Group #83340 — subgroups

Subgroups — #83340

Order 384.

348 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 149 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×#533	C₂	1	1	yes	maximal, characteristic
2	#5470	C₂	1	1	yes	maximal, characteristic
3	#5081	C₂	1	4	yes	maximal, complemented
4	#5064	C₂	1	4	yes	maximal, complemented
5	#4998	C₂	1	4	yes	maximal
6	C₂×#803	C₂	1	1	yes	maximal, characteristic
7	#3162	—	3	1	no	maximal, Sylow
8	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₃×#125	C₂²	1	2	yes	—
10	C₃×C₂²⋊Q₈	—	2	2	no	—
11	C₃×C₄×Q₈	—	2	2	no	—
12	C₃×#117	C₂²	1	2	yes	—
13	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
14	#926	C₂²	1	2	yes	—
15	#924	—	2	2	no	—
16	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
17	#870	C₂²	1	2	yes	—
18	#869	C₂²	1	2	yes	—
19	C₄×Dic₆	—	2	2	no	—
20	#808	C₂²	1	8	yes	—
21	#807	C₂²	1	8	yes	—
22	#803	C₂²	1	2	yes	—
23	#803	C₂²	1	2	yes	—
24	#533	—	3	1	no	—
25	#533	D₃	1	1	yes	characteristic
26	#501	—	3	4	no	—
27	#501	—	3	4	no	—
28	#427	—	3	4	no	—
29	C₂×#97	—	3	1	no	—
30	C₂×C₃×Q₈	—	2	2	no	—
31	C₂²×C₃×C₄	D₄	1	2	yes	—
32	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂×Dic₆	—	2	2	no	—
34	C₃×C₄⋊C₄	—	2	4	no	—
35	C₃×C₄⋊C₄	—	2	2	no	—
36	C₃×C₄⋊C₄	C₂³	1	4	yes	—
37	C₃×C₂²⋊C₄	—	2	2	no	—
38	C₃×C₄²	—	2	2	no	—
39	C₃×C₄²	D₄	1	2	yes	—
40	C₃×C₄²	C₂³	1	1	yes	characteristic
41	C₃×C₄²	C₂³	1	1	yes	characteristic
42	#217	—	2	2	no	—
43	C₄⋊Dic₃	—	2	2	no	—
44	C₄⋊Dic₃	C₂³	1	4	yes	—
45	Dic₃⋊C₄	—	2	4	no	—
46	C₄×Dic₃	—	2	2	no	—
47	C₂×C₃⋊C₈	Q₈	1	8	yes	complemented
48	C₂×C₃⋊C₈	C₂³	1	4	yes	—
49	C₂²×C₈	—	3	2	no	—
50	C₄⋊Q₈	—	3	2	no	—
51	C₄⋊Q₈	C₂×D₃	1	2	yes	—
52	#125	—	3	2	no	—
53	#125	C₂×D₃	1	2	yes	—
54	C₂²⋊Q₈	—	2	2	no	—
55	C₂²⋊Q₈	—	6	2	no	—
56	C₄×Q₈	—	2	2	no	—
57	C₄×Q₈	—	6	2	no	—
58	#117	—	3	2	no	—
59	#117	C₂×D₃	1	2	yes	—
60	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
61	#107	—	3	8	no	—
62	#106	—	3	8	no	—
63	#97	—	3	2	no	—
64	#97	—	3	2	no	—
65	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
66	C₃×Q₈	—	4	4	no	—
67	C₂×C₃×C₄	—	2	4	no	—
68	C₂×C₃×C₄	—	4	2	no	—
69	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
70	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
71	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
72	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
73	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
74	C₂×Dic₃	—	2	4	no	—
75	Dic₆	—	4	4	no	—
76	C₃⋊C₈	C₂×Q₈	1	8	yes	—
77	C₂×Q₈	—	2	2	no	—
78	C₂×Q₈	—	6	2	no	—
79	C₂²×C₄	#67	1	2	yes	—
80	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
81	C₂×C₈	—	3	8	no	—
82	C₂×C₈	—	3	4	no	—
83	C₄⋊C₄	—	2	4	no	—
84	C₄⋊C₄	—	2	2	no	—
85	C₄⋊C₄	—	3	4	no	—
86	C₄⋊C₄	—	6	4	no	—
87	C₄⋊C₄	—	6	2	no	—
88	C₄⋊C₄	C₂²×D₃	1	4	yes	—
89	C₂²⋊C₄	—	2	2	no	—
90	C₂²⋊C₄	—	6	2	no	—
91	C₄²	—	2	2	no	—
92	C₄²	—	6	2	no	—
93	C₄²	#67	1	2	yes	—
94	C₄²	C₂²×D₃	1	1	yes	characteristic
95	C₄²	C₂²×D₃	1	1	yes	characteristic
96	C₂²×C₃	—	2	2	no	—
97	C₂²×C₃	C₄⋊Q₈	1	2	yes	—
98	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
99	C₃×C₄	—	2	2	no	—
100	C₃×C₄	—	2	2	no	—
101	C₃×C₄	—	4	4	no	—
102	C₃×C₄	C₄⋊Q₈	1	2	yes	—
103	C₃×C₄	C₂²×Q₈	1	2	yes	—
104	Dic₃	—	4	4	no	—
105	C₂³	C₂×#67	1	1	yes	characteristic
106	Q₈	—	4	4	no	—
107	Q₈	—	12	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	6	4	no	—
111	C₂×C₄	D₃×Q₈	1	4	yes	—
112	C₂×C₄	C₂×#67	1	2	yes	—
113	C₂×C₄	C₂×#67	1	2	yes	—
114	C₂×C₄	C₂×#67	1	1	yes	characteristic
115	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
116	C₈	—	3	8	no	—
117	C₂×C₃	—	2	2	no	—
118	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
119	C₂×C₃	#575	1	2	yes	—
120	C₂²	—	2	2	no	—
121	C₂²	#944	1	2	yes	—
122	C₂²	C₂²×#67	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	4	4	no	—
126	C₄	—	12	4	no	—
127	C₄	#944	1	2	yes	—
128	C₄	C₂×D₃×Q₈	1	2	yes	—
129	C₃	#3162	1	1	yes	characteristic, complemented, Sylow
130	C₂	—	2	2	no	—
131	C₂	C₂×#944	1	1	yes	characteristic
132	C₂	#5822	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#533

C₂

yes

maximal, characteristic

#5470

C₂

yes

maximal, characteristic

#5081

C₂

yes

maximal, complemented

#5064

C₂

yes

maximal, complemented

#4998

C₂

yes

maximal

C₂×#803

C₂

yes

maximal, characteristic

#3162

—

maximal, Sylow

C₃×C₄⋊Q₈

C₂²

yes

—

C₃×#125

C₂²

yes

—

C₃×C₂²⋊Q₈

—

C₃×C₄×Q₈

—

C₃×#117

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

#926

C₂²

yes

—

#924

—

C₂²×C₃⋊C₈

C₂²

yes

—

#870

C₂²

yes

—

#869

C₂²

yes

—

C₄×Dic₆

—

#808

C₂²

yes

—

#807

C₂²

yes

—

#803

C₂²

yes

—

#803

C₂²

yes

—

#533

—

#533

D₃

yes

—

—

—

C₂×#97

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂×Dic₆

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂³

yes

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₃×C₄²

D₄

yes

—

C₃×C₄²

C₂³

yes

characteristic

C₃×C₄²

C₂³

yes

characteristic

#217

—

C₄⋊Dic₃

—

C₄⋊Dic₃

C₂³

yes

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

Q₈

yes

complemented

C₂×C₃⋊C₈

C₂³

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃

yes

—

#125

—

#125

C₂×D₃

yes

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

C₂×D₃

yes

—

C₂×C₄²

C₂×D₃

yes

—

—

—

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

C₂⁴

yes

characteristic

C₂×Dic₃

—

Dic₆

—

C₃⋊C₈

C₂×Q₈

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

#67

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×D₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

#67

yes

—

C₄²

C₂²×D₃

yes

characteristic

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

—

C₂²×C₃

C₄⋊Q₈

yes

—

C₂²×C₃

C₂²×D₄

yes

characteristic

C₃×C₄

—

100

C₃×C₄

—

101

C₃×C₄

—

102

C₃×C₄

C₄⋊Q₈

yes

—

103

C₃×C₄

C₂²×Q₈

yes

—

104

Dic₃

—

105

C₂³

C₂×#67

yes

characteristic

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

D₃×Q₈

yes

—

112

C₂×C₄

C₂×#67

yes

—

113

C₂×C₄

C₂×#67

yes

—

114

C₂×C₄

C₂×#67

yes

characteristic

115

C₂×C₄

C₂³×D₃

yes

characteristic

116

C₈

—

117

C₂×C₃

—

118

C₂×C₃

C₂×C₄⋊Q₈

yes

characteristic

119

C₂×C₃

#575

yes

—

120

C₂²

—

121

C₂²

#944

yes

—

122

C₂²

C₂²×#67

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

#944

yes

—

128

C₄

C₂×D₃×Q₈

yes

—

129

C₃

#3162

yes

characteristic, complemented, Sylow

130

C₂

—

131

C₂

C₂×#944

yes

characteristic

132

C₂

#5822

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×#533	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂³×C₃	C₂×D₄
hypercenter	#533	D₃
hypocenter	C₃	#3162
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂×C₄

C₂×#67

commutator, cosocle

C₂×C₃×C₄

C₂⁴

Frattini

C₂×C₄

C₂³×D₃

Fitting

C₃×#533

C₂

socle

C₂²×C₃

C₂²×D₄

prime radical

C₂³×C₃

C₂×D₄

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
3. #5081	C₂	semidirect
4. #5064	C₂	semidirect
47. C₂×C₃⋊C₈	Q₈	semidirect
129. C₃	#3162	semidirect

H ≅ G/N

type

3. #5081

C₂

semidirect

4. #5064

C₂

semidirect

47. C₂×C₃⋊C₈

Q₈

semidirect

129. C₃

#3162

semidirect

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

subgroup	index
C₃×#533	2
#5470	2
#5081	2
#5064	2
#4998	2
C₂×#803	2
#3162	3

subgroup

index

Subgroups — #83340

Conjugacy classes of subgroups

Complements

Maximal subgroups