Group #68474 — subgroups

Subgroups — #68474

Order 384.

146 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 51 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	#5090	C₂	1	1	yes	maximal, characteristic
2	#5080	C₂	1	1	yes	maximal, characteristic
3	#4850	C₂	1	1	yes	maximal, characteristic
4	#4831	C₂	1	1	yes	maximal, characteristic
5	C₃×#332	C₂	1	1	yes	maximal, characteristic
6	#4481	C₂	1	1	yes	maximal, characteristic
7	#4479	C₂	1	1	yes	maximal, characteristic
8	#1692	—	3	1	no	maximal, Sylow
9	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₂×#216	—	2	1	no	—
11	#869	C₂²	1	1	yes	characteristic
12	C₄×Dic₆	C₂²	1	1	yes	characteristic
13	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	#835	—	2	1	no	—
16	#818	—	2	1	no	—
17	#816	—	2	1	no	—
18	#815	—	2	1	no	—
19	Dic₃⋊C₈	—	2	1	no	—
20	#808	—	2	1	no	—
21	#808	—	2	1	no	—
22	C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	#498	—	3	1	no	—
25	#477	—	3	1	no	—
26	#462	—	3	1	no	—
27	#446	—	3	1	no	—
28	#335	—	3	1	no	—
29	#333	—	3	1	no	—
30	#332	D₃	1	1	yes	characteristic
31	C₂×C₃×Q₈	—	2	1	no	—
32	C₂×Dic₆	—	2	1	no	—
33	C₂×Dic₆	D₄	1	1	yes	characteristic
34	C₂×C₃×C₈	—	2	1	no	—
35	C₂×C₃×C₈	—	2	1	no	—
36	C₃×C₄⋊C₄	—	2	1	no	—
37	C₃×C₄⋊C₄	—	2	1	no	—
38	C₃×C₄²	C₂³	1	1	yes	characteristic
39	#216	—	4	2	no	—
40	C₄⋊Dic₃	—	2	1	no	—
41	C₄⋊Dic₃	—	2	1	no	—
42	C₄⋊Dic₃	D₄	1	1	yes	characteristic
43	Dic₃⋊C₄	—	2	1	no	—
44	C₄×Dic₃	—	2	1	no	—
45	C₂×C₃⋊C₈	—	2	1	no	—
46	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
47	C₂×C₃⋊C₈	D₄	1	1	yes	characteristic
48	C₂×Q₁₆	—	6	1	no	—
49	C₄⋊Q₈	—	3	1	no	—
50	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
51	C₄×Q₈	—	3	1	no	—
52	#107	—	6	1	no	—
53	#106	—	6	1	no	—
54	#106	—	6	1	no	—
55	C₄⋊C₈	—	3	1	no	—
56	C₄⋊C₈	—	6	1	no	—
57	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
58	C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
59	#103	—	6	1	no	—
60	#103	—	6	1	no	—
61	#97	—	6	1	no	—
62	C₄×C₈	—	3	1	no	—
63	C₃×Q₈	—	4	1	no	—
64	C₃×Q₈	—	4	1	no	—
65	C₂×C₃×C₄	—	4	1	no	—
66	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
67	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
68	C₂×C₃×C₄	#41	1	1	yes	characteristic
69	C₂×Dic₃	—	2	1	no	—
70	C₂×Dic₃	—	2	1	no	—
71	C₂×Dic₃	—	4	1	no	—
72	Dic₆	—	2	1	no	—
73	Dic₆	—	2	1	no	—
74	Dic₆	—	8	1	no	—
75	C₃×C₈	—	4	1	no	—
76	C₃×C₈	—	4	1	no	—
77	C₃⋊C₈	—	2	1	no	—
78	C₃⋊C₈	—	4	1	no	—
79	C₃⋊C₈	Q₁₆	1	2	yes	complemented
80	C₂×Q₈	—	2	1	no	—
81	C₂×Q₈	—	3	1	no	—
82	C₂×Q₈	—	6	1	no	—
83	Q₁₆	—	12	2	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	2	1	no	—
86	C₂×C₈	—	3	1	no	—
87	C₂×C₈	—	3	1	no	—
88	C₂×C₈	—	6	1	no	—
89	C₄⋊C₄	—	2	1	no	—
90	C₄⋊C₄	—	2	1	no	—
91	C₄⋊C₄	—	3	1	no	—
92	C₄⋊C₄	—	6	1	no	—
93	C₄⋊C₄	—	6	1	no	—
94	C₄⋊C₄	—	6	1	no	—
95	C₄²	—	6	1	no	—
96	C₄²	C₂²×D₃	1	1	yes	characteristic
97	C₂²×C₃	#121	1	1	yes	characteristic
98	C₃×C₄	—	2	1	no	—
99	C₃×C₄	—	8	1	no	—
100	C₃×C₄	C₂×Q₁₆	1	1	yes	characteristic
101	C₃×C₄	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₃×C₄	#137	1	1	yes	characteristic
103	C₃×C₄	#137	1	1	yes	characteristic
104	Dic₃	—	4	1	no	—
105	Dic₃	—	4	1	no	—
106	Dic₃	—	8	1	no	—
107	Q₈	—	4	1	no	—
108	Q₈	—	4	1	no	—
109	Q₈	—	6	1	no	—
110	Q₈	—	6	1	no	—
111	Q₈	—	24	1	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	6	1	no	—
114	C₂×C₄	—	6	1	no	—
115	C₂×C₄	—	12	1	no	—
116	C₂×C₄	#235	1	1	yes	characteristic
117	C₂×C₄	D₃×D₄	1	1	yes	characteristic
118	C₂×C₄	D₃×D₄	1	1	yes	characteristic
119	C₈	—	3	2	no	—
120	C₈	—	4	1	no	—
121	C₈	—	4	1	no	—
122	C₈	—	6	1	no	—
123	C₈	—	12	1	no	—
124	C₂×C₃	#462	1	1	yes	characteristic
125	C₂×C₃	#468	1	1	yes	characteristic
126	C₂×C₃	#471	1	1	yes	characteristic
127	C₂²	#884	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	8	1	no	—
130	C₄	—	12	1	no	—
131	C₄	—	12	1	no	—
132	C₄	—	24	1	no	—
133	C₄	#911	1	1	yes	characteristic
134	C₄	#915	1	1	yes	characteristic
135	C₄	#915	1	1	yes	characteristic
136	C₄	D₃×Q₁₆	1	1	yes	characteristic
137	C₃	#1692	1	1	yes	characteristic, complemented, Sylow
138	C₂	#4754	1	1	yes	characteristic
139	C₂	#4780	1	1	yes	characteristic
140	C₂	#4795	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#5090

C₂

yes

maximal, characteristic

#5080

C₂

yes

maximal, characteristic

#4850

C₂

yes

maximal, characteristic

#4831

C₂

yes

maximal, characteristic

C₃×#332

C₂

yes

maximal, characteristic

#4481

C₂

yes

maximal, characteristic

#4479

C₂

yes

maximal, characteristic

#1692

—

maximal, Sylow

C₃×C₄⋊Q₈

C₂²

yes

characteristic

C₂×#216

—

#869

C₂²

yes

characteristic

C₄×Dic₆

C₂²

yes

characteristic

C₃×C₄⋊C₈

C₂²

yes

characteristic

C₃×C₄⋊C₈

C₂²

yes

—

—

—

—

Dic₃⋊C₈

—

#808

—

#808

—

C₄⋊(C₃⋊C₈)

C₂²

yes

characteristic

C₄×C₃⋊C₈

C₂²

yes

—

—

—

—

—

—

D₃

yes

characteristic

C₂×C₃×Q₈

—

C₂×Dic₆

—

C₂×Dic₆

D₄

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂³

yes

characteristic

#216

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

D₄

yes

characteristic

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

D₄

yes

characteristic

C₂×C₃⋊C₈

D₄

yes

characteristic

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃

yes

characteristic

C₄×Q₈

—

#107

—

#106

—

#106

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃

yes

characteristic

C₄⋊C₈

C₂×D₃

yes

—

—

—

C₄×C₈

—

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

#41

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₃⋊C₈

Q₁₆

yes

complemented

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

#121

yes

characteristic

C₃×C₄

—

C₃×C₄

—

100

C₃×C₄

C₂×Q₁₆

yes

characteristic

101

C₃×C₄

AGL(1,ℤ/8)

yes

characteristic

102

C₃×C₄

#137

yes

characteristic

103

C₃×C₄

#137

yes

characteristic

104

Dic₃

—

105

Dic₃

—

106

Dic₃

—

107

Q₈

—

108

Q₈

—

109

Q₈

—

110

Q₈

—

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

#235

yes

characteristic

117

C₂×C₄

D₃×D₄

yes

characteristic

118

C₂×C₄

D₃×D₄

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₂×C₃

#462

yes

characteristic

125

C₂×C₃

#468

yes

characteristic

126

C₂×C₃

#471

yes

characteristic

127

C₂²

#884

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#911

yes

characteristic

134

C₄

#915

yes

characteristic

135

C₄

#915

yes

characteristic

136

C₄

D₃×Q₁₆

yes

characteristic

137

C₃

#1692

yes

characteristic, complemented, Sylow

138

C₂

#4754

yes

characteristic

139

C₂

#4780

yes

characteristic

140

C₂

#4795

yes

characteristic

	subgroup	G/H
center	C₂²	#884
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#332	C₂
socle, prime radical	C₂²×C₃	#121
hypercenter	#332	D₃
hypocenter	C₃	#1692
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #332

subgroup

G/H

center

C₂²

#884

commutator, cosocle

C₃×C₄²

C₂³

Frattini

C₄²

C₂²×D₃

C₂

C₂²×C₃

D₃

C₃

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #332

N	H ≅ G/N	type
79. C₃⋊C₈	Q₁₆	semidirect
137. C₃	#1692	semidirect

H ≅ G/N

type

79. C₃⋊C₈

Q₁₆

semidirect

137. C₃

#1692

semidirect

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

subgroup	index
#5090	2
#5080	2
#4850	2
#4831	2
C₃×#332	2
#4481	2
#4479	2
#1692	3

subgroup

index

Subgroups — #68474

Conjugacy classes of subgroups

Complements

Maximal subgroups