Subgroups — #10624618

Order 729.

93 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#6894	C₃	1	1	yes	maximal, characteristic
2	#6894	C₃	1	1	yes	maximal, characteristic
3	C₉×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic
4	#6887	C₃	1	1	yes	maximal, characteristic
5	#6861	C₃	1	1	yes	maximal, characteristic
6	#6861	C₃	1	1	yes	maximal, characteristic
7	#6861	C₃	1	1	yes	maximal, characteristic
8	#6860	C₃	1	1	yes	maximal, characteristic
9	#6860	C₃	1	1	yes	maximal, characteristic
10	#6860	C₃	1	1	yes	maximal, characteristic
11	#6858	C₃	1	1	yes	maximal, characteristic
12	#6858	C₃	1	1	yes	maximal, characteristic
13	#6858	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	C₉⋊C₉	—	3	1	no	—
19	C₉⋊C₉	—	3	1	no	—
20	C₉⋊C₉	—	3	1	no	—
21	C₉⋊C₉	—	3	1	no	—
22	C₉⋊C₉	—	3	1	no	—
23	C₉⋊C₉	—	3	1	no	—
24	C₉⋊C₉	—	3	1	no	—
25	C₉⋊C₉	—	3	1	no	—
26	C₉⋊C₉	—	3	1	no	—
27	C₉⋊C₉	—	3	1	no	—
28	C₃²⋊C₉	C₃²	1	1	yes	characteristic
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	C₃²⋊C₉	C₃²	1	1	yes	characteristic
36	C₃²⋊C₉	C₃²	1	1	yes	characteristic
37	C₉²	—	3	1	no	—
38	C₉²	—	3	1	no	—
39	C₃³	C₃³	1	1	yes	characteristic
40	3₋¹⁺²	—	3	3	no	—
41	3₋¹⁺²	—	9	1	no	—
42	3₋¹⁺²	—	9	1	no	—
43	C₃×C₉	—	3	1	no	—
44	C₃×C₉	—	3	3	no	—
45	C₃×C₉	—	3	1	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	H(3,𝔽₃)	1	1	yes	characteristic
57	C₃×C₉	H(3,𝔽₃)	1	1	yes	characteristic
58	C₃×C₉	H(3,𝔽₃)	1	1	yes	characteristic
59	C₃²	—	3	1	no	—
60	C₃²	—	3	1	no	—
61	C₃²	—	3	1	no	—
62	C₃²	—	3	1	no	—
63	C₃²	C₃×H(3,𝔽₃)	1	1	yes	characteristic
64	C₉	—	3	1	no	—
65	C₉	—	3	1	no	—
66	C₉	—	3	3	no	—
67	C₉	—	9	1	no	—
68	C₉	—	9	1	no	—
69	C₉	—	9	1	no	—
70	C₉	—	9	1	no	—
71	C₉	—	9	1	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	#733	1	3	yes	—
79	C₃	—	9	1	no	—
80	C₃	C₃×#733	1	1	yes	characteristic
81	C₃	#6908	1	1	yes	characteristic
82	C₃	#6913	1	1	yes	characteristic
83	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 ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃³ ▷ C₃² ▷ C₁

Maximal subgroups

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

subgroup	index
#6894	3
C₉×3₋¹⁺²	3
#6887	3
#6861	3
#6860	3
#6858	3