Symbolism, Imagery, Allegory
You know the old chestnut about how dogs start to look like their owners? Well, in Oliver Twist, dogs start to act like their owners.
Sikes’s dog (whose name is Bull’s-Eye, but we only hear the name mentioned once or twice) is like Sikes’s shadow. He has some of the same personality flaws as Sikes, including a violent temper. Because the dog is always with him, it’s tempting to read the dog as a kind of stand-in for some part of Sikes’s psyche.
The only time the dog leaves him is when Sikes is almost out of his mind with guilt after having killed Nancy—maybe the dog represents Sikes’s violent and criminal impulses? The dog even kills itself by jumping after Sikes off of the roof, and smashing its head on the rocks below. The dog can’t exist without Sikes, because he’s a part of him.