Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
13906 conjugacy classes of subgroups; only the 4258 normal subgroups are shown.
This group has 13906 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 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| prime radical | C24×D8 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C24×Q16 | C2 | semidirect |
| 2. C24×SD16 | C2 | semidirect |
| 4. C23×#124 | C2 | semidirect |
| 5. C22×#495 | C2 | direct (N × H) |
| 6. C23×C4×C8 | C2 | semidirect |
| 10. C23×Q16 | C22 | semidirect |
| 12. C23×SD16 | C22 | semidirect |
| 18. C22×#124 | C22 | semidirect |
| 21. C2×#495 | C22 | direct (N × H) |
| 22. C22×C4×C8 | C22 | semidirect |
| 36. C22×Q16 | C23 | semidirect |
| 38. C22×SD16 | C23 | semidirect |
| 46. C2×#124 | C23 | semidirect |
| 50. #495 | C23 | direct (N × H) |
| 51. C2×C4×C8 | C23 | semidirect |
| 75. C2×Q16 | C24 | semidirect |
| 77. C2×SD16 | C24 | semidirect |
| 85. #124 | C24 | semidirect |
| 89. C4×C8 | C24 | semidirect |
| 130. C23 | #495 | direct (N × H) |
| 154. C22 | C2×#495 | direct (N × H) |
| 164. C2 | C22×#495 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C24×Q16 | 2 |
| C24×SD16 | 2 |
| C24×D8 | 2 |
| C23×#124 | 2 |
| C22×#495 | 2 |
| C23×C4×C8 | 2 |