Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
2934 conjugacy classes of subgroups; only the 1098 normal subgroups are shown.
This group has 2934 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 | C23×C4 | C24 |
| commutator | C22 | C24×C8 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C24 | C23×C4 |
| prime radical | C22×#127 | C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. #23355 | C2 | direct (N × H) |
| 3. C2×#1869 | C2 | semidirect |
| 4. C2×#1839 | C2 | semidirect |
| 14. #1869 | C22 | semidirect |
| 15. #1839 | C22 | semidirect |
| 22. C2×#127 | C8 | semidirect |
| 43. C22×D4 | C2×C8 | semidirect |
| 57. #127 | C2×C8 | semidirect |
| 67. C2×C42 | C2×C8 | semidirect |
| 81. C2×D4 | C22×C8 | semidirect |
| 107. C42 | C22×C8 | semidirect |
| 151. C2 | #23355 | 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 |
| #23355 | 2 |
| C2×#1869 | 2 |
| C2×#1839 | 2 |