Group #36712323 — subgroups

Subgroups — Q₈×Dic₇₁₂ #36712323

Order 22784.

460 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 175 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×Dic₃₅₆	C₂	1	2	yes	maximal
2	#26022527	C₂	1	3	yes	maximal
3	#26022405	C₂	1	6	yes	maximal
4	C₄×Dic₇₁₂	C₂	1	3	yes	maximal
5	Q₈×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
6	Q₈×Dic₁₇₈	—	2	2	no	—
7	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
8	#19152003	C₂²	1	6	yes	—
9	#19151894	—	2	6	no	—
10	C₄×Dic₃₅₆	C₂²	1	6	yes	—
11	#19151424	C₂²	1	6	yes	—
12	#19151424	C₂²	1	3	yes	—
13	#19151403	C₂²	1	6	yes	—
14	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₈₉	C₂²	1	3	yes	—
16	C₄×C₁₆×C₈₉	C₂²	1	3	yes	—
17	C₂×Dic₃₅₆	C₂³	1	2	yes	—
18	#15668012	—	2	6	no	—
19	#15668011	—	4	6	no	—
20	Q₈×Dic₈₉	—	4	2	no	—
21	C₄×Dic₁₇₈	—	2	6	no	—
22	Dic₇₁₂	Q₈	1	4	yes	direct product factor
23	#15667946	—	2	6	no	—
24	#15667946	C₂³	1	6	yes	—
25	#15667937	—	2	6	no	—
26	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
27	C₂×C₁₆×C₈₉	C₂³	1	3	yes	—
28	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
29	C₄⋊C₈×C₈₉	D₄	1	3	yes	—
30	C₄×C₈×C₈₉	C₂³	1	3	yes	—
31	C₂×Dic₁₇₈	—	2	2	no	—
32	C₂×Dic₁₇₈	—	4	6	no	—
33	Dic₃₅₆	—	2	2	no	—
34	Dic₃₅₆	C₂×Q₈	1	4	yes	—
35	C₄⋊Dic₈₉	—	2	6	no	—
36	C₄⋊Dic₈₉	—	4	6	no	—
37	#13083553	—	4	6	no	—
38	C₄×Dic₈₉	—	4	6	no	—
39	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
40	C₂×C₈×C₈₉	C₂×D₄	1	3	yes	—
41	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
42	C₄⋊C₄×C₈₉	D₈	1	3	yes	—
43	C₄²×C₈₉	C₂×D₄	1	3	yes	—
44	C₁₆×C₈₉	—	2	3	no	—
45	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
46	Q₈×C₈₉	Q₃₂	1	4	yes	complemented
47	C₂×C₄×C₈₉	C₂×D₈	1	3	yes	—
48	C₂×C₄×C₈₉	C₂×D₈	1	3	yes	—
49	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
50	C₂×Dic₈₉	—	4	2	no	—
51	C₂×Dic₈₉	—	4	6	no	—
52	Dic₁₇₈	—	2	4	no	—
53	Dic₁₇₈	—	4	2	no	—
54	Dic₁₇₈	—	16	6	no	—
55	C₈×C₈₉	—	2	3	no	—
56	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
57	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
58	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
59	C₄×C₈₉	—	2	3	no	—
60	C₄×C₈₉	C₂×Q₃₂	1	6	yes	—
61	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
62	C₄×C₈₉	#578	1	1	yes	characteristic
63	Dic₈₉	—	4	4	no	—
64	Dic₈₉	—	8	6	no	—
65	Q₈×Q₃₂	—	89	1	no	maximal, Sylow
66	C₂×C₈₉	Q₈×D₈	1	1	yes	characteristic
67	C₂×C₈₉	C₂²×Q₃₂	1	1	yes	characteristic
68	C₂×C₈₉	#3415	1	1	yes	characteristic
69	Q₈×Q₁₆	—	89	2	no	—
70	#2252	—	89	3	no	—
71	#2227	—	89	6	no	—
72	Q₈×C₁₆	D₈₉	1	1	yes	characteristic
73	C₄×Q₃₂	—	89	3	no	—
74	C₈₉	Q₈×Q₃₂	1	1	yes	characteristic, complemented, Sylow
75	Q₈²	—	178	2	no	—
76	C₂×Q₃₂	—	89	1	no	—
77	#500	—	89	6	no	—
78	#477	—	178	6	no	—
79	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
80	C₄×Q₁₆	—	89	6	no	—
81	#366	—	89	3	no	—
82	#366	—	89	6	no	—
83	C₄⋊C₁₆	C₂×D₈₉	1	3	yes	—
84	#358	—	89	6	no	—
85	C₄×C₁₆	C₂×D₈₉	1	3	yes	—
86	C₂×Q₁₆	—	89	2	no	—
87	C₄⋊Q₈	—	356	6	no	—
88	C₄⋊Q₈	—	178	6	no	—
89	C₄×Q₈	—	356	2	no	—
90	C₄×Q₈	—	178	6	no	—
91	C₄×Q₈	D₃₅₆	1	1	yes	characteristic
92	Q₃₂	—	89	4	no	—
93	C₂×C₁₆	C₂²×D₈₉	1	3	yes	—
94	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
95	#107	—	89	6	no	—
96	#107	—	178	6	no	—
97	C₄⋊C₈	D₃₅₆	1	3	yes	—
98	#103	—	178	6	no	—
99	C₄×C₈	C₂²×D₈₉	1	3	yes	—
100	C₂×Q₈	—	356	6	no	—
101	C₂×Q₈	—	178	2	no	—
102	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
103	Q₁₆	—	89	4	no	—
104	Q₁₆	—	178	2	no	—
105	C₂×C₈	C₂×D₃₅₆	1	3	yes	—
106	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
107	C₄⋊C₄	—	356	6	no	—
108	C₄⋊C₄	—	356	6	no	—
109	C₄⋊C₄	—	178	6	no	—
110	C₄⋊C₄	D₇₁₂	1	3	yes	—
111	C₄²	—	356	6	no	—
112	C₄²	C₂×D₃₅₆	1	3	yes	—
113	C₁₆	—	2	3	no	—
114	C₁₆	Q₈×D₈₉	1	2	yes	—
115	Q₈	—	356	2	no	—
116	Q₈	—	1424	6	no	—
117	Q₈	—	178	4	no	—
118	Q₈	Dic₇₁₂	1	4	yes	direct product factor
119	C₂×C₄	—	356	2	no	—
120	C₂×C₄	—	356	6	no	—
121	C₂×C₄	C₂×D₇₁₂	1	3	yes	—
122	C₂×C₄	C₂×D₇₁₂	1	3	yes	—
123	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
124	C₈	—	2	3	no	—
125	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
126	C₈	#15668059	1	1	yes	characteristic
127	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	356	4	no	—
130	C₄	—	712	6	no	—
131	C₄	C₂×Dic₇₁₂	1	6	yes	—
132	C₄	Q₈×D₃₅₆	1	1	yes	characteristic
133	C₄	#19152415	1	1	yes	characteristic
134	C₂	Q₈×D₇₁₂	1	1	yes	characteristic
135	C₂	C₂²×Dic₇₁₂	1	1	yes	characteristic
136	C₂	#26033393	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₃₅₆

C₂

yes

maximal

#26022527

C₂

yes

maximal

#26022405

C₂

yes

maximal

C₄×Dic₇₁₂

C₂

yes

maximal

Q₈×C₁₆×C₈₉

C₂

yes

maximal, characteristic

Q₈×Dic₁₇₈

—

C₂×Dic₇₁₂

C₂²

yes

characteristic

#19152003

C₂²

yes

—

#19151894

—

C₄×Dic₃₅₆

C₂²

yes

—

#19151424

C₂²

yes

—

#19151424

C₂²

yes

—

#19151403

C₂²

yes

—

C₈×Q₈×C₈₉

C₂²

yes

characteristic

C₄⋊C₁₆×C₈₉

C₂²

yes

—

C₄×C₁₆×C₈₉

C₂²

yes

—

C₂×Dic₃₅₆

C₂³

yes

—

#15668012

—

#15668011

—

Q₈×Dic₈₉

—

C₄×Dic₁₇₈

—

Dic₇₁₂

Q₈

yes

direct product factor

#15667946

—

#15667946

C₂³

yes

—

#15667937

—

C₄×Q₈×C₈₉

D₄

yes

characteristic

C₂×C₁₆×C₈₉

C₂³

yes

—

C₂×C₁₆×C₈₉

C₂³

yes

characteristic

C₄⋊C₈×C₈₉

D₄

yes

—

C₄×C₈×C₈₉

C₂³

yes

—

C₂×Dic₁₇₈

—

C₂×Dic₁₇₈

—

Dic₃₅₆

—

Dic₃₅₆

C₂×Q₈

yes

—

C₄⋊Dic₈₉

—

C₄⋊Dic₈₉

—

#13083553

—

C₄×Dic₈₉

—

C₂×Q₈×C₈₉

D₈

yes

characteristic

C₂×C₈×C₈₉

C₂×D₄

yes

—

C₂×C₈×C₈₉

C₂⁴

yes

characteristic

C₄⋊C₄×C₈₉

D₈

yes

—

C₄²×C₈₉

C₂×D₄

yes

—

C₁₆×C₈₉

—

C₁₆×C₈₉

C₂×Q₈

yes

—

Q₈×C₈₉

Q₃₂

yes

complemented

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×C₄×C₈₉

C₂×D₈

yes

—

C₂×C₄×C₈₉

C₂²×D₄

yes

characteristic

C₂×Dic₈₉

—

C₂×Dic₈₉

—

Dic₁₇₈

—

Dic₁₇₈

—

Dic₁₇₈

—

C₈×C₈₉

—

C₈×C₈₉

C₂²×Q₈

yes

characteristic

C₈×C₈₉

2₋¹⁺⁴

yes

characteristic

C₂²×C₈₉

C₂²×D₈

yes

characteristic

C₄×C₈₉

—

C₄×C₈₉

C₂×Q₃₂

yes

—

C₄×C₈₉

D₄×Q₈

yes

characteristic

C₄×C₈₉

#578

yes

characteristic

Dic₈₉

—

Dic₈₉

—

Q₈×Q₃₂

—

maximal, Sylow

C₂×C₈₉

Q₈×D₈

yes

characteristic

C₂×C₈₉

C₂²×Q₃₂

yes

characteristic

C₂×C₈₉

#3415

yes

characteristic

Q₈×Q₁₆

—

#2252

—

#2227

—

Q₈×C₁₆

D₈₉

yes

characteristic

C₄×Q₃₂

—

C₈₉

Q₈×Q₃₂

yes

characteristic, complemented, Sylow

Q₈²

—

178

—

C₂×Q₃₂

—

#500

—

#477

—

178

—

C₈×Q₈

C₂×D₈₉

yes

characteristic

C₄×Q₁₆

—

#366

—

#366

—

C₄⋊C₁₆

C₂×D₈₉

yes

—

#358

—

C₄×C₁₆

C₂×D₈₉

yes

—

C₂×Q₁₆

—

C₄⋊Q₈

—

356

—

C₄⋊Q₈

—

178

—

C₄×Q₈

—

356

—

C₄×Q₈

—

178

—

C₄×Q₈

D₃₅₆

yes

characteristic

Q₃₂

—

C₂×C₁₆

C₂²×D₈₉

yes

—

C₂×C₁₆

C₂²×D₈₉

yes

characteristic

#107

—

#107

—

178

—

C₄⋊C₈

D₃₅₆

yes

—

#103

—

178

—

C₄×C₈

C₂²×D₈₉

yes

—

100

C₂×Q₈

—

356

—

101

C₂×Q₈

—

178

—

102

C₂×Q₈

D₇₁₂

yes

characteristic

103

Q₁₆

—

104

Q₁₆

—

178

—

105

C₂×C₈

C₂×D₃₅₆

yes

—

106

C₂×C₈

C₂³×D₈₉

yes

characteristic

107

C₄⋊C₄

—

356

—

108

C₄⋊C₄

—

356

—

109

C₄⋊C₄

—

178

—

110

C₄⋊C₄

D₇₁₂

yes

—

111

C₄²

—

356

—

112

C₄²

C₂×D₃₅₆

yes

—

113

C₁₆

—

114

C₁₆

Q₈×D₈₉

yes

—

115

Q₈

—

356

—

116

Q₈

—

1424

—

117

Q₈

—

178

—

118

Q₈

Dic₇₁₂

yes

direct product factor

119

C₂×C₄

—

356

—

120

C₂×C₄

—

356

—

121

C₂×C₄

C₂×D₇₁₂

yes

—

122

C₂×C₄

C₂×D₇₁₂

yes

—

123

C₂×C₄

C₂²×D₃₅₆

yes

characteristic

124

C₈

—

125

C₈

C₂×Q₈×D₈₉

yes

characteristic

126

C₈

#15668059

yes

characteristic

127

C₂²

C₂²×D₇₁₂

yes

characteristic

128

C₄

—

129

C₄

—

356

—

130

C₄

—

712

—

131

C₄

C₂×Dic₇₁₂

yes

—

132

C₄

Q₈×D₃₅₆

yes

characteristic

133

C₄

#19152415

yes

characteristic

134

C₂

Q₈×D₇₁₂

yes

characteristic

135

C₂

C₂²×Dic₇₁₂

yes

characteristic

136

C₂

#26033393

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	Q₈×C₁₆×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	C₂²×D₈
hypercenter	Q₈×C₁₆	D₈₉
hypocenter	C₈₉	Q₈×Q₃₂
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆

subgroup

G/H

center

C₂²

C₂²×D₇₁₂

commutator, cosocle

C₂×C₈×C₈₉

C₂⁴

Frattini

C₂×C₈

C₂³×D₈₉

Fitting

Q₈×C₁₆×C₈₉

C₂

socle, prime radical

C₂²×C₈₉

C₂²×D₈

hypercenter

Q₈×C₁₆

D₈₉

hypocenter

C₈₉

Q₈×Q₃₂

LCS

G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ Q₈×C₁₆

N	H ≅ G/N	type
22. Dic₇₁₂	Q₈	direct (N × H)
46. Q₈×C₈₉	Q₃₂	semidirect
74. C₈₉	Q₈×Q₃₂	semidirect
118. Q₈	Dic₇₁₂	direct (N × H)

H ≅ G/N

type

22. Dic₇₁₂

Q₈

direct (N × H)

46. Q₈×C₈₉

Q₃₂

semidirect

74. C₈₉

Q₈×Q₃₂

semidirect

118. Q₈

Dic₇₁₂

direct (N × H)

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

subgroup	index
Q₈×Dic₃₅₆	2
#26022527	2
#26022405	2
C₄×Dic₇₁₂	2
Q₈×C₁₆×C₈₉	2
Q₈×Q₃₂	89

subgroup

index

Q₈×Dic₃₅₆

#26022527

#26022405

C₄×Dic₇₁₂

Q₈×C₁₆×C₈₉

Q₈×Q₃₂

Subgroups — Q₈×Dic₇₁₂ #36712323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×Dic712 #36712323

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×Dic₇₁₂ #36712323