Group #18068 — subgroups

Subgroups — #18068

Order 256.

293 conjugacy classes of subgroups; only the 163 normal subgroups are shown.

normal subgroups only

This group has 293 conjugacy classes of subgroups — too many to draw. Only its 163 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	#3170	C₂	1	1	yes	maximal, characteristic
2	#2962	C₂	1	2	yes	maximal
3	#1591	C₂	1	2	yes	maximal, complemented
4	#1584	C₂	1	4	yes	maximal, complemented
5	C₈×Q₁₆	C₂	1	2	yes	maximal, complemented
6	C₂×#335	C₂	1	1	yes	maximal, characteristic
7	#1481	C₂	1	2	yes	maximal
8	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
9	#533	C₂²	1	2	yes	—
10	#500	C₄	1	1	yes	characteristic
11	#494	C₄	1	1	yes	characteristic
12	#487	C₄	1	2	yes	—
13	#467	C₄	1	2	yes	—
14	#467	C₄	1	2	yes	—
15	#446	—	2	4	no	—
16	#446	C₂²	1	4	yes	—
17	C₈×Q₈	C₂²	1	4	yes	—
18	C₄×Q₁₆	C₂²	1	2	yes	—
19	C₄×Q₁₆	C₂²	1	2	yes	—
20	#431	C₂²	1	1	yes	characteristic
21	#424	—	2	2	no	—
22	C₂×C₄⋊C₈	C₂²	1	2	yes	—
23	#405	—	2	2	no	—
24	C₄×M₁₆	C₂²	1	1	yes	characteristic
25	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
26	#335	C₂²	1	1	yes	characteristic
27	#335	C₂²	1	2	yes	—
28	#335	C₂²	1	1	yes	characteristic
29	Q₈⋊C₈	C₂²	1	4	yes	—
30	Q₈⋊C₈	C₂²	1	4	yes	—
31	#322	C₂²	1	2	yes	—
32	C₈²	C₂²	1	2	yes	—
33	C₂×Q₁₆	C₂×C₄	1	2	yes	—
34	C₂×Q₁₆	C₂×C₄	1	2	yes	—
35	C₂×M₁₆	D₄	1	1	yes	characteristic
36	C₂×M₁₆	D₄	1	1	yes	characteristic
37	C₂²×C₈	—	2	2	no	—
38	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
40	C₄⋊Q₈	C₂×C₄	1	2	yes	—
41	C₄⋊Q₈	C₂×C₄	1	2	yes	—
42	#125	—	2	2	no	—
43	C₂²⋊Q₈	—	2	2	no	—
44	C₂²⋊Q₈	C₂×C₄	1	4	yes	—
45	C₄×Q₈	—	2	2	no	—
46	C₄×Q₈	C₂³	1	4	yes	—
47	#117	—	2	2	no	—
48	C₂×C₄²	C₂³	1	1	yes	characteristic
49	#107	C₂×C₄	1	2	yes	—
50	#107	C₂×C₄	1	2	yes	—
51	C₄⋊C₈	—	2	4	no	—
52	C₄⋊C₈	—	2	4	no	—
53	C₄⋊C₈	—	2	2	no	—
54	C₄⋊C₈	—	2	1	no	—
55	C₄⋊C₈	C₂³	1	2	yes	—
56	C₄⋊C₈	C₂³	1	2	yes	—
57	#103	C₂×C₄	1	4	yes	—
58	#103	C₂×C₄	1	4	yes	—
59	C₂²⋊C₈	—	2	1	no	—
60	#97	—	2	4	no	—
61	#97	D₄	1	2	yes	—
62	C₄×C₈	—	2	4	no	—
63	C₄×C₈	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈	C₂³	1	2	yes	—
66	C₄×C₈	C₂³	1	1	yes	characteristic
67	C₄×C₈	C₂³	1	1	yes	characteristic
68	C₄×C₈	C₂³	1	2	yes	—
69	C₂×Q₈	—	2	2	no	—
70	C₂×Q₈	C₂²×C₄	1	4	yes	—
71	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄	#41	1	1	yes	characteristic
74	Q₁₆	—	2	4	no	—
75	Q₁₆	—	4	2	no	—
76	M₁₆	—	2	2	no	—
77	M₁₆	Q₁₆	1	4	yes	complemented
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	4	1	no	—
83	C₂×C₈	—	4	4	no	—
84	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₈	C₂×D₄	1	2	yes	—
89	C₂×C₈	C₂×D₄	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	2	4	no	—
94	C₄⋊C₄	—	2	4	no	—
95	C₄⋊C₄	C₂²×C₄	1	4	yes	—
96	C₂²⋊C₄	—	2	4	no	—
97	C₄²	—	2	4	no	—
98	C₄²	C₂²×C₄	1	1	yes	characteristic
99	C₄²	C₂×D₄	1	1	yes	characteristic
100	C₄²	#41	1	1	yes	characteristic
101	C₄²	C₂⁴	1	1	yes	characteristic
102	C₂³	C₄×D₄	1	1	yes	characteristic
103	Q₈	—	2	4	no	—
104	Q₈	—	2	4	no	—
105	Q₈	—	8	2	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	C₄×D₄	1	1	yes	characteristic
112	C₂×C₄	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄	C₂×Q₁₆	1	2	yes	—
115	C₂×C₄	#135	1	2	yes	—
116	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
117	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
118	C₂×C₄	C₂×#41	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	2	1	no	—
121	C₈	—	2	1	no	—
122	C₈	—	2	2	no	—
123	C₈	—	4	2	no	—
124	C₈	—	4	2	no	—
125	C₈	—	4	2	no	—
126	C₈	#131	1	2	yes	—
127	C₈	C₂×Q₁₆	1	4	yes	—
128	C₂²	—	2	1	no	—
129	C₂²	—	2	1	no	—
130	C₂²	C₄×Q₁₆	1	2	yes	—
131	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	4	no	—
136	C₄	—	4	4	no	—
137	C₄	C₄×Q₁₆	1	2	yes	—
138	C₄	C₂×#131	1	1	yes	characteristic
139	C₄	#568	1	1	yes	characteristic
140	C₄	C₂²×Q₁₆	1	1	yes	characteristic
141	C₄	C₂×#135	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	#2935	1	1	yes	characteristic
144	C₂	C₂×C₄×Q₁₆	1	1	yes	characteristic
145	C₂	#2957	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3170

C₂

yes

maximal, characteristic

#2962

C₂

yes

maximal

#1591

C₂

yes

maximal, complemented

#1584

C₂

yes

maximal, complemented

C₈×Q₁₆

C₂

yes

maximal, complemented

C₂×#335

C₂

yes

maximal, characteristic

#1481

C₂

yes

maximal

C₈×M₁₆

C₂

yes

maximal, characteristic

#533

C₂²

yes

—

#500

C₄

yes

characteristic

#494

C₄

yes

characteristic

#487

C₄

yes

—

#467

C₄

yes

—

#467

C₄

yes

—

#446

—

#446

C₂²

yes

—

C₈×Q₈

C₂²

yes

—

C₄×Q₁₆

C₂²

yes

—

C₄×Q₁₆

C₂²

yes

—

#431

C₂²

yes

characteristic

#424

—

C₂×C₄⋊C₈

C₂²

yes

—

#405

—

C₄×M₁₆

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#335

C₂²

yes

characteristic

#335

C₂²

yes

—

#335

C₂²

yes

characteristic

Q₈⋊C₈

C₂²

yes

—

Q₈⋊C₈

C₂²

yes

—

#322

C₂²

yes

—

C₈²

C₂²

yes

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂×M₁₆

D₄

yes

characteristic

C₂×M₁₆

D₄

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×C₄

yes

characteristic

C₂²×C₈

C₂×C₄

yes

characteristic

C₄⋊Q₈

C₂×C₄

yes

—

C₄⋊Q₈

C₂×C₄

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂×C₄

yes

—

C₄×Q₈

—

C₄×Q₈

C₂³

yes

—

#117

—

C₂×C₄²

C₂³

yes

characteristic

#107

C₂×C₄

yes

—

#107

C₂×C₄

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂³

yes

—

C₄⋊C₈

C₂³

yes

—

#103

C₂×C₄

yes

—

#103

C₂×C₄

yes

—

C₂²⋊C₈

—

#97

—

#97

D₄

yes

—

C₄×C₈

—

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂×C₄

yes

characteristic

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

—

C₂×Q₈

—

C₂×Q₈

C₂²×C₄

yes

—

C₂²×C₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

#41

yes

characteristic

Q₁₆

—

Q₁₆

—

M₁₆

—

M₁₆

Q₁₆

yes

complemented

C₂×C₈

—

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

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×C₄

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×C₄

yes

characteristic

C₄²

C₂×D₄

yes

characteristic

100

C₄²

#41

yes

characteristic

101

C₄²

C₂⁴

yes

characteristic

102

C₂³

C₄×D₄

yes

characteristic

103

Q₈

—

104

Q₈

—

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₄×D₄

yes

characteristic

112

C₂×C₄

C₄×D₄

yes

characteristic

113

C₂×C₄

C₄×D₄

yes

characteristic

114

C₂×C₄

C₂×Q₁₆

yes

—

115

C₂×C₄

#135

yes

—

116

C₂×C₄

C₂³×C₄

yes

characteristic

117

C₂×C₄

C₂²×D₄

yes

characteristic

118

C₂×C₄

C₂×#41

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

#131

yes

—

127

C₈

C₂×Q₁₆

yes

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₄×Q₁₆

yes

—

131

C₂²

C₂×C₄×D₄

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

C₄×Q₁₆

yes

—

138

C₄

C₂×#131

yes

characteristic

139

C₄

#568

yes

characteristic

140

C₄

C₂²×Q₁₆

yes

characteristic

141

C₄

C₂×#135

yes

characteristic

142

C₂

—

143

C₂

#2935

yes

characteristic

144

C₂

C₂×C₄×Q₁₆

yes

characteristic

145

C₂

#2957

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄
Frattini, cosocle	C₄²	C₂⁴
socle	C₂²	C₂×C₄×D₄
prime radical	C₂³	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center

C₂×C₄

C₂²×D₄

commutator

C₂×C₄

C₂³×C₄

Frattini, cosocle

C₄²

C₂⁴

socle

C₂²

C₂×C₄×D₄

prime radical

C₂³

C₄×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
3. #1591	C₂	semidirect
4. #1584	C₂	semidirect
5. C₈×Q₁₆	C₂	semidirect
77. M₁₆	Q₁₆	semidirect

H ≅ G/N

type

3. #1591

C₂

semidirect

4. #1584

C₂

semidirect

5. C₈×Q₁₆

C₂

semidirect

77. M₁₆

Q₁₆

semidirect

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

subgroup	index
#3170	2
#2962	2
#1591	2
#1584	2
C₈×Q₁₆	2
C₂×#335	2
#1481	2
C₈×M₁₆	2

subgroup

index

C₈×Q₁₆

C₈×M₁₆

Subgroups — #18068

Conjugacy classes of subgroups

Complements

Maximal subgroups