Group #14517837 — subgroups

Subgroups — Q₈×C₉⋊H(3,𝔽₃) #14517837

Order 1944.

498 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 234 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	C₄×C₉⋊H(3,𝔽₃)	C₂	1	3	yes	maximal
2	C₃×Q₈×3₋¹⁺²	C₃	1	3	yes	maximal, complemented
3	C₃×Q₈×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	C₃²×Q₈×C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	Q₈×C₃²⋊C₉	C₃	1	6	yes	maximal, complemented
6	Q₈×C₃²⋊C₉	C₃	1	2	yes	maximal, complemented
7	C₂×C₉⋊H(3,𝔽₃)	C₂²	1	1	yes	characteristic
8	C₃×C₄×3₋¹⁺²	C₂×C₃	1	9	yes	—
9	C₃×C₄×H(3,𝔽₃)	C₂×C₃	1	3	yes	—
10	C₃²×C₄×C₉	C₂×C₃	1	3	yes	—
11	C₄×C₃²⋊C₉	C₂×C₃	1	18	yes	—
12	C₄×C₃²⋊C₉	C₂×C₃	1	6	yes	—
13	C₉⋊H(3,𝔽₃)	Q₈	1	1	yes	characteristic, direct product factor, Sylow
14	C₃³×Q₈	C₃²	1	3	yes	—
15	C₃³×Q₈	C₃²	1	1	yes	characteristic
16	Q₈×3₋¹⁺²	—	3	9	no	—
17	Q₈×H(3,𝔽₃)	—	3	3	no	—
18	C₃×Q₈×C₉	—	3	3	no	—
19	C₃×Q₈×C₉	C₃²	1	2	yes	—
20	C₃×Q₈×C₉	C₃²	1	6	yes	—
21	C₃×Q₈×C₉	C₃²	1	1	yes	characteristic
22	C₂×C₃×3₋¹⁺²	C₂²×C₃	1	3	yes	—
23	C₂×C₃×H(3,𝔽₃)	C₂²×C₃	1	1	yes	characteristic
24	C₂×C₃²×C₉	C₂²×C₃	1	1	yes	characteristic
25	C₂×C₃²⋊C₉	C₂²×C₃	1	6	yes	—
26	C₂×C₃²⋊C₉	C₂²×C₃	1	2	yes	—
27	C₃³×C₄	C₂×C₃²	1	9	yes	—
28	C₃³×C₄	C₂×C₃²	1	3	yes	—
29	C₄×3₋¹⁺²	—	3	27	no	—
30	C₄×H(3,𝔽₃)	—	3	9	no	—
31	C₃×C₄×C₉	—	3	9	no	—
32	C₃×C₄×C₉	C₂×C₃²	1	6	yes	—
33	C₃×C₄×C₉	C₂×C₃²	1	18	yes	—
34	C₃×C₄×C₉	C₂×C₃²	1	3	yes	—
35	C₃×3₋¹⁺²	C₃×Q₈	1	3	yes	complemented
36	C₃×H(3,𝔽₃)	C₃×Q₈	1	1	yes	characteristic
37	C₃²×C₉	C₃×Q₈	1	1	yes	characteristic, complemented
38	C₃²⋊C₉	C₃×Q₈	1	6	yes	complemented
39	C₃²⋊C₉	C₃×Q₈	1	2	yes	complemented
40	C₃²×Q₈	—	3	1	no	—
41	C₃²×Q₈	—	3	2	no	—
42	C₃²×Q₈	—	3	3	no	—
43	C₃²×Q₈	—	3	3	no	—
44	C₃²×Q₈	—	3	6	no	—
45	C₃²×Q₈	3₋¹⁺²	1	3	yes	complemented
46	C₃²×Q₈	C₃³	1	1	yes	characteristic
47	Q₈×C₉	—	3	2	no	—
48	Q₈×C₉	—	3	6	no	—
49	Q₈×C₉	H(3,𝔽₃)	1	3	yes	complemented
50	C₂×C₃³	C₂²×C₃²	1	3	yes	—
51	C₂×C₃³	C₂²×C₃²	1	1	yes	characteristic
52	C₂×3₋¹⁺²	—	3	9	no	—
53	C₂×H(3,𝔽₃)	—	3	3	no	—
54	C₂×C₃×C₉	—	3	3	no	—
55	C₂×C₃×C₉	C₂²×C₃²	1	2	yes	—
56	C₂×C₃×C₉	C₂²×C₃²	1	6	yes	—
57	C₂×C₃×C₉	C₂²×C₃²	1	1	yes	characteristic
58	C₃²×C₄	—	3	3	no	—
59	C₃²×C₄	—	3	6	no	—
60	C₃²×C₄	—	3	9	no	—
61	C₃²×C₄	—	3	9	no	—
62	C₃²×C₄	—	3	18	no	—
63	C₃²×C₄	C₂×3₋¹⁺²	1	9	yes	—
64	C₃²×C₄	C₂×C₃³	1	3	yes	—
65	C₄×C₉	—	3	6	no	—
66	C₄×C₉	—	3	18	no	—
67	C₄×C₉	C₂×H(3,𝔽₃)	1	9	yes	—
68	C₃³	C₃²×Q₈	1	3	yes	—
69	C₃³	C₃²×Q₈	1	1	yes	characteristic
70	3₋¹⁺²	—	3	9	no	—
71	H(3,𝔽₃)	—	3	3	no	—
72	C₃×C₉	—	3	3	no	—
73	C₃×C₉	C₃²×Q₈	1	2	yes	—
74	C₃×C₉	C₃²×Q₈	1	6	yes	—
75	C₃×C₉	C₃²×Q₈	1	1	yes	characteristic
76	C₃×Q₈	—	3	3	no	—
77	C₃×Q₈	—	9	3	no	—
78	C₃×Q₈	C₃×H(3,𝔽₃)	1	1	yes	characteristic
79	C₃×Q₈	C₃×3₋¹⁺²	1	1	yes	characteristic
80	C₃×Q₈	#737	1	2	yes	—
81	C₂×C₃²	—	3	1	no	—
82	C₂×C₃²	—	3	2	no	—
83	C₂×C₃²	—	3	3	no	—
84	C₂×C₃²	—	3	3	no	—
85	C₂×C₃²	—	3	6	no	—
86	C₂×C₃²	C₂²×3₋¹⁺²	1	3	yes	—
87	C₂×C₃²	C₂²×C₃³	1	1	yes	characteristic
88	C₂×C₉	—	3	2	no	—
89	C₂×C₉	—	3	6	no	—
90	C₂×C₉	C₂²×H(3,𝔽₃)	1	3	yes	—
91	C₃×C₄	—	3	9	no	—
92	C₃×C₄	—	9	9	no	—
93	C₃×C₄	C₂×C₃×H(3,𝔽₃)	1	3	yes	—
94	C₃×C₄	C₂×C₃×3₋¹⁺²	1	3	yes	—
95	C₃×C₄	C₂×#737	1	6	yes	—
96	C₃²	—	3	1	no	—
97	C₃²	—	3	2	no	—
98	C₃²	—	3	3	no	—
99	C₃²	—	3	3	no	—
100	C₃²	—	3	6	no	—
101	C₃²	Q₈×3₋¹⁺²	1	3	yes	complemented
102	C₃²	C₃³×Q₈	1	1	yes	characteristic
103	C₉	—	3	2	no	—
104	C₉	—	3	6	no	—
105	C₉	Q₈×H(3,𝔽₃)	1	3	yes	complemented
106	Q₈	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic, direct product factor, Sylow
107	C₂×C₃	—	3	3	no	—
108	C₂×C₃	—	9	3	no	—
109	C₂×C₃	C₂²×C₃×H(3,𝔽₃)	1	1	yes	characteristic
110	C₂×C₃	C₂²×C₃×3₋¹⁺²	1	1	yes	characteristic
111	C₂×C₃	C₂²×#737	1	2	yes	—
112	C₄	C₂×C₉⋊H(3,𝔽₃)	1	3	yes	—
113	C₃	—	3	3	no	—
114	C₃	—	9	3	no	—
115	C₃	C₃×Q₈×H(3,𝔽₃)	1	1	yes	characteristic
116	C₃	C₃×Q₈×3₋¹⁺²	1	1	yes	characteristic
117	C₃	Q₈×#737	1	2	yes	—
118	C₂	C₂²×C₉⋊H(3,𝔽₃)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₉⋊H(3,𝔽₃)

C₂

yes

maximal

C₃×Q₈×3₋¹⁺²

C₃

yes

maximal, complemented

C₃×Q₈×H(3,𝔽₃)

C₃

yes

maximal, characteristic

C₃²×Q₈×C₉

C₃

yes

maximal, characteristic, complemented

Q₈×C₃²⋊C₉

C₃

yes

maximal, complemented

Q₈×C₃²⋊C₉

C₃

yes

maximal, complemented

C₂×C₉⋊H(3,𝔽₃)

C₂²

yes

characteristic

C₃×C₄×3₋¹⁺²

C₂×C₃

yes

—

C₃×C₄×H(3,𝔽₃)

C₂×C₃

yes

—

C₃²×C₄×C₉

C₂×C₃

yes

—

C₄×C₃²⋊C₉

C₂×C₃

yes

—

C₄×C₃²⋊C₉

C₂×C₃

yes

—

C₉⋊H(3,𝔽₃)

Q₈

yes

characteristic, direct product factor, Sylow

C₃³×Q₈

C₃²

yes

—

C₃³×Q₈

C₃²

yes

characteristic

Q₈×3₋¹⁺²

—

Q₈×H(3,𝔽₃)

—

C₃×Q₈×C₉

—

C₃×Q₈×C₉

C₃²

yes

—

C₃×Q₈×C₉

C₃²

yes

—

C₃×Q₈×C₉

C₃²

yes

characteristic

C₂×C₃×3₋¹⁺²

C₂²×C₃

yes

—

C₂×C₃×H(3,𝔽₃)

C₂²×C₃

yes

characteristic

C₂×C₃²×C₉

C₂²×C₃

yes

characteristic

C₂×C₃²⋊C₉

C₂²×C₃

yes

—

C₂×C₃²⋊C₉

C₂²×C₃

yes

—

C₃³×C₄

C₂×C₃²

yes

—

C₃³×C₄

C₂×C₃²

yes

—

C₄×3₋¹⁺²

—

C₄×H(3,𝔽₃)

—

C₃×C₄×C₉

—

C₃×C₄×C₉

C₂×C₃²

yes

—

C₃×C₄×C₉

C₂×C₃²

yes

—

C₃×C₄×C₉

C₂×C₃²

yes

—

C₃×3₋¹⁺²

C₃×Q₈

yes

complemented

C₃×H(3,𝔽₃)

C₃×Q₈

yes

characteristic

C₃²×C₉

C₃×Q₈

yes

characteristic, complemented

C₃²⋊C₉

C₃×Q₈

yes

complemented

C₃²⋊C₉

C₃×Q₈

yes

complemented

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

3₋¹⁺²

yes

complemented

C₃²×Q₈

C₃³

yes

characteristic

Q₈×C₉

—

Q₈×C₉

—

Q₈×C₉

H(3,𝔽₃)

yes

complemented

C₂×C₃³

C₂²×C₃²

yes

—

C₂×C₃³

C₂²×C₃²

yes

characteristic

C₂×3₋¹⁺²

—

C₂×H(3,𝔽₃)

—

C₂×C₃×C₉

—

C₂×C₃×C₉

C₂²×C₃²

yes

—

C₂×C₃×C₉

C₂²×C₃²

yes

—

C₂×C₃×C₉

C₂²×C₃²

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×3₋¹⁺²

yes

—

C₃²×C₄

C₂×C₃³

yes

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂×H(3,𝔽₃)

yes

—

C₃³

C₃²×Q₈

yes

—

C₃³

C₃²×Q₈

yes

characteristic

3₋¹⁺²

—

H(3,𝔽₃)

—

C₃×C₉

—

C₃×C₉

C₃²×Q₈

yes

—

C₃×C₉

C₃²×Q₈

yes

—

C₃×C₉

C₃²×Q₈

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

C₃×H(3,𝔽₃)

yes

characteristic

C₃×Q₈

C₃×3₋¹⁺²

yes

characteristic

C₃×Q₈

#737

yes

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

C₂²×3₋¹⁺²

yes

—

C₂×C₃²

C₂²×C₃³

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₂²×H(3,𝔽₃)

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂×C₃×H(3,𝔽₃)

yes

—

C₃×C₄

C₂×C₃×3₋¹⁺²

yes

—

C₃×C₄

C₂×#737

yes

—

C₃²

—

C₃²

—

C₃²

—

C₃²

—

100

C₃²

—

101

C₃²

Q₈×3₋¹⁺²

yes

complemented

102

C₃²

C₃³×Q₈

yes

characteristic

103

C₉

—

104

C₉

—

105

C₉

Q₈×H(3,𝔽₃)

yes

complemented

106

Q₈

C₉⋊H(3,𝔽₃)

yes

characteristic, direct product factor, Sylow

107

C₂×C₃

—

108

C₂×C₃

—

109

C₂×C₃

C₂²×C₃×H(3,𝔽₃)

yes

characteristic

110

C₂×C₃

C₂²×C₃×3₋¹⁺²

yes

characteristic

111

C₂×C₃

C₂²×#737

yes

—

112

C₄

C₂×C₉⋊H(3,𝔽₃)

yes

—

113

C₃

—

114

C₃

—

115

C₃

C₃×Q₈×H(3,𝔽₃)

yes

characteristic

116

C₃

C₃×Q₈×3₋¹⁺²

yes

characteristic

117

C₃

Q₈×#737

yes

—

118

C₂

C₂²×C₉⋊H(3,𝔽₃)

yes

characteristic

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

subgroup

G/H

center, commutator, Frattini, socle, cosocle

C₂×C₃²

C₂²×C₃³

prime radical

C₂×C₃×H(3,𝔽₃)

C₂²×C₃

LCS

G ▷ C₂×C₃² ▷ C₁

UCS

C₁ ◁ C₂×C₃² ◁ G

N	H ≅ G/N	type
2. C₃×Q₈×3₋¹⁺²	C₃	semidirect
4. C₃²×Q₈×C₉	C₃	semidirect
5. Q₈×C₃²⋊C₉	C₃	semidirect
6. Q₈×C₃²⋊C₉	C₃	semidirect
13. C₉⋊H(3,𝔽₃)	Q₈	direct (N × H)
35. C₃×3₋¹⁺²	C₃×Q₈	semidirect
37. C₃²×C₉	C₃×Q₈	semidirect
38. C₃²⋊C₉	C₃×Q₈	semidirect
39. C₃²⋊C₉	C₃×Q₈	semidirect
45. C₃²×Q₈	3₋¹⁺²	semidirect
49. Q₈×C₉	H(3,𝔽₃)	semidirect
101. C₃²	Q₈×3₋¹⁺²	semidirect
105. C₉	Q₈×H(3,𝔽₃)	semidirect
106. Q₈	C₉⋊H(3,𝔽₃)	direct (N × H)

H ≅ G/N

type

2. C₃×Q₈×3₋¹⁺²

C₃

semidirect

4. C₃²×Q₈×C₉

C₃

semidirect

5. Q₈×C₃²⋊C₉

C₃

semidirect

6. Q₈×C₃²⋊C₉

C₃

semidirect

13. C₉⋊H(3,𝔽₃)

Q₈

direct (N × H)

35. C₃×3₋¹⁺²

C₃×Q₈

semidirect

37. C₃²×C₉

C₃×Q₈

semidirect

38. C₃²⋊C₉

C₃×Q₈

semidirect

39. C₃²⋊C₉

C₃×Q₈

semidirect

45. C₃²×Q₈

3₋¹⁺²

semidirect

49. Q₈×C₉

H(3,𝔽₃)

semidirect

101. C₃²

Q₈×3₋¹⁺²

semidirect

105. C₉

Q₈×H(3,𝔽₃)

semidirect

106. Q₈

C₉⋊H(3,𝔽₃)

direct (N × H)

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

subgroup	index
C₄×C₉⋊H(3,𝔽₃)	2
C₃×Q₈×3₋¹⁺²	3
C₃×Q₈×H(3,𝔽₃)	3
C₃²×Q₈×C₉	3
Q₈×C₃²⋊C₉	3

subgroup

index

C₄×C₉⋊H(3,𝔽₃)

C₃×Q₈×3₋¹⁺²

C₃×Q₈×H(3,𝔽₃)

C₃²×Q₈×C₉

Q₈×C₃²⋊C₉

Subgroups — Q₈×C₉⋊H(3,𝔽₃) #14517837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q8×C9⋊H(3,𝔽3) #14517837

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₉⋊H(3,𝔽₃) #14517837