Group #83198 — subgroups

Subgroups — #83198

Order 384.

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

normal subgroups only

This group has 380 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	C₂×C₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	#5470	C₂	1	1	yes	maximal, characteristic
3	#5225	C₂	1	4	yes	maximal
4	#5082	C₂	1	4	yes	maximal, complemented
5	#5079	C₂	1	4	yes	maximal, complemented
6	C₄×#208	C₂	1	1	yes	maximal, characteristic
7	#3139	—	3	1	no	maximal, Sylow
8	C₂²×C₃×Q₈	C₂²	1	2	yes	—
9	C₃×C₄⋊Q₈	—	2	2	no	—
10	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₃×C₄⋊Q₈	C₂²	1	2	yes	—
12	C₂×C₃×C₄⋊C₄	—	2	2	no	—
13	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
14	#926	C₂²	1	2	yes	—
15	#924	—	2	2	no	—
16	C₂×#208	C₂²	1	2	yes	—
17	#870	C₂²	1	2	yes	—
18	#869	C₂²	1	2	yes	—
19	C₄×Dic₆	—	2	2	no	—
20	#835	C₂²	1	16	yes	—
21	#803	C₂²	1	2	yes	—
22	C₄×C₃⋊C₈	C₂²	1	2	yes	—
23	#533	—	3	1	no	—
24	C₂×C₄⋊Q₈	D₃	1	1	yes	characteristic
25	#491	—	3	4	no	—
26	#487	—	3	4	no	—
27	#419	—	3	4	no	—
28	C₄×M₁₆	—	3	1	no	—
29	C₂×C₃×Q₈	—	2	2	no	—
30	C₂×C₃×Q₈	—	2	8	no	—
31	C₂×C₃×Q₈	C₂³	1	4	yes	—
32	C₂²×C₃×C₄	—	2	2	no	—
33	C₂²×C₃×C₄	D₄	1	2	yes	—
34	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
35	C₂×Dic₆	—	2	2	no	—
36	C₃×C₄⋊C₄	—	2	4	no	—
37	C₃×C₄⋊C₄	—	2	4	no	—
38	C₃×C₄²	D₄	1	2	yes	—
39	C₃×C₄²	C₂³	1	1	yes	characteristic
40	C₃×C₄²	C₂³	1	1	yes	characteristic
41	#217	—	2	2	no	—
42	C₄⋊Dic₃	—	2	2	no	—
43	C₄⋊Dic₃	C₂³	1	4	yes	—
44	Dic₃⋊C₄	—	2	4	no	—
45	C₄×Dic₃	—	2	2	no	—
46	#208	—	2	4	no	—
47	C₂×C₃⋊C₈	C₂³	1	4	yes	—
48	C₂²×Q₈	C₂×D₃	1	2	yes	—
49	C₂×M₁₆	—	3	2	no	—
50	C₄⋊Q₈	—	2	2	no	—
51	C₄⋊Q₈	—	3	2	no	—
52	C₄⋊Q₈	C₂×D₃	1	2	yes	—
53	C₄⋊Q₈	C₂×D₃	1	2	yes	—
54	#125	—	3	2	no	—
55	C₂²⋊Q₈	—	6	2	no	—
56	C₄×Q₈	—	6	2	no	—
57	#117	—	3	2	no	—
58	C₂×C₄⋊C₄	—	2	2	no	—
59	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
60	#103	—	3	16	no	—
61	#97	—	3	2	no	—
62	C₄×C₈	—	3	2	no	—
63	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
64	C₃×Q₈	—	2	8	no	—
65	C₃×Q₈	—	4	4	no	—
66	C₂×C₃×C₄	—	2	4	no	—
67	C₂×C₃×C₄	—	2	4	no	—
68	C₂×C₃×C₄	—	4	4	no	—
69	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
70	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
71	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
72	C₂×C₃×C₄	#41	1	4	yes	—
73	C₂×C₃×C₄	C₂⁴	1	1	yes	characteristic
74	C₂×Dic₃	—	2	4	no	—
75	Dic₆	—	4	4	no	—
76	C₃⋊C₈	—	2	4	no	—
77	C₂×Q₈	—	2	2	no	—
78	C₂×Q₈	—	2	8	no	—
79	C₂×Q₈	—	6	2	no	—
80	C₂×Q₈	C₂²×D₃	1	4	yes	—
81	C₂²×C₄	—	2	2	no	—
82	C₂²×C₄	#67	1	2	yes	—
83	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
84	M₁₆	—	6	4	no	—
85	C₂×C₈	—	3	4	no	—
86	C₄⋊C₄	—	2	4	no	—
87	C₄⋊C₄	—	2	4	no	—
88	C₄⋊C₄	—	3	4	no	—
89	C₄⋊C₄	—	6	4	no	—
90	C₄⋊C₄	—	6	2	no	—
91	C₂²⋊C₄	—	6	2	no	—
92	C₄²	—	6	2	no	—
93	C₄²	#67	1	2	yes	—
94	C₄²	C₂²×D₃	1	1	yes	characteristic
95	C₄²	C₂²×D₃	1	1	yes	characteristic
96	C₂²×C₃	—	2	2	no	—
97	C₂²×C₃	#124	1	2	yes	—
98	C₂²×C₃	C₂²×D₄	1	1	yes	characteristic
99	C₃×C₄	—	2	2	no	—
100	C₃×C₄	—	4	4	no	—
101	C₃×C₄	#124	1	2	yes	—
102	C₃×C₄	#137	1	4	yes	—
103	C₃×C₄	C₂×#41	1	2	yes	—
104	Dic₃	—	4	4	no	—
105	C₂³	C₂×#67	1	1	yes	characteristic
106	Q₈	—	2	8	no	—
107	Q₈	—	4	4	no	—
108	Q₈	—	12	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	4	no	—
111	C₂×C₄	—	4	4	no	—
112	C₂×C₄	—	6	4	no	—
113	C₂×C₄	#237	1	4	yes	—
114	C₂×C₄	C₂×#67	1	2	yes	—
115	C₂×C₄	C₂×#67	1	1	yes	characteristic
116	C₂×C₄	C₂×#67	1	2	yes	—
117	C₂×C₄	C₂³×D₃	1	1	yes	characteristic
118	C₈	—	6	4	no	—
119	C₂×C₃	—	2	2	no	—
120	C₂×C₃	C₂×#124	1	1	yes	characteristic
121	C₂×C₃	C₂×#137	1	2	yes	—
122	C₂²	—	2	2	no	—
123	C₂²	#936	1	2	yes	—
124	C₂²	C₂²×#67	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	4	4	no	—
127	C₄	—	12	4	no	—
128	C₄	#936	1	2	yes	—
129	C₄	#942	1	4	yes	—
130	C₄	C₂×#237	1	2	yes	—
131	C₃	#3139	1	1	yes	characteristic, complemented, Sylow
132	C₂	—	2	2	no	—
133	C₂	C₂×#936	1	1	yes	characteristic
134	C₂	C₂×#942	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×C₄⋊Q₈

C₂

yes

maximal, characteristic

#5470

C₂

yes

maximal, characteristic

#5225

C₂

yes

maximal

#5082

C₂

yes

maximal, complemented

#5079

C₂

yes

maximal, complemented

C₄×#208

C₂

yes

maximal, characteristic

#3139

—

maximal, Sylow

C₂²×C₃×Q₈

C₂²

yes

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

C₂²

yes

—

C₃×C₄⋊Q₈

C₂²

yes

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄²

C₂²

yes

characteristic

#926

C₂²

yes

—

#924

—

C₂×#208

C₂²

yes

—

#870

C₂²

yes

—

#869

C₂²

yes

—

C₄×Dic₆

—

#835

C₂²

yes

—

#803

C₂²

yes

—

C₄×C₃⋊C₈

C₂²

yes

—

#533

—

C₂×C₄⋊Q₈

D₃

yes

—

—

—

C₄×M₁₆

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₂³

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂×Dic₆

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

D₄

yes

—

C₃×C₄²

C₂³

yes

characteristic

C₃×C₄²

C₂³

yes

characteristic

#217

—

C₄⋊Dic₃

—

C₄⋊Dic₃

C₂³

yes

—

Dic₃⋊C₄

—

C₄×Dic₃

—

#208

—

C₂×C₃⋊C₈

C₂³

yes

—

C₂²×Q₈

C₂×D₃

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃

yes

—

C₄⋊Q₈

C₂×D₃

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃

yes

characteristic

#103

—

#97

—

C₄×C₈

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×C₃×C₄

C₂⁴

yes

characteristic

C₂×Dic₃

—

Dic₆

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²×D₃

yes

—

C₂²×C₄

—

C₂²×C₄

#67

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

M₁₆

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

#67

yes

—

C₄²

C₂²×D₃

yes

characteristic

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

—

C₂²×C₃

#124

yes

—

C₂²×C₃

C₂²×D₄

yes

characteristic

C₃×C₄

—

100

C₃×C₄

—

101

C₃×C₄

#124

yes

—

102

C₃×C₄

#137

yes

—

103

C₃×C₄

C₂×#41

yes

—

104

Dic₃

—

105

C₂³

C₂×#67

yes

characteristic

106

Q₈

—

107

Q₈

—

108

Q₈

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

#237

yes

—

114

C₂×C₄

C₂×#67

yes

—

115

C₂×C₄

C₂×#67

yes

characteristic

116

C₂×C₄

C₂×#67

yes

—

117

C₂×C₄

C₂³×D₃

yes

characteristic

118

C₈

—

119

C₂×C₃

—

120

C₂×C₃

C₂×#124

yes

characteristic

121

C₂×C₃

C₂×#137

yes

—

122

C₂²

—

123

C₂²

#936

yes

—

124

C₂²

C₂²×#67

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

#936

yes

—

129

C₄

#942

yes

—

130

C₄

C₂×#237

yes

—

131

C₃

#3139

yes

characteristic, complemented, Sylow

132

C₂

—

133

C₂

C₂×#936

yes

characteristic

134

C₂

C₂×#942

yes

—

	subgroup	G/H
center	C₂²	C₂²×#67
commutator, cosocle	C₂×C₃×C₄	C₂⁴
Frattini	C₂×C₄	C₂³×D₃
Fitting	C₂×C₃×C₄⋊Q₈	C₂
socle	C₂²×C₃	C₂²×D₄
prime radical	C₂³×C₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₃
hypocenter	C₃	#3139
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂²×#67

commutator, cosocle

C₂×C₃×C₄

C₂⁴

Frattini

C₂×C₄

C₂³×D₃

Fitting

C₂×C₃×C₄⋊Q₈

C₂

socle

C₂²×C₃

C₂²×D₄

prime radical

C₂³×C₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₃

hypocenter

C₃

#3139

LCS

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
4. #5082	C₂	semidirect
5. #5079	C₂	semidirect
131. C₃	#3139	semidirect

H ≅ G/N

type

4. #5082

C₂

semidirect

5. #5079

C₂

semidirect

131. C₃

#3139

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
#5470	2
#5225	2
#5082	2
#5079	2
C₄×#208	2
#3139	3

subgroup

index

C₂×C₃×C₄⋊Q₈

Subgroups — #83198

Conjugacy classes of subgroups

Complements

Maximal subgroups