The group of rotations of Euclidean space in $N$ dimensions is the special orthogonal group $\text{SO}(N)$. It is simple and all its Lie subgroups are (semi)simple as well.
The conformal group of Euclidean space is simple too; it's the indefinite special orthogonal group $\text{SO}(N+1,1)$. But one of its subgroups is the Euclidean isometry group, also called the inhomogeneous special orthogonal group $\text{ISO}(N)$. This subgroup is not simple: $\text{ISO}(N) = \mathbb R^N \rtimes \text{SO}(N)$.
The above was just an example. In general, is there any way I can know if a simple group will or will not have nonsimple subgroups?