Group #68089 — subgroups

Subgroups — #68089

Order 384.

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

normal subgroups only

This group has 148 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	C₄×Dic₁₂	C₂	1	1	yes	maximal, characteristic
2	#4679	C₂	1	2	yes	maximal
3	C₃×#322	C₂	1	1	yes	maximal, characteristic
4	#4457	C₂	1	1	yes	maximal, characteristic
5	#4456	C₂	1	2	yes	maximal
6	#1593	—	3	1	no	maximal, Sylow
7	C₂×Dic₁₂	C₄	1	1	yes	characteristic
8	C₄×Dic₆	C₂²	1	2	yes	—
9	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
10	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×C₈	C₂²	1	1	yes	characteristic
12	#818	C₄	1	1	yes	characteristic
13	#816	C₄	1	2	yes	—
14	#815	—	2	2	no	—
15	Dic₃⋊C₈	—	2	2	no	—
16	C₄⋊(C₃⋊C₈)	C₂²	1	2	yes	—
17	#446	—	3	2	no	—
18	C₄×Q₁₆	—	3	1	no	—
19	#334	—	3	1	no	—
20	Q₈⋊C₈	—	3	2	no	—
21	#322	D₃	1	1	yes	characteristic
22	C₂×Dic₆	C₂×C₄	1	2	yes	—
23	C₂×C₃×C₈	—	2	1	no	—
24	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
25	C₂×C₃×C₈	C₂×C₄	1	1	yes	characteristic
26	C₂×C₃×C₈	D₄	1	2	yes	—
27	C₃×C₄²	C₂³	1	1	yes	characteristic
28	C₄⋊Dic₃	C₂×C₄	1	2	yes	—
29	Dic₃⋊C₄	—	2	2	no	—
30	C₄×Dic₃	—	2	2	no	—
31	C₂×C₃⋊C₈	—	2	2	no	—
32	Dic₁₂	—	2	2	no	—
33	C₂×Q₁₆	—	3	1	no	—
34	C₄×Q₈	—	3	2	no	—
35	#107	—	3	1	no	—
36	C₄⋊C₈	—	3	2	no	—
37	C₄⋊C₈	—	6	2	no	—
38	#103	—	3	2	no	—
39	#97	—	6	2	no	—
40	C₄×C₈	C₂×D₃	1	1	yes	characteristic
41	C₄×C₈	C₂×D₃	1	1	yes	characteristic
42	C₄×C₈	C₂×D₃	1	1	yes	characteristic
43	C₂×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
45	C₂×C₃×C₄	#41	1	1	yes	characteristic
46	C₂×Dic₃	—	2	2	no	—
47	C₂×Dic₃	—	2	2	no	—
48	Dic₆	—	2	2	no	—
49	Dic₆	—	2	2	no	—
50	C₃×C₈	—	2	1	no	—
51	C₃×C₈	—	2	2	no	—
52	C₃×C₈	—	4	1	no	—
53	C₃×C₈	M₁₆	1	2	yes	—
54	C₃⋊C₈	—	4	2	no	—
55	C₂×Q₈	—	3	2	no	—
56	Q₁₆	—	6	2	no	—
57	C₂×C₈	—	2	1	no	—
58	C₂×C₈	—	6	2	no	—
59	C₂×C₈	C₄×D₃	1	1	yes	characteristic
60	C₂×C₈	C₄×D₃	1	1	yes	characteristic
61	C₂×C₈	D₁₂	1	2	yes	—
62	C₄⋊C₄	—	3	2	no	—
63	C₄⋊C₄	—	6	2	no	—
64	C₄²	—	6	2	no	—
65	C₄²	C₂²×D₃	1	1	yes	characteristic
66	C₂²×C₃	C₄×D₄	1	1	yes	characteristic
67	C₃×C₄	—	2	1	no	—
68	C₃×C₄	C₂×M₁₆	1	1	yes	characteristic
69	C₃×C₄	#131	1	1	yes	characteristic
70	C₃×C₄	#137	1	2	yes	—
71	Dic₃	—	4	2	no	—
72	Dic₃	—	4	2	no	—
73	Q₈	—	6	2	no	—
74	Q₈	—	6	2	no	—
75	C₂×C₄	—	6	2	no	—
76	C₂×C₄	—	6	2	no	—
77	C₂×C₄	C₂×C₄×D₃	1	1	yes	characteristic
78	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
79	C₂×C₄	#235	1	1	yes	characteristic
80	C₈	—	2	1	no	—
81	C₈	—	2	2	no	—
82	C₈	—	4	1	no	—
83	C₈	—	12	2	no	—
84	C₈	ΓL(1,𝔽₂₅)	1	2	yes	—
85	C₂×C₃	#436	1	1	yes	characteristic
86	C₂×C₃	#441	1	1	yes	characteristic
87	C₂×C₃	#444	1	1	yes	characteristic
88	C₂²	C₄×D₁₂	1	1	yes	characteristic
89	C₄	—	2	1	no	—
90	C₄	—	12	2	no	—
91	C₄	—	12	2	no	—
92	C₄	C₂×ΓL(1,𝔽₂₅)	1	1	yes	characteristic
93	C₄	#901	1	1	yes	characteristic
94	C₄	#909	1	2	yes	—
95	C₃	#1593	1	1	yes	characteristic, complemented, Sylow
96	C₂	#4688	1	1	yes	characteristic
97	C₂	#4716	1	1	yes	characteristic
98	C₂	#4717	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₁₂

C₂

yes

maximal, characteristic

#4679

C₂

yes

maximal

C₃×#322

C₂

yes

maximal, characteristic

#4457

C₂

yes

maximal, characteristic

#4456

C₂

yes

maximal

#1593

—

maximal, Sylow

C₂×Dic₁₂

C₄

yes

characteristic

C₄×Dic₆

C₂²

yes

—

C₃×C₄×C₈

C₂²

yes

characteristic

C₃×C₄×C₈

C₂²

yes

characteristic

C₃×C₄×C₈

C₂²

yes

characteristic

#818

C₄

yes

characteristic

#816

C₄

yes

—

#815

—

Dic₃⋊C₈

—

C₄⋊(C₃⋊C₈)

C₂²

yes

—

#446

—

C₄×Q₁₆

—

#334

—

Q₈⋊C₈

—

#322

D₃

yes

characteristic

C₂×Dic₆

C₂×C₄

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

C₂×C₄

yes

characteristic

C₂×C₃×C₈

C₂×C₄

yes

characteristic

C₂×C₃×C₈

D₄

yes

—

C₃×C₄²

C₂³

yes

characteristic

C₄⋊Dic₃

C₂×C₄

yes

—

Dic₃⋊C₄

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

Dic₁₂

—

C₂×Q₁₆

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

#97

—

C₄×C₈

C₂×D₃

yes

characteristic

C₄×C₈

C₂×D₃

yes

characteristic

C₄×C₈

C₂×D₃

yes

characteristic

C₂×C₃×C₄

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

#41

yes

characteristic

C₂×Dic₃

—

C₂×Dic₃

—

Dic₆

—

Dic₆

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

—

C₃×C₈

M₁₆

yes

—

C₃⋊C₈

—

C₂×Q₈

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₃

yes

characteristic

C₂×C₈

C₄×D₃

yes

characteristic

C₂×C₈

D₁₂

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂²×D₃

yes

characteristic

C₂²×C₃

C₄×D₄

yes

characteristic

C₃×C₄

—

C₃×C₄

C₂×M₁₆

yes

C₃×C₄

yes

C₃×C₄

yes

—

Dic₃

—

Dic₃

—

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×D₃

yes

characteristic

C₂×C₄

C₂×D₁₂

yes

characteristic

C₂×C₄

#235

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

ΓL(1,𝔽₂₅)

yes

—

C₂×C₃

yes

C₂×C₃

yes

C₂×C₃

yes

C₂²

C₄×D₁₂

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₂×ΓL(1,𝔽₂₅)

yes

C₄

yes

C₄

yes

—

C₃

#1593

yes

characteristic, complemented, Sylow

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂×C₄	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₄
Frattini	C₄²	C₂²×D₃
Fitting	C₃×#322	C₂
socle, prime radical	C₂²×C₃	C₄×D₄
cosocle	C₃×C₄²	C₂³
hypercenter	#322	D₃
hypocenter	C₃	#1593
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

subgroup

G/H

center

C₂×C₄

C₂×D₁₂

commutator

C₂×C₃×C₄

C₂²×C₄

Frattini

C₄²

C₂²×D₃

Fitting

C₃×#322

C₂

socle, prime radical

C₂²×C₃

C₄×D₄

cosocle

C₃×C₄²

C₂³

D₃

C₃

G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #322

N	H ≅ G/N	type
95. C₃	#1593	semidirect

H ≅ G/N

type

95. C₃

#1593

semidirect

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

subgroup	index
C₄×Dic₁₂	2
#4679	2
C₃×#322	2
#4457	2
#4456	2
#1593	3

subgroup

index

C₄×Dic₁₂

Subgroups — #68089

Conjugacy classes of subgroups

Complements

Maximal subgroups