Group #13145232 — subgroups

Subgroups — H(3,𝔽₃)×#491 #13145232

Order 1728.

495 conjugacy classes of subgroups; only the 175 normal subgroups are shown.

normal subgroups only

This group has 495 conjugacy classes of subgroups — too many to draw. Only its 175 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)×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	H(3,𝔽₃)×#125	C₂	1	1	yes	maximal, characteristic
3	H(3,𝔽₃)×#103	C₂	1	4	yes	maximal
4	H(3,𝔽₃)×#97	C₂	1	1	yes	maximal, characteristic
5	C₃²×#491	C₃	1	4	yes	maximal, complemented
6	C₂×Q₈×H(3,𝔽₃)	C₂²	1	2	yes	—
7	C₂×C₈×H(3,𝔽₃)	C₂²	1	2	yes	—
8	C₄⋊C₄×H(3,𝔽₃)	—	2	2	no	—
9	C₄⋊C₄×H(3,𝔽₃)	—	2	2	no	—
10	C₄⋊C₄×H(3,𝔽₃)	C₂²	1	2	yes	—
11	C₄²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
12	C₃²×C₄⋊Q₈	C₂×C₃	1	4	yes	—
13	C₃²×#125	C₂×C₃	1	4	yes	—
14	C₃²×#103	C₂×C₃	1	16	yes	—
15	C₃²×#97	C₂×C₃	1	4	yes	—
16	Q₈×H(3,𝔽₃)	—	2	4	no	—
17	C₂×C₄×H(3,𝔽₃)	—	2	2	no	—
18	C₂×C₄×H(3,𝔽₃)	—	2	2	no	—
19	C₂×C₄×H(3,𝔽₃)	D₄	1	2	yes	—
20	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
21	C₈×H(3,𝔽₃)	—	2	2	no	—
22	C₃×#491	—	3	4	no	—
23	C₃×#491	C₃²	1	1	yes	characteristic
24	C₂×C₃²×Q₈	C₂²×C₃	1	8	yes	—
25	C₂×C₃²×C₈	C₂²×C₃	1	8	yes	—
26	C₃²×C₄⋊C₄	—	2	8	no	—
27	C₃²×C₄⋊C₄	—	2	8	no	—
28	C₃²×C₄⋊C₄	C₂²×C₃	1	8	yes	—
29	C₃²×C₄²	C₂²×C₃	1	4	yes	—
30	C₂²×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
31	C₄×H(3,𝔽₃)	—	2	2	no	—
32	C₄×H(3,𝔽₃)	—	4	2	no	—
33	C₄×H(3,𝔽₃)	—	4	2	no	—
34	C₄×H(3,𝔽₃)	#41	1	2	yes	—
35	C₃×C₄⋊Q₈	—	3	4	no	—
36	C₃×C₄⋊Q₈	C₂×C₃²	1	1	yes	characteristic
37	C₃×#125	—	3	4	no	—
38	C₃×#125	C₂×C₃²	1	1	yes	characteristic
39	C₃×#103	—	3	16	no	—
40	C₃×#103	C₂×C₃²	1	4	yes	—
41	C₃×#97	—	3	4	no	—
42	C₃×#97	C₂×C₃²	1	1	yes	characteristic
43	C₃²×Q₈	—	2	16	no	—
44	C₂×C₃²×C₄	—	2	8	no	—
45	C₂×C₃²×C₄	—	2	8	no	—
46	C₂×C₃²×C₄	C₃×D₄	1	8	yes	—
47	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
48	C₃²×C₈	—	2	8	no	—
49	#491	H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
50	C₂×H(3,𝔽₃)	#124	1	1	yes	characteristic
51	C₂×H(3,𝔽₃)	#137	1	2	yes	—
52	C₂×C₃×Q₈	—	3	8	no	—
53	C₂×C₃×Q₈	C₂²×C₃²	1	2	yes	—
54	C₂×C₃×C₈	—	3	8	no	—
55	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
56	C₃×C₄⋊C₄	—	2	2	no	—
57	C₃×C₄⋊C₄	—	2	2	no	—
58	C₃×C₄⋊C₄	—	3	8	no	—
59	C₃×C₄⋊C₄	—	6	8	no	—
60	C₃×C₄⋊C₄	—	6	8	no	—
61	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
62	C₃×C₄²	—	3	4	no	—
63	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
64	C₂²×C₃²	C₂×C₃×D₄	1	4	yes	—
65	C₃²×C₄	—	2	8	no	—
66	C₃²×C₄	—	4	8	no	—
67	C₃²×C₄	—	4	8	no	—
68	C₃²×C₄	C₃×#41	1	8	yes	—
69	C₄⋊Q₈	C₂×H(3,𝔽₃)	1	1	yes	characteristic
70	#125	C₂×H(3,𝔽₃)	1	1	yes	characteristic
71	#103	C₂×H(3,𝔽₃)	1	4	yes	—
72	#97	C₂×H(3,𝔽₃)	1	1	yes	characteristic
73	H(3,𝔽₃)	#491	1	1	yes	characteristic, direct product factor, Sylow
74	C₃×Q₈	—	2	4	no	—
75	C₃×Q₈	—	6	16	no	—
76	C₂×C₃×C₄	—	2	2	no	—
77	C₂×C₃×C₄	—	2	2	no	—
78	C₂×C₃×C₄	—	3	8	no	—
79	C₂×C₃×C₄	—	3	4	no	—
80	C₂×C₃×C₄	—	6	8	no	—
81	C₂×C₃×C₄	—	6	8	no	—
82	C₂×C₃×C₄	C₃²×D₄	1	2	yes	—
83	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
84	C₃×C₈	—	2	2	no	—
85	C₃×C₈	—	6	8	no	—
86	C₂×C₃²	C₃×#124	1	4	yes	—
87	C₂×C₃²	C₃×#137	1	8	yes	—
88	C₂×Q₈	C₂²×H(3,𝔽₃)	1	2	yes	—
89	C₂×C₈	C₂²×H(3,𝔽₃)	1	2	yes	—
90	C₄⋊C₄	—	2	2	no	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	C₂²×H(3,𝔽₃)	1	2	yes	—
93	C₄²	C₂²×H(3,𝔽₃)	1	1	yes	characteristic
94	C₂²×C₃	—	3	4	no	—
95	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
96	C₃×C₄	—	2	2	no	—
97	C₃×C₄	—	3	8	no	—
98	C₃×C₄	—	4	2	no	—
99	C₃×C₄	—	4	2	no	—
100	C₃×C₄	—	6	8	no	—
101	C₃×C₄	—	12	8	no	—
102	C₃×C₄	—	12	8	no	—
103	C₃×C₄	C₃²×#41	1	2	yes	—
104	C₃²	C₃×#491	1	4	yes	complemented
105	Q₈	—	2	4	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	D₄×H(3,𝔽₃)	1	2	yes	—
109	C₂×C₄	C₂³×H(3,𝔽₃)	1	1	yes	characteristic
110	C₈	—	2	2	no	—
111	C₂×C₃	—	3	4	no	—
112	C₂×C₃	—	3	8	no	—
113	C₂×C₃	C₃²×#124	1	1	yes	characteristic
114	C₂×C₃	C₃²×#137	1	2	yes	—
115	C₂²	C₂×D₄×H(3,𝔽₃)	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	4	2	no	—
118	C₄	—	4	2	no	—
119	C₄	#41×H(3,𝔽₃)	1	2	yes	—
120	C₃	—	3	4	no	—
121	C₃	C₃²×#491	1	1	yes	characteristic
122	C₂	H(3,𝔽₃)×#124	1	1	yes	characteristic
123	C₂	H(3,𝔽₃)×#137	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

H(3,𝔽₃)×C₄⋊Q₈

C₂

yes

maximal, characteristic

H(3,𝔽₃)×#125

C₂

yes

maximal, characteristic

H(3,𝔽₃)×#103

C₂

yes

maximal

H(3,𝔽₃)×#97

C₂

yes

maximal, characteristic

C₃²×#491

C₃

yes

maximal, complemented

C₂×Q₈×H(3,𝔽₃)

C₂²

yes

—

C₂×C₈×H(3,𝔽₃)

C₂²

yes

—

C₄⋊C₄×H(3,𝔽₃)

—

C₄⋊C₄×H(3,𝔽₃)

—

C₄⋊C₄×H(3,𝔽₃)

C₂²

yes

—

C₄²×H(3,𝔽₃)

C₂²

yes

characteristic

C₃²×C₄⋊Q₈

C₂×C₃

yes

—

C₃²×#125

C₂×C₃

yes

—

C₃²×#103

C₂×C₃

yes

—

C₃²×#97

C₂×C₃

yes

—

Q₈×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

D₄

yes

—

C₂×C₄×H(3,𝔽₃)

C₂³

yes

characteristic

C₈×H(3,𝔽₃)

—

C₃×#491

—

C₃×#491

C₃²

yes

characteristic

C₂×C₃²×Q₈

C₂²×C₃

yes

—

C₂×C₃²×C₈

C₂²×C₃

yes

—

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

C₂²×C₃

yes

—

C₃²×C₄²

C₂²×C₃

yes

—

C₂²×H(3,𝔽₃)

C₂×D₄

yes

characteristic

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

#41

yes

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

C₂×C₃²

yes

characteristic

C₃×#125

—

C₃×#125

C₂×C₃²

yes

characteristic

C₃×#103

—

C₃×#103

C₂×C₃²

yes

—

C₃×#97

—

C₃×#97

C₂×C₃²

yes

characteristic

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₃×D₄

yes

—

C₂×C₃²×C₄

C₂³×C₃

yes

—

C₃²×C₈

—

#491

H(3,𝔽₃)

yes

characteristic, direct product factor, Sylow

C₂×H(3,𝔽₃)

#124

yes

characteristic

C₂×H(3,𝔽₃)

#137

yes

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₂²×C₃²

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²×C₃²

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂²×C₃²

yes

—

C₃×C₄²

—

C₃×C₄²

C₂²×C₃²

yes

characteristic

C₂²×C₃²

C₂×C₃×D₄

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₃×#41

yes

—

C₄⋊Q₈

C₂×H(3,𝔽₃)

yes

characteristic

#125

C₂×H(3,𝔽₃)

yes

characteristic

#103

C₂×H(3,𝔽₃)

yes

—

#97

C₂×H(3,𝔽₃)

yes

characteristic

H(3,𝔽₃)

#491

yes

characteristic, direct product factor, Sylow

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₃²×D₄

yes

—

C₂×C₃×C₄

C₂³×C₃²

yes

characteristic

C₃×C₈

—

C₃×C₈

—

C₂×C₃²

C₃×#124

yes

—

C₂×C₃²

C₃×#137

yes

—

C₂×Q₈

C₂²×H(3,𝔽₃)

yes

—

C₂×C₈

C₂²×H(3,𝔽₃)

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂²×H(3,𝔽₃)

yes

—

C₄²

C₂²×H(3,𝔽₃)

yes

characteristic

C₂²×C₃

—

C₂²×C₃

C₂×C₃²×D₄

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

100

C₃×C₄

—

101

C₃×C₄

—

102

C₃×C₄

—

103

C₃×C₄

C₃²×#41

yes

—

104

C₃²

C₃×#491

yes

complemented

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

D₄×H(3,𝔽₃)

yes

—

109

C₂×C₄

C₂³×H(3,𝔽₃)

yes

characteristic

110

C₈

—

111

C₂×C₃

—

112

C₂×C₃

—

113

C₂×C₃

C₃²×#124

yes

characteristic

114

C₂×C₃

C₃²×#137

yes

—

115

C₂²

C₂×D₄×H(3,𝔽₃)

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

#41×H(3,𝔽₃)

yes

—

120

C₃

—

121

C₃

C₃²×#491

yes

characteristic

122

C₂

H(3,𝔽₃)×#124

yes

characteristic

123

C₂

H(3,𝔽₃)×#137

yes

—

	subgroup	G/H
center, socle	C₂²×C₃	C₂×C₃²×D₄
commutator, Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
prime radical	C₂²×H(3,𝔽₃)	C₂×D₄
LCS	G ▷ C₂×C₃×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃ ◁ C₄²×H(3,𝔽₃) ◁ G

subgroup

G/H

center, socle

C₂²×C₃

C₂×C₃²×D₄

commutator, Frattini, cosocle

C₂×C₃×C₄

C₂³×C₃²

prime radical

C₂²×H(3,𝔽₃)

C₂×D₄

LCS

G ▷ C₂×C₃×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₃ ◁ C₄²×H(3,𝔽₃) ◁ G

N	H ≅ G/N	type
5. C₃²×#491	C₃	semidirect
49. #491	H(3,𝔽₃)	direct (N × H)
73. H(3,𝔽₃)	#491	direct (N × H)
104. C₃²	C₃×#491	semidirect

H ≅ G/N

type

5. C₃²×#491

C₃

semidirect

49. #491

H(3,𝔽₃)

direct (N × H)

73. H(3,𝔽₃)

#491

direct (N × H)

104. C₃²

C₃×#491

semidirect

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

subgroup	index
H(3,𝔽₃)×C₄⋊Q₈	2
H(3,𝔽₃)×#125	2
H(3,𝔽₃)×#103	2
H(3,𝔽₃)×#97	2
C₃²×#491	3

subgroup

index

H(3,𝔽₃)×C₄⋊Q₈

H(3,𝔽₃)×#125

H(3,𝔽₃)×#103

H(3,𝔽₃)×#97

C₃²×#491

Subgroups — H(3,𝔽₃)×#491 #13145232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽3)×#491 #13145232

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(3,𝔽₃)×#491 #13145232