"on the ground" or "above the ground"

That depends on the context. There are 3 separate positions. One is under, one is on, and the other is above. What is the full context of what you want to say.

For example, a snake lives under ground. When it's not in its home, it moves on the ground.

However, some hornets ansd wasps build a nest under gound, but when not in their homes, fly over the ground.

In addition to what Ragman posted, in the common vernacular of American English, we typically say under ground or above ground. We don't use "the" in between those words.