Group #68604 — subgroups

Subgroups — #68604

Order 384.

260 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 121 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×#815	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×Dic₃	C₂	1	1	yes	maximal, characteristic
5	#4932	C₂	1	1	yes	maximal, characteristic
6	C₃×#336	C₂	1	1	yes	maximal, characteristic
7	#4524	C₂	1	1	yes	maximal, characteristic
8	#1768	—	3	1	no	maximal, Sylow
9	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
12	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
14	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₃⋊#95	C₄	1	2	yes	—
17	C₃⋊#95	C₄	1	2	yes	—
18	#815	—	2	2	no	—
19	Dic₃⋊C₈	C₄	1	4	yes	complemented
20	Dic₃⋊C₈	C₄	1	4	yes	complemented
21	C₈×Dic₃	—	2	2	no	—
22	C₂×C₄⋊C₈	—	3	1	no	—
23	C₂×C₄⋊C₈	—	3	1	no	—
24	C₂×#97	—	3	1	no	—
25	C₂×C₄×C₈	—	3	1	no	—
26	#376	—	3	1	no	—
27	#336	—	3	1	no	—
28	#336	D₃	1	1	yes	characteristic
29	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
30	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
31	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×Dic₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×Dic₃	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₈	—	2	1	no	—
35	C₂×C₃×C₈	—	2	1	no	—
36	C₂×C₃×C₈	—	2	1	no	—
37	C₂×C₃×C₈	—	2	1	no	—
38	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
39	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
40	C₃×C₄²	—	2	1	no	—
41	C₃×C₄²	—	2	1	no	—
42	C₄×Dic₃	C₂×C₄	1	2	yes	—
43	C₄×Dic₃	C₂×C₄	1	2	yes	—
44	C₄×Dic₃	D₄	1	2	yes	—
45	C₄×Dic₃	Q₈	1	2	yes	—
46	C₂×C₃⋊C₈	—	2	1	no	—
47	C₂×C₃⋊C₈	—	2	1	no	—
48	C₂×C₃⋊C₈	—	2	1	no	—
49	C₂×C₃⋊C₈	—	2	1	no	—
50	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
51	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
52	C₂²×C₈	—	3	1	no	—
53	C₂²×C₈	—	3	1	no	—
54	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
55	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
56	C₂×C₄²	—	3	1	no	—
57	C₂×C₄²	—	3	1	no	—
58	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
59	C₄⋊C₈	—	3	4	no	—
60	C₄⋊C₈	—	3	4	no	—
61	#97	—	6	2	no	—
62	C₄×C₈	—	6	2	no	—
63	#95	—	3	2	no	—
64	#95	—	3	2	no	—
65	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
66	C₂×C₃×C₄	—	2	2	no	—
67	C₂×C₃×C₄	—	2	2	no	—
68	C₂×C₃×C₄	—	2	2	no	—
69	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₃×C₄	#41	1	1	yes	characteristic
74	C₂×C₃×C₄	#41	1	1	yes	characteristic
75	C₂×Dic₃	—	2	2	no	—
76	C₂×Dic₃	—	2	2	no	—
77	C₂×Dic₃	—	2	2	no	—
78	C₂×Dic₃	—	2	2	no	—
79	C₂×Dic₃	C₄²	1	2	yes	—
80	C₂×Dic₃	C₄²	1	2	yes	—
81	C₂×Dic₃	C₄⋊C₄	1	4	yes	—
82	C₃×C₈	—	2	2	no	—
83	C₃×C₈	—	2	2	no	—
84	C₃⋊C₈	—	2	2	no	—
85	C₃⋊C₈	—	2	2	no	—
86	C₂²×C₄	—	3	1	no	—
87	C₂²×C₄	—	3	1	no	—
88	C₂²×C₄	—	3	2	no	—
89	C₂²×C₄	C₄×D₃	1	2	yes	—
90	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	1	no	—
93	C₂×C₈	—	2	1	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	3	2	no	—
96	C₂×C₈	—	3	2	no	—
97	C₂×C₈	—	6	1	no	—
98	C₂×C₈	—	6	1	no	—
99	C₂×C₈	—	6	1	no	—
100	C₂×C₈	—	6	1	no	—
101	C₂×C₈	C₄×D₃	1	2	yes	—
102	C₂×C₈	C₄×D₃	1	2	yes	—
103	C₄²	—	2	1	no	—
104	C₄²	—	2	1	no	—
105	C₄²	—	3	2	no	—
106	C₄²	—	3	2	no	—
107	C₄²	—	3	2	no	—
108	C₄²	—	3	2	no	—
109	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
110	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
111	C₂²×C₃	#117	1	1	yes	characteristic
112	C₂²×C₃	#131	1	2	yes	—
113	C₂²×C₃	#131	1	2	yes	—
114	C₃×C₄	—	4	2	no	—
115	C₃×C₄	C₄×D₄	1	1	yes	characteristic
116	C₃×C₄	C₄×D₄	1	1	yes	characteristic
117	C₃×C₄	C₄×Q₈	1	1	yes	characteristic
118	C₃×C₄	C₄×Q₈	1	1	yes	characteristic
119	Dic₃	—	2	2	no	—
120	Dic₃	—	2	2	no	—
121	Dic₃	—	2	4	no	—
122	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	3	2	no	—
127	C₂×C₄	—	3	2	no	—
128	C₂×C₄	—	3	4	no	—
129	C₂×C₄	—	6	2	no	—
130	C₂×C₄	—	6	2	no	—
131	C₂×C₄	—	6	2	no	—
132	C₂×C₄	—	6	2	no	—
133	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
135	C₂×C₄	D₃×D₄	1	1	yes	characteristic
136	C₂×C₄	#237	1	1	yes	characteristic
137	C₂×C₄	#237	1	1	yes	characteristic
138	C₂×C₄	D₃×Q₈	1	1	yes	characteristic
139	C₈	—	2	2	no	—
140	C₈	—	2	2	no	—
141	C₈	—	6	2	no	—
142	C₈	—	6	2	no	—
143	C₂×C₃	C₄×C₄⋊C₄	1	1	yes	characteristic
144	C₂×C₃	#405	1	1	yes	characteristic
145	C₂×C₃	#405	1	1	yes	characteristic
146	C₂×C₃	#424	1	2	yes	—
147	C₂×C₃	#433	1	2	yes	—
148	C₂²	C₄²×D₃	1	1	yes	characteristic
149	C₂²	#877	1	1	yes	characteristic
150	C₂²	D₃×C₄⋊C₄	1	1	yes	characteristic
151	C₂²	#901	1	2	yes	—
152	C₂²	#907	1	2	yes	—
153	C₄	—	4	2	no	—
154	C₄	—	6	2	no	—
155	C₄	—	6	2	no	—
156	C₄	—	6	4	no	—
157	C₄	#67⋊C₄	1	1	yes	characteristic
158	C₄	#67⋊C₄	1	1	yes	characteristic
159	C₄	#887	1	1	yes	characteristic
160	C₄	#887	1	1	yes	characteristic
161	C₃	#1768	1	1	yes	characteristic, complemented, Sylow
162	C₂	#4649	1	1	yes	characteristic
163	C₂	#4689	1	1	yes	characteristic
164	C₂	#4708	1	1	yes	characteristic
165	C₂	#4720	1	2	yes	—
166	C₂	#4839	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#815

C₂

yes

maximal, characteristic

C₂×Dic₃⋊C₈

C₂

yes

maximal, characteristic

C₂×Dic₃⋊C₈

C₂

yes

maximal, characteristic

C₂×C₈×Dic₃

C₂

yes

maximal, characteristic

#4932

C₂

yes

maximal, characteristic

C₃×#336

C₂

yes

maximal, characteristic

#4524

C₂

yes

maximal, characteristic

#1768

—

maximal, Sylow

C₂²×C₃×C₈

C₂²

yes

characteristic

C₂²×C₃×C₈

C₂²

yes

characteristic

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄×Dic₃

C₂²

yes

characteristic

C₂×C₄×Dic₃

C₂²

yes

characteristic

C₂²×C₃⋊C₈

C₂²

yes

characteristic

C₂²×C₃⋊C₈

C₂²

yes

characteristic

C₃⋊#95

C₄

yes

—

C₃⋊#95

C₄

yes

—

#815

—

Dic₃⋊C₈

C₄

yes

complemented

Dic₃⋊C₈

C₄

yes

complemented

C₈×Dic₃

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

#376

—

#336

—

#336

D₃

yes

characteristic

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

C₂×C₄

yes

characteristic

C₂²×Dic₃

C₂×C₄

yes

characteristic

C₂²×Dic₃

C₂×C₄

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₄

yes

—

C₂×C₃×C₈

C₂×C₄

yes

—

C₃×C₄²

—

C₃×C₄²

—

C₄×Dic₃

C₂×C₄

yes

—

C₄×Dic₃

C₂×C₄

yes

—

C₄×Dic₃

D₄

yes

—

C₄×Dic₃

Q₈

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

C₂×C₄

yes

—

C₂×C₃⋊C₈

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃

yes

characteristic

C₂²×C₈

C₂×D₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

#95

—

#95

—

C₂³×C₃

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

C₂×C₃×C₄

yes

C₂×C₃×C₄

yes

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

C₄²

yes

—

C₂×Dic₃

C₄²

yes

—

C₂×Dic₃

C₄⋊C₄

yes

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₄×D₃

yes

—

102

C₂×C₈

C₄×D₃

yes

—

103

C₄²

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₂²×C₃

C₂×C₄²

yes

characteristic

110

C₂²×C₃

C₂×C₄⋊C₄

yes

characteristic

111

C₂²×C₃

#117

yes

characteristic

112

C₂²×C₃

#131

yes

—

113

C₂²×C₃

#131

yes

—

114

C₃×C₄

—

115

C₃×C₄

C₄×D₄

yes

characteristic

116

C₃×C₄

C₄×D₄

yes

characteristic

117

C₃×C₄

C₄×Q₈

yes

characteristic

118

C₃×C₄

C₄×Q₈

yes

characteristic

119

Dic₃

—

120

Dic₃

—

121

Dic₃

—

122

C₂³

C₂×C₄×D₃

yes

characteristic

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₂×C₄×D₃

yes

characteristic

134

C₂×C₄

C₂×C₄×D₃

yes

characteristic

135

C₂×C₄

D₃×D₄

yes

characteristic

136

C₂×C₄

#237

yes

characteristic

137

C₂×C₄

#237

yes

characteristic

138

C₂×C₄

D₃×Q₈

yes

characteristic

139

C₈

—

140

C₈

—

141

C₈

—

142

C₈

—

143

C₂×C₃

C₄×C₄⋊C₄

yes

characteristic

144

C₂×C₃

#405

yes

characteristic

145

C₂×C₃

#405

yes

characteristic

146

C₂×C₃

#424

yes

—

147

C₂×C₃

#433

yes

—

148

C₂²

C₄²×D₃

yes

characteristic

149

C₂²

#877

yes

characteristic

150

C₂²

D₃×C₄⋊C₄

yes

characteristic

151

C₂²

#901

yes

—

152

C₂²

#907

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

#67⋊C₄

yes

characteristic

158

C₄

#67⋊C₄

yes

characteristic

159

C₄

#887

yes

characteristic

160

C₄

#887

yes

characteristic

161

C₃

#1768

yes

characteristic, complemented, Sylow

162

C₂

#4649

yes

characteristic

163

C₂

#4689

yes

characteristic

164

C₂

#4708

yes

characteristic

165

C₂

#4720

yes

—

166

C₂

#4839

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂×C₄²
Fitting	C₃×#336	C₂
socle, prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#336	D₃
hypocenter	C₃	#1768
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃

commutator

C₂²×C₃

C₂×C₄²

Fitting

C₃×#336

C₂

socle, prime radical

C₂³×C₃

C₂²×C₄

cosocle

C₂²×C₃×C₄

C₂³

D₃

C₃

G ▷ C₂²×C₃ ▷ C₃

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
19. Dic₃⋊C₈	C₄	semidirect
20. Dic₃⋊C₈	C₄	semidirect
161. C₃	#1768	semidirect

H ≅ G/N

type

19. Dic₃⋊C₈

C₄

semidirect

20. Dic₃⋊C₈

C₄

semidirect

161. C₃

#1768

semidirect

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

subgroup	index
C₂×#815	2
C₂×Dic₃⋊C₈	2
C₂×C₈×Dic₃	2
#4932	2
C₃×#336	2
#4524	2
#1768	3

subgroup

index

C₂×#815

C₂×Dic₃⋊C₈

C₂×C₈×Dic₃

Subgroups — #68604

Conjugacy classes of subgroups

Complements

Maximal subgroups