Finite Group Databasebeta

Subgroups — #14552237

Order 2187.

1070 conjugacy classes of subgroups; only the 287 normal subgroups are shown.

normal subgroups only

This group has 1070 conjugacy classes of subgroups — too many to draw. Only its 287 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×#6893	C₃	1	1	yes	maximal, characteristic
2	C₃×C₉⋊H(3,𝔽₃)	C₃	1	2	yes	maximal
3	C₃×C₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
4	#10624570	C₃	1	6	yes	maximal, complemented
5	#10624569	C₃	1	3	yes	maximal, complemented
6	#10624568	C₃	1	6	yes	maximal, complemented
7	#10624567	C₃	1	3	yes	maximal, complemented
8	#10624558	C₃	1	2	yes	maximal, complemented
9	#10624558	C₃	1	4	yes	maximal, complemented
10	#10624558	C₃	1	6	yes	maximal, complemented
11	#10624549	C₃	1	2	yes	maximal, complemented
12	#10624549	C₃	1	3	yes	maximal, complemented
13	C₃×#6855	C₃	1	1	yes	maximal, characteristic, complemented
14	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
15	C₃²×3₋¹⁺²	C₃²	1	6	yes	—
16	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
17	C₃³×C₉	C₃²	1	1	yes	characteristic
18	C₃³×C₉	C₃²	1	1	yes	characteristic
19	C₃³×C₉	C₃²	1	2	yes	—
20	C₃³×C₉	C₃²	1	3	yes	—
21	#6893	—	3	9	no	—
22	C₉⋊H(3,𝔽₃)	—	3	18	no	—
23	C₉×H(3,𝔽₃)	—	3	9	no	—
24	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
25	C₃×C₃²⋊C₉	C₃²	1	4	yes	—
26	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
27	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
28	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
29	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
30	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
31	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
32	#6855	C₃²	1	1	yes	characteristic
33	#6855	C₃²	1	2	yes	—
34	#6855	C₃²	1	1	yes	characteristic
35	#6855	C₃²	1	2	yes	—
36	#6855	C₃²	1	1	yes	characteristic
37	#6855	C₃²	1	2	yes	—
38	#6855	C₃²	1	6	yes	—
39	#6855	C₃²	1	6	yes	—
40	#6855	C₃²	1	12	yes	—
41	#6855	C₃²	1	12	yes	—
42	#6855	C₃²	1	12	yes	—
43	#6855	C₃²	1	12	yes	—
44	#6855	C₃²	1	6	yes	—
45	#6855	C₃²	1	6	yes	—
46	C₃⁴	C₃³	1	3	yes	—
47	C₃⁴	C₃³	1	1	yes	characteristic
48	C₃×3₋¹⁺²	—	3	18	no	—
49	C₃×3₋¹⁺²	—	3	6	no	—
50	C₃×3₋¹⁺²	—	3	6	no	—
51	C₃×3₋¹⁺²	—	3	12	no	—
52	C₃×3₋¹⁺²	—	3	18	no	—
53	C₃×3₋¹⁺²	—	9	12	no	—
54	C₃×H(3,𝔽₃)	—	3	3	no	—
55	C₃×H(3,𝔽₃)	—	3	6	no	—
56	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
57	C₃²×C₉	—	3	9	no	—
58	C₃²×C₉	—	3	3	no	—
59	C₃²×C₉	—	3	6	no	—
60	C₃²×C₉	—	3	3	no	—
61	C₃²×C₉	—	3	3	no	—
62	C₃²×C₉	—	3	6	no	—
63	C₃²×C₉	—	3	3	no	—
64	C₃²×C₉	—	3	9	no	—
65	C₃²×C₉	—	3	6	no	—
66	C₃²×C₉	—	9	2	no	—
67	C₃²×C₉	—	9	6	no	—
68	C₃²×C₉	—	9	4	no	—
69	C₃²×C₉	C₃³	1	1	yes	characteristic
70	C₃²×C₉	C₃³	1	2	yes	—
71	C₃²×C₉	C₃³	1	1	yes	characteristic
72	C₃²×C₉	C₃³	1	1	yes	characteristic
73	C₃²×C₉	C₃³	1	2	yes	—
74	C₃²×C₉	C₃³	1	2	yes	—
75	C₃²×C₉	C₃³	1	6	yes	—
76	C₃²×C₉	C₃³	1	6	yes	—
77	C₃²×C₉	C₃³	1	12	yes	—
78	C₃²×C₉	C₃³	1	1	yes	characteristic
79	C₃²×C₉	C₃³	1	2	yes	—
80	C₃²⋊C₉	—	3	9	no	—
81	C₃²⋊C₉	—	3	9	no	—
82	C₃²⋊C₉	—	3	6	no	—
83	C₃²⋊C₉	—	3	18	no	—
84	C₃²⋊C₉	—	3	18	no	—
85	C₃²⋊C₉	—	3	12	no	—
86	C₃²⋊C₉	—	3	9	no	—
87	C₃²⋊C₉	—	3	18	no	—
88	C₃³	—	3	1	no	—
89	C₃³	—	3	2	no	—
90	C₃³	—	3	4	no	—
91	C₃³	—	3	2	no	—
92	C₃³	—	3	3	no	—
93	C₃³	—	3	3	no	—
94	C₃³	—	3	6	no	—
95	C₃³	—	3	6	no	—
96	C₃³	—	3	6	no	—
97	C₃³	—	3	12	no	—
98	C₃³	C₃×H(3,𝔽₃)	1	9	yes	—
99	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
100	C₃³	#737	1	3	yes	—
101	C₃³	#737	1	6	yes	—
102	C₃³	C₃⁴	1	1	yes	characteristic
103	3₋¹⁺²	—	9	54	no	—
104	3₋¹⁺²	—	9	18	no	—
105	H(3,𝔽₃)	—	3	27	no	—
106	C₃×C₉	—	3	1	no	—
107	C₃×C₉	—	3	2	no	—
108	C₃×C₉	—	3	2	no	—
109	C₃×C₉	—	3	4	no	—
110	C₃×C₉	—	3	1	no	—
111	C₃×C₉	—	3	2	no	—
112	C₃×C₉	—	3	1	no	—
113	C₃×C₉	—	3	6	no	—
114	C₃×C₉	—	3	1	no	—
115	C₃×C₉	—	3	2	no	—
116	C₃×C₉	—	3	2	no	—
117	C₃×C₉	—	3	6	no	—
118	C₃×C₉	—	3	12	no	—
119	C₃×C₉	—	3	1	no	—
120	C₃×C₉	—	3	2	no	—
121	C₃×C₉	—	3	2	no	—
122	C₃×C₉	—	3	4	no	—
123	C₃×C₉	—	3	1	no	—
124	C₃×C₉	—	3	1	no	—
125	C₃×C₉	—	3	6	no	—
126	C₃×C₉	—	3	2	no	—
127	C₃×C₉	—	3	2	no	—
128	C₃×C₉	—	3	6	no	—
129	C₃×C₉	—	3	12	no	—
130	C₃×C₉	—	3	12	no	—
131	C₃×C₉	—	3	4	no	—
132	C₃×C₉	—	3	4	no	—
133	C₃×C₉	—	3	2	no	—
134	C₃×C₉	—	3	2	no	—
135	C₃×C₉	—	3	12	no	—
136	C₃×C₉	—	3	24	no	—
137	C₃×C₉	—	9	9	no	—
138	C₃×C₉	—	9	27	no	—
139	C₃×C₉	—	9	18	no	—
140	C₃×C₉	—	9	9	no	—
141	C₃×C₉	#737	1	3	yes	—
142	C₃×C₉	#737	1	6	yes	—
143	C₃²	—	3	3	no	—
144	C₃²	—	3	6	no	—
145	C₃²	—	3	3	no	—
146	C₃²	—	3	6	no	—
147	C₃²	—	3	6	no	—
148	C₃²	—	3	12	no	—
149	C₃²	—	3	9	no	—
150	C₃²	—	3	9	no	—
151	C₃²	—	3	18	no	—
152	C₃²	—	9	3	no	—
153	C₃²	—	9	6	no	—
154	C₃²	—	9	6	no	—
155	C₃²	—	9	12	no	—
156	C₃²	#6893	1	9	yes	—
157	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
158	C₃²	C₃×#737	1	1	yes	characteristic
159	C₃²	C₃×#737	1	2	yes	—
160	C₃²	C₃×#737	1	1	yes	characteristic
161	C₃²	C₃×#737	1	2	yes	—
162	C₃²	3₋¹⁺⁴	1	4	yes	—
163	C₃²	3₋¹⁺⁴	1	2	yes	—
164	C₉	—	3	3	no	—
165	C₉	—	3	6	no	—
166	C₉	—	9	1	no	—
167	C₉	—	9	2	no	—
168	C₉	—	9	6	no	—
169	C₉	—	9	1	no	—
170	C₉	—	9	1	no	—
171	C₉	—	9	6	no	—
172	C₉	—	9	2	no	—
173	C₉	—	9	2	no	—
174	C₉	—	9	12	no	—
175	C₃	—	3	9	no	—
176	C₃	—	9	9	no	—
177	C₃	C₃×#6893	1	1	yes	characteristic
178	C₃	#10624877	1	1	yes	characteristic
179	C₃	#10624891	1	2	yes	—
180	C₃	#10624894	1	1	yes	characteristic
181	C₃	#10624894	1	2	yes	—
182	C₃	#10624904	1	2	yes	—
183	C₃	#10624907	1	4	yes	—

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

Complements

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

N	H ≅ G/N	type
4. #10624570	C₃	semidirect
5. #10624569	C₃	semidirect
6. #10624568	C₃	semidirect
7. #10624567	C₃	semidirect
8. #10624558	C₃	semidirect
9. #10624558	C₃	semidirect
10. #10624558	C₃	semidirect
11. #10624549	C₃	semidirect
12. #10624549	C₃	semidirect
13. C₃×#6855	C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6893	3
C₃×C₉⋊H(3,𝔽₃)	3
C₃×C₉×H(3,𝔽₃)	3
#10624570	3
#10624569	3
#10624568	3
#10624567	3
#10624558	3
#10624549	3
C₃×#6855	3