Subgroups — D₃³ #6325

Order 216.

162 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1113	C₂	1	1	yes	maximal, characteristic, complemented
2	D₃×C₃⋊D₃	C₂	1	3	yes	maximal, complemented
3	C₃×D₃²	C₂	1	3	yes	maximal, complemented
4	C₂×D₃²	—	3	3	no	maximal
5	C₃⋊(C₃⋊D₃)	C₂²	1	1	yes	characteristic, complemented
6	C₃×C₃⋊D₃	C₂²	1	3	yes	complemented
7	C₃²×D₃	C₂²	1	3	yes	complemented
8	C₂×C₃⋊D₃	—	3	3	no	—
9	C₂×C₃×D₃	—	3	6	no	—
10	D₃²	—	3	3	no	—
11	D₃²	—	3	6	no	—
12	D₃²	—	6	3	no	—
13	D₃²	D₃	1	3	yes	direct product factor
14	C₃³	C₂³	1	1	yes	characteristic, complemented, Sylow
15	C₂²×D₃	—	9	3	no	—
16	C₂×C₃²	—	3	3	no	—
17	C₃⋊D₃	—	3	3	no	—
18	C₃⋊D₃	—	6	3	no	—
19	C₃⋊D₃	—	12	1	no	—
20	C₃⋊D₃	C₂×D₃	1	3	yes	complemented
21	C₃×D₃	—	2	3	no	—
22	C₃×D₃	—	3	6	no	—
23	C₃×D₃	—	6	3	no	—
24	C₃×D₃	C₂×D₃	1	6	yes	complemented
25	C₂²×C₃	—	9	3	no	—
26	C₂×D₃	—	3	6	no	—
27	C₂×D₃	—	9	3	no	—
28	C₂×D₃	—	9	6	no	—
29	C₂×D₃	—	9	3	no	—
30	C₂×D₃	—	18	3	no	—
31	C₃²	—	2	3	no	—
32	C₃²	—	4	1	no	—
33	C₃²	C₂²×D₃	1	3	yes	complemented
34	C₂³	—	27	1	no	Sylow
35	C₂×C₃	—	3	6	no	—
36	C₂×C₃	—	6	3	no	—
37	C₂×C₃	—	9	3	no	—
38	D₃	—	3	6	no	—
39	D₃	—	6	3	no	—
40	D₃	—	9	3	no	—
41	D₃	—	18	3	no	—
42	D₃	—	36	1	no	—
43	D₃	D₃²	1	3	yes	direct product factor
44	C₂²	—	9	3	no	—
45	C₂²	—	27	3	no	—
46	C₂²	—	27	1	no	—
47	C₃	—	2	3	no	—
48	C₃	—	4	1	no	—
49	C₃	C₂×D₃²	1	3	yes	complemented
50	C₂	—	3	3	no	—
51	C₂	—	9	3	no	—
52	C₂	—	27	1	no	—

	subgroup	G/H
center, Frattini, hypercenter	C₁	G
commutator, Fitting, socle, cosocle, hypocenter	C₃³	C₂³
LCS	G ▷ 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.