1. The almost Hermitian structures belong to the most important and meaningful differential-geometrical structures. The existence of Gray-Brown 3-vector cross products [1] in Cayley algebra gives a set of substantive examples of almost Hermitian manifolds. As it is well known, every 3-vector cross product on Cayley algebra induces a 1vector cross product (or, what is the same in this case, an almost Hermitian structure) on its six-dimensional oriented submanifold [2], [3]. Such almost Hermitian structures (in particular, K¨ahlerian, nearly K¨ahlerian, quasi K¨ahlerian, Hermitian, special Hermitian etc) were studied by a number of outstanding geometers: E. Calabi, A. Gray, V. F. Kirichenko, K. Sekigawa and L. Vranchen. We recall that an almost Hermitian manifold is a 2n-dimensional manifold M2n with a Riemannian metric g = ⟨·, ·⟩ and an almost complex structure J. Moreover, the following condition must hold ⟨JX, JY ⟩ = ⟨X, Y ⟩ , X, Y ∈ ℵ(M2n), where ℵ(M2n) is the module of smooth vector fields on M2n. An almost Hermitian manifold is Hermitian, if its almost complex structure is integrable [4]. 2. In the present work, we consider six-dimensional Hermitian planar submanifolds of Cayley algebra. We present the following results. Theorem 1. If a six-dimensional Hermitian planar submanifold of Cayley algebra satisfies the U-Kenmotsu hypersurfaces axiom, then it is K¨ahlerian. Theorem 2. A symmetric non-K¨ahlerian Hermitian six-dimensional submanifold of Ricci type does not admit totally umbilical Kenmotsu hypersurfaces. Theorem 3. If a six-dimensional Hermitian planar submanifold of general type of Cayley algebra satisfies the 1-cosymplectic hypersurfaces axiom, then it is K¨ahlerian. Theorem 4. The Hermitian structure on a 6-dimensional planar submanifold of Cayley algebra is stable if and only if such submanifold is totally geodesic. Theorem 5. The quasi-Sasakian structure on a totally umbilical hypersurface of a six-dimensional Hermitian planar submanifold of Cayley algebra is Sasakian.