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Call a compact Riemannian manifold M a strongly unstable manifold if it is not the range or domain of a nonconstant stable harmonic map and also the homotopy class of any map to or from M contains elements of arbitrarily small energy. If M is isometrically immersed in Euclidean space, then a condition on the second fundamental form of M is given which implies M is strongly unstable. As compact isotropy irreducible homogeneous spaces have "standard" immersions into Euclidean space this allows a complete list of the strongly unstable compact irreducible symmetric spaces to be made.

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