Subgroup Lattices of Groups of Small Order

group

D

4

highlight:

none

cyclic subgroups

normal subgroups

center

The subgroup lattice of a group is the Hasse diagram of the subgroups under the partial ordering of set inclusion. This Demonstration displays the subgroup lattice for each of the groups (up to isomorphism) of orders 2 through 12. You can highlight the cyclic subgroups, the normal subgroups, or the center of the group. Moving the cursor over a subgroup displays a description of the subgroup.

Details

A subgroup

H

of a group

G

is normal in

G

if

gH=Hg

for all elements

g

in

G

. The center

Z(G)

of a group

G

is the normal subgroup consisting of those elements in

G

that commute with all elements in

G

. A group

G

is Abelian iff

Z(G)=G

.

External Links

Hasse Diagram of Power Sets

Group (Wolfram MathWorld)

Abelian Group (Wolfram MathWorld)

Symmetric Group (Wolfram MathWorld)

Alternating Group (Wolfram MathWorld)

Dihedral Group (Wolfram MathWorld)

The Fundamental Theorem of Finite Abelian Groups

Number of Finite Groups

Guessing the Symmetry Group

Permutation Lattice

Permanent Citation

Marc Brodie

"Subgroup Lattices of Groups of Small Order"
http://demonstrations.wolfram.com/SubgroupLatticesOfGroupsOfSmallOrder/
Wolfram Demonstrations Project
Published: August 10, 2012