homework and exercises - Simple Quantum Mechanics Question about The Commutator of Translation Operators

Say there is $\hat{J} = \exp[-i \hat{p} l/ \hbar]$ and $\hat{U}= \exp[-i\hat{H}t/ \hbar]$, where $\hat{H}$ is time-independent.

Can we say anything about $[\hat{J},\hat{U}]$? Is it zero? How do we show this?

For example if $\hat{H} = \hat{p}^2 /2m + m\omega^2 \hat{x}^2/2$.