What could be the elementary divisors of subgroup of $\mathbb{Z}^2$

Question

What can be the elementary divisors of subgroup $H \le \mathbb{Z}^2$ of index $36$?

I can't see what's the connection between the index and the elementary divisors?

As far as I know, elementary divisors are based on the basis of free group, and has nothing to do with the index. For a given basis, the elementary divisors of subgroup are natural numbers so that each number divides the next, and multiplying the basis by those numbers gives us a new basis for the second group.

What's the connection?

Answer 1

Hint: The question can be rephrased as follows:

What are the possible abelian groups of order $36$?

and then applied to $\mathbb{Z}^2/H$.