Finite Group Databasebeta

Subgroups — #14547231

Order 2187.

341 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 56 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×#6895	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624826	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624825	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624812	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624812	C₃	1	1	yes	maximal, characteristic, complemented
6	#10624805	C₃	1	1	yes	maximal, characteristic
7	C₃×H(3,𝔽₃)⋊C₉	C₃	1	1	yes	maximal, characteristic
8	#10624619	C₃	1	1	yes	maximal, characteristic, complemented
9	#10624615	C₃	1	1	yes	maximal, characteristic, complemented
10	#10624605	C₃	1	1	yes	maximal, characteristic, complemented
11	#10624602	C₃	1	1	yes	maximal, characteristic
12	#10624600	C₃	1	1	yes	maximal, characteristic, complemented
13	#10624593	C₃	1	1	yes	maximal, characteristic
14	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
15	C₃³×C₉	C₃²	1	1	yes	characteristic
16	#6895	—	3	3	no	—
17	#6895	—	3	3	no	—
18	#6895	—	3	3	no	—
19	#6891	C₃²	1	1	yes	characteristic
20	#6891	C₃²	1	1	yes	characteristic
21	#6891	C₃²	1	1	yes	characteristic
22	#6891	C₃²	1	1	yes	characteristic
23	#6891	C₃²	1	1	yes	characteristic
24	#6891	C₃²	1	1	yes	characteristic
25	C₃×C₉⋊C₉	—	3	3	no	—
26	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
27	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
28	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
29	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
30	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
31	3₋¹⁺²⋊C₉	—	3	3	no	—
32	3₋¹⁺²⋊C₉	—	3	3	no	—
33	3₋¹⁺²⋊C₉	—	3	3	no	—
34	3₋¹⁺²⋊C₉	—	3	3	no	—
35	H(3,𝔽₃)⋊C₉	—	3	3	no	—
36	#6866	—	3	3	no	—
37	#6862	—	3	3	no	—
38	#6861	—	3	3	no	—
39	#6859	—	3	3	no	—
40	#6859	—	3	3	no	—
41	#6858	—	3	3	no	—
42	#6858	—	3	3	no	—
43	C₃⁴	—	3	1	no	—
44	C₃⁴	C₃³	1	1	yes	characteristic
45	C₃×3₋¹⁺²	—	3	1	no	—
46	C₃×3₋¹⁺²	—	3	1	no	—
47	C₃×3₋¹⁺²	—	3	1	no	—
48	C₃×3₋¹⁺²	—	3	1	no	—
49	C₃×3₋¹⁺²	—	3	1	no	—
50	C₃×3₋¹⁺²	—	3	1	no	—
51	C₃×H(3,𝔽₃)	—	3	1	no	—
52	C₃×H(3,𝔽₃)	—	3	3	no	—
53	C₃×H(3,𝔽₃)	—	3	1	no	—
54	C₃×H(3,𝔽₃)	—	3	1	no	—
55	C₃×H(3,𝔽₃)	—	9	2	no	—
56	C₃²×C₉	—	3	1	no	—
57	C₃²×C₉	—	3	1	no	—
58	C₃²×C₉	—	3	1	no	—
59	C₃²×C₉	—	3	1	no	—
60	C₃²×C₉	—	3	1	no	—
61	C₃²×C₉	—	3	1	no	—
62	C₃²×C₉	—	3	1	no	—
63	C₃²×C₉	—	3	1	no	—
64	C₃²×C₉	—	9	3	no	—
65	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
66	C₉⋊C₉	—	3	27	no	—
67	C₃²⋊C₉	—	3	1	no	—
68	C₃²⋊C₉	—	3	1	no	—
69	C₃²⋊C₉	—	3	1	no	—
70	C₃²⋊C₉	—	3	1	no	—
71	C₃²⋊C₉	—	3	1	no	—
72	C₃²⋊C₉	—	3	1	no	—
73	C₃²⋊C₉	—	3	1	no	—
74	C₃²⋊C₉	—	3	1	no	—
75	C₃²⋊C₉	—	3	1	no	—
76	C₃²⋊C₉	—	3	1	no	—
77	C₃²⋊C₉	—	3	1	no	—
78	C₃²⋊C₉	—	3	1	no	—
79	C₃²⋊C₉	—	3	1	no	—
80	C₃²⋊C₉	—	3	1	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	H(3,𝔽₃)	1	3	yes	—
98	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
99	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
100	C₃³	—	3	3	no	—
101	C₃³	—	3	2	no	—
102	C₃³	—	3	1	no	—
103	C₃³	—	9	1	no	—
104	C₃³	—	9	1	no	—
105	C₃³	—	9	3	no	—
106	C₃³	—	9	1	no	—
107	C₃³	—	9	3	no	—
108	C₃³	—	27	1	no	—
109	C₃³	—	27	1	no	—
110	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
111	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
112	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
113	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
114	3₋¹⁺²	—	9	3	no	—
115	3₋¹⁺²	—	9	3	no	—
116	3₋¹⁺²	—	9	2	no	—
117	3₋¹⁺²	—	9	2	no	—
118	3₋¹⁺²	—	9	1	no	—
119	3₋¹⁺²	—	9	1	no	—
120	3₋¹⁺²	—	27	1	no	—
121	3₋¹⁺²	—	27	1	no	—
122	H(3,𝔽₃)	—	3	3	no	—
123	H(3,𝔽₃)	—	9	3	no	—
124	H(3,𝔽₃)	—	9	3	no	—
125	H(3,𝔽₃)	—	9	2	no	—
126	C₃×C₉	—	3	3	no	—
127	C₃×C₉	—	3	3	no	—
128	C₃×C₉	—	3	3	no	—
129	C₃×C₉	—	3	3	no	—
130	C₃×C₉	—	9	3	no	—
131	C₃×C₉	—	9	3	no	—
132	C₃×C₉	—	9	3	no	—
133	C₃×C₉	—	9	3	no	—
134	C₃×C₉	—	9	2	no	—
135	C₃×C₉	—	9	3	no	—
136	C₃×C₉	—	9	1	no	—
137	C₃×C₉	—	9	3	no	—
138	C₃×C₉	—	9	1	no	—
139	C₃×C₉	—	9	1	no	—
140	C₃×C₉	—	9	1	no	—
141	C₃×C₉	—	9	1	no	—
142	C₃×C₉	—	9	1	no	—
143	C₃×C₉	—	9	1	no	—
144	C₃×C₉	—	9	1	no	—
145	C₃×C₉	—	9	1	no	—
146	C₃×C₉	—	27	1	no	—
147	C₃×C₉	—	27	1	no	—
148	C₃²	—	3	1	no	—
149	C₃²	—	3	1	no	—
150	C₃²	—	3	1	no	—
151	C₃²	—	3	1	no	—
152	C₃²	—	3	1	no	—
153	C₃²	—	3	1	no	—
154	C₃²	—	3	1	no	—
155	C₃²	—	3	1	no	—
156	C₃²	—	3	1	no	—
157	C₃²	—	3	1	no	—
158	C₃²	—	3	1	no	—
159	C₃²	—	3	1	no	—
160	C₃²	—	3	1	no	—
161	C₃²	—	3	1	no	—
162	C₃²	—	9	6	no	—
163	C₃²	—	9	3	no	—
164	C₃²	—	9	3	no	—
165	C₃²	—	27	9	no	—
166	C₃²	—	27	1	no	—
167	C₃²	—	27	1	no	—
168	C₃²	—	27	1	no	—
169	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
170	C₃²	#6908	1	1	yes	characteristic
171	C₃²	#6913	1	3	yes	—
172	C₃²	#6913	1	2	yes	—
173	C₉	—	9	3	no	—
174	C₉	—	9	3	no	—
175	C₉	—	9	3	no	—
176	C₉	—	9	3	no	—
177	C₉	—	27	1	no	—
178	C₉	—	27	1	no	—
179	C₉	—	27	1	no	—
180	C₉	—	27	1	no	—
181	C₉	—	27	1	no	—
182	C₉	—	27	1	no	—
183	C₉	—	27	1	no	—
184	C₉	—	27	1	no	—
185	C₃	—	3	3	no	—
186	C₃	—	3	2	no	—
187	C₃	—	3	1	no	—
188	C₃	—	9	1	no	—
189	C₃	—	9	1	no	—
190	C₃	—	27	3	no	—
191	C₃	#10624739	1	1	yes	characteristic
192	C₃	#10624746	1	1	yes	characteristic
193	C₃	#10624756	1	1	yes	characteristic
194	C₃	#10624759	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
1. C₃×#6895	C₃	semidirect
2. #10624826	C₃	semidirect
3. #10624825	C₃	semidirect
4. #10624812	C₃	semidirect
5. #10624812	C₃	semidirect
8. #10624619	C₃	semidirect
9. #10624615	C₃	semidirect
10. #10624605	C₃	semidirect
12. #10624600	C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6895	3
#10624826	3
#10624825	3
#10624812	3
#10624805	3
C₃×H(3,𝔽₃)⋊C₉	3
#10624619	3
#10624615	3
#10624605	3
#10624602	3
#10624600	3
#10624593	3