Let $C\_1,C\_2,\ldots,C\_n$ be the conjugacy classes of a finite group $G$. It is possible that there are two non-conjugate subgroups $K \leq G$ and $H \leq G$ such that $|H \cap C\_i|=|K \cap C\_i|$ for all $i = 1,\ldots,n$?



Determining conjugacy class of a subgroup from the sizes of its intersections with the conjugacy classes
Yes, this is precisely the definition of a *Gassmann triple*. Your condition is equivalent to the permutation representations $\mathbb{C}[G/K]$ and $\mathbb{C}[G/H]$ being isomorphic. I believe that the smallest example (in terms of $[G:K]$) is $G=SL\_3(\mathbb{F}\_2)$, $K$ the stabilizer of a line in $\mathbb{F}\_2^3$ and $H$ the stabilizer of a plane.


