Finite Group Databasebeta

Subgroups — #14547478

Order 2187.

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

normal subgroups only

This group has 439 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	#10624897	C₃	1	1	yes	maximal, characteristic, complemented
2	#10624820	C₃	1	2	yes	maximal, complemented
3	#10624779	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624740	C₃	1	1	yes	maximal, characteristic
5	#10624731	C₃	1	2	yes	maximal, complemented
6	C₃×3₋¹⁺²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×H(3,𝔽₃)⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
8	#10624609	C₃	1	2	yes	maximal, complemented
9	#10624605	C₃	1	2	yes	maximal, complemented
10	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
11	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
12	C₃³×C₉	C₃²	1	1	yes	characteristic
13	C₃×#732	—	3	2	no	—
14	C₃×#732	—	3	1	no	—
15	C₃×#731	—	3	6	no	—
16	C₉⋊H(3,𝔽₃)	—	3	6	no	—
17	C₉⋊H(3,𝔽₃)	—	3	2	no	—
18	#6891	—	3	2	no	—
19	#6891	C₃²	1	2	yes	—
20	#6891	C₃²	1	2	yes	—
21	C₃²⋊H(3,𝔽₃)	C₃²	1	2	yes	—
22	C₉×H(3,𝔽₃)	—	3	1	no	—
23	C₃×C₃²⋊C₉	—	3	1	no	—
24	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
25	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
26	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
27	3₋¹⁺²⋊C₉	—	3	1	no	—
28	3₋¹⁺²⋊C₉	—	3	2	no	—
29	H(3,𝔽₃)⋊C₉	—	3	2	no	—
30	H(3,𝔽₃)⋊C₉	—	3	1	no	—
31	H(3,𝔽₃)⋊C₉	—	3	2	no	—
32	H(3,𝔽₃)⋊C₉	—	3	1	no	—
33	H(3,𝔽₃)⋊C₉	—	3	2	no	—
34	H(3,𝔽₃)⋊C₉	—	3	4	no	—
35	#6858	—	3	6	no	—
36	#6857	—	3	6	no	—
37	C₃⁴	—	3	1	no	—
38	C₃⁴	C₃³	1	1	yes	characteristic
39	C₃×3₋¹⁺²	—	3	6	no	—
40	C₃×3₋¹⁺²	—	3	1	no	—
41	C₃×3₋¹⁺²	—	3	3	no	—
42	C₃×3₋¹⁺²	—	3	2	no	—
43	C₃×3₋¹⁺²	—	3	2	no	—
44	C₃×3₋¹⁺²	—	9	2	no	—
45	C₃×3₋¹⁺²	H(3,𝔽₃)	1	6	yes	—
46	C₃×H(3,𝔽₃)	—	3	3	no	—
47	C₃×H(3,𝔽₃)	—	3	2	no	—
48	C₃×H(3,𝔽₃)	—	3	2	no	—
49	C₃×H(3,𝔽₃)	—	3	2	no	—
50	C₃×H(3,𝔽₃)	—	3	2	no	—
51	C₃×H(3,𝔽₃)	—	3	2	no	—
52	C₃×H(3,𝔽₃)	—	9	2	no	—
53	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	2	yes	—
54	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	1	yes	characteristic
55	C₃²×C₉	—	3	2	no	—
56	C₃²×C₉	—	3	3	no	—
57	C₃²×C₉	—	3	1	no	—
58	C₃²×C₉	—	3	1	no	—
59	C₃²×C₉	—	3	1	no	—
60	C₃²×C₉	—	3	1	no	—
61	C₃²×C₉	—	3	1	no	—
62	C₃²×C₉	—	3	2	no	—
63	C₃²×C₉	—	9	3	no	—
64	#732	—	9	6	no	—
65	#732	—	9	3	no	—
66	#731	—	9	18	no	—
67	C₃²⋊C₉	—	3	6	no	—
68	C₃²⋊C₉	—	3	6	no	—
69	C₃²⋊C₉	—	3	6	no	—
70	C₃²⋊C₉	—	3	2	no	—
71	C₃²⋊C₉	—	3	2	no	—
72	C₃²⋊C₉	—	3	2	no	—
73	C₃²⋊C₉	—	3	2	no	—
74	C₃²⋊C₉	—	3	2	no	—
75	C₃²⋊C₉	—	3	2	no	—
76	C₃²⋊C₉	—	3	2	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	1	no	—
82	C₃²⋊C₉	H(3,𝔽₃)	1	1	yes	characteristic
83	C₃²⋊C₉	H(3,𝔽₃)	1	2	yes	—
84	C₃³	—	3	2	no	—
85	C₃³	—	3	1	no	—
86	C₃³	—	3	1	no	—
87	C₃³	—	3	1	no	—
88	C₃³	—	3	2	no	—
89	C₃³	—	3	2	no	—
90	C₃³	—	9	1	no	—
91	C₃³	—	9	1	no	—
92	C₃³	—	9	3	no	—
93	C₃³	—	9	2	no	—
94	C₃³	—	9	1	no	—
95	C₃³	—	9	2	no	—
96	C₃³	—	27	1	no	—
97	C₃³	—	27	1	no	—
98	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
99	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
100	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
101	3₋¹⁺²	—	3	3	no	—
102	3₋¹⁺²	—	9	2	no	—
103	3₋¹⁺²	—	9	6	no	—
104	3₋¹⁺²	—	9	6	no	—
105	3₋¹⁺²	—	9	6	no	—
106	3₋¹⁺²	—	27	6	no	—
107	H(3,𝔽₃)	—	3	2	no	—
108	H(3,𝔽₃)	—	3	2	no	—
109	H(3,𝔽₃)	—	3	2	no	—
110	H(3,𝔽₃)	—	3	2	no	—
111	H(3,𝔽₃)	—	9	6	no	—
112	H(3,𝔽₃)	—	9	6	no	—
113	H(3,𝔽₃)	—	9	4	no	—
114	H(3,𝔽₃)	—	9	2	no	—
115	H(3,𝔽₃)	—	9	2	no	—
116	H(3,𝔽₃)	—	9	4	no	—
117	H(3,𝔽₃)	—	27	2	no	—
118	H(3,𝔽₃)	#731	1	3	yes	complemented
119	C₃×C₉	—	3	6	no	—
120	C₃×C₉	—	3	6	no	—
121	C₃×C₉	—	3	1	no	—
122	C₃×C₉	—	3	2	no	—
123	C₃×C₉	—	9	4	no	—
124	C₃×C₉	—	9	1	no	—
125	C₃×C₉	—	9	1	no	—
126	C₃×C₉	—	9	3	no	—
127	C₃×C₉	—	9	3	no	—
128	C₃×C₉	—	9	3	no	—
129	C₃×C₉	—	9	1	no	—
130	C₃×C₉	—	9	2	no	—
131	C₃×C₉	—	9	2	no	—
132	C₃×C₉	—	9	1	no	—
133	C₃×C₉	—	9	2	no	—
134	C₃×C₉	—	27	1	no	—
135	C₃×C₉	—	27	3	no	—
136	C₃×C₉	—	27	2	no	—
137	C₃²	—	3	1	no	—
138	C₃²	—	3	1	no	—
139	C₃²	—	3	1	no	—
140	C₃²	—	3	1	no	—
141	C₃²	—	3	1	no	—
142	C₃²	—	3	1	no	—
143	C₃²	—	3	3	no	—
144	C₃²	—	3	2	no	—
145	C₃²	—	3	2	no	—
146	C₃²	—	3	2	no	—
147	C₃²	—	3	2	no	—
148	C₃²	—	9	2	no	—
149	C₃²	—	9	4	no	—
150	C₃²	—	9	1	no	—
151	C₃²	—	9	2	no	—
152	C₃²	—	9	1	no	—
153	C₃²	—	9	1	no	—
154	C₃²	—	9	1	no	—
155	C₃²	—	9	2	no	—
156	C₃²	—	9	2	no	—
157	C₃²	—	9	2	no	—
158	C₃²	—	9	2	no	—
159	C₃²	—	27	6	no	—
160	C₃²	—	27	3	no	—
161	C₃²	—	27	2	no	—
162	C₃²	—	27	2	no	—
163	C₃²	—	27	2	no	—
164	C₃²	—	27	2	no	—
165	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
166	C₃²	C₃×#731	1	1	yes	characteristic
167	C₃²	#6908	1	1	yes	characteristic
168	C₃²	#6910	1	2	yes	—
169	C₃²	#6913	1	2	yes	—
170	C₉	—	9	3	no	—
171	C₉	—	9	3	no	—
172	C₉	—	9	1	no	—
173	C₉	—	9	6	no	—
174	C₉	—	9	2	no	—
175	C₉	—	27	1	no	—
176	C₉	—	27	2	no	—
177	C₉	—	27	2	no	—
178	C₃	—	3	2	no	—
179	C₃	—	3	1	no	—
180	C₃	—	3	1	no	—
181	C₃	—	3	2	no	—
182	C₃	—	9	2	no	—
183	C₃	—	9	3	no	—
184	C₃	—	27	2	no	—
185	C₃	—	81	2	no	—
186	C₃	#10624745	1	1	yes	characteristic
187	C₃	#10624746	1	1	yes	characteristic
188	C₃	UT(4,𝔽₃)	1	1	yes	characteristic
189	C₃	#10624763	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	#10624740	C₃
LCS	G ▷ C₃⁴ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. #10624897	C₃	semidirect
2. #10624820	C₃	semidirect
3. #10624779	C₃	semidirect
5. #10624731	C₃	semidirect
6. C₃×3₋¹⁺²⋊C₉	C₃	semidirect
7. C₃×H(3,𝔽₃)⋊C₉	C₃	semidirect
8. #10624609	C₃	semidirect
9. #10624605	C₃	semidirect
118. H(3,𝔽₃)	#731	semidirect

Maximal subgroups

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

subgroup	index
#10624897	3
#10624820	3
#10624779	3
#10624740	3
#10624731	3
C₃×3₋¹⁺²⋊C₉	3
C₃×H(3,𝔽₃)⋊C₉	3
#10624609	3
#10624605	3