Neutron scattering experiments were performed on single crystals of layered cobalt-oxides La2-xCax-CoO4 (LCCO) to characterize the charge and spin orders in a wide hole-doping range of 0.3 <= x <= 0.8. For a commensurate value of x = 0.5 in (H, 0, L) plane, two types of superlattice reflections concomitantly appear at low temperature; one corresponds to a checkerboard charge ordered pattern of Co2+/ Co3+ ions and the other is magnetic in origin. Further, the latter magnefic-superlattice peaks show two types of symmetry in the reflections, suggesting anti ferromagnetic-stacking (AF-S) and ferromagnetic-stacking (F-S) patterns of spins along the c direction. From the hole-doping dependence, the in-plane correlation lengths of both charge and spin orders are found to give a maximum at x = 0.5. These features are the same with those of x = 0.5 in Lal(1-x)Sri(1+x)MnO(4) (LSMO), a typical checkerboard and spin ordered compound. However, in (H, H, L) plane, we found a magnetic scattering peak at Q = (0.25, 0.25, 0.5) position below T-N. This magnetic peak can not be understood by considering the Co2+ spin configuration, suggesting that this peak is originated from Co3+ spin order. By analyzing these superlattice reflections, we found that they are originated from high-spin state of Co3+ spin order.