Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
3630 conjugacy classes of subgroups; only the 1098 normal subgroups are shown.
This group has 3630 conjugacy classes of subgroups — too many to draw. Only its 1098 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 3. #14688 | C2 | semidirect |
| 4. #14676 | C2 | semidirect |
| 5. #14486 | C2 | semidirect |
| 6. #13872 | C2 | semidirect |
| 11. C4×#127 | C4 | semidirect |
| 31. C2×C4×D4 | C2×C4 | semidirect |
| 41. #390 | C2×C4 | semidirect |
| 43. #384 | C2×C4 | semidirect |
| 49. C43 | C2×C4 | semidirect |
| 69. #127 | C4⋊C4 | semidirect |
| 107. C2×D4 | C2×C4⋊C4 | semidirect |
| 137. C42 | C2×C4⋊C4 | semidirect |
| 192. C4 | #2380 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2380 | 2 |
| C2×C4×#127 | 2 |
| #14688 | 2 |
| #14676 | 2 |
| #14486 | 2 |
| #13872 | 2 |