Finite Group DatabasebetaFinite Group DatabasebetaOrder 1440.
288 conjugacy classes of subgroups; only the 56 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 56 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | #10624030 |
| commutator, Fitting, cosocle | C32×C4×C5 | C23 |
| Frattini, hypercenter | C4 | D32×D5 |
| socle | C2×C32×C5 | C2×D4 |
| hypocenter | C32×C5 | #135 |
| 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. #10624196 | C2 | semidirect |
| 2. #10624149 | C2 | semidirect |
| 3. #10623826 | C2 | semidirect |
| 4. #10623794 | C2 | semidirect |
| 5. C5×#3716 | C2 | semidirect |
| 6. C3×#6657 | C2 | semidirect |
| 7. C3×#6652 | C2 | semidirect |
| 34. #6657 | D3 | semidirect |
| 35. #6652 | D3 | semidirect |
| 50. #3716 | D5 | semidirect |
| 137. C32×C5 | #135 | semidirect |
| 214. C3×C5 | #912 | semidirect |
| 215. C3×C5 | #916 | semidirect |
| 239. C32 | #4062 | semidirect |
| 262. C5 | #63936 | semidirect |
| 271. C3 | #91276 | semidirect |
| 272. C3 | #91497 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #10624196 | 2 |
| #10624149 | 2 |
| #10623826 | 2 |
| #10623794 | 2 |
| C5×#3716 | 2 |
| C3×#6657 | 2 |
| C3×#6652 | 2 |
| #91497 | 3 |
| #91276 | 3 |
| #63936 | 5 |