Group #13144204 — subgroups

Subgroups — D₁₆×C₃⋊D₉ #13144204

Order 1728.

490 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 97 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	D₈×C₃⋊D₉	C₂	1	2	yes	maximal, complemented
2	C₃×C₉×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	#11720563	C₂	1	2	yes	maximal, complemented
4	#11720507	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₆×C₃⋊D₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃⋊D₃×D₁₆	—	3	1	no	maximal
7	D₉×D₁₆	—	3	3	no	maximal
8	D₄×C₃⋊D₉	—	2	2	no	—
9	C₃×C₉×D₈	C₂²	1	2	yes	complemented
10	#88544	—	2	2	no	—
11	#88525	C₂²	1	2	yes	complemented
12	C₈×C₃⋊D₉	C₂²	1	1	yes	characteristic
13	C₃×C₉×C₁₆	C₂²	1	1	yes	characteristic, complemented
14	C₃⋊(C₉⋊C₁₆)	C₂²	1	1	yes	characteristic, complemented
15	D₈×C₃⋊D₃	—	3	2	no	—
16	C₃²×D₁₆	D₃	1	1	yes	characteristic
17	#63653	—	3	2	no	—
18	#63626	—	3	1	no	—
19	C₁₆×C₃⋊D₃	—	3	1	no	—
20	D₈×D₉	—	3	6	no	—
21	C₉×D₁₆	D₃	1	3	yes	complemented
22	#63385	—	3	6	no	—
23	D₁₄₄	—	3	3	no	—
24	C₁₆×D₉	—	3	3	no	—
25	C₂²×C₃⋊D₉	—	4	2	no	—
26	C₃×D₄×C₉	—	2	2	no	—
27	#6233	—	4	2	no	—
28	C₄⋊(C₃⋊D₉)	—	2	2	no	—
29	C₄×C₃⋊D₉	D₄	1	1	yes	characteristic
30	C₃×C₈×C₉	C₂³	1	1	yes	characteristic
31	C₃⋊(C₉⋊C₈)	D₄	1	1	yes	characteristic
32	D₃×D₁₆	—	9	3	no	—
33	D₃×D₁₆	—	9	1	no	—
34	D₄×C₃⋊D₃	—	6	2	no	—
35	C₃²×D₈	C₂×D₃	1	2	yes	—
36	#3753	—	6	2	no	—
37	#3745	—	3	2	no	—
38	C₈×C₃⋊D₃	—	3	1	no	—
39	D₄×D₉	—	6	6	no	—
40	C₃²×C₁₆	C₂×D₃	1	1	yes	characteristic
41	C₃⋊(C₃⋊C₁₆)	—	3	1	no	—
42	C₉×D₈	C₂×D₃	1	6	yes	complemented
43	#3673	—	6	6	no	—
44	D₇₂	—	3	6	no	—
45	C₈×D₉	—	3	3	no	—
46	C₉×C₁₆	C₂×D₃	1	3	yes	complemented
47	C₉⋊C₁₆	—	3	3	no	—
48	C₂²×C₃×C₉	—	4	2	no	—
49	C₂×C₃⋊D₉	—	4	2	no	—
50	C₂×C₃⋊D₉	—	8	4	no	—
51	C₂×C₃⋊D₉	D₈	1	1	yes	characteristic
52	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
53	C₃⋊Dic₉	D₈	1	1	yes	characteristic
54	D₃×D₈	—	9	6	no	—
55	D₃×D₈	—	9	2	no	—
56	C₃×D₁₆	D₉	1	3	yes	complemented
57	C₃×D₁₆	C₃⋊D₃	1	1	yes	characteristic
58	#826	—	9	6	no	—
59	#826	—	9	2	no	—
60	D₄₈	—	9	3	no	—
61	D₄₈	—	9	1	no	—
62	D₃×C₁₆	—	9	3	no	—
63	D₃×C₁₆	—	9	1	no	—
64	C₂²×C₃⋊D₃	—	12	2	no	—
65	C₃²×D₄	—	2	2	no	—
66	#638	—	12	2	no	—
67	C₄⋊(C₃⋊D₃)	—	6	2	no	—
68	C₄×C₃⋊D₃	—	3	1	no	—
69	C₂²×D₉	—	12	6	no	—
70	C₃²×C₈	C₂²×D₃	1	1	yes	characteristic
71	C₃⋊(C₃⋊C₈)	—	3	1	no	—
72	D₄×C₉	—	2	6	no	—
73	#611	—	12	6	no	—
74	D₃₆	—	6	6	no	—
75	C₄×D₉	—	3	3	no	—
76	C₈×C₉	C₂²×D₃	1	3	yes	—
77	C₉⋊C₈	—	3	3	no	—
78	C₂×D₁₆	—	27	1	no	Sylow
79	C₂×C₃×C₉	—	8	2	no	—
80	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
81	C₃⋊D₉	—	8	2	no	—
82	C₃⋊D₉	D₁₆	1	2	yes	direct product factor
83	D₃×D₄	—	18	2	no	—
84	D₃×D₄	—	18	6	no	—
85	C₃×D₈	C₂×D₉	1	6	yes	complemented
86	C₃×D₈	C₂×C₃⋊D₃	1	2	yes	—
87	#213	—	18	2	no	—
88	#213	—	18	6	no	—
89	D₂₄	—	9	6	no	—
90	D₂₄	—	9	2	no	—
91	D₃×C₈	—	9	3	no	—
92	D₃×C₈	—	9	1	no	—
93	C₃×C₁₆	C₂×D₉	1	3	yes	complemented
94	C₃×C₁₆	C₂×C₃⋊D₃	1	1	yes	characteristic
95	C₃⋊C₁₆	—	9	3	no	—
96	C₃⋊C₁₆	—	9	1	no	—
97	C₂²×C₃²	—	4	2	no	—
98	C₂×C₃⋊D₃	—	3	1	no	—
99	C₂×C₃⋊D₃	—	12	2	no	—
100	C₂×C₃⋊D₃	—	24	4	no	—
101	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
102	C₃⋊Dic₃	—	3	1	no	—
103	C₂²×C₉	—	4	6	no	—
104	C₂×D₉	—	3	3	no	—
105	C₂×D₉	—	12	6	no	—
106	C₂×D₉	—	24	12	no	—
107	C₄×C₉	D₃×D₄	1	3	yes	—
108	Dic₉	—	3	3	no	—
109	C₂×D₈	—	27	2	no	—
110	D₁₆	—	27	2	no	—
111	D₁₆	—	27	1	no	—
112	D₁₆	C₃⋊D₉	1	1	yes	characteristic, direct product factor
113	C₂×C₁₆	—	27	1	no	—
114	C₃×C₉	C₂×D₁₆	1	1	yes	characteristic, complemented, Sylow
115	C₂²×D₃	—	36	2	no	—
116	C₂²×D₃	—	36	6	no	—
117	C₃×D₄	—	2	6	no	—
118	C₃×D₄	—	2	2	no	—
119	#67	—	36	2	no	—
120	#67	—	36	6	no	—
121	D₁₂	—	18	6	no	—
122	D₁₂	—	18	2	no	—
123	C₄×D₃	—	9	3	no	—
124	C₄×D₃	—	9	1	no	—
125	C₃×C₈	C₂²×D₉	1	3	yes	—
126	C₃×C₈	C₂²×C₃⋊D₃	1	1	yes	characteristic
127	C₃⋊C₈	—	9	3	no	—
128	C₃⋊C₈	—	9	1	no	—
129	C₂×C₃²	—	8	2	no	—
130	C₂×C₃²	D₃×D₈	1	1	yes	characteristic
131	C₃⋊D₃	—	3	2	no	—
132	C₃⋊D₃	—	24	2	no	—
133	C₂×C₉	—	8	6	no	—
134	C₂×C₉	D₃×D₈	1	3	yes	—
135	D₉	—	3	6	no	—
136	D₉	—	24	6	no	—
137	C₂×D₄	—	54	2	no	—
138	D₈	—	27	2	no	—
139	D₈	—	54	2	no	—
140	D₈	C₂×C₃⋊D₉	1	2	yes	complemented
141	C₂×C₈	—	27	1	no	—
142	C₁₆	—	27	1	no	—
143	C₁₆	C₂×C₃⋊D₉	1	1	yes	characteristic, complemented
144	C₂²×C₃	—	4	2	no	—
145	C₂²×C₃	—	4	6	no	—
146	C₂×D₃	—	9	1	no	—
147	C₂×D₃	—	9	3	no	—
148	C₂×D₃	—	36	2	no	—
149	C₂×D₃	—	36	6	no	—
150	C₂×D₃	—	72	4	no	—
151	C₂×D₃	—	72	12	no	—
152	C₃×C₄	D₄×D₉	1	3	yes	—
153	C₃×C₄	D₄×C₃⋊D₃	1	1	yes	characteristic
154	Dic₃	—	9	1	no	—
155	Dic₃	—	9	3	no	—
156	C₃²	D₃×D₁₆	1	1	yes	characteristic
157	C₉	D₃×D₁₆	1	3	yes	complemented
158	C₂³	—	108	2	no	—
159	D₄	—	2	2	no	—
160	D₄	—	54	2	no	—
161	D₄	—	108	2	no	—
162	C₂×C₄	—	27	1	no	—
163	C₈	—	27	1	no	—
164	C₈	C₂²×C₃⋊D₉	1	1	yes	characteristic
165	C₂×C₃	—	8	2	no	—
166	C₂×C₃	—	8	6	no	—
167	C₂×C₃	D₈×D₉	1	3	yes	—
168	C₂×C₃	D₈×C₃⋊D₃	1	1	yes	characteristic
169	D₃	—	9	2	no	—
170	D₃	—	9	6	no	—
171	D₃	—	72	2	no	—
172	D₃	—	72	6	no	—
173	C₂²	—	4	2	no	—
174	C₂²	—	27	1	no	—
175	C₂²	—	108	2	no	—
176	C₂²	—	216	4	no	—
177	C₄	—	27	1	no	—
178	C₄	D₄×C₃⋊D₉	1	1	yes	characteristic
179	C₃	D₉×D₁₆	1	3	yes	complemented
180	C₃	C₃⋊D₃×D₁₆	1	1	yes	characteristic
181	C₂	—	8	2	no	—
182	C₂	—	27	2	no	—
183	C₂	—	216	2	no	—
184	C₂	D₈×C₃⋊D₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₈×C₃⋊D₉

C₂

yes

maximal, complemented

C₃×C₉×D₁₆

C₂

yes

maximal, characteristic, complemented

#11720563

C₂

yes

maximal, complemented

#11720507

C₂

yes

maximal, characteristic, complemented

C₁₆×C₃⋊D₉

C₂

yes

maximal, characteristic, complemented

C₃⋊D₃×D₁₆

—

maximal

D₉×D₁₆

—

maximal

D₄×C₃⋊D₉

—

C₃×C₉×D₈

C₂²

yes

complemented

#88544

—

#88525

C₂²

yes

complemented

C₈×C₃⋊D₉

C₂²

yes

characteristic

C₃×C₉×C₁₆

C₂²

yes

characteristic, complemented

C₃⋊(C₉⋊C₁₆)

C₂²

yes

characteristic, complemented

D₈×C₃⋊D₃

—

C₃²×D₁₆

D₃

yes

characteristic

#63653

—

#63626

—

C₁₆×C₃⋊D₃

—

D₈×D₉

—

C₉×D₁₆

D₃

yes

complemented

#63385

—

D₁₄₄

—

C₁₆×D₉

—

C₂²×C₃⋊D₉

—

C₃×D₄×C₉

—

#6233

—

C₄⋊(C₃⋊D₉)

—

C₄×C₃⋊D₉

D₄

yes

characteristic

C₃×C₈×C₉

C₂³

yes

characteristic

C₃⋊(C₉⋊C₈)

D₄

yes

characteristic

D₃×D₁₆

—

D₃×D₁₆

—

D₄×C₃⋊D₃

—

C₃²×D₈

C₂×D₃

yes

—

#3753

—

#3745

—

C₈×C₃⋊D₃

—

D₄×D₉

—

C₃²×C₁₆

C₂×D₃

yes

characteristic

C₃⋊(C₃⋊C₁₆)

—

C₉×D₈

C₂×D₃

yes

complemented

#3673

—

D₇₂

—

C₈×D₉

—

C₉×C₁₆

C₂×D₃

yes

complemented

C₉⋊C₁₆

—

C₂²×C₃×C₉

—

C₂×C₃⋊D₉

—

C₂×C₃⋊D₉

—

C₂×C₃⋊D₉

D₈

yes

characteristic

C₃×C₄×C₉

C₂×D₄

yes

characteristic

C₃⋊Dic₉

D₈

yes

characteristic

D₃×D₈

—

D₃×D₈

—

C₃×D₁₆

D₉

yes

complemented

C₃×D₁₆

C₃⋊D₃

yes

characteristic

#826

—

#826

—

D₄₈

—

D₄₈

—

D₃×C₁₆

—

D₃×C₁₆

—

C₂²×C₃⋊D₃

—

C₃²×D₄

—

#638

—

C₄⋊(C₃⋊D₃)

—

C₄×C₃⋊D₃

—

C₂²×D₉

—

C₃²×C₈

C₂²×D₃

yes

characteristic

C₃⋊(C₃⋊C₈)

—

D₄×C₉

—

#611

—

D₃₆

—

C₄×D₉

—

C₈×C₉

C₂²×D₃

yes

—

C₉⋊C₈

—

C₂×D₁₆

—

Sylow

C₂×C₃×C₉

—

C₂×C₃×C₉

C₂×D₈

yes

characteristic

C₃⋊D₉

—

C₃⋊D₉

D₁₆

yes

direct product factor

D₃×D₄

—

D₃×D₄

—

C₃×D₈

C₂×D₉

yes

complemented

C₃×D₈

C₂×C₃⋊D₃

yes

—

#213

—

#213

—

D₂₄

—

D₂₄

—

D₃×C₈

—

D₃×C₈

—

C₃×C₁₆

C₂×D₉

yes

complemented

C₃×C₁₆

C₂×C₃⋊D₃

yes

characteristic

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₂²×C₃²

—

C₂×C₃⋊D₃

—

C₂×C₃⋊D₃

—

100

C₂×C₃⋊D₃

—

101

C₃²×C₄

D₃×D₄

yes

characteristic

102

C₃⋊Dic₃

—

103

C₂²×C₉

—

104

C₂×D₉

—

105

C₂×D₉

—

106

C₂×D₉

—

107

C₄×C₉

D₃×D₄

yes

—

108

Dic₉

—

109

C₂×D₈

—

110

D₁₆

—

111

D₁₆

—

112

D₁₆

C₃⋊D₉

yes

characteristic, direct product factor

113

C₂×C₁₆

—

114

C₃×C₉

C₂×D₁₆

yes

characteristic, complemented, Sylow

115

C₂²×D₃

—

116

C₂²×D₃

—

117

C₃×D₄

—

118

C₃×D₄

—

119

#67

—

120

#67

—

121

D₁₂

—

122

D₁₂

—

123

C₄×D₃

—

124

C₄×D₃

—

125

C₃×C₈

C₂²×D₉

yes

—

126

C₃×C₈

C₂²×C₃⋊D₃

yes

characteristic

127

C₃⋊C₈

—

128

C₃⋊C₈

—

129

C₂×C₃²

—

130

C₂×C₃²

D₃×D₈

yes

characteristic

131

C₃⋊D₃

—

132

C₃⋊D₃

—

133

C₂×C₉

—

134

C₂×C₉

D₃×D₈

yes

—

135

D₉

—

136

D₉

—

137

C₂×D₄

—

138

D₈

—

139

D₈

—

140

D₈

C₂×C₃⋊D₉

yes

complemented

141

C₂×C₈

—

142

C₁₆

—

143

C₁₆

C₂×C₃⋊D₉

yes

characteristic, complemented

144

C₂²×C₃

—

145

C₂²×C₃

—

146

C₂×D₃

—

147

C₂×D₃

—

148

C₂×D₃

—

149

C₂×D₃

—

150

C₂×D₃

—

151

C₂×D₃

—

152

C₃×C₄

D₄×D₉

yes

—

153

C₃×C₄

D₄×C₃⋊D₃

yes

characteristic

154

Dic₃

—

155

Dic₃

—

156

C₃²

D₃×D₁₆

yes

characteristic

157

C₉

D₃×D₁₆

yes

complemented

158

C₂³

—

108

—

159

D₄

—

160

D₄

—

161

D₄

—

108

—

162

C₂×C₄

—

163

C₈

—

164

C₈

C₂²×C₃⋊D₉

yes

characteristic

165

C₂×C₃

—

166

C₂×C₃

—

167

C₂×C₃

D₈×D₉

yes

—

168

C₂×C₃

D₈×C₃⋊D₃

yes

characteristic

169

D₃

—

170

D₃

—

171

D₃

—

172

D₃

—

173

C₂²

—

174

C₂²

—

175

C₂²

—

108

—

176

C₂²

—

216

—

177

C₄

—

178

C₄

D₄×C₃⋊D₉

yes

characteristic

179

C₃

D₉×D₁₆

yes

complemented

180

C₃

C₃⋊D₃×D₁₆

yes

characteristic

181

C₂

—

182

C₂

—

183

C₂

—

216

—

184

C₂

D₈×C₃⋊D₉

yes

characteristic

	subgroup	G/H
center	C₂	D₈×C₃⋊D₉
commutator, cosocle	C₃×C₈×C₉	C₂³
Frattini	C₃×C₈	C₂²×C₃⋊D₃
Fitting	C₃×C₉×D₁₆	C₂
socle	C₂×C₃²	D₃×D₈
hypercenter	D₁₆	C₃⋊D₉
hypocenter	C₃×C₉	C₂×D₁₆
LCS	G ▷ C₃×C₈×C₉ ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₁₆

subgroup

G/H

center

C₂

D₈×C₃⋊D₉

commutator, cosocle

C₃×C₈×C₉

C₂³

Frattini

C₃×C₈

C₂²×C₃⋊D₃

Fitting

C₃×C₉×D₁₆

C₂

socle

C₂×C₃²

D₃×D₈

hypercenter

D₁₆

C₃⋊D₉

hypocenter

C₃×C₉

C₂×D₁₆

LCS

G ▷ C₃×C₈×C₉ ▷ C₃×C₄×C₉ ▷ C₂×C₃×C₉ ▷ C₃×C₉

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ D₁₆

N	H ≅ G/N	type
1. D₈×C₃⋊D₉	C₂	semidirect
2. C₃×C₉×D₁₆	C₂	semidirect
3. #11720563	C₂	semidirect
4. #11720507	C₂	semidirect
5. C₁₆×C₃⋊D₉	C₂	semidirect
9. C₃×C₉×D₈	C₂²	semidirect
11. #88525	C₂²	semidirect
13. C₃×C₉×C₁₆	C₂²	semidirect
14. C₃⋊(C₉⋊C₁₆)	C₂²	semidirect
21. C₉×D₁₆	D₃	semidirect
42. C₉×D₈	C₂×D₃	semidirect
46. C₉×C₁₆	C₂×D₃	semidirect
56. C₃×D₁₆	D₉	semidirect
82. C₃⋊D₉	D₁₆	direct (N × H)
85. C₃×D₈	C₂×D₉	semidirect
93. C₃×C₁₆	C₂×D₉	semidirect
112. D₁₆	C₃⋊D₉	direct (N × H)
114. C₃×C₉	C₂×D₁₆	semidirect
140. D₈	C₂×C₃⋊D₉	semidirect
143. C₁₆	C₂×C₃⋊D₉	semidirect
157. C₉	D₃×D₁₆	semidirect
179. C₃	D₉×D₁₆	semidirect

H ≅ G/N

type

1. D₈×C₃⋊D₉

C₂

semidirect

2. C₃×C₉×D₁₆

C₂

semidirect

3. #11720563

C₂

semidirect

4. #11720507

C₂

semidirect

5. C₁₆×C₃⋊D₉

C₂

semidirect

9. C₃×C₉×D₈

C₂²

semidirect

11. #88525

C₂²

semidirect

13. C₃×C₉×C₁₆

C₂²

semidirect

14. C₃⋊(C₉⋊C₁₆)

C₂²

semidirect

21. C₉×D₁₆

D₃

semidirect

42. C₉×D₈

C₂×D₃

semidirect

46. C₉×C₁₆

C₂×D₃

semidirect

56. C₃×D₁₆

D₉

semidirect

82. C₃⋊D₉

D₁₆

direct (N × H)

85. C₃×D₈

C₂×D₉

semidirect

93. C₃×C₁₆

C₂×D₉

semidirect

112. D₁₆

C₃⋊D₉

direct (N × H)

114. C₃×C₉

C₂×D₁₆

semidirect

140. D₈

C₂×C₃⋊D₉

semidirect

143. C₁₆

C₂×C₃⋊D₉

semidirect

157. C₉

D₃×D₁₆

semidirect

179. C₃

D₉×D₁₆

semidirect

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

subgroup	index
D₈×C₃⋊D₉	2
C₃×C₉×D₁₆	2
#11720563	2
#11720507	2
C₁₆×C₃⋊D₉	2
C₃⋊D₃×D₁₆	3
D₉×D₁₆	3

subgroup

index

D₈×C₃⋊D₉

C₃×C₉×D₁₆

#11720563

#11720507

C₁₆×C₃⋊D₉

C₃⋊D₃×D₁₆

D₉×D₁₆

Subgroups — D₁₆×C₃⋊D₉ #13144204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D16×C3⋊D9 #13144204

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₁₆×C₃⋊D₉ #13144204