Group #13136053 — subgroups

Subgroups — 3₋¹⁺²×#350 #13136053

Order 1728.

224 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 119 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×#131	C₂	1	1	yes	maximal, characteristic
2	3₋¹⁺²×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₆×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#350	C₃	1	1	yes	maximal, characteristic
5	C₉×#350	C₃	1	3	yes	maximal, complemented
6	#41×3₋¹⁺²	C₄	1	1	yes	characteristic
7	M₁₆×3₋¹⁺²	—	2	1	no	—
8	M₁₆×3₋¹⁺²	C₄	1	1	yes	characteristic
9	C₂×C₈×3₋¹⁺²	—	2	1	no	—
10	C₂×C₈×3₋¹⁺²	C₂²	1	1	yes	characteristic
11	C₁₆×3₋¹⁺²	—	2	1	no	—
12	C₁₆×3₋¹⁺²	—	2	1	no	—
13	C₃²×#131	C₂×C₃	1	1	yes	characteristic
14	C₃²×M₃₂	C₂×C₃	1	1	yes	characteristic
15	C₂×C₃²×C₁₆	C₂×C₃	1	1	yes	characteristic
16	C₉×#131	C₂×C₃	1	3	yes	—
17	C₉×M₃₂	C₂×C₃	1	3	yes	complemented
18	C₂×C₉×C₁₆	C₂×C₃	1	3	yes	complemented
19	Q₈×3₋¹⁺²	C₈	1	1	yes	characteristic
20	D₄×3₋¹⁺²	—	2	1	no	—
21	D₄×3₋¹⁺²	C₈	1	1	yes	characteristic
22	C₂×C₄×3₋¹⁺²	—	2	1	no	—
23	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
24	C₈×3₋¹⁺²	—	2	1	no	—
25	C₈×3₋¹⁺²	D₄	1	1	yes	characteristic
26	C₈×3₋¹⁺²	D₄	1	1	yes	characteristic
27	C₃×#350	—	3	1	no	—
28	C₃×#350	C₃²	1	1	yes	characteristic
29	C₃²×#41	C₃×C₄	1	1	yes	characteristic
30	C₃²×M₁₆	—	2	1	no	—
31	C₃²×M₁₆	C₃×C₄	1	1	yes	characteristic
32	C₂×C₃²×C₈	—	2	1	no	—
33	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
34	C₉×#41	C₃×C₄	1	3	yes	—
35	C₃²×C₁₆	—	2	1	no	—
36	C₃²×C₁₆	—	2	1	no	—
37	C₉×M₁₆	—	2	3	no	—
38	C₉×M₁₆	C₃×C₄	1	3	yes	—
39	C₂×C₈×C₉	—	2	3	no	—
40	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
41	C₉×C₁₆	—	2	3	no	—
42	C₉×C₁₆	—	2	3	no	—
43	C₂²×3₋¹⁺²	—	2	1	no	—
44	C₂²×3₋¹⁺²	M₁₆	1	1	yes	characteristic
45	C₄×3₋¹⁺²	—	2	1	no	—
46	C₄×3₋¹⁺²	C₂²⋊C₄	1	1	yes	characteristic
47	C₄×3₋¹⁺²	C₂×C₈	1	1	yes	characteristic
48	C₃×#131	—	3	1	no	—
49	C₃×#131	C₂×C₃²	1	1	yes	characteristic
50	C₃×M₃₂	—	3	1	no	—
51	C₃×M₃₂	C₂×C₃²	1	1	yes	characteristic
52	C₂×C₃×C₁₆	—	3	1	no	—
53	C₂×C₃×C₁₆	C₂×C₃²	1	1	yes	characteristic
54	C₃²×Q₈	C₃×C₈	1	1	yes	characteristic
55	C₃²×D₄	—	2	1	no	—
56	C₃²×D₄	C₃×C₈	1	1	yes	characteristic
57	C₂×C₃²×C₄	—	2	1	no	—
58	C₂×C₃²×C₄	C₂×C₃×C₄	1	1	yes	characteristic
59	C₃²×C₈	—	2	1	no	—
60	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
61	C₃²×C₈	C₃×D₄	1	1	yes	characteristic
62	Q₈×C₉	C₃×C₈	1	3	yes	—
63	D₄×C₉	—	2	3	no	—
64	D₄×C₉	C₃×C₈	1	3	yes	—
65	C₂×C₄×C₉	—	2	3	no	—
66	C₂×C₄×C₉	C₂×C₃×C₄	1	3	yes	—
67	C₈×C₉	—	2	3	no	—
68	C₈×C₉	C₃×D₄	1	3	yes	—
69	C₈×C₉	C₃×D₄	1	3	yes	—
70	#350	3₋¹⁺²	1	1	yes	characteristic, direct product factor, Sylow
71	C₂×3₋¹⁺²	—	2	1	no	—
72	C₂×3₋¹⁺²	—	4	1	no	—
73	C₂×3₋¹⁺²	C₂²⋊C₈	1	1	yes	characteristic
74	C₃×#41	—	3	1	no	—
75	C₃×#41	C₃²×C₄	1	1	yes	characteristic
76	C₃×M₁₆	—	2	1	no	—
77	C₃×M₁₆	—	3	1	no	—
78	C₃×M₁₆	—	6	1	no	—
79	C₃×M₁₆	C₃²×C₄	1	1	yes	characteristic
80	C₂×C₃×C₈	—	2	1	no	—
81	C₂×C₃×C₈	—	3	1	no	—
82	C₂×C₃×C₈	—	6	1	no	—
83	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
84	C₃×C₁₆	—	2	1	no	—
85	C₃×C₁₆	—	2	1	no	—
86	C₃×C₁₆	—	6	1	no	—
87	C₃×C₁₆	—	6	1	no	—
88	C₂²×C₃²	—	2	1	no	—
89	C₂²×C₃²	C₃×M₁₆	1	1	yes	characteristic
90	C₃²×C₄	—	2	1	no	—
91	C₃²×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
92	C₃²×C₄	C₂×C₃×C₈	1	1	yes	characteristic
93	C₂²×C₉	—	2	3	no	—
94	C₂²×C₉	C₃×M₁₆	1	3	yes	—
95	C₄×C₉	—	2	3	no	—
96	C₄×C₉	C₃×C₂²⋊C₄	1	3	yes	—
97	C₄×C₉	C₂×C₃×C₈	1	3	yes	—
98	#131	C₂×3₋¹⁺²	1	1	yes	characteristic
99	M₃₂	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
100	C₂×C₁₆	C₂×3₋¹⁺²	1	1	yes	characteristic, complemented
101	3₋¹⁺²	#350	1	1	yes	characteristic, direct product factor, Sylow
102	C₃×Q₈	—	3	1	no	—
103	C₃×Q₈	C₃²×C₈	1	1	yes	characteristic
104	C₃×D₄	—	2	1	no	—
105	C₃×D₄	—	3	1	no	—
106	C₃×D₄	—	6	1	no	—
107	C₃×D₄	C₃²×C₈	1	1	yes	characteristic
108	C₂×C₃×C₄	—	2	1	no	—
109	C₂×C₃×C₄	—	3	1	no	—
110	C₂×C₃×C₄	—	6	1	no	—
111	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
112	C₃×C₈	—	2	1	no	—
113	C₃×C₈	—	3	1	no	—
114	C₃×C₈	—	3	1	no	—
115	C₃×C₈	—	6	1	no	—
116	C₃×C₈	C₃²×D₄	1	1	yes	characteristic
117	C₃×C₈	C₃²×D₄	1	1	yes	characteristic
118	C₂×C₃²	—	2	1	no	—
119	C₂×C₃²	—	4	1	no	—
120	C₂×C₃²	C₃×C₂²⋊C₈	1	1	yes	characteristic
121	C₂×C₉	—	2	3	no	—
122	C₂×C₉	—	4	3	no	—
123	C₂×C₉	C₃×C₂²⋊C₈	1	3	yes	—
124	#41	C₄×3₋¹⁺²	1	1	yes	characteristic
125	M₁₆	—	2	1	no	—
126	M₁₆	C₄×3₋¹⁺²	1	1	yes	characteristic
127	C₂×C₈	—	2	1	no	—
128	C₂×C₈	C₂²×3₋¹⁺²	1	1	yes	characteristic
129	C₁₆	—	2	1	no	—
130	C₁₆	—	2	1	no	—
131	C₂²×C₃	—	2	1	no	—
132	C₂²×C₃	—	3	1	no	—
133	C₂²×C₃	—	6	1	no	—
134	C₂²×C₃	C₃²×M₁₆	1	1	yes	characteristic
135	C₃×C₄	—	2	1	no	—
136	C₃×C₄	—	3	1	no	—
137	C₃×C₄	—	3	1	no	—
138	C₃×C₄	—	6	1	no	—
139	C₃×C₄	C₃²×C₂²⋊C₄	1	1	yes	characteristic
140	C₃×C₄	C₂×C₃²×C₈	1	1	yes	characteristic
141	C₃²	C₃×#350	1	1	yes	characteristic
142	C₉	C₃×#350	1	3	yes	complemented
143	Q₈	C₈×3₋¹⁺²	1	1	yes	characteristic
144	D₄	—	2	1	no	—
145	D₄	C₈×3₋¹⁺²	1	1	yes	characteristic
146	C₂×C₄	—	2	1	no	—
147	C₂×C₄	C₂×C₄×3₋¹⁺²	1	1	yes	characteristic
148	C₈	—	2	1	no	—
149	C₈	D₄×3₋¹⁺²	1	1	yes	characteristic
150	C₈	D₄×3₋¹⁺²	1	1	yes	characteristic
151	C₂×C₃	—	2	1	no	—
152	C₂×C₃	—	3	1	no	—
153	C₂×C₃	—	4	1	no	—
154	C₂×C₃	—	6	1	no	—
155	C₂×C₃	—	12	1	no	—
156	C₂×C₃	C₃²×C₂²⋊C₈	1	1	yes	characteristic
157	C₂²	—	2	1	no	—
158	C₂²	M₁₆×3₋¹⁺²	1	1	yes	characteristic
159	C₄	—	2	1	no	—
160	C₄	C₂²⋊C₄×3₋¹⁺²	1	1	yes	characteristic
161	C₄	C₂×C₈×3₋¹⁺²	1	1	yes	characteristic
162	C₃	—	3	1	no	—
163	C₃	C₃²×#350	1	1	yes	characteristic
164	C₂	—	2	1	no	—
165	C₂	—	4	1	no	—
166	C₂	3₋¹⁺²×C₂²⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

3₋¹⁺²×#131

C₂

yes

maximal, characteristic

3₋¹⁺²×M₃₂

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×3₋¹⁺²

C₂

yes

maximal, characteristic, complemented

C₃²×#350

C₃

yes

maximal, characteristic

C₉×#350

C₃

yes

maximal, complemented

#41×3₋¹⁺²

C₄

yes

characteristic

M₁₆×3₋¹⁺²

—

M₁₆×3₋¹⁺²

C₄

yes

characteristic

C₂×C₈×3₋¹⁺²

—

C₂×C₈×3₋¹⁺²

C₂²

yes

characteristic

C₁₆×3₋¹⁺²

—

C₁₆×3₋¹⁺²

—

C₃²×#131

C₂×C₃

yes

characteristic

C₃²×M₃₂

C₂×C₃

yes

characteristic

C₂×C₃²×C₁₆

C₂×C₃

yes

characteristic

C₉×#131

C₂×C₃

yes

—

C₉×M₃₂

C₂×C₃

yes

complemented

C₂×C₉×C₁₆

C₂×C₃

yes

complemented

Q₈×3₋¹⁺²

C₈

yes

characteristic

D₄×3₋¹⁺²

—

D₄×3₋¹⁺²

C₈

yes

characteristic

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

characteristic

C₈×3₋¹⁺²

—

C₈×3₋¹⁺²

D₄

yes

characteristic

C₈×3₋¹⁺²

D₄

yes

characteristic

C₃×#350

—

C₃×#350

C₃²

yes

characteristic

C₃²×#41

C₃×C₄

yes

characteristic

C₃²×M₁₆

—

C₃²×M₁₆

C₃×C₄

yes

characteristic

C₂×C₃²×C₈

—

C₂×C₃²×C₈

C₂²×C₃

yes

characteristic

C₉×#41

C₃×C₄

yes

—

C₃²×C₁₆

—

C₃²×C₁₆

—

C₉×M₁₆

—

C₉×M₁₆

C₃×C₄

yes

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₂²×C₃

yes

—

C₉×C₁₆

—

C₉×C₁₆

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

M₁₆

yes

characteristic

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

C₂²⋊C₄

yes

characteristic

C₄×3₋¹⁺²

C₂×C₈

yes

characteristic

C₃×#131

—

C₃×#131

C₂×C₃²

yes

characteristic

C₃×M₃₂

—

C₃×M₃₂

C₂×C₃²

yes

characteristic

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

C₂×C₃²

yes

characteristic

C₃²×Q₈

C₃×C₈

yes

characteristic

C₃²×D₄

—

C₃²×D₄

C₃×C₈

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×C₃×C₄

yes

characteristic

C₃²×C₈

—

C₃²×C₈

C₃×D₄

yes

characteristic

C₃²×C₈

C₃×D₄

yes

characteristic

Q₈×C₉

C₃×C₈

yes

—

D₄×C₉

—

D₄×C₉

C₃×C₈

yes

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

—

C₈×C₉

—

C₈×C₉

C₃×D₄

yes

—

C₈×C₉

C₃×D₄

yes

—

#350

3₋¹⁺²

yes

characteristic, direct product factor, Sylow

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₂²⋊C₈

yes

characteristic

C₃×#41

—

C₃×#41

C₃²×C₄

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

—

C₃×M₁₆

—

C₃×M₁₆

C₃²×C₄

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²×C₃²

yes

characteristic

C₃×C₁₆

—

C₃×C₁₆

—

C₃×C₁₆

—

C₃×C₁₆

—

C₂²×C₃²

—

C₂²×C₃²

C₃×M₁₆

yes

characteristic

C₃²×C₄

—

C₃²×C₄

C₃×C₂²⋊C₄

yes

characteristic

C₃²×C₄

C₂×C₃×C₈

yes

characteristic

C₂²×C₉

—

C₂²×C₉

C₃×M₁₆

yes

—

C₄×C₉

—

C₄×C₉

C₃×C₂²⋊C₄

yes

—

C₄×C₉

C₂×C₃×C₈

yes

—

#131

C₂×3₋¹⁺²

yes

characteristic

M₃₂

C₂×3₋¹⁺²

yes

characteristic, complemented

100

C₂×C₁₆

C₂×3₋¹⁺²

yes

characteristic, complemented

101

3₋¹⁺²

#350

yes

characteristic, direct product factor, Sylow

102

C₃×Q₈

—

103

C₃×Q₈

C₃²×C₈

yes

characteristic

104

C₃×D₄

—

105

C₃×D₄

—

106

C₃×D₄

—

107

C₃×D₄

C₃²×C₈

yes

characteristic

108

C₂×C₃×C₄

—

109

C₂×C₃×C₄

—

110

C₂×C₃×C₄

—

111

C₂×C₃×C₄

C₂×C₃²×C₄

yes

characteristic

112

C₃×C₈

—

113

C₃×C₈

—

114

C₃×C₈

—

115

C₃×C₈

—

116

C₃×C₈

C₃²×D₄

yes

characteristic

117

C₃×C₈

C₃²×D₄

yes

characteristic

118

C₂×C₃²

—

119

C₂×C₃²

—

120

C₂×C₃²

C₃×C₂²⋊C₈

yes

characteristic

121

C₂×C₉

—

122

C₂×C₉

—

123

C₂×C₉

C₃×C₂²⋊C₈

yes

—

124

#41

C₄×3₋¹⁺²

yes

characteristic

125

M₁₆

—

126

M₁₆

C₄×3₋¹⁺²

yes

characteristic

127

C₂×C₈

—

128

C₂×C₈

C₂²×3₋¹⁺²

yes

characteristic

129

C₁₆

—

130

C₁₆

—

131

C₂²×C₃

—

132

C₂²×C₃

—

133

C₂²×C₃

—

134

C₂²×C₃

C₃²×M₁₆

yes

characteristic

135

C₃×C₄

—

136

C₃×C₄

—

137

C₃×C₄

—

138

C₃×C₄

—

139

C₃×C₄

C₃²×C₂²⋊C₄

yes

characteristic

140

C₃×C₄

C₂×C₃²×C₈

yes

characteristic

141

C₃²

C₃×#350

yes

characteristic

142

C₉

C₃×#350

yes

complemented

143

Q₈

C₈×3₋¹⁺²

yes

characteristic

144

D₄

—

145

D₄

C₈×3₋¹⁺²

yes

characteristic

146

C₂×C₄

—

147

C₂×C₄

C₂×C₄×3₋¹⁺²

yes

characteristic

148

C₈

—

149

C₈

D₄×3₋¹⁺²

yes

characteristic

150

C₈

D₄×3₋¹⁺²

yes

characteristic

151

C₂×C₃

—

152

C₂×C₃

—

153

C₂×C₃

—

154

C₂×C₃

—

155

C₂×C₃

—

156

C₂×C₃

C₃²×C₂²⋊C₈

yes

characteristic

157

C₂²

—

158

C₂²

M₁₆×3₋¹⁺²

yes

characteristic

159

C₄

—

160

C₄

C₂²⋊C₄×3₋¹⁺²

yes

characteristic

161

C₄

C₂×C₈×3₋¹⁺²

yes

characteristic

162

C₃

—

163

C₃

C₃²×#350

yes

characteristic

164

C₂

—

165

C₂

—

166

C₂

3₋¹⁺²×C₂²⋊C₈

yes

characteristic

	subgroup	G/H
center	C₃×C₈	C₃²×D₄
commutator	C₃×C₄	C₂×C₃²×C₈
Frattini, cosocle	C₂×C₃×C₈	C₂²×C₃²
socle	C₂×C₃	C₃²×C₂²⋊C₈
prime radical	C₃²×#41	C₃×C₄
LCS	G ▷ C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₃×C₈ ◁ C₂×C₈×3₋¹⁺² ◁ G

subgroup

G/H

center

C₃×C₈

C₃²×D₄

commutator

C₃×C₄

C₂×C₃²×C₈

Frattini, cosocle

C₂×C₃×C₈

C₂²×C₃²

socle

C₂×C₃

C₃²×C₂²⋊C₈

prime radical

C₃²×#41

C₃×C₄

LCS

G ▷ C₃×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₃×C₈ ◁ C₂×C₈×3₋¹⁺² ◁ G

N	H ≅ G/N	type
2. 3₋¹⁺²×M₃₂	C₂	semidirect
3. C₂×C₁₆×3₋¹⁺²	C₂	semidirect
5. C₉×#350	C₃	semidirect
17. C₉×M₃₂	C₂×C₃	semidirect
18. C₂×C₉×C₁₆	C₂×C₃	semidirect
70. #350	3₋¹⁺²	direct (N × H)
99. M₃₂	C₂×3₋¹⁺²	semidirect
100. C₂×C₁₆	C₂×3₋¹⁺²	semidirect
101. 3₋¹⁺²	#350	direct (N × H)
142. C₉	C₃×#350	semidirect

H ≅ G/N

type

2. 3₋¹⁺²×M₃₂

C₂

semidirect

3. C₂×C₁₆×3₋¹⁺²

C₂

semidirect

5. C₉×#350

C₃

semidirect

17. C₉×M₃₂

C₂×C₃

semidirect

18. C₂×C₉×C₁₆

C₂×C₃

semidirect

70. #350

3₋¹⁺²

direct (N × H)

99. M₃₂

C₂×3₋¹⁺²

semidirect

100. C₂×C₁₆

C₂×3₋¹⁺²

semidirect

101. 3₋¹⁺²

#350

direct (N × H)

142. C₉

C₃×#350

semidirect

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

subgroup	index
3₋¹⁺²×#131	2
3₋¹⁺²×M₃₂	2
C₂×C₁₆×3₋¹⁺²	2
C₃²×#350	3
C₉×#350	3

subgroup

index

3₋¹⁺²×#131

3₋¹⁺²×M₃₂

C₂×C₁₆×3₋¹⁺²

C₃²×#350

C₉×#350

Subgroups — 3₋¹⁺²×#350 #13136053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3−1+2×#350 #13136053

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — 3₋¹⁺²×#350 #13136053