Subgroups — #10619538

Order 648.

84 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#66328	C₂	1	1	yes	maximal, characteristic
2	#66316	C₂	1	1	yes	maximal, characteristic
3	(C₃⋊Dic₃)⋊C₉	C₂	1	1	yes	maximal, characteristic
4	#6296	—	3	1	no	maximal
5	#6189	—	3	1	no	maximal
6	C₂×C₃²⋊C₉	C₂²	1	1	yes	characteristic
7	C₃×C₃⋊Dic₃	—	3	1	no	—
8	C₃×C₃⋊Dic₃	—	3	1	no	—
9	C₃×C₃⋊Dic₃	D₃	1	1	yes	characteristic
10	C₃⋊Dic₉	D₃	1	1	yes	characteristic
11	C₉×Dic₃	—	3	1	no	—
12	C₃×Dic₉	—	3	1	no	—
13	C₃×Dic₉	—	6	1	no	—
14	C₃²⋊C₉	Q₈	1	1	yes	characteristic, complemented, Sylow
15	#627	—	9	1	no	—
16	#627	—	9	1	no	—
17	#627	—	9	1	no	—
18	Dic₁₈	—	9	1	no	—
19	C₂×C₃³	C₂×D₃	1	1	yes	characteristic
20	C₂×C₃×C₉	—	2	1	no	—
21	C₂×C₃×C₉	C₂×D₃	1	1	yes	characteristic
22	C₃⋊Dic₃	—	3	1	no	—
23	C₃⋊Dic₃	—	3	1	no	—
24	C₃⋊Dic₃	D₉	1	1	yes	characteristic
25	C₃×Dic₃	—	3	1	no	—
26	C₃×Dic₃	—	9	1	no	—
27	C₃×Dic₃	—	9	1	no	—
28	C₃×Dic₃	—	9	1	no	—
29	C₃×Dic₃	—	9	1	no	—
30	C₃×Dic₃	—	18	1	no	—
31	C₃×Dic₃	—	18	1	no	—
32	C₃×Dic₃	—	18	1	no	—
33	C₄×C₉	—	9	1	no	—
34	Dic₉	—	3	1	no	—
35	Dic₉	—	6	1	no	—
36	Dic₉	—	9	1	no	—
37	C₃³	Dic₆	1	1	yes	characteristic
38	C₃×C₉	—	2	1	no	—
39	C₃×C₉	Dic₆	1	1	yes	characteristic, complemented
40	Dic₆	—	27	1	no	—
41	Dic₆	—	27	1	no	—
42	Dic₆	—	27	1	no	—
43	C₂×C₃²	—	2	1	no	—
44	C₂×C₃²	—	6	1	no	—
45	C₂×C₃²	—	12	1	no	—
46	C₂×C₃²	C₂×D₉	1	1	yes	characteristic
47	C₂×C₃²	D₃²	1	1	yes	characteristic
48	C₂×C₉	—	3	1	no	—
49	C₂×C₉	—	12	1	no	—
50	C₃×C₄	—	9	1	no	—
51	C₃×C₄	—	27	1	no	—
52	C₃×C₄	—	27	1	no	—
53	Dic₃	—	3	1	no	—
54	Dic₃	—	9	1	no	—
55	Dic₃	—	9	1	no	—
56	Dic₃	—	9	1	no	—
57	Dic₃	—	9	1	no	—
58	Dic₃	—	18	1	no	—
59	Dic₃	—	18	1	no	—
60	Dic₃	—	18	1	no	—
61	C₃²	—	2	1	no	—
62	C₃²	—	6	1	no	—
63	C₃²	—	12	1	no	—
64	C₃²	Dic₁₈	1	1	yes	characteristic, complemented
65	C₃²	#627	1	1	yes	characteristic
66	C₉	—	3	1	no	—
67	C₉	—	12	1	no	—
68	Q₈	—	81	1	no	Sylow
69	C₂×C₃	—	2	1	no	—
70	C₂×C₃	—	6	1	no	—
71	C₂×C₃	—	6	1	no	—
72	C₂×C₃	D₃×D₉	1	1	yes	characteristic
73	C₂×C₃	((C₃⋊D₃)⋊C₃)⋊C₂	1	1	yes	characteristic
74	C₄	—	9	1	no	—
75	C₄	—	27	1	no	—
76	C₄	—	27	1	no	—
77	C₃	—	2	1	no	—
78	C₃	—	6	1	no	—
79	C₃	—	6	1	no	—
80	C₃	#6189	1	1	yes	characteristic
81	C₃	#6196	1	1	yes	characteristic
82	C₂	((C₃⋊D₃)⋊C₉)⋊C₂	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂	((C₃⋊D₃)⋊C₉)⋊C₂
commutator, Fitting, cosocle	C₂×C₃²⋊C₉	C₂²
Frattini, socle	C₂×C₃²	D₃²
prime radical	C₂×C₃³	C₂×D₃
hypocenter	C₃²⋊C₉	Q₈
LCS	G ▷ C₂×C₃²⋊C₉ ▷ C₃²⋊C₉
UCS	C₁ ◁ C₂

Complements

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

N	H ≅ G/N	type
14. C₃²⋊C₉	Q₈	semidirect
39. C₃×C₉	Dic₆	semidirect
64. C₃²	Dic₁₈	semidirect

Maximal subgroups

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

subgroup	index
#66328	2
#66316	2
(C₃⋊Dic₃)⋊C₉	2
#6296	3
#6189	3