7170 conjugacy classes of subgroups — too many to draw. Only its 4138 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 | C2 | C22×C43 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C2 ▷ 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. #13749 | C2 | semidirect |
| 4. C42×C22⋊C4 | C2 | semidirect |
| 8. #2273 | C4 | direct (N × H) |
| 9. C2×C4×C22⋊C4 | C4 | semidirect |
| 14. C2×C43 | C22 | semidirect |
| 15. C4×#95 | C22 | semidirect |
| 16. C4×#95 | C22 | semidirect |
| 27. C22×C42 | C2×C4 | semidirect |
| 31. C4×C22⋊C4 | C2×C4 | semidirect |
| 32. C4×C22⋊C4 | C2×C4 | semidirect |
| 33. C4×C22⋊C4 | C2×C4 | semidirect |
| 34. C2×#95 | C2×C4 | semidirect |
| 48. C23×C4 | C42 | semidirect |
| 58. C2×C22⋊C4 | C42 | semidirect |
| 68. C2×C42 | C22×C4 | semidirect |
| 70. #95 | C22×C4 | semidirect |
| 95. C22×C4 | C2×C42 | semidirect |
| 102. C22⋊C4 | C2×C42 | semidirect |
| 114. C23 | C43 | semidirect |
| 147. C4 | #2273 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2273 | 2 |
| C22×C4×C22⋊C4 | 2 |
| #13749 | 2 |
| C42×C22⋊C4 | 2 |