Let K1,K2 be cones. We say that K1 is a subcone of K2 if ExtK1⊂ExtK2. Furthermore, if K1≠K2, K1 is called a proper subcone; if dimK1=dimK2, K1 is called a non-degenerate subcone. We first prove that every n-dimensional indecomposable cone, n⩾3, contains a non-degenerate indecomposable subcone which has no more than 2n-2 extremals. Then we construct for each n⩾3 an n-dimensional indecomposable cone with exactly 2n-2 extremals such that each of its proper non-degenerate subcones is decomposable.
