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We study the smallest, as well as the largest numbers of congruences oflattices of an arbitrary finite cardinality $n$. Continuing the work of Freeseand Cz\' edli, we prove that the third, fourth and fifth largest numbers ofcongruences of an $n$--element lattice are: $5\cdot 2^{n-5}$ if $n\geq 5$,respectively $2^{n-3}$ and $7\cdot 2^{n-6}$ if $n\geq 6$. We also determine thestructures of the $n$--element lattices having $5\cdot 2^{n-5}$, respectively$2^{n-3}$ congruences, along with the structures of their congruence lattices.

Some Extremal Values of the Number of Congruences of a Finite Lattice