Group #44503245 — subgroups

Subgroups — SD₁₆⋊C₈×C₂₄₁ #44503245

Order 30848.

176 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 100 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	C₈×Q₈×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×SD₁₆×C₂₄₁	C₂	1	1	yes	maximal, characteristic
3	C₈×D₄×C₂₄₁	C₂	1	1	yes	maximal, characteristic
4	#335×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
5	Q₈⋊C₈×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄⋊C₈×C₂₄₁	C₂	1	1	yes	maximal, characteristic
7	#322×C₂₄₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×SD₁₆×C₂₄₁	C₄	1	1	yes	characteristic
9	C₂²×C₈×C₂₄₁	—	2	1	no	—
10	C₄×Q₈×C₂₄₁	C₂²	1	1	yes	characteristic
11	C₄×D₄×C₂₄₁	C₂²	1	1	yes	characteristic
12	#106×C₂₄₁	C₄	1	1	yes	characteristic
13	C₄⋊C₈×C₂₄₁	—	2	1	no	—
14	C₄⋊C₈×C₂₄₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₂₄₁	C₂²	1	1	yes	characteristic
16	#103×C₂₄₁	C₄	1	1	yes	characteristic
17	#102×C₂₄₁	C₄	1	1	yes	characteristic
18	C₂²⋊C₈×C₂₄₁	—	2	1	no	—
19	C₄×C₈×C₂₄₁	—	2	1	no	—
20	C₄×C₈×C₂₄₁	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₂₄₁	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₂₄₁	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₂₄₁	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄×C₂₄₁	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₄₁	—	2	1	no	—
26	SD₁₆×C₂₄₁	C₈	1	4	yes	complemented
27	C₂×C₈×C₂₄₁	—	2	1	no	—
28	C₂×C₈×C₂₄₁	—	2	1	no	—
29	C₂×C₈×C₂₄₁	—	2	1	no	—
30	C₂×C₈×C₂₄₁	—	4	2	no	—
31	C₂×C₈×C₂₄₁	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₂₄₁	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₂₄₁	D₄	1	1	yes	characteristic
34	C₂×C₈×C₂₄₁	D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₂₄₁	—	2	1	no	—
36	C₄⋊C₄×C₂₄₁	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₄×C₂₄₁	C₂×C₄	1	1	yes	characteristic
38	C₂²⋊C₄×C₂₄₁	—	2	1	no	—
39	C₄²×C₂₄₁	—	2	1	no	—
40	C₄²×C₂₄₁	C₂³	1	1	yes	characteristic
41	C₂³×C₂₄₁	—	2	1	no	—
42	Q₈×C₂₄₁	—	2	1	no	—
43	Q₈×C₂₄₁	C₂×C₈	1	2	yes	complemented
44	D₄×C₂₄₁	—	2	1	no	—
45	D₄×C₂₄₁	C₂×C₈	1	2	yes	—
46	C₂×C₄×C₂₄₁	—	2	1	no	—
47	C₂×C₄×C₂₄₁	—	2	1	no	—
48	C₂×C₄×C₂₄₁	—	2	1	no	—
49	C₂×C₄×C₂₄₁	—	4	1	no	—
50	C₂×C₄×C₂₄₁	—	4	1	no	—
51	C₂×C₄×C₂₄₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₂₄₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₄₁	#41	1	1	yes	characteristic
54	C₈×C₂₄₁	—	2	1	no	—
55	C₈×C₂₄₁	—	2	1	no	—
56	C₈×C₂₄₁	—	2	1	no	—
57	C₈×C₂₄₁	—	4	1	no	—
58	C₈×C₂₄₁	—	4	1	no	—
59	C₈×C₂₄₁	—	4	1	no	—
60	C₈×C₂₄₁	C₂×C₈	1	2	yes	complemented
61	C₂²×C₂₄₁	—	2	2	no	—
62	C₂²×C₂₄₁	—	4	1	no	—
63	C₂²×C₂₄₁	—	4	1	no	—
64	C₂²×C₂₄₁	C₄×D₄	1	1	yes	characteristic
65	C₄×C₂₄₁	—	2	2	no	—
66	C₄×C₂₄₁	—	2	1	no	—
67	C₄×C₂₄₁	—	4	1	no	—
68	C₄×C₂₄₁	—	4	1	no	—
69	C₄×C₂₄₁	C₂²×C₈	1	1	yes	characteristic
70	C₄×C₂₄₁	#131	1	1	yes	characteristic
71	C₄×C₂₄₁	AGL(1,ℤ/8)	1	1	yes	characteristic
72	C₄×C₂₄₁	#137	1	1	yes	characteristic
73	C₂×C₂₄₁	—	4	2	no	—
74	C₂×C₂₄₁	C₈×D₄	1	1	yes	characteristic
75	C₂×C₂₄₁	#440	1	1	yes	characteristic
76	C₂×C₂₄₁	#444	1	1	yes	characteristic
77	C₂₄₁	SD₁₆⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
78	SD₁₆⋊C₈	C₂₄₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₈×Q₈	C₂×C₂₄₁	1	1	yes	characteristic, complemented
80	C₄×SD₁₆	C₂×C₂₄₁	1	1	yes	characteristic
81	C₈×D₄	C₂×C₂₄₁	1	1	yes	characteristic
82	#335	C₂×C₂₄₁	1	1	yes	characteristic, complemented
83	Q₈⋊C₈	C₂×C₂₄₁	1	1	yes	characteristic, complemented
84	D₄⋊C₈	C₂×C₂₄₁	1	1	yes	characteristic
85	#322	C₂×C₂₄₁	1	1	yes	characteristic, complemented
86	C₂×SD₁₆	C₄×C₂₄₁	1	1	yes	characteristic
87	C₂²×C₈	—	2	1	no	—
88	C₄×Q₈	C₂²×C₂₄₁	1	1	yes	characteristic
89	C₄×D₄	C₂²×C₂₄₁	1	1	yes	characteristic
90	#106	C₄×C₂₄₁	1	1	yes	characteristic
91	C₄⋊C₈	—	2	1	no	—
92	C₄⋊C₈	C₂²×C₂₄₁	1	1	yes	characteristic
93	C₄⋊C₈	C₂²×C₂₄₁	1	1	yes	characteristic
94	#103	C₄×C₂₄₁	1	1	yes	characteristic
95	#102	C₄×C₂₄₁	1	1	yes	characteristic
96	C₂²⋊C₈	—	2	1	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	C₂²×C₂₄₁	1	1	yes	characteristic
99	C₄×C₈	C₂²×C₂₄₁	1	1	yes	characteristic
100	C₄×C₈	C₂²×C₂₄₁	1	1	yes	characteristic
101	C₂×Q₈	C₂×C₄×C₂₄₁	1	1	yes	characteristic
102	C₂×D₄	C₂×C₄×C₂₄₁	1	1	yes	characteristic
103	C₂²×C₄	—	2	1	no	—
104	SD₁₆	C₈×C₂₄₁	1	4	yes	complemented
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	C₂×C₄×C₂₄₁	1	1	yes	characteristic
110	C₂×C₈	C₂×C₄×C₂₄₁	1	1	yes	characteristic
111	C₂×C₈	D₄×C₂₄₁	1	1	yes	characteristic
112	C₂×C₈	D₄×C₂₄₁	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	C₂×C₄×C₂₄₁	1	1	yes	characteristic
115	C₄⋊C₄	C₂×C₄×C₂₄₁	1	1	yes	characteristic
116	C₂²⋊C₄	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	C₂³×C₂₄₁	1	1	yes	characteristic
119	C₂³	—	2	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	C₂×C₈×C₂₄₁	1	2	yes	complemented
122	D₄	—	2	1	no	—
123	D₄	C₂×C₈×C₂₄₁	1	2	yes	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	C₂²×C₄×C₂₄₁	1	1	yes	characteristic
130	C₂×C₄	C₂×D₄×C₂₄₁	1	1	yes	characteristic
131	C₂×C₄	#41×C₂₄₁	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	4	1	no	—
136	C₈	—	4	1	no	—
137	C₈	—	4	1	no	—
138	C₈	C₂×C₈×C₂₄₁	1	2	yes	complemented
139	C₂²	—	2	2	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	1	no	—
142	C₂²	C₄×D₄×C₂₄₁	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	2	1	no	—
145	C₄	—	4	1	no	—
146	C₄	—	4	1	no	—
147	C₄	C₂²×C₈×C₂₄₁	1	1	yes	characteristic
148	C₄	#131×C₂₄₁	1	1	yes	characteristic
149	C₄	AGL(1,ℤ/8)×C₂₄₁	1	1	yes	characteristic
150	C₄	#137×C₂₄₁	1	1	yes	characteristic
151	C₂	—	4	2	no	—
152	C₂	C₈×D₄×C₂₄₁	1	1	yes	characteristic
153	C₂	#440×C₂₄₁	1	1	yes	characteristic
154	C₂	#444×C₂₄₁	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Q₈×C₂₄₁

C₂

yes

maximal, characteristic, complemented

C₄×SD₁₆×C₂₄₁

C₂

yes

maximal, characteristic

C₈×D₄×C₂₄₁

C₂

yes

maximal, characteristic

#335×C₂₄₁

C₂

yes

maximal, characteristic, complemented

Q₈⋊C₈×C₂₄₁

C₂

yes

maximal, characteristic, complemented

D₄⋊C₈×C₂₄₁

C₂

yes

maximal, characteristic

#322×C₂₄₁

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₂₄₁

C₄

yes

characteristic

C₂²×C₈×C₂₄₁

—

C₄×Q₈×C₂₄₁

C₂²

yes

characteristic

C₄×D₄×C₂₄₁

C₂²

yes

characteristic

#106×C₂₄₁

C₄

yes

characteristic

C₄⋊C₈×C₂₄₁

—

C₄⋊C₈×C₂₄₁

C₂²

yes

characteristic

C₄⋊C₈×C₂₄₁

C₂²

yes

characteristic

#103×C₂₄₁

C₄

yes

characteristic

#102×C₂₄₁

C₄

yes

characteristic

C₂²⋊C₈×C₂₄₁

—

C₄×C₈×C₂₄₁

—

C₄×C₈×C₂₄₁

C₂²

yes

characteristic

C₄×C₈×C₂₄₁

C₂²

yes

characteristic

C₄×C₈×C₂₄₁

C₂²

yes

characteristic

C₂×Q₈×C₂₄₁

C₂×C₄

yes

characteristic

C₂×D₄×C₂₄₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₄₁

—

SD₁₆×C₂₄₁

C₈

yes

complemented

C₂×C₈×C₂₄₁

—

C₂×C₈×C₂₄₁

—

C₂×C₈×C₂₄₁

—

C₂×C₈×C₂₄₁

—

C₂×C₈×C₂₄₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₄₁

C₂×C₄

yes

characteristic

C₂×C₈×C₂₄₁

D₄

yes

characteristic

C₂×C₈×C₂₄₁

D₄

yes

characteristic

C₄⋊C₄×C₂₄₁

—

C₄⋊C₄×C₂₄₁

C₂×C₄

yes

characteristic

C₄⋊C₄×C₂₄₁

C₂×C₄

yes

characteristic

C₂²⋊C₄×C₂₄₁

—

C₄²×C₂₄₁

—

C₄²×C₂₄₁

C₂³

yes

characteristic

C₂³×C₂₄₁

—

Q₈×C₂₄₁

—

Q₈×C₂₄₁

C₂×C₈

yes

complemented

D₄×C₂₄₁

—

D₄×C₂₄₁

C₂×C₈

yes

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

—

C₂×C₄×C₂₄₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₄₁

C₂×D₄

yes

characteristic

C₂×C₄×C₂₄₁

#41

yes

characteristic

C₈×C₂₄₁

—

C₈×C₂₄₁

—

C₈×C₂₄₁

—

C₈×C₂₄₁

—

C₈×C₂₄₁

—

C₈×C₂₄₁

—

C₈×C₂₄₁

C₂×C₈

yes

complemented

C₂²×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

—

C₂²×C₂₄₁

C₄×D₄

yes

characteristic

C₄×C₂₄₁

—

C₄×C₂₄₁

—

C₄×C₂₄₁

—

C₄×C₂₄₁

—

C₄×C₂₄₁

C₂²×C₈

yes

C₄×C₂₄₁

yes

C₄×C₂₄₁

yes

C₄×C₂₄₁

yes

C₂×C₂₄₁

—

C₂×C₂₄₁

C₈×D₄

yes

C₂×C₂₄₁

yes

C₂×C₂₄₁

yes

C₂₄₁

SD₁₆⋊C₈

yes

characteristic, direct product factor, Sylow

SD₁₆⋊C₈

C₂₄₁

yes

maximal, characteristic, direct product factor, Sylow

C₈×Q₈

C₂×C₂₄₁

yes

characteristic, complemented

C₄×SD₁₆

C₂×C₂₄₁

yes

characteristic

C₈×D₄

C₂×C₂₄₁

yes

characteristic

#335

C₂×C₂₄₁

yes

characteristic, complemented

Q₈⋊C₈

C₂×C₂₄₁

yes

characteristic, complemented

D₄⋊C₈

C₂×C₂₄₁

yes

characteristic

#322

C₂×C₂₄₁

yes

characteristic, complemented

C₂×SD₁₆

C₄×C₂₄₁

yes

characteristic

C₂²×C₈

—

C₄×Q₈

C₂²×C₂₄₁

yes

characteristic

C₄×D₄

C₂²×C₂₄₁

yes

characteristic

#106

C₄×C₂₄₁

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₂₄₁

yes

characteristic

C₄⋊C₈

C₂²×C₂₄₁

yes

characteristic

#103

C₄×C₂₄₁

yes

characteristic

#102

C₄×C₂₄₁

yes

characteristic

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂²×C₂₄₁

yes

characteristic

C₄×C₈

C₂²×C₂₄₁

yes

characteristic

100

C₄×C₈

C₂²×C₂₄₁

yes

characteristic

101

C₂×Q₈

C₂×C₄×C₂₄₁

yes

characteristic

102

C₂×D₄

C₂×C₄×C₂₄₁

yes

characteristic

103

C₂²×C₄

—

104

SD₁₆

C₈×C₂₄₁

yes

complemented

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂×C₄×C₂₄₁

yes

characteristic

110

C₂×C₈

C₂×C₄×C₂₄₁

yes

characteristic

111

C₂×C₈

D₄×C₂₄₁

yes

characteristic

112

C₂×C₈

D₄×C₂₄₁

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

C₂×C₄×C₂₄₁

yes

characteristic

115

C₄⋊C₄

C₂×C₄×C₂₄₁

yes

characteristic

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₂³×C₂₄₁

yes

characteristic

119

C₂³

—

120

Q₈

—

121

Q₈

C₂×C₈×C₂₄₁

yes

complemented

122

D₄

—

123

D₄

C₂×C₈×C₂₄₁

yes

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂²×C₄×C₂₄₁

yes

characteristic

130

C₂×C₄

C₂×D₄×C₂₄₁

yes

characteristic

131

C₂×C₄

#41×C₂₄₁

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

C₂×C₈×C₂₄₁

yes

complemented

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₄×D₄×C₂₄₁

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

C₂²×C₈×C₂₄₁

yes

characteristic

148

C₄

#131×C₂₄₁

yes

characteristic

149

C₄

AGL(1,ℤ/8)×C₂₄₁

yes

characteristic

150

C₄

#137×C₂₄₁

yes

characteristic

151

C₂

—

152

C₂

C₈×D₄×C₂₄₁

yes

characteristic

153

C₂

#440×C₂₄₁

yes

characteristic

154

C₂

#444×C₂₄₁

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₂₄₁	C₂×D₄
commutator	C₄	C₂²×C₈×C₂₄₁
Frattini, cosocle	C₄²	C₂³×C₂₄₁
socle	C₂²×C₂₄₁	C₄×D₄
prime radical	C₂×D₄×C₂₄₁	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₄₁ ◁ C₄×C₈×C₂₄₁ ◁ G

subgroup

G/H

center

C₂×C₄×C₂₄₁

C₂×D₄

commutator

C₄

C₂²×C₈×C₂₄₁

Frattini, cosocle

C₄²

C₂³×C₂₄₁

socle

C₂²×C₂₄₁

C₄×D₄

prime radical

C₂×D₄×C₂₄₁

C₂×C₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₂₄₁ ◁ C₄×C₈×C₂₄₁ ◁ G

N	H ≅ G/N	type
1. C₈×Q₈×C₂₄₁	C₂	semidirect
4. #335×C₂₄₁	C₂	semidirect
5. Q₈⋊C₈×C₂₄₁	C₂	semidirect
7. #322×C₂₄₁	C₂	semidirect
26. SD₁₆×C₂₄₁	C₈	semidirect
43. Q₈×C₂₄₁	C₂×C₈	semidirect
60. C₈×C₂₄₁	C₂×C₈	semidirect
77. C₂₄₁	SD₁₆⋊C₈	direct (N × H)
78. SD₁₆⋊C₈	C₂₄₁	direct (N × H)
79. C₈×Q₈	C₂×C₂₄₁	semidirect
82. #335	C₂×C₂₄₁	semidirect
83. Q₈⋊C₈	C₂×C₂₄₁	semidirect
85. #322	C₂×C₂₄₁	semidirect
104. SD₁₆	C₈×C₂₄₁	semidirect
121. Q₈	C₂×C₈×C₂₄₁	semidirect
138. C₈	C₂×C₈×C₂₄₁	semidirect

H ≅ G/N

type

1. C₈×Q₈×C₂₄₁

C₂

semidirect

4. #335×C₂₄₁

C₂

semidirect

5. Q₈⋊C₈×C₂₄₁

C₂

semidirect

7. #322×C₂₄₁

C₂

semidirect

26. SD₁₆×C₂₄₁

C₈

semidirect

43. Q₈×C₂₄₁

C₂×C₈

semidirect

60. C₈×C₂₄₁

C₂×C₈

semidirect

77. C₂₄₁

SD₁₆⋊C₈

direct (N × H)

78. SD₁₆⋊C₈

C₂₄₁

direct (N × H)

79. C₈×Q₈

C₂×C₂₄₁

semidirect

82. #335

C₂×C₂₄₁

semidirect

83. Q₈⋊C₈

C₂×C₂₄₁

semidirect

85. #322

C₂×C₂₄₁

semidirect

104. SD₁₆

C₈×C₂₄₁

semidirect

121. Q₈

C₂×C₈×C₂₄₁

semidirect

138. C₈

C₂×C₈×C₂₄₁

semidirect

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

subgroup	index
C₈×Q₈×C₂₄₁	2
C₄×SD₁₆×C₂₄₁	2
C₈×D₄×C₂₄₁	2
#335×C₂₄₁	2
Q₈⋊C₈×C₂₄₁	2
D₄⋊C₈×C₂₄₁	2
#322×C₂₄₁	2
SD₁₆⋊C₈	241

subgroup

index

C₈×Q₈×C₂₄₁

C₄×SD₁₆×C₂₄₁

C₈×D₄×C₂₄₁

#335×C₂₄₁

Q₈⋊C₈×C₂₄₁

D₄⋊C₈×C₂₄₁

#322×C₂₄₁

SD₁₆⋊C₈

241

Subgroups — SD₁₆⋊C₈×C₂₄₁ #44503245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — SD16⋊C8×C241 #44503245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — SD₁₆⋊C₈×C₂₄₁ #44503245