Group #69068 — subgroups

Subgroups — #69068

Order 384.

126 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 71 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	C₈×Dic₆	C₂	1	1	yes	maximal, characteristic
2	C₃×#346	C₂	1	1	yes	maximal, characteristic
3	#4502	C₂	1	1	yes	maximal, characteristic
4	Dic₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	Dic₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	Dic₃×C₁₆	C₂	1	1	yes	maximal, characteristic
7	C₄⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
8	#2183	—	3	1	no	maximal, Sylow
9	C₄×Dic₆	C₄	1	1	yes	characteristic
10	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₃×C₁₆	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	Dic₃⋊C₈	C₄	1	1	yes	characteristic
14	Dic₃⋊C₈	C₄	1	1	yes	characteristic
15	C₈×Dic₃	C₂²	1	1	yes	characteristic
16	C₈×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₄⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
20	C₈×Q₈	—	3	1	no	—
21	C₄⋊C₁₆	—	3	1	no	—
22	C₄⋊C₁₆	—	3	1	no	—
23	C₄⋊C₁₆	—	3	1	no	—
24	#346	—	3	1	no	—
25	#346	D₃	1	1	yes	characteristic
26	C₄×C₁₆	—	3	1	no	—
27	C₂×Dic₆	C₈	1	1	yes	characteristic
28	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₃×C₈	C₂³	1	1	yes	characteristic
30	C₃×C₄²	C₂×C₄	1	1	yes	characteristic
31	C₄⋊Dic₃	C₈	1	1	yes	characteristic
32	Dic₃⋊C₄	C₈	1	1	yes	characteristic
33	Dic₃⋊C₄	C₈	1	1	yes	characteristic
34	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
35	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
36	C₂×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
37	C₂×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
38	C₃×C₁₆	—	2	1	no	—
39	C₃×C₁₆	Q₈	1	2	yes	complemented
40	C₃⋊C₁₆	—	2	1	no	—
41	C₃⋊C₁₆	—	2	1	no	—
42	C₄×Q₈	—	3	1	no	—
43	C₂×C₁₆	—	3	1	no	—
44	C₂×C₁₆	—	3	1	no	—
45	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
46	C₂×C₁₆	C₂×D₃	1	1	yes	characteristic
47	C₄⋊C₈	—	3	1	no	—
48	C₄⋊C₈	—	3	1	no	—
49	C₄⋊C₈	—	3	1	no	—
50	C₄×C₈	—	3	1	no	—
51	C₄×C₈	—	3	1	no	—
52	C₄×C₈	C₂×D₃	1	1	yes	characteristic
53	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
54	C₂×C₃×C₄	C₂×C₈	1	1	yes	characteristic
55	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
56	C₂×Dic₃	C₂×C₈	1	1	yes	characteristic
57	C₂×Dic₃	C₂×C₈	1	1	yes	characteristic
58	C₂×Dic₃	C₂×C₈	1	1	yes	characteristic
59	C₂×Dic₃	C₂×C₈	1	1	yes	characteristic
60	Dic₆	—	2	2	no	—
61	C₃×C₈	—	2	1	no	—
62	C₃×C₈	C₂×Q₈	1	1	yes	characteristic
63	C₃×C₈	#41	1	1	yes	characteristic
64	C₃⋊C₈	—	2	1	no	—
65	C₃⋊C₈	—	2	1	no	—
66	C₂×Q₈	—	3	1	no	—
67	C₂×C₈	—	3	1	no	—
68	C₂×C₈	—	3	1	no	—
69	C₂×C₈	C₄×D₃	1	1	yes	characteristic
70	C₂×C₈	C₂²×D₃	1	1	yes	characteristic
71	C₄⋊C₄	—	3	1	no	—
72	C₄⋊C₄	—	3	1	no	—
73	C₄⋊C₄	—	3	1	no	—
74	C₄²	—	3	1	no	—
75	C₄²	—	3	1	no	—
76	C₄²	C₄×D₃	1	1	yes	characteristic
77	C₁₆	—	2	1	no	—
78	C₁₆	—	6	1	no	—
79	C₁₆	—	6	1	no	—
80	C₁₆	Dic₆	1	2	yes	complemented
81	C₂²×C₃	C₂²×C₈	1	1	yes	characteristic
82	C₃×C₄	—	2	1	no	—
83	C₃×C₄	—	2	1	no	—
84	C₃×C₄	C₄×Q₈	1	1	yes	characteristic
85	C₃×C₄	#131	1	1	yes	characteristic
86	Dic₃	—	2	1	no	—
87	Dic₃	—	2	1	no	—
88	Dic₃	—	2	1	no	—
89	Dic₃	M₃₂	1	2	yes	—
90	Q₈	—	6	2	no	—
91	C₂×C₄	—	3	1	no	—
92	C₂×C₄	—	3	1	no	—
93	C₂×C₄	—	3	1	no	—
94	C₂×C₄	—	3	1	no	—
95	C₂×C₄	D₃×C₈	1	1	yes	characteristic
96	C₂×C₄	D₃×C₈	1	1	yes	characteristic
97	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
98	C₈	—	2	1	no	—
99	C₈	—	6	1	no	—
100	C₈	—	6	1	no	—
101	C₈	C₂×Dic₆	1	1	yes	characteristic
102	C₈	#235	1	1	yes	characteristic
103	C₂×C₃	C₈×Q₈	1	1	yes	characteristic
104	C₂×C₃	C₂×M₃₂	1	1	yes	characteristic
105	C₂×C₃	#504	1	1	yes	characteristic
106	C₂²	C₂×D₃×C₈	1	1	yes	characteristic
107	C₄	—	2	1	no	—
108	C₄	—	2	1	no	—
109	C₄	—	3	2	no	—
110	C₄	—	6	1	no	—
111	C₄	—	6	1	no	—
112	C₄	—	6	1	no	—
113	C₄	C₄×Dic₆	1	1	yes	characteristic
114	C₄	#901	1	1	yes	characteristic
115	C₃	#2183	1	1	yes	characteristic, complemented, Sylow
116	C₂	C₈×Dic₆	1	1	yes	characteristic
117	C₂	D₃×M₃₂	1	1	yes	characteristic
118	C₂	#4907	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₆

C₂

yes

maximal, characteristic

C₃×#346

C₂

yes

maximal, characteristic

#4502

C₂

yes

maximal, characteristic

Dic₃⋊C₁₆

C₂

yes

maximal, characteristic

Dic₃⋊C₁₆

C₂

yes

maximal, characteristic

Dic₃×C₁₆

C₂

yes

maximal, characteristic

C₄⋊(C₃⋊C₁₆)

C₂

yes

maximal, characteristic

#2183

—

maximal, Sylow

C₄×Dic₆

C₄

yes

characteristic

C₂×C₃×C₁₆

C₂²

yes

characteristic

C₂×C₃×C₁₆

C₂²

yes

characteristic

C₃×C₄×C₈

C₂²

yes

characteristic

Dic₃⋊C₈

C₄

yes

characteristic

Dic₃⋊C₈

C₄

yes

characteristic

C₈×Dic₃

C₂²

yes

characteristic

C₈×Dic₃

C₂²

yes

characteristic

C₂×C₃⋊C₁₆

C₂²

yes

characteristic

C₂×C₃⋊C₁₆

C₂²

yes

characteristic

C₄⋊(C₃⋊C₈)

C₄

yes

characteristic

C₈×Q₈

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

#346

—

#346

D₃

yes

characteristic

C₄×C₁₆

—

C₂×Dic₆

C₈

yes

characteristic

C₂×C₃×C₈

C₂×C₄

yes

characteristic

C₂×C₃×C₈

C₂³

yes

characteristic

C₃×C₄²

C₂×C₄

yes

characteristic

C₄⋊Dic₃

C₈

yes

characteristic

Dic₃⋊C₄

C₈

yes

characteristic

Dic₃⋊C₄

C₈

yes

characteristic

C₄×Dic₃

C₂×C₄

yes

characteristic

C₄×Dic₃

C₂×C₄

yes

characteristic

C₂×C₃⋊C₈

C₂×C₄

yes

characteristic

C₂×C₃⋊C₈

C₂×C₄

yes

characteristic

C₃×C₁₆

—

C₃×C₁₆

Q₈

yes

complemented

C₃⋊C₁₆

—

C₃⋊C₁₆

—

C₄×Q₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₃

yes

characteristic

C₂×C₁₆

C₂×D₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₃

yes

characteristic

C₂×C₃×C₄

C₂×C₈

yes

characteristic

C₂×C₃×C₄

C₂×C₈

yes

characteristic

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×Dic₃

C₂×C₈

yes

characteristic

C₂×Dic₃

C₂×C₈

yes

characteristic

C₂×Dic₃

C₂×C₈

yes

characteristic

C₂×Dic₃

C₂×C₈

yes

characteristic

Dic₆

—

C₃×C₈

—

C₃×C₈

C₂×Q₈

yes

characteristic

C₃×C₈

#41

yes

characteristic

C₃⋊C₈

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₃

yes

characteristic

C₂×C₈

C₂²×D₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₄×D₃

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

Dic₆

yes

complemented

C₂²×C₃

C₂²×C₈

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₄×Q₈

yes

characteristic

C₃×C₄

#131

yes

characteristic

Dic₃

—

Dic₃

—

Dic₃

—

Dic₃

M₃₂

yes

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₃×C₈

yes

characteristic

C₂×C₄

D₃×C₈

yes

characteristic

C₂×C₄

C₂×C₄×D₃

yes

characteristic

C₈

—

C₈

—

100

C₈

—

101

C₈

C₂×Dic₆

yes

characteristic

102

C₈

#235

yes

characteristic

103

C₂×C₃

C₈×Q₈

yes

characteristic

104

C₂×C₃

C₂×M₃₂

yes

characteristic

105

C₂×C₃

#504

yes

characteristic

106

C₂²

C₂×D₃×C₈

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

C₄×Dic₆

yes

characteristic

114

C₄

#901

yes

characteristic

115

C₃

#2183

yes

characteristic, complemented, Sylow

116

C₂

C₈×Dic₆

yes

characteristic

117

C₂

D₃×M₃₂

yes

characteristic

118

C₂

#4907

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₃
commutator, socle, prime radical	C₂²×C₃	C₂²×C₈
Fitting	C₃×#346	C₂
cosocle	C₂×C₃×C₈	C₂³
hypercenter	#346	D₃
hypocenter	C₃	#2183
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₈ ◁ #346

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₃

commutator, socle, prime radical

C₂²×C₃

C₂²×C₈

Fitting

C₃×#346

C₂

cosocle

C₂×C₃×C₈

C₂³

D₃

C₃

G ▷ C₂²×C₃ ▷ C₃

C₁ ◁ C₂×C₈ ◁ #346

N	H ≅ G/N	type
39. C₃×C₁₆	Q₈	semidirect
80. C₁₆	Dic₆	semidirect
115. C₃	#2183	semidirect

H ≅ G/N

type

39. C₃×C₁₆

Q₈

semidirect

80. C₁₆

Dic₆

semidirect

115. C₃

#2183

semidirect

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

subgroup	index
C₈×Dic₆	2
C₃×#346	2
#4502	2
Dic₃⋊C₁₆	2
Dic₃×C₁₆	2
C₄⋊(C₃⋊C₁₆)	2
#2183	3

subgroup

index

C₈×Dic₆

C₃×#346

#4502

Dic₃⋊C₁₆

Dic₃×C₁₆

C₄⋊(C₃⋊C₁₆)

#2183

Subgroups — #69068

Conjugacy classes of subgroups

Complements

Maximal subgroups