C₂×#11724469 #13181404

order	1728
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³×C₃³

Aliases

	subgroup	G/H
center	C₂²	C₂×C₃⋊(C₃⋊S₄)
commutator, cosocle, hypocenter	C₃²×SL(2,𝔽₃)	C₂³
Frattini	C₂	C₂²×C₃⋊(C₃⋊S₄)
Fitting	C₂×C₃²×#41	D₃
socle	C₂²×C₃²	C₂×S₄
prime radical	C₂×C₃²×#41⋊C₃	C₂
hypercenter	C₂×C₄	C₃⋊(C₃⋊S₄)
LCS	G ▷ C₃²×SL(2,𝔽₃)
UCS	C₁ ◁ C₂² ◁ C₂×C₄

1406 conjugacy classes of subgroups; only the 199 normal subgroups are shown.

normal subgroups only

This group has 1406 conjugacy classes of subgroups — too many to draw. Only its 199 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

μ_t(G) = 576

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(1728, 47467);