Group #18072 — subgroups

Subgroups — #18072

Order 256.

289 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 289 conjugacy classes of subgroups — too many to draw. Only its 155 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	#1593	C₂	1	2	yes	maximal, complemented
4	#1587	C₂	1	4	yes	maximal, complemented
5	#1579	C₂	1	2	yes	maximal, complemented
6	C₂×#334	C₂	1	1	yes	maximal, characteristic
7	#1481	C₂	1	2	yes	maximal
8	#1451	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	#334	C₂²	1	1	yes	characteristic
27	#334	C₂²	1	2	yes	—
28	#334	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	#322	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	2	yes	—
36	C₂²×C₈	—	2	2	no	—
37	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
38	C₂²×C₈	C₂×C₄	1	1	yes	characteristic
39	C₄⋊Q₈	C₂×C₄	1	2	yes	—
40	C₄⋊Q₈	C₂×C₄	1	2	yes	—
41	#125	—	2	2	no	—
42	C₂²⋊Q₈	—	2	2	no	—
43	C₂²⋊Q₈	C₂×C₄	1	4	yes	—
44	C₄×Q₈	—	2	2	no	—
45	C₄×Q₈	C₂³	1	4	yes	—
46	#117	—	2	2	no	—
47	C₂×C₄²	C₂³	1	1	yes	characteristic
48	#107	C₂×C₄	1	2	yes	—
49	#107	C₂×C₄	1	2	yes	—
50	C₄⋊C₈	—	2	4	no	—
51	C₄⋊C₈	—	2	4	no	—
52	C₄⋊C₈	—	2	2	no	—
53	C₄⋊C₈	—	2	1	no	—
54	C₄⋊C₈	C₂³	1	2	yes	—
55	C₄⋊C₈	C₂³	1	2	yes	—
56	#103	C₂×C₄	1	4	yes	—
57	#103	C₂×C₄	1	4	yes	—
58	C₂²⋊C₈	—	2	1	no	—
59	#97	—	2	4	no	—
60	#97	D₄	1	2	yes	—
61	C₄×C₈	—	2	4	no	—
62	C₄×C₈	C₂×C₄	1	1	yes	characteristic
63	C₄×C₈	C₂×C₄	1	1	yes	characteristic
64	C₄×C₈	C₂³	1	2	yes	—
65	C₄×C₈	C₂³	1	1	yes	characteristic
66	C₄×C₈	C₂³	1	1	yes	characteristic
67	C₄×C₈	C₂³	1	2	yes	—
68	C₂×Q₈	—	2	2	no	—
69	C₂×Q₈	C₂²×C₄	1	4	yes	—
70	C₂²×C₄	C₂²×C₄	1	1	yes	characteristic
71	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂²×C₄	#41	1	1	yes	characteristic
73	Q₁₆	—	2	4	no	—
74	Q₁₆	—	4	2	no	—
75	M₁₆	—	2	4	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	2	2	no	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	4	1	no	—
81	C₂×C₈	—	4	4	no	—
82	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
83	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₈	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₈	C₂×D₄	1	4	yes	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	2	4	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	C₂²×C₄	1	4	yes	—
93	C₂²⋊C₄	—	2	4	no	—
94	C₄²	—	2	4	no	—
95	C₄²	C₂²×C₄	1	1	yes	characteristic
96	C₄²	C₂×D₄	1	1	yes	characteristic
97	C₄²	#41	1	1	yes	characteristic
98	C₄²	C₂⁴	1	1	yes	characteristic
99	C₂³	C₄×D₄	1	1	yes	characteristic
100	Q₈	—	2	4	no	—
101	Q₈	—	2	4	no	—
102	Q₈	—	8	2	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	C₄×D₄	1	1	yes	characteristic
109	C₂×C₄	C₄×D₄	1	1	yes	characteristic
110	C₂×C₄	C₄×D₄	1	1	yes	characteristic
111	C₂×C₄	#137	1	4	yes	—
112	C₂×C₄	C₂³×C₄	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄	C₂×#41	1	1	yes	characteristic
115	C₈	—	2	1	no	—
116	C₈	—	2	1	no	—
117	C₈	—	2	1	no	—
118	C₈	—	2	4	no	—
119	C₈	—	4	2	no	—
120	C₈	—	4	2	no	—
121	C₈	—	4	2	no	—
122	C₈	#131	1	2	yes	—
123	C₂²	—	2	1	no	—
124	C₂²	—	2	1	no	—
125	C₂²	#441	1	2	yes	—
126	C₂²	C₂×C₄×D₄	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	2	1	no	—
129	C₄	—	2	1	no	—
130	C₄	—	4	4	no	—
131	C₄	—	4	4	no	—
132	C₄	#441	1	2	yes	—
133	C₄	C₂×#131	1	1	yes	characteristic
134	C₄	#568	1	1	yes	characteristic
135	C₄	C₂×#137	1	2	yes	—
136	C₂	—	2	2	no	—
137	C₂	#2935	1	1	yes	characteristic
138	C₂	C₂×#441	1	1	yes	characteristic
139	C₂	#2957	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3170

C₂

yes

maximal, characteristic

#2962

C₂

yes

maximal

#1593

C₂

yes

maximal, complemented

#1587

C₂

yes

maximal, complemented

#1579

C₂

yes

maximal, complemented

C₂×#334

C₂

yes

maximal, characteristic

#1481

C₂

yes

maximal

#1451

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

#334

C₂²

yes

characteristic

#334

C₂²

yes

—

#334

C₂²

yes

characteristic

Q₈⋊C₈

C₂²

yes

—

Q₈⋊C₈

C₂²

yes

—

#322

C₂²

yes

—

#322

C₂²

yes

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂×Q₁₆

C₂×C₄

yes

—

C₂×M₁₆

D₄

yes

—

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

—

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

—

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

C₄²

#41

yes

characteristic

C₄²

C₂⁴

yes

characteristic

C₂³

C₄×D₄

yes

characteristic

100

Q₈

—

101

Q₈

—

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

C₄×D₄

yes

characteristic

109

C₂×C₄

C₄×D₄

yes

characteristic

110

C₂×C₄

C₄×D₄

yes

characteristic

111

C₂×C₄

#137

yes

—

112

C₂×C₄

C₂³×C₄

yes

characteristic

113

C₂×C₄

C₂²×D₄

yes

characteristic

114

C₂×C₄

C₂×#41

yes

characteristic

115

C₈

—

116

C₈

—

117

C₈

—

118

C₈

—

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

#131

yes

—

123

C₂²

—

124

C₂²

—

125

C₂²

#441

yes

—

126

C₂²

C₂×C₄×D₄

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#441

yes

—

133

C₄

C₂×#131

yes

characteristic

134

C₄

#568

yes

characteristic

135

C₄

C₂×#137

yes

—

136

C₂

—

137

C₂

#2935

yes

characteristic

138

C₂

C₂×#441

yes

characteristic

139

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. #1593	C₂	semidirect
4. #1587	C₂	semidirect
5. #1579	C₂	semidirect

H ≅ G/N

type

3. #1593

C₂

semidirect

4. #1587

C₂

semidirect

5. #1579

C₂

semidirect

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

subgroup	index
#3170	2
#2962	2
#1593	2
#1587	2
#1579	2
C₂×#334	2
#1481	2
#1451	2

subgroup

index

Subgroups — #18072

Conjugacy classes of subgroups

Complements

Maximal subgroups