Your set = { 1, 2, 1/2 }
on * operator 2 * 2 = 4 not in Set. Hence not closed => not Group => no question of finite subset.
Besides, Necessary condition for a subgroup is,
order(Subgroup) = order(Group) / k , k = {1,2,3,...}
Here order(G) = 3
Hence order(S) can be only 1 as 1 = 3 / 3
Example you chose is, Subgroup = { 1, 1/2 }, hence order(S) = 2
which does not divide 3. Hence not subgroup by LaGrange's theorem.