Group #68839 — subgroups

Subgroups — #68839

Order 384.

224 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 89 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×#818	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#4984	C₂	1	2	yes	maximal
4	C₃×#340	C₂	1	1	yes	maximal, characteristic
5	#4529	C₂	1	2	yes	maximal
6	#2087	—	3	1	no	maximal, Sylow
7	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
8	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₃	—	2	2	no	—
10	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
12	C₂²×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₃⋊#95	—	2	2	no	—
14	#818	—	2	2	no	—
15	Dic₃⋊C₈	—	2	2	no	—
16	C₂×#107	—	3	1	no	—
17	C₂×C₄⋊C₈	—	3	1	no	—
18	#389	—	3	2	no	—
19	#340	—	3	2	no	—
20	#340	D₃	1	1	yes	characteristic
21	C₂²×C₃×C₄	—	2	2	no	—
22	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
23	C₂²×Dic₃	—	2	2	no	—
24	C₂²×Dic₃	D₄	1	1	yes	characteristic
25	C₂²×Dic₃	D₄	1	1	yes	characteristic
26	C₂×C₃×C₈	—	2	2	no	—
27	C₂×C₃×C₈	D₄	1	2	yes	—
28	C₃×C₄⋊C₄	—	2	2	no	—
29	C₃×C₄⋊C₄	Q₈	1	4	yes	—
30	C₄⋊Dic₃	—	2	2	no	—
31	C₄⋊Dic₃	—	2	2	no	—
32	C₄⋊Dic₃	—	4	4	no	—
33	C₄×Dic₃	—	2	1	no	—
34	C₄×Dic₃	—	2	1	no	—
35	C₂×C₃⋊C₈	—	2	1	no	—
36	C₂×C₃⋊C₈	—	2	2	no	—
37	C₂²×C₈	—	3	1	no	—
38	C₂²×C₈	C₂×D₃	1	1	yes	characteristic
39	C₂×C₄⋊C₄	—	3	2	no	—
40	C₂×C₄⋊C₄	—	6	2	no	—
41	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
42	C₂×C₄²	—	3	1	no	—
43	#107	—	6	2	no	—
44	C₄⋊C₈	—	6	2	no	—
45	#95	—	6	2	no	—
46	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₃×C₄	—	2	4	no	—
48	C₂×C₃×C₄	—	4	2	no	—
49	C₂×C₃×C₄	—	4	2	no	—
50	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
51	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
52	C₂×C₃×C₄	#41	1	2	yes	—
53	C₂×C₃×C₄	#41	1	1	yes	characteristic
54	C₂×Dic₃	—	2	1	no	—
55	C₂×Dic₃	—	2	1	no	—
56	C₂×Dic₃	—	2	1	no	—
57	C₂×Dic₃	—	2	2	no	—
58	C₂×Dic₃	—	4	2	no	—
59	C₂×Dic₃	—	4	2	no	—
60	C₂×Dic₃	—	4	2	no	—
61	C₂×Dic₃	Q₁₆	1	2	yes	—
62	C₃×C₈	—	2	2	no	—
63	C₃⋊C₈	—	4	1	no	—
64	C₂²×C₄	—	2	2	no	—
65	C₂²×C₄	—	3	1	no	—
66	C₂²×C₄	—	3	1	no	—
67	C₂²×C₄	—	6	2	no	—
68	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
69	C₂×C₈	—	2	2	no	—
70	C₂×C₈	—	6	1	no	—
71	C₂×C₈	—	6	2	no	—
72	C₂×C₈	D₁₂	1	2	yes	—
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	6	2	no	—
75	C₄⋊C₄	—	6	2	no	—
76	C₄⋊C₄	—	12	4	no	—
77	C₄⋊C₄	Dic₆	1	4	yes	—
78	C₄²	—	6	1	no	—
79	C₄²	—	6	1	no	—
80	C₂²×C₃	#121	1	1	yes	characteristic
81	C₂²×C₃	C₂²⋊Q₈	1	2	yes	—
82	C₂²×C₃	C₂×Q₁₆	1	1	yes	characteristic
83	C₂²×C₃	#135	1	1	yes	characteristic
84	C₂²×C₃	AGL(1,ℤ/8)	1	2	yes	—
85	C₃×C₄	—	4	4	no	—
86	C₃×C₄	#123	1	1	yes	characteristic
87	C₃×C₄	#125	1	2	yes	—
88	C₃×C₄	C₄⋊Q₈	1	1	yes	characteristic
89	Dic₃	—	2	2	no	—
90	Dic₃	—	4	1	no	—
91	Dic₃	—	8	2	no	—
92	C₂³	D₃×D₄	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	—	3	2	no	—
95	C₂×C₄	—	4	2	no	—
96	C₂×C₄	—	4	2	no	—
97	C₂×C₄	—	6	1	no	—
98	C₂×C₄	—	6	1	no	—
99	C₂×C₄	—	6	1	no	—
100	C₂×C₄	—	6	2	no	—
101	C₂×C₄	—	12	2	no	—
102	C₂×C₄	—	12	2	no	—
103	C₂×C₄	—	12	2	no	—
104	C₂×C₄	C₂×Dic₆	1	2	yes	—
105	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
106	C₂×C₄	#237	1	2	yes	—
107	C₂×C₄	#239	1	1	yes	characteristic
108	C₈	—	2	2	no	—
109	C₈	—	12	1	no	—
110	C₂×C₃	#398	1	1	yes	characteristic
111	C₂×C₃	#467	1	1	yes	characteristic
112	C₂×C₃	#469	1	1	yes	characteristic
113	C₂×C₃	#477	1	2	yes	—
114	C₂×C₃	#478	1	2	yes	—
115	C₂²	#878	1	2	yes	—
116	C₂²	#895	1	1	yes	characteristic
117	C₂²	#912	1	1	yes	characteristic
118	C₂²	#914	1	2	yes	—
119	C₂²	D₃×Q₁₆	1	1	yes	characteristic
120	C₄	—	4	4	no	—
121	C₄	—	6	2	no	—
122	C₄	—	12	1	no	—
123	C₄	—	24	2	no	—
124	C₄	#869	1	1	yes	characteristic
125	C₄	#886	1	1	yes	characteristic
126	C₄	#890	1	2	yes	—
127	C₃	#2087	1	1	yes	characteristic, complemented, Sylow
128	C₂	#4661	1	1	yes	characteristic
129	C₂	#4766	1	2	yes	—
130	C₂	#4793	1	2	yes	—
131	C₂	#4865	1	1	yes	characteristic
132	C₂	#4887	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#818

C₂

yes

maximal, characteristic

C₂×Dic₃⋊C₈

C₂

yes

maximal, characteristic

#4984

C₂

yes

maximal

C₃×#340

C₂

yes

maximal, characteristic

#4529

C₂

yes

maximal

#2087

—

maximal, Sylow

C₂²×C₃×C₈

C₂²

yes

characteristic

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

characteristic

C₂²×C₃⋊C₈

C₂²

yes

characteristic

C₃⋊#95

—

#818

—

Dic₃⋊C₈

—

C₂×#107

—

C₂×C₄⋊C₈

—

#389

—

#340

—

#340

D₃

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₂²×Dic₃

D₄

yes

characteristic

C₂²×Dic₃

D₄

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

Q₈

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

—

#107

—

C₄⋊C₈

—

#95

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×C₃×C₄

#41

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

Q₁₆

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₁₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

Dic₆

yes

—

C₄²

—

C₄²

—

C₂²×C₃

#121

yes

characteristic

C₂²×C₃

C₂²⋊Q₈

yes

—

C₂²×C₃

C₂×Q₁₆

yes

C₂²×C₃

yes

C₂²×C₃

yes

—

C₃×C₄

—

C₃×C₄

#123

yes

characteristic

C₃×C₄

#125

yes

—

C₃×C₄

C₄⋊Q₈

yes

characteristic

Dic₃

—

Dic₃

—

Dic₃

—

C₂³

D₃×D₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×Dic₆

yes

—

105

C₂×C₄

C₂×D₁₂

yes

characteristic

106

C₂×C₄

#237

yes

—

107

C₂×C₄

#239

yes

characteristic

108

C₈

—

109

C₈

—

110

C₂×C₃

#398

yes

characteristic

111

C₂×C₃

#467

yes

characteristic

112

C₂×C₃

#469

yes

characteristic

113

C₂×C₃

#477

yes

—

114

C₂×C₃

#478

yes

—

115

C₂²

#878

yes

—

116

C₂²

#895

yes

characteristic

117

C₂²

#912

yes

characteristic

118

C₂²

#914

yes

—

119

C₂²

D₃×Q₁₆

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

#869

yes

characteristic

125

C₄

#886

yes

characteristic

126

C₄

#890

yes

—

127

C₃

#2087

yes

characteristic, complemented, Sylow

128

C₂

#4661

yes

characteristic

129

C₂

#4766

yes

—

130

C₂

#4793

yes

—

131

C₂

#4865

yes

characteristic

132

C₂

#4887

yes

characteristic

	subgroup	G/H
center	C₂³	D₃×D₄
commutator, cosocle	C₂²×C₃×C₄	C₂³
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#340	C₂
socle, prime radical	C₂³×C₃	C₂×D₄
hypercenter	#340	D₃
hypocenter	C₃	#2087
LCS	G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₃×D₄

commutator, cosocle

C₂²×C₃×C₄

C₂³

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₃×#340

C₂

socle, prime radical

C₂³×C₃

C₂×D₄

D₃

C₃

G ▷ C₂²×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
127. C₃	#2087	semidirect

H ≅ G/N

type

127. C₃

#2087

semidirect

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

subgroup	index
C₂×#818	2
C₂×Dic₃⋊C₈	2
#4984	2
C₃×#340	2
#4529	2
#2087	3

subgroup

index

C₂×Dic₃⋊C₈

Subgroups — #68839

Conjugacy classes of subgroups

Complements

Maximal subgroups