For a locally compact group G we consider the algebra CD(G) of convolution dominated operators on L2(G): An operator A:L2(G)→L2(G) is called convolution dominated if there exists a∈L1(G) such that for all f∈L2(G) |Af(x)|≤a⋆|f|(x)for almost all x∈G. In the case of discrete groups those operators can be dealt with quite sufficiently if the group in question is rigidly symmetric. For non-discrete groups we investigate the subalgebra of regular convolution dominated operatorsCDreg(G). For amenable G which is rigidly symmetric as a discrete group we show that any element of CDreg(G) is invertible in CDreg(G) if it is invertible as a bounded operator on L2(G). We give an example of a symmetric group E for which the convolution dominated operators are not inverse-closed in the bounded operators on L2(E).