Group #19654 — subgroups

Subgroups — #19654

Order 256.

399 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 139 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	#3227	C₂	1	2	yes	maximal, complemented
2	#3206	C₂	1	2	yes	maximal, complemented
3	C₂×#498	C₂	1	1	yes	maximal, characteristic, complemented
4	#3147	C₂	1	1	yes	maximal, characteristic
5	#1716	C₂	1	2	yes	maximal, complemented
6	#1715	C₂	1	2	yes	maximal, complemented
7	#1708	C₂	1	2	yes	maximal, complemented
8	#1705	C₂	1	2	yes	maximal, complemented
9	#1450	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#135	—	2	2	no	—
11	#532	C₂²	1	1	yes	characteristic
12	#532	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Q₈	C₂²	1	2	yes	—
14	#498	—	2	2	no	—
15	#498	C₂²	1	2	yes	—
16	#498	C₂²	1	2	yes	—
17	#498	C₂²	1	4	yes	—
18	#495	—	2	2	no	—
19	#494	C₂²	1	1	yes	characteristic
20	#493	C₂²	1	1	yes	characteristic
21	#492	C₂²	1	1	yes	characteristic
22	#492	C₂²	1	1	yes	characteristic
23	#492	C₂²	1	4	yes	—
24	#487	C₂²	1	4	yes	—
25	#486	C₂²	1	4	yes	—
26	#481	—	2	4	no	—
27	#476	—	2	4	no	—
28	#461	—	2	4	no	—
29	#448	—	2	4	no	—
30	#431	C₂²	1	2	yes	—
31	C₂×#106	—	2	2	no	—
32	C₂×C₄×C₈	C₂²	1	1	yes	characteristic
33	#322	C₂²	1	2	yes	—
34	C₈²	C₂²	1	2	yes	—
35	C₂×#41	—	2	1	no	—
36	C₂×#41	—	2	1	no	—
37	C₂²×Q₈	—	2	2	no	—
38	#135	—	4	8	no	—
39	C₂×Q₁₆	—	2	2	no	—
40	C₂×SD₁₆	—	2	2	no	—
41	C₂×SD₁₆	—	2	2	no	—
42	C₂×SD₁₆	—	2	8	no	—
43	C₂×D₈	—	2	2	no	—
44	C₂²×C₈	D₄	1	2	yes	—
45	C₄⋊Q₈	—	2	2	no	—
46	C₄⋊Q₈	—	2	2	no	—
47	C₄⋊Q₈	—	2	2	no	—
48	C₄⋊Q₈	C₂³	1	4	yes	—
49	C₄⋊Q₈	C₂³	1	1	yes	characteristic
50	C₄⋊Q₈	C₂³	1	1	yes	characteristic
51	#127	C₂³	1	1	yes	characteristic
52	#127	C₂³	1	1	yes	characteristic
53	#124	—	2	1	no	—
54	#124	—	2	1	no	—
55	#121	—	2	2	no	—
56	#121	—	2	2	no	—
57	C₄×D₄	—	2	2	no	—
58	C₄×D₄	—	2	2	no	—
59	C₂×C₄⋊C₄	—	2	2	no	—
60	C₂×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄²	C₂³	1	1	yes	characteristic
62	#106	—	2	4	no	—
63	#106	—	2	4	no	—
64	#106	—	2	8	no	—
65	C₄⋊C₈	D₄	1	4	yes	—
66	#103	—	2	8	no	—
67	#102	—	2	8	no	—
68	C₂²⋊C₈	D₄	1	4	yes	—
69	C₄×C₈	D₄	1	2	yes	—
70	C₄×C₈	C₂³	1	1	yes	characteristic
71	C₄×C₈	C₂³	1	4	yes	—
72	C₄×C₈	C₂³	1	1	yes	characteristic
73	#41	—	4	4	no	—
74	#41	—	4	4	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	2	1	no	—
77	C₂×Q₈	—	2	1	no	—
78	C₂×Q₈	—	4	2	no	—
79	C₂×Q₈	—	8	4	no	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	1	no	—
83	C₂×D₄	—	2	1	no	—
84	C₂×D₄	—	4	1	no	—
85	C₂×D₄	—	4	1	no	—
86	C₂²×C₄	—	4	2	no	—
87	C₂²×C₄	—	4	1	no	—
88	C₂²×C₄	—	4	1	no	—
89	C₂²×C₄	C₂×D₄	1	2	yes	—
90	C₂²×C₄	C₂×D₄	1	1	yes	characteristic
91	Q₁₆	—	4	4	no	—
92	SD₁₆	—	4	4	no	—
93	SD₁₆	—	4	4	no	—
94	SD₁₆	—	8	8	no	—
95	D₈	—	4	4	no	—
96	C₂×C₈	SD₁₆	1	8	yes	complemented
97	C₂×C₈	C₂×D₄	1	8	yes	—
98	C₂×C₈	C₂×D₄	1	2	yes	—
99	C₂×C₈	C₂×D₄	1	2	yes	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	4	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	4	2	no	—
105	C₄⋊C₄	—	4	2	no	—
106	C₂²⋊C₄	—	4	2	no	—
107	C₂²⋊C₄	—	4	2	no	—
108	C₄²	C₂×D₄	1	1	yes	characteristic
109	C₄²	C₂×D₄	1	2	yes	—
110	C₄²	C₂⁴	1	1	yes	characteristic
111	C₂³	—	4	1	no	—
112	C₂³	—	4	1	no	—
113	C₂³	#127	1	1	yes	characteristic
114	Q₈	—	4	2	no	—
115	Q₈	—	4	2	no	—
116	Q₈	—	4	8	no	—
117	Q₈	—	8	4	no	—
118	D₄	—	4	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	4	no	—
121	D₄	—	4	4	no	—
122	D₄	—	8	2	no	—
123	D₄	—	8	2	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	4	no	—
128	C₂×C₄	—	8	4	no	—
129	C₂×C₄	—	8	2	no	—
130	C₂×C₄	—	8	2	no	—
131	C₂×C₄	#127	1	2	yes	—
132	C₂×C₄	#127	1	1	yes	characteristic
133	C₂×C₄	C₂×SD₁₆	1	4	yes	—
134	C₂×C₄	C₂²×D₄	1	1	yes	characteristic
135	C₂×C₄	C₂²×D₄	1	2	yes	—
136	C₈	—	2	8	no	—
137	C₈	C₂×SD₁₆	1	4	yes	—
138	C₈	C₂×SD₁₆	1	4	yes	—
139	C₂²	—	2	2	no	—
140	C₂²	—	8	1	no	—
141	C₂²	—	8	1	no	—
142	C₂²	—	8	2	no	—
143	C₂²	—	8	2	no	—
144	C₂²	#492	1	2	yes	—
145	C₂²	C₂×#127	1	1	yes	characteristic
146	C₄	—	2	2	no	—
147	C₄	—	8	1	no	—
148	C₄	—	8	1	no	—
149	C₄	—	8	4	no	—
150	C₄	#492	1	2	yes	—
151	C₄	C₂²×SD₁₆	1	2	yes	—
152	C₄	#577	1	4	yes	—
153	C₂	—	2	2	no	—
154	C₂	—	16	1	no	—
155	C₂	—	16	1	no	—
156	C₂	C₂×#492	1	1	yes	characteristic
157	C₂	#3154	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#3227

C₂

yes

maximal, complemented

#3206

C₂

yes

maximal, complemented

C₂×#498

C₂

yes

maximal, characteristic, complemented

#3147

C₂

yes

maximal, characteristic

#1716

C₂

yes

maximal, complemented

#1715

C₂

yes

maximal, complemented

#1708

C₂

yes

maximal, complemented

#1705

C₂

yes

maximal, complemented

#1450

C₂

yes

maximal, characteristic, complemented

C₂×#135

—

#532

C₂²

yes

characteristic

#532

C₂²

yes

characteristic

C₂×C₄⋊Q₈

C₂²

yes

—

#498

—

#498

C₂²

yes

—

#498

C₂²

yes

—

#498

C₂²

yes

—

#495

—

#494

C₂²

yes

characteristic

#493

C₂²

yes

characteristic

#492

C₂²

yes

characteristic

#492

C₂²

yes

characteristic

#492

C₂²

yes

—

#487

C₂²

yes

—

#486

C₂²

yes

—

—

—

—

—

C₂²

yes

—

C₂×#106

—

C₂×C₄×C₈

C₂²

yes

characteristic

#322

C₂²

yes

—

C₈²

C₂²

yes

—

C₂×#41

—

C₂×#41

—

C₂²×Q₈

—

#135

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₂×D₈

—

C₂²×C₈

D₄

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂³

yes

—

C₄⋊Q₈

C₂³

yes

characteristic

C₄⋊Q₈

C₂³

yes

characteristic

#127

C₂³

yes

characteristic

#127

C₂³

yes

—

—

—

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂³

yes

—

—

—

C₄⋊C₈

D₄

yes

—

#103

—

#102

—

C₂²⋊C₈

D₄

yes

—

C₄×C₈

D₄

yes

—

C₄×C₈

C₂³

yes

characteristic

C₄×C₈

C₂³

yes

—

C₄×C₈

C₂³

yes

characteristic

#41

—

#41

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄

yes

—

C₂²×C₄

C₂×D₄

yes

characteristic

Q₁₆

—

SD₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

C₂×C₈

SD₁₆

yes

complemented

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

C₂×C₈

C₂×D₄

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₂²⋊C₄

—

107

C₂²⋊C₄

—

108

C₄²

C₂×D₄

yes

characteristic

109

C₄²

C₂×D₄

yes

—

110

C₄²

C₂⁴

yes

characteristic

111

C₂³

—

112

C₂³

—

113

C₂³

#127

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

117

Q₈

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

#127

yes

—

132

C₂×C₄

#127

yes

characteristic

133

C₂×C₄

C₂×SD₁₆

yes

—

134

C₂×C₄

C₂²×D₄

yes

characteristic

135

C₂×C₄

C₂²×D₄

yes

—

136

C₈

—

137

C₈

C₂×SD₁₆

yes

—

138

C₈

C₂×SD₁₆

yes

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

#492

yes

—

145

C₂²

C₂×#127

yes

characteristic

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

#492

yes

—

151

C₄

C₂²×SD₁₆

yes

—

152

C₄

#577

yes

—

153

C₂

—

154

C₂

—

155

C₂

—

156

C₂

C₂×#492

yes

characteristic

157

C₂

#3154

yes

—

	subgroup	G/H
center, socle	C₂²	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁴
prime radical	#3147	C₂
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

C₂

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂² ◁ C₂×C₄² ◁ G

LECS

G ▷ C₄² ▷ C₂² ▷ C₁

N	H ≅ G/N	type
1. #3227	C₂	semidirect
2. #3206	C₂	semidirect
3. C₂×#498	C₂	semidirect
5. #1716	C₂	semidirect
6. #1715	C₂	semidirect
7. #1708	C₂	semidirect
8. #1705	C₂	semidirect
9. #1450	C₂	semidirect
96. C₂×C₈	SD₁₆	semidirect

H ≅ G/N

type

1. #3227

C₂

semidirect

2. #3206

C₂

semidirect

3. C₂×#498

C₂

semidirect

5. #1716

C₂

semidirect

6. #1715

C₂

semidirect

7. #1708

C₂

semidirect

8. #1705

C₂

semidirect

9. #1450

C₂

semidirect

96. C₂×C₈

SD₁₆

semidirect

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

subgroup	index
#3227	2
#3206	2
C₂×#498	2
#3147	2
#1716	2
#1715	2
#1708	2
#1705	2
#1450	2

subgroup

index

Subgroups — #19654

Conjugacy classes of subgroups

Complements

Maximal subgroups