To classify organisms, we use the five-kingdom system of classification. Obviously, the crayfish is not an amoeba-like organism or a single-cell organism. It is clearly not a plant, because it does not manufacture its own food, has a nerve cord and can move. Therefore it must be an animal. It is actually relatively complex as animals go. I know its classification to phylum level: Kingdom: Animalia - Animals.
Subkindgom: Metazoa - Animals with a digestive tract.
Phylum: Chordata - Animals with a neural tube. The crayfish, as a crustacean, is not a vertebrate, but it is still quite a complex animal.
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The lineage of crayfish as follows: Kingdom Animalia, Phylum Arthropoda, Subphylum Crustacea, Class Malacostraca, Order Decapoda. The living crayfish of the world are distributed into 3 different families, Astacidae, Parastacidae, and Cambaridae. Cambaridae contains by far the largest number of species, with at least 300 hundred.
Crayfish belong to class Crustacea, which also includes lobsters, crabs, shrimp, waterfleas, sow bugs, barnacles, and many others.