I know that subnetting helps in dividing the internet part and physical address + host part.

Like, a class C prefix uses 3 octets for network part and 1 byte for host part. I'm reading this book and they keep referring that subnets are used in the internet.

But how do the router happen to change the last part of the public IP address to send request for a particular host in the local network? 

Does the router have control over the last octet to change it by itself? Or is the subnetting technology used for local network only?

And does the ISP have any control over how we use subnet? Like my public IP is 49.244.218.112. Does it change the last part 112 (if it were using Class C) for different hosts in a network?