Group #69816 — subgroups

Subgroups — #69816

Order 384.

256 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 141 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×C₃×#97	C₂	1	1	yes	maximal, characteristic
2	C₂×#818	C₂	1	2	yes	maximal
3	C₂×#817	C₂	1	2	yes	maximal
4	#4936	C₂	1	1	yes	maximal, characteristic
5	#4935	C₂	1	1	yes	maximal, characteristic
6	#1847	—	3	1	no	maximal, Sylow
7	C₂²×C₃×C₈	C₂²	1	2	yes	—
8	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
9	C₂×C₄⋊Dic₃	—	2	2	no	—
10	C₂×C₄⋊Dic₃	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊Dic₃	C₂²	1	1	yes	characteristic
13	C₃×#97	C₄	1	4	yes	complemented
14	C₃⋊#95	—	2	2	no	—
15	#818	—	2	4	no	—
16	#817	—	2	4	no	—
17	C₂×#107	—	3	2	no	—
18	C₂×#106	—	3	2	no	—
19	C₂×#97	D₃	1	1	yes	characteristic
20	#384	—	3	1	no	—
21	#382	—	3	1	no	—
22	C₂²×C₃×C₄	D₄	1	2	yes	—
23	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
24	C₂²×Dic₃	—	2	1	no	—
25	C₂²×Dic₃	—	2	1	no	—
26	C₂²×Dic₃	—	2	2	no	—
27	C₂×C₃×C₈	C₂×C₄	1	4	yes	—
28	C₂×C₃×C₈	D₄	1	4	yes	—
29	C₂×C₃×C₈	Q₈	1	4	yes	—
30	C₃×C₄²	C₂×C₄	1	2	yes	—
31	C₃×C₄²	Q₈	1	2	yes	—
32	C₄⋊Dic₃	—	2	2	no	—
33	C₄⋊Dic₃	—	2	2	no	—
34	C₄⋊Dic₃	—	2	2	no	—
35	C₄⋊Dic₃	—	2	2	no	—
36	C₄⋊Dic₃	—	4	4	no	—
37	C₂²×C₈	C₂×D₃	1	2	yes	—
38	C₂×C₄⋊C₄	—	3	1	no	—
39	C₂×C₄⋊C₄	—	3	1	no	—
40	C₂×C₄⋊C₄	—	3	2	no	—
41	C₂×C₄⋊C₄	—	6	2	no	—
42	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
43	#107	—	6	4	no	—
44	#106	—	6	4	no	—
45	#97	Dic₃	1	4	yes	complemented
46	#95	—	6	2	no	—
47	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
48	C₂×C₃×C₄	—	2	2	no	—
49	C₂×C₃×C₄	—	2	2	no	—
50	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
51	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
53	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
54	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
55	C₂×Dic₃	—	4	2	no	—
56	C₂×Dic₃	—	4	1	no	—
57	C₂×Dic₃	—	4	1	no	—
58	C₂×Dic₃	—	4	4	no	—
59	C₂×Dic₃	—	4	2	no	—
60	C₂×Dic₃	—	4	2	no	—
61	C₃×C₈	C₄⋊C₄	1	8	yes	complemented
62	C₂²×C₄	—	6	1	no	—
63	C₂²×C₄	—	6	1	no	—
64	C₂²×C₄	—	6	2	no	—
65	C₂²×C₄	D₁₂	1	2	yes	—
66	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
67	C₂×C₈	Dic₆	1	4	yes	—
68	C₂×C₈	D₁₂	1	4	yes	—
69	C₂×C₈	C₂×Dic₃	1	4	yes	—
70	C₄⋊C₄	—	6	2	no	—
71	C₄⋊C₄	—	6	2	no	—
72	C₄⋊C₄	—	6	2	no	—
73	C₄⋊C₄	—	6	2	no	—
74	C₄⋊C₄	—	12	4	no	—
75	C₄²	Dic₆	1	2	yes	—
76	C₄²	C₂×Dic₃	1	2	yes	—
77	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₃	#127	1	1	yes	characteristic
79	C₂²×C₃	C₄⋊Q₈	1	1	yes	characteristic
80	C₂²×C₃	AGL(1,ℤ/8)	1	2	yes	—
81	C₂²×C₃	#137	1	2	yes	—
82	C₃×C₄	—	2	4	no	—
83	C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
84	C₃×C₄	C₄⋊Q₈	1	2	yes	—
85	Dic₃	—	8	2	no	—
86	Dic₃	—	8	1	no	—
87	Dic₃	—	8	1	no	—
88	C₂³	C₂×D₁₂	1	1	yes	characteristic
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	12	2	no	—
92	C₂×C₄	—	12	1	no	—
93	C₂×C₄	—	12	1	no	—
94	C₂×C₄	—	12	4	no	—
95	C₂×C₄	—	12	2	no	—
96	C₂×C₄	—	12	2	no	—
97	C₂×C₄	C₄⋊Dic₃	1	4	yes	—
98	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
99	C₂×C₄	C₂×Dic₆	1	2	yes	—
100	C₂×C₄	C₂×D₁₂	1	2	yes	—
101	C₂×C₄	C₂²×Dic₃	1	1	yes	characteristic
102	C₈	C₄⋊Dic₃	1	8	yes	complemented
103	C₂×C₃	#384	1	1	yes	characteristic
104	C₂×C₃	#428	1	2	yes	—
105	C₂×C₃	#496	1	1	yes	characteristic
106	C₂×C₃	#497	1	1	yes	characteristic
107	C₂×C₃	#501	1	2	yes	—
108	C₂²	#869	1	1	yes	characteristic
109	C₂²	#874	1	1	yes	characteristic
110	C₂²	#908	1	2	yes	—
111	C₂²	#909	1	2	yes	—
112	C₂²	C₂×C₄⋊Dic₃	1	1	yes	characteristic
113	C₄	—	2	4	no	—
114	C₄	—	24	2	no	—
115	C₄	—	24	1	no	—
116	C₄	—	24	1	no	—
117	C₄	#869	1	2	yes	—
118	C₄	C₂×C₄⋊Dic₃	1	2	yes	—
119	C₃	#1847	1	1	yes	characteristic, complemented, Sylow
120	C₂	#4702	1	2	yes	—
121	C₂	#4712	1	1	yes	characteristic
122	C₂	#4715	1	1	yes	characteristic
123	C₂	#4935	1	1	yes	characteristic
124	C₂	#5121	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃×#97

C₂

yes

maximal, characteristic

C₂×#818

C₂

yes

maximal

C₂×#817

C₂

yes

maximal

#4936

C₂

yes

maximal, characteristic

#4935

C₂

yes

maximal, characteristic

#1847

—

maximal, Sylow

C₂²×C₃×C₈

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂×C₄⋊Dic₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

C₂²

yes

characteristic

C₃×#97

C₄

yes

—

—

—

—

—

C₂×#97

D₃

yes

characteristic

#384

—

#382

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₂²×Dic₃

—

C₂²×Dic₃

—

C₂×C₃×C₈

C₂×C₄

yes

—

C₂×C₃×C₈

D₄

yes

—

C₂×C₃×C₈

Q₈

yes

—

C₃×C₄²

C₂×C₄

yes

—

C₃×C₄²

Q₈

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂²×C₈

C₂×D₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃

yes

—

—

Dic₃

yes

complemented

#95

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊C₄

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₈

C₄⋊C₄

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₁₂

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

Dic₆

yes

—

C₂×C₈

D₁₂

yes

—

C₂×C₈

C₂×Dic₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₆

yes

—

C₄²

C₂×Dic₃

yes

—

C₂²×C₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃

#127

yes

characteristic

C₂²×C₃

C₄⋊Q₈

yes

characteristic

C₂²×C₃

AGL(1,ℤ/8)

yes

—

C₂²×C₃

#137

yes

—

C₃×C₄

—

C₃×C₄

C₂×C₄⋊C₄

yes

—

C₃×C₄

C₄⋊Q₈

yes

—

Dic₃

—

Dic₃

—

Dic₃

—

C₂³

C₂×D₁₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₃

yes

—

C₂×C₄

C₂×Dic₆

yes

characteristic

C₂×C₄

C₂×Dic₆

yes

—

100

C₂×C₄

C₂×D₁₂

yes

—

101

C₂×C₄

C₂²×Dic₃

yes

characteristic

102

C₈

C₄⋊Dic₃

yes

complemented

103

C₂×C₃

#384

yes

characteristic

104

C₂×C₃

#428

yes

—

105

C₂×C₃

#496

yes

characteristic

106

C₂×C₃

#497

yes

characteristic

107

C₂×C₃

#501

yes

—

108

C₂²

#869

yes

characteristic

109

C₂²

#874

yes

characteristic

110

C₂²

#908

yes

—

111

C₂²

#909

yes

—

112

C₂²

C₂×C₄⋊Dic₃

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

#869

yes

—

118

C₄

C₂×C₄⋊Dic₃

yes

—

119

C₃

#1847

yes

characteristic, complemented, Sylow

120

C₂

#4702

yes

—

121

C₂

#4712

yes

characteristic

122

C₂

#4715

yes

characteristic

123

C₂

#4935

yes

characteristic

124

C₂

#5121

yes

—

	subgroup	G/H
center	C₂³	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₂×C₃×#97	C₂
socle, prime radical	C₂³×C₃	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×#97	D₃
hypocenter	C₃	#1847
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₁₂

commutator

C₂×C₃×C₄

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₂×C₃×#97

C₂

socle, prime radical

C₂³×C₃

C₂×D₄

cosocle

C₂²×C₃×C₄

C₂³

hypercenter

C₂×#97

D₃

hypocenter

C₃

#1847

LCS

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
13. C₃×#97	C₄	semidirect
45. #97	Dic₃	semidirect
61. C₃×C₈	C₄⋊C₄	semidirect
102. C₈	C₄⋊Dic₃	semidirect
119. C₃	#1847	semidirect

H ≅ G/N

type

13. C₃×#97

C₄

semidirect

45. #97

Dic₃

semidirect

61. C₃×C₈

C₄⋊C₄

semidirect

102. C₈

C₄⋊Dic₃

semidirect

119. C₃

#1847

semidirect

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

subgroup	index
C₂×C₃×#97	2
C₂×#818	2
C₂×#817	2
#4936	2
#4935	2
#1847	3

subgroup

index

C₂×C₃×#97

Subgroups — #69816

Conjugacy classes of subgroups

Complements

Maximal subgroups