Finite Group Databasebeta

Subgroups — #14550839

Order 2187.

413 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 59 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×#6908	C₃	1	1	yes	maximal, characteristic
2	#10624872	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624872	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624688	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624487	C₃	1	1	yes	maximal, characteristic, complemented
6	#10624487	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624487	C₃	1	1	yes	maximal, characteristic, complemented
8	#10624487	C₃	1	1	yes	maximal, characteristic, complemented
9	#10624487	C₃	1	1	yes	maximal, characteristic, complemented
10	#10624486	C₃	1	1	yes	maximal, characteristic, complemented
11	#10624485	C₃	1	1	yes	maximal, characteristic, complemented
12	#10624485	C₃	1	1	yes	maximal, characteristic, complemented
13	#10624484	C₃	1	1	yes	maximal, characteristic, complemented
14	C₃×#737	—	3	1	no	—
15	C₃³×C₉	C₃²	1	1	yes	characteristic
16	#6908	—	3	6	no	—
17	#6908	—	3	3	no	—
18	C₃×#733	C₃²	1	1	yes	characteristic
19	C₃×#733	C₃²	1	1	yes	characteristic
20	C₃×#732	C₃²	1	1	yes	characteristic
21	C₃×#731	—	3	1	no	—
22	C₃×AΣL(1,𝔽₂₇)	—	3	1	no	—
23	#6895	—	3	1	no	—
24	#6895	—	3	1	no	—
25	#6895	C₃²	1	1	yes	characteristic
26	#6895	C₃²	1	1	yes	characteristic
27	#6895	C₃²	1	1	yes	characteristic
28	#6895	C₃²	1	1	yes	characteristic
29	#6895	C₃²	1	1	yes	characteristic
30	#6895	C₃²	1	1	yes	characteristic
31	#6894	—	3	6	no	—
32	#6894	—	3	6	no	—
33	#6891	—	3	1	no	—
34	#6891	—	3	1	no	—
35	C₉×3₋¹⁺²	—	3	3	no	—
36	C₉×3₋¹⁺²	—	3	3	no	—
37	#6887	—	3	1	no	—
38	#6887	—	3	1	no	—
39	#6887	—	3	1	no	—
40	#6868	—	3	1	no	—
41	#6867	C₃²	1	1	yes	characteristic
42	#6867	C₃²	1	1	yes	characteristic
43	#6867	C₃²	1	1	yes	characteristic
44	#6866	—	3	1	no	—
45	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
46	#737	—	3	6	no	—
47	#737	—	3	3	no	—
48	#737	—	9	2	no	—
49	#737	—	9	2	no	—
50	#737	—	9	1	no	—
51	#737	—	9	1	no	—
52	C₃×3₋¹⁺²	—	3	3	no	—
53	C₃×3₋¹⁺²	—	3	3	no	—
54	C₃×3₋¹⁺²	—	3	1	no	—
55	C₃×3₋¹⁺²	—	3	1	no	—
56	C₃×3₋¹⁺²	—	9	2	no	—
57	C₃×H(3,𝔽₃)	—	3	1	no	—
58	C₃²×C₉	—	3	1	no	—
59	C₃²×C₉	—	3	1	no	—
60	C₃²×C₉	—	3	2	no	—
61	C₃²×C₉	—	3	2	no	—
62	C₃²×C₉	—	3	2	no	—
63	C₃²×C₉	—	3	1	no	—
64	C₃²×C₉	—	3	2	no	—
65	C₃²×C₉	—	3	1	no	—
66	C₃²×C₉	—	9	1	no	—
67	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
68	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
69	C₃²×C₉	C₃³	1	1	yes	characteristic
70	#733	—	3	3	no	—
71	#733	—	3	3	no	—
72	#732	—	3	3	no	—
73	#731	—	3	9	no	—
74	AΣL(1,𝔽₂₇)	—	3	9	no	—
75	C₉⋊C₉	—	3	3	no	—
76	C₉⋊C₉	—	3	3	no	—
77	C₉⋊C₉	—	3	3	no	—
78	C₉⋊C₉	—	3	3	no	—
79	C₉⋊C₉	—	3	3	no	—
80	C₉⋊C₉	—	3	3	no	—
81	C₉⋊C₉	—	3	9	no	—
82	C₉⋊C₉	—	3	9	no	—
83	C₉⋊C₉	—	3	6	no	—
84	C₉⋊C₉	—	3	6	no	—
85	C₉⋊C₉	—	9	2	no	—
86	C₃²⋊C₉	—	3	6	no	—
87	C₃²⋊C₉	—	3	6	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	C₉²	—	3	3	no	—
98	C₉²	—	3	3	no	—
99	C₉²	—	9	1	no	—
100	C₃³	—	3	1	no	—
101	C₃³	—	3	1	no	—
102	C₃³	—	3	1	no	—
103	C₃³	—	3	1	no	—
104	C₃³	—	9	3	no	—
105	C₃³	—	9	1	no	—
106	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
107	3₋¹⁺²	—	3	3	no	—
108	3₋¹⁺²	—	3	3	no	—
109	3₋¹⁺²	—	9	9	no	—
110	3₋¹⁺²	—	9	9	no	—
111	3₋¹⁺²	—	9	1	no	—
112	3₋¹⁺²	—	9	1	no	—
113	3₋¹⁺²	—	9	1	no	—
114	3₋¹⁺²	—	9	1	no	—
115	3₋¹⁺²	—	9	6	no	—
116	3₋¹⁺²	—	9	6	no	—
117	3₋¹⁺²	—	9	6	no	—
118	H(3,𝔽₃)	—	3	3	no	—
119	H(3,𝔽₃)	—	9	1	no	—
120	H(3,𝔽₃)	—	9	1	no	—
121	C₃×C₉	—	3	3	no	—
122	C₃×C₉	—	3	3	no	—
123	C₃×C₉	—	3	3	no	—
124	C₃×C₉	—	3	3	no	—
125	C₃×C₉	—	3	3	no	—
126	C₃×C₉	—	3	3	no	—
127	C₃×C₉	—	3	1	no	—
128	C₃×C₉	—	3	1	no	—
129	C₃×C₉	—	3	1	no	—
130	C₃×C₉	—	3	1	no	—
131	C₃×C₉	—	3	1	no	—
132	C₃×C₉	—	3	1	no	—
133	C₃×C₉	—	3	1	no	—
134	C₃×C₉	—	3	1	no	—
135	C₃×C₉	—	9	1	no	—
136	C₃×C₉	—	9	1	no	—
137	C₃×C₉	—	9	2	no	—
138	C₃×C₉	—	9	2	no	—
139	C₃×C₉	—	9	2	no	—
140	C₃×C₉	—	9	2	no	—
141	C₃×C₉	—	9	6	no	—
142	C₃×C₉	—	9	1	no	—
143	C₃×C₉	—	9	3	no	—
144	C₃×C₉	—	9	1	no	—
145	C₃×C₉	—	9	1	no	—
146	C₃×C₉	—	9	1	no	—
147	C₃×C₉	AΣL(1,𝔽₂₇)	1	2	yes	—
148	C₃×C₉	AΣL(1,𝔽₂₇)	1	2	yes	—
149	C₃×C₉	AΣL(1,𝔽₂₇)	1	2	yes	—
150	C₃×C₉	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
151	C₃×C₉	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
152	C₃×C₉	AΣL(1,𝔽₂₇)	1	1	yes	characteristic
153	C₃×C₉	#731	1	2	yes	—
154	C₃×C₉	#731	1	1	yes	characteristic
155	C₃²	—	3	3	no	—
156	C₃²	—	3	3	no	—
157	C₃²	—	9	1	no	—
158	C₃²	—	9	3	no	—
159	C₃²	—	9	3	no	—
160	C₃²	—	9	2	no	—
161	C₃²	—	9	2	no	—
162	C₃²	—	9	1	no	—
163	C₃²	—	9	3	no	—
164	C₃²	—	27	3	no	—
165	C₃²	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
166	C₃²	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
167	C₃²	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
168	C₃²	C₃×#731	1	1	yes	characteristic
169	C₉	—	9	3	no	—
170	C₉	—	9	3	no	—
171	C₉	—	9	3	no	—
172	C₉	—	9	3	no	—
173	C₉	—	9	3	no	—
174	C₉	—	9	3	no	—
175	C₉	—	9	1	no	—
176	C₉	—	9	1	no	—
177	C₉	—	9	3	no	—
178	C₉	—	9	3	no	—
179	C₉	—	27	1	no	—
180	C₉	—	27	1	no	—
181	C₉	—	27	1	no	—
182	C₉	#6857	1	2	yes	—
183	C₉	#6857	1	2	yes	—
184	C₉	#6857	1	2	yes	—
185	C₉	#6857	1	2	yes	—
186	C₉	#6857	1	1	yes	characteristic
187	C₃	—	3	1	no	—
188	C₃	—	3	1	no	—
189	C₃	—	3	1	no	—
190	C₃	—	3	1	no	—
191	C₃	—	9	3	no	—
192	C₃	—	27	3	no	—
193	C₃	C₃×#6857	1	1	yes	characteristic

	subgroup	G/H
center	C₉	#6857
commutator, Frattini, cosocle	C₃²×C₉	C₃³
socle	C₃	C₃×#6857
prime radical	C₃×#6908	C₃
LCS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉ ◁ C₃²×C₉ ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #10624872	C₃	semidirect
3. #10624872	C₃	semidirect
4. #10624688	C₃	semidirect
5. #10624487	C₃	semidirect
6. #10624487	C₃	semidirect
7. #10624487	C₃	semidirect
8. #10624487	C₃	semidirect
9. #10624487	C₃	semidirect
10. #10624486	C₃	semidirect
11. #10624485	C₃	semidirect
12. #10624485	C₃	semidirect
13. #10624484	C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6908	3
#10624872	3
#10624688	3
#10624487	3
#10624486	3
#10624485	3
#10624484	3