Group #10682 — subgroups

Subgroups — #10682

Order 256.

197 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 197 conjugacy classes of subgroups — too many to draw. Only its 77 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	#1990	C₂	1	1	yes	maximal, characteristic
2	#1990	C₂	1	1	yes	maximal, characteristic
3	#1956	C₂	1	1	yes	maximal, characteristic
4	#1834	C₂	1	1	yes	maximal, characteristic
5	#1834	C₂	1	1	yes	maximal, characteristic
6	#1834	C₂	1	1	yes	maximal, characteristic
7	#1273	C₂	1	1	yes	maximal, characteristic
8	C₂×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×#103	—	2	2	no	—
10	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
13	#384	C₂²	1	1	yes	characteristic
14	#384	C₂²	1	1	yes	characteristic
15	#384	C₂²	1	1	yes	characteristic
16	#340	—	2	2	no	—
17	#340	—	2	2	no	—
18	#340	—	2	2	no	—
19	#340	—	2	2	no	—
20	#340	—	2	2	no	—
21	C₂²×Q₈	—	2	1	no	—
22	C₂²×C₈	D₄	1	2	yes	—
23	C₂²×C₈	D₄	1	2	yes	—
24	C₂²×C₈	D₄	1	2	yes	—
25	C₄⋊Q₈	—	2	1	no	—
26	C₄⋊Q₈	—	2	1	no	—
27	C₄⋊Q₈	—	2	1	no	—
28	C₄⋊Q₈	Q₈	1	2	yes	—
29	C₂×C₄⋊C₄	—	2	1	no	—
30	C₂×C₄⋊C₄	—	2	1	no	—
31	C₂×C₄⋊C₄	—	2	1	no	—
32	C₂×C₄⋊C₄	—	2	1	no	—
33	C₂×C₄⋊C₄	—	2	1	no	—
34	C₂×C₄⋊C₄	—	2	1	no	—
35	C₂×C₄⋊C₄	—	2	1	no	—
36	C₂×C₄⋊C₄	—	2	1	no	—
37	C₂×C₄⋊C₄	—	2	1	no	—
38	C₂×C₄⋊C₄	—	2	1	no	—
39	C₂×C₄⋊C₄	—	2	1	no	—
40	C₂×C₄²	C₂³	1	1	yes	characteristic
41	#103	—	4	4	no	—
42	C₄×C₈	—	2	1	no	—
43	C₄×C₈	—	2	1	no	—
44	C₄×C₈	—	2	1	no	—
45	C₄×C₈	—	2	1	no	—
46	C₄×C₈	—	2	1	no	—
47	C₄×C₈	—	2	1	no	—
48	C₂×Q₈	—	2	2	no	—
49	C₂×Q₈	—	4	1	no	—
50	C₂×Q₈	—	4	2	no	—
51	C₂×Q₈	—	4	2	no	—
52	C₂²×C₄	—	4	1	no	—
53	C₂²×C₄	—	4	1	no	—
54	C₂²×C₄	—	4	1	no	—
55	C₂²×C₄	—	4	1	no	—
56	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	—	2	2	no	—
68	C₄⋊C₄	—	2	2	no	—
69	C₄⋊C₄	—	2	2	no	—
70	C₄⋊C₄	—	4	1	no	—
71	C₄⋊C₄	—	4	1	no	—
72	C₄⋊C₄	—	4	2	no	—
73	C₄⋊C₄	—	4	2	no	—
74	C₄⋊C₄	—	4	2	no	—
75	C₄⋊C₄	—	4	2	no	—
76	C₄⋊C₄	—	4	2	no	—
77	C₄⋊C₄	—	4	2	no	—
78	C₄⋊C₄	—	4	2	no	—
79	C₄⋊C₄	—	4	2	no	—
80	C₄⋊C₄	—	4	2	no	—
81	C₄²	C₂×Q₈	1	1	yes	characteristic
82	C₄²	#41	1	1	yes	characteristic
83	C₄²	#41	1	1	yes	characteristic
84	C₄²	#41	1	1	yes	characteristic
85	C₂³	#127	1	1	yes	characteristic
86	Q₈	—	4	4	no	—
87	Q₈	—	8	2	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	8	1	no	—
90	C₂×C₄	—	8	1	no	—
91	C₂×C₄	—	8	1	no	—
92	C₂×C₄	—	8	1	no	—
93	C₂×C₄	—	8	1	no	—
94	C₂×C₄	—	8	1	no	—
95	C₂×C₄	—	8	1	no	—
96	C₂×C₄	—	8	1	no	—
97	C₂×C₄	—	8	1	no	—
98	C₂×C₄	—	8	1	no	—
99	C₂×C₄	—	8	1	no	—
100	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
101	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
102	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄	#123	1	1	yes	characteristic
104	C₂×C₄	#123	1	1	yes	characteristic
105	C₂×C₄	#123	1	1	yes	characteristic
106	C₂×C₄	#135	1	2	yes	—
107	C₂×C₄	#135	1	2	yes	—
108	C₂×C₄	#135	1	2	yes	—
109	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
110	C₂×C₄	#137	1	2	yes	—
111	C₂×C₄	#137	1	2	yes	—
112	C₈	—	4	2	no	—
113	C₈	—	4	2	no	—
114	C₈	—	4	2	no	—
115	C₂²	#399	1	1	yes	characteristic
116	C₂²	#495	1	1	yes	characteristic
117	C₂²	#495	1	1	yes	characteristic
118	C₂²	#495	1	1	yes	characteristic
119	C₂²	#496	1	1	yes	characteristic
120	C₂²	#497	1	1	yes	characteristic
121	C₂²	#497	1	1	yes	characteristic
122	C₄	—	8	2	no	—
123	C₄	—	16	1	no	—
124	C₄	—	16	1	no	—
125	C₄	—	16	1	no	—
126	C₄	#478	1	2	yes	—
127	C₄	#479	1	2	yes	—
128	C₄	#479	1	2	yes	—
129	C₄	#482	1	2	yes	—
130	C₄	#485	1	2	yes	—
131	C₄	#485	1	2	yes	—
132	C₂	#1711	1	1	yes	characteristic
133	C₂	#1718	1	1	yes	characteristic
134	C₂	#1719	1	1	yes	characteristic
135	C₂	#1720	1	1	yes	characteristic
136	C₂	#2092	1	1	yes	characteristic
137	C₂	#2097	1	1	yes	characteristic
138	C₂	#2097	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1990

C₂

yes

maximal, characteristic

#1990

C₂

yes

maximal, characteristic

#1956

C₂

yes

maximal, characteristic

#1834

C₂

yes

maximal, characteristic

#1834

C₂

yes

maximal, characteristic

#1834

C₂

yes

maximal, characteristic

#1273

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈

C₂²

yes

characteristic

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

characteristic

#384

C₂²

yes

—

—

—

—

—

C₂²×Q₈

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Q₈

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×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

characteristic

#103

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

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

—

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

C₂×Q₈

yes

C₄²

yes

C₄²

yes

C₄²

yes

C₂³

yes

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

C₂²⋊Q₈

yes

characteristic

101

C₂×C₄

C₂²⋊Q₈

yes

characteristic

102

C₂×C₄

C₂²⋊Q₈

yes

characteristic

103

C₂×C₄

#123

yes

characteristic

104

C₂×C₄

#123

yes

characteristic

105

C₂×C₄

#123

yes

characteristic

106

C₂×C₄

#135

yes

—

107

C₂×C₄

#135

yes

—

108

C₂×C₄

#135

yes

—

109

C₂×C₄

AGL(1,ℤ/8)

yes

—

110

C₂×C₄

#137

yes

—

111

C₂×C₄

#137

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₂²

#399

yes

characteristic

116

C₂²

#495

yes

characteristic

117

C₂²

#495

yes

characteristic

118

C₂²

#495

yes

characteristic

119

C₂²

#496

yes

characteristic

120

C₂²

#497

yes

characteristic

121

C₂²

#497

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#478

yes

—

127

C₄

#479

yes

—

128

C₄

#479

yes

—

129

C₄

#482

yes

—

130

C₄

#485

yes

—

131

C₄

#485

yes

—

132

C₂

#1711

yes

characteristic

133

C₂

#1718

yes

characteristic

134

C₂

#1719

yes

characteristic

135

C₂

#1720

yes

characteristic

136

C₂

#2092

yes

characteristic

137

C₂

#2097

yes

characteristic

138

C₂

#2097

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂³
LCS	G ▷ C₂×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, Frattini, cosocle

C₂×C₄²

C₂³

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ 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
#1990	2
#1956	2
#1834	2
#1273	2

subgroup

index

Subgroups — #10682

Conjugacy classes of subgroups

Maximal subgroups