Finite Group Databasebeta

Subgroups — #14548450

Order 2187.

360 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 50 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	#10624897	C₃	1	1	yes	maximal, characteristic
2	#10624744	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624738	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624673	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624671	C₃	1	1	yes	maximal, characteristic, complemented
6	#10624659	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624648	C₃	1	1	yes	maximal, characteristic, complemented
8	#10624646	C₃	1	1	yes	maximal, characteristic, complemented
9	#10624632	C₃	1	1	yes	maximal, characteristic
10	#10624614	C₃	1	1	yes	maximal, characteristic, complemented
11	#10624602	C₃	1	1	yes	maximal, characteristic
12	#10624594	C₃	1	1	yes	maximal, characteristic, complemented
13	#10624590	C₃	1	1	yes	maximal, characteristic
14	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
15	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
16	C₃×#733	—	3	3	no	—
17	C₃×#733	—	3	3	no	—
18	C₉⋊H(3,𝔽₃)	—	3	3	no	—
19	C₉⋊H(3,𝔽₃)	—	3	3	no	—
20	C₉⋊H(3,𝔽₃)	—	3	2	no	—
21	#6891	—	3	2	no	—
22	#6891	C₃²	1	1	yes	characteristic
23	#6891	C₃²	1	1	yes	characteristic
24	#6891	C₃²	1	1	yes	characteristic
25	#6891	C₃²	1	1	yes	characteristic
26	#6891	C₃²	1	1	yes	characteristic
27	#6891	C₃²	1	1	yes	characteristic
28	#6891	C₃²	1	1	yes	characteristic
29	#6891	C₃²	1	1	yes	characteristic
30	#6891	C₃²	1	1	yes	characteristic
31	C₉×H(3,𝔽₃)	—	3	1	no	—
32	C₃×C₃²⋊C₉	—	3	1	no	—
33	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
34	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
35	#6861	—	3	1	no	—
36	#6861	—	3	2	no	—
37	#6861	—	3	1	no	—
38	#6861	—	3	2	no	—
39	#6860	—	3	3	no	—
40	#6860	—	3	3	no	—
41	#6860	—	3	1	no	—
42	#6860	—	3	2	no	—
43	#6859	—	3	1	no	—
44	#6859	—	3	2	no	—
45	#6859	—	3	1	no	—
46	#6859	—	3	2	no	—
47	#6858	—	3	3	no	—
48	#6858	—	3	3	no	—
49	#6857	—	3	1	no	—
50	#6857	—	3	2	no	—
51	C₃⁴	—	3	1	no	—
52	C₃⁴	C₃³	1	1	yes	characteristic
53	C₃×3₋¹⁺²	—	3	6	no	—
54	C₃×3₋¹⁺²	—	3	1	no	—
55	C₃×3₋¹⁺²	—	3	1	no	—
56	C₃×3₋¹⁺²	—	3	1	no	—
57	C₃×3₋¹⁺²	—	3	1	no	—
58	C₃×3₋¹⁺²	—	3	1	no	—
59	C₃×3₋¹⁺²	—	3	1	no	—
60	C₃×3₋¹⁺²	—	3	1	no	—
61	C₃×3₋¹⁺²	—	3	1	no	—
62	C₃×3₋¹⁺²	—	3	1	no	—
63	C₃×3₋¹⁺²	—	3	1	no	—
64	C₃×3₋¹⁺²	—	9	3	no	—
65	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
66	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
67	C₃×H(3,𝔽₃)	—	3	1	no	—
68	C₃×H(3,𝔽₃)	—	3	1	no	—
69	C₃×H(3,𝔽₃)	—	9	3	no	—
70	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	1	yes	characteristic
71	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	2	yes	—
72	C₃²×C₉	—	3	1	no	—
73	C₃²×C₉	—	3	3	no	—
74	C₃²×C₉	—	3	1	no	—
75	C₃²×C₉	—	3	1	no	—
76	#733	—	9	9	no	—
77	#733	—	9	9	no	—
78	C₃²⋊C₉	—	3	6	no	—
79	C₃²⋊C₉	—	3	6	no	—
80	C₃²⋊C₉	—	3	6	no	—
81	C₃²⋊C₉	—	3	1	no	—
82	C₃²⋊C₉	—	3	1	no	—
83	C₃²⋊C₉	—	3	1	no	—
84	C₃²⋊C₉	—	3	1	no	—
85	C₃²⋊C₉	—	3	1	no	—
86	C₃²⋊C₉	—	3	1	no	—
87	C₃²⋊C₉	—	3	1	no	—
88	C₃²⋊C₉	—	3	1	no	—
89	C₃²⋊C₉	—	3	1	no	—
90	C₃²⋊C₉	—	3	1	no	—
91	C₃²⋊C₉	—	3	1	no	—
92	C₃²⋊C₉	—	3	1	no	—
93	C₃²⋊C₉	—	3	1	no	—
94	C₃²⋊C₉	—	3	1	no	—
95	C₃²⋊C₉	—	3	1	no	—
96	C₃²⋊C₉	—	3	1	no	—
97	C₃²⋊C₉	—	3	1	no	—
98	C₃²⋊C₉	—	3	1	no	—
99	C₃²⋊C₉	—	3	1	no	—
100	C₃²⋊C₉	—	3	1	no	—
101	C₃²⋊C₉	—	3	1	no	—
102	C₃²⋊C₉	—	3	1	no	—
103	C₃²⋊C₉	—	3	1	no	—
104	C₃²⋊C₉	—	3	1	no	—
105	C₃²⋊C₉	—	3	1	no	—
106	C₃²⋊C₉	—	3	1	no	—
107	C₃²⋊C₉	—	3	1	no	—
108	C₃²⋊C₉	—	3	1	no	—
109	C₃²⋊C₉	—	3	1	no	—
110	C₃²⋊C₉	—	3	1	no	—
111	C₃²⋊C₉	—	3	1	no	—
112	C₃²⋊C₉	—	3	1	no	—
113	C₃²⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
114	C₃²⋊C₉	H(3,𝔽₃)	1	2	yes	—
115	C₃³	—	3	1	no	—
116	C₃³	—	3	2	no	—
117	C₃³	—	9	1	no	—
118	C₃³	—	9	1	no	—
119	C₃³	—	9	1	no	—
120	C₃³	—	9	1	no	—
121	C₃³	—	9	3	no	—
122	C₃³	—	9	3	no	—
123	C₃³	—	27	1	no	—
124	C₃³	—	27	1	no	—
125	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
126	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
127	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
128	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
129	3₋¹⁺²	—	3	3	no	—
130	3₋¹⁺²	—	9	2	no	—
131	3₋¹⁺²	—	9	3	no	—
132	3₋¹⁺²	—	9	3	no	—
133	3₋¹⁺²	—	9	3	no	—
134	3₋¹⁺²	—	9	3	no	—
135	3₋¹⁺²	—	9	3	no	—
136	3₋¹⁺²	—	9	3	no	—
137	3₋¹⁺²	—	9	3	no	—
138	3₋¹⁺²	—	27	6	no	—
139	3₋¹⁺²	—	27	1	no	—
140	3₋¹⁺²	—	27	1	no	—
141	3₋¹⁺²	—	27	1	no	—
142	3₋¹⁺²	—	27	1	no	—
143	H(3,𝔽₃)	—	3	3	no	—
144	H(3,𝔽₃)	—	9	3	no	—
145	H(3,𝔽₃)	—	9	3	no	—
146	H(3,𝔽₃)	—	9	2	no	—
147	C₃×C₉	—	3	3	no	—
148	C₃×C₉	—	3	3	no	—
149	C₃×C₉	—	3	3	no	—
150	C₃×C₉	—	3	1	no	—
151	C₃×C₉	—	3	2	no	—
152	C₃×C₉	—	9	2	no	—
153	C₃×C₉	—	9	1	no	—
154	C₃×C₉	—	9	3	no	—
155	C₃×C₉	—	9	1	no	—
156	C₃×C₉	—	9	1	no	—
157	C₃×C₉	—	9	1	no	—
158	C₃×C₉	—	9	1	no	—
159	C₃×C₉	—	9	1	no	—
160	C₃×C₉	—	9	1	no	—
161	C₃×C₉	—	9	1	no	—
162	C₃×C₉	—	9	1	no	—
163	C₃×C₉	—	27	1	no	—
164	C₃×C₉	—	27	3	no	—
165	C₃²	—	3	1	no	—
166	C₃²	—	3	1	no	—
167	C₃²	—	3	1	no	—
168	C₃²	—	3	1	no	—
169	C₃²	—	3	1	no	—
170	C₃²	—	3	3	no	—
171	C₃²	—	3	1	no	—
172	C₃²	—	3	1	no	—
173	C₃²	—	3	1	no	—
174	C₃²	—	3	1	no	—
175	C₃²	—	3	1	no	—
176	C₃²	—	3	1	no	—
177	C₃²	—	3	1	no	—
178	C₃²	—	3	1	no	—
179	C₃²	—	3	1	no	—
180	C₃²	—	3	1	no	—
181	C₃²	—	3	1	no	—
182	C₃²	—	9	1	no	—
183	C₃²	—	9	1	no	—
184	C₃²	—	9	1	no	—
185	C₃²	—	9	2	no	—
186	C₃²	—	9	2	no	—
187	C₃²	—	9	2	no	—
188	C₃²	—	9	2	no	—
189	C₃²	—	27	3	no	—
190	C₃²	—	27	9	no	—
191	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
192	C₉	—	9	3	no	—
193	C₉	—	9	1	no	—
194	C₉	—	9	3	no	—
195	C₉	—	9	3	no	—
196	C₉	—	9	2	no	—
197	C₉	—	27	1	no	—
198	C₉	—	27	1	no	—
199	C₉	—	27	1	no	—
200	C₉	—	27	1	no	—
201	C₉	—	27	1	no	—
202	C₉	—	27	1	no	—
203	C₉	—	27	1	no	—
204	C₉	—	27	1	no	—
205	C₃	—	9	1	no	—
206	C₃	—	9	1	no	—
207	C₃	—	9	1	no	—
208	C₃	—	9	1	no	—
209	C₃	—	9	3	no	—
210	C₃	—	27	2	no	—
211	C₃	UT(4,𝔽₃)	1	1	yes	characteristic
212	C₃	#10624755	1	1	yes	characteristic
213	C₃	#10624758	1	1	yes	characteristic
214	C₃	#10624764	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. #10624744	C₃	semidirect
3. #10624738	C₃	semidirect
4. #10624673	C₃	semidirect
5. #10624671	C₃	semidirect
6. #10624659	C₃	semidirect
7. #10624648	C₃	semidirect
8. #10624646	C₃	semidirect
10. #10624614	C₃	semidirect
12. #10624594	C₃	semidirect

Maximal subgroups

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

subgroup	index
#10624897	3
#10624744	3
#10624738	3
#10624673	3
#10624671	3
#10624659	3
#10624648	3
#10624646	3
#10624632	3
#10624614	3
#10624602	3
#10624594	3
#10624590	3