Finite Group DatabasebetaFinite Group DatabasebetaOrder 1440.
624 conjugacy classes of subgroups; only the 154 normal subgroups are shown.
This group has 624 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C2 | C22×D3×D15 |
| commutator, socle, cosocle | C2×C32×C5 | C24 |
| Fitting | C32×C5×Q8 | C22 |
| hypercenter | Q8 | D3×D15 |
| hypocenter | C32×C5 | 2−1+4 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #10624287 | C2 | semidirect |
| 2. C3×Q8×D15 | C2 | semidirect |
| 3. C3×C5×D3×Q8 | C2 | semidirect |
| 4. #10624201 | C2 | semidirect |
| 5. #10624199 | C2 | semidirect |
| 6. #10624198 | C2 | semidirect |
| 7. #10624197 | C2 | semidirect |
| 29. Q8×D15 | D3 | semidirect |
| 32. C5×D3×Q8 | D3 | semidirect |
| 48. C3×D3×Q8 | D5 | semidirect |
| 132. D3×Q8 | D15 | semidirect |
| 137. C32×C5 | 2−1+4 | semidirect |
| 214. C3×C5 | #1007 | semidirect |
| 215. C3×C5 | #1007 | semidirect |
| 237. C32 | #4147 | semidirect |
| 260. C5 | #64316 | semidirect |
| 269. C3 | #92029 | semidirect |
| 270. C3 | #92098 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624287 | 2 |
| C3×Q8×D15 | 2 |
| C3×C5×D3×Q8 | 2 |
| #10624201 | 2 |
| #10624199 | 2 |
| #10624198 | 2 |
| #10624197 | 2 |
| #92098 | 3 |
| #92029 | 3 |
| #64316 | 5 |