Finite Group Databasebeta

Subgroups — #14552103

Order 2187.

1070 conjugacy classes of subgroups; only the 458 normal subgroups are shown.

normal subgroups only

This group has 1070 conjugacy classes of subgroups — too many to draw. Only its 458 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	C₃×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	C₃×C₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	#10624560	C₃	1	6	yes	maximal, complemented
4	#10624558	C₃	1	3	yes	maximal, complemented
5	#10624557	C₃	1	12	yes	maximal, complemented
6	#10624556	C₃	1	4	yes	maximal, complemented
7	#10624556	C₃	1	6	yes	maximal, complemented
8	#10624555	C₃	1	2	yes	maximal
9	#10624548	C₃	1	2	yes	maximal, complemented
10	C₉×C₃²⋊C₉	C₃	1	2	yes	maximal, complemented
11	C₃×#6855	C₃	1	1	yes	maximal, characteristic, complemented
12	C₃²×3₋¹⁺²	C₃²	1	3	yes	—
13	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
14	C₃³×C₉	C₃²	1	2	yes	—
15	C₃³×C₉	C₃²	1	3	yes	—
16	C₃³×C₉	C₃²	1	1	yes	characteristic
17	C₃³×C₉	C₃²	1	1	yes	characteristic
18	C₉⋊H(3,𝔽₃)	C₉	1	27	yes	complemented
19	C₉×H(3,𝔽₃)	—	3	9	no	—
20	C₃×C₉⋊C₉	C₃²	1	6	yes	—
21	C₃×C₉⋊C₉	C₃²	1	12	yes	—
22	C₃×C₃²⋊C₉	C₃²	1	4	yes	—
23	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
24	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
25	C₃×C₃²⋊C₉	C₃²	1	12	yes	—
26	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
27	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
28	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
29	C₃×C₉²	C₃²	1	6	yes	—
30	#6855	C₃²	1	2	yes	—
31	#6855	C₃²	1	4	yes	—
32	#6855	C₃²	1	1	yes	characteristic
33	#6855	C₃²	1	2	yes	—
34	#6855	C₃²	1	6	yes	—
35	#6855	C₃²	1	6	yes	—
36	#6855	C₃²	1	6	yes	—
37	#6855	C₃²	1	6	yes	—
38	#6855	C₃²	1	12	yes	—
39	#6855	C₃²	1	12	yes	—
40	C₃⁴	C₃³	1	3	yes	—
41	C₃⁴	C₃³	1	1	yes	characteristic
42	C₃×3₋¹⁺²	—	9	3	no	—
43	C₃×3₋¹⁺²	—	9	6	no	—
44	C₃×3₋¹⁺²	C₃×C₉	1	27	yes	complemented
45	C₃×H(3,𝔽₃)	—	3	3	no	—
46	C₃×H(3,𝔽₃)	—	3	6	no	—
47	C₃×H(3,𝔽₃)	C₃×C₉	1	9	yes	—
48	C₃²×C₉	—	3	6	no	—
49	C₃²×C₉	—	3	9	no	—
50	C₃²×C₉	—	9	1	no	—
51	C₃²×C₉	—	9	1	no	—
52	C₃²×C₉	—	9	3	no	—
53	C₃²×C₉	—	9	2	no	—
54	C₃²×C₉	—	9	6	no	—
55	C₃²×C₉	—	9	2	no	—
56	C₃²×C₉	—	9	2	no	—
57	C₃²×C₉	—	9	4	no	—
58	C₃²×C₉	C₃×C₉	1	9	yes	complemented
59	C₃²×C₉	3₋¹⁺²	1	9	yes	complemented
60	C₃²×C₉	C₃³	1	2	yes	—
61	C₃²×C₉	C₃³	1	6	yes	—
62	C₃²×C₉	C₃³	1	4	yes	—
63	C₃²×C₉	C₃³	1	12	yes	—
64	C₃²×C₉	C₃³	1	1	yes	characteristic
65	C₃²×C₉	C₃³	1	2	yes	—
66	C₃²×C₉	C₃³	1	6	yes	—
67	C₃²×C₉	C₃³	1	1	yes	characteristic
68	C₃²×C₉	C₃³	1	2	yes	—
69	C₉⋊C₉	—	9	12	no	—
70	C₉⋊C₉	—	9	6	no	—
71	C₃²⋊C₉	—	3	6	no	—
72	C₃²⋊C₉	—	3	12	no	—
73	C₃²⋊C₉	—	3	18	no	—
74	C₃²⋊C₉	—	9	6	no	—
75	C₃²⋊C₉	—	9	12	no	—
76	C₃²⋊C₉	C₃×C₉	1	54	yes	complemented
77	C₃²⋊C₉	C₃×C₉	1	18	yes	complemented
78	C₉²	—	9	6	no	—
79	C₃³	—	3	1	no	—
80	C₃³	—	3	2	no	—
81	C₃³	—	3	2	no	—
82	C₃³	—	3	4	no	—
83	C₃³	—	3	3	no	—
84	C₃³	—	3	6	no	—
85	C₃³	—	3	3	no	—
86	C₃³	—	3	12	no	—
87	C₃³	—	3	6	no	—
88	C₃³	—	3	6	no	—
89	C₃³	C₃²×C₉	1	9	yes	—
90	C₃³	C₃²×C₉	1	3	yes	—
91	C₃³	C₃×3₋¹⁺²	1	3	yes	—
92	C₃³	#737	1	6	yes	—
93	C₃³	C₃⁴	1	1	yes	characteristic
94	3₋¹⁺²	—	9	27	no	—
95	H(3,𝔽₃)	—	3	27	no	—
96	C₃×C₉	—	3	2	no	—
97	C₃×C₉	—	3	6	no	—
98	C₃×C₉	—	3	2	no	—
99	C₃×C₉	—	3	1	no	—
100	C₃×C₉	—	3	1	no	—
101	C₃×C₉	—	3	2	no	—
102	C₃×C₉	—	3	4	no	—
103	C₃×C₉	—	3	4	no	—
104	C₃×C₉	—	3	2	no	—
105	C₃×C₉	—	3	12	no	—
106	C₃×C₉	—	3	6	no	—
107	C₃×C₉	—	3	12	no	—
108	C₃×C₉	—	3	2	no	—
109	C₃×C₉	—	3	4	no	—
110	C₃×C₉	—	3	12	no	—
111	C₃×C₉	—	3	2	no	—
112	C₃×C₉	—	3	6	no	—
113	C₃×C₉	—	3	4	no	—
114	C₃×C₉	—	3	4	no	—
115	C₃×C₉	—	3	24	no	—
116	C₃×C₉	—	3	12	no	—
117	C₃×C₉	—	3	4	no	—
118	C₃×C₉	—	3	4	no	—
119	C₃×C₉	—	9	9	no	—
120	C₃×C₉	—	9	9	no	—
121	C₃×C₉	—	9	27	no	—
122	C₃×C₉	—	9	18	no	—
123	C₃×C₉	C₃²×C₉	1	6	yes	—
124	C₃×C₉	C₃²×C₉	1	18	yes	—
125	C₃×C₉	C₃²×C₉	1	3	yes	—
126	C₃×C₉	C₃×3₋¹⁺²	1	6	yes	—
127	C₃×C₉	C₃×3₋¹⁺²	1	3	yes	—
128	C₃²	—	3	3	no	—
129	C₃²	—	3	6	no	—
130	C₃²	—	3	3	no	—
131	C₃²	—	3	6	no	—
132	C₃²	—	3	6	no	—
133	C₃²	—	3	12	no	—
134	C₃²	—	3	9	no	—
135	C₃²	—	3	9	no	—
136	C₃²	—	3	18	no	—
137	C₃²	—	9	3	no	—
138	C₃²	—	9	6	no	—
139	C₃²	—	9	6	no	—
140	C₃²	—	9	12	no	—
141	C₃²	C₉×3₋¹⁺²	1	9	yes	—
142	C₃²	C₃³×C₉	1	1	yes	characteristic
143	C₃²	C₃²×3₋¹⁺²	1	1	yes	characteristic
144	C₃²	C₃×#737	1	2	yes	—
145	C₃²	H(5,𝔽₃)	1	1	yes	characteristic
146	C₃²	3₋¹⁺⁴	1	2	yes	—
147	C₃²	3₋¹⁺⁴	1	2	yes	—
148	C₃²	3₋¹⁺⁴	1	4	yes	—
149	C₉	—	3	6	no	—
150	C₉	—	3	3	no	—
151	C₉	—	3	18	no	—
152	C₉	—	3	3	no	—
153	C₉	—	3	6	no	—
154	C₉	—	9	6	no	—
155	C₉	—	9	12	no	—
156	C₉	—	9	2	no	—
157	C₉	—	9	4	no	—
158	C₃	—	3	9	no	—
159	C₃	—	9	9	no	—
160	C₃	C₃×C₉×3₋¹⁺²	1	1	yes	characteristic
161	C₃	#10624862	1	1	yes	characteristic
162	C₃	#10624864	1	2	yes	—
163	C₃	#10624883	1	1	yes	characteristic
164	C₃	#10624887	1	2	yes	—
165	C₃	#10624888	1	2	yes	—
166	C₃	#10624891	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
3. #10624560	C₃	semidirect
4. #10624558	C₃	semidirect
5. #10624557	C₃	semidirect
6. #10624556	C₃	semidirect
7. #10624556	C₃	semidirect
9. #10624548	C₃	semidirect
10. C₉×C₃²⋊C₉	C₃	semidirect
11. C₃×#6855	C₃	semidirect
18. C₉⋊H(3,𝔽₃)	C₉	semidirect
44. C₃×3₋¹⁺²	C₃×C₉	semidirect
58. C₃²×C₉	C₃×C₉	semidirect
59. C₃²×C₉	3₋¹⁺²	semidirect
76. C₃²⋊C₉	C₃×C₉	semidirect
77. C₃²⋊C₉	C₃×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉⋊H(3,𝔽₃)	3
C₃×C₉×H(3,𝔽₃)	3
#10624560	3
#10624558	3
#10624557	3
#10624556	3
#10624555	3
#10624548	3
C₉×C₃²⋊C₉	3
C₃×#6855	3