Subgroups — #4174⋊C₂ #66347

Order 324.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#4183	C₂	1	1	yes	maximal, characteristic, complemented
2	#4175	C₂	1	1	yes	maximal, characteristic, complemented
3	#4174	C₂	1	1	yes	maximal, characteristic, complemented
4	D₃×C₃⋊D₃	—	3	1	no	maximal
5	((C₃⋊D₃)⋊C₃)⋊C₂	—	3	1	no	maximal
6	AΣL(1,𝔽₂₇)	C₂²	1	1	yes	characteristic, complemented, Sylow
7	C₃⋊(C₃⋊D₃)	D₃	1	1	yes	characteristic, complemented
8	C₃×C₃⋊D₃	—	3	1	no	—
9	C₃²×D₃	—	3	1	no	—
10	#272	D₃	1	1	yes	characteristic, complemented
11	AGL(1,ℤ/9)	—	6	1	no	—
12	(C₃⋊D₃)⋊C₃	—	3	1	no	—
13	(C₃⋊D₃)⋊C₃	—	3	1	no	—
14	C₂×C₃⋊D₃	—	9	1	no	—
15	D₃²	—	9	1	no	—
16	D₃²	—	9	3	no	—
17	D₃²	—	9	1	no	—
18	C₃³	C₂×D₃	1	1	yes	characteristic, complemented
19	3₋¹⁺²	—	2	1	no	—
20	H(3,𝔽₃)	C₂×D₃	1	1	yes	characteristic, complemented
21	C₂×C₃²	—	9	1	no	—
22	C₃⋊D₃	—	3	1	no	—
23	C₃⋊D₃	—	3	1	no	—
24	C₃⋊D₃	—	3	1	no	—
25	C₃⋊D₃	—	9	1	no	—
26	C₃⋊D₃	—	9	3	no	—
27	C₃×D₃	—	3	3	no	—
28	C₃×D₃	—	3	1	no	—
29	C₃×D₃	—	9	1	no	—
30	C₃×D₃	—	9	1	no	—
31	C₃×D₃	—	9	3	no	—
32	C₃×D₃	—	9	1	no	—
33	D₉	—	6	1	no	—
34	C₂×D₃	—	27	1	no	—
35	C₂×D₃	—	27	1	no	—
36	C₂×D₃	—	27	3	no	—
37	C₂×D₃	—	27	1	no	—
38	C₃²	—	3	3	no	—
39	C₃²	—	3	1	no	—
40	C₃²	—	3	1	no	—
41	C₃²	D₃²	1	1	yes	characteristic
42	C₉	—	6	1	no	—
43	C₂×C₃	—	9	3	no	—
44	C₂×C₃	—	9	1	no	—
45	C₂×C₃	—	27	1	no	—
46	C₂×C₃	—	27	1	no	—
47	D₃	—	3	1	no	—
48	D₃	—	9	1	no	—
49	D₃	—	9	3	no	—
50	D₃	—	9	1	no	—
51	D₃	—	9	1	no	—
52	D₃	—	27	1	no	—
53	D₃	—	27	3	no	—
54	D₃	—	27	1	no	—
55	C₂²	—	81	1	no	Sylow
56	C₃	—	3	1	no	—
57	C₃	—	3	3	no	—
58	C₃	—	9	1	no	—
59	C₃	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic
60	C₂	—	9	1	no	—
61	C₂	—	27	1	no	—
62	C₂	—	27	1	no	—

	subgroup	G/H
center, hypercenter	C₁	G
commutator, Fitting, cosocle, hypocenter	AΣL(1,𝔽₂₇)	C₂²
Frattini	C₃²	D₃²
socle	C₃	((C₃⋊D₃)⋊C₃)⋊C₂
LCS	G ▷ AΣL(1,𝔽₂₇)
UCS	C₁

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. #4183	C₂	semidirect
2. #4175	C₂	semidirect
3. #4174	C₂	semidirect
6. AΣL(1,𝔽₂₇)	C₂²	semidirect
7. C₃⋊(C₃⋊D₃)	D₃	semidirect
10. #272	D₃	semidirect
18. C₃³	C₂×D₃	semidirect
20. H(3,𝔽₃)	C₂×D₃	semidirect

Maximal subgroups

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

subgroup	index
#4183	2
#4175	2
#4174	2
D₃×C₃⋊D₃	3
((C₃⋊D₃)⋊C₃)⋊C₂	3

Subgroups — #4174⋊C2 #66347

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #4174⋊C₂ #66347