Finite Group Databasebeta

Subgroups — #14549792

Order 2187.

386 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 74 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×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
2	#10624556	C₃	1	2	yes	maximal, complemented
3	#10624549	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624451	C₃	1	2	yes	maximal, complemented
5	#10624451	C₃	1	1	yes	maximal, characteristic, complemented
6	#10624450	C₃	1	2	yes	maximal, complemented
7	#10624450	C₃	1	1	yes	maximal, characteristic, complemented
8	#10624449	C₃	1	1	yes	maximal, characteristic
9	#10624449	C₃	1	2	yes	maximal
10	C₃²×3₋¹⁺²	—	3	1	no	—
11	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
12	C₃³×C₉	C₃²	1	1	yes	characteristic
13	C₉⋊H(3,𝔽₃)	—	3	3	no	—
14	C₉⋊H(3,𝔽₃)	—	3	3	no	—
15	C₉⋊H(3,𝔽₃)	—	3	3	no	—
16	C₃×C₉⋊C₉	—	3	2	no	—
17	C₃×C₃²⋊C₉	—	3	2	no	—
18	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
19	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
20	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
21	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
22	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
23	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
24	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
25	#6855	—	3	2	no	—
26	#6855	—	3	1	no	—
27	#6855	—	3	4	no	—
28	C₃⁴	—	3	1	no	—
29	C₃⁴	C₃³	1	1	yes	characteristic
30	C₃×3₋¹⁺²	—	3	9	no	—
31	C₃×3₋¹⁺²	—	9	3	no	—
32	C₃×3₋¹⁺²	—	9	3	no	—
33	C₃×3₋¹⁺²	—	9	3	no	—
34	C₃×H(3,𝔽₃)	—	3	1	no	—
35	C₃×H(3,𝔽₃)	—	3	1	no	—
36	C₃×H(3,𝔽₃)	—	9	2	no	—
37	C₃×H(3,𝔽₃)	—	9	1	no	—
38	C₃×H(3,𝔽₃)	3₋¹⁺²	1	3	yes	—
39	C₃²×C₉	—	3	1	no	—
40	C₃²×C₉	—	3	1	no	—
41	C₃²×C₉	—	3	2	no	—
42	C₃²×C₉	—	3	1	no	—
43	C₃²×C₉	—	3	1	no	—
44	C₃²×C₉	—	3	1	no	—
45	C₃²×C₉	—	3	3	no	—
46	C₃²×C₉	—	3	1	no	—
47	C₃²×C₉	—	3	2	no	—
48	C₃²×C₉	—	3	2	no	—
49	C₃²×C₉	—	3	2	no	—
50	C₃²×C₉	—	9	1	no	—
51	C₃²×C₉	—	9	1	no	—
52	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
53	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
54	C₉⋊C₉	—	9	6	no	—
55	C₃²⋊C₉	—	3	18	no	—
56	C₃²⋊C₉	—	3	1	no	—
57	C₃²⋊C₉	—	3	1	no	—
58	C₃²⋊C₉	—	3	1	no	—
59	C₃²⋊C₉	—	3	1	no	—
60	C₃²⋊C₉	—	3	6	no	—
61	C₃²⋊C₉	—	9	2	no	—
62	C₃²⋊C₉	—	9	2	no	—
63	C₃²⋊C₉	—	9	2	no	—
64	C₃²⋊C₉	—	9	1	no	—
65	C₃²⋊C₉	3₋¹⁺²	1	3	yes	—
66	C₃²⋊C₉	3₋¹⁺²	1	3	yes	—
67	C₃³	—	3	2	no	—
68	C₃³	—	3	2	no	—
69	C₃³	—	3	2	no	—
70	C₃³	—	3	1	no	—
71	C₃³	—	3	1	no	—
72	C₃³	—	3	2	no	—
73	C₃³	—	3	1	no	—
74	C₃³	—	3	1	no	—
75	C₃³	—	3	3	no	—
76	C₃³	—	9	2	no	—
77	C₃³	—	9	1	no	—
78	C₃³	—	9	2	no	—
79	C₃³	—	9	2	no	—
80	C₃³	—	9	1	no	—
81	C₃³	—	9	2	no	—
82	C₃³	—	9	1	no	—
83	C₃³	—	9	1	no	—
84	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
85	C₃³	C₃×3₋¹⁺²	1	1	yes	characteristic
86	C₃³	C₃×3₋¹⁺²	1	1	yes	characteristic
87	C₃³	C₃×3₋¹⁺²	1	1	yes	characteristic
88	3₋¹⁺²	—	9	27	no	—
89	H(3,𝔽₃)	—	9	3	no	—
90	H(3,𝔽₃)	—	9	3	no	—
91	H(3,𝔽₃)	—	9	3	no	—
92	C₃×C₉	—	3	1	no	—
93	C₃×C₉	—	3	1	no	—
94	C₃×C₉	—	3	1	no	—
95	C₃×C₉	—	3	2	no	—
96	C₃×C₉	—	3	3	no	—
97	C₃×C₉	—	3	3	no	—
98	C₃×C₉	—	3	2	no	—
99	C₃×C₉	—	3	6	no	—
100	C₃×C₉	—	3	2	no	—
101	C₃×C₉	—	3	2	no	—
102	C₃×C₉	—	9	3	no	—
103	C₃×C₉	—	9	3	no	—
104	C₃×C₉	—	9	3	no	—
105	C₃×C₉	—	9	2	no	—
106	C₃×C₉	—	9	2	no	—
107	C₃×C₉	—	9	2	no	—
108	C₃×C₉	—	9	1	no	—
109	C₃×C₉	—	9	1	no	—
110	C₃×C₉	—	9	1	no	—
111	C₃×C₉	—	9	1	no	—
112	C₃×C₉	—	9	1	no	—
113	C₃×C₉	—	9	1	no	—
114	C₃×C₉	—	9	2	no	—
115	C₃×C₉	—	9	2	no	—
116	C₃×C₉	—	9	2	no	—
117	C₃×C₉	—	9	2	no	—
118	C₃×C₉	—	9	2	no	—
119	C₃×C₉	—	9	2	no	—
120	C₃×C₉	—	9	2	no	—
121	C₃×C₉	—	9	1	no	—
122	C₃×C₉	—	9	2	no	—
123	C₃×C₉	—	9	2	no	—
124	C₃×C₉	—	9	2	no	—
125	C₃×C₉	—	9	2	no	—
126	C₃×C₉	—	9	2	no	—
127	C₃×C₉	—	9	2	no	—
128	C₃×C₉	—	9	1	no	—
129	C₃×C₉	#732	1	3	yes	—
130	C₃²	—	3	1	no	—
131	C₃²	—	3	1	no	—
132	C₃²	—	3	1	no	—
133	C₃²	—	3	1	no	—
134	C₃²	—	3	1	no	—
135	C₃²	—	3	1	no	—
136	C₃²	—	3	2	no	—
137	C₃²	—	3	2	no	—
138	C₃²	—	3	2	no	—
139	C₃²	—	9	2	no	—
140	C₃²	—	9	1	no	—
141	C₃²	—	9	1	no	—
142	C₃²	—	9	2	no	—
143	C₃²	—	9	2	no	—
144	C₃²	—	9	1	no	—
145	C₃²	—	9	3	no	—
146	C₃²	—	9	3	no	—
147	C₃²	—	9	6	no	—
148	C₃²	—	27	2	no	—
149	C₃²	—	27	1	no	—
150	C₃²	—	27	1	no	—
151	C₃²	—	27	2	no	—
152	C₃²	—	27	2	no	—
153	C₃²	—	27	1	no	—
154	C₃²	#6891	1	1	yes	characteristic
155	C₃²	C₃×#732	1	1	yes	characteristic
156	C₃²	#6908	1	2	yes	—
157	C₃²	#6909	1	1	yes	characteristic
158	C₃²	#6910	1	1	yes	characteristic
159	C₃²	#6913	1	2	yes	—
160	C₃²	#6913	1	1	yes	characteristic
161	C₃²	#6913	1	2	yes	—
162	C₃²	#6913	1	2	yes	—
163	C₉	—	3	3	no	—
164	C₉	—	9	2	no	—
165	C₉	—	9	3	no	—
166	C₉	—	9	3	no	—
167	C₉	—	9	6	no	—
168	C₉	—	27	2	no	—
169	C₉	—	27	2	no	—
170	C₉	—	27	2	no	—
171	C₉	—	27	1	no	—
172	C₃	—	9	1	no	—
173	C₃	—	9	1	no	—
174	C₃	—	9	1	no	—
175	C₃	—	27	3	no	—
176	C₃	#10624601	1	2	yes	—
177	C₃	#10624625	1	1	yes	characteristic
178	C₃	#10624638	1	1	yes	characteristic
179	C₃	#10624651	1	2	yes	—
180	C₃	#10624672	1	2	yes	—
181	C₃	#10624677	1	1	yes	characteristic
182	C₃	#10624774	1	1	yes	characteristic
183	C₃	#10624777	1	1	yes	characteristic
184	C₃	#10624780	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. #10624556	C₃	semidirect
3. #10624549	C₃	semidirect
4. #10624451	C₃	semidirect
5. #10624451	C₃	semidirect
6. #10624450	C₃	semidirect
7. #10624450	C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×C₉⋊H(3,𝔽₃)	3
#10624556	3
#10624549	3
#10624451	3
#10624450	3
#10624449	3