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Yes. You can be in E-3 status and pursue a green card; it just means that your international travel might be limited during the process.

I got a green card while on E-3. Make sure your visa doesn't expire while you are in a no-travel period, as you might not be able to renew it if you have a pending change of status (or so I hear - was not applicable in my case).

I moved from e3 to green card. There is a no-travel period between petition and granting, but it’s possible