Group #13153669 — subgroups

Subgroups — #13153669

Order 1728.

876 conjugacy classes of subgroups; only the 452 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 452 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	Q₈×3₋¹⁺²⋊C₄	C₂	1	8	yes	maximal, complemented
3	#63488⋊C₃	C₂	1	6	yes	maximal
4	C₃×#5812	—	3	1	no	maximal
5	#10589914	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂×Q₈×3₋¹⁺²	C₄	1	8	yes	complemented
7	C₂×Q₈×3₋¹⁺²	C₂²	1	4	yes	—
8	C₂²×C₄×3₋¹⁺²	C₂²	1	3	yes	—
9	#3676⋊C₃	C₂²	1	4	yes	—
10	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	12	yes	—
11	C₄×3₋¹⁺²⋊C₄	C₂²	1	12	yes	—
12	C₂²×C₃²×Q₈	D₃	1	1	yes	characteristic
13	C₃×Q₈×Dic₃	—	3	8	no	—
14	C₃×#926	—	3	6	no	—
15	C₂²×Q₈×C₉	—	2	1	no	—
16	C₂²×Q₈×C₉	C₂×C₃	1	1	yes	characteristic
17	Q₈×Dic₉	C₂×C₃	1	8	yes	complemented
18	#63488	C₂×C₃	1	6	yes	—
19	C₂³×3₋¹⁺²	C₂³	1	1	yes	characteristic
20	Q₈×3₋¹⁺²	C₂×C₄	1	16	yes	—
21	C₂×C₄×3₋¹⁺²	C₂×C₄	1	12	yes	—
22	C₂×C₄×3₋¹⁺²	C₂³	1	6	yes	—
23	C₂×3₋¹⁺²⋊C₄	C₂³	1	8	yes	—
24	C₃×#519	—	9	1	no	—
25	#5812	—	3	1	no	—
26	C₂×C₃²×Q₈	Dic₃	1	8	yes	—
27	C₂×C₃²×Q₈	C₂×D₃	1	4	yes	—
28	C₂²×C₃²×C₄	C₂×D₃	1	3	yes	—
29	C₃×#217	—	3	4	no	—
30	C₃×C₄⋊Dic₃	—	3	12	no	—
31	C₃×C₄×Dic₃	—	3	12	no	—
32	C₂×Q₈×C₉	—	2	8	no	—
33	C₂×Q₈×C₉	—	2	4	no	—
34	C₂×Q₈×C₉	C₃×C₄	1	8	yes	complemented
35	C₂×Q₈×C₉	C₂²×C₃	1	4	yes	—
36	C₂²×C₄×C₉	—	2	3	no	—
37	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
38	#3676	C₂²×C₃	1	4	yes	—
39	C₄⋊Dic₉	C₂²×C₃	1	12	yes	—
40	C₄×Dic₉	C₂²×C₃	1	12	yes	—
41	C₂²×3₋¹⁺²	—	2	2	no	—
42	C₂²×3₋¹⁺²	C₂²×C₄	1	2	yes	—
43	C₂²×3₋¹⁺²	C₂⁴	1	1	yes	characteristic
44	C₄×3₋¹⁺²	C₂²×C₄	1	12	yes	—
45	3₋¹⁺²⋊C₄	—	2	8	no	—
46	C₂²×C₃×Q₈	—	3	1	no	—
47	C₂²×C₃×Q₈	C₃×D₃	1	1	yes	characteristic
48	C₃×C₄×Q₈	—	9	8	no	—
49	C₃×#117	—	9	6	no	—
50	Q₈×Dic₃	—	3	8	no	—
51	#926	—	3	6	no	—
52	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
53	C₃²×Q₈	C₂×Dic₃	1	16	yes	—
54	C₂×C₃²×C₄	C₂×Dic₃	1	12	yes	—
55	C₂×C₃²×C₄	C₂²×D₃	1	6	yes	—
56	C₂×C₃×Dic₃	—	3	8	no	—
57	C₂³×C₉	—	2	1	no	—
58	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
59	Q₈×C₉	—	2	16	no	—
60	Q₈×C₉	C₂×C₃×C₄	1	16	yes	—
61	C₂×C₄×C₉	—	2	12	no	—
62	C₂×C₄×C₉	—	2	6	no	—
63	C₂×C₄×C₉	C₂×C₃×C₄	1	12	yes	—
64	C₂×C₄×C₉	C₂³×C₃	1	6	yes	—
65	C₂×Dic₉	C₂³×C₃	1	8	yes	—
66	#519	—	9	1	no	Sylow
67	C₂×3₋¹⁺²	—	2	2	no	—
68	C₂×3₋¹⁺²	C₂³×C₄	1	1	yes	characteristic
69	C₂×3₋¹⁺²	2₋¹⁺⁴	1	2	yes	—
70	C₂×C₃×Q₈	—	3	8	no	—
71	C₂×C₃×Q₈	—	3	4	no	—
72	C₂×C₃×Q₈	C₃×Dic₃	1	8	yes	—
73	C₂×C₃×Q₈	C₂×C₃×D₃	1	4	yes	—
74	C₂²×C₃×C₄	—	3	3	no	—
75	C₂²×C₃×C₄	C₂×C₃×D₃	1	3	yes	—
76	C₃×C₄⋊C₄	—	9	12	no	—
77	C₃×C₂²⋊C₄	—	9	4	no	—
78	C₃×C₄²	—	9	12	no	—
79	#217	—	3	4	no	—
80	C₄⋊Dic₃	—	3	12	no	—
81	C₄×Dic₃	—	3	12	no	—
82	C₂²×C₃²	—	2	2	no	—
83	C₂²×C₃²	C₂²×Dic₃	1	2	yes	—
84	C₂²×C₃²	C₂³×D₃	1	1	yes	characteristic
85	C₃²×C₄	C₂²×Dic₃	1	12	yes	—
86	C₃×Dic₃	—	6	8	no	—
87	C₂²×C₉	—	2	4	no	—
88	C₂²×C₉	—	2	2	no	—
89	C₂²×C₉	—	2	1	no	—
90	C₂²×C₉	—	2	2	no	—
91	C₂²×C₉	C₂²×C₃×C₄	1	2	yes	—
92	C₂²×C₉	C₂⁴×C₃	1	1	yes	characteristic
93	C₄×C₉	—	2	12	no	—
94	C₄×C₉	C₂²×C₃×C₄	1	12	yes	—
95	Dic₉	—	2	8	no	—
96	C₂²×Q₈	AGL(1,ℤ/9)	1	1	yes	characteristic
97	C₄×Q₈	—	9	8	no	—
98	#117	—	9	6	no	—
99	3₋¹⁺²	#519	1	1	yes	characteristic, complemented, Sylow
100	C₂³×C₃	—	3	1	no	—
101	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
102	C₃×Q₈	—	3	16	no	—
103	C₃×Q₈	C₂×C₃×Dic₃	1	16	yes	—
104	C₂×C₃×C₄	—	3	12	no	—
105	C₂×C₃×C₄	—	3	6	no	—
106	C₂×C₃×C₄	—	9	8	no	—
107	C₂×C₃×C₄	C₂×C₃×Dic₃	1	12	yes	—
108	C₂×C₃×C₄	C₂²×C₃×D₃	1	6	yes	—
109	C₂×Dic₃	—	3	8	no	—
110	C₂×C₃²	—	2	2	no	—
111	C₂×C₃²	#1007	1	2	yes	—
112	C₂×C₃²	C₂³×Dic₃	1	1	yes	characteristic
113	C₂×C₉	—	2	4	no	—
114	C₂×C₉	—	2	2	no	—
115	C₂×C₉	—	2	2	no	—
116	C₂×C₉	—	2	1	no	—
117	C₂×C₉	C₂³×C₃×C₄	1	1	yes	characteristic
118	C₂×C₉	C₃×2₋¹⁺⁴	1	2	yes	—
119	C₂×Q₈	3₋¹⁺²⋊C₄	1	8	yes	complemented
120	C₂×Q₈	C₂×AGL(1,ℤ/9)	1	4	yes	—
121	C₂²×C₄	C₂×AGL(1,ℤ/9)	1	3	yes	—
122	C₄⋊C₄	—	9	12	no	—
123	C₂²⋊C₄	—	9	4	no	—
124	C₄²	—	9	12	no	—
125	C₂²×C₃	—	2	2	no	—
126	C₂²×C₃	—	3	1	no	—
127	C₂²×C₃	—	3	2	no	—
128	C₂²×C₃	—	6	2	no	—
129	C₂²×C₃	C₂²×C₃×Dic₃	1	2	yes	—
130	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
131	C₃×C₄	—	3	12	no	—
132	C₃×C₄	—	18	8	no	—
133	C₃×C₄	C₂²×C₃×Dic₃	1	12	yes	—
134	Dic₃	—	6	8	no	—
135	C₃²	#5812	1	1	yes	characteristic
136	C₉	—	2	1	no	—
137	C₉	C₃×#519	1	1	yes	characteristic, complemented
138	C₂³	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
139	Q₈	C₂×3₋¹⁺²⋊C₄	1	16	yes	—
140	C₂×C₄	—	9	8	no	—
141	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	12	yes	—
142	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	6	yes	—
143	C₂×C₃	—	2	2	no	—
144	C₂×C₃	—	3	2	no	—
145	C₂×C₃	—	3	1	no	—
146	C₂×C₃	—	6	2	no	—
147	C₂×C₃	C₃×#1007	1	2	yes	—
148	C₂×C₃	C₂³×C₃×Dic₃	1	1	yes	characteristic
149	C₂²	—	2	2	no	—
150	C₂²	C₂²×3₋¹⁺²⋊C₄	1	2	yes	—
151	C₂²	C₂³×AGL(1,ℤ/9)	1	1	yes	characteristic
152	C₄	—	18	8	no	—
153	C₄	C₂²×3₋¹⁺²⋊C₄	1	12	yes	—
154	C₃	—	3	1	no	—
155	C₃	C₃×#5812	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#11722630	1	2	yes	—
158	C₂	C₂³×3₋¹⁺²⋊C₄	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Q₈×3₋¹⁺²

C₂

yes

maximal, characteristic

Q₈×3₋¹⁺²⋊C₄

C₂

yes

maximal, complemented

#63488⋊C₃

C₂

yes

—

C₃

yes

maximal, characteristic, complemented

C₂×Q₈×3₋¹⁺²

C₄

yes

complemented

C₂×Q₈×3₋¹⁺²

C₂²

yes

—

C₂²×C₄×3₋¹⁺²

C₂²

yes

—

#3676⋊C₃

C₂²

yes

—

C₄⋊(3₋¹⁺²⋊C₄)

C₂²

yes

—

C₄×3₋¹⁺²⋊C₄

C₂²

yes

—

C₂²×C₃²×Q₈

D₃

yes

characteristic

C₃×Q₈×Dic₃

—

C₃×#926

—

C₂²×Q₈×C₉

—

C₂²×Q₈×C₉

C₂×C₃

yes

characteristic

Q₈×Dic₉

C₂×C₃

yes

complemented

#63488

C₂×C₃

yes

—

C₂³×3₋¹⁺²

C₂³

yes

characteristic

Q₈×3₋¹⁺²

C₂×C₄

yes

—

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

—

C₂×C₄×3₋¹⁺²

C₂³

yes

—

C₂×3₋¹⁺²⋊C₄

C₂³

yes

—

C₃×#519

—

#5812

—

C₂×C₃²×Q₈

Dic₃

yes

—

C₂×C₃²×Q₈

C₂×D₃

yes

—

C₂²×C₃²×C₄

C₂×D₃

yes

—

C₃×#217

—

C₃×C₄⋊Dic₃

—

C₃×C₄×Dic₃

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

C₃×C₄

yes

complemented

C₂×Q₈×C₉

C₂²×C₃

yes

—

C₂²×C₄×C₉

—

C₂²×C₄×C₉

C₂²×C₃

yes

—

#3676

C₂²×C₃

yes

—

C₄⋊Dic₉

C₂²×C₃

yes

—

C₄×Dic₉

C₂²×C₃

yes

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

C₂²×C₄

yes

—

C₂²×3₋¹⁺²

C₂⁴

yes

characteristic

C₄×3₋¹⁺²

C₂²×C₄

yes

—

3₋¹⁺²⋊C₄

—

C₂²×C₃×Q₈

—

C₂²×C₃×Q₈

C₃×D₃

yes

characteristic

C₃×C₄×Q₈

—

C₃×#117

—

Q₈×Dic₃

—

#926

—

C₂³×C₃²

C₂²×D₃

yes

characteristic

C₃²×Q₈

C₂×Dic₃

yes

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃×Dic₃

—

C₂³×C₉

—

C₂³×C₉

C₂³×C₃

yes

characteristic

Q₈×C₉

—

Q₈×C₉

C₂×C₃×C₄

yes

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

—

C₂×C₄×C₉

C₂³×C₃

yes

—

C₂×Dic₉

C₂³×C₃

yes

—

#519

—

Sylow

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₂³×C₄

yes

characteristic

C₂×3₋¹⁺²

2₋¹⁺⁴

yes

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₃×Dic₃

yes

—

C₂×C₃×Q₈

C₂×C₃×D₃

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₃×D₃

yes

—

C₃×C₄⋊C₄

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

#217

—

C₄⋊Dic₃

—

C₄×Dic₃

—

C₂²×C₃²

—

C₂²×C₃²

C₂²×Dic₃

yes

—

C₂²×C₃²

C₂³×D₃

yes

characteristic

C₃²×C₄

C₂²×Dic₃

yes

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂²×C₃×C₄

yes

—

C₂²×C₉

C₂⁴×C₃

yes

characteristic

C₄×C₉

—

C₄×C₉

C₂²×C₃×C₄

yes

—

Dic₉

—

C₂²×Q₈

AGL(1,ℤ/9)

yes

characteristic

C₄×Q₈

—

#117

—

3₋¹⁺²

#519

yes

characteristic, complemented, Sylow

100

C₂³×C₃

—

101

C₂³×C₃

C₂²×C₃×D₃

yes

characteristic

102

C₃×Q₈

—

103

C₃×Q₈

C₂×C₃×Dic₃

yes

—

104

C₂×C₃×C₄

—

105

C₂×C₃×C₄

—

106

C₂×C₃×C₄

—

107

C₂×C₃×C₄

C₂×C₃×Dic₃

yes

—

108

C₂×C₃×C₄

C₂²×C₃×D₃

yes

—

109

C₂×Dic₃

—

110

C₂×C₃²

—

111

C₂×C₃²

#1007

yes

—

112

C₂×C₃²

C₂³×Dic₃

yes

characteristic

113

C₂×C₉

—

114

C₂×C₉

—

115

C₂×C₉

—

116

C₂×C₉

—

117

C₂×C₉

C₂³×C₃×C₄

yes

characteristic

118

C₂×C₉

C₃×2₋¹⁺⁴

yes

—

119

C₂×Q₈

3₋¹⁺²⋊C₄

yes

complemented

120

C₂×Q₈

C₂×AGL(1,ℤ/9)

yes

—

121

C₂²×C₄

C₂×AGL(1,ℤ/9)

yes

—

122

C₄⋊C₄

—

123

C₂²⋊C₄

—

124

C₄²

—

125

C₂²×C₃

—

126

C₂²×C₃

—

127

C₂²×C₃

—

128

C₂²×C₃

—

129

C₂²×C₃

C₂²×C₃×Dic₃

yes

—

130

C₂²×C₃

C₂³×C₃×D₃

yes

characteristic

131

C₃×C₄

—

132

C₃×C₄

—

133

C₃×C₄

C₂²×C₃×Dic₃

yes

—

134

Dic₃

—

135

C₃²

#5812

yes

characteristic

136

C₉

—

137

C₉

C₃×#519

yes

characteristic, complemented

138

C₂³

C₂²×AGL(1,ℤ/9)

yes

characteristic

139

Q₈

C₂×3₋¹⁺²⋊C₄

yes

—

140

C₂×C₄

—

141

C₂×C₄

C₂×3₋¹⁺²⋊C₄

yes

—

142

C₂×C₄

C₂²×AGL(1,ℤ/9)

yes

—

143

C₂×C₃

—

144

C₂×C₃

—

145

C₂×C₃

—

146

C₂×C₃

—

147

C₂×C₃

C₃×#1007

yes

—

148

C₂×C₃

C₂³×C₃×Dic₃

yes

characteristic

149

C₂²

—

150

C₂²

C₂²×3₋¹⁺²⋊C₄

yes

—

151

C₂²

C₂³×AGL(1,ℤ/9)

yes

characteristic

152

C₄

—

153

C₄

C₂²×3₋¹⁺²⋊C₄

yes

—

154

C₃

—

155

C₃

C₃×#5812

yes

characteristic

156

C₂

—

157

C₂

#11722630

yes

—

158

C₂

C₂³×3₋¹⁺²⋊C₄

yes

characteristic

	subgroup	G/H
center	C₂²	C₂³×AGL(1,ℤ/9)
commutator	C₂×C₉	C₂³×C₃×C₄
Frattini, socle	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₂²×Q₈×3₋¹⁺²	C₂
prime radical	C₂³×C₃²	C₂²×D₃
cosocle	C₂²×C₉	C₂⁴×C₃
hypercenter	C₂²×Q₈	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#519
LCS	G ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

subgroup

G/H

center

C₂²

C₂³×AGL(1,ℤ/9)

commutator

C₂×C₉

C₂³×C₃×C₄

Frattini, socle

C₂²×C₃

C₂³×C₃×D₃

Fitting

C₂²×Q₈×3₋¹⁺²

C₂

prime radical

C₂³×C₃²

C₂²×D₃

cosocle

C₂²×C₉

C₂⁴×C₃

C₂²×Q₈

C₉

G ▷ C₂×C₉ ▷ C₉

C₁ ◁ C₂² ◁ C₂²×Q₈

N	H ≅ G/N	type
2. Q₈×3₋¹⁺²⋊C₄	C₂	semidirect
5. #10589914	C₃	semidirect
6. C₂×Q₈×3₋¹⁺²	C₄	semidirect
17. Q₈×Dic₉	C₂×C₃	semidirect
34. C₂×Q₈×C₉	C₃×C₄	semidirect
99. 3₋¹⁺²	#519	semidirect
119. C₂×Q₈	3₋¹⁺²⋊C₄	semidirect
137. C₉	C₃×#519	semidirect

H ≅ G/N

type

2. Q₈×3₋¹⁺²⋊C₄

C₂

semidirect

5. #10589914

C₃

semidirect

6. C₂×Q₈×3₋¹⁺²

C₄

semidirect

17. Q₈×Dic₉

C₂×C₃

semidirect

34. C₂×Q₈×C₉

C₃×C₄

semidirect

99. 3₋¹⁺²

#519

semidirect

119. C₂×Q₈

3₋¹⁺²⋊C₄

semidirect

137. C₉

C₃×#519

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
Q₈×3₋¹⁺²⋊C₄	2
#63488⋊C₃	2
C₃×#5812	3
#10589914	3

subgroup

index

C₂²×Q₈×3₋¹⁺²

Q₈×3₋¹⁺²⋊C₄

#63488⋊C₃

C₃×#5812

#10589914

Subgroups — #13153669

Conjugacy classes of subgroups

Complements

Maximal subgroups