Let $\alpha = (\alpha_1,\alpha_2,\dots)$ be a partition of the positive integer $n$, that is, a nonincreasing sequence of nonnegative integers $\alpha_j$, with only finitely many nonzero terms, whose sum is $n$. A subpartition of $\alpha$ is a partition $\beta = (\beta_1,\beta_2,\dots)$ such that $\beta_j \le \alpha_j$ for all $j\ge1$. How many distinct subpartitions can $\alpha$ have? I am interested in as good an upper bound as possible, in terms of $n$, that holds for every partition of $n$ (that is, I'm interested in bounds that hold in the "worst case").

Every subpartition of $\alpha$ is a partition of some number $0\le m\le n$, and so a trivial upper bound (using the notation $p(m)$ for the number of partitions of $m$) is $$ \sum_{m=0}^n p(m) \le (n+1)p(n) \ll e^{\pi\sqrt{2n/3}} $$ by the Hardy–Ramanujan asymptotics for $p(n)$. However, I suspect the truth is quite a bit smaller than this. References to existing bounds would be optimal, but arguments that improve this trivial bound substantially (including improvements to the constant in the exponent) are very welcome.

[Note that we are literally counting partitions here, not weighting the partitions differently according to their shape. (The motivation for my question comes from trying to estimate the number of isomorphism classes of subgroups of a finite $p$-group.) Asking a similar question but where partitions are weighted by a Plancherel measure, while not relevant to this question, is probably still interesting (and, as far as I know, an open problem); there my suspicion is that this upper bound cannot be substantially improved—the intuition being that "typical" partitions $\alpha$ of $n$ give the worst case for the upper bound, and most typical partitions of $(1-\epsilon)n$ might in fact be subpartitions of $\alpha$.]