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The maximal possible symmetry is realized by the quaternionic projective space $\\mathbb{H}P^n$, which is flat and has the symmetry algebra $\\mathfrak{sl}(n+1,\\mathbb{H})$ of dimension $4n^2+8n+3$. For non-flat almost quaternionic manifolds we compute the next biggest (submaximal) symmetry dimension. We show that it is equal to $4n^2-4n+9$ for $n>1$ (it is equal to 8 for $n=1$). 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