Group #14517827 — subgroups

Subgroups — D₄×#6912 #14517827

Order 1944.

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

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 198 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×#6912	C₂	1	2	yes	maximal, complemented
2	C₄×#6912	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₄×3₋¹⁺²	C₃	1	3	yes	maximal
4	C₃×D₄×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
5	D₄×#733	C₃	1	9	yes	maximal, complemented
6	C₂×#6912	—	2	2	no	—
7	C₂×#6912	C₂²	1	1	yes	characteristic
8	C₂²×C₃×3₋¹⁺²	C₂×C₃	1	6	yes	—
9	C₂²×C₃×3₋¹⁺²	C₂×C₃	1	2	yes	—
10	C₃×C₄×3₋¹⁺²	C₂×C₃	1	3	yes	—
11	C₃×C₄×3₋¹⁺²	C₂×C₃	1	1	yes	characteristic
12	C₂²×#733	C₂×C₃	1	18	yes	complemented
13	C₄×#733	C₂×C₃	1	9	yes	complemented
14	#6912	D₄	1	1	yes	characteristic, direct product factor, Sylow
15	C₃³×D₄	C₃²	1	1	yes	characteristic
16	D₄×3₋¹⁺²	—	3	6	no	—
17	D₄×3₋¹⁺²	—	3	3	no	—
18	D₄×3₋¹⁺²	C₃²	1	9	yes	—
19	C₃×D₄×C₉	—	3	3	no	—
20	C₃×D₄×C₉	C₃²	1	3	yes	—
21	C₂×C₃×3₋¹⁺²	—	2	2	no	—
22	C₂×C₃×3₋¹⁺²	—	2	6	no	—
23	C₂×C₃×3₋¹⁺²	C₂²×C₃	1	3	yes	—
24	C₂×C₃×3₋¹⁺²	C₂²×C₃	1	1	yes	characteristic
25	C₂×#733	—	2	18	no	—
26	C₂×#733	C₂²×C₃	1	9	yes	—
27	C₂²×C₃³	C₂×C₃²	1	2	yes	—
28	C₃³×C₄	C₂×C₃²	1	1	yes	characteristic
29	C₂²×3₋¹⁺²	—	3	12	no	—
30	C₂²×3₋¹⁺²	—	3	6	no	—
31	C₂²×3₋¹⁺²	C₂×C₃²	1	18	yes	—
32	C₂²×C₃×C₉	—	3	6	no	—
33	C₂²×C₃×C₉	C₂×C₃²	1	6	yes	—
34	C₄×3₋¹⁺²	—	3	6	no	—
35	C₄×3₋¹⁺²	—	3	3	no	—
36	C₄×3₋¹⁺²	C₂×C₃²	1	9	yes	—
37	C₃×C₄×C₉	—	3	3	no	—
38	C₃×C₄×C₉	C₂×C₃²	1	3	yes	—
39	C₃×3₋¹⁺²	C₃×D₄	1	3	yes	—
40	C₃×3₋¹⁺²	C₃×D₄	1	1	yes	characteristic
41	#733	C₃×D₄	1	9	yes	complemented
42	C₃²×D₄	—	9	1	no	—
43	C₃²×D₄	H(3,𝔽₃)	1	3	yes	—
44	C₃²×D₄	C₃³	1	1	yes	characteristic
45	D₄×C₉	—	3	9	no	—
46	D₄×C₉	—	3	3	no	—
47	C₂×C₃³	—	2	2	no	—
48	C₂×C₃³	C₂²×C₃²	1	1	yes	characteristic
49	C₂×3₋¹⁺²	—	2	18	no	—
50	C₂×3₋¹⁺²	—	3	6	no	—
51	C₂×3₋¹⁺²	—	3	3	no	—
52	C₂×3₋¹⁺²	—	6	6	no	—
53	C₂×3₋¹⁺²	—	6	12	no	—
54	C₂×3₋¹⁺²	C₂²×C₃²	1	9	yes	—
55	C₂×C₃×C₉	—	2	6	no	—
56	C₂×C₃×C₉	—	3	3	no	—
57	C₂×C₃×C₉	—	6	6	no	—
58	C₂×C₃×C₉	C₂²×C₃²	1	3	yes	—
59	C₂²×C₃²	—	9	2	no	—
60	C₂²×C₃²	C₂×H(3,𝔽₃)	1	6	yes	—
61	C₂²×C₃²	C₂×C₃³	1	2	yes	—
62	C₃²×C₄	—	9	1	no	—
63	C₃²×C₄	C₂×H(3,𝔽₃)	1	3	yes	—
64	C₃²×C₄	C₂×C₃³	1	1	yes	characteristic
65	C₂²×C₉	—	3	18	no	—
66	C₂²×C₉	—	3	6	no	—
67	C₄×C₉	—	3	9	no	—
68	C₄×C₉	—	3	3	no	—
69	C₃³	C₃²×D₄	1	1	yes	characteristic
70	3₋¹⁺²	—	3	6	no	—
71	3₋¹⁺²	—	3	3	no	—
72	3₋¹⁺²	C₃²×D₄	1	9	yes	—
73	C₃×C₉	—	3	3	no	—
74	C₃×C₉	C₃²×D₄	1	3	yes	—
75	C₃×D₄	—	3	3	no	—
76	C₃×D₄	—	3	1	no	—
77	C₃×D₄	C₃×H(3,𝔽₃)	1	1	yes	characteristic
78	C₂×C₃²	—	2	6	no	—
79	C₂×C₃²	—	2	2	no	—
80	C₂×C₃²	—	9	1	no	—
81	C₂×C₃²	—	18	2	no	—
82	C₂×C₃²	C₂²×H(3,𝔽₃)	1	3	yes	—
83	C₂×C₃²	C₂²×C₃³	1	1	yes	characteristic
84	C₂×C₉	—	3	9	no	—
85	C₂×C₉	—	3	3	no	—
86	C₂×C₉	—	6	6	no	—
87	C₂×C₉	—	6	18	no	—
88	C₂²×C₃	—	3	6	no	—
89	C₂²×C₃	—	3	2	no	—
90	C₂²×C₃	C₂×C₃×H(3,𝔽₃)	1	2	yes	—
91	C₃×C₄	—	3	3	no	—
92	C₃×C₄	—	3	1	no	—
93	C₃×C₄	C₂×C₃×H(3,𝔽₃)	1	1	yes	characteristic
94	C₃²	—	9	1	no	—
95	C₃²	D₄×H(3,𝔽₃)	1	3	yes	—
96	C₃²	C₃³×D₄	1	1	yes	characteristic
97	C₉	—	3	9	no	—
98	C₉	—	3	3	no	—
99	D₄	#6912	1	1	yes	characteristic, direct product factor, Sylow
100	C₂×C₃	—	2	2	no	—
101	C₂×C₃	—	3	3	no	—
102	C₂×C₃	—	3	1	no	—
103	C₂×C₃	—	6	6	no	—
104	C₂×C₃	—	6	2	no	—
105	C₂×C₃	C₂²×C₃×H(3,𝔽₃)	1	1	yes	characteristic
106	C₂²	C₂×#6912	1	2	yes	complemented
107	C₄	C₂×#6912	1	1	yes	characteristic, complemented
108	C₃	—	3	3	no	—
109	C₃	—	3	1	no	—
110	C₃	C₃×D₄×H(3,𝔽₃)	1	1	yes	characteristic
111	C₂	—	2	2	no	—
112	C₂	C₂²×#6912	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#6912

C₂

yes

maximal, complemented

C₄×#6912

C₂

yes

maximal, characteristic, complemented

C₃×D₄×3₋¹⁺²

C₃

yes

maximal

C₃×D₄×3₋¹⁺²

C₃

yes

maximal, characteristic

D₄×#733

C₃

yes

maximal, complemented

C₂×#6912

—

C₂×#6912

C₂²

yes

characteristic

C₂²×C₃×3₋¹⁺²

C₂×C₃

yes

—

C₂²×C₃×3₋¹⁺²

C₂×C₃

yes

—

C₃×C₄×3₋¹⁺²

C₂×C₃

yes

—

C₃×C₄×3₋¹⁺²

C₂×C₃

yes

characteristic

C₂²×#733

C₂×C₃

yes

complemented

C₄×#733

C₂×C₃

yes

complemented

#6912

D₄

yes

characteristic, direct product factor, Sylow

C₃³×D₄

C₃²

yes

characteristic

D₄×3₋¹⁺²

—

D₄×3₋¹⁺²

—

D₄×3₋¹⁺²

C₃²

yes

—

C₃×D₄×C₉

—

C₃×D₄×C₉

C₃²

yes

—

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

—

C₂×C₃×3₋¹⁺²

C₂²×C₃

yes

—

C₂×C₃×3₋¹⁺²

C₂²×C₃

yes

characteristic

C₂×#733

—

C₂×#733

C₂²×C₃

yes

—

C₂²×C₃³

C₂×C₃²

yes

—

C₃³×C₄

C₂×C₃²

yes

characteristic

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

—

C₂²×3₋¹⁺²

C₂×C₃²

yes

—

C₂²×C₃×C₉

—

C₂²×C₃×C₉

C₂×C₃²

yes

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

C₂×C₃²

yes

—

C₃×C₄×C₉

—

C₃×C₄×C₉

C₂×C₃²

yes

—

C₃×3₋¹⁺²

C₃×D₄

yes

—

C₃×3₋¹⁺²

C₃×D₄

yes

characteristic

#733

C₃×D₄

yes

complemented

C₃²×D₄

—

C₃²×D₄

H(3,𝔽₃)

yes

—

C₃²×D₄

C₃³

yes

characteristic

D₄×C₉

—

D₄×C₉

—

C₂×C₃³

—

C₂×C₃³

C₂²×C₃²

yes

characteristic

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

—

C₂×3₋¹⁺²

C₂²×C₃²

yes

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₂²×C₃²

yes

—

C₂²×C₃²

—

C₂²×C₃²

C₂×H(3,𝔽₃)

yes

—

C₂²×C₃²

C₂×C₃³

yes

—

C₃²×C₄

—

C₃²×C₄

C₂×H(3,𝔽₃)

yes

—

C₃²×C₄

C₂×C₃³

yes

characteristic

C₂²×C₉

—

C₂²×C₉

—

C₄×C₉

—

C₄×C₉

—

C₃³

C₃²×D₄

yes

characteristic

3₋¹⁺²

—

3₋¹⁺²

—

3₋¹⁺²

C₃²×D₄

yes

—

C₃×C₉

—

C₃×C₉

C₃²×D₄

yes

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

C₃×H(3,𝔽₃)

yes

characteristic

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₂²×H(3,𝔽₃)

yes

—

C₂×C₃²

C₂²×C₃³

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂×C₃×H(3,𝔽₃)

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂×C₃×H(3,𝔽₃)

yes

characteristic

C₃²

—

C₃²

D₄×H(3,𝔽₃)

yes

—

C₃²

C₃³×D₄

yes

characteristic

C₉

—

C₉

—

D₄

#6912

yes

characteristic, direct product factor, Sylow

100

C₂×C₃

—

101

C₂×C₃

—

102

C₂×C₃

—

103

C₂×C₃

—

104

C₂×C₃

—

105

C₂×C₃

C₂²×C₃×H(3,𝔽₃)

yes

characteristic

106

C₂²

C₂×#6912

yes

complemented

107

C₄

C₂×#6912

yes

characteristic, complemented

108

C₃

—

109

C₃

—

110

C₃

C₃×D₄×H(3,𝔽₃)

yes

characteristic

111

C₂

—

112

C₂

C₂²×#6912

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₃	C₂²×C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₂×C₃²	C₂²×C₃³
prime radical	C₃³×D₄	C₃²
LCS	G ▷ C₂×C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₂×C₃ ◁ C₃³×D₄ ◁ G

subgroup

G/H

center, socle

C₂×C₃

C₂²×C₃×H(3,𝔽₃)

commutator, Frattini, cosocle

C₂×C₃²

C₂²×C₃³

prime radical

C₃³×D₄

C₃²

LCS

G ▷ C₂×C₃² ▷ C₃ ▷ C₁

UCS

C₁ ◁ C₂×C₃ ◁ C₃³×D₄ ◁ G

N	H ≅ G/N	type
1. C₂²×#6912	C₂	semidirect
2. C₄×#6912	C₂	semidirect
5. D₄×#733	C₃	semidirect
12. C₂²×#733	C₂×C₃	semidirect
13. C₄×#733	C₂×C₃	semidirect
14. #6912	D₄	direct (N × H)
41. #733	C₃×D₄	semidirect
99. D₄	#6912	direct (N × H)
106. C₂²	C₂×#6912	semidirect
107. C₄	C₂×#6912	semidirect

H ≅ G/N

type

1. C₂²×#6912

C₂

semidirect

2. C₄×#6912

C₂

semidirect

5. D₄×#733

C₃

semidirect

12. C₂²×#733

C₂×C₃

semidirect

13. C₄×#733

C₂×C₃

semidirect

14. #6912

D₄

direct (N × H)

41. #733

C₃×D₄

semidirect

99. D₄

#6912

direct (N × H)

106. C₂²

C₂×#6912

semidirect

107. C₄

C₂×#6912

semidirect

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

subgroup	index
C₂²×#6912	2
C₄×#6912	2
C₃×D₄×3₋¹⁺²	3
D₄×#733	3

subgroup

index

C₂²×#6912

C₄×#6912

C₃×D₄×3₋¹⁺²

D₄×#733

Subgroups — D₄×#6912 #14517827

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D4×#6912 #14517827

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₄×#6912 #14517827