An all-dielectric periodic array is proposed to form plasmon lattice resonances (PLR). In the array, Si nanopillars are on top of SiO2 nanopillars, and SiO2 nanopillars are on top of quartz substrates. The simulated results show that the line-width of the PLR can be as small as 3.3 nm. This can be attributed to the coupling between the Mie resonances of Si nanopillars and the diffracted waves. While the PLR can't be formed by the periodic Si nanopillar array directly sitting on quartz substrates. The diameter and height of Si nanopillars, the period of the array and the height of SiO2 nanopillars have significant impacts on the PLR. This work extends the application of PLR.