Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
7878 conjugacy classes of subgroups; only the 1706 normal subgroups are shown.
This group has 7878 conjugacy classes of subgroups — too many to draw. Only its 1706 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 | C23 | C24×C4 |
| Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2867 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C4×#127 | C2 | semidirect |
| 3. #14713 | C2 | semidirect |
| 4. #14692 | C2 | semidirect |
| 5. C44 | C2 | semidirect |
| 8. #2867 | C4 | direct (N × H) |
| 19. C2×#127 | C2×C4 | semidirect |
| 29. C43 | C2×C4 | semidirect |
| 30. C43 | D4 | semidirect |
| 68. C42 | C4×D4 | semidirect |
| 69. C42 | #127 | semidirect |
| 93. C4 | C4×#127 | semidirect |
| 94. C4 | #2867 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#2867 | 2 |
| C2×C4×#127 | 2 |
| #14713 | 2 |
| #14692 | 2 |
| C44 | 2 |