Group #19632 — subgroups

Subgroups — #19632

Order 256.

375 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 375 conjugacy classes of subgroups — too many to draw. Only its 143 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	#3230	C₂	1	2	yes	maximal
2	#3170	C₂	1	1	yes	maximal, characteristic
3	#3165	C₂	1	2	yes	maximal
4	C₂×#494	C₂	1	1	yes	maximal, characteristic
5	#1723	C₂	1	4	yes	maximal, complemented
6	#1713	C₂	1	2	yes	maximal, complemented
7	#1707	C₂	1	2	yes	maximal, complemented
8	C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆	—	2	2	no	—
10	#533	C₂²	1	2	yes	—
11	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
12	#501	—	2	4	no	—
13	#500	C₂²	1	4	yes	—
14	#500	C₂²	1	1	yes	characteristic
15	#500	C₂²	1	4	yes	—
16	#498	C₂²	1	4	yes	—
17	#494	—	2	2	no	—
18	#494	C₂²	1	2	yes	—
19	#494	C₂²	1	2	yes	—
20	#494	C₂²	1	1	yes	characteristic
21	#487	C₂²	1	2	yes	—
22	#487	C₂²	1	4	yes	—
23	#484	—	2	4	no	—
24	#477	—	2	4	no	—
25	#467	—	2	2	no	—
26	C₄×Q₁₆	—	2	2	no	—
27	#431	C₂²	1	1	yes	characteristic
28	#428	—	2	4	no	—
29	C₄×M₁₆	C₂²	1	1	yes	characteristic
30	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
31	#322	C₂²	1	2	yes	—
32	C₈²	C₂²	1	2	yes	—
33	C₂²×Q₈	—	2	2	no	—
34	C₂×Q₁₆	—	2	4	no	—
35	C₂×Q₁₆	—	2	4	no	—
36	C₂×Q₁₆	—	2	2	no	—
37	C₂×Q₁₆	—	4	8	no	—
38	C₂×M₁₆	D₄	1	2	yes	—
39	C₂²×C₈	D₄	1	2	yes	—
40	C₄⋊Q₈	—	2	2	no	—
41	C₄⋊Q₈	—	2	2	no	—
42	C₄⋊Q₈	—	2	2	no	—
43	C₄⋊Q₈	C₂³	1	4	yes	—
44	C₄⋊Q₈	C₂³	1	2	yes	—
45	C₄⋊Q₈	C₂³	1	2	yes	—
46	#125	—	2	2	no	—
47	C₂²⋊Q₈	—	2	2	no	—
48	C₂²⋊Q₈	—	2	2	no	—
49	C₄×Q₈	—	2	2	no	—
50	C₄×Q₈	—	2	2	no	—
51	#117	—	2	2	no	—
52	C₂×C₄⋊C₄	—	2	2	no	—
53	C₂×C₄⋊C₄	—	2	2	no	—
54	C₂×C₄²	C₂³	1	1	yes	characteristic
55	#107	—	2	8	no	—
56	#107	—	2	2	no	—
57	#107	—	2	8	no	—
58	#106	—	2	8	no	—
59	C₄⋊C₈	D₄	1	2	yes	—
60	#103	—	2	4	no	—
61	#103	—	2	8	no	—
62	C₂²⋊C₈	D₄	1	2	yes	—
63	#97	D₄	1	2	yes	—
64	C₄×C₈	D₄	1	2	yes	—
65	C₄×C₈	C₂³	1	1	yes	characteristic
66	C₄×C₈	C₂³	1	2	yes	—
67	C₄×C₈	C₂³	1	2	yes	—
68	C₄×C₈	C₂³	1	1	yes	characteristic
69	C₂×Q₈	—	2	2	no	—
70	C₂×Q₈	—	2	2	no	—
71	C₂×Q₈	—	2	4	no	—
72	C₂×Q₈	—	4	8	no	—
73	C₂×Q₈	—	4	2	no	—
74	C₂²×C₄	—	4	2	no	—
75	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
78	Q₁₆	—	4	8	no	—
79	Q₁₆	—	4	4	no	—
80	Q₁₆	—	4	8	no	—
81	M₁₆	Q₁₆	1	8	yes	complemented
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	C₂×D₄	1	2	yes	—
84	C₂×C₈	C₂×D₄	1	4	yes	—
85	C₂×C₈	C₂×D₄	1	4	yes	—
86	C₂×C₈	C₂×D₄	1	2	yes	—
87	C₄⋊C₄	—	2	4	no	—
88	C₄⋊C₄	—	2	2	no	—
89	C₄⋊C₄	—	2	2	no	—
90	C₄⋊C₄	—	2	4	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	4	2	no	—
94	C₄⋊C₄	—	4	2	no	—
95	C₄⋊C₄	—	4	2	no	—
96	C₄⋊C₄	—	4	2	no	—
97	C₂²⋊C₄	—	4	2	no	—
98	C₄²	—	4	2	no	—
99	C₄²	C₂×D₄	1	1	yes	characteristic
100	C₄²	C₂×D₄	1	1	yes	characteristic
101	C₄²	C₂×D₄	1	1	yes	characteristic
102	C₄²	C₂⁴	1	1	yes	characteristic
103	C₂³	#127	1	1	yes	characteristic
104	Q₈	—	4	8	no	—
105	Q₈	—	4	4	no	—
106	Q₈	—	4	4	no	—
107	Q₈	—	8	4	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	—	4	4	no	—
113	C₂×C₄	—	8	4	no	—
114	C₂×C₄	#127	1	1	yes	characteristic
115	C₂×C₄	#127	1	1	yes	characteristic
116	C₂×C₄	#127	1	1	yes	characteristic
117	C₂×C₄	C₂×Q₁₆	1	4	yes	—
118	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
119	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
121	C₈	—	2	2	no	—
122	C₈	—	2	4	no	—
123	C₈	C₂×Q₁₆	1	8	yes	complemented
124	C₈	AGL(1,ℤ/8)	1	4	yes	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#494	1	2	yes	—
128	C₂²	C₂×#127	1	1	yes	characteristic
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	8	4	no	—
132	C₄	—	8	2	no	—
133	C₄	—	8	2	no	—
134	C₄	#494	1	2	yes	—
135	C₄	C₂²×Q₁₆	1	2	yes	—
136	C₄	C₂×AGL(1,ℤ/8)	1	2	yes	—
137	C₄	#578	1	2	yes	—
138	C₂	—	2	2	no	—
139	C₂	C₂×#494	1	1	yes	characteristic
140	C₂	#3151	1	1	yes	characteristic
141	C₂	#3156	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3230

C₂

yes

maximal

#3170

C₂

yes

maximal, characteristic

#3165

C₂

yes

maximal

C₂×#494

C₂

yes

maximal, characteristic

#1723

C₂

yes

maximal, complemented

#1713

C₂

yes

maximal, complemented

#1707

C₂

yes

maximal, complemented

C₈×M₁₆

C₂

yes

maximal, characteristic

C₂²×Q₁₆

—

#533

C₂²

yes

—

C₂×C₄⋊Q₈

C₂²

yes

—

#501

—

#500

C₂²

yes

—

#500

C₂²

yes

characteristic

#500

C₂²

yes

—

#498

C₂²

yes

—

#494

—

#494

C₂²

yes

—

#494

C₂²

yes

—

#494

C₂²

yes

characteristic

#487

C₂²

yes

—

#487

C₂²

yes

—

#484

—

#477

—

#467

—

C₄×Q₁₆

—

#431

C₂²

yes

characteristic

#428

—

C₄×M₁₆

C₂²

yes

characteristic

C₂×C₄×C₈

C₂²

yes

characteristic

#322

C₂²

yes

—

C₈²

C₂²

yes

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

D₄

yes

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

—

—

—

—

C₄⋊C₈

D₄

yes

—

#103

—

#103

—

C₂²⋊C₈

D₄

yes

—

#97

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

Q₁₆

—

Q₁₆

—

M₁₆

Q₁₆

yes

complemented

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₄

yes

characteristic

100

C₄²

C₂×D₄

yes

characteristic

101

C₄²

C₂×D₄

yes

characteristic

102

C₄²

C₂⁴

yes

characteristic

103

C₂³

#127

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

#127

yes

characteristic

115

C₂×C₄

#127

yes

characteristic

116

C₂×C₄

#127

yes

characteristic

117

C₂×C₄

C₂×Q₁₆

yes

—

118

C₂×C₄

C₂²×D₄

yes

characteristic

119

C₂×C₄

C₂²×D₄

yes

characteristic

120

C₂×C₄

C₂²×D₄

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

C₂×Q₁₆

yes

complemented

124

C₈

AGL(1,ℤ/8)

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

#494

yes

—

128

C₂²

C₂×#127

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

#494

yes

—

135

C₄

C₂²×Q₁₆

yes

—

136

C₄

C₂×AGL(1,ℤ/8)

yes

—

137

C₄

#578

yes

—

138

C₂

—

139

C₂

C₂×#494

yes

characteristic

140

C₂

#3151

yes

characteristic

141

C₂

#3156

yes

characteristic

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	C₂³	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle

C₂²

C₂×#127

commutator, Frattini, cosocle

C₄²

C₂⁴

C₂³

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
5. #1723	C₂	semidirect
6. #1713	C₂	semidirect
7. #1707	C₂	semidirect
81. M₁₆	Q₁₆	semidirect
123. C₈	C₂×Q₁₆	semidirect

H ≅ G/N

type

5. #1723

C₂

semidirect

6. #1713

C₂

semidirect

7. #1707

C₂

semidirect

81. M₁₆

Q₁₆

semidirect

123. C₈

C₂×Q₁₆

semidirect

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

subgroup	index
#3230	2
#3170	2
#3165	2
C₂×#494	2
#1723	2
#1713	2
#1707	2
C₈×M₁₆	2

subgroup

index

C₈×M₁₆

Subgroups — #19632

Conjugacy classes of subgroups

Complements

Maximal subgroups