We develop a theory of Ennola duality for subgroups of finite groups of Lie type, relating subgroups of twisted and untwisted groups of the same type. Roughly speaking, one finds that subgroups H of GUd(q)\documentclass[12pt]{minimal}
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\begin{document}$${\mathrm {GU}}_d(q)$$\end{document} correspond to subgroups of GLd(-q)\documentclass[12pt]{minimal}
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\begin{document}$${\mathrm {GL}}_d(-q)$$\end{document}, where -q\documentclass[12pt]{minimal}
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\begin{document}$$-q$$\end{document} is interpreted modulo |H|. Analogous results for types other than A\documentclass[12pt]{minimal}
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\begin{document}$${\mathrm A}$$\end{document} are established, including for those exceptional types where the maximal subgroups are known, although the result for type D\documentclass[12pt]{minimal}
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\begin{document}$${\mathrm D}$$\end{document} is still conjectural. Let M denote the Gram matrix of a non-zero orthogonal form for a real, irreducible representation of a finite group, and consider α=det(M)\documentclass[12pt]{minimal}
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\begin{document}$$\alpha =\sqrt{\det (M)}$$\end{document}. If the representation has twice odd dimension, we conjecture that α\documentclass[12pt]{minimal}
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\begin{document}$$\alpha $$\end{document} lies in some cyclotomic field. This does not hold for representations of dimension a multiple of 4, with a specific example of the Janko group J1\documentclass[12pt]{minimal}
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\begin{document}$${\mathrm J_1}$$\end{document} in dimension 56 given. (This tallies with Ennola duality for representations, where type D2n\documentclass[12pt]{minimal}
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\begin{document}$${\mathrm D_{2n}}$$\end{document} has no Ennola duality with 2D2n\documentclass[12pt]{minimal}
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\begin{document}$${}^2\mathrm D_{2n}$$\end{document}.)
