Subgroups — #10619564

Order 648.

89 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4179	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#4178	C₂	1	1	yes	maximal, characteristic, complemented
3	#66332	C₂	1	1	yes	maximal, characteristic, complemented
4	#6200	—	3	1	no	maximal
5	#6195	—	3	1	no	maximal
6	C₂×#732	C₂²	1	1	yes	characteristic
7	#4179	—	2	1	no	—
8	#4178	—	2	1	no	—
9	C₂×#272	D₃	1	1	yes	characteristic
10	C₂×C₃⋊D₉	D₃	1	1	yes	characteristic, complemented
11	C₂×(C₃⋊D₃)⋊C₃	—	3	1	no	—
12	C₂×D₃×C₉	—	3	1	no	—
13	(C₃⋊Dic₃)⋊C₃	—	3	1	no	—
14	(C₃⋊Dic₃)⋊C₃	—	6	1	no	—
15	C₃×Dic₉	—	3	1	no	—
16	#732	D₄	1	1	yes	characteristic, complemented, Sylow
17	#626	—	9	1	no	—
18	#625	—	9	1	no	—
19	#611	—	9	1	no	—
20	C₂×H(3,𝔽₃)	—	2	1	no	—
21	C₂×H(3,𝔽₃)	C₂×D₃	1	1	yes	characteristic
22	C₂×C₃×C₉	C₂×D₃	1	1	yes	characteristic
23	#272	—	2	1	no	—
24	C₃⋊D₉	—	2	1	no	—
25	(C₃⋊D₃)⋊C₃	—	6	1	no	—
26	D₃×C₉	—	6	1	no	—
27	C₂×C₃⋊D₃	—	3	1	no	—
28	C₂×C₃×D₃	—	3	1	no	—
29	C₂×C₃×D₃	—	9	1	no	—
30	C₂×C₃×D₃	—	18	1	no	—
31	C₃⋊Dic₃	—	3	1	no	—
32	C₃⋊Dic₃	—	6	1	no	—
33	C₃×Dic₃	—	9	1	no	—
34	C₂²×C₉	—	9	1	no	—
35	C₂×D₉	—	18	1	no	—
36	Dic₉	—	3	1	no	—
37	H(3,𝔽₃)	—	2	1	no	—
38	H(3,𝔽₃)	#67	1	1	yes	characteristic
39	C₃×C₉	#67	1	1	yes	characteristic, complemented
40	#67	—	27	1	no	—
41	#67	—	27	1	no	—
42	D₁₂	—	27	1	no	—
43	C₂×C₃²	—	6	1	no	—
44	C₂×C₃²	—	12	1	no	—
45	C₂×C₃²	D₃²	1	1	yes	characteristic
46	C₃⋊D₃	—	6	1	no	—
47	C₃×D₃	—	6	1	no	—
48	C₃×D₃	—	18	1	no	—
49	C₃×D₃	—	18	1	no	—
50	C₂×C₉	—	6	1	no	—
51	C₂×C₉	—	18	1	no	—
52	D₉	—	18	1	no	—
53	C₂²×C₃	—	9	1	no	—
54	C₂²×C₃	—	27	1	no	—
55	C₂×D₃	—	3	1	no	—
56	C₂×D₃	—	9	1	no	—
57	C₂×D₃	—	18	1	no	—
58	C₂×D₃	—	27	1	no	—
59	C₃×C₄	—	27	1	no	—
60	Dic₃	—	9	1	no	—
61	Dic₃	—	54	1	no	—
62	Dic₃	—	54	1	no	—
63	C₃²	—	6	1	no	—
64	C₃²	—	12	1	no	—
65	C₃²	#625	1	1	yes	characteristic
66	C₉	—	6	1	no	—
67	D₄	—	81	1	no	Sylow
68	C₂×C₃	—	3	1	no	—
69	C₂×C₃	—	18	1	no	—
70	C₂×C₃	—	18	1	no	—
71	C₂×C₃	—	36	1	no	—
72	C₂×C₃	—	54	1	no	—
73	C₂×C₃	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic
74	D₃	—	6	1	no	—
75	D₃	—	18	1	no	—
76	D₃	—	18	1	no	—
77	D₃	—	54	1	no	—
78	C₂²	—	9	1	no	—
79	C₂²	—	27	1	no	—
80	C₄	—	27	1	no	—
81	C₃	—	3	1	no	—
82	C₃	—	18	1	no	—
83	C₃	—	36	1	no	—
84	C₃	#6198	1	1	yes	characteristic
85	C₂	—	18	1	no	—
86	C₂	—	54	1	no	—
87	C₂	#4178⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	#4178⋊C₂
commutator, Fitting, cosocle	C₂×#732	C₂²
Frattini	C₂×C₃²	D₃²
socle	C₂×C₃	((C₃⋊D₃)⋊C₃)⋊C₂
hypocenter	#732	D₄
LCS	G ▷ C₂×#732 ▷ #732
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
1. C₂×#4179	C₂	semidirect
2. C₂×#4178	C₂	semidirect
3. #66332	C₂	semidirect
10. C₂×C₃⋊D₉	D₃	semidirect
16. #732	D₄	semidirect
39. C₃×C₉	#67	semidirect

Maximal subgroups

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

subgroup	index
C₂×#4179	2
C₂×#4178	2
#66332	2
#6200	3
#6195	3