I just started this game and spent an hour getting everything together and going through the tutorial. I never saw an autosave status appear or when I opened the menu, I didn't have the option to save my gave anywhere and I looked. Online mode was way too laggy(my server is wired) for me and caused me to die in a battle where I couldn't even defend myself so I tried switching and that didn't allow me so I had to exit the game, where I now have restart everything, even editing my character. It's not going to make me stop the game but it will cause me if there isn't a proper UX that allows players to save or switch between the modes.
The game does autosave automatically - but offline mode and online mode are two separate modes of gameplay so they don't cross over naturally.
You can move a character from online to offline (https://inflexiongames.zendesk.com/hc/en-us/articles/26890830706203-How-to-Copy-your-Online-Mode-Character-to-Offline-Mode)
If you use the same mode, your game will be automatically saved. You just can't switch between them at will.