diff --git a/Lib/test/test_hashlib.py b/Lib/test/test_hashlib.py
index 46088e5..9204b44 100644
--- a/Lib/test/test_hashlib.py
+++ b/Lib/test/test_hashlib.py
@@ -27,6 +27,11 @@
 py_hashlib = import_fresh_module('hashlib', blocked=['_hashlib'])
 
 try:
+    from _hashlib import HASH
+except ImportError:
+    HASH = None
+
+try:
     import _blake2
 except ImportError:
     _blake2 = None
@@ -386,6 +391,9 @@
         constructors = self.constructors_to_test[name]
         for hash_object_constructor in constructors:
             m = hash_object_constructor()
+            if HASH is not None and isinstance(m, HASH):
+                # _hashopenssl's variant does not have extra SHA3 attributes
+                continue
             self.assertEqual(capacity + rate, 1600)
             self.assertEqual(m._capacity_bits, capacity)
             self.assertEqual(m._rate_bits, rate)
@@ -985,6 +993,10 @@
                 hashlib.scrypt(b'password', salt=b'salt', n=2, r=8, p=1,
                                dklen=dklen)
 
+    def test_normalized_name(self):
+        self.assertNotIn("blake2b512", hashlib.algorithms_available)
+        self.assertNotIn("sha3-512", hashlib.algorithms_available)
+
 
 if __name__ == "__main__":
     unittest.main()