Group #13141207 — subgroups

Subgroups — C₈×C₉×SL(2,𝔽₃) #13141207

Order 1728.

186 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 73 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×SL(2,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₃×C₈×SL(2,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₈×Q₈×C₉	C₃	1	1	yes	maximal, characteristic, complemented
4	C₈×Q₈⋊C₉	C₃	1	2	yes	maximal, complemented
5	C₂×C₉×SL(2,𝔽₃)	C₄	1	1	yes	characteristic
6	C₂×C₃×C₈×C₉	—	4	1	no	maximal
7	C₃×C₄×SL(2,𝔽₃)	C₂×C₃	1	1	yes	characteristic
8	C₄×Q₈×C₉	C₂×C₃	1	1	yes	characteristic
9	C₄×Q₈⋊C₉	C₂×C₃	1	2	yes	—
10	C₉×C₄⋊C₈	—	3	1	no	—
11	C₄×C₈×C₉	—	3	1	no	—
12	C₂×C₃×C₄×C₉	—	4	1	no	—
13	C₉×SL(2,𝔽₃)	C₈	1	1	yes	characteristic, direct product factor
14	C₃×C₈×C₉	—	4	2	no	—
15	C₃×C₈×Q₈	C₃²	1	1	yes	characteristic
16	C₈×SL(2,𝔽₃)	C₉	1	3	yes	direct product factor
17	C₂×C₃×SL(2,𝔽₃)	C₃×C₄	1	1	yes	characteristic
18	C₂×C₃²×C₈	—	4	1	no	—
19	C₂×Q₈×C₉	C₃×C₄	1	1	yes	characteristic
20	C₂×Q₈⋊C₉	C₃×C₄	1	2	yes	—
21	C₂×C₈×C₉	—	3	1	no	—
22	C₂×C₈×C₉	—	4	2	no	—
23	C₂×C₈×C₉	A₄	1	1	yes	characteristic
24	C₉×C₄⋊C₄	—	3	1	no	—
25	C₄²×C₉	—	3	1	no	—
26	C₂²×C₃×C₉	—	4	1	no	—
27	C₃×C₄×C₉	—	4	1	no	—
28	C₃×C₄×C₉	—	4	1	no	—
29	C₃×C₄×Q₈	C₂×C₃²	1	1	yes	characteristic
30	C₄×SL(2,𝔽₃)	C₂×C₉	1	3	yes	—
31	C₃×C₄⋊C₈	—	3	1	no	—
32	C₃×C₄×C₈	—	3	1	no	—
33	C₂×C₃²×C₄	—	4	1	no	—
34	C₃×SL(2,𝔽₃)	C₃×C₈	1	1	yes	characteristic
35	C₃²×C₈	—	4	2	no	—
36	Q₈×C₉	—	3	1	no	—
37	Q₈×C₉	C₃×C₈	1	1	yes	characteristic, complemented
38	C₂×C₄×C₉	—	3	1	no	—
39	C₂×C₄×C₉	—	3	1	no	—
40	C₂×C₄×C₉	—	4	2	no	—
41	C₂×C₄×C₉	C₂×A₄	1	1	yes	characteristic
42	Q₈⋊C₉	C₃×C₈	1	2	yes	complemented
43	C₈×C₉	—	4	4	no	—
44	C₈×C₉	—	6	1	no	—
45	C₈×C₉	SL(2,𝔽₃)	1	2	yes	direct product factor
46	C₈×Q₈	C₃×C₉	1	1	yes	characteristic, complemented, Sylow
47	C₂×C₃×C₉	—	4	1	no	—
48	C₂×C₃×C₉	—	4	1	no	—
49	C₂×C₃×C₉	—	4	1	no	—
50	C₂×C₃×Q₈	C₃²×C₄	1	1	yes	characteristic
51	C₂×SL(2,𝔽₃)	C₄×C₉	1	3	yes	—
52	C₂×C₃×C₈	—	3	1	no	—
53	C₂×C₃×C₈	—	4	3	no	—
54	C₂×C₃×C₈	C₃×A₄	1	1	yes	characteristic
55	C₃×C₄⋊C₄	—	3	1	no	—
56	C₃×C₄²	—	3	1	no	—
57	C₂²×C₃²	—	4	1	no	—
58	C₃²×C₄	—	4	1	no	—
59	C₃²×C₄	—	4	1	no	—
60	C₂²×C₉	—	4	2	no	—
61	C₂²×C₉	C₄×A₄	1	1	yes	characteristic
62	C₄×C₉	—	3	1	no	—
63	C₄×C₉	—	3	1	no	—
64	C₄×C₉	—	4	2	no	—
65	C₄×C₉	—	4	2	no	—
66	C₄×C₉	—	6	1	no	—
67	C₄×C₉	C₂×SL(2,𝔽₃)	1	1	yes	characteristic
68	C₄×C₉	#41⋊C₃	1	1	yes	characteristic
69	C₄×Q₈	C₂×C₃×C₉	1	1	yes	characteristic
70	C₄⋊C₈	—	3	1	no	—
71	C₄×C₈	—	3	1	no	—
72	C₃×C₉	—	4	1	no	Sylow
73	C₃×Q₈	—	3	1	no	—
74	C₃×Q₈	C₃²×C₈	1	1	yes	characteristic
75	C₂×C₃×C₄	—	3	1	no	—
76	C₂×C₃×C₄	—	3	1	no	—
77	C₂×C₃×C₄	—	4	3	no	—
78	C₂×C₃×C₄	C₂×C₃×A₄	1	1	yes	characteristic
79	SL(2,𝔽₃)	C₈×C₉	1	3	yes	direct product factor
80	C₃×C₈	—	4	6	no	—
81	C₃×C₈	—	6	1	no	—
82	C₃×C₈	C₃×SL(2,𝔽₃)	1	2	yes	—
83	C₂×C₃²	—	4	1	no	—
84	C₂×C₃²	—	4	1	no	—
85	C₂×C₃²	—	4	1	no	—
86	C₂×C₉	—	4	2	no	—
87	C₂×C₉	—	4	2	no	—
88	C₂×C₉	—	4	2	no	—
89	C₂×C₉	C₄×SL(2,𝔽₃)	1	1	yes	characteristic
90	C₂×C₉	C₈×A₄	1	1	yes	characteristic
91	C₂×C₉	#131⋊C₃	1	1	yes	characteristic
92	C₂×Q₈	C₃×C₄×C₉	1	1	yes	characteristic
93	C₂×C₈	—	3	1	no	—
94	C₂×C₈	C₉×A₄	1	1	yes	characteristic
95	C₄⋊C₄	—	3	1	no	—
96	C₄²	—	3	1	no	—
97	C₂²×C₃	—	4	3	no	—
98	C₂²×C₃	C₃×C₄×A₄	1	1	yes	characteristic
99	C₃×C₄	—	3	1	no	—
100	C₃×C₄	—	3	1	no	—
101	C₃×C₄	—	4	3	no	—
102	C₃×C₄	—	4	3	no	—
103	C₃×C₄	—	6	1	no	—
104	C₃×C₄	C₂×C₃×SL(2,𝔽₃)	1	1	yes	characteristic
105	C₃×C₄	C₃×#41⋊C₃	1	1	yes	characteristic
106	C₃²	—	4	1	no	—
107	C₉	—	4	2	no	—
108	C₉	C₈×SL(2,𝔽₃)	1	1	yes	characteristic, direct product factor
109	Q₈	—	3	1	no	—
110	Q₈	C₃×C₈×C₉	1	1	yes	characteristic, complemented
111	C₂×C₄	—	3	1	no	—
112	C₂×C₄	—	3	1	no	—
113	C₂×C₄	C₂×C₉×A₄	1	1	yes	characteristic
114	C₈	—	6	1	no	—
115	C₈	C₉×SL(2,𝔽₃)	1	2	yes	direct product factor
116	C₂×C₃	—	4	3	no	—
117	C₂×C₃	—	4	3	no	—
118	C₂×C₃	—	4	3	no	—
119	C₂×C₃	C₃×C₄×SL(2,𝔽₃)	1	1	yes	characteristic
120	C₂×C₃	C₃×C₈×A₄	1	1	yes	characteristic
121	C₂×C₃	C₃×#131⋊C₃	1	1	yes	characteristic
122	C₂²	C₄×C₉×A₄	1	1	yes	characteristic
123	C₄	—	3	1	no	—
124	C₄	—	3	1	no	—
125	C₄	—	6	1	no	—
126	C₄	C₂×C₉×SL(2,𝔽₃)	1	1	yes	characteristic
127	C₄	C₉×#41⋊C₃	1	1	yes	characteristic
128	C₃	—	4	3	no	—
129	C₃	C₃×C₈×SL(2,𝔽₃)	1	1	yes	characteristic
130	C₂	C₄×C₉×SL(2,𝔽₃)	1	1	yes	characteristic
131	C₂	C₈×C₉×A₄	1	1	yes	characteristic
132	C₂	C₉×#131⋊C₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×C₉×SL(2,𝔽₃)

C₂

yes

maximal, characteristic

C₃×C₈×SL(2,𝔽₃)

C₃

yes

maximal, characteristic

C₈×Q₈×C₉

C₃

yes

maximal, characteristic, complemented

C₈×Q₈⋊C₉

C₃

yes

maximal, complemented

C₂×C₉×SL(2,𝔽₃)

C₄

yes

characteristic

C₂×C₃×C₈×C₉

—

maximal

C₃×C₄×SL(2,𝔽₃)

C₂×C₃

yes

characteristic

C₄×Q₈×C₉

C₂×C₃

yes

characteristic

C₄×Q₈⋊C₉

C₂×C₃

yes

—

C₉×C₄⋊C₈

—

C₄×C₈×C₉

—

C₂×C₃×C₄×C₉

—

C₉×SL(2,𝔽₃)

C₈

yes

characteristic, direct product factor

C₃×C₈×C₉

—

C₃×C₈×Q₈

C₃²

yes

characteristic

C₈×SL(2,𝔽₃)

C₉

yes

direct product factor

C₂×C₃×SL(2,𝔽₃)

C₃×C₄

yes

characteristic

C₂×C₃²×C₈

—

C₂×Q₈×C₉

C₃×C₄

yes

characteristic

C₂×Q₈⋊C₉

C₃×C₄

yes

—

C₂×C₈×C₉

—

C₂×C₈×C₉

—

C₂×C₈×C₉

A₄

yes

characteristic

C₉×C₄⋊C₄

—

C₄²×C₉

—

C₂²×C₃×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×Q₈

C₂×C₃²

yes

characteristic

C₄×SL(2,𝔽₃)

C₂×C₉

yes

—

C₃×C₄⋊C₈

—

C₃×C₄×C₈

—

C₂×C₃²×C₄

—

C₃×SL(2,𝔽₃)

C₃×C₈

yes

characteristic

C₃²×C₈

—

Q₈×C₉

—

Q₈×C₉

C₃×C₈

yes

characteristic, complemented

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×A₄

yes

characteristic

Q₈⋊C₉

C₃×C₈

yes

complemented

C₈×C₉

—

C₈×C₉

—

C₈×C₉

SL(2,𝔽₃)

yes

direct product factor

C₈×Q₈

C₃×C₉

yes

characteristic, complemented, Sylow

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×Q₈

C₃²×C₄

yes

characteristic

C₂×SL(2,𝔽₃)

C₄×C₉

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₃×A₄

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₂²×C₃²

—

C₃²×C₄

—

C₃²×C₄

—

C₂²×C₉

—

C₂²×C₉

C₄×A₄

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₂×SL(2,𝔽₃)

yes

characteristic

C₄×C₉

#41⋊C₃

yes

characteristic

C₄×Q₈

C₂×C₃×C₉

yes

characteristic

C₄⋊C₈

—

C₄×C₈

—

C₃×C₉

—

Sylow

C₃×Q₈

—

C₃×Q₈

C₃²×C₈

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃×A₄

yes

characteristic

SL(2,𝔽₃)

C₈×C₉

yes

direct product factor

C₃×C₈

—

C₃×C₈

—

C₃×C₈

C₃×SL(2,𝔽₃)

yes

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₃²

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₄×SL(2,𝔽₃)

yes

characteristic

C₂×C₉

C₈×A₄

yes

characteristic

C₂×C₉

#131⋊C₃

yes

characteristic

C₂×Q₈

C₃×C₄×C₉

yes

characteristic

C₂×C₈

—

C₂×C₈

C₉×A₄

yes

characteristic

C₄⋊C₄

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

C₃×C₄×A₄

yes

characteristic

C₃×C₄

—

100

C₃×C₄

—

101

C₃×C₄

—

102

C₃×C₄

—

103

C₃×C₄

—

104

C₃×C₄

C₂×C₃×SL(2,𝔽₃)

yes

characteristic

105

C₃×C₄

C₃×#41⋊C₃

yes

characteristic

106

C₃²

—

107

C₉

—

108

C₉

C₈×SL(2,𝔽₃)

yes

characteristic, direct product factor

109

Q₈

—

110

Q₈

C₃×C₈×C₉

yes

characteristic, complemented

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×C₉×A₄

yes

characteristic

114

C₈

—

115

C₈

C₉×SL(2,𝔽₃)

yes

direct product factor

116

C₂×C₃

—

117

C₂×C₃

—

118

C₂×C₃

—

119

C₂×C₃

C₃×C₄×SL(2,𝔽₃)

yes

characteristic

120

C₂×C₃

C₃×C₈×A₄

yes

characteristic

121

C₂×C₃

C₃×#131⋊C₃

yes

characteristic

122

C₂²

C₄×C₉×A₄

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

C₂×C₉×SL(2,𝔽₃)

yes

characteristic

127

C₄

C₉×#41⋊C₃

yes

characteristic

128

C₃

—

129

C₃

C₃×C₈×SL(2,𝔽₃)

yes

characteristic

130

C₂

C₄×C₉×SL(2,𝔽₃)

yes

characteristic

131

C₂

C₈×C₉×A₄

yes

characteristic

132

C₂

C₉×#131⋊C₃

yes

characteristic

	subgroup	G/H
center, hypercenter	C₂×C₈×C₉	A₄
commutator, hypocenter	Q₈	C₃×C₈×C₉
Frattini	C₂×C₃×C₄	C₂×C₃×A₄
Fitting	C₈×Q₈×C₉	C₃
socle	C₂²×C₃	C₃×C₄×A₄
prime radical	C₂×C₃×SL(2,𝔽₃)	C₃×C₄
cosocle	C₃×C₄×Q₈	C₂×C₃²
LCS	G ▷ Q₈
UCS	C₁ ◁ C₂×C₈×C₉

subgroup

G/H

center, hypercenter

C₂×C₈×C₉

A₄

commutator, hypocenter

Q₈

C₃×C₈×C₉

Frattini

C₂×C₃×C₄

C₂×C₃×A₄

Fitting

C₈×Q₈×C₉

C₃

socle

C₂²×C₃

C₃×C₄×A₄

prime radical

C₂×C₃×SL(2,𝔽₃)

C₃×C₄

cosocle

C₃×C₄×Q₈

C₂×C₃²

LCS

G ▷ Q₈

UCS

C₁ ◁ C₂×C₈×C₉

N	H ≅ G/N	type
3. C₈×Q₈×C₉	C₃	semidirect
4. C₈×Q₈⋊C₉	C₃	semidirect
13. C₉×SL(2,𝔽₃)	C₈	direct (N × H)
16. C₈×SL(2,𝔽₃)	C₉	direct (N × H)
37. Q₈×C₉	C₃×C₈	semidirect
42. Q₈⋊C₉	C₃×C₈	semidirect
45. C₈×C₉	SL(2,𝔽₃)	direct (N × H)
46. C₈×Q₈	C₃×C₉	semidirect
79. SL(2,𝔽₃)	C₈×C₉	direct (N × H)
108. C₉	C₈×SL(2,𝔽₃)	direct (N × H)
110. Q₈	C₃×C₈×C₉	semidirect
115. C₈	C₉×SL(2,𝔽₃)	direct (N × H)

H ≅ G/N

type

3. C₈×Q₈×C₉

C₃

semidirect

4. C₈×Q₈⋊C₉

C₃

semidirect

13. C₉×SL(2,𝔽₃)

C₈

direct (N × H)

16. C₈×SL(2,𝔽₃)

C₉

direct (N × H)

37. Q₈×C₉

C₃×C₈

semidirect

42. Q₈⋊C₉

C₃×C₈

semidirect

45. C₈×C₉

SL(2,𝔽₃)

direct (N × H)

46. C₈×Q₈

C₃×C₉

semidirect

79. SL(2,𝔽₃)

C₈×C₉

direct (N × H)

108. C₉

C₈×SL(2,𝔽₃)

direct (N × H)

110. Q₈

C₃×C₈×C₉

semidirect

115. C₈

C₉×SL(2,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×SL(2,𝔽₃)	2
C₃×C₈×SL(2,𝔽₃)	3
C₈×Q₈×C₉	3
C₈×Q₈⋊C₉	3
C₂×C₃×C₈×C₉	4

subgroup

index

C₄×C₉×SL(2,𝔽₃)

C₃×C₈×SL(2,𝔽₃)

C₈×Q₈×C₉

C₈×Q₈⋊C₉

C₂×C₃×C₈×C₉

Subgroups — C₈×C₉×SL(2,𝔽₃) #13141207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C8×C9×SL(2,𝔽3) #13141207

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×C₉×SL(2,𝔽₃) #13141207