Subgroups — #88812

Order 432.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#634	C₂	1	2	yes	maximal
2	C₃×C₃⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	#3719	—	3	2	no	maximal
4	#3718	—	3	1	no	maximal
5	C₃³×C₄	C₂²	1	1	yes	characteristic
6	C₃×C₃⋊Dic₃	—	2	2	no	—
7	#634	D₃	1	2	yes	—
8	C₃×Dic₆	—	3	2	no	—
9	C₃×Dic₆	—	3	2	no	—
10	C₃×Dic₆	—	6	2	no	—
11	C₃⋊(C₃⋊C₈)	D₃	1	1	yes	characteristic
12	C₃×C₃⋊C₈	—	3	2	no	—
13	C₃×C₃⋊C₈	—	6	1	no	—
14	C₂×C₃³	D₄	1	1	yes	characteristic
15	#216	—	9	2	no	—
16	Dic₁₂	—	9	1	no	—
17	C₃²×C₄	—	2	1	no	—
18	C₃²×C₄	—	2	2	no	—
19	C₃²×C₄	—	4	1	no	—
20	C₃²×C₄	C₂×D₃	1	1	yes	characteristic
21	C₃²×C₄	C₂×D₃	1	2	yes	—
22	C₃⋊Dic₃	—	2	2	no	—
23	C₃×Dic₃	—	6	4	no	—
24	C₃×Dic₃	—	6	2	no	—
25	C₃×Dic₃	—	6	2	no	—
26	C₃³	Q₁₆	1	1	yes	characteristic, complemented, Sylow
27	C₃×Q₈	—	9	2	no	—
28	Dic₆	—	3	2	no	—
29	Dic₆	—	3	2	no	—
30	Dic₆	—	6	2	no	—
31	C₃×C₈	—	9	1	no	—
32	C₃⋊C₈	—	3	2	no	—
33	C₃⋊C₈	—	6	1	no	—
34	C₂×C₃²	—	2	1	no	—
35	C₂×C₃²	—	2	2	no	—
36	C₂×C₃²	—	4	1	no	—
37	C₂×C₃²	D₁₂	1	1	yes	characteristic
38	C₂×C₃²	#67	1	2	yes	—
39	Q₁₆	—	27	1	no	Sylow
40	C₃×C₄	—	2	1	no	—
41	C₃×C₄	—	2	2	no	—
42	C₃×C₄	—	4	1	no	—
43	C₃×C₄	—	18	2	no	—
44	C₃×C₄	D₃²	1	1	yes	characteristic
45	C₃×C₄	D₃²	1	2	yes	—
46	Dic₃	—	6	4	no	—
47	Dic₃	—	6	2	no	—
48	Dic₃	—	6	2	no	—
49	C₃²	—	2	1	no	—
50	C₃²	—	2	2	no	—
51	C₃²	—	4	1	no	—
52	C₃²	Dic₁₂	1	1	yes	characteristic, complemented
53	C₃²	#216	1	2	yes	complemented
54	Q₈	—	9	2	no	—
55	C₈	—	9	1	no	—
56	C₂×C₃	—	2	2	no	—
57	C₂×C₃	—	2	1	no	—
58	C₂×C₃	—	4	1	no	—
59	C₂×C₃	#625	1	1	yes	characteristic
60	C₂×C₃	#626	1	2	yes	—
61	C₄	—	18	2	no	—
62	C₄	#1113	1	1	yes	characteristic
63	C₃	—	2	1	no	—
64	C₃	—	2	2	no	—
65	C₃	—	4	1	no	—
66	C₃	#3718	1	1	yes	characteristic, complemented
67	C₃	#3719	1	2	yes	complemented
68	C₂	#6295	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#6295
commutator, Fitting, cosocle	C₃³×C₄	C₂²
Frattini, hypercenter	C₄	#1113
socle, prime radical	C₂×C₃³	D₄
hypocenter	C₃³	Q₁₆
LCS	G ▷ C₃³×C₄ ▷ C₂×C₃³ ▷ C₃³
UCS	C₁ ◁ C₂ ◁ C₄

Complements

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

N	H ≅ G/N	type
26. C₃³	Q₁₆	semidirect
52. C₃²	Dic₁₂	semidirect
53. C₃²	#216	semidirect
66. C₃	#3718	semidirect
67. C₃	#3719	semidirect

Maximal subgroups

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

subgroup	index
C₃×#634	2
C₃×C₃⋊(C₃⋊C₈)	2
#3719	3
#3718	3