Finite Group Databasebeta

Subgroups — #14554456

Order 2187.

785 conjugacy classes of subgroups; only the 278 normal subgroups are shown.

normal subgroups only

This group has 785 conjugacy classes of subgroups — too many to draw. Only its 278 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#731	C₃	1	1	yes	maximal, characteristic
2	#10624886	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624883	C₃	1	1	yes	maximal, characteristic
4	C₃×#6895	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624863	C₃	1	1	yes	maximal, characteristic, complemented
6	#10624803	C₃	1	18	yes	maximal, complemented
7	#10624800	C₃	1	9	yes	maximal, complemented
8	#10624699	C₃	1	2	yes	maximal, complemented
9	#10624699	C₃	1	2	yes	maximal, complemented
10	#10624698	C₃	1	2	yes	maximal
11	C₃×#6867	C₃	1	2	yes	maximal, complemented
12	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
13	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
14	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
15	C₃³×C₉	C₃²	1	1	yes	characteristic
16	C₃×#731	—	3	9	no	—
17	C₃×#731	C₃²	1	3	yes	—
18	C₃×#731	C₃²	1	6	yes	—
19	#6895	—	3	3	no	—
20	#6895	—	3	6	no	—
21	#6895	—	3	3	no	—
22	#6894	—	3	6	no	—
23	#6894	C₃²	1	9	yes	complemented
24	C₉⋊H(3,𝔽₃)	—	3	6	no	—
25	C₉⋊H(3,𝔽₃)	C₃²	1	9	yes	—
26	#6891	—	3	3	no	—
27	C₉×3₋¹⁺²	—	3	6	no	—
28	C₉×3₋¹⁺²	C₃²	1	9	yes	complemented
29	#6887	—	3	3	no	—
30	C₃×C₉⋊C₉	C₃²	1	3	yes	complemented
31	C₃×C₉⋊C₉	C₃²	1	6	yes	complemented
32	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
33	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
34	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
35	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
36	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
37	3₋¹⁺²⋊C₉	C₃²	1	18	yes	complemented
38	3₋¹⁺²⋊C₉	C₃²	1	18	yes	complemented
39	H(3,𝔽₃)⋊C₉	C₃²	1	18	yes	—
40	#6867	C₃²	1	6	yes	complemented
41	#6867	C₃²	1	12	yes	complemented
42	C₃⁴	—	3	1	no	—
43	C₃⁴	C₃³	1	1	yes	characteristic
44	C₃×3₋¹⁺²	—	3	2	no	—
45	C₃×3₋¹⁺²	—	3	6	no	—
46	C₃×3₋¹⁺²	—	3	2	no	—
47	C₃×3₋¹⁺²	—	3	9	no	—
48	C₃×3₋¹⁺²	—	3	3	no	—
49	C₃×3₋¹⁺²	—	3	6	no	—
50	C₃×3₋¹⁺²	3₋¹⁺²	1	9	yes	complemented
51	C₃×3₋¹⁺²	C₃³	1	3	yes	—
52	C₃×3₋¹⁺²	C₃³	1	3	yes	—
53	C₃×H(3,𝔽₃)	—	3	3	no	—
54	C₃×H(3,𝔽₃)	—	3	6	no	—
55	C₃×H(3,𝔽₃)	—	3	2	no	—
56	C₃×H(3,𝔽₃)	C₃³	1	3	yes	—
57	C₃²×C₉	—	3	1	no	—
58	C₃²×C₉	—	3	1	no	—
59	C₃²×C₉	—	3	2	no	—
60	C₃²×C₉	—	3	1	no	—
61	C₃²×C₉	—	3	3	no	—
62	C₃²×C₉	—	3	6	no	—
63	C₃²×C₉	—	3	2	no	—
64	C₃²×C₉	—	3	2	no	—
65	C₃²×C₉	—	3	2	no	—
66	C₃²×C₉	C₃³	1	1	yes	characteristic
67	C₃²×C₉	C₃³	1	2	yes	—
68	C₃²×C₉	C₃³	1	6	yes	—
69	C₃²×C₉	C₃³	1	3	yes	—
70	#731	—	3	9	no	—
71	#731	—	3	18	no	—
72	C₉⋊C₉	—	3	9	no	—
73	C₉⋊C₉	—	3	18	no	—
74	C₉⋊C₉	—	3	18	no	—
75	C₉⋊C₉	—	3	27	no	—
76	C₃²⋊C₉	—	3	2	no	—
77	C₃²⋊C₉	—	3	2	no	—
78	C₃²⋊C₉	—	3	2	no	—
79	C₃²⋊C₉	—	3	2	no	—
80	C₃²⋊C₉	—	3	2	no	—
81	C₃²⋊C₉	—	3	2	no	—
82	C₃²⋊C₉	—	3	4	no	—
83	C₃²⋊C₉	—	3	2	no	—
84	C₃²⋊C₉	—	3	18	no	—
85	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
86	C₃²⋊C₉	H(3,𝔽₃)	1	6	yes	—
87	C₃²⋊C₉	C₃³	1	6	yes	—
88	C₃²⋊C₉	C₃³	1	6	yes	—
89	C₃²⋊C₉	C₃³	1	6	yes	—
90	C₉²	—	3	9	no	—
91	C₃³	—	3	3	no	—
92	C₃³	—	9	1	no	—
93	C₃³	—	9	2	no	—
94	C₃³	—	9	3	no	—
95	C₃³	—	27	1	no	—
96	C₃³	—	27	2	no	—
97	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
98	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
99	C₃³	C₃×3₋¹⁺²	1	3	yes	—
100	C₃³	#737	1	6	yes	—
101	C₃³	C₃⁴	1	1	yes	characteristic
102	3₋¹⁺²	—	3	6	no	—
103	3₋¹⁺²	—	3	18	no	—
104	3₋¹⁺²	—	3	3	no	—
105	3₋¹⁺²	—	3	6	no	—
106	3₋¹⁺²	—	9	6	no	—
107	3₋¹⁺²	—	27	9	no	—
108	3₋¹⁺²	C₃×3₋¹⁺²	1	9	yes	—
109	H(3,𝔽₃)	—	3	3	no	—
110	H(3,𝔽₃)	—	3	6	no	—
111	H(3,𝔽₃)	—	9	6	no	—
112	C₃×C₉	—	3	3	no	—
113	C₃×C₉	—	3	9	no	—
114	C₃×C₉	—	3	6	no	—
115	C₃×C₉	—	3	6	no	—
116	C₃×C₉	—	3	6	no	—
117	C₃×C₉	—	3	6	no	—
118	C₃×C₉	—	3	1	no	—
119	C₃×C₉	—	3	2	no	—
120	C₃×C₉	—	3	3	no	—
121	C₃×C₉	—	3	1	no	—
122	C₃×C₉	—	3	2	no	—
123	C₃×C₉	—	3	2	no	—
124	C₃×C₉	—	3	4	no	—
125	C₃×C₉	—	9	3	no	—
126	C₃×C₉	—	9	3	no	—
127	C₃×C₉	—	9	6	no	—
128	C₃×C₉	—	9	3	no	—
129	C₃×C₉	—	9	6	no	—
130	C₃×C₉	—	27	2	no	—
131	C₃×C₉	—	27	2	no	—
132	C₃×C₉	—	27	2	no	—
133	C₃×C₉	C₃×H(3,𝔽₃)	1	9	yes	—
134	C₃²	—	3	1	no	—
135	C₃²	—	3	2	no	—
136	C₃²	—	3	1	no	—
137	C₃²	—	3	2	no	—
138	C₃²	—	3	2	no	—
139	C₃²	—	3	2	no	—
140	C₃²	—	3	2	no	—
141	C₃²	—	9	3	no	—
142	C₃²	—	9	6	no	—
143	C₃²	—	9	3	no	—
144	C₃²	—	9	3	no	—
145	C₃²	—	9	6	no	—
146	C₃²	—	27	9	no	—
147	C₃²	C₉⋊H(3,𝔽₃)	1	3	yes	—
148	C₃²	C₉⋊H(3,𝔽₃)	1	6	yes	—
149	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
150	C₃²	C₃²×3₋¹⁺²	1	1	yes	characteristic
151	C₃²	C₃×#737	1	2	yes	—
152	C₉	—	3	9	no	—
153	C₉	—	3	9	no	—
154	C₉	—	9	1	no	—
155	C₉	—	9	2	no	—
156	C₉	—	9	3	no	—
157	C₉	—	9	6	no	—
158	C₉	—	9	6	no	—
159	C₉	—	9	6	no	—
160	C₉	—	9	6	no	—
161	C₃	—	3	3	no	—
162	C₃	—	3	1	no	—
163	C₃	—	3	2	no	—
164	C₃	—	3	6	no	—
165	C₃	—	27	3	no	—
166	C₃	C₃×C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
167	C₃	#10624931	1	1	yes	characteristic
168	C₃	#10624935	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. #10624886	C₃	semidirect
4. C₃×#6895	C₃	semidirect
5. #10624863	C₃	semidirect
6. #10624803	C₃	semidirect
7. #10624800	C₃	semidirect
8. #10624699	C₃	semidirect
9. #10624699	C₃	semidirect
11. C₃×#6867	C₃	semidirect
23. #6894	C₃²	semidirect
28. C₉×3₋¹⁺²	C₃²	semidirect
30. C₃×C₉⋊C₉	C₃²	semidirect
31. C₃×C₉⋊C₉	C₃²	semidirect
37. 3₋¹⁺²⋊C₉	C₃²	semidirect
38. 3₋¹⁺²⋊C₉	C₃²	semidirect
40. #6867	C₃²	semidirect
41. #6867	C₃²	semidirect
50. C₃×3₋¹⁺²	3₋¹⁺²	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#731	3
#10624886	3
#10624883	3
C₃×#6895	3
#10624863	3
#10624803	3
#10624800	3
#10624699	3
#10624698	3
C₃×#6867	3