Abstract This paper deals with the structure of the controllable set of a multimodal system. We define a maximal component of the controllable set, and we investigate the controllable set as the union of its maximal components. We show that for each positive integer k, state dimension n ⩾ 3, and control dimension m ⩽ n − 1 there is a multimodal system whose controllable set S ( L) is the union of exactly k maximal subspaces of R n , and this system has k as bound on the number of iterations necessary to reach any state in S ( L) from zero. We also show the above holds with k = ∞. We show that for each state dimension n and each control dimension m, there is a completely controllable multimodal system having bound 2 n - 2 m.