Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
3294 conjugacy classes of subgroups; only the 1530 normal subgroups are shown.
This group has 3294 conjugacy classes of subgroups — too many to draw. Only its 1530 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2×C43 | C22 |
| commutator | C2 | C2×C42×C8 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C24 | C23×C4 |
| prime radical | C25 | C22×C4 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. C2×C42×C8 | C2 | semidirect |
| 4. C4×#336 | C2 | semidirect |
| 13. C4×C22⋊C8 | C4 | direct (N × H) |
| 23. C22×C42 | C8 | semidirect |
| 33. C2×C4×C8 | C2×C4 | semidirect |
| 37. #336 | C2×C4 | semidirect |
| 71. C22⋊C8 | C42 | direct (N × H) |
| 90. C22×C4 | C4×C8 | semidirect |
| 105. C2×C8 | C2×C42 | semidirect |
| 109. C42 | C22⋊C8 | direct (N × H) |
| 146. C22 | C42×C8 | semidirect |
| 156. C4 | C4×C22⋊C8 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C43 | 2 |
| C2×C4×C22⋊C8 | 2 |
| C2×C42×C8 | 2 |
| C4×#336 | 2 |