Let $(X,d)$ be a metric space and ${Iso}(X,d)$ the associated isometry group. We study the subadditivity of the minimal displacement function $f:{Iso}(X,d)\to {R}$ for different metric spaces. When $(X,d)$ is ultrametric, we prove that the minimal displacement function is subadditive. We show, by a simple algebraic argument, that subadditivity does not hold for the direct isometry group of the hyperbolic plane. The same argument can be used for other metric spaces.