Finite Group DatabasebetaFinite Group DatabasebetaOrder 480.
332 conjugacy classes of subgroups; only the 108 normal subgroups are shown.
This group has 332 conjugacy classes of subgroups — too many to draw. Only its 108 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 | H(5,𝔽2) |
| 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. #6821 | C2 | semidirect |
| 2. C5×#239 | C2 | semidirect |
| 3. C3×#713 | C2 | semidirect |
| 4. #6777 | C2 | semidirect |
| 5. D3×D20 | C2 | semidirect |
| 6. D5×D12 | C2 | semidirect |
| 7. #6771 | C2 | semidirect |
| 9. C2×D3×D5 | C22 | semidirect |
| 10. C3×C5×Q8 | C22 | semidirect |
| 11. D60 | C22 | semidirect |
| 12. C4×D15 | 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. #1204 | C22 | semidirect |
| 18. #1203 | C22 | semidirect |
| 19. #1202 | C22 | semidirect |
| 20. #1201 | C22 | semidirect |
| 24. #713 | D3 | semidirect |
| 39. #239 | D5 | semidirect |
| 45. C5×Q8 | C2×D3 | semidirect |
| 52. D20 | C2×D3 | semidirect |
| 54. C4×D5 | C2×D3 | semidirect |
| 62. C3×Q8 | C2×D5 | semidirect |
| 69. D12 | C2×D5 | semidirect |
| 71. C4×D3 | C2×D5 | semidirect |
| 88. C3×C5 | H(5,𝔽2) | semidirect |
| 103. Q8 | D3×D5 | semidirect |
| 117. C5 | #1009 | semidirect |
| 126. C3 | #4149 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #6821 | 2 |
| C5×#239 | 2 |
| C3×#713 | 2 |
| #6777 | 2 |
| D3×D20 | 2 |
| D5×D12 | 2 |
| #6771 | 2 |
| #4149 | 3 |
| #1009 | 5 |