Subgroups — PSU(3,𝔽₂) #644

Order 72.

14 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#157	C₂	1	3	yes	maximal
2	C₃⋊D₃	C₂²	1	1	yes	characteristic
3	C₃²	Q₈	1	1	yes	characteristic, complemented, Sylow
4	Q₈	—	9	1	no	maximal, Sylow
5	D₃	—	12	1	no	—
6	C₄	—	9	3	no	—
7	C₃	—	4	1	no	—
8	C₂	—	9	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, prime radical, cosocle	C₃⋊D₃	C₂²
Fitting, socle, hypocenter	C₃²	Q₈
LCS	G ▷ C₃⋊D₃ ▷ C₃²
UCS	C₁

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. C₃²	Q₈	semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#157	2
Q₈	9