Group #68074 — subgroups

Subgroups — #68074

Order 384.

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

normal subgroups only

This group has 178 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	#4685	C₂	1	1	yes	maximal, characteristic
2	#4685	C₂	1	1	yes	maximal, characteristic
3	#4685	C₂	1	1	yes	maximal, characteristic
4	C₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
5	#4455	C₂	1	1	yes	maximal, characteristic, complemented
6	#4454	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#1454	—	3	1	no	maximal, Sylow
9	C₄²×D₃	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
13	(C₂×D₃)⋊C₈	C₄	1	2	yes	—
14	(C₂×D₃)⋊C₈	C₄	1	2	yes	—
15	(C₂×D₃)⋊C₈	C₄	1	2	yes	—
16	Dic₃⋊C₈	C₄	1	2	yes	—
17	Dic₃⋊C₈	C₄	1	2	yes	—
18	Dic₃⋊C₈	C₄	1	2	yes	—
19	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
20	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
21	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
22	#431	—	3	1	no	—
23	#431	—	3	1	no	—
24	#431	—	3	1	no	—
25	#322	—	3	1	no	—
26	#322	—	3	1	no	—
27	#322	D₃	1	1	yes	characteristic, complemented
28	C₈²	—	3	1	no	—
29	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄×D₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
34	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
35	C₃×C₄²	C₂³	1	1	yes	characteristic
36	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
37	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
38	C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
40	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
41	C₂×C₃⋊C₈	C₂×C₄	1	2	yes	—
42	C₂×C₄²	—	3	1	no	—
43	C₄⋊C₈	—	3	2	no	—
44	C₄⋊C₈	—	3	2	no	—
45	C₄⋊C₈	—	3	2	no	—
46	C₂²⋊C₈	—	3	2	no	—
47	C₂²⋊C₈	—	3	2	no	—
48	C₂²⋊C₈	—	3	2	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	C₄×C₈	C₂×D₃	1	1	yes	characteristic
54	C₄×C₈	C₂×D₃	1	1	yes	characteristic
55	C₂²×D₃	C₄²	1	1	yes	characteristic
56	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₂×Dic₃	C₄²	1	1	yes	characteristic
60	C₂×Dic₃	C₄²	1	1	yes	characteristic
61	C₂×Dic₃	C₄²	1	1	yes	characteristic
62	C₄×D₃	—	2	2	no	—
63	C₄×D₃	—	2	2	no	—
64	C₄×D₃	—	2	2	no	—
65	C₃×C₈	—	2	2	no	—
66	C₃×C₈	—	2	2	no	—
67	C₃×C₈	—	2	2	no	—
68	C₃⋊C₈	—	2	2	no	—
69	C₃⋊C₈	—	2	2	no	—
70	C₃⋊C₈	—	2	2	no	—
71	C₂²×C₄	—	3	1	no	—
72	C₂²×C₄	—	3	1	no	—
73	C₂²×C₄	—	3	1	no	—
74	C₂×C₈	—	3	2	no	—
75	C₂×C₈	—	3	2	no	—
76	C₂×C₈	—	3	2	no	—
77	C₂×C₈	C₄×D₃	1	2	yes	—
78	C₂×C₈	C₄×D₃	1	2	yes	—
79	C₂×C₈	C₄×D₃	1	2	yes	—
80	C₄²	—	3	1	no	—
81	C₄²	—	3	1	no	—
82	C₄²	—	3	1	no	—
83	C₄²	C₂²×D₃	1	1	yes	characteristic
84	C₂²×C₃	C₂×C₄²	1	1	yes	characteristic
85	C₂×D₃	—	2	1	no	—
86	C₂×D₃	—	2	1	no	—
87	C₂×D₃	—	2	1	no	—
88	C₃×C₄	#131	1	2	yes	—
89	C₃×C₄	#131	1	2	yes	—
90	C₃×C₄	#131	1	2	yes	—
91	Dic₃	—	2	1	no	—
92	Dic₃	—	2	1	no	—
93	Dic₃	—	2	1	no	—
94	C₂³	—	3	1	no	—
95	C₂×C₄	—	3	1	no	—
96	C₂×C₄	—	3	1	no	—
97	C₂×C₄	—	3	1	no	—
98	C₂×C₄	—	6	2	no	—
99	C₂×C₄	—	6	2	no	—
100	C₂×C₄	—	6	2	no	—
101	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
102	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
103	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
104	C₈	—	2	2	no	—
105	C₈	—	2	2	no	—
106	C₈	—	2	2	no	—
107	C₈	—	6	2	no	—
108	C₈	—	6	2	no	—
109	C₈	—	6	2	no	—
110	C₂×C₃	#405	1	1	yes	characteristic
111	C₂×C₃	#405	1	1	yes	characteristic
112	C₂×C₃	#405	1	1	yes	characteristic
113	D₃	—	4	1	no	—
114	C₂²	—	6	1	no	—
115	C₂²	—	6	1	no	—
116	C₂²	—	6	1	no	—
117	C₂²	C₄²×D₃	1	1	yes	characteristic
118	C₄	—	6	1	no	—
119	C₄	—	6	1	no	—
120	C₄	—	6	1	no	—
121	C₄	#901	1	2	yes	—
122	C₄	#907	1	2	yes	—
123	C₄	#907	1	2	yes	—
124	C₃	#1454	1	1	yes	characteristic, complemented, Sylow
125	C₂	—	12	1	no	—
126	C₂	#4689	1	1	yes	characteristic
127	C₂	#4708	1	1	yes	characteristic
128	C₂	#4708	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#4685

C₂

yes

maximal, characteristic

#4685

C₂

yes

maximal, characteristic

#4685

C₂

yes

maximal, characteristic

C₃×#322

C₂

yes

maximal, characteristic, complemented

#4455

C₂

yes

maximal, characteristic, complemented

#4454

C₂

yes

maximal, characteristic, complemented

C₈×C₃⋊C₈

C₂

yes

maximal, characteristic, complemented

#1454

—

maximal, Sylow

C₄²×D₃

C₂²

yes

characteristic

C₃×C₄×C₈

C₂²

yes

characteristic

C₃×C₄×C₈

C₂²

yes

characteristic

C₃×C₄×C₈

C₂²

yes

characteristic

(C₂×D₃)⋊C₈

C₄

yes

—

(C₂×D₃)⋊C₈

C₄

yes

—

(C₂×D₃)⋊C₈

C₄

yes

—

Dic₃⋊C₈

C₄

yes

—

Dic₃⋊C₈

C₄

yes

—

Dic₃⋊C₈

C₄

yes

—

C₄×C₃⋊C₈

C₂²

yes

characteristic

C₄×C₃⋊C₈

C₂²

yes

characteristic

C₄×C₃⋊C₈

C₂²

yes

—

—

—

—

—

D₃

yes

characteristic, complemented

C₈²

—

C₂×C₄×D₃

C₂×C₄

yes

characteristic

C₂×C₄×D₃

C₂×C₄

yes

characteristic

C₂×C₄×D₃

C₂×C₄

yes

characteristic

C₂×C₃×C₈

C₂×C₄

yes

—

C₂×C₃×C₈

C₂×C₄

yes

—

C₂×C₃×C₈

C₂×C₄

yes

—

C₃×C₄²

C₂³

yes

characteristic

C₄×Dic₃

C₂×C₄

yes

characteristic

C₄×Dic₃

C₂×C₄

yes

characteristic

C₄×Dic₃

C₂×C₄

yes

characteristic

C₂×C₃⋊C₈

C₂×C₄

yes

—

C₂×C₃⋊C₈

C₂×C₄

yes

—

C₂×C₃⋊C₈

C₂×C₄

yes

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₃

yes

characteristic

C₄×C₈

C₂×D₃

yes

characteristic

C₄×C₈

C₂×D₃

yes

characteristic

C₂²×D₃

C₄²

yes

characteristic

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×Dic₃

C₄²

yes

characteristic

C₂×Dic₃

C₄²

yes

characteristic

C₂×Dic₃

C₄²

yes

characteristic

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₃

yes

—

C₂×C₈

C₄×D₃

yes

—

C₂×C₈

C₄×D₃

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

C₂×C₄²

yes

characteristic

C₂×D₃

—

C₂×D₃

—

C₂×D₃

—

C₃×C₄

#131

yes

—

C₃×C₄

#131

yes

—

C₃×C₄

#131

yes

—

Dic₃

—

Dic₃

—

Dic₃

—

C₂³

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×C₄×D₃

yes

characteristic

102

C₂×C₄

C₂×C₄×D₃

yes

characteristic

103

C₂×C₄

C₂×C₄×D₃

yes

characteristic

104

C₈

—

105

C₈

—

106

C₈

—

107

C₈

—

108

C₈

—

109

C₈

—

110

C₂×C₃

#405

yes

characteristic

111

C₂×C₃

#405

yes

characteristic

112

C₂×C₃

#405

yes

characteristic

113

D₃

—

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

C₄²×D₃

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

#901

yes

—

122

C₄

#907

yes

—

123

C₄

#907

yes

—

124

C₃

#1454

yes

characteristic, complemented, Sylow

125

C₂

—

126

C₂

#4689

yes

characteristic

127

C₂

#4708

yes

characteristic

128

C₂

#4708

yes

characteristic

	subgroup	G/H
center, Frattini	C₄²	C₂²×D₃
commutator, socle	C₂²×C₃	C₂×C₄²
Fitting	C₃×#322	C₂
prime radical	C₂²×D₃	C₄²
cosocle	C₃×C₄²	C₂³
hypercenter	#322	D₃
hypocenter	C₃	#1454
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₄² ◁ #322

subgroup

G/H

center, Frattini

C₄²

C₂²×D₃

commutator, socle

C₂²×C₃

C₂×C₄²

Fitting

C₃×#322

C₂

prime radical

C₂²×D₃

C₄²

cosocle

C₃×C₄²

C₂³

D₃

C₃

G ▷ C₂²×C₃ ▷ C₃

C₁ ◁ C₄² ◁ #322

N	H ≅ G/N	type
4. C₃×#322	C₂	semidirect
5. #4455	C₂	semidirect
6. #4454	C₂	semidirect
7. C₈×C₃⋊C₈	C₂	semidirect
27. #322	D₃	semidirect
124. C₃	#1454	semidirect

H ≅ G/N

type

4. C₃×#322

C₂

semidirect

5. #4455

C₂

semidirect

6. #4454

C₂

semidirect

7. C₈×C₃⋊C₈

C₂

semidirect

27. #322

D₃

semidirect

124. C₃

#1454

semidirect

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

subgroup	index
#4685	2
C₃×#322	2
#4455	2
#4454	2
C₈×C₃⋊C₈	2
#1454	3

subgroup

index

C₈×C₃⋊C₈

Subgroups — #68074

Conjugacy classes of subgroups

Complements

Maximal subgroups