Finite Group DatabasebetaFinite Group DatabasebetaOrder 720.
176 conjugacy classes of subgroups; only the 46 normal subgroups are shown.
This group has 176 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C2 | D32×D5 |
| commutator, Fitting, socle, cosocle | C2×C32×C5 | C23 |
| hypocenter | C32×C5 | #41 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #67154 | C2 | semidirect |
| 2. #67152 | C2 | semidirect |
| 3. #67151 | C2 | semidirect |
| 4. C5×D3×Dic3 | C2 | semidirect |
| 5. #67137 | C2 | semidirect |
| 6. C3×#1204 | C2 | semidirect |
| 7. C3×#1203 | C2 | semidirect |
| 24. #1204 | D3 | semidirect |
| 26. #1203 | D3 | semidirect |
| 44. D3×Dic3 | D5 | semidirect |
| 64. C32×C5 | #41 | semidirect |
| 117. C3×C5 | #237 | semidirect |
| 118. C3×C5 | #239 | semidirect |
| 140. C32 | #709 | semidirect |
| 158. C5 | #3796 | semidirect |
| 167. C3 | #6769 | semidirect |
| 168. C3 | #6779 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #67154 | 2 |
| #67152 | 2 |
| #67151 | 2 |
| C5×D3×Dic3 | 2 |
| #67137 | 2 |
| C3×#1204 | 2 |
| C3×#1203 | 2 |
| #6779 | 3 |
| #6769 | 3 |
| #3796 | 5 |