Finite Group DatabasebetaFinite Group DatabasebetaOrder 1440.
1078 conjugacy classes of subgroups; only the 296 normal subgroups are shown.
This group has 1078 conjugacy classes of subgroups — too many to draw. Only its 296 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C23×C3 | D3×D5 |
| commutator, cosocle | C2×C3×C5 | C24×C3 |
| Frattini | C2 | C22×C3×D3×D5 |
| Fitting, socle | C23×C32×C5 | C22 |
| hypocenter | C3×C5 | C22×C3×D4 |
| LCS | G ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C23×C3 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C23×C3×D15 | C2 | semidirect |
| 2. C23×C32×D5 | C2 | semidirect |
| 3. C22×C3×C5×Dic3 | C2 | semidirect |
| 4. C2×C3×#1202 | C2 | direct (N × H) |
| 6. C22×#1202 | C3 | direct (N × H) |
| 9. C22×C3×D15 | C22 | semidirect |
| 11. C22×C32×D5 | C22 | semidirect |
| 12. C2×C3×C5×Dic3 | C22 | semidirect |
| 13. C3×#1202 | C22 | direct (N × H) |
| 15. C23×D15 | C2×C3 | semidirect |
| 18. C23×C3×D5 | D3 | semidirect |
| 19. C23×C3×D5 | C2×C3 | semidirect |
| 21. C22×C5×Dic3 | C2×C3 | semidirect |
| 23. C2×#1202 | C2×C3 | direct (N × H) |
| 24. C22×C32×C5 | D4 | semidirect |
| 27. C2×C3×D15 | C23 | semidirect |
| 29. C2×C32×D5 | C23 | semidirect |
| 30. C3×C5×Dic3 | C23 | semidirect |
| 35. C22×C3×Dic3 | D5 | semidirect |
| 40. C22×D15 | C22×C3 | semidirect |
| 47. C22×C3×D5 | C2×D3 | semidirect |
| 48. C22×C3×D5 | C22×C3 | semidirect |
| 50. C2×C5×Dic3 | C22×C3 | semidirect |
| 52. #1202 | C22×C3 | direct (N × H) |
| 55. C2×C32×C5 | C2×D4 | semidirect |
| 60. C23×D5 | C3×D3 | semidirect |
| 69. C2×C3×Dic3 | C2×D5 | semidirect |
| 72. C22×C3×C5 | #67 | semidirect |
| 73. C22×C3×C5 | C3×D4 | semidirect |
| 77. C2×D15 | C23×C3 | semidirect |
| 84. C2×C3×D5 | C22×D3 | semidirect |
| 85. C2×C3×D5 | C23×C3 | semidirect |
| 87. C5×Dic3 | C23×C3 | semidirect |
| 96. C22×Dic3 | C3×D5 | semidirect |
| 97. C32×C5 | C22×D4 | semidirect |
| 102. C22×D5 | C2×C3×D3 | semidirect |
| 107. C22×C32 | D20 | semidirect |
| 111. C3×Dic3 | C22×D5 | semidirect |
| 115. C2×C3×C5 | C2×#67 | semidirect |
| 116. C2×C3×C5 | C2×C3×D4 | semidirect |
| 140. C2×Dic3 | C2×C3×D5 | semidirect |
| 141. C22×C5 | C3×#67 | semidirect |
| 146. C2×D5 | C22×C3×D3 | semidirect |
| 149. C2×C32 | C2×D20 | semidirect |
| 157. C3×C5 | C22×#67 | semidirect |
| 158. C3×C5 | C22×C3×D4 | semidirect |
| 169. C22×C3 | #1202 | direct (N × H) |
| 170. C22×C3 | C3×D20 | semidirect |
| 176. Dic3 | C22×C3×D5 | semidirect |
| 177. C2×C5 | C2×C3×#67 | semidirect |
| 181. C32 | C22×D20 | semidirect |
| 195. C2×C3 | C2×#1202 | direct (N × H) |
| 196. C2×C3 | C2×C3×D20 | semidirect |
| 200. C5 | C22×C3×#67 | semidirect |
| 205. C22 | C3×#1202 | direct (N × H) |
| 209. C3 | C22×#1202 | direct (N × H) |
| 210. C3 | C22×C3×D20 | semidirect |
| 213. C2 | C2×C3×#1202 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C23×C3×D15 | 2 |
| C23×C32×D5 | 2 |
| C22×C3×C5×Dic3 | 2 |
| C2×C3×#1202 | 2 |
| C22×C3×D20 | 3 |
| C22×#1202 | 3 |
| C22×C3×#67 | 5 |