Element structure of M16

To describe subgroups, we use the defining presentation given here:

$$M_{16} = \langle a,x \mid a^8 = x^2 = e, xax = a^5 \rangle$$

with $$e$$ denoting the identity element.

This is a group of order 16, with elements:

$$\! \{ e, a, a^2, a^3, a^4, a^5, a^6, a^7, x, ax, a^2x, a^3x, a^4x, a^5x, a^6x, a^7x \}$$

Conjugacy class structure


Automorphism class structure


Endomorphism structure
The bubbles here correspond to equivalence classes under automorphisms. The arrows here indicate that there is an endomorphism taking an element in one bubble to an element in the other bubble. To avoid confusion, we do not draw all arrows, we draw enough that taking the transitive closure generates all possible arrows.



Directed power graph
Below is a collapsed version of the directed power graph of the group. Here, each node represents an equivalence class of elements under the relation of generating a given cyclic subgroup. An edge from one node to occurs if some element in the latter node is the square of some element in the former node. We remove the loop at the identity element.



1-isomorphism
The group is 1-isomorphic to direct product of Z8 and Z2. In particular, the directed power graphs of the two groups are isomorphic as graphs. This 1-isomorphism is a special case of a general version of the Baer correspondence, which in turn is a special case of a Lazard correspondence. See proof of generalized Baer construction of Lie ring for class two 2-group with a suitable cocycle.

For more on the specific 1-isomorphism, refer Baer correspondence between M16 and direct product of Z8 and Z2.