Group #13154292 — subgroups

Subgroups — #13154292

Order 1728.

1166 conjugacy classes of subgroups; only the 235 normal subgroups are shown.

normal subgroups only

This group has 1166 conjugacy classes of subgroups — too many to draw. Only its 235 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	#11721861	C₂	1	4	yes	maximal
3	#11721756	C₂	1	4	yes	maximal
4	#11721755	C₂	1	4	yes	maximal
5	C₄×#6229	C₂	1	2	yes	maximal
6	C₃×#5722	—	3	4	no	maximal
7	C₂×Q₈×H(3,𝔽₃)	C₂²	1	2	yes	—
8	C₂×#6229	C₂²	1	4	yes	—
9	C₂×#6229	C₂²	1	2	yes	—
10	C₄⋊C₄×H(3,𝔽₃)	C₂²	1	4	yes	—
11	C₄²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
12	C₄⋊#1084	C₂²	1	2	yes	—
13	#88540	C₂²	1	8	yes	—
14	#88540	C₂²	1	4	yes	—
15	C₄×#1084	C₂²	1	4	yes	—
16	C₄×#1084	C₂²	1	4	yes	—
17	C₃²×C₄⋊Q₈	D₃	1	4	yes	—
18	C₃×#944	—	3	16	no	—
19	C₃×#888	—	3	16	no	—
20	C₃×#887	—	3	16	no	—
21	C₃×C₄×Dic₆	—	3	8	no	—
22	Q₈×H(3,𝔽₃)	—	4	2	no	—
23	C₂×C₄×H(3,𝔽₃)	C₂³	1	4	yes	—
24	C₂×C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
25	C₂×C₄×H(3,𝔽₃)	C₂³	1	2	yes	—
26	C₂×#1084	C₂³	1	4	yes	—
27	C₂×#1084	C₂³	1	4	yes	—
28	#6229	—	4	4	no	—
29	#6229	Q₈	1	8	yes	complemented
30	C₃×Q₈²	—	9	1	no	—
31	#5722	—	3	4	no	—
32	C₂×C₃²×Q₈	C₂×D₃	1	8	yes	—
33	C₂×C₃×Dic₆	—	3	8	no	—
34	C₂×C₃×Dic₆	—	3	16	no	—
35	C₃²×C₄⋊C₄	C₂×D₃	1	16	yes	—
36	C₃²×C₄²	C₂×D₃	1	4	yes	—
37	C₃×C₄⋊Dic₃	—	3	8	no	—
38	C₃×Dic₃⋊C₄	—	3	32	no	—
39	C₃×Dic₃⋊C₄	—	3	16	no	—
40	C₃×C₄×Dic₃	—	3	16	no	—
41	C₃×C₄×Dic₃	—	3	16	no	—
42	C₂²×H(3,𝔽₃)	C₂⁴	1	1	yes	characteristic
43	C₄×H(3,𝔽₃)	—	2	4	no	—
44	C₄×H(3,𝔽₃)	—	2	1	no	—
45	C₄×H(3,𝔽₃)	C₂×Q₈	1	4	yes	—
46	#1084	—	2	4	no	—
47	#1084	C₂×Q₈	1	8	yes	—
48	C₃×C₄⋊Q₈	—	3	4	no	—
49	C₃×C₄⋊Q₈	—	9	4	no	—
50	C₃×C₄⋊Q₈	—	9	4	no	—
51	C₃×C₄⋊Q₈	C₃⋊D₃	1	1	yes	characteristic
52	C₃×C₄×Q₈	—	9	4	no	—
53	C₃×C₄×Q₈	—	9	2	no	—
54	#944	—	3	16	no	—
55	#888	—	3	16	no	—
56	#887	—	3	16	no	—
57	C₄×Dic₆	—	3	8	no	—
58	C₃²×Q₈	—	4	8	no	—
59	C₂×C₃²×C₄	C₂²×D₃	1	16	yes	—
60	C₂×C₃²×C₄	C₂²×D₃	1	4	yes	—
61	C₂×C₃²×C₄	C₂²×D₃	1	8	yes	—
62	C₂×C₃×Dic₃	—	3	16	no	—
63	C₂×C₃×Dic₃	—	3	16	no	—
64	C₃×Dic₆	—	3	32	no	—
65	C₃×Dic₆	—	12	16	no	—
66	Q₈²	—	9	1	no	Sylow
67	C₂×H(3,𝔽₃)	C₂²×Q₈	1	2	yes	—
68	C₂×H(3,𝔽₃)	H(5,𝔽₂)	1	1	yes	characteristic
69	C₂×C₃×Q₈	—	3	8	no	—
70	C₂×C₃×Q₈	—	9	2	no	—
71	C₂×C₃×Q₈	—	9	4	no	—
72	C₂×C₃×Q₈	C₂×C₃⋊D₃	1	2	yes	—
73	C₂×Dic₆	—	3	8	no	—
74	C₂×Dic₆	—	3	16	no	—
75	C₃×C₄⋊C₄	—	3	16	no	—
76	C₃×C₄⋊C₄	—	9	8	no	—
77	C₃×C₄⋊C₄	—	9	4	no	—
78	C₃×C₄⋊C₄	—	9	2	no	—
79	C₃×C₄⋊C₄	C₂×C₃⋊D₃	1	4	yes	—
80	C₃×C₄²	—	3	4	no	—
81	C₃×C₄²	—	9	4	no	—
82	C₃×C₄²	—	9	4	no	—
83	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
84	C₄⋊Dic₃	—	3	8	no	—
85	Dic₃⋊C₄	—	3	16	no	—
86	Dic₃⋊C₄	—	3	32	no	—
87	C₄×Dic₃	—	3	16	no	—
88	C₄×Dic₃	—	3	16	no	—
89	C₂²×C₃²	C₂³×D₃	1	4	yes	—
90	C₃²×C₄	—	2	16	no	—
91	C₃²×C₄	—	2	4	no	—
92	C₃²×C₄	D₃×Q₈	1	16	yes	—
93	C₃×Dic₃	—	3	32	no	—
94	C₃×Dic₃	—	6	16	no	—
95	C₄⋊Q₈	—	9	4	no	—
96	C₄⋊Q₈	—	9	4	no	—
97	C₄⋊Q₈	#272	1	1	yes	characteristic
98	C₄×Q₈	—	9	4	no	—
99	C₄×Q₈	—	9	2	no	—
100	H(3,𝔽₃)	Q₈²	1	1	yes	characteristic, complemented, Sylow
101	C₃×Q₈	—	4	2	no	—
102	C₃×Q₈	—	9	8	no	—
103	C₃×Q₈	—	12	8	no	—
104	C₃×Q₈	—	36	4	no	—
105	C₂×C₃×C₄	—	3	8	no	—
106	C₂×C₃×C₄	—	3	16	no	—
107	C₂×C₃×C₄	—	3	4	no	—
108	C₂×C₃×C₄	—	9	4	no	—
109	C₂×C₃×C₄	—	9	4	no	—
110	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	4	yes	—
111	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
112	C₂×C₃×C₄	C₂²×C₃⋊D₃	1	2	yes	—
113	C₂×Dic₃	—	3	16	no	—
114	C₂×Dic₃	—	3	16	no	—
115	Dic₆	—	3	32	no	—
116	Dic₆	—	12	16	no	—
117	C₂×C₃²	#1004	1	4	yes	—
118	C₂×C₃²	C₂×D₃×Q₈	1	8	yes	—
119	C₂×Q₈	—	9	2	no	—
120	C₂×Q₈	—	9	4	no	—
121	C₂×Q₈	C₂×#272	1	2	yes	—
122	C₄⋊C₄	—	9	8	no	—
123	C₄⋊C₄	—	9	4	no	—
124	C₄⋊C₄	—	9	2	no	—
125	C₄⋊C₄	C₂×#272	1	4	yes	—
126	C₄²	—	9	4	no	—
127	C₄²	—	9	4	no	—
128	C₄²	C₂×#272	1	1	yes	characteristic
129	C₂²×C₃	—	3	4	no	—
130	C₂²×C₃	C₂³×C₃⋊D₃	1	1	yes	characteristic
131	C₃×C₄	—	2	1	no	—
132	C₃×C₄	—	2	4	no	—
133	C₃×C₄	—	3	16	no	—
134	C₃×C₄	—	6	16	no	—
135	C₃×C₄	—	6	4	no	—
136	C₃×C₄	—	9	8	no	—
137	C₃×C₄	—	18	4	no	—
138	C₃×C₄	Q₈×C₃⋊D₃	1	4	yes	—
139	Dic₃	—	3	32	no	—
140	Dic₃	—	6	16	no	—
141	C₃²	#5722	1	4	yes	complemented
142	Q₈	—	4	2	no	—
143	Q₈	—	9	8	no	—
144	Q₈	—	36	4	no	—
145	C₂×C₄	—	9	4	no	—
146	C₂×C₄	—	9	4	no	—
147	C₂×C₄	C₂²×#272	1	4	yes	—
148	C₂×C₄	C₂²×#272	1	1	yes	characteristic
149	C₂×C₄	C₂²×#272	1	2	yes	—
150	C₂×C₃	—	3	4	no	—
151	C₂×C₃	—	3	8	no	—
152	C₂×C₃	#64361	1	1	yes	characteristic
153	C₂×C₃	C₂×Q₈×C₃⋊D₃	1	2	yes	—
154	C₂²	C₂³×#272	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	2	4	no	—
157	C₄	—	9	8	no	—
158	C₄	—	18	4	no	—
159	C₄	Q₈×#272	1	4	yes	—
160	C₃	—	3	4	no	—
161	C₃	#10596193	1	1	yes	characteristic
162	C₂	#11722664	1	1	yes	characteristic
163	C₂	C₂×Q₈×#272	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

H(3,𝔽₃)×C₄⋊Q₈

C₂

yes

maximal, characteristic

#11721861

C₂

yes

maximal

#11721756

C₂

yes

maximal

#11721755

C₂

yes

maximal

C₄×#6229

C₂

yes

maximal

C₃×#5722

—

maximal

C₂×Q₈×H(3,𝔽₃)

C₂²

yes

—

C₂×#6229

C₂²

yes

—

C₂×#6229

C₂²

yes

—

C₄⋊C₄×H(3,𝔽₃)

C₂²

yes

—

C₄²×H(3,𝔽₃)

C₂²

yes

characteristic

C₄⋊#1084

C₂²

yes

—

#88540

C₂²

yes

—

#88540

C₂²

yes

—

C₄×#1084

C₂²

yes

—

C₄×#1084

C₂²

yes

—

C₃²×C₄⋊Q₈

D₃

yes

—

C₃×#944

—

C₃×#888

—

C₃×#887

—

C₃×C₄×Dic₆

—

Q₈×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

C₂³

yes

—

C₂×C₄×H(3,𝔽₃)

C₂³

yes

characteristic

C₂×C₄×H(3,𝔽₃)

C₂³

yes

—

C₂×#1084

C₂³

yes

—

C₂×#1084

C₂³

yes

—

#6229

—

#6229

Q₈

yes

complemented

C₃×Q₈²

—

#5722

—

C₂×C₃²×Q₈

C₂×D₃

yes

—

C₂×C₃×Dic₆

—

C₂×C₃×Dic₆

—

C₃²×C₄⋊C₄

C₂×D₃

yes

—

C₃²×C₄²

C₂×D₃

yes

—

C₃×C₄⋊Dic₃

—

C₃×Dic₃⋊C₄

—

C₃×Dic₃⋊C₄

—

C₃×C₄×Dic₃

—

C₃×C₄×Dic₃

—

C₂²×H(3,𝔽₃)

C₂⁴

yes

characteristic

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

C₂×Q₈

yes

—

#1084

—

#1084

C₂×Q₈

yes

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

C₃⋊D₃

yes

characteristic

C₃×C₄×Q₈

—

C₃×C₄×Q₈

—

#944

—

#888

—

#887

—

C₄×Dic₆

—

C₃²×Q₈

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃²×C₄

C₂²×D₃

yes

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃×Dic₆

—

Q₈²

—

Sylow

C₂×H(3,𝔽₃)

C₂²×Q₈

yes

—

C₂×H(3,𝔽₃)

H(5,𝔽₂)

yes

characteristic

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₂×C₃⋊D₃

yes

—

C₂×Dic₆

—

C₂×Dic₆

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₂×C₃⋊D₃

yes

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₄⋊Dic₃

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂²×C₃²

C₂³×D₃

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

D₃×Q₈

yes

—

C₃×Dic₃

—

C₃×Dic₃

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#272

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

100

H(3,𝔽₃)

Q₈²

yes

characteristic, complemented, Sylow

101

C₃×Q₈

—

102

C₃×Q₈

—

103

C₃×Q₈

—

104

C₃×Q₈

—

105

C₂×C₃×C₄

—

106

C₂×C₃×C₄

—

107

C₂×C₃×C₄

—

108

C₂×C₃×C₄

—

109

C₂×C₃×C₄

—

110

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

111

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

characteristic

112

C₂×C₃×C₄

C₂²×C₃⋊D₃

yes

—

113

C₂×Dic₃

—

114

C₂×Dic₃

—

115

Dic₆

—

116

Dic₆

—

117

C₂×C₃²

#1004

yes

—

118

C₂×C₃²

C₂×D₃×Q₈

yes

—

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

C₂×#272

yes

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

C₂×#272

yes

—

126

C₄²

—

127

C₄²

—

128

C₄²

C₂×#272

yes

characteristic

129

C₂²×C₃

—

130

C₂²×C₃

C₂³×C₃⋊D₃

yes

characteristic

131

C₃×C₄

—

132

C₃×C₄

—

133

C₃×C₄

—

134

C₃×C₄

—

135

C₃×C₄

—

136

C₃×C₄

—

137

C₃×C₄

—

138

C₃×C₄

Q₈×C₃⋊D₃

yes

—

139

Dic₃

—

140

Dic₃

—

141

C₃²

#5722

yes

complemented

142

Q₈

—

143

Q₈

—

144

Q₈

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

C₂²×#272

yes

—

148

C₂×C₄

C₂²×#272

yes

characteristic

149

C₂×C₄

C₂²×#272

yes

—

150

C₂×C₃

—

151

C₂×C₃

—

152

C₂×C₃

#64361

yes

characteristic

153

C₂×C₃

C₂×Q₈×C₃⋊D₃

yes

—

154

C₂²

C₂³×#272

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

Q₈×#272

yes

—

160

C₃

—

161

C₃

#10596193

yes

characteristic

162

C₂

#11722664

yes

characteristic

163

C₂

C₂×Q₈×#272

yes

—

	subgroup	G/H
center, Frattini, socle	C₂²×C₃	C₂³×C₃⋊D₃
commutator, prime radical, cosocle	C₂²×H(3,𝔽₃)	C₂⁴
Fitting	H(3,𝔽₃)×C₄⋊Q₈	C₂
hypercenter	C₃×C₄⋊Q₈	C₃⋊D₃
hypocenter	H(3,𝔽₃)	Q₈²
LCS	G ▷ C₂²×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄⋊Q₈

subgroup

G/H

center, Frattini, socle

C₂²×C₃

C₂³×C₃⋊D₃

commutator, prime radical, cosocle

C₂²×H(3,𝔽₃)

C₂⁴

Fitting

H(3,𝔽₃)×C₄⋊Q₈

C₂

hypercenter

C₃×C₄⋊Q₈

C₃⋊D₃

hypocenter

H(3,𝔽₃)

Q₈²

LCS

G ▷ C₂²×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂²×C₃ ◁ C₃×C₄⋊Q₈

N	H ≅ G/N	type
29. #6229	Q₈	semidirect
100. H(3,𝔽₃)	Q₈²	semidirect
141. C₃²	#5722	semidirect

H ≅ G/N

type

29. #6229

Q₈

semidirect

100. H(3,𝔽₃)

Q₈²

semidirect

141. C₃²

#5722

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
#11721861	2
#11721756	2
#11721755	2
C₄×#6229	2
C₃×#5722	3

subgroup

index

H(3,𝔽₃)×C₄⋊Q₈

Subgroups — #13154292

Conjugacy classes of subgroups

Complements

Maximal subgroups