Group #69392 — subgroups

Subgroups — #69392

Order 384.

260 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 149 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	C₃×C₄³	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄⋊(C₃⋊C₈)	C₂	1	2	yes	maximal
3	#4524	C₂	1	4	yes	maximal
4	#1831	—	3	1	no	maximal, Sylow
5	C₂×C₃×C₄²	C₄	1	2	yes	—
6	C₂×C₃×C₄²	C₄	1	1	yes	characteristic
7	C₂×C₃×C₄²	C₄	1	1	yes	characteristic
8	C₂×C₃×C₄²	C₂²	1	2	yes	—
9	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
10	C₂²×C₃⋊C₈	C₂²	1	4	yes	—
11	C₄⋊(C₃⋊C₈)	—	2	4	no	—
12	C₂×C₄⋊C₈	—	3	2	no	—
13	C₄³	D₃	1	1	yes	characteristic
14	#336	—	3	4	no	—
15	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
17	C₂²×C₃×C₄	C₂×C₄	1	2	yes	—
18	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
20	C₃×C₄²	—	2	4	no	—
21	C₃×C₄²	—	2	2	no	—
22	C₃×C₄²	—	2	4	no	—
23	C₃×C₄²	C₈	1	4	yes	complemented
24	C₃×C₄²	D₄	1	2	yes	—
25	C₃×C₄²	Q₈	1	2	yes	—
26	C₂×C₃⋊C₈	—	2	4	no	—
27	C₂×C₃⋊C₈	—	2	8	no	—
28	C₂²×C₈	—	3	4	no	—
29	C₂×C₄²	Dic₃	1	2	yes	—
30	C₂×C₄²	Dic₃	1	1	yes	characteristic
31	C₂×C₄²	Dic₃	1	1	yes	characteristic
32	C₂×C₄²	C₂×D₃	1	2	yes	—
33	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
34	C₄⋊C₈	—	6	4	no	—
35	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₃×C₄	—	2	2	no	—
37	C₂×C₃×C₄	—	2	2	no	—
38	C₂×C₃×C₄	—	2	2	no	—
39	C₂×C₃×C₄	—	2	2	no	—
40	C₂×C₃×C₄	—	2	2	no	—
41	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
42	C₂×C₃×C₄	C₄⋊C₄	1	2	yes	—
43	C₂×C₃×C₄	C₂×C₈	1	4	yes	—
44	C₂×C₃×C₄	C₂×C₈	1	2	yes	—
45	C₂×C₃×C₄	M₁₆	1	4	yes	—
46	C₂×C₃×C₄	M₁₆	1	2	yes	—
47	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
48	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
49	C₂×C₃×C₄	#41	1	4	yes	—
50	C₃⋊C₈	—	4	4	no	—
51	C₂²×C₄	C₂×Dic₃	1	2	yes	—
52	C₂²×C₄	C₂×Dic₃	1	2	yes	—
53	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
54	C₂²×C₄	C₂×Dic₃	1	1	yes	characteristic
55	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
56	C₂×C₈	—	6	4	no	—
57	C₂×C₈	—	6	8	no	—
58	C₄²	—	2	4	no	—
59	C₄²	—	2	2	no	—
60	C₄²	—	2	4	no	—
61	C₄²	C₃⋊C₈	1	4	yes	complemented
62	C₄²	Dic₆	1	2	yes	—
63	C₄²	D₁₂	1	2	yes	—
64	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
65	C₂²×C₃	#117	1	2	yes	—
66	C₂²×C₃	C₂²×C₈	1	1	yes	characteristic
67	C₂²×C₃	C₂×M₁₆	1	1	yes	characteristic
68	C₂²×C₃	C₂×M₁₆	1	2	yes	—
69	C₃×C₄	—	2	2	no	—
70	C₃×C₄	—	2	4	no	—
71	C₃×C₄	—	2	4	no	—
72	C₃×C₄	C₄⋊C₈	1	4	yes	—
73	C₃×C₄	#124	1	2	yes	—
74	C₃×C₄	#125	1	2	yes	—
75	C₂³	C₂²×Dic₃	1	1	yes	characteristic
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	2	no	—
78	C₂×C₄	—	2	2	no	—
79	C₂×C₄	—	2	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	C₂×C₃⋊C₈	1	4	yes	—
82	C₂×C₄	C₂×C₃⋊C₈	1	2	yes	—
83	C₂×C₄	#208	1	4	yes	—
84	C₂×C₄	#208	1	2	yes	—
85	C₂×C₄	C₄⋊Dic₃	1	2	yes	—
86	C₂×C₄	C₄⋊Dic₃	1	2	yes	—
87	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
88	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
89	C₂×C₄	#235	1	4	yes	—
90	C₈	—	12	4	no	—
91	C₂×C₃	#382	1	1	yes	characteristic
92	C₂×C₃	C₂×C₄⋊C₈	1	1	yes	characteristic
93	C₂×C₃	#423	1	1	yes	characteristic
94	C₂×C₃	#431	1	2	yes	—
95	C₂×C₃	#432	1	2	yes	—
96	C₂²	C₂²×C₃⋊C₈	1	1	yes	characteristic
97	C₂²	C₂×#208	1	1	yes	characteristic
98	C₂²	C₂×#208	1	2	yes	—
99	C₂²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
100	C₂²	#926	1	2	yes	—
101	C₄	—	2	2	no	—
102	C₄	—	2	4	no	—
103	C₄	—	2	4	no	—
104	C₄	C₄⋊(C₃⋊C₈)	1	4	yes	—
105	C₄	#870	1	2	yes	—
106	C₄	#875	1	2	yes	—
107	C₃	#1831	1	1	yes	characteristic, complemented, Sylow
108	C₂	C₂×C₄⋊(C₃⋊C₈)	1	1	yes	characteristic
109	C₂	#4924	1	1	yes	characteristic
110	C₂	#4925	1	2	yes	—
111	C₂	#4926	1	2	yes	—
112	C₂	#4936	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×C₄³

C₂

yes

maximal, characteristic

C₂×C₄⋊(C₃⋊C₈)

C₂

yes

maximal

#4524

C₂

yes

maximal

#1831

—

maximal, Sylow

C₂×C₃×C₄²

C₄

yes

—

C₂×C₃×C₄²

C₄

yes

characteristic

C₂×C₃×C₄²

C₄

yes

characteristic

C₂×C₃×C₄²

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂²×C₃⋊C₈

C₂²

yes

—

C₄⋊(C₃⋊C₈)

—

C₂×C₄⋊C₈

—

C₄³

D₃

yes

characteristic

#336

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂×C₄

yes

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

C₂³

yes

characteristic

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₈

yes

complemented

C₃×C₄²

D₄

yes

—

C₃×C₄²

Q₈

yes

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂²×C₈

—

C₂×C₄²

Dic₃

yes

—

C₂×C₄²

Dic₃

yes

characteristic

C₂×C₄²

Dic₃

yes

characteristic

C₂×C₄²

C₂×D₃

yes

—

C₂×C₄²

C₂×D₃

yes

characteristic

C₄⋊C₈

—

C₂³×C₃

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊C₄

yes

—

C₂×C₃×C₄

C₄⋊C₄

yes

—

C₂×C₃×C₄

C₂×C₈

yes

—

C₂×C₃×C₄

C₂×C₈

yes

—

C₂×C₃×C₄

M₁₆

yes

—

C₂×C₃×C₄

M₁₆

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

characteristic

C₂×C₃×C₄

#41

yes

—

C₃⋊C₈

—

C₂²×C₄

C₂×Dic₃

yes

—

C₂²×C₄

C₂×Dic₃

yes

—

C₂²×C₄

C₂×Dic₃

yes

characteristic

C₂²×C₄

C₂×Dic₃

yes

characteristic

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₃⋊C₈

yes

complemented

C₄²

Dic₆

yes

—

C₄²

D₁₂

yes

—

C₂²×C₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃

#117

yes

—

C₂²×C₃

C₂²×C₈

yes

characteristic

C₂²×C₃

C₂×M₁₆

yes

characteristic

C₂²×C₃

C₂×M₁₆

yes

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₄⋊C₈

yes

—

C₃×C₄

#124

yes

—

C₃×C₄

#125

yes

—

C₂³

C₂²×Dic₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₃⋊C₈

yes

—

C₂×C₄

C₂×C₃⋊C₈

yes

—

C₂×C₄

#208

yes

—

C₂×C₄

#208

yes

—

C₂×C₄

C₄⋊Dic₃

yes

—

C₂×C₄

C₄⋊Dic₃

yes

—

C₂×C₄

C₂×Dic₆

yes

characteristic

C₂×C₄

C₂×D₁₂

yes

characteristic

C₂×C₄

#235

yes

—

C₈

—

C₂×C₃

#382

yes

characteristic

C₂×C₃

C₂×C₄⋊C₈

yes

C₂×C₃

yes

C₂×C₃

yes

—

C₂×C₃

#432

yes

—

C₂²

C₂²×C₃⋊C₈

yes

C₂²

yes

C₂²

yes

—

C₂²

C₂×C₄⋊Dic₃

yes

characteristic

100

C₂²

#926

yes

—

101

C₄

—

102

C₄

—

103

C₄

—

104

C₄

C₄⋊(C₃⋊C₈)

yes

—

105

C₄

#870

yes

—

106

C₄

#875

yes

—

107

C₃

#1831

yes

characteristic, complemented, Sylow

108

C₂

C₂×C₄⋊(C₃⋊C₈)

yes

characteristic

109

C₂

#4924

yes

characteristic

110

C₂

#4925

yes

—

111

C₂

#4926

yes

—

112

C₂

#4936

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃
commutator	C₂²×C₃	C₂²×C₈
Fitting	C₃×C₄³	C₂
socle, prime radical	C₂³×C₃	C₂²×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₄³	D₃
hypocenter	C₃	#1831
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃

commutator

C₂²×C₃

C₂²×C₈

Fitting

C₃×C₄³

C₂

socle, prime radical

C₂³×C₃

C₂²×C₄

cosocle

C₂²×C₃×C₄

C₂³

hypercenter

C₄³

D₃

C₃

G ▷ C₂²×C₃ ▷ C₃

C₁ ◁ C₂²×C₄ ◁ C₄³

N	H ≅ G/N	type
23. C₃×C₄²	C₈	semidirect
61. C₄²	C₃⋊C₈	semidirect
107. C₃	#1831	semidirect

H ≅ G/N

type

23. C₃×C₄²

C₈

semidirect

61. C₄²

C₃⋊C₈

semidirect

107. C₃

#1831

semidirect

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

subgroup	index
C₃×C₄³	2
C₂×C₄⋊(C₃⋊C₈)	2
#4524	2
#1831	3

subgroup

index

C₃×C₄³

C₂×C₄⋊(C₃⋊C₈)

#4524

#1831

Subgroups — #69392

Conjugacy classes of subgroups

Complements

Maximal subgroups