Group #11721861 — subgroups

Subgroups — #11721861

Order 864.

374 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 92 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	C₂×Q₈×H(3,𝔽₃)	C₂	1	1	yes	maximal, characteristic
2	C₂×#6229	C₂	1	1	yes	maximal, characteristic
3	#88540	C₂	1	4	yes	maximal
4	C₄×#1084	C₂	1	1	yes	maximal, characteristic
5	C₃×#944	—	3	1	no	maximal
6	C₃×#944	—	3	3	no	maximal
7	Q₈×H(3,𝔽₃)	—	2	2	no	—
8	C₂×C₄×H(3,𝔽₃)	C₂²	1	2	yes	—
9	C₂×C₄×H(3,𝔽₃)	C₂²	1	1	yes	characteristic
10	C₂×#1084	C₂²	1	2	yes	—
11	C₂×#1084	C₂²	1	2	yes	—
12	#6229	—	2	2	no	—
13	C₂×C₃²×Q₈	D₃	1	1	yes	characteristic
14	C₂×C₃²×Q₈	D₃	1	3	yes	—
15	C₂×C₃×Dic₆	—	3	1	no	—
16	C₂×C₃×Dic₆	—	3	3	no	—
17	C₃×Dic₃⋊C₄	—	3	4	no	—
18	C₃×Dic₃⋊C₄	—	3	12	no	—
19	C₃×C₄×Dic₃	—	3	1	no	—
20	C₃×C₄×Dic₃	—	3	3	no	—
21	C₂²×H(3,𝔽₃)	C₂³	1	1	yes	characteristic
22	C₄×H(3,𝔽₃)	—	2	2	no	—
23	C₄×H(3,𝔽₃)	D₄	1	2	yes	—
24	#1084	—	2	2	no	—
25	#1084	Q₈	1	4	yes	complemented
26	C₃×C₄⋊Q₈	—	9	1	no	—
27	#944	—	3	1	no	—
28	#944	—	3	3	no	—
29	C₃²×Q₈	—	2	2	no	—
30	C₃²×Q₈	—	2	6	no	—
31	C₂×C₃²×C₄	C₂×D₃	1	2	yes	—
32	C₂×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
33	C₂×C₃²×C₄	C₂×D₃	1	6	yes	—
34	C₂×C₃²×C₄	C₂×D₃	1	3	yes	—
35	C₂×C₃×Dic₃	—	3	2	no	—
36	C₂×C₃×Dic₃	—	3	6	no	—
37	C₂×C₃×Dic₃	—	3	2	no	—
38	C₂×C₃×Dic₃	—	3	6	no	—
39	C₃×Dic₆	—	6	2	no	—
40	C₃×Dic₆	—	6	6	no	—
41	C₂×H(3,𝔽₃)	C₂×D₄	1	1	yes	characteristic
42	C₂×H(3,𝔽₃)	C₂×Q₈	1	2	yes	—
43	C₂×C₃×Q₈	—	3	1	no	—
44	C₂×C₃×Q₈	—	6	3	no	—
45	C₂×C₃×Q₈	—	9	1	no	—
46	C₂×C₃×Q₈	C₃⋊D₃	1	1	yes	characteristic
47	C₂×Dic₆	—	3	1	no	—
48	C₂×Dic₆	—	3	3	no	—
49	C₃×C₄⋊C₄	—	9	4	no	—
50	C₃×C₄²	—	9	1	no	—
51	Dic₃⋊C₄	—	3	4	no	—
52	Dic₃⋊C₄	—	3	12	no	—
53	C₄×Dic₃	—	3	1	no	—
54	C₄×Dic₃	—	3	3	no	—
55	C₂²×C₃²	C₂²×D₃	1	1	yes	characteristic
56	C₂²×C₃²	C₂²×D₃	1	3	yes	—
57	C₃²×C₄	—	2	2	no	—
58	C₃²×C₄	—	2	6	no	—
59	C₃²×C₄	#67	1	2	yes	—
60	C₃²×C₄	#67	1	6	yes	—
61	C₃×Dic₃	—	3	4	no	—
62	C₃×Dic₃	—	3	12	no	—
63	C₃×Dic₃	—	6	2	no	—
64	C₃×Dic₃	—	6	6	no	—
65	C₄⋊Q₈	—	9	1	no	Sylow
66	H(3,𝔽₃)	C₄⋊Q₈	1	1	yes	characteristic, complemented, Sylow
67	C₃×Q₈	—	2	2	no	—
68	C₃×Q₈	—	6	2	no	—
69	C₃×Q₈	—	6	6	no	—
70	C₃×Q₈	—	18	2	no	—
71	C₂×C₃×C₄	—	3	1	no	—
72	C₂×C₃×C₄	—	3	2	no	—
73	C₂×C₃×C₄	—	6	3	no	—
74	C₂×C₃×C₄	—	6	6	no	—
75	C₂×C₃×C₄	—	9	2	no	—
76	C₂×C₃×C₄	—	9	2	no	—
77	C₂×C₃×C₄	C₂×C₃⋊D₃	1	2	yes	—
78	C₂×C₃×C₄	C₂×C₃⋊D₃	1	1	yes	characteristic
79	C₂×Dic₃	—	3	2	no	—
80	C₂×Dic₃	—	3	6	no	—
81	C₂×Dic₃	—	3	2	no	—
82	C₂×Dic₃	—	3	6	no	—
83	Dic₆	—	6	2	no	—
84	Dic₆	—	6	6	no	—
85	C₂×C₃²	D₃×Q₈	1	2	yes	—
86	C₂×C₃²	D₃×Q₈	1	6	yes	—
87	C₂×C₃²	C₂×#67	1	1	yes	characteristic
88	C₂×C₃²	C₂×#67	1	3	yes	—
89	C₂×Q₈	—	9	1	no	—
90	C₂×Q₈	#272	1	1	yes	characteristic
91	C₄⋊C₄	—	9	4	no	—
92	C₄²	—	9	1	no	—
93	C₂²×C₃	—	3	1	no	—
94	C₂²×C₃	—	6	3	no	—
95	C₂²×C₃	C₂²×C₃⋊D₃	1	1	yes	characteristic
96	C₃×C₄	—	2	2	no	—
97	C₃×C₄	—	3	2	no	—
98	C₃×C₄	—	6	2	no	—
99	C₃×C₄	—	6	6	no	—
100	C₃×C₄	—	6	6	no	—
101	C₃×C₄	—	9	4	no	—
102	C₃×C₄	—	18	2	no	—
103	C₃×C₄	#638	1	2	yes	—
104	Dic₃	—	3	4	no	—
105	Dic₃	—	3	12	no	—
106	Dic₃	—	6	2	no	—
107	Dic₃	—	6	6	no	—
108	C₃²	#944	1	1	yes	characteristic, complemented
109	C₃²	#944	1	3	yes	complemented
110	Q₈	—	2	2	no	—
111	Q₈	—	18	2	no	—
112	C₂×C₄	—	9	2	no	—
113	C₂×C₄	—	9	2	no	—
114	C₂×C₄	C₂×#272	1	2	yes	—
115	C₂×C₄	C₂×#272	1	1	yes	characteristic
116	C₂×C₃	—	3	2	no	—
117	C₂×C₃	—	3	1	no	—
118	C₂×C₃	—	6	6	no	—
119	C₂×C₃	—	6	3	no	—
120	C₂×C₃	Q₈×C₃⋊D₃	1	2	yes	—
121	C₂×C₃	C₂×#638	1	1	yes	characteristic
122	C₂²	C₂²×#272	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	9	4	no	—
125	C₄	—	18	2	no	—
126	C₄	#6235	1	2	yes	—
127	C₃	—	3	1	no	—
128	C₃	—	6	3	no	—
129	C₃	#64153	1	1	yes	characteristic
130	C₂	Q₈×#272	1	2	yes	—
131	C₂	C₂×#6235	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×H(3,𝔽₃)

C₂

yes

maximal, characteristic

C₂×#6229

C₂

yes

maximal, characteristic

#88540

C₂

yes

maximal

C₄×#1084

C₂

yes

maximal, characteristic

—

—

Q₈×H(3,𝔽₃)

—

C₂×C₄×H(3,𝔽₃)

C₂²

yes

—

C₂×C₄×H(3,𝔽₃)

C₂²

yes

characteristic

C₂×#1084

C₂²

yes

—

C₂×#1084

C₂²

yes

—

#6229

—

C₂×C₃²×Q₈

D₃

yes

characteristic

C₂×C₃²×Q₈

D₃

yes

—

C₂×C₃×Dic₆

—

C₂×C₃×Dic₆

—

C₃×Dic₃⋊C₄

—

C₃×Dic₃⋊C₄

—

C₃×C₄×Dic₃

—

C₃×C₄×Dic₃

—

C₂²×H(3,𝔽₃)

C₂³

yes

characteristic

C₄×H(3,𝔽₃)

—

C₄×H(3,𝔽₃)

D₄

yes

—

#1084

—

#1084

Q₈

yes

complemented

C₃×C₄⋊Q₈

—

#944

—

#944

—

C₃²×Q₈

—

C₃²×Q₈

—

C₂×C₃²×C₄

C₂×D₃

yes

—

C₂×C₃²×C₄

C₂×D₃

yes

characteristic

C₂×C₃²×C₄

C₂×D₃

yes

—

C₂×C₃²×C₄

C₂×D₃

yes

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₂×C₃×Dic₃

—

C₃×Dic₆

—

C₃×Dic₆

—

C₂×H(3,𝔽₃)

C₂×D₄

yes

characteristic

C₂×H(3,𝔽₃)

C₂×Q₈

yes

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

C₃⋊D₃

yes

characteristic

C₂×Dic₆

—

C₂×Dic₆

—

C₃×C₄⋊C₄

—

C₃×C₄²

—

Dic₃⋊C₄

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂²×C₃²

C₂²×D₃

yes

characteristic

C₂²×C₃²

C₂²×D₃

yes

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

#67

yes

—

C₃²×C₄

#67

yes

—

C₃×Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₃×Dic₃

—

C₄⋊Q₈

—

Sylow

H(3,𝔽₃)

C₄⋊Q₈

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×C₄

C₂×C₃⋊D₃

yes

—

C₂×C₃×C₄

C₂×C₃⋊D₃

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₂×C₃²

D₃×Q₈

yes

—

C₂×C₃²

D₃×Q₈

yes

—

C₂×C₃²

C₂×#67

yes

characteristic

C₂×C₃²

C₂×#67

yes

—

C₂×Q₈

—

C₂×Q₈

#272

yes

characteristic

C₄⋊C₄

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂²×C₃⋊D₃

yes

characteristic

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

100

C₃×C₄

—

101

C₃×C₄

—

102

C₃×C₄

—

103

C₃×C₄

#638

yes

—

104

Dic₃

—

105

Dic₃

—

106

Dic₃

—

107

Dic₃

—

108

C₃²

#944

yes

characteristic, complemented

109

C₃²

#944

yes

complemented

110

Q₈

—

111

Q₈

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×#272

yes

—

115

C₂×C₄

C₂×#272

yes

characteristic

116

C₂×C₃

—

117

C₂×C₃

—

118

C₂×C₃

—

119

C₂×C₃

—

120

C₂×C₃

Q₈×C₃⋊D₃

yes

—

121

C₂×C₃

C₂×#638

yes

characteristic

122

C₂²

C₂²×#272

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#6235

yes

—

127

C₃

—

128

C₃

—

129

C₃

#64153

yes

characteristic

130

C₂

Q₈×#272

yes

—

131

C₂

C₂×#6235

yes

characteristic

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

subgroup

G/H

center, Frattini, socle

C₂²×C₃

C₂²×C₃⋊D₃

commutator, prime radical, cosocle

C₂²×H(3,𝔽₃)

C₂³

Fitting

C₂×Q₈×H(3,𝔽₃)

C₂

hypercenter

C₂×C₃×Q₈

C₃⋊D₃

hypocenter

H(3,𝔽₃)

C₄⋊Q₈

LCS

G ▷ C₂²×H(3,𝔽₃) ▷ H(3,𝔽₃)

UCS

C₁ ◁ C₂²×C₃ ◁ C₂×C₃×Q₈

N	H ≅ G/N	type
25. #1084	Q₈	semidirect
66. H(3,𝔽₃)	C₄⋊Q₈	semidirect
108. C₃²	#944	semidirect
109. C₃²	#944	semidirect

H ≅ G/N

type

25. #1084

Q₈

semidirect

66. H(3,𝔽₃)

C₄⋊Q₈

semidirect

108. C₃²

#944

semidirect

109. C₃²

#944

semidirect

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

subgroup	index
C₂×Q₈×H(3,𝔽₃)	2
C₂×#6229	2
#88540	2
C₄×#1084	2
C₃×#944	3

subgroup

index

C₂×Q₈×H(3,𝔽₃)

Subgroups — #11721861

Conjugacy classes of subgroups

Complements

Maximal subgroups