Subgroups — #3757⋊C₃ #88519

Order 432.

151 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×(C₃⋊Dic₃)⋊C₃	C₂	1	2	yes	maximal, complemented
3	#3757	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×#217	—	3	1	no	maximal
5	C₂²×H(3,𝔽₃)	—	2	2	no	—
6	C₂²×H(3,𝔽₃)	C₄	1	2	yes	complemented
7	C₂²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
8	(C₃⋊Dic₃)⋊C₃	—	2	2	no	—
9	C₂³×C₃²	—	2	1	no	—
10	C₂³×C₃²	D₃	1	1	yes	characteristic
11	C₂³×C₃²	C₂×C₃	1	1	yes	characteristic
12	C₂×C₃⋊Dic₃	C₂×C₃	1	2	yes	complemented
13	C₂×C₃×Dic₃	—	3	2	no	—
14	C₂×H(3,𝔽₃)	—	2	2	no	—
15	C₂×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
16	C₂×H(3,𝔽₃)	D₄	1	2	yes	—
17	C₃×C₂²⋊C₄	—	9	1	no	—
18	#217	—	3	1	no	—
19	#217	—	9	1	no	—
20	C₂²×C₃²	—	2	4	no	—
21	C₂²×C₃²	—	2	2	no	—
22	C₂²×C₃²	—	2	2	no	—
23	C₂²×C₃²	—	2	2	no	—
24	C₂²×C₃²	—	2	1	no	—
25	C₂²×C₃²	Dic₃	1	2	yes	complemented
26	C₂²×C₃²	C₃×C₄	1	2	yes	complemented
27	C₂²×C₃²	C₂×D₃	1	1	yes	characteristic
28	C₂²×C₃²	C₂²×C₃	1	1	yes	characteristic
29	C₃⋊Dic₃	—	2	2	no	—
30	C₃×Dic₃	—	6	2	no	—
31	H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
32	C₂³×C₃	—	3	1	no	—
33	C₂³×C₃	—	3	1	no	—
34	C₂³×C₃	—	6	1	no	—
35	C₂³×C₃	C₃×D₃	1	1	yes	characteristic
36	C₂×C₃×C₄	—	9	2	no	—
37	C₂×Dic₃	—	3	2	no	—
38	C₂×Dic₃	—	9	2	no	—
39	C₂×C₃²	—	2	4	no	—
40	C₂×C₃²	—	2	2	no	—
41	C₂×C₃²	—	2	2	no	—
42	C₂×C₃²	—	2	1	no	—
43	C₂×C₃²	—	2	2	no	—
44	C₂×C₃²	C₂×Dic₃	1	1	yes	characteristic
45	C₂×C₃²	#67	1	2	yes	—
46	C₂×C₃²	C₂×C₃×C₄	1	1	yes	characteristic
47	C₂×C₃²	C₃×D₄	1	2	yes	—
48	C₂²⋊C₄	—	9	1	no	Sylow
49	C₂²×C₃	—	2	2	no	—
50	C₂²×C₃	—	3	2	no	—
51	C₂²×C₃	—	3	2	no	—
52	C₂²×C₃	—	3	1	no	—
53	C₂²×C₃	—	3	1	no	—
54	C₂²×C₃	—	6	2	no	—
55	C₂²×C₃	—	6	2	no	—
56	C₂²×C₃	—	6	4	no	—
57	C₂²×C₃	—	6	2	no	—
58	C₂²×C₃	—	6	1	no	—
59	C₂²×C₃	C₃×Dic₃	1	2	yes	—
60	C₂²×C₃	C₂×C₃×D₃	1	1	yes	characteristic
61	C₃×C₄	—	18	2	no	—
62	Dic₃	—	6	2	no	—
63	Dic₃	—	18	2	no	—
64	C₃²	—	2	1	no	—
65	C₃²	#217	1	1	yes	characteristic, complemented
66	C₃²	C₃×C₂²⋊C₄	1	1	yes	characteristic, complemented
67	C₂³	(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
68	C₂×C₄	—	9	2	no	—
69	C₂×C₃	—	2	2	no	—
70	C₂×C₃	—	3	1	no	—
71	C₂×C₃	—	3	1	no	—
72	C₂×C₃	—	3	2	no	—
73	C₂×C₃	—	3	2	no	—
74	C₂×C₃	—	6	2	no	—
75	C₂×C₃	—	6	2	no	—
76	C₂×C₃	—	6	4	no	—
77	C₂×C₃	—	6	1	no	—
78	C₂×C₃	—	6	2	no	—
79	C₂×C₃	C₂×C₃×Dic₃	1	1	yes	characteristic
80	C₂×C₃	C₃×#67	1	2	yes	—
81	C₂²	—	2	2	no	—
82	C₂²	(C₃⋊Dic₃)⋊C₃	1	2	yes	complemented
83	C₂²	C₂×(C₃⋊D₃)⋊C₃	1	1	yes	characteristic
84	C₄	—	18	2	no	—
85	C₃	—	3	1	no	—
86	C₃	—	3	1	no	—
87	C₃	—	6	1	no	—
88	C₃	C₃×#217	1	1	yes	characteristic
89	C₂	—	2	2	no	—
90	C₂	C₂×(C₃⋊Dic₃)⋊C₃	1	1	yes	characteristic
91	C₂	#638⋊C₃	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×(C₃⋊D₃)⋊C₃
commutator	C₂×C₃²	C₂×C₃×C₄
Frattini, socle	C₂²×C₃	C₂×C₃×D₃
Fitting, prime radical	C₂³×H(3,𝔽₃)	C₂
cosocle	C₂²×C₃²	C₂²×C₃
hypercenter	C₂³	(C₃⋊D₃)⋊C₃
hypocenter	C₃²	C₃×C₂²⋊C₄
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂² ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂×(C₃⋊Dic₃)⋊C₃	C₂	semidirect
3. #3757	C₃	semidirect
6. C₂²×H(3,𝔽₃)	C₄	semidirect
12. C₂×C₃⋊Dic₃	C₂×C₃	semidirect
25. C₂²×C₃²	Dic₃	semidirect
26. C₂²×C₃²	C₃×C₄	semidirect
31. H(3,𝔽₃)	C₂²⋊C₄	semidirect
65. C₃²	#217	semidirect
66. C₃²	C₃×C₂²⋊C₄	semidirect
82. C₂²	(C₃⋊Dic₃)⋊C₃	semidirect

Maximal subgroups

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

subgroup	index
C₂³×H(3,𝔽₃)	2
C₂×(C₃⋊Dic₃)⋊C₃	2
#3757	3
C₃×#217	3

Subgroups — #3757⋊C3 #88519

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #3757⋊C₃ #88519