Group #72487 — subgroups

Subgroups — C₃×#1772 #72487

Order 384.

232 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 232 conjugacy classes of subgroups — too many to draw. Only its 152 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×#108	C₂	1	1	yes	maximal, characteristic
2	C₃×#427	C₂	1	1	yes	maximal, characteristic
3	C₃×#405	C₂	1	2	yes	maximal
4	C₂×C₃×C₄×C₈	C₂	1	1	yes	maximal, characteristic
5	C₃×#339	C₂	1	2	yes	maximal
6	#1772	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
7	C₂×C₃×M₁₆	C₂²	1	2	yes	—
8	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₈	C₂²	1	1	yes	characteristic
10	C₃×#117	C₂²	1	2	yes	—
11	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
12	C₃×#108	C₄	1	4	yes	complemented
13	C₃×#107	C₄	1	2	yes	complemented
14	C₃×#106	C₄	1	2	yes	complemented
15	C₃×#97	—	2	2	no	—
16	C₃×C₄×C₈	—	2	2	no	—
17	C₃×C₄×C₈	C₄	1	4	yes	complemented
18	C₂×#108	C₂×C₃	1	1	yes	characteristic
19	#427	C₂×C₃	1	1	yes	characteristic
20	#405	C₂×C₃	1	2	yes	—
21	C₂×C₄×C₈	C₂×C₃	1	1	yes	characteristic
22	#339	C₂×C₃	1	2	yes	—
23	C₂²×C₃×C₄	—	2	1	no	—
24	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
25	C₃×M₁₆	—	2	2	no	—
26	C₃×M₁₆	C₂×C₄	1	4	yes	—
27	C₂×C₃×C₈	—	2	2	no	—
28	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
30	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
31	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
32	C₂×C₃×C₈	C₂×C₄	1	2	yes	—
33	C₂×C₃×C₈	D₄	1	1	yes	characteristic
34	C₂×C₃×C₈	D₄	1	1	yes	characteristic
35	C₂×C₃×C₈	Q₈	1	2	yes	—
36	C₃×C₄⋊C₄	C₂×C₄	1	4	yes	—
37	C₃×C₂²⋊C₄	—	2	2	no	—
38	C₃×C₄²	—	2	2	no	—
39	C₃×C₄²	—	2	1	no	—
40	C₃×C₄²	C₂×C₄	1	2	yes	—
41	C₂×M₁₆	C₂²×C₃	1	2	yes	—
42	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
43	C₂²×C₈	C₂²×C₃	1	1	yes	characteristic
44	#117	C₂²×C₃	1	2	yes	—
45	C₂×C₄²	C₂²×C₃	1	1	yes	characteristic
46	#108	C₃×C₄	1	4	yes	complemented
47	#107	C₃×C₄	1	2	yes	complemented
48	#106	C₃×C₄	1	2	yes	complemented
49	#97	—	2	2	no	—
50	C₄×C₈	—	2	2	no	—
51	C₄×C₈	C₃×C₄	1	4	yes	complemented
52	C₂³×C₃	#41	1	1	yes	characteristic
53	C₂×C₃×C₄	—	2	4	no	—
54	C₂×C₃×C₄	—	2	2	no	—
55	C₂×C₃×C₄	—	2	4	no	—
56	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
59	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
61	C₂×C₃×C₄	#41	1	1	yes	characteristic
62	C₃×C₈	—	2	4	no	—
63	C₃×C₈	C₄²	1	2	yes	—
64	C₃×C₈	C₄²	1	2	yes	—
65	C₃×C₈	C₄⋊C₄	1	2	yes	—
66	C₃×C₈	C₄⋊C₄	1	2	yes	—
67	C₂²×C₄	—	2	1	no	—
68	C₂²×C₄	C₂³×C₃	1	1	yes	characteristic
69	M₁₆	—	2	2	no	—
70	M₁₆	C₂×C₃×C₄	1	4	yes	—
71	C₂×C₈	—	2	2	no	—
72	C₂×C₈	C₂×C₃×C₄	1	2	yes	—
73	C₂×C₈	C₂×C₃×C₄	1	1	yes	characteristic
74	C₂×C₈	C₂×C₃×C₄	1	1	yes	characteristic
75	C₂×C₈	C₂×C₃×C₄	1	2	yes	—
76	C₂×C₈	C₂×C₃×C₄	1	2	yes	—
77	C₂×C₈	C₃×D₄	1	1	yes	characteristic
78	C₂×C₈	C₃×D₄	1	1	yes	characteristic
79	C₂×C₈	C₃×Q₈	1	2	yes	—
80	C₄⋊C₄	C₂×C₃×C₄	1	4	yes	—
81	C₂²⋊C₄	—	2	2	no	—
82	C₄²	—	2	2	no	—
83	C₄²	—	2	1	no	—
84	C₄²	C₂×C₃×C₄	1	2	yes	—
85	C₂²×C₃	—	2	2	no	—
86	C₂²×C₃	#117	1	1	yes	characteristic
87	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
88	C₂²×C₃	C₄×Q₈	1	1	yes	characteristic
89	C₃×C₄	—	2	4	no	—
90	C₃×C₄	—	4	4	no	—
91	C₃×C₄	C₂×C₄²	1	1	yes	characteristic
92	C₃×C₄	C₂×C₄⋊C₄	1	1	yes	characteristic
93	C₃×C₄	C₄×D₄	1	1	yes	characteristic
94	C₃×C₄	C₄×Q₈	1	1	yes	characteristic
95	C₂³	C₃×#41	1	1	yes	characteristic
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
100	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
101	C₂×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
102	C₂×C₄	C₂×C₃×D₄	1	1	yes	characteristic
103	C₂×C₄	C₂×C₃×Q₈	1	1	yes	characteristic
104	C₂×C₄	C₃×#41	1	1	yes	characteristic
105	C₈	—	2	4	no	—
106	C₈	C₃×C₄²	1	2	yes	—
107	C₈	C₃×C₄²	1	2	yes	—
108	C₈	C₃×C₄⋊C₄	1	2	yes	—
109	C₈	C₃×C₄⋊C₄	1	2	yes	—
110	C₂×C₃	—	2	1	no	—
111	C₂×C₃	—	2	1	no	—
112	C₂×C₃	C₄×C₄⋊C₄	1	1	yes	characteristic
113	C₂×C₃	#443	1	2	yes	—
114	C₂²	—	2	2	no	—
115	C₂²	C₃×#117	1	1	yes	characteristic
116	C₂²	C₃×C₄×D₄	1	1	yes	characteristic
117	C₂²	C₃×C₄×Q₈	1	1	yes	characteristic
118	C₄	—	2	4	no	—
119	C₄	—	4	4	no	—
120	C₄	C₂×C₃×C₄²	1	1	yes	characteristic
121	C₄	C₂×C₃×C₄⋊C₄	1	1	yes	characteristic
122	C₄	C₃×C₄×D₄	1	1	yes	characteristic
123	C₄	C₃×C₄×Q₈	1	1	yes	characteristic
124	C₃	#1772	1	1	yes	characteristic, direct product factor, Sylow
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	C₃×C₄×C₄⋊C₄	1	1	yes	characteristic
128	C₂	C₃×#443	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#108

C₂

yes

maximal, characteristic

C₃×#427

C₂

yes

maximal, characteristic

C₃×#405

C₂

yes

maximal

C₂×C₃×C₄×C₈

C₂

yes

maximal, characteristic

C₃×#339

C₂

yes

maximal

#1772

C₃

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₃×M₁₆

C₂²

yes

—

C₂²×C₃×C₈

C₂²

yes

characteristic

C₂²×C₃×C₈

C₂²

yes

characteristic

C₃×#117

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₃×#108

C₄

yes

complemented

C₃×#107

C₄

yes

complemented

C₃×#106

C₄

yes

complemented

C₃×#97

—

C₃×C₄×C₈

—

C₃×C₄×C₈

C₄

yes

complemented

C₂×#108

C₂×C₃

yes

characteristic

#427

C₂×C₃

yes

characteristic

#405

C₂×C₃

yes

—

C₂×C₄×C₈

C₂×C₃

yes

characteristic

#339

C₂×C₃

yes

—

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₃×M₁₆

—

C₃×M₁₆

C₂×C₄

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₄

yes

—

C₂×C₃×C₈

C₂×C₄

yes

characteristic

C₂×C₃×C₈

C₂×C₄

yes

characteristic

C₂×C₃×C₈

C₂×C₄

yes

—

C₂×C₃×C₈

C₂×C₄

yes

—

C₂×C₃×C₈

D₄

yes

characteristic

C₂×C₃×C₈

D₄

yes

characteristic

C₂×C₃×C₈

Q₈

yes

—

C₃×C₄⋊C₄

C₂×C₄

yes

—

C₃×C₂²⋊C₄

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂×C₄

yes

—

C₂×M₁₆

C₂²×C₃

yes

—

C₂²×C₈

C₂²×C₃

yes

characteristic

C₂²×C₈

C₂²×C₃

yes

characteristic

#117

C₂²×C₃

yes

—

C₂×C₄²

C₂²×C₃

yes

characteristic

#108

C₃×C₄

yes

complemented

#107

C₃×C₄

yes

complemented

#106

C₃×C₄

yes

complemented

#97

—

C₄×C₈

—

C₄×C₈

C₃×C₄

yes

complemented

C₂³×C₃

#41

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

characteristic

C₂×C₃×C₄

#41

yes

characteristic

C₃×C₈

—

C₃×C₈

C₄²

yes

—

C₃×C₈

C₄²

yes

—

C₃×C₈

C₄⋊C₄

yes

—

C₃×C₈

C₄⋊C₄

yes

—

C₂²×C₄

—

C₂²×C₄

C₂³×C₃

yes

characteristic

M₁₆

—

M₁₆

C₂×C₃×C₄

yes

—

C₂×C₈

—

C₂×C₈

C₂×C₃×C₄

yes

—

C₂×C₈

C₂×C₃×C₄

yes

characteristic

C₂×C₈

C₂×C₃×C₄

yes

characteristic

C₂×C₈

C₂×C₃×C₄

yes

—

C₂×C₈

C₂×C₃×C₄

yes

—

C₂×C₈

C₃×D₄

yes

characteristic

C₂×C₈

C₃×D₄

yes

characteristic

C₂×C₈

C₃×Q₈

yes

—

C₄⋊C₄

C₂×C₃×C₄

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂×C₃×C₄

yes

—

C₂²×C₃

—

C₂²×C₃

#117

yes

characteristic

C₂²×C₃

C₄×D₄

yes

characteristic

C₂²×C₃

C₄×Q₈

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂×C₄²

yes

characteristic

C₃×C₄

C₂×C₄⋊C₄

yes

characteristic

C₃×C₄

C₄×D₄

yes

characteristic

C₃×C₄

C₄×Q₈

yes

characteristic

C₂³

C₃×#41

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²×C₃×C₄

yes

characteristic

100

C₂×C₄

C₂²×C₃×C₄

yes

characteristic

101

C₂×C₄

C₂²×C₃×C₄

yes

characteristic

102

C₂×C₄

C₂×C₃×D₄

yes

characteristic

103

C₂×C₄

C₂×C₃×Q₈

yes

characteristic

104

C₂×C₄

C₃×#41

yes

characteristic

105

C₈

—

106

C₈

C₃×C₄²

yes

—

107

C₈

C₃×C₄²

yes

—

108

C₈

C₃×C₄⋊C₄

yes

—

109

C₈

C₃×C₄⋊C₄

yes

—

110

C₂×C₃

—

111

C₂×C₃

—

112

C₂×C₃

C₄×C₄⋊C₄

yes

characteristic

113

C₂×C₃

#443

yes

—

114

C₂²

—

115

C₂²

C₃×#117

yes

characteristic

116

C₂²

C₃×C₄×D₄

yes

characteristic

117

C₂²

C₃×C₄×Q₈

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

C₂×C₃×C₄²

yes

characteristic

121

C₄

C₂×C₃×C₄⋊C₄

yes

characteristic

122

C₄

C₃×C₄×D₄

yes

characteristic

123

C₄

C₃×C₄×Q₈

yes

characteristic

124

C₃

#1772

yes

characteristic, direct product factor, Sylow

125

C₂

—

126

C₂

—

127

C₂

C₃×C₄×C₄⋊C₄

yes

characteristic

128

C₂

C₃×#443

yes

—

	subgroup	G/H
center	C₂×C₃×C₈	D₄
commutator	C₄	C₂×C₃×C₄²
Frattini, cosocle	C₂²×C₄	C₂³×C₃
socle	C₂²×C₃	C₄×D₄
prime radical	C₂³×C₃	#41
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₈ ◁ C₂²×C₃×C₈ ◁ G

subgroup

G/H

center

C₂×C₃×C₈

D₄

commutator

C₄

C₂×C₃×C₄²

Frattini, cosocle

C₂²×C₄

C₂³×C₃

socle

C₂²×C₃

C₄×D₄

C₂³×C₃

G ▷ C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₃×C₈ ◁ C₂²×C₃×C₈ ◁ G

N	H ≅ G/N	type
6. #1772	C₃	direct (N × H)
12. C₃×#108	C₄	semidirect
13. C₃×#107	C₄	semidirect
14. C₃×#106	C₄	semidirect
17. C₃×C₄×C₈	C₄	semidirect
46. #108	C₃×C₄	semidirect
47. #107	C₃×C₄	semidirect
48. #106	C₃×C₄	semidirect
51. C₄×C₈	C₃×C₄	semidirect
124. C₃	#1772	direct (N × H)

H ≅ G/N

type

6. #1772

C₃

direct (N × H)

12. C₃×#108

C₄

semidirect

13. C₃×#107

C₄

semidirect

14. C₃×#106

C₄

semidirect

17. C₃×C₄×C₈

C₄

semidirect

46. #108

C₃×C₄

semidirect

47. #107

C₃×C₄

semidirect

48. #106

C₃×C₄

semidirect

51. C₄×C₈

C₃×C₄

semidirect

124. C₃

#1772

direct (N × H)

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

subgroup	index
C₂×C₃×#108	2
C₃×#427	2
C₃×#405	2
C₂×C₃×C₄×C₈	2
C₃×#339	2
#1772	3

subgroup

index

C₂×C₃×#108

C₂×C₃×C₄×C₈

Subgroups — C₃×#1772 #72487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×#1772 #72487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#1772 #72487