Group #14517864 — subgroups

Subgroups — C₃×C₃⋊C₈×3₋¹⁺² #14517864

Order 1944.

622 conjugacy classes of subgroups; only the 224 normal subgroups are shown.

normal subgroups only

This group has 622 conjugacy classes of subgroups — too many to draw. Only its 224 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	C₃²×C₄×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₃³×C₃⋊C₈	C₃	1	1	yes	maximal, characteristic
3	C₃×C₈×3₋¹⁺²	—	3	1	no	maximal
4	C₃⋊C₈×3₋¹⁺²	C₃	1	9	yes	maximal, direct product factor
5	C₃×C₉×C₃⋊C₈	C₃	1	3	yes	maximal, complemented
6	C₂×C₃²×3₋¹⁺²	C₄	1	1	yes	characteristic
7	C₃⁴×C₄	C₂×C₃	1	1	yes	characteristic
8	C₃×C₄×3₋¹⁺²	—	2	1	no	—
9	C₃×C₄×3₋¹⁺²	—	2	9	no	—
10	C₃×C₄×3₋¹⁺²	—	2	3	no	—
11	C₃×C₄×3₋¹⁺²	D₃	1	1	yes	characteristic
12	C₃×C₄×3₋¹⁺²	C₂×C₃	1	9	yes	—
13	C₃²×C₄×C₉	C₂×C₃	1	3	yes	—
14	C₃²×3₋¹⁺²	C₈	1	1	yes	characteristic, complemented, Sylow
15	C₃³×C₈	—	3	1	no	—
16	C₃²×C₃⋊C₈	—	3	3	no	—
17	C₃²×C₃⋊C₈	C₃²	1	3	yes	—
18	C₃²×C₃⋊C₈	C₃²	1	1	yes	characteristic
19	C₈×3₋¹⁺²	—	3	9	no	—
20	C₃×C₈×C₉	—	3	3	no	—
21	C₉×C₃⋊C₈	C₃²	1	9	yes	complemented
22	C₂×C₃⁴	C₃×C₄	1	1	yes	characteristic
23	C₂×C₃×3₋¹⁺²	—	2	1	no	—
24	C₂×C₃×3₋¹⁺²	—	2	9	no	—
25	C₂×C₃×3₋¹⁺²	—	2	3	no	—
26	C₂×C₃×3₋¹⁺²	Dic₃	1	1	yes	characteristic
27	C₂×C₃×3₋¹⁺²	C₃×C₄	1	9	yes	—
28	C₂×C₃²×C₉	C₃×C₄	1	3	yes	—
29	C₃³×C₄	—	2	3	no	—
30	C₃³×C₄	—	2	1	no	—
31	C₃³×C₄	—	3	3	no	—
32	C₃³×C₄	—	6	3	no	—
33	C₃³×C₄	C₃×D₃	1	1	yes	characteristic
34	C₃³×C₄	C₂×C₃²	1	1	yes	characteristic
35	C₃³×C₄	C₂×C₃²	1	3	yes	—
36	C₄×3₋¹⁺²	—	2	9	no	—
37	C₄×3₋¹⁺²	—	2	27	no	—
38	C₄×3₋¹⁺²	C₃×D₃	1	9	yes	—
39	C₃×C₄×C₉	—	2	3	no	—
40	C₃×C₄×C₉	—	2	9	no	—
41	C₃×C₄×C₉	C₃×D₃	1	3	yes	—
42	C₃×C₄×C₉	C₂×C₃²	1	9	yes	—
43	C₃⁴	C₃×C₈	1	1	yes	characteristic
44	C₃×3₋¹⁺²	—	2	1	no	—
45	C₃×3₋¹⁺²	—	2	9	no	—
46	C₃×3₋¹⁺²	—	2	3	no	—
47	C₃×3₋¹⁺²	C₃⋊C₈	1	1	yes	characteristic, direct product factor
48	C₃×3₋¹⁺²	C₃×C₈	1	9	yes	complemented
49	C₃²×C₉	C₃×C₈	1	3	yes	complemented
50	C₃²×C₈	—	3	1	no	—
51	C₃²×C₈	—	3	3	no	—
52	C₃²×C₈	—	9	3	no	—
53	C₃×C₃⋊C₈	—	3	3	no	—
54	C₃×C₃⋊C₈	3₋¹⁺²	1	3	yes	direct product factor
55	C₃×C₃⋊C₈	C₃³	1	1	yes	characteristic
56	C₈×C₉	—	3	9	no	—
57	C₂×C₃³	—	2	3	no	—
58	C₂×C₃³	—	2	1	no	—
59	C₂×C₃³	—	3	3	no	—
60	C₂×C₃³	—	6	3	no	—
61	C₂×C₃³	C₃×Dic₃	1	1	yes	characteristic
62	C₂×C₃³	C₃²×C₄	1	1	yes	characteristic
63	C₂×C₃³	C₃²×C₄	1	3	yes	—
64	C₂×3₋¹⁺²	—	2	9	no	—
65	C₂×3₋¹⁺²	—	2	27	no	—
66	C₂×3₋¹⁺²	C₃×Dic₃	1	9	yes	—
67	C₂×C₃×C₉	—	2	3	no	—
68	C₂×C₃×C₉	—	2	9	no	—
69	C₂×C₃×C₉	C₃×Dic₃	1	3	yes	—
70	C₂×C₃×C₉	C₃²×C₄	1	9	yes	—
71	C₃²×C₄	—	2	3	no	—
72	C₃²×C₄	—	2	1	no	—
73	C₃²×C₄	—	2	3	no	—
74	C₃²×C₄	—	3	3	no	—
75	C₃²×C₄	—	3	3	no	—
76	C₃²×C₄	—	6	9	no	—
77	C₃²×C₄	—	6	3	no	—
78	C₃²×C₄	—	6	3	no	—
79	C₃²×C₄	C₂×3₋¹⁺²	1	3	yes	—
80	C₃²×C₄	C₃²×D₃	1	3	yes	—
81	C₃²×C₄	C₃²×D₃	1	1	yes	characteristic
82	C₃²×C₄	C₂×C₃³	1	1	yes	characteristic
83	C₄×C₉	—	2	9	no	—
84	C₄×C₉	C₃²×D₃	1	9	yes	—
85	C₃³	—	2	3	no	—
86	C₃³	—	2	1	no	—
87	C₃³	—	3	3	no	—
88	C₃³	—	6	3	no	—
89	C₃³	C₃×C₃⋊C₈	1	1	yes	characteristic
90	C₃³	C₃²×C₈	1	1	yes	characteristic
91	C₃³	C₃²×C₈	1	3	yes	—
92	3₋¹⁺²	—	2	9	no	—
93	3₋¹⁺²	—	2	27	no	—
94	3₋¹⁺²	C₃×C₃⋊C₈	1	9	yes	direct product factor
95	C₃×C₉	—	2	3	no	—
96	C₃×C₉	—	2	9	no	—
97	C₃×C₉	C₃×C₃⋊C₈	1	3	yes	complemented
98	C₃×C₉	C₃²×C₈	1	9	yes	complemented
99	C₃×C₈	—	3	3	no	—
100	C₃×C₈	—	3	1	no	—
101	C₃×C₈	—	9	3	no	—
102	C₃⋊C₈	C₃×3₋¹⁺²	1	1	yes	characteristic, direct product factor
103	C₂×C₃²	—	2	3	no	—
104	C₂×C₃²	—	2	1	no	—
105	C₂×C₃²	—	2	3	no	—
106	C₂×C₃²	—	3	3	no	—
107	C₂×C₃²	—	3	3	no	—
108	C₂×C₃²	—	6	9	no	—
109	C₂×C₃²	—	6	3	no	—
110	C₂×C₃²	—	6	3	no	—
111	C₂×C₃²	C₄×3₋¹⁺²	1	3	yes	—
112	C₂×C₃²	C₃²×Dic₃	1	3	yes	—
113	C₂×C₃²	C₃²×Dic₃	1	1	yes	characteristic
114	C₂×C₃²	C₃³×C₄	1	1	yes	characteristic
115	C₂×C₉	—	2	9	no	—
116	C₂×C₉	C₃²×Dic₃	1	9	yes	—
117	C₃×C₄	—	2	3	no	—
118	C₃×C₄	—	2	1	no	—
119	C₃×C₄	—	3	3	no	—
120	C₃×C₄	—	6	3	no	—
121	C₃×C₄	D₃×3₋¹⁺²	1	3	yes	—
122	C₃×C₄	C₂×C₃×3₋¹⁺²	1	1	yes	characteristic
123	C₃×C₄	C₃³×D₃	1	1	yes	characteristic
124	C₃²	—	2	3	no	—
125	C₃²	—	2	1	no	—
126	C₃²	—	2	3	no	—
127	C₃²	—	3	3	no	—
128	C₃²	—	3	3	no	—
129	C₃²	—	6	3	no	—
130	C₃²	—	6	9	no	—
131	C₃²	—	6	3	no	—
132	C₃²	C₈×3₋¹⁺²	1	3	yes	complemented
133	C₃²	C₃²×C₃⋊C₈	1	3	yes	—
134	C₃²	C₃²×C₃⋊C₈	1	1	yes	characteristic
135	C₃²	C₃³×C₈	1	1	yes	characteristic
136	C₉	—	2	9	no	—
137	C₉	C₃²×C₃⋊C₈	1	9	yes	complemented
138	C₈	—	3	1	no	Sylow
139	C₂×C₃	—	2	3	no	—
140	C₂×C₃	—	2	1	no	—
141	C₂×C₃	—	3	3	no	—
142	C₂×C₃	—	6	3	no	—
143	C₂×C₃	Dic₃×3₋¹⁺²	1	3	yes	—
144	C₂×C₃	C₃×C₄×3₋¹⁺²	1	1	yes	characteristic
145	C₂×C₃	C₃³×Dic₃	1	1	yes	characteristic
146	C₄	C₃×D₃×3₋¹⁺²	1	1	yes	characteristic
147	C₃	—	2	3	no	—
148	C₃	—	2	1	no	—
149	C₃	—	3	3	no	—
150	C₃	—	6	3	no	—
151	C₃	C₃⋊C₈×3₋¹⁺²	1	3	yes	direct product factor
152	C₃	C₃×C₈×3₋¹⁺²	1	1	yes	characteristic, complemented
153	C₃	C₃³×C₃⋊C₈	1	1	yes	characteristic
154	C₂	C₃×Dic₃×3₋¹⁺²	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₄×3₋¹⁺²

C₂

yes

maximal, characteristic

C₃³×C₃⋊C₈

C₃

yes

maximal, characteristic

C₃×C₈×3₋¹⁺²

—

maximal

C₃⋊C₈×3₋¹⁺²

C₃

yes

maximal, direct product factor

C₃×C₉×C₃⋊C₈

C₃

yes

maximal, complemented

C₂×C₃²×3₋¹⁺²

C₄

yes

characteristic

C₃⁴×C₄

C₂×C₃

yes

characteristic

C₃×C₄×3₋¹⁺²

—

C₃×C₄×3₋¹⁺²

—

C₃×C₄×3₋¹⁺²

—

C₃×C₄×3₋¹⁺²

D₃

yes

characteristic

C₃×C₄×3₋¹⁺²

C₂×C₃

yes

—

C₃²×C₄×C₉

C₂×C₃

yes

—

C₃²×3₋¹⁺²

C₈

yes

characteristic, complemented, Sylow

C₃³×C₈

—

C₃²×C₃⋊C₈

—

C₃²×C₃⋊C₈

C₃²

yes

—

C₃²×C₃⋊C₈

C₃²

yes

characteristic

C₈×3₋¹⁺²

—

C₃×C₈×C₉

—

C₉×C₃⋊C₈

C₃²

yes

complemented

C₂×C₃⁴

C₃×C₄

yes

characteristic

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

Dic₃

yes

characteristic

C₂×C₃×3₋¹⁺²

C₃×C₄

yes

—

C₂×C₃²×C₉

C₃×C₄

yes

—

C₃³×C₄

—

C₃³×C₄

—

C₃³×C₄

—

C₃³×C₄

—

C₃³×C₄

C₃×D₃

yes

characteristic

C₃³×C₄

C₂×C₃²

yes

characteristic

C₃³×C₄

C₂×C₃²

yes

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

C₃×D₃

yes

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

C₃×D₃

yes

—

C₃×C₄×C₉

C₂×C₃²

yes

—

C₃⁴

C₃×C₈

yes

characteristic

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

—

C₃×3₋¹⁺²

C₃⋊C₈

yes

characteristic, direct product factor

C₃×3₋¹⁺²

C₃×C₈

yes

complemented

C₃²×C₉

C₃×C₈

yes

complemented

C₃²×C₈

—

C₃²×C₈

—

C₃²×C₈

—

C₃×C₃⋊C₈

—

C₃×C₃⋊C₈

3₋¹⁺²

yes

direct product factor

C₃×C₃⋊C₈

C₃³

yes

characteristic

C₈×C₉

—

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

C₃×Dic₃

yes

characteristic

C₂×C₃³

C₃²×C₄

yes

characteristic

C₂×C₃³

C₃²×C₄

yes

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₃×Dic₃

yes

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₃×Dic₃

yes

—

C₂×C₃×C₉

C₃²×C₄

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×3₋¹⁺²

yes

—

C₃²×C₄

C₃²×D₃

yes

—

C₃²×C₄

C₃²×D₃

yes

characteristic

C₃²×C₄

C₂×C₃³

yes

characteristic

C₄×C₉

—

C₄×C₉

C₃²×D₃

yes

—

C₃³

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×C₃⋊C₈

yes

characteristic

C₃³

C₃²×C₈

yes

characteristic

C₃³

C₃²×C₈

yes

—

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃×C₃⋊C₈

yes

direct product factor

C₃×C₉

—

C₃×C₉

—

C₃×C₉

C₃×C₃⋊C₈

yes

complemented

C₃×C₉

C₃²×C₈

yes

complemented

C₃×C₈

—

100

C₃×C₈

—

101

C₃×C₈

—

102

C₃⋊C₈

C₃×3₋¹⁺²

yes

characteristic, direct product factor

103

C₂×C₃²

—

104

C₂×C₃²

—

105

C₂×C₃²

—

106

C₂×C₃²

—

107

C₂×C₃²

—

108

C₂×C₃²

—

109

C₂×C₃²

—

110

C₂×C₃²

—

111

C₂×C₃²

C₄×3₋¹⁺²

yes

—

112

C₂×C₃²

C₃²×Dic₃

yes

—

113

C₂×C₃²

C₃²×Dic₃

yes

characteristic

114

C₂×C₃²

C₃³×C₄

yes

characteristic

115

C₂×C₉

—

116

C₂×C₉

C₃²×Dic₃

yes

—

117

C₃×C₄

—

118

C₃×C₄

—

119

C₃×C₄

—

120

C₃×C₄

—

121

C₃×C₄

D₃×3₋¹⁺²

yes

—

122

C₃×C₄

C₂×C₃×3₋¹⁺²

yes

characteristic

123

C₃×C₄

C₃³×D₃

yes

characteristic

124

C₃²

—

125

C₃²

—

126

C₃²

—

127

C₃²

—

128

C₃²

—

129

C₃²

—

130

C₃²

—

131

C₃²

—

132

C₃²

C₈×3₋¹⁺²

yes

complemented

133

C₃²

C₃²×C₃⋊C₈

yes

—

134

C₃²

C₃²×C₃⋊C₈

yes

characteristic

135

C₃²

C₃³×C₈

yes

characteristic

136

C₉

—

137

C₉

C₃²×C₃⋊C₈

yes

complemented

138

C₈

—

Sylow

139

C₂×C₃

—

140

C₂×C₃

—

141

C₂×C₃

—

142

C₂×C₃

—

143

C₂×C₃

Dic₃×3₋¹⁺²

yes

—

144

C₂×C₃

C₃×C₄×3₋¹⁺²

yes

characteristic

145

C₂×C₃

C₃³×Dic₃

yes

characteristic

146

C₄

C₃×D₃×3₋¹⁺²

yes

characteristic

147

C₃

—

148

C₃

—

149

C₃

—

150

C₃

—

151

C₃

C₃⋊C₈×3₋¹⁺²

yes

direct product factor

152

C₃

C₃×C₈×3₋¹⁺²

yes

characteristic, complemented

153

C₃

C₃³×C₃⋊C₈

yes

characteristic

154

C₂

C₃×Dic₃×3₋¹⁺²

yes

characteristic

	subgroup	G/H
center	C₃²×C₄	C₃²×D₃
commutator	C₃²	C₃³×C₈
Frattini	C₃×C₄	C₃³×D₃
Fitting	C₃²×C₄×3₋¹⁺²	C₂
socle	C₂×C₃³	C₃²×C₄
prime radical	C₂×C₃⁴	C₃×C₄
cosocle	C₃²×C₄	C₂×C₃³
hypercenter	C₃×C₄×3₋¹⁺²	D₃
hypocenter	C₃	C₃×C₈×3₋¹⁺²
LCS	G ▷ C₃² ▷ C₃
UCS	C₁ ◁ C₃²×C₄ ◁ C₃×C₄×3₋¹⁺²

subgroup

G/H

center

C₃²×C₄

C₃²×D₃

commutator

C₃²

C₃³×C₈

Frattini

C₃×C₄

C₃³×D₃

Fitting

C₃²×C₄×3₋¹⁺²

C₂

socle

C₂×C₃³

C₃²×C₄

prime radical

C₂×C₃⁴

C₃×C₄

cosocle

C₃²×C₄

C₂×C₃³

hypercenter

C₃×C₄×3₋¹⁺²

D₃

hypocenter

C₃

C₃×C₈×3₋¹⁺²

LCS

G ▷ C₃² ▷ C₃

UCS

C₁ ◁ C₃²×C₄ ◁ C₃×C₄×3₋¹⁺²

N	H ≅ G/N	type
4. C₃⋊C₈×3₋¹⁺²	C₃	direct (N × H)
5. C₃×C₉×C₃⋊C₈	C₃	semidirect
14. C₃²×3₋¹⁺²	C₈	semidirect
21. C₉×C₃⋊C₈	C₃²	semidirect
47. C₃×3₋¹⁺²	C₃⋊C₈	direct (N × H)
48. C₃×3₋¹⁺²	C₃×C₈	semidirect
49. C₃²×C₉	C₃×C₈	semidirect
54. C₃×C₃⋊C₈	3₋¹⁺²	direct (N × H)
94. 3₋¹⁺²	C₃×C₃⋊C₈	direct (N × H)
97. C₃×C₉	C₃×C₃⋊C₈	semidirect
98. C₃×C₉	C₃²×C₈	semidirect
102. C₃⋊C₈	C₃×3₋¹⁺²	direct (N × H)
132. C₃²	C₈×3₋¹⁺²	semidirect
137. C₉	C₃²×C₃⋊C₈	semidirect
151. C₃	C₃⋊C₈×3₋¹⁺²	direct (N × H)
152. C₃	C₃×C₈×3₋¹⁺²	semidirect

H ≅ G/N

type

4. C₃⋊C₈×3₋¹⁺²

C₃

direct (N × H)

5. C₃×C₉×C₃⋊C₈

C₃

semidirect

14. C₃²×3₋¹⁺²

C₈

semidirect

21. C₉×C₃⋊C₈

C₃²

semidirect

47. C₃×3₋¹⁺²

C₃⋊C₈

direct (N × H)

48. C₃×3₋¹⁺²

C₃×C₈

semidirect

49. C₃²×C₉

C₃×C₈

semidirect

54. C₃×C₃⋊C₈

3₋¹⁺²

direct (N × H)

94. 3₋¹⁺²

C₃×C₃⋊C₈

direct (N × H)

97. C₃×C₉

C₃×C₃⋊C₈

semidirect

98. C₃×C₉

C₃²×C₈

semidirect

102. C₃⋊C₈

C₃×3₋¹⁺²

direct (N × H)

132. C₃²

C₈×3₋¹⁺²

semidirect

137. C₉

C₃²×C₃⋊C₈

semidirect

151. C₃

C₃⋊C₈×3₋¹⁺²

direct (N × H)

152. C₃

C₃×C₈×3₋¹⁺²

semidirect

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

subgroup	index
C₃²×C₄×3₋¹⁺²	2
C₃³×C₃⋊C₈	3
C₃×C₈×3₋¹⁺²	3
C₃⋊C₈×3₋¹⁺²	3
C₃×C₉×C₃⋊C₈	3

subgroup

index

C₃²×C₄×3₋¹⁺²

C₃³×C₃⋊C₈

C₃×C₈×3₋¹⁺²

C₃⋊C₈×3₋¹⁺²

C₃×C₉×C₃⋊C₈

Subgroups — C₃×C₃⋊C₈×3₋¹⁺² #14517864

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×C3⋊C8×3−1+2 #14517864

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₃⋊C₈×3₋¹⁺² #14517864