Group #13135642 — subgroups

Subgroups — #10588223⋊C₃ #13135642

Order 1728.

226 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 86 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	3₋¹⁺²×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₄⋊(3₋¹⁺²⋊C₈)	C₂	1	2	yes	maximal
3	C₃×#4546	—	3	1	no	maximal
4	#10588223	C₃	1	1	yes	maximal, characteristic, complemented
5	C₂×Q₈×3₋¹⁺²	C₄	1	2	yes	—
6	C₄⋊C₄×3₋¹⁺²	—	2	2	no	—
7	C₄²×3₋¹⁺²	C₂²	1	1	yes	characteristic
8	C₂×3₋¹⁺²⋊C₈	—	2	2	no	—
9	C₃²×C₄⋊Q₈	D₃	1	1	yes	characteristic
10	C₃×C₄⋊(C₃⋊C₈)	—	3	2	no	—
11	C₉×C₄⋊Q₈	—	2	1	no	—
12	C₉×C₄⋊Q₈	C₂×C₃	1	1	yes	characteristic
13	C₄⋊(C₉⋊C₈)	C₂×C₃	1	2	yes	—
14	Q₈×3₋¹⁺²	—	4	2	no	—
15	C₂×C₄×3₋¹⁺²	—	2	2	no	—
16	C₂×C₄×3₋¹⁺²	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×3₋¹⁺²	D₄	1	2	yes	—
18	3₋¹⁺²⋊C₈	—	4	2	no	—
19	C₃×#333	—	9	1	no	—
20	#4546	—	3	1	no	—
21	C₂×C₃²×Q₈	Dic₃	1	2	yes	—
22	C₃²×C₄⋊C₄	—	2	2	no	—
23	C₃²×C₄²	C₂×D₃	1	1	yes	characteristic
24	C₂×C₃×C₃⋊C₈	—	6	2	no	—
25	C₂×Q₈×C₉	—	2	2	no	—
26	C₂×Q₈×C₉	C₃×C₄	1	2	yes	—
27	C₉×C₄⋊C₄	—	2	4	no	—
28	C₉×C₄⋊C₄	—	2	2	no	—
29	C₄²×C₉	—	2	1	no	—
30	C₄²×C₉	C₂²×C₃	1	1	yes	characteristic
31	C₂×C₉⋊C₈	—	2	2	no	—
32	C₂²×3₋¹⁺²	C₂²⋊C₄	1	1	yes	characteristic
33	C₄×3₋¹⁺²	—	2	1	no	—
34	C₄×3₋¹⁺²	—	4	2	no	—
35	C₄×3₋¹⁺²	SD₁₆	1	2	yes	—
36	C₄×3₋¹⁺²	Q₁₆	1	2	yes	—
37	C₃×C₄⋊Q₈	—	3	1	no	—
38	C₃×C₄⋊Q₈	C₃×D₃	1	1	yes	characteristic
39	C₃×C₄⋊C₈	—	9	2	no	—
40	C₄⋊(C₃⋊C₈)	—	3	2	no	—
41	C₃²×Q₈	—	4	2	no	—
42	C₂×C₃²×C₄	—	2	2	no	—
43	C₂×C₃²×C₄	C₂×Dic₃	1	1	yes	characteristic
44	C₂×C₃²×C₄	#67	1	2	yes	—
45	C₃×C₃⋊C₈	—	12	2	no	—
46	Q₈×C₉	—	4	2	no	—
47	Q₈×C₉	—	4	4	no	—
48	C₂×C₄×C₉	—	2	4	no	—
49	C₂×C₄×C₉	—	2	2	no	—
50	C₂×C₄×C₉	—	2	1	no	—
51	C₂×C₄×C₉	—	2	2	no	—
52	C₂×C₄×C₉	C₂×C₃×C₄	1	1	yes	characteristic
53	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
54	C₉⋊C₈	—	4	2	no	—
55	#333	—	9	1	no	Sylow
56	C₂×3₋¹⁺²	#100	1	1	yes	characteristic
57	C₂×3₋¹⁺²	#103	1	2	yes	—
58	C₂×C₃×Q₈	—	3	2	no	—
59	C₂×C₃×Q₈	C₃×Dic₃	1	2	yes	—
60	C₂×C₃×C₈	—	18	2	no	—
61	C₃×C₄⋊C₄	—	2	2	no	—
62	C₃×C₄⋊C₄	—	6	2	no	—
63	C₃×C₄²	—	3	1	no	—
64	C₃×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
65	C₂×C₃⋊C₈	—	6	2	no	—
66	C₂²×C₃²	#217	1	1	yes	characteristic
67	C₃²×C₄	—	2	1	no	—
68	C₃²×C₄	—	4	2	no	—
69	C₃²×C₄	#215	1	2	yes	—
70	C₃²×C₄	#216	1	2	yes	—
71	C₂²×C₉	—	2	1	no	—
72	C₂²×C₉	C₃×C₂²⋊C₄	1	1	yes	characteristic
73	C₄×C₉	—	2	2	no	—
74	C₄×C₉	—	2	1	no	—
75	C₄×C₉	—	2	2	no	—
76	C₄×C₉	—	2	2	no	—
77	C₄×C₉	—	4	4	no	—
78	C₄×C₉	—	4	2	no	—
79	C₄×C₉	C₃×SD₁₆	1	2	yes	—
80	C₄×C₉	C₃×Q₁₆	1	2	yes	—
81	C₄⋊Q₈	AGL(1,ℤ/9)	1	1	yes	characteristic
82	C₄⋊C₈	—	9	2	no	—
83	3₋¹⁺²	#333	1	1	yes	characteristic, complemented, Sylow
84	C₃×Q₈	—	4	2	no	—
85	C₃×Q₈	—	12	2	no	—
86	C₂×C₃×C₄	—	2	2	no	—
87	C₂×C₃×C₄	—	3	1	no	—
88	C₂×C₃×C₄	—	3	2	no	—
89	C₂×C₃×C₄	—	6	2	no	—
90	C₂×C₃×C₄	C₂×C₃×Dic₃	1	1	yes	characteristic
91	C₂×C₃×C₄	C₃×#67	1	2	yes	—
92	C₃×C₈	—	36	2	no	—
93	C₃⋊C₈	—	12	2	no	—
94	C₂×C₃²	#833	1	1	yes	characteristic
95	C₂×C₃²	#835	1	2	yes	—
96	C₂×C₉	—	2	1	no	—
97	C₂×C₉	—	2	2	no	—
98	C₂×C₉	C₃×#100	1	1	yes	characteristic
99	C₂×C₉	C₃×#103	1	2	yes	—
100	C₂×Q₈	3₋¹⁺²⋊C₄	1	2	yes	—
101	C₂×C₈	—	18	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄²	C₂×AGL(1,ℤ/9)	1	1	yes	characteristic
104	C₂²×C₃	—	3	1	no	—
105	C₂²×C₃	C₃×#217	1	1	yes	characteristic
106	C₃×C₄	—	2	1	no	—
107	C₃×C₄	—	3	2	no	—
108	C₃×C₄	—	3	2	no	—
109	C₃×C₄	—	4	2	no	—
110	C₃×C₄	—	6	1	no	—
111	C₃×C₄	—	12	2	no	—
112	C₃×C₄	C₃×#215	1	2	yes	—
113	C₃×C₄	C₃×#216	1	2	yes	—
114	C₃²	#4546	1	1	yes	characteristic
115	C₉	—	2	1	no	—
116	C₉	C₃×#333	1	1	yes	characteristic, complemented
117	Q₈	—	4	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	C₂×3₋¹⁺²⋊C₄	1	1	yes	characteristic
120	C₂×C₄	#611⋊C₃	1	2	yes	—
121	C₈	—	36	2	no	—
122	C₂×C₃	—	3	1	no	—
123	C₂×C₃	—	3	2	no	—
124	C₂×C₃	C₃×#833	1	1	yes	characteristic
125	C₂×C₃	C₃×#835	1	2	yes	—
126	C₂²	#3676⋊C₃	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	4	2	no	—
129	C₄	#3674⋊C₃	1	2	yes	—
130	C₄	#3675⋊C₃	1	2	yes	—
131	C₃	—	3	1	no	—
132	C₃	C₃×#4546	1	1	yes	characteristic
133	C₂	#63391⋊C₃	1	1	yes	characteristic
134	C₂	#63395⋊C₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

3₋¹⁺²×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₄⋊(3₋¹⁺²⋊C₈)

C₂

yes

—

C₃

yes

maximal, characteristic, complemented

C₂×Q₈×3₋¹⁺²

C₄

yes

—

C₄⋊C₄×3₋¹⁺²

—

C₄²×3₋¹⁺²

C₂²

yes

characteristic

C₂×3₋¹⁺²⋊C₈

—

C₃²×C₄⋊Q₈

D₃

yes

characteristic

C₃×C₄⋊(C₃⋊C₈)

—

C₉×C₄⋊Q₈

—

C₉×C₄⋊Q₈

C₂×C₃

yes

characteristic

C₄⋊(C₉⋊C₈)

C₂×C₃

yes

—

Q₈×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

—

C₂×C₄×3₋¹⁺²

C₂×C₄

yes

characteristic

C₂×C₄×3₋¹⁺²

D₄

yes

—

3₋¹⁺²⋊C₈

—

C₃×#333

—

#4546

—

C₂×C₃²×Q₈

Dic₃

yes

—

C₃²×C₄⋊C₄

—

C₃²×C₄²

C₂×D₃

yes

characteristic

C₂×C₃×C₃⋊C₈

—

C₂×Q₈×C₉

—

C₂×Q₈×C₉

C₃×C₄

yes

—

C₉×C₄⋊C₄

—

C₉×C₄⋊C₄

—

C₄²×C₉

—

C₄²×C₉

C₂²×C₃

yes

characteristic

C₂×C₉⋊C₈

—

C₂²×3₋¹⁺²

C₂²⋊C₄

yes

characteristic

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

—

C₄×3₋¹⁺²

SD₁₆

yes

—

C₄×3₋¹⁺²

Q₁₆

yes

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

C₃×D₃

yes

characteristic

C₃×C₄⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×Dic₃

yes

characteristic

C₂×C₃²×C₄

#67

yes

—

C₃×C₃⋊C₈

—

Q₈×C₉

—

Q₈×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

—

C₂×C₄×C₉

C₂×C₃×C₄

yes

characteristic

C₂×C₄×C₉

C₃×D₄

yes

—

C₉⋊C₈

—

—

C₂×3₋¹⁺²

yes

C₂×3₋¹⁺²

yes

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₃×Dic₃

yes

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

C₃×C₄²

C₂×C₃×D₃

yes

characteristic

C₂×C₃⋊C₈

—

C₂²×C₃²

#217

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

#215

yes

—

C₃²×C₄

#216

yes

—

C₂²×C₉

—

C₂²×C₉

C₃×C₂²⋊C₄

yes

characteristic

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

—

C₄×C₉

C₃×SD₁₆

yes

—

C₄×C₉

C₃×Q₁₆

yes

—

C₄⋊Q₈

AGL(1,ℤ/9)

yes

characteristic

C₄⋊C₈

—

3₋¹⁺²

#333

yes

characteristic, complemented, Sylow

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃×Dic₃

yes

characteristic

C₂×C₃×C₄

C₃×#67

yes

—

C₃×C₈

—

C₃⋊C₈

—

C₂×C₃²

#833

yes

characteristic

C₂×C₃²

#835

yes

—

C₂×C₉

—

C₂×C₉

—

C₂×C₉

C₃×#100

yes

characteristic

C₂×C₉

C₃×#103

yes

—

100

C₂×Q₈

3₋¹⁺²⋊C₄

yes

—

101

C₂×C₈

—

102

C₄⋊C₄

—

103

C₄²

C₂×AGL(1,ℤ/9)

yes

characteristic

104

C₂²×C₃

—

105

C₂²×C₃

C₃×#217

yes

characteristic

106

C₃×C₄

—

107

C₃×C₄

—

108

C₃×C₄

—

109

C₃×C₄

—

110

C₃×C₄

—

111

C₃×C₄

—

112

C₃×C₄

C₃×#215

yes

—

113

C₃×C₄

C₃×#216

yes

—

114

C₃²

#4546

yes

characteristic

115

C₉

—

116

C₉

C₃×#333

yes

characteristic, complemented

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

C₂×3₋¹⁺²⋊C₄

yes

characteristic

120

C₂×C₄

#611⋊C₃

yes

—

121

C₈

—

122

C₂×C₃

—

123

C₂×C₃

—

124

C₂×C₃

C₃×#833

yes

characteristic

125

C₂×C₃

C₃×#835

yes

—

126

C₂²

#3676⋊C₃

yes

characteristic

127

C₄

—

128

C₄

—

129

C₄

#3674⋊C₃

yes

—

130

C₄

#3675⋊C₃

yes

—

131

C₃

—

132

C₃

C₃×#4546

yes

characteristic

133

C₂

#63391⋊C₃

yes

characteristic

134

C₂

#63395⋊C₃

yes

—

	subgroup	G/H
center	C₂²	#3676⋊C₃
commutator	C₂×C₄×C₉	C₂×C₃×C₄
Frattini	C₃×C₄²	C₂×C₃×D₃
Fitting	3₋¹⁺²×C₄⋊Q₈	C₂
socle	C₂²×C₃	C₃×#217
prime radical	C₂²×C₃²	#217
cosocle	C₄²×C₉	C₂²×C₃
hypercenter	C₄⋊Q₈	AGL(1,ℤ/9)
hypocenter	C₉	C₃×#333
LCS	G ▷ C₂×C₄×C₉ ▷ C₂²×C₉ ▷ C₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

#3676⋊C₃

commutator

C₂×C₄×C₉

C₂×C₃×C₄

Frattini

C₃×C₄²

C₂×C₃×D₃

Fitting

3₋¹⁺²×C₄⋊Q₈

C₂

C₂²×C₃

C₂²×C₃²

C₄²×C₉

C₂²×C₃

C₄⋊Q₈

C₉

G ▷ C₂×C₄×C₉ ▷ C₂²×C₉ ▷ C₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
4. #10588223	C₃	semidirect
83. 3₋¹⁺²	#333	semidirect
116. C₉	C₃×#333	semidirect

H ≅ G/N

type

4. #10588223

C₃

semidirect

83. 3₋¹⁺²

#333

semidirect

116. C₉

C₃×#333

semidirect

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

subgroup	index
3₋¹⁺²×C₄⋊Q₈	2
C₄⋊(3₋¹⁺²⋊C₈)	2
C₃×#4546	3
#10588223	3

subgroup

index

3₋¹⁺²×C₄⋊Q₈

C₄⋊(3₋¹⁺²⋊C₈)

C₃×#4546

#10588223

Subgroups — #10588223⋊C₃ #13135642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588223⋊C3 #13135642

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #10588223⋊C₃ #13135642