Group #88711 — subgroups

Subgroups — D₃×D₄×C₉ #88711

Order 432.

178 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 75 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	C₂²×D₃×C₉	C₂	1	2	yes	maximal, complemented
2	C₃×D₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₉×#67	C₂	1	2	yes	maximal, complemented
4	C₉×D₁₂	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄×D₃×C₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×D₃×D₄	C₃	1	1	yes	maximal, characteristic
7	C₂×D₄×C₉	—	3	1	no	maximal
8	C₂²×C₃×C₉	C₂²	1	2	yes	complemented
9	C₂×D₃×C₉	—	2	4	no	—
10	C₂×D₃×C₉	C₂²	1	1	yes	characteristic
11	C₂×D₃×C₉	C₂²	1	2	yes	complemented
12	C₃×C₄×C₉	C₂²	1	1	yes	characteristic, complemented
13	C₉×Dic₃	C₂²	1	1	yes	characteristic, complemented
14	C₂²×C₃×D₃	C₂×C₃	1	2	yes	—
15	C₃²×D₄	C₂×C₃	1	1	yes	characteristic
16	C₃×#67	C₂×C₃	1	2	yes	—
17	C₃×D₁₂	C₂×C₃	1	1	yes	characteristic
18	C₃×C₄×D₃	C₂×C₃	1	1	yes	characteristic
19	C₂³×C₉	—	3	2	no	—
20	D₄×C₉	—	2	1	no	—
21	D₄×C₉	—	3	2	no	—
22	D₄×C₉	—	3	1	no	—
23	D₄×C₉	D₃	1	1	yes	characteristic, direct product factor
24	C₂×C₄×C₉	—	3	1	no	—
25	C₂×C₃×C₉	—	2	2	no	—
26	C₂×C₃×C₉	C₂³	1	1	yes	characteristic
27	D₃×C₉	—	2	2	no	—
28	D₃×C₉	D₄	1	2	yes	direct product factor
29	C₂×C₃×D₄	—	3	1	no	—
30	D₃×D₄	C₉	1	1	yes	characteristic, direct product factor
31	C₂²×C₃²	C₂²×C₃	1	2	yes	—
32	C₂×C₃×D₃	—	2	4	no	—
33	C₂×C₃×D₃	C₂²×C₃	1	1	yes	characteristic
34	C₂×C₃×D₃	C₂²×C₃	1	2	yes	—
35	C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
36	C₃×Dic₃	C₂²×C₃	1	1	yes	characteristic
37	C₂²×C₉	—	2	2	no	—
38	C₂²×C₉	—	3	1	no	—
39	C₂²×C₉	—	3	2	no	—
40	C₂²×C₉	—	6	4	no	—
41	C₂²×C₉	C₂×D₃	1	2	yes	complemented
42	C₄×C₉	—	2	1	no	—
43	C₄×C₉	—	3	1	no	—
44	C₄×C₉	C₂×D₃	1	1	yes	characteristic, complemented
45	C₃×C₉	C₂×D₄	1	1	yes	characteristic, complemented, Sylow
46	C₂³×C₃	—	3	2	no	—
47	C₂²×D₃	C₂×C₉	1	2	yes	complemented
48	C₃×D₄	—	2	1	no	—
49	C₃×D₄	—	3	2	no	—
50	C₃×D₄	—	3	1	no	—
51	C₃×D₄	C₂×C₉	1	1	yes	characteristic, complemented
52	C₃×D₄	C₃×D₃	1	1	yes	characteristic
53	C₂×C₃×C₄	—	3	1	no	—
54	#67	C₂×C₉	1	2	yes	complemented
55	D₁₂	C₂×C₉	1	1	yes	characteristic, complemented
56	C₄×D₃	C₂×C₉	1	1	yes	characteristic, complemented
57	C₂×C₃²	—	2	2	no	—
58	C₂×C₃²	C₂³×C₃	1	1	yes	characteristic
59	C₃×D₃	—	2	2	no	—
60	C₃×D₃	C₃×D₄	1	2	yes	—
61	C₂×C₉	—	2	2	no	—
62	C₂×C₉	—	2	1	no	—
63	C₂×C₉	—	3	2	no	—
64	C₂×C₉	—	4	2	no	—
65	C₂×C₉	—	6	2	no	—
66	C₂×C₉	C₂²×D₃	1	1	yes	characteristic
67	C₂×D₄	—	3	1	no	Sylow
68	C₂²×C₃	—	2	2	no	—
69	C₂²×C₃	—	3	1	no	—
70	C₂²×C₃	—	3	2	no	—
71	C₂²×C₃	—	6	4	no	—
72	C₂²×C₃	C₂²×C₉	1	2	yes	complemented
73	C₂²×C₃	C₂×C₃×D₃	1	2	yes	—
74	C₂×D₃	—	2	4	no	—
75	C₂×D₃	C₂²×C₉	1	1	yes	characteristic
76	C₂×D₃	C₂²×C₉	1	2	yes	complemented
77	C₃×C₄	—	2	1	no	—
78	C₃×C₄	—	3	1	no	—
79	C₃×C₄	C₂²×C₉	1	1	yes	characteristic, complemented
80	C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
81	Dic₃	C₂²×C₉	1	1	yes	characteristic, complemented
82	C₃²	C₂×C₃×D₄	1	1	yes	characteristic
83	C₉	—	2	1	no	—
84	C₉	D₃×D₄	1	1	yes	characteristic, direct product factor
85	C₂³	—	3	2	no	—
86	D₄	—	3	2	no	—
87	D₄	—	3	1	no	—
88	D₄	D₃×C₉	1	1	yes	characteristic, direct product factor
89	C₂×C₄	—	3	1	no	—
90	C₂×C₃	—	2	2	no	—
91	C₂×C₃	—	2	2	no	—
92	C₂×C₃	—	2	1	no	—
93	C₂×C₃	—	3	2	no	—
94	C₂×C₃	—	4	2	no	—
95	C₂×C₃	—	6	2	no	—
96	C₂×C₃	C₂³×C₉	1	1	yes	characteristic
97	C₂×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
98	D₃	—	2	2	no	—
99	D₃	D₄×C₉	1	2	yes	direct product factor
100	C₂²	—	3	1	no	—
101	C₂²	—	3	2	no	—
102	C₂²	—	6	4	no	—
103	C₂²	C₂×D₃×C₉	1	2	yes	complemented
104	C₄	—	3	1	no	—
105	C₄	C₂×D₃×C₉	1	1	yes	characteristic, complemented
106	C₃	—	2	1	no	—
107	C₃	C₂×D₄×C₉	1	1	yes	characteristic, complemented
108	C₃	C₃×D₃×D₄	1	1	yes	characteristic
109	C₂	—	2	2	no	—
110	C₂	—	3	2	no	—
111	C₂	—	6	2	no	—
112	C₂	C₂²×D₃×C₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₃×C₉

C₂

yes

maximal, complemented

C₃×D₄×C₉

C₂

yes

maximal, characteristic, complemented

C₉×#67

C₂

yes

maximal, complemented

C₉×D₁₂

C₂

yes

maximal, characteristic, complemented

C₄×D₃×C₉

C₂

yes

maximal, characteristic, complemented

C₃×D₃×D₄

C₃

yes

maximal, characteristic

C₂×D₄×C₉

—

maximal

C₂²×C₃×C₉

C₂²

yes

complemented

C₂×D₃×C₉

—

C₂×D₃×C₉

C₂²

yes

characteristic

C₂×D₃×C₉

C₂²

yes

complemented

C₃×C₄×C₉

C₂²

yes

characteristic, complemented

C₉×Dic₃

C₂²

yes

characteristic, complemented

C₂²×C₃×D₃

C₂×C₃

yes

—

C₃²×D₄

C₂×C₃

yes

characteristic

C₃×#67

C₂×C₃

yes

—

C₃×D₁₂

C₂×C₃

yes

characteristic

C₃×C₄×D₃

C₂×C₃

yes

characteristic

C₂³×C₉

—

D₄×C₉

—

D₄×C₉

—

D₄×C₉

—

D₄×C₉

D₃

yes

characteristic, direct product factor

C₂×C₄×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₂³

yes

characteristic

D₃×C₉

—

D₃×C₉

D₄

yes

direct product factor

C₂×C₃×D₄

—

D₃×D₄

C₉

yes

characteristic, direct product factor

C₂²×C₃²

C₂²×C₃

yes

—

C₂×C₃×D₃

—

C₂×C₃×D₃

C₂²×C₃

yes

characteristic

C₂×C₃×D₃

C₂²×C₃

yes

—

C₃²×C₄

C₂²×C₃

yes

characteristic

C₃×Dic₃

C₂²×C₃

yes

characteristic

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

—

C₂²×C₉

C₂×D₃

yes

complemented

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂×D₃

yes

characteristic, complemented

C₃×C₉

C₂×D₄

yes

characteristic, complemented, Sylow

C₂³×C₃

—

C₂²×D₃

C₂×C₉

yes

complemented

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

C₂×C₉

yes

characteristic, complemented

C₃×D₄

C₃×D₃

yes

characteristic

C₂×C₃×C₄

—

#67

C₂×C₉

yes

complemented

D₁₂

C₂×C₉

yes

characteristic, complemented

C₄×D₃

C₂×C₉

yes

characteristic, complemented

C₂×C₃²

—

C₂×C₃²

C₂³×C₃

yes

characteristic

C₃×D₃

—

C₃×D₃

C₃×D₄

yes

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₂²×D₃

yes

characteristic

C₂×D₄

—

Sylow

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂²×C₉

yes

complemented

C₂²×C₃

C₂×C₃×D₃

yes

—

C₂×D₃

—

C₂×D₃

C₂²×C₉

yes

characteristic

C₂×D₃

C₂²×C₉

yes

complemented

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂²×C₉

yes

characteristic, complemented

C₃×C₄

C₂×C₃×D₃

yes

characteristic

Dic₃

C₂²×C₉

yes

characteristic, complemented

C₃²

C₂×C₃×D₄

yes

characteristic

C₉

—

C₉

D₃×D₄

yes

characteristic, direct product factor

C₂³

—

D₄

—

D₄

—

D₄

D₃×C₉

yes

characteristic, direct product factor

C₂×C₄

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

C₂×C₃

C₂³×C₉

yes

characteristic

C₂×C₃

C₂²×C₃×D₃

yes

characteristic

D₃

—

D₃

D₄×C₉

yes

direct product factor

100

C₂²

—

101

C₂²

—

102

C₂²

—

103

C₂²

C₂×D₃×C₉

yes

complemented

104

C₄

—

105

C₄

C₂×D₃×C₉

yes

characteristic, complemented

106

C₃

—

107

C₃

C₂×D₄×C₉

yes

characteristic, complemented

108

C₃

C₃×D₃×D₄

yes

characteristic

109

C₂

—

110

C₂

—

111

C₂

—

112

C₂

C₂²×D₃×C₉

yes

characteristic

	subgroup	G/H
center	C₂×C₉	C₂²×D₃
commutator	C₂×C₃	C₂³×C₉
Frattini	C₂×C₃	C₂²×C₃×D₃
Fitting	C₃×D₄×C₉	C₂
socle, cosocle	C₂×C₃²	C₂³×C₃
prime radical	C₃×D₃×D₄	C₃
hypercenter	D₄×C₉	D₃
hypocenter	C₃	C₂×D₄×C₉
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₉ ◁ D₄×C₉

subgroup

G/H

center

C₂×C₉

C₂²×D₃

commutator

C₂×C₃

C₂³×C₉

Frattini

C₂×C₃

C₂²×C₃×D₃

Fitting

C₃×D₄×C₉

C₂

socle, cosocle

C₂×C₃²

C₂³×C₃

prime radical

C₃×D₃×D₄

C₃

hypercenter

D₄×C₉

D₃

hypocenter

C₃

C₂×D₄×C₉

LCS

G ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₉ ◁ D₄×C₉

N	H ≅ G/N	type
1. C₂²×D₃×C₉	C₂	semidirect
2. C₃×D₄×C₉	C₂	semidirect
3. C₉×#67	C₂	semidirect
4. C₉×D₁₂	C₂	semidirect
5. C₄×D₃×C₉	C₂	semidirect
8. C₂²×C₃×C₉	C₂²	semidirect
11. C₂×D₃×C₉	C₂²	semidirect
12. C₃×C₄×C₉	C₂²	semidirect
13. C₉×Dic₃	C₂²	semidirect
23. D₄×C₉	D₃	direct (N × H)
28. D₃×C₉	D₄	direct (N × H)
30. D₃×D₄	C₉	direct (N × H)
41. C₂²×C₉	C₂×D₃	semidirect
44. C₄×C₉	C₂×D₃	semidirect
45. C₃×C₉	C₂×D₄	semidirect
47. C₂²×D₃	C₂×C₉	semidirect
51. C₃×D₄	C₂×C₉	semidirect
54. #67	C₂×C₉	semidirect
55. D₁₂	C₂×C₉	semidirect
56. C₄×D₃	C₂×C₉	semidirect
72. C₂²×C₃	C₂²×C₉	semidirect
76. C₂×D₃	C₂²×C₉	semidirect
79. C₃×C₄	C₂²×C₉	semidirect
81. Dic₃	C₂²×C₉	semidirect
84. C₉	D₃×D₄	direct (N × H)
88. D₄	D₃×C₉	direct (N × H)
99. D₃	D₄×C₉	direct (N × H)
103. C₂²	C₂×D₃×C₉	semidirect
105. C₄	C₂×D₃×C₉	semidirect
107. C₃	C₂×D₄×C₉	semidirect

H ≅ G/N

type

1. C₂²×D₃×C₉

C₂

semidirect

2. C₃×D₄×C₉

C₂

semidirect

3. C₉×#67

C₂

semidirect

4. C₉×D₁₂

C₂

semidirect

5. C₄×D₃×C₉

C₂

semidirect

8. C₂²×C₃×C₉

C₂²

semidirect

11. C₂×D₃×C₉

C₂²

semidirect

12. C₃×C₄×C₉

C₂²

semidirect

13. C₉×Dic₃

C₂²

semidirect

23. D₄×C₉

D₃

direct (N × H)

28. D₃×C₉

D₄

direct (N × H)

30. D₃×D₄

C₉

direct (N × H)

41. C₂²×C₉

C₂×D₃

semidirect

44. C₄×C₉

C₂×D₃

semidirect

45. C₃×C₉

C₂×D₄

semidirect

47. C₂²×D₃

C₂×C₉

semidirect

51. C₃×D₄

C₂×C₉

semidirect

54. #67

C₂×C₉

semidirect

55. D₁₂

C₂×C₉

semidirect

56. C₄×D₃

C₂×C₉

semidirect

72. C₂²×C₃

C₂²×C₉

semidirect

76. C₂×D₃

C₂²×C₉

semidirect

79. C₃×C₄

C₂²×C₉

semidirect

81. Dic₃

C₂²×C₉

semidirect

84. C₉

D₃×D₄

direct (N × H)

88. D₄

D₃×C₉

direct (N × H)

99. D₃

D₄×C₉

direct (N × H)

103. C₂²

C₂×D₃×C₉

semidirect

105. C₄

C₂×D₃×C₉

semidirect

107. C₃

C₂×D₄×C₉

semidirect

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

subgroup	index
C₂²×D₃×C₉	2
C₃×D₄×C₉	2
C₉×#67	2
C₉×D₁₂	2
C₄×D₃×C₉	2
C₃×D₃×D₄	3
C₂×D₄×C₉	3

subgroup

index

C₂²×D₃×C₉

C₃×D₄×C₉

C₉×#67

C₉×D₁₂

C₄×D₃×C₉

C₃×D₃×D₄

C₂×D₄×C₉

Subgroups — D₃×D₄×C₉ #88711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D3×D4×C9 #88711

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃×D₄×C₉ #88711