Subgroups — C₃⋊PSU(3,𝔽₂) #6324

Order 216.

30 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⋊#157	C₂	1	2	yes	maximal
2	C₃×#157	C₂	1	1	yes	maximal, characteristic
3	PSU(3,𝔽₂)	—	3	1	no	maximal
4	C₃×C₃⋊D₃	C₂²	1	1	yes	characteristic
5	#157	—	3	2	no	—
6	#157	D₃	1	1	yes	characteristic
7	C₃³	Q₈	1	1	yes	characteristic, complemented, Sylow
8	Dic₆	—	9	1	no	maximal
9	C₃⋊D₃	C₂×D₃	1	1	yes	characteristic
10	C₃×D₃	—	12	1	no	—
11	C₃×C₄	—	9	1	no	—
12	Dic₃	—	9	2	no	—
13	C₃²	—	4	1	no	—
14	C₃²	—	8	1	no	—
15	C₃²	Dic₆	1	1	yes	characteristic, complemented
16	Q₈	—	27	1	no	Sylow
17	C₂×C₃	—	9	1	no	—
18	D₃	—	12	1	no	—
19	C₄	—	9	1	no	—
20	C₄	—	27	2	no	—
21	C₃	—	4	1	no	—
22	C₃	—	8	1	no	—
23	C₃	PSU(3,𝔽₂)	1	1	yes	characteristic, complemented
24	C₂	—	9	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, prime radical, cosocle	C₃×C₃⋊D₃	C₂²
Fitting, socle, hypocenter	C₃³	Q₈
LCS	G ▷ C₃×C₃⋊D₃ ▷ C₃³
UCS	C₁

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

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