The full passage you are referring to says: 'Then comes the end, when He delivers the kingdom to God the Father, when He puts an end to all rule and all authority and power.25 For He must reign till He has put all enemies under His feet.26 The last enemy that will be destroyed is death.27 For 'He has put all things under His feet.' But when He says 'all things are put under Him,' it is evident that He who put all things under Him is excepted.28 Now when all things are made subject to Him, then the Son Himself will also be subject to Him who put all things under Him, that God may be all in all' (1 Corinthaisn 15:24-28).
This passage is describing the end of the Son's mediatorial ministry of redemption, not the end of the Son Himself. When God took on human flesh, it was forever. We will never cease to see the Father by looking at the Son. God will forever be manifested in the flesh.
It would be more accurate to say that the sonship is fulfilled. The key to understanding this passage is differentiating between the Son and His mission. The mission is completed but the Son lives forever. Jesus said, 'I am He who lives, and was dead, and behold, I am alive forevermore. Amen' (Revelation 1:18).
Another way of thinking about this passage is to see the Son as God's humanity, which is currently involved in the process of redeeming man. By this process, everything is brought into subjection to the Son by the power of the Spirit (the Spirit puts everything put under His feet). Once that process is completed, the redemptive purposes of the Son (God's humanity) will be subjected to the larger purposes of the eternal Spirit.