Group #74355 — subgroups

Subgroups — #74355

Order 384.

472 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 197 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×#890	C₂	1	2	yes	maximal
2	C₃×#384	C₂	1	1	yes	maximal, characteristic
3	#4984	C₂	1	4	yes	maximal
4	#4977	C₂	1	2	yes	maximal
5	#4972	C₂	1	4	yes	maximal
6	#4935	C₂	1	1	yes	maximal, characteristic
7	C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic
8	#2859	—	3	1	no	maximal, Sylow
9	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
10	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₃	C₂²	1	4	yes	—
13	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₃	C₂²	1	4	yes	—
15	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
16	C₂×C₄×Dic₃	C₂²	1	2	yes	—
17	C₂×C₄×Dic₃	C₂²	1	2	yes	—
18	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
20	#890	—	2	8	no	—
21	C₃⋊#95	C₂²	1	8	yes	—
22	C₂×#125	—	3	2	no	—
23	#389	—	3	4	no	—
24	#389	—	3	4	no	—
25	#384	—	3	1	no	—
26	#384	D₃	1	1	yes	characteristic
27	#382	—	3	2	no	—
28	C₄³	—	3	1	no	—
29	C₂²×C₃×C₄	C₂³	1	4	yes	—
30	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
31	C₂²×C₃×C₄	C₂³	1	2	yes	—
32	C₂²×Dic₃	C₂³	1	4	yes	—
33	C₂²×Dic₃	C₂³	1	1	yes	characteristic
34	C₂²×Dic₃	C₂³	1	1	yes	characteristic
35	C₂²×Dic₃	C₂³	1	2	yes	—
36	C₃×C₄⋊C₄	—	2	4	no	—
37	C₃×C₄⋊C₄	—	2	8	no	—
38	C₃×C₄²	Q₈	1	4	yes	—
39	C₄⋊Dic₃	—	2	4	no	—
40	C₄⋊Dic₃	—	2	8	no	—
41	C₄⋊Dic₃	—	2	8	no	—
42	Dic₃⋊C₄	—	2	8	no	—
43	C₄×Dic₃	—	2	4	no	—
44	C₄×Dic₃	—	2	4	no	—
45	C₄×Dic₃	D₄	1	4	yes	—
46	C₄×Dic₃	Q₈	1	4	yes	—
47	#125	—	6	8	no	—
48	C₂×C₄⋊C₄	—	3	2	no	—
49	C₂×C₄⋊C₄	—	3	4	no	—
50	C₂×C₄⋊C₄	—	3	4	no	—
51	C₂×C₄⋊C₄	—	3	4	no	—
52	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
53	C₂×C₄⋊C₄	C₂×D₃	1	4	yes	—
54	C₂×C₄²	—	3	1	no	—
55	C₂×C₄²	—	3	1	no	—
56	C₂×C₄²	—	3	2	no	—
57	C₂×C₄²	—	3	2	no	—
58	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
59	#95	—	3	8	no	—
60	C₂³×C₃	C₂⁴	1	1	yes	characteristic
61	C₂×C₃×C₄	—	2	4	no	—
62	C₂×C₃×C₄	—	2	8	no	—
63	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
64	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
65	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
66	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
67	C₂×C₃×C₄	#41	1	4	yes	—
68	C₂×C₃×C₄	#41	1	4	yes	—
69	C₂×Dic₃	—	2	2	no	—
70	C₂×Dic₃	—	2	2	no	—
71	C₂×Dic₃	—	2	2	no	—
72	C₂×Dic₃	—	2	2	no	—
73	C₂×Dic₃	—	2	4	no	—
74	C₂×Dic₃	—	2	8	no	—
75	C₂×Dic₃	C₂×D₄	1	2	yes	—
76	C₂×Dic₃	C₂×D₄	1	2	yes	—
77	C₂×Dic₃	C₂×Q₈	1	4	yes	—
78	C₂²×C₄	—	3	2	no	—
79	C₂²×C₄	—	3	1	no	—
80	C₂²×C₄	—	3	1	no	—
81	C₂²×C₄	—	3	4	no	—
82	C₂²×C₄	C₂²×D₃	1	4	yes	—
83	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
84	C₂²×C₄	C₂²×D₃	1	2	yes	—
85	C₄⋊C₄	—	2	4	no	—
86	C₄⋊C₄	—	2	8	no	—
87	C₄⋊C₄	—	6	8	no	—
88	C₄⋊C₄	—	6	8	no	—
89	C₄⋊C₄	—	6	4	no	—
90	C₄⋊C₄	—	6	8	no	—
91	C₄²	—	3	4	no	—
92	C₄²	—	3	4	no	—
93	C₄²	—	6	4	no	—
94	C₄²	—	6	4	no	—
95	C₄²	Dic₆	1	4	yes	—
96	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
97	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
98	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
99	C₂²×C₃	C₂×#41	1	2	yes	—
100	C₂²×C₃	C₂×#41	1	2	yes	—
101	C₃×C₄	—	4	4	no	—
102	C₃×C₄	#125	1	8	yes	complemented
103	C₃×C₄	C₄⋊Q₈	1	4	yes	—
104	Dic₃	—	2	2	no	—
105	Dic₃	—	2	2	no	—
106	Dic₃	—	2	4	no	—
107	Dic₃	—	4	4	no	—
108	C₂³	C₂³×D₃	1	1	yes	characteristic
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	8	no	—
111	C₂×C₄	—	3	2	no	—
112	C₂×C₄	—	3	2	no	—
113	C₂×C₄	—	3	4	no	—
114	C₂×C₄	—	6	2	no	—
115	C₂×C₄	—	6	2	no	—
116	C₂×C₄	—	6	2	no	—
117	C₂×C₄	—	6	2	no	—
118	C₂×C₄	—	6	4	no	—
119	C₂×C₄	—	6	8	no	—
120	C₂×C₄	C₂×Dic₆	1	2	yes	—
121	C₂×C₄	C₂×Dic₆	1	4	yes	—
122	C₂×C₄	D₃×D₄	1	2	yes	—
123	C₂×C₄	#237	1	4	yes	—
124	C₂×C₄	D₃×Q₈	1	2	yes	—
125	C₂×C₄	#239	1	4	yes	—
126	C₂×C₃	C₂×#125	1	2	yes	—
127	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
128	C₂×C₃	#532	1	1	yes	characteristic
129	C₂×C₃	#532	1	1	yes	characteristic
130	C₂×C₃	#533	1	2	yes	—
131	C₂²	C₂²×Dic₆	1	1	yes	characteristic
132	C₂²	C₂×D₃×D₄	1	1	yes	characteristic
133	C₂²	C₂×#237	1	2	yes	—
134	C₂²	C₂×D₃×Q₈	1	1	yes	characteristic
135	C₂²	C₂×#239	1	2	yes	—
136	C₄	—	4	4	no	—
137	C₄	—	6	2	no	—
138	C₄	—	6	2	no	—
139	C₄	—	6	4	no	—
140	C₄	—	12	4	no	—
141	C₄	#888	1	4	yes	—
142	C₄	#890	1	8	yes	complemented
143	C₃	#2859	1	1	yes	characteristic, complemented, Sylow
144	C₂	C₂×#888	1	1	yes	characteristic
145	C₂	C₂×#890	1	2	yes	—
146	C₂	#5716	1	1	yes	characteristic
147	C₂	#5725	1	1	yes	characteristic
148	C₂	#5728	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#890

C₂

yes

maximal

C₃×#384

C₂

yes

maximal, characteristic

#4984

C₂

yes

maximal

#4977

C₂

yes

maximal

#4972

C₂

yes

maximal

#4935

C₂

yes

maximal, characteristic

C₄²×Dic₃

C₂

yes

maximal, characteristic

#2859

—

maximal, Sylow

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×Dic₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

characteristic

C₂×C₄×Dic₃

C₂²

yes

characteristic

#890

—

C₃⋊#95

C₂²

yes

—

—

—

—

—

D₃

yes

characteristic

#382

—

C₄³

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

characteristic

C₂²×Dic₃

C₂³

yes

characteristic

C₂²×Dic₃

C₂³

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

Q₈

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

D₄

yes

—

C₄×Dic₃

Q₈

yes

—

#125

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃

yes

characteristic

#95

—

C₂³×C₃

C₂⁴

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

C₂×D₄

yes

—

C₂×Dic₃

C₂×D₄

yes

—

C₂×Dic₃

C₂×Q₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂²×C₄

C₂²×D₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₄²

Dic₆

yes

—

C₂²×C₃

C₂²×D₄

yes

characteristic

C₂²×C₃

C₂²×Q₈

yes

characteristic

C₂²×C₃

C₂²×Q₈

yes

characteristic

C₂²×C₃

C₂×#41

yes

—

100

C₂²×C₃

C₂×#41

yes

—

101

C₃×C₄

—

102

C₃×C₄

#125

yes

complemented

103

C₃×C₄

C₄⋊Q₈

yes

—

104

Dic₃

—

105

Dic₃

—

106

Dic₃

—

107

Dic₃

—

108

C₂³

C₂³×D₃

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂×Dic₆

yes

—

121

C₂×C₄

C₂×Dic₆

yes

—

122

C₂×C₄

D₃×D₄

yes

—

123

C₂×C₄

#237

yes

—

124

C₂×C₄

D₃×Q₈

yes

—

125

C₂×C₄

#239

yes

—

126

C₂×C₃

C₂×#125

yes

—

127

C₂×C₃

C₂×C₄⋊Q₈

yes

characteristic

128

C₂×C₃

#532

yes

characteristic

129

C₂×C₃

#532

yes

characteristic

130

C₂×C₃

#533

yes

—

131

C₂²

C₂²×Dic₆

yes

characteristic

132

C₂²

C₂×D₃×D₄

yes

characteristic

133

C₂²

C₂×#237

yes

—

134

C₂²

C₂×D₃×Q₈

yes

characteristic

135

C₂²

C₂×#239

yes

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#888

yes

—

142

C₄

#890

yes

complemented

143

C₃

#2859

yes

characteristic, complemented, Sylow

144

C₂

C₂×#888

yes

characteristic

145

C₂

C₂×#890

yes

—

146

C₂

#5716

yes

characteristic

147

C₂

#5725

yes

characteristic

148

C₂

#5728

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂³×C₃	C₂⁴
Fitting	C₃×#384	C₂
hypercenter	#384	D₃
hypocenter	C₃	#2859
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃

commutator, socle, prime radical, cosocle

C₂³×C₃

C₂⁴

C₂

D₃

C₃

G ▷ C₂³×C₃ ▷ C₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
102. C₃×C₄	#125	semidirect
142. C₄	#890	semidirect
143. C₃	#2859	semidirect

H ≅ G/N

type

102. C₃×C₄

#125

semidirect

142. C₄

#890

semidirect

143. C₃

#2859

semidirect

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

subgroup	index
C₂×#890	2
C₃×#384	2
#4984	2
#4977	2
#4972	2
#4935	2
C₄²×Dic₃	2
#2859	3

subgroup

index

C₄²×Dic₃

Subgroups — #74355

Conjugacy classes of subgroups

Complements

Maximal subgroups