Unplug the router end of the patch cable first.
The RJ45 system relies on a spring tab holding the plug in which is released by compressing the plastic tab sticking out parallel to the cable. This may well be partially covered by a shroud.
Option two. Can you get at the RJ45 socket from the other (in)side? If it's a network card, remove it from the motherboard connecting strip & deal with it accordingly. If, as I suspect, it's actually of the motherboard or you have a laptop/small form factor then this can be a challenge. I've only had to do this once when my cat managed to chew through the CAT5 cable (I know ~ bad pun) too close for me to get a grip on the cable.
Do not be tempted to pull the end of the network cable — you may well be left with the RJ45 plug buried in the socket.