The following solution is based on using Stream API
- Count the frequencies of elements from missin each list value ofmapsand collect the frequency into some object (e.g. a list)
- Sort the new object by the frequency in the reverse order and then by the key of initial map (note: the key may need to be converted into int to provide the expected output: 1, 4, 11;  comparing the keys as String returns the order 1, 11, 4)
- Build the resulting map using Collectors.toMapwithLinkedHashMap::newsupplier
List<String> miss = List.of("3", "7");
Map<String, List<String>> maps = Map.of(
     "1", List.of("0", "3", "6"),
     "4", List.of("2", "3", "4"),
     "6", List.of("0", "3", "7"),
    "11", List.of("1", "3", "6"),
    "17", List.of("2", "6", "11")
);
Map<String, List<String>> sorted = maps.entrySet()
    .stream()
    .map(e -> Arrays.asList(e, 
        e.getValue().stream()
            .mapToInt(i -> (int) miss.stream().filter(i::equals).count())
            .sum()
    ))
    .sorted(Comparator
        .<List>comparingInt(ee -> (int) ee.get(1)).reversed()
        .thenComparingInt(ee -> Integer.parseInt(((Map.Entry<String, List<String>>) ee.get(0)).getKey()))
    )
    .map(ee -> (Map.Entry<String, List<String>>) ee.get(0))
    .collect(Collectors.toMap(
        Map.Entry::getKey,
        Map.Entry::getValue,
        (v1, v2) -> v1,
        LinkedHashMap::new
    ));
System.out.println(sorted);
Output:
{6=[0, 3, 7], 1=[0, 3, 6], 4=[2, 3, 4], 11=[1, 3, 6], 17=[2, 6, 11]}