Group #21698 — subgroups

Subgroups — #21698

Order 256.

507 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 507 conjugacy classes of subgroups — too many to draw. Only its 195 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×#494	C₂	1	1	yes	maximal, characteristic
2	C₂×#441	C₂	1	2	yes	maximal
3	C₂×C₄×Q₁₆	C₂	1	2	yes	maximal
4	#2327	C₂	1	2	yes	maximal
5	#1971	C₂	1	2	yes	maximal
6	#1969	C₂	1	2	yes	maximal
7	#1960	C₂	1	2	yes	maximal
8	#1846	C₂	1	1	yes	maximal, characteristic
9	#1726	C₂	1	1	yes	maximal, characteristic
10	C₂²×Q₁₆	C₂²	1	2	yes	—
11	C₂²×Q₁₆	C₂²	1	2	yes	—
12	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₂×C₄×Q₈	C₂²	1	4	yes	—
14	#494	C₄	1	8	yes	complemented
15	#441	—	2	8	no	—
16	C₄×Q₁₆	—	2	8	no	—
17	C₂×#107	C₂²	1	2	yes	—
18	C₂×#106	C₂²	1	2	yes	—
19	C₂×#103	C₂²	1	4	yes	—
20	C₂×#103	C₂²	1	4	yes	—
21	C₂×#97	C₂²	1	1	yes	characteristic
22	C₂×#97	C₂²	1	1	yes	characteristic
23	C₂×#97	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
27	#391	C₂²	1	4	yes	—
28	#387	—	2	8	no	—
29	#382	C₂²	1	2	yes	—
30	#376	C₂²	1	1	yes	characteristic
31	C₂²×Q₈	C₂³	1	4	yes	—
32	C₂×Q₁₆	—	2	4	no	—
33	C₂×Q₁₆	—	2	4	no	—
34	C₂×Q₁₆	—	2	4	no	—
35	C₂×Q₁₆	—	2	4	no	—
36	C₂×Q₁₆	C₂×C₄	1	8	yes	—
37	C₂×Q₁₆	C₂×C₄	1	8	yes	—
38	C₂²×C₈	C₂³	1	1	yes	characteristic
39	C₂²×C₈	C₂³	1	1	yes	characteristic
40	C₂²×C₈	C₂³	1	1	yes	characteristic
41	C₂²×C₈	C₂³	1	1	yes	characteristic
42	C₄⋊Q₈	—	2	4	no	—
43	C₄⋊Q₈	C₂×C₄	1	8	yes	—
44	C₄×Q₈	—	2	8	no	—
45	C₄×Q₈	—	2	8	no	—
46	C₂×C₄⋊C₄	—	2	4	no	—
47	C₂×C₄⋊C₄	—	2	4	no	—
48	C₂×C₄⋊C₄	C₂³	1	4	yes	—
49	C₂×C₄²	—	2	4	no	—
50	C₂×C₄²	C₂³	1	1	yes	characteristic
51	C₂×C₄²	C₂³	1	1	yes	characteristic
52	C₂×C₄²	C₂³	1	1	yes	characteristic
53	#107	—	2	4	no	—
54	#106	—	2	4	no	—
55	#103	—	2	8	no	—
56	#103	—	2	8	no	—
57	#97	—	2	2	no	—
58	#97	—	2	2	no	—
59	#97	D₄	1	4	yes	—
60	C₄×C₈	—	2	2	no	—
61	C₄×C₈	C₂×C₄	1	4	yes	—
62	C₄×C₈	D₄	1	4	yes	—
63	#95	—	2	4	no	—
64	#95	—	2	8	no	—
65	#95	D₄	1	4	yes	—
66	C₂×Q₈	—	2	4	no	—
67	C₂×Q₈	—	2	4	no	—
68	C₂×Q₈	—	2	4	no	—
69	C₂×Q₈	—	2	4	no	—
70	C₂×Q₈	—	2	4	no	—
71	C₂×Q₈	C₂²×C₄	1	8	yes	—
72	C₂²×C₄	—	2	4	no	—
73	C₂²×C₄	—	2	4	no	—
74	C₂²×C₄	C₂×D₄	1	2	yes	—
75	C₂²×C₄	C₂×D₄	1	2	yes	—
76	C₂²×C₄	C₂×D₄	1	2	yes	—
77	C₂²×C₄	C₂⁴	1	1	yes	characteristic
78	Q₁₆	—	2	16	no	—
79	Q₁₆	—	2	16	no	—
80	C₂×C₈	—	2	1	no	—
81	C₂×C₈	—	2	1	no	—
82	C₂×C₈	—	2	1	no	—
83	C₂×C₈	—	2	1	no	—
84	C₂×C₈	C₂²×C₄	1	2	yes	—
85	C₂×C₈	C₂²×C₄	1	2	yes	—
86	C₂×C₈	C₂×D₄	1	2	yes	—
87	C₂×C₈	C₂×D₄	1	2	yes	—
88	C₂×C₈	C₂×D₄	1	2	yes	—
89	C₂×C₈	C₂×D₄	1	2	yes	—
90	C₂×C₈	#41	1	2	yes	—
91	C₂×C₈	#41	1	2	yes	—
92	C₄⋊C₄	—	2	16	no	—
93	C₄⋊C₄	—	2	4	no	—
94	C₄⋊C₄	—	2	4	no	—
95	C₄⋊C₄	—	4	8	no	—
96	C₄²	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₄²	—	4	8	no	—
99	C₄²	C₂²×C₄	1	2	yes	—
100	C₄²	C₂×D₄	1	2	yes	—
101	C₄²	C₂×D₄	1	2	yes	—
102	C₄²	#41	1	2	yes	—
103	C₂³	C₂²×D₄	1	1	yes	characteristic
104	Q₈	—	2	8	no	—
105	Q₈	—	2	8	no	—
106	Q₈	—	4	4	no	—
107	Q₈	—	4	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	8	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	—	4	4	no	—
114	C₂×C₄	—	4	4	no	—
115	C₂×C₄	—	4	4	no	—
116	C₂×C₄	—	4	8	no	—
117	C₂×C₄	C₄×D₄	1	4	yes	—
118	C₂×C₄	#127	1	4	yes	—
119	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
121	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
122	C₂×C₄	C₂×#41	1	1	yes	characteristic
123	C₂×C₄	C₂×#41	1	1	yes	characteristic
124	C₂×C₄	C₂×#41	1	1	yes	characteristic
125	C₈	—	2	2	no	—
126	C₈	—	2	2	no	—
127	C₈	C₄×D₄	1	4	yes	—
128	C₈	C₄×D₄	1	4	yes	—
129	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
130	C₂²	C₂×#127	1	1	yes	characteristic
131	C₂²	#532	1	1	yes	characteristic
132	C₂²	#578	1	4	yes	—
133	C₄	—	2	4	no	—
134	C₄	—	2	4	no	—
135	C₄	—	4	8	no	—
136	C₄	—	4	2	no	—
137	C₄	—	8	4	no	—
138	C₄	C₂×C₄×D₄	1	1	yes	characteristic
139	C₄	C₂×C₄×D₄	1	1	yes	characteristic
140	C₄	#532	1	1	yes	characteristic
141	C₄	#532	1	1	yes	characteristic
142	C₂	C₄×#127	1	1	yes	characteristic
143	C₂	#2950	1	2	yes	—
144	C₂	#3156	1	2	yes	—
145	C₂	#3177	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#494

C₂

yes

maximal, characteristic

C₂×#441

C₂

yes

maximal

C₂×C₄×Q₁₆

C₂

yes

maximal

#2327

C₂

yes

maximal

#1971

C₂

yes

maximal

#1969

C₂

yes

maximal

#1960

C₂

yes

maximal

#1846

C₂

yes

maximal, characteristic

#1726

C₂

yes

maximal, characteristic

C₂²×Q₁₆

C₂²

yes

—

C₂²×Q₁₆

C₂²

yes

—

C₂×C₄⋊Q₈

C₂²

yes

—

C₂×C₄×Q₈

C₂²

yes

—

#494

C₄

yes

complemented

#441

—

C₄×Q₁₆

—

C₂×#107

C₂²

yes

—

C₂×#106

C₂²

yes

—

C₂×#103

C₂²

yes

—

C₂×#103

C₂²

yes

—

C₂×#97

C₂²

yes

characteristic

C₂×#97

C₂²

yes

characteristic

C₂×#97

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#391

C₂²

yes

—

#387

—

#382

C₂²

yes

—

#376

C₂²

yes

characteristic

C₂²×Q₈

C₂³

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₂²×C₈

C₂³

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

C₂×C₄

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂³

yes

—

C₂×C₄²

—

C₂×C₄²

C₂³

yes

characteristic

C₂×C₄²

C₂³

yes

characteristic

C₂×C₄²

C₂³

yes

—

—

—

—

—

—

D₄

yes

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

—

C₄×C₈

D₄

yes

—

#95

—

#95

—

#95

D₄

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×C₄

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂⁴

yes

characteristic

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂²×C₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

#41

yes

—

C₂×C₈

#41

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×C₄

yes

—

100

C₄²

C₂×D₄

yes

—

101

C₄²

C₂×D₄

yes

—

102

C₄²

#41

yes

—

103

C₂³

C₂²×D₄

yes

characteristic

104

Q₈

—

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

C₄×D₄

yes

—

118

C₂×C₄

#127

yes

—

119

C₂×C₄

C₂³×C₄

yes

characteristic

120

C₂×C₄

C₂²×D₄

yes

characteristic

121

C₂×C₄

C₂²×D₄

yes

characteristic

122

C₂×C₄

C₂×#41

yes

characteristic

123

C₂×C₄

C₂×#41

yes

characteristic

124

C₂×C₄

C₂×#41

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

C₄×D₄

yes

—

128

C₈

C₄×D₄

yes

—

129

C₂²

C₂×C₄×D₄

yes

characteristic

130

C₂²

C₂×#127

yes

characteristic

131

C₂²

#532

yes

characteristic

132

C₂²

#578

yes

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

C₂×C₄×D₄

yes

characteristic

139

C₄

C₂×C₄×D₄

yes

characteristic

140

C₄

#532

yes

characteristic

141

C₄

#532

yes

characteristic

142

C₂

C₄×#127

yes

characteristic

143

C₂

#2950

yes

—

144

C₂

#3156

yes

—

145

C₂

#3177

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂³

C₂²×D₄

commutator

C₂×C₄

C₂³×C₄

Frattini, cosocle

C₂²×C₄

C₂⁴

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³ ◁ #376 ◁ G

LECS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

N	H ≅ G/N	type
14. #494	C₄	semidirect

H ≅ G/N

type

14. #494

C₄

semidirect

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

subgroup	index
C₂×#494	2
C₂×#441	2
C₂×C₄×Q₁₆	2
#2327	2
#1971	2
#1969	2
#1960	2
#1846	2
#1726	2

subgroup

index

C₂×C₄×Q₁₆

Subgroups — #21698

Conjugacy classes of subgroups

Complements

Maximal subgroups