Determine the type of Cuboid Introduction:

I'm a twisty puzzles collector. Here you can see my current collection of ± 300 puzzles.

I think everyone knows the regular Rubik's Cube (3x3x3 Cube), which is a NxNxN Cube. There are also Cuboids (block-shaped puzzles), which come in different forms, perhaps better explained by SuperAntionioVivaldi here:


Regular Domino Cuboids (like the 2x2x3; 2x3x3; 3x3x4; etc.) - They come in the form of NxNx(N+O) or Nx(N+O)x(N+O), which have either two odd dimensions and an even, or two even and an odd.
Shapeshifter Cuboids (like the 2x2x4; 3x3x5; 3x3x9; 4x4x6; etc.) - They come in the form of NxNx(N+P), which as the name suggests, shapeshifts (in all directions). All three dimensions are either odd or even.
Floppy Cuboids (like the 1x3x3; 2x4x4; etc.) - They come in the form of Nx(N+P)x(N+P), which are almost the same as the Shapeshifters, but with so-called Floppy Parities. 
Brick Cuboids (like the 2x3x4; 3x4x5; 2x3x5; etc.) - They come in the form of Nx(N+O)x(N+P), which just like the Regular Domino Cuboids have either two odd dimensions and an even, or two even and an odd; but don't have any of the same dimensions.
Ultimate Shapeshifters (like the 2x4x6; 3x5x7; 2x4x10; etc.) - They come in the form of Nx(N+O)x(N+R), and shapeshift in any direction. All three dimensions are either odd or even; but don't have any of the same dimensions.


Challenge:

Input:

A positive integer n with the following restriction: 8 &lt;= n &lt;= 125.
n can be uniquely decoded as the product of three values (the dimensions), of which each is between 2 and 5 inclusive.

The reason I've restricted it to 2-5 is to prevent duplicated inputs (like 1x2x4 = 8 and 2x2x2 = 8), even though there are many lower/higher order Cuboids out there. This also means there are no test cases for Ultimate Shapeshifters.

Output / Test cases:

These are all the cases your program/function should support, ranging from edge lengths 2 through 5 in every possible three-dimensional configuration:

Input   Cuboid/Cube   Type/Output
8       2x2x2         Cube
12      2x2x3         Regular Domino Cuboid
16      2x2x4         Shapeshifter Cuboid
20      2x2x5         Regular Domino Cuboid
18      2x3x3         Regular Domino Cuboid
24      2x3x4         Brick Cuboid
30      2x3x5         Brick Cuboid
32      2x4x4         Floppy Cuboid
40      2x4x5         Brick Cuboid
50      2x5x5         Regular Domino Cuboid
27      3x3x3         Cube
36      3x3x4         Regular Domino Cuboid
45      3x3x5         Shapeshifter Cuboid
48      3x4x4         Regular Domino Cuboid
60      3x4x5         Brick Cuboid
75      3x5x5         Floppy Cuboid
64      4x4x4         Cube
80      4x4x5         Regular Domino Cuboid
100     4x5x5         Regular Domino Cuboid
125     5x5x5         Cube


Challenge rules:


Any non-Cube/non-Cuboid input within the 8-125 range should result in 'none' as output.
The output format is your own choice. I think the most reasonable is integers, like 0 = 'none'; 1 = Cube; 2 = Regular Domino Cuboid; 3 = Shapeshifter Cuboid; 4 = Floppy Cuboid; 5 = Brick Cuboid. Any other output format is also fine, as long as you specify which one you've used.


General rules:


This is code-golf, so shortest answer in bytes wins.
Don't let code-golf languages discourage you from posting answers with non-codegolfing languages. Try to come up with an as short as possible answer for 'any' programming language.
Standard rules apply for your answer, so you are allowed to use STDIN/STDOUT, functions/method with the proper parameters, full programs. Your call.
Default Loopholes are forbidden. (NOTE: Since I don't know if there is a smart formula for the input-to-output conversion, it is allowed to hardcode the answers based on the input.)
If possible, please add a link with a test for your code.
Also, please add an explanation if necessary.
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.