Group #102105 — subgroups

Subgroups — #102105

Order 512.

528 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 182 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	C₄×#494	C₂	1	1	yes	maximal, characteristic
2	#9870	C₂	1	4	yes	maximal
3	#8806	C₂	1	1	yes	maximal, characteristic
4	C₄×C₈²	C₂	1	1	yes	maximal, characteristic
5	C₂×#494	C₄	1	1	yes	characteristic
6	C₂×C₄×Q₁₆	—	2	2	no	—
7	C₄×C₄⋊Q₈	C₂²	1	2	yes	—
8	#1969	—	2	2	no	—
9	#1956	C₄	1	2	yes	—
10	#1845	C₄	1	1	yes	characteristic
11	#1842	C₂²	1	2	yes	—
12	C₄²×C₈	C₂²	1	2	yes	—
13	C₄²×C₈	C₂²	1	1	yes	characteristic
14	C₂×#335	—	2	2	no	—
15	C₂×C₈²	—	2	2	no	—
16	#1279	—	2	8	no	—
17	C₂²×Q₁₆	—	2	2	no	—
18	C₂×C₄⋊Q₈	C₂×C₄	1	2	yes	—
19	C₂×C₄×Q₈	—	2	2	no	—
20	#494	—	2	2	no	—
21	#494	C₈	1	4	yes	complemented
22	C₄×Q₁₆	—	4	4	no	—
23	C₄×Q₁₆	—	4	4	no	—
24	C₂×#107	—	2	2	no	—
25	C₂×C₄⋊C₈	—	2	2	no	—
26	C₂×C₄⋊C₈	—	2	4	no	—
27	C₂×#103	—	2	4	no	—
28	C₂×C₄×C₈	—	2	2	no	—
29	C₂×C₄×C₈	—	2	2	no	—
30	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄×C₈	D₄	1	2	yes	—
33	C₂×C₄×C₈	D₄	1	4	yes	—
34	C₂×C₄×C₈	D₄	1	2	yes	—
35	#391	—	2	2	no	—
36	#389	—	2	4	no	—
37	#384	C₂×C₄	1	2	yes	—
38	C₄×C₄⋊C₄	—	2	4	no	—
39	C₄³	C₂³	1	1	yes	characteristic
40	#335	—	4	4	no	—
41	C₈²	—	2	8	no	—
42	C₂²×Q₈	—	2	2	no	—
43	C₂×Q₁₆	—	2	8	no	—
44	C₂×Q₁₆	—	4	4	no	—
45	C₂×Q₁₆	—	4	4	no	—
46	C₂²×C₈	—	2	4	no	—
47	C₂²×C₈	—	4	2	no	—
48	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
49	C₂²×C₈	C₂²⋊C₄	1	2	yes	—
50	C₄⋊Q₈	—	2	2	no	—
51	C₄⋊Q₈	—	2	4	no	—
52	C₄⋊Q₈	C₂×C₈	1	4	yes	—
53	C₄×Q₈	—	4	4	no	—
54	C₄×Q₈	—	4	4	no	—
55	C₂×C₄⋊C₄	—	2	4	no	—
56	C₂×C₄⋊C₄	—	2	2	no	—
57	C₂×C₄⋊C₄	—	2	4	no	—
58	C₂×C₄⋊C₄	—	4	2	no	—
59	C₂×C₄²	—	4	2	no	—
60	C₂×C₄²	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
62	C₂×C₄²	C₂×D₄	1	1	yes	characteristic
63	C₂×C₄²	C₂×D₄	1	2	yes	—
64	C₂×C₄²	#41	1	2	yes	—
65	#107	—	4	4	no	—
66	C₄⋊C₈	—	4	4	no	—
67	C₄⋊C₈	—	4	8	no	—
68	#103	—	4	8	no	—
69	C₄×C₈	—	2	8	no	—
70	C₄×C₈	—	2	8	no	—
71	C₄×C₈	—	2	4	no	—
72	C₄×C₈	—	2	2	no	—
73	C₄×C₈	—	2	4	no	—
74	C₄×C₈	C₂×C₈	1	2	yes	—
75	C₄×C₈	M₁₆	1	2	yes	—
76	C₄×C₈	D₈	1	4	yes	—
77	C₄×C₈	SD₁₆	1	4	yes	—
78	C₄×C₈	Q₁₆	1	8	yes	complemented
79	#95	—	4	4	no	—
80	C₂×Q₈	—	2	4	no	—
81	C₂×Q₈	—	4	2	no	—
82	C₂×Q₈	—	4	4	no	—
83	C₂×Q₈	—	4	4	no	—
84	C₂²×C₄	—	4	2	no	—
85	C₂²×C₄	—	4	2	no	—
86	C₂²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
87	C₂²×C₄	C₄×D₄	1	2	yes	—
88	C₂²×C₄	#121	1	2	yes	—
89	C₂²×C₄	#124	1	1	yes	characteristic
90	C₂²×C₄	#127	1	1	yes	characteristic
91	Q₁₆	—	4	16	no	—
92	C₂×C₈	—	2	8	no	—
93	C₂×C₈	—	2	8	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	8	no	—
97	C₂×C₈	—	8	2	no	—
98	C₂×C₈	—	8	4	no	—
99	C₂×C₈	C₂²⋊C₈	1	4	yes	—
100	C₂×C₈	#102	1	4	yes	—
101	C₂×C₈	#102	1	4	yes	—
102	C₂×C₈	#104	1	4	yes	—
103	C₄⋊C₄	—	2	8	no	—
104	C₄⋊C₄	—	4	4	no	—
105	C₄⋊C₄	—	4	4	no	—
106	C₄⋊C₄	—	4	8	no	—
107	C₄⋊C₄	—	8	4	no	—
108	C₄²	—	2	2	no	—
109	C₄²	—	2	4	no	—
110	C₄²	—	8	2	no	—
111	C₄²	—	8	2	no	—
112	C₄²	C₂²×C₈	1	1	yes	characteristic
113	C₄²	C₂×M₁₆	1	1	yes	characteristic
114	C₄²	#131	1	2	yes	—
115	C₄²	C₂×D₈	1	2	yes	—
116	C₄²	C₂×SD₁₆	1	2	yes	—
117	C₄²	C₂×Q₁₆	1	4	yes	—
118	C₄²	#135	1	4	yes	—
119	C₂³	#390	1	1	yes	characteristic
120	Q₈	—	4	8	no	—
121	Q₈	—	8	4	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	2	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	4	4	no	—
126	C₂×C₄	—	8	2	no	—
127	C₂×C₄	—	8	2	no	—
128	C₂×C₄	—	8	2	no	—
129	C₂×C₄	—	8	4	no	—
130	C₂×C₄	C₂×C₂²⋊C₈	1	1	yes	characteristic
131	C₂×C₄	#408	1	1	yes	characteristic
132	C₂×C₄	C₂×#102	1	2	yes	—
133	C₂×C₄	C₂×#104	1	2	yes	—
134	C₂×C₄	C₈×D₄	1	2	yes	—
135	C₂×C₄	#436	1	2	yes	—
136	C₂×C₄	C₄×Q₁₆	1	4	yes	—
137	C₂×C₄	#443	1	4	yes	—
138	C₂×C₄	#467	1	4	yes	—
139	C₂×C₄	#486	1	2	yes	—
140	C₂×C₄	#492	1	1	yes	characteristic
141	C₂×C₄	#493	1	1	yes	characteristic
142	C₂×C₄	#494	1	2	yes	—
143	C₂×C₄	#495	1	2	yes	—
144	C₈	—	2	16	no	—
145	C₈	—	2	4	no	—
146	C₈	—	16	2	no	—
147	C₈	D₄⋊C₈	1	8	yes	—
148	C₂²	#1954	1	1	yes	characteristic
149	C₂²	#1957	1	1	yes	characteristic
150	C₂²	#1962	1	1	yes	characteristic
151	C₂²	#1969	1	2	yes	—
152	C₂²	#1974	1	2	yes	—
153	C₄	—	2	4	no	—
154	C₄	—	2	2	no	—
155	C₄	—	8	4	no	—
156	C₄	—	16	2	no	—
157	C₄	C₂×D₄⋊C₈	1	2	yes	—
158	C₄	#1492	1	2	yes	—
159	C₄	C₈×Q₁₆	1	4	yes	—
160	C₄	#1591	1	4	yes	—
161	C₄	#1707	1	2	yes	—
162	C₄	#1709	1	1	yes	characteristic
163	C₄	#1713	1	1	yes	characteristic
164	C₂	#9638	1	1	yes	characteristic
165	C₂	#9656	1	2	yes	—
166	C₂	#9825	1	1	yes	characteristic
167	C₂	#9845	1	2	yes	—
168	C₂	#9856	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#494

C₂

yes

maximal, characteristic

#9870

C₂

yes

maximal

#8806

C₂

yes

maximal, characteristic

C₄×C₈²

C₂

yes

maximal, characteristic

C₂×#494

C₄

yes

characteristic

C₂×C₄×Q₁₆

—

C₄×C₄⋊Q₈

C₂²

yes

—

#1969

—

#1956

C₄

yes

—

#1845

C₄

yes

characteristic

#1842

C₂²

yes

—

C₄²×C₈

C₂²

yes

—

C₄²×C₈

C₂²

yes

characteristic

C₂×#335

—

C₂×C₈²

—

#1279

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₄×Q₈

—

#494

—

#494

C₈

yes

complemented

C₄×Q₁₆

—

C₄×Q₁₆

—

C₂×#107

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#103

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

C₂×C₄

yes

characteristic

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

—

C₂×C₄×C₈

D₄

yes

—

#391

—

#389

—

#384

C₂×C₄

yes

—

C₄×C₄⋊C₄

—

C₄³

C₂³

yes

characteristic

#335

—

C₈²

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₂²×C₈

C₂²⋊C₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×C₈

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂²×C₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

characteristic

C₂×C₄²

C₂×D₄

yes

—

C₂×C₄²

#41

yes

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₈

yes

—

C₄×C₈

M₁₆

yes

—

C₄×C₈

D₈

yes

—

C₄×C₈

SD₁₆

yes

—

C₄×C₈

Q₁₆

yes

complemented

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₄×D₄

yes

—

C₂²×C₄

#121

yes

—

C₂²×C₄

yes

C₂²×C₄

yes

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₈

yes

—

100

C₂×C₈

#102

yes

—

101

C₂×C₈

#102

yes

—

102

C₂×C₈

#104

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₄²

C₂²×C₈

yes

characteristic

113

C₄²

C₂×M₁₆

yes

characteristic

114

C₄²

#131

yes

—

115

C₄²

C₂×D₈

yes

—

116

C₄²

C₂×SD₁₆

yes

—

117

C₄²

C₂×Q₁₆

yes

—

118

C₄²

#135

yes

—

119

C₂³

#390

yes

characteristic

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂×C₂²⋊C₈

yes

characteristic

131

C₂×C₄

#408

yes

characteristic

132

C₂×C₄

C₂×#102

yes

—

133

C₂×C₄

C₂×#104

yes

—

134

C₂×C₄

C₈×D₄

yes

—

135

C₂×C₄

#436

yes

—

136

C₂×C₄

C₄×Q₁₆

yes

—

137

C₂×C₄

#443

yes

—

138

C₂×C₄

#467

yes

—

139

C₂×C₄

#486

yes

—

140

C₂×C₄

#492

yes

characteristic

141

C₂×C₄

#493

yes

characteristic

142

C₂×C₄

#494

yes

—

143

C₂×C₄

#495

yes

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

D₄⋊C₈

yes

—

148

C₂²

#1954

yes

characteristic

149

C₂²

#1957

yes

characteristic

150

C₂²

#1962

yes

characteristic

151

C₂²

#1969

yes

—

152

C₂²

#1974

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

C₂×D₄⋊C₈

yes

—

158

C₄

#1492

yes

—

159

C₄

C₈×Q₁₆

yes

—

160

C₄

#1591

yes

—

161

C₄

#1707

yes

—

162

C₄

#1709

yes

characteristic

163

C₄

#1713

yes

characteristic

164

C₂

#9638

yes

characteristic

165

C₂

#9656

yes

—

166

C₂

#9825

yes

characteristic

167

C₂

#9845

yes

—

168

C₂

#9856

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#390
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

subgroup

G/H

center

C₂²×C₄

#127

commutator

C₄²

C₂²×C₈

Frattini, cosocle

C₄³

C₂³

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G

LECS

G ▷ C₄³ ▷ C₂³ ▷ C₁

N	H ≅ G/N	type
21. #494	C₈	semidirect
78. C₄×C₈	Q₁₆	semidirect

H ≅ G/N

type

21. #494

C₈

semidirect

78. C₄×C₈

Q₁₆

semidirect

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

subgroup	index
C₄×#494	2
#9870	2
#8806	2
C₄×C₈²	2

subgroup

index

C₄×#494

#9870

#8806

C₄×C₈²

Subgroups — #102105

Conjugacy classes of subgroups

Complements

Maximal subgroups