Group #14517302 — subgroups

Subgroups — 3₋¹⁺²×Dic₁₈ #14517302

Order 1944.

219 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 219 conjugacy classes of subgroups — too many to draw. Only its 84 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₉×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	3₋¹⁺²×Dic₉	C₂	1	2	yes	maximal
3	Dic₆×3₋¹⁺²	—	3	1	no	maximal
4	C₃²×Dic₁₈	C₃	1	1	yes	maximal, characteristic
5	C₉×Dic₁₈	C₃	1	3	yes	maximal, complemented
6	C₂×C₉×3₋¹⁺²	C₂²	1	1	yes	characteristic
7	C₃×C₄×3₋¹⁺²	D₃	1	1	yes	characteristic
8	C₃²×C₄×C₉	C₂×C₃	1	1	yes	characteristic
9	Dic₃×3₋¹⁺²	—	3	2	no	—
10	C₃²×Dic₉	C₂×C₃	1	2	yes	—
11	C₄×C₉⋊C₉	—	2	3	no	—
12	C₄×C₃²⋊C₉	—	2	1	no	—
13	C₄×C₉²	C₂×C₃	1	3	yes	—
14	C₉×Dic₉	C₂×C₃	1	6	yes	—
15	C₉×3₋¹⁺²	Q₈	1	1	yes	characteristic, complemented, Sylow
16	C₃²×Dic₆	—	3	1	no	—
17	Q₈×3₋¹⁺²	—	9	1	no	—
18	C₉×Dic₆	—	3	3	no	—
19	C₃×Dic₁₈	—	3	1	no	—
20	C₃×Dic₁₈	C₃²	1	1	yes	characteristic
21	C₂×C₃×3₋¹⁺²	C₂×D₃	1	1	yes	characteristic
22	C₂×C₃²×C₉	C₂²×C₃	1	1	yes	characteristic
23	C₂×C₉⋊C₉	—	2	3	no	—
24	C₂×C₃²⋊C₉	—	2	1	no	—
25	C₂×C₉²	C₂²×C₃	1	3	yes	—
26	C₃³×C₄	C₃×D₃	1	1	yes	characteristic
27	C₃²×Dic₃	—	3	2	no	—
28	C₄×3₋¹⁺²	—	2	1	no	—
29	C₄×3₋¹⁺²	—	2	3	no	—
30	C₄×3₋¹⁺²	—	9	2	no	—
31	C₄×3₋¹⁺²	D₉	1	1	yes	characteristic
32	C₃×C₄×C₉	—	2	3	no	—
33	C₃×C₄×C₉	—	2	1	no	—
34	C₃×C₄×C₉	—	3	1	no	—
35	C₃×C₄×C₉	—	6	1	no	—
36	C₃×C₄×C₉	C₃×D₃	1	3	yes	—
37	C₃×C₄×C₉	C₂×C₃²	1	1	yes	characteristic
38	C₉×Dic₃	—	3	6	no	—
39	C₃×Dic₉	—	3	2	no	—
40	C₃×Dic₉	C₂×C₃²	1	2	yes	—
41	C₃×3₋¹⁺²	Dic₆	1	1	yes	characteristic
42	C₃²×C₉	C₃×Q₈	1	1	yes	characteristic
43	C₉⋊C₉	—	2	3	no	—
44	C₃²⋊C₉	—	2	1	no	—
45	C₉²	C₃×Q₈	1	3	yes	complemented
46	C₃²×Q₈	—	9	1	no	—
47	C₃×Dic₆	—	3	1	no	—
48	C₃×Dic₆	—	9	1	no	—
49	Q₈×C₉	—	9	3	no	—
50	Dic₁₈	3₋¹⁺²	1	1	yes	characteristic, direct product factor
51	C₂×C₃³	C₂×C₃×D₃	1	1	yes	characteristic
52	C₂×3₋¹⁺²	—	2	1	no	—
53	C₂×3₋¹⁺²	—	2	3	no	—
54	C₂×3₋¹⁺²	C₂×D₉	1	1	yes	characteristic
55	C₂×C₃×C₉	—	2	3	no	—
56	C₂×C₃×C₉	—	2	1	no	—
57	C₂×C₃×C₉	—	3	1	no	—
58	C₂×C₃×C₉	—	6	1	no	—
59	C₂×C₃×C₉	C₂×C₃×D₃	1	3	yes	—
60	C₂×C₃×C₉	C₂²×C₃²	1	1	yes	characteristic
61	C₃²×C₄	—	2	1	no	—
62	C₃²×C₄	—	3	1	no	—
63	C₃²×C₄	—	6	1	no	—
64	C₃²×C₄	—	9	2	no	—
65	C₃²×C₄	C₃×D₉	1	1	yes	characteristic
66	C₃²×C₄	C₃²×D₃	1	1	yes	characteristic
67	C₃×Dic₃	—	3	2	no	—
68	C₃×Dic₃	—	9	2	no	—
69	C₄×C₉	—	2	3	no	—
70	C₄×C₉	—	2	1	no	—
71	C₄×C₉	—	6	3	no	—
72	C₄×C₉	—	6	1	no	—
73	C₄×C₉	—	9	6	no	—
74	C₄×C₉	C₃×D₉	1	3	yes	—
75	C₄×C₉	C₂×3₋¹⁺²	1	1	yes	characteristic
76	Dic₉	C₂×3₋¹⁺²	1	2	yes	—
77	C₃³	C₃×Dic₆	1	1	yes	characteristic
78	3₋¹⁺²	—	2	1	no	—
79	3₋¹⁺²	—	2	3	no	—
80	3₋¹⁺²	Dic₁₈	1	1	yes	characteristic, direct product factor
81	C₃×C₉	—	2	3	no	—
82	C₃×C₉	—	2	1	no	—
83	C₃×C₉	—	3	1	no	—
84	C₃×C₉	—	6	1	no	—
85	C₃×C₉	C₃×Dic₆	1	3	yes	—
86	C₃×C₉	C₃²×Q₈	1	1	yes	characteristic
87	C₃×Q₈	—	9	1	no	—
88	C₃×Q₈	—	27	1	no	—
89	Dic₆	—	3	1	no	—
90	C₂×C₃²	—	2	1	no	—
91	C₂×C₃²	—	3	1	no	—
92	C₂×C₃²	—	6	1	no	—
93	C₂×C₃²	C₂×C₃×D₉	1	1	yes	characteristic
94	C₂×C₃²	C₂×C₃²×D₃	1	1	yes	characteristic
95	C₂×C₉	—	2	3	no	—
96	C₂×C₉	—	2	1	no	—
97	C₂×C₉	—	6	3	no	—
98	C₂×C₉	—	6	1	no	—
99	C₂×C₉	C₂×C₃×D₉	1	3	yes	—
100	C₂×C₉	C₂²×3₋¹⁺²	1	1	yes	characteristic
101	C₃×C₄	—	2	1	no	—
102	C₃×C₄	—	3	1	no	—
103	C₃×C₄	—	6	1	no	—
104	C₃×C₄	—	9	2	no	—
105	C₃×C₄	—	27	2	no	—
106	C₃×C₄	C₃²×D₉	1	1	yes	characteristic
107	C₃×C₄	D₃×3₋¹⁺²	1	1	yes	characteristic
108	Dic₃	—	3	2	no	—
109	C₃²	—	2	1	no	—
110	C₃²	—	3	1	no	—
111	C₃²	—	6	1	no	—
112	C₃²	C₃×Dic₁₈	1	1	yes	characteristic
113	C₃²	C₃²×Dic₆	1	1	yes	characteristic
114	C₉	—	2	3	no	—
115	C₉	—	2	1	no	—
116	C₉	—	6	3	no	—
117	C₉	—	6	1	no	—
118	C₉	C₃×Dic₁₈	1	3	yes	complemented
119	C₉	Q₈×3₋¹⁺²	1	1	yes	characteristic, complemented
120	Q₈	—	9	1	no	Sylow
121	C₂×C₃	—	2	1	no	—
122	C₂×C₃	—	3	1	no	—
123	C₂×C₃	—	6	1	no	—
124	C₂×C₃	C₂×C₃²×D₉	1	1	yes	characteristic
125	C₂×C₃	C₂×D₃×3₋¹⁺²	1	1	yes	characteristic
126	C₄	—	9	2	no	—
127	C₄	D₉×3₋¹⁺²	1	1	yes	characteristic
128	C₃	—	2	1	no	—
129	C₃	—	3	1	no	—
130	C₃	—	6	1	no	—
131	C₃	C₃²×Dic₁₈	1	1	yes	characteristic
132	C₃	Dic₆×3₋¹⁺²	1	1	yes	characteristic
133	C₂	C₂×D₉×3₋¹⁺²	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₉×3₋¹⁺²

C₂

yes

maximal, characteristic

3₋¹⁺²×Dic₉

C₂

yes

maximal

Dic₆×3₋¹⁺²

—

maximal

C₃²×Dic₁₈

C₃

yes

maximal, characteristic

C₉×Dic₁₈

C₃

yes

maximal, complemented

C₂×C₉×3₋¹⁺²

C₂²

yes

characteristic

C₃×C₄×3₋¹⁺²

D₃

yes

characteristic

C₃²×C₄×C₉

C₂×C₃

yes

characteristic

Dic₃×3₋¹⁺²

—

C₃²×Dic₉

C₂×C₃

yes

—

C₄×C₉⋊C₉

—

C₄×C₃²⋊C₉

—

C₄×C₉²

C₂×C₃

yes

—

C₉×Dic₉

C₂×C₃

yes

—

C₉×3₋¹⁺²

Q₈

yes

characteristic, complemented, Sylow

C₃²×Dic₆

—

Q₈×3₋¹⁺²

—

C₉×Dic₆

—

C₃×Dic₁₈

—

C₃×Dic₁₈

C₃²

yes

characteristic

C₂×C₃×3₋¹⁺²

C₂×D₃

yes

characteristic

C₂×C₃²×C₉

C₂²×C₃

yes

characteristic

C₂×C₉⋊C₉

—

C₂×C₃²⋊C₉

—

C₂×C₉²

C₂²×C₃

yes

—

C₃³×C₄

C₃×D₃

yes

characteristic

C₃²×Dic₃

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

D₉

yes

characteristic

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

C₃×D₃

yes

—

C₃×C₄×C₉

C₂×C₃²

yes

characteristic

C₉×Dic₃

—

C₃×Dic₉

—

C₃×Dic₉

C₂×C₃²

yes

—

C₃×3₋¹⁺²

Dic₆

yes

characteristic

C₃²×C₉

C₃×Q₈

yes

characteristic

C₉⋊C₉

—

C₃²⋊C₉

—

C₉²

C₃×Q₈

yes

complemented

C₃²×Q₈

—

C₃×Dic₆

—

C₃×Dic₆

—

Q₈×C₉

—

Dic₁₈

3₋¹⁺²

yes

characteristic, direct product factor

C₂×C₃³

C₂×C₃×D₃

yes

characteristic

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₂×D₉

yes

characteristic

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₂×C₃×D₃

yes

—

C₂×C₃×C₉

C₂²×C₃²

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₃×D₉

yes

characteristic

C₃²×C₄

C₃²×D₃

yes

characteristic

C₃×Dic₃

—

C₃×Dic₃

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₃×D₉

yes

—

C₄×C₉

C₂×3₋¹⁺²

yes

characteristic

Dic₉

C₂×3₋¹⁺²

yes

—

C₃³

C₃×Dic₆

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

Dic₁₈

yes

characteristic, direct product factor

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×Dic₆

yes

—

C₃×C₉

C₃²×Q₈

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

Dic₆

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₂×C₃×D₉

yes

characteristic

C₂×C₃²

C₂×C₃²×D₃

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₂×C₃×D₉

yes

—

100

C₂×C₉

C₂²×3₋¹⁺²

yes

characteristic

101

C₃×C₄

—

102

C₃×C₄

—

103

C₃×C₄

—

104

C₃×C₄

—

105

C₃×C₄

—

106

C₃×C₄

C₃²×D₉

yes

characteristic

107

C₃×C₄

D₃×3₋¹⁺²

yes

characteristic

108

Dic₃

—

109

C₃²

—

110

C₃²

—

111

C₃²

—

112

C₃²

C₃×Dic₁₈

yes

characteristic

113

C₃²

C₃²×Dic₆

yes

characteristic

114

C₉

—

115

C₉

—

116

C₉

—

117

C₉

—

118

C₉

C₃×Dic₁₈

yes

complemented

119

C₉

Q₈×3₋¹⁺²

yes

characteristic, complemented

120

Q₈

—

Sylow

121

C₂×C₃

—

122

C₂×C₃

—

123

C₂×C₃

—

124

C₂×C₃

C₂×C₃²×D₉

yes

characteristic

125

C₂×C₃

C₂×D₃×3₋¹⁺²

yes

characteristic

126

C₄

—

127

C₄

D₉×3₋¹⁺²

yes

characteristic

128

C₃

—

129

C₃

—

130

C₃

—

131

C₃

C₃²×Dic₁₈

yes

characteristic

132

C₃

Dic₆×3₋¹⁺²

yes

characteristic

133

C₂

C₂×D₉×3₋¹⁺²

yes

characteristic

	subgroup	G/H
center	C₂×C₃	C₂×C₃²×D₉
commutator, cosocle	C₂×C₃×C₉	C₂²×C₃²
Frattini, socle	C₂×C₃²	C₂×C₃²×D₃
Fitting	C₄×C₉×3₋¹⁺²	C₂
prime radical	C₂×C₃³	C₂×C₃×D₃
hypercenter	C₄×3₋¹⁺²	D₉
hypocenter	C₉	Q₈×3₋¹⁺²
LCS	G ▷ C₂×C₃×C₉ ▷ C₉
UCS	C₁ ◁ C₂×C₃ ◁ C₄×3₋¹⁺²

subgroup

G/H

center

C₂×C₃

C₂×C₃²×D₉

commutator, cosocle

C₂×C₃×C₉

C₂²×C₃²

Frattini, socle

C₂×C₃²

C₂×C₃²×D₃

Fitting

C₄×C₉×3₋¹⁺²

C₂

prime radical

C₂×C₃³

C₂×C₃×D₃

hypercenter

C₄×3₋¹⁺²

D₉

hypocenter

C₉

Q₈×3₋¹⁺²

LCS

G ▷ C₂×C₃×C₉ ▷ C₉

UCS

C₁ ◁ C₂×C₃ ◁ C₄×3₋¹⁺²

N	H ≅ G/N	type
5. C₉×Dic₁₈	C₃	semidirect
15. C₉×3₋¹⁺²	Q₈	semidirect
45. C₉²	C₃×Q₈	semidirect
50. Dic₁₈	3₋¹⁺²	direct (N × H)
80. 3₋¹⁺²	Dic₁₈	direct (N × H)
118. C₉	C₃×Dic₁₈	semidirect
119. C₉	Q₈×3₋¹⁺²	semidirect

H ≅ G/N

type

5. C₉×Dic₁₈

C₃

semidirect

15. C₉×3₋¹⁺²

Q₈

semidirect

45. C₉²

C₃×Q₈

semidirect

50. Dic₁₈

3₋¹⁺²

direct (N × H)

80. 3₋¹⁺²

Dic₁₈

direct (N × H)

118. C₉

C₃×Dic₁₈

semidirect

119. C₉

Q₈×3₋¹⁺²

semidirect

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

subgroup	index
C₄×C₉×3₋¹⁺²	2
3₋¹⁺²×Dic₉	2
Dic₆×3₋¹⁺²	3
C₃²×Dic₁₈	3
C₉×Dic₁₈	3

subgroup

index

C₄×C₉×3₋¹⁺²

3₋¹⁺²×Dic₉

Dic₆×3₋¹⁺²

C₃²×Dic₁₈

C₉×Dic₁₈

Subgroups — 3₋¹⁺²×Dic₁₈ #14517302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3−1+2×Dic18 #14517302

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×Dic₁₈ #14517302