Thanks for your hint!
After second thought, x+x+x+x=0 not only implies order(x)=4, but also indicates x=0 or order(x)=2. Therefore, the candidate group will be:
Z2+Z2+Z2+Z2, which contains (0,0,0,0) or other elements with the order of 2
Z2+Z2+Z4, which contains (0,0,0) or other element with the order of 2 or 4
Z4+Z4, which contains (0,0) or other elements with the order of 2 or 4