Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2022 conjugacy classes of subgroups; only the 658 normal subgroups are shown.
This group has 2022 conjugacy classes of subgroups — too many to draw. Only its 658 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×#466 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C23×#107 | C2 | semidirect |
| 3. C2×#2233 | C2 | direct (N × H) |
| 4. C22×#367 | C2 | semidirect |
| 5. C22×#357 | C2 | semidirect |
| 6. C22×C22⋊C16 | C2 | semidirect |
| 18. C22×#107 | C22 | semidirect |
| 20. #2233 | C22 | direct (N × H) |
| 21. C2×#367 | C22 | semidirect |
| 22. C2×#357 | C22 | semidirect |
| 23. C2×C22⋊C16 | C22 | semidirect |
| 54. C2×#107 | C23 | semidirect |
| 56. #367 | C23 | semidirect |
| 57. #357 | C23 | semidirect |
| 58. C22⋊C16 | C23 | semidirect |
| 201. C22 | #2233 | direct (N × H) |
| 215. C2 | C2×#2233 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#466 | 2 |
| C23×#107 | 2 |
| C2×#2233 | 2 |
| C22×#367 | 2 |
| C22×#357 | 2 |
| C22×C22⋊C16 | 2 |