Group #11721807 — subgroups

Subgroups — #11721807

Order 864.

330 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 62 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	#88748	C₂	1	1	yes	maximal, characteristic
2	Q₈×#272	C₂	1	1	yes	maximal, characteristic, complemented
3	Q₁₆×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic, complemented
4	#88550	C₂	1	1	yes	maximal, characteristic, complemented
5	#88549	C₂	1	1	yes	maximal, characteristic, complemented
6	#88528	C₂	1	1	yes	maximal, characteristic, complemented
7	#88527	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#918	—	3	4	no	maximal
9	Q₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
10	Q₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
11	C₄⋊#272	—	2	1	no	—
12	C₄⋊#272	C₂²	1	1	yes	characteristic
13	C₄×#272	—	2	1	no	—
14	C₄×#272	—	2	1	no	—
15	C₄×#272	C₂²	1	1	yes	characteristic
16	#6229	—	2	1	no	—
17	#6229	C₂²	1	1	yes	characteristic
18	C₈×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
19	#6180	C₂²	1	1	yes	characteristic
20	C₃×#239	—	3	4	no	—
21	C₃×D₃×Q₈	—	3	4	no	—
22	C₃²×Q₁₆	D₃	1	4	yes	complemented
23	C₃×#216	—	3	4	no	—
24	C₃×#215	—	3	4	no	—
25	C₃×#204	—	3	4	no	—
26	C₃×ΓL(1,𝔽₂₅)	—	3	4	no	—
27	C₂×#272	—	2	1	no	—
28	C₂×#272	D₄	1	1	yes	characteristic
29	C₄×H(3,𝔽₃)	—	2	1	no	—
30	C₄×H(3,𝔽₃)	—	2	1	no	—
31	C₄×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
32	#1084	—	2	1	no	—
33	#1084	D₄	1	1	yes	characteristic
34	C₃×#137	—	9	1	no	—
35	#918	—	3	4	no	—
36	C₃²×Q₈	C₂×D₃	1	4	yes	—
37	C₃²×Q₈	C₂×D₃	1	4	yes	—
38	C₃×D₁₂	—	3	4	no	—
39	C₃×D₁₂	—	6	4	no	—
40	C₃×C₄×D₃	—	3	4	no	—
41	C₃×C₄×D₃	—	6	4	no	—
42	C₃×C₄×D₃	—	6	4	no	—
43	C₃×Dic₆	—	3	4	no	—
44	C₃×Dic₆	—	6	4	no	—
45	C₃²×C₈	C₂×D₃	1	4	yes	—
46	C₃×C₃⋊C₈	—	3	4	no	—
47	C₂×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
48	#272	—	2	1	no	—
49	#272	—	4	1	no	—
50	C₃×#41	—	9	1	no	—
51	C₂×C₃×Q₈	—	9	1	no	—
52	#239	—	3	4	no	—
53	D₃×Q₈	—	3	4	no	—
54	C₃×Q₁₆	—	3	4	no	—
55	C₃×Q₁₆	—	9	1	no	—
56	C₃×Q₁₆	C₃⋊D₃	1	1	yes	characteristic
57	C₃×SD₁₆	—	9	1	no	—
58	C₃×SD₁₆	—	9	1	no	—
59	C₃×M₁₆	—	9	1	no	—
60	#216	—	3	4	no	—
61	#215	—	3	4	no	—
62	#204	—	3	4	no	—
63	ΓL(1,𝔽₂₅)	—	3	4	no	—
64	C₂×C₃×D₃	—	3	4	no	—
65	C₂×C₃×D₃	—	6	4	no	—
66	C₃²×C₄	—	2	4	no	—
67	C₃²×C₄	—	2	4	no	—
68	C₃²×C₄	C₂²×D₃	1	4	yes	—
69	C₃×Dic₃	—	3	4	no	—
70	C₃×Dic₃	—	6	4	no	—
71	#137	—	9	1	no	Sylow
72	H(3,𝔽₃)	#137	1	1	yes	characteristic, complemented, Sylow
73	C₃×Q₈	—	3	4	no	—
74	C₃×Q₈	—	3	4	no	—
75	C₃×Q₈	—	9	1	no	—
76	C₃×Q₈	—	18	1	no	—
77	C₃×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
78	C₃×Q₈	C₂×C₃⋊D₃	1	1	yes	characteristic
79	C₃×D₄	—	9	1	no	—
80	C₃×D₄	—	18	1	no	—
81	C₂×C₃×C₄	—	9	1	no	—
82	C₂×C₃×C₄	—	18	1	no	—
83	C₂×C₃×C₄	—	18	1	no	—
84	D₁₂	—	3	4	no	—
85	D₁₂	—	6	4	no	—
86	C₄×D₃	—	3	4	no	—
87	C₄×D₃	—	6	4	no	—
88	C₄×D₃	—	6	4	no	—
89	Dic₆	—	3	4	no	—
90	Dic₆	—	6	4	no	—
91	C₃×C₈	—	3	4	no	—
92	C₃×C₈	—	9	1	no	—
93	C₃×C₈	C₂×C₃⋊D₃	1	1	yes	characteristic
94	C₃⋊C₈	—	3	4	no	—
95	C₂×C₃²	D₃×D₄	1	4	yes	—
96	C₃×D₃	—	6	4	no	—
97	C₃×D₃	—	12	4	no	—
98	#41	—	9	1	no	—
99	C₂×Q₈	—	9	1	no	—
100	Q₁₆	—	9	1	no	—
101	Q₁₆	#272	1	1	yes	characteristic, complemented
102	SD₁₆	—	9	1	no	—
103	SD₁₆	—	9	1	no	—
104	M₁₆	—	9	1	no	—
105	C₂²×C₃	—	9	1	no	—
106	C₂²×C₃	—	18	1	no	—
107	C₂×D₃	—	3	4	no	—
108	C₂×D₃	—	6	4	no	—
109	C₃×C₄	—	2	1	no	—
110	C₃×C₄	—	2	1	no	—
111	C₃×C₄	—	3	4	no	—
112	C₃×C₄	—	6	4	no	—
113	C₃×C₄	—	6	4	no	—
114	C₃×C₄	—	9	1	no	—
115	C₃×C₄	—	18	1	no	—
116	C₃×C₄	C₂²×C₃⋊D₃	1	1	yes	characteristic
117	Dic₃	—	3	4	no	—
118	Dic₃	—	6	4	no	—
119	C₃²	#918	1	4	yes	complemented
120	Q₈	—	9	1	no	—
121	Q₈	—	18	1	no	—
122	Q₈	C₂×#272	1	1	yes	characteristic
123	Q₈	C₂×#272	1	1	yes	characteristic
124	D₄	—	9	1	no	—
125	D₄	—	18	1	no	—
126	C₂×C₄	—	9	1	no	—
127	C₂×C₄	—	18	1	no	—
128	C₂×C₄	—	18	1	no	—
129	C₈	—	9	1	no	—
130	C₈	C₂×#272	1	1	yes	characteristic
131	C₂×C₃	—	3	4	no	—
132	C₂×C₃	—	18	1	no	—
133	C₂×C₃	—	36	1	no	—
134	C₂×C₃	D₄×C₃⋊D₃	1	1	yes	characteristic
135	D₃	—	6	4	no	—
136	D₃	—	12	4	no	—
137	C₂²	—	9	1	no	—
138	C₂²	—	18	1	no	—
139	C₄	—	2	1	no	—
140	C₄	—	2	1	no	—
141	C₄	—	9	1	no	—
142	C₄	—	18	1	no	—
143	C₄	C₂²×#272	1	1	yes	characteristic
144	C₃	—	3	4	no	—
145	C₃	#64127	1	1	yes	characteristic
146	C₂	—	18	1	no	—
147	C₂	—	36	1	no	—
148	C₂	D₄×#272	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#88748

C₂

yes

maximal, characteristic

Q₈×#272

C₂

yes

maximal, characteristic, complemented

Q₁₆×H(3,𝔽₃)

C₂

yes

maximal, characteristic, complemented

#88550

C₂

yes

maximal, characteristic, complemented

#88549

C₂

yes

maximal, characteristic, complemented

#88528

C₂

yes

maximal, characteristic, complemented

#88527

C₂

yes

maximal, characteristic, complemented

C₃×#918

—

maximal

Q₈×H(3,𝔽₃)

C₂²

yes

characteristic

Q₈×H(3,𝔽₃)

C₂²

yes

characteristic

C₄⋊#272

—

C₄⋊#272

C₂²

yes

—

—

C₂²

yes

characteristic

#6229

—

#6229

C₂²

yes

characteristic

C₈×H(3,𝔽₃)

C₂²

yes

characteristic

#6180

C₂²

yes

characteristic

C₃×#239

—

C₃×D₃×Q₈

—

C₃²×Q₁₆

D₃

yes

—

—

—

C₃×ΓL(1,𝔽₂₅)

—

C₂×#272

—

C₂×#272

D₄

yes

characteristic

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

C₂³

yes

characteristic

#1084

—

#1084

D₄

yes

characteristic

C₃×#137

—

#918

—

C₃²×Q₈

C₂×D₃

yes

—

C₃²×Q₈

C₂×D₃

yes

—

C₃×D₁₂

—

C₃×D₁₂

—

C₃×C₄×D₃

—

C₃×C₄×D₃

—

C₃×C₄×D₃

—

C₃×Dic₆

—

C₃×Dic₆

—

C₃²×C₈

C₂×D₃

yes

—

C₃×C₃⋊C₈

—

C₂×H(3,𝔽₃)

C₂×D₄

yes

characteristic

#272

—

#272

—

C₃×#41

—

C₂×C₃×Q₈

—

#239

—

D₃×Q₈

—

C₃×Q₁₆

—

C₃×Q₁₆

—

C₃×Q₁₆

C₃⋊D₃

yes

characteristic

C₃×SD₁₆

—

C₃×SD₁₆

—

C₃×M₁₆

—

#216

—

#215

—

#204

—

ΓL(1,𝔽₂₅)

—

C₂×C₃×D₃

—

C₂×C₃×D₃

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂²×D₃

yes

—

C₃×Dic₃

—

C₃×Dic₃

—

—

yes

characteristic, complemented, Sylow

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

C₂×C₃⋊D₃

yes

characteristic

C₃×Q₈

C₂×C₃⋊D₃

yes

characteristic

C₃×D₄

—

C₃×D₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

D₁₂

—

D₁₂

—

C₄×D₃

—

C₄×D₃

—

C₄×D₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

C₂×C₃⋊D₃

yes

characteristic

C₃⋊C₈

—

C₂×C₃²

D₃×D₄

yes

—

C₃×D₃

—

C₃×D₃

—

#41

—

C₂×Q₈

—

100

Q₁₆

—

101

Q₁₆

#272

yes

characteristic, complemented

102

SD₁₆

—

103

SD₁₆

—

104

M₁₆

—

105

C₂²×C₃

—

106

C₂²×C₃

—

107

C₂×D₃

—

108

C₂×D₃

—

109

C₃×C₄

—

110

C₃×C₄

—

111

C₃×C₄

—

112

C₃×C₄

—

113

C₃×C₄

—

114

C₃×C₄

—

115

C₃×C₄

—

116

C₃×C₄

C₂²×C₃⋊D₃

yes

characteristic

117

Dic₃

—

118

Dic₃

—

119

C₃²

#918

yes

complemented

120

Q₈

—

121

Q₈

—

122

Q₈

C₂×#272

yes

characteristic

123

Q₈

C₂×#272

yes

characteristic

124

D₄

—

125

D₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₈

—

130

C₈

C₂×#272

yes

characteristic

131

C₂×C₃

—

132

C₂×C₃

—

133

C₂×C₃

—

134

C₂×C₃

D₄×C₃⋊D₃

yes

characteristic

135

D₃

—

136

D₃

—

137

C₂²

—

138

C₂²

—

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

C₂²×#272

yes

characteristic

144

C₃

—

145

C₃

#64127

yes

characteristic

146

C₂

—

147

C₂

—

148

C₂

D₄×#272

yes

characteristic

	subgroup	G/H
center, socle	C₂×C₃	D₄×C₃⋊D₃
commutator, cosocle	C₄×H(3,𝔽₃)	C₂³
Frattini	C₃×C₄	C₂²×C₃⋊D₃
Fitting	Q₁₆×H(3,𝔽₃)	C₂
prime radical	#88748	C₂
hypercenter	C₃×Q₁₆	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#137
LCS	G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×Q₁₆

subgroup

G/H

center, socle

C₂×C₃

D₄×C₃⋊D₃

commutator, cosocle

C₄×H(3,𝔽₃)

C₂³

Frattini

C₃×C₄

C₂²×C₃⋊D₃

Fitting

Q₁₆×H(3,𝔽₃)

C₂

prime radical

#88748

C₂

hypercenter

C₃×Q₁₆

C₃⋊D₃

G ▷ C₄×H(3,𝔽₃) ▷ C₂×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂×C₃ ◁ C₃×C₄ ◁ C₃×Q₁₆

N	H ≅ G/N	type
2. Q₈×#272	C₂	semidirect
3. Q₁₆×H(3,𝔽₃)	C₂	semidirect
4. #88550	C₂	semidirect
5. #88549	C₂	semidirect
6. #88528	C₂	semidirect
7. #88527	C₂	semidirect
22. C₃²×Q₁₆	D₃	semidirect
72. H(3,𝔽₃)	#137	semidirect
101. Q₁₆	#272	semidirect
119. C₃²	#918	semidirect

H ≅ G/N

type

2. Q₈×#272

C₂

semidirect

3. Q₁₆×H(3,𝔽₃)

C₂

semidirect

4. #88550

C₂

semidirect

5. #88549

C₂

semidirect

6. #88528

C₂

semidirect

7. #88527

C₂

semidirect

22. C₃²×Q₁₆

D₃

semidirect

72. H(3,𝔽₃)

#137

semidirect

101. Q₁₆

#272

semidirect

119. C₃²

#918

semidirect

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

subgroup	index
#88748	2
Q₈×#272	2
Q₁₆×H(3,𝔽₃)	2
#88550	2
#88549	2
#88528	2
#88527	2
C₃×#918	3

subgroup

index

Q₁₆×H(3,𝔽₃)

Subgroups — #11721807

Conjugacy classes of subgroups

Complements

Maximal subgroups