This was a problem in the Scottish Book posed by Ulam. It was solved  by 

F. J. Freniche, The Number of Nonisomorphic Boolean subalgebras of a Power Set, Proc. Amer. Math. Soc., 91 (1984) 199-201.

the number of non isomorphic sub algebras is $2^{2^k}$ for any cardinal $k$