Finite Group DatabasebetaFinite Group DatabasebetaOrder 720.
120 conjugacy classes of subgroups; only the 50 normal subgroups are shown.
This group has 120 conjugacy classes of subgroups — too many to draw. Only its 50 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, hypercenter | C22 | D3×D15 |
| commutator | C2×C32×C5 | C2×C4 |
| Fitting, socle, prime radical, cosocle | C22×C32×C5 | C22 |
| hypocenter | C32×C5 | C4⋊C4 |
| LCS | G ▷ C2×C32×C5 ▷ C32×C5 | |
| UCS | C1 ◁ C22 | |
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 7. C3⋊Dic15 | C4 | semidirect |
| 37. C32×C5 | C4⋊C4 | semidirect |
| 70. C3×C5 | Dic3⋊C4 | semidirect |
| 71. C3×C5 | Dic3⋊C4 | semidirect |
| 85. C32 | #683 | semidirect |
| 98. C5 | #3724 | semidirect |
| 104. C3 | #6669 | semidirect |
| 105. C3 | #6712 | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C2×C3⋊Dic15 | 2 |
| C2×C3×Dic15 | 2 |
| C2×C3×C5×Dic3 | 2 |
| #6712 | 3 |
| #6669 | 3 |
| #3724 | 5 |