TITLE: Advice on finding counterexamples
QUESTION [15 upvotes]: I am reaching out for specific advice on how one should go about finding counterexamples. It seems almost every time I've ever attempted a "find a counterexample" problem, I have to cheat by asking a friend, or I have to use a computer. That being said, I've encountered another problem which I can look up the solution to, and probably understand why it is a counter example after it is given to me, but I am helpless when it comes to actually finding it. That being said, I am asking if someone could post a counterexample to the following problem and explain to me their thought process. This would help me so much. 
Problem: Give an example of two subgroups $H$ and $K$ of a group $G$ whose union $H\cup K$ is not a subgroup of $G$.
How far I've gotten: I know that a nonempty subset $H$ of a finite group $G$ is a subgroup if and only if $a,b\in H$ implies $ab\in H$. Using this we can know that  If $a,b\in H\cup K$, but $ab\not \in H\cup K$, then $H$ is not a subgroup of $G$. So I know what a counterexample should look like if I see one, but how do I find it? 
Thanks so much.

REPLY [8 votes]: If you don’t find a counterexample using the approaches already suggested, try to prove the theorem, and see where you get stuck: this may well give you a better handle on the properties that a counterexample will have to have, and that’s invaluable information when you’re searching for a counterexample or trying to construct one from scratch. This doesn’t really arise in the case under consideration, because counterexamples are very easy to come by: it’s almost harder not to find one. In more complicated settings, however, the technique is very helpful.