Group #13146770 — subgroups

Subgroups — C₉×#5406 #13146770

Order 1728.

200 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 200 conjugacy classes of subgroups — too many to draw. Only its 39 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	C₉×#986	C₂	1	1	yes	maximal, characteristic
2	C₉×#984	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#131⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#5406	C₃	1	1	yes	maximal, characteristic
5	C₉×#471	—	3	1	no	maximal
6	C₉×#41⋊C₃	C₂²	1	1	yes	characteristic
7	C₉×GL(2,𝔽₃)	—	2	1	no	—
8	C₉×#226	—	2	1	no	—
9	C₉×#204	—	4	1	no	maximal
10	C₃×#986	C₂×C₃	1	1	yes	characteristic
11	C₃×#984	C₂×C₃	1	1	yes	characteristic
12	C₃×#131⋊C₃	C₂×C₃	1	1	yes	characteristic
13	C₉×#137	—	3	1	no	—
14	C₉×AGL(1,ℤ/8)	—	3	1	no	—
15	C₂×C₉×SD₁₆	—	3	1	no	—
16	C₉×#131	D₃	1	1	yes	characteristic, complemented
17	#131⋊C₉	—	2	1	no	—
18	C₉×#108	—	3	1	no	—
19	C₉×#101	—	3	1	no	—
20	C₉×#100	—	3	1	no	—
21	C₉×D₁₂	—	4	1	no	—
22	C₉×Dic₆	—	4	1	no	—
23	C₉×SL(2,𝔽₃)	D₄	1	1	yes	characteristic
24	C₃×C₈×C₉	—	4	1	no	—
25	#5406	C₉	1	1	yes	characteristic, direct product factor
26	C₃×#471	—	3	1	no	—
27	C₃×#41⋊C₃	C₂²×C₃	1	1	yes	characteristic
28	C₃×GL(2,𝔽₃)	—	2	1	no	—
29	C₃×#226	—	2	1	no	—
30	C₃×#204	—	4	1	no	—
31	C₉×#41	C₂×D₃	1	1	yes	characteristic
32	C₂×Q₈×C₉	—	3	1	no	—
33	C₂×D₄×C₉	—	3	1	no	—
34	#41⋊C₉	—	2	1	no	—
35	C₉×Q₁₆	—	6	1	no	—
36	C₉×SD₁₆	—	6	1	no	—
37	C₉×SD₁₆	—	6	1	no	—
38	C₉×SD₁₆	—	6	1	no	—
39	C₉×SD₁₆	—	6	1	no	—
40	C₉×D₈	—	6	1	no	—
41	C₉×M₁₆	—	3	1	no	—
42	C₉×M₁₆	—	3	1	no	—
43	C₉×M₁₆	—	3	1	no	—
44	C₂×C₈×C₉	—	3	1	no	—
45	C₂×D₃×C₉	—	8	1	no	—
46	C₃×C₄×C₉	—	4	1	no	—
47	C₉×Dic₃	—	8	1	no	—
48	#986	C₂×C₉	1	1	yes	characteristic
49	#984	C₂×C₉	1	1	yes	characteristic, complemented
50	C₃×#137	—	3	1	no	—
51	C₃×AGL(1,ℤ/8)	—	3	1	no	—
52	C₂×C₃×SD₁₆	—	3	1	no	—
53	C₃×#131	C₃×D₃	1	1	yes	characteristic
54	#131⋊C₃	—	2	1	no	—
55	#131⋊C₃	C₂×C₉	1	1	yes	characteristic, complemented
56	C₃×#108	—	3	1	no	—
57	C₃×#101	—	3	1	no	—
58	C₃×#100	—	3	1	no	—
59	C₃×D₁₂	—	4	1	no	—
60	C₃×Dic₆	—	4	1	no	—
61	C₃×SL(2,𝔽₃)	C₃×D₄	1	1	yes	characteristic
62	C₂³×C₉	—	6	1	no	—
63	C₃²×C₈	—	4	1	no	—
64	Q₈×C₉	—	6	1	no	—
65	Q₈×C₉	—	6	1	no	—
66	Q₈×C₉	D₁₂	1	1	yes	characteristic
67	D₄×C₉	—	3	1	no	—
68	D₄×C₉	—	6	1	no	—
69	D₄×C₉	—	6	1	no	—
70	C₂×C₄×C₉	—	3	1	no	—
71	C₂×C₄×C₉	—	6	1	no	—
72	Q₈⋊C₉	—	2	1	no	—
73	C₈×C₉	—	3	1	no	—
74	C₈×C₉	—	6	1	no	—
75	C₈×C₉	—	6	1	no	—
76	C₈×C₉	—	8	1	no	—
77	C₈×C₉	S₄	1	1	yes	characteristic
78	#471	—	3	1	no	Sylow
79	C₂×C₃×C₉	—	4	1	no	—
80	D₃×C₉	—	16	1	no	—
81	C₃×#41	C₂×C₃×D₃	1	1	yes	characteristic
82	C₂×C₃×Q₈	—	3	1	no	—
83	C₂×C₃×D₄	—	3	1	no	—
84	#41⋊C₃	—	2	1	no	—
85	#41⋊C₃	C₂²×C₉	1	1	yes	characteristic
86	GL(2,𝔽₃)	—	2	1	no	—
87	#226	—	2	1	no	—
88	C₃×Q₁₆	—	6	1	no	—
89	C₃×SD₁₆	—	6	1	no	—
90	C₃×SD₁₆	—	6	1	no	—
91	C₃×SD₁₆	—	6	1	no	—
92	C₃×SD₁₆	—	6	1	no	—
93	C₃×D₈	—	6	1	no	—
94	C₃×M₁₆	—	3	1	no	—
95	C₃×M₁₆	—	3	1	no	—
96	C₃×M₁₆	—	3	1	no	—
97	C₂×C₃×C₈	—	3	1	no	—
98	#204	—	4	1	no	—
99	C₂×C₃×D₃	—	8	1	no	—
100	C₃²×C₄	—	4	1	no	—
101	C₃×Dic₃	—	8	1	no	—
102	C₂²×C₉	—	3	1	no	—
103	C₂²×C₉	—	12	1	no	—
104	C₂²×C₉	—	24	1	no	—
105	C₄×C₉	—	3	1	no	—
106	C₄×C₉	—	8	1	no	—
107	C₄×C₉	—	12	1	no	—
108	C₄×C₉	C₂×S₄	1	1	yes	characteristic
109	#137	—	3	1	no	—
110	AGL(1,ℤ/8)	—	3	1	no	—
111	C₂×SD₁₆	—	3	1	no	—
112	#131	D₃×C₉	1	1	yes	characteristic, complemented
113	#108	—	3	1	no	—
114	#101	—	3	1	no	—
115	#100	—	3	1	no	—
116	C₃×C₉	—	4	1	no	Sylow
117	C₂³×C₃	—	6	1	no	—
118	C₃×Q₈	—	6	1	no	—
119	C₃×Q₈	—	6	1	no	—
120	C₃×Q₈	C₃×D₁₂	1	1	yes	characteristic
121	C₃×D₄	—	3	1	no	—
122	C₃×D₄	—	6	1	no	—
123	C₃×D₄	—	6	1	no	—
124	C₂×C₃×C₄	—	3	1	no	—
125	C₂×C₃×C₄	—	6	1	no	—
126	D₁₂	—	4	1	no	—
127	Dic₆	—	4	1	no	—
128	SL(2,𝔽₃)	—	2	1	no	—
129	SL(2,𝔽₃)	D₄×C₉	1	1	yes	characteristic
130	C₃×C₈	—	3	1	no	—
131	C₃×C₈	—	4	1	no	—
132	C₃×C₈	—	6	1	no	—
133	C₃×C₈	—	6	1	no	—
134	C₃×C₈	—	8	1	no	—
135	C₃×C₈	C₃×S₄	1	1	yes	characteristic
136	C₂×C₃²	—	4	1	no	—
137	C₃×D₃	—	16	1	no	—
138	C₂×C₉	—	6	1	no	—
139	C₂×C₉	—	8	1	no	—
140	C₂×C₉	—	24	1	no	—
141	C₂×C₉	C₄⋊S₄	1	1	yes	characteristic
142	#41	C₂×D₃×C₉	1	1	yes	characteristic
143	C₂×Q₈	—	3	1	no	—
144	C₂×D₄	—	3	1	no	—
145	Q₁₆	—	6	1	no	—
146	SD₁₆	—	6	1	no	—
147	SD₁₆	—	6	1	no	—
148	SD₁₆	—	6	1	no	—
149	SD₁₆	—	6	1	no	—
150	D₈	—	6	1	no	—
151	M₁₆	—	3	1	no	—
152	M₁₆	—	3	1	no	—
153	M₁₆	—	3	1	no	—
154	C₂×C₈	—	3	1	no	—
155	C₂²×C₃	—	3	1	no	—
156	C₂²×C₃	—	12	1	no	—
157	C₂²×C₃	—	24	1	no	—
158	C₂×D₃	—	8	1	no	—
159	C₃×C₄	—	3	1	no	—
160	C₃×C₄	—	4	1	no	—
161	C₃×C₄	—	8	1	no	—
162	C₃×C₄	—	12	1	no	—
163	C₃×C₄	C₂×C₃×S₄	1	1	yes	characteristic
164	Dic₃	—	8	1	no	—
165	C₃²	—	4	1	no	—
166	C₉	—	8	1	no	—
167	C₉	#5406	1	1	yes	characteristic, direct product factor
168	C₂³	—	6	1	no	—
169	Q₈	—	6	1	no	—
170	Q₈	—	6	1	no	—
171	Q₈	C₉×D₁₂	1	1	yes	characteristic
172	D₄	—	3	1	no	—
173	D₄	—	6	1	no	—
174	D₄	—	6	1	no	—
175	C₂×C₄	—	3	1	no	—
176	C₂×C₄	—	6	1	no	—
177	C₈	—	3	1	no	—
178	C₈	—	6	1	no	—
179	C₈	—	6	1	no	—
180	C₈	C₉×S₄	1	1	yes	characteristic
181	C₂×C₃	—	4	1	no	—
182	C₂×C₃	—	6	1	no	—
183	C₂×C₃	—	8	1	no	—
184	C₂×C₃	—	24	1	no	—
185	C₂×C₃	C₃×C₄⋊S₄	1	1	yes	characteristic
186	D₃	—	16	1	no	—
187	C₂²	—	3	1	no	—
188	C₂²	—	12	1	no	—
189	C₂²	—	24	1	no	—
190	C₄	—	3	1	no	—
191	C₄	—	12	1	no	—
192	C₄	C₂×C₉×S₄	1	1	yes	characteristic
193	C₃	—	4	1	no	—
194	C₃	—	8	1	no	—
195	C₃	C₃×#5406	1	1	yes	characteristic
196	C₂	—	6	1	no	—
197	C₂	—	24	1	no	—
198	C₂	C₉×C₄⋊S₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₉×#986

C₂

yes

maximal, characteristic

C₉×#984

C₂

yes

maximal, characteristic, complemented

C₉×#131⋊C₃

C₂

yes

maximal, characteristic, complemented

C₃×#5406

C₃

yes

maximal, characteristic

C₉×#471

—

maximal

C₉×#41⋊C₃

C₂²

yes

characteristic

C₉×GL(2,𝔽₃)

—

—

—

C₂×C₃

yes

characteristic

C₃×#984

C₂×C₃

yes

characteristic

C₃×#131⋊C₃

C₂×C₃

yes

characteristic

C₉×#137

—

C₉×AGL(1,ℤ/8)

—

C₂×C₉×SD₁₆

—

C₉×#131

D₃

yes

characteristic, complemented

—

—

—

—

C₉×D₁₂

—

C₉×Dic₆

—

C₉×SL(2,𝔽₃)

D₄

yes

characteristic

C₃×C₈×C₉

—

#5406

C₉

yes

characteristic, direct product factor

C₃×#471

—

C₃×#41⋊C₃

C₂²×C₃

yes

characteristic

C₃×GL(2,𝔽₃)

—

C₃×#226

—

C₃×#204

—

C₉×#41

C₂×D₃

yes

characteristic

C₂×Q₈×C₉

—

C₂×D₄×C₉

—

#41⋊C₉

—

C₉×Q₁₆

—

C₉×SD₁₆

—

C₉×SD₁₆

—

C₉×SD₁₆

—

C₉×SD₁₆

—

C₉×D₈

—

C₉×M₁₆

—

C₉×M₁₆

—

C₉×M₁₆

—

C₂×C₈×C₉

—

C₂×D₃×C₉

—

C₃×C₄×C₉

—

C₉×Dic₃

—

#986

C₂×C₉

yes

characteristic

#984

C₂×C₉

yes

characteristic, complemented

C₃×#137

—

C₃×AGL(1,ℤ/8)

—

C₂×C₃×SD₁₆

—

C₃×#131

C₃×D₃

yes

characteristic

#131⋊C₃

—

#131⋊C₃

C₂×C₉

yes

characteristic, complemented

C₃×#108

—

C₃×#101

—

C₃×#100

—

C₃×D₁₂

—

C₃×Dic₆

—

C₃×SL(2,𝔽₃)

C₃×D₄

yes

characteristic

C₂³×C₉

—

C₃²×C₈

—

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

D₁₂

yes

characteristic

D₄×C₉

—

D₄×C₉

—

D₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

Q₈⋊C₉

—

C₈×C₉

—

C₈×C₉

—

C₈×C₉

—

C₈×C₉

—

C₈×C₉

S₄

yes

characteristic

#471

—

Sylow

C₂×C₃×C₉

—

D₃×C₉

—

C₃×#41

C₂×C₃×D₃

yes

characteristic

C₂×C₃×Q₈

—

C₂×C₃×D₄

—

#41⋊C₃

—

#41⋊C₃

C₂²×C₉

yes

characteristic

GL(2,𝔽₃)

—

#226

—

C₃×Q₁₆

—

C₃×SD₁₆

—

C₃×SD₁₆

—

C₃×SD₁₆

—

C₃×SD₁₆

—

C₃×D₈

—

C₃×M₁₆

—

C₃×M₁₆

—

C₃×M₁₆

—

C₂×C₃×C₈

—

#204

—

C₂×C₃×D₃

—

100

C₃²×C₄

—

101

C₃×Dic₃

—

102

C₂²×C₉

—

103

C₂²×C₉

—

104

C₂²×C₉

—

105

C₄×C₉

—

106

C₄×C₉

—

107

C₄×C₉

—

108

C₄×C₉

C₂×S₄

yes

characteristic

109

#137

—

110

AGL(1,ℤ/8)

—

111

C₂×SD₁₆

—

112

#131

D₃×C₉

yes

characteristic, complemented

113

#108

—

114

#101

—

115

#100

—

116

C₃×C₉

—

Sylow

117

C₂³×C₃

—

118

C₃×Q₈

—

119

C₃×Q₈

—

120

C₃×Q₈

C₃×D₁₂

yes

characteristic

121

C₃×D₄

—

122

C₃×D₄

—

123

C₃×D₄

—

124

C₂×C₃×C₄

—

125

C₂×C₃×C₄

—

126

D₁₂

—

127

Dic₆

—

128

SL(2,𝔽₃)

—

129

SL(2,𝔽₃)

D₄×C₉

yes

characteristic

130

C₃×C₈

—

131

C₃×C₈

—

132

C₃×C₈

—

133

C₃×C₈

—

134

C₃×C₈

—

135

C₃×C₈

C₃×S₄

yes

characteristic

136

C₂×C₃²

—

137

C₃×D₃

—

138

C₂×C₉

—

139

C₂×C₉

—

140

C₂×C₉

—

141

C₂×C₉

C₄⋊S₄

yes

characteristic

142

#41

C₂×D₃×C₉

yes

characteristic

143

C₂×Q₈

—

144

C₂×D₄

—

145

Q₁₆

—

146

SD₁₆

—

147

SD₁₆

—

148

SD₁₆

—

149

SD₁₆

—

150

D₈

—

151

M₁₆

—

152

M₁₆

—

153

M₁₆

—

154

C₂×C₈

—

155

C₂²×C₃

—

156

C₂²×C₃

—

157

C₂²×C₃

—

158

C₂×D₃

—

159

C₃×C₄

—

160

C₃×C₄

—

161

C₃×C₄

—

162

C₃×C₄

—

163

C₃×C₄

C₂×C₃×S₄

yes

characteristic

164

Dic₃

—

165

C₃²

—

166

C₉

—

167

C₉

#5406

yes

characteristic, direct product factor

168

C₂³

—

169

Q₈

—

170

Q₈

—

171

Q₈

C₉×D₁₂

yes

characteristic

172

D₄

—

173

D₄

—

174

D₄

—

175

C₂×C₄

—

176

C₂×C₄

—

177

C₈

—

178

C₈

—

179

C₈

—

180

C₈

C₉×S₄

yes

characteristic

181

C₂×C₃

—

182

C₂×C₃

—

183

C₂×C₃

—

184

C₂×C₃

—

185

C₂×C₃

C₃×C₄⋊S₄

yes

characteristic

186

D₃

—

187

C₂²

—

188

C₂²

—

189

C₂²

—

190

C₄

—

191

C₄

—

192

C₄

C₂×C₉×S₄

yes

characteristic

193

C₃

—

194

C₃

—

195

C₃

C₃×#5406

yes

characteristic

196

C₂

—

197

C₂

—

198

C₂

C₉×C₄⋊S₄

yes

characteristic

	subgroup	G/H
center	C₂×C₉	C₄⋊S₄
commutator	#41⋊C₃	C₂²×C₉
Frattini	C₃×C₄	C₂×C₃×S₄
Fitting	C₉×#131	D₃
socle	C₂×C₃	C₃×C₄⋊S₄
prime radical	C₃×#986	C₂×C₃
cosocle	C₃×#41⋊C₃	C₂²×C₃
hypercenter	C₈×C₉	S₄
hypocenter	SL(2,𝔽₃)	D₄×C₉
LCS	G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)
UCS	C₁ ◁ C₂×C₉ ◁ C₄×C₉ ◁ C₈×C₉

subgroup

G/H

center

C₂×C₉

C₄⋊S₄

commutator

#41⋊C₃

C₂²×C₉

Frattini

C₃×C₄

C₂×C₃×S₄

Fitting

C₉×#131

D₃

socle

C₂×C₃

C₃×C₄⋊S₄

prime radical

C₃×#986

C₂×C₃

cosocle

C₃×#41⋊C₃

C₂²×C₃

hypercenter

C₈×C₉

S₄

hypocenter

SL(2,𝔽₃)

D₄×C₉

LCS

G ▷ #41⋊C₃ ▷ SL(2,𝔽₃)

UCS

C₁ ◁ C₂×C₉ ◁ C₄×C₉ ◁ C₈×C₉

N	H ≅ G/N	type
2. C₉×#984	C₂	semidirect
3. C₉×#131⋊C₃	C₂	semidirect
16. C₉×#131	D₃	semidirect
25. #5406	C₉	direct (N × H)
49. #984	C₂×C₉	semidirect
55. #131⋊C₃	C₂×C₉	semidirect
112. #131	D₃×C₉	semidirect
167. C₉	#5406	direct (N × H)

H ≅ G/N

type

2. C₉×#984

C₂

semidirect

3. C₉×#131⋊C₃

C₂

semidirect

16. C₉×#131

D₃

semidirect

25. #5406

C₉

direct (N × H)

49. #984

C₂×C₉

semidirect

55. #131⋊C₃

C₂×C₉

semidirect

112. #131

D₃×C₉

semidirect

167. C₉

#5406

direct (N × H)

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

subgroup	index
C₉×#986	2
C₉×#984	2
C₉×#131⋊C₃	2
C₃×#5406	3
C₉×#471	3
C₉×#204	4

subgroup

index

C₉×#131⋊C₃

Subgroups — C₉×#5406 #13146770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C9×#5406 #13146770

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉×#5406 #13146770