Finite Group DatabasebetaFinite Group DatabasebetaOrder 512.
498 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 498 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | D16 |
| commutator | C16 | C23×C4 |
| Frattini, cosocle | C2×C16 | C24 |
| socle | C23 | C2×D16 |
| prime radical | C22×D32 | C2 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 2. C2×C4×D16 | C2 | semidirect |
| 3. C4×D32 | C2 | direct (N × H) |
| 4. C2×#1423 | C2 | semidirect |
| 5. C2×#1415 | C2 | semidirect |
| 6. C2×C4×C32 | C2 | semidirect |
| 9. C2×D32 | C4 | direct (N × H) |
| 14. C4×D16 | C22 | semidirect |
| 18. #1423 | C22 | semidirect |
| 19. #1415 | C22 | semidirect |
| 20. C4×C32 | C22 | semidirect |
| 25. C2×D16 | C2×C4 | semidirect |
| 34. D32 | C2×C4 | direct (N × H) |
| 36. C2×C32 | C2×C4 | semidirect |
| 59. D16 | C22×C4 | semidirect |
| 72. C32 | C22×C4 | semidirect |
| 114. C2×C4 | D32 | direct (N × H) |
| 136. C4 | C2×D32 | direct (N × H) |
| 140. C2 | C4×D32 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×D32 | 2 |
| C2×C4×D16 | 2 |
| C4×D32 | 2 |
| C2×#1423 | 2 |
| C2×#1415 | 2 |
| C2×C4×C32 | 2 |