Group #71826 — subgroups

Subgroups — #71826

Order 384.

232 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 101 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×#533	C₂	1	1	yes	maximal, characteristic
2	#4924	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#4535	C₂	1	4	yes	maximal, complemented
5	#1493	—	3	1	no	maximal, Sylow
6	C₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
8	C₃×#125	—	2	1	no	—
9	C₃×C₂²⋊Q₈	—	2	1	no	—
10	C₃×C₂²⋊Q₈	C₄	1	2	yes	—
11	C₃×C₄×Q₈	—	2	1	no	—
12	C₃×C₄×Q₈	C₂²	1	2	yes	—
13	C₃×#117	—	2	1	no	—
14	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
15	C₂×#208	—	2	1	no	—
16	C₂²×C₃⋊C₈	—	2	1	no	—
17	C₄⋊(C₃⋊C₈)	—	2	1	no	—
18	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
19	C₄×C₃⋊C₈	—	2	1	no	—
20	C₄×C₃⋊C₈	C₂²	1	2	yes	—
21	#533	D₃	1	1	yes	characteristic
22	#423	—	3	1	no	—
23	C₂×C₄×C₈	—	3	1	no	—
24	Q₈⋊C₈	—	3	4	no	—
25	C₂×C₃×Q₈	—	2	1	no	—
26	C₂×C₃×Q₈	C₂×C₄	1	2	yes	—
27	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
29	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
30	C₃×C₄⋊C₄	—	2	1	no	—
31	C₃×C₄⋊C₄	—	2	1	no	—
32	C₃×C₄⋊C₄	—	2	1	no	—
33	C₃×C₄⋊C₄	—	2	2	no	—
34	C₃×C₄⋊C₄	—	2	2	no	—
35	C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
36	C₃×C₂²⋊C₄	—	2	2	no	—
37	C₃×C₄²	—	2	2	no	—
38	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
39	C₃×C₄²	D₄	1	1	yes	characteristic
40	C₃×C₄²	D₄	1	1	yes	characteristic
41	C₃×C₄²	C₂³	1	1	yes	characteristic
42	#208	—	4	2	no	—
43	C₂×C₃⋊C₈	—	2	4	no	—
44	C₂×C₃⋊C₈	—	2	2	no	—
45	C₂×C₃⋊C₈	—	2	2	no	—
46	C₂×M₁₆	—	6	1	no	—
47	C₂²×C₈	—	6	1	no	—
48	C₄⋊Q₈	Dic₃	1	1	yes	characteristic
49	C₄⋊Q₈	Dic₃	1	1	yes	characteristic
50	#125	—	2	1	no	—
51	C₂²⋊Q₈	—	2	1	no	—
52	C₂²⋊Q₈	Dic₃	1	2	yes	—
53	C₄×Q₈	—	2	1	no	—
54	C₄×Q₈	C₂×D₃	1	2	yes	—
55	#117	—	2	1	no	—
56	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
57	C₄⋊C₈	—	3	2	no	—
58	C₄⋊C₈	—	6	1	no	—
59	C₄×C₈	—	3	2	no	—
60	C₄×C₈	—	6	1	no	—
61	C₂³×C₃	C₂²⋊C₄	1	1	yes	characteristic
62	C₃×Q₈	—	2	4	no	—
63	C₃×Q₈	—	4	2	no	—
64	C₂×C₃×C₄	—	2	2	no	—
65	C₂×C₃×C₄	—	2	2	no	—
66	C₂×C₃×C₄	—	2	2	no	—
67	C₂×C₃×C₄	—	2	2	no	—
68	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
69	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₃×C₄	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₃×C₄	SD₁₆	1	2	yes	—
72	C₂×C₃×C₄	Q₁₆	1	2	yes	—
73	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
75	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
76	C₃⋊C₈	—	2	4	no	—
77	C₃⋊C₈	—	4	2	no	—
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	C₂×Dic₃	1	2	yes	—
80	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
81	C₂²×C₄	#67	1	1	yes	characteristic
82	C₂²×C₄	#67	1	1	yes	characteristic
83	M₁₆	—	12	2	no	—
84	C₂×C₈	—	6	4	no	—
85	C₂×C₈	—	6	2	no	—
86	C₂×C₈	—	6	2	no	—
87	C₄⋊C₄	—	2	1	no	—
88	C₄⋊C₄	—	2	1	no	—
89	C₄⋊C₄	—	2	1	no	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	C₂×Dic₃	1	2	yes	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	—	2	2	no	—
95	C₄²	C₂×Dic₃	1	1	yes	characteristic
96	C₄²	#67	1	1	yes	characteristic
97	C₄²	#67	1	1	yes	characteristic
98	C₄²	C₂²×D₃	1	1	yes	characteristic
99	C₂²×C₃	—	2	1	no	—
100	C₂²×C₃	—	2	1	no	—
101	C₂²×C₃	#103	1	2	yes	—
102	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
103	C₃×C₄	—	2	1	no	—
104	C₃×C₄	—	2	1	no	—
105	C₃×C₄	—	4	2	no	—
106	C₃×C₄	—	4	2	no	—
107	C₃×C₄	#103	1	2	yes	—
108	C₃×C₄	#104	1	2	yes	—
109	C₃×C₄	#131	1	2	yes	—
110	C₃×C₄	C₂×SD₁₆	1	1	yes	characteristic
111	C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
112	C₂³	#217	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	4	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	#215	1	2	yes	—
120	C₂×C₄	#216	1	2	yes	—
121	C₂×C₄	#217	1	1	yes	characteristic
122	C₂×C₄	#217	1	1	yes	characteristic
123	C₂×C₄	#217	1	1	yes	characteristic
124	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
125	C₂×C₄	C₂×#67	1	1	yes	characteristic
126	C₂×C₄	C₂×#67	1	1	yes	characteristic
127	C₈	—	6	4	no	—
128	C₈	—	12	2	no	—
129	C₂×C₃	—	2	2	no	—
130	C₂×C₃	#408	1	1	yes	characteristic
131	C₂×C₃	C₂×#103	1	1	yes	characteristic
132	C₂×C₃	C₂×#104	1	1	yes	characteristic
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	#835	1	2	yes	—
136	C₂²	C₂×#217	1	1	yes	characteristic
137	C₄	—	2	1	no	—
138	C₄	—	2	1	no	—
139	C₄	—	4	2	no	—
140	C₄	—	4	2	no	—
141	C₄	#835	1	2	yes	—
142	C₄	#837	1	2	yes	—
143	C₄	C₂×#215	1	1	yes	characteristic
144	C₄	C₂×#216	1	1	yes	characteristic
145	C₄	#948	1	2	yes	—
146	C₃	#1493	1	1	yes	characteristic, complemented, Sylow
147	C₂	—	2	2	no	—
148	C₂	C₂×#835	1	1	yes	characteristic
149	C₂	#5234	1	1	yes	characteristic
150	C₂	C₂×#837	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#533

C₂

yes

maximal, characteristic

#4924

C₂

yes

maximal, characteristic

C₂×C₄×C₃⋊C₈

C₂

yes

maximal, characteristic

#4535

C₂

yes

maximal, complemented

#1493

—

maximal, Sylow

C₃×C₄⋊Q₈

C₄

yes

characteristic

C₃×C₄⋊Q₈

C₄

yes

characteristic

C₃×#125

—

C₃×C₂²⋊Q₈

—

C₃×C₂²⋊Q₈

C₄

yes

—

C₃×C₄×Q₈

—

C₃×C₄×Q₈

C₂²

yes

—

C₃×#117

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×#208

—

C₂²×C₃⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

C₂²

yes

—

C₄×C₃⋊C₈

—

C₄×C₃⋊C₈

C₂²

yes

—

#533

D₃

yes

characteristic

#423

—

C₂×C₄×C₈

—

Q₈⋊C₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

D₄

yes

characteristic

C₂²×C₃×C₄

D₄

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂×C₄

yes

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×C₄

yes

characteristic

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

D₄

yes

characteristic

C₃×C₄²

C₂³

yes

characteristic

#208

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×M₁₆

—

C₂²×C₈

—

C₄⋊Q₈

Dic₃

yes

characteristic

C₄⋊Q₈

Dic₃

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

Dic₃

yes

—

C₄×Q₈

—

C₄×Q₈

C₂×D₃

yes

—

#117

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₂³×C₃

C₂²⋊C₄

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

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

C₂²⋊C₄

yes

characteristic

C₂×C₃×C₄

SD₁₆

yes

—

C₂×C₃×C₄

Q₁₆

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₃⋊C₈

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₃

yes

—

C₂²×C₄

C₂×Dic₃

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₃

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×Dic₃

yes

C₄²

yes

C₄²

yes

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

—

100

C₂²×C₃

—

101

C₂²×C₃

#103

yes

—

102

C₂²×C₃

C₂×C₂²⋊C₄

yes

characteristic

103

C₃×C₄

—

104

C₃×C₄

—

105

C₃×C₄

—

106

C₃×C₄

—

107

C₃×C₄

#103

yes

—

108

C₃×C₄

#104

yes

—

109

C₃×C₄

#131

yes

—

110

C₃×C₄

C₂×SD₁₆

yes

characteristic

111

C₃×C₄

C₂×Q₁₆

yes

characteristic

112

C₂³

#217

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

#215

yes

—

120

C₂×C₄

#216

yes

—

121

C₂×C₄

#217

yes

characteristic

122

C₂×C₄

#217

yes

characteristic

123

C₂×C₄

#217

yes

characteristic

124

C₂×C₄

C₂²×Dic₃

yes

characteristic

125

C₂×C₄

C₂×#67

yes

characteristic

126

C₂×C₄

C₂×#67

yes

characteristic

127

C₈

—

128

C₈

—

129

C₂×C₃

—

130

C₂×C₃

#408

yes

characteristic

131

C₂×C₃

C₂×#103

yes

characteristic

132

C₂×C₃

C₂×#104

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

#835

yes

—

136

C₂²

C₂×#217

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#835

yes

—

142

C₄

#837

yes

—

143

C₄

C₂×#215

yes

characteristic

144

C₄

C₂×#216

yes

characteristic

145

C₄

#948

yes

—

146

C₃

#1493

yes

characteristic, complemented, Sylow

147

C₂

—

148

C₂

C₂×#835

yes

characteristic

149

C₂

#5234

yes

characteristic

150

C₂

C₂×#837

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#533	C₂
socle	C₂²×C₃	C₂×C₂²⋊C₄
prime radical	C₂³×C₃	C₂²⋊C₄
cosocle	C₃×C₄²	C₂³
hypercenter	#533	D₃
hypocenter	C₃	#1493
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×#67

commutator

C₂×C₃×C₄

C₂²×C₄

Frattini

C₄²

C₂²×D₃

Fitting

C₃×#533

C₂

socle

C₂²×C₃

C₂×C₂²⋊C₄

prime radical

C₂³×C₃

C₂²⋊C₄

cosocle

C₃×C₄²

C₂³

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
4. #4535	C₂	semidirect
146. C₃	#1493	semidirect

H ≅ G/N

type

4. #4535

C₂

semidirect

146. C₃

#1493

semidirect

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

subgroup	index
C₃×#533	2
#4924	2
C₂×C₄×C₃⋊C₈	2
#4535	2
#1493	3

subgroup

index

C₂×C₄×C₃⋊C₈

Subgroups — #71826

Conjugacy classes of subgroups

Complements

Maximal subgroups