Group #69576 — subgroups

Subgroups — #69576

Order 384.

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

normal subgroups only

This group has 280 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×#384	C₂	1	1	yes	maximal, characteristic
2	C₂×#808	C₂	1	4	yes	maximal
3	#4935	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×C₃⋊C₈	C₂	1	1	yes	maximal, characteristic
5	#1844	—	3	1	no	maximal, Sylow
6	C₂×C₃×C₄⋊C₄	—	2	2	no	—
7	C₂×C₃×C₄⋊C₄	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	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
12	#808	—	2	8	no	—
13	C₄×C₃⋊C₈	C₄	1	4	yes	complemented
14	C₂×#106	—	3	4	no	—
15	C₂×C₄×C₈	—	3	1	no	—
16	#384	—	3	1	no	—
17	#384	D₃	1	1	yes	characteristic
18	C₂²×C₃×C₄	—	2	2	no	—
19	C₂²×C₃×C₄	D₄	1	2	yes	—
20	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
21	C₂²×Dic₃	—	2	2	no	—
22	C₃×C₄⋊C₄	—	2	4	no	—
23	C₃×C₄⋊C₄	—	4	4	no	—
24	C₃×C₄²	C₂×C₄	1	2	yes	—
25	C₃×C₄²	Q₈	1	2	yes	—
26	C₄⋊Dic₃	—	2	4	no	—
27	C₄⋊Dic₃	—	4	4	no	—
28	C₂×C₃⋊C₈	C₂×C₄	1	4	yes	—
29	C₂×C₃⋊C₈	D₄	1	4	yes	—
30	C₂×C₃⋊C₈	Q₈	1	4	yes	—
31	C₂²×C₈	—	3	2	no	—
32	C₂×C₄⋊C₄	—	2	2	no	—
33	C₂×C₄⋊C₄	—	3	2	no	—
34	C₂×C₄⋊C₄	—	6	2	no	—
35	C₂×C₄⋊C₄	C₂×D₃	1	2	yes	—
36	C₂×C₄²	C₂×D₃	1	1	yes	characteristic
37	#106	—	6	8	no	—
38	C₄×C₈	—	3	4	no	—
39	C₂³×C₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₃×C₄	—	4	2	no	—
41	C₂×C₃×C₄	—	4	4	no	—
42	C₂×C₃×C₄	C₄⋊C₄	1	4	yes	—
43	C₂×C₃×C₄	SD₁₆	1	4	yes	—
44	C₂×C₃×C₄	SD₁₆	1	4	yes	—
45	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₃×C₄	C₂×D₄	1	2	yes	—
47	C₂×C₃×C₄	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₃×C₄	C₂×Q₈	1	2	yes	—
49	C₂×Dic₃	—	4	2	no	—
50	C₂×Dic₃	—	4	4	no	—
51	C₃⋊C₈	C₄⋊C₄	1	8	yes	complemented
52	C₂²×C₄	—	2	2	no	—
53	C₂²×C₄	—	6	2	no	—
54	C₂²×C₄	#67	1	2	yes	—
55	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
56	C₂×C₈	—	3	4	no	—
57	C₂×C₈	—	3	4	no	—
58	C₂×C₈	—	3	4	no	—
59	C₄⋊C₄	—	2	4	no	—
60	C₄⋊C₄	—	4	4	no	—
61	C₄⋊C₄	—	6	4	no	—
62	C₄⋊C₄	—	12	4	no	—
63	C₄²	Dic₆	1	2	yes	—
64	C₄²	C₄×D₃	1	2	yes	—
65	C₂²×C₃	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₃	#127	1	1	yes	characteristic
67	C₂²×C₃	C₄⋊Q₈	1	1	yes	characteristic
68	C₂²×C₃	C₂×SD₁₆	1	2	yes	—
69	C₂²×C₃	C₂×SD₁₆	1	2	yes	—
70	C₃×C₄	—	8	2	no	—
71	C₃×C₄	#106	1	8	yes	complemented
72	C₃×C₄	C₂×C₄⋊C₄	1	2	yes	—
73	C₃×C₄	C₄⋊Q₈	1	2	yes	—
74	Dic₃	—	8	2	no	—
75	C₂³	C₂×#67	1	1	yes	characteristic
76	C₂×C₄	—	4	4	no	—
77	C₂×C₄	—	4	2	no	—
78	C₂×C₄	—	12	2	no	—
79	C₂×C₄	—	12	4	no	—
80	C₂×C₄	Dic₃⋊C₄	1	4	yes	—
81	C₂×C₄	#214	1	4	yes	—
82	C₂×C₄	#215	1	4	yes	—
83	C₂×C₄	C₂×Dic₆	1	1	yes	characteristic
84	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
85	C₂×C₄	D₃×D₄	1	2	yes	—
86	C₂×C₄	D₃×Q₈	1	2	yes	—
87	C₈	—	3	8	no	—
88	C₂×C₃	#384	1	1	yes	characteristic
89	C₂×C₃	C₂×#106	1	2	yes	—
90	C₂×C₃	#492	1	1	yes	characteristic
91	C₂×C₃	#492	1	1	yes	characteristic
92	C₂×C₃	#498	1	2	yes	—
93	C₂²	C₂×Dic₃⋊C₄	1	1	yes	characteristic
94	C₂²	C₂×#214	1	2	yes	—
95	C₂²	#940	1	1	yes	characteristic
96	C₂²	C₂×#215	1	2	yes	—
97	C₂²	#944	1	1	yes	characteristic
98	C₄	—	8	2	no	—
99	C₄	—	24	2	no	—
100	C₄	#808	1	8	yes	complemented
101	C₄	#888	1	2	yes	—
102	C₄	D₃×C₄⋊C₄	1	2	yes	—
103	C₃	#1844	1	1	yes	characteristic, complemented, Sylow
104	C₂	C₂×#808	1	2	yes	—
105	C₂	#4975	1	1	yes	characteristic
106	C₂	#5076	1	1	yes	characteristic
107	C₂	#5080	1	2	yes	—
108	C₂	#5085	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃×#384

C₂

yes

maximal, characteristic

C₂×#808

C₂

yes

maximal

#4935

C₂

yes

maximal, characteristic

C₂×C₄×C₃⋊C₈

C₂

yes

maximal, characteristic

#1844

—

maximal, Sylow

C₂×C₃×C₄⋊C₄

—

C₂×C₃×C₄⋊C₄

C₂²

yes

—

C₂×C₃×C₄²

C₂²

yes

characteristic

C₂×C₄⋊Dic₃

—

C₂×C₄⋊Dic₃

C₂²

yes

—

C₂²×C₃⋊C₈

C₂²

yes

—

#808

—

C₄×C₃⋊C₈

C₄

yes

complemented

C₂×#106

—

C₂×C₄×C₈

—

#384

—

#384

D₃

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

—

C₃×C₄⋊C₄

—

C₃×C₄⋊C₄

—

C₃×C₄²

C₂×C₄

yes

—

C₃×C₄²

Q₈

yes

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₂×C₃⋊C₈

C₂×C₄

yes

—

C₂×C₃⋊C₈

D₄

yes

—

C₂×C₃⋊C₈

Q₈

yes

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃

yes

—

C₂×C₄²

C₂×D₃

yes

characteristic

#106

—

C₄×C₈

—

C₂³×C₃

C₂×D₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₄⋊C₄

yes

—

C₂×C₃×C₄

SD₁₆

yes

—

C₂×C₃×C₄

SD₁₆

yes

—

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

—

C₂×C₃×C₄

C₂×Q₈

yes

characteristic

C₂×C₃×C₄

C₂×Q₈

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃⋊C₈

C₄⋊C₄

yes

complemented

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

—

C₄⋊C₄

—

C₄²

Dic₆

yes

—

C₄²

C₄×D₃

yes

—

C₂²×C₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃

#127

yes

characteristic

C₂²×C₃

C₄⋊Q₈

yes

characteristic

C₂²×C₃

C₂×SD₁₆

yes

—

C₂²×C₃

C₂×SD₁₆

yes

—

C₃×C₄

—

C₃×C₄

#106

yes

complemented

C₃×C₄

C₂×C₄⋊C₄

yes

—

C₃×C₄

C₄⋊Q₈

yes

—

Dic₃

—

C₂³

C₂×#67

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Dic₃⋊C₄

yes

—

C₂×C₄

#214

yes

—

C₂×C₄

#215

yes

—

C₂×C₄

C₂×Dic₆

yes

characteristic

C₂×C₄

C₂×C₄×D₃

yes

characteristic

C₂×C₄

D₃×D₄

yes

—

C₂×C₄

D₃×Q₈

yes

—

C₈

—

C₂×C₃

#384

yes

characteristic

C₂×C₃

C₂×#106

yes

—

C₂×C₃

yes

C₂×C₃

yes

C₂×C₃

yes

—

C₂²

C₂×Dic₃⋊C₄

yes

characteristic

C₂²

C₂×#214

yes

—

C₂²

#940

yes

characteristic

C₂²

C₂×#215

yes

—

C₂²

#944

yes

characteristic

C₄

—

C₄

—

100

C₄

#808

yes

complemented

101

C₄

#888

yes

—

102

C₄

D₃×C₄⋊C₄

yes

—

103

C₃

#1844

yes

characteristic, complemented, Sylow

104

C₂

C₂×#808

yes

—

105

C₂

#4975

yes

characteristic

106

C₂

#5076

yes

characteristic

107

C₂

#5080

yes

—

108

C₂

#5085

yes

characteristic

	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	#1844
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. C₄×C₃⋊C₈	C₄	semidirect
51. C₃⋊C₈	C₄⋊C₄	semidirect
71. C₃×C₄	#106	semidirect
100. C₄	#808	semidirect
103. C₃	#1844	semidirect

H ≅ G/N

type

13. C₄×C₃⋊C₈

C₄

semidirect

51. C₃⋊C₈

C₄⋊C₄

semidirect

71. C₃×C₄

#106

semidirect

100. C₄

#808

semidirect

103. C₃

#1844

semidirect

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

subgroup	index
C₃×#384	2
C₂×#808	2
#4935	2
C₂×C₄×C₃⋊C₈	2
#1844	3

subgroup

index

C₂×C₄×C₃⋊C₈

Subgroups — #69576

Conjugacy classes of subgroups

Complements

Maximal subgroups