Finite Group DatabasebetaFinite Group DatabasebetaOrder 480.
124 conjugacy classes of subgroups; only the 38 normal subgroups are shown.
This group has 124 conjugacy classes of subgroups — too many to draw. Only its 38 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center | C2 | #6790 |
| commutator, cosocle | C3×C4×C5 | C23 |
| Frattini | C4 | C2×D3×D5 |
| Fitting | C3×C5×Q8 | C22 |
| socle | C2×C3×C5 | C2×D4 |
| hypercenter | Q8 | D3×D5 |
| hypocenter | C3×C5 | #135 |
| LCS | G ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5 | |
| UCS | C1 ◁ C2 ◁ C4 ◁ Q8 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. #6821 | C2 | semidirect |
| 2. #6766 | C2 | semidirect |
| 3. C5×#216 | C2 | semidirect |
| 4. C3×#688 | C2 | semidirect |
| 5. #6660 | C2 | semidirect |
| 6. #6653 | C2 | semidirect |
| 7. #6648 | C2 | semidirect |
| 25. #688 | D3 | semidirect |
| 38. #216 | D5 | semidirect |
| 83. C3×C5 | #135 | semidirect |
| 110. C5 | #916 | semidirect |
| 118. C3 | #4065 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| #6821 | 2 |
| #6766 | 2 |
| C5×#216 | 2 |
| C3×#688 | 2 |
| #6660 | 2 |
| #6653 | 2 |
| #6648 | 2 |
| #4065 | 3 |
| #916 | 5 |