Finite Group DatabasebetaFinite Group DatabasebetaOrder 960.
560 conjugacy classes of subgroups; only the 187 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 187 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 | C23×D15 |
| commutator, socle, cosocle | C22×C3×C5 | C24 |
| Fitting | C3×C4×C5×Q8 | C2 |
| prime radical | C2×D60 | C22 |
| hypercenter | C4×Q8 | D15 |
| hypocenter | C3×C5 | D4×Q8 |
| LCS | G ▷ C22×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C2×Q8×D15 | C2 | semidirect |
| 2. C3×C4×C5×Q8 | C2 | semidirect |
| 3. #91786 | C2 | semidirect |
| 5. #91758 | C2 | semidirect |
| 19. C4×C5×Q8 | D3 | semidirect |
| 24. C3×C5×Q8 | D4 | semidirect |
| 29. D60 | Q8 | direct (N × H) |
| 33. C3×C4×Q8 | D5 | semidirect |
| 51. C2×D15 | C2×Q8 | semidirect |
| 54. C3×C4×C5 | C2×Q8 | semidirect |
| 67. C5×Q8 | D12 | semidirect |
| 78. C4×Q8 | D15 | semidirect |
| 85. C3×Q8 | D20 | semidirect |
| 99. C4×C5 | D3×Q8 | semidirect |
| 111. C3×C5 | D4×Q8 | semidirect |
| 118. C3×C4 | Q8×D5 | semidirect |
| 126. Q8 | D60 | direct (N × H) |
| 137. C5 | Q8×D12 | semidirect |
| 145. C4 | Q8×D15 | semidirect |
| 146. C3 | Q8×D20 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×Q8×D15 | 2 |
| C3×C4×C5×Q8 | 2 |
| #91786 | 2 |
| C4×D60 | 2 |
| #91758 | 2 |
| Q8×D20 | 3 |
| Q8×D12 | 5 |