Types of isomorphism between groups

Question

I found this problem: find the types of isomorphism between abelian groups of order 144. Up until now I only found that G, an abelian group of 144 elements, can only be isomorphic with either Z2 x Z2 x Z2 x Z2 x Z3 x Z3, Z2 x Z8 x Z3 x Z3, Z4 x Z4 x Z3 x Z3, Z2 x Z2 x Z4 x Z3 x Z3 or Z16 x Z3 x Z3, and again all of those but with Z9 instead of Z3 x Z3, but I don't know anything about the morphisms between them.

Answer 1

The possible abelian groups of order $144=2^4 \cdot 3^2$ correspond to the combinations of additive partitions of the exponents $4$ and $2$. There are thus $5 \cdot 2=10$ possible abelian groups of order $144$. Just complete the table below. No two entries are isomorphic. $$ \matrix{ & 2 & 1+1 \\ 4 & C_{16} \times C_9 & C_{16} \times C_3 \times C_3 \\ 3 + 1 & C_{8} \times C_ 2 \times C_9 & C_{16} \times C_3 \times C_3 \\ 2 + 2 \\ 2 + 1 + 1 \\ 1 + 1 + 1 + 1 & & C_ 2 \times C_ 2 \times C_ 2 \times C_ 2 \times C_ 3 \times C_ 3\\ } $$ Note that you can combine $2$-factors with $3$-factors, but won't get anything new. For instance $$ C_{8} \times C_{18} \cong C_{8} \times C_2 \times C_9 \cong C_{2} \times C_{72} $$