Group #69489 — subgroups

Subgroups — #69489

Order 384.

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

normal subgroups only

This group has 292 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×C₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
4	#4529	C₂	1	4	yes	maximal
5	#1987	—	3	1	no	maximal, Sylow
6	C₂²×Dic₆	—	2	1	no	—
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	C₂²	1	2	yes	—
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	C₄⋊(C₃⋊C₈)	—	2	2	no	—
17	C₂×C₄⋊Q₈	—	3	1	no	—
18	C₂×C₄⋊C₈	—	3	1	no	—
19	C₄×C₄⋊C₄	D₃	1	1	yes	characteristic
20	#340	—	3	4	no	—
21	C₂²×C₃×C₄	—	2	2	no	—
22	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
23	C₂²×C₃×C₄	D₄	1	1	yes	characteristic
24	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
25	C₂²×Dic₃	—	2	2	no	—
26	C₂×Dic₆	—	2	4	no	—
27	C₂×Dic₆	—	4	4	no	—
28	C₃×C₄⋊C₄	—	2	2	no	—
29	C₃×C₄⋊C₄	Q₈	1	4	yes	complemented
30	C₃×C₄²	—	4	2	no	—
31	C₃×C₄²	C₂×C₄	1	2	yes	—
32	C₃×C₄²	D₄	1	2	yes	—
33	C₄⋊Dic₃	—	2	4	no	—
34	C₄⋊Dic₃	—	2	2	no	—
35	C₄⋊Dic₃	C₂×C₄	1	4	yes	—
36	C₂×C₃⋊C₈	—	2	2	no	—
37	C₂×C₃⋊C₈	—	2	4	no	—
38	C₂²×Q₈	—	6	1	no	—
39	C₂²×C₈	—	3	2	no	—
40	C₄⋊Q₈	—	3	4	no	—
41	C₄⋊Q₈	—	6	2	no	—
42	C₂×C₄⋊C₄	—	3	2	no	—
43	C₂×C₄⋊C₄	—	6	1	no	—
44	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
45	C₂×C₄²	—	2	1	no	—
46	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
47	C₄⋊C₈	—	6	2	no	—
48	#95	—	2	1	no	—
49	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₃×C₄	—	2	2	no	—
51	C₂×C₃×C₄	—	2	4	no	—
52	C₂×C₃×C₄	—	4	2	no	—
53	C₂×C₃×C₄	—	4	2	no	—
54	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
55	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
56	C₂×C₃×C₄	SD₁₆	1	2	yes	—
57	C₂×C₃×C₄	Q₁₆	1	2	yes	—
58	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
61	C₂×C₃×C₄	#41	1	2	yes	—
62	C₂×Dic₃	—	2	4	no	—
63	C₂×Dic₃	—	4	4	no	—
64	Dic₆	—	4	8	no	—
65	C₃⋊C₈	—	4	2	no	—
66	C₂×Q₈	—	6	4	no	—
67	C₂×Q₈	—	12	4	no	—
68	C₂²×C₄	—	2	2	no	—
69	C₂²×C₄	—	6	2	no	—
70	C₂²×C₄	#67	1	1	yes	characteristic
71	C₂²×C₄	#67	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
73	C₂×C₈	—	6	2	no	—
74	C₂×C₈	—	6	4	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	3	4	no	—
77	C₄⋊C₄	—	6	4	no	—
78	C₄⋊C₄	—	6	2	no	—
79	C₄⋊C₄	Dic₆	1	4	yes	complemented
80	C₄²	—	4	2	no	—
81	C₄²	C₄×D₃	1	2	yes	—
82	C₄²	D₁₂	1	2	yes	—
83	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₃	C₂²⋊Q₈	1	2	yes	—
85	C₂²×C₃	C₂×SD₁₆	1	1	yes	characteristic
86	C₂²×C₃	C₂×Q₁₆	1	1	yes	characteristic
87	C₂²×C₃	AGL(1,ℤ/8)	1	2	yes	—
88	C₃×C₄	—	2	2	no	—
89	C₃×C₄	—	4	4	no	—
90	C₃×C₄	#103	1	4	yes	—
91	C₃×C₄	C₄×Q₈	1	2	yes	—
92	C₃×C₄	#124	1	1	yes	characteristic
93	C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
94	Dic₃	—	4	4	no	—
95	C₂³	C₂×#67	1	1	yes	characteristic
96	Q₈	—	12	8	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	4	2	no	—
101	C₂×C₄	—	6	4	no	—
102	C₂×C₄	—	12	4	no	—
103	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
104	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
105	C₂×C₄	#214	1	2	yes	—
106	C₂×C₄	#216	1	2	yes	—
107	C₂×C₄	C₂×Dic₆	1	2	yes	—
108	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
109	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
110	C₂×C₄	#235	1	2	yes	—
111	C₈	—	12	2	no	—
112	C₂×C₃	#391	1	1	yes	characteristic
113	C₂×C₃	C₂×#103	1	1	yes	characteristic
114	C₂×C₃	#418	1	1	yes	characteristic
115	C₂×C₃	#476	1	2	yes	—
116	C₂×C₃	#477	1	2	yes	—
117	C₂²	#924	1	2	yes	—
118	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
119	C₂²	C₂×#214	1	1	yes	characteristic
120	C₂²	C₂×#216	1	1	yes	characteristic
121	C₂²	#949	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	4	4	no	—
124	C₄	—	12	4	no	—
125	C₄	#810	1	4	yes	—
126	C₄	C₄×Dic₆	1	2	yes	—
127	C₄	#869	1	1	yes	characteristic
128	C₄	#875	1	1	yes	characteristic
129	C₃	#1987	1	1	yes	characteristic, complemented, Sylow
130	C₂	#4929	1	1	yes	characteristic
131	C₂	C₂×#810	1	1	yes	characteristic
132	C₂	#5001	1	1	yes	characteristic
133	C₂	#5008	1	2	yes	—
134	C₂	#5021	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×C₄⋊(C₃⋊C₈)

C₂

yes

maximal, characteristic

#4529

C₂

yes

maximal

#1987

—

maximal, Sylow

C₂²×Dic₆

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂²×C₃⋊C₈

C₂²

yes

—

#869

—

#869

C₄

yes

complemented

C₃×#95

—

C₄⋊(C₃⋊C₈)

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

D₃

yes

characteristic

#340

—

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

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

Q₈

yes

complemented

C₃×C₄²

—

C₃×C₄²

C₂×C₄

yes

—

C₃×C₄²

D₄

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

C₂×C₄

yes

—

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

—

#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

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

C₃⋊C₈

—

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

—

C₄⋊C₄

Dic₆

yes

complemented

C₄²

—

C₄²

C₄×D₃

yes

—

C₄²

D₁₂

yes

—

C₂²×C₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃

C₂²⋊Q₈

yes

—

C₂²×C₃

C₂×SD₁₆

yes

characteristic

C₂²×C₃

C₂×Q₁₆

yes

characteristic

C₂²×C₃

AGL(1,ℤ/8)

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

#103

yes

—

C₃×C₄

C₄×Q₈

yes

—

C₃×C₄

#124

yes

characteristic

C₃×C₄

C₄⋊Q₈

yes

characteristic

Dic₃

—

C₂³

C₂×#67

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

(C₂×D₃)⋊C₄

yes

—

104

C₂×C₄

(C₂×D₃)⋊C₄

yes

—

105

C₂×C₄

#214

yes

—

106

C₂×C₄

#216

yes

—

107

C₂×C₄

C₂×Dic₆

yes

—

108

C₂×C₄

C₂×C₄×D₃

yes

characteristic

109

C₂×C₄

C₂×D₁₂

yes

characteristic

110

C₂×C₄

#235

yes

—

111

C₈

—

112

C₂×C₃

#391

yes

characteristic

113

C₂×C₃

C₂×#103

yes

characteristic

114

C₂×C₃

#418

yes

characteristic

115

C₂×C₃

#476

yes

—

116

C₂×C₃

#477

yes

—

117

C₂²

#924

yes

—

118

C₂²

C₂×(C₂×D₃)⋊C₄

yes

characteristic

119

C₂²

C₂×#214

yes

characteristic

120

C₂²

C₂×#216

yes

characteristic

121

C₂²

#949

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#810

yes

—

126

C₄

C₄×Dic₆

yes

—

127

C₄

#869

yes

characteristic

128

C₄

#875

yes

characteristic

129

C₃

#1987

yes

characteristic, complemented, Sylow

130

C₂

#4929

yes

characteristic

131

C₂

C₂×#810

yes

characteristic

132

C₂

#5001

yes

characteristic

133

C₂

#5008

yes

—

134

C₂

#5021

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	#1987
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

#1987

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
29. C₃×C₄⋊C₄	Q₈	semidirect
79. C₄⋊C₄	Dic₆	semidirect
129. C₃	#1987	semidirect

H ≅ G/N

type

14. #869

C₄

semidirect

29. C₃×C₄⋊C₄

Q₈

semidirect

79. C₄⋊C₄

Dic₆

semidirect

129. C₃

#1987

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×C₄⋊(C₃⋊C₈)	2
#4529	2
#1987	3

subgroup

index

C₂×#869

C₃×C₄×C₄⋊C₄

C₂×C₄⋊(C₃⋊C₈)

#4529

#1987

Subgroups — #69489

Conjugacy classes of subgroups

Complements

Maximal subgroups