Group #10631132 — subgroups

Subgroups — C₃×#12525 #10631132

Order 768.

340 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 340 conjugacy classes of subgroups — too many to draw. Only its 102 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	C₃×#2246	C₂	1	2	yes	maximal, complemented
2	C₃×#2240	C₂	1	2	yes	maximal, complemented
3	C₃×#1712	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#1706	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	#12525	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×D₁₆	—	2	4	no	—
8	C₃×#500	C₂²	1	1	yes	characteristic
9	C₃×#493	—	2	1	no	—
10	C₃×#493	—	2	1	no	—
11	C₃×#493	C₂²	1	1	yes	characteristic
12	C₃×#493	C₂²	1	2	yes	—
13	C₃×#492	—	2	1	no	—
14	C₃×#486	—	2	1	no	—
15	C₃×#357	—	2	4	no	—
16	C₃×C₄×C₁₆	C₂²	1	2	yes	—
17	C₃×C₈²	C₂²	1	1	yes	characteristic
18	#2246	C₂×C₃	1	2	yes	complemented
19	#2240	C₂×C₃	1	2	yes	complemented
20	#1712	C₂×C₃	1	1	yes	characteristic, complemented
21	#1706	C₂×C₃	1	1	yes	characteristic, complemented
22	#1312	C₂×C₃	1	1	yes	characteristic, complemented
23	C₂×C₃×SD₁₆	—	4	2	no	—
24	C₂×C₃×D₈	—	2	2	no	—
25	C₂×C₃×D₈	—	2	2	no	—
26	C₂×C₃×D₈	—	2	2	no	—
27	C₂×C₃×D₈	—	4	2	no	—
28	C₂×C₃×D₈	—	4	2	no	—
29	C₃×C₄⋊Q₈	—	2	1	no	—
30	C₃×#127	—	2	1	no	—
31	C₃×#127	—	2	2	no	—
32	C₃×D₁₆	—	4	8	no	—
33	C₂×C₃×C₁₆	D₄	1	4	yes	—
34	C₃×#107	—	2	2	no	—
35	C₃×#102	—	4	2	no	—
36	C₃×C₄×C₈	D₄	1	1	yes	characteristic
37	C₃×C₄×C₈	D₄	1	1	yes	characteristic
38	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
39	C₂×D₁₆	—	2	4	no	—
40	#500	C₂²×C₃	1	1	yes	characteristic
41	#493	—	2	1	no	—
42	#493	—	2	1	no	—
43	#493	C₂²×C₃	1	1	yes	characteristic
44	#493	C₂²×C₃	1	2	yes	—
45	#492	—	2	1	no	—
46	#486	—	2	1	no	—
47	#357	—	2	4	no	—
48	C₄×C₁₆	C₂²×C₃	1	2	yes	—
49	C₈²	C₂²×C₃	1	1	yes	characteristic
50	C₂×C₃×Q₈	—	4	1	no	—
51	C₂×C₃×D₄	—	4	1	no	—
52	C₂×C₃×D₄	—	4	2	no	—
53	C₂×C₃×D₄	—	4	1	no	—
54	C₂×C₃×D₄	—	4	2	no	—
55	C₂×C₃×D₄	—	4	2	no	—
56	C₂×C₃×D₄	—	4	1	no	—
57	C₃×SD₁₆	—	8	4	no	—
58	C₃×D₈	—	4	4	no	—
59	C₃×D₈	—	4	4	no	—
60	C₃×D₈	—	4	4	no	—
61	C₃×D₈	—	8	4	no	—
62	C₃×D₈	—	8	4	no	—
63	C₂×C₃×C₈	D₈	1	2	yes	—
64	C₂×C₃×C₈	D₈	1	2	yes	—
65	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
66	C₃×C₄⋊C₄	—	4	1	no	—
67	C₃×C₄⋊C₄	—	4	1	no	—
68	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
69	C₃×C₁₆	—	2	4	no	—
70	C₂×SD₁₆	—	4	2	no	—
71	C₂×D₈	—	2	2	no	—
72	C₂×D₈	—	2	2	no	—
73	C₂×D₈	—	2	2	no	—
74	C₂×D₈	—	4	2	no	—
75	C₂×D₈	—	4	2	no	—
76	C₄⋊Q₈	—	2	1	no	—
77	#127	—	2	1	no	—
78	#127	—	2	2	no	—
79	D₁₆	—	4	8	no	—
80	C₂×C₁₆	C₃×D₄	1	4	yes	—
81	#107	—	2	2	no	—
82	#102	—	4	2	no	—
83	C₄×C₈	C₃×D₄	1	1	yes	characteristic
84	C₄×C₈	C₃×D₄	1	1	yes	characteristic
85	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
86	C₂³×C₃	—	8	1	no	—
87	C₂³×C₃	—	8	2	no	—
88	C₃×Q₈	—	8	2	no	—
89	C₃×D₄	—	8	2	no	—
90	C₃×D₄	—	8	2	no	—
91	C₃×D₄	—	8	2	no	—
92	C₃×D₄	—	8	4	no	—
93	C₃×D₄	—	8	2	no	—
94	C₃×D₄	—	8	4	no	—
95	C₃×D₄	—	8	2	no	—
96	C₂×C₃×C₄	—	8	1	no	—
97	C₂×C₃×C₄	#127	1	1	yes	characteristic
98	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
99	C₂×C₃×C₄	C₂×D₈	1	1	yes	characteristic
100	C₃×C₈	—	2	2	no	—
101	C₃×C₈	D₁₆	1	4	yes	complemented
102	C₃×C₈	AGL(1,ℤ/8)	1	4	yes	—
103	C₂×Q₈	—	4	1	no	—
104	C₂×D₄	—	4	1	no	—
105	C₂×D₄	—	4	2	no	—
106	C₂×D₄	—	4	1	no	—
107	C₂×D₄	—	4	2	no	—
108	C₂×D₄	—	4	2	no	—
109	C₂×D₄	—	4	1	no	—
110	SD₁₆	—	8	4	no	—
111	D₈	—	4	4	no	—
112	D₈	—	4	4	no	—
113	D₈	—	4	4	no	—
114	D₈	—	8	4	no	—
115	D₈	—	8	4	no	—
116	C₂×C₈	C₃×D₈	1	2	yes	—
117	C₂×C₈	C₃×D₈	1	2	yes	—
118	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
119	C₄⋊C₄	—	4	1	no	—
120	C₄⋊C₄	—	4	1	no	—
121	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
122	C₁₆	—	2	4	no	—
123	C₂²×C₃	—	16	1	no	—
124	C₂²×C₃	—	16	2	no	—
125	C₂²×C₃	—	16	2	no	—
126	C₂²×C₃	—	16	1	no	—
127	C₂²×C₃	—	16	2	no	—
128	C₂²×C₃	—	16	1	no	—
129	C₂²×C₃	#493	1	1	yes	characteristic
130	C₃×C₄	—	16	1	no	—
131	C₃×C₄	#496	1	2	yes	—
132	C₃×C₄	C₂×D₁₆	1	2	yes	—
133	C₃×C₄	#509	1	2	yes	—
134	C₂³	—	8	1	no	—
135	C₂³	—	8	2	no	—
136	Q₈	—	8	2	no	—
137	D₄	—	8	4	no	—
138	D₄	—	8	2	no	—
139	D₄	—	8	4	no	—
140	D₄	—	8	2	no	—
141	D₄	—	8	2	no	—
142	D₄	—	8	2	no	—
143	D₄	—	8	2	no	—
144	C₂×C₄	—	8	1	no	—
145	C₂×C₄	C₃×#127	1	1	yes	characteristic
146	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
147	C₂×C₄	C₂×C₃×D₈	1	1	yes	characteristic
148	C₈	—	2	2	no	—
149	C₈	C₃×D₁₆	1	4	yes	complemented
150	C₈	C₃×AGL(1,ℤ/8)	1	4	yes	—
151	C₂×C₃	—	32	2	no	—
152	C₂×C₃	—	32	1	no	—
153	C₂×C₃	#1721	1	1	yes	characteristic
154	C₂×C₃	#2246	1	1	yes	characteristic
155	C₂×C₃	#2250	1	1	yes	characteristic
156	C₂²	—	16	2	no	—
157	C₂²	—	16	1	no	—
158	C₂²	—	16	2	no	—
159	C₂²	—	16	1	no	—
160	C₂²	—	16	1	no	—
161	C₂²	—	16	2	no	—
162	C₂²	C₃×#493	1	1	yes	characteristic
163	C₄	—	16	1	no	—
164	C₄	C₃×#496	1	2	yes	—
165	C₄	C₂×C₃×D₁₆	1	2	yes	—
166	C₄	C₃×#509	1	2	yes	—
167	C₃	#12525	1	1	yes	characteristic, direct product factor, Sylow
168	C₂	—	32	2	no	—
169	C₂	—	32	1	no	—
170	C₂	C₃×#1721	1	1	yes	characteristic
171	C₂	C₃×#2246	1	1	yes	characteristic
172	C₂	C₃×#2250	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#2246

C₂

yes

maximal, complemented

C₃×#2240

C₂

yes

maximal, complemented

C₃×#1712

C₂

yes

maximal, characteristic, complemented

C₃×#1706

C₂

yes

maximal, characteristic, complemented

C₃×#1312

C₂

yes

maximal, characteristic, complemented

#12525

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₃×D₁₆

—

C₃×#500

C₂²

yes

—

—

C₂²

yes

characteristic

C₃×#493

C₂²

yes

—

C₃×#492

—

C₃×#486

—

C₃×#357

—

C₃×C₄×C₁₆

C₂²

yes

—

C₃×C₈²

C₂²

yes

characteristic

#2246

C₂×C₃

yes

complemented

#2240

C₂×C₃

yes

complemented

#1712

C₂×C₃

yes

characteristic, complemented

#1706

C₂×C₃

yes

characteristic, complemented

#1312

C₂×C₃

yes

characteristic, complemented

C₂×C₃×SD₁₆

—

C₂×C₃×D₈

—

C₂×C₃×D₈

—

C₂×C₃×D₈

—

C₂×C₃×D₈

—

C₂×C₃×D₈

—

C₃×C₄⋊Q₈

—

C₃×#127

—

C₃×#127

—

C₃×D₁₆

—

C₂×C₃×C₁₆

D₄

yes

—

C₃×#107

—

C₃×#102

—

C₃×C₄×C₈

D₄

yes

characteristic

C₃×C₄×C₈

D₄

yes

characteristic

C₃×C₄×C₈

C₂³

yes

characteristic

C₂×D₁₆

—

#500

C₂²×C₃

yes

—

—

C₂²×C₃

yes

characteristic

#493

C₂²×C₃

yes

—

#492

—

#486

—

#357

—

C₄×C₁₆

C₂²×C₃

yes

—

C₈²

C₂²×C₃

yes

characteristic

C₂×C₃×Q₈

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₃×SD₁₆

—

C₃×D₈

—

C₃×D₈

—

C₃×D₈

—

C₃×D₈

—

C₃×D₈

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

D₈

yes

—

C₂×C₃×C₈

C₂×D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×D₄

yes

characteristic

C₃×C₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

#127

—

D₁₆

—

C₂×C₁₆

C₃×D₄

yes

—

#107

—

#102

—

C₄×C₈

C₃×D₄

yes

characteristic

C₄×C₈

C₃×D₄

yes

characteristic

C₄×C₈

C₂³×C₃

yes

characteristic

C₂³×C₃

—

C₂³×C₃

—

C₃×Q₈

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

#127

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

characteristic

C₂×C₃×C₄

C₂×D₈

yes

characteristic

100

C₃×C₈

—

101

C₃×C₈

D₁₆

yes

complemented

102

C₃×C₈

AGL(1,ℤ/8)

yes

—

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

SD₁₆

—

111

D₈

—

112

D₈

—

113

D₈

—

114

D₈

—

115

D₈

—

116

C₂×C₈

C₃×D₈

yes

—

117

C₂×C₈

C₃×D₈

yes

—

118

C₂×C₈

C₂×C₃×D₄

yes

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄²

C₂×C₃×D₄

yes

characteristic

122

C₁₆

—

123

C₂²×C₃

—

124

C₂²×C₃

—

125

C₂²×C₃

—

126

C₂²×C₃

—

127

C₂²×C₃

—

128

C₂²×C₃

—

129

C₂²×C₃

#493

yes

characteristic

130

C₃×C₄

—

131

C₃×C₄

#496

yes

—

132

C₃×C₄

C₂×D₁₆

yes

—

133

C₃×C₄

#509

yes

—

134

C₂³

—

135

C₂³

—

136

Q₈

—

137

D₄

—

138

D₄

—

139

D₄

—

140

D₄

—

141

D₄

—

142

D₄

—

143

D₄

—

144

C₂×C₄

—

145

C₂×C₄

C₃×#127

yes

characteristic

146

C₂×C₄

C₂×C₃×D₈

yes

characteristic

147

C₂×C₄

C₂×C₃×D₈

yes

characteristic

148

C₈

—

149

C₈

C₃×D₁₆

yes

complemented

150

C₈

C₃×AGL(1,ℤ/8)

yes

—

151

C₂×C₃

—

152

C₂×C₃

—

153

C₂×C₃

#1721

yes

characteristic

154

C₂×C₃

#2246

yes

characteristic

155

C₂×C₃

#2250

yes

characteristic

156

C₂²

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

—

162

C₂²

C₃×#493

yes

characteristic

163

C₄

—

164

C₄

C₃×#496

yes

—

165

C₄

C₂×C₃×D₁₆

yes

—

166

C₄

C₃×#509

yes

—

167

C₃

#12525

yes

characteristic, direct product factor, Sylow

168

C₂

—

169

C₂

—

170

C₂

C₃×#1721

yes

characteristic

171

C₂

C₃×#2246

yes

characteristic

172

C₂

C₃×#2250

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₃	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₃
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₈² ◁ G

subgroup

G/H

center, socle

C₂²×C₃

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₃

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₈² ◁ G

N	H ≅ G/N	type
1. C₃×#2246	C₂	semidirect
2. C₃×#2240	C₂	semidirect
3. C₃×#1712	C₂	semidirect
4. C₃×#1706	C₂	semidirect
5. C₃×#1312	C₂	semidirect
6. #12525	C₃	direct (N × H)
18. #2246	C₂×C₃	semidirect
19. #2240	C₂×C₃	semidirect
20. #1712	C₂×C₃	semidirect
21. #1706	C₂×C₃	semidirect
22. #1312	C₂×C₃	semidirect
101. C₃×C₈	D₁₆	semidirect
149. C₈	C₃×D₁₆	semidirect
167. C₃	#12525	direct (N × H)

H ≅ G/N

type

1. C₃×#2246

C₂

semidirect

2. C₃×#2240

C₂

semidirect

3. C₃×#1712

C₂

semidirect

4. C₃×#1706

C₂

semidirect

5. C₃×#1312

C₂

semidirect

6. #12525

C₃

direct (N × H)

18. #2246

C₂×C₃

semidirect

19. #2240

C₂×C₃

semidirect

20. #1712

C₂×C₃

semidirect

21. #1706

C₂×C₃

semidirect

22. #1312

C₂×C₃

semidirect

101. C₃×C₈

D₁₆

semidirect

149. C₈

C₃×D₁₆

semidirect

167. C₃

#12525

direct (N × H)

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

subgroup	index
C₃×#2246	2
C₃×#2240	2
C₃×#1712	2
C₃×#1706	2
C₃×#1312	2
#12525	3

subgroup

index

Subgroups — C₃×#12525 #10631132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#12525 #10631132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#12525 #10631132