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Open your Contacts app to the Phone view. Tap the More or ⋮ button. Tap Settings or Manage Contacts. Tap Move device contacts to or Copy. Tap Phone in the From list. Tap your Google account in the To list. Tap Copy or OK. Visit contacts.google.com in your browser. Sign in with your Google account. Find your newly-added contacts.
Your contacts app should just be displaying the contacts that are stored on your device's memory. Note that the terminology in this section will vary a lot depending on your phone manufacturer. The features discussed may not be available on all devices.   The wording on this feature will vary a lot depending on your device. Look for a utility that allows you to transfer contacts from one account to another. If you don't have the ability to copy contacts to your Google account, you can still export your contacts as a file and them import them to Google. If you are prompted to select the account you want to move contacts from, select your phone's storage. Select your Google account in the list of accounts you can move your contacts to. This will ensure that they reappear when you sign back in with your Google account, and that they can be accessed from contacts.google.com. Your contacts will begin copying to your Google account. This may take a few moments if you are copying a lot of contacts. You can verify that your contacts were successfully added here. Sign in with the same Google account you copied the contacts to. If you see your contacts from your phone here, they have been safely backed up to Google. You may need to wait a little while for the contacts to sync.