Finite Group DatabasebetaFinite Group DatabasebetaOrder 720.
172 conjugacy classes of subgroups; only the 64 normal subgroups are shown.
This group has 172 conjugacy classes of subgroups — too many to draw. Only its 64 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. C3×D60 | C2 | semidirect |
| 2. C3×C4×C5×D3 | C2 | semidirect |
| 3. C32×Dic10 | C2 | semidirect |
| 4. C3×#1203 | C2 | semidirect |
| 5. C3×#1200 | C2 | semidirect |
| 7. #6769 | C3 | direct (N × H) |
| 15. D60 | C2×C3 | semidirect |
| 16. C4×C5×D3 | C2×C3 | semidirect |
| 21. C3×Dic10 | D3 | semidirect |
| 22. C3×Dic10 | C2×C3 | semidirect |
| 23. #1203 | C2×C3 | semidirect |
| 24. #1200 | C2×C3 | semidirect |
| 33. C3×C4×D3 | D5 | semidirect |
| 48. C32×C5 | #41 | semidirect |
| 53. Dic10 | C3×D3 | semidirect |
| 76. C4×D3 | C3×D5 | semidirect |
| 87. C3×C5 | #239 | semidirect |
| 88. C3×C5 | C3×#41 | semidirect |
| 104. C32 | #709 | semidirect |
| 117. C5 | C3×#239 | semidirect |
| 124. C3 | #6769 | direct (N × H) |
| 125. C3 | C3×#709 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C3×D60 | 2 |
| C3×C4×C5×D3 | 2 |
| C32×Dic10 | 2 |
| C3×#1203 | 2 |
| C3×#1200 | 2 |
| C3×#709 | 3 |
| #6769 | 3 |
| C3×#239 | 5 |