Group #13141253 — subgroups

Subgroups — C₃×C₈×Q₈⋊C₉ #13141253

Order 1728.

222 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 222 conjugacy classes of subgroups — too many to draw. Only its 106 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	C₃×C₄×Q₈⋊C₉	C₂	1	1	yes	maximal, characteristic
2	C₃²×C₈×Q₈	C₃	1	1	yes	maximal, characteristic
3	C₈×Q₈⋊C₉	C₃	1	3	yes	maximal, direct product factor
4	C₂×C₃×Q₈⋊C₉	C₄	1	1	yes	characteristic
5	C₂×C₃×C₈×C₉	—	4	1	no	maximal
6	C₃²×C₄×Q₈	C₂×C₃	1	1	yes	characteristic
7	C₃²×C₄⋊C₈	—	3	1	no	—
8	C₃²×C₄×C₈	—	3	1	no	—
9	C₄×Q₈⋊C₉	C₂×C₃	1	3	yes	—
10	C₂×C₃×C₄×C₉	—	4	1	no	—
11	C₃×Q₈⋊C₉	C₈	1	1	yes	characteristic, direct product factor
12	C₃×C₈×C₉	—	4	2	no	—
13	C₃×C₈×Q₈	C₉	1	3	yes	complemented
14	C₃×C₈×Q₈	C₃²	1	1	yes	characteristic
15	C₂×C₃²×Q₈	C₃×C₄	1	1	yes	characteristic
16	C₂×C₃²×C₈	—	3	1	no	—
17	C₂×C₃²×C₈	A₄	1	1	yes	characteristic
18	C₃²×C₄⋊C₄	—	3	1	no	—
19	C₃²×C₄²	—	3	1	no	—
20	C₂×Q₈⋊C₉	C₃×C₄	1	3	yes	—
21	C₂×C₈×C₉	—	4	3	no	—
22	C₂²×C₃×C₉	—	4	1	no	—
23	C₃×C₄×C₉	—	4	1	no	—
24	C₃×C₄×C₉	—	4	1	no	—
25	C₃×C₄×Q₈	C₂×C₉	1	3	yes	—
26	C₃×C₄×Q₈	C₂×C₃²	1	1	yes	characteristic
27	C₃×C₄⋊C₈	—	3	3	no	—
28	C₃×C₄⋊C₈	—	3	1	no	—
29	C₃×C₄×C₈	—	3	3	no	—
30	C₃×C₄×C₈	—	3	1	no	—
31	C₃²×Q₈	—	3	1	no	—
32	C₃²×Q₈	C₃×C₈	1	1	yes	characteristic
33	C₂×C₃²×C₄	—	3	1	no	—
34	C₂×C₃²×C₄	—	3	1	no	—
35	C₂×C₃²×C₄	C₂×A₄	1	1	yes	characteristic
36	C₃²×C₈	—	6	1	no	—
37	C₃²×C₈	SL(2,𝔽₃)	1	2	yes	—
38	C₂×C₄×C₉	—	4	3	no	—
39	Q₈⋊C₉	C₃×C₈	1	3	yes	direct product factor
40	C₈×C₉	—	4	6	no	—
41	C₈×Q₈	C₃×C₉	1	1	yes	characteristic, complemented, Sylow
42	C₂×C₃×C₉	—	4	1	no	—
43	C₂×C₃×C₉	—	4	1	no	—
44	C₂×C₃×C₉	—	4	1	no	—
45	C₂×C₃×Q₈	C₄×C₉	1	3	yes	—
46	C₂×C₃×Q₈	C₃²×C₄	1	1	yes	characteristic
47	C₂×C₃×C₈	—	3	3	no	—
48	C₂×C₃×C₈	—	3	1	no	—
49	C₂×C₃×C₈	C₂²⋊C₉	1	3	yes	—
50	C₂×C₃×C₈	C₃×A₄	1	1	yes	characteristic
51	C₃×C₄⋊C₄	—	3	3	no	—
52	C₃×C₄⋊C₄	—	3	1	no	—
53	C₃×C₄²	—	3	3	no	—
54	C₃×C₄²	—	3	1	no	—
55	C₂²×C₃²	C₄×A₄	1	1	yes	characteristic
56	C₃²×C₄	—	3	1	no	—
57	C₃²×C₄	—	3	1	no	—
58	C₃²×C₄	—	6	1	no	—
59	C₃²×C₄	C₂×SL(2,𝔽₃)	1	1	yes	characteristic
60	C₃²×C₄	#41⋊C₃	1	1	yes	characteristic
61	C₂²×C₉	—	4	3	no	—
62	C₄×C₉	—	4	3	no	—
63	C₄×C₉	—	4	3	no	—
64	C₄×Q₈	C₂×C₃×C₉	1	1	yes	characteristic
65	C₄⋊C₈	—	3	1	no	—
66	C₄×C₈	—	3	1	no	—
67	C₃×C₉	—	4	1	no	Sylow
68	C₃×Q₈	—	3	3	no	—
69	C₃×Q₈	—	3	1	no	—
70	C₃×Q₈	C₈×C₉	1	3	yes	complemented
71	C₃×Q₈	C₃²×C₈	1	1	yes	characteristic
72	C₂×C₃×C₄	—	3	3	no	—
73	C₂×C₃×C₄	—	3	1	no	—
74	C₂×C₃×C₄	—	3	3	no	—
75	C₂×C₃×C₄	—	3	1	no	—
76	C₂×C₃×C₄	C₂×C₂²⋊C₉	1	3	yes	—
77	C₂×C₃×C₄	C₂×C₃×A₄	1	1	yes	characteristic
78	C₃×C₈	—	6	3	no	—
79	C₃×C₈	—	6	1	no	—
80	C₃×C₈	Q₈⋊C₉	1	6	yes	direct product factor
81	C₃×C₈	C₃×SL(2,𝔽₃)	1	2	yes	—
82	C₂×C₃²	C₄×SL(2,𝔽₃)	1	1	yes	characteristic
83	C₂×C₃²	C₈×A₄	1	1	yes	characteristic
84	C₂×C₃²	#131⋊C₃	1	1	yes	characteristic
85	C₂×C₉	—	4	3	no	—
86	C₂×C₉	—	4	3	no	—
87	C₂×C₉	—	4	3	no	—
88	C₂×Q₈	C₃×C₄×C₉	1	1	yes	characteristic
89	C₂×C₈	—	3	1	no	—
90	C₂×C₈	C₃×C₂²⋊C₉	1	1	yes	characteristic
91	C₄⋊C₄	—	3	1	no	—
92	C₄²	—	3	1	no	—
93	C₂²×C₃	C₄×C₂²⋊C₉	1	3	yes	—
94	C₂²×C₃	C₃×C₄×A₄	1	1	yes	characteristic
95	C₃×C₄	—	3	3	no	—
96	C₃×C₄	—	3	3	no	—
97	C₃×C₄	—	3	1	no	—
98	C₃×C₄	—	3	1	no	—
99	C₃×C₄	—	6	3	no	—
100	C₃×C₄	—	6	1	no	—
101	C₃×C₄	C₂×Q₈⋊C₉	1	3	yes	—
102	C₃×C₄	#41⋊C₉	1	3	yes	—
103	C₃×C₄	C₂×C₃×SL(2,𝔽₃)	1	1	yes	characteristic
104	C₃×C₄	C₃×#41⋊C₃	1	1	yes	characteristic
105	C₃²	C₈×SL(2,𝔽₃)	1	1	yes	characteristic
106	C₉	—	4	3	no	—
107	Q₈	—	3	1	no	—
108	Q₈	C₃×C₈×C₉	1	1	yes	characteristic, complemented
109	C₂×C₄	—	3	1	no	—
110	C₂×C₄	—	3	1	no	—
111	C₂×C₄	C₂×C₃×C₂²⋊C₉	1	1	yes	characteristic
112	C₈	—	6	1	no	—
113	C₈	C₃×Q₈⋊C₉	1	2	yes	direct product factor
114	C₂×C₃	C₄×Q₈⋊C₉	1	3	yes	—
115	C₂×C₃	C₈×C₂²⋊C₉	1	3	yes	—
116	C₂×C₃	#131⋊C₉	1	3	yes	—
117	C₂×C₃	C₃×C₄×SL(2,𝔽₃)	1	1	yes	characteristic
118	C₂×C₃	C₃×C₈×A₄	1	1	yes	characteristic
119	C₂×C₃	C₃×#131⋊C₃	1	1	yes	characteristic
120	C₂²	C₃×C₄×C₂²⋊C₉	1	1	yes	characteristic
121	C₄	—	3	1	no	—
122	C₄	—	3	1	no	—
123	C₄	—	6	1	no	—
124	C₄	C₂×C₃×Q₈⋊C₉	1	1	yes	characteristic
125	C₄	C₃×#41⋊C₉	1	1	yes	characteristic
126	C₃	C₈×Q₈⋊C₉	1	3	yes	direct product factor
127	C₃	C₃×C₈×SL(2,𝔽₃)	1	1	yes	characteristic
128	C₂	C₃×C₄×Q₈⋊C₉	1	1	yes	characteristic
129	C₂	C₃×C₈×C₂²⋊C₉	1	1	yes	characteristic
130	C₂	C₃×#131⋊C₉	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₄×Q₈⋊C₉

C₂

yes

maximal, characteristic

C₃²×C₈×Q₈

C₃

yes

maximal, characteristic

C₈×Q₈⋊C₉

C₃

yes

maximal, direct product factor

C₂×C₃×Q₈⋊C₉

C₄

yes

characteristic

C₂×C₃×C₈×C₉

—

maximal

C₃²×C₄×Q₈

C₂×C₃

yes

characteristic

C₃²×C₄⋊C₈

—

C₃²×C₄×C₈

—

C₄×Q₈⋊C₉

C₂×C₃

yes

—

C₂×C₃×C₄×C₉

—

C₃×Q₈⋊C₉

C₈

yes

characteristic, direct product factor

C₃×C₈×C₉

—

C₃×C₈×Q₈

C₉

yes

complemented

C₃×C₈×Q₈

C₃²

yes

characteristic

C₂×C₃²×Q₈

C₃×C₄

yes

characteristic

C₂×C₃²×C₈

—

C₂×C₃²×C₈

A₄

yes

characteristic

C₃²×C₄⋊C₄

—

C₃²×C₄²

—

C₂×Q₈⋊C₉

C₃×C₄

yes

—

C₂×C₈×C₉

—

C₂²×C₃×C₉

—

C₃×C₄×C₉

—

C₃×C₄×C₉

—

C₃×C₄×Q₈

C₂×C₉

yes

—

C₃×C₄×Q₈

C₂×C₃²

yes

characteristic

C₃×C₄⋊C₈

—

C₃×C₄⋊C₈

—

C₃×C₄×C₈

—

C₃×C₄×C₈

—

C₃²×Q₈

—

C₃²×Q₈

C₃×C₈

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×A₄

yes

characteristic

C₃²×C₈

—

C₃²×C₈

SL(2,𝔽₃)

yes

—

C₂×C₄×C₉

—

Q₈⋊C₉

C₃×C₈

yes

direct product factor

C₈×C₉

—

C₈×Q₈

C₃×C₉

yes

characteristic, complemented, Sylow

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×C₉

—

C₂×C₃×Q₈

C₄×C₉

yes

—

C₂×C₃×Q₈

C₃²×C₄

yes

characteristic

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂²⋊C₉

yes

—

C₂×C₃×C₈

C₃×A₄

yes

characteristic

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

—

C₂²×C₃²

C₄×A₄

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×SL(2,𝔽₃)

yes

characteristic

C₃²×C₄

#41⋊C₃

yes

characteristic

C₂²×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×Q₈

C₂×C₃×C₉

yes

characteristic

C₄⋊C₈

—

C₄×C₈

—

C₃×C₉

—

Sylow

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

C₈×C₉

yes

complemented

C₃×Q₈

C₃²×C₈

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₂²⋊C₉

yes

—

C₂×C₃×C₄

C₂×C₃×A₄

yes

characteristic

C₃×C₈

—

C₃×C₈

—

C₃×C₈

Q₈⋊C₉

yes

direct product factor

C₃×C₈

C₃×SL(2,𝔽₃)

yes

—

C₂×C₃²

C₄×SL(2,𝔽₃)

yes

characteristic

C₂×C₃²

C₈×A₄

yes

characteristic

C₂×C₃²

#131⋊C₃

yes

characteristic

C₂×C₉

—

C₂×C₉

—

C₂×C₉

—

C₂×Q₈

C₃×C₄×C₉

yes

characteristic

C₂×C₈

—

C₂×C₈

C₃×C₂²⋊C₉

yes

characteristic

C₄⋊C₄

—

C₄²

—

C₂²×C₃

C₄×C₂²⋊C₉

yes

—

C₂²×C₃

C₃×C₄×A₄

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

100

C₃×C₄

—

101

C₃×C₄

C₂×Q₈⋊C₉

yes

—

102

C₃×C₄

#41⋊C₉

yes

—

103

C₃×C₄

C₂×C₃×SL(2,𝔽₃)

yes

characteristic

104

C₃×C₄

C₃×#41⋊C₃

yes

characteristic

105

C₃²

C₈×SL(2,𝔽₃)

yes

characteristic

106

C₉

—

107

Q₈

—

108

Q₈

C₃×C₈×C₉

yes

characteristic, complemented

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

C₂×C₃×C₂²⋊C₉

yes

characteristic

112

C₈

—

113

C₈

C₃×Q₈⋊C₉

yes

direct product factor

114

C₂×C₃

C₄×Q₈⋊C₉

yes

—

115

C₂×C₃

C₈×C₂²⋊C₉

yes

—

116

C₂×C₃

#131⋊C₉

yes

—

117

C₂×C₃

C₃×C₄×SL(2,𝔽₃)

yes

characteristic

118

C₂×C₃

C₃×C₈×A₄

yes

characteristic

119

C₂×C₃

C₃×#131⋊C₃

yes

characteristic

120

C₂²

C₃×C₄×C₂²⋊C₉

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

C₂×C₃×Q₈⋊C₉

yes

characteristic

125

C₄

C₃×#41⋊C₉

yes

characteristic

126

C₃

C₈×Q₈⋊C₉

yes

direct product factor

127

C₃

C₃×C₈×SL(2,𝔽₃)

yes

characteristic

128

C₂

C₃×C₄×Q₈⋊C₉

yes

characteristic

129

C₂

C₃×C₈×C₂²⋊C₉

yes

characteristic

130

C₂

C₃×#131⋊C₉

yes

characteristic

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

subgroup

G/H

center, hypercenter

C₂×C₃²×C₈

A₄

commutator, hypocenter

Q₈

C₃×C₈×C₉

Frattini

C₂×C₃×C₄

C₂×C₃×A₄

Fitting

C₃²×C₈×Q₈

C₃

socle, prime radical

C₂²×C₃²

C₄×A₄

cosocle

C₃×C₄×Q₈

C₂×C₃²

LCS

G ▷ Q₈

UCS

C₁ ◁ C₂×C₃²×C₈

N	H ≅ G/N	type
3. C₈×Q₈⋊C₉	C₃	direct (N × H)
11. C₃×Q₈⋊C₉	C₈	direct (N × H)
13. C₃×C₈×Q₈	C₉	semidirect
39. Q₈⋊C₉	C₃×C₈	direct (N × H)
41. C₈×Q₈	C₃×C₉	semidirect
70. C₃×Q₈	C₈×C₉	semidirect
80. C₃×C₈	Q₈⋊C₉	direct (N × H)
108. Q₈	C₃×C₈×C₉	semidirect
113. C₈	C₃×Q₈⋊C₉	direct (N × H)
126. C₃	C₈×Q₈⋊C₉	direct (N × H)

H ≅ G/N

type

3. C₈×Q₈⋊C₉

C₃

direct (N × H)

11. C₃×Q₈⋊C₉

C₈

direct (N × H)

13. C₃×C₈×Q₈

C₉

semidirect

39. Q₈⋊C₉

C₃×C₈

direct (N × H)

41. C₈×Q₈

C₃×C₉

semidirect

70. C₃×Q₈

C₈×C₉

semidirect

80. C₃×C₈

Q₈⋊C₉

direct (N × H)

108. Q₈

C₃×C₈×C₉

semidirect

113. C₈

C₃×Q₈⋊C₉

direct (N × H)

126. C₃

C₈×Q₈⋊C₉

direct (N × H)

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

subgroup	index
C₃×C₄×Q₈⋊C₉	2
C₃²×C₈×Q₈	3
C₈×Q₈⋊C₉	3
C₂×C₃×C₈×C₉	4

subgroup

index

C₃×C₄×Q₈⋊C₉

C₃²×C₈×Q₈

C₈×Q₈⋊C₉

C₂×C₃×C₈×C₉

Subgroups — C₃×C₈×Q₈⋊C₉ #13141253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C3×C8×Q8⋊C9 #13141253

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₈×Q₈⋊C₉ #13141253