Group #14518915 — subgroups

Subgroups — H(3,𝔽₃)×#634 #14518915

Order 1944.

972 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 972 conjugacy classes of subgroups — too many to draw. Only its 147 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	C₃²×C₄×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	H(3,𝔽₃)×C₃⋊Dic₃	C₂	1	2	yes	maximal
3	C₃²×#634	C₃	1	4	yes	maximal, complemented
4	Dic₆×H(3,𝔽₃)	—	3	4	no	maximal
5	C₂×C₃²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
6	C₃⁴×C₄	C₂×C₃	1	4	yes	—
7	C₃²×C₃⋊Dic₃	C₂×C₃	1	8	yes	—
8	C₃×C₄×H(3,𝔽₃)	—	2	16	no	—
9	C₃×C₄×H(3,𝔽₃)	D₃	1	4	yes	—
10	Dic₃×H(3,𝔽₃)	—	3	8	no	—
11	C₃²×H(3,𝔽₃)	Q₈	1	1	yes	characteristic, complemented, Sylow
12	C₃×#634	—	3	4	no	—
13	C₃×#634	C₃²	1	1	yes	characteristic
14	C₃²×Dic₆	—	3	16	no	—
15	Q₈×H(3,𝔽₃)	—	9	1	no	—
16	C₂×C₃⁴	C₂²×C₃	1	4	yes	—
17	C₂×C₃×H(3,𝔽₃)	—	2	16	no	—
18	C₂×C₃×H(3,𝔽₃)	C₂×D₃	1	4	yes	—
19	C₃³×C₄	—	2	16	no	—
20	C₃³×C₄	—	3	4	no	—
21	C₃³×C₄	—	6	16	no	—
22	C₃³×C₄	C₃×D₃	1	16	yes	—
23	C₃³×C₄	C₂×C₃²	1	1	yes	characteristic
24	C₃×C₃⋊Dic₃	—	3	8	no	—
25	C₃×C₃⋊Dic₃	C₂×C₃²	1	2	yes	—
26	C₃²×Dic₃	—	3	32	no	—
27	C₄×H(3,𝔽₃)	—	2	24	no	—
28	C₄×H(3,𝔽₃)	—	2	16	no	—
29	C₄×H(3,𝔽₃)	—	9	2	no	—
30	C₄×H(3,𝔽₃)	C₃⋊D₃	1	1	yes	characteristic
31	C₃⁴	C₃×Q₈	1	4	yes	complemented
32	C₃×H(3,𝔽₃)	—	2	16	no	—
33	C₃×H(3,𝔽₃)	Dic₆	1	4	yes	complemented
34	C₃²×Q₈	—	9	4	no	—
35	#634	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor
36	C₃×Dic₆	—	3	4	no	—
37	C₃×Dic₆	—	9	16	no	—
38	C₂×C₃³	—	2	16	no	—
39	C₂×C₃³	—	3	4	no	—
40	C₂×C₃³	—	6	16	no	—
41	C₂×C₃³	C₂×C₃×D₃	1	16	yes	—
42	C₂×C₃³	C₂²×C₃²	1	1	yes	characteristic
43	C₂×H(3,𝔽₃)	—	2	24	no	—
44	C₂×H(3,𝔽₃)	—	2	16	no	—
45	C₂×H(3,𝔽₃)	C₂×C₃⋊D₃	1	1	yes	characteristic
46	C₃²×C₄	—	2	4	no	—
47	C₃²×C₄	—	2	16	no	—
48	C₃²×C₄	—	3	16	no	—
49	C₃²×C₄	—	6	16	no	—
50	C₃²×C₄	—	6	32	no	—
51	C₃²×C₄	—	6	16	no	—
52	C₃²×C₄	—	9	8	no	—
53	C₃²×C₄	C₂×H(3,𝔽₃)	1	1	yes	characteristic
54	C₃²×C₄	C₃²×D₃	1	4	yes	—
55	C₃²×C₄	C₃×C₃⋊D₃	1	4	yes	—
56	C₃⋊Dic₃	C₂×H(3,𝔽₃)	1	2	yes	—
57	C₃×Dic₃	—	3	8	no	—
58	C₃×Dic₃	—	9	32	no	—
59	C₃³	—	2	16	no	—
60	C₃³	—	3	4	no	—
61	C₃³	—	6	16	no	—
62	C₃³	C₃×Dic₆	1	16	yes	complemented
63	C₃³	C₃²×Q₈	1	1	yes	characteristic
64	H(3,𝔽₃)	—	2	24	no	—
65	H(3,𝔽₃)	—	2	16	no	—
66	H(3,𝔽₃)	#634	1	1	yes	characteristic, direct product factor
67	C₃×Q₈	—	9	1	no	—
68	C₃×Q₈	—	27	4	no	—
69	Dic₆	—	3	4	no	—
70	C₂×C₃²	—	2	4	no	—
71	C₂×C₃²	—	2	16	no	—
72	C₂×C₃²	—	3	16	no	—
73	C₂×C₃²	—	6	16	no	—
74	C₂×C₃²	—	6	32	no	—
75	C₂×C₃²	—	6	16	no	—
76	C₂×C₃²	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
77	C₂×C₃²	C₂×C₃²×D₃	1	4	yes	—
78	C₂×C₃²	C₂×C₃×C₃⋊D₃	1	4	yes	—
79	C₃×C₄	—	2	4	no	—
80	C₃×C₄	—	3	4	no	—
81	C₃×C₄	—	6	16	no	—
82	C₃×C₄	—	9	2	no	—
83	C₃×C₄	—	27	8	no	—
84	C₃×C₄	D₃×H(3,𝔽₃)	1	4	yes	—
85	C₃×C₄	C₃²×C₃⋊D₃	1	1	yes	characteristic
86	Dic₃	—	3	8	no	—
87	C₃²	—	2	4	no	—
88	C₃²	—	2	16	no	—
89	C₃²	—	3	16	no	—
90	C₃²	—	6	16	no	—
91	C₃²	—	6	32	no	—
92	C₃²	—	6	16	no	—
93	C₃²	Q₈×H(3,𝔽₃)	1	1	yes	characteristic, complemented
94	C₃²	C₃²×Dic₆	1	4	yes	—
95	C₃²	C₃×#634	1	4	yes	complemented
96	Q₈	—	9	1	no	Sylow
97	C₂×C₃	—	2	4	no	—
98	C₂×C₃	—	3	4	no	—
99	C₂×C₃	—	6	16	no	—
100	C₂×C₃	C₂×D₃×H(3,𝔽₃)	1	4	yes	—
101	C₂×C₃	C₂×C₃²×C₃⋊D₃	1	1	yes	characteristic
102	C₄	—	9	2	no	—
103	C₄	C₃⋊D₃×H(3,𝔽₃)	1	1	yes	characteristic
104	C₃	—	2	4	no	—
105	C₃	—	3	4	no	—
106	C₃	—	6	16	no	—
107	C₃	Dic₆×H(3,𝔽₃)	1	4	yes	complemented
108	C₃	C₃²×#634	1	1	yes	characteristic
109	C₂	C₂×C₃⋊D₃×H(3,𝔽₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×C₄×H(3,𝔽₃)

C₂

yes

maximal, characteristic

H(3,𝔽₃)×C₃⋊Dic₃

C₂

yes

maximal

C₃²×#634

C₃

yes

maximal, complemented

Dic₆×H(3,𝔽₃)

—

maximal

C₂×C₃²×H(3,𝔽₃)

C₂²

yes

characteristic

C₃⁴×C₄

C₂×C₃

yes

—

C₃²×C₃⋊Dic₃

C₂×C₃

yes

—

C₃×C₄×H(3,𝔽₃)

—

C₃×C₄×H(3,𝔽₃)

D₃

yes

—

Dic₃×H(3,𝔽₃)

—

C₃²×H(3,𝔽₃)

Q₈

yes

characteristic, complemented, Sylow

C₃×#634

—

C₃×#634

C₃²

yes

characteristic

C₃²×Dic₆

—

Q₈×H(3,𝔽₃)

—

C₂×C₃⁴

C₂²×C₃

yes

—

C₂×C₃×H(3,𝔽₃)

—

C₂×C₃×H(3,𝔽₃)

C₂×D₃

yes

—

C₃³×C₄

—

C₃³×C₄

—

C₃³×C₄

—

C₃³×C₄

C₃×D₃

yes

—

C₃³×C₄

C₂×C₃²

yes

characteristic

C₃×C₃⋊Dic₃

—

C₃×C₃⋊Dic₃

C₂×C₃²

yes

—

C₃²×Dic₃

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

C₃⋊D₃

yes

characteristic

C₃⁴

C₃×Q₈

yes

complemented

C₃×H(3,𝔽₃)

—

C₃×H(3,𝔽₃)

Dic₆

yes

complemented

C₃²×Q₈

—

#634

H(3,𝔽₃)

yes

characteristic, direct product factor

C₃×Dic₆

—

C₃×Dic₆

—

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

—

C₂×C₃³

C₂×C₃×D₃

yes

—

C₂×C₃³

C₂²×C₃²

yes

characteristic

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

C₂×C₃⋊D₃

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×H(3,𝔽₃)

yes

characteristic

C₃²×C₄

C₃²×D₃

yes

—

C₃²×C₄

C₃×C₃⋊D₃

yes

—

C₃⋊Dic₃

C₂×H(3,𝔽₃)

yes

—

C₃×Dic₃

—

C₃×Dic₃

—

C₃³

—

C₃³

—

C₃³

—

C₃³

C₃×Dic₆

yes

complemented

C₃³

C₃²×Q₈

yes

—

—

yes

characteristic, direct product factor

C₃×Q₈

—

C₃×Q₈

—

Dic₆

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₂²×H(3,𝔽₃)

yes

characteristic

C₂×C₃²

C₂×C₃²×D₃

yes

—

C₂×C₃²

C₂×C₃×C₃⋊D₃

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

D₃×H(3,𝔽₃)

yes

—

C₃×C₄

C₃²×C₃⋊D₃

yes

characteristic

Dic₃

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

C₃²

Q₈×H(3,𝔽₃)

yes

characteristic, complemented

C₃²

C₃²×Dic₆

yes

—

C₃²

C₃×#634

yes

complemented

Q₈

—

Sylow

C₂×C₃

—

C₂×C₃

—

C₂×C₃

—

100

C₂×C₃

C₂×D₃×H(3,𝔽₃)

yes

—

101

C₂×C₃

C₂×C₃²×C₃⋊D₃

yes

characteristic

102

C₄

—

103

C₄

C₃⋊D₃×H(3,𝔽₃)

yes

characteristic

104

C₃

—

105

C₃

—

106

C₃

—

107

C₃

Dic₆×H(3,𝔽₃)

yes

complemented

108

C₃

C₃²×#634

yes

characteristic

109

C₂

C₂×C₃⋊D₃×H(3,𝔽₃)

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₃	C₂×C₃²×C₃⋊D₃
commutator, socle, cosocle	C₂×C₃³	C₂²×C₃²
Fitting	C₃²×C₄×H(3,𝔽₃)	C₂
prime radical	C₂×C₃²×H(3,𝔽₃)	C₂²
hypercenter	C₄×H(3,𝔽₃)	C₃⋊D₃
hypocenter	C₃²	Q₈×H(3,𝔽₃)
LCS	G ▷ C₂×C₃³ ▷ C₃²
UCS	C₁ ◁ C₂×C₃ ◁ C₄×H(3,𝔽₃)

subgroup

G/H

center, Frattini

C₂×C₃

C₂×C₃²×C₃⋊D₃

commutator, socle, cosocle

C₂×C₃³

C₂²×C₃²

Fitting

C₃²×C₄×H(3,𝔽₃)

C₂

prime radical

C₂×C₃²×H(3,𝔽₃)

C₂²

hypercenter

C₄×H(3,𝔽₃)

C₃⋊D₃

hypocenter

C₃²

Q₈×H(3,𝔽₃)

LCS

G ▷ C₂×C₃³ ▷ C₃²

UCS

C₁ ◁ C₂×C₃ ◁ C₄×H(3,𝔽₃)

N	H ≅ G/N	type
3. C₃²×#634	C₃	semidirect
11. C₃²×H(3,𝔽₃)	Q₈	semidirect
31. C₃⁴	C₃×Q₈	semidirect
33. C₃×H(3,𝔽₃)	Dic₆	semidirect
35. #634	H(3,𝔽₃)	direct (N × H)
62. C₃³	C₃×Dic₆	semidirect
66. H(3,𝔽₃)	#634	direct (N × H)
93. C₃²	Q₈×H(3,𝔽₃)	semidirect
95. C₃²	C₃×#634	semidirect
107. C₃	Dic₆×H(3,𝔽₃)	semidirect

H ≅ G/N

type

3. C₃²×#634

C₃

semidirect

11. C₃²×H(3,𝔽₃)

Q₈

semidirect

31. C₃⁴

C₃×Q₈

semidirect

33. C₃×H(3,𝔽₃)

Dic₆

semidirect

35. #634

H(3,𝔽₃)

direct (N × H)

62. C₃³

C₃×Dic₆

semidirect

66. H(3,𝔽₃)

#634

direct (N × H)

93. C₃²

Q₈×H(3,𝔽₃)

semidirect

95. C₃²

C₃×#634

semidirect

107. C₃

Dic₆×H(3,𝔽₃)

semidirect

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

subgroup	index
C₃²×C₄×H(3,𝔽₃)	2
H(3,𝔽₃)×C₃⋊Dic₃	2
C₃²×#634	3
Dic₆×H(3,𝔽₃)	3

subgroup

index

C₃²×C₄×H(3,𝔽₃)

H(3,𝔽₃)×C₃⋊Dic₃

C₃²×#634

Dic₆×H(3,𝔽₃)

Subgroups — H(3,𝔽₃)×#634 #14518915

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽3)×#634 #14518915

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×#634 #14518915