Finite Group DatabasebetaFinite Group DatabasebetaOrder 480.
312 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 312 conjugacy classes of subgroups — too many to draw. Only its 122 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×D5 |
| commutator, socle, cosocle | C2×C3×C5 | C24 |
| Fitting | C3×C5×Q8 | C22 |
| prime radical | C2×D3×D5 | C22 |
| hypercenter | Q8 | D3×D5 |
| hypocenter | C3×C5 | C22×Q8 |
| LCS | G ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C2 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. Q8×D15 | C2 | semidirect |
| 2. C5×D3×Q8 | C2 | semidirect |
| 3. C3×Q8×D5 | C2 | semidirect |
| 5. D15⋊Q8 | C2 | semidirect |
| 6. D3×Dic10 | C2 | semidirect |
| 7. D5×Dic6 | C2 | semidirect |
| 10. C3×C5×Q8 | C22 | semidirect |
| 12. Dic30 | C22 | semidirect |
| 14. C5×Dic6 | C22 | semidirect |
| 16. C3×Dic10 | C22 | semidirect |
| 17. #1204 | C22 | semidirect |
| 26. Q8×D5 | D3 | direct (N × H) |
| 32. D3×D5 | Q8 | direct (N × H) |
| 41. D3×Q8 | D5 | direct (N × H) |
| 46. C5×Q8 | C2×D3 | semidirect |
| 55. Dic10 | C2×D3 | semidirect |
| 58. D15 | C2×Q8 | semidirect |
| 59. C3×D5 | C2×Q8 | semidirect |
| 60. C5×D3 | C2×Q8 | semidirect |
| 63. C3×Q8 | C2×D5 | semidirect |
| 72. Dic6 | C2×D5 | semidirect |
| 88. C3×C5 | C22×Q8 | semidirect |
| 100. D5 | D3×Q8 | direct (N × H) |
| 106. Q8 | D3×D5 | direct (N × H) |
| 116. D3 | Q8×D5 | direct (N × H) |
| 117. C5 | C2×D3×Q8 | semidirect |
| 126. C3 | C2×Q8×D5 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×D15 | 2 |
| C5×D3×Q8 | 2 |
| C3×Q8×D5 | 2 |
| C4×D3×D5 | 2 |
| D15⋊Q8 | 2 |
| D3×Dic10 | 2 |
| D5×Dic6 | 2 |
| C2×Q8×D5 | 3 |
| C2×D3×Q8 | 5 |