Group #83023 — subgroups

Subgroups — #83023

Order 384.

444 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 237 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×C₃×#125	C₂	1	1	yes	maximal, characteristic
2	#4984	C₂	1	4	yes	maximal
3	#4984	C₂	1	2	yes	maximal
4	#4980	C₂	1	4	yes	maximal
5	Dic₃×C₄⋊C₄	C₂	1	2	yes	maximal
6	#4935	C₂	1	1	yes	maximal, characteristic
7	#4932	C₂	1	1	yes	maximal, characteristic
8	#2330	—	3	1	no	maximal, Sylow
9	C₃×#125	C₄	1	8	yes	complemented
10	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₃×C₄⋊C₄	C₂²	1	4	yes	—
12	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₂×C₄⋊Dic₃	C₂²	1	4	yes	—
15	C₂×C₄⋊Dic₃	C₂²	1	4	yes	—
16	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×Dic₃	C₂²	1	4	yes	—
18	C₂×C₄×Dic₃	C₂²	1	1	yes	characteristic
19	C₃⋊#95	C₂²	1	2	yes	—
20	C₃⋊#95	C₂²	1	2	yes	—
21	C₃⋊#95	C₂²	1	4	yes	—
22	C₃⋊#95	C₂²	1	4	yes	—
23	C₂×#125	D₃	1	1	yes	characteristic
24	#389	—	3	4	no	—
25	#389	—	3	2	no	—
26	#387	—	3	4	no	—
27	#384	—	3	1	no	—
28	C₄×C₄⋊C₄	—	3	2	no	—
29	#376	—	3	1	no	—
30	C₂²×C₃×C₄	C₂³	1	4	yes	—
31	C₂²×C₃×C₄	C₂³	1	2	yes	—
32	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂²×Dic₃	C₂³	1	1	yes	characteristic
34	C₂²×Dic₃	C₂³	1	1	yes	characteristic
35	C₂²×Dic₃	C₂³	1	4	yes	—
36	C₂²×Dic₃	C₂³	1	2	yes	—
37	C₃×C₄⋊C₄	C₂×C₄	1	16	yes	—
38	C₃×C₄⋊C₄	C₂×C₄	1	8	yes	—
39	C₃×C₄²	C₂×C₄	1	4	yes	—
40	C₄⋊Dic₃	—	2	4	no	—
41	C₄⋊Dic₃	—	4	4	no	—
42	C₄⋊Dic₃	—	4	4	no	—
43	C₄×Dic₃	—	2	2	no	—
44	C₄×Dic₃	—	4	4	no	—
45	C₄×Dic₃	Q₈	1	4	yes	—
46	#125	Dic₃	1	8	yes	complemented
47	C₂×C₄⋊C₄	—	3	2	no	—
48	C₂×C₄⋊C₄	—	3	4	no	—
49	C₂×C₄⋊C₄	—	3	4	no	—
50	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
51	C₂×C₄⋊C₄	C₂×D₃	1	4	yes	—
52	C₂×C₄²	—	3	1	no	—
53	C₂×C₄²	—	3	1	no	—
54	C₂×C₄²	—	3	4	no	—
55	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
56	#95	—	3	2	no	—
57	#95	—	3	2	no	—
58	#95	—	3	4	no	—
59	#95	—	3	4	no	—
60	C₂³×C₃	C₂⁴	1	1	yes	characteristic
61	C₂×C₃×C₄	—	2	4	no	—
62	C₂×C₃×C₄	—	2	2	no	—
63	C₂×C₃×C₄	—	2	4	no	—
64	C₂×C₃×C₄	—	2	2	no	—
65	C₂×C₃×C₄	C₂²×C₄	1	4	yes	—
66	C₂×C₃×C₄	C₂²×C₄	1	8	yes	—
67	C₂×C₃×C₄	C₂²×C₄	1	2	yes	—
68	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
69	C₂×C₃×C₄	#41	1	2	yes	—
70	C₂×Dic₃	—	2	2	no	—
71	C₂×Dic₃	—	2	2	no	—
72	C₂×Dic₃	—	2	1	no	—
73	C₂×Dic₃	—	2	1	no	—
74	C₂×Dic₃	—	2	4	no	—
75	C₂×Dic₃	—	2	4	no	—
76	C₂×Dic₃	—	2	2	no	—
77	C₂×Dic₃	—	2	4	no	—
78	C₂×Dic₃	—	2	2	no	—
79	C₂×Dic₃	C₂×Q₈	1	4	yes	—
80	C₂²×C₄	—	3	2	no	—
81	C₂²×C₄	—	3	1	no	—
82	C₂²×C₄	—	3	1	no	—
83	C₂²×C₄	—	3	4	no	—
84	C₂²×C₄	C₂²×D₃	1	4	yes	—
85	C₂²×C₄	C₂²×D₃	1	2	yes	—
86	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
87	C₄⋊C₄	—	6	4	no	—
88	C₄⋊C₄	—	12	4	no	—
89	C₄⋊C₄	—	12	4	no	—
90	C₄⋊C₄	C₂×Dic₃	1	16	yes	—
91	C₄⋊C₄	C₂×Dic₃	1	8	yes	—
92	C₄²	—	3	4	no	—
93	C₄²	—	6	2	no	—
94	C₄²	—	12	4	no	—
95	C₄²	C₂×Dic₃	1	4	yes	—
96	C₂²×C₃	C₂³×C₄	1	1	yes	characteristic
97	C₂²×C₃	C₂²×Q₈	1	1	yes	characteristic
98	C₂²×C₃	C₂×#41	1	1	yes	characteristic
99	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
100	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
101	C₃×C₄	—	2	8	no	—
102	C₃×C₄	—	2	4	no	—
103	C₃×C₄	C₄×Q₈	1	4	yes	—
104	Dic₃	—	2	4	no	—
105	Dic₃	—	4	1	no	—
106	Dic₃	—	4	4	no	—
107	Dic₃	—	4	1	no	—
108	C₂³	C₂³×D₃	1	1	yes	characteristic
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	3	4	no	—
114	C₂×C₄	—	6	2	no	—
115	C₂×C₄	—	6	2	no	—
116	C₂×C₄	—	6	1	no	—
117	C₂×C₄	—	6	1	no	—
118	C₂×C₄	—	6	4	no	—
119	C₂×C₄	—	6	4	no	—
120	C₂×C₄	—	6	2	no	—
121	C₂×C₄	—	6	4	no	—
122	C₂×C₄	—	6	2	no	—
123	C₂×C₄	D₃×Q₈	1	2	yes	—
124	C₂×C₄	#239	1	2	yes	—
125	C₂×C₄	C₂²×Dic₃	1	4	yes	—
126	C₂×C₄	C₂²×Dic₃	1	8	yes	—
127	C₂×C₄	C₂²×Dic₃	1	2	yes	—
128	C₂×C₃	C₂×C₄×Q₈	1	1	yes	characteristic
129	C₂×C₃	#520	1	2	yes	—
130	C₂×C₃	#540	1	1	yes	characteristic
131	C₂×C₃	#541	1	1	yes	characteristic
132	C₂×C₃	#544	1	2	yes	—
133	C₂²	C₂×D₃×Q₈	1	1	yes	characteristic
134	C₂²	C₂×#239	1	1	yes	characteristic
135	C₂²	#1009	1	2	yes	—
136	C₂²	#1010	1	2	yes	—
137	C₂²	C₂³×Dic₃	1	1	yes	characteristic
138	C₄	—	2	8	no	—
139	C₄	—	2	4	no	—
140	C₄	—	6	4	no	—
141	C₄	—	12	1	no	—
142	C₄	—	12	4	no	—
143	C₄	—	12	1	no	—
144	C₄	Q₈×Dic₃	1	4	yes	—
145	C₃	#2330	1	1	yes	characteristic, complemented, Sylow
146	C₂	#5689	1	2	yes	—
147	C₂	#5697	1	1	yes	characteristic
148	C₂	#5698	1	1	yes	characteristic
149	C₂	C₂×Q₈×Dic₃	1	1	yes	characteristic
150	C₂	#5827	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#125

C₂

yes

maximal, characteristic

#4984

C₂

yes

maximal

#4984

C₂

yes

maximal

#4980

C₂

yes

maximal

Dic₃×C₄⋊C₄

C₂

yes

maximal

#4935

C₂

yes

maximal, characteristic

#4932

C₂

yes

maximal, characteristic

#2330

—

maximal, Sylow

C₃×#125

C₄

yes

complemented

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×C₄×Dic₃

C₂²

yes

characteristic

C₂×C₄×Dic₃

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

characteristic

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₂×#125

D₃

yes

—

—

—

—

C₄×C₄⋊C₄

—

#376

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

C₂³

yes

characteristic

C₂²×Dic₃

C₂³

yes

characteristic

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₃×C₄⋊C₄

C₂×C₄

yes

—

C₃×C₄⋊C₄

C₂×C₄

yes

—

C₃×C₄²

C₂×C₄

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

Q₈

yes

—

#125

Dic₃

yes

complemented

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×D₃

yes

—

—

—

—

C₂³×C₃

C₂⁴

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

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×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

C₂×Q₈

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₃

yes

—

C₄⋊C₄

C₂×Dic₃

yes

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₂×Dic₃

yes

—

C₂²×C₃

C₂³×C₄

yes

characteristic

C₂²×C₃

C₂²×Q₈

yes

characteristic

C₂²×C₃

C₂×#41

yes

characteristic

C₂²×C₃

H(5,𝔽₂)

yes

—

100

C₂²×C₃

2₋¹⁺⁴

yes

—

101

C₃×C₄

—

102

C₃×C₄

—

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

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

D₃×Q₈

yes

—

124

C₂×C₄

#239

yes

—

125

C₂×C₄

C₂²×Dic₃

yes

—

126

C₂×C₄

C₂²×Dic₃

yes

—

127

C₂×C₄

C₂²×Dic₃

yes

—

128

C₂×C₃

C₂×C₄×Q₈

yes

characteristic

129

C₂×C₃

#520

yes

—

130

C₂×C₃

#540

yes

characteristic

131

C₂×C₃

#541

yes

characteristic

132

C₂×C₃

#544

yes

—

133

C₂²

C₂×D₃×Q₈

yes

characteristic

134

C₂²

C₂×#239

yes

characteristic

135

C₂²

#1009

yes

—

136

C₂²

#1010

yes

—

137

C₂²

C₂³×Dic₃

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

Q₈×Dic₃

yes

—

145

C₃

#2330

yes

characteristic, complemented, Sylow

146

C₂

#5689

yes

—

147

C₂

#5697

yes

characteristic

148

C₂

#5698

yes

characteristic

149

C₂

C₂×Q₈×Dic₃

yes

characteristic

150

C₂

#5827

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator	C₂²×C₃	C₂³×C₄
Fitting	C₂×C₃×#125	C₂
socle, prime radical, cosocle	C₂³×C₃	C₂⁴
hypercenter	C₂×#125	D₃
hypocenter	C₃	#2330
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×#125

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃

commutator

C₂²×C₃

C₂³×C₄

Fitting

C₂×C₃×#125

C₂

socle, prime radical, cosocle

C₂³×C₃

C₂⁴

D₃

C₃

G ▷ C₂²×C₃ ▷ C₃

C₁ ◁ C₂³ ◁ C₂×#125

N	H ≅ G/N	type
9. C₃×#125	C₄	semidirect
46. #125	Dic₃	semidirect
145. C₃	#2330	semidirect

H ≅ G/N

type

9. C₃×#125

C₄

semidirect

46. #125

Dic₃

semidirect

145. C₃

#2330

semidirect

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

subgroup	index
C₂×C₃×#125	2
#4984	2
#4980	2
Dic₃×C₄⋊C₄	2
#4935	2
#4932	2
#2330	3

subgroup

index

C₂×C₃×#125

Dic₃×C₄⋊C₄

Subgroups — #83023

Conjugacy classes of subgroups

Complements

Maximal subgroups