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. #10624193 | C2 | semidirect |
| 2. C5×D3×D12 | C2 | semidirect |
| 3. #10623821 | C2 | semidirect |
| 4. #10623805 | C2 | semidirect |
| 5. #10623764 | C2 | semidirect |
| 6. C3×#6654 | C2 | semidirect |
| 7. C3×#6650 | C2 | semidirect |
| 11. #67180 | C22 | semidirect |
| 13. C3×D60 | C22 | semidirect |
| 19. C3×C3⋊#168 | C22 | semidirect |
| 20. C32×#168 | C22 | semidirect |
| 34. #6654 | D3 | semidirect |
| 36. #6650 | D3 | semidirect |
| 46. D3×D12 | D5 | semidirect |
| 58. D60 | C2×D3 | semidirect |
| 73. C3⋊#168 | C2×D3 | semidirect |
| 76. C3×#168 | C2×D3 | semidirect |
| 77. C3×#168 | C2×D3 | semidirect |
| 137. C32×C5 | AGL(1,ℤ/8) | semidirect |
| 148. #168 | D32 | semidirect |
| 216. C3×C5 | #908 | semidirect |
| 217. C3×C5 | #911 | semidirect |
| 242. C32 | #4078 | semidirect |
| 266. C5 | #63795 | semidirect |
| 276. C3 | #91304 | semidirect |
| 277. C3 | #91494 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624193 | 2 |
| C5×D3×D12 | 2 |
| #10623821 | 2 |
| #10623805 | 2 |
| #10623764 | 2 |
| C3×#6654 | 2 |
| C3×#6650 | 2 |
| #91494 | 3 |
| #91304 | 3 |
| #63795 | 5 |