Group #19931 — subgroups

Subgroups — #19931

Order 256.

277 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 277 conjugacy classes of subgroups — too many to draw. Only its 109 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	#3387	C₂	1	2	yes	maximal
2	#3363	C₂	1	2	yes	maximal
3	#1707	C₂	1	1	yes	maximal, characteristic
4	#1692	C₂	1	2	yes	maximal
5	#1684	C₂	1	2	yes	maximal
6	#1648	C₂	1	2	yes	maximal
7	#1633	C₂	1	2	yes	maximal
8	#1591	C₂	1	1	yes	maximal, characteristic
9	C₈×Q₁₆	C₂	1	1	yes	maximal, characteristic
10	Q₈²	C₂²	1	2	yes	—
11	#557	C₂²	1	2	yes	—
12	#501	—	2	2	no	—
13	#500	C₂²	1	1	yes	characteristic
14	#498	C₂²	1	2	yes	—
15	#494	C₂²	1	1	yes	characteristic
16	#487	—	2	2	no	—
17	#487	C₂²	1	2	yes	—
18	#479	—	2	2	no	—
19	#477	—	2	2	no	—
20	#477	C₂²	1	2	yes	—
21	#477	C₂²	1	2	yes	—
22	#475	—	2	2	no	—
23	#475	C₂²	1	2	yes	—
24	#462	—	2	2	no	—
25	#462	C₂²	1	2	yes	—
26	#446	C₂²	1	2	yes	—
27	C₈×Q₈	C₂²	1	2	yes	—
28	#441	—	2	2	no	—
29	C₄×Q₁₆	—	2	2	no	—
30	C₄×Q₁₆	C₂²	1	1	yes	characteristic
31	C₄×Q₁₆	C₂²	1	1	yes	characteristic
32	#335	C₂²	1	1	yes	characteristic
33	#335	C₂²	1	1	yes	characteristic
34	#333	C₂²	1	2	yes	—
35	#332	C₂²	1	2	yes	—
36	Q₈⋊C₈	C₂²	1	2	yes	—
37	Q₈⋊C₈	C₂²	1	2	yes	—
38	C₈²	C₂²	1	1	yes	characteristic
39	C₂×Q₁₆	—	2	2	no	—
40	C₂×Q₁₆	—	4	1	no	—
41	C₂×Q₁₆	D₄	1	1	yes	characteristic
42	C₂×Q₁₆	D₄	1	1	yes	characteristic
43	C₄⋊Q₈	—	2	2	no	—
44	C₄⋊Q₈	—	2	2	no	—
45	C₄⋊Q₈	—	2	2	no	—
46	C₄⋊Q₈	—	4	2	no	—
47	C₄⋊Q₈	C₂³	1	2	yes	—
48	C₄⋊Q₈	C₂³	1	2	yes	—
49	#125	—	2	2	no	—
50	#125	—	4	2	no	—
51	C₄×Q₈	—	2	2	no	—
52	C₄×Q₈	—	2	2	no	—
53	C₄×Q₈	—	2	2	no	—
54	C₄×Q₈	—	2	2	no	—
55	C₄×Q₈	C₂³	1	2	yes	—
56	C₄×Q₈	C₂³	1	2	yes	—
57	#107	—	2	2	no	—
58	#107	—	2	2	no	—
59	#107	—	4	1	no	—
60	#107	D₄	1	1	yes	characteristic
61	#107	D₄	1	1	yes	characteristic
62	#106	—	2	2	no	—
63	#106	—	2	2	no	—
64	#106	—	2	2	no	—
65	C₄⋊C₈	—	2	2	no	—
66	C₄⋊C₈	—	2	2	no	—
67	C₄⋊C₈	C₂³	1	2	yes	—
68	C₄⋊C₈	C₂³	1	2	yes	—
69	#103	—	2	2	no	—
70	#103	—	2	2	no	—
71	#103	—	2	2	no	—
72	#103	—	2	2	no	—
73	#103	—	2	2	no	—
74	#103	—	2	2	no	—
75	#103	D₄	1	2	yes	—
76	#103	D₄	1	2	yes	—
77	#97	—	2	2	no	—
78	C₄×C₈	—	2	2	no	—
79	C₄×C₈	C₂³	1	1	yes	characteristic
80	C₄×C₈	C₂³	1	1	yes	characteristic
81	C₄×C₈	C₂³	1	1	yes	characteristic
82	C₂×Q₈	—	2	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	—	4	2	no	—
85	C₂×Q₈	C₂×D₄	1	2	yes	—
86	C₂×Q₈	C₂×D₄	1	2	yes	—
87	Q₁₆	—	4	1	no	—
88	Q₁₆	—	4	4	no	—
89	Q₁₆	—	8	2	no	—
90	Q₁₆	Q₁₆	1	4	yes	complemented
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈	C₂×D₄	1	1	yes	characteristic
98	C₄⋊C₄	—	2	2	no	—
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	2	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	4	2	no	—
109	C₄⋊C₄	C₂×D₄	1	2	yes	—
110	C₄⋊C₄	C₂×D₄	1	2	yes	—
111	C₄²	—	2	2	no	—
112	C₄²	—	2	2	no	—
113	C₄²	—	4	2	no	—
114	C₄²	—	4	2	no	—
115	C₄²	C₂⁴	1	1	yes	characteristic
116	Q₈	—	2	2	no	—
117	Q₈	—	2	2	no	—
118	Q₈	—	2	2	no	—
119	Q₈	—	4	2	no	—
120	Q₈	—	4	2	no	—
121	Q₈	—	8	2	no	—
122	Q₈	—	16	2	no	—
123	Q₈	C₂×Q₁₆	1	4	yes	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
132	C₂×C₄	H(5,𝔽₂)	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	2	1	no	—
135	C₈	—	2	1	no	—
136	C₈	—	2	1	no	—
137	C₈	—	4	2	no	—
138	C₈	—	4	2	no	—
139	C₈	C₂×Q₁₆	1	2	yes	—
140	C₂²	D₄²	1	1	yes	characteristic
141	C₄	—	2	4	no	—
142	C₄	—	2	1	no	—
143	C₄	—	4	2	no	—
144	C₄	—	4	2	no	—
145	C₄	—	4	2	no	—
146	C₄	—	8	2	no	—
147	C₄	—	8	2	no	—
148	C₄	C₂²×Q₁₆	1	1	yes	characteristic
149	C₄	#575	1	1	yes	characteristic
150	C₄	#578	1	1	yes	characteristic
151	C₄	#578	1	1	yes	characteristic
152	C₂	#3277	1	1	yes	characteristic
153	C₂	D₄×Q₁₆	1	1	yes	characteristic
154	C₂	#3291	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#3387

C₂

yes

maximal

#3363

C₂

yes

maximal

#1707

C₂

yes

maximal, characteristic

#1692

C₂

yes

maximal

#1684

C₂

yes

maximal

#1648

C₂

yes

maximal

#1633

C₂

yes

maximal

#1591

C₂

yes

maximal, characteristic

C₈×Q₁₆

C₂

yes

maximal, characteristic

Q₈²

C₂²

yes

—

#557

C₂²

yes

—

#501

—

#500

C₂²

yes

characteristic

#498

C₂²

yes

—

#494

C₂²

yes

characteristic

#487

—

#487

C₂²

yes

—

#479

—

#477

—

#477

C₂²

yes

—

#477

C₂²

yes

—

#475

—

#475

C₂²

yes

—

#462

—

#462

C₂²

yes

—

#446

C₂²

yes

—

C₈×Q₈

C₂²

yes

—

#441

—

C₄×Q₁₆

—

C₄×Q₁₆

C₂²

yes

characteristic

C₄×Q₁₆

C₂²

yes

characteristic

#335

C₂²

yes

characteristic

#335

C₂²

yes

characteristic

#333

C₂²

yes

—

#332

C₂²

yes

—

Q₈⋊C₈

C₂²

yes

—

Q₈⋊C₈

C₂²

yes

—

C₈²

C₂²

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

D₄

yes

characteristic

C₂×Q₁₆

D₄

yes

characteristic

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

—

#125

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂³

yes

—

C₄×Q₈

C₂³

yes

—

—

—

—

D₄

yes

characteristic

#107

D₄

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂³

yes

—

C₄⋊C₈

C₂³

yes

—

—

—

—

—

—

—

D₄

yes

—

#103

D₄

yes

—

#97

—

C₄×C₈

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₄

yes

—

C₂×Q₈

C₂×D₄

yes

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

Q₁₆

yes

complemented

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₂×C₈

C₂×D₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₄⋊C₄

C₂×D₄

yes

—

110

C₄⋊C₄

C₂×D₄

yes

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₄²

C₂⁴

yes

characteristic

116

Q₈

—

117

Q₈

—

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

123

Q₈

C₂×Q₁₆

yes

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

C₂²×D₄

yes

characteristic

131

C₂×C₄

C₂²×D₄

yes

characteristic

132

C₂×C₄

H(5,𝔽₂)

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

—

139

C₈

C₂×Q₁₆

yes

—

140

C₂²

D₄²

yes

characteristic

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

C₂²×Q₁₆

yes

characteristic

149

C₄

#575

yes

characteristic

150

C₄

#578

yes

characteristic

151

C₄

#578

yes

characteristic

152

C₂

#3277

yes

characteristic

153

C₂

D₄×Q₁₆

yes

characteristic

154

C₂

#3291

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

subgroup

G/H

center, socle, prime radical

C₂²

D₄²

commutator, Frattini, cosocle

C₄²

C₂⁴

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
90. Q₁₆	Q₁₆	semidirect

H ≅ G/N

type

90. Q₁₆

Q₁₆

semidirect

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

subgroup	index
#3387	2
#3363	2
#1707	2
#1692	2
#1684	2
#1648	2
#1633	2
#1591	2
C₈×Q₁₆	2

subgroup

index

C₈×Q₁₆

Subgroups — #19931

Conjugacy classes of subgroups

Complements

Maximal subgroups