Subgroups — C₃²×#6869 #14554118

Order 2187.

2330 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×C₂₇⋊C₃	C₃	1	3	yes	maximal, complemented
3	C₃×#6869	C₃	1	36	yes	maximal, direct product factor
4	C₃⁵	C₉	1	1	yes	characteristic
5	C₃³×C₉	—	3	8	no	—
6	C₃³×C₉	—	3	1	no	—
7	C₃³×C₉	—	3	16	no	—
8	C₃³×C₉	—	3	2	no	—
9	C₃³×C₉	C₉	1	2	yes	—
10	C₃³×C₉	C₉	1	24	yes	—
11	C₃³×C₉	C₃²	1	12	yes	—
12	C₃³×C₉	C₃²	1	1	yes	characteristic
13	C₃×C₂₇⋊C₃	—	3	24	no	—
14	C₃×C₂₇⋊C₃	C₃²	1	36	yes	complemented
15	C₃²×C₂₇	—	3	3	no	—
16	#6869	C₃²	1	81	yes	direct product factor
17	C₃⁴	—	3	27	no	—
18	C₃⁴	—	3	8	no	—
19	C₃⁴	—	3	1	no	—
20	C₃⁴	C₃×C₉	1	12	yes	—
21	C₃⁴	C₃×C₉	1	1	yes	characteristic
22	C₃²×C₉	—	3	16	no	—
23	C₃²×C₉	—	3	2	no	—
24	C₃²×C₉	—	3	144	no	—
25	C₃²×C₉	—	3	48	no	—
26	C₃²×C₉	—	3	24	no	—
27	C₃²×C₉	—	3	12	no	—
28	C₃²×C₉	—	3	24	no	—
29	C₃²×C₉	—	3	48	no	—
30	C₃²×C₉	—	3	24	no	—
31	C₃²×C₉	C₃×C₉	1	24	yes	—
32	C₃²×C₉	C₃×C₉	1	72	yes	—
33	C₃²×C₉	C₃×C₉	1	8	yes	—
34	C₃²×C₉	H(3,𝔽₃)	1	8	yes	—
35	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
36	C₃²×C₉	3₋¹⁺²	1	2	yes	—
37	C₃²×C₉	3₋¹⁺²	1	16	yes	—
38	C₃²×C₉	C₃³	1	9	yes	—
39	C₃²×C₉	C₃³	1	4	yes	—
40	C₂₇⋊C₃	—	3	72	no	—
41	C₂₇⋊C₃	C₃³	1	27	yes	complemented
42	C₃×C₂₇	—	3	36	no	—
43	C₃³	—	3	108	no	—
44	C₃³	—	3	216	no	—
45	C₃³	—	3	27	no	—
46	C₃³	—	3	9	no	—
47	C₃³	—	3	24	no	—
48	C₃³	—	3	12	no	—
49	C₃³	C₃²⋊C₉	1	8	yes	—
50	C₃³	C₃²⋊C₉	1	1	yes	characteristic
51	C₃³	C₃²×C₉	1	9	yes	—
52	C₃³	C₃²×C₉	1	4	yes	—
53	C₃×C₉	—	3	48	no	—
54	C₃×C₉	—	3	24	no	—
55	C₃×C₉	—	3	9	no	—
56	C₃×C₉	—	3	18	no	—
57	C₃×C₉	—	3	144	no	—
58	C₃×C₉	—	3	72	no	—
59	C₃×C₉	C₃²⋊C₉	1	48	yes	—
60	C₃×C₉	C₃²⋊C₉	1	16	yes	—
61	C₃×C₉	C₃²⋊C₉	1	8	yes	—
62	C₃×C₉	C₃²×C₉	1	18	yes	—
63	C₃×C₉	C₃²×C₉	1	8	yes	—
64	C₃×C₉	C₃×H(3,𝔽₃)	1	4	yes	—
65	C₃×C₉	C₃×H(3,𝔽₃)	1	8	yes	—
66	C₃×C₉	C₃×3₋¹⁺²	1	16	yes	—
67	C₃×C₉	C₃×3₋¹⁺²	1	8	yes	—
68	C₃×C₉	C₃⁴	1	1	yes	characteristic
69	C₂₇	—	3	27	no	—
70	C₃²	—	3	27	no	—
71	C₃²	—	3	216	no	—
72	C₃²	—	3	9	no	—
73	C₃²	—	3	108	no	—
74	C₃²	—	3	24	no	—
75	C₃²	—	3	12	no	—
76	C₃²	#6869	1	9	yes	direct product factor
77	C₃²	C₃×C₃²⋊C₉	1	8	yes	—
78	C₃²	C₃×C₃²⋊C₉	1	4	yes	—
79	C₃²	C₃³×C₉	1	1	yes	characteristic
80	C₉	—	3	18	no	—
81	C₉	C₃×C₃²⋊C₉	1	16	yes	—
82	C₉	C₃×C₃²⋊C₉	1	8	yes	—
83	C₉	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
84	C₉	C₃²×3₋¹⁺²	1	2	yes	—
85	C₃	—	3	27	no	—
86	C₃	—	3	1	no	—
87	C₃	—	3	8	no	—
88	C₃	C₃×#6869	1	12	yes	direct product factor
89	C₃	C₃²×C₃²⋊C₉	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₃²×C₂₇⋊C₃	C₃	semidirect
3. C₃×#6869	C₃	direct (N × H)
14. C₃×C₂₇⋊C₃	C₃²	semidirect
16. #6869	C₃²	direct (N × H)
41. C₂₇⋊C₃	C₃³	semidirect
76. C₃²	#6869	direct (N × H)
88. C₃	C₃×#6869	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×C₂₇⋊C₃	3
C₃×#6869	3

Subgroups — C32×#6869 #14554118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃²×#6869 #14554118