Group #8290 — subgroups

Subgroups — #8290

Order 256.

205 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 205 conjugacy classes of subgroups — too many to draw. Only its 115 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	C₄×#106	C₂	1	1	yes	maximal, characteristic
2	#1770	C₂	1	1	yes	maximal, characteristic
3	C₈×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
4	C₂×#334	C₂	1	1	yes	maximal, characteristic
5	C₂×#322	C₂	1	1	yes	maximal, characteristic
6	#1279	C₂	1	1	yes	maximal, characteristic
7	#1279	C₂	1	1	yes	maximal, characteristic
8	C₂×#106	C₄	1	2	yes	—
9	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊C₈	C₂²	1	1	yes	characteristic
11	C₂×#97	—	2	1	no	—
12	C₂×C₄×C₈	—	2	1	no	—
13	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
17	C₄×C₄⋊C₄	C₂²	1	1	yes	characteristic
18	#340	C₄	1	2	yes	—
19	#336	—	2	1	no	—
20	#336	—	2	1	no	—
21	#334	C₄	1	4	yes	complemented
22	#322	C₄	1	4	yes	complemented
23	C₂²×C₈	—	2	1	no	—
24	C₂²×C₈	—	2	1	no	—
25	C₂²×C₈	—	2	1	no	—
26	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈	D₄	1	1	yes	characteristic
29	C₂²×C₈	D₄	1	1	yes	characteristic
30	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
31	C₂×C₄⋊C₄	C₂×C₄	1	2	yes	—
32	C₂×C₄²	—	2	1	no	—
33	C₂×C₄²	—	2	1	no	—
34	C₂×C₄²	C₂³	1	1	yes	characteristic
35	#106	—	2	4	no	—
36	C₄⋊C₈	—	2	1	no	—
37	C₄⋊C₈	—	2	1	no	—
38	C₄⋊C₈	C₂×C₄	1	2	yes	—
39	C₄⋊C₈	C₂×C₄	1	2	yes	—
40	#97	—	4	2	no	—
41	C₄×C₈	—	2	1	no	—
42	C₄×C₈	—	4	2	no	—
43	C₄×C₈	C₂×C₄	1	2	yes	—
44	C₄×C₈	C₂×C₄	1	2	yes	—
45	C₄×C₈	C₂×C₄	1	2	yes	—
46	C₄×C₈	C₂×C₄	1	2	yes	—
47	C₄×C₈	Q₈	1	2	yes	—
48	#95	—	2	1	no	—
49	#95	—	2	1	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	C₂×D₄	1	1	yes	characteristic
54	C₂²×C₄	#41	1	1	yes	characteristic
55	C₂×C₈	—	2	2	no	—
56	C₂×C₈	—	2	2	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	2	1	no	—
59	C₂×C₈	—	2	2	no	—
60	C₂×C₈	—	2	2	no	—
61	C₂×C₈	—	2	2	no	—
62	C₂×C₈	—	4	2	no	—
63	C₂×C₈	—	4	1	no	—
64	C₂×C₈	—	4	1	no	—
65	C₂×C₈	—	4	1	no	—
66	C₂×C₈	—	4	1	no	—
67	C₂×C₈	C₄²	1	2	yes	—
68	C₂×C₈	C₄²	1	2	yes	—
69	C₂×C₈	C₄⋊C₄	1	2	yes	—
70	C₂×C₈	C₄⋊C₄	1	2	yes	—
71	C₂×C₈	SD₁₆	1	2	yes	—
72	C₂×C₈	SD₁₆	1	2	yes	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	2	2	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	M₁₆	1	4	yes	—
77	C₄²	—	4	1	no	—
78	C₄²	—	4	1	no	—
79	C₄²	—	4	1	no	—
80	C₄²	—	4	1	no	—
81	C₄²	C₂²×C₄	1	1	yes	characteristic
82	C₄²	C₂²×C₄	1	1	yes	characteristic
83	C₄²	C₂×Q₈	1	1	yes	characteristic
84	C₄²	#41	1	1	yes	characteristic
85	C₂³	C₄×D₄	1	1	yes	characteristic
86	C₂×C₄	—	2	2	no	—
87	C₂×C₄	—	2	4	no	—
88	C₂×C₄	—	4	2	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	C₂×C₄²	1	1	yes	characteristic
94	C₂×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
95	C₂×C₄	#117	1	1	yes	characteristic
96	C₂×C₄	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
98	C₂×C₄	C₄×Q₈	1	1	yes	characteristic
99	C₂×C₄	C₂×M₁₆	1	2	yes	—
100	C₂×C₄	#131	1	2	yes	—
101	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
103	C₂×C₄	#135	1	1	yes	characteristic
104	C₂×C₄	#135	1	1	yes	characteristic
105	C₈	—	2	2	no	—
106	C₈	—	2	2	no	—
107	C₈	—	2	2	no	—
108	C₈	—	4	2	no	—
109	C₈	—	4	2	no	—
110	C₈	—	4	2	no	—
111	C₈	#106	1	4	yes	complemented
112	C₂²	C₄×C₄⋊C₄	1	1	yes	characteristic
113	C₂²	#435	1	2	yes	—
114	C₂²	C₄×SD₁₆	1	1	yes	characteristic
115	C₂²	C₄×SD₁₆	1	1	yes	characteristic
116	C₂²	#444	1	2	yes	—
117	C₄	—	2	1	no	—
118	C₄	—	2	1	no	—
119	C₄	—	4	4	no	—
120	C₄	—	8	2	no	—
121	C₄	C₄×M₁₆	1	1	yes	characteristic
122	C₄	#405	1	1	yes	characteristic
123	C₄	C₂×#106	1	1	yes	characteristic
124	C₄	#427	1	1	yes	characteristic
125	C₄	C₄×SD₁₆	1	1	yes	characteristic
126	C₄	C₄×SD₁₆	1	1	yes	characteristic
127	C₄	#446	1	2	yes	—
128	C₂	#1582	1	2	yes	—
129	C₂	#1583	1	2	yes	—
130	C₂	#1771	1	1	yes	characteristic
131	C₂	#1773	1	1	yes	characteristic
132	C₂	C₄×#106	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#106

C₂

yes

maximal, characteristic

#1770

C₂

yes

maximal, characteristic

C₈×C₄⋊C₄

C₂

yes

maximal, characteristic

C₂×#334

C₂

yes

maximal, characteristic

C₂×#322

C₂

yes

maximal, characteristic

#1279

C₂

yes

maximal, characteristic

#1279

C₂

yes

maximal, characteristic

C₂×#106

C₄

yes

—

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊C₈

C₂²

yes

characteristic

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

characteristic

C₄×C₄⋊C₄

C₂²

yes

characteristic

#340

C₄

yes

—

#336

—

#336

—

#334

C₄

yes

complemented

#322

C₄

yes

complemented

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

D₄

yes

characteristic

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄⋊C₄

C₂×C₄

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×C₄

yes

—

C₄⋊C₈

C₂×C₄

yes

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

Q₈

yes

—

#95

—

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄²

yes

—

C₂×C₈

C₄²

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

C₄⋊C₄

yes

—

C₂×C₈

SD₁₆

yes

—

C₂×C₈

SD₁₆

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

M₁₆

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×Q₈

yes

characteristic

C₄²

#41

yes

characteristic

C₂³

C₄×D₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄²

yes

characteristic

C₂×C₄

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄

#117

yes

characteristic

C₂×C₄

C₄×D₄

yes

characteristic

C₂×C₄

C₄×Q₈

yes

characteristic

C₂×C₄

C₄×Q₈

yes

characteristic

C₂×C₄

C₂×M₁₆

yes

—

100

C₂×C₄

#131

yes

—

101

C₂×C₄

C₂×SD₁₆

yes

characteristic

102

C₂×C₄

C₂×SD₁₆

yes

characteristic

103

C₂×C₄

#135

yes

characteristic

104

C₂×C₄

#135

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

#106

yes

complemented

112

C₂²

C₄×C₄⋊C₄

yes

characteristic

113

C₂²

#435

yes

—

114

C₂²

C₄×SD₁₆

yes

characteristic

115

C₂²

C₄×SD₁₆

yes

characteristic

116

C₂²

#444

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₄×M₁₆

yes

characteristic

122

C₄

#405

yes

characteristic

123

C₄

C₂×#106

yes

characteristic

124

C₄

#427

yes

characteristic

125

C₄

C₄×SD₁₆

yes

characteristic

126

C₄

C₄×SD₁₆

yes

characteristic

127

C₄

#446

yes

—

128

C₂

#1582

yes

—

129

C₂

#1583

yes

—

130

C₂

#1771

yes

characteristic

131

C₂

#1773

yes

characteristic

132

C₂

C₄×#106

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₄
commutator	C₂×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄²	C₂³
socle, prime radical	C₂³	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂²×C₄

C₂×D₄

commutator

C₂×C₄

C₂×C₄²

Frattini, cosocle

C₂×C₄²

C₂³

socle, prime radical

C₂³

C₄×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
21. #334	C₄	semidirect
22. #322	C₄	semidirect
111. C₈	#106	semidirect

H ≅ G/N

type

21. #334

C₄

semidirect

22. #322

C₄

semidirect

111. C₈

#106

semidirect

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

subgroup	index
C₄×#106	2
#1770	2
C₈×C₄⋊C₄	2
C₂×#334	2
C₂×#322	2
#1279	2

subgroup

index

C₈×C₄⋊C₄

Subgroups — #8290

Conjugacy classes of subgroups

Complements

Maximal subgroups