Group #21458675 — subgroups

Subgroups — Q₈×#11758201 #21458675

Order 7808.

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×D₂₄₄	C₂	1	1	yes	maximal, characteristic
2	#16840158	C₂	1	3	yes	maximal, complemented
3	#16840049	C₂	1	3	yes	maximal
4	#16840047	C₂	1	3	yes	maximal, complemented
5	Q₈×#92418	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#11758201	C₂	1	3	yes	maximal
7	Q₈²×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×Q₈×D₆₁	—	2	1	no	—
9	C₂×#11758201	C₂²	1	1	yes	characteristic
10	#14519777	C₂²	1	3	yes	—
11	#14519757	—	2	3	no	—
12	C₄×D₂₄₄	C₂²	1	3	yes	—
13	#14519708	C₂²	1	6	yes	—
14	#14519708	C₂²	1	3	yes	—
15	C₄⋊#92418	C₂²	1	3	yes	—
16	#14519701	C₂²	1	3	yes	—
17	#14519697	C₂²	1	3	yes	—
18	C₄×#92418	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₆₁	—	2	3	no	—
20	C₄⋊Q₈×C₆₁	C₂²	1	3	yes	—
21	C₄×Q₈×C₆₁	—	2	1	no	—
22	C₄×Q₈×C₆₁	C₂²	1	3	yes	—
23	C₄×Q₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₆₁	—	2	3	no	—
25	C₂×Q₈×C₆₁	D₄	1	1	yes	characteristic
26	C₂×Q₈×C₆₁	C₂³	1	1	yes	characteristic
27	Q₈×D₆₁	—	4	4	no	—
28	C₂×D₂₄₄	C₂³	1	1	yes	characteristic
29	C₂×C₄×D₆₁	—	2	3	no	—
30	C₂×Dic₁₂₂	—	2	3	no	—
31	C₄⋊C₄×C₆₁	—	2	3	no	—
32	C₄⋊C₄×C₆₁	—	2	3	no	—
33	C₄⋊C₄×C₆₁	D₄	1	3	yes	—
34	C₄⋊C₄×C₆₁	C₂³	1	3	yes	—
35	C₄²×C₆₁	—	2	3	no	—
36	C₄²×C₆₁	C₂³	1	3	yes	—
37	#11758201	Q₈	1	4	yes	direct product factor
38	#11758198	—	2	3	no	—
39	C₄⋊Dic₆₁	—	2	3	no	—
40	C₄⋊Dic₆₁	C₂³	1	3	yes	—
41	C₂×#92418	C₂³	1	3	yes	—
42	C₂×#92418	C₂³	1	1	yes	characteristic
43	C₂²×D₆₁	—	2	1	no	—
44	Q₈×C₆₁	—	2	1	no	—
45	Q₈×C₆₁	—	8	3	no	—
46	Q₈×C₆₁	SD₁₆	1	4	yes	complemented
47	Q₈×C₆₁	C₂×Q₈	1	2	yes	complemented
48	C₂×C₄×C₆₁	—	2	1	no	—
49	C₂×C₄×C₆₁	—	2	3	no	—
50	C₂×C₄×C₆₁	C₂×D₄	1	3	yes	—
51	C₂×C₄×C₆₁	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₆₁	—	2	3	no	—
54	D₂₄₄	—	2	1	no	—
55	D₂₄₄	C₂×Q₈	1	2	yes	—
56	C₄×D₆₁	—	4	6	no	—
57	Dic₁₂₂	—	4	6	no	—
58	#92418	—	2	3	no	—
59	#92418	C₂×Q₈	1	2	yes	complemented
60	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
61	C₂×D₆₁	—	2	2	no	—
62	C₂×D₆₁	—	4	1	no	—
63	C₂×D₆₁	—	4	1	no	—
64	C₄×C₆₁	—	2	2	no	—
65	C₄×C₆₁	—	2	3	no	—
66	C₄×C₆₁	—	4	3	no	—
67	C₄×C₆₁	C₂×SD₁₆	1	6	yes	—
68	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₆₁	2₋¹⁺⁴	1	1	yes	characteristic
70	Dic₆₁	—	4	3	no	—
71	Q₈×SD₁₆	—	61	1	no	maximal, Sylow
72	C₂×C₆₁	D₄×Q₈	1	1	yes	characteristic
73	C₂×C₆₁	C₂²×SD₁₆	1	1	yes	characteristic
74	C₂×C₆₁	#577	1	1	yes	characteristic
75	D₆₁	—	4	2	no	—
76	Q₈²	D₆₁	1	1	yes	characteristic, complemented
77	D₄×Q₈	—	61	1	no	—
78	#498	—	61	3	no	—
79	#476	—	61	3	no	—
80	#475	—	61	3	no	—
81	C₈×Q₈	—	61	1	no	—
82	C₄×SD₁₆	—	61	3	no	—
83	C₆₁	Q₈×SD₁₆	1	1	yes	characteristic, complemented, Sylow
84	C₂²×Q₈	—	122	1	no	—
85	C₂×SD₁₆	—	61	1	no	—
86	C₄⋊Q₈	—	2	3	no	—
87	C₄⋊Q₈	—	61	3	no	—
88	C₄⋊Q₈	C₂×D₆₁	1	3	yes	—
89	C₂²⋊Q₈	—	122	3	no	—
90	C₄×Q₈	—	2	1	no	—
91	C₄×Q₈	C₂×D₆₁	1	3	yes	—
92	C₄×Q₈	C₂×D₆₁	1	1	yes	characteristic
93	C₄×D₄	—	61	3	no	—
94	#106	—	61	3	no	—
95	#106	—	61	6	no	—
96	C₄⋊C₈	—	61	3	no	—
97	#103	—	61	3	no	—
98	#102	—	61	3	no	—
99	C₄×C₈	—	61	3	no	—
100	C₂×Q₈	—	2	3	no	—
101	C₂×Q₈	—	122	3	no	—
102	C₂×Q₈	—	244	4	no	—
103	C₂×Q₈	C₂²⋊D₆₁	1	1	yes	characteristic
104	C₂×Q₈	C₂²×D₆₁	1	1	yes	characteristic
105	C₂×D₄	—	61	1	no	—
106	C₂²×C₄	—	122	3	no	—
107	SD₁₆	—	61	4	no	—
108	C₂×C₈	—	61	1	no	—
109	C₂×C₈	—	61	3	no	—
110	C₄⋊C₄	—	2	3	no	—
111	C₄⋊C₄	—	2	3	no	—
112	C₄⋊C₄	—	61	3	no	—
113	C₄⋊C₄	—	122	3	no	—
114	C₄⋊C₄	C₂²⋊D₆₁	1	3	yes	—
115	C₄⋊C₄	C₂²×D₆₁	1	3	yes	—
116	C₂²⋊C₄	—	122	3	no	—
117	C₄²	—	2	3	no	—
118	C₄²	C₂²×D₆₁	1	3	yes	—
119	C₂³	—	122	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	—	8	3	no	—
122	Q₈	—	244	6	no	—
123	Q₈	#11758201	1	4	yes	direct product factor
124	Q₈	Q₈×D₆₁	1	2	yes	complemented
125	D₄	—	61	2	no	—
126	D₄	—	122	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	3	no	—
129	C₂×C₄	—	122	3	no	—
130	C₂×C₄	—	244	6	no	—
131	C₂×C₄	C₂×C₂²⋊D₆₁	1	3	yes	—
132	C₂×C₄	C₂×C₂²⋊D₆₁	1	3	yes	—
133	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
134	C₈	—	61	2	no	—
135	C₈	—	122	3	no	—
136	C₂²	—	122	2	no	—
137	C₂²	—	244	1	no	—
138	C₂²	—	244	1	no	—
139	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
140	C₄	—	2	3	no	—
141	C₄	—	2	2	no	—
142	C₄	—	4	3	no	—
143	C₄	—	244	3	no	—
144	C₄	C₂×#11758201	1	6	yes	—
145	C₄	C₂×Q₈×D₆₁	1	1	yes	characteristic
146	C₄	#14519825	1	1	yes	characteristic
147	C₂	—	244	2	no	—
148	C₂	Q₈×C₂²⋊D₆₁	1	1	yes	characteristic
149	C₂	C₂²×#11758201	1	1	yes	characteristic
150	C₂	#16840573	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×D₂₄₄

C₂

yes

maximal, characteristic

#16840158

C₂

yes

maximal, complemented

#16840049

C₂

yes

maximal

#16840047

C₂

yes

maximal, complemented

Q₈×#92418

C₂

yes

maximal, characteristic, complemented

C₄×#11758201

C₂

yes

maximal

Q₈²×C₆₁

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×D₆₁

—

C₂×#11758201

C₂²

yes

characteristic

#14519777

C₂²

yes

—

#14519757

—

C₄×D₂₄₄

C₂²

yes

—

#14519708

C₂²

yes

—

#14519708

C₂²

yes

—

C₄⋊#92418

C₂²

yes

—

#14519701

C₂²

yes

—

#14519697

C₂²

yes

—

C₄×#92418

C₂²

yes

—

C₄⋊Q₈×C₆₁

—

C₄⋊Q₈×C₆₁

C₂²

yes

—

C₄×Q₈×C₆₁

—

C₄×Q₈×C₆₁

C₂²

yes

—

C₄×Q₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

D₄

yes

characteristic

C₂×Q₈×C₆₁

C₂³

yes

characteristic

Q₈×D₆₁

—

C₂×D₂₄₄

C₂³

yes

characteristic

C₂×C₄×D₆₁

—

C₂×Dic₁₂₂

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

D₄

yes

—

C₄⋊C₄×C₆₁

C₂³

yes

—

C₄²×C₆₁

—

C₄²×C₆₁

C₂³

yes

—

#11758201

Q₈

yes

direct product factor

#11758198

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

C₂×#92418

C₂³

yes

—

C₂×#92418

C₂³

yes

characteristic

C₂²×D₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

—

Q₈×C₆₁

SD₁₆

yes

complemented

Q₈×C₆₁

C₂×Q₈

yes

complemented

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

D₂₄₄

—

D₂₄₄

C₂×Q₈

yes

—

C₄×D₆₁

—

Dic₁₂₂

—

#92418

—

#92418

C₂×Q₈

yes

complemented

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂×D₆₁

—

C₂×D₆₁

—

C₂×D₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

C₂×SD₁₆

yes

—

C₄×C₆₁

C₂²×Q₈

yes

characteristic

C₄×C₆₁

2₋¹⁺⁴

yes

characteristic

Dic₆₁

—

Q₈×SD₁₆

—

maximal, Sylow

C₂×C₆₁

D₄×Q₈

yes

characteristic

C₂×C₆₁

C₂²×SD₁₆

yes

characteristic

C₂×C₆₁

#577

yes

characteristic

D₆₁

—

Q₈²

D₆₁

yes

characteristic, complemented

D₄×Q₈

—

#498

—

#476

—

#475

—

C₈×Q₈

—

C₄×SD₁₆

—

C₆₁

Q₈×SD₁₆

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

122

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₁

yes

—

C₂²⋊Q₈

—

122

—

C₄×Q₈

—

C₄×Q₈

C₂×D₆₁

yes

—

C₄×Q₈

C₂×D₆₁

yes

characteristic

C₄×D₄

—

#106

—

#106

—

C₄⋊C₈

—

#103

—

#102

—

C₄×C₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

122

—

102

C₂×Q₈

—

244

—

103

C₂×Q₈

C₂²⋊D₆₁

yes

characteristic

104

C₂×Q₈

C₂²×D₆₁

yes

characteristic

105

C₂×D₄

—

106

C₂²×C₄

—

122

—

107

SD₁₆

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

122

—

114

C₄⋊C₄

C₂²⋊D₆₁

yes

—

115

C₄⋊C₄

C₂²×D₆₁

yes

—

116

C₂²⋊C₄

—

122

—

117

C₄²

—

118

C₄²

C₂²×D₆₁

yes

—

119

C₂³

—

122

—

120

Q₈

—

121

Q₈

—

122

Q₈

—

244

—

123

Q₈

#11758201

yes

direct product factor

124

Q₈

Q₈×D₆₁

yes

complemented

125

D₄

—

126

D₄

—

122

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

122

—

130

C₂×C₄

—

244

—

131

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

132

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

133

C₂×C₄

C₂³×D₆₁

yes

characteristic

134

C₈

—

135

C₈

—

122

—

136

C₂²

—

122

—

137

C₂²

—

244

—

138

C₂²

—

244

—

139

C₂²

C₂²×C₂²⋊D₆₁

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

244

—

144

C₄

C₂×#11758201

yes

—

145

C₄

C₂×Q₈×D₆₁

yes

characteristic

146

C₄

#14519825

yes

characteristic

147

C₂

—

244

—

148

C₂

Q₈×C₂²⋊D₆₁

yes

characteristic

149

C₂

C₂²×#11758201

yes

characteristic

150

C₂

#16840573

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₆₁
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	Q₈²×C₆₁	C₂
socle	C₂²×C₆₁	C₂²×D₄
hypercenter	Q₈²	D₆₁
hypocenter	C₆₁	Q₈×SD₁₆
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₆₁

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

Q₈²×C₆₁

C₂

socle

C₂²×C₆₁

C₂²×D₄

hypercenter

Q₈²

D₆₁

hypocenter

C₆₁

Q₈×SD₁₆

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

N	H ≅ G/N	type
2. #16840158	C₂	semidirect
4. #16840047	C₂	semidirect
5. Q₈×#92418	C₂	semidirect
7. Q₈²×C₆₁	C₂	semidirect
37. #11758201	Q₈	direct (N × H)
46. Q₈×C₆₁	SD₁₆	semidirect
47. Q₈×C₆₁	C₂×Q₈	semidirect
59. #92418	C₂×Q₈	semidirect
76. Q₈²	D₆₁	semidirect
83. C₆₁	Q₈×SD₁₆	semidirect
123. Q₈	#11758201	direct (N × H)
124. Q₈	Q₈×D₆₁	semidirect

H ≅ G/N

type

2. #16840158

C₂

semidirect

4. #16840047

C₂

semidirect

5. Q₈×#92418

C₂

semidirect

7. Q₈²×C₆₁

C₂

semidirect

37. #11758201

Q₈

direct (N × H)

46. Q₈×C₆₁

SD₁₆

semidirect

47. Q₈×C₆₁

C₂×Q₈

semidirect

59. #92418

C₂×Q₈

semidirect

76. Q₈²

D₆₁

semidirect

83. C₆₁

Q₈×SD₁₆

semidirect

123. Q₈

#11758201

direct (N × H)

124. Q₈

Q₈×D₆₁

semidirect

Subgroups — Q₈×#11758201 #21458675

Conjugacy classes of subgroups

Complements

Subgroups — Q8×#11758201 #21458675

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×#11758201 #21458675