Nonlinear lattices (NLs) arise nowadays in many physical applications, such as nonlinear optics and models of Bose-Einstein condensates. In the latter case, NLs appear due to periodic modulation of the local nonlinearity strength by means of properly patterned magnetic or optical field. This setting is modeled by the equation of the nonlinear Schrödinger type with a periodically modulated coefficient in front of cubic term. It is known that this model supports simplest (single-peaked, spatially symmetric) solitons. These objects are stable in some interval of the respective chemical potential. In our study we address the following two issues: (i) do there exist more complex solitons in this model, and (ii) if yes, which of them are stable? Addressing the former issue, we have found that the model supports a plethora of complex localized modes, that can be coded by means of bi-infinite words of alphabet with an infinite number of symbols. Addressing the latter issue, we have found that a majority of complex nonlinear modes are unstable. However, there are two stable species: (a) single-peaked fundamental solitons, and (b) a new species of subfundamental solitons, namely, narrow spatially antisymmetric modes, which are squeezed, essentially, into a single NL cell. The stability of these modes is predicted, in a certain region of values of the chemical potential, by a variational approach, and has been checked by means of the linear stability analysis, as well as by direct numerical simulations.