Group #24920323 — subgroups

Subgroups — Q₈×#13050578 #24920323

Order 10624.

370 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 149 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	Q₈×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic, complemented
2	#19128097	C₂	1	3	yes	maximal, complemented
3	#19127991	C₂	1	3	yes	maximal
4	#19127983	C₂	1	3	yes	maximal, complemented
5	Q₈×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#13050578	C₂	1	3	yes	maximal
7	D₄×Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic
8	C₂×#13050578	C₂²	1	1	yes	characteristic
9	#14581792	C₂²	1	3	yes	—
10	#14581791	—	2	3	no	—
11	Q₈×Dic₈₃	—	2	1	no	—
12	C₄×Dic₁₆₆	C₂²	1	3	yes	—
13	#14581723	C₂²	1	3	yes	—
14	#14581723	C₂²	1	6	yes	—
15	C₄⋊(C₈₃⋊C₈)	C₂²	1	3	yes	—
16	#14581715	C₂²	1	3	yes	—
17	#14581713	C₂²	1	3	yes	—
18	C₄×C₈₃⋊C₈	C₂²	1	3	yes	—
19	C₂²×Q₈×C₈₃	—	2	1	no	—
20	C₄⋊Q₈×C₈₃	C₂²	1	3	yes	—
21	C₂²⋊Q₈×C₈₃	—	2	3	no	—
22	C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₈₃	C₂²	1	3	yes	—
24	C₂×Dic₁₆₆	—	2	3	no	—
25	C₂×Dic₁₆₆	C₂³	1	1	yes	characteristic
26	#13050578	Q₈	1	4	yes	direct product factor
27	C₂×C₈₃⋊C₈	C₂³	1	3	yes	—
28	C₂×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
29	C₄⋊Dic₈₃	—	2	3	no	—
30	C₄⋊Dic₈₃	C₂³	1	3	yes	—
31	Dic₈₃⋊C₄	—	2	3	no	—
32	C₄×Dic₈₃	—	2	3	no	—
33	C₂×Q₈×C₈₃	—	2	3	no	—
34	C₂×Q₈×C₈₃	—	4	4	no	—
35	C₂×Q₈×C₈₃	D₄	1	1	yes	characteristic
36	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
37	C₂²×C₄×C₈₃	—	2	3	no	—
38	C₄⋊C₄×C₈₃	—	2	3	no	—
39	C₄⋊C₄×C₈₃	D₄	1	3	yes	—
40	C₄⋊C₄×C₈₃	C₂³	1	3	yes	—
41	C₂²⋊C₄×C₈₃	—	2	3	no	—
42	C₄²×C₈₃	C₂³	1	3	yes	—
43	C₂³×C₈₃	—	2	1	no	—
44	Q₈×C₈₃	—	4	6	no	—
45	Q₈×C₈₃	SD₁₆	1	4	yes	complemented
46	D₄×C₈₃	—	2	1	no	—
47	D₄×C₈₃	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₈₃	—	2	3	no	—
49	C₂×C₄×C₈₃	—	4	6	no	—
50	C₂×C₄×C₈₃	C₂×D₄	1	3	yes	—
51	C₂×C₄×C₈₃	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₈₃	—	2	1	no	—
54	C₂×Dic₈₃	—	2	3	no	—
55	Dic₁₆₆	—	2	1	no	—
56	Dic₁₆₆	—	8	3	no	—
57	Dic₁₆₆	C₂×Q₈	1	2	yes	complemented
58	C₈₃⋊C₈	—	2	3	no	—
59	C₈₃⋊C₈	C₂×Q₈	1	2	yes	complemented
60	C₂²×C₈₃	—	2	2	no	—
61	C₂²×C₈₃	—	4	1	no	—
62	C₂²×C₈₃	—	4	1	no	—
63	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
64	C₄×C₈₃	—	2	3	no	—
65	C₄×C₈₃	—	4	3	no	—
66	C₄×C₈₃	C₂×SD₁₆	1	6	yes	—
67	C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
68	C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
69	Dic₈₃	—	2	2	no	—
70	Dic₈₃	—	4	3	no	—
71	C₂×C₈₃	—	4	2	no	—
72	C₂×C₈₃	D₄×Q₈	1	1	yes	characteristic
73	C₂×C₈₃	C₂²×SD₁₆	1	1	yes	characteristic
74	C₂×C₈₃	#577	1	1	yes	characteristic
75	Q₈×SD₁₆	—	83	1	no	maximal, Sylow
76	C₈₃	Q₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
77	Q₈²	—	83	1	no	—
78	D₄×Q₈	D₈₃	1	1	yes	characteristic
79	#498	—	83	3	no	—
80	#476	—	83	3	no	—
81	#475	—	83	3	no	—
82	C₈×Q₈	—	83	1	no	—
83	C₄×SD₁₆	—	83	3	no	—
84	C₂²×Q₈	—	2	1	no	—
85	C₂×SD₁₆	—	83	1	no	—
86	C₄⋊Q₈	—	83	3	no	—
87	C₄⋊Q₈	—	166	3	no	—
88	C₄⋊Q₈	C₂×D₈₃	1	3	yes	—
89	C₂²⋊Q₈	—	2	3	no	—
90	C₄×Q₈	—	83	3	no	—
91	C₄×Q₈	—	166	1	no	—
92	C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
93	C₄×D₄	C₂×D₈₃	1	3	yes	—
94	#106	—	83	3	no	—
95	#106	—	83	6	no	—
96	C₄⋊C₈	—	83	3	no	—
97	#103	—	83	3	no	—
98	#102	—	83	3	no	—
99	C₄×C₈	—	83	3	no	—
100	C₂×Q₈	—	2	3	no	—
101	C₂×Q₈	—	4	4	no	—
102	C₂×Q₈	—	83	1	no	—
103	C₂×Q₈	—	166	3	no	—
104	C₂×Q₈	C₂²⋊D₈₃	1	1	yes	characteristic
105	C₂×D₄	C₂²×D₈₃	1	1	yes	characteristic
106	C₂²×C₄	—	2	3	no	—
107	SD₁₆	—	83	4	no	—
108	C₂×C₈	—	83	1	no	—
109	C₂×C₈	—	83	3	no	—
110	C₄⋊C₄	—	2	3	no	—
111	C₄⋊C₄	—	83	3	no	—
112	C₄⋊C₄	—	166	3	no	—
113	C₄⋊C₄	—	166	3	no	—
114	C₄⋊C₄	C₂²⋊D₈₃	1	3	yes	—
115	C₄⋊C₄	C₂²×D₈₃	1	3	yes	—
116	C₂²⋊C₄	—	2	3	no	—
117	C₄²	—	166	3	no	—
118	C₄²	C₂²×D₈₃	1	3	yes	—
119	C₂³	—	2	1	no	—
120	Q₈	—	4	6	no	—
121	Q₈	—	83	2	no	—
122	Q₈	—	664	3	no	—
123	Q₈	—	166	1	no	—
124	Q₈	#13050578	1	4	yes	direct product factor
125	D₄	—	2	1	no	—
126	D₄	Q₈×D₈₃	1	2	yes	—
127	C₂×C₄	—	2	3	no	—
128	C₂×C₄	—	4	6	no	—
129	C₂×C₄	—	166	3	no	—
130	C₂×C₄	—	166	1	no	—
131	C₂×C₄	C₂×C₂²⋊D₈₃	1	3	yes	—
132	C₂×C₄	C₂×C₂²⋊D₈₃	1	3	yes	—
133	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
134	C₈	—	83	2	no	—
135	C₈	—	166	3	no	—
136	C₂²	—	2	2	no	—
137	C₂²	—	4	1	no	—
138	C₂²	—	4	1	no	—
139	C₂²	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
140	C₄	—	2	3	no	—
141	C₄	—	4	3	no	—
142	C₄	—	332	3	no	—
143	C₄	—	166	2	no	—
144	C₄	C₂×#13050578	1	6	yes	—
145	C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
146	C₄	#14581840	1	1	yes	characteristic
147	C₂	—	4	2	no	—
148	C₂	Q₈×C₂²⋊D₈₃	1	1	yes	characteristic
149	C₂	C₂²×#13050578	1	1	yes	characteristic
150	C₂	#19128509	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₆₆

C₂

yes

maximal, characteristic, complemented

#19128097

C₂

yes

maximal, complemented

#19127991

C₂

yes

maximal

#19127983

C₂

yes

maximal, complemented

Q₈×C₈₃⋊C₈

C₂

yes

maximal, characteristic, complemented

C₄×#13050578

C₂

yes

maximal

D₄×Q₈×C₈₃

C₂

yes

maximal, characteristic

C₂×#13050578

C₂²

yes

characteristic

#14581792

C₂²

yes

—

#14581791

—

Q₈×Dic₈₃

—

C₄×Dic₁₆₆

C₂²

yes

—

#14581723

C₂²

yes

—

#14581723

C₂²

yes

—

C₄⋊(C₈₃⋊C₈)

C₂²

yes

—

#14581715

C₂²

yes

—

#14581713

C₂²

yes

—

C₄×C₈₃⋊C₈

C₂²

yes

—

C₂²×Q₈×C₈₃

—

C₄⋊Q₈×C₈₃

C₂²

yes

—

C₂²⋊Q₈×C₈₃

—

C₄×Q₈×C₈₃

C₂²

yes

characteristic

C₄×D₄×C₈₃

C₂²

yes

—

C₂×Dic₁₆₆

—

C₂×Dic₁₆₆

C₂³

yes

characteristic

#13050578

Q₈

yes

direct product factor

C₂×C₈₃⋊C₈

C₂³

yes

—

C₂×C₈₃⋊C₈

C₂³

yes

characteristic

C₄⋊Dic₈₃

—

C₄⋊Dic₈₃

C₂³

yes

—

Dic₈₃⋊C₄

—

C₄×Dic₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

—

C₂×Q₈×C₈₃

D₄

yes

characteristic

C₂×D₄×C₈₃

C₂³

yes

characteristic

C₂²×C₄×C₈₃

—

C₄⋊C₄×C₈₃

—

C₄⋊C₄×C₈₃

D₄

yes

—

C₄⋊C₄×C₈₃

C₂³

yes

—

C₂²⋊C₄×C₈₃

—

C₄²×C₈₃

C₂³

yes

—

C₂³×C₈₃

—

Q₈×C₈₃

—

Q₈×C₈₃

SD₁₆

yes

complemented

D₄×C₈₃

—

D₄×C₈₃

C₂×Q₈

yes

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂×D₄

yes

—

C₂×C₄×C₈₃

C₂⁴

yes

characteristic

C₂×Dic₈₃

—

C₂×Dic₈₃

—

Dic₁₆₆

—

Dic₁₆₆

—

Dic₁₆₆

C₂×Q₈

yes

complemented

C₈₃⋊C₈

—

C₈₃⋊C₈

C₂×Q₈

yes

complemented

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

—

C₂²×C₈₃

C₂²×D₄

yes

characteristic

C₄×C₈₃

—

C₄×C₈₃

—

C₄×C₈₃

C₂×SD₁₆

yes

—

C₄×C₈₃

C₂²×Q₈

yes

characteristic

C₄×C₈₃

2₋¹⁺⁴

yes

characteristic

Dic₈₃

—

Dic₈₃

—

C₂×C₈₃

—

C₂×C₈₃

D₄×Q₈

yes

characteristic

C₂×C₈₃

C₂²×SD₁₆

yes

characteristic

C₂×C₈₃

#577

yes

characteristic

Q₈×SD₁₆

—

maximal, Sylow

C₈₃

Q₈×SD₁₆

yes

characteristic, complemented, Sylow

Q₈²

—

D₄×Q₈

D₈₃

yes

—

—

—

C₈×Q₈

—

C₄×SD₁₆

—

C₂²×Q₈

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

166

—

C₄⋊Q₈

C₂×D₈₃

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

166

—

C₄×Q₈

C₂×D₈₃

yes

characteristic

C₄×D₄

C₂×D₈₃

yes

—

#106

—

#106

—

C₄⋊C₈

—

#103

—

#102

—

C₄×C₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

166

—

104

C₂×Q₈

C₂²⋊D₈₃

yes

characteristic

105

C₂×D₄

C₂²×D₈₃

yes

characteristic

106

C₂²×C₄

—

107

SD₁₆

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

166

—

113

C₄⋊C₄

—

166

—

114

C₄⋊C₄

C₂²⋊D₈₃

yes

—

115

C₄⋊C₄

C₂²×D₈₃

yes

—

116

C₂²⋊C₄

—

117

C₄²

—

166

—

118

C₄²

C₂²×D₈₃

yes

—

119

C₂³

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

664

—

123

Q₈

—

166

—

124

Q₈

#13050578

yes

direct product factor

125

D₄

—

126

D₄

Q₈×D₈₃

yes

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

166

—

130

C₂×C₄

—

166

—

131

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

132

C₂×C₄

C₂×C₂²⋊D₈₃

yes

—

133

C₂×C₄

C₂³×D₈₃

yes

characteristic

134

C₈

—

135

C₈

—

166

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

C₂²×C₂²⋊D₈₃

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

332

—

143

C₄

—

166

—

144

C₄

C₂×#13050578

yes

—

145

C₄

C₂×Q₈×D₈₃

yes

characteristic

146

C₄

#14581840

yes

characteristic

147

C₂

—

148

C₂

Q₈×C₂²⋊D₈₃

yes

characteristic

149

C₂

C₂²×#13050578

yes

characteristic

150

C₂

#19128509

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₈₃
commutator, cosocle	C₂×C₄×C₈₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	D₄×Q₈×C₈₃	C₂
socle	C₂²×C₈₃	C₂²×D₄
prime radical	C₂×D₄×C₈₃	C₂³
hypercenter	D₄×Q₈	D₈₃
hypocenter	C₈₃	Q₈×SD₁₆
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₈₃

commutator, cosocle

C₂×C₄×C₈₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₈₃

Fitting

D₄×Q₈×C₈₃

C₂

socle

C₂²×C₈₃

C₂²×D₄

prime radical

C₂×D₄×C₈₃

C₂³

hypercenter

D₄×Q₈

D₈₃

hypocenter

C₈₃

Q₈×SD₁₆

LCS

G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

N	H ≅ G/N	type
1. Q₈×Dic₁₆₆	C₂	semidirect
2. #19128097	C₂	semidirect
4. #19127983	C₂	semidirect
5. Q₈×C₈₃⋊C₈	C₂	semidirect
26. #13050578	Q₈	direct (N × H)
45. Q₈×C₈₃	SD₁₆	semidirect
57. Dic₁₆₆	C₂×Q₈	semidirect
59. C₈₃⋊C₈	C₂×Q₈	semidirect
76. C₈₃	Q₈×SD₁₆	semidirect
124. Q₈	#13050578	direct (N × H)

H ≅ G/N

type

1. Q₈×Dic₁₆₆

C₂

semidirect

2. #19128097

C₂

semidirect

4. #19127983

C₂

semidirect

5. Q₈×C₈₃⋊C₈

C₂

semidirect

26. #13050578

Q₈

direct (N × H)

45. Q₈×C₈₃

SD₁₆

semidirect

57. Dic₁₆₆

C₂×Q₈

semidirect

59. C₈₃⋊C₈

C₂×Q₈

semidirect

76. C₈₃

Q₈×SD₁₆

semidirect

124. Q₈

#13050578

direct (N × H)

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

subgroup	index
Q₈×Dic₁₆₆	2
#19128097	2
#19127991	2
#19127983	2
Q₈×C₈₃⋊C₈	2
C₄×#13050578	2
D₄×Q₈×C₈₃	2
Q₈×SD₁₆	83

subgroup

index

Q₈×Dic₁₆₆

Q₈×C₈₃⋊C₈

D₄×Q₈×C₈₃

Q₈×SD₁₆

Subgroups — Q₈×#13050578 #24920323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×#13050578 #24920323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#13050578 #24920323