Group #7682 — subgroups

Subgroups — #7682

Order 256.

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

normal subgroups only

This group has 224 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	#2862	C₂	1	1	yes	maximal, characteristic
2	#1767	C₂	1	1	yes	maximal, characteristic
3	#1744	C₂	1	1	yes	maximal, characteristic
4	#1272	C₂	1	4	yes	maximal
5	C₂×C₄⋊Q₈	C₄	1	1	yes	characteristic
6	C₂×C₄⋊Q₈	C₄	1	1	yes	characteristic
7	C₂×C₄⋊C₈	—	2	1	no	—
8	C₂×#97	—	2	1	no	—
9	C₂×C₄×C₈	—	2	1	no	—
10	#391	—	2	1	no	—
11	#391	—	2	1	no	—
12	#391	C₄	1	2	yes	—
13	#391	C₂²	1	2	yes	—
14	#382	—	2	1	no	—
15	#382	—	2	1	no	—
16	C₄³	C₂²	1	1	yes	characteristic
17	#336	—	2	1	no	—
18	#336	C₂²	1	2	yes	—
19	#336	C₂²	1	2	yes	—
20	C₂²×Q₈	—	2	1	no	—
21	C₂²×Q₈	C₂×C₄	1	2	yes	—
22	C₂²×C₈	—	2	2	no	—
23	C₂²×C₈	—	2	2	no	—
24	C₄⋊Q₈	—	4	2	no	—
25	C₄⋊Q₈	—	4	2	no	—
26	C₂×C₄⋊C₄	—	2	1	no	—
27	C₂×C₄⋊C₄	—	2	1	no	—
28	C₂×C₄⋊C₄	—	2	1	no	—
29	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
30	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄²	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄²	D₄	1	1	yes	characteristic
33	C₂×C₄²	D₄	1	1	yes	characteristic
34	C₂×C₄²	D₄	1	1	yes	characteristic
35	C₂×C₄²	D₄	1	1	yes	characteristic
36	C₂×C₄²	C₂³	1	1	yes	characteristic
37	C₄⋊C₈	—	2	4	no	—
38	#97	—	4	2	no	—
39	C₄×C₈	—	2	4	no	—
40	#95	—	2	2	no	—
41	#95	—	2	2	no	—
42	#95	—	2	2	no	—
43	#95	—	2	2	no	—
44	C₂×Q₈	—	2	4	no	—
45	C₂×Q₈	—	4	2	no	—
46	C₂×Q₈	—	4	2	no	—
47	C₂×Q₈	—	4	2	no	—
48	C₂×Q₈	—	4	2	no	—
49	C₂×Q₈	—	4	2	no	—
50	C₂²×C₄	—	2	2	no	—
51	C₂²×C₄	—	2	2	no	—
52	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
53	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
54	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
55	C₂²×C₄	C₂²⋊C₄	1	1	yes	characteristic
56	C₂²×C₄	C₂²×C₄	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	4	no	—
60	C₂×C₈	—	2	4	no	—
61	C₂×C₈	—	4	2	no	—
62	C₂×C₈	—	4	2	no	—
63	C₂×C₈	—	4	2	no	—
64	C₂×C₈	—	4	2	no	—
65	C₄⋊C₄	—	4	2	no	—
66	C₄⋊C₄	—	4	2	no	—
67	C₄⋊C₄	—	4	2	no	—
68	C₄⋊C₄	—	4	2	no	—
69	C₄²	—	2	2	no	—
70	C₄²	—	2	2	no	—
71	C₄²	—	2	2	no	—
72	C₄²	—	2	2	no	—
73	C₄²	—	2	2	no	—
74	C₄²	—	4	1	no	—
75	C₄²	SD₁₆	1	2	yes	—
76	C₄²	Q₁₆	1	2	yes	—
77	C₂³	C₂×C₂²⋊C₄	1	1	yes	characteristic
78	Q₈	—	8	4	no	—
79	Q₈	—	8	4	no	—
80	C₂×C₄	—	2	1	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	2	1	no	—
88	C₂×C₄	—	2	1	no	—
89	C₂×C₄	—	2	1	no	—
90	C₂×C₄	—	2	1	no	—
91	C₂×C₄	—	2	1	no	—
92	C₂×C₄	—	2	1	no	—
93	C₂×C₄	—	4	2	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	4	2	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	#100	1	2	yes	—
100	C₂×C₄	#103	1	2	yes	—
101	C₂×C₄	#103	1	2	yes	—
102	C₂×C₄	#104	1	2	yes	—
103	C₂×C₄	#104	1	2	yes	—
104	C₂×C₄	#131	1	2	yes	—
105	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
106	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
107	C₂×C₄	#137	1	2	yes	—
108	C₈	—	4	4	no	—
109	C₈	—	4	4	no	—
110	C₂²	#408	1	1	yes	characteristic
111	C₂²	#410	1	1	yes	characteristic
112	C₂²	C₂×#100	1	1	yes	characteristic
113	C₂²	C₂×#103	1	1	yes	characteristic
114	C₂²	#419	1	1	yes	characteristic
115	C₂²	C₂×#104	1	1	yes	characteristic
116	C₂²	C₂×#104	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	2	1	no	—
121	C₄	—	2	1	no	—
122	C₄	—	2	2	no	—
123	C₄	—	4	1	no	—
124	C₄	—	8	2	no	—
125	C₄	—	8	2	no	—
126	C₄	#329	1	4	yes	—
127	C₂	#1469	1	1	yes	characteristic
128	C₂	#1485	1	1	yes	characteristic
129	C₂	#1493	1	1	yes	characteristic
130	C₂	#1503	1	1	yes	characteristic
131	C₂	#1506	1	1	yes	characteristic
132	C₂	C₂×#329	1	1	yes	characteristic
133	C₂	#1548	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2862

C₂

yes

maximal, characteristic

#1767

C₂

yes

maximal, characteristic

#1744

C₂

yes

maximal, characteristic

#1272

C₂

yes

maximal

C₂×C₄⋊Q₈

C₄

yes

characteristic

C₂×C₄⋊Q₈

C₄

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

#391

—

#391

—

#391

C₄

yes

—

#391

C₂²

yes

—

#382

—

#382

—

C₄³

C₂²

yes

characteristic

#336

—

#336

C₂²

yes

—

#336

C₂²

yes

—

C₂²×Q₈

—

C₂²×Q₈

C₂×C₄

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

C₂×C₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

#97

—

C₄×C₈

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

C₂²⋊C₄

yes

characteristic

C₂²×C₄

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

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

SD₁₆

yes

—

C₄²

Q₁₆

yes

—

C₂³

C₂×C₂²⋊C₄

yes

characteristic

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

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#100

yes

—

100

C₂×C₄

#103

yes

—

101

C₂×C₄

#103

yes

—

102

C₂×C₄

#104

yes

—

103

C₂×C₄

#104

yes

—

104

C₂×C₄

#131

yes

—

105

C₂×C₄

C₂×SD₁₆

yes

characteristic

106

C₂×C₄

C₂×Q₁₆

yes

characteristic

107

C₂×C₄

#137

yes

—

108

C₈

—

109

C₈

—

110

C₂²

#408

yes

characteristic

111

C₂²

#410

yes

characteristic

112

C₂²

C₂×#100

yes

characteristic

113

C₂²

C₂×#103

yes

characteristic

114

C₂²

#419

yes

characteristic

115

C₂²

C₂×#104

yes

characteristic

116

C₂²

C₂×#104

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#329

yes

—

127

C₂

#1469

yes

characteristic

128

C₂

#1485

yes

characteristic

129

C₂

#1493

yes

characteristic

130

C₂

#1503

yes

characteristic

131

C₂

#1506

yes

characteristic

132

C₂

C₂×#329

yes

characteristic

133

C₂

#1548

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂×C₂²⋊C₄

commutator

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³ ◁ C₄³ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

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

subgroup	index
#2862	2
#1767	2
#1744	2
#1272	2

subgroup

index

Subgroups — #7682

Conjugacy classes of subgroups

Maximal subgroups