Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
664 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 664 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #493 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C2×#493 | C22 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×#2247 | C2 | semidirect |
| 3. C2×#2241 | C2 | semidirect |
| 6. C2×#1707 | C2 | semidirect |
| 7. #12530 | C2 | direct (N × H) |
| 8. C2×#1312 | C2 | semidirect |
| 22. #2247 | C22 | semidirect |
| 24. #2241 | C22 | semidirect |
| 34. #1707 | C22 | semidirect |
| 36. #1312 | C22 | semidirect |
| 296. C2 | #12530 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2248 | 2 |
| C2×#2247 | 2 |
| C2×#2241 | 2 |
| C2×#2240 | 2 |
| C2×#1709 | 2 |
| C2×#1707 | 2 |
| #12530 | 2 |
| C2×#1312 | 2 |