Why is my iPhone 8 Plus not showing software update?

You can update to iOS 16.7.10 and no further. In the case that you are trying to update to that, make sure you meet all the over the air update requirements written by Community member Lawrence Finch below.

• You need at least 10 GB of free storage for iOS 16 and later or 6 GB for iOS 15 or lower; more is better
• You must be connected to a Wi-Fi network (and not a personal hotspot) unless it is an iPhone that supports cellular 5G (12, 13, 14, 15 or SE 3rd edition) and has a 5G network connection
• The phone must be charged to at least 50%, and preferably plugged in so it stays above 50% during the update
• The phone must not be in Low Power Mode
• The phone must not be connected through VPN, and it may be necessary to delete any VPN apps and VPN profiles
• The phone cannot be in Airplane mode and must have a cellular data connection
• You cannot still have a beta of a previous version installed on the phone
• You cannot have a previous version that was downloaded, but not installed - go to Settings/General/iPhone Storage to look; if you find an iOS installer, delete it

If you can't meet all of them you can still update using your computer.

Your iPhone 8+ is nominally capable of running iOS 16.

”Jump Updating” (i.e. 14 to 16) is sometimes more reliably performed w/ a cable and computer:

Update your iPhone or iPad - Apple Support

Also see this:

If your iPhone or iPad won't update - Apple Support

You can update an iPhone 8 Plus to the latest release of iOS 16 – currently iOS 16.7.10.

About iOS 16 Updates - Apple Support

You cannot update an iPhone 8 Plus to iOS 17 or higher. For the iPhone 8 Plus, which originally ran iOS 11, iOS 16 is the end of the line.