Finite Group Databasebeta

Subgroups — #14552239

Order 2187.

1151 conjugacy classes of subgroups; only the 287 normal subgroups are shown.

normal subgroups only

This group has 1151 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×#6898	C₃	1	3	yes	maximal, complemented
2	C₃²×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic
3	#10624569	C₃	1	6	yes	maximal, complemented
4	#10624566	C₃	1	12	yes	maximal, complemented
5	#10624557	C₃	1	6	yes	maximal, complemented
6	#10624556	C₃	1	2	yes	maximal, complemented
7	#10624556	C₃	1	4	yes	maximal, complemented
8	#10624548	C₃	1	2	yes	maximal, complemented
9	#10624547	C₃	1	1	yes	maximal, characteristic
10	#10624547	C₃	1	2	yes	maximal
11	C₉×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
12	C₃⁵	C₃²	1	1	yes	characteristic
13	C₃³×C₉	C₃²	1	1	yes	characteristic
14	C₃³×C₉	C₃²	1	2	yes	—
15	#6898	—	3	27	no	—
16	C₃×C₉⋊C₉	C₃²	1	6	yes	complemented
17	C₃×C₉⋊C₉	C₃²	1	6	yes	complemented
18	C₃×C₉⋊C₉	C₃²	1	12	yes	complemented
19	C₃×C₃²⋊C₉	—	3	9	no	—
20	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
21	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
22	C₃×C₃²⋊C₉	C₃²	1	4	yes	—
23	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
24	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
25	C₃×C₃²⋊C₉	C₃²	1	3	yes	—
26	C₃×C₃²⋊C₉	C₃²	1	6	yes	—
27	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
28	C₃×C₃²⋊C₉	C₃²	1	4	yes	—
29	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
30	C₃×C₃²⋊C₉	C₃²	1	12	yes	—
31	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
32	C₃×C₉²	C₃²	1	3	yes	complemented
33	#6855	C₃²	1	6	yes	complemented
34	#6855	C₃²	1	12	yes	complemented
35	#6855	C₃²	1	24	yes	complemented
36	#6855	C₃²	1	12	yes	complemented
37	C₃⁴	—	3	3	no	—
38	C₃⁴	—	3	6	no	—
39	C₃⁴	—	3	3	no	—
40	C₃⁴	—	3	6	no	—
41	C₃⁴	—	3	3	no	—
42	C₃⁴	—	9	2	no	—
43	C₃⁴	—	9	4	no	—
44	C₃⁴	C₃³	1	1	yes	characteristic
45	C₃⁴	C₃³	1	2	yes	—
46	C₃⁴	C₃³	1	1	yes	characteristic
47	C₃²×C₉	—	3	3	no	—
48	C₃²×C₉	—	3	6	no	—
49	C₃²×C₉	—	3	3	no	—
50	C₃²×C₉	—	3	6	no	—
51	C₃²×C₉	—	9	2	no	—
52	C₃²×C₉	—	9	4	no	—
53	C₃²×C₉	C₃³	1	2	yes	—
54	C₃²×C₉	C₃³	1	4	yes	—
55	C₃²×C₉	C₃³	1	3	yes	—
56	C₃²×C₉	C₃³	1	6	yes	—
57	C₃²×C₉	C₃³	1	6	yes	—
58	C₃²×C₉	C₃³	1	12	yes	—
59	C₃²×C₉	C₃³	1	1	yes	characteristic
60	C₃²×C₉	C₃³	1	2	yes	—
61	C₉⋊C₉	—	3	18	no	—
62	C₉⋊C₉	—	3	36	no	—
63	C₉⋊C₉	—	3	18	no	—
64	C₃²⋊C₉	—	3	6	no	—
65	C₃²⋊C₉	—	3	9	no	—
66	C₃²⋊C₉	—	3	18	no	—
67	C₃²⋊C₉	—	3	18	no	—
68	C₃²⋊C₉	—	3	3	no	—
69	C₃²⋊C₉	—	3	6	no	—
70	C₃²⋊C₉	—	3	12	no	—
71	C₃²⋊C₉	—	3	6	no	—
72	C₃²⋊C₉	—	3	6	no	—
73	C₃²⋊C₉	—	3	12	no	—
74	C₃²⋊C₉	—	9	12	no	—
75	C₃²⋊C₉	H(3,𝔽₃)	1	9	yes	—
76	C₉²	—	3	9	no	—
77	C₃³	—	3	1	no	—
78	C₃³	—	3	2	no	—
79	C₃³	—	3	2	no	—
80	C₃³	—	3	4	no	—
81	C₃³	—	3	27	no	—
82	C₃³	—	3	2	no	—
83	C₃³	—	3	1	no	—
84	C₃³	—	3	2	no	—
85	C₃³	—	3	2	no	—
86	C₃³	—	3	2	no	—
87	C₃³	—	3	1	no	—
88	C₃³	—	3	2	no	—
89	C₃³	—	3	2	no	—
90	C₃³	—	3	4	no	—
91	C₃³	—	3	4	no	—
92	C₃³	—	3	2	no	—
93	C₃³	—	3	4	no	—
94	C₃³	—	3	4	no	—
95	C₃³	—	3	4	no	—
96	C₃³	—	9	9	no	—
97	C₃³	—	9	18	no	—
98	C₃³	—	9	18	no	—
99	C₃³	—	9	36	no	—
100	C₃³	—	9	18	no	—
101	C₃³	—	9	9	no	—
102	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
103	C₃³	#737	1	6	yes	—
104	C₃³	#737	1	3	yes	—
105	C₃³	#737	1	6	yes	—
106	C₃³	#737	1	3	yes	—
107	C₃³	C₃⁴	1	1	yes	characteristic
108	C₃×C₉	—	3	1	no	—
109	C₃×C₉	—	3	2	no	—
110	C₃×C₉	—	3	2	no	—
111	C₃×C₉	—	3	4	no	—
112	C₃×C₉	—	3	3	no	—
113	C₃×C₉	—	3	6	no	—
114	C₃×C₉	—	3	4	no	—
115	C₃×C₉	—	3	2	no	—
116	C₃×C₉	—	3	6	no	—
117	C₃×C₉	—	3	12	no	—
118	C₃×C₉	—	3	3	no	—
119	C₃×C₉	—	3	6	no	—
120	C₃×C₉	—	3	6	no	—
121	C₃×C₉	—	3	12	no	—
122	C₃×C₉	—	3	6	no	—
123	C₃×C₉	—	3	12	no	—
124	C₃×C₉	—	3	2	no	—
125	C₃×C₉	—	3	4	no	—
126	C₃×C₉	—	3	12	no	—
127	C₃×C₉	—	3	24	no	—
128	C₃×C₉	—	3	4	no	—
129	C₃×C₉	—	3	8	no	—
130	C₃×C₉	—	9	9	no	—
131	C₃×C₉	—	9	18	no	—
132	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
133	C₃×C₉	C₃×H(3,𝔽₃)	1	6	yes	—
134	C₃²	—	3	3	no	—
135	C₃²	—	3	6	no	—
136	C₃²	—	3	3	no	—
137	C₃²	—	3	6	no	—
138	C₃²	—	3	6	no	—
139	C₃²	—	3	12	no	—
140	C₃²	—	3	3	no	—
141	C₃²	—	3	6	no	—
142	C₃²	—	3	3	no	—
143	C₃²	—	3	6	no	—
144	C₃²	—	3	6	no	—
145	C₃²	—	3	12	no	—
146	C₃²	—	9	81	no	—
147	C₃²	—	9	1	no	—
148	C₃²	—	9	2	no	—
149	C₃²	—	9	4	no	—
150	C₃²	—	9	2	no	—
151	C₃²	—	9	2	no	—
152	C₃²	—	9	4	no	—
153	C₃²	—	9	2	no	—
154	C₃²	—	9	2	no	—
155	C₃²	—	9	4	no	—
156	C₃²	—	9	4	no	—
157	C₃²	#6898	1	9	yes	complemented
158	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
159	C₃²	C₃×#737	1	1	yes	characteristic
160	C₃²	C₃×#737	1	2	yes	—
161	C₃²	C₃×#737	1	1	yes	characteristic
162	C₃²	C₃×#737	1	2	yes	—
163	C₃²	3₋¹⁺⁴	1	2	yes	—
164	C₃²	3₋¹⁺⁴	1	4	yes	—
165	C₉	—	3	3	no	—
166	C₉	—	3	6	no	—
167	C₉	—	9	3	no	—
168	C₉	—	9	6	no	—
169	C₉	—	9	4	no	—
170	C₉	—	9	2	no	—
171	C₉	—	9	6	no	—
172	C₉	—	9	12	no	—
173	C₃	—	3	9	no	—
174	C₃	—	9	3	no	—
175	C₃	—	9	6	no	—
176	C₃	#10624869	1	1	yes	characteristic
177	C₃	C₃×#6898	1	1	yes	characteristic
178	C₃	#10624890	1	2	yes	—
179	C₃	#10624894	1	1	yes	characteristic
180	C₃	#10624896	1	2	yes	—
181	C₃	#10624902	1	2	yes	—
182	C₃	#10624906	1	4	yes	—

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₃³	C₃⁴
prime radical	C₃⁵	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
1. C₃×#6898	C₃	semidirect
3. #10624569	C₃	semidirect
4. #10624566	C₃	semidirect
5. #10624557	C₃	semidirect
6. #10624556	C₃	semidirect
7. #10624556	C₃	semidirect
8. #10624548	C₃	semidirect
11. C₉×C₃²⋊C₉	C₃	semidirect
16. C₃×C₉⋊C₉	C₃²	semidirect
17. C₃×C₉⋊C₉	C₃²	semidirect
18. C₃×C₉⋊C₉	C₃²	semidirect
32. C₃×C₉²	C₃²	semidirect
33. #6855	C₃²	semidirect
34. #6855	C₃²	semidirect
35. #6855	C₃²	semidirect
36. #6855	C₃²	semidirect
157. C₃²	#6898	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6898	3
C₃²×C₃²⋊C₉	3
#10624569	3
#10624566	3
#10624557	3
#10624556	3
#10624548	3
#10624547	3
C₉×C₃²⋊C₉	3