Finite Group DatabasebetaFinite Group DatabasebetaOrder 1440.
486 conjugacy classes of subgroups; only the 88 normal subgroups are shown.
This group has 486 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C2×D3×D15 |
| commutator, socle, cosocle | C22×C32×C5 | C23 |
| Fitting | C23×C32×C5 | C22 |
| hypercenter | C23 | D3×D15 |
| hypocenter | C32×C5 | #127 |
| LCS | G ▷ C22×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C22 ◁ C23 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×#67184 | C2 | semidirect |
| 2. C2×C3×#1220 | C2 | semidirect |
| 3. C2×C3×C5×#67 | C2 | semidirect |
| 4. C2×#67152 | C2 | semidirect |
| 5. C2×#67151 | C2 | semidirect |
| 6. C2×#67150 | C2 | semidirect |
| 7. Dic3×Dic15 | C2 | semidirect |
| 30. C2×#1220 | D3 | semidirect |
| 32. C2×C5×#67 | D3 | semidirect |
| 52. C3⋊Dic15 | D4 | semidirect |
| 53. C3×Dic15 | D4 | semidirect |
| 54. C3×C5×Dic3 | D4 | semidirect |
| 58. C2×C3×#67 | D5 | semidirect |
| 159. Dic15 | #67 | semidirect |
| 161. C5×Dic3 | #67 | semidirect |
| 172. C2×#67 | D15 | semidirect |
| 177. C32×C5 | #127 | semidirect |
| 206. C3×Dic3 | C22⋊D5 | semidirect |
| 288. C3×C5 | #940 | semidirect |
| 289. C3×C5 | #940 | semidirect |
| 329. Dic3 | #1220 | semidirect |
| 337. C32 | #4086 | semidirect |
| 370. C5 | #63979 | semidirect |
| 388. C3 | #91567 | semidirect |
| 389. C3 | #91829 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×#67184 | 2 |
| C2×C3×#1220 | 2 |
| C2×C3×C5×#67 | 2 |
| C2×#67152 | 2 |
| C2×#67151 | 2 |
| C2×#67150 | 2 |
| Dic3×Dic15 | 2 |
| #91829 | 3 |
| #91567 | 3 |
| #63979 | 5 |