Group #8677 — subgroups

Subgroups — #8677

Order 256.

183 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 183 conjugacy classes of subgroups — too many to draw. Only its 79 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	#1846	C₂	1	1	yes	maximal, characteristic
3	#1564	C₂	1	1	yes	maximal, characteristic
4	#1518	C₂	1	2	yes	maximal
5	#1286	C₂	1	2	yes	maximal
6	#533	C₂²	1	1	yes	characteristic
7	#446	—	2	1	no	—
8	#446	C₄	1	2	yes	complemented
9	C₈×Q₈	—	2	1	no	—
10	C₈×Q₈	C₄	1	2	yes	complemented
11	#431	C₂²	1	2	yes	—
12	C₂×#107	—	2	1	no	—
13	C₂×#106	—	2	1	no	—
14	#424	—	2	1	no	—
15	#423	C₂²	1	1	yes	characteristic
16	#405	—	2	1	no	—
17	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
18	#382	C₂²	1	2	yes	—
19	#335	—	2	1	no	—
20	#334	—	2	1	no	—
21	#328	—	2	4	no	—
22	C₂×M₁₆	—	2	1	no	—
23	C₂²×C₈	D₄	1	1	yes	characteristic
24	C₂²×C₈	D₄	1	1	yes	characteristic
25	C₄⋊Q₈	Q₈	1	1	yes	characteristic
26	C₄⋊Q₈	Q₈	1	1	yes	characteristic
27	#125	—	2	1	no	—
28	C₂²⋊Q₈	—	2	1	no	—
29	C₂²⋊Q₈	D₄	1	2	yes	—
30	C₄×Q₈	—	2	1	no	—
31	C₄×Q₈	C₂×C₄	1	2	yes	—
32	#117	—	2	1	no	—
33	C₂×C₄⋊C₄	—	2	2	no	—
34	C₂×C₄²	C₂³	1	1	yes	characteristic
35	#107	—	4	2	no	—
36	#106	—	4	2	no	—
37	C₄⋊C₈	—	2	2	no	—
38	C₄⋊C₈	—	2	2	no	—
39	C₄⋊C₈	—	2	1	no	—
40	C₄⋊C₈	—	2	2	no	—
41	C₄⋊C₈	C₂×C₄	1	2	yes	—
42	C₂²⋊C₈	—	2	2	no	—
43	#97	—	2	2	no	—
44	C₄×C₈	—	2	2	no	—
45	C₄×C₈	—	2	2	no	—
46	C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈	D₄	1	1	yes	characteristic
49	C₄×C₈	D₄	1	1	yes	characteristic
50	#95	—	2	2	no	—
51	#95	—	2	2	no	—
52	C₂×Q₈	—	2	1	no	—
53	C₂×Q₈	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄	—	4	2	no	—
55	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄	#41	1	1	yes	characteristic
58	M₁₆	—	4	1	no	—
59	M₁₆	—	4	1	no	—
60	C₂×C₈	—	2	1	no	—
61	C₂×C₈	—	2	1	no	—
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	4	2	no	—
64	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
65	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
66	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
67	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
68	C₂×C₈	SD₁₆	1	2	yes	—
69	C₂×C₈	Q₁₆	1	2	yes	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	—	2	2	no	—
72	C₄⋊C₄	—	2	1	no	—
73	C₄⋊C₄	—	2	1	no	—
74	C₄⋊C₄	—	2	1	no	—
75	C₄⋊C₄	—	4	2	no	—
76	C₄⋊C₄	—	4	2	no	—
77	C₄⋊C₄	C₄⋊C₄	1	2	yes	—
78	C₂²⋊C₄	—	2	2	no	—
79	C₄²	—	2	2	no	—
80	C₄²	C₂²×C₄	1	1	yes	characteristic
81	C₄²	C₂×D₄	1	1	yes	characteristic
82	C₄²	C₂×Q₈	1	1	yes	characteristic
83	C₄²	#41	1	1	yes	characteristic
84	C₂³	#121	1	1	yes	characteristic
85	Q₈	—	4	2	no	—
86	Q₈	—	4	2	no	—
87	C₂×C₄	—	2	2	no	—
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	4	1	no	—
90	C₂×C₄	—	4	1	no	—
91	C₂×C₄	—	8	2	no	—
92	C₂×C₄	—	8	2	no	—
93	C₂×C₄	—	8	2	no	—
94	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
96	C₂×C₄	#117	1	1	yes	characteristic
97	C₂×C₄	#121	1	1	yes	characteristic
98	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
99	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
100	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
101	C₂×C₄	C₂×Q₁₆	1	1	yes	characteristic
102	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄	#137	1	1	yes	characteristic
104	C₈	—	2	1	no	—
105	C₈	—	2	1	no	—
106	C₈	—	4	2	no	—
107	C₈	—	8	2	no	—
108	C₈	#103	1	2	yes	—
109	C₈	#103	1	2	yes	—
110	C₂²	—	2	1	no	—
111	C₂²	—	2	1	no	—
112	C₂²	#380	1	1	yes	characteristic
113	C₂²	#460	1	1	yes	characteristic
114	C₂²	#462	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	2	1	no	—
117	C₄	—	2	1	no	—
118	C₄	—	2	1	no	—
119	C₄	—	4	2	no	—
120	C₄	—	4	2	no	—
121	C₄	—	16	2	no	—
122	C₄	C₂×#103	1	1	yes	characteristic
123	C₄	#417	1	1	yes	characteristic
124	C₄	#475	1	1	yes	characteristic
125	C₄	#477	1	1	yes	characteristic
126	C₂	—	2	1	no	—
127	C₂	—	2	1	no	—
128	C₂	#1803	1	1	yes	characteristic
129	C₂	#1812	1	1	yes	characteristic
130	C₂	#1814	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2960

C₂

yes

maximal, characteristic

#1846

C₂

yes

maximal, characteristic

#1564

C₂

yes

maximal, characteristic

#1518

C₂

yes

maximal

#1286

C₂

yes

maximal

#533

C₂²

yes

characteristic

#446

—

#446

C₄

yes

complemented

C₈×Q₈

—

C₈×Q₈

C₄

yes

complemented

#431

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

#405

—

C₂×C₄×C₈

C₂²

yes

characteristic

#382

C₂²

yes

—

#335

—

#334

—

#328

—

C₂×M₁₆

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄⋊Q₈

Q₈

yes

characteristic

C₄⋊Q₈

Q₈

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₄

yes

—

C₄×Q₈

—

C₄×Q₈

C₂×C₄

yes

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

characteristic

#107

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

C₄⋊C₄

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

M₁₆

—

M₁₆

—

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

C₂×C₈

SD₁₆

yes

—

C₂×C₈

Q₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×Q₈

yes

C₄²

yes

C₂³

yes

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄

yes

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊Q₈

yes

characteristic

C₂×C₄

C₂²⋊Q₈

yes

characteristic

100

C₂×C₄

C₂×SD₁₆

yes

characteristic

101

C₂×C₄

C₂×Q₁₆

yes

characteristic

102

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

103

C₂×C₄

#137

yes

characteristic

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

#103

yes

—

109

C₈

#103

yes

—

110

C₂²

—

111

C₂²

—

112

C₂²

#380

yes

characteristic

113

C₂²

#460

yes

characteristic

114

C₂²

#462

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

122

C₄

C₂×#103

yes

characteristic

123

C₄

#417

yes

characteristic

124

C₄

#475

yes

characteristic

125

C₄

#477

yes

characteristic

126

C₂

—

127

C₂

—

128

C₂

#1803

yes

characteristic

129

C₂

#1812

yes

characteristic

130

C₂

#1814

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#380
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	C₂³	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

#380

commutator

C₄²

C₂²×C₄

Frattini, cosocle

C₂×C₄²

C₂³

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
8. #446	C₄	semidirect
10. C₈×Q₈	C₄	semidirect

H ≅ G/N

type

8. #446

C₄

semidirect

10. C₈×Q₈

C₄

semidirect

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

subgroup	index
#2960	2
#1846	2
#1564	2
#1518	2
#1286	2

subgroup

index

Subgroups — #8677

Conjugacy classes of subgroups

Complements

Maximal subgroups