Finite Group DatabasebetaFinite Group DatabasebetaOrder 1440.
492 conjugacy classes of subgroups; only the 165 normal subgroups are shown.
This group has 492 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C22 | C22×D45 |
| commutator, cosocle | C2×C5×C9 | C24 |
| Frattini | C2×C3 | C23×D15 |
| Fitting | C2×C5×Q8×C9 | C2 |
| socle | C22×C3×C5 | C22×D3 |
| hypercenter | C2×Q8 | D45 |
| hypocenter | C5×C9 | C2×#41 |
| LCS | G ▷ C2×C5×C9 ▷ C5×C9 | |
| UCS | C1 ◁ C22 ◁ C2×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×C5×Q8×C9 | C2 | semidirect |
| 2. #10623713 | C2 | direct (N × H) |
| 3. C2×D180 | C2 | semidirect |
| 4. C2×C4×D45 | C2 | semidirect |
| 7. C5×Q8×C9 | C22 | semidirect |
| 10. D180 | C22 | semidirect |
| 11. C4×D45 | C22 | semidirect |
| 23. C2×Q8×C9 | D5 | semidirect |
| 34. C2×C5×C9 | #41 | semidirect |
| 37. C2×C5×Q8 | D9 | semidirect |
| 42. Q8×C9 | C2×D5 | semidirect |
| 56. C5×C9 | C2×#41 | semidirect |
| 58. C5×Q8 | C2×D9 | semidirect |
| 83. C2×C9 | #713 | semidirect |
| 87. C2×Q8 | D45 | semidirect |
| 96. C2×C5 | #3701 | semidirect |
| 99. C9 | C2×#713 | semidirect |
| 101. Q8 | C2×D45 | semidirect |
| 109. C5 | C2×#3701 | semidirect |
| 117. C2 | #10623713 | direct (N × H) |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C5×Q8×C9 | 2 |
| #10623713 | 2 |
| C2×D180 | 2 |
| C2×C4×D45 | 2 |
| C2×#6821 | 3 |
| C2×#3701 | 5 |