This labeled graph is given, I need to find the number of its spanning trees.

The number of spanning trees of the following graph is 3

and the number of spanning trees of this is 8

So as a result the number of spanning trees of the graph is $3^2 \times 8^3 = 4608$ while it is not the correct answer.
Your reasoning looks correct. And, according to Sage, 4608 is indeed the number of spanning trees.