Group #13136047 — subgroups

Subgroups — 3₋¹⁺²×C₂²⋊C₁₆ #13136047

Order 1728.

264 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 147 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	C₂²×C₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₆×3₋¹⁺²	C₂	1	2	yes	maximal, complemented
3	C₃²×C₂²⋊C₁₆	C₃	1	1	yes	maximal, characteristic
4	C₉×C₂²⋊C₁₆	C₃	1	3	yes	maximal, complemented
5	C₂²×C₄×3₋¹⁺²	C₄	1	1	yes	characteristic
6	C₂×C₈×3₋¹⁺²	—	2	2	no	—
7	C₂×C₈×3₋¹⁺²	C₄	1	1	yes	characteristic
8	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
9	C₁₆×3₋¹⁺²	—	2	2	no	—
10	C₂²×C₃²×C₈	C₂×C₃	1	1	yes	characteristic
11	C₂×C₃²×C₁₆	C₂×C₃	1	2	yes	—
12	C₂²×C₈×C₉	C₂×C₃	1	3	yes	—
13	C₂×C₉×C₁₆	C₂×C₃	1	6	yes	complemented
14	C₂³×3₋¹⁺²	C₈	1	1	yes	characteristic
15	C₂×C₄×3₋¹⁺²	—	2	1	no	—
16	C₂×C₄×3₋¹⁺²	—	2	1	no	—
17	C₂×C₄×3₋¹⁺²	C₈	1	1	yes	characteristic
18	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
19	C₈×3₋¹⁺²	—	2	1	no	—
20	C₈×3₋¹⁺²	D₄	1	2	yes	—
21	C₃×C₂²⋊C₁₆	—	3	1	no	—
22	C₃×C₂²⋊C₁₆	C₃²	1	1	yes	characteristic
23	C₂²×C₃²×C₄	C₃×C₄	1	1	yes	characteristic
24	C₂×C₃²×C₈	—	2	2	no	—
25	C₂×C₃²×C₈	C₃×C₄	1	1	yes	characteristic
26	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
27	C₂²×C₄×C₉	C₃×C₄	1	3	yes	—
28	C₃²×C₁₆	—	2	2	no	—
29	C₂×C₈×C₉	—	2	6	no	—
30	C₂×C₈×C₉	C₃×C₄	1	3	yes	—
31	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
32	C₉×C₁₆	—	2	6	no	—
33	C₂²×3₋¹⁺²	—	2	1	no	—
34	C₂²×3₋¹⁺²	—	2	1	no	—
35	C₂²×3₋¹⁺²	C₁₆	1	2	yes	complemented
36	C₂²×3₋¹⁺²	C₂×C₈	1	1	yes	characteristic
37	C₄×3₋¹⁺²	—	2	1	no	—
38	C₄×3₋¹⁺²	C₂²⋊C₄	1	1	yes	characteristic
39	C₄×3₋¹⁺²	M₁₆	1	1	yes	characteristic
40	C₂²×C₃×C₈	—	3	1	no	—
41	C₂²×C₃×C₈	C₂×C₃²	1	1	yes	characteristic
42	C₂×C₃×C₁₆	—	3	2	no	—
43	C₂×C₃×C₁₆	C₂×C₃²	1	2	yes	—
44	C₂³×C₃²	C₃×C₈	1	1	yes	characteristic
45	C₂×C₃²×C₄	—	2	1	no	—
46	C₂×C₃²×C₄	—	2	1	no	—
47	C₂×C₃²×C₄	C₃×C₈	1	1	yes	characteristic
48	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
49	C₂³×C₉	C₃×C₈	1	3	yes	—
50	C₃²×C₈	—	2	1	no	—
51	C₃²×C₈	C₃×D₄	1	2	yes	—
52	C₂×C₄×C₉	—	2	3	no	—
53	C₂×C₄×C₉	—	2	3	no	—
54	C₂×C₄×C₉	C₃×C₈	1	3	yes	—
55	C₂×C₄×C₉	C₂×C₃×C₄	1	3	yes	—
56	C₈×C₉	—	2	3	no	—
57	C₈×C₉	C₃×D₄	1	6	yes	—
58	C₂²⋊C₁₆	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
59	C₂×3₋¹⁺²	—	2	2	no	—
60	C₂×3₋¹⁺²	C₂²⋊C₈	1	1	yes	characteristic
61	C₂×3₋¹⁺²	C₂×C₁₆	1	1	yes	characteristic
62	C₂×3₋¹⁺²	M₃₂	1	1	yes	characteristic
63	C₂²×C₃×C₄	—	3	1	no	—
64	C₂²×C₃×C₄	C₃²×C₄	1	1	yes	characteristic
65	C₂×C₃×C₈	—	2	2	no	—
66	C₂×C₃×C₈	—	3	1	no	—
67	C₂×C₃×C₈	—	3	1	no	—
68	C₂×C₃×C₈	—	6	2	no	—
69	C₂×C₃×C₈	C₃²×C₄	1	1	yes	characteristic
70	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
71	C₃×C₁₆	—	2	2	no	—
72	C₃×C₁₆	—	6	2	no	—
73	C₂²×C₃²	—	2	1	no	—
74	C₂²×C₃²	—	2	1	no	—
75	C₂²×C₃²	C₃×C₁₆	1	2	yes	—
76	C₂²×C₃²	C₂×C₃×C₈	1	1	yes	characteristic
77	C₃²×C₄	—	2	1	no	—
78	C₃²×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
79	C₃²×C₄	C₃×M₁₆	1	1	yes	characteristic
80	C₂²×C₉	—	2	3	no	—
81	C₂²×C₉	—	2	3	no	—
82	C₂²×C₉	C₃×C₁₆	1	6	yes	complemented
83	C₂²×C₉	C₂×C₃×C₈	1	3	yes	—
84	C₄×C₉	—	2	3	no	—
85	C₄×C₉	C₃×C₂²⋊C₄	1	3	yes	—
86	C₄×C₉	C₃×M₁₆	1	3	yes	—
87	C₂²×C₈	C₂×3₋¹⁺²	1	1	yes	characteristic
88	C₂×C₁₆	C₂×3₋¹⁺²	1	2	yes	complemented
89	3₋¹⁺²	C₂²⋊C₁₆	1	1	yes	characteristic, direct product factor, Sylow
90	C₂³×C₃	—	3	1	no	—
91	C₂³×C₃	C₃²×C₈	1	1	yes	characteristic
92	C₂×C₃×C₄	—	2	1	no	—
93	C₂×C₃×C₄	—	2	1	no	—
94	C₂×C₃×C₄	—	3	1	no	—
95	C₂×C₃×C₄	—	3	1	no	—
96	C₂×C₃×C₄	—	6	1	no	—
97	C₂×C₃×C₄	—	6	1	no	—
98	C₂×C₃×C₄	C₃²×C₈	1	1	yes	characteristic
99	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
100	C₃×C₈	—	2	1	no	—
101	C₃×C₈	—	3	2	no	—
102	C₃×C₈	—	6	1	no	—
103	C₃×C₈	C₃²×D₄	1	2	yes	—
104	C₂×C₃²	—	2	2	no	—
105	C₂×C₃²	C₃×C₂²⋊C₈	1	1	yes	characteristic
106	C₂×C₃²	C₂×C₃×C₁₆	1	1	yes	characteristic
107	C₂×C₃²	C₃×M₃₂	1	1	yes	characteristic
108	C₂×C₉	—	2	6	no	—
109	C₂×C₉	C₃×C₂²⋊C₈	1	3	yes	—
110	C₂×C₉	C₂×C₃×C₁₆	1	3	yes	—
111	C₂×C₉	C₃×M₃₂	1	3	yes	—
112	C₂²×C₄	C₄×3₋¹⁺²	1	1	yes	characteristic
113	C₂×C₈	—	2	2	no	—
114	C₂×C₈	C₄×3₋¹⁺²	1	1	yes	characteristic
115	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
116	C₁₆	—	2	2	no	—
117	C₂²×C₃	—	2	1	no	—
118	C₂²×C₃	—	2	1	no	—
119	C₂²×C₃	—	3	2	no	—
120	C₂²×C₃	—	3	1	no	—
121	C₂²×C₃	—	6	1	no	—
122	C₂²×C₃	—	6	1	no	—
123	C₂²×C₃	C₃²×C₁₆	1	2	yes	—
124	C₂²×C₃	C₂×C₃²×C₈	1	1	yes	characteristic
125	C₃×C₄	—	2	1	no	—
126	C₃×C₄	—	3	1	no	—
127	C₃×C₄	—	3	1	no	—
128	C₃×C₄	—	6	1	no	—
129	C₃×C₄	C₃²×C₂²⋊C₄	1	1	yes	characteristic
130	C₃×C₄	C₃²×M₁₆	1	1	yes	characteristic
131	C₃²	C₃×C₂²⋊C₁₆	1	1	yes	characteristic
132	C₉	C₃×C₂²⋊C₁₆	1	3	yes	complemented
133	C₂³	C₈×3₋¹⁺²	1	1	yes	characteristic
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	2	1	no	—
136	C₂×C₄	C₈×3₋¹⁺²	1	1	yes	characteristic
137	C₂×C₄	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
138	C₈	—	2	1	no	—
139	C₈	D₄×3₋¹⁺²	1	2	yes	—
140	C₂×C₃	—	2	2	no	—
141	C₂×C₃	—	3	1	no	—
142	C₂×C₃	—	3	1	no	—
143	C₂×C₃	—	3	1	no	—
144	C₂×C₃	—	6	2	no	—
145	C₂×C₃	C₃²×C₂²⋊C₈	1	1	yes	characteristic
146	C₂×C₃	C₂×C₃²×C₁₆	1	1	yes	characteristic
147	C₂×C₃	C₃²×M₃₂	1	1	yes	characteristic
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₁₆×3₋¹⁺²	1	2	yes	complemented
151	C₂²	C₂×C₈×3₋¹⁺²	1	1	yes	characteristic
152	C₄	—	2	1	no	—
153	C₄	C₂²⋊C₄×3₋¹⁺²	1	1	yes	characteristic
154	C₄	M₁₆×3₋¹⁺²	1	1	yes	characteristic
155	C₃	—	3	1	no	—
156	C₃	C₃²×C₂²⋊C₁₆	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	3₋¹⁺²×C₂²⋊C₈	1	1	yes	characteristic
159	C₂	C₂×C₁₆×3₋¹⁺²	1	1	yes	characteristic
160	C₂	3₋¹⁺²×M₃₂	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×3₋¹⁺²

C₂

yes

maximal, characteristic

C₂×C₁₆×3₋¹⁺²

C₂

yes

maximal, complemented

C₃²×C₂²⋊C₁₆

C₃

yes

maximal, characteristic

C₉×C₂²⋊C₁₆

C₃

yes

maximal, complemented

C₂²×C₄×3₋¹⁺²

C₄

yes

characteristic

C₂×C₈×3₋¹⁺²

—

C₂×C₈×3₋¹⁺²

C₄

yes

characteristic

C₂×C₈×3₋¹⁺²

C₂²

yes

characteristic

C₁₆×3₋¹⁺²

—

C₂²×C₃²×C₈

C₂×C₃

yes

characteristic

C₂×C₃²×C₁₆

C₂×C₃

yes

—

C₂²×C₈×C₉

C₂×C₃

yes

—

C₂×C₉×C₁₆

C₂×C₃

yes

complemented

C₂³×3₋¹⁺²

C₈

yes

characteristic

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

C₈

yes

characteristic

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

characteristic

C₈×3₋¹⁺²

—

C₈×3₋¹⁺²

D₄

yes

—

C₃×C₂²⋊C₁₆

—

C₃×C₂²⋊C₁₆

C₃²

yes

characteristic

C₂²×C₃²×C₄

C₃×C₄

yes

characteristic

C₂×C₃²×C₈

—

C₂×C₃²×C₈

C₃×C₄

yes

characteristic

C₂×C₃²×C₈

C₂²×C₃

yes

characteristic

C₂²×C₄×C₉

C₃×C₄

yes

—

C₃²×C₁₆

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₃×C₄

yes

—

C₂×C₈×C₉

C₂²×C₃

yes

—

C₉×C₁₆

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

C₁₆

yes

complemented

C₂²×3₋¹⁺²

C₂×C₈

yes

characteristic

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

C₂²⋊C₄

yes

characteristic

C₄×3₋¹⁺²

M₁₆

yes

characteristic

C₂²×C₃×C₈

—

C₂²×C₃×C₈

C₂×C₃²

yes

characteristic

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

C₂×C₃²

yes

—

C₂³×C₃²

C₃×C₈

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₃×C₈

yes

characteristic

C₂×C₃²×C₄

C₂×C₃×C₄

yes

characteristic

C₂³×C₉

C₃×C₈

yes

—

C₃²×C₈

—

C₃²×C₈

C₃×D₄

yes

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₃×C₈

yes

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

—

C₈×C₉

—

C₈×C₉

C₃×D₄

yes

—

C₂²⋊C₁₆

3₋¹⁺²

yes

characteristic, direct product factor, Sylow

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₂²⋊C₈

yes

characteristic

C₂×3₋¹⁺²

C₂×C₁₆

yes

characteristic

C₂×3₋¹⁺²

M₃₂

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₃²×C₄

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₃²×C₄

yes

characteristic

C₂×C₃×C₈

C₂²×C₃²

yes

characteristic

C₃×C₁₆

—

C₃×C₁₆

—

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₃×C₁₆

yes

—

C₂²×C₃²

C₂×C₃×C₈

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₃×C₂²⋊C₄

yes

characteristic

C₃²×C₄

C₃×M₁₆

yes

characteristic

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₃×C₁₆

yes

complemented

C₂²×C₉

C₂×C₃×C₈

yes

—

C₄×C₉

—

C₄×C₉

C₃×C₂²⋊C₄

yes

—

C₄×C₉

C₃×M₁₆

yes

—

C₂²×C₈

C₂×3₋¹⁺²

yes

characteristic

C₂×C₁₆

C₂×3₋¹⁺²

yes

complemented

3₋¹⁺²

C₂²⋊C₁₆

yes

characteristic, direct product factor, Sylow

C₂³×C₃

—

C₂³×C₃

C₃²×C₈

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₃²×C₈

yes

characteristic

C₂×C₃×C₄

C₂×C₃²×C₄

yes

characteristic

100

C₃×C₈

—

101

C₃×C₈

—

102

C₃×C₈

—

103

C₃×C₈

C₃²×D₄

yes

—

104

C₂×C₃²

—

105

C₂×C₃²

C₃×C₂²⋊C₈

yes

characteristic

106

C₂×C₃²

C₂×C₃×C₁₆

yes

characteristic

107

C₂×C₃²

C₃×M₃₂

yes

characteristic

108

C₂×C₉

—

109

C₂×C₉

C₃×C₂²⋊C₈

yes

—

110

C₂×C₉

C₂×C₃×C₁₆

yes

—

111

C₂×C₉

C₃×M₃₂

yes

—

112

C₂²×C₄

C₄×3₋¹⁺²

yes

characteristic

113

C₂×C₈

—

114

C₂×C₈

C₄×3₋¹⁺²

yes

characteristic

115

C₂×C₈

C₂²×3₋¹⁺²

yes

characteristic

116

C₁₆

—

117

C₂²×C₃

—

118

C₂²×C₃

—

119

C₂²×C₃

—

120

C₂²×C₃

—

121

C₂²×C₃

—

122

C₂²×C₃

—

123

C₂²×C₃

C₃²×C₁₆

yes

—

124

C₂²×C₃

C₂×C₃²×C₈

yes

characteristic

125

C₃×C₄

—

126

C₃×C₄

—

127

C₃×C₄

—

128

C₃×C₄

—

129

C₃×C₄

C₃²×C₂²⋊C₄

yes

characteristic

130

C₃×C₄

C₃²×M₁₆

yes

characteristic

131

C₃²

C₃×C₂²⋊C₁₆

yes

characteristic

132

C₉

C₃×C₂²⋊C₁₆

yes

complemented

133

C₂³

C₈×3₋¹⁺²

yes

characteristic

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

C₈×3₋¹⁺²

yes

characteristic

137

C₂×C₄

C₂×C₄×3₋¹⁺²

yes

characteristic

138

C₈

—

139

C₈

D₄×3₋¹⁺²

yes

—

140

C₂×C₃

—

141

C₂×C₃

—

142

C₂×C₃

—

143

C₂×C₃

—

144

C₂×C₃

—

145

C₂×C₃

C₃²×C₂²⋊C₈

yes

characteristic

146

C₂×C₃

C₂×C₃²×C₁₆

yes

characteristic

147

C₂×C₃

C₃²×M₃₂

yes

characteristic

148

C₂²

—

149

C₂²

—

150

C₂²

C₁₆×3₋¹⁺²

yes

complemented

151

C₂²

C₂×C₈×3₋¹⁺²

yes

characteristic

152

C₄

—

153

C₄

C₂²⋊C₄×3₋¹⁺²

yes

characteristic

154

C₄

M₁₆×3₋¹⁺²

yes

characteristic

155

C₃

—

156

C₃

C₃²×C₂²⋊C₁₆

yes

characteristic

157

C₂

—

158

C₂

3₋¹⁺²×C₂²⋊C₈

yes

characteristic

159

C₂

C₂×C₁₆×3₋¹⁺²

yes

characteristic

160

C₂

3₋¹⁺²×M₃₂

yes

characteristic

	subgroup	G/H
center, Frattini, cosocle	C₂×C₃×C₈	C₂²×C₃²
commutator	C₂×C₃	C₂×C₃²×C₁₆
socle	C₂²×C₃	C₂×C₃²×C₈
prime radical	C₂³×C₃²	C₃×C₈
LCS	G ▷ C₂×C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₈ ◁ G

subgroup

G/H

center, Frattini, cosocle

C₂×C₃×C₈

C₂²×C₃²

commutator

C₂×C₃

C₂×C₃²×C₁₆

socle

C₂²×C₃

C₂×C₃²×C₈

prime radical

C₂³×C₃²

C₃×C₈

LCS

G ▷ C₂×C₃ ▷ C₁

UCS

C₁ ◁ C₂×C₃×C₈ ◁ G

N	H ≅ G/N	type
2. C₂×C₁₆×3₋¹⁺²	C₂	semidirect
4. C₉×C₂²⋊C₁₆	C₃	semidirect
13. C₂×C₉×C₁₆	C₂×C₃	semidirect
35. C₂²×3₋¹⁺²	C₁₆	semidirect
58. C₂²⋊C₁₆	3₋¹⁺²	direct (N × H)
82. C₂²×C₉	C₃×C₁₆	semidirect
88. C₂×C₁₆	C₂×3₋¹⁺²	semidirect
89. 3₋¹⁺²	C₂²⋊C₁₆	direct (N × H)
132. C₉	C₃×C₂²⋊C₁₆	semidirect
150. C₂²	C₁₆×3₋¹⁺²	semidirect

H ≅ G/N

type

2. C₂×C₁₆×3₋¹⁺²

C₂

semidirect

4. C₉×C₂²⋊C₁₆

C₃

semidirect

13. C₂×C₉×C₁₆

C₂×C₃

semidirect

35. C₂²×3₋¹⁺²

C₁₆

semidirect

58. C₂²⋊C₁₆

3₋¹⁺²

direct (N × H)

82. C₂²×C₉

C₃×C₁₆

semidirect

88. C₂×C₁₆

C₂×3₋¹⁺²

semidirect

89. 3₋¹⁺²

C₂²⋊C₁₆

direct (N × H)

132. C₉

C₃×C₂²⋊C₁₆

semidirect

150. C₂²

C₁₆×3₋¹⁺²

semidirect

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

subgroup	index
C₂²×C₈×3₋¹⁺²	2
C₂×C₁₆×3₋¹⁺²	2
C₃²×C₂²⋊C₁₆	3
C₉×C₂²⋊C₁₆	3

subgroup

index

C₂²×C₈×3₋¹⁺²

C₂×C₁₆×3₋¹⁺²

C₃²×C₂²⋊C₁₆

C₉×C₂²⋊C₁₆

Subgroups — 3₋¹⁺²×C₂²⋊C₁₆ #13136047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3−1+2×C22⋊C16 #13136047

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×C₂²⋊C₁₆ #13136047