Smith Normal Form and classification of factor groups according to the theorem of finitely generated abelian groups With reference to this question, it was mentioned in the comments that these problems could be solved using the Smith Normal Form. However, I am unable to extract an exact method from the Wikipedia article alone, and further searching yields few general examples. I am asking for an example where the algorithm is used to classify a factor group.

OK, here is an example. Find the structure of the group
$${\mathbb Z}^3/\langle (6,6,4),\,(6,12,8) \rangle.$$
Write down the matrix:
$$\left( \begin{array}{rrr} 6&6&4\\6&12&8 \end{array}\right)$$
There are many different ways to reduce this to SNF. Here is one way (hoping I haven't miscalculated):
Subtract Column 3 from Column 2
Interchange Columns 1 and 2
Subtract 2 times Row 1 from Row 2
Subtract 3 times Column 1 from Column 2    
Subtract 2 times Column 1 from Column 3
Negate Row 2

giving SNF:
$$\left( \begin{array}{rrr} 2&0&0\\0&6&0 \end{array}\right)$$
It is important to remember that the number of cyclic factors should be equal to the number of generators of the original group (although, in general, some of these factors could have order $1$ and hence be trivial). So if there are fewer subgroup generators than group generators, we have to adjoin copies of ${\mathbb Z}$. So, in this example, the group is isomorphic to
$${\mathbb Z}_2 \oplus {\mathbb Z}_6 \oplus {\mathbb Z}$$
