Group #11768886 — subgroups

Subgroups — C₃²×#1837 #11768886

Order 1152.

564 conjugacy classes of subgroups; only the 336 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 336 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×#423	C₂	1	2	yes	maximal
2	C₂×C₃²×C₄×C₈	C₂	1	1	yes	maximal, characteristic
3	C₃²×#341	C₂	1	4	yes	maximal
4	C₃×#1837	C₃	1	4	yes	maximal, direct product factor
5	C₂×C₃²×M₁₆	C₂²	1	4	yes	—
6	C₂²×C₃²×C₈	C₂²	1	2	yes	—
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₃²×C₄⋊C₈	—	2	4	no	—
9	C₃²×C₄×C₈	C₄	1	2	yes	—
10	C₃²×C₄×C₈	C₄	1	2	yes	—
11	C₃×#423	C₂×C₃	1	8	yes	—
12	C₂×C₃×C₄×C₈	C₂×C₃	1	4	yes	—
13	C₃×#341	C₂×C₃	1	16	yes	—
14	C₂²×C₃²×C₄	Q₈	1	2	yes	—
15	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
16	C₃²×M₁₆	—	2	8	no	—
17	C₂×C₃²×C₈	—	2	4	no	—
18	C₂×C₃²×C₈	—	2	4	no	—
19	C₂×C₃²×C₈	C₂×C₄	1	4	yes	—
20	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
21	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
22	C₃²×C₄²	D₄	1	2	yes	—
23	#1837	C₃²	1	1	yes	characteristic, direct product factor, Sylow
24	C₂×C₃×M₁₆	C₂²×C₃	1	16	yes	—
25	C₂²×C₃×C₈	C₂²×C₃	1	8	yes	—
26	C₂×C₃×C₄²	C₂²×C₃	1	4	yes	—
27	C₃×C₄⋊C₈	—	2	16	no	—
28	C₃×C₄×C₈	C₃×C₄	1	8	yes	—
29	C₃×C₄×C₈	C₃×C₄	1	8	yes	—
30	C₂³×C₃²	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₃²×C₄	—	2	4	no	—
32	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
33	C₂×C₃²×C₄	C₄⋊C₄	1	2	yes	—
34	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
36	C₂×C₃²×C₄	#41	1	4	yes	—
37	C₃²×C₈	—	2	4	no	—
38	C₃²×C₈	—	4	4	no	—
39	#423	C₂×C₃²	1	2	yes	—
40	C₂×C₄×C₈	C₂×C₃²	1	1	yes	characteristic
41	#341	C₂×C₃²	1	4	yes	—
42	C₂²×C₃×C₄	C₃×Q₈	1	8	yes	—
43	C₂²×C₃×C₄	C₂³×C₃	1	4	yes	—
44	C₃×M₁₆	—	2	32	no	—
45	C₂×C₃×C₈	—	2	16	no	—
46	C₂×C₃×C₈	—	2	16	no	—
47	C₂×C₃×C₈	C₂×C₃×C₄	1	16	yes	—
48	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
49	C₃×C₄²	C₂×C₃×C₄	1	4	yes	—
50	C₃×C₄²	C₃×D₄	1	8	yes	—
51	C₂²×C₃²	—	2	2	no	—
52	C₂²×C₃²	C₂×C₄⋊C₄	1	1	yes	characteristic
53	C₂²×C₃²	#125	1	2	yes	—
54	C₃²×C₄	—	2	2	no	—
55	C₃²×C₄	#108	1	4	yes	—
56	C₃²×C₄	#117	1	2	yes	—
57	C₃²×C₄	#124	1	2	yes	—
58	C₂×M₁₆	C₂²×C₃²	1	4	yes	—
59	C₂²×C₈	C₂²×C₃²	1	2	yes	—
60	C₂×C₄²	C₂²×C₃²	1	1	yes	characteristic
61	C₄⋊C₈	—	2	4	no	—
62	C₄×C₈	C₃²×C₄	1	2	yes	—
63	C₄×C₈	C₃²×C₄	1	2	yes	—
64	C₂³×C₃	C₂×C₃×Q₈	1	4	yes	—
65	C₂×C₃×C₄	—	2	16	no	—
66	C₂×C₃×C₄	C₃×C₄⋊C₄	1	8	yes	—
67	C₂×C₃×C₄	C₃×C₄⋊C₄	1	8	yes	—
68	C₂×C₃×C₄	C₂²×C₃×C₄	1	4	yes	—
69	C₂×C₃×C₄	C₂×C₃×D₄	1	4	yes	—
70	C₂×C₃×C₄	C₃×#41	1	16	yes	—
71	C₃×C₈	—	2	16	no	—
72	C₃×C₈	—	4	16	no	—
73	C₂×C₃²	—	2	2	no	—
74	C₂×C₃²	#382	1	1	yes	characteristic
75	C₂×C₃²	C₂×#108	1	2	yes	—
76	C₂²×C₄	C₃²×Q₈	1	2	yes	—
77	C₂²×C₄	C₂³×C₃²	1	1	yes	characteristic
78	M₁₆	—	2	8	no	—
79	C₂×C₈	—	2	4	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	C₂×C₃²×C₄	1	4	yes	—
82	C₄²	C₂×C₃²×C₄	1	1	yes	characteristic
83	C₄²	C₂×C₃²×C₄	1	1	yes	characteristic
84	C₄²	C₃²×D₄	1	2	yes	—
85	C₂²×C₃	—	2	8	no	—
86	C₂²×C₃	C₂×C₃×C₄⋊C₄	1	4	yes	—
87	C₂²×C₃	C₃×#125	1	8	yes	—
88	C₃×C₄	—	2	8	no	—
89	C₃×C₄	C₃×#108	1	16	yes	—
90	C₃×C₄	C₃×#117	1	8	yes	—
91	C₃×C₄	C₃×#124	1	8	yes	—
92	C₃²	#1837	1	1	yes	characteristic, direct product factor, Sylow
93	C₂³	C₂×C₃²×Q₈	1	1	yes	characteristic
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	C₃²×C₄⋊C₄	1	2	yes	—
96	C₂×C₄	C₃²×C₄⋊C₄	1	2	yes	—
97	C₂×C₄	C₂²×C₃²×C₄	1	1	yes	characteristic
98	C₂×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
99	C₂×C₄	C₃²×#41	1	4	yes	—
100	C₈	—	2	4	no	—
101	C₈	—	4	4	no	—
102	C₂×C₃	—	2	8	no	—
103	C₂×C₃	C₃×#382	1	4	yes	—
104	C₂×C₃	C₂×C₃×#108	1	8	yes	—
105	C₂²	—	2	2	no	—
106	C₂²	C₂×C₃²×C₄⋊C₄	1	1	yes	characteristic
107	C₂²	C₃²×#125	1	2	yes	—
108	C₄	—	2	2	no	—
109	C₄	C₃²×#108	1	4	yes	—
110	C₄	C₃²×#117	1	2	yes	—
111	C₄	C₃²×#124	1	2	yes	—
112	C₃	C₃×#1837	1	4	yes	direct product factor
113	C₂	—	2	2	no	—
114	C₂	C₃²×#382	1	1	yes	characteristic
115	C₂	C₂×C₃²×#108	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₃²×#423

C₂

yes

maximal

C₂×C₃²×C₄×C₈

C₂

yes

maximal, characteristic

C₃²×#341

C₂

yes

maximal

C₃×#1837

C₃

yes

maximal, direct product factor

C₂×C₃²×M₁₆

C₂²

yes

—

C₂²×C₃²×C₈

C₂²

yes

—

C₂×C₃²×C₄²

C₂²

yes

characteristic

C₃²×C₄⋊C₈

—

C₃²×C₄×C₈

C₄

yes

—

C₃²×C₄×C₈

C₄

yes

—

C₃×#423

C₂×C₃

yes

—

C₂×C₃×C₄×C₈

C₂×C₃

yes

—

C₃×#341

C₂×C₃

yes

—

C₂²×C₃²×C₄

Q₈

yes

—

C₂²×C₃²×C₄

C₂³

yes

characteristic

C₃²×M₁₆

—

C₂×C₃²×C₈

—

C₂×C₃²×C₈

—

C₂×C₃²×C₈

C₂×C₄

yes

—

C₃²×C₄²

C₂×C₄

yes

characteristic

C₃²×C₄²

C₂×C₄

yes

characteristic

C₃²×C₄²

D₄

yes

—

#1837

C₃²

yes

characteristic, direct product factor, Sylow

C₂×C₃×M₁₆

C₂²×C₃

yes

—

C₂²×C₃×C₈

C₂²×C₃

yes

—

C₂×C₃×C₄²

C₂²×C₃

yes

—

C₃×C₄⋊C₈

—

C₃×C₄×C₈

C₃×C₄

yes

—

C₃×C₄×C₈

C₃×C₄

yes

—

C₂³×C₃²

C₂×Q₈

yes

characteristic

C₂×C₃²×C₄

—

C₂×C₃²×C₄

C₄⋊C₄

yes

—

C₂×C₃²×C₄

C₄⋊C₄

yes

—

C₂×C₃²×C₄

C₂²×C₄

yes

characteristic

C₂×C₃²×C₄

C₂×D₄

yes

characteristic

C₂×C₃²×C₄

#41

yes

—

C₃²×C₈

—

C₃²×C₈

—

#423

C₂×C₃²

yes

—

C₂×C₄×C₈

C₂×C₃²

yes

characteristic

#341

C₂×C₃²

yes

—

C₂²×C₃×C₄

C₃×Q₈

yes

—

C₂²×C₃×C₄

C₂³×C₃

yes

—

C₃×M₁₆

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₃×C₄

yes

—

C₃×C₄²

C₂×C₃×C₄

yes

—

C₃×C₄²

C₂×C₃×C₄

yes

—

C₃×C₄²

C₃×D₄

yes

—

C₂²×C₃²

—

C₂²×C₃²

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃²

#125

yes

—

C₃²×C₄

—

C₃²×C₄

#108

yes

—

C₃²×C₄

#117

yes

—

C₃²×C₄

#124

yes

—

C₂×M₁₆

C₂²×C₃²

yes

—

C₂²×C₈

C₂²×C₃²

yes

—

C₂×C₄²

C₂²×C₃²

yes

characteristic

C₄⋊C₈

—

C₄×C₈

C₃²×C₄

yes

—

C₄×C₈

C₃²×C₄

yes

—

C₂³×C₃

C₂×C₃×Q₈

yes

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₃×C₄⋊C₄

yes

—

C₂×C₃×C₄

C₃×C₄⋊C₄

yes

—

C₂×C₃×C₄

C₂²×C₃×C₄

yes

—

C₂×C₃×C₄

C₂×C₃×D₄

yes

—

C₂×C₃×C₄

C₃×#41

yes

—

C₃×C₈

—

C₃×C₈

—

C₂×C₃²

—

C₂×C₃²

#382

yes

characteristic

C₂×C₃²

C₂×#108

yes

—

C₂²×C₄

C₃²×Q₈

yes

—

C₂²×C₄

C₂³×C₃²

yes

characteristic

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₃²×C₄

yes

—

C₄²

C₂×C₃²×C₄

yes

characteristic

C₄²

C₂×C₃²×C₄

yes

characteristic

C₄²

C₃²×D₄

yes

—

C₂²×C₃

—

C₂²×C₃

C₂×C₃×C₄⋊C₄

yes

—

C₂²×C₃

C₃×#125

yes

—

C₃×C₄

—

C₃×C₄

C₃×#108

yes

—

C₃×C₄

C₃×#117

yes

—

C₃×C₄

C₃×#124

yes

—

C₃²

#1837

yes

characteristic, direct product factor, Sylow

C₂³

C₂×C₃²×Q₈

yes

characteristic

C₂×C₄

—

C₂×C₄

C₃²×C₄⋊C₄

yes

—

C₂×C₄

C₃²×C₄⋊C₄

yes

—

C₂×C₄

C₂²×C₃²×C₄

yes

characteristic

C₂×C₄

C₂×C₃²×D₄

yes

characteristic

C₂×C₄

C₃²×#41

yes

—

100

C₈

—

101

C₈

—

102

C₂×C₃

—

103

C₂×C₃

C₃×#382

yes

—

104

C₂×C₃

C₂×C₃×#108

yes

—

105

C₂²

—

106

C₂²

C₂×C₃²×C₄⋊C₄

yes

characteristic

107

C₂²

C₃²×#125

yes

—

108

C₄

—

109

C₄

C₃²×#108

yes

—

110

C₄

C₃²×#117

yes

—

111

C₄

C₃²×#124

yes

—

112

C₃

C₃×#1837

yes

direct product factor

113

C₂

—

114

C₂

C₃²×#382

yes

characteristic

115

C₂

C₂×C₃²×#108

yes

—

	subgroup	G/H
center	C₂×C₃²×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₃²×C₄
Frattini, cosocle	C₂²×C₄	C₂³×C₃²
socle	C₂²×C₃²	C₂×C₄⋊C₄
prime radical	C₂³×C₃²	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃²×C₄ ◁ C₂×C₃²×C₄² ◁ G

subgroup

G/H

center

C₂×C₃²×C₄

C₂×D₄

commutator

C₂×C₄

C₂²×C₃²×C₄

Frattini, cosocle

C₂²×C₄

C₂³×C₃²

socle

C₂²×C₃²

C₂×C₄⋊C₄

prime radical

C₂³×C₃²

C₂×Q₈

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₃²×C₄ ◁ C₂×C₃²×C₄² ◁ G

N	H ≅ G/N	type
4. C₃×#1837	C₃	direct (N × H)
23. #1837	C₃²	direct (N × H)
92. C₃²	#1837	direct (N × H)
112. C₃	C₃×#1837	direct (N × H)

H ≅ G/N

type

4. C₃×#1837

C₃

direct (N × H)

23. #1837

C₃²

direct (N × H)

92. C₃²

#1837

direct (N × H)

112. C₃

C₃×#1837

direct (N × H)

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

subgroup	index
C₃²×#423	2
C₂×C₃²×C₄×C₈	2
C₃²×#341	2
C₃×#1837	3

subgroup

index

C₃²×#423

C₂×C₃²×C₄×C₈

C₃²×#341

C₃×#1837

Subgroups — C₃²×#1837 #11768886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C32×#1837 #11768886

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#1837 #11768886