Group #13135995 — subgroups

Subgroups — H(3,𝔽₃)×#331 #13135995

Order 1728.

462 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 133 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	H(3,𝔽₃)×#127	C₂	1	1	yes	maximal, characteristic
2	H(3,𝔽₃)×C₄⋊C₈	C₂	1	2	yes	maximal, complemented
3	C₃²×#331	C₃	1	4	yes	maximal, complemented
4	C₂×D₄×H(3,𝔽₃)	—	2	2	no	—
5	C₂×D₄×H(3,𝔽₃)	C₄	1	2	yes	—
6	C₂×C₈×H(3,𝔽₃)	—	2	2	no	—
7	C₄²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
8	C₃²×#127	C₂×C₃	1	4	yes	—
9	C₃²×C₄⋊C₈	C₂×C₃	1	8	yes	complemented
10	C₂³×H(3,𝔽₃)	—	2	2	no	—
11	D₄×H(3,𝔽₃)	—	4	2	no	—
12	D₄×H(3,𝔽₃)	—	4	2	no	—
13	D₄×H(3,𝔽₃)	—	4	2	no	—
14	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄×H(3,𝔽₃)	D₄	1	2	yes	—
16	C₈×H(3,𝔽₃)	—	4	2	no	—
17	C₃×#331	—	3	4	no	—
18	C₃×#331	C₃²	1	1	yes	characteristic
19	C₂×C₃²×D₄	—	2	8	no	—
20	C₂×C₃²×D₄	C₃×C₄	1	8	yes	—
21	C₂×C₃²×C₈	—	2	8	no	—
22	C₃²×C₄²	C₂²×C₃	1	4	yes	—
23	C₂²×H(3,𝔽₃)	—	4	2	no	—
24	C₂²×H(3,𝔽₃)	—	4	4	no	—
25	C₂²×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
26	C₄×H(3,𝔽₃)	—	2	1	no	—
27	C₄×H(3,𝔽₃)	D₈	1	2	yes	—
28	C₄×H(3,𝔽₃)	SD₁₆	1	2	yes	—
29	C₃×#127	—	3	4	no	—
30	C₃×#127	C₂×C₃²	1	1	yes	characteristic
31	C₃×C₄⋊C₈	—	3	8	no	—
32	C₃×C₄⋊C₈	C₂×C₃²	1	2	yes	—
33	C₂³×C₃²	—	2	8	no	—
34	C₃²×D₄	—	4	8	no	—
35	C₃²×D₄	—	4	8	no	—
36	C₃²×D₄	—	4	8	no	—
37	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
38	C₂×C₃²×C₄	C₃×D₄	1	8	yes	—
39	C₃²×C₈	—	4	8	no	—
40	#331	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
41	C₂×H(3,𝔽₃)	—	8	2	no	—
42	C₂×H(3,𝔽₃)	#100	1	1	yes	characteristic
43	C₂×H(3,𝔽₃)	#102	1	2	yes	—
44	C₂×C₃×D₄	—	2	2	no	—
45	C₂×C₃×D₄	—	3	8	no	—
46	C₂×C₃×D₄	—	6	8	no	—
47	C₂×C₃×D₄	C₃²×C₄	1	2	yes	—
48	C₂×C₃×C₈	—	2	2	no	—
49	C₂×C₃×C₈	—	6	8	no	—
50	C₃×C₄²	—	3	4	no	—
51	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
52	C₂²×C₃²	—	4	8	no	—
53	C₂²×C₃²	—	4	16	no	—
54	C₂²×C₃²	C₃×C₂²⋊C₄	1	4	yes	—
55	C₃²×C₄	—	2	4	no	—
56	C₃²×C₄	C₃×D₈	1	8	yes	—
57	C₃²×C₄	C₃×SD₁₆	1	8	yes	—
58	#127	C₂×H(3,𝔽₃)	1	1	yes	characteristic
59	C₄⋊C₈	C₂×H(3,𝔽₃)	1	2	yes	complemented
60	H(3,𝔽₃)	#331	1	1	yes	characteristic, direct product factor, Sylow
61	C₂³×C₃	—	2	2	no	—
62	C₂³×C₃	—	6	8	no	—
63	C₃×D₄	—	4	2	no	—
64	C₃×D₄	—	4	2	no	—
65	C₃×D₄	—	4	2	no	—
66	C₃×D₄	—	12	8	no	—
67	C₃×D₄	—	12	8	no	—
68	C₃×D₄	—	12	8	no	—
69	C₂×C₃×C₄	—	3	4	no	—
70	C₂×C₃×C₄	—	3	8	no	—
71	C₂×C₃×C₄	C₂×C₃²×C₄	1	1	yes	characteristic
72	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
73	C₃×C₈	—	4	2	no	—
74	C₃×C₈	—	12	8	no	—
75	C₂×C₃²	—	8	8	no	—
76	C₂×C₃²	C₃×#100	1	4	yes	—
77	C₂×C₃²	C₃×#102	1	8	yes	—
78	C₂×D₄	—	2	2	no	—
79	C₂×D₄	C₄×H(3,𝔽₃)	1	2	yes	—
80	C₂×C₈	—	2	2	no	—
81	C₄²	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
82	C₂²×C₃	—	3	4	no	—
83	C₂²×C₃	—	4	2	no	—
84	C₂²×C₃	—	4	4	no	—
85	C₂²×C₃	—	12	16	no	—
86	C₂²×C₃	—	12	8	no	—
87	C₂²×C₃	C₃²×C₂²⋊C₄	1	1	yes	characteristic
88	C₃×C₄	—	2	1	no	—
89	C₃×C₄	—	3	8	no	—
90	C₃×C₄	—	3	8	no	—
91	C₃×C₄	—	6	4	no	—
92	C₃×C₄	C₃²×D₈	1	2	yes	—
93	C₃×C₄	C₃²×SD₁₆	1	2	yes	—
94	C₃²	C₃×#331	1	4	yes	complemented
95	C₂³	—	2	2	no	—
96	D₄	—	4	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	C₂×C₄	C₂×C₄×H(3,𝔽₃)	1	1	yes	characteristic
100	C₂×C₄	D₄×H(3,𝔽₃)	1	2	yes	—
101	C₈	—	4	2	no	—
102	C₂×C₃	—	3	4	no	—
103	C₂×C₃	—	3	8	no	—
104	C₂×C₃	—	8	2	no	—
105	C₂×C₃	—	24	8	no	—
106	C₂×C₃	C₃²×#100	1	1	yes	characteristic
107	C₂×C₃	C₃²×#102	1	2	yes	—
108	C₂²	—	4	2	no	—
109	C₂²	—	4	4	no	—
110	C₂²	C₂²⋊C₄×H(3,𝔽₃)	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	D₈×H(3,𝔽₃)	1	2	yes	—
113	C₄	SD₁₆×H(3,𝔽₃)	1	2	yes	—
114	C₃	—	3	4	no	—
115	C₃	C₃²×#331	1	1	yes	characteristic
116	C₂	—	8	2	no	—
117	C₂	H(3,𝔽₃)×#100	1	1	yes	characteristic
118	C₂	H(3,𝔽₃)×#102	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

H(3,𝔽₃)×#127

C₂

yes

maximal, characteristic

H(3,𝔽₃)×C₄⋊C₈

C₂

yes

maximal, complemented

C₃²×#331

C₃

yes

maximal, complemented

C₂×D₄×H(3,𝔽₃)

—

C₂×D₄×H(3,𝔽₃)

C₄

yes

—

C₂×C₈×H(3,𝔽₃)

—

C₄²×H(3,𝔽₃)

C₂²

yes

characteristic

C₃²×#127

C₂×C₃

yes

—

C₃²×C₄⋊C₈

C₂×C₃

yes

complemented

C₂³×H(3,𝔽₃)

—

D₄×H(3,𝔽₃)

—

D₄×H(3,𝔽₃)

—

D₄×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

C₂×C₄

yes

characteristic

C₂×C₄×H(3,𝔽₃)

D₄

yes

—

C₈×H(3,𝔽₃)

—

C₃×#331

—

C₃×#331

C₃²

yes

characteristic

C₂×C₃²×D₄

—

C₂×C₃²×D₄

C₃×C₄

yes

—

C₂×C₃²×C₈

—

C₃²×C₄²

C₂²×C₃

yes

—

C₂²×H(3,𝔽₃)

—

C₂²×H(3,𝔽₃)

—

C₂²×H(3,𝔽₃)

C₂²⋊C₄

yes

characteristic

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

D₈

yes

—

C₄×H(3,𝔽₃)

SD₁₆

yes

—

C₃×#127

—

C₃×#127

C₂×C₃²

yes

characteristic

C₃×C₄⋊C₈

—

C₃×C₄⋊C₈

C₂×C₃²

yes

—

C₂³×C₃²

—

C₃²×D₄

—

C₃²×D₄

—

C₃²×D₄

—

C₂×C₃²×C₄

C₂×C₃×C₄

yes

—

C₂×C₃²×C₄

C₃×D₄

yes

—

C₃²×C₈

—

#331

H(3,𝔽₃)

yes

characteristic, direct product factor, Sylow

C₂×H(3,𝔽₃)

—

C₂×H(3,𝔽₃)

#100

yes

characteristic

C₂×H(3,𝔽₃)

#102

yes

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

—

C₂×C₃×D₄

C₃²×C₄

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄²

—

C₃×C₄²

C₂²×C₃²

yes

characteristic

C₂²×C₃²

—

C₂²×C₃²

—

C₂²×C₃²

C₃×C₂²⋊C₄

yes

—

C₃²×C₄

—

C₃²×C₄

C₃×D₈

yes

—

C₃²×C₄

C₃×SD₁₆

yes

—

#127

C₂×H(3,𝔽₃)

yes

characteristic

C₄⋊C₈

C₂×H(3,𝔽₃)

yes

complemented

H(3,𝔽₃)

#331

yes

characteristic, direct product factor, Sylow

C₂³×C₃

—

C₂³×C₃

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃²×C₄

yes

characteristic

C₂×C₃×C₄

C₃²×D₄

yes

—

C₃×C₈

—

C₃×C₈

—

C₂×C₃²

—

C₂×C₃²

C₃×#100

yes

—

C₂×C₃²

C₃×#102

yes

—

C₂×D₄

—

C₂×D₄

C₄×H(3,𝔽₃)

yes

—

C₂×C₈

—

C₄²

C₂²×H(3,𝔽₃)

yes

characteristic

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₃²×C₂²⋊C₄

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₃²×D₈

yes

—

C₃×C₄

C₃²×SD₁₆

yes

—

C₃²

C₃×#331

yes

complemented

C₂³

—

D₄

—

D₄

—

D₄

—

C₂×C₄

C₂×C₄×H(3,𝔽₃)

yes

characteristic

100

C₂×C₄

D₄×H(3,𝔽₃)

yes

—

101

C₈

—

102

C₂×C₃

—

103

C₂×C₃

—

104

C₂×C₃

—

105

C₂×C₃

—

106

C₂×C₃

C₃²×#100

yes

characteristic

107

C₂×C₃

C₃²×#102

yes

—

108

C₂²

—

109

C₂²

—

110

C₂²

C₂²⋊C₄×H(3,𝔽₃)

yes

characteristic

111

C₄

—

112

C₄

D₈×H(3,𝔽₃)

yes

—

113

C₄

SD₁₆×H(3,𝔽₃)

yes

—

114

C₃

—

115

C₃

C₃²×#331

yes

characteristic

116

C₂

—

117

C₂

H(3,𝔽₃)×#100

yes

characteristic

118

C₂

H(3,𝔽₃)×#102

yes

—

	subgroup	G/H
center, socle	C₂²×C₃	C₃²×C₂²⋊C₄
commutator	C₂×C₃×C₄	C₂×C₃²×C₄
Frattini, cosocle	C₃×C₄²	C₂²×C₃²
prime radical	H(3,𝔽₃)×#127	C₂
LCS	G ▷ C₂×C₃×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₄²×H(3,𝔽₃) ◁ G

subgroup

G/H

center, socle

C₂²×C₃

C₃²×C₂²⋊C₄

commutator

C₂×C₃×C₄

C₂×C₃²×C₄

Frattini, cosocle

C₃×C₄²

C₂²×C₃²

prime radical

H(3,𝔽₃)×#127

C₂

LCS

G ▷ C₂×C₃×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₃ ◁ C₄²×H(3,𝔽₃) ◁ G

N	H ≅ G/N	type
2. H(3,𝔽₃)×C₄⋊C₈	C₂	semidirect
3. C₃²×#331	C₃	semidirect
9. C₃²×C₄⋊C₈	C₂×C₃	semidirect
40. #331	H(3,𝔽₃)	direct (N × H)
59. C₄⋊C₈	C₂×H(3,𝔽₃)	semidirect
60. H(3,𝔽₃)	#331	direct (N × H)
94. C₃²	C₃×#331	semidirect

H ≅ G/N

type

2. H(3,𝔽₃)×C₄⋊C₈

C₂

semidirect

3. C₃²×#331

C₃

semidirect

9. C₃²×C₄⋊C₈

C₂×C₃

semidirect

40. #331

H(3,𝔽₃)

direct (N × H)

59. C₄⋊C₈

C₂×H(3,𝔽₃)

semidirect

60. H(3,𝔽₃)

#331

direct (N × H)

94. C₃²

C₃×#331

semidirect

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

subgroup	index
H(3,𝔽₃)×#127	2
H(3,𝔽₃)×C₄⋊C₈	2
C₃²×#331	3

subgroup

index

H(3,𝔽₃)×#127

H(3,𝔽₃)×C₄⋊C₈

C₃²×#331

Subgroups — H(3,𝔽₃)×#331 #13135995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽3)×#331 #13135995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×#331 #13135995