A hexagonal nanostructure formed by seven core shell nanocylinders filled with different dielectric cores is investigated. The surface plasmon resonance in such a hexagonal nanostructure under conditions of different illumination wavelengths, dielectric cores, angles of incidence, and thicknesses of silver shells is studied by use of the finite element method. Simulation results show that the resonant wavelength is redshifted as the dielectric constant and the size of the core increase. The peak resonant wavelength and the local field enhancement are approximately proportional to the radius of the dielectric core. Additionally, the surface plasmon field excited by TM-polarized light at the incident angle of theta=15 degrees is exactly a linear combination of those excited at incident angles of theta=0 degrees and 30 degrees, confirming the linear nature of the surface plasmon resonance in a nanostructure formed by linear media.