Finite Group Databasebeta

Subgroups — #14550411

Order 2187.

542 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 107 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	#10624862	C₃	1	1	yes	maximal, characteristic
3	#10624699	C₃	1	2	yes	maximal, complemented
4	#10624699	C₃	1	2	yes	maximal, complemented
5	#10624699	C₃	1	2	yes	maximal, complemented
6	#10624689	C₃	1	2	yes	maximal
7	C₃×#6868	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃×#6867	C₃	1	1	yes	maximal, characteristic, complemented
9	#10624549	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
11	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
12	C₃³×C₉	C₃²	1	1	yes	characteristic
13	#6893	—	3	6	no	—
14	#6893	C₉	1	9	yes	complemented
15	C₉×H(3,𝔽₃)	—	3	3	no	—
16	C₉×H(3,𝔽₃)	—	3	3	no	—
17	C₉×H(3,𝔽₃)	—	3	3	no	—
18	C₃×C₃²⋊C₉	—	3	3	no	—
19	C₃×C₃²⋊C₉	—	3	3	no	—
20	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
21	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
22	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
23	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
24	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
25	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
26	3₋¹⁺²⋊C₉	—	3	6	no	—
27	3₋¹⁺²⋊C₉	—	3	6	no	—
28	3₋¹⁺²⋊C₉	—	3	6	no	—
29	#6868	—	3	3	no	—
30	#6867	—	3	3	no	—
31	#6866	—	3	6	no	—
32	#6855	—	3	3	no	—
33	C₃⁴	H(3,𝔽₃)	1	3	yes	—
34	C₃⁴	C₃³	1	1	yes	characteristic
35	C₃×3₋¹⁺²	—	3	4	no	—
36	C₃×3₋¹⁺²	—	3	2	no	—
37	C₃×3₋¹⁺²	—	3	2	no	—
38	C₃×3₋¹⁺²	—	3	2	no	—
39	C₃×3₋¹⁺²	—	3	4	no	—
40	C₃×3₋¹⁺²	—	3	4	no	—
41	C₃×3₋¹⁺²	—	3	4	no	—
42	C₃×3₋¹⁺²	C₃×C₉	1	6	yes	complemented
43	C₃×H(3,𝔽₃)	—	3	1	no	—
44	C₃×H(3,𝔽₃)	—	3	1	no	—
45	C₃×H(3,𝔽₃)	—	3	1	no	—
46	C₃×H(3,𝔽₃)	—	3	2	no	—
47	C₃×H(3,𝔽₃)	—	3	2	no	—
48	C₃×H(3,𝔽₃)	—	3	2	no	—
49	C₃×H(3,𝔽₃)	C₃×C₉	1	3	yes	—
50	C₃×H(3,𝔽₃)	3₋¹⁺²	1	6	yes	—
51	C₃²×C₉	—	3	1	no	—
52	C₃²×C₉	—	3	1	no	—
53	C₃²×C₉	—	3	1	no	—
54	C₃²×C₉	—	3	1	no	—
55	C₃²×C₉	—	3	2	no	—
56	C₃²×C₉	—	3	2	no	—
57	C₃²×C₉	—	3	2	no	—
58	C₃²×C₉	—	3	2	no	—
59	C₃²×C₉	—	3	2	no	—
60	C₃²×C₉	—	3	9	no	—
61	C₃²×C₉	—	3	1	no	—
62	C₃²×C₉	—	3	1	no	—
63	C₃²×C₉	—	3	2	no	—
64	C₃²×C₉	—	3	2	no	—
65	C₃²×C₉	—	3	2	no	—
66	C₃²×C₉	C₃×C₉	1	3	yes	complemented
67	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
68	C₃²⋊C₉	—	3	18	no	—
69	C₃²⋊C₉	—	3	27	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	2	no	—
78	C₃²⋊C₉	—	3	2	no	—
79	C₃²⋊C₉	—	3	2	no	—
80	C₃²⋊C₉	—	3	2	no	—
81	C₃²⋊C₉	—	3	2	no	—
82	C₃²⋊C₉	—	3	2	no	—
83	C₃²⋊C₉	—	3	2	no	—
84	C₃²⋊C₉	—	3	2	no	—
85	C₃²⋊C₉	—	3	2	no	—
86	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
87	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
88	C₃³	—	3	3	no	—
89	C₃³	—	3	6	no	—
90	C₃³	—	9	1	no	—
91	C₃³	—	9	2	no	—
92	C₃³	—	9	3	no	—
93	C₃³	—	9	6	no	—
94	C₃³	C₃²⋊C₉	1	9	yes	—
95	C₃³	C₃²×C₉	1	1	yes	characteristic
96	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
97	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
98	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
99	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
100	C₃³	C₃×3₋¹⁺²	1	2	yes	—
101	C₃³	#737	1	2	yes	—
102	C₃³	#737	1	2	yes	—
103	C₃³	#737	1	2	yes	—
104	3₋¹⁺²	—	3	6	no	—
105	3₋¹⁺²	—	3	6	no	—
106	3₋¹⁺²	—	3	6	no	—
107	3₋¹⁺²	—	9	4	no	—
108	3₋¹⁺²	—	9	4	no	—
109	3₋¹⁺²	—	9	4	no	—
110	H(3,𝔽₃)	—	3	3	no	—
111	H(3,𝔽₃)	—	3	3	no	—
112	H(3,𝔽₃)	—	3	3	no	—
113	H(3,𝔽₃)	—	3	6	no	—
114	H(3,𝔽₃)	—	3	6	no	—
115	H(3,𝔽₃)	—	3	6	no	—
116	C₃×C₉	—	3	3	no	—
117	C₃×C₉	—	3	6	no	—
118	C₃×C₉	—	3	3	no	—
119	C₃×C₉	—	3	3	no	—
120	C₃×C₉	—	3	3	no	—
121	C₃×C₉	—	3	3	no	—
122	C₃×C₉	—	3	3	no	—
123	C₃×C₉	—	3	3	no	—
124	C₃×C₉	—	9	1	no	—
125	C₃×C₉	—	9	2	no	—
126	C₃×C₉	—	9	2	no	—
127	C₃×C₉	—	9	2	no	—
128	C₃×C₉	—	9	2	no	—
129	C₃×C₉	—	9	2	no	—
130	C₃×C₉	—	9	4	no	—
131	C₃×C₉	—	9	3	no	—
132	C₃×C₉	—	9	6	no	—
133	C₃×C₉	—	9	2	no	—
134	C₃×C₉	—	9	2	no	—
135	C₃×C₉	—	9	2	no	—
136	C₃×C₉	—	27	1	no	—
137	C₃×C₉	—	27	1	no	—
138	C₃×C₉	—	27	9	no	—
139	C₃×C₉	—	27	2	no	—
140	C₃×C₉	—	27	2	no	—
141	C₃²	—	3	1	no	—
142	C₃²	—	3	1	no	—
143	C₃²	—	3	1	no	—
144	C₃²	—	3	1	no	—
145	C₃²	—	3	2	no	—
146	C₃²	—	3	2	no	—
147	C₃²	—	3	2	no	—
148	C₃²	—	3	2	no	—
149	C₃²	—	3	9	no	—
150	C₃²	—	9	1	no	—
151	C₃²	—	9	2	no	—
152	C₃²	—	9	2	no	—
153	C₃²	—	9	2	no	—
154	C₃²	—	9	2	no	—
155	C₃²	—	9	9	no	—
156	C₃²	—	9	3	no	—
157	C₃²	—	9	6	no	—
158	C₃²	—	27	6	no	—
159	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
160	C₃²	C₉×H(3,𝔽₃)	1	1	yes	characteristic
161	C₃²	C₉×H(3,𝔽₃)	1	1	yes	characteristic
162	C₃²	C₉×H(3,𝔽₃)	1	1	yes	characteristic
163	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
164	C₃²	C₉⋊H(3,𝔽₃)	1	2	yes	—
165	C₃²	C₉⋊H(3,𝔽₃)	1	2	yes	—
166	C₃²	C₉⋊H(3,𝔽₃)	1	2	yes	—
167	C₃²	#6893	1	2	yes	—
168	C₉	—	9	3	no	—
169	C₉	—	9	6	no	—
170	C₉	—	9	3	no	—
171	C₉	—	9	3	no	—
172	C₉	—	9	3	no	—
173	C₉	—	27	2	no	—
174	C₉	—	27	2	no	—
175	C₉	—	27	2	no	—
176	C₃	—	3	1	no	—
177	C₃	—	3	2	no	—
178	C₃	—	3	1	no	—
179	C₃	—	3	1	no	—
180	C₃	—	3	1	no	—
181	C₃	—	3	2	no	—
182	C₃	—	3	2	no	—
183	C₃	—	3	2	no	—
184	C₃	—	9	9	no	—
185	C₃	#10624555	1	1	yes	characteristic
186	C₃	#10624748	1	1	yes	characteristic
187	C₃	#10624750	1	2	yes	—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator	C₃³	C₃²×C₉
Frattini, cosocle	C₃⁴	C₃³
prime radical	C₃²×H(3,𝔽₃)	C₃²
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #10624699	C₃	semidirect
4. #10624699	C₃	semidirect
5. #10624699	C₃	semidirect
7. C₃×#6868	C₃	semidirect
8. C₃×#6867	C₃	semidirect
9. #10624549	C₃	semidirect
14. #6893	C₉	semidirect
42. C₃×3₋¹⁺²	C₃×C₉	semidirect
66. C₃²×C₉	C₃×C₉	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
#10624862	3
#10624699	3
#10624689	3
C₃×#6868	3
C₃×#6867	3
#10624549	3