On the basis of the family of quasifiliform Lie algebra laws of dimension 9 of 16 parameters and 17 constraints, this paper is devoted to identify the invariants that completelyclassify the algebras over the complex numbers except for isomorphism. It is proved thatthe nullification of certain parameters or of parameter expressions divides the family intosubfamilies such that any couple of them is nonisomorphic and any quasifiliform Liealgebra of dimension 9 is isomorphic to one of them. The iterative and exhaustive computation with Maple provides the classification, which divides the original family into263 subfamilies, composed of 157 simple algebras, 77 families depending on 1 parameter,24 families depending on 2 parameters, and 5 families depending on 3 parameters.
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