Group #8246 — subgroups

Subgroups — #8246

Order 256.

188 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 85 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	#2960	C₂	1	1	yes	maximal, characteristic
2	#1544	C₂	1	1	yes	maximal, characteristic
3	#1526	C₂	1	1	yes	maximal, characteristic
4	#1451	C₂	1	1	yes	maximal, characteristic
5	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
6	#1299	C₂	1	1	yes	maximal, characteristic
7	#1299	C₂	1	1	yes	maximal, characteristic
8	#533	C₂²	1	1	yes	characteristic
9	#446	—	2	1	no	—
10	#446	—	2	1	no	—
11	C₈×Q₈	—	2	1	no	—
12	C₈×Q₈	—	2	1	no	—
13	#431	C₂²	1	1	yes	characteristic
14	#431	C₂²	1	1	yes	characteristic
15	#424	C₄	1	2	yes	—
16	#423	C₂²	1	1	yes	characteristic
17	#405	C₄	1	2	yes	—
18	C₄×M₁₆	C₂²	1	1	yes	characteristic
19	C₄×M₁₆	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
21	#332	C₄	1	2	yes	—
22	#332	C₄	1	2	yes	—
23	#330	C₄	1	4	yes	—
24	#322	—	2	1	no	—
25	#322	—	2	1	no	—
26	#322	—	2	1	no	—
27	C₈²	—	2	1	no	—
28	C₂×M₁₆	—	2	1	no	—
29	C₂×M₁₆	—	2	1	no	—
30	C₂×M₁₆	C₂×C₄	1	2	yes	—
31	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
33	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
34	C₄⋊Q₈	C₂×C₄	1	1	yes	characteristic
35	#125	C₂×C₄	1	2	yes	—
36	C₂²⋊Q₈	—	2	1	no	—
37	C₂²⋊Q₈	—	2	1	no	—
38	C₄×Q₈	—	2	1	no	—
39	C₄×Q₈	—	2	1	no	—
40	#117	C₂×C₄	1	2	yes	—
41	C₂×C₄²	C₂³	1	1	yes	characteristic
42	C₄⋊C₈	—	2	2	no	—
43	C₄⋊C₈	—	2	2	no	—
44	C₄⋊C₈	—	2	1	no	—
45	C₄⋊C₈	—	2	1	no	—
46	C₄⋊C₈	C₂×C₄	1	2	yes	—
47	C₄⋊C₈	C₂×C₄	1	2	yes	—
48	C₂²⋊C₈	—	2	1	no	—
49	C₂²⋊C₈	—	2	1	no	—
50	#97	—	2	2	no	—
51	#97	C₂×C₄	1	2	yes	—
52	#97	C₂×C₄	1	2	yes	—
53	C₄×C₈	—	2	2	no	—
54	C₄×C₈	—	2	1	no	—
55	C₄×C₈	—	2	1	no	—
56	C₄×C₈	—	2	1	no	—
57	C₄×C₈	—	2	1	no	—
58	C₄×C₈	D₄	1	1	yes	characteristic
59	C₄×C₈	D₄	1	1	yes	characteristic
60	C₄×C₈	D₄	1	1	yes	characteristic
61	C₄×C₈	D₄	1	1	yes	characteristic
62	C₂×Q₈	—	2	1	no	—
63	C₂×Q₈	—	2	1	no	—
64	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄	#41	1	1	yes	characteristic
66	C₂²×C₄	#41	1	1	yes	characteristic
67	M₁₆	—	2	4	no	—
68	M₁₆	—	4	2	no	—
69	M₁₆	—	4	2	no	—
70	C₂×C₈	—	2	2	no	—
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	—	4	1	no	—
77	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
78	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
79	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
80	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	2	1	no	—
84	C₄⋊C₄	—	2	1	no	—
85	C₄⋊C₄	C₄²	1	2	yes	—
86	C₄⋊C₄	C₄²	1	2	yes	—
87	C₂²⋊C₄	—	2	2	no	—
88	C₄²	—	2	2	no	—
89	C₄²	C₂²×C₄	1	1	yes	characteristic
90	C₄²	C₂²×C₄	1	1	yes	characteristic
91	C₄²	C₂×D₄	1	1	yes	characteristic
92	C₄²	C₂×D₄	1	1	yes	characteristic
93	C₂³	#117	1	1	yes	characteristic
94	Q₈	—	4	2	no	—
95	Q₈	—	4	2	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
103	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄	C₄×D₄	1	1	yes	characteristic
105	C₂×C₄	C₄×D₄	1	1	yes	characteristic
106	C₂×C₄	C₄×D₄	1	1	yes	characteristic
107	C₂×C₄	C₄×D₄	1	1	yes	characteristic
108	C₈	—	2	4	no	—
109	C₈	—	2	1	no	—
110	C₈	—	2	1	no	—
111	C₈	—	2	1	no	—
112	C₈	—	4	2	no	—
113	C₈	—	4	2	no	—
114	C₈	—	4	2	no	—
115	C₈	—	4	2	no	—
116	C₈	#101	1	2	yes	—
117	C₂²	—	2	1	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	C₄×C₂²⋊C₄	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	4	2	no	—
125	C₄	—	4	2	no	—
126	C₄	C₂×#101	1	1	yes	characteristic
127	C₄	#413	1	1	yes	characteristic
128	C₄	#443	1	2	yes	—
129	C₄	#444	1	2	yes	—
130	C₂	—	4	1	no	—
131	C₂	C₄×#101	1	1	yes	characteristic
132	C₂	#1764	1	1	yes	characteristic
133	C₂	#1765	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2960

C₂

yes

maximal, characteristic

#1544

C₂

yes

maximal, characteristic

#1526

C₂

yes

maximal, characteristic

#1451

C₂

yes

maximal, characteristic

C₈×M₁₆

C₂

yes

maximal, characteristic

#1299

C₂

yes

maximal, characteristic

#1299

C₂

yes

maximal, characteristic

#533

C₂²

yes

characteristic

#446

—

#446

—

C₈×Q₈

—

C₈×Q₈

—

#431

C₂²

yes

characteristic

#431

C₂²

yes

characteristic

#424

C₄

yes

—

#423

C₂²

yes

characteristic

#405

C₄

yes

—

C₄×M₁₆

C₂²

yes

characteristic

C₄×M₁₆

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#332

C₄

yes

—

#332

C₄

yes

—

#330

C₄

yes

—

#322

—

#322

—

#322

—

C₈²

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×C₄

yes

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₄⋊Q₈

C₂×C₄

yes

characteristic

C₄⋊Q₈

C₂×C₄

yes

characteristic

#125

C₂×C₄

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

C₂×C₄

yes

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂×C₄

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

C₂×C₄

yes

—

#97

C₂×C₄

yes

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

yes

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₄²

yes

—

C₄⋊C₄

C₄²

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₂³

#117

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

C₂×C₄²

yes

characteristic

103

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

104

C₂×C₄

C₄×D₄

yes

characteristic

105

C₂×C₄

C₄×D₄

yes

characteristic

106

C₂×C₄

C₄×D₄

yes

characteristic

107

C₂×C₄

C₄×D₄

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

—

116

C₈

#101

yes

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₄×C₂²⋊C₄

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

C₂×#101

yes

characteristic

127

C₄

#413

yes

characteristic

128

C₄

#443

yes

—

129

C₄

#444

yes

—

130

C₂

—

131

C₂

C₄×#101

yes

characteristic

132

C₂

#1764

yes

characteristic

133

C₂

#1765

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
socle	C₂²	C₄×C₂²⋊C₄
prime radical	C₂³	#117
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

C₂×C₂²⋊C₄

commutator

C₂×C₄

C₂×C₄²

Frattini, cosocle

C₂×C₄²

C₂³

socle

C₂²

C₄×C₂²⋊C₄

C₂³

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ 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
#2960	2
#1544	2
#1526	2
#1451	2
C₈×M₁₆	2
#1299	2

subgroup

index

C₈×M₁₆

Subgroups — #8246

Conjugacy classes of subgroups

Maximal subgroups