Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
6630 conjugacy classes of subgroups; only the 2202 normal subgroups are shown.
This group has 6630 conjugacy classes of subgroups — too many to draw. Only its 2202 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C42 | C23 |
| commutator | C22 | C23×C42 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C42×D4 | C2 | semidirect |
| 3. C4×#384 | C2 | semidirect |
| 4. C4×#390 | C2 | semidirect |
| 5. C44 | C2 | semidirect |
| 9. C4×#127 | C4 | direct (N × H) |
| 25. C2×C4×D4 | C2×C4 | semidirect |
| 36. #390 | C2×C4 | semidirect |
| 37. #384 | C2×C4 | semidirect |
| 42. C43 | C2×C4 | semidirect |
| 43. C43 | D4 | semidirect |
| 55. #127 | C42 | direct (N × H) |
| 82. C2×D4 | C2×C42 | semidirect |
| 105. C42 | C2×C42 | semidirect |
| 106. C42 | C4×D4 | semidirect |
| 107. C42 | #127 | direct (N × H) |
| 143. C4 | C42×D4 | semidirect |
| 144. C4 | C4×#127 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C4×#127 | 2 |
| C2×C42×D4 | 2 |
| C4×#384 | 2 |
| C4×#390 | 2 |
| C44 | 2 |