Group #83246 — subgroups

Subgroups — #83246

Order 384.

448 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 165 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×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#870	C₂	1	1	yes	maximal, characteristic
3	#5230	C₂	1	4	yes	maximal
4	#4983	C₂	1	4	yes	maximal
5	#4977	C₂	1	2	yes	maximal
6	#4975	C₂	1	2	yes	maximal
7	#4932	C₂	1	1	yes	maximal, characteristic
8	#2660	—	3	1	no	maximal, Sylow
9	C₂²×C₃×Q₈	C₂²	1	2	yes	—
10	C₃×C₄⋊Q₈	—	2	4	no	—
11	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
12	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
13	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₃⋊C₄	C₂²	1	2	yes	—
17	C₂×Dic₃⋊C₄	C₂²	1	2	yes	—
18	C₂×Dic₃⋊C₄	C₂²	1	8	yes	—
19	C₂×C₄×Dic₃	C₂²	1	2	yes	—
20	#870	—	2	4	no	—
21	C₃⋊#95	C₂²	1	2	yes	—
22	C₃⋊#95	C₂²	1	2	yes	—
23	C₃⋊#95	C₂²	1	8	yes	—
24	C₂×C₄⋊Q₈	D₃	1	1	yes	characteristic
25	C₂×#125	—	3	1	no	—
26	#398	—	3	4	no	—
27	#395	—	3	4	no	—
28	#384	—	3	2	no	—
29	#382	—	3	2	no	—
30	#376	—	3	1	no	—
31	C₂×C₃×Q₈	—	2	4	no	—
32	C₂×C₃×Q₈	—	4	4	no	—
33	C₂²×C₃×C₄	C₂³	1	4	yes	—
34	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
35	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
36	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
37	C₂²×Dic₃	C₂³	1	2	yes	—
38	C₂²×Dic₃	C₂³	1	2	yes	—
39	C₂²×Dic₃	C₂³	1	4	yes	—
40	C₃×C₄⋊C₄	—	2	4	no	—
41	C₃×C₄⋊C₄	—	2	4	no	—
42	C₃×C₄²	D₄	1	4	yes	—
43	C₄⋊Dic₃	—	2	2	no	—
44	C₄⋊Dic₃	—	2	2	no	—
45	Dic₃⋊C₄	—	2	4	no	—
46	Dic₃⋊C₄	—	2	4	no	—
47	Dic₃⋊C₄	—	4	8	no	—
48	C₄×Dic₃	Q₈	1	8	yes	complemented
49	C₂²×Q₈	C₂×D₃	1	2	yes	—
50	C₄⋊Q₈	—	2	4	no	—
51	#125	—	6	4	no	—
52	C₂×C₄⋊C₄	—	3	1	no	—
53	C₂×C₄⋊C₄	—	3	1	no	—
54	C₂×C₄⋊C₄	—	3	2	no	—
55	C₂×C₄⋊C₄	—	3	2	no	—
56	C₂×C₄⋊C₄	—	3	8	no	—
57	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
58	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
59	C₂×C₄²	—	3	2	no	—
60	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
61	#95	—	3	2	no	—
62	#95	—	3	2	no	—
63	#95	—	3	8	no	—
64	C₂³×C₃	C₂⁴	1	1	yes	characteristic
65	C₃×Q₈	—	4	8	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	—	2	8	no	—
70	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
71	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
72	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
73	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
74	C₂×Dic₃	—	2	4	no	—
75	C₂×Dic₃	—	2	4	no	—
76	C₂×Dic₃	—	2	2	no	—
77	C₂×Dic₃	—	2	2	no	—
78	C₂×Dic₃	—	2	4	no	—
79	C₂×Dic₃	—	2	4	no	—
80	C₂×Dic₃	C₂×Q₈	1	8	yes	—
81	C₂×Q₈	—	2	4	no	—
82	C₂×Q₈	—	4	4	no	—
83	C₂²×C₄	—	3	4	no	—
84	C₂²×C₄	—	3	2	no	—
85	C₂²×C₄	—	3	2	no	—
86	C₂²×C₄	C₂²×D₃	1	4	yes	—
87	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
88	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
89	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
90	C₄⋊C₄	—	2	4	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	6	4	no	—
93	C₄⋊C₄	—	6	4	no	—
94	C₄⋊C₄	—	6	2	no	—
95	C₄⋊C₄	—	6	2	no	—
96	C₄⋊C₄	—	12	8	no	—
97	C₄²	—	3	8	no	—
98	C₄²	#67	1	4	yes	—
99	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
100	C₂²×C₃	C₂²×Q₈	1	2	yes	—
101	C₂²×C₃	H(5,𝔽₂)	1	2	yes	—
102	C₂²×C₃	2₋¹⁺⁴	1	2	yes	—
103	C₃×C₄	—	2	2	no	—
104	C₃×C₄	—	2	2	no	—
105	C₃×C₄	—	4	4	no	—
106	C₃×C₄	C₄⋊Q₈	1	4	yes	—
107	Dic₃	—	2	8	no	—
108	Dic₃	—	4	2	no	—
109	Dic₃	—	4	2	no	—
110	C₂³	C₂³×D₃	1	1	yes	characteristic
111	Q₈	—	4	8	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	2	8	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	3	8	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	2	no	—
122	C₂×C₄	—	6	2	no	—
123	C₂×C₄	D₃×Q₈	1	4	yes	—
124	C₂×C₄	C₂×#67	1	2	yes	—
125	C₂×C₄	C₂×#67	1	2	yes	—
126	C₂×C₄	C₂×#67	1	2	yes	—
127	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
128	C₂×C₃	#535	1	1	yes	characteristic
129	C₂×C₃	#536	1	1	yes	characteristic
130	C₂×C₃	#544	1	4	yes	—
131	C₂²	#1004	1	2	yes	—
132	C₂²	C₂×D₃×Q₈	1	2	yes	—
133	C₂²	#1007	1	2	yes	—
134	C₂²	C₂²×#67	1	1	yes	characteristic
135	C₄	—	2	2	no	—
136	C₄	—	2	2	no	—
137	C₄	—	4	4	no	—
138	C₄	—	6	8	no	—
139	C₄	—	12	2	no	—
140	C₄	—	12	2	no	—
141	C₄	#944	1	4	yes	—
142	C₃	#2660	1	1	yes	characteristic, complemented, Sylow
143	C₂	#5729	1	4	yes	—
144	C₂	#5805	1	1	yes	characteristic
145	C₂	C₂×#944	1	1	yes	characteristic
146	C₂	#5816	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×#870

C₂

yes

maximal, characteristic

#5230

C₂

yes

maximal

#4983

C₂

yes

maximal

#4977

C₂

yes

maximal

#4975

C₂

yes

maximal

#4932

C₂

yes

maximal, characteristic

#2660

—

maximal, Sylow

C₂²×C₃×Q₈

C₂²

yes

—

C₃×C₄⋊Q₈

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

characteristic

C₂×Dic₃⋊C₄

C₂²

yes

—

C₂×Dic₃⋊C₄

C₂²

yes

—

C₂×Dic₃⋊C₄

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

—

#870

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₃⋊#95

C₂²

yes

—

C₂×C₄⋊Q₈

D₃

yes

—

—

—

—

—

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂²×C₃×C₄

C₂³

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₂²×Dic₃

C₂³

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

D₄

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

Q₈

yes

complemented

C₂²×Q₈

C₂×D₃

yes

—

C₄⋊Q₈

—

#125

—

C₂×C₄⋊C₄

—

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×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×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

C₂×Q₈

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂²×C₄

C₂²×D₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

#67

yes

—

C₂²×C₃

C₂²×D₄

yes

characteristic

100

C₂²×C₃

C₂²×Q₈

yes

—

101

C₂²×C₃

H(5,𝔽₂)

yes

—

102

C₂²×C₃

2₋¹⁺⁴

yes

—

103

C₃×C₄

—

104

C₃×C₄

—

105

C₃×C₄

—

106

C₃×C₄

C₄⋊Q₈

yes

—

107

Dic₃

—

108

Dic₃

—

109

Dic₃

—

110

C₂³

C₂³×D₃

yes

characteristic

111

Q₈

—

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

C₂×#67

yes

—

125

C₂×C₄

C₂×#67

yes

—

126

C₂×C₄

C₂×#67

yes

—

127

C₂×C₃

C₂×C₄⋊Q₈

yes

characteristic

128

C₂×C₃

#535

yes

characteristic

129

C₂×C₃

#536

yes

characteristic

130

C₂×C₃

#544

yes

—

131

C₂²

#1004

yes

—

132

C₂²

C₂×D₃×Q₈

yes

—

133

C₂²

#1007

yes

—

134

C₂²

C₂²×#67

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

#944

yes

—

142

C₃

#2660

yes

characteristic, complemented, Sylow

143

C₂

#5729

yes

—

144

C₂

#5805

yes

characteristic

145

C₂

C₂×#944

yes

characteristic

146

C₂

#5816

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₃
commutator, socle, prime radical, cosocle	C₂³×C₃	C₂⁴
Fitting	C₂×C₃×C₄⋊Q₈	C₂
hypercenter	C₂×C₄⋊Q₈	D₃
hypocenter	C₃	#2660
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₃

commutator, socle, prime radical, cosocle

C₂³×C₃

C₂⁴

Fitting

C₂×C₃×C₄⋊Q₈

C₂

hypercenter

C₂×C₄⋊Q₈

D₃

C₃

G ▷ C₂³×C₃ ▷ C₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
48. C₄×Dic₃	Q₈	semidirect
142. C₃	#2660	semidirect

H ≅ G/N

type

48. C₄×Dic₃

Q₈

semidirect

142. C₃

#2660

semidirect

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

subgroup	index
C₂×C₃×C₄⋊Q₈	2
C₂×#870	2
#5230	2
#4983	2
#4977	2
#4975	2
#4932	2
#2660	3

subgroup

index

C₂×C₃×C₄⋊Q₈

Subgroups — #83246

Conjugacy classes of subgroups

Complements

Maximal subgroups