Finite Group DatabasebetaFinite Group DatabasebetaOrder 1440.
292 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | D3×#1203 |
| commutator, Fitting, cosocle | C32×C4×C5 | C23 |
| Frattini, hypercenter | C4 | D32×D5 |
| socle | C2×C32×C5 | C2×D4 |
| hypocenter | C32×C5 | AGL(1,ℤ/8) |
| LCS | G ▷ C32×C4×C5 ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 ◁ C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. D3×D60 | C2 | semidirect |
| 2. C5×#3795 | C2 | semidirect |
| 3. #10623823 | C2 | semidirect |
| 4. #10623800 | C2 | semidirect |
| 5. #10623765 | C2 | semidirect |
| 6. C3×#6654 | C2 | semidirect |
| 7. C3×#6650 | C2 | semidirect |
| 12. C5×#634 | C22 | semidirect |
| 14. C3×C5×D12 | C22 | semidirect |
| 19. C3×C3⋊#168 | C22 | semidirect |
| 20. C32×#168 | C22 | semidirect |
| 35. #6654 | D3 | semidirect |
| 36. #6650 | D3 | semidirect |
| 47. #3795 | D5 | semidirect |
| 65. C5×D12 | C2×D3 | semidirect |
| 73. C3⋊#168 | C2×D3 | semidirect |
| 76. C3×#168 | C2×D3 | semidirect |
| 77. C3×#168 | C2×D3 | semidirect |
| 94. #634 | C2×D5 | semidirect |
| 96. C3×D12 | C2×D5 | semidirect |
| 137. C32×C5 | AGL(1,ℤ/8) | semidirect |
| 148. #168 | D32 | semidirect |
| 184. D12 | D3×D5 | semidirect |
| 216. C3×C5 | #908 | semidirect |
| 217. C3×C5 | #911 | semidirect |
| 242. C32 | #4095 | semidirect |
| 266. C5 | #63795 | semidirect |
| 276. C3 | #91305 | semidirect |
| 277. C3 | #91480 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| D3×D60 | 2 |
| C5×#3795 | 2 |
| #10623823 | 2 |
| #10623800 | 2 |
| #10623765 | 2 |
| C3×#6654 | 2 |
| C3×#6650 | 2 |
| #91480 | 3 |
| #91305 | 3 |
| #63795 | 5 |