Group #83252 — subgroups

Subgroups — #83252

Order 384.

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

normal subgroups only

This group has 504 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×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#869	C₂	1	1	yes	maximal, characteristic
3	#5229	C₂	1	4	yes	maximal
4	#4977	C₂	1	4	yes	maximal
5	#4973	C₂	1	4	yes	maximal
6	C₄²×Dic₃	C₂	1	1	yes	maximal, characteristic
7	#2862	—	3	1	no	maximal, Sylow
8	C₂²×C₃×Q₈	C₂²	1	2	yes	—
9	C₂²×Dic₆	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	2	yes	—
15	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
16	C₂×C₄×Dic₃	C₂²	1	2	yes	—
17	C₂×C₄×Dic₃	C₂²	1	2	yes	—
18	C₂×C₄×Dic₃	C₂²	1	2	yes	—
19	#869	—	2	4	no	—
20	C₃⋊#95	C₂²	1	16	yes	—
21	C₂×C₄⋊Q₈	—	3	1	no	—
22	C₂×C₄⋊Q₈	D₃	1	1	yes	characteristic
23	#391	—	3	4	no	—
24	#391	—	3	4	no	—
25	#382	—	3	4	no	—
26	C₄³	—	3	1	no	—
27	C₂×C₃×Q₈	—	2	4	no	—
28	C₂×C₃×Q₈	—	2	8	no	—
29	C₂²×C₃×C₄	C₂³	1	4	yes	—
30	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
31	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
32	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
33	C₂²×Dic₃	C₂³	1	4	yes	—
34	C₂²×Dic₃	C₂³	1	4	yes	—
35	C₂×Dic₆	—	2	4	no	—
36	C₂×Dic₆	—	2	8	no	—
37	C₃×C₄⋊C₄	—	2	4	no	—
38	C₃×C₄⋊C₄	—	2	4	no	—
39	C₃×C₄²	D₄	1	4	yes	—
40	C₄⋊Dic₃	—	2	4	no	—
41	C₄⋊Dic₃	—	2	4	no	—
42	C₄×Dic₃	—	2	4	no	—
43	C₄×Dic₃	—	2	4	no	—
44	C₄×Dic₃	Q₈	1	8	yes	complemented
45	C₂²×Q₈	—	3	2	no	—
46	C₂²×Q₈	C₂×D₃	1	2	yes	—
47	C₄⋊Q₈	—	2	4	no	—
48	C₄⋊Q₈	—	6	4	no	—
49	C₂×C₄⋊C₄	—	3	2	no	—
50	C₂×C₄⋊C₄	—	3	2	no	—
51	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
52	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
53	C₂×C₄²	—	3	2	no	—
54	C₂×C₄²	—	3	2	no	—
55	C₂×C₄²	—	3	2	no	—
56	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
57	#95	—	3	16	no	—
58	C₂³×C₃	C₂⁴	1	1	yes	characteristic
59	C₃×Q₈	—	4	8	no	—
60	C₂×C₃×C₄	—	2	4	no	—
61	C₂×C₃×C₄	—	2	8	no	—
62	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
63	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
64	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
65	C₂×C₃×C₄	C₂×Q₈	1	4	yes	—
66	C₂×C₃×C₄	#41	1	4	yes	—
67	C₂×C₃×C₄	#41	1	4	yes	—
68	C₂×Dic₃	—	2	4	no	—
69	C₂×Dic₃	—	2	4	no	—
70	C₂×Dic₃	—	2	4	no	—
71	C₂×Dic₃	—	2	8	no	—
72	C₂×Dic₃	C₂×Q₈	1	8	yes	—
73	Dic₆	—	4	8	no	—
74	C₂×Q₈	—	2	8	no	—
75	C₂×Q₈	—	2	4	no	—
76	C₂×Q₈	—	6	8	no	—
77	C₂×Q₈	—	6	4	no	—
78	C₂²×C₄	—	3	4	no	—
79	C₂²×C₄	—	3	4	no	—
80	C₂²×C₄	C₂²×D₃	1	4	yes	—
81	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
82	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
83	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
84	C₄⋊C₄	—	2	4	no	—
85	C₄⋊C₄	—	2	4	no	—
86	C₄⋊C₄	—	6	4	no	—
87	C₄⋊C₄	—	6	4	no	—
88	C₄²	—	3	8	no	—
89	C₄²	—	6	4	no	—
90	C₄²	—	6	4	no	—
91	C₄²	#67	1	4	yes	—
92	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
93	C₂²×C₃	C₂²×Q₈	1	2	yes	—
94	C₂²×C₃	C₂×#41	1	2	yes	—
95	C₂²×C₃	C₂×#41	1	2	yes	—
96	C₃×C₄	—	4	4	no	—
97	C₃×C₄	#124	1	4	yes	—
98	C₃×C₄	#124	1	4	yes	—
99	C₃×C₄	C₄⋊Q₈	1	4	yes	—
100	Dic₃	—	2	8	no	—
101	Dic₃	—	4	4	no	—
102	C₂³	C₂³×D₃	1	1	yes	characteristic
103	Q₈	—	4	8	no	—
104	Q₈	—	12	8	no	—
105	C₂×C₄	—	2	8	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	3	8	no	—
108	C₂×C₄	—	6	4	no	—
109	C₂×C₄	—	6	4	no	—
110	C₂×C₄	—	6	8	no	—
111	C₂×C₄	—	6	4	no	—
112	C₂×C₄	#237	1	4	yes	—
113	C₂×C₄	D₃×Q₈	1	4	yes	—
114	C₂×C₄	#239	1	4	yes	—
115	C₂×C₄	C₂×#67	1	2	yes	—
116	C₂×C₄	C₂×#67	1	2	yes	—
117	C₂×C₄	C₂×#67	1	2	yes	—
118	C₂×C₃	C₂×#124	1	1	yes	characteristic
119	C₂×C₃	C₂×#124	1	1	yes	characteristic
120	C₂×C₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
121	C₂×C₃	#533	1	4	yes	—
122	C₂²	C₂×#237	1	2	yes	—
123	C₂²	C₂×D₃×Q₈	1	2	yes	—
124	C₂²	C₂×#239	1	2	yes	—
125	C₂²	C₂²×#67	1	1	yes	characteristic
126	C₄	—	4	4	no	—
127	C₄	—	6	8	no	—
128	C₄	—	12	4	no	—
129	C₄	#936	1	4	yes	—
130	C₄	#944	1	4	yes	—
131	C₄	#947	1	4	yes	—
132	C₃	#2862	1	1	yes	characteristic, complemented, Sylow
133	C₂	#5728	1	4	yes	—
134	C₂	C₂×#936	1	1	yes	characteristic
135	C₂	C₂×#944	1	1	yes	characteristic
136	C₂	C₂×#947	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₂×#869

C₂

yes

maximal, characteristic

#5229

C₂

yes

maximal

#4977

C₂

yes

maximal

#4973

C₂

yes

maximal

C₄²×Dic₃

C₂

yes

maximal, characteristic

#2862

—

maximal, Sylow

C₂²×C₃×Q₈

C₂²

yes

—

C₂²×Dic₆

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

—

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

—

C₂×C₄×Dic₃

C₂²

yes

—

#869

—

C₃⋊#95

C₂²

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

D₃

yes

—

—

—

C₄³

—

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

—

C₂×Dic₆

—

C₂×Dic₆

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

D₄

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

Q₈

yes

complemented

C₂²×Q₈

—

C₂²×Q₈

C₂×D₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

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

characteristic

#95

—

C₂³×C₃

C₂⁴

yes

characteristic

C₃×Q₈

—

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×C₃×C₄

#41

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

C₂×Q₈

yes

—

Dic₆

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

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

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#67

yes

—

C₂²×C₃

C₂²×D₄

yes

characteristic

C₂²×C₃

C₂²×Q₈

yes

—

C₂²×C₃

C₂×#41

yes

—

C₂²×C₃

C₂×#41

yes

—

C₃×C₄

—

C₃×C₄

#124

yes

—

C₃×C₄

#124

yes

—

C₃×C₄

C₄⋊Q₈

yes

—

100

Dic₃

—

101

Dic₃

—

102

C₂³

C₂³×D₃

yes

characteristic

103

Q₈

—

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

#237

yes

—

113

C₂×C₄

D₃×Q₈

yes

—

114

C₂×C₄

#239

yes

—

115

C₂×C₄

C₂×#67

yes

—

116

C₂×C₄

C₂×#67

yes

—

117

C₂×C₄

C₂×#67

yes

—

118

C₂×C₃

C₂×#124

yes

characteristic

119

C₂×C₃

C₂×#124

yes

characteristic

120

C₂×C₃

C₂×C₄⋊Q₈

yes

characteristic

121

C₂×C₃

#533

yes

—

122

C₂²

C₂×#237

yes

—

123

C₂²

C₂×D₃×Q₈

yes

—

124

C₂²

C₂×#239

yes

—

125

C₂²

C₂²×#67

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

#936

yes

—

130

C₄

#944

yes

—

131

C₄

#947

yes

—

132

C₃

#2862

yes

characteristic, complemented, Sylow

133

C₂

#5728

yes

—

134

C₂

C₂×#936

yes

characteristic

135

C₂

C₂×#944

yes

characteristic

136

C₂

C₂×#947

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	#2862
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
44. C₄×Dic₃	Q₈	semidirect
132. C₃	#2862	semidirect

H ≅ G/N

type

44. C₄×Dic₃

Q₈

semidirect

132. C₃

#2862

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×#869	2
#5229	2
#4977	2
#4973	2
C₄²×Dic₃	2
#2862	3

subgroup

index

C₂×C₃×C₄⋊Q₈

C₄²×Dic₃

Subgroups — #83252

Conjugacy classes of subgroups

Complements

Maximal subgroups