but airport utility can't connect to Airport Extreme.
The most common cause of this is incorrect IPv6 settings on your Mac and/or the AirPort Extreme.
Yosemite....if that is what you are using....has a number of network bugs, so one single solution will not solve all problems. But, the iPv6 correction should help keep the AirPort from disappearing in AirPort Utility.
On each Mac......
Open System Preferences (gear icon on the dock)
Open Network
Click on WiFi on the left panel to highlight it
Click Advanced at the lower right of the window
Click the TCP/IP tab
Make sure that the setting for Configure IPv6 is set to Link Local Only
Click OK
Click Apply
With the network window still open
Click on Ethernet on the left panel to highlight it
Click Advanced
Click the TCP/IP tab
Make sure that the settting for Configure IPv6 is set to Link Local Only
Click OK
Click Apply
Close the windows and restart the Mac
Then, repeat the same thing on each Mac that might be running Lion, Mountain Lion, Mavericks or Yosemite
Post back when that task is done, and we will check the IPv6 settings on the AirPort Extreme.