Finite Group DatabasebetaFinite Group DatabasebetaOrder 1440.
732 conjugacy classes of subgroups; only the 93 normal subgroups are shown.
This group has 732 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, hypercenter | C22 | A4×D15 |
| commutator, cosocle, hypocenter | C22×C3×C5 | C23×C3 |
| Frattini | C1 | G |
| Fitting, socle | C24×C3×C5 | C2×C3 |
| LCS | G ▷ C22×C3×C5 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×C3×C5×A4 | C2 | semidirect |
| 2. C2×A4×D15 | C2 | direct (N × H) |
| 3. C24×D15 | C3 | semidirect |
| 6. C2×C3×C5×A4 | C22 | semidirect |
| 7. A4×D15 | C22 | direct (N × H) |
| 9. C24×C3×C5 | C2×C3 | semidirect |
| 13. C23×D15 | C2×C3 | semidirect |
| 15. C22×C5×A4 | D3 | semidirect |
| 19. C3×C5×A4 | C23 | semidirect |
| 21. C22×C3×A4 | D5 | semidirect |
| 25. C23×C3×C5 | C22×C3 | semidirect |
| 33. C22×D15 | A4 | direct (N × H) |
| 34. C22×D15 | C22×C3 | semidirect |
| 37. C2×C5×A4 | C2×D3 | semidirect |
| 43. C24×C5 | C3×D3 | semidirect |
| 49. C2×C3×A4 | C2×D5 | semidirect |
| 57. C22×C3×C5 | C2×A4 | semidirect |
| 58. C22×C3×C5 | C23×C3 | semidirect |
| 64. C2×D15 | C2×A4 | direct (N × H) |
| 67. C5×A4 | C22×D3 | semidirect |
| 68. C24×C3 | C3×D5 | semidirect |
| 75. C22×A4 | D15 | direct (N × H) |
| 79. C23×C5 | C2×C3×D3 | semidirect |
| 91. C3×A4 | C22×D5 | semidirect |
| 97. C2×C3×C5 | C22×A4 | semidirect |
| 99. D15 | C22×A4 | direct (N × H) |
| 104. C23×C3 | C2×C3×D5 | semidirect |
| 116. C2×A4 | C2×D15 | direct (N × H) |
| 121. C22×C5 | D3×A4 | semidirect |
| 122. C22×C5 | C22×C3×D3 | semidirect |
| 135. C24 | C3×D15 | semidirect |
| 138. C3×C5 | C23×A4 | semidirect |
| 146. C22×C3 | D5×A4 | semidirect |
| 147. C22×C3 | C22×C3×D5 | semidirect |
| 155. A4 | C22×D15 | direct (N × H) |
| 158. C2×C5 | C2×D3×A4 | semidirect |
| 173. C23 | C2×C3×D15 | semidirect |
| 179. C2×C3 | C2×D5×A4 | semidirect |
| 182. C5 | C22×D3×A4 | semidirect |
| 193. C22 | A4×D15 | direct (N × H) |
| 194. C22 | C22×C3×D15 | semidirect |
| 197. C3 | C22×D5×A4 | semidirect |
| 202. C2 | C2×A4×D15 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×C3×C5×A4 | 2 |
| C2×A4×D15 | 2 |
| C24×D15 | 3 |
| C22×D5×A4 | 3 |
| C22×C3×D15 | 4 |
| C22×D3×A4 | 5 |