Group #69486 — subgroups

Subgroups — #69486

Order 384.

324 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₂×#869	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₂×#810	C₂	1	4	yes	maximal
4	C₂×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	#1981	—	3	1	no	maximal, Sylow
6	C₂²×Dic₆	C₂²	1	2	yes	—
7	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
8	C₂×C₃×C₄²	—	2	1	no	—
9	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃	—	2	1	no	—
11	C₂×C₄⋊Dic₃	—	2	1	no	—
12	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
13	#869	—	2	2	no	—
14	#869	C₄	1	4	yes	complemented
15	C₃×#95	—	2	1	no	—
16	#810	—	2	8	no	—
17	C₄⋊(C₃⋊C₈)	—	2	2	no	—
18	C₂×C₄⋊Q₈	—	3	1	no	—
19	C₂×C₄⋊C₈	—	3	1	no	—
20	C₂×#103	—	3	4	no	—
21	C₄×C₄⋊C₄	D₃	1	1	yes	characteristic
22	C₂²×C₃×C₄	—	2	2	no	—
23	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
24	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
25	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
26	C₂²×Dic₃	—	2	2	no	—
27	C₂×Dic₆	—	2	4	no	—
28	C₂×Dic₆	—	2	4	no	—
29	C₂×Dic₆	—	2	2	no	—
30	C₂×Dic₆	C₂×C₄	1	4	yes	—
31	C₃×C₄⋊C₄	—	2	2	no	—
32	C₃×C₄⋊C₄	D₄	1	4	yes	—
33	C₃×C₄²	—	4	2	no	—
34	C₃×C₄²	C₂×C₄	1	2	yes	—
35	C₃×C₄²	D₄	1	2	yes	—
36	C₄⋊Dic₃	—	2	4	no	—
37	C₄⋊Dic₃	—	2	4	no	—
38	C₂×C₃⋊C₈	—	2	2	no	—
39	C₂×C₃⋊C₈	—	2	4	no	—
40	C₂²×Q₈	—	3	2	no	—
41	C₂²×C₈	—	3	2	no	—
42	C₄⋊Q₈	—	3	4	no	—
43	C₄⋊Q₈	—	6	2	no	—
44	C₂×C₄⋊C₄	—	6	1	no	—
45	C₂×C₄⋊C₄	—	6	1	no	—
46	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
47	C₂×C₄²	—	2	1	no	—
48	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
49	C₄⋊C₈	—	6	2	no	—
50	#103	—	6	8	no	—
51	#95	—	2	1	no	—
52	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₃×C₄	—	2	2	no	—
54	C₂×C₃×C₄	—	2	4	no	—
55	C₂×C₃×C₄	—	4	2	no	—
56	C₂×C₃×C₄	—	4	2	no	—
57	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
58	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
59	C₂×C₃×C₄	SD₁₆	1	2	yes	—
60	C₂×C₃×C₄	Q₁₆	1	2	yes	—
61	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
63	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
64	C₂×C₃×C₄	#41	1	2	yes	—
65	C₂×Dic₃	—	2	4	no	—
66	C₂×Dic₃	—	4	4	no	—
67	Dic₆	—	2	8	no	—
68	Dic₆	—	4	2	no	—
69	Dic₆	—	4	2	no	—
70	C₃⋊C₈	—	4	2	no	—
71	C₂×Q₈	—	3	4	no	—
72	C₂×Q₈	—	6	2	no	—
73	C₂×Q₈	—	6	4	no	—
74	C₂×Q₈	—	6	4	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	—	6	2	no	—
77	C₂²×C₄	#67	1	1	yes	characteristic
78	C₂²×C₄	#67	1	1	yes	characteristic
79	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
80	C₂×C₈	—	6	4	no	—
81	C₂×C₈	—	6	2	no	—
82	C₄⋊C₄	—	2	2	no	—
83	C₄⋊C₄	—	6	4	no	—
84	C₄⋊C₄	—	6	4	no	—
85	C₄⋊C₄	D₁₂	1	4	yes	—
86	C₄²	—	4	2	no	—
87	C₄²	C₄×D₃	1	2	yes	—
88	C₄²	D₁₂	1	2	yes	—
89	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
90	C₂²×C₃	#121	1	2	yes	—
91	C₂²×C₃	C₂×SD₁₆	1	1	yes	characteristic
92	C₂²×C₃	C₂×Q₁₆	1	1	yes	characteristic
93	C₂²×C₃	#137	1	2	yes	—
94	C₃×C₄	—	2	2	no	—
95	C₃×C₄	—	4	4	no	—
96	C₃×C₄	#103	1	4	yes	complemented
97	C₃×C₄	C₄×D₄	1	2	yes	—
98	C₃×C₄	#124	1	1	yes	characteristic
99	C₃×C₄	#127	1	1	yes	characteristic
100	Dic₃	—	4	4	no	—
101	C₂³	C₂×#67	1	1	yes	characteristic
102	Q₈	—	6	8	no	—
103	Q₈	—	12	2	no	—
104	Q₈	—	12	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	6	4	no	—
110	C₂×C₄	—	12	4	no	—
111	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
112	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
113	C₂×C₄	#214	1	2	yes	—
114	C₂×C₄	#216	1	2	yes	—
115	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
116	C₂×C₄	C₂×D₁₂	1	2	yes	—
117	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
118	C₂×C₄	#235	1	2	yes	—
119	C₈	—	12	2	no	—
120	C₂×C₃	#390	1	1	yes	characteristic
121	C₂×C₃	C₂×#103	1	1	yes	characteristic
122	C₂×C₃	#419	1	1	yes	characteristic
123	C₂×C₃	#461	1	2	yes	—
124	C₂×C₃	#462	1	2	yes	—
125	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
126	C₂²	#930	1	2	yes	—
127	C₂²	C₂×#214	1	1	yes	characteristic
128	C₂²	C₂×#216	1	1	yes	characteristic
129	C₂²	#951	1	2	yes	—
130	C₄	—	2	2	no	—
131	C₄	—	4	4	no	—
132	C₄	—	12	4	no	—
133	C₄	#810	1	4	yes	complemented
134	C₄	C₄×D₁₂	1	2	yes	—
135	C₄	#874	1	1	yes	characteristic
136	C₄	#875	1	1	yes	characteristic
137	C₃	#1981	1	1	yes	characteristic, complemented, Sylow
138	C₂	#4939	1	1	yes	characteristic
139	C₂	C₂×#810	1	1	yes	characteristic
140	C₂	#5004	1	1	yes	characteristic
141	C₂	#5019	1	2	yes	—
142	C₂	#5031	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#869

C₂

yes

maximal, characteristic

C₃×C₄×C₄⋊C₄

C₂

yes

maximal, characteristic

C₂×#810

C₂

yes

maximal

C₂×C₄⋊(C₃⋊C₈)

C₂

yes

maximal, characteristic

#1981

—

maximal, Sylow

C₂²×Dic₆

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

—

C₂²×C₃⋊C₈

C₂²

yes

—

#869

—

#869

C₄

yes

complemented

C₃×#95

—

#810

—

C₄⋊(C₃⋊C₈)

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₂×#103

—

C₄×C₄⋊C₄

D₃

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

D₄

yes

characteristic

C₂²×C₃×C₄

D₄

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

C₂×C₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

D₄

yes

—

C₃×C₄²

—

C₃×C₄²

C₂×C₄

yes

—

C₃×C₄²

D₄

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

#103

—

#95

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

SD₁₆

yes

—

C₂×C₃×C₄

Q₁₆

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

Dic₆

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

D₁₂

yes

—

C₄²

—

C₄²

C₄×D₃

yes

—

C₄²

D₁₂

yes

—

C₂²×C₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃

#121

yes

—

C₂²×C₃

C₂×SD₁₆

yes

characteristic

C₂²×C₃

C₂×Q₁₆

yes

characteristic

C₂²×C₃

#137

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#103

yes

complemented

C₃×C₄

C₄×D₄

yes

—

C₃×C₄

yes

C₃×C₄

yes

100

Dic₃

—

101

C₂³

C₂×#67

yes

characteristic

102

Q₈

—

103

Q₈

—

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

(C₂×D₃)⋊C₄

yes

—

112

C₂×C₄

(C₂×D₃)⋊C₄

yes

—

113

C₂×C₄

#214

yes

—

114

C₂×C₄

#216

yes

—

115

C₂×C₄

C₂×C₄×D₃

yes

characteristic

116

C₂×C₄

C₂×D₁₂

yes

—

117

C₂×C₄

C₂×D₁₂

yes

characteristic

118

C₂×C₄

#235

yes

—

119

C₈

—

120

C₂×C₃

#390

yes

characteristic

121

C₂×C₃

C₂×#103

yes

characteristic

122

C₂×C₃

#419

yes

characteristic

123

C₂×C₃

#461

yes

—

124

C₂×C₃

#462

yes

—

125

C₂²

C₂×(C₂×D₃)⋊C₄

yes

characteristic

126

C₂²

#930

yes

—

127

C₂²

C₂×#214

yes

characteristic

128

C₂²

C₂×#216

yes

characteristic

129

C₂²

#951

yes

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#810

yes

complemented

134

C₄

C₄×D₁₂

yes

—

135

C₄

#874

yes

characteristic

136

C₄

#875

yes

characteristic

137

C₃

#1981

yes

characteristic, complemented, Sylow

138

C₂

#4939

yes

characteristic

139

C₂

C₂×#810

yes

characteristic

140

C₂

#5004

yes

characteristic

141

C₂

#5019

yes

—

142

C₂

#5031

yes

—

	subgroup	G/H
center	C₂³	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×C₄×C₄⋊C₄	C₂
socle, prime radical	C₂³×C₃	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₄×C₄⋊C₄	D₃
hypocenter	C₃	#1981
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂×#67

commutator

C₂×C₃×C₄

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₃×C₄×C₄⋊C₄

C₂

socle, prime radical

C₂³×C₃

C₂×D₄

cosocle

C₂²×C₃×C₄

C₂³

hypercenter

C₄×C₄⋊C₄

D₃

hypocenter

C₃

#1981

LCS

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₄×C₄⋊C₄

N	H ≅ G/N	type
14. #869	C₄	semidirect
96. C₃×C₄	#103	semidirect
133. C₄	#810	semidirect
137. C₃	#1981	semidirect

H ≅ G/N

type

14. #869

C₄

semidirect

96. C₃×C₄

#103

semidirect

133. C₄

#810

semidirect

137. C₃

#1981

semidirect

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

subgroup	index
C₂×#869	2
C₃×C₄×C₄⋊C₄	2
C₂×#810	2
C₂×C₄⋊(C₃⋊C₈)	2
#1981	3

subgroup

index

C₂×#869

C₃×C₄×C₄⋊C₄

C₂×#810

C₂×C₄⋊(C₃⋊C₈)

#1981

Subgroups — #69486

Conjugacy classes of subgroups

Complements

Maximal subgroups