Subgroups — C₂²×#272 #6276

Order 216.

176 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#272	C₂	1	6	yes	maximal, direct product factor
3	C₂²×C₃×D₃	—	3	4	no	maximal
4	C₂×H(3,𝔽₃)	C₂²	1	3	yes	complemented
5	#272	C₂²	1	4	yes	direct product factor
6	C₂²×C₃²	D₃	1	4	yes	complemented
7	C₂×C₃×D₃	—	3	24	no	—
8	H(3,𝔽₃)	C₂³	1	1	yes	characteristic, complemented, Sylow
9	C₂³×C₃	—	9	1	no	—
10	C₂²×D₃	—	3	4	no	—
11	C₂×C₃²	C₂×D₃	1	12	yes	complemented
12	C₃×D₃	—	3	16	no	—
13	C₂²×C₃	—	3	4	no	—
14	C₂²×C₃	—	9	6	no	—
15	C₂²×C₃	C₃⋊D₃	1	1	yes	characteristic
16	C₂×D₃	—	3	24	no	—
17	C₃²	C₂²×D₃	1	4	yes	complemented
18	C₂³	—	9	1	no	Sylow
19	C₂×C₃	—	3	12	no	—
20	C₂×C₃	—	9	4	no	—
21	C₂×C₃	C₂×C₃⋊D₃	1	3	yes	—
22	D₃	—	3	16	no	—
23	C₂²	—	9	6	no	—
24	C₂²	#272	1	1	yes	characteristic, direct product factor
25	C₃	—	3	4	no	—
26	C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
27	C₂	—	9	4	no	—
28	C₂	C₂×#272	1	3	yes	direct product factor

	subgroup	G/H
center, socle, hypercenter	C₂²×C₃	C₃⋊D₃
commutator, cosocle, hypocenter	H(3,𝔽₃)	C₂³
Frattini	C₃	C₂²×C₃⋊D₃
Fitting	C₂²×H(3,𝔽₃)	C₂
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃

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.