Group #13141469 — subgroups

Subgroups — #10588362⋊C₃ #13141469

Order 1728.

316 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 132 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	#63431⋊C₃	C₂	1	2	yes	maximal
2	C₄×C₈×3₋¹⁺²	C₂	1	1	yes	maximal, characteristic
3	#63378⋊C₃	C₂	1	4	yes	maximal
4	C₃×#4682	—	3	1	no	maximal
5	#10588362	C₃	1	1	yes	maximal, characteristic, complemented
6	C₂×3₋¹⁺²⋊Q₈	—	2	2	no	—
7	C₂×C₈×3₋¹⁺²	C₂²	1	2	yes	—
8	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
9	C₄⋊(3₋¹⁺²⋊C₄)	—	2	2	no	—
10	C₄⋊(3₋¹⁺²⋊C₄)	C₂²	1	4	yes	—
11	C₃×#869	—	3	2	no	—
12	C₃²×C₄×C₈	D₃	1	1	yes	characteristic
13	C₃×#818	—	3	4	no	—
14	#63431	C₂×C₃	1	2	yes	—
15	C₄×C₈×C₉	—	2	1	no	—
16	C₄×C₈×C₉	C₂×C₃	1	1	yes	characteristic
17	#63378	C₂×C₃	1	4	yes	—
18	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
19	C₂×C₄×3₋¹⁺²	D₄	1	1	yes	characteristic
20	C₂×C₄×3₋¹⁺²	C₂³	1	1	yes	characteristic
21	C₂×3₋¹⁺²⋊C₄	—	2	4	no	—
22	3₋¹⁺²⋊Q₈	—	4	4	no	—
23	C₈×3₋¹⁺²	Q₈	1	4	yes	complemented
24	C₃×#500	—	9	1	no	—
25	#4682	—	3	1	no	—
26	C₂×C₃×Dic₆	—	6	2	no	—
27	C₂×C₃²×C₈	C₂×D₃	1	2	yes	—
28	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
29	C₃×C₄⋊Dic₃	—	3	4	no	—
30	C₃×C₄⋊Dic₃	—	6	2	no	—
31	C₂×Dic₁₈	—	2	2	no	—
32	C₂×C₈×C₉	—	2	2	no	—
33	C₂×C₈×C₉	C₂²×C₃	1	2	yes	—
34	C₄²×C₉	—	2	1	no	—
35	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
36	C₄⋊Dic₉	—	2	2	no	—
37	C₄⋊Dic₉	C₂²×C₃	1	4	yes	—
38	C₂²×3₋¹⁺²	C₂×D₄	1	1	yes	characteristic
39	C₄×3₋¹⁺²	D₈	1	2	yes	—
40	C₄×3₋¹⁺²	Q₁₆	1	2	yes	—
41	C₄×3₋¹⁺²	C₂×Q₈	1	2	yes	—
42	3₋¹⁺²⋊C₄	—	4	4	no	—
43	C₃×C₄⋊Q₈	—	9	2	no	—
44	#869	—	3	2	no	—
45	C₃×#107	—	9	4	no	—
46	C₃×C₄×C₈	—	3	1	no	—
47	C₃×C₄×C₈	C₃×D₃	1	1	yes	characteristic
48	#818	—	3	4	no	—
49	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
50	C₂×C₃²×C₄	D₁₂	1	1	yes	characteristic
51	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
52	C₂×C₃×Dic₃	—	6	4	no	—
53	C₃×Dic₆	—	12	4	no	—
54	C₃²×C₈	Dic₆	1	4	yes	—
55	C₂×C₄×C₉	—	2	1	no	—
56	C₂×C₄×C₉	—	2	1	no	—
57	C₂×C₄×C₉	—	2	1	no	—
58	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
59	C₂×C₄×C₉	C₃×D₄	1	1	yes	characteristic
60	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
61	C₂×Dic₉	—	2	4	no	—
62	Dic₁₈	—	4	4	no	—
63	C₈×C₉	—	2	4	no	—
64	C₈×C₉	C₃×Q₈	1	4	yes	complemented
65	#500	—	9	1	no	Sylow
66	C₂×3₋¹⁺²	C₄⋊Q₈	1	1	yes	characteristic
67	C₂×3₋¹⁺²	C₂×D₈	1	1	yes	characteristic
68	C₂×3₋¹⁺²	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×C₃×Q₈	—	18	2	no	—
70	C₂×Dic₆	—	6	2	no	—
71	C₂×C₃×C₈	—	3	2	no	—
72	C₂×C₃×C₈	C₂×C₃×D₃	1	2	yes	—
73	C₃×C₄⋊C₄	—	9	4	no	—
74	C₃×C₄⋊C₄	—	18	2	no	—
75	C₃×C₄²	—	3	1	no	—
76	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
77	C₄⋊Dic₃	—	3	4	no	—
78	C₄⋊Dic₃	—	6	2	no	—
79	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
80	C₃²×C₄	D₂₄	1	2	yes	—
81	C₃²×C₄	Dic₁₂	1	2	yes	—
82	C₃²×C₄	C₂×Dic₆	1	2	yes	—
83	C₃×Dic₃	—	12	4	no	—
84	C₂²×C₉	—	2	1	no	—
85	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
86	C₄×C₉	—	2	2	no	—
87	C₄×C₉	—	2	2	no	—
88	C₄×C₉	—	2	2	no	—
89	C₄×C₉	C₃×D₈	1	2	yes	—
90	C₄×C₉	C₃×Q₁₆	1	2	yes	—
91	C₄×C₉	C₂×C₃×Q₈	1	2	yes	—
92	Dic₉	—	4	4	no	—
93	C₄⋊Q₈	—	9	2	no	—
94	#107	—	9	4	no	—
95	C₄×C₈	AGL(1,ℤ/9)	1	1	yes	characteristic
96	3₋¹⁺²	#500	1	1	yes	characteristic, complemented, Sylow
97	C₃×Q₈	—	36	4	no	—
98	C₂×C₃×C₄	—	3	1	no	—
99	C₂×C₃×C₄	—	3	1	no	—
100	C₂×C₃×C₄	—	3	1	no	—
101	C₂×C₃×C₄	—	18	4	no	—
102	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
103	C₂×C₃×C₄	C₃×D₁₂	1	1	yes	characteristic
104	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
105	C₂×Dic₃	—	6	4	no	—
106	Dic₆	—	12	4	no	—
107	C₃×C₈	—	3	4	no	—
108	C₃×C₈	C₃×Dic₆	1	4	yes	—
109	C₂×C₃²	#869	1	1	yes	characteristic
110	C₂×C₃²	C₂×D₂₄	1	1	yes	characteristic
111	C₂×C₃²	C₂×Dic₁₂	1	1	yes	characteristic
112	C₂×C₉	—	2	1	no	—
113	C₂×C₉	—	2	1	no	—
114	C₂×C₉	—	2	1	no	—
115	C₂×C₉	C₃×C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₉	C₂×C₃×D₈	1	1	yes	characteristic
117	C₂×C₉	C₂×C₃×Q₁₆	1	1	yes	characteristic
118	C₂×Q₈	—	18	2	no	—
119	C₂×C₈	C₂×AGL(1,ℤ/9)	1	2	yes	—
120	C₄⋊C₄	—	9	4	no	—
121	C₄⋊C₄	—	18	2	no	—
122	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
123	C₂²×C₃	—	3	1	no	—
124	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
125	C₃×C₄	—	3	2	no	—
126	C₃×C₄	—	3	2	no	—
127	C₃×C₄	—	3	2	no	—
128	C₃×C₄	—	36	4	no	—
129	C₃×C₄	C₃×D₂₄	1	2	yes	—
130	C₃×C₄	C₃×Dic₁₂	1	2	yes	—
131	C₃×C₄	C₂×C₃×Dic₆	1	2	yes	—
132	Dic₃	—	12	4	no	—
133	C₃²	#4682	1	1	yes	characteristic
134	C₉	—	2	1	no	—
135	C₉	C₃×#500	1	1	yes	characteristic, complemented
136	Q₈	—	36	4	no	—
137	C₂×C₄	—	18	4	no	—
138	C₂×C₄	C₄⋊AGL(1,ℤ/9)	1	1	yes	characteristic
139	C₂×C₄	C₄⋊AGL(1,ℤ/9)	1	1	yes	characteristic
140	C₂×C₄	C₂²×AGL(1,ℤ/9)	1	1	yes	characteristic
141	C₈	3₋¹⁺²⋊Q₈	1	4	yes	complemented
142	C₂×C₃	—	3	1	no	—
143	C₂×C₃	—	3	1	no	—
144	C₂×C₃	—	3	1	no	—
145	C₂×C₃	C₃×#869	1	1	yes	characteristic
146	C₂×C₃	C₂×C₃×D₂₄	1	1	yes	characteristic
147	C₂×C₃	C₂×C₃×Dic₁₂	1	1	yes	characteristic
148	C₂²	C₂×C₄⋊AGL(1,ℤ/9)	1	1	yes	characteristic
149	C₄	—	36	4	no	—
150	C₄	Dic₃₆⋊C₃	1	2	yes	—
151	C₄	D₇₂⋊C₃	1	2	yes	—
152	C₄	C₂×3₋¹⁺²⋊Q₈	1	2	yes	—
153	C₃	—	3	1	no	—
154	C₃	C₃×#4682	1	1	yes	characteristic
155	C₂	#63431⋊C₃	1	1	yes	characteristic
156	C₂	C₂×Dic₃₆⋊C₃	1	1	yes	characteristic
157	C₂	C₂×D₇₂⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#63431⋊C₃

C₂

yes

maximal

C₄×C₈×3₋¹⁺²

C₂

yes

maximal, characteristic

#63378⋊C₃

C₂

yes

—

C₃

yes

maximal, characteristic, complemented

C₂×3₋¹⁺²⋊Q₈

—

C₂×C₈×3₋¹⁺²

C₂²

yes

—

C₄²×3₋¹⁺²

C₂²

yes

characteristic

C₄⋊(3₋¹⁺²⋊C₄)

—

C₄⋊(3₋¹⁺²⋊C₄)

C₂²

yes

—

C₃×#869

—

C₃²×C₄×C₈

D₃

yes

characteristic

C₃×#818

—

#63431

C₂×C₃

yes

—

C₄×C₈×C₉

—

C₄×C₈×C₉

C₂×C₃

yes

characteristic

#63378

C₂×C₃

yes

—

C₂×C₄×3₋¹⁺²

D₄

yes

characteristic

C₂×C₄×3₋¹⁺²

D₄

yes

characteristic

C₂×C₄×3₋¹⁺²

C₂³

yes

characteristic

C₂×3₋¹⁺²⋊C₄

—

3₋¹⁺²⋊Q₈

—

C₈×3₋¹⁺²

Q₈

yes

complemented

C₃×#500

—

#4682

—

C₂×C₃×Dic₆

—

C₂×C₃²×C₈

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₃×C₄⋊Dic₃

—

C₃×C₄⋊Dic₃

—

C₂×Dic₁₈

—

C₂×C₈×C₉

—

C₂×C₈×C₉

C₂²×C₃

yes

—

C₄²×C₉

—

C₄²×C₉

C₂²×C₃

yes

characteristic

C₄⋊Dic₉

—

C₄⋊Dic₉

C₂²×C₃

yes

—

C₂²×3₋¹⁺²

C₂×D₄

yes

characteristic

C₄×3₋¹⁺²

D₈

yes

—

C₄×3₋¹⁺²

Q₁₆

yes

—

C₄×3₋¹⁺²

C₂×Q₈

yes

—

3₋¹⁺²⋊C₄

—

C₃×C₄⋊Q₈

—

#869

—

C₃×#107

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₃×D₃

yes

characteristic

#818

—

C₂×C₃²×C₄

D₁₂

yes

characteristic

C₂×C₃²×C₄

D₁₂

yes

characteristic

C₂×C₃²×C₄

C₂²×D₃

yes

characteristic

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃²×C₈

Dic₆

yes

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₃×D₄

yes

characteristic

C₂×C₄×C₉

C₃×D₄

yes

characteristic

C₂×C₄×C₉

C₂³×C₃

yes

characteristic

C₂×Dic₉

—

Dic₁₈

—

C₈×C₉

—

C₈×C₉

C₃×Q₈

yes

complemented

#500

—

Sylow

C₂×3₋¹⁺²

C₄⋊Q₈

yes

characteristic

C₂×3₋¹⁺²

C₂×D₈

yes

characteristic

C₂×3₋¹⁺²

C₂×Q₁₆

yes

characteristic

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃×D₃

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×C₃×D₃

yes

characteristic

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂²×C₃²

C₂×D₁₂

yes

characteristic

C₃²×C₄

D₂₄

yes

—

C₃²×C₄

Dic₁₂

yes

—

C₃²×C₄

C₂×Dic₆

yes

—

C₃×Dic₃

—

C₂²×C₉

—

C₂²×C₉

C₂×C₃×D₄

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₃×D₈

yes

—

C₄×C₉

C₃×Q₁₆

yes

—

C₄×C₉

C₂×C₃×Q₈

yes

—

Dic₉

—

C₄⋊Q₈

—

#107

—

C₄×C₈

AGL(1,ℤ/9)

yes

characteristic

3₋¹⁺²

#500

yes

characteristic, complemented, Sylow

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

100

C₂×C₃×C₄

—

101

C₂×C₃×C₄

—

102

C₂×C₃×C₄

C₃×D₁₂

yes

characteristic

103

C₂×C₃×C₄

C₃×D₁₂

yes

characteristic

104

C₂×C₃×C₄

C₂²×C₃×D₃

yes

characteristic

105

C₂×Dic₃

—

106

Dic₆

—

107

C₃×C₈

—

108

C₃×C₈

C₃×Dic₆

yes

—

109

C₂×C₃²

#869

yes

characteristic

110

C₂×C₃²

C₂×D₂₄

yes

characteristic

111

C₂×C₃²

C₂×Dic₁₂

yes

characteristic

112

C₂×C₉

—

113

C₂×C₉

—

114

C₂×C₉

—

115

C₂×C₉

C₃×C₄⋊Q₈

yes

characteristic

116

C₂×C₉

C₂×C₃×D₈

yes

characteristic

117

C₂×C₉

C₂×C₃×Q₁₆

yes

characteristic

118

C₂×Q₈

—

119

C₂×C₈

C₂×AGL(1,ℤ/9)

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄²

C₂×AGL(1,ℤ/9)

yes

characteristic

123

C₂²×C₃

—

124

C₂²×C₃

C₂×C₃×D₁₂

yes

characteristic

125

C₃×C₄

—

126

C₃×C₄

—

127

C₃×C₄

—

128

C₃×C₄

—

129

C₃×C₄

C₃×D₂₄

yes

—

130

C₃×C₄

C₃×Dic₁₂

yes

—

131

C₃×C₄

C₂×C₃×Dic₆

yes

—

132

Dic₃

—

133

C₃²

#4682

yes

characteristic

134

C₉

—

135

C₉

C₃×#500

yes

characteristic, complemented

136

Q₈

—

137

C₂×C₄

—

138

C₂×C₄

C₄⋊AGL(1,ℤ/9)

yes

characteristic

139

C₂×C₄

C₄⋊AGL(1,ℤ/9)

yes

characteristic

140

C₂×C₄

C₂²×AGL(1,ℤ/9)

yes

characteristic

141

C₈

3₋¹⁺²⋊Q₈

yes

complemented

142

C₂×C₃

—

143

C₂×C₃

—

144

C₂×C₃

—

145

C₂×C₃

C₃×#869

yes

characteristic

146

C₂×C₃

C₂×C₃×D₂₄

yes

characteristic

147

C₂×C₃

C₂×C₃×Dic₁₂

yes

characteristic

148

C₂²

C₂×C₄⋊AGL(1,ℤ/9)

yes

characteristic

149

C₄

—

150

C₄

Dic₃₆⋊C₃

yes

—

151

C₄

D₇₂⋊C₃

yes

—

152

C₄

C₂×3₋¹⁺²⋊Q₈

yes

—

153

C₃

—

154

C₃

C₃×#4682

yes

characteristic

155

C₂

#63431⋊C₃

yes

characteristic

156

C₂

C₂×Dic₃₆⋊C₃

yes

characteristic

157

C₂

C₂×D₇₂⋊C₃

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₄⋊AGL(1,ℤ/9)
commutator, cosocle	C₂×C₄×C₉	C₂³×C₃
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₄×C₈×3₋¹⁺²	C₂
socle	C₂²×C₃	C₂×C₃×D₁₂
prime radical	C₂²×C₃²	C₂×D₁₂
hypercenter	C₄×C₈	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#500
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊AGL(1,ℤ/9)

commutator, cosocle

C₂×C₄×C₉

C₂³×C₃

Frattini

C₂×C₃×C₄

C₂²×C₃×D₃

Fitting

C₄×C₈×3₋¹⁺²

C₂

socle

C₂²×C₃

C₂×C₃×D₁₂

prime radical

C₂²×C₃²

C₂×D₁₂

C₄×C₈

C₉

G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
5. #10588362	C₃	semidirect
23. C₈×3₋¹⁺²	Q₈	semidirect
64. C₈×C₉	C₃×Q₈	semidirect
96. 3₋¹⁺²	#500	semidirect
135. C₉	C₃×#500	semidirect
141. C₈	3₋¹⁺²⋊Q₈	semidirect

H ≅ G/N

type

5. #10588362

C₃

semidirect

23. C₈×3₋¹⁺²

Q₈

semidirect

64. C₈×C₉

C₃×Q₈

semidirect

96. 3₋¹⁺²

#500

semidirect

135. C₉

C₃×#500

semidirect

141. C₈

3₋¹⁺²⋊Q₈

semidirect

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

subgroup	index
#63431⋊C₃	2
C₄×C₈×3₋¹⁺²	2
#63378⋊C₃	2
C₃×#4682	3
#10588362	3

subgroup

index

C₄×C₈×3₋¹⁺²

Subgroups — #10588362⋊C₃ #13141469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588362⋊C3 #13141469

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588362⋊C₃ #13141469