Group #11768765 — subgroups

Subgroups — C₃²×#1716 #11768765

Order 1152.

528 conjugacy classes of subgroups; only the 240 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 240 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	C₃²×#498	C₂	1	1	yes	maximal, characteristic
2	C₃²×#498	C₂	1	2	yes	maximal
3	C₃²×#494	C₂	1	1	yes	maximal, characteristic
4	C₃²×#487	C₂	1	2	yes	maximal
5	C₃²×#322	C₂	1	1	yes	maximal, characteristic
6	C₃×#1716	C₃	1	4	yes	maximal, direct product factor
7	C₂×C₃²×Q₁₆	—	2	2	no	—
8	C₃²×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₃²×C₄⋊Q₈	C₂²	1	2	yes	—
10	C₃²×C₄⋊Q₈	C₂²	1	1	yes	characteristic
11	C₃²×#106	—	2	4	no	—
12	C₃²×#106	—	2	2	no	—
13	C₃²×#103	—	2	4	no	—
14	C₃²×C₄×C₈	C₂²	1	2	yes	—
15	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
16	C₃×#498	C₂×C₃	1	8	yes	—
17	C₃×#498	C₂×C₃	1	4	yes	—
18	C₃×#494	C₂×C₃	1	4	yes	—
19	C₃×#487	C₂×C₃	1	8	yes	—
20	C₃×#322	C₂×C₃	1	4	yes	—
21	C₂×C₃²×Q₈	—	2	1	no	—
22	C₂×C₃²×Q₈	—	2	1	no	—
23	C₂×C₃²×Q₈	—	2	2	no	—
24	C₃²×Q₁₆	—	4	4	no	—
25	C₂×C₃²×C₈	D₄	1	4	yes	—
26	C₂×C₃²×C₈	D₄	1	2	yes	—
27	C₃²×C₄⋊C₄	—	2	2	no	—
28	C₃²×C₄⋊C₄	—	2	2	no	—
29	C₃²×C₄⋊C₄	—	2	2	no	—
30	C₃²×C₄⋊C₄	—	2	2	no	—
31	C₃²×C₄²	C₂³	1	1	yes	characteristic
32	#1716	C₃²	1	1	yes	characteristic, direct product factor, Sylow
33	C₂×C₃×Q₁₆	—	2	8	no	—
34	C₃×C₄⋊Q₈	C₂²×C₃	1	4	yes	—
35	C₃×C₄⋊Q₈	C₂²×C₃	1	8	yes	—
36	C₃×C₄⋊Q₈	C₂²×C₃	1	4	yes	—
37	C₃×#106	—	2	16	no	—
38	C₃×#106	—	2	8	no	—
39	C₃×#103	—	2	16	no	—
40	C₃×C₄×C₈	C₂²×C₃	1	8	yes	—
41	C₃×C₄×C₈	C₂²×C₃	1	4	yes	—
42	C₃²×Q₈	—	4	2	no	—
43	C₃²×Q₈	—	4	2	no	—
44	C₃²×Q₈	—	4	4	no	—
45	C₂×C₃²×C₄	—	4	1	no	—
46	C₂×C₃²×C₄	—	4	1	no	—
47	C₂×C₃²×C₄	—	4	2	no	—
48	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
49	C₂×C₃²×C₄	C₂×D₄	1	2	yes	—
50	C₃²×C₈	—	2	4	no	—
51	C₃²×C₈	SD₁₆	1	4	yes	—
52	#498	C₂×C₃²	1	2	yes	—
53	#498	C₂×C₃²	1	1	yes	characteristic
54	#494	C₂×C₃²	1	1	yes	characteristic
55	#487	C₂×C₃²	1	2	yes	—
56	#322	C₂×C₃²	1	1	yes	characteristic
57	C₂×C₃×Q₈	—	2	8	no	—
58	C₂×C₃×Q₈	—	2	4	no	—
59	C₂×C₃×Q₈	—	2	4	no	—
60	C₃×Q₁₆	—	4	16	no	—
61	C₂×C₃×C₈	C₃×D₄	1	16	yes	—
62	C₂×C₃×C₈	C₃×D₄	1	8	yes	—
63	C₃×C₄⋊C₄	—	2	8	no	—
64	C₃×C₄⋊C₄	—	2	8	no	—
65	C₃×C₄⋊C₄	—	2	8	no	—
66	C₃×C₄⋊C₄	—	2	8	no	—
67	C₃×C₄²	C₂³×C₃	1	4	yes	—
68	C₂²×C₃²	#127	1	1	yes	characteristic
69	C₃²×C₄	—	8	1	no	—
70	C₃²×C₄	—	8	1	no	—
71	C₃²×C₄	—	8	2	no	—
72	C₃²×C₄	C₂×SD₁₆	1	2	yes	—
73	C₃²×C₄	#137	1	4	yes	—
74	C₂×Q₁₆	—	2	2	no	—
75	C₄⋊Q₈	C₂²×C₃²	1	1	yes	characteristic
76	C₄⋊Q₈	C₂²×C₃²	1	2	yes	—
77	C₄⋊Q₈	C₂²×C₃²	1	1	yes	characteristic
78	#106	—	2	4	no	—
79	#106	—	2	2	no	—
80	#103	—	2	4	no	—
81	C₄×C₈	C₂²×C₃²	1	2	yes	—
82	C₄×C₈	C₂²×C₃²	1	1	yes	characteristic
83	C₃×Q₈	—	4	8	no	—
84	C₃×Q₈	—	4	8	no	—
85	C₃×Q₈	—	4	16	no	—
86	C₂×C₃×C₄	—	4	4	no	—
87	C₂×C₃×C₄	—	4	4	no	—
88	C₂×C₃×C₄	—	4	8	no	—
89	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
90	C₂×C₃×C₄	C₂×C₃×D₄	1	8	yes	—
91	C₃×C₈	—	2	16	no	—
92	C₃×C₈	C₃×SD₁₆	1	16	yes	—
93	C₂×C₃²	#492	1	1	yes	characteristic
94	C₂×C₃²	#497	1	2	yes	—
95	C₂×Q₈	—	2	2	no	—
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	2	1	no	—
98	Q₁₆	—	4	4	no	—
99	C₂×C₈	C₃²×D₄	1	4	yes	—
100	C₂×C₈	C₃²×D₄	1	2	yes	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄²	C₂³×C₃²	1	1	yes	characteristic
106	C₂²×C₃	C₃×#127	1	4	yes	—
107	C₃×C₄	—	8	4	no	—
108	C₃×C₄	—	8	4	no	—
109	C₃×C₄	—	8	8	no	—
110	C₃×C₄	C₂×C₃×SD₁₆	1	8	yes	—
111	C₃×C₄	C₃×#137	1	16	yes	—
112	C₃²	#1716	1	1	yes	characteristic, direct product factor, Sylow
113	Q₈	—	4	2	no	—
114	Q₈	—	4	2	no	—
115	Q₈	—	4	4	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	4	1	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
120	C₂×C₄	C₂×C₃²×D₄	1	2	yes	—
121	C₈	—	2	4	no	—
122	C₈	C₃²×SD₁₆	1	4	yes	—
123	C₂×C₃	C₃×#492	1	4	yes	—
124	C₂×C₃	C₃×#497	1	8	yes	—
125	C₂²	C₃²×#127	1	1	yes	characteristic
126	C₄	—	8	1	no	—
127	C₄	—	8	1	no	—
128	C₄	—	8	2	no	—
129	C₄	C₂×C₃²×SD₁₆	1	2	yes	—
130	C₄	C₃²×#137	1	4	yes	—
131	C₃	C₃×#1716	1	4	yes	direct product factor
132	C₂	C₃²×#492	1	1	yes	characteristic
133	C₂	C₃²×#497	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#498

C₂

yes

maximal, characteristic

C₃²×#498

C₂

yes

maximal

C₃²×#494

C₂

yes

maximal, characteristic

C₃²×#487

C₂

yes

maximal

C₃²×#322

C₂

yes

maximal, characteristic

C₃×#1716

C₃

yes

maximal, direct product factor

C₂×C₃²×Q₁₆

—

C₃²×C₄⋊Q₈

C₂²

yes

characteristic

C₃²×C₄⋊Q₈

C₂²

yes

—

C₃²×C₄⋊Q₈

C₂²

yes

characteristic

C₃²×#106

—

C₃²×#106

—

C₃²×#103

—

C₃²×C₄×C₈

C₂²

yes

—

C₃²×C₄×C₈

C₂²

yes

characteristic

C₃×#498

C₂×C₃

yes

—

C₃×#498

C₂×C₃

yes

—

C₃×#494

C₂×C₃

yes

—

C₃×#487

C₂×C₃

yes

—

C₃×#322

C₂×C₃

yes

—

C₂×C₃²×Q₈

—

C₂×C₃²×Q₈

—

C₂×C₃²×Q₈

—

C₃²×Q₁₆

—

C₂×C₃²×C₈

D₄

yes

—

C₂×C₃²×C₈

D₄

yes

—

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

—

C₃²×C₄⋊C₄

—

C₃²×C₄²

C₂³

yes

characteristic

#1716

C₃²

yes

characteristic, direct product factor, Sylow

C₂×C₃×Q₁₆

—

C₃×C₄⋊Q₈

C₂²×C₃

yes

—

C₃×C₄⋊Q₈

C₂²×C₃

yes

—

C₃×C₄⋊Q₈

C₂²×C₃

yes

—

C₃×#106

—

C₃×#106

—

C₃×#103

—

C₃×C₄×C₈

C₂²×C₃

yes

—

C₃×C₄×C₈

C₂²×C₃

yes

—

C₃²×Q₈

—

C₃²×Q₈

—

C₃²×Q₈

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃²×C₄

C₂×D₄

yes

—

C₃²×C₈

—

C₃²×C₈

SD₁₆

yes

—

#498

C₂×C₃²

yes

—

#498

C₂×C₃²

yes

characteristic

#494

C₂×C₃²

yes

characteristic

#487

C₂×C₃²

yes

—

#322

C₂×C₃²

yes

characteristic

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

—

C₃×Q₁₆

—

C₂×C₃×C₈

C₃×D₄

yes

—

C₂×C₃×C₈

C₃×D₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂³×C₃

yes

—

C₂²×C₃²

#127

yes

characteristic

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

—

C₃²×C₄

C₂×SD₁₆

yes

—

C₃²×C₄

#137

yes

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂²×C₃²

yes

characteristic

C₄⋊Q₈

C₂²×C₃²

yes

—

C₄⋊Q₈

C₂²×C₃²

yes

—

—

—

C₄×C₈

C₂²×C₃²

yes

—

C₄×C₈

C₂²×C₃²

yes

characteristic

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×C₃×D₄

yes

—

C₂×C₃×C₄

C₂×C₃×D₄

yes

—

C₃×C₈

—

C₃×C₈

C₃×SD₁₆

yes

—

C₂×C₃²

#492

yes

characteristic

C₂×C₃²

#497

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

Q₁₆

—

C₂×C₈

C₃²×D₄

yes

—

100

C₂×C₈

C₃²×D₄

yes

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄²

C₂³×C₃²

yes

characteristic

106

C₂²×C₃

C₃×#127

yes

—

107

C₃×C₄

—

108

C₃×C₄

—

109

C₃×C₄

—

110

C₃×C₄

C₂×C₃×SD₁₆

yes

—

111

C₃×C₄

C₃×#137

yes

—

112

C₃²

#1716

yes

characteristic, direct product factor, Sylow

113

Q₈

—

114

Q₈

—

115

Q₈

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂×C₃²×D₄

yes

characteristic

120

C₂×C₄

C₂×C₃²×D₄

yes

—

121

C₈

—

122

C₈

C₃²×SD₁₆

yes

—

123

C₂×C₃

C₃×#492

yes

—

124

C₂×C₃

C₃×#497

yes

—

125

C₂²

C₃²×#127

yes

characteristic

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

C₂×C₃²×SD₁₆

yes

—

130

C₄

C₃²×#137

yes

—

131

C₃

C₃×#1716

yes

direct product factor

132

C₂

C₃²×#492

yes

characteristic

133

C₂

C₃²×#497

yes

—

	subgroup	G/H
center, socle, prime radical	C₂²×C₃²	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₃²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₃²

#127

commutator, Frattini, cosocle

C₄²

C₂³×C₃²

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₃² ◁ C₃²×C₄² ◁ G

N	H ≅ G/N	type
6. C₃×#1716	C₃	direct (N × H)
32. #1716	C₃²	direct (N × H)
112. C₃²	#1716	direct (N × H)
131. C₃	C₃×#1716	direct (N × H)

H ≅ G/N

type

6. C₃×#1716

C₃

direct (N × H)

32. #1716

C₃²

direct (N × H)

112. C₃²

#1716

direct (N × H)

131. C₃

C₃×#1716

direct (N × H)

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

subgroup	index
C₃²×#498	2
C₃²×#494	2
C₃²×#487	2
C₃²×#322	2
C₃×#1716	3

subgroup

index

C₃²×#498

C₃²×#494

C₃²×#487

C₃²×#322

C₃×#1716

Subgroups — C₃²×#1716 #11768765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C32×#1716 #11768765

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1716 #11768765