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We develop a procedure for the complete computational enumeration of lattice$3$-polytopes of width larger than one, up to any given number of latticepoints. We also implement an algorithm for doing this and enumerate those withat most eleven lattice points (there are 216,453 of them).  In order to achieve this we prove that if $P$ is a lattice 3-polytope ofwidth larger than one and with at least seven lattice points then it fits inone of three categories that we call boxed, spiked and merged.  Boxed polytopes have at most 11 lattice points; in particular they arefinitely many, and we enumerate them completely with computer help. Spikedpolytopes are infinitely many but admit a quite precise description (andenumeration). Merged polytopes are computed as a union (merging) of twopolytopes of width larger than one and strictly smaller number of latticepoints.

Enumeration of lattice 3-polytopes by their number of lattice points