Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
986 conjugacy classes of subgroups; only the 334 normal subgroups are shown.
This group has 986 conjugacy classes of subgroups — too many to draw. Only its 334 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #127 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×#127 |
| prime radical | C2×#127 | C23 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C4×#498 | C2 | semidirect |
| 5. #9819 | C2 | semidirect |
| 6. #8813 | C2 | semidirect |
| 7. C4×C82 | C2 | semidirect |
| 22. #1705 | C4 | direct (N × H) |
| 31. #498 | C2×C4 | semidirect |
| 53. C82 | C2×C4 | semidirect |
| 86. C4×C8 | SD16 | semidirect |
| 148. C8 | C4×SD16 | semidirect |
| 159. C4 | #1705 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C4×#498 | 2 |
| C4×#492 | 2 |
| C2×#1705 | 2 |
| #9820 | 2 |
| #9819 | 2 |
| #8813 | 2 |
| C4×C82 | 2 |