Subgroups — #10619565

Order 648.

95 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#4185	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#4179	C₂	1	1	yes	maximal, characteristic, complemented
3	#66325	C₂	1	1	yes	maximal, characteristic, complemented
4	#6198	—	3	1	no	maximal
5	#6192	—	3	1	no	maximal
6	C₂×#732	C₂²	1	1	yes	characteristic
7	#4185	—	2	1	no	—
8	#4179	—	2	1	no	—
9	C₂×C₃⋊D₉	D₃	1	1	yes	characteristic, complemented
10	C₂×(C₃⋊D₃)⋊C₃	—	3	1	no	—
11	C₂×(C₃⋊D₃)⋊C₃	—	3	1	no	—
12	C₂×(C₃⋊D₃)⋊C₃	—	6	1	no	—
13	C₂×C₃×D₉	—	3	1	no	—
14	#1084	D₃	1	1	yes	characteristic, complemented
15	C₉×Dic₃	—	3	1	no	—
16	#732	D₄	1	1	yes	characteristic, complemented, Sylow
17	#626	—	9	1	no	—
18	#626	—	9	1	no	—
19	D₃₆	—	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	C₃⋊D₉	—	2	1	no	—
24	(C₃⋊D₃)⋊C₃	—	6	2	no	—
25	(C₃⋊D₃)⋊C₃	—	6	1	no	—
26	(C₃⋊D₃)⋊C₃	—	6	1	no	—
27	C₃×D₉	—	6	1	no	—
28	C₂×C₃⋊D₃	—	3	1	no	—
29	C₂×C₃⋊D₃	—	3	1	no	—
30	C₂×C₃⋊D₃	—	6	1	no	—
31	C₂×C₃×D₃	—	9	1	no	—
32	C₂×C₃×D₃	—	9	1	no	—
33	C₃×Dic₃	—	3	1	no	—
34	C₃×Dic₃	—	9	1	no	—
35	C₂×D₉	—	3	1	no	—
36	C₂×D₉	—	18	1	no	—
37	C₄×C₉	—	9	1	no	—
38	H(3,𝔽₃)	—	2	1	no	—
39	H(3,𝔽₃)	D₁₂	1	1	yes	characteristic
40	C₃×C₉	#67	1	1	yes	characteristic, complemented
41	#67	—	27	1	no	—
42	#67	—	27	1	no	—
43	D₁₂	—	27	1	no	—
44	C₂×C₃²	—	3	1	no	—
45	C₂×C₃²	—	12	1	no	—
46	C₂×C₃²	D₃²	1	1	yes	characteristic
47	C₃⋊D₃	—	6	1	no	—
48	C₃⋊D₃	—	6	2	no	—
49	C₃⋊D₃	—	6	1	no	—
50	C₃×D₃	—	18	1	no	—
51	C₃×D₃	—	18	1	no	—
52	C₂×C₉	—	6	1	no	—
53	D₉	—	6	1	no	—
54	D₉	—	18	1	no	—
55	C₂²×C₃	—	27	1	no	—
56	C₂²×C₃	—	27	1	no	—
57	C₂×D₃	—	9	1	no	—
58	C₂×D₃	—	9	1	no	—
59	C₂×D₃	—	27	1	no	—
60	C₂×D₃	—	54	1	no	—
61	C₂×D₃	—	54	1	no	—
62	C₃×C₄	—	9	1	no	—
63	Dic₃	—	3	1	no	—
64	Dic₃	—	18	1	no	—
65	C₃²	—	3	1	no	—
66	C₃²	—	12	1	no	—
67	C₃²	#626	1	1	yes	characteristic
68	C₉	—	6	1	no	—
69	D₄	—	81	1	no	Sylow
70	C₂×C₃	—	3	1	no	—
71	C₂×C₃	—	18	1	no	—
72	C₂×C₃	—	18	1	no	—
73	C₂×C₃	—	54	1	no	—
74	C₂×C₃	—	54	1	no	—
75	C₂×C₃	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic
76	D₃	—	18	1	no	—
77	D₃	—	18	1	no	—
78	D₃	—	54	1	no	—
79	D₃	—	54	2	no	—
80	D₃	—	54	1	no	—
81	C₂²	—	27	1	no	—
82	C₂²	—	27	1	no	—
83	C₄	—	9	1	no	—
84	C₃	—	3	1	no	—
85	C₃	—	18	1	no	—
86	C₃	—	18	1	no	—
87	C₃	#6200	1	1	yes	characteristic
88	C₂	—	54	1	no	—
89	C₂	—	54	1	no	—
90	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×#4185	C₂	semidirect
2. C₂×#4179	C₂	semidirect
3. #66325	C₂	semidirect
9. C₂×C₃⋊D₉	D₃	semidirect
14. #1084	D₃	semidirect
16. #732	D₄	semidirect
40. 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×#4185	2
C₂×#4179	2
#66325	2
#6198	3
#6192	3