Group #8933 — subgroups

Subgroups — #8933

Order 256.

179 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 179 conjugacy classes of subgroups — too many to draw. Only its 71 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	#2962	C₂	1	1	yes	maximal, characteristic
2	#1950	C₂	1	1	yes	maximal, characteristic
3	#1571	C₂	1	1	yes	maximal, characteristic
4	#1542	C₂	1	1	yes	maximal, characteristic
5	#1493	C₂	1	1	yes	maximal, characteristic
6	#1278	C₂	1	1	yes	maximal, characteristic
7	#1277	C₂	1	1	yes	maximal, characteristic
8	#533	C₂²	1	1	yes	characteristic
9	#446	—	2	2	no	—
10	#446	—	2	1	no	—
11	C₈×Q₈	—	2	1	no	—
12	#431	C₂²	1	1	yes	characteristic
13	C₂×#108	—	2	1	no	—
14	#424	—	2	1	no	—
15	#423	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	#405	—	2	1	no	—
19	C₄×M₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
21	#341	—	2	1	no	—
22	#335	—	2	1	no	—
23	#335	—	2	1	no	—
24	#333	C₄	1	1	yes	characteristic
25	#333	C₄	1	1	yes	characteristic
26	#332	C₄	1	1	yes	characteristic
27	#332	C₄	1	1	yes	characteristic
28	Q₈⋊C₈	—	2	2	no	—
29	C₂×M₁₆	—	2	1	no	—
30	C₂×M₁₆	D₄	1	1	yes	characteristic
31	C₂×M₁₆	D₄	1	1	yes	characteristic
32	C₂²×C₈	—	2	1	no	—
33	C₂²×C₈	—	2	1	no	—
34	C₂²×C₈	—	2	1	no	—
35	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
36	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
37	#125	—	2	1	no	—
38	C₂²⋊Q₈	—	2	1	no	—
39	C₂²⋊Q₈	Q₈	1	2	yes	—
40	C₄×Q₈	—	2	1	no	—
41	C₄×Q₈	D₄	1	2	yes	—
42	#117	—	2	1	no	—
43	C₂×C₄²	C₂³	1	1	yes	characteristic
44	#108	—	4	2	no	—
45	C₄⋊C₈	—	2	2	no	—
46	C₄⋊C₈	—	2	2	no	—
47	C₄⋊C₈	—	2	1	no	—
48	C₄⋊C₈	—	2	1	no	—
49	C₄⋊C₈	—	2	1	no	—
50	C₄⋊C₈	—	2	1	no	—
51	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
52	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₄⋊C₈	C₂×C₄	1	1	yes	characteristic
55	C₄⋊C₈	D₄	1	1	yes	characteristic
56	C₄⋊C₈	D₄	1	1	yes	characteristic
57	C₂²⋊C₈	—	2	1	no	—
58	#97	—	2	2	no	—
59	#97	—	2	1	no	—
60	C₄×C₈	—	2	2	no	—
61	C₄×C₈	—	2	1	no	—
62	C₄×C₈	—	2	1	no	—
63	C₄×C₈	—	2	1	no	—
64	C₄×C₈	—	2	1	no	—
65	C₂×Q₈	—	2	1	no	—
66	C₂×Q₈	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄	C₂×Q₈	1	1	yes	characteristic
69	C₂²×C₄	#41	1	1	yes	characteristic
70	M₁₆	—	2	1	no	—
71	M₁₆	—	2	1	no	—
72	M₁₆	—	2	1	no	—
73	M₁₆	—	4	1	no	—
74	M₁₆	—	4	1	no	—
75	M₁₆	Q₁₆	1	2	yes	—
76	C₂×C₈	—	2	1	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	2	1	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	4	1	no	—
86	C₂×C₈	—	4	1	no	—
87	C₂×C₈	—	4	1	no	—
88	C₂×C₈	—	4	1	no	—
89	C₂×C₈	—	4	1	no	—
90	C₂×C₈	—	4	1	no	—
91	C₂×C₈	—	4	1	no	—
92	C₄⋊C₄	—	2	1	no	—
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
98	C₂²⋊C₄	—	2	2	no	—
99	C₄²	—	2	2	no	—
100	C₄²	C₂²×C₄	1	1	yes	characteristic
101	C₄²	C₂×D₄	1	1	yes	characteristic
102	C₄²	C₂×D₄	1	1	yes	characteristic
103	C₄²	#41	1	1	yes	characteristic
104	C₂³	C₂²⋊Q₈	1	1	yes	characteristic
105	Q₈	—	4	2	no	—
106	Q₈	—	4	1	no	—
107	Q₈	—	4	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
114	C₂×C₄	C₄×D₄	1	1	yes	characteristic
115	C₂×C₄	C₄×D₄	1	1	yes	characteristic
116	C₂×C₄	#120	1	1	yes	characteristic
117	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
118	C₂×C₄	#123	1	1	yes	characteristic
119	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
120	C₂×C₄	#135	1	1	yes	characteristic
121	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
122	C₂×C₄	#137	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	—	4	2	no	—
125	C₈	—	4	1	no	—
126	C₈	—	4	1	no	—
127	C₈	—	4	1	no	—
128	C₈	—	4	1	no	—
129	C₈	—	4	1	no	—
130	C₈	—	4	1	no	—
131	C₈	—	4	1	no	—
132	C₈	—	8	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	#385	1	1	yes	characteristic
136	C₂²	#477	1	1	yes	characteristic
137	C₂²	#479	1	1	yes	characteristic
138	C₄	—	2	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	—	4	2	no	—
142	C₄	—	4	2	no	—
143	C₄	C₄×Q₁₆	1	1	yes	characteristic
144	C₄	#440	1	1	yes	characteristic
145	C₄	#443	1	1	yes	characteristic
146	C₄	#444	1	1	yes	characteristic
147	C₄	#449	1	1	yes	characteristic
148	C₄	#451	1	1	yes	characteristic
149	C₂	—	2	1	no	—
150	C₂	—	2	1	no	—
151	C₂	#1858	1	1	yes	characteristic
152	C₂	#1863	1	1	yes	characteristic
153	C₂	#1866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2962

C₂

yes

maximal, characteristic

#1950

C₂

yes

maximal, characteristic

#1571

C₂

yes

maximal, characteristic

#1542

C₂

yes

maximal, characteristic

#1493

C₂

yes

maximal, characteristic

#1278

C₂

yes

maximal, characteristic

#1277

C₂

yes

maximal, characteristic

#533

C₂²

yes

characteristic

#446

—

#446

—

C₈×Q₈

—

#431

C₂²

yes

—

—

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

#405

—

C₄×M₁₆

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

—

—

—

C₄

yes

characteristic

#333

C₄

yes

characteristic

#332

C₄

yes

characteristic

#332

C₄

yes

characteristic

Q₈⋊C₈

—

C₂×M₁₆

—

C₂×M₁₆

D₄

yes

characteristic

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

C₂×C₄

yes

characteristic

C₄⋊Q₈

C₂×C₄

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

Q₈

yes

—

C₄×Q₈

—

C₄×Q₈

D₄

yes

—

#117

—

C₂×C₄²

C₂³

yes

characteristic

#108

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

characteristic

C₄⋊C₈

C₂×C₄

yes

characteristic

C₄⋊C₈

C₂×C₄

yes

characteristic

C₄⋊C₈

C₂×C₄

yes

characteristic

C₄⋊C₈

D₄

yes

characteristic

C₄⋊C₈

D₄

yes

characteristic

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

C₄⋊C₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×Q₈

yes

characteristic

C₂²×C₄

#41

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

Q₁₆

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

yes

—

C₂²⋊C₄

—

C₄²

—

100

C₄²

C₂²×C₄

yes

characteristic

101

C₄²

C₂×D₄

yes

characteristic

102

C₄²

C₂×D₄

yes

characteristic

103

C₄²

#41

yes

characteristic

104

C₂³

C₂²⋊Q₈

yes

characteristic

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

114

C₂×C₄

C₄×D₄

yes

characteristic

115

C₂×C₄

C₄×D₄

yes

characteristic

116

C₂×C₄

#120

yes

characteristic

117

C₂×C₄

C₂²⋊Q₈

yes

characteristic

118

C₂×C₄

#123

yes

characteristic

119

C₂×C₄

C₂×Q₁₆

yes

characteristic

120

C₂×C₄

#135

yes

characteristic

121

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

122

C₂×C₄

#137

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₈

—

133

C₂²

—

134

C₂²

—

135

C₂²

#385

yes

characteristic

136

C₂²

#477

yes

characteristic

137

C₂²

#479

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

C₄×Q₁₆

yes

characteristic

144

C₄

#440

yes

characteristic

145

C₄

#443

yes

characteristic

146

C₄

#444

yes

characteristic

147

C₄

#449

yes

characteristic

148

C₄

#451

yes

characteristic

149

C₂

—

150

C₂

—

151

C₂

#1858

yes

characteristic

152

C₂

#1863

yes

characteristic

153

C₂

#1866

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#120
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	#385
prime radical	C₂³	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

#120

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

socle

C₂²

#385

prime radical

C₂³

C₂²⋊Q₈

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

subgroup	index
#2962	2
#1950	2
#1571	2
#1542	2
#1493	2
#1278	2
#1277	2

subgroup

index

Subgroups — #8933

Conjugacy classes of subgroups

Maximal subgroups