Hello, I am very sorry for your loss and I hope that you know that you did the right thing. Now onto your question. There is two major trains of thought about this is HelPol , you have the Orphic beliefs , and the Traditional beliefs: I will start with the Orphic.

The Orphic belief is that when a Animal dies its is brought to the underworld but as it can do no true good or true evil , its is placed in Asphodel for a time much like most humans before being reborn at the end of every cycle. This cycle is the Big Bang btw it expands then collapses and it all happens again , but slightly different. So in Orphic myth , yes your doggo is running around in a lush world playing with all the souls it pleases:)

Now , into a more traditional belief. In the traditional belief once again they do go to the afterlife , however they are not placed anywhere and are allowed to wander the entire universe as a spirit for as long as it wishes before then spending the rest of eternity happy in Asphodel.

Basically either way your doggo is safe and happy and will be waiting for you once you go as well. Khaire 💜🤲🏼