Subgroups — #10624802

Order 729.

111 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#733	C₃	1	1	yes	maximal, characteristic
2	#6894	C₃	1	1	yes	maximal, characteristic
3	#6894	C₃	1	1	yes	maximal, characteristic
4	C₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
5	C₃×C₉⋊C₉	C₃	1	1	yes	maximal, characteristic
6	3₋¹⁺²⋊C₉	C₃	1	1	yes	maximal, characteristic
7	3₋¹⁺²⋊C₉	C₃	1	1	yes	maximal, characteristic
8	3₋¹⁺²⋊C₉	C₃	1	1	yes	maximal, characteristic
9	3₋¹⁺²⋊C₉	C₃	1	1	yes	maximal, characteristic
10	3₋¹⁺²⋊C₉	C₃	1	1	yes	maximal, characteristic
11	3₋¹⁺²⋊C₉	C₃	1	1	yes	maximal, characteristic
12	#6868	C₃	1	1	yes	maximal, characteristic
13	#6868	C₃	1	1	yes	maximal, characteristic
14	C₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
15	C₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
16	C₃×3₋¹⁺²	C₃²	1	1	yes	characteristic
17	C₃²×C₉	C₃²	1	1	yes	characteristic
18	C₃²×C₉	C₃²	1	1	yes	characteristic
19	C₃²×C₉	C₃²	1	1	yes	characteristic
20	C₃²×C₉	C₃²	1	1	yes	characteristic
21	#733	—	3	3	no	—
22	C₉⋊C₉	—	3	1	no	—
23	C₉⋊C₉	—	3	2	no	—
24	C₉⋊C₉	—	3	2	no	—
25	C₉⋊C₉	—	3	1	no	—
26	C₉⋊C₉	—	3	1	no	—
27	C₉⋊C₉	—	3	2	no	—
28	C₉⋊C₉	—	3	2	no	—
29	C₃²⋊C₉	C₃²	1	1	yes	characteristic
30	C₃²⋊C₉	C₃²	1	1	yes	characteristic
31	C₃²⋊C₉	C₃²	1	1	yes	characteristic
32	C₃²⋊C₉	C₃²	1	1	yes	characteristic
33	C₃²⋊C₉	C₃²	1	1	yes	characteristic
34	C₃²⋊C₉	C₃²	1	1	yes	characteristic
35	C₉²	—	3	1	no	—
36	C₃³	C₃³	1	1	yes	characteristic
37	3₋¹⁺²	—	3	2	no	—
38	3₋¹⁺²	—	3	1	no	—
39	3₋¹⁺²	—	3	1	no	—
40	3₋¹⁺²	—	3	2	no	—
41	3₋¹⁺²	—	3	2	no	—
42	3₋¹⁺²	3₋¹⁺²	1	3	yes	—
43	C₃×C₉	—	3	1	no	—
44	C₃×C₉	—	3	1	no	—
45	C₃×C₉	—	3	3	no	—
46	C₃×C₉	—	3	1	no	—
47	C₃×C₉	—	3	1	no	—
48	C₃×C₉	—	3	1	no	—
49	C₃×C₉	—	3	1	no	—
50	C₃×C₉	—	3	1	no	—
51	C₃×C₉	—	3	1	no	—
52	C₃×C₉	—	3	1	no	—
53	C₃×C₉	—	3	1	no	—
54	C₃×C₉	—	3	1	no	—
55	C₃×C₉	—	3	1	no	—
56	C₃×C₉	—	3	1	no	—
57	C₃×C₉	—	3	2	no	—
58	C₃×C₉	—	9	1	no	—
59	C₃×C₉	—	9	1	no	—
60	C₃×C₉	H(3,𝔽₃)	1	1	yes	characteristic
61	C₃×C₉	H(3,𝔽₃)	1	2	yes	—
62	C₃²	—	3	1	no	—
63	C₃²	—	3	1	no	—
64	C₃²	—	3	1	no	—
65	C₃²	C₃×H(3,𝔽₃)	1	1	yes	characteristic
66	C₃²	C₃×3₋¹⁺²	1	1	yes	characteristic
67	C₃²	#737	1	2	yes	—
68	C₉	—	3	1	no	—
69	C₉	—	3	2	no	—
70	C₉	—	3	3	no	—
71	C₉	—	3	3	no	—
72	C₉	—	9	1	no	—
73	C₉	—	9	1	no	—
74	C₉	—	9	1	no	—
75	C₉	—	9	1	no	—
76	C₉	—	9	1	no	—
77	C₉	—	9	1	no	—
78	C₉	—	9	1	no	—
79	C₉	—	9	1	no	—
80	C₉	—	9	1	no	—
81	C₉	—	9	1	no	—
82	C₃	—	3	1	no	—
83	C₃	—	3	2	no	—
84	C₃	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
85	C₃	#6908	1	1	yes	characteristic
86	C₃	#6912	1	1	yes	characteristic
87	C₃	#6913	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃×H(3,𝔽₃)
commutator, Frattini, prime radical, cosocle	C₃³	C₃³
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ C₁

Maximal subgroups

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

subgroup	index
C₃×#733	3
#6894	3
C₉×3₋¹⁺²	3
C₃×C₉⋊C₉	3
3₋¹⁺²⋊C₉	3
#6868	3