diff --git a/src/circuit/Gate.js b/src/circuit/Gate.js
index bafe9020..0e07856e 100644
--- a/src/circuit/Gate.js
+++ b/src/circuit/Gate.js
@@ -180,6 +180,12 @@ class Gate {
          * @type {undefined | !function(!int) : !int}
          */
         this.knownBitPermutationFunc = undefined;
+        /**
+         * When a permutation factors into sub-permutations over subsets of the bits, this map indexes each bit with
+         * an id for the group that it belongs to.
+         * @type {!Array.<!int>}
+         */
+        this.knownBitPermutationGroupMasks = undefined;
         /**
          * Indicates that a gate is a phasing function, and that it phases each computation basis state by the amount
          * returned by this function.
@@ -278,6 +284,7 @@ class Gate {
         g.knownPermutationFuncTakingInputs = this.knownPermutationFuncTakingInputs;
         g.customColumnContextProvider = this.customColumnContextProvider;
         g.customDisableReasonFinder = this.customDisableReasonFinder;
+        g.knownBitPermutationGroupMasks = this.knownBitPermutationGroupMasks;
         return g;
     }
 
@@ -568,11 +575,12 @@ class GateBuilder {
      * Provides a function equivalent to how the gate rearranges wires, for checking in tests if the gate's behavior is
      * correct.
      * @param {!function(!int) : !int} knownBitPermutationFunc Returns the output of the permutation for a
-     * given input, assuming the gate is exactly sized to the overall circuit.
+     *     given input, assuming the gate is exactly sized to the overall circuit.
      * @returns {!GateBuilder}
      */
     setKnownEffectToBitPermutation(knownBitPermutationFunc) {
         this.gate.knownBitPermutationFunc = knownBitPermutationFunc;
+        this.gate.knownBitPermutationGroupMasks = permutationGrouping(knownBitPermutationFunc, this.gate.height);
         this.gate._isDefinitelyUnitary = true;
         this.gate._stableDuration = Infinity;
         this.gate._hasNoEffect = false;
@@ -928,4 +936,28 @@ class GateBuilder {
     }
 }
 
+/**
+ * @param {!function(!int) : !int} knownBitPermutationFunc Returns the output of the permutation for a
+ *     given input, assuming the gate is exactly sized to the overall circuit.
+ * @param {!int} height
+ * @returns {!Array.<!int>}
+ */
+function permutationGrouping(knownBitPermutationFunc, height) {
+    let seen = new Set();
+    let result = [];
+    for (let i = 0; i < height; i++) {
+        let mask = 0;
+        let j = i;
+        while (!seen.has(j)) {
+            seen.add(j);
+            mask |= 1 << j;
+            j = knownBitPermutationFunc(j);
+        }
+        if (mask !== 0) {
+            result.push(mask);
+        }
+    }
+    return result;
+}
+
 export {Gate, GateBuilder}
diff --git a/src/circuit/GateColumn.js b/src/circuit/GateColumn.js
index fbaca1db..f809ebb8 100644
--- a/src/circuit/GateColumn.js
+++ b/src/circuit/GateColumn.js
@@ -110,6 +110,21 @@ class GateColumn {
         return false;
     }
 
+    /**
+     * @returns {!int}
+     */
+    controlMask() {
+        let mask = 0;
+        for (let i = 0; i < this.gates.length; i++) {
+            if (this.gates[i] !== undefined &&
+                    this.gates[i].definitelyHasNoEffect() &&
+                    this.gates[i].isControl()) {
+                mask |= 1 << i;
+            }
+        }
+        return mask;
+    }
+
     /**
      * @param {!int} inputMeasureMask
      * @param {!int} row
@@ -170,25 +185,41 @@ class GateColumn {
         let g = this.gates[row];
         let mask = ((1 << g.height) - 1) << row;
         let maskMeasured = mask & inputMeasureMask;
-        if (maskMeasured !== 0) {
+        if (maskMeasured !== 0 && g.knownBitPermutationFunc === undefined) {
             // Don't try to superpose measured qubits.
             if (g.effectMightCreateSuperpositions()) {
                 return "no\nremix\n(sorry)";
             }
 
+            // Don't try to mix measured and coherent qubits, or coherently mix measured qubits.
             if (g.effectMightPermutesStates()) {
-                // Only permutations that respect bit boundaries can be performed on mixed qubits.
-                if (maskMeasured !== mask && (g.knownBitPermutationFunc === undefined ||
-                                              this.hasMeasuredControl(inputMeasureMask))) {
+                if (maskMeasured !== mask || this.hasCoherentControl(inputMeasureMask)) {
                     return "no\nremix\n(sorry)";
                 }
+            }
+        }
 
-                // Permutations affecting classical states can't have quantum controls.
-                if (this.hasCoherentControl(inputMeasureMask)) {
+        // Check permutation subgroups for bad mixing of measured and coherent qubits.
+        if (g.knownBitPermutationGroupMasks !== undefined) {
+            for (let maskGroup of g.knownBitPermutationGroupMasks) {
+                let isSingleton = ((maskGroup - 1) & maskGroup) === 0;
+                if (isSingleton) {
+                    continue;
+                }
+
+                maskGroup <<= row;
+                let hasCoherentQubits = (maskGroup & inputMeasureMask) !== maskGroup;
+                let hasMeasuredQubits = (maskGroup & inputMeasureMask) !== 0;
+                let coherentControl = this.hasCoherentControl(inputMeasureMask);
+                let controlled = this.hasControl(inputMeasureMask);
+                let coherentControlledMixingOfMeasured = hasMeasuredQubits && coherentControl;
+                let controlledMixingOfCoherentAndMeasured = hasCoherentQubits && hasMeasuredQubits && controlled;
+                if (coherentControlledMixingOfMeasured || controlledMixingOfCoherentAndMeasured) {
                     return "no\nremix\n(sorry)";
                 }
             }
         }
+
         return undefined;
     }
 
diff --git a/test/circuit/CircuitDefinition.test.js b/test/circuit/CircuitDefinition.test.js
index 66970dcf..36d076c9 100644
--- a/test/circuit/CircuitDefinition.test.js
+++ b/test/circuit/CircuitDefinition.test.js
@@ -901,6 +901,66 @@ suite.test("gateAtLocIsDisabledReason", () => {
     bad(1, 0, `MR`, ['R', Gates.Detectors.XDetectControlClear]);
     bad(1, 0, `MR`, ['R', Gates.Detectors.YDetectControlClear]);
     good(1, 0, `MR`, ['R', Gates.Detectors.ZDetectControlClear]);
+
+    // Permutation sub-groups.
+    good(2, 0, `--P-
+                --/-
+                --/-
+                --/-
+                H-●-`, ['P', Gates.InterleaveBitsGates.InterleaveBitsGateFamily]);
+    good(2, 0, `-MP-
+                --/-
+                --/-
+                -M/-
+                H-●-`, ['P', Gates.InterleaveBitsGates.InterleaveBitsGateFamily]);
+    good(2, 0, `--P-
+                --/-
+                --/-
+                -M/-
+                HM●-`, ['P', Gates.InterleaveBitsGates.InterleaveBitsGateFamily]);
+    good(2, 0, `-MP-
+                --/-
+                --/-
+                -M/-
+                HM●-`, ['P', Gates.InterleaveBitsGates.InterleaveBitsGateFamily]);
+    good(2, 0, `--P-
+                -M/-
+                -M/-
+                --/-
+                HM●-`, ['P', Gates.InterleaveBitsGates.InterleaveBitsGateFamily]);
+    good(2, 0, `-MP-
+                -M/-
+                -M/-
+                -M/-
+                HM●-`, ['P', Gates.InterleaveBitsGates.InterleaveBitsGateFamily]);
+    bad(2, 0, `--P-
+               -M/-
+               -M/-
+               --/-
+               H-●-`, ['P', Gates.InterleaveBitsGates.InterleaveBitsGateFamily]);
+    bad(2, 0, `--P-
+               -M/-
+               --/-
+               --/-
+               HM●-`, ['P', Gates.InterleaveBitsGates.InterleaveBitsGateFamily]);
+
+    // Non-trivial subgroup.
+    good(2, 0, `--P-
+                -M/-
+                -M/-
+                --/-
+                -M/-
+                --/-
+                --/-
+                HM●-`, ['P', Gates.InterleaveBitsGates.InterleaveBitsGateFamily]);
+    bad(2, 0, `--P-
+               -M/-
+               --/-
+               -M/-
+               -M/-
+               --/-
+               --/-
+               HM●-`, ['P', Gates.InterleaveBitsGates.InterleaveBitsGateFamily]);
 });
 
 suite.test("gateAtLocIsDisabledReason_controls", () => {