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It puts out 65W of heat and it's an LGA1151 socket.
You can get something like this for it:
https://www.quietpc.com/mugen-5-pcghWhich will take the CPU fan down to 15 dBA at peak.
There's no mention of a GPU, so I'm guessing you're using the onboard on the motherboard https://www.gigabyte.com/Motherboard/GA-Z170-HD3P-rev-10
I think you should be fine to get a big case that could take a larger CPU heatsink and achieve total silence.
The only other fan in your system appears to be the PSU:
https://www.evga.com/products/product.aspx?pn=120-G1-0650-RXSomething like this: https://www.quietpc.com/fsp-hyper-m?product=5375
This is only 600W, but without an optical drive or GPU... you don't need 650W.
With a new CPU heatsink and a new PSU and a case, the entire system would be under 20dBA (basically virtually unnoticeable).
I got a custom Quiet PC from https://www.quietpc.com/ :)
It's totally silent.
However this is only achieved because there are no fans anywhere in it (no case fans, no CPU fan, and no GPU card)... if you add a fan you will have a hum from it. The hotter the components get (CPU, GPU, motherboard) and the less cold air that can pass over them... the faster the fans run and the worse the noise.
Yes you can change the case, but there is a natural tension between the volume of air that can pass through the case (holes) and the increase of noise associated (sound is carried by air passing through holes).
I'm basically saying: Yes, change your case and it will make a difference... but if your CPU and GPU produce a lot of heat then you'll never get it to silent so don't be too spendy on the case unless you want to go all-in and tackle the root cause which will be the choice of components.