Solution:

There are $2^8 = 256$ subsets in all. The ones to exclude are those contained in $\{1,2,3,4,5\}$ (there are $2^5 = 32$) or contained in $\{4,5,6,7,8\}$ (another $32$). A subset of both is exactly a subset of the intersection $\{4, 5\},$ and there are $2^2 = 4$ of those.

By inclusion-exclusion, $32 + 32 - 4 = 60$ subsets fail, so $256 - 60 = 196$ subsets have the required property.