Group #13135686 — subgroups

Subgroups — 3₋¹⁺²⋊#338 #13135686

Order 1728.

292 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 114 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	C₂×M₁₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂×#3667⋊C₃	C₂	1	2	yes	maximal
3	C₃×C₃⋊#338	—	3	1	no	maximal
4	C₉⋊#338	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
6	M₁₆×3₋¹⁺²	—	2	2	no	—
7	C₂×C₈×3₋¹⁺²	C₄	1	2	yes	—
8	#3667⋊C₃	—	2	2	no	—
9	#3667⋊C₃	—	2	2	no	—
10	C₂×3₋¹⁺²⋊C₈	C₄	1	4	yes	—
11	C₂×C₃²×M₁₆	D₃	1	1	yes	characteristic
12	C₂×C₃×#208	—	3	2	no	—
13	C₂×C₉×M₁₆	—	2	1	no	—
14	C₂×C₉×M₁₆	C₂×C₃	1	1	yes	characteristic
15	C₂×#3667	C₂×C₃	1	2	yes	—
16	C₂³×3₋¹⁺²	Q₈	1	1	yes	characteristic
17	C₂×C₄×3₋¹⁺²	C₂×C₄	1	2	yes	—
18	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
20	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
21	C₈×3₋¹⁺²	—	2	2	no	—
22	3₋¹⁺²⋊C₈	—	2	4	no	—
23	C₃×#338	—	9	1	no	—
24	C₃⋊#338	—	3	1	no	—
25	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
26	C₃²×M₁₆	—	2	2	no	—
27	C₂×C₃²×C₈	Dic₃	1	2	yes	—
28	C₃×#208	—	6	2	no	—
29	C₃×#208	—	6	2	no	—
30	C₂×C₃×C₃⋊C₈	—	3	4	no	—
31	C₂²×C₄×C₉	—	2	1	no	—
32	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
33	C₉×M₁₆	—	2	4	no	—
34	C₉×M₁₆	—	2	2	no	—
35	C₂×C₈×C₉	—	2	2	no	—
36	C₂×C₈×C₉	C₃×C₄	1	2	yes	—
37	#3667	—	2	2	no	—
38	#3667	—	2	2	no	—
39	C₂×C₉⋊C₈	C₃×C₄	1	4	yes	—
40	C₂²×3₋¹⁺²	—	4	1	no	—
41	C₂²×3₋¹⁺²	C₄⋊C₄	1	2	yes	—
42	C₂²×3₋¹⁺²	C₄⋊C₄	1	1	yes	characteristic
43	C₄×3₋¹⁺²	C₄²	1	1	yes	characteristic
44	C₄×3₋¹⁺²	C₂²⋊C₄	1	2	yes	—
45	C₄×3₋¹⁺²	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₃×M₁₆	—	3	1	no	—
47	C₂×C₃×M₁₆	—	9	2	no	—
48	C₂×C₃×M₁₆	C₃×D₃	1	1	yes	characteristic
49	C₂×#208	—	3	2	no	—
50	C₂³×C₃²	Dic₆	1	1	yes	characteristic
51	C₂×C₃²×C₄	C₄×D₃	1	2	yes	—
52	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
53	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
54	C₂×C₃²×C₄	#67	1	2	yes	—
55	C₂³×C₉	—	2	1	no	—
56	C₂³×C₉	C₃×Q₈	1	1	yes	characteristic
57	C₃²×C₈	—	2	2	no	—
58	C₃×C₃⋊C₈	—	6	4	no	—
59	C₂×C₄×C₉	—	2	2	no	—
60	C₂×C₄×C₉	—	2	1	no	—
61	C₂×C₄×C₉	—	2	1	no	—
62	C₂×C₄×C₉	—	2	2	no	—
63	C₂×C₄×C₉	C₂×C₃×C₄	1	2	yes	—
64	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
65	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
66	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
67	C₈×C₉	—	2	4	no	—
68	C₈×C₉	—	2	2	no	—
69	C₉⋊C₈	—	2	4	no	—
70	#338	—	9	1	no	Sylow
71	C₂×3₋¹⁺²	—	2	1	no	—
72	C₂×3₋¹⁺²	—	2	2	no	—
73	C₂×3₋¹⁺²	#95	1	1	yes	characteristic
74	C₂²×C₃×C₄	—	3	1	no	—
75	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
76	C₃×M₁₆	—	2	2	no	—
77	C₃×M₁₆	—	6	2	no	—
78	C₃×M₁₆	—	18	2	no	—
79	C₃×M₁₆	—	18	2	no	—
80	C₂×C₃×C₈	—	3	2	no	—
81	C₂×C₃×C₈	—	9	4	no	—
82	C₂×C₃×C₈	C₃×Dic₃	1	2	yes	—
83	#208	—	6	2	no	—
84	#208	—	6	2	no	—
85	C₂×C₃⋊C₈	—	3	4	no	—
86	C₂²×C₃²	—	4	1	no	—
87	C₂²×C₃²	Dic₃⋊C₄	1	2	yes	—
88	C₂²×C₃²	C₄⋊Dic₃	1	1	yes	characteristic
89	C₃²×C₄	C₄×Dic₃	1	1	yes	characteristic
90	C₃²×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
91	C₃²×C₄	#217	1	1	yes	characteristic
92	C₂²×C₉	—	2	2	no	—
93	C₂²×C₉	—	2	1	no	—
94	C₂²×C₉	—	4	1	no	—
95	C₂²×C₉	—	4	2	no	—
96	C₂²×C₉	C₃×C₄⋊C₄	1	2	yes	—
97	C₂²×C₉	C₃×C₄⋊C₄	1	1	yes	characteristic
98	C₄×C₉	—	2	1	no	—
99	C₄×C₉	—	2	2	no	—
100	C₄×C₉	—	2	1	no	—
101	C₄×C₉	C₃×C₄²	1	1	yes	characteristic
102	C₄×C₉	C₃×C₂²⋊C₄	1	2	yes	—
103	C₄×C₉	C₃×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×M₁₆	—	9	2	no	—
105	C₂×M₁₆	AGL(1,ℤ/9)	1	1	yes	characteristic
106	3₋¹⁺²	#338	1	1	yes	characteristic, complemented, Sylow
107	C₂³×C₃	—	3	1	no	—
108	C₂³×C₃	C₃×Dic₆	1	1	yes	characteristic
109	C₂×C₃×C₄	—	3	2	no	—
110	C₂×C₃×C₄	—	3	1	no	—
111	C₂×C₃×C₄	—	3	1	no	—
112	C₂×C₃×C₄	—	3	2	no	—
113	C₂×C₃×C₄	C₃×C₄×D₃	1	2	yes	—
114	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
115	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
116	C₂×C₃×C₄	C₃×#67	1	2	yes	—
117	C₃×C₈	—	2	2	no	—
118	C₃×C₈	—	6	2	no	—
119	C₃×C₈	—	18	4	no	—
120	C₃⋊C₈	—	6	4	no	—
121	C₂×C₃²	—	2	2	no	—
122	C₂×C₃²	—	2	1	no	—
123	C₂×C₃²	C₃⋊#95	1	1	yes	characteristic
124	C₂×C₉	—	2	2	no	—
125	C₂×C₉	—	2	2	no	—
126	C₂×C₉	—	2	1	no	—
127	C₂×C₉	—	2	1	no	—
128	C₂×C₉	—	4	2	no	—
129	C₂×C₉	C₃×#95	1	1	yes	characteristic
130	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
131	M₁₆	—	2	2	no	—
132	M₁₆	—	18	2	no	—
133	M₁₆	—	18	2	no	—
134	C₂×C₈	—	9	4	no	—
135	C₂×C₈	3₋¹⁺²⋊C₄	1	2	yes	—
136	C₂²×C₃	—	3	2	no	—
137	C₂²×C₃	—	3	1	no	—
138	C₂²×C₃	—	4	1	no	—
139	C₂²×C₃	—	12	1	no	—
140	C₂²×C₃	C₃×Dic₃⋊C₄	1	2	yes	—
141	C₂²×C₃	C₃×C₄⋊Dic₃	1	1	yes	characteristic
142	C₃×C₄	—	3	1	no	—
143	C₃×C₄	—	3	2	no	—
144	C₃×C₄	—	3	1	no	—
145	C₃×C₄	C₃×C₄×Dic₃	1	1	yes	characteristic
146	C₃×C₄	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
147	C₃×C₄	C₃×#217	1	1	yes	characteristic
148	C₃²	C₃⋊#338	1	1	yes	characteristic
149	C₉	—	2	1	no	—
150	C₉	C₃×#338	1	1	yes	characteristic, complemented
151	C₂³	3₋¹⁺²⋊Q₈	1	1	yes	characteristic
152	C₂×C₄	C₄×AGL(1,ℤ/9)	1	2	yes	—
153	C₂×C₄	C₄⋊AGL(1,ℤ/9)	1	1	yes	characteristic
154	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	1	yes	characteristic
155	C₂×C₄	#611⋊C₃	1	2	yes	—
156	C₈	—	2	2	no	—
157	C₈	—	18	4	no	—
158	C₂×C₃	—	2	2	no	—
159	C₂×C₃	—	2	1	no	—
160	C₂×C₃	—	3	1	no	—
161	C₂×C₃	—	6	2	no	—
162	C₂×C₃	—	6	1	no	—
163	C₂×C₃	C₃×C₃⋊#95	1	1	yes	characteristic
164	C₂²	—	4	1	no	—
165	C₂²	(3₋¹⁺²⋊C₄)⋊C₄	1	2	yes	—
166	C₂²	C₄⋊(3₋¹⁺²⋊C₄)	1	1	yes	characteristic
167	C₄	C₄×3₋¹⁺²⋊C₄	1	1	yes	characteristic
168	C₄	(C₂×AGL(1,ℤ/9))⋊C₄	1	2	yes	—
169	C₄	#3676⋊C₃	1	1	yes	characteristic
170	C₃	—	3	1	no	—
171	C₃	C₃×C₃⋊#338	1	1	yes	characteristic
172	C₂	—	2	2	no	—
173	C₂	—	2	1	no	—
174	C₂	3₋¹⁺²⋊#95	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₁₆×3₋¹⁺²

C₂

yes

maximal, characteristic

C₂×#3667⋊C₃

C₂

yes

maximal

C₃×C₃⋊#338

—

maximal

C₉⋊#338

C₃

yes

maximal, characteristic, complemented

C₂²×C₄×3₋¹⁺²

C₂²

yes

characteristic

M₁₆×3₋¹⁺²

—

C₂×C₈×3₋¹⁺²

C₄

yes

—

#3667⋊C₃

—

#3667⋊C₃

—

C₂×3₋¹⁺²⋊C₈

C₄

yes

—

C₂×C₃²×M₁₆

D₃

yes

characteristic

C₂×C₃×#208

—

C₂×C₉×M₁₆

—

C₂×C₉×M₁₆

C₂×C₃

yes

characteristic

C₂×#3667

C₂×C₃

yes

—

C₂³×3₋¹⁺²

Q₈

yes

characteristic

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

—

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

characteristic

C₂×C₄×3₋¹⁺²

D₄

yes

characteristic

C₂×C₄×3₋¹⁺²

D₄

yes

—

C₈×3₋¹⁺²

—

3₋¹⁺²⋊C₈

—

C₃×#338

—

C₃⋊#338

—

C₂²×C₃²×C₄

C₂×D₃

yes

characteristic

C₃²×M₁₆

—

C₂×C₃²×C₈

Dic₃

yes

—

C₃×#208

—

C₃×#208

—

C₂×C₃×C₃⋊C₈

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₂²×C₃

yes

characteristic

C₉×M₁₆

—

C₉×M₁₆

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₃×C₄

yes

—

#3667

—

#3667

—

C₂×C₉⋊C₈

C₃×C₄

yes

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

C₄⋊C₄

yes

—

C₂²×3₋¹⁺²

C₄⋊C₄

yes

characteristic

C₄×3₋¹⁺²

C₄²

yes

characteristic

C₄×3₋¹⁺²

C₂²⋊C₄

yes

—

C₄×3₋¹⁺²

C₂²⋊C₄

yes

characteristic

C₂×C₃×M₁₆

—

C₂×C₃×M₁₆

—

C₂×C₃×M₁₆

C₃×D₃

yes

characteristic

C₂×#208

—

C₂³×C₃²

Dic₆

yes

characteristic

C₂×C₃²×C₄

C₄×D₃

yes

—

C₂×C₃²×C₄

D₁₂

yes

characteristic

C₂×C₃²×C₄

C₂×Dic₃

yes

characteristic

C₂×C₃²×C₄

#67

yes

—

C₂³×C₉

—

C₂³×C₉

C₃×Q₈

yes

characteristic

C₃²×C₈

—

C₃×C₃⋊C₈

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

characteristic

C₂×C₄×C₉

C₃×D₄

yes

characteristic

C₂×C₄×C₉

C₃×D₄

yes

—

C₈×C₉

—

C₈×C₉

—

C₉⋊C₈

—

#338

—

Sylow

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

#95

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₃×D₃

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₃×Dic₃

yes

—

#208

—

#208

—

C₂×C₃⋊C₈

—

C₂²×C₃²

—

C₂²×C₃²

Dic₃⋊C₄

yes

—

C₂²×C₃²

C₄⋊Dic₃

yes

characteristic

C₃²×C₄

C₄×Dic₃

yes

characteristic

C₃²×C₄

(C₂×D₃)⋊C₄

yes

—

C₃²×C₄

#217

yes

characteristic

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₃×C₄⋊C₄

yes

—

C₂²×C₉

C₃×C₄⋊C₄

yes

characteristic

C₄×C₉

—

C₄×C₉

—

100

C₄×C₉

—

101

C₄×C₉

C₃×C₄²

yes

characteristic

102

C₄×C₉

C₃×C₂²⋊C₄

yes

—

103

C₄×C₉

C₃×C₂²⋊C₄

yes

characteristic

104

C₂×M₁₆

—

105

C₂×M₁₆

AGL(1,ℤ/9)

yes

characteristic

106

3₋¹⁺²

#338

yes

characteristic, complemented, Sylow

107

C₂³×C₃

—

108

C₂³×C₃

C₃×Dic₆

yes

characteristic

109

C₂×C₃×C₄

—

110

C₂×C₃×C₄

—

111

C₂×C₃×C₄

—

112

C₂×C₃×C₄

—

113

C₂×C₃×C₄

C₃×C₄×D₃

yes

—

114

C₂×C₃×C₄

C₃×D₁₂

yes

characteristic

115

C₂×C₃×C₄

C₂×C₃×Dic₃

yes

characteristic

116

C₂×C₃×C₄

C₃×#67

yes

—

117

C₃×C₈

—

118

C₃×C₈

—

119

C₃×C₈

—

120

C₃⋊C₈

—

121

C₂×C₃²

—

122

C₂×C₃²

—

123

C₂×C₃²

C₃⋊#95

yes

characteristic

124

C₂×C₉

—

125

C₂×C₉

—

126

C₂×C₉

—

127

C₂×C₉

—

128

C₂×C₉

—

129

C₂×C₉

C₃×#95

yes

characteristic

130

C₂²×C₄

C₂×AGL(1,ℤ/9)

yes

characteristic

131

M₁₆

—

132

M₁₆

—

133

M₁₆

—

134

C₂×C₈

—

135

C₂×C₈

3₋¹⁺²⋊C₄

yes

—

136

C₂²×C₃

—

137

C₂²×C₃

—

138

C₂²×C₃

—

139

C₂²×C₃

—

140

C₂²×C₃

C₃×Dic₃⋊C₄

yes

—

141

C₂²×C₃

C₃×C₄⋊Dic₃

yes

characteristic

142

C₃×C₄

—

143

C₃×C₄

—

144

C₃×C₄

—

145

C₃×C₄

C₃×C₄×Dic₃

yes

characteristic

146

C₃×C₄

C₃×(C₂×D₃)⋊C₄

yes

—

147

C₃×C₄

C₃×#217

yes

characteristic

148

C₃²

C₃⋊#338

yes

characteristic

149

C₉

—

150

C₉

C₃×#338

yes

characteristic, complemented

151

C₂³

3₋¹⁺²⋊Q₈

yes

characteristic

152

C₂×C₄

C₄×AGL(1,ℤ/9)

yes

—

153

C₂×C₄

C₄⋊AGL(1,ℤ/9)

yes

characteristic

154

C₂×C₄

C₂×3₋¹⁺²⋊C₄

yes

characteristic

155

C₂×C₄

#611⋊C₃

yes

—

156

C₈

—

157

C₈

—

158

C₂×C₃

—

159

C₂×C₃

—

160

C₂×C₃

—

161

C₂×C₃

—

162

C₂×C₃

—

163

C₂×C₃

C₃×C₃⋊#95

yes

characteristic

164

C₂²

—

165

C₂²

(3₋¹⁺²⋊C₄)⋊C₄

yes

—

166

C₂²

C₄⋊(3₋¹⁺²⋊C₄)

yes

characteristic

167

C₄

C₄×3₋¹⁺²⋊C₄

yes

characteristic

168

C₄

(C₂×AGL(1,ℤ/9))⋊C₄

yes

—

169

C₄

#3676⋊C₃

yes

characteristic

170

C₃

—

171

C₃

C₃×C₃⋊#338

yes

characteristic

172

C₂

—

173

C₂

—

174

C₂

3₋¹⁺²⋊#95

yes

characteristic

	subgroup	G/H
center	C₄	C₄×3₋¹⁺²⋊C₄
commutator	C₄×C₉	C₃×C₄²
Frattini	C₂²×C₃×C₄	C₂×C₃×D₃
Fitting	C₂×M₁₆×3₋¹⁺²	C₂
socle	C₂×C₃	C₃×C₃⋊#95
prime radical	C₂³×C₃²	Dic₆
cosocle	C₂²×C₄×C₉	C₂²×C₃
hypercenter	C₂×M₁₆	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#338
LCS	G ▷ C₄×C₉ ▷ C₉
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆

subgroup

G/H

center

C₄

C₄×3₋¹⁺²⋊C₄

commutator

C₄×C₉

C₃×C₄²

Frattini

C₂²×C₃×C₄

C₂×C₃×D₃

Fitting

C₂×M₁₆×3₋¹⁺²

C₂

socle

C₂×C₃

C₃×C₃⋊#95

prime radical

C₂³×C₃²

Dic₆

cosocle

C₂²×C₄×C₉

C₂²×C₃

C₂×M₁₆

C₉

G ▷ C₄×C₉ ▷ C₉

C₁ ◁ C₄ ◁ C₂×M₁₆

N	H ≅ G/N	type
4. C₉⋊#338	C₃	semidirect
106. 3₋¹⁺²	#338	semidirect
150. C₉	C₃×#338	semidirect

H ≅ G/N

type

4. C₉⋊#338

C₃

semidirect

106. 3₋¹⁺²

#338

semidirect

150. C₉

C₃×#338

semidirect

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

subgroup	index
C₂×M₁₆×3₋¹⁺²	2
C₂×#3667⋊C₃	2
C₃×C₃⋊#338	3
C₉⋊#338	3

subgroup

index

C₂×M₁₆×3₋¹⁺²

C₂×#3667⋊C₃

C₃×C₃⋊#338

C₉⋊#338

Subgroups — 3₋¹⁺²⋊#338 #13135686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3−1+2⋊#338 #13135686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²⋊#338 #13135686