Determine the type of Cuboid
05AB1E, 26 21 bytes

None: 0
 Cube: 1
 Regular Domino Cuboid: 2
 Shapeshifter Cuboid: 3
 Brick Cuboid: 4
 Floppy Cuboid: 5

•S3X@I¨%÷'•5L¦3ãPÙIkè


Try it online!
or as a Test suite

Explanation

•S3X@I¨%÷'•            # push base-214 compression of the number 123224454212324512210
           5L¦         # push the list [2,3,4,5]
              3ã       # cartesian product with repetion of size 3
                P      # product of each sublist
                 Ù     # remove duplicates
                  Ik   # get the index of input in that list (-1 if non-existant)
                    è  # get the element at this index in the above number


The only place I see that we could save bytes here is find a better way of generating the number 123224454212324512210. 

It is only 1-off from a prime, so one possible save would be to find the index of that prime and generate the index in less than 9 bytes.
I don't know how well the pi-function works for 21-digit primes but that could be a possibility.