Group #10630695 — subgroups

Subgroups — C₃×#12088 #10630695

Order 768.

198 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 90 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	C₃×#2176	C₂	1	1	yes	maximal, characteristic
2	C₃×#1593	C₂	1	2	yes	maximal
3	C₃×#1368	C₂	1	1	yes	maximal, characteristic
4	C₃×#1334	C₂	1	2	yes	maximal
5	C₃×#1313	C₂	1	1	yes	maximal, characteristic
6	#12088	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×Q₃₂	C₄	1	1	yes	characteristic
8	C₃×#446	—	2	2	no	—
9	C₃×C₄×Q₁₆	C₂²	1	2	yes	—
10	C₃×#367	C₄	1	1	yes	characteristic
11	C₃×#358	C₄	1	2	yes	—
12	C₃×#346	C₂²	1	1	yes	characteristic
13	C₃×#346	C₂²	1	1	yes	characteristic
14	C₃×#334	C₂²	1	2	yes	—
15	C₃×Q₈⋊C₈	—	2	2	no	—
16	C₃×#322	C₂²	1	1	yes	characteristic
17	#2176	C₂×C₃	1	1	yes	characteristic
18	#1593	C₂×C₃	1	2	yes	—
19	#1368	C₂×C₃	1	1	yes	characteristic
20	#1334	C₂×C₃	1	2	yes	—
21	#1313	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×Q₁₆	C₂×C₄	1	2	yes	—
23	C₃×C₄×Q₈	—	2	2	no	—
24	C₃×Q₃₂	—	2	2	no	—
25	C₂×C₃×C₁₆	—	2	1	no	—
26	C₂×C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
27	C₂×C₃×C₁₆	C₂×C₄	1	1	yes	characteristic
28	C₃×#107	C₂×C₄	1	2	yes	—
29	C₃×C₄⋊C₈	—	2	2	no	—
30	C₃×C₄⋊C₈	—	4	2	no	—
31	C₃×#103	—	2	2	no	—
32	C₃×#97	—	4	2	no	—
33	C₃×C₄×C₈	D₄	1	2	yes	—
34	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
35	C₂×Q₃₂	C₃×C₄	1	1	yes	characteristic
36	#446	—	2	2	no	—
37	C₄×Q₁₆	C₂²×C₃	1	2	yes	—
38	#367	C₃×C₄	1	1	yes	characteristic
39	#358	C₃×C₄	1	2	yes	—
40	#346	C₂²×C₃	1	1	yes	characteristic
41	#346	C₂²×C₃	1	1	yes	characteristic
42	#334	C₂²×C₃	1	2	yes	—
43	Q₈⋊C₈	—	2	2	no	—
44	#322	C₂²×C₃	1	1	yes	characteristic
45	C₂×C₃×Q₈	—	2	2	no	—
46	C₃×Q₁₆	—	2	2	no	—
47	C₃×Q₁₆	—	2	2	no	—
48	C₂×C₃×C₈	—	2	2	no	—
49	C₂×C₃×C₈	—	4	2	no	—
50	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₃×C₈	#41	1	1	yes	characteristic
52	C₃×C₄⋊C₄	—	2	2	no	—
53	C₃×C₄⋊C₄	—	4	2	no	—
54	C₃×C₄²	—	4	2	no	—
55	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
56	C₃×C₁₆	—	2	1	no	—
57	C₃×C₁₆	—	4	1	no	—
58	C₃×C₁₆	M₁₆	1	2	yes	—
59	C₂×Q₁₆	C₂×C₃×C₄	1	2	yes	—
60	C₄×Q₈	—	2	2	no	—
61	Q₃₂	—	2	2	no	—
62	C₂×C₁₆	—	2	1	no	—
63	C₂×C₁₆	C₂×C₃×C₄	1	1	yes	characteristic
64	C₂×C₁₆	C₂×C₃×C₄	1	1	yes	characteristic
65	#107	C₂×C₃×C₄	1	2	yes	—
66	C₄⋊C₈	—	2	2	no	—
67	C₄⋊C₈	—	4	2	no	—
68	#103	—	2	2	no	—
69	#97	—	4	2	no	—
70	C₄×C₈	C₃×D₄	1	2	yes	—
71	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
72	C₃×Q₈	—	4	2	no	—
73	C₃×Q₈	—	4	2	no	—
74	C₂×C₃×C₄	—	4	2	no	—
75	C₂×C₃×C₄	—	4	2	no	—
76	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
77	C₂×C₃×C₄	AGL(1,ℤ/8)	1	2	yes	—
78	C₃×C₈	—	2	1	no	—
79	C₃×C₈	—	4	2	no	—
80	C₃×C₈	—	8	2	no	—
81	C₃×C₈	C₂×M₁₆	1	1	yes	characteristic
82	C₃×C₈	#131	1	1	yes	characteristic
83	C₂×Q₈	—	2	2	no	—
84	Q₁₆	—	2	2	no	—
85	Q₁₆	—	2	2	no	—
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	4	2	no	—
88	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
89	C₂×C₈	C₃×#41	1	1	yes	characteristic
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	4	2	no	—
92	C₄²	—	4	2	no	—
93	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
94	C₁₆	—	2	1	no	—
95	C₁₆	—	4	1	no	—
96	C₁₆	C₃×M₁₆	1	2	yes	—
97	C₂²×C₃	#442	1	1	yes	characteristic
98	C₃×C₄	—	2	2	no	—
99	C₃×C₄	—	8	2	no	—
100	C₃×C₄	—	8	2	no	—
101	C₃×C₄	#436	1	1	yes	characteristic
102	C₃×C₄	#444	1	1	yes	characteristic
103	Q₈	—	4	2	no	—
104	Q₈	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
108	C₂×C₄	C₃×AGL(1,ℤ/8)	1	2	yes	—
109	C₈	—	2	1	no	—
110	C₈	—	4	2	no	—
111	C₈	—	8	2	no	—
112	C₈	C₂×C₃×M₁₆	1	1	yes	characteristic
113	C₈	C₃×#131	1	1	yes	characteristic
114	C₂×C₃	#1594	1	1	yes	characteristic
115	C₂×C₃	#2180	1	2	yes	—
116	C₂²	C₃×#442	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	8	2	no	—
119	C₄	—	8	2	no	—
120	C₄	C₃×#436	1	1	yes	characteristic
121	C₄	C₃×#444	1	1	yes	characteristic
122	C₃	#12088	1	1	yes	characteristic, direct product factor, Sylow
123	C₂	C₃×#1594	1	1	yes	characteristic
124	C₂	C₃×#2180	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#2176

C₂

yes

maximal, characteristic

C₃×#1593

C₂

yes

maximal

C₃×#1368

C₂

yes

maximal, characteristic

C₃×#1334

C₂

yes

maximal

C₃×#1313

C₂

yes

maximal, characteristic

#12088

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₃×Q₃₂

C₄

yes

characteristic

C₃×#446

—

C₃×C₄×Q₁₆

C₂²

yes

—

C₃×#367

C₄

yes

characteristic

C₃×#358

C₄

yes

—

C₃×#346

C₂²

yes

characteristic

C₃×#346

C₂²

yes

characteristic

C₃×#334

C₂²

yes

—

C₃×Q₈⋊C₈

—

C₃×#322

C₂²

yes

characteristic

#2176

C₂×C₃

yes

characteristic

#1593

C₂×C₃

yes

—

#1368

C₂×C₃

yes

characteristic

#1334

C₂×C₃

yes

—

#1313

C₂×C₃

yes

characteristic

C₂×C₃×Q₁₆

C₂×C₄

yes

—

C₃×C₄×Q₈

—

C₃×Q₃₂

—

C₂×C₃×C₁₆

—

C₂×C₃×C₁₆

C₂×C₄

yes

characteristic

C₂×C₃×C₁₆

C₂×C₄

yes

characteristic

C₃×#107

C₂×C₄

yes

—

C₃×C₄⋊C₈

—

C₃×C₄⋊C₈

—

C₃×#103

—

C₃×#97

—

C₃×C₄×C₈

D₄

yes

—

C₃×C₄×C₈

C₂³

yes

characteristic

C₂×Q₃₂

C₃×C₄

yes

characteristic

#446

—

C₄×Q₁₆

C₂²×C₃

yes

—

#367

C₃×C₄

yes

characteristic

#358

C₃×C₄

yes

—

#346

C₂²×C₃

yes

characteristic

#346

C₂²×C₃

yes

characteristic

#334

C₂²×C₃

yes

—

Q₈⋊C₈

—

#322

C₂²×C₃

yes

characteristic

C₂×C₃×Q₈

—

C₃×Q₁₆

—

C₃×Q₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²×C₄

yes

characteristic

C₂×C₃×C₈

#41

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×D₄

yes

characteristic

C₃×C₁₆

—

C₃×C₁₆

—

C₃×C₁₆

M₁₆

yes

—

C₂×Q₁₆

C₂×C₃×C₄

yes

—

C₄×Q₈

—

Q₃₂

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₃×C₄

yes

characteristic

C₂×C₁₆

C₂×C₃×C₄

yes

characteristic

#107

C₂×C₃×C₄

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#97

—

C₄×C₈

C₃×D₄

yes

—

C₄×C₈

C₂³×C₃

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄×D₄

yes

characteristic

C₂×C₃×C₄

AGL(1,ℤ/8)

yes

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

C₂×M₁₆

yes

characteristic

C₃×C₈

#131

yes

characteristic

C₂×Q₈

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₃×C₄

yes

characteristic

C₂×C₈

C₃×#41

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂×C₃×D₄

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

C₃×M₁₆

yes

—

C₂²×C₃

#442

yes

characteristic

C₃×C₄

—

C₃×C₄

—

100

C₃×C₄

—

101

C₃×C₄

#436

yes

characteristic

102

C₃×C₄

#444

yes

characteristic

103

Q₈

—

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₃×C₄×D₄

yes

characteristic

108

C₂×C₄

C₃×AGL(1,ℤ/8)

yes

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

C₂×C₃×M₁₆

yes

characteristic

113

C₈

C₃×#131

yes

characteristic

114

C₂×C₃

#1594

yes

characteristic

115

C₂×C₃

#2180

yes

—

116

C₂²

C₃×#442

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

C₃×#436

yes

characteristic

121

C₄

C₃×#444

yes

characteristic

122

C₃

#12088

yes

characteristic, direct product factor, Sylow

123

C₂

C₃×#1594

yes

characteristic

124

C₂

C₃×#2180

yes

—

	subgroup	G/H
center, socle, prime radical	C₂²×C₃	#442
commutator	C₂×C₈	C₂²×C₃×C₄
Frattini, cosocle	C₄×C₈	C₂³×C₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#322 ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₃

#442

commutator

C₂×C₈

C₂²×C₃×C₄

Frattini, cosocle

C₄×C₈

C₂³×C₃

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×#322 ◁ G

N	H ≅ G/N	type
6. #12088	C₃	direct (N × H)
122. C₃	#12088	direct (N × H)

H ≅ G/N

type

6. #12088

C₃

direct (N × H)

122. C₃

#12088

direct (N × H)

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

subgroup	index
C₃×#2176	2
C₃×#1593	2
C₃×#1368	2
C₃×#1334	2
C₃×#1313	2
#12088	3

subgroup

index

Subgroups — C₃×#12088 #10630695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#12088 #10630695

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#12088 #10630695