Finite Group DatabasebetaFinite Group DatabasebetaOrder 1440.
402 conjugacy classes of subgroups; only the 84 normal subgroups are shown.
This group has 402 conjugacy classes of subgroups — too many to draw. Only its 84 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 | C2×C32×C4×C5 | C22 |
| hypercenter | C2×C4 | D3×D15 |
| hypocenter | C32×C5 | #121 |
| LCS | G ▷ C22×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C22 ◁ C2×C4 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C3×C5×D12 | C2 | semidirect |
| 2. C2×#67150 | C2 | semidirect |
| 3. C2×#67150 | C2 | semidirect |
| 4. C2×D3×Dic15 | C2 | semidirect |
| 5. #10623906 | C2 | semidirect |
| 6. C3×#6714 | C2 | semidirect |
| 7. (C2×D3)⋊Dic15 | C2 | semidirect |
| 26. C2×C5×D12 | D3 | semidirect |
| 31. #6714 | D3 | semidirect |
| 46. C2×C3×C5×D3 | D4 | semidirect |
| 49. C3⋊Dic15 | D4 | semidirect |
| 52. C2×C3×D12 | D5 | semidirect |
| 127. C2×C5×D3 | #67 | semidirect |
| 148. C2×D12 | D15 | semidirect |
| 157. C32×C5 | #121 | semidirect |
| 178. C2×C3×D3 | C22⋊D5 | semidirect |
| 252. C3×C5 | #884 | semidirect |
| 253. C3×C5 | #939 | semidirect |
| 274. C2×D3 | #1220 | semidirect |
| 292. C32 | #4085 | semidirect |
| 319. C5 | #63885 | semidirect |
| 335. C3 | #91453 | semidirect |
| 336. C3 | #91828 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3×C5×D12 | 2 |
| C2×#67150 | 2 |
| C2×D3×Dic15 | 2 |
| #10623906 | 2 |
| C3×#6714 | 2 |
| (C2×D3)⋊Dic15 | 2 |
| #91828 | 3 |
| #91453 | 3 |
| #63885 | 5 |