Let $R$ be a commutative Noetherian ring, $\fa$ an ideal of $R$, and $M$ a minimax $R$-module. We prove that the local cohomology modules $H_{\fa}^j(M)$ are $\fa$-cominimax; that is, $\Ext_R^i(R/{\fa},H_{\fa}^j(M))$ is minimax for all $i$ and $j$ in the following cases: (a) $\Dim R/{\fa}=1$; (b) $\Cd(\fa)=1$, where cd is the cohomological dimension of $\fa$ in $R$; (c) $\Dim R \leq 2$. In these cases we also prove that the Bass numbers and the Betti numbers of $H_{\fa}^j(M)$ are finite.
