Subgroups — C₃×#3757 #88848

Order 432.

332 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₃³	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₃⋊Dic₃	C₂	1	2	yes	maximal, complemented
3	#3757	C₃	1	1	yes	maximal, characteristic, direct product factor
4	C₃×#217	—	3	4	no	maximal
5	C₂²×C₃³	—	2	2	no	—
6	C₂²×C₃³	C₄	1	2	yes	complemented
7	C₂²×C₃³	C₂²	1	1	yes	characteristic
8	C₃×C₃⋊Dic₃	—	2	2	no	—
9	C₂³×C₃²	—	2	4	no	—
10	C₂³×C₃²	D₃	1	4	yes	—
11	C₂³×C₃²	C₂×C₃	1	1	yes	characteristic
12	C₂×C₃⋊Dic₃	C₂×C₃	1	2	yes	complemented
13	C₂×C₃×Dic₃	—	3	8	no	—
14	C₂×C₃³	—	2	2	no	—
15	C₂×C₃³	C₂×C₄	1	1	yes	characteristic
16	C₂×C₃³	D₄	1	2	yes	—
17	C₃×C₂²⋊C₄	—	9	1	no	—
18	#217	—	3	4	no	—
19	C₂²×C₃²	—	2	16	no	—
20	C₂²×C₃²	—	2	8	no	—
21	C₂²×C₃²	—	2	2	no	—
22	C₂²×C₃²	—	2	8	no	—
23	C₂²×C₃²	—	2	4	no	—
24	C₂²×C₃²	Dic₃	1	8	yes	complemented
25	C₂²×C₃²	C₃×C₄	1	2	yes	complemented
26	C₂²×C₃²	C₂×D₃	1	4	yes	—
27	C₂²×C₃²	C₂²×C₃	1	1	yes	characteristic
28	C₃⋊Dic₃	—	2	2	no	—
29	C₃×Dic₃	—	6	8	no	—
30	C₃³	C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
31	C₂³×C₃	—	2	4	no	—
32	C₂³×C₃	C₃×D₃	1	4	yes	—
33	C₂³×C₃	C₃⋊D₃	1	1	yes	characteristic
34	C₂×C₃×C₄	—	9	2	no	—
35	C₂×Dic₃	—	3	8	no	—
36	C₂×C₃²	—	2	16	no	—
37	C₂×C₃²	—	2	8	no	—
38	C₂×C₃²	—	2	2	no	—
39	C₂×C₃²	—	2	4	no	—
40	C₂×C₃²	—	2	8	no	—
41	C₂×C₃²	C₂×Dic₃	1	4	yes	—
42	C₂×C₃²	#67	1	8	yes	—
43	C₂×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
44	C₂×C₃²	C₃×D₄	1	2	yes	—
45	C₂²⋊C₄	—	9	1	no	Sylow
46	C₂²×C₃	—	2	16	no	—
47	C₂²×C₃	—	2	8	no	—
48	C₂²×C₃	—	2	2	no	—
49	C₂²×C₃	—	2	8	no	—
50	C₂²×C₃	—	2	4	no	—
51	C₂²×C₃	C₃×Dic₃	1	8	yes	complemented
52	C₂²×C₃	C₃⋊Dic₃	1	2	yes	complemented
53	C₂²×C₃	C₂×C₃×D₃	1	4	yes	—
54	C₂²×C₃	C₂×C₃⋊D₃	1	1	yes	characteristic
55	C₃×C₄	—	18	2	no	—
56	Dic₃	—	6	8	no	—
57	C₃²	—	2	4	no	—
58	C₃²	#217	1	4	yes	complemented
59	C₃²	C₃×C₂²⋊C₄	1	1	yes	characteristic, complemented
60	C₂³	C₃×C₃⋊D₃	1	1	yes	characteristic
61	C₂×C₄	—	9	2	no	—
62	C₂×C₃	—	2	16	no	—
63	C₂×C₃	—	2	8	no	—
64	C₂×C₃	—	2	2	no	—
65	C₂×C₃	—	2	4	no	—
66	C₂×C₃	—	2	8	no	—
67	C₂×C₃	C₂×C₃×Dic₃	1	4	yes	—
68	C₂×C₃	C₃×#67	1	8	yes	—
69	C₂×C₃	C₂×C₃⋊Dic₃	1	1	yes	characteristic
70	C₂×C₃	#638	1	2	yes	—
71	C₂²	—	2	2	no	—
72	C₂²	C₃×C₃⋊Dic₃	1	2	yes	complemented
73	C₂²	C₂×C₃×C₃⋊D₃	1	1	yes	characteristic
74	C₄	—	18	2	no	—
75	C₃	—	2	4	no	—
76	C₃	C₃×#217	1	4	yes	complemented
77	C₃	#3757	1	1	yes	characteristic, direct product factor
78	C₂	—	2	2	no	—
79	C₂	C₂×C₃×C₃⋊Dic₃	1	1	yes	characteristic
80	C₂	C₃×#638	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂×C₃⋊D₃
commutator	C₂×C₃²	C₂×C₃×C₄
Frattini	C₂²	C₂×C₃×C₃⋊D₃
Fitting, prime radical	C₂³×C₃³	C₂
socle	C₂²×C₃³	C₂²
cosocle	C₂²×C₃²	C₂²×C₃
hypercenter	C₂³×C₃	C₃⋊D₃
hypocenter	C₃²	C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂²×C₃ ◁ C₂³×C₃

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₃⋊Dic₃	C₂	semidirect
3. #3757	C₃	direct (N × H)
6. C₂²×C₃³	C₄	semidirect
12. C₂×C₃⋊Dic₃	C₂×C₃	semidirect
24. C₂²×C₃²	Dic₃	semidirect
25. C₂²×C₃²	C₃×C₄	semidirect
30. C₃³	C₂²⋊C₄	semidirect
51. C₂²×C₃	C₃×Dic₃	semidirect
52. C₂²×C₃	C₃⋊Dic₃	semidirect
58. C₃²	#217	semidirect
59. C₃²	C₃×C₂²⋊C₄	semidirect
72. C₂²	C₃×C₃⋊Dic₃	semidirect
76. C₃	C₃×#217	semidirect
77. C₃	#3757	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₃³	2
C₂×C₃×C₃⋊Dic₃	2
#3757	3
C₃×#217	3

Subgroups — C3×#3757 #88848

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#3757 #88848