Group #76602 — subgroups

Subgroups — #76602

Order 384.

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

normal subgroups only

This group has 368 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×#391	C₂	1	1	yes	maximal, characteristic
2	#5229	C₂	1	2	yes	maximal
3	#4986	C₂	1	2	yes	maximal
4	#4928	C₂	1	2	yes	maximal
5	#4661	C₂	1	4	yes	maximal
6	#4655	C₂	1	4	yes	maximal
7	#2809	—	3	1	no	maximal, Sylow
8	C₂²×C₃×Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
10	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
11	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
12	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
13	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
15	C₂×Dic₃⋊C₄	C₂²	1	4	yes	—
16	C₂×C₄×Dic₃	C₂²	1	2	yes	—
17	C₃×#95	C₂²	1	4	yes	—
18	C₃⋊#95	C₂²	1	2	yes	—
19	C₃⋊#95	C₂²	1	4	yes	—
20	C₃⋊#95	C₂²	1	4	yes	—
21	C₃⋊#95	C₂²	1	4	yes	—
22	#400	—	3	2	no	—
23	#398	—	3	4	no	—
24	#391	—	3	2	no	—
25	#391	D₃	1	1	yes	characteristic
26	#389	—	3	2	no	—
27	#387	—	3	4	no	—
28	C₂×C₃×Q₈	—	4	1	no	—
29	C₂×C₃×Q₈	—	4	1	no	—
30	C₂×C₃×Q₈	Q₈	1	4	yes	—
31	C₂²×C₃×C₄	C₂³	1	2	yes	—
32	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂²×C₃×C₄	C₂³	1	2	yes	—
34	C₂²×C₃×C₄	C₂³	1	2	yes	—
35	C₂²×Dic₃	C₂³	1	4	yes	—
36	C₂²×Dic₃	C₂³	1	4	yes	—
37	C₃×C₄⋊C₄	—	4	1	no	—
38	C₃×C₄²	—	4	1	no	—
39	C₄⋊Dic₃	—	2	4	no	—
40	C₄⋊Dic₃	—	4	2	no	—
41	C₄⋊Dic₃	—	4	2	no	—
42	C₄⋊Dic₃	—	4	2	no	—
43	Dic₃⋊C₄	—	2	8	no	—
44	C₄×Dic₃	—	4	2	no	—
45	C₂²×Q₈	C₂×D₃	1	1	yes	characteristic
46	C₂×C₄⋊C₄	—	3	4	no	—
47	C₂×C₄⋊C₄	—	3	2	no	—
48	C₂×C₄⋊C₄	—	3	2	no	—
49	C₂×C₄⋊C₄	—	3	2	no	—
50	C₂×C₄⋊C₄	—	3	2	no	—
51	C₂×C₄⋊C₄	C₂×D₃	1	1	yes	characteristic
52	C₂×C₄²	—	3	2	no	—
53	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
54	#95	—	3	2	no	—
55	#95	—	3	4	no	—
56	#95	—	3	4	no	—
57	#95	—	3	4	no	—
58	#95	C₂×D₃	1	4	yes	—
59	C₂³×C₃	C₂⁴	1	1	yes	characteristic
60	C₃×Q₈	—	4	4	no	—
61	C₂×C₃×C₄	—	2	2	no	—
62	C₂×C₃×C₄	—	2	2	no	—
63	C₂×C₃×C₄	—	2	2	no	—
64	C₂×C₃×C₄	—	2	1	no	—
65	C₂×C₃×C₄	—	2	1	no	—
66	C₂×C₃×C₄	—	2	2	no	—
67	C₂×C₃×C₄	—	2	2	no	—
68	C₂×C₃×C₄	—	2	4	no	—
69	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
70	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
71	C₂×C₃×C₄	#41	1	4	yes	—
72	C₂×Dic₃	—	2	4	no	—
73	C₂×Dic₃	—	2	4	no	—
74	C₂×Dic₃	—	2	4	no	—
75	C₂×Dic₃	—	2	4	no	—
76	C₂×Dic₃	—	2	4	no	—
77	C₂×Dic₃	—	2	4	no	—
78	C₂×Q₈	—	4	1	no	—
79	C₂×Q₈	—	4	1	no	—
80	C₂×Q₈	Dic₆	1	4	yes	—
81	C₂²×C₄	—	3	4	no	—
82	C₂²×C₄	—	3	4	no	—
83	C₂²×C₄	C₂²×D₃	1	2	yes	—
84	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
85	C₂²×C₄	C₂²×D₃	1	2	yes	—
86	C₂²×C₄	C₂²×D₃	1	2	yes	—
87	C₄⋊C₄	—	4	1	no	—
88	C₄⋊C₄	—	6	8	no	—
89	C₄⋊C₄	—	6	4	no	—
90	C₄⋊C₄	—	12	2	no	—
91	C₄⋊C₄	—	12	2	no	—
92	C₄⋊C₄	—	12	2	no	—
93	C₄²	—	4	1	no	—
94	C₄²	—	12	2	no	—
95	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
96	C₂²×C₃	C₂×#41	1	2	yes	—
97	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
98	C₂²×C₃	H(5,𝔽₂)	1	1	yes	characteristic
99	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
100	C₃×C₄	—	2	4	no	—
101	C₃×C₄	—	2	4	no	—
102	C₃×C₄	—	2	2	no	—
103	C₃×C₄	—	4	2	no	—
104	Dic₃	—	4	4	no	—
105	Dic₃	—	4	4	no	—
106	C₂³	C₂³×D₃	1	1	yes	characteristic
107	Q₈	—	4	4	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	6	4	no	—
117	C₂×C₄	—	6	4	no	—
118	C₂×C₄	—	6	4	no	—
119	C₂×C₄	—	6	4	no	—
120	C₂×C₄	—	6	4	no	—
121	C₂×C₄	—	6	4	no	—
122	C₂×C₄	C₂×Dic₆	1	2	yes	—
123	C₂×C₄	C₂×Dic₆	1	4	yes	—
124	C₂×C₄	#239	1	4	yes	—
125	C₂×C₃	#539	1	2	yes	—
126	C₂×C₃	#543	1	1	yes	characteristic
127	C₂×C₃	#555	1	1	yes	characteristic
128	C₂×C₃	#557	1	2	yes	—
129	C₂×C₃	#563	1	1	yes	characteristic
130	C₂²	C₂²×Dic₆	1	1	yes	characteristic
131	C₂²	#1004	1	1	yes	characteristic
132	C₂²	#1004	1	1	yes	characteristic
133	C₂²	C₂×#239	1	2	yes	—
134	C₂²	#1010	1	2	yes	—
135	C₄	—	2	4	no	—
136	C₄	—	2	2	no	—
137	C₄	—	2	4	no	—
138	C₄	—	4	2	no	—
139	C₄	—	12	4	no	—
140	C₄	—	12	4	no	—
141	C₃	#2809	1	1	yes	characteristic, complemented, Sylow
142	C₂	#5483	1	1	yes	characteristic
143	C₂	#5570	1	2	yes	—
144	C₂	#5645	1	2	yes	—
145	C₂	#5672	1	1	yes	characteristic
146	C₂	#5734	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#391

C₂

yes

maximal, characteristic

#5229

C₂

yes

maximal

#4986

C₂

yes

maximal

#4928

C₂

yes

maximal

#4661

C₂

yes

maximal

#4655

C₂

yes

maximal

#2809

—

maximal, Sylow

C₂²×C₃×Q₈

C₂²

yes

characteristic

C₂×C₃×C₄⋊C₄

C₂²

yes

characteristic

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

—

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

—

C₃×#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

—

—

—

D₃

yes

characteristic

#389

—

#387

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

Q₈

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂²×Q₈

C₂×D₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

C₂×D₃

yes

—

—

—

—

C₂×D₃

yes

—

C₂³×C₃

C₂⁴

yes

characteristic

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

Dic₆

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₂²×C₃

C₂²×Q₈

yes

characteristic

C₂²×C₃

C₂×#41

yes

—

C₂²×C₃

yes

C₂²×C₃

yes

C₂²×C₃

2₋¹⁺⁴

yes

—

100

C₃×C₄

—

101

C₃×C₄

—

102

C₃×C₄

—

103

C₃×C₄

—

104

Dic₃

—

105

Dic₃

—

106

C₂³

C₂³×D₃

yes

characteristic

107

Q₈

—

108

C₂×C₄

—

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

—

121

C₂×C₄

—

122

C₂×C₄

C₂×Dic₆

yes

—

123

C₂×C₄

C₂×Dic₆

yes

—

124

C₂×C₄

#239

yes

—

125

C₂×C₃

#539

yes

—

126

C₂×C₃

#543

yes

characteristic

127

C₂×C₃

#555

yes

characteristic

128

C₂×C₃

#557

yes

—

129

C₂×C₃

#563

yes

characteristic

130

C₂²

C₂²×Dic₆

yes

characteristic

131

C₂²

#1004

yes

characteristic

132

C₂²

#1004

yes

characteristic

133

C₂²

C₂×#239

yes

—

134

C₂²

#1010

yes

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₃

#2809

yes

characteristic, complemented, Sylow

142

C₂

#5483

yes

characteristic

143

C₂

#5570

yes

—

144

C₂

#5645

yes

—

145

C₂

#5672

yes

characteristic

146

C₂

#5734

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂³×C₃	C₂⁴
Fitting	C₃×#391	C₂
hypercenter	#391	D₃
hypocenter	C₃	#2809
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ #391

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 ◁ #391

N	H ≅ G/N	type
141. C₃	#2809	semidirect

H ≅ G/N

type

141. C₃

#2809

semidirect

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

subgroup	index
C₃×#391	2
#5229	2
#4986	2
#4928	2
#4661	2
#4655	2
#2809	3

subgroup

index

Subgroups — #76602

Conjugacy classes of subgroups

Complements

Maximal subgroups