Group #9840 — subgroups

Subgroups — #9840

Order 256.

239 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 239 conjugacy classes of subgroups — too many to draw. Only its 91 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×#487	C₂	1	1	yes	maximal, characteristic
2	#1990	C₂	1	1	yes	maximal, characteristic
3	#1990	C₂	1	1	yes	maximal, characteristic
4	#1990	C₂	1	1	yes	maximal, characteristic
5	#1956	C₂	1	1	yes	maximal, characteristic
6	#1834	C₂	1	1	yes	maximal, characteristic
7	C₂×#322	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	#487	—	2	2	no	—
11	#487	C₄	1	4	yes	complemented
12	C₂×#103	—	2	2	no	—
13	C₂×#103	—	2	2	no	—
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
17	#384	C₂²	1	1	yes	characteristic
18	#384	C₂²	1	1	yes	characteristic
19	#340	—	2	2	no	—
20	#340	—	2	2	no	—
21	#340	—	2	2	no	—
22	#340	—	2	2	no	—
23	#322	—	2	2	no	—
24	C₂²×Q₈	—	2	1	no	—
25	C₂²×Q₈	—	2	1	no	—
26	C₂²×C₈	D₄	1	2	yes	—
27	C₂²×C₈	D₄	1	2	yes	—
28	C₂²×C₈	D₄	1	2	yes	—
29	C₄⋊Q₈	—	2	1	no	—
30	C₄⋊Q₈	—	2	1	no	—
31	C₄⋊Q₈	—	2	1	no	—
32	C₄⋊Q₈	—	2	1	no	—
33	C₄⋊Q₈	—	2	1	no	—
34	C₄⋊Q₈	—	2	1	no	—
35	C₄⋊Q₈	C₂×C₄	1	2	yes	—
36	C₄⋊Q₈	C₂×C₄	1	2	yes	—
37	C₂×C₄⋊C₄	—	2	1	no	—
38	C₂×C₄⋊C₄	—	2	1	no	—
39	C₂×C₄⋊C₄	—	2	1	no	—
40	C₂×C₄⋊C₄	—	2	1	no	—
41	C₂×C₄⋊C₄	—	2	1	no	—
42	C₂×C₄⋊C₄	—	2	1	no	—
43	C₂×C₄⋊C₄	—	2	1	no	—
44	C₂×C₄⋊C₄	—	2	1	no	—
45	C₂×C₄⋊C₄	—	2	1	no	—
46	C₂×C₄⋊C₄	—	2	1	no	—
47	C₂×C₄²	C₂³	1	1	yes	characteristic
48	#103	—	2	8	no	—
49	#103	—	4	4	no	—
50	C₄×C₈	—	2	1	no	—
51	C₄×C₈	—	2	1	no	—
52	C₄×C₈	—	2	1	no	—
53	C₄×C₈	—	2	1	no	—
54	C₄×C₈	C₂×C₄	1	2	yes	—
55	C₄×C₈	D₄	1	2	yes	—
56	C₂×Q₈	—	2	2	no	—
57	C₂×Q₈	—	2	2	no	—
58	C₂×Q₈	—	4	1	no	—
59	C₂×Q₈	—	4	1	no	—
60	C₂×Q₈	—	4	2	no	—
61	C₂×Q₈	—	4	2	no	—
62	C₂×Q₈	—	4	2	no	—
63	C₂×Q₈	—	4	2	no	—
64	C₂²×C₄	—	4	1	no	—
65	C₂²×C₄	—	4	1	no	—
66	C₂²×C₄	—	4	1	no	—
67	C₂²×C₄	—	4	1	no	—
68	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
69	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	2	2	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	2	2	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	C₂²⋊C₄	1	4	yes	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	2	no	—
84	C₄⋊C₄	—	4	1	no	—
85	C₄⋊C₄	—	4	1	no	—
86	C₄⋊C₄	—	4	1	no	—
87	C₄⋊C₄	—	4	1	no	—
88	C₄⋊C₄	—	4	2	no	—
89	C₄⋊C₄	—	4	2	no	—
90	C₄⋊C₄	—	4	2	no	—
91	C₄⋊C₄	—	4	2	no	—
92	C₄⋊C₄	—	4	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄²	C₂²×C₄	1	1	yes	characteristic
95	C₄²	C₂×D₄	1	1	yes	characteristic
96	C₄²	#41	1	1	yes	characteristic
97	C₄²	#41	1	1	yes	characteristic
98	C₂³	#127	1	1	yes	characteristic
99	Q₈	—	4	4	no	—
100	Q₈	—	4	4	no	—
101	Q₈	—	8	2	no	—
102	Q₈	—	8	2	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	8	1	no	—
106	C₂×C₄	—	8	1	no	—
107	C₂×C₄	—	8	1	no	—
108	C₂×C₄	—	8	1	no	—
109	C₂×C₄	—	8	1	no	—
110	C₂×C₄	—	8	1	no	—
111	C₂×C₄	—	8	1	no	—
112	C₂×C₄	—	8	1	no	—
113	C₂×C₄	—	8	1	no	—
114	C₂×C₄	—	8	1	no	—
115	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
116	C₂×C₄	C₄×D₄	1	1	yes	characteristic
117	C₂×C₄	C₄×D₄	1	1	yes	characteristic
118	C₂×C₄	#121	1	1	yes	characteristic
119	C₂×C₄	#121	1	1	yes	characteristic
120	C₂×C₄	#124	1	1	yes	characteristic
121	C₂×C₄	#135	1	2	yes	—
122	C₂×C₄	#135	1	2	yes	—
123	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
124	C₂×C₄	#137	1	2	yes	—
125	C₂×C₄	#137	1	2	yes	—
126	C₂×C₄	#137	1	2	yes	—
127	C₈	—	2	4	no	—
128	C₈	—	4	2	no	—
129	C₈	—	4	2	no	—
130	C₂²	#390	1	1	yes	characteristic
131	C₂²	#495	1	1	yes	characteristic
132	C₂²	#495	1	1	yes	characteristic
133	C₂²	#496	1	1	yes	characteristic
134	C₂²	#497	1	1	yes	characteristic
135	C₂²	#497	1	1	yes	characteristic
136	C₂²	#497	1	1	yes	characteristic
137	C₄	—	8	2	no	—
138	C₄	—	8	2	no	—
139	C₄	—	16	1	no	—
140	C₄	—	16	1	no	—
141	C₄	#416	1	2	yes	—
142	C₄	#440	1	2	yes	—
143	C₄	#441	1	2	yes	—
144	C₄	#468	1	2	yes	—
145	C₄	#470	1	2	yes	—
146	C₄	#488	1	2	yes	—
147	C₂	#1718	1	1	yes	characteristic
148	C₂	#1718	1	1	yes	characteristic
149	C₂	#1719	1	1	yes	characteristic
150	C₂	#1720	1	1	yes	characteristic
151	C₂	#1958	1	1	yes	characteristic
152	C₂	#1971	1	1	yes	characteristic
153	C₂	#1973	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#487

C₂

yes

maximal, characteristic

#1990

C₂

yes

maximal, characteristic

#1990

C₂

yes

maximal, characteristic

#1990

C₂

yes

maximal, characteristic

#1956

C₂

yes

maximal, characteristic

#1834

C₂

yes

maximal, characteristic

C₂×#322

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂²

yes

characteristic

C₂×C₄⋊Q₈

C₂²

yes

characteristic

#487

—

#487

C₄

yes

complemented

C₂×#103

—

C₂×#103

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#384

C₂²

yes

characteristic

#384

C₂²

yes

—

—

—

—

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×C₄

yes

—

C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

characteristic

#103

—

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

C₄²

yes

C₄²

yes

C₂³

yes

Q₈

—

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

116

C₂×C₄

C₄×D₄

yes

characteristic

117

C₂×C₄

C₄×D₄

yes

characteristic

118

C₂×C₄

#121

yes

characteristic

119

C₂×C₄

#121

yes

characteristic

120

C₂×C₄

#124

yes

characteristic

121

C₂×C₄

#135

yes

—

122

C₂×C₄

#135

yes

—

123

C₂×C₄

AGL(1,ℤ/8)

yes

—

124

C₂×C₄

#137

yes

—

125

C₂×C₄

#137

yes

—

126

C₂×C₄

#137

yes

—

127

C₈

—

128

C₈

—

129

C₈

—

130

C₂²

#390

yes

characteristic

131

C₂²

#495

yes

characteristic

132

C₂²

#495

yes

characteristic

133

C₂²

#496

yes

characteristic

134

C₂²

#497

yes

characteristic

135

C₂²

#497

yes

characteristic

136

C₂²

#497

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#416

yes

—

142

C₄

#440

yes

—

143

C₄

#441

yes

—

144

C₄

#468

yes

—

145

C₄

#470

yes

—

146

C₄

#488

yes

—

147

C₂

#1718

yes

characteristic

148

C₂

#1718

yes

characteristic

149

C₂

#1719

yes

characteristic

150

C₂

#1720

yes

characteristic

151

C₂

#1958

yes

characteristic

152

C₂

#1971

yes

characteristic

153

C₂

#1973

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#127
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

#127

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
11. #487	C₄	semidirect

H ≅ G/N

type

11. #487

C₄

semidirect

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

subgroup	index
C₂×#487	2
#1990	2
#1956	2
#1834	2
C₂×#322	2

subgroup

index

Subgroups — #9840

Conjugacy classes of subgroups

Complements

Maximal subgroups