Group #69579 — subgroups

Subgroups — #69579

Order 384.

332 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 137 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×#869	C₂	1	1	yes	maximal, characteristic
2	C₃×#384	C₂	1	1	yes	maximal, characteristic
3	C₂×#810	C₂	1	4	yes	maximal
4	C₂×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#1956	—	3	1	no	maximal, Sylow
6	C₂²×Dic₆	C₂²	1	2	yes	—
7	C₂×C₃×C₄⋊C₄	—	2	2	no	—
8	C₂×C₃×C₄⋊C₄	C₂²	1	2	yes	—
9	C₂×C₃×C₄²	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₃	—	2	2	no	—
11	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
12	#869	—	2	2	no	—
13	#869	C₄	1	4	yes	complemented
14	#810	—	2	8	no	—
15	C₄×C₃⋊C₈	—	2	2	no	—
16	C₂×C₄⋊Q₈	—	3	1	no	—
17	C₂×#103	—	3	4	no	—
18	C₂×C₄×C₈	—	3	1	no	—
19	#384	D₃	1	1	yes	characteristic
20	C₂²×C₃×C₄	—	2	2	no	—
21	C₂²×C₃×C₄	D₄	1	2	yes	—
22	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
23	C₂²×Dic₃	—	2	2	no	—
24	C₂×Dic₆	—	2	4	no	—
25	C₂×Dic₆	—	2	4	no	—
26	C₂×Dic₆	—	2	2	no	—
27	C₂×Dic₆	C₂×C₄	1	4	yes	—
28	C₃×C₄⋊C₄	—	2	4	no	—
29	C₃×C₄⋊C₄	—	4	4	no	—
30	C₃×C₄²	C₂×C₄	1	2	yes	—
31	C₃×C₄²	D₄	1	2	yes	—
32	C₄⋊Dic₃	—	2	8	no	—
33	C₂×C₃⋊C₈	—	2	2	no	—
34	C₂×C₃⋊C₈	—	2	2	no	—
35	C₂×C₃⋊C₈	D₄	1	4	yes	—
36	C₂²×Q₈	—	3	2	no	—
37	C₂²×C₈	—	3	2	no	—
38	C₄⋊Q₈	—	3	4	no	—
39	C₄⋊Q₈	—	6	2	no	—
40	C₂×C₄⋊C₄	—	2	2	no	—
41	C₂×C₄⋊C₄	—	6	2	no	—
42	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
43	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
44	#103	—	6	8	no	—
45	C₄×C₈	—	6	2	no	—
46	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₃×C₄	—	4	2	no	—
48	C₂×C₃×C₄	—	4	4	no	—
49	C₂×C₃×C₄	C₂²⋊C₄	1	4	yes	—
50	C₂×C₃×C₄	SD₁₆	1	4	yes	—
51	C₂×C₃×C₄	Q₁₆	1	4	yes	—
52	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
54	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
55	C₂×C₃×C₄	#41	1	2	yes	—
56	C₂×Dic₃	—	2	4	no	—
57	C₂×Dic₃	—	4	2	no	—
58	C₂×Dic₃	—	4	2	no	—
59	Dic₆	—	2	8	no	—
60	Dic₆	—	4	4	no	—
61	C₃⋊C₈	—	2	4	no	—
62	C₂×Q₈	—	3	4	no	—
63	C₂×Q₈	—	6	2	no	—
64	C₂×Q₈	—	6	4	no	—
65	C₂×Q₈	—	6	4	no	—
66	C₂²×C₄	—	2	2	no	—
67	C₂²×C₄	—	6	2	no	—
68	C₂²×C₄	#67	1	2	yes	—
69	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
70	C₂×C₈	—	3	4	no	—
71	C₂×C₈	—	6	2	no	—
72	C₂×C₈	—	6	2	no	—
73	C₄⋊C₄	—	2	4	no	—
74	C₄⋊C₄	—	4	4	no	—
75	C₄⋊C₄	—	6	8	no	—
76	C₄²	C₄×D₃	1	2	yes	—
77	C₄²	D₁₂	1	2	yes	—
78	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₃	#124	1	1	yes	characteristic
80	C₂²×C₃	#127	1	1	yes	characteristic
81	C₂²×C₃	C₂×SD₁₆	1	2	yes	—
82	C₂²×C₃	C₂×Q₁₆	1	2	yes	—
83	C₃×C₄	—	8	2	no	—
84	C₃×C₄	#103	1	8	yes	complemented
85	C₃×C₄	C₄×D₄	1	2	yes	—
86	C₃×C₄	#121	1	2	yes	—
87	Dic₃	—	4	4	no	—
88	C₂³	C₂×#67	1	1	yes	characteristic
89	Q₈	—	6	8	no	—
90	Q₈	—	12	4	no	—
91	C₂×C₄	—	4	4	no	—
92	C₂×C₄	—	4	2	no	—
93	C₂×C₄	—	6	4	no	—
94	C₂×C₄	—	12	2	no	—
95	C₂×C₄	—	12	2	no	—
96	C₂×C₄	(C₂×D₃)⋊C₄	1	4	yes	—
97	C₂×C₄	#214	1	4	yes	—
98	C₂×C₄	#216	1	4	yes	—
99	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
100	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
101	C₂×C₄	D₃×D₄	1	2	yes	—
102	C₂×C₄	#239	1	2	yes	—
103	C₈	—	6	4	no	—
104	C₂×C₃	#390	1	1	yes	characteristic
105	C₂×C₃	C₂×#103	1	2	yes	—
106	C₂×C₃	#487	1	2	yes	—
107	C₂×C₃	#492	1	1	yes	characteristic
108	C₂×C₃	#494	1	1	yes	characteristic
109	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
110	C₂²	C₂×#214	1	2	yes	—
111	C₂²	#940	1	1	yes	characteristic
112	C₂²	C₂×#216	1	2	yes	—
113	C₂²	#947	1	1	yes	characteristic
114	C₄	—	8	2	no	—
115	C₄	—	12	4	no	—
116	C₄	#810	1	8	yes	complemented
117	C₄	#893	1	2	yes	—
118	C₄	#895	1	2	yes	—
119	C₃	#1956	1	1	yes	characteristic, complemented, Sylow
120	C₂	C₂×#810	1	2	yes	—
121	C₂	#4988	1	1	yes	characteristic
122	C₂	#5076	1	1	yes	characteristic
123	C₂	#5092	1	1	yes	characteristic
124	C₂	#5093	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#869

C₂

yes

maximal, characteristic

C₃×#384

C₂

yes

maximal, characteristic

C₂×#810

C₂

yes

maximal

C₂×C₄×C₃⋊C₈

C₂

yes

maximal, characteristic

#1956

—

maximal, Sylow

C₂²×Dic₆

C₂²

yes

—

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂²×C₃⋊C₈

C₂²

yes

—

#869

—

#869

C₄

yes

complemented

#810

—

C₄×C₃⋊C₈

—

C₂×C₄⋊Q₈

—

C₂×#103

—

C₂×C₄×C₈

—

#384

D₃

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

—

C₂×Dic₆

C₂×C₄

yes

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×C₄

yes

—

C₃×C₄²

D₄

yes

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

D₄

yes

—

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

C₂×D₃

yes

characteristic

#103

—

C₄×C₈

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

SD₁₆

yes

—

C₂×C₃×C₄

Q₁₆

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#67

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₄×D₃

yes

—

C₄²

D₁₂

yes

—

C₂²×C₃

C₂×C₂²⋊C₄

yes

C₂²×C₃

yes

C₂²×C₃

yes

C₂²×C₃

C₂×SD₁₆

yes

—

C₂²×C₃

C₂×Q₁₆

yes

—

C₃×C₄

—

C₃×C₄

#103

yes

complemented

C₃×C₄

C₄×D₄

yes

—

C₃×C₄

#121

yes

—

Dic₃

—

C₂³

C₂×#67

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

(C₂×D₃)⋊C₄

yes

—

C₂×C₄

#214

yes

—

C₂×C₄

#216

yes

—

C₂×C₄

C₂×C₄×D₃

yes

characteristic

100

C₂×C₄

C₂×D₁₂

yes

characteristic

101

C₂×C₄

D₃×D₄

yes

—

102

C₂×C₄

#239

yes

—

103

C₈

—

104

C₂×C₃

#390

yes

characteristic

105

C₂×C₃

C₂×#103

yes

—

106

C₂×C₃

#487

yes

—

107

C₂×C₃

#492

yes

characteristic

108

C₂×C₃

#494

yes

characteristic

109

C₂²

C₂×(C₂×D₃)⋊C₄

yes

characteristic

110

C₂²

C₂×#214

yes

—

111

C₂²

#940

yes

characteristic

112

C₂²

C₂×#216

yes

—

113

C₂²

#947

yes

characteristic

114

C₄

—

115

C₄

—

116

C₄

#810

yes

complemented

117

C₄

#893

yes

—

118

C₄

#895

yes

—

119

C₃

#1956

yes

characteristic, complemented, Sylow

120

C₂

C₂×#810

yes

—

121

C₂

#4988

yes

characteristic

122

C₂

#5076

yes

characteristic

123

C₂

#5092

yes

characteristic

124

C₂

#5093

yes

—

	subgroup	G/H
center	C₂³	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃
Fitting	C₃×#384	C₂
socle, prime radical	C₂³×C₃	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	#384	D₃
hypocenter	C₃	#1956
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×#67

commutator

C₂×C₃×C₄

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃

Fitting

C₃×#384

C₂

socle, prime radical

C₂³×C₃

C₂×D₄

cosocle

C₂²×C₃×C₄

C₂³

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
13. #869	C₄	semidirect
84. C₃×C₄	#103	semidirect
116. C₄	#810	semidirect
119. C₃	#1956	semidirect

H ≅ G/N

type

13. #869

C₄

semidirect

84. C₃×C₄

#103

semidirect

116. C₄

#810

semidirect

119. C₃

#1956

semidirect

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

subgroup	index
C₂×#869	2
C₃×#384	2
C₂×#810	2
C₂×C₄×C₃⋊C₈	2
#1956	3

subgroup

index

C₂×C₄×C₃⋊C₈

Subgroups — #69579

Conjugacy classes of subgroups

Complements

Maximal subgroups