A generalization of Per Alexandersson's comment is to take the isomorphism class of any graph $G$. There is exactly one such unlabelled graph, so it is obviously harder to count the labelled objects.

For Per Alexandersson's comment, $G$ is a path with $n$ vertices, in which case there are $n!$ labelled versions of $G$. If $G$ is a cycle with $n$ vertices, there are $\frac{(n-1)!}{2}$ labelled versions of $G$. If $G$ is $K_{n,n}$, there are $\frac{1}{2}\binom{2n}{n}$ labelled versions of $G$. These examples are quite structured, making it fairly easy to count the labelled versions, but for a random graph $G$ on $n$ vertices, it will likely be difficult to count the number of labeled versions of $G$.

A generalization of Per Alexandersson's comment is to take the isomorphism class of any graph (or digraph) $G$. There is exactly one such unlabelled graph, so it is obviously harder to count the labelled objects.

For Per Alexandersson's comment, $G$ is a directed path with $n$ vertices, in which case there are $n!$ labelled versions of $G$. If $G$ is a cycle with $n$ vertices, there are $\frac{(n-1)!}{2}$ labelled versions of $G$. If $G$ is $K_{n,n}$, there are $\frac{1}{2}\binom{2n}{n}$ labelled versions of $G$. These examples are quite structured, making it fairly easy to count the labelled versions, but for a random graph $G$ on $n$ vertices, it will likely be difficult to count the number of labeled versions of $G$.