Group #13143610 — subgroups

Subgroups — #13143610

Order 1728.

624 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 164 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×Q₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
2	C₂×(3₋¹⁺²⋊C₄)⋊C₄	C₂	1	3	yes	maximal
3	3₋¹⁺²⋊#95	C₂	1	3	yes	maximal
4	C₃×#5230	—	3	1	no	maximal
5	#10588910	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×3₋¹⁺²	—	2	6	no	—
7	C₂²×C₄×3₋¹⁺²	C₂²	1	3	yes	—
8	C₂²×3₋¹⁺²⋊C₄	C₂²	1	1	yes	characteristic
9	C₂²×3₋¹⁺²⋊C₄	C₂²	1	3	yes	—
10	(3₋¹⁺²⋊C₄)⋊C₄	—	2	6	no	—
11	C₂²×C₃²×Q₈	D₃	1	1	yes	characteristic
12	C₂×C₃×Dic₃⋊C₄	—	3	3	no	—
13	C₂²×Q₈×C₉	—	2	1	no	—
14	C₂²×Q₈×C₉	C₂×C₃	1	1	yes	characteristic
15	C₃×C₃⋊#95	—	3	3	no	—
16	C₂×Dic₉⋊C₄	C₂×C₃	1	3	yes	—
17	C₉⋊#95	C₂×C₃	1	3	yes	—
18	C₂³×3₋¹⁺²	C₂³	1	1	yes	characteristic
19	Q₈×3₋¹⁺²	—	2	8	no	—
20	C₂×C₄×3₋¹⁺²	—	2	6	no	—
21	C₂×C₄×3₋¹⁺²	D₄	1	6	yes	—
22	C₂×3₋¹⁺²⋊C₄	—	2	3	no	—
23	C₂×3₋¹⁺²⋊C₄	—	2	3	no	—
24	C₂×3₋¹⁺²⋊C₄	—	2	3	no	—
25	C₂×3₋¹⁺²⋊C₄	Q₈	1	6	yes	complemented
26	C₃×#395	—	9	1	no	—
27	#5230	—	3	1	no	—
28	C₂×C₃²×Q₈	—	2	6	no	—
29	C₂²×C₃²×C₄	C₂×D₃	1	3	yes	—
30	C₂²×C₃×Dic₃	—	3	3	no	—
31	C₂²×C₃×Dic₃	—	3	1	no	—
32	C₃×Dic₃⋊C₄	—	6	6	no	—
33	C₂×Q₈×C₉	—	2	12	no	—
34	C₂×Q₈×C₉	—	2	6	no	—
35	C₂²×C₄×C₉	—	2	3	no	—
36	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
37	C₂²×Dic₉	C₂²×C₃	1	1	yes	characteristic
38	C₂²×Dic₉	C₂²×C₃	1	3	yes	—
39	Dic₉⋊C₄	—	2	6	no	—
40	C₂²×3₋¹⁺²	C₂×D₄	1	3	yes	—
41	C₂²×3₋¹⁺²	C₂×Q₈	1	3	yes	—
42	C₂²×3₋¹⁺²	#41	1	1	yes	characteristic
43	C₄×3₋¹⁺²	—	2	6	no	—
44	3₋¹⁺²⋊C₄	—	2	6	no	—
45	3₋¹⁺²⋊C₄	—	4	1	no	—
46	C₂²×C₃×Q₈	—	3	1	no	—
47	C₂²×C₃×Q₈	C₃×D₃	1	1	yes	characteristic
48	C₂×C₃×C₄⋊C₄	—	9	3	no	—
49	C₂×Dic₃⋊C₄	—	3	3	no	—
50	C₃×#95	—	9	3	no	—
51	C₃⋊#95	—	3	3	no	—
52	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
53	C₃²×Q₈	—	2	8	no	—
54	C₂×C₃²×C₄	—	2	6	no	—
55	C₂×C₃²×C₄	#67	1	6	yes	—
56	C₂×C₃×Dic₃	—	3	6	no	—
57	C₂×C₃×Dic₃	—	6	3	no	—
58	C₂×C₃×Dic₃	—	6	3	no	—
59	C₂×C₃×Dic₃	—	6	3	no	—
60	C₂³×C₉	—	2	1	no	—
61	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
62	Q₈×C₉	—	2	16	no	—
63	Q₈×C₉	—	2	8	no	—
64	C₂×C₄×C₉	—	2	12	no	—
65	C₂×C₄×C₉	—	2	6	no	—
66	C₂×C₄×C₉	—	2	6	no	—
67	C₂×C₄×C₉	C₃×D₄	1	6	yes	—
68	C₂×Dic₉	—	2	3	no	—
69	C₂×Dic₉	—	2	3	no	—
70	C₂×Dic₉	—	2	3	no	—
71	C₂×Dic₉	C₃×Q₈	1	6	yes	complemented
72	#395	—	9	1	no	Sylow
73	C₂×3₋¹⁺²	#120	1	1	yes	characteristic
74	C₂×3₋¹⁺²	C₂²⋊Q₈	1	3	yes	—
75	C₂×3₋¹⁺²	C₄⋊Q₈	1	3	yes	—
76	C₂×C₃×Q₈	—	2	6	no	—
77	C₂×C₃×Q₈	—	6	6	no	—
78	C₂²×C₃×C₄	—	3	3	no	—
79	C₂²×C₃×C₄	—	9	3	no	—
80	C₂²×C₃×C₄	—	9	1	no	—
81	C₂²×C₃×C₄	C₂×C₃×D₃	1	3	yes	—
82	C₂²×Dic₃	—	3	3	no	—
83	C₂²×Dic₃	—	3	1	no	—
84	C₃×C₄⋊C₄	—	18	6	no	—
85	Dic₃⋊C₄	—	6	6	no	—
86	C₂²×C₃²	D₃×Q₈	1	3	yes	—
87	C₂²×C₃²	#239	1	1	yes	characteristic
88	C₂²×C₃²	C₂×#67	1	3	yes	—
89	C₃²×C₄	—	2	6	no	—
90	C₃×Dic₃	—	6	6	no	—
91	C₃×Dic₃	—	12	1	no	—
92	C₂²×C₉	—	2	3	no	—
93	C₂²×C₉	—	2	1	no	—
94	C₂²×C₉	—	2	3	no	—
95	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
96	C₂²×C₉	C₂×C₃×Q₈	1	3	yes	—
97	C₂²×C₉	C₃×#41	1	1	yes	characteristic
98	C₄×C₉	—	2	12	no	—
99	C₄×C₉	—	2	6	no	—
100	Dic₉	—	2	6	no	—
101	Dic₉	—	4	1	no	—
102	C₂²×Q₈	AGL(1,ℤ/9)	1	1	yes	characteristic
103	C₂×C₄⋊C₄	—	9	3	no	—
104	#95	—	9	3	no	—
105	3₋¹⁺²	#395	1	1	yes	characteristic, complemented, Sylow
106	C₂³×C₃	—	3	1	no	—
107	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
108	C₃×Q₈	—	2	8	no	—
109	C₃×Q₈	—	6	8	no	—
110	C₂×C₃×C₄	—	2	6	no	—
111	C₂×C₃×C₄	—	3	6	no	—
112	C₂×C₃×C₄	—	6	6	no	—
113	C₂×C₃×C₄	—	9	6	no	—
114	C₂×C₃×C₄	—	18	3	no	—
115	C₂×C₃×C₄	—	18	3	no	—
116	C₂×C₃×C₄	—	18	3	no	—
117	C₂×C₃×C₄	C₃×#67	1	6	yes	—
118	C₂×Dic₃	—	3	6	no	—
119	C₂×Dic₃	—	6	3	no	—
120	C₂×Dic₃	—	6	3	no	—
121	C₂×Dic₃	—	6	3	no	—
122	C₂×C₃²	#944	1	3	yes	—
123	C₂×C₃²	#946	1	3	yes	—
124	C₂×C₃²	#953	1	1	yes	characteristic
125	C₂×C₉	—	2	3	no	—
126	C₂×C₉	—	2	3	no	—
127	C₂×C₉	—	2	1	no	—
128	C₂×C₉	C₃×#120	1	1	yes	characteristic
129	C₂×C₉	C₃×C₂²⋊Q₈	1	3	yes	—
130	C₂×C₉	C₃×C₄⋊Q₈	1	3	yes	—
131	C₂×Q₈	—	2	6	no	—
132	C₂²×C₄	—	9	3	no	—
133	C₂²×C₄	—	9	1	no	—
134	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	3	yes	—
135	C₄⋊C₄	—	18	6	no	—
136	C₂²×C₃	—	3	3	no	—
137	C₂²×C₃	—	3	1	no	—
138	C₂²×C₃	—	3	3	no	—
139	C₂²×C₃	C₃×D₃×Q₈	1	3	yes	—
140	C₂²×C₃	C₃×#239	1	1	yes	characteristic
141	C₂²×C₃	C₂×C₃×#67	1	3	yes	—
142	C₃×C₄	—	2	6	no	—
143	C₃×C₄	—	6	6	no	—
144	C₃×C₄	—	18	6	no	—
145	C₃×C₄	—	36	1	no	—
146	Dic₃	—	6	6	no	—
147	Dic₃	—	12	1	no	—
148	C₃²	#5230	1	1	yes	characteristic
149	C₉	—	2	1	no	—
150	C₉	C₃×#395	1	1	yes	characteristic, complemented
151	C₂³	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
152	Q₈	—	2	8	no	—
153	C₂×C₄	—	2	6	no	—
154	C₂×C₄	—	9	6	no	—
155	C₂×C₄	—	18	3	no	—
156	C₂×C₄	—	18	3	no	—
157	C₂×C₄	—	18	3	no	—
158	C₂×C₄	#611⋊C₃	1	6	yes	—
159	C₂×C₃	—	3	3	no	—
160	C₂×C₃	—	3	3	no	—
161	C₂×C₃	—	3	1	no	—
162	C₂×C₃	C₃×#944	1	3	yes	—
163	C₂×C₃	C₃×#946	1	3	yes	—
164	C₂×C₃	C₃×#953	1	1	yes	characteristic
165	C₂²	Q₈×AGL(1,ℤ/9)	1	3	yes	—
166	C₂²	#88724	1	1	yes	characteristic
167	C₂²	C₂×#611⋊C₃	1	3	yes	—
168	C₄	—	2	6	no	—
169	C₄	—	18	6	no	—
170	C₄	—	36	1	no	—
171	C₃	—	3	1	no	—
172	C₃	C₃×#5230	1	1	yes	characteristic
173	C₂	#63506⋊C₃	1	3	yes	—
174	C₂	#63508⋊C₃	1	3	yes	—
175	C₂	#11721704	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₈×3₋¹⁺²

C₂

yes

maximal, characteristic

C₂×(3₋¹⁺²⋊C₄)⋊C₄

C₂

yes

maximal

3₋¹⁺²⋊#95

C₂

yes

—

C₃

yes

maximal, characteristic, complemented

C₂×Q₈×3₋¹⁺²

—

C₂²×C₄×3₋¹⁺²

C₂²

yes

—

C₂²×3₋¹⁺²⋊C₄

C₂²

yes

characteristic

C₂²×3₋¹⁺²⋊C₄

C₂²

yes

—

(3₋¹⁺²⋊C₄)⋊C₄

—

C₂²×C₃²×Q₈

D₃

yes

characteristic

C₂×C₃×Dic₃⋊C₄

—

C₂²×Q₈×C₉

—

C₂²×Q₈×C₉

C₂×C₃

yes

characteristic

C₃×C₃⋊#95

—

C₂×Dic₉⋊C₄

C₂×C₃

yes

—

C₉⋊#95

C₂×C₃

yes

—

C₂³×3₋¹⁺²

C₂³

yes

characteristic

Q₈×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

D₄

yes

—

C₂×3₋¹⁺²⋊C₄

—

C₂×3₋¹⁺²⋊C₄

—

C₂×3₋¹⁺²⋊C₄

—

C₂×3₋¹⁺²⋊C₄

Q₈

yes

complemented

C₃×#395

—

#5230

—

C₂×C₃²×Q₈

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₂²×C₃×Dic₃

—

C₂²×C₃×Dic₃

—

C₃×Dic₃⋊C₄

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₂²×C₃

yes

—

C₂²×Dic₉

C₂²×C₃

yes

characteristic

C₂²×Dic₉

C₂²×C₃

yes

—

Dic₉⋊C₄

—

C₂²×3₋¹⁺²

C₂×D₄

yes

—

C₂²×3₋¹⁺²

C₂×Q₈

yes

—

C₂²×3₋¹⁺²

#41

yes

characteristic

C₄×3₋¹⁺²

—

3₋¹⁺²⋊C₄

—

3₋¹⁺²⋊C₄

—

C₂²×C₃×Q₈

—

C₂²×C₃×Q₈

C₃×D₃

yes

characteristic

C₂×C₃×C₄⋊C₄

—

C₂×Dic₃⋊C₄

—

C₃×#95

—

C₃⋊#95

—

C₂³×C₃²

C₂²×D₃

yes

characteristic

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₂³×C₉

—

C₂³×C₉

C₂³×C₃

yes

characteristic

Q₈×C₉

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₃×D₄

yes

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

—

C₂×Dic₉

C₃×Q₈

yes

—

C₂×3₋¹⁺²

yes

C₂×3₋¹⁺²

C₂²⋊Q₈

yes

—

C₂×3₋¹⁺²

C₄⋊Q₈

yes

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₃×D₃

yes

—

C₂²×Dic₃

—

C₂²×Dic₃

—

C₃×C₄⋊C₄

—

Dic₃⋊C₄

—

C₂²×C₃²

D₃×Q₈

yes

—

C₂²×C₃²

#239

yes

characteristic

C₂²×C₃²

C₂×#67

yes

—

C₃²×C₄

—

C₃×Dic₃

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂×C₃×D₄

yes

—

C₂²×C₉

C₂×C₃×Q₈

yes

—

C₂²×C₉

C₃×#41

yes

characteristic

C₄×C₉

—

C₄×C₉

—

100

Dic₉

—

101

Dic₉

—

102

C₂²×Q₈

AGL(1,ℤ/9)

yes

characteristic

103

C₂×C₄⋊C₄

—

104

#95

—

105

3₋¹⁺²

#395

yes

characteristic, complemented, Sylow

106

C₂³×C₃

—

107

C₂³×C₃

C₂²×C₃×D₃

yes

characteristic

108

C₃×Q₈

—

109

C₃×Q₈

—

110

C₂×C₃×C₄

—

111

C₂×C₃×C₄

—

112

C₂×C₃×C₄

—

113

C₂×C₃×C₄

—

114

C₂×C₃×C₄

—

115

C₂×C₃×C₄

—

116

C₂×C₃×C₄

—

117

C₂×C₃×C₄

C₃×#67

yes

—

118

C₂×Dic₃

—

119

C₂×Dic₃

—

120

C₂×Dic₃

—

121

C₂×Dic₃

—

122

C₂×C₃²

#944

yes

—

123

C₂×C₃²

#946

yes

—

124

C₂×C₃²

#953

yes

characteristic

125

C₂×C₉

—

126

C₂×C₉

—

127

C₂×C₉

—

128

C₂×C₉

C₃×#120

yes

characteristic

129

C₂×C₉

C₃×C₂²⋊Q₈

yes

—

130

C₂×C₉

C₃×C₄⋊Q₈

yes

—

131

C₂×Q₈

—

132

C₂²×C₄

—

133

C₂²×C₄

—

134

C₂²×C₄

C₂×AGL(1,ℤ/9)

yes

—

135

C₄⋊C₄

—

136

C₂²×C₃

—

137

C₂²×C₃

—

138

C₂²×C₃

—

139

C₂²×C₃

C₃×D₃×Q₈

yes

—

140

C₂²×C₃

C₃×#239

yes

characteristic

141

C₂²×C₃

C₂×C₃×#67

yes

—

142

C₃×C₄

—

143

C₃×C₄

—

144

C₃×C₄

—

145

C₃×C₄

—

146

Dic₃

—

147

Dic₃

—

148

C₃²

#5230

yes

characteristic

149

C₉

—

150

C₉

C₃×#395

yes

characteristic, complemented

151

C₂³

C₂²×AGL(1,ℤ/9)

yes

characteristic

152

Q₈

—

153

C₂×C₄

—

154

C₂×C₄

—

155

C₂×C₄

—

156

C₂×C₄

—

157

C₂×C₄

—

158

C₂×C₄

#611⋊C₃

yes

—

159

C₂×C₃

—

160

C₂×C₃

—

161

C₂×C₃

—

162

C₂×C₃

C₃×#944

yes

—

163

C₂×C₃

C₃×#946

yes

—

164

C₂×C₃

C₃×#953

yes

characteristic

165

C₂²

Q₈×AGL(1,ℤ/9)

yes

—

166

C₂²

#88724

yes

characteristic

167

C₂²

C₂×#611⋊C₃

yes

—

168

C₄

—

169

C₄

—

170

C₄

—

171

C₃

—

172

C₃

C₃×#5230

yes

characteristic

173

C₂

#63506⋊C₃

yes

—

174

C₂

#63508⋊C₃

yes

—

175

C₂

#11721704

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²×AGL(1,ℤ/9)
commutator, cosocle	C₂³×C₉	C₂³×C₃
Frattini, socle	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₂²×Q₈×3₋¹⁺²	C₂
prime radical	C₂³×C₃²	C₂²×D₃
hypercenter	C₂²×Q₈	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#395
LCS	G ▷ C₂³×C₉ ▷ C₉
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

subgroup

G/H

center

C₂³

C₂²×AGL(1,ℤ/9)

commutator, cosocle

C₂³×C₉

C₂³×C₃

Frattini, socle

C₂³×C₃

C₂²×C₃×D₃

Fitting

C₂²×Q₈×3₋¹⁺²

C₂

prime radical

C₂³×C₃²

C₂²×D₃

C₂²×Q₈

C₉

G ▷ C₂³×C₉ ▷ C₉

C₁ ◁ C₂³ ◁ C₂²×Q₈

N	H ≅ G/N	type
5. #10588910	C₃	semidirect
25. C₂×3₋¹⁺²⋊C₄	Q₈	semidirect
71. C₂×Dic₉	C₃×Q₈	semidirect
105. 3₋¹⁺²	#395	semidirect
150. C₉	C₃×#395	semidirect

H ≅ G/N

type

5. #10588910

C₃

semidirect

25. C₂×3₋¹⁺²⋊C₄

Q₈

semidirect

71. C₂×Dic₉

C₃×Q₈

semidirect

105. 3₋¹⁺²

#395

semidirect

150. C₉

C₃×#395

semidirect

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

subgroup	index
C₂²×Q₈×3₋¹⁺²	2
C₂×(3₋¹⁺²⋊C₄)⋊C₄	2
3₋¹⁺²⋊#95	2
C₃×#5230	3
#10588910	3

subgroup

index

C₂²×Q₈×3₋¹⁺²

C₂×(3₋¹⁺²⋊C₄)⋊C₄

3₋¹⁺²⋊#95

C₃×#5230

#10588910

Subgroups — #13143610

Conjugacy classes of subgroups

Complements

Maximal subgroups