There is a difference between maximal and maximum. A subgroup is maximal if there is no proper subgroup that contains it. I've rarely heard the term maximum applied to a subgroup, but what it should mean is that every other subgroup is contained in it. For example, in Z8, the subgroup {0, 2, 4, 6} is a "maximum subgroup." This is the same as the meaning of maximum applied to sets of real numbers: the maximum of the set [0, 3] is 3, because every other element of the set is less than it.
Note that a group might not have a maximum subgroup - for example, Z2 X Z2 - just as the set (0,3) does not have a maximum. (However, any group does have maximal subgroups.)
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