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Top of Page Molecular Structures of Organic Compounds - Hierarchy of Point Groups
Hierarchy of Point Groups:

The question which point group (i.e. which molecular conformation or geometry) is 'more symmetrical' than another can only be answered if one point group is a sub-group of the other group. Otherwise, the decision if for example a C4 symmetrical molecule is more symmetrical than one of the C6 point group would involve purely aesthetic considerations.

The table below gives a full correlation matrix between the 32 crystallographic point groups, where the initial high-symmetry groups are listed in the first column, and correlations with 'less symmetrical' groups are indicated in the corresponding row ('X' and '+'). A table of this kind gives an indication which reduction of symmetry may be expected from rotation, vibration, or substitution of a molecular geometry.

For example, as indicated by dark gray shading ('X'), the Th point group is a sub-group of Oh (see first row of matrix). Light grey shadings ('+') indicate, that the correlated group is correlated to another sub-group of the initial high-symmetry group (e.g. the Th and D2h groups are both sub-groups of Oh, whereas D2h is also a sub-group of Th).
Symmetry and Point Groups
Symmetry Elements of a 3D-Object
Top of Page Hierarchy of Point Groups - Symmetry Correlations
The only groups not having any correlation to Cs (= S1), Ci (= S2), or any of the Sn (S4 or S6) groups are Cn (n ≥ 1), Dn (n ≥ 2), T, O, and I (not a crystallographic point group). Therefore, these (and only these!) point groups must be chiral (bold face symbols in the first column, see also 'Chiral and Polar Point Groups').

 
Initial high-symmetry point group (first column)
 
Correlated point group (first row)
X
Correlation available between point groups
+
Correlation available between point groups (implicit correlation to sub-group of initial group)
.
No Correlation between point groups
C1
Ci
Cs
C2
C3
C4
C6
S4
S6
C2h
C3h
C4h
C6h
C2v
C3v
C4v
C6v
D2
D3
D4
D6
D2d
D3d
D2h
D3h
D4h
D6h
T
Th
Td
O
Oh
Oh
+
+
+
+
+
+
.
+
+
+
.
+
.
+
+
+
.
+
+
+
.
+
X
+
.
X
.
+
X
X
X
O
+
.
.
+
+
+
.
.
.
.
.
.
.
.
.
.
.
+
X
X
.
.
.
.
.
.
.
X
.
.
.
Td
+
.
+
+
+
.
.
+
.
.
.
.
.
+
X
.
.
+
.
.
.
X
.
.
.
.
.
X
.
.
.
Th
+
+
+
+
+
.
.
.
X
+
.
.
.
+
.
.
.
+
.
.
.
.
.
X
.
.
.
X
.
.
.
T
+
.
.
+
X
.
.
.
.
.
.
.
.
.
.
.
.
X
.
.
.
.
.
.
.
.
.
.
.
.
.
D6h
+
+
+
+
+
.
+
.
+
+
+
.
X
+
+
.
X
+
+
.
X
.
X
X
X
.
.
.
.
.
.
D4h
+
+
+
+
.
+
.
+
.
+
.
X
.
+
.
X
.
+
.
X
.
X
.
X
.
.
.
.
.
.
.
D3h
+
.
+
+
+
.
.
.
.
.
X
.
.
X
X
.
.
.
X
.
.
.
.
.
.
.
.
.
.
.
.
D2h
+
+
+
+
.
.
.
.
.
X
.
.
.
X
.
.
.
X
.
.
.
.
.
.
.
.
.
.
.
.
.
D3d
+
+
+
+
+
.
.
.
X
X
.
.
.
.
X
.
.
.
X
.
.
.
.
.
.
.
.
.
.
.
.
D2d
+
.
+
+
.
.
.
X
.
.
.
.
.
X
.
.
.
X
.
.
.
.
.
.
.
.
.
.
.
.
.
D6
+
.
.
+
+
.
X
.
.
.
.
.
.
.
.
.
.
X
X
.
.
.
.
.
.
.
.
.
.
.
.
D4
+
.
.
+
.
X
.
.
.
.
.
.
.
.
.
.
.
X
.
.
.
.
.
.
.
.
.
.
.
.
.
D3
+
.
.
X
X
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
D2
+
.
.
X
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
C6v
+
.
+
+
+
.
X
.
.
.
.
.
.
X
X
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
C4v
+
.
+
+
.
X
.
.
.
.
.
.
.
X
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
C3v
+
.
X
.
X
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
C2v
+
.
X
X
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
C6h
+
+
+
+
+
.
X
.
X
X
X
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
C4h
+
+
+
+
.
X
.
X
.
X
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
C3h
+
.
X
.
X
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
C2h
+
X
X
X
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
S6
+
X
.
.
X
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
S4
+
.
.
X
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
C6
+
.
.
X
X
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
C4
+
.
.
X
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
C3
X
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
C2
X
.
.
.
.
.
.
.
.
.
.
.
.
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.
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.
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Cs
X
.
.
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Ci
X
.
.
.
.
.
.
.
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.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
C1
.
.
.
.
.
.
.
.
.
.
.
.
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Top of Page Hierarchy of Point Groups - Graphical Representation
The following schemes provide an graphical overview on the correlations amongst the various crystallographic point groups. Point groups are arranged with decreasing symmetry, and correlations are indicated by arrows. For the sub-groups of the high-symmetry cubic point groups the correlation is continued in the second scheme.

Point Group Correlation Scheme

Point Group C3 Point Group D2h Point Group D2 Point Group C3v Point Group S6 Point Group D3 Point Group D2d Point Group T Point Group D4 Point Group D3d Point Group Th Point Group Td Point Group O Point Group D4h Point Group Oh
Point Group Correlation Scheme

Point Group Sn Point Group Dnd Point Group Cn Point Group Cnh Point Group Cnv Point Group Dn Point Group Dnh Point Group Ci Point Group Cs Point Group C1

For more information on other research topics, please refer to the complete list of publications and to the gallery of graphics and animations.

© Copyright PD Dr. S. Immel