Finite Group Databasebeta

Subgroups — #14550933

Order 2187.

306 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 47 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, complemented
2	#10624766	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624734	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624549	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624483	C₃	1	1	yes	maximal, characteristic, complemented
6	#10624483	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624482	C₃	1	1	yes	maximal, characteristic, complemented
8	#10624482	C₃	1	1	yes	maximal, characteristic, complemented
9	#10624481	C₃	1	1	yes	maximal, characteristic, complemented
10	#10624481	C₃	1	1	yes	maximal, characteristic, complemented
11	#10624481	C₃	1	1	yes	maximal, characteristic, complemented
12	#10624481	C₃	1	1	yes	maximal, characteristic, complemented
13	#10624480	C₃	1	1	yes	maximal, characteristic, complemented
14	C₃×#737	—	3	1	no	—
15	C₃³×C₉	C₃²	1	1	yes	characteristic
16	#6908	—	3	9	no	—
17	C₃×#733	C₃²	1	1	yes	characteristic
18	C₃×#733	C₃²	1	1	yes	characteristic
19	C₃×#733	C₃²	1	1	yes	characteristic
20	C₃×#733	C₃²	1	1	yes	characteristic
21	C₃×#732	C₃²	1	1	yes	characteristic
22	C₃×#732	C₃²	1	1	yes	characteristic
23	C₃×#731	—	3	1	no	—
24	C₃×#731	—	3	1	no	—
25	C₃×AΣL(1,𝔽₂₇)	—	3	1	no	—
26	C₃×AΣL(1,𝔽₂₇)	—	3	1	no	—
27	#6895	—	3	1	no	—
28	#6893	—	3	1	no	—
29	C₃×C₃²⋊C₉	—	3	1	no	—
30	#6868	C₃²	1	1	yes	characteristic
31	#6868	C₃²	1	1	yes	characteristic
32	#6868	C₃²	1	1	yes	characteristic
33	#6867	—	3	1	no	—
34	#6866	—	3	1	no	—
35	#6855	—	3	1	no	—
36	#6855	—	3	1	no	—
37	#6855	C₃²	1	1	yes	characteristic
38	#6855	C₃²	1	1	yes	characteristic
39	#6855	C₃²	1	1	yes	characteristic
40	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
41	#737	—	3	9	no	—
42	#737	—	9	3	no	—
43	#737	—	9	3	no	—
44	C₃×3₋¹⁺²	—	3	1	no	—
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	—	9	2	no	—
49	C₃×H(3,𝔽₃)	—	3	1	no	—
50	C₃×H(3,𝔽₃)	—	3	1	no	—
51	C₃²×C₉	—	3	3	no	—
52	C₃²×C₉	—	3	1	no	—
53	C₃²×C₉	—	3	2	no	—
54	C₃²×C₉	—	3	1	no	—
55	C₃²×C₉	—	3	1	no	—
56	C₃²×C₉	—	3	1	no	—
57	C₃²×C₉	—	9	1	no	—
58	C₃²×C₉	—	9	1	no	—
59	C₃²×C₉	—	9	1	no	—
60	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
61	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
62	C₃²×C₉	C₃³	1	1	yes	characteristic
63	#733	—	3	3	no	—
64	#733	—	3	3	no	—
65	#733	—	9	1	no	—
66	#733	—	9	1	no	—
67	#732	—	3	3	no	—
68	#732	—	9	1	no	—
69	#731	—	3	9	no	—
70	#731	—	9	3	no	—
71	AΣL(1,𝔽₂₇)	—	3	9	no	—
72	AΣL(1,𝔽₂₇)	—	9	3	no	—
73	C₉⋊C₉	—	9	1	no	—
74	C₉⋊C₉	—	9	2	no	—
75	C₃²⋊C₉	—	3	1	no	—
76	C₃²⋊C₉	—	3	1	no	—
77	C₃²⋊C₉	—	3	1	no	—
78	C₃²⋊C₉	—	9	3	no	—
79	C₃²⋊C₉	—	9	1	no	—
80	C₃²⋊C₉	—	9	2	no	—
81	C₃³	—	3	1	no	—
82	C₃³	—	3	1	no	—
83	C₃³	—	3	1	no	—
84	C₃³	—	9	1	no	—
85	C₃³	—	9	1	no	—
86	C₃³	—	9	1	no	—
87	C₃³	—	9	1	no	—
88	C₃³	—	9	1	no	—
89	C₃³	AΣL(1,𝔽₂₇)	1	3	yes	—
90	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
91	3₋¹⁺²	—	3	3	no	—
92	3₋¹⁺²	—	3	3	no	—
93	3₋¹⁺²	—	9	18	no	—
94	3₋¹⁺²	—	9	1	no	—
95	3₋¹⁺²	—	9	1	no	—
96	3₋¹⁺²	—	9	1	no	—
97	3₋¹⁺²	—	9	1	no	—
98	3₋¹⁺²	—	9	1	no	—
99	3₋¹⁺²	—	9	1	no	—
100	3₋¹⁺²	—	9	1	no	—
101	3₋¹⁺²	—	9	1	no	—
102	3₋¹⁺²	—	9	1	no	—
103	3₋¹⁺²	—	9	1	no	—
104	H(3,𝔽₃)	—	3	3	no	—
105	H(3,𝔽₃)	—	9	1	no	—
106	H(3,𝔽₃)	—	9	1	no	—
107	H(3,𝔽₃)	—	9	1	no	—
108	H(3,𝔽₃)	—	9	1	no	—
109	H(3,𝔽₃)	—	9	1	no	—
110	C₃×C₉	—	3	1	no	—
111	C₃×C₉	—	3	1	no	—
112	C₃×C₉	—	3	1	no	—
113	C₃×C₉	—	3	1	no	—
114	C₃×C₉	—	3	1	no	—
115	C₃×C₉	—	3	1	no	—
116	C₃×C₉	—	3	1	no	—
117	C₃×C₉	—	3	1	no	—
118	C₃×C₉	—	3	1	no	—
119	C₃×C₉	—	3	1	no	—
120	C₃×C₉	—	3	1	no	—
121	C₃×C₉	—	9	3	no	—
122	C₃×C₉	—	9	6	no	—
123	C₃×C₉	—	9	9	no	—
124	C₃×C₉	—	9	1	no	—
125	C₃×C₉	—	9	1	no	—
126	C₃×C₉	—	9	1	no	—
127	C₃×C₉	—	9	1	no	—
128	C₃×C₉	—	9	1	no	—
129	C₃×C₉	—	9	1	no	—
130	C₃×C₉	—	9	1	no	—
131	C₃×C₉	—	9	1	no	—
132	C₃×C₉	—	9	1	no	—
133	C₃×C₉	—	9	1	no	—
134	C₃×C₉	—	9	1	no	—
135	C₃×C₉	—	9	1	no	—
136	C₃×C₉	—	9	1	no	—
137	C₃×C₉	—	9	1	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	#732	1	1	yes	characteristic
144	C₃×C₉	#732	1	2	yes	—
145	C₃²	—	3	1	no	—
146	C₃²	—	3	1	no	—
147	C₃²	—	3	1	no	—
148	C₃²	—	3	3	no	—
149	C₃²	—	9	3	no	—
150	C₃²	—	9	3	no	—
151	C₃²	—	9	3	no	—
152	C₃²	—	9	1	no	—
153	C₃²	—	9	1	no	—
154	C₃²	—	9	1	no	—
155	C₃²	—	9	3	no	—
156	C₃²	—	27	1	no	—
157	C₃²	—	27	1	no	—
158	C₃²	—	27	1	no	—
159	C₃²	—	27	1	no	—
160	C₃²	—	27	1	no	—
161	C₃²	—	27	1	no	—
162	C₃²	—	27	1	no	—
163	C₃²	C₃×AΣL(1,𝔽₂₇)	1	1	yes	characteristic
164	C₃²	C₃×#732	1	1	yes	characteristic
165	C₃²	#6909	1	1	yes	characteristic
166	C₃²	#6911	1	1	yes	characteristic
167	C₉	—	3	1	no	—
168	C₉	—	3	2	no	—
169	C₉	—	9	1	no	—
170	C₉	—	9	1	no	—
171	C₉	—	9	3	no	—
172	C₉	—	9	3	no	—
173	C₉	—	27	1	no	—
174	C₉	—	27	1	no	—
175	C₉	—	27	1	no	—
176	C₉	—	27	1	no	—
177	C₉	—	27	1	no	—
178	C₉	—	27	1	no	—
179	C₉	—	27	1	no	—
180	C₉	—	27	1	no	—
181	C₃	—	3	1	no	—
182	C₃	—	3	1	no	—
183	C₃	—	3	1	no	—
184	C₃	—	9	3	no	—
185	C₃	—	27	3	no	—
186	C₃	—	81	1	no	—
187	C₃	#10624596	1	1	yes	characteristic
188	C₃	#10624841	1	1	yes	characteristic
189	C₃	#10624845	1	1	yes	characteristic
190	C₃	#10624849	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃×#732
commutator, Frattini, cosocle	C₃²×C₉	C₃³
LCS	G ▷ C₃²×C₉ ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ 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
1. C₃×#6908	C₃	semidirect
2. #10624766	C₃	semidirect
3. #10624734	C₃	semidirect
4. #10624549	C₃	semidirect
5. #10624483	C₃	semidirect
6. #10624483	C₃	semidirect
7. #10624482	C₃	semidirect
8. #10624482	C₃	semidirect
9. #10624481	C₃	semidirect
10. #10624481	C₃	semidirect
11. #10624481	C₃	semidirect
12. #10624481	C₃	semidirect
13. #10624480	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
#10624766	3
#10624734	3
#10624549	3
#10624483	3
#10624482	3
#10624481	3
#10624480	3