Subgroups — C₂×((C₃⋊D₃)⋊C₃)⋊C₂ #6265

Order 216.

122 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#272	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×(C₃⋊D₃)⋊C₃	C₂	1	2	yes	maximal, complemented
3	((C₃⋊D₃)⋊C₃)⋊C₂	C₂	1	4	yes	maximal, direct product factor
4	C₂×D₃²	—	3	2	no	maximal
5	C₂×H(3,𝔽₃)	C₂²	1	1	yes	characteristic, complemented
6	#272	C₂²	1	2	yes	complemented
7	(C₃⋊D₃)⋊C₃	C₂²	1	4	yes	complemented
8	C₂×C₃⋊D₃	D₃	1	2	yes	complemented
9	C₂×C₃×D₃	—	3	2	no	—
10	C₂×C₃×D₃	—	3	2	no	—
11	C₂×C₃×D₃	—	6	1	no	—
12	D₃²	—	3	8	no	—
13	H(3,𝔽₃)	C₂³	1	1	yes	characteristic, complemented, Sylow
14	C₂²×D₃	—	9	1	no	—
15	C₂²×D₃	—	9	2	no	—
16	C₂×C₃²	—	2	1	no	—
17	C₂×C₃²	C₂×D₃	1	2	yes	complemented
18	C₃⋊D₃	C₂×D₃	1	4	yes	complemented
19	C₃×D₃	—	3	4	no	—
20	C₃×D₃	—	3	4	no	—
21	C₃×D₃	—	6	2	no	—
22	C₂²×C₃	—	9	2	no	—
23	C₂²×C₃	—	9	1	no	—
24	C₂×D₃	—	3	2	no	—
25	C₂×D₃	—	3	2	no	—
26	C₂×D₃	—	6	1	no	—
27	C₂×D₃	—	9	2	no	—
28	C₂×D₃	—	9	4	no	—
29	C₂×D₃	—	9	8	no	—
30	C₃²	—	2	1	no	—
31	C₃²	C₂²×D₃	1	2	yes	complemented
32	C₂³	—	27	1	no	Sylow
33	C₂×C₃	—	3	2	no	—
34	C₂×C₃	—	6	1	no	—
35	C₂×C₃	—	9	4	no	—
36	C₂×C₃	—	9	2	no	—
37	C₂×C₃	D₃²	1	1	yes	characteristic
38	D₃	—	3	4	no	—
39	D₃	—	3	4	no	—
40	D₃	—	6	2	no	—
41	D₃	—	9	4	no	—
42	C₂²	—	9	1	no	—
43	C₂²	—	9	2	no	—
44	C₂²	—	27	4	no	—
45	C₃	—	3	2	no	—
46	C₃	—	6	1	no	—
47	C₃	C₂×D₃²	1	1	yes	characteristic
48	C₂	—	9	2	no	—
49	C₂	—	9	4	no	—
50	C₂	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic, direct product factor

	subgroup	G/H
center, hypercenter	C₂	((C₃⋊D₃)⋊C₃)⋊C₂
commutator, cosocle, hypocenter	H(3,𝔽₃)	C₂³
Frattini	C₃	C₂×D₃²
Fitting	C₂×H(3,𝔽₃)	C₂²
socle	C₂×C₃	D₃²
LCS	G ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂

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.