Finite Group Databasebeta

Subgroups — #14550693

Order 2187.

522 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 80 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×#6909	C₃	1	1	yes	maximal, characteristic
2	#10624883	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624699	C₃	1	2	yes	maximal, complemented
4	#10624699	C₃	1	2	yes	maximal, complemented
5	#10624698	C₃	1	1	yes	maximal, characteristic, complemented
6	#10624698	C₃	1	2	yes	maximal, complemented
7	C₃×H(3,𝔽₃)⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃×H(3,𝔽₃)⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
9	#10624558	C₃	1	2	yes	maximal, complemented
10	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
11	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
12	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
13	C₃³×C₉	C₃²	1	1	yes	characteristic
14	#6909	—	3	6	no	—
15	#6909	—	3	3	no	—
16	C₃×AΣL(1,𝔽₂₇)	—	3	3	no	—
17	C₉⋊H(3,𝔽₃)	—	3	3	no	—
18	C₉⋊H(3,𝔽₃)	—	3	6	no	—
19	#6891	—	3	3	no	—
20	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
21	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
22	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
23	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
24	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
25	3₋¹⁺²⋊C₉	—	3	6	no	—
26	3₋¹⁺²⋊C₉	—	3	6	no	—
27	H(3,𝔽₃)⋊C₉	—	3	3	no	—
28	H(3,𝔽₃)⋊C₉	—	3	3	no	—
29	H(3,𝔽₃)⋊C₉	—	3	6	no	—
30	H(3,𝔽₃)⋊C₉	—	3	3	no	—
31	#6855	—	3	6	no	—
32	C₃⁴	—	3	1	no	—
33	C₃⁴	—	3	1	no	—
34	C₃⁴	C₃³	1	1	yes	characteristic
35	C₃×3₋¹⁺²	—	3	2	no	—
36	C₃×3₋¹⁺²	—	3	2	no	—
37	C₃×3₋¹⁺²	—	3	9	no	—
38	C₃×3₋¹⁺²	—	3	2	no	—
39	C₃×3₋¹⁺²	—	3	2	no	—
40	C₃×3₋¹⁺²	—	3	2	no	—
41	C₃×3₋¹⁺²	—	3	2	no	—
42	C₃×3₋¹⁺²	—	3	2	no	—
43	C₃×3₋¹⁺²	—	3	2	no	—
44	C₃×3₋¹⁺²	—	3	2	no	—
45	C₃×3₋¹⁺²	—	3	2	no	—
46	C₃×3₋¹⁺²	—	3	2	no	—
47	C₃×3₋¹⁺²	3₋¹⁺²	1	6	yes	complemented
48	C₃×H(3,𝔽₃)	—	3	2	no	—
49	C₃×H(3,𝔽₃)	—	3	1	no	—
50	C₃×H(3,𝔽₃)	—	3	2	no	—
51	C₃×H(3,𝔽₃)	—	3	1	no	—
52	C₃×H(3,𝔽₃)	—	3	2	no	—
53	C₃×H(3,𝔽₃)	—	3	2	no	—
54	C₃×H(3,𝔽₃)	—	3	1	no	—
55	C₃×H(3,𝔽₃)	—	3	1	no	—
56	C₃×H(3,𝔽₃)	—	3	1	no	—
57	C₃×H(3,𝔽₃)	—	3	2	no	—
58	C₃×H(3,𝔽₃)	—	3	2	no	—
59	C₃×H(3,𝔽₃)	—	3	1	no	—
60	C₃×H(3,𝔽₃)	—	3	2	no	—
61	C₃×H(3,𝔽₃)	—	3	2	no	—
62	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
63	C₃×H(3,𝔽₃)	3₋¹⁺²	1	3	yes	complemented
64	C₃²×C₉	—	3	2	no	—
65	C₃²×C₉	—	3	2	no	—
66	C₃²×C₉	—	3	1	no	—
67	C₃²×C₉	—	3	1	no	—
68	C₃²×C₉	—	3	1	no	—
69	C₃²×C₉	—	3	2	no	—
70	C₃²×C₉	—	3	1	no	—
71	C₃²×C₉	—	3	2	no	—
72	C₃²×C₉	—	9	3	no	—
73	C₃²×C₉	H(3,𝔽₃)	1	6	yes	—
74	AΣL(1,𝔽₂₇)	—	3	27	no	—
75	C₃²⋊C₉	—	3	18	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	2	no	—
83	C₃²⋊C₉	—	3	2	no	—
84	C₃²⋊C₉	—	3	1	no	—
85	C₃²⋊C₉	—	3	2	no	—
86	C₃²⋊C₉	—	3	2	no	—
87	C₃²⋊C₉	—	3	2	no	—
88	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
89	C₃³	—	3	3	no	—
90	C₃³	—	3	3	no	—
91	C₃³	—	9	1	no	—
92	C₃³	—	9	1	no	—
93	C₃³	—	9	1	no	—
94	C₃³	—	9	1	no	—
95	C₃³	—	9	1	no	—
96	C₃³	—	9	2	no	—
97	C₃³	—	9	1	no	—
98	C₃³	—	9	1	no	—
99	C₃³	—	27	1	no	—
100	C₃³	—	27	1	no	—
101	C₃³	—	27	1	no	—
102	C₃³	—	27	1	no	—
103	C₃³	—	27	1	no	—
104	C₃³	—	27	1	no	—
105	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
106	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
107	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
108	C₃³	C₃×3₋¹⁺²	1	2	yes	—
109	C₃³	C₃×3₋¹⁺²	1	1	yes	characteristic
110	C₃³	#737	1	1	yes	characteristic
111	C₃³	#737	1	2	yes	—
112	C₃³	#737	1	2	yes	—
113	C₃³	#737	1	1	yes	characteristic
114	3₋¹⁺²	—	3	6	no	—
115	3₋¹⁺²	—	3	6	no	—
116	3₋¹⁺²	—	3	6	no	—
117	3₋¹⁺²	—	9	2	no	—
118	3₋¹⁺²	—	9	2	no	—
119	3₋¹⁺²	—	9	2	no	—
120	3₋¹⁺²	—	9	2	no	—
121	3₋¹⁺²	—	9	2	no	—
122	3₋¹⁺²	—	9	2	no	—
123	3₋¹⁺²	—	27	9	no	—
124	H(3,𝔽₃)	—	3	3	no	—
125	H(3,𝔽₃)	—	3	6	no	—
126	H(3,𝔽₃)	—	3	3	no	—
127	H(3,𝔽₃)	—	3	3	no	—
128	H(3,𝔽₃)	—	3	3	no	—
129	H(3,𝔽₃)	—	9	2	no	—
130	H(3,𝔽₃)	—	9	2	no	—
131	H(3,𝔽₃)	—	9	2	no	—
132	H(3,𝔽₃)	—	9	2	no	—
133	H(3,𝔽₃)	—	9	2	no	—
134	H(3,𝔽₃)	—	9	2	no	—
135	C₃×C₉	—	3	6	no	—
136	C₃×C₉	—	3	6	no	—
137	C₃×C₉	—	3	3	no	—
138	C₃×C₉	—	3	6	no	—
139	C₃×C₉	—	9	4	no	—
140	C₃×C₉	—	9	1	no	—
141	C₃×C₉	—	9	2	no	—
142	C₃×C₉	—	9	2	no	—
143	C₃×C₉	—	9	2	no	—
144	C₃×C₉	—	9	3	no	—
145	C₃×C₉	—	9	3	no	—
146	C₃×C₉	—	9	6	no	—
147	C₃×C₉	—	9	6	no	—
148	C₃×C₉	—	9	1	no	—
149	C₃×C₉	—	9	2	no	—
150	C₃×C₉	—	9	2	no	—
151	C₃×C₉	—	27	1	no	—
152	C₃×C₉	—	27	1	no	—
153	C₃×C₉	—	27	2	no	—
154	C₃²	—	3	2	no	—
155	C₃²	—	3	1	no	—
156	C₃²	—	3	1	no	—
157	C₃²	—	3	2	no	—
158	C₃²	—	3	1	no	—
159	C₃²	—	3	1	no	—
160	C₃²	—	3	2	no	—
161	C₃²	—	3	1	no	—
162	C₃²	—	3	1	no	—
163	C₃²	—	9	2	no	—
164	C₃²	—	9	1	no	—
165	C₃²	—	9	1	no	—
166	C₃²	—	9	1	no	—
167	C₃²	—	9	2	no	—
168	C₃²	—	9	2	no	—
169	C₃²	—	9	3	no	—
170	C₃²	—	9	3	no	—
171	C₃²	—	9	3	no	—
172	C₃²	—	9	3	no	—
173	C₃²	—	9	3	no	—
174	C₃²	—	27	3	no	—
175	C₃²	—	27	3	no	—
176	C₃²	—	27	6	no	—
177	C₃²	—	27	3	no	—
178	C₃²	—	27	3	no	—
179	C₃²	—	27	1	no	—
180	C₃²	—	27	1	no	—
181	C₃²	—	27	1	no	—
182	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
183	C₃²	#6891	1	1	yes	characteristic
184	C₃²	C₉⋊H(3,𝔽₃)	1	2	yes	—
185	C₃²	C₉⋊H(3,𝔽₃)	1	2	yes	—
186	C₃²	C₉⋊H(3,𝔽₃)	1	2	yes	—
187	C₃²	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
188	C₃²	C₉⋊H(3,𝔽₃)	1	2	yes	—
189	C₃²	#6893	1	1	yes	characteristic
190	C₃²	#6893	1	1	yes	characteristic
191	C₉	—	9	6	no	—
192	C₉	—	9	6	no	—
193	C₉	—	9	3	no	—
194	C₉	—	9	3	no	—
195	C₉	—	27	1	no	—
196	C₉	—	27	2	no	—
197	C₉	—	27	2	no	—
198	C₃	—	3	1	no	—
199	C₃	—	3	1	no	—
200	C₃	—	3	2	no	—
201	C₃	—	3	1	no	—
202	C₃	—	3	1	no	—
203	C₃	—	3	2	no	—
204	C₃	—	3	2	no	—
205	C₃	—	3	2	no	—
206	C₃	—	27	3	no	—
207	C₃	—	27	3	no	—
208	C₃	#10624563	1	1	yes	characteristic
209	C₃	#10624748	1	1	yes	characteristic
210	C₃	#10624752	1	1	yes	characteristic
211	C₃	#10624753	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	C₃×#6909	C₃
LCS	G ▷ C₃⁴ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #10624883	C₃	semidirect
3. #10624699	C₃	semidirect
4. #10624699	C₃	semidirect
5. #10624698	C₃	semidirect
6. #10624698	C₃	semidirect
7. C₃×H(3,𝔽₃)⋊C₉	C₃	semidirect
8. C₃×H(3,𝔽₃)⋊C₉	C₃	semidirect
9. #10624558	C₃	semidirect
47. C₃×3₋¹⁺²	3₋¹⁺²	semidirect
63. C₃×H(3,𝔽₃)	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
C₃×#6909	3
#10624883	3
#10624699	3
#10624698	3
C₃×H(3,𝔽₃)⋊C₉	3
#10624558	3