Group #72226200 — subgroups

Subgroups — #72226200

Order 3072.

846 conjugacy classes of subgroups; only the 383 normal subgroups are shown.

normal subgroups only

This group has 846 conjugacy classes of subgroups — too many to draw. Only its 383 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×#122211	C₂	1	1	yes	maximal, characteristic
2	#72225823	C₂	1	3	yes	maximal
3	#72225815	C₂	1	3	yes	maximal
4	#72214006	—	3	1	no	maximal, Sylow
5	#10711613	—	2	6	no	—
6	#10706823	—	2	6	no	—
7	#10688988	C₂²	1	3	yes	—
8	#10688954	C₂²	1	1	yes	characteristic
9	C₃×C₈×M₃₂	C₂²	1	3	yes	—
10	C₃×#7052	C₄	1	3	yes	complemented
11	C₃×C₁₆²	C₄	1	1	yes	characteristic, complemented
12	#135792	—	3	3	no	—
13	#135758	—	3	3	no	—
14	#122211	D₃	1	1	yes	characteristic
15	C₃×C₄×M₃₂	D₄	1	6	yes	—
16	C₂×C₃×C₈²	C₂³	1	1	yes	characteristic
17	C₂×#4513	—	4	12	no	—
18	#69792	—	2	3	no	—
19	#69764	—	2	3	no	—
20	#69730	—	2	6	no	—
21	C₃×#1311	Q₈	1	6	yes	—
22	C₃×C₈×C₁₆	C₂×C₄	1	3	yes	—
23	C₃×C₈×C₁₆	C₂×C₄	1	3	yes	—
24	#67513	—	4	12	no	—
25	#12903	—	6	6	no	—
26	#12020	—	6	6	no	—
27	C₈×M₃₂	C₂×D₃	1	3	yes	—
28	#8822	—	3	3	no	—
29	#8813	—	3	1	no	—
30	#7052	Dic₃	1	3	yes	complemented
31	C₁₆²	Dic₃	1	1	yes	characteristic, complemented
32	C₂×C₃×M₃₂	SD₁₆	1	12	yes	—
33	C₂×C₃×C₄×C₈	C₂×D₄	1	3	yes	—
34	C₂×#819	—	4	6	no	—
35	C₂×#817	—	4	6	no	—
36	#4935	—	4	1	no	—
37	#4924	—	4	3	no	—
38	C₃×#346	SD₁₆	1	12	yes	—
39	C₃×C₄×C₁₆	C₄⋊C₄	1	6	yes	—
40	C₃×C₄×C₁₆	C₄⋊C₄	1	6	yes	—
41	C₃×C₈²	C₂²×C₄	1	1	yes	characteristic
42	C₃×C₈²	C₂×Q₈	1	3	yes	—
43	#4513	—	8	24	no	—
44	#4457	—	4	12	no	—
45	C₂×#365	—	12	12	no	—
46	C₄×M₃₂	D₁₂	1	6	yes	—
47	#1848	—	6	3	no	—
48	#1844	—	6	3	no	—
49	#1566	—	6	6	no	—
50	C₂×C₈²	C₂²×D₃	1	1	yes	characteristic
51	#1368	—	12	12	no	—
52	#1311	Dic₆	1	6	yes	—
53	C₈×C₁₆	C₂×Dic₃	1	3	yes	—
54	C₈×C₁₆	C₂×Dic₃	1	3	yes	—
55	C₂²×C₃×C₈	C₂×SD₁₆	1	6	yes	—
56	C₂×C₃×C₄²	#127	1	1	yes	characteristic
57	C₂×C₄⋊Dic₃	—	8	3	no	—
58	C₂×#208	—	8	3	no	—
59	C₃×M₃₂	—	2	24	no	—
60	C₂×C₃×C₁₆	#106	1	12	yes	—
61	C₂×C₃×C₁₆	#106	1	12	yes	—
62	C₃×C₄×C₈	C₂×C₄⋊C₄	1	3	yes	—
63	C₃×C₄×C₈	C₄⋊Q₈	1	3	yes	—
64	C₃×C₄×C₈	C₂×SD₁₆	1	6	yes	—
65	#819	—	8	12	no	—
66	#817	—	8	12	no	—
67	C₄⋊(C₃⋊C₈)	—	8	6	no	—
68	C₂×M₃₂	#204	1	12	yes	—
69	C₂×#108	—	12	6	no	—
70	C₂×#106	—	12	6	no	—
71	#423	—	12	3	no	—
72	C₂×C₄×C₈	C₂×D₁₂	1	3	yes	—
73	#384	—	12	1	no	—
74	#365	—	24	24	no	—
75	#346	#204	1	12	yes	—
76	C₄×C₁₆	C₄⋊Dic₃	1	6	yes	—
77	C₄×C₁₆	C₄⋊Dic₃	1	6	yes	—
78	#334	—	12	12	no	—
79	C₈²	C₂×Dic₆	1	3	yes	—
80	C₈²	C₂²×Dic₃	1	1	yes	characteristic
81	C₂²×C₃×C₄	#492	1	3	yes	—
82	C₂²×Dic₃	—	16	1	no	—
83	C₂×C₃×C₈	—	2	12	no	—
84	C₂×C₃×C₈	C₂×#106	1	6	yes	—
85	C₂×C₃×C₈	#498	1	6	yes	—
86	C₃×C₄²	#384	1	1	yes	characteristic
87	C₃×C₄²	#492	1	3	yes	—
88	C₄⋊Dic₃	—	16	6	no	—
89	#208	—	16	6	no	—
90	C₂×C₃⋊C₈	—	16	3	no	—
91	C₃×C₁₆	—	2	12	no	—
92	C₃×C₁₆	#365	1	24	yes	complemented
93	C₂×M₁₆	—	24	3	no	—
94	C₂²×C₈	C₂×#204	1	6	yes	—
95	C₂×C₄⋊C₄	—	24	3	no	—
96	C₂×C₄²	#874	1	1	yes	characteristic
97	M₃₂	—	2	24	no	—
98	C₂×C₁₆	#817	1	12	yes	—
99	C₂×C₁₆	#817	1	12	yes	—
100	#108	—	24	12	no	—
101	#106	—	24	12	no	—
102	C₄⋊C₈	—	24	6	no	—
103	C₄×C₈	#869	1	3	yes	—
104	C₄×C₈	C₂×#204	1	6	yes	—
105	C₄×C₈	C₂×C₄⋊Dic₃	1	3	yes	—
106	C₂³×C₃	#1705	1	1	yes	characteristic
107	C₂×C₃×C₄	—	2	6	no	—
108	C₂×C₃×C₄	#1705	1	3	yes	—
109	C₂×C₃×C₄	#1844	1	3	yes	—
110	C₂×Dic₃	—	32	3	no	—
111	C₃×C₈	—	2	6	no	—
112	C₃×C₈	C₂×#365	1	12	yes	—
113	C₃⋊C₈	—	32	3	no	—
114	C₂²×C₄	—	48	1	no	—
115	C₂²×C₄	#4693	1	3	yes	—
116	M₁₆	—	48	6	no	—
117	C₂×C₈	—	2	12	no	—
118	C₂×C₈	—	48	3	no	—
119	C₂×C₈	#4680	1	6	yes	—
120	C₂×C₈	C₂×#817	1	6	yes	—
121	C₄⋊C₄	—	48	6	no	—
122	C₄²	#4693	1	3	yes	—
123	C₄²	#4935	1	1	yes	characteristic
124	C₁₆	—	2	12	no	—
125	C₁₆	#4513	1	24	yes	complemented
126	C₂²×C₃	—	2	3	no	—
127	C₂²×C₃	#8813	1	1	yes	characteristic
128	C₃×C₄	—	2	3	no	—
129	C₃×C₄	#12903	1	6	yes	—
130	Dic₃	—	64	1	no	—
131	C₂³	#68057	1	1	yes	characteristic
132	C₂×C₄	—	2	6	no	—
133	C₂×C₄	—	96	3	no	—
134	C₂×C₄	#68057	1	3	yes	—
135	C₂×C₄	#69764	1	3	yes	—
136	C₈	—	2	6	no	—
137	C₈	—	96	3	no	—
138	C₈	C₂×#4513	1	12	yes	—
139	C₂×C₃	—	4	1	no	—
140	C₂×C₃	#135758	1	3	yes	—
141	C₂²	—	2	3	no	—
142	C₂²	#10688954	1	1	yes	characteristic
143	C₄	—	2	3	no	—
144	C₄	—	192	1	no	—
145	C₄	#10711613	1	6	yes	—
146	C₃	#72214006	1	1	yes	characteristic, complemented, Sylow
147	C₂	—	4	1	no	—
148	C₂	#72225815	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#122211

C₂

yes

maximal, characteristic

#72225823

C₂

yes

maximal

#72225815

C₂

yes

—

—

—

C₂²

yes

—

#10688954

C₂²

yes

characteristic

C₃×C₈×M₃₂

C₂²

yes

—

C₃×#7052

C₄

yes

complemented

C₃×C₁₆²

C₄

yes

characteristic, complemented

#135792

—

#135758

—

#122211

D₃

yes

characteristic

C₃×C₄×M₃₂

D₄

yes

—

C₂×C₃×C₈²

C₂³

yes

—

—

—

—

Q₈

yes

—

C₃×C₈×C₁₆

C₂×C₄

yes

—

C₃×C₈×C₁₆

C₂×C₄

yes

—

#67513

—

#12903

—

#12020

—

C₈×M₃₂

C₂×D₃

yes

—

#8822

—

#8813

—

#7052

Dic₃

yes

complemented

C₁₆²

Dic₃

yes

characteristic, complemented

C₂×C₃×M₃₂

SD₁₆

yes

—

C₂×C₃×C₄×C₈

C₂×D₄

yes

—

—

—

—

—

SD₁₆

yes

—

C₃×C₄×C₁₆

C₄⋊C₄

yes

—

C₃×C₄×C₁₆

C₄⋊C₄

yes

—

C₃×C₈²

C₂²×C₄

yes

characteristic

C₃×C₈²

C₂×Q₈

yes

—

#4513

—

#4457

—

C₂×#365

—

C₄×M₃₂

D₁₂

yes

—

#1848

—

#1844

—

#1566

—

C₂×C₈²

C₂²×D₃

yes

characteristic

#1368

—

#1311

Dic₆

yes

—

C₈×C₁₆

C₂×Dic₃

yes

—

C₈×C₁₆

C₂×Dic₃

yes

—

C₂²×C₃×C₈

C₂×SD₁₆

yes

—

C₂×C₃×C₄²

#127

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×#208

—

C₃×M₃₂

—

C₂×C₃×C₁₆

#106

yes

—

C₂×C₃×C₁₆

#106

yes

—

C₃×C₄×C₈

C₂×C₄⋊C₄

yes

—

C₃×C₄×C₈

C₄⋊Q₈

yes

—

C₃×C₄×C₈

C₂×SD₁₆

yes

—

#819

—

#817

—

C₄⋊(C₃⋊C₈)

—

C₂×M₃₂

#204

yes

—

C₂×#108

—

C₂×#106

—

#423

—

C₂×C₄×C₈

C₂×D₁₂

yes

—

—

—

yes

—

C₄×C₁₆

C₄⋊Dic₃

yes

—

C₄×C₁₆

C₄⋊Dic₃

yes

—

#334

—

C₈²

C₂×Dic₆

yes

—

C₈²

C₂²×Dic₃

yes

characteristic

C₂²×C₃×C₄

#492

yes

—

C₂²×Dic₃

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×#106

yes

—

C₂×C₃×C₈

#498

yes

—

C₃×C₄²

#384

yes

characteristic

C₃×C₄²

#492

yes

—

C₄⋊Dic₃

—

#208

—

C₂×C₃⋊C₈

—

C₃×C₁₆

—

C₃×C₁₆

#365

yes

complemented

C₂×M₁₆

—

C₂²×C₈

C₂×#204

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

#874

yes

characteristic

M₃₂

—

C₂×C₁₆

#817

yes

—

C₂×C₁₆

#817

yes

—

100

#108

—

101

#106

—

102

C₄⋊C₈

—

103

C₄×C₈

#869

yes

—

104

C₄×C₈

C₂×#204

yes

—

105

C₄×C₈

C₂×C₄⋊Dic₃

yes

—

106

C₂³×C₃

#1705

yes

characteristic

107

C₂×C₃×C₄

—

108

C₂×C₃×C₄

#1705

yes

—

109

C₂×C₃×C₄

#1844

yes

—

110

C₂×Dic₃

—

111

C₃×C₈

—

112

C₃×C₈

C₂×#365

yes

—

113

C₃⋊C₈

—

114

C₂²×C₄

—

115

C₂²×C₄

#4693

yes

—

116

M₁₆

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

#4680

yes

—

120

C₂×C₈

C₂×#817

yes

—

121

C₄⋊C₄

—

122

C₄²

#4693

yes

—

123

C₄²

#4935

yes

characteristic

124

C₁₆

—

125

C₁₆

#4513

yes

complemented

126

C₂²×C₃

—

127

C₂²×C₃

#8813

yes

characteristic

128

C₃×C₄

—

129

C₃×C₄

#12903

yes

—

130

Dic₃

—

131

C₂³

#68057

yes

characteristic

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

#68057

yes

—

135

C₂×C₄

#69764

yes

—

136

C₈

—

137

C₈

—

138

C₈

C₂×#4513

yes

—

139

C₂×C₃

—

140

C₂×C₃

#135758

yes

—

141

C₂²

—

142

C₂²

#10688954

yes

characteristic

143

C₄

—

144

C₄

—

192

—

145

C₄

#10711613

yes

—

146

C₃

#72214006

yes

characteristic, complemented, Sylow

147

C₂

—

148

C₂

#72225815

yes

—

	subgroup	G/H
center	C₂²	#10688954
commutator	C₃×C₈²	C₂²×C₄
Frattini	C₂×C₈²	C₂²×D₃
Fitting	C₃×#122211	C₂
socle	C₂²×C₃	#8813
prime radical	C₂³×C₃	#1705
cosocle	C₂×C₃×C₈²	C₂³
hypercenter	#122211	D₃
hypocenter	C₃	#72214006
LCS	G ▷ C₃×C₈² ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ #122211

subgroup

G/H

center

C₂²

#10688954

commutator

C₃×C₈²

C₂²×C₄

Frattini

C₂×C₈²

C₂²×D₃

C₂

C₂²×C₃

C₂³×C₃

C₂×C₃×C₈²

C₂³

D₃

C₃

G ▷ C₃×C₈² ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈² ◁ #122211

N	H ≅ G/N	type
10. C₃×#7052	C₄	semidirect
11. C₃×C₁₆²	C₄	semidirect
30. #7052	Dic₃	semidirect
31. C₁₆²	Dic₃	semidirect
92. C₃×C₁₆	#365	semidirect
125. C₁₆	#4513	semidirect
146. C₃	#72214006	semidirect

H ≅ G/N

type

10. C₃×#7052

C₄

semidirect

11. C₃×C₁₆²

C₄

semidirect

30. #7052

Dic₃

semidirect

31. C₁₆²

Dic₃

semidirect

92. C₃×C₁₆

#365

semidirect

125. C₁₆

#4513

semidirect

146. C₃

#72214006

semidirect

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

subgroup	index
C₃×#122211	2
#72225823	2
#72225815	2
#72214006	3

subgroup

index

Subgroups — #72226200

Conjugacy classes of subgroups

Complements

Maximal subgroups