Group #12822 — subgroups

Subgroups — #12822

Order 256.

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

normal subgroups only

This group has 200 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	C₂×#443	C₂	1	1	yes	maximal, characteristic
2	C₂×#364	C₂	1	1	yes	maximal, characteristic, complemented
3	#2143	C₂	1	2	yes	maximal
4	#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	#1388	C₂	1	2	yes	maximal, complemented
6	C₂×#135	C₂²	1	1	yes	characteristic
7	C₂×#131	—	2	1	no	—
8	C₂×M₃₂	C₂²	1	2	yes	—
9	#443	—	2	1	no	—
10	#443	—	2	1	no	—
11	#443	C₄	1	2	yes	—
12	#443	C₄	1	2	yes	—
13	C₂×#108	—	2	1	no	—
14	C₂×#108	C₂²	1	2	yes	—
15	C₂×#108	C₂²	1	1	yes	characteristic
16	#423	—	2	1	no	—
17	C₂×#104	—	2	1	no	—
18	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
19	#364	—	2	1	no	—
20	#364	—	2	1	no	—
21	#359	—	2	4	no	—
22	C₂×#41	—	2	1	no	—
23	#135	—	2	1	no	—
24	#135	—	2	1	no	—
25	#135	C₂×C₄	1	2	yes	—
26	#135	D₄	1	2	yes	—
27	C₂×Q₁₆	Q₈	1	1	yes	characteristic
28	C₂×SD₁₆	—	2	1	no	—
29	C₂×D₈	Q₈	1	1	yes	characteristic
30	#131	—	2	2	no	—
31	#131	—	2	2	no	—
32	#131	—	4	1	no	—
33	#131	—	4	1	no	—
34	C₂×M₁₆	—	2	1	no	—
35	C₂×M₁₆	—	2	2	no	—
36	C₂×M₁₆	—	4	1	no	—
37	C₂²×C₈	—	4	1	no	—
38	C₂²×C₈	D₄	1	1	yes	characteristic
39	C₂²×C₈	C₂³	1	1	yes	characteristic
40	C₂×C₄²	D₄	1	1	yes	characteristic
41	M₃₂	—	2	2	no	—
42	M₃₂	—	2	2	no	—
43	C₂×C₁₆	—	2	2	no	—
44	#108	—	2	1	no	—
45	#108	—	2	2	no	—
46	#108	—	2	2	no	—
47	#108	—	4	1	no	—
48	#108	—	4	1	no	—
49	#108	C₂×C₄	1	2	yes	—
50	C₄⋊C₈	—	4	2	no	—
51	#104	—	2	2	no	—
52	#104	—	2	2	no	—
53	C₄×C₈	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈	D₄	1	1	yes	characteristic
56	C₄×C₈	D₄	1	1	yes	characteristic
57	#41	—	2	2	no	—
58	#41	—	2	2	no	—
59	#41	—	4	1	no	—
60	#41	—	4	1	no	—
61	C₂×Q₈	—	2	1	no	—
62	C₂×D₄	—	2	1	no	—
63	C₂×D₄	—	4	1	no	—
64	C₂²×C₄	—	2	1	no	—
65	C₂²×C₄	—	4	1	no	—
66	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
67	Q₁₆	—	2	1	no	—
68	Q₁₆	C₄⋊C₄	1	2	yes	—
69	SD₁₆	—	2	2	no	—
70	SD₁₆	—	2	2	no	—
71	D₈	—	2	1	no	—
72	D₈	C₄⋊C₄	1	2	yes	—
73	M₁₆	—	2	2	no	—
74	M₁₆	—	4	1	no	—
75	M₁₆	—	4	2	no	—
76	M₁₆	—	4	2	no	—
77	M₁₆	—	4	2	no	—
78	M₁₆	—	4	2	no	—
79	C₂×C₈	—	4	1	no	—
80	C₂×C₈	—	4	2	no	—
81	C₂×C₈	—	4	2	no	—
82	C₂×C₈	—	4	2	no	—
83	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈	D₈	1	1	yes	characteristic
86	C₂×C₈	D₈	1	1	yes	characteristic
87	C₂×C₈	SD₁₆	1	1	yes	characteristic
88	C₂×C₈	SD₁₆	1	1	yes	characteristic
89	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈	C₂×Q₈	1	1	yes	characteristic
93	C₂×C₈	#41	1	1	yes	characteristic
94	C₂×C₈	#41	1	1	yes	characteristic
95	C₄²	—	2	1	no	—
96	C₄²	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²	C₂²⋊C₄	1	1	yes	characteristic
98	C₁₆	—	4	2	no	—
99	C₂³	—	4	1	no	—
100	C₂³	#137	1	1	yes	characteristic
101	Q₈	—	2	2	no	—
102	Q₈	—	4	1	no	—
103	D₄	—	2	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	4	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	—	4	2	no	—
113	C₂×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄	#121	1	1	yes	characteristic
115	C₂×C₄	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₈	—	4	2	no	—
120	C₈	—	8	2	no	—
121	C₈	#102	1	1	yes	characteristic
122	C₈	#102	1	1	yes	characteristic
123	C₈	#102	1	1	yes	characteristic
124	C₈	#102	1	1	yes	characteristic
125	C₈	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₈	#117	1	1	yes	characteristic
127	C₈	#121	1	1	yes	characteristic
128	C₈	C₂²⋊Q₈	1	1	yes	characteristic
129	C₂²	—	4	1	no	—
130	C₂²	—	4	2	no	—
131	C₂²	—	8	2	no	—
132	C₂²	#417	1	1	yes	characteristic
133	C₂²	#461	1	1	yes	characteristic
134	C₂²	#474	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	4	1	no	—
137	C₄	—	4	2	no	—
138	C₄	#380	1	1	yes	characteristic
139	C₄	C₂×#102	1	1	yes	characteristic
140	C₄	#459	1	1	yes	characteristic
141	C₄	#476	1	1	yes	characteristic
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	—	8	2	no	—
146	C₂	#1802	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#443

C₂

yes

maximal, characteristic

C₂×#364

C₂

yes

maximal, characteristic, complemented

#2143

C₂

yes

maximal

#1848

C₂

yes

maximal, characteristic, complemented

#1388

C₂

yes

maximal, complemented

C₂×#135

C₂²

yes

characteristic

C₂×#131

—

C₂×M₃₂

C₂²

yes

—

#443

—

#443

—

#443

C₄

yes

—

#443

C₄

yes

—

C₂×#108

—

C₂×#108

C₂²

yes

—

C₂×#108

C₂²

yes

characteristic

#423

—

C₂×#104

—

C₂×C₄×C₈

C₂²

yes

—

—

—

C₂×#41

—

#135

—

#135

—

#135

C₂×C₄

yes

—

#135

D₄

yes

—

C₂×Q₁₆

Q₈

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

Q₈

yes

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

D₄

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂×C₄²

D₄

yes

characteristic

M₃₂

—

M₃₂

—

C₂×C₁₆

—

—

—

—

—

—

C₂×C₄

yes

—

C₄⋊C₈

—

#104

—

#104

—

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

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

C₄⋊C₄

yes

—

SD₁₆

—

SD₁₆

—

D₈

—

D₈

C₄⋊C₄

yes

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

C₂²⋊C₄

yes

characteristic

C₂×C₈

D₈

yes

characteristic

C₂×C₈

D₈

yes

characteristic

C₂×C₈

SD₁₆

yes

characteristic

C₂×C₈

SD₁₆

yes

characteristic

C₂×C₈

C₂²×C₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×Q₈

yes

C₂×C₈

yes

C₂×C₈

yes

C₄²

—

C₄²

C₂²⋊C₄

yes

characteristic

C₄²

C₂²⋊C₄

yes

characteristic

C₁₆

—

C₂³

—

100

C₂³

#137

yes

characteristic

101

Q₈

—

102

Q₈

—

103

D₄

—

104

D₄

—

105

D₄

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×C₂²⋊C₄

yes

characteristic

114

C₂×C₄

#121

yes

characteristic

115

C₂×C₄

C₂²⋊Q₈

yes

characteristic

116

C₂×C₄

C₂×D₈

yes

characteristic

117

C₂×C₄

C₂×SD₁₆

yes

characteristic

118

C₂×C₄

AGL(1,ℤ/8)

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

#102

yes

characteristic

122

C₈

#102

yes

characteristic

123

C₈

#102

yes

characteristic

124

C₈

#102

yes

characteristic

125

C₈

C₂×C₄⋊C₄

yes

characteristic

126

C₈

#117

yes

characteristic

127

C₈

#121

yes

characteristic

128

C₈

C₂²⋊Q₈

yes

characteristic

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

#417

yes

characteristic

133

C₂²

#461

yes

characteristic

134

C₂²

#474

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#380

yes

characteristic

139

C₄

C₂×#102

yes

characteristic

140

C₄

#459

yes

characteristic

141

C₄

#476

yes

characteristic

142

C₂

—

143

C₂

—

144

C₂

—

145

C₂

—

146

C₂

#1802

yes

characteristic

	subgroup	G/H
center	C₄	#459
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₂²×C₈	C₂³
socle	C₂	#1802
prime radical	C₂×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center

C₄

#459

commutator

C₂×C₈

C₂²×C₄

Frattini, cosocle

C₂²×C₈

C₂³

C₂

C₂²

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G

LECS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
2. C₂×#364	C₂	semidirect
4. #1848	C₂	semidirect
5. #1388	C₂	semidirect

H ≅ G/N

type

2. C₂×#364

C₂

semidirect

4. #1848

C₂

semidirect

5. #1388

C₂

semidirect

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

subgroup	index
C₂×#443	2
C₂×#364	2
#2143	2
#1848	2
#1388	2

subgroup

index

Subgroups — #12822

Conjugacy classes of subgroups

Complements

Maximal subgroups