If by "not invaded", you mean that a war hasn't been fought on our territory since 1812, you'd be wrong. There was this little thing called "The Civil War" (an oxymoron if there ever was one) that killed more Americans (north and south) than any other war we've ever fought, perhaps as many as all the others put together.

There was also the attack on Pearl Harbor. Admittedly, it wasn't an invasion and Hawaii wasn't a state at the time, but it was a US territory and some 3000 people were killed including quite a few civilians.

As for invasion, two of Alaska's Aleutian Islands were invaded and occupied by Japanese troops in 1942. They were Kiska and Attu. The Japanese withdrew from Kiska before our invasion to take it back, but it took a very costly invasion to knock them off of Attu. I spend a year on a neighboring island (Shemya) and can tell you that it's truly a miserable place.

And let's not forget 9-11...