Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
1882 conjugacy classes of subgroups; only the 1234 normal subgroups are shown.
This group has 1882 conjugacy classes of subgroups — too many to draw. Only its 1234 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×C8 | C22 |
| commutator | C2 | C2×C42×C8 |
| Frattini, cosocle | C23×C4 | C24 |
| socle, prime radical | C24 | C23×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 10. #1744 | C4 | semidirect |
| 12. C42×C8 | C4 | semidirect |
| 26. #376 | C8 | direct (N × H) |
| 73. C42 | C4×C8 | semidirect |
| 93. C8 | #376 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#1744 | 2 |
| C2×C42×C8 | 2 |
| C4×#376 | 2 |
| C8×#95 | 2 |