By means of the Hirota bilinear method and symbolic computation, high-order lump-type solutions and a kind of interaction solutions are presented for a (3+1)-dimensional nonlinear evolution equation. The high-order lump-type solutions of the associated Hirota bilinear equation are presented, which is a kind of positive quartic-quadratic-function solution. At the same time, the interaction solutions can also be obtained, which are linear combination solutions of quartic-quadratic-functions and hyperbolic cosine functions. Physical properties and dynamical structures of two classes of the presented solutions are demonstrated in detail by their graphs. Copyright © 2019 Chinese Physical Society and IOP Publishing Ltd.