Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2068 conjugacy classes of subgroups; only the 1538 normal subgroups are shown.
This group has 2068 conjugacy classes of subgroups — too many to draw. Only its 1538 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C24 |
| commutator | C22 | C23×C42 |
| Frattini, cosocle | C42 | C25 |
| socle | C23 | C22×C42 |
| prime radical | C25 | C42 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #17338 | C2 | direct (N × H) |
| 3. C2×#1453 | C2 | semidirect |
| 12. #1453 | C22 | semidirect |
| 13. C2×#322 | C22 | semidirect |
| 14. C2×C82 | C22 | semidirect |
| 33. #322 | C23 | semidirect |
| 34. C82 | C23 | semidirect |
| 125. C2 | #17338 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2920 | 2 |
| #17338 | 2 |
| C2×#1453 | 2 |