Finite Group DatabasebetaFinite Group DatabasebetaOrder 480.
328 conjugacy classes of subgroups; only the 110 normal subgroups are shown.
This group has 328 conjugacy classes of subgroups — too many to draw. Only its 110 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 |
| hypercenter | Q8 | D3×D5 |
| hypocenter | C3×C5 | C2×#41 |
| 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×#239 | C2 | semidirect |
| 3. C3×#713 | C2 | semidirect |
| 4. #6777 | C2 | semidirect |
| 5. C4×D3×D5 | C2 | semidirect |
| 6. #6772 | C2 | semidirect |
| 7. #6765 | C2 | semidirect |
| 10. C3×C5×Q8 | C22 | semidirect |
| 12. Dic30 | C22 | semidirect |
| 13. C5×D12 | C22 | semidirect |
| 14. C4×C5×D3 | C22 | semidirect |
| 15. C3×D20 | C22 | semidirect |
| 16. C3×C4×D5 | C22 | semidirect |
| 17. #1201 | C22 | semidirect |
| 19. D3×Dic5 | C22 | semidirect |
| 20. D5×Dic3 | C22 | semidirect |
| 23. #713 | D3 | semidirect |
| 38. #239 | D5 | semidirect |
| 45. C5×Q8 | C2×D3 | semidirect |
| 50. D20 | C2×D3 | semidirect |
| 54. C4×D5 | C2×D3 | semidirect |
| 58. D15 | #41 | semidirect |
| 62. C3×Q8 | C2×D5 | semidirect |
| 67. D12 | C2×D5 | semidirect |
| 71. C4×D3 | C2×D5 | semidirect |
| 88. C3×C5 | C2×#41 | semidirect |
| 103. Q8 | D3×D5 | semidirect |
| 117. C5 | D3×#41 | semidirect |
| 126. C3 | D5×#41 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| Q8×D15 | 2 |
| C5×#239 | 2 |
| C3×#713 | 2 |
| #6777 | 2 |
| C4×D3×D5 | 2 |
| #6772 | 2 |
| #6765 | 2 |
| D5×#41 | 3 |
| D3×#41 | 5 |