Group #13135641 — subgroups

Subgroups — #10588224⋊C₃ #13135641

Order 1728.

226 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 86 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	3₋¹⁺²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(3₋¹⁺²⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₄⋊(3₋¹⁺²⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	C₃×#4547	—	3	1	no	maximal
5	#10588224	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×3₋¹⁺²	—	2	1	no	—
7	C₄⋊C₄×3₋¹⁺²	—	2	1	no	—
8	C₄⋊C₄×3₋¹⁺²	C₄	1	2	yes	—
9	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
10	C₂×3₋¹⁺²⋊C₈	—	2	1	no	—
11	C₂×3₋¹⁺²⋊C₈	—	2	1	no	—
12	C₃²×C₄⋊Q₈	D₃	1	1	yes	characteristic
13	C₃×C₄⋊(C₃⋊C₈)	—	3	1	no	—
14	C₃×C₄⋊(C₃⋊C₈)	—	3	1	no	—
15	C₉×C₄⋊Q₈	—	2	1	no	—
16	C₉×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
17	C₄⋊(C₉⋊C₈)	C₂×C₃	1	1	yes	characteristic
18	C₄⋊(C₉⋊C₈)	C₂×C₃	1	1	yes	characteristic
19	Q₈×3₋¹⁺²	—	4	1	no	—
20	Q₈×3₋¹⁺²	—	4	1	no	—
21	C₂×C₄×3₋¹⁺²	—	2	2	no	—
22	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
23	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
24	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
25	3₋¹⁺²⋊C₈	—	4	1	no	—
26	3₋¹⁺²⋊C₈	—	4	1	no	—
27	C₃×#332	—	9	1	no	—
28	#4547	—	3	1	no	—
29	C₂×C₃²×Q₈	—	2	1	no	—
30	C₃²×C₄⋊C₄	—	2	1	no	—
31	C₃²×C₄⋊C₄	Dic₃	1	2	yes	—
32	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
33	C₂×C₃×C₃⋊C₈	—	6	1	no	—
34	C₂×C₃×C₃⋊C₈	—	6	1	no	—
35	C₂×Q₈×C₉	—	2	2	no	—
36	C₂×Q₈×C₉	—	2	1	no	—
37	C₉×C₄⋊C₄	—	2	2	no	—
38	C₉×C₄⋊C₄	—	2	2	no	—
39	C₉×C₄⋊C₄	—	2	1	no	—
40	C₉×C₄⋊C₄	C₃×C₄	1	2	yes	—
41	C₄²×C₉	—	2	1	no	—
42	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
43	C₂×C₉⋊C₈	—	2	1	no	—
44	C₂×C₉⋊C₈	—	2	1	no	—
45	C₂²×3₋¹⁺²	C₂²⋊C₄	1	1	yes	characteristic
46	C₄×3₋¹⁺²	—	2	1	no	—
47	C₄×3₋¹⁺²	—	4	2	no	—
48	C₄×3₋¹⁺²	D₈	1	1	yes	characteristic
49	C₄×3₋¹⁺²	SD₁₆	1	1	yes	characteristic
50	C₄×3₋¹⁺²	SD₁₆	1	1	yes	characteristic
51	C₄×3₋¹⁺²	Q₁₆	1	1	yes	characteristic
52	C₃×C₄⋊Q₈	—	3	1	no	—
53	C₃×C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic
54	C₃×C₄⋊C₈	—	9	1	no	—
55	C₃×C₄⋊C₈	—	9	1	no	—
56	C₄⋊(C₃⋊C₈)	—	3	1	no	—
57	C₄⋊(C₃⋊C₈)	—	3	1	no	—
58	C₃²×Q₈	—	4	1	no	—
59	C₃²×Q₈	—	4	1	no	—
60	C₂×C₃²×C₄	—	2	2	no	—
61	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
62	C₂×C₃²×C₄	#67	1	1	yes	characteristic
63	C₂×C₃²×C₄	#67	1	1	yes	characteristic
64	C₃×C₃⋊C₈	—	12	1	no	—
65	C₃×C₃⋊C₈	—	12	1	no	—
66	Q₈×C₉	—	4	2	no	—
67	Q₈×C₉	—	4	2	no	—
68	Q₈×C₉	—	4	1	no	—
69	Q₈×C₉	—	4	1	no	—
70	C₂×C₄×C₉	—	2	4	no	—
71	C₂×C₄×C₉	—	2	2	no	—
72	C₂×C₄×C₉	—	2	1	no	—
73	C₂×C₄×C₉	—	2	1	no	—
74	C₂×C₄×C₉	—	2	1	no	—
75	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
76	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
77	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
78	C₉⋊C₈	—	4	1	no	—
79	C₉⋊C₈	—	4	1	no	—
80	#332	—	9	1	no	Sylow
81	C₂×3₋¹⁺²	#101	1	1	yes	characteristic
82	C₂×3₋¹⁺²	#102	1	1	yes	characteristic
83	C₂×3₋¹⁺²	#103	1	1	yes	characteristic
84	C₂×C₃×Q₈	—	2	1	no	—
85	C₂×C₃×Q₈	—	6	1	no	—
86	C₂×C₃×C₈	—	18	1	no	—
87	C₂×C₃×C₈	—	18	1	no	—
88	C₃×C₄⋊C₄	—	2	1	no	—
89	C₃×C₄⋊C₄	—	3	2	no	—
90	C₃×C₄⋊C₄	—	6	1	no	—
91	C₃×C₄⋊C₄	C₃×Dic₃	1	2	yes	—
92	C₃×C₄²	—	3	1	no	—
93	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
94	C₂×C₃⋊C₈	—	6	1	no	—
95	C₂×C₃⋊C₈	—	6	1	no	—
96	C₂²×C₃²	#217	1	1	yes	characteristic
97	C₃²×C₄	—	2	1	no	—
98	C₃²×C₄	—	4	2	no	—
99	C₃²×C₄	#213	1	1	yes	characteristic
100	C₃²×C₄	#214	1	1	yes	characteristic
101	C₃²×C₄	#215	1	1	yes	characteristic
102	C₃²×C₄	#216	1	1	yes	characteristic
103	C₂²×C₉	—	2	1	no	—
104	C₂²×C₉	C₃×C₂²⋊C₄	1	1	yes	characteristic
105	C₄×C₉	—	2	2	no	—
106	C₄×C₉	—	2	1	no	—
107	C₄×C₉	—	2	1	no	—
108	C₄×C₉	—	2	1	no	—
109	C₄×C₉	—	2	1	no	—
110	C₄×C₉	—	2	1	no	—
111	C₄×C₉	—	4	4	no	—
112	C₄×C₉	—	4	2	no	—
113	C₄×C₉	C₃×D₈	1	1	yes	characteristic
114	C₄×C₉	C₃×SD₁₆	1	1	yes	characteristic
115	C₄×C₉	C₃×SD₁₆	1	1	yes	characteristic
116	C₄×C₉	C₃×Q₁₆	1	1	yes	characteristic
117	C₄⋊Q₈	AGL(1,ℤ/9)	1	1	yes	characteristic
118	C₄⋊C₈	—	9	1	no	—
119	C₄⋊C₈	—	9	1	no	—
120	3₋¹⁺²	#332	1	1	yes	characteristic, complemented, Sylow
121	C₃×Q₈	—	4	1	no	—
122	C₃×Q₈	—	4	1	no	—
123	C₃×Q₈	—	12	1	no	—
124	C₃×Q₈	—	12	1	no	—
125	C₂×C₃×C₄	—	2	2	no	—
126	C₂×C₃×C₄	—	3	1	no	—
127	C₂×C₃×C₄	—	3	1	no	—
128	C₂×C₃×C₄	—	3	1	no	—
129	C₂×C₃×C₄	—	6	2	no	—
130	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
131	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
132	C₂×C₃×C₄	C₃×#67	1	1	yes	characteristic
133	C₃×C₈	—	36	1	no	—
134	C₃×C₈	—	36	1	no	—
135	C₃⋊C₈	—	12	1	no	—
136	C₃⋊C₈	—	12	1	no	—
137	C₂×C₃²	#832	1	1	yes	characteristic
138	C₂×C₃²	#835	1	1	yes	characteristic
139	C₂×C₃²	#836	1	1	yes	characteristic
140	C₂×C₉	—	2	1	no	—
141	C₂×C₉	—	2	1	no	—
142	C₂×C₉	—	2	1	no	—
143	C₂×C₉	C₃×#101	1	1	yes	characteristic
144	C₂×C₉	C₃×#102	1	1	yes	characteristic
145	C₂×C₉	C₃×#103	1	1	yes	characteristic
146	C₂×Q₈	—	2	1	no	—
147	C₂×C₈	—	18	1	no	—
148	C₂×C₈	—	18	1	no	—
149	C₄⋊C₄	—	2	1	no	—
150	C₄⋊C₄	3₋¹⁺²⋊C₄	1	2	yes	—
151	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
152	C₂²×C₃	—	3	1	no	—
153	C₂²×C₃	C₃×#217	1	1	yes	characteristic
154	C₃×C₄	—	2	1	no	—
155	C₃×C₄	—	3	1	no	—
156	C₃×C₄	—	3	1	no	—
157	C₃×C₄	—	3	1	no	—
158	C₃×C₄	—	3	1	no	—
159	C₃×C₄	—	4	2	no	—
160	C₃×C₄	—	6	1	no	—
161	C₃×C₄	—	12	2	no	—
162	C₃×C₄	C₃×#213	1	1	yes	characteristic
163	C₃×C₄	C₃×#214	1	1	yes	characteristic
164	C₃×C₄	C₃×#215	1	1	yes	characteristic
165	C₃×C₄	C₃×#216	1	1	yes	characteristic
166	C₃²	#4547	1	1	yes	characteristic
167	C₉	—	2	1	no	—
168	C₉	C₃×#332	1	1	yes	characteristic, complemented
169	Q₈	—	4	1	no	—
170	Q₈	—	4	1	no	—
171	C₂×C₄	—	2	2	no	—
172	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	1	yes	characteristic
173	C₂×C₄	#611⋊C₃	1	1	yes	characteristic
174	C₂×C₄	#611⋊C₃	1	1	yes	characteristic
175	C₈	—	36	1	no	—
176	C₈	—	36	1	no	—
177	C₂×C₃	—	3	1	no	—
178	C₂×C₃	—	3	1	no	—
179	C₂×C₃	—	3	1	no	—
180	C₂×C₃	C₃×#832	1	1	yes	characteristic
181	C₂×C₃	C₃×#835	1	1	yes	characteristic
182	C₂×C₃	C₃×#836	1	1	yes	characteristic
183	C₂²	#3676⋊C₃	1	1	yes	characteristic
184	C₄	—	2	1	no	—
185	C₄	—	4	2	no	—
186	C₄	#3672⋊C₃	1	1	yes	characteristic
187	C₄	#3673⋊C₃	1	1	yes	characteristic
188	C₄	#3674⋊C₃	1	1	yes	characteristic
189	C₄	#3675⋊C₃	1	1	yes	characteristic
190	C₃	—	3	1	no	—
191	C₃	C₃×#4547	1	1	yes	characteristic
192	C₂	#63392⋊C₃	1	1	yes	characteristic
193	C₂	#63394⋊C₃	1	1	yes	characteristic
194	C₂	#63395⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

3₋¹⁺²×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₄⋊(3₋¹⁺²⋊C₈)

C₂

yes

maximal, characteristic

C₄⋊(3₋¹⁺²⋊C₈)

C₂

yes

maximal, characteristic

C₃×#4547

—

maximal

#10588224

C₃

yes

maximal, characteristic, complemented

C₂×Q₈×3₋¹⁺²

—

C₄⋊C₄×3₋¹⁺²

—

C₄⋊C₄×3₋¹⁺²

C₄

yes

—

C₄²×3₋¹⁺²

C₂²

yes

characteristic

C₂×3₋¹⁺²⋊C₈

—

C₂×3₋¹⁺²⋊C₈

—

C₃²×C₄⋊Q₈

D₃

yes

characteristic

C₃×C₄⋊(C₃⋊C₈)

—

C₃×C₄⋊(C₃⋊C₈)

—

C₉×C₄⋊Q₈

—

C₉×C₄⋊Q₈

C₂×C₃

yes

characteristic

C₄⋊(C₉⋊C₈)

C₂×C₃

yes

characteristic

C₄⋊(C₉⋊C₈)

C₂×C₃

yes

characteristic

Q₈×3₋¹⁺²

—

Q₈×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

characteristic

C₂×C₄×3₋¹⁺²

D₄

yes

characteristic

C₂×C₄×3₋¹⁺²

D₄

yes

characteristic

3₋¹⁺²⋊C₈

—

3₋¹⁺²⋊C₈

—

C₃×#332

—

#4547

—

C₂×C₃²×Q₈

—

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

Dic₃

yes

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₂×C₃×C₃⋊C₈

—

C₂×C₃×C₃⋊C₈

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

C₃×C₄

yes

—

C₄²×C₉

—

C₄²×C₉

C₂²×C₃

yes

characteristic

C₂×C₉⋊C₈

—

C₂×C₉⋊C₈

—

C₂²×3₋¹⁺²

C₂²⋊C₄

yes

characteristic

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

D₈

yes

characteristic

C₄×3₋¹⁺²

SD₁₆

yes

characteristic

C₄×3₋¹⁺²

SD₁₆

yes

characteristic

C₄×3₋¹⁺²

Q₁₆

yes

characteristic

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

C₃×D₃

yes

characteristic

C₃×C₄⋊C₈

—

C₃×C₄⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₃²×Q₈

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×Dic₃

yes

C₂×C₃²×C₄

yes

C₂×C₃²×C₄

yes

C₃×C₃⋊C₈

—

C₃×C₃⋊C₈

—

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

characteristic

C₂×C₄×C₉

C₃×D₄

yes

characteristic

C₂×C₄×C₉

C₃×D₄

yes

characteristic

C₉⋊C₈

—

C₉⋊C₈

—

—

C₂×3₋¹⁺²

yes

C₂×3₋¹⁺²

yes

C₂×3₋¹⁺²

yes

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₃×Dic₃

yes

—

C₃×C₄²

—

C₃×C₄²

C₂×C₃×D₃

yes

characteristic

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

#217

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

#213

yes

characteristic

100

C₃²×C₄

#214

yes

characteristic

101

C₃²×C₄

#215

yes

characteristic

102

C₃²×C₄

#216

yes

characteristic

103

C₂²×C₉

—

104

C₂²×C₉

C₃×C₂²⋊C₄

yes

characteristic

105

C₄×C₉

—

106

C₄×C₉

—

107

C₄×C₉

—

108

C₄×C₉

—

109

C₄×C₉

—

110

C₄×C₉

—

111

C₄×C₉

—

112

C₄×C₉

—

113

C₄×C₉

C₃×D₈

yes

characteristic

114

C₄×C₉

C₃×SD₁₆

yes

characteristic

115

C₄×C₉

C₃×SD₁₆

yes

characteristic

116

C₄×C₉

C₃×Q₁₆

yes

characteristic

117

C₄⋊Q₈

AGL(1,ℤ/9)

yes

characteristic

118

C₄⋊C₈

—

119

C₄⋊C₈

—

120

3₋¹⁺²

#332

yes

characteristic, complemented, Sylow

121

C₃×Q₈

—

122

C₃×Q₈

—

123

C₃×Q₈

—

124

C₃×Q₈

—

125

C₂×C₃×C₄

—

126

C₂×C₃×C₄

—

127

C₂×C₃×C₄

—

128

C₂×C₃×C₄

—

129

C₂×C₃×C₄

—

130

C₂×C₃×C₄

C₂×C₃×Dic₃

yes

characteristic

131

C₂×C₃×C₄

C₃×#67

yes

characteristic

132

C₂×C₃×C₄

C₃×#67

yes

characteristic

133

C₃×C₈

—

134

C₃×C₈

—

135

C₃⋊C₈

—

136

C₃⋊C₈

—

137

C₂×C₃²

#832

yes

characteristic

138

C₂×C₃²

#835

yes

characteristic

139

C₂×C₃²

#836

yes

characteristic

140

C₂×C₉

—

141

C₂×C₉

—

142

C₂×C₉

—

143

C₂×C₉

C₃×#101

yes

characteristic

144

C₂×C₉

C₃×#102

yes

characteristic

145

C₂×C₉

C₃×#103

yes

characteristic

146

C₂×Q₈

—

147

C₂×C₈

—

148

C₂×C₈

—

149

C₄⋊C₄

—

150

C₄⋊C₄

3₋¹⁺²⋊C₄

yes

—

151

C₄²

C₂×AGL(1,ℤ/9)

yes

characteristic

152

C₂²×C₃

—

153

C₂²×C₃

C₃×#217

yes

characteristic

154

C₃×C₄

—

155

C₃×C₄

—

156

C₃×C₄

—

157

C₃×C₄

—

158

C₃×C₄

—

159

C₃×C₄

—

160

C₃×C₄

—

161

C₃×C₄

—

162

C₃×C₄

C₃×#213

yes

characteristic

163

C₃×C₄

C₃×#214

yes

characteristic

164

C₃×C₄

C₃×#215

yes

characteristic

165

C₃×C₄

C₃×#216

yes

characteristic

166

C₃²

#4547

yes

characteristic

167

C₉

—

168

C₉

C₃×#332

yes

characteristic, complemented

169

Q₈

—

170

Q₈

—

171

C₂×C₄

—

172

C₂×C₄

C₂×3₋¹⁺²⋊C₄

yes

characteristic

173

C₂×C₄

#611⋊C₃

yes

characteristic

174

C₂×C₄

#611⋊C₃

yes

characteristic

175

C₈

—

176

C₈

—

177

C₂×C₃

—

178

C₂×C₃

—

179

C₂×C₃

—

180

C₂×C₃

C₃×#832

yes

characteristic

181

C₂×C₃

C₃×#835

yes

characteristic

182

C₂×C₃

C₃×#836

yes

characteristic

183

C₂²

#3676⋊C₃

yes

characteristic

184

C₄

—

185

C₄

—

186

C₄

#3672⋊C₃

yes

characteristic

187

C₄

#3673⋊C₃

yes

characteristic

188

C₄

#3674⋊C₃

yes

characteristic

189

C₄

#3675⋊C₃

yes

characteristic

190

C₃

—

191

C₃

C₃×#4547

yes

characteristic

192

C₂

#63392⋊C₃

yes

characteristic

193

C₂

#63394⋊C₃

yes

characteristic

194

C₂

#63395⋊C₃

yes

characteristic

	subgroup	G/H
center	C₂²	#3676⋊C₃
commutator	C₂×C₄×C₉	C₂×C₃×C₄
Frattini	C₃×C₄²	C₂×C₃×D₃
Fitting	3₋¹⁺²×C₄⋊Q₈	C₂
socle	C₂²×C₃	C₃×#217
prime radical	C₂²×C₃²	#217
cosocle	C₄²×C₉	C₂²×C₃
hypercenter	C₄⋊Q₈	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#332
LCS	G ▷ C₂×C₄×C₉ ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

#3676⋊C₃

commutator

C₂×C₄×C₉

C₂×C₃×C₄

Frattini

C₃×C₄²

C₂×C₃×D₃

Fitting

3₋¹⁺²×C₄⋊Q₈

C₂

C₂²×C₃

C₂²×C₃²

C₄²×C₉

C₂²×C₃

C₄⋊Q₈

C₉

G ▷ C₂×C₄×C₉ ▷ C₂²×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
5. #10588224	C₃	semidirect
120. 3₋¹⁺²	#332	semidirect
168. C₉	C₃×#332	semidirect

H ≅ G/N

type

5. #10588224

C₃

semidirect

120. 3₋¹⁺²

#332

semidirect

168. C₉

C₃×#332

semidirect

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

subgroup	index
3₋¹⁺²×C₄⋊Q₈	2
C₄⋊(3₋¹⁺²⋊C₈)	2
C₃×#4547	3
#10588224	3

subgroup

index

3₋¹⁺²×C₄⋊Q₈

C₄⋊(3₋¹⁺²⋊C₈)

C₃×#4547

#10588224

Subgroups — #10588224⋊C₃ #13135641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588224⋊C3 #13135641

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588224⋊C₃ #13135641