Admirals (ranks O-7 through O-10) are a subset of flag officers in that flag officer is generic for admirals and generals both. As some others have mentioned,

Admirals (ranks O-7 through O-10) are a subset of flag officers in that flag officer is generic for admirals and generals both. As some others have mentioned, there are joint billets that don't count against these quotas. DIRNSA is one example, and one with some name recognition perhaps. Currently it's Admiral Rogers, though another recent person in this billet was General Alexander. NATO commands like SACEUR are another place where you'd find flag officers not included in these quotas.