Group #71939 — subgroups

Subgroups — #71939

Order 384.

228 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 65 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×Q₈²	C₂	1	1	yes	maximal, characteristic, complemented
2	#5085	C₂	1	1	yes	maximal, characteristic
3	#5027	C₂	1	2	yes	maximal
4	#4546	C₂	1	1	yes	maximal, characteristic, complemented
5	#4535	C₂	1	2	yes	maximal, complemented
6	#1623	—	3	1	no	maximal, Sylow
7	C₃×C₄⋊Q₈	—	2	2	no	—
8	C₃×C₄⋊Q₈	—	2	2	no	—
9	C₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
10	C₃×C₄×Q₈	—	2	2	no	—
11	C₃×C₄×Q₈	C₂²	1	2	yes	—
12	C₂×#215	—	2	2	no	—
13	C₂×#215	—	2	2	no	—
14	#874	C₂²	1	1	yes	characteristic
15	#809	—	2	2	no	—
16	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
17	C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
18	Q₈²	D₃	1	1	yes	characteristic, complemented
19	#492	—	3	1	no	—
20	#460	—	3	2	no	—
21	#333	—	3	1	no	—
22	Q₈⋊C₈	—	3	2	no	—
23	C₂×C₃×Q₈	—	2	2	no	—
24	C₂×C₃×Q₈	D₄	1	2	yes	—
25	C₂×C₃×Q₈	D₄	1	2	yes	—
26	C₂×D₁₂	—	2	2	no	—
27	C₂×D₁₂	—	2	1	no	—
28	C₃×C₄⋊C₄	—	2	4	no	—
29	C₃×C₄⋊C₄	—	2	2	no	—
30	C₃×C₄⋊C₄	—	2	2	no	—
31	C₃×C₄⋊C₄	D₄	1	2	yes	—
32	C₃×C₄²	—	2	2	no	—
33	C₃×C₄²	—	2	2	no	—
34	C₃×C₄²	C₂³	1	1	yes	characteristic
35	#215	—	4	4	no	—
36	#215	—	4	4	no	—
37	C₂×C₃⋊C₈	—	2	1	no	—
38	C₂×C₃⋊C₈	—	2	2	no	—
39	C₂×SD₁₆	—	6	2	no	—
40	C₂×SD₁₆	—	6	2	no	—
41	C₄⋊Q₈	—	2	2	no	—
42	C₄⋊Q₈	—	2	2	no	—
43	C₄⋊Q₈	C₂×D₃	1	1	yes	characteristic
44	#127	—	3	1	no	—
45	C₄×Q₈	—	2	2	no	—
46	C₄×Q₈	C₂×D₃	1	2	yes	—
47	C₄⋊C₈	—	3	2	no	—
48	#102	—	6	2	no	—
49	C₄×C₈	—	3	1	no	—
50	C₂²×D₃	—	4	1	no	—
51	C₃×Q₈	—	2	2	no	—
52	C₃×Q₈	—	4	2	no	—
53	C₃×Q₈	—	8	2	no	—
54	C₃×Q₈	SD₁₆	1	4	yes	complemented
55	C₂×C₃×C₄	—	2	2	no	—
56	C₂×C₃×C₄	—	2	2	no	—
57	C₂×C₃×C₄	—	2	2	no	—
58	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
60	D₁₂	—	4	2	no	—
61	D₁₂	—	4	2	no	—
62	D₁₂	—	4	2	no	—
63	C₃⋊C₈	—	2	2	no	—
64	C₃⋊C₈	—	4	2	no	—
65	C₂×Q₈	—	2	2	no	—
66	C₂×Q₈	#67	1	2	yes	—
67	C₂×Q₈	#67	1	2	yes	—
68	C₂×D₄	—	6	2	no	—
69	C₂×D₄	—	6	1	no	—
70	SD₁₆	—	12	4	no	—
71	SD₁₆	—	12	4	no	—
72	C₂×C₈	—	6	1	no	—
73	C₂×C₈	—	6	2	no	—
74	C₄⋊C₄	—	2	4	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	#67	1	2	yes	—
78	C₄²	—	2	2	no	—
79	C₄²	—	2	2	no	—
80	C₄²	C₂²×D₃	1	1	yes	characteristic
81	C₂²×C₃	#120	1	1	yes	characteristic
82	C₂×D₃	—	8	2	no	—
83	C₂×D₃	—	8	1	no	—
84	C₃×C₄	—	2	4	no	—
85	C₃×C₄	—	2	1	no	—
86	C₃×C₄	—	4	2	no	—
87	C₃×C₄	—	4	2	no	—
88	C₃×C₄	C₂×SD₁₆	1	2	yes	—
89	C₃×C₄	#137	1	2	yes	—
90	C₂³	—	12	1	no	—
91	Q₈	—	2	2	no	—
92	Q₈	—	4	2	no	—
93	Q₈	—	8	2	no	—
94	Q₈	#215	1	4	yes	complemented
95	D₄	—	12	2	no	—
96	D₄	—	12	2	no	—
97	D₄	—	12	2	no	—
98	C₂×C₄	—	2	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	C₂×#67	1	1	yes	characteristic
102	C₂×C₄	C₂×#67	1	2	yes	—
103	C₈	—	6	2	no	—
104	C₈	—	12	2	no	—
105	C₂×C₃	#448	1	2	yes	—
106	C₂×C₃	#453	1	1	yes	characteristic
107	D₃	—	16	1	no	—
108	C₂²	—	24	2	no	—
109	C₂²	—	24	1	no	—
110	C₂²	#953	1	1	yes	characteristic
111	C₄	—	2	4	no	—
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	4	2	no	—
115	C₄	C₂×#215	1	2	yes	—
116	C₄	#942	1	2	yes	—
117	C₃	#1623	1	1	yes	characteristic, complemented, Sylow
118	C₂	—	48	1	no	—
119	C₂	#5227	1	2	yes	—
120	C₂	#5241	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×Q₈²

C₂

yes

maximal, characteristic, complemented

#5085

C₂

yes

maximal, characteristic

#5027

C₂

yes

maximal

#4546

C₂

yes

maximal, characteristic, complemented

#4535

C₂

yes

maximal, complemented

#1623

—

maximal, Sylow

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

—

C₃×C₄⋊Q₈

C₂²

yes

characteristic

C₃×C₄×Q₈

—

C₃×C₄×Q₈

C₂²

yes

—

C₂×#215

—

C₂×#215

—

#874

C₂²

yes

characteristic

#809

—

C₄⋊(C₃⋊C₈)

C₂²

yes

—

C₄×C₃⋊C₈

C₂²

yes

characteristic

Q₈²

D₃

yes

characteristic, complemented

#492

—

#460

—

#333

—

Q₈⋊C₈

—

C₂×C₃×Q₈

—

C₂×C₃×Q₈

D₄

yes

—

C₂×C₃×Q₈

D₄

yes

—

C₂×D₁₂

—

C₂×D₁₂

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

D₄

yes

—

C₃×C₄²

—

C₃×C₄²

—

C₃×C₄²

C₂³

yes

characteristic

#215

—

#215

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃

yes

characteristic

#127

—

C₄×Q₈

—

C₄×Q₈

C₂×D₃

yes

—

C₄⋊C₈

—

#102

—

C₄×C₈

—

C₂²×D₃

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

—

C₃×Q₈

SD₁₆

yes

complemented

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

D₁₂

—

D₁₂

—

D₁₂

—

C₃⋊C₈

—

C₃⋊C₈

—

C₂×Q₈

—

C₂×Q₈

#67

yes

—

C₂×Q₈

#67

yes

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

#67

yes

—

C₄²

—

C₄²

—

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

#120

yes

characteristic

C₂×D₃

—

C₂×D₃

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

—

C₃×C₄

C₂×SD₁₆

yes

—

C₃×C₄

#137

yes

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

Q₈

#215

yes

complemented

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

C₂×#67

yes

characteristic

102

C₂×C₄

C₂×#67

yes

—

103

C₈

—

104

C₈

—

105

C₂×C₃

#448

yes

—

106

C₂×C₃

#453

yes

characteristic

107

D₃

—

108

C₂²

—

109

C₂²

—

110

C₂²

#953

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

C₂×#215

yes

—

116

C₄

#942

yes

—

117

C₃

#1623

yes

characteristic, complemented, Sylow

118

C₂

—

119

C₂

#5227

yes

—

120

C₂

#5241

yes

characteristic

	subgroup	G/H
center	C₂²	#953
commutator, cosocle	C₃×C₄²	C₂³
Frattini	C₄²	C₂²×D₃
Fitting	C₃×Q₈²	C₂
socle	C₂²×C₃	#120
prime radical	#874	C₂²
hypercenter	Q₈²	D₃
hypocenter	C₃	#1623
LCS	G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

subgroup

G/H

center

C₂²

#953

commutator, cosocle

C₃×C₄²

C₂³

Frattini

C₄²

C₂²×D₃

Fitting

C₃×Q₈²

C₂

C₂²×C₃

C₂²

Q₈²

D₃

hypocenter

C₃

#1623

LCS

G ▷ C₃×C₄² ▷ C₂²×C₃ ▷ C₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ Q₈²

N	H ≅ G/N	type
1. C₃×Q₈²	C₂	semidirect
4. #4546	C₂	semidirect
5. #4535	C₂	semidirect
18. Q₈²	D₃	semidirect
54. C₃×Q₈	SD₁₆	semidirect
94. Q₈	#215	semidirect
117. C₃	#1623	semidirect

H ≅ G/N

type

1. C₃×Q₈²

C₂

semidirect

4. #4546

C₂

semidirect

5. #4535

C₂

semidirect

18. Q₈²

D₃

semidirect

54. C₃×Q₈

SD₁₆

semidirect

94. Q₈

#215

semidirect

117. C₃

#1623

semidirect

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

subgroup	index
C₃×Q₈²	2
#5085	2
#5027	2
#4546	2
#4535	2
#1623	3

subgroup

index

C₃×Q₈²

Subgroups — #71939

Conjugacy classes of subgroups

Complements

Maximal subgroups