Group #13135924 — subgroups

Subgroups — #13135924

Order 1728.

352 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 352 conjugacy classes of subgroups — too many to draw. Only its 115 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	C₄⋊#6180	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#6180	C₂	1	1	yes	maximal, characteristic
4	C₃×#4547	—	3	4	no	maximal
5	C₂×Q₈×H(3,𝔽₃)	—	2	1	no	—
6	C₄⋊C₄×H(3,𝔽₃)	—	2	1	no	—
7	C₄⋊C₄×H(3,𝔽₃)	C₄	1	2	yes	—
8	C₄²×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
9	C₂×#6180	—	2	1	no	—
10	C₂×#6180	—	2	1	no	—
11	C₃²×C₄⋊Q₈	D₃	1	4	yes	—
12	C₃×C₄⋊(C₃⋊C₈)	—	3	4	no	—
13	C₃×C₄⋊(C₃⋊C₈)	—	3	4	no	—
14	Q₈×H(3,𝔽₃)	—	4	1	no	—
15	Q₈×H(3,𝔽₃)	—	4	1	no	—
16	C₂×C₄×H(3,𝔽₃)	—	2	2	no	—
17	C₂×C₄×H(3,𝔽₃)	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
19	C₂×C₄×H(3,𝔽₃)	D₄	1	1	yes	characteristic
20	#6180	—	4	1	no	—
21	#6180	—	4	1	no	—
22	C₃×#332	—	9	1	no	—
23	#4547	—	3	4	no	—
24	C₂×C₃²×Q₈	—	2	4	no	—
25	C₃²×C₄⋊C₄	—	2	4	no	—
26	C₃²×C₄⋊C₄	Dic₃	1	8	yes	—
27	C₃²×C₄²	C₂×D₃	1	4	yes	—
28	C₂×C₃×C₃⋊C₈	—	6	4	no	—
29	C₂×C₃×C₃⋊C₈	—	6	4	no	—
30	C₂²×H(3,𝔽₃)	C₂²⋊C₄	1	1	yes	characteristic
31	C₄×H(3,𝔽₃)	—	2	1	no	—
32	C₄×H(3,𝔽₃)	—	4	2	no	—
33	C₄×H(3,𝔽₃)	D₈	1	1	yes	characteristic
34	C₄×H(3,𝔽₃)	SD₁₆	1	1	yes	characteristic
35	C₄×H(3,𝔽₃)	SD₁₆	1	1	yes	characteristic
36	C₄×H(3,𝔽₃)	Q₁₆	1	1	yes	characteristic
37	C₃×C₄⋊Q₈	—	3	4	no	—
38	C₃×C₄⋊Q₈	C₃⋊D₃	1	1	yes	characteristic
39	C₃×C₄⋊C₈	—	9	1	no	—
40	C₃×C₄⋊C₈	—	9	1	no	—
41	C₄⋊(C₃⋊C₈)	—	3	4	no	—
42	C₄⋊(C₃⋊C₈)	—	3	4	no	—
43	C₃²×Q₈	—	4	4	no	—
44	C₃²×Q₈	—	4	4	no	—
45	C₂×C₃²×C₄	—	2	8	no	—
46	C₂×C₃²×C₄	C₂×Dic₃	1	4	yes	—
47	C₂×C₃²×C₄	#67	1	4	yes	—
48	C₂×C₃²×C₄	#67	1	4	yes	—
49	C₃×C₃⋊C₈	—	12	4	no	—
50	C₃×C₃⋊C₈	—	12	4	no	—
51	#332	—	9	1	no	Sylow
52	C₂×H(3,𝔽₃)	#101	1	1	yes	characteristic
53	C₂×H(3,𝔽₃)	#102	1	1	yes	characteristic
54	C₂×H(3,𝔽₃)	#103	1	1	yes	characteristic
55	C₂×C₃×Q₈	—	2	1	no	—
56	C₂×C₃×Q₈	—	6	4	no	—
57	C₂×C₃×C₈	—	18	1	no	—
58	C₂×C₃×C₈	—	18	1	no	—
59	C₃×C₄⋊C₄	—	2	1	no	—
60	C₃×C₄⋊C₄	—	3	8	no	—
61	C₃×C₄⋊C₄	—	6	4	no	—
62	C₃×C₄⋊C₄	C₃⋊Dic₃	1	2	yes	—
63	C₃×C₄²	—	3	4	no	—
64	C₃×C₄²	C₂×C₃⋊D₃	1	1	yes	characteristic
65	C₂×C₃⋊C₈	—	6	4	no	—
66	C₂×C₃⋊C₈	—	6	4	no	—
67	C₂²×C₃²	#217	1	4	yes	—
68	C₃²×C₄	—	2	4	no	—
69	C₃²×C₄	—	4	8	no	—
70	C₃²×C₄	#213	1	4	yes	—
71	C₃²×C₄	#214	1	4	yes	—
72	C₃²×C₄	#215	1	4	yes	—
73	C₃²×C₄	#216	1	4	yes	—
74	C₄⋊Q₈	#272	1	1	yes	characteristic
75	C₄⋊C₈	—	9	1	no	—
76	C₄⋊C₈	—	9	1	no	—
77	H(3,𝔽₃)	#332	1	1	yes	characteristic, complemented, Sylow
78	C₃×Q₈	—	4	1	no	—
79	C₃×Q₈	—	4	1	no	—
80	C₃×Q₈	—	12	4	no	—
81	C₃×Q₈	—	12	4	no	—
82	C₂×C₃×C₄	—	2	2	no	—
83	C₂×C₃×C₄	—	3	4	no	—
84	C₂×C₃×C₄	—	3	4	no	—
85	C₂×C₃×C₄	—	3	4	no	—
86	C₂×C₃×C₄	—	6	8	no	—
87	C₂×C₃×C₄	C₂×C₃⋊Dic₃	1	1	yes	characteristic
88	C₂×C₃×C₄	#638	1	1	yes	characteristic
89	C₂×C₃×C₄	#638	1	1	yes	characteristic
90	C₃×C₈	—	36	1	no	—
91	C₃×C₈	—	36	1	no	—
92	C₃⋊C₈	—	12	4	no	—
93	C₃⋊C₈	—	12	4	no	—
94	C₂×C₃²	#832	1	4	yes	—
95	C₂×C₃²	#835	1	4	yes	—
96	C₂×C₃²	#836	1	4	yes	—
97	C₂×Q₈	—	2	1	no	—
98	C₂×C₈	—	18	1	no	—
99	C₂×C₈	—	18	1	no	—
100	C₄⋊C₄	—	2	1	no	—
101	C₄⋊C₄	#1084	1	2	yes	—
102	C₄²	C₂×#272	1	1	yes	characteristic
103	C₂²×C₃	—	3	4	no	—
104	C₂²×C₃	#3757	1	1	yes	characteristic
105	C₃×C₄	—	2	1	no	—
106	C₃×C₄	—	3	4	no	—
107	C₃×C₄	—	3	4	no	—
108	C₃×C₄	—	3	4	no	—
109	C₃×C₄	—	3	4	no	—
110	C₃×C₄	—	4	2	no	—
111	C₃×C₄	—	6	4	no	—
112	C₃×C₄	—	12	8	no	—
113	C₃×C₄	#3753	1	1	yes	characteristic
114	C₃×C₄	#3754	1	1	yes	characteristic
115	C₃×C₄	#3755	1	1	yes	characteristic
116	C₃×C₄	#3756	1	1	yes	characteristic
117	C₃²	#4547	1	4	yes	complemented
118	Q₈	—	4	1	no	—
119	Q₈	—	4	1	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	C₂×#1084	1	1	yes	characteristic
122	C₂×C₄	#6235	1	1	yes	characteristic
123	C₂×C₄	#6235	1	1	yes	characteristic
124	C₈	—	36	1	no	—
125	C₈	—	36	1	no	—
126	C₂×C₃	—	3	4	no	—
127	C₂×C₃	—	3	4	no	—
128	C₂×C₃	—	3	4	no	—
129	C₂×C₃	#63659	1	1	yes	characteristic
130	C₂×C₃	#63662	1	1	yes	characteristic
131	C₂×C₃	#63663	1	1	yes	characteristic
132	C₂²	#88552	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	4	2	no	—
135	C₄	#88545	1	1	yes	characteristic
136	C₄	#88546	1	1	yes	characteristic
137	C₄	#88549	1	1	yes	characteristic
138	C₄	#88550	1	1	yes	characteristic
139	C₃	—	3	4	no	—
140	C₃	#10589413	1	1	yes	characteristic
141	C₂	#11720578	1	1	yes	characteristic
142	C₂	#11720583	1	1	yes	characteristic
143	C₂	#11720584	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

H(3,𝔽₃)×C₄⋊Q₈

C₂

yes

maximal, characteristic

C₄⋊#6180

C₂

yes

maximal, characteristic

C₄⋊#6180

C₂

yes

maximal, characteristic

C₃×#4547

—

maximal

C₂×Q₈×H(3,𝔽₃)

—

C₄⋊C₄×H(3,𝔽₃)

—

C₄⋊C₄×H(3,𝔽₃)

C₄

yes

—

C₄²×H(3,𝔽₃)

C₂²

yes

characteristic

C₂×#6180

—

C₂×#6180

—

C₃²×C₄⋊Q₈

D₃

yes

—

C₃×C₄⋊(C₃⋊C₈)

—

C₃×C₄⋊(C₃⋊C₈)

—

Q₈×H(3,𝔽₃)

—

Q₈×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

C₂×C₄

yes

characteristic

C₂×C₄×H(3,𝔽₃)

D₄

yes

characteristic

C₂×C₄×H(3,𝔽₃)

D₄

yes

—

—

—

—

C₂×C₃²×Q₈

—

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

Dic₃

yes

—

C₃²×C₄²

C₂×D₃

yes

—

C₂×C₃×C₃⋊C₈

—

C₂×C₃×C₃⋊C₈

—

C₂²×H(3,𝔽₃)

C₂²⋊C₄

yes

characteristic

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

D₈

yes

characteristic

C₄×H(3,𝔽₃)

SD₁₆

yes

characteristic

C₄×H(3,𝔽₃)

SD₁₆

yes

characteristic

C₄×H(3,𝔽₃)

Q₁₆

yes

characteristic

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

C₃⋊D₃

yes

characteristic

C₃×C₄⋊C₈

—

C₃×C₄⋊C₈

—

C₄⋊(C₃⋊C₈)

—

C₄⋊(C₃⋊C₈)

—

C₃²×Q₈

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×Dic₃

yes

—

C₂×C₃²×C₄

#67

yes

—

C₂×C₃²×C₄

#67

yes

—

C₃×C₃⋊C₈

—

C₃×C₃⋊C₈

—

—

C₂×H(3,𝔽₃)

yes

C₂×H(3,𝔽₃)

yes

C₂×H(3,𝔽₃)

yes

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

C₃⋊Dic₃

yes

—

C₃×C₄²

—

C₃×C₄²

C₂×C₃⋊D₃

yes

characteristic

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₃²

#217

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

#213

yes

—

C₃²×C₄

#214

yes

—

C₃²×C₄

#215

yes

—

C₃²×C₄

#216

yes

—

C₄⋊Q₈

#272

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

H(3,𝔽₃)

#332

yes

characteristic, complemented, Sylow

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃⋊Dic₃

yes

C₂×C₃×C₄

yes

C₂×C₃×C₄

yes

C₃×C₈

—

C₃×C₈

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×C₃²

#832

yes

—

C₂×C₃²

#835

yes

—

C₂×C₃²

#836

yes

—

C₂×Q₈

—

C₂×C₈

—

C₂×C₈

—

100

C₄⋊C₄

—

101

C₄⋊C₄

#1084

yes

—

102

C₄²

C₂×#272

yes

characteristic

103

C₂²×C₃

—

104

C₂²×C₃

#3757

yes

characteristic

105

C₃×C₄

—

106

C₃×C₄

—

107

C₃×C₄

—

108

C₃×C₄

—

109

C₃×C₄

—

110

C₃×C₄

—

111

C₃×C₄

—

112

C₃×C₄

—

113

C₃×C₄

#3753

yes

characteristic

114

C₃×C₄

#3754

yes

characteristic

115

C₃×C₄

#3755

yes

characteristic

116

C₃×C₄

#3756

yes

characteristic

117

C₃²

#4547

yes

complemented

118

Q₈

—

119

Q₈

—

120

C₂×C₄

—

121

C₂×C₄

C₂×#1084

yes

characteristic

122

C₂×C₄

#6235

yes

characteristic

123

C₂×C₄

#6235

yes

characteristic

124

C₈

—

125

C₈

—

126

C₂×C₃

—

127

C₂×C₃

—

128

C₂×C₃

—

129

C₂×C₃

#63659

yes

characteristic

130

C₂×C₃

#63662

yes

characteristic

131

C₂×C₃

#63663

yes

characteristic

132

C₂²

#88552

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

#88545

yes

characteristic

136

C₄

#88546

yes

characteristic

137

C₄

#88549

yes

characteristic

138

C₄

#88550

yes

characteristic

139

C₃

—

140

C₃

#10589413

yes

characteristic

141

C₂

#11720578

yes

characteristic

142

C₂

#11720583

yes

characteristic

143

C₂

#11720584

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₃	#3757
commutator	C₂×C₄×H(3,𝔽₃)	C₂×C₄
Frattini	C₃×C₄²	C₂×C₃⋊D₃
Fitting	H(3,𝔽₃)×C₄⋊Q₈	C₂
prime radical	C₂²×H(3,𝔽₃)	C₂²⋊C₄
cosocle	C₄²×H(3,𝔽₃)	C₂²
hypercenter	C₃×C₄⋊Q₈	C₃⋊D₃
hypocenter	H(3,𝔽₃)	#332
LCS	G ▷ C₂×C₄×H(3,𝔽₃) ▷ C₂²×H(3,𝔽₃) ▷ H(3,𝔽₃)
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄⋊Q₈

subgroup

G/H

center, socle

C₂²×C₃

#3757

commutator

C₂×C₄×H(3,𝔽₃)

C₂×C₄

Frattini

C₃×C₄²

C₂×C₃⋊D₃

Fitting

H(3,𝔽₃)×C₄⋊Q₈

C₂

prime radical

C₂²×H(3,𝔽₃)

C₂²⋊C₄

cosocle

C₄²×H(3,𝔽₃)

C₂²

hypercenter

C₃×C₄⋊Q₈

C₃⋊D₃

G ▷ C₂×C₄×H(3,𝔽₃) ▷ C₂²×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂²×C₃ ◁ C₃×C₄² ◁ C₃×C₄⋊Q₈

N	H ≅ G/N	type
77. H(3,𝔽₃)	#332	semidirect
117. C₃²	#4547	semidirect

H ≅ G/N

type

77. H(3,𝔽₃)

#332

semidirect

117. C₃²

#4547

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
C₄⋊#6180	2
C₃×#4547	3

subgroup

index

H(3,𝔽₃)×C₄⋊Q₈

C₄⋊#6180

C₃×#4547

Subgroups — #13135924

Conjugacy classes of subgroups

Complements

Maximal subgroups