Group #69684 — subgroups

Subgroups — #69684

Order 384.

254 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 254 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₃×#391	C₂	1	1	yes	maximal, characteristic
2	C₂×#835	C₂	1	2	yes	maximal
3	C₂×#807	C₂	1	2	yes	maximal
4	#4935	C₂	1	1	yes	maximal, characteristic
5	#4524	C₂	1	1	yes	maximal, characteristic
6	#2095	—	3	1	no	maximal, Sylow
7	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃	—	2	2	no	—
11	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
13	C₃×#95	—	2	2	no	—
14	#835	—	2	4	no	—
15	#807	—	2	4	no	—
16	C₂×#107	—	3	2	no	—
17	C₂×#103	—	3	2	no	—
18	#391	D₃	1	1	yes	characteristic
19	#384	—	3	1	no	—
20	#336	—	3	1	no	—
21	C₂×C₃×Q₈	—	2	1	no	—
22	C₂×C₃×Q₈	—	2	2	no	—
23	C₂×C₃×Q₈	—	2	2	no	—
24	C₂×C₃×Q₈	Q₈	1	2	yes	—
25	C₂²×C₃×C₄	—	2	1	no	—
26	C₂²×C₃×C₄	—	2	1	no	—
27	C₂²×C₃×C₄	D₄	1	2	yes	—
28	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
29	C₂²×Dic₃	—	2	2	no	—
30	C₃×C₄⋊C₄	—	2	2	no	—
31	C₃×C₄²	—	2	2	no	—
32	C₄⋊Dic₃	—	2	2	no	—
33	C₄⋊Dic₃	—	2	2	no	—
34	C₄⋊Dic₃	—	4	4	no	—
35	C₂×C₃⋊C₈	—	2	2	no	—
36	C₂×C₃⋊C₈	—	2	2	no	—
37	C₂×C₃⋊C₈	D₄	1	4	yes	—
38	C₂²×Q₈	C₂×D₃	1	1	yes	characteristic
39	C₂²×C₈	—	3	2	no	—
40	C₂×C₄⋊C₄	—	3	2	no	—
41	C₂×C₄⋊C₄	—	6	2	no	—
42	C₂×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
43	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
44	#107	—	6	4	no	—
45	#103	—	6	4	no	—
46	#95	—	2	2	no	—
47	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
48	C₃×Q₈	—	2	4	no	—
49	C₃×Q₈	—	4	2	no	—
50	C₂×C₃×C₄	—	2	2	no	—
51	C₂×C₃×C₄	—	2	2	no	—
52	C₂×C₃×C₄	—	2	2	no	—
53	C₂×C₃×C₄	—	4	1	no	—
54	C₂×C₃×C₄	—	4	1	no	—
55	C₂×C₃×C₄	—	4	1	no	—
56	C₂×C₃×C₄	—	4	2	no	—
57	C₂×C₃×C₄	Q₁₆	1	4	yes	—
58	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
59	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₃×C₄	#41	1	1	yes	characteristic
61	C₂×C₃×C₄	#41	1	2	yes	—
62	C₂×Dic₃	—	4	2	no	—
63	C₂×Dic₃	—	4	4	no	—
64	C₃⋊C₈	—	2	4	no	—
65	C₂×Q₈	—	2	1	no	—
66	C₂×Q₈	—	2	2	no	—
67	C₂×Q₈	—	2	2	no	—
68	C₂×Q₈	Dic₆	1	2	yes	—
69	C₂²×C₄	—	2	1	no	—
70	C₂²×C₄	—	2	1	no	—
71	C₂²×C₄	—	6	2	no	—
72	C₂²×C₄	#67	1	2	yes	—
73	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
74	C₂×C₈	—	3	4	no	—
75	C₂×C₈	—	6	2	no	—
76	C₂×C₈	—	6	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄⋊C₄	—	6	2	no	—
79	C₄⋊C₄	—	6	2	no	—
80	C₄⋊C₄	—	12	4	no	—
81	C₄²	—	2	2	no	—
82	C₂²×C₃	C₂²⋊Q₈	1	1	yes	characteristic
83	C₂²×C₃	#123	1	1	yes	characteristic
84	C₂²×C₃	#127	1	1	yes	characteristic
85	C₂²×C₃	C₂×Q₁₆	1	2	yes	—
86	C₂²×C₃	AGL(1,ℤ/8)	1	2	yes	—
87	C₃×C₄	—	2	4	no	—
88	C₃×C₄	—	4	2	no	—
89	C₃×C₄	—	8	1	no	—
90	C₃×C₄	C₂²⋊Q₈	1	2	yes	—
91	C₃×C₄	#123	1	2	yes	—
92	Dic₃	—	8	2	no	—
93	C₂³	C₂×#67	1	1	yes	characteristic
94	Q₈	—	2	4	no	—
95	Q₈	—	4	2	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	4	1	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	4	1	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	12	2	no	—
104	C₂×C₄	—	12	4	no	—
105	C₂×C₄	#216	1	4	yes	—
106	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
107	C₂×C₄	#235	1	1	yes	characteristic
108	C₂×C₄	D₃×D₄	1	2	yes	—
109	C₂×C₄	#237	1	2	yes	—
110	C₈	—	6	4	no	—
111	C₂×C₃	#399	1	1	yes	characteristic
112	C₂×C₃	#477	1	2	yes	—
113	C₂×C₃	#484	1	2	yes	—
114	C₂×C₃	#494	1	1	yes	characteristic
115	C₂×C₃	#496	1	1	yes	characteristic
116	C₂²	#924	1	1	yes	characteristic
117	C₂²	#935	1	1	yes	characteristic
118	C₂²	#940	1	1	yes	characteristic
119	C₂²	C₂×#216	1	2	yes	—
120	C₂²	#949	1	2	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	4	2	no	—
123	C₄	—	8	1	no	—
124	C₄	—	24	2	no	—
125	C₄	#878	1	2	yes	—
126	C₄	#883	1	2	yes	—
127	C₃	#2095	1	1	yes	characteristic, complemented, Sylow
128	C₂	#4949	1	1	yes	characteristic
129	C₂	#5021	1	2	yes	—
130	C₂	#5044	1	2	yes	—
131	C₂	#5058	1	1	yes	characteristic
132	C₂	#5092	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#391

C₂

yes

maximal, characteristic

C₂×#835

C₂

yes

maximal

C₂×#807

C₂

yes

maximal

#4935

C₂

yes

maximal, characteristic

#4524

C₂

yes

maximal, characteristic

#2095

—

maximal, Sylow

C₂²×C₃×Q₈

C₂²

yes

characteristic

C₂×C₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂²×C₃⋊C₈

C₂²

yes

—

C₃×#95

—

—

—

—

—

D₃

yes

characteristic

#384

—

#336

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

Q₈

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

D₄

yes

—

C₂²×Q₈

C₂×D₃

yes

characteristic

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

characteristic

C₂×C₄²

C₂×D₃

yes

—

—

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

Q₁₆

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

C₂×C₃×C₄

yes

C₂×C₃×C₄

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#67

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₂²×C₃

C₂²⋊Q₈

yes

C₂²×C₃

yes

C₂²×C₃

yes

C₂²×C₃

C₂×Q₁₆

yes

—

C₂²×C₃

AGL(1,ℤ/8)

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂²⋊Q₈

yes

—

C₃×C₄

#123

yes

—

Dic₃

—

C₂³

C₂×#67

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

#216

yes

—

106

C₂×C₄

C₂×Dic₆

yes

characteristic

107

C₂×C₄

#235

yes

characteristic

108

C₂×C₄

D₃×D₄

yes

—

109

C₂×C₄

#237

yes

—

110

C₈

—

111

C₂×C₃

#399

yes

characteristic

112

C₂×C₃

#477

yes

—

113

C₂×C₃

#484

yes

—

114

C₂×C₃

#494

yes

characteristic

115

C₂×C₃

#496

yes

characteristic

116

C₂²

#924

yes

characteristic

117

C₂²

#935

yes

characteristic

118

C₂²

#940

yes

characteristic

119

C₂²

C₂×#216

yes

—

120

C₂²

#949

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#878

yes

—

126

C₄

#883

yes

—

127

C₃

#2095

yes

characteristic, complemented, Sylow

128

C₂

#4949

yes

characteristic

129

C₂

#5021

yes

—

130

C₂

#5044

yes

—

131

C₂

#5058

yes

characteristic

132

C₂

#5092

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×#67
commutator, cosocle	C₂²×C₃×C₄	C₂³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#391	C₂
socle, prime radical	C₂³×C₃	C₂×D₄
hypercenter	#391	D₃
hypocenter	C₃	#2095
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×#67

commutator, cosocle

C₂²×C₃×C₄

C₂³

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₃×#391

C₂

socle, prime radical

C₂³×C₃

C₂×D₄

D₃

C₃

G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
127. C₃	#2095	semidirect

H ≅ G/N

type

127. C₃

#2095

semidirect

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

subgroup	index
C₃×#391	2
C₂×#835	2
C₂×#807	2
#4935	2
#4524	2
#2095	3

subgroup

index

Subgroups — #69684

Conjugacy classes of subgroups

Complements

Maximal subgroups