Let the weights of the puppies be, in order, A,B,C,D,E. Then the largest value of the ten is A+B=258 and the smallest is D+E=228. The sum of all ten values is 4(A+B+C+D+E)=2424, so A+B+C+D+E=606 and hence C=120.

Now the second largest value is 252 and this is A+C, so that A=132 and B=126. Similarly D=118 and E=110.

The answer is 132.