Group #83707 — subgroups

Subgroups — #83707

Order 384.

336 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 141 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×Q₈²	C₂	1	1	yes	maximal, characteristic
2	#5570	C₂	1	1	yes	maximal, characteristic
3	#5093	C₂	1	3	yes	maximal
4	#5091	C₂	1	3	yes	maximal
5	#5031	C₂	1	3	yes	maximal
6	C₄×#216	C₂	1	3	yes	maximal
7	Q₈×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
8	#3363	—	3	1	no	maximal, Sylow
9	C₃×C₄⋊Q₈	—	2	3	no	—
10	C₃×C₄⋊Q₈	C₂²	1	3	yes	—
11	C₃×C₄×Q₈	—	2	1	no	—
12	C₃×C₄×Q₈	C₂²	1	3	yes	—
13	C₃×C₄×Q₈	C₂²	1	1	yes	characteristic
14	Q₈×Dic₃	—	2	1	no	—
15	C₂×#216	C₂²	1	3	yes	—
16	#890	—	2	3	no	—
17	#869	C₂²	1	3	yes	—
18	C₄×Dic₆	C₂²	1	3	yes	—
19	#835	C₂²	1	1	yes	characteristic
20	#810	C₂²	1	3	yes	—
21	#810	C₂²	1	6	yes	—
22	#807	C₂²	1	3	yes	—
23	C₄⋊(C₃⋊C₈)	C₂²	1	3	yes	—
24	C₄×C₃⋊C₈	C₂²	1	3	yes	—
25	Q₈²	D₃	1	1	yes	characteristic
26	#557	—	3	1	no	—
27	#487	—	3	3	no	—
28	#477	—	3	3	no	—
29	#462	—	3	3	no	—
30	C₈×Q₈	—	3	1	no	—
31	C₄×Q₁₆	—	3	3	no	—
32	C₂×C₃×Q₈	—	2	3	no	—
33	C₂×C₃×Q₈	D₄	1	1	yes	characteristic
34	C₂×C₃×Q₈	C₂³	1	1	yes	characteristic
35	C₂×Dic₆	C₂³	1	3	yes	—
36	C₃×C₄⋊C₄	—	2	3	no	—
37	C₃×C₄⋊C₄	—	2	3	no	—
38	C₃×C₄⋊C₄	D₄	1	3	yes	—
39	C₃×C₄⋊C₄	C₂³	1	3	yes	—
40	C₃×C₄²	—	2	3	no	—
41	C₃×C₄²	C₂³	1	3	yes	—
42	#216	—	2	6	no	—
43	C₄⋊Dic₃	—	2	3	no	—
44	C₄⋊Dic₃	—	2	3	no	—
45	C₄⋊Dic₃	C₂³	1	1	yes	characteristic
46	Dic₃⋊C₄	—	2	3	no	—
47	C₄×Dic₃	—	2	3	no	—
48	C₂×C₃⋊C₈	C₂³	1	1	yes	characteristic
49	C₂×C₃⋊C₈	C₂³	1	3	yes	—
50	C₂×Q₁₆	—	3	3	no	—
51	C₄⋊Q₈	—	2	3	no	—
52	C₄⋊Q₈	—	3	3	no	—
53	C₄⋊Q₈	C₂×D₃	1	3	yes	—
54	#125	—	6	3	no	—
55	C₄×Q₈	—	2	1	no	—
56	C₄×Q₈	—	3	3	no	—
57	C₄×Q₈	—	6	1	no	—
58	C₄×Q₈	C₂×D₃	1	3	yes	—
59	C₄×Q₈	C₂×D₃	1	1	yes	characteristic
60	#107	—	3	3	no	—
61	C₄⋊C₈	—	3	3	no	—
62	#103	—	3	1	no	—
63	#103	—	3	3	no	—
64	#103	—	3	6	no	—
65	C₄×C₈	—	3	3	no	—
66	C₃×Q₈	—	2	1	no	—
67	C₃×Q₈	—	8	3	no	—
68	C₃×Q₈	Q₁₆	1	4	yes	complemented
69	C₃×Q₈	#41	1	2	yes	—
70	C₂×C₃×C₄	—	2	1	no	—
71	C₂×C₃×C₄	—	2	3	no	—
72	C₂×C₃×C₄	C₂×D₄	1	3	yes	—
73	C₂×C₃×C₄	C₂×D₄	1	3	yes	—
74	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
75	C₂×Dic₃	—	2	1	no	—
76	C₂×Dic₃	—	2	3	no	—
77	Dic₆	—	2	3	no	—
78	Dic₆	—	2	3	no	—
79	C₃⋊C₈	—	2	3	no	—
80	C₃⋊C₈	—	2	1	no	—
81	C₂×Q₈	—	2	3	no	—
82	C₂×Q₈	—	3	3	no	—
83	C₂×Q₈	#67	1	1	yes	characteristic
84	C₂×Q₈	C₂²×D₃	1	1	yes	characteristic
85	Q₁₆	—	6	6	no	—
86	C₂×C₈	—	3	1	no	—
87	C₂×C₈	—	3	3	no	—
88	C₄⋊C₄	—	2	3	no	—
89	C₄⋊C₄	—	2	3	no	—
90	C₄⋊C₄	—	3	1	no	—
91	C₄⋊C₄	—	6	3	no	—
92	C₄⋊C₄	—	6	3	no	—
93	C₄⋊C₄	—	6	3	no	—
94	C₄⋊C₄	#67	1	3	yes	—
95	C₄⋊C₄	C₂²×D₃	1	3	yes	—
96	C₄²	—	2	3	no	—
97	C₄²	—	6	3	no	—
98	C₄²	C₂²×D₃	1	3	yes	—
99	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
100	C₃×C₄	—	2	2	no	—
101	C₃×C₄	—	2	3	no	—
102	C₃×C₄	—	4	3	no	—
103	C₃×C₄	C₂×Q₁₆	1	6	yes	—
104	C₃×C₄	C₂×#41	1	1	yes	characteristic
105	C₃×C₄	2₋¹⁺⁴	1	1	yes	characteristic
106	Dic₃	—	4	3	no	—
107	Dic₃	—	4	1	no	—
108	Q₈	—	2	1	no	—
109	Q₈	—	6	3	no	—
110	Q₈	—	6	3	no	—
111	Q₈	—	8	3	no	—
112	Q₈	#216	1	4	yes	complemented
113	Q₈	#239	1	2	yes	—
114	C₂×C₄	—	2	1	no	—
115	C₂×C₄	—	2	3	no	—
116	C₂×C₄	—	6	1	no	—
117	C₂×C₄	—	6	3	no	—
118	C₂×C₄	C₂×#67	1	3	yes	—
119	C₂×C₄	C₂×#67	1	3	yes	—
120	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
121	C₈	—	6	3	no	—
122	C₈	—	6	1	no	—
123	C₂×C₃	#548	1	1	yes	characteristic
124	C₂×C₃	C₂²×Q₁₆	1	1	yes	characteristic
125	C₂×C₃	#577	1	1	yes	characteristic
126	C₂²	C₂²×#67	1	1	yes	characteristic
127	C₄	—	2	2	no	—
128	C₄	—	2	3	no	—
129	C₄	—	4	3	no	—
130	C₄	—	12	3	no	—
131	C₄	—	12	1	no	—
132	C₄	C₂×#216	1	6	yes	—
133	C₄	C₂×#239	1	1	yes	characteristic
134	C₄	#1007	1	1	yes	characteristic
135	C₃	#3363	1	1	yes	characteristic, complemented, Sylow
136	C₂	C₂²×#216	1	1	yes	characteristic
137	C₂	#5817	1	1	yes	characteristic
138	C₂	#5836	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×Q₈²

C₂

yes

maximal, characteristic

#5570

C₂

yes

maximal, characteristic

#5093

C₂

yes

maximal

#5091

C₂

yes

maximal

#5031

C₂

yes

maximal

C₄×#216

C₂

yes

maximal

Q₈×C₃⋊C₈

C₂

yes

maximal, characteristic

#3363

—

maximal, Sylow

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

C₂²

yes

—

C₃×C₄×Q₈

—

C₃×C₄×Q₈

C₂²

yes

—

C₃×C₄×Q₈

C₂²

yes

characteristic

Q₈×Dic₃

—

C₂×#216

C₂²

yes

—

#890

—

#869

C₂²

yes

—

C₄×Dic₆

C₂²

yes

—

#835

C₂²

yes

characteristic

#810

C₂²

yes

—

#810

C₂²

yes

—

#807

C₂²

yes

—

C₄⋊(C₃⋊C₈)

C₂²

yes

—

C₄×C₃⋊C₈

C₂²

yes

—

Q₈²

D₃

yes

—

—

—

—

C₈×Q₈

—

C₄×Q₁₆

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

D₄

yes

characteristic

C₂×C₃×Q₈

C₂³

yes

characteristic

C₂×Dic₆

C₂³

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

D₄

yes

—

C₃×C₄⋊C₄

C₂³

yes

—

C₃×C₄²

—

C₃×C₄²

C₂³

yes

—

#216

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

C₂³

yes

characteristic

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

C₂³

yes

characteristic

C₂×C₃⋊C₈

C₂³

yes

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃

yes

—

#125

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₃

yes

—

C₄×Q₈

C₂×D₃

yes

characteristic

#107

—

C₄⋊C₈

—

#103

—

#103

—

#103

—

C₄×C₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

Q₁₆

yes

complemented

C₃×Q₈

#41

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂⁴

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

#67

yes

characteristic

C₂×Q₈

C₂²×D₃

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

#67

yes

—

C₄⋊C₄

C₂²×D₃

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₃

yes

—

C₂²×C₃

C₂²×D₄

yes

characteristic

100

C₃×C₄

—

101

C₃×C₄

—

102

C₃×C₄

—

103

C₃×C₄

C₂×Q₁₆

yes

—

104

C₃×C₄

C₂×#41

yes

characteristic

105

C₃×C₄

2₋¹⁺⁴

yes

characteristic

106

Dic₃

—

107

Dic₃

—

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

Q₈

#216

yes

complemented

113

Q₈

#239

yes

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

C₂×#67

yes

—

119

C₂×C₄

C₂×#67

yes

—

120

C₂×C₄

C₂³×D₃

yes

characteristic

121

C₈

—

122

C₈

—

123

C₂×C₃

#548

yes

characteristic

124

C₂×C₃

C₂²×Q₁₆

yes

characteristic

125

C₂×C₃

#577

yes

characteristic

126

C₂²

C₂²×#67

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

C₂×#216

yes

—

133

C₄

C₂×#239

yes

characteristic

134

C₄

#1007

yes

characteristic

135

C₃

#3363

yes

characteristic, complemented, Sylow

136

C₂

C₂²×#216

yes

characteristic

137

C₂

#5817

yes

characteristic

138

C₂

#5836

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₃×Q₈²	C₂
socle, prime radical	C₂²×C₃	C₂²×D₄
hypercenter	Q₈²	D₃
hypocenter	C₃	#3363
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

subgroup

G/H

center

C₂²

C₂²×#67

commutator, cosocle

C₂×C₃×C₄

C₂⁴

Frattini

C₂×C₄

C₂³×D₃

Fitting

C₃×Q₈²

C₂

socle, prime radical

C₂²×C₃

C₂²×D₄

hypercenter

Q₈²

D₃

hypocenter

C₃

#3363

LCS

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

N	H ≅ G/N	type
68. C₃×Q₈	Q₁₆	semidirect
112. Q₈	#216	semidirect
135. C₃	#3363	semidirect

H ≅ G/N

type

68. C₃×Q₈

Q₁₆

semidirect

112. Q₈

#216

semidirect

135. C₃

#3363

semidirect

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

subgroup	index
C₃×Q₈²	2
#5570	2
#5093	2
#5091	2
#5031	2
C₄×#216	2
Q₈×C₃⋊C₈	2
#3363	3

subgroup

index

C₃×Q₈²

Q₈×C₃⋊C₈

Subgroups — #83707

Conjugacy classes of subgroups

Complements

Maximal subgroups