Subgroups — C₂²×#66332 #13047479

Order 1296.

432 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#732	C₂	1	1	yes	maximal, characteristic
2	C₂×#66332	C₂	1	6	yes	maximal, direct product factor
3	C₂²×(C₃⋊Dic₃)⋊C₃	—	3	3	no	maximal
4	C₂²×C₃×Dic₉	—	3	1	no	maximal
5	C₂²×#732	C₄	1	4	yes	complemented
6	C₂²×#732	C₂²	1	3	yes	—
7	#66332	C₂²	1	4	yes	direct product factor
8	C₂³×H(3,𝔽₃)	D₃	1	3	yes	—
9	C₂³×C₃×C₉	D₃	1	1	yes	characteristic
10	C₂×(C₃⋊Dic₃)⋊C₃	—	3	18	no	—
11	C₂×C₃×Dic₉	—	3	6	no	—
12	C₂×#732	C₂×C₄	1	6	yes	complemented
13	C₂×#732	C₂³	1	1	yes	characteristic
14	C₂²×C₃⋊Dic₃	—	3	3	no	—
15	C₂²×C₃×Dic₃	—	9	1	no	—
16	C₂²×Dic₉	—	3	1	no	—
17	C₂²×H(3,𝔽₃)	Dic₃	1	12	yes	complemented
18	C₂²×H(3,𝔽₃)	C₂×D₃	1	9	yes	—
19	C₂²×C₃×C₉	Dic₃	1	4	yes	complemented
20	C₂²×C₃×C₉	C₂×D₃	1	3	yes	—
21	(C₃⋊Dic₃)⋊C₃	—	3	12	no	—
22	C₃×Dic₉	—	3	4	no	—
23	#732	C₂²×C₄	1	1	yes	characteristic, complemented, Sylow
24	C₂³×C₃²	—	3	3	no	—
25	C₂³×C₃²	C₃⋊D₃	1	1	yes	characteristic
26	C₂×C₃⋊Dic₃	—	3	18	no	—
27	C₂×C₃×Dic₃	—	9	6	no	—
28	C₂³×C₉	—	3	1	no	—
29	C₂×Dic₉	—	3	6	no	—
30	C₂×H(3,𝔽₃)	C₂×Dic₃	1	18	yes	complemented
31	C₂×H(3,𝔽₃)	C₂²×D₃	1	3	yes	—
32	C₂×C₃×C₉	C₂×Dic₃	1	6	yes	complemented
33	C₂×C₃×C₉	C₂²×D₃	1	1	yes	characteristic
34	C₂²×C₃×C₄	—	27	1	no	—
35	C₂²×Dic₃	—	9	1	no	—
36	C₂²×Dic₃	—	27	3	no	—
37	C₂²×C₃²	—	3	12	no	—
38	C₂²×C₃²	—	3	9	no	—
39	C₂²×C₃²	C₃⋊Dic₃	1	4	yes	—
40	C₂²×C₃²	C₂×C₃⋊D₃	1	3	yes	—
41	C₃⋊Dic₃	—	3	12	no	—
42	C₃×Dic₃	—	9	4	no	—
43	C₂²×C₉	—	3	4	no	—
44	C₂²×C₉	—	3	3	no	—
45	Dic₉	—	3	4	no	—
46	H(3,𝔽₃)	C₂²×Dic₃	1	3	yes	complemented
47	C₃×C₉	C₂²×Dic₃	1	1	yes	characteristic, complemented
48	C₂³×C₃	—	3	1	no	—
49	C₂³×C₃	—	9	3	no	—
50	C₂³×C₃	#272	1	1	yes	characteristic
51	C₂×C₃×C₄	—	27	6	no	—
52	C₂×Dic₃	—	9	6	no	—
53	C₂×Dic₃	—	27	18	no	—
54	C₂×C₃²	—	3	18	no	—
55	C₂×C₃²	—	3	3	no	—
56	C₂×C₃²	C₂×C₃⋊Dic₃	1	6	yes	—
57	C₂×C₃²	C₂²×C₃⋊D₃	1	1	yes	characteristic
58	C₂×C₉	—	3	6	no	—
59	C₂×C₉	—	3	1	no	—
60	C₂²×C₄	—	27	1	no	Sylow
61	C₂²×C₃	—	3	4	no	—
62	C₂²×C₃	—	3	3	no	—
63	C₂²×C₃	—	9	12	no	—
64	C₂²×C₃	—	9	9	no	—
65	C₂²×C₃	#1084	1	4	yes	—
66	C₂²×C₃	C₂×#272	1	3	yes	—
67	C₃×C₄	—	27	4	no	—
68	Dic₃	—	9	4	no	—
69	Dic₃	—	27	12	no	—
70	C₃²	—	3	3	no	—
71	C₃²	C₂²×C₃⋊Dic₃	1	1	yes	characteristic
72	C₉	—	3	1	no	—
73	C₂³	#4185	1	1	yes	characteristic
74	C₂×C₄	—	27	6	no	—
75	C₂×C₃	—	3	6	no	—
76	C₂×C₃	—	3	1	no	—
77	C₂×C₃	—	9	18	no	—
78	C₂×C₃	—	9	3	no	—
79	C₂×C₃	C₂×#1084	1	6	yes	—
80	C₂×C₃	C₂²×#272	1	1	yes	characteristic
81	C₂²	#66332	1	4	yes	direct product factor
82	C₂²	C₂×#4185	1	3	yes	—
83	C₄	—	27	4	no	—
84	C₃	—	3	1	no	—
85	C₃	—	9	3	no	—
86	C₃	C₂²×#1084	1	1	yes	characteristic
87	C₂	C₂×#66332	1	6	yes	direct product factor
88	C₂	C₂²×#4185	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³	#4185
commutator, hypocenter	#732	C₂²×C₄
Frattini	C₂×C₃²	C₂²×C₃⋊D₃
Fitting, prime radical	C₂³×#732	C₂
socle	C₂³×C₃	#272
cosocle	C₂×#732	C₂³
LCS	G ▷ #732
UCS	C₁ ◁ C₂³

Complements

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

N	H ≅ G/N	type
2. C₂×#66332	C₂	direct (N × H)
5. C₂²×#732	C₄	semidirect
7. #66332	C₂²	direct (N × H)
12. C₂×#732	C₂×C₄	semidirect
17. C₂²×H(3,𝔽₃)	Dic₃	semidirect
19. C₂²×C₃×C₉	Dic₃	semidirect
23. #732	C₂²×C₄	semidirect
30. C₂×H(3,𝔽₃)	C₂×Dic₃	semidirect
32. C₂×C₃×C₉	C₂×Dic₃	semidirect
46. H(3,𝔽₃)	C₂²×Dic₃	semidirect
47. C₃×C₉	C₂²×Dic₃	semidirect
81. C₂²	#66332	direct (N × H)
87. C₂	C₂×#66332	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×#732	2
C₂×#66332	2
C₂²×(C₃⋊Dic₃)⋊C₃	3
C₂²×C₃×Dic₉	3

Subgroups — C22×#66332 #13047479

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#66332 #13047479