Group #8678 — subgroups

Subgroups — #8678

Order 256.

209 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 209 conjugacy classes of subgroups — too many to draw. Only its 71 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	#2931	C₂	1	1	yes	maximal, characteristic
2	#1835	C₂	1	1	yes	maximal, characteristic, complemented
3	#1564	C₂	1	1	yes	maximal, characteristic, complemented
4	#1517	C₂	1	2	yes	maximal
5	#1286	C₂	1	2	yes	maximal, complemented
6	C₂×#131	—	2	1	no	—
7	#532	C₂²	1	1	yes	characteristic
8	#436	—	2	1	no	—
9	#436	C₄	1	2	yes	complemented
10	C₈×D₄	—	2	1	no	—
11	C₈×D₄	C₄	1	2	yes	complemented
12	#431	C₂²	1	2	yes	—
13	#423	C₂²	1	1	yes	characteristic
14	#408	—	2	1	no	—
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	#382	C₂²	1	2	yes	—
17	#340	—	2	1	no	—
18	#340	—	2	1	no	—
19	#335	—	2	1	no	—
20	#334	—	2	1	no	—
21	#327	—	2	4	no	—
22	C₂×#41	—	2	1	no	—
23	#131	—	4	4	no	—
24	C₂×M₁₆	—	2	1	no	—
25	C₂×M₁₆	—	2	2	no	—
26	C₂²×C₈	—	2	2	no	—
27	C₂²×C₈	D₄	1	1	yes	characteristic
28	C₂²×C₈	D₄	1	1	yes	characteristic
29	C₄⋊Q₈	Q₈	1	1	yes	characteristic
30	#127	Q₈	1	1	yes	characteristic
31	#124	—	2	1	no	—
32	#121	—	2	1	no	—
33	#121	D₄	1	2	yes	—
34	C₄×D₄	—	2	1	no	—
35	C₄×D₄	C₂×C₄	1	2	yes	—
36	C₂×C₄⋊C₄	—	2	2	no	—
37	C₂×C₄²	C₂³	1	1	yes	characteristic
38	C₄⋊C₈	—	2	1	no	—
39	C₄⋊C₈	—	2	2	no	—
40	C₄⋊C₈	C₂×C₄	1	2	yes	—
41	C₂²⋊C₈	—	2	2	no	—
42	C₂²⋊C₈	—	2	2	no	—
43	C₂²⋊C₈	—	2	2	no	—
44	C₄×C₈	—	2	2	no	—
45	C₄×C₈	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈	C₂×C₄	1	1	yes	characteristic
47	C₄×C₈	D₄	1	1	yes	characteristic
48	C₄×C₈	D₄	1	1	yes	characteristic
49	#95	—	2	2	no	—
50	#95	—	2	2	no	—
51	#41	—	4	2	no	—
52	#41	—	4	2	no	—
53	C₂×Q₈	—	2	1	no	—
54	C₂×D₄	—	2	2	no	—
55	C₂×D₄	—	2	1	no	—
56	C₂×D₄	—	2	1	no	—
57	C₂×D₄	C₄⋊C₄	1	2	yes	—
58	C₂²×C₄	—	2	2	no	—
59	C₂²×C₄	—	4	2	no	—
60	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄	#41	1	1	yes	characteristic
63	M₁₆	—	4	4	no	—
64	M₁₆	—	4	2	no	—
65	C₂×C₈	—	2	2	no	—
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	—	2	2	no	—
68	C₂×C₈	—	4	4	no	—
69	C₂×C₈	—	4	2	no	—
70	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
73	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
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	C₂²⋊C₄	—	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	—	2	2	no	—
85	C₂³	#121	1	1	yes	characteristic
86	Q₈	—	4	2	no	—
87	D₄	—	4	2	no	—
88	D₄	—	4	2	no	—
89	D₄	—	4	4	no	—
90	D₄	—	4	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	4	1	no	—
93	C₂×C₄	—	4	1	no	—
94	C₂×C₄	—	4	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	8	2	no	—
97	C₂×C₄	—	8	2	no	—
98	C₂×C₄	—	8	2	no	—
99	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
100	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₄	#117	1	1	yes	characteristic
102	C₂×C₄	#121	1	1	yes	characteristic
103	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
104	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
105	C₂×C₄	#135	1	2	yes	—
106	C₂×C₄	AGL(1,ℤ/8)	1	2	yes	—
107	C₈	—	2	1	no	—
108	C₈	—	2	1	no	—
109	C₈	—	2	1	no	—
110	C₈	—	2	1	no	—
111	C₈	—	4	2	no	—
112	C₈	—	8	2	no	—
113	C₂²	—	2	1	no	—
114	C₂²	—	2	1	no	—
115	C₂²	—	4	2	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	4	2	no	—
118	C₂²	#380	1	1	yes	characteristic
119	C₂²	#463	1	2	yes	—
120	C₄	—	2	1	no	—
121	C₄	—	2	1	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	4	2	no	—
125	C₄	—	16	2	no	—
126	C₄	#416	1	1	yes	characteristic
127	C₄	#418	1	1	yes	characteristic
128	C₄	#478	1	2	yes	—
129	C₂	—	2	2	no	—
130	C₂	—	8	2	no	—
131	C₂	#1804	1	1	yes	characteristic
132	C₂	#1812	1	1	yes	characteristic
133	C₂	#1814	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2931

C₂

yes

maximal, characteristic

#1835

C₂

yes

maximal, characteristic, complemented

#1564

C₂

yes

maximal, characteristic, complemented

#1517

C₂

yes

maximal

#1286

C₂

yes

maximal, complemented

C₂×#131

—

#532

C₂²

yes

characteristic

#436

—

#436

C₄

yes

complemented

C₈×D₄

—

C₈×D₄

C₄

yes

complemented

#431

C₂²

yes

—

#423

C₂²

yes

characteristic

#408

—

C₂×C₄×C₈

C₂²

yes

characteristic

#382

C₂²

yes

—

—

—

—

—

—

C₂×#41

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₄⋊Q₈

Q₈

yes

characteristic

#127

Q₈

yes

—

—

D₄

yes

—

C₄×D₄

—

C₄×D₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

D₄

yes

characteristic

C₄×C₈

D₄

yes

—

—

—

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₄⋊C₄

yes

—

C₂²×C₄

—

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

—

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

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

C₄²

C₂×Q₈

yes

characteristic

C₄²

#41

yes

characteristic

C₂³

—

C₂³

#121

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

100

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

101

C₂×C₄

#117

yes

characteristic

102

C₂×C₄

#121

yes

characteristic

103

C₂×C₄

C₂²⋊Q₈

yes

characteristic

104

C₂×C₄

C₂²⋊Q₈

yes

characteristic

105

C₂×C₄

#135

yes

—

106

C₂×C₄

AGL(1,ℤ/8)

yes

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

#380

yes

characteristic

119

C₂²

#463

yes

—

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#416

yes

characteristic

127

C₄

#418

yes

characteristic

128

C₄

#478

yes

—

129

C₂

—

130

C₂

—

131

C₂

#1804

yes

characteristic

132

C₂

#1812

yes

characteristic

133

C₂

#1814

yes

characteristic

	subgroup	G/H
center, socle	C₂²	#380
commutator	C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄²	C₂³
prime radical	#532	C₂²
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

C₂²

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
2. #1835	C₂	semidirect
3. #1564	C₂	semidirect
5. #1286	C₂	semidirect
9. #436	C₄	semidirect
11. C₈×D₄	C₄	semidirect

H ≅ G/N

type

2. #1835

C₂

semidirect

3. #1564

C₂

semidirect

5. #1286

C₂

semidirect

9. #436

C₄

semidirect

11. C₈×D₄

C₄

semidirect

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

subgroup	index
#2931	2
#1835	2
#1564	2
#1517	2
#1286	2

subgroup

index

Subgroups — #8678

Conjugacy classes of subgroups

Complements

Maximal subgroups