Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1758 conjugacy classes of subgroups; only the 658 normal subgroups are shown.
This group has 1758 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 | C23×D8 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×#499 | C2 | semidirect |
| 3. C2×#2242 | C2 | direct (N × H) |
| 4. C22×#358 | C2 | semidirect |
| 6. C22×C4×C16 | C2 | semidirect |
| 14. C2×#499 | C22 | semidirect |
| 20. #2242 | C22 | direct (N × H) |
| 21. C2×#358 | C22 | semidirect |
| 23. C2×C4×C16 | C22 | semidirect |
| 47. #499 | C23 | semidirect |
| 55. #358 | C23 | semidirect |
| 57. C4×C16 | C23 | semidirect |
| 175. C22 | #2242 | direct (N × H) |
| 186. C2 | C2×#2242 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×#499 | 2 |
| C22×#495 | 2 |
| C2×#2242 | 2 |
| C22×#358 | 2 |
| C22×#357 | 2 |
| C22×C4×C16 | 2 |