Finite Group DatabasebetaFinite Group DatabasebetaOrder 720.
176 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C32×C4×C5 | C2 | semidirect |
| 2. C3×#1220 | C2 | semidirect |
| 3. C3×D60 | C2 | semidirect |
| 4. C3×C4×D15 | C2 | semidirect |
| 5. C3×Dic30 | C2 | semidirect |
| 6. #6817 | C3 | direct (N × H) |
| 15. C2×C3×C4×C5 | D3 | semidirect |
| 16. C2×C3×C4×C5 | C2×C3 | semidirect |
| 17. #1220 | C2×C3 | semidirect |
| 18. D60 | C2×C3 | semidirect |
| 19. C4×D15 | C2×C3 | semidirect |
| 20. Dic30 | C2×C3 | semidirect |
| 29. C2×C32×C4 | D5 | semidirect |
| 49. C32×C5 | #41 | semidirect |
| 50. C2×C4×C5 | C3×D3 | semidirect |
| 73. C2×C3×C4 | C3×D5 | semidirect |
| 74. C2×C3×C4 | D15 | semidirect |
| 89. C3×C5 | #235 | semidirect |
| 90. C3×C5 | C3×#41 | semidirect |
| 107. C32 | #709 | semidirect |
| 112. C2×C4 | C3×D15 | semidirect |
| 121. C5 | C3×#235 | semidirect |
| 128. C3 | C3×#709 | semidirect |
| 129. C3 | #6817 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C32×C4×C5 | 2 |
| C3×#1220 | 2 |
| C3×D60 | 2 |
| C3×C4×D15 | 2 |
| C3×Dic30 | 2 |
| #6817 | 3 |
| C3×#709 | 3 |
| C3×#235 | 5 |