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In Dummit and Foote´s Abstract Algebra, when talking about the lattice of subgroups of $A_4$, the authors make the statement that, unlike virtaully all groups, $A_4$ has a planar lattice? My question is

What do they mean when they say virtually all groups? Is there a reference for this statement?

On a somewhat different note, are there classes of graphs which can be realized as the subgroup lattice of some group?

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When I put subgroup lattice planar in Google, this paper is among the first results. –  Martin Sleziak Jun 18 '12 at 12:56

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up vote 4 down vote accepted

Starr–Turner (2004) classifies abelian groups with planar subgroup lattices, and Schmidt (2006) and Bohanon–Reid (2006) complete the classification. Dummit–Foote made this claim prior to this work, so it may have just been the observation that fairly small groups have non-planar lattices, so that it would intuitively be pretty hard for a large group to have a planar lattice.

James Wilson's catalog of groups by lattice (from 2004-2005) is mostly in tact at his new webpage.

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