Subgroups — C₃²×#6868 #14554111

Order 2187.

2600 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃⁴×C₉	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₃²⋊C₉	C₃	1	3	yes	maximal
3	C₃×#6868	C₃	1	36	yes	maximal, direct product factor
4	C₃⁵	C₃²	1	1	yes	characteristic
5	C₃³×C₉	—	3	8	no	—
6	C₃³×C₉	—	3	1	no	—
7	C₃³×C₉	—	3	18	no	—
8	C₃³×C₉	—	3	3	no	—
9	C₃³×C₉	C₉	1	27	yes	complemented
10	C₃³×C₉	C₃²	1	12	yes	—
11	C₃×C₃²⋊C₉	—	3	24	no	—
12	C₃×C₃²⋊C₉	C₃²	1	36	yes	—
13	#6868	C₃²	1	81	yes	direct product factor
14	C₃⁴	—	3	9	no	—
15	C₃⁴	—	3	18	no	—
16	C₃⁴	C₃×C₉	1	9	yes	—
17	C₃⁴	H(3,𝔽₃)	1	8	yes	—
18	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
19	C₃⁴	3₋¹⁺²	1	18	yes	—
20	C₃⁴	C₃³	1	4	yes	—
21	C₃²×C₉	—	3	216	no	—
22	C₃²×C₉	—	3	27	no	—
23	C₃²×C₉	—	3	24	no	—
24	C₃²×C₉	—	3	12	no	—
25	C₃²×C₉	—	3	72	no	—
26	C₃²×C₉	—	3	36	no	—
27	C₃²×C₉	—	9	27	no	—
28	C₃²×C₉	C₃×C₉	1	108	yes	complemented
29	C₃²×C₉	C₃³	1	9	yes	—
30	C₃²⋊C₉	—	3	72	no	—
31	C₃²⋊C₉	C₃³	1	27	yes	—
32	C₃³	—	3	81	no	—
33	C₃³	—	3	162	no	—
34	C₃³	—	3	24	no	—
35	C₃³	—	3	12	no	—
36	C₃³	—	3	48	no	—
37	C₃³	—	3	24	no	—
38	C₃³	C₃²⋊C₉	1	72	yes	—
39	C₃³	C₃²⋊C₉	1	9	yes	—
40	C₃³	#732	1	9	yes	—
41	C₃³	#733	1	18	yes	—
42	C₃³	C₃²×C₉	1	12	yes	—
43	C₃³	C₃×H(3,𝔽₃)	1	8	yes	—
44	C₃³	C₃×H(3,𝔽₃)	1	4	yes	—
45	C₃³	C₃×3₋¹⁺²	1	24	yes	—
46	C₃³	C₃⁴	1	1	yes	characteristic
47	C₃×C₉	—	3	9	no	—
48	C₃×C₉	—	3	18	no	—
49	C₃×C₉	—	3	216	no	—
50	C₃×C₉	—	3	108	no	—
51	C₃×C₉	—	3	27	no	—
52	C₃×C₉	—	9	108	no	—
53	C₃×C₉	C₃²×C₉	1	27	yes	complemented
54	C₃²	—	3	8	no	—
55	C₃²	—	3	1	no	—
56	C₃²	—	3	16	no	—
57	C₃²	—	3	2	no	—
58	C₃²	—	3	216	no	—
59	C₃²	—	3	36	no	—
60	C₃²	—	3	72	no	—
61	C₃²	#6868	1	81	yes	direct product factor
62	C₃²	C₃×C₃²⋊C₉	1	24	yes	—
63	C₃²	C₃×C₃²⋊C₉	1	12	yes	—
64	C₃²	C₃×#732	1	12	yes	—
65	C₃²	C₃×#733	1	24	yes	—
66	C₃²	C₃³×C₉	1	1	yes	characteristic
67	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
68	C₃²	C₃²×3₋¹⁺²	1	2	yes	—
69	C₉	—	3	27	no	—
70	C₉	—	9	27	no	—
71	C₃	—	3	1	no	—
72	C₃	—	3	2	no	—
73	C₃	—	3	24	no	—
74	C₃	C₃×#6868	1	36	yes	direct product factor
75	C₃	C₃²×C₃²⋊C₉	1	1	yes	characteristic
76	C₃	C₃²×#732	1	1	yes	characteristic
77	C₃	C₃²×#733	1	2	yes	—

	subgroup	G/H
center, socle	C₃⁴	H(3,𝔽₃)
commutator	C₃²	C₃³×C₉
Frattini, cosocle	C₃³	C₃⁴
prime radical	C₃⁵	C₃²
LCS	G ▷ C₃² ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃⁴ ◁ C₃⁵ ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₃×#6868	C₃	direct (N × H)
9. C₃³×C₉	C₉	semidirect
13. #6868	C₃²	direct (N × H)
28. C₃²×C₉	C₃×C₉	semidirect
53. C₃×C₉	C₃²×C₉	semidirect
61. C₃²	#6868	direct (N × H)
74. C₃	C₃×#6868	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃⁴×C₉	3
C₃²×C₃²⋊C₉	3
C₃×#6868	3

Subgroups — C32×#6868 #14554111

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#6868 #14554111