Group #11721676 — subgroups

Subgroups — #63505⋊C₃ #11721676

Order 864.

199 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 199 conjugacy classes of subgroups — too many to draw. Only its 67 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	C₂×Q₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic, complemented
2	#3697⋊C₃	C₂	1	1	yes	maximal, characteristic
3	#3675⋊C₃	C₂	1	2	yes	maximal, complemented
4	#3674⋊C₃	C₂	1	2	yes	maximal, complemented
5	#3667⋊C₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#942	—	3	1	no	maximal
7	#63505	C₃	1	1	yes	maximal, characteristic, complemented
8	Q₈×3₋¹⁺²	—	2	1	no	—
9	Q₈×3₋¹⁺²	C₂²	1	2	yes	—
10	C₂×C₄×3₋¹⁺²	—	2	1	no	—
11	C₂×C₄×3₋¹⁺²	C₂²	1	1	yes	characteristic
12	#611⋊C₃	—	2	1	no	—
13	C₄⋊AGL(1,ℤ/9)	C₂²	1	1	yes	characteristic
14	C₄×AGL(1,ℤ/9)	—	2	1	no	—
15	3₋¹⁺²⋊Q₈	C₂²	1	1	yes	characteristic
16	3₋¹⁺²⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃²×Q₈	D₃	1	1	yes	characteristic
18	C₃×#235	—	3	1	no	—
19	C₃×#216	—	3	2	no	—
20	C₃×#215	—	3	2	no	—
21	C₃×#208	—	3	1	no	—
22	C₂×Q₈×C₉	—	2	1	no	—
23	C₂×Q₈×C₉	C₂×C₃	1	1	yes	characteristic, complemented
24	#3697	C₂×C₃	1	1	yes	characteristic
25	#3675	C₂×C₃	1	2	yes	complemented
26	#3674	C₂×C₃	1	2	yes	complemented
27	#3667	C₂×C₃	1	1	yes	characteristic, complemented
28	C₂²×3₋¹⁺²	D₄	1	1	yes	characteristic
29	C₂×AGL(1,ℤ/9)	—	2	1	no	—
30	C₄×3₋¹⁺²	—	2	2	no	—
31	C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
32	C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
33	3₋¹⁺²⋊C₄	—	2	1	no	—
34	C₃×#137	—	9	1	no	—
35	#942	—	3	1	no	—
36	C₃²×Q₈	—	2	1	no	—
37	C₃²×Q₈	C₂×D₃	1	2	yes	—
38	C₂×C₃²×C₄	—	2	1	no	—
39	C₂×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
40	C₃×#67	—	6	1	no	—
41	C₃×D₁₂	—	3	1	no	—
42	C₃×C₄×D₃	—	6	1	no	—
43	C₃×Dic₆	—	3	1	no	—
44	C₃×C₃⋊C₈	—	3	2	no	—
45	Q₈×C₉	—	2	2	no	—
46	Q₈×C₉	—	2	1	no	—
47	Q₈×C₉	—	2	2	no	—
48	Q₈×C₉	C₂²×C₃	1	2	yes	—
49	C₂×C₄×C₉	—	2	2	no	—
50	C₂×C₄×C₉	—	2	1	no	—
51	C₂×C₄×C₉	—	2	1	no	—
52	C₂×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
53	#611	—	2	1	no	—
54	D₃₆	C₂²×C₃	1	1	yes	characteristic
55	C₄×D₉	—	2	1	no	—
56	Dic₁₈	C₂²×C₃	1	1	yes	characteristic
57	C₉⋊C₈	C₂²×C₃	1	2	yes	—
58	C₂×3₋¹⁺²	—	2	1	no	—
59	C₂×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
60	AGL(1,ℤ/9)	—	4	1	no	—
61	C₃×#41	—	9	1	no	—
62	C₂×C₃×Q₈	—	3	1	no	—
63	C₂×C₃×Q₈	C₃×D₃	1	1	yes	characteristic
64	#235	—	3	1	no	—
65	C₃×Q₁₆	—	9	2	no	—
66	C₃×SD₁₆	—	9	2	no	—
67	C₃×M₁₆	—	9	1	no	—
68	#216	—	3	2	no	—
69	#215	—	3	2	no	—
70	#208	—	3	1	no	—
71	C₂²×C₃²	#67	1	1	yes	characteristic
72	C₂×C₃×D₃	—	6	1	no	—
73	C₃²×C₄	—	2	2	no	—
74	C₃²×C₄	#67	1	1	yes	characteristic
75	C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
76	C₃×Dic₃	—	6	1	no	—
77	C₂²×C₉	—	2	1	no	—
78	C₂²×C₉	C₃×D₄	1	1	yes	characteristic
79	C₂×D₉	—	2	1	no	—
80	C₄×C₉	—	2	4	no	—
81	C₄×C₉	—	2	2	no	—
82	C₄×C₉	—	2	1	no	—
83	C₄×C₉	—	2	1	no	—
84	C₄×C₉	C₃×D₄	1	1	yes	characteristic
85	C₄×C₉	C₂³×C₃	1	1	yes	characteristic
86	Dic₉	—	2	1	no	—
87	#137	—	9	1	no	Sylow
88	3₋¹⁺²	#137	1	1	yes	characteristic, complemented, Sylow
89	C₃×Q₈	—	2	1	no	—
90	C₃×Q₈	—	3	2	no	—
91	C₃×Q₈	—	6	1	no	—
92	C₃×Q₈	—	9	1	no	—
93	C₃×Q₈	C₂×C₃×D₃	1	2	yes	—
94	C₃×D₄	—	9	1	no	—
95	C₃×D₄	—	18	1	no	—
96	C₂×C₃×C₄	—	2	1	no	—
97	C₂×C₃×C₄	—	3	1	no	—
98	C₂×C₃×C₄	—	6	1	no	—
99	C₂×C₃×C₄	—	18	1	no	—
100	C₂×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
101	#67	—	6	1	no	—
102	D₁₂	—	3	1	no	—
103	C₄×D₃	—	6	1	no	—
104	Dic₆	—	3	1	no	—
105	C₃×C₈	—	9	2	no	—
106	C₃⋊C₈	—	3	2	no	—
107	C₂×C₃²	—	2	1	no	—
108	C₂×C₃²	C₂×#67	1	1	yes	characteristic
109	C₃×D₃	—	12	1	no	—
110	C₂×C₉	—	2	2	no	—
111	C₂×C₉	—	2	1	no	—
112	C₂×C₉	—	2	1	no	—
113	C₂×C₉	C₂×C₃×D₄	1	1	yes	characteristic
114	D₉	—	4	1	no	—
115	#41	—	9	1	no	—
116	C₂×Q₈	AGL(1,ℤ/9)	1	1	yes	characteristic, complemented
117	Q₁₆	—	9	2	no	—
118	SD₁₆	—	9	2	no	—
119	M₁₆	—	9	1	no	—
120	C₂²×C₃	—	3	1	no	—
121	C₂²×C₃	—	18	1	no	—
122	C₂²×C₃	C₃×#67	1	1	yes	characteristic
123	C₂×D₃	—	6	1	no	—
124	C₃×C₄	—	2	2	no	—
125	C₃×C₄	—	3	1	no	—
126	C₃×C₄	—	3	1	no	—
127	C₃×C₄	—	6	2	no	—
128	C₃×C₄	—	18	1	no	—
129	C₃×C₄	C₃×#67	1	1	yes	characteristic
130	C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
131	Dic₃	—	6	1	no	—
132	C₃²	#942	1	1	yes	characteristic
133	C₉	—	2	1	no	—
134	C₉	C₃×#137	1	1	yes	characteristic, complemented
135	Q₈	—	2	1	no	—
136	Q₈	—	9	1	no	—
137	Q₈	C₂×AGL(1,ℤ/9)	1	2	yes	—
138	D₄	—	9	1	no	—
139	D₄	—	18	1	no	—
140	C₂×C₄	—	2	1	no	—
141	C₂×C₄	—	18	1	no	—
142	C₂×C₄	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
143	C₈	—	9	2	no	—
144	C₂×C₃	—	2	1	no	—
145	C₂×C₃	—	3	1	no	—
146	C₂×C₃	—	6	1	no	—
147	C₂×C₃	—	36	1	no	—
148	C₂×C₃	C₂×C₃×#67	1	1	yes	characteristic
149	D₃	—	12	1	no	—
150	C₂²	—	18	1	no	—
151	C₂²	#611⋊C₃	1	1	yes	characteristic
152	C₄	—	2	2	no	—
153	C₄	—	18	1	no	—
154	C₄	#611⋊C₃	1	1	yes	characteristic
155	C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
156	C₃	—	3	1	no	—
157	C₃	C₃×#942	1	1	yes	characteristic
158	C₂	—	2	1	no	—
159	C₂	—	36	1	no	—
160	C₂	C₂×#611⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×3₋¹⁺²

C₂

yes

maximal, characteristic, complemented

#3697⋊C₃

C₂

yes

maximal, characteristic

#3675⋊C₃

C₂

yes

maximal, complemented

#3674⋊C₃

C₂

yes

maximal, complemented

#3667⋊C₃

C₂

yes

maximal, characteristic, complemented

C₃×#942

—

maximal

#63505

C₃

yes

maximal, characteristic, complemented

Q₈×3₋¹⁺²

—

Q₈×3₋¹⁺²

C₂²

yes

—

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

C₂²

yes

characteristic

#611⋊C₃

—

C₄⋊AGL(1,ℤ/9)

C₂²

yes

characteristic

C₄×AGL(1,ℤ/9)

—

3₋¹⁺²⋊Q₈

C₂²

yes

characteristic

3₋¹⁺²⋊C₈

C₂²

yes

—

C₂×C₃²×Q₈

D₃

yes

—

—

—

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

C₂×C₃

yes

characteristic, complemented

#3697

C₂×C₃

yes

characteristic

#3675

C₂×C₃

yes

complemented

#3674

C₂×C₃

yes

complemented

#3667

C₂×C₃

yes

characteristic, complemented

C₂²×3₋¹⁺²

D₄

yes

characteristic

C₂×AGL(1,ℤ/9)

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

D₄

yes

characteristic

C₄×3₋¹⁺²

C₂³

yes

characteristic

3₋¹⁺²⋊C₄

—

C₃×#137

—

#942

—

C₃²×Q₈

—

C₃²×Q₈

C₂×D₃

yes

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×D₃

yes

characteristic

C₃×#67

—

C₃×D₁₂

—

C₃×C₄×D₃

—

C₃×Dic₆

—

C₃×C₃⋊C₈

—

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

C₂²×C₃

yes

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂²×C₃

yes

characteristic

#611

—

D₃₆

C₂²×C₃

yes

characteristic

C₄×D₉

—

Dic₁₈

C₂²×C₃

yes

characteristic

C₉⋊C₈

C₂²×C₃

yes

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₂×D₄

yes

characteristic

AGL(1,ℤ/9)

—

C₃×#41

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₃×D₃

yes

characteristic

#235

—

C₃×Q₁₆

—

C₃×SD₁₆

—

C₃×M₁₆

—

#216

—

#215

—

#208

—

C₂²×C₃²

#67

yes

characteristic

C₂×C₃×D₃

—

C₃²×C₄

—

C₃²×C₄

#67

yes

characteristic

C₃²×C₄

C₂²×D₃

yes

characteristic

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

C₃×D₄

yes

characteristic

C₂×D₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₃×D₄

yes

characteristic

C₄×C₉

C₂³×C₃

yes

characteristic

Dic₉

—

#137

—

Sylow

3₋¹⁺²

#137

yes

characteristic, complemented, Sylow

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

C₂×C₃×D₃

yes

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

100

C₂×C₃×C₄

C₂×C₃×D₃

yes

characteristic

101

#67

—

102

D₁₂

—

103

C₄×D₃

—

104

Dic₆

—

105

C₃×C₈

—

106

C₃⋊C₈

—

107

C₂×C₃²

—

108

C₂×C₃²

C₂×#67

yes

characteristic

109

C₃×D₃

—

110

C₂×C₉

—

111

C₂×C₉

—

112

C₂×C₉

—

113

C₂×C₉

C₂×C₃×D₄

yes

characteristic

114

D₉

—

115

#41

—

116

C₂×Q₈

AGL(1,ℤ/9)

yes

characteristic, complemented

117

Q₁₆

—

118

SD₁₆

—

119

M₁₆

—

120

C₂²×C₃

—

121

C₂²×C₃

—

122

C₂²×C₃

C₃×#67

yes

characteristic

123

C₂×D₃

—

124

C₃×C₄

—

125

C₃×C₄

—

126

C₃×C₄

—

127

C₃×C₄

—

128

C₃×C₄

—

129

C₃×C₄

C₃×#67

yes

characteristic

130

C₃×C₄

C₂²×C₃×D₃

yes

characteristic

131

Dic₃

—

132

C₃²

#942

yes

characteristic

133

C₉

—

134

C₉

C₃×#137

yes

characteristic, complemented

135

Q₈

—

136

Q₈

—

137

Q₈

C₂×AGL(1,ℤ/9)

yes

—

138

D₄

—

139

D₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

C₂×AGL(1,ℤ/9)

yes

characteristic

143

C₈

—

144

C₂×C₃

—

145

C₂×C₃

—

146

C₂×C₃

—

147

C₂×C₃

—

148

C₂×C₃

C₂×C₃×#67

yes

characteristic

149

D₃

—

150

C₂²

—

151

C₂²

#611⋊C₃

yes

characteristic

152

C₄

—

153

C₄

—

154

C₄

#611⋊C₃

yes

characteristic

155

C₄

C₂²×AGL(1,ℤ/9)

yes

characteristic

156

C₃

—

157

C₃

C₃×#942

yes

characteristic

158

C₂

—

159

C₂

—

160

C₂

C₂×#611⋊C₃

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#611⋊C₃
commutator, cosocle	C₄×C₉	C₂³×C₃
Frattini	C₃×C₄	C₂²×C₃×D₃
Fitting	C₂×Q₈×3₋¹⁺²	C₂
socle	C₂×C₃	C₂×C₃×#67
prime radical	#3697⋊C₃	C₂
hypercenter	C₂×Q₈	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#137
LCS	G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

subgroup

G/H

center

C₂

C₂×#611⋊C₃

commutator, cosocle

C₄×C₉

C₂³×C₃

Frattini

C₃×C₄

C₂²×C₃×D₃

Fitting

C₂×Q₈×3₋¹⁺²

C₂

socle

C₂×C₃

C₂×C₃×#67

C₂

C₂×Q₈

C₉

G ▷ C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×Q₈

N	H ≅ G/N	type
1. C₂×Q₈×3₋¹⁺²	C₂	semidirect
3. #3675⋊C₃	C₂	semidirect
4. #3674⋊C₃	C₂	semidirect
5. #3667⋊C₃	C₂	semidirect
7. #63505	C₃	semidirect
23. C₂×Q₈×C₉	C₂×C₃	semidirect
25. #3675	C₂×C₃	semidirect
26. #3674	C₂×C₃	semidirect
27. #3667	C₂×C₃	semidirect
88. 3₋¹⁺²	#137	semidirect
116. C₂×Q₈	AGL(1,ℤ/9)	semidirect
134. C₉	C₃×#137	semidirect

H ≅ G/N

type

1. C₂×Q₈×3₋¹⁺²

C₂

semidirect

3. #3675⋊C₃

C₂

semidirect

4. #3674⋊C₃

C₂

semidirect

5. #3667⋊C₃

C₂

semidirect

7. #63505

C₃

semidirect

23. C₂×Q₈×C₉

C₂×C₃

semidirect

25. #3675

C₂×C₃

semidirect

26. #3674

C₂×C₃

semidirect

27. #3667

C₂×C₃

semidirect

88. 3₋¹⁺²

#137

semidirect

116. C₂×Q₈

AGL(1,ℤ/9)

semidirect

134. C₉

C₃×#137

semidirect

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

subgroup	index
C₂×Q₈×3₋¹⁺²	2
#3697⋊C₃	2
#3675⋊C₃	2
#3674⋊C₃	2
#3667⋊C₃	2
C₃×#942	3
#63505	3

subgroup

index

C₂×Q₈×3₋¹⁺²

Subgroups — #63505⋊C₃ #11721676

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #63505⋊C3 #11721676

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #63505⋊C₃ #11721676