Finite Group Databasebeta

Subgroups — #14549833

Order 2187.

383 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 71 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×#6856	C₃	1	1	yes	maximal, characteristic
2	#10624555	C₃	1	2	yes	maximal, complemented
3	#10624548	C₃	1	1	yes	maximal, characteristic, complemented
4	C₉×C₃²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624451	C₃	1	1	yes	maximal, characteristic, complemented
6	#10624451	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624451	C₃	1	2	yes	maximal, complemented
8	#10624449	C₃	1	2	yes	maximal, complemented
9	#10624449	C₃	1	2	yes	maximal, complemented
10	C₃²×H(3,𝔽₃)	C₃²	1	2	yes	—
11	C₃³×C₉	C₃²	1	1	yes	characteristic
12	C₃×C₉⋊C₉	—	3	1	no	—
13	C₃×C₃²⋊C₉	—	3	2	no	—
14	C₃×C₃²⋊C₉	—	3	2	no	—
15	C₃×C₃²⋊C₉	—	3	2	no	—
16	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
17	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
18	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
19	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
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×C₉²	—	3	1	no	—
24	#6856	—	3	3	no	—
25	#6855	—	3	1	no	—
26	#6855	—	3	1	no	—
27	#6855	—	3	1	no	—
28	#6855	—	3	1	no	—
29	C₃⁴	—	3	2	no	—
30	C₃⁴	C₃³	1	1	yes	characteristic
31	C₃×H(3,𝔽₃)	—	3	6	no	—
32	C₃×H(3,𝔽₃)	—	3	2	no	—
33	C₃×H(3,𝔽₃)	—	3	4	no	—
34	C₃×H(3,𝔽₃)	—	9	2	no	—
35	C₃×H(3,𝔽₃)	—	9	2	no	—
36	C₃²×C₉	—	3	1	no	—
37	C₃²×C₉	—	3	1	no	—
38	C₃²×C₉	—	3	1	no	—
39	C₃²×C₉	—	3	3	no	—
40	C₃²×C₉	—	3	3	no	—
41	C₃²×C₉	—	3	2	no	—
42	C₃²×C₉	—	3	1	no	—
43	C₃²×C₉	—	3	2	no	—
44	C₃²×C₉	—	3	2	no	—
45	C₃²×C₉	—	9	2	no	—
46	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
47	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
48	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
49	C₉⋊C₉	—	9	3	no	—
50	C₃²⋊C₉	—	3	2	no	—
51	C₃²⋊C₉	—	3	3	no	—
52	C₃²⋊C₉	—	3	3	no	—
53	C₃²⋊C₉	—	3	1	no	—
54	C₃²⋊C₉	—	3	2	no	—
55	C₃²⋊C₉	—	9	6	no	—
56	C₃²⋊C₉	—	9	6	no	—
57	C₃²⋊C₉	—	9	6	no	—
58	C₃²⋊C₉	—	9	2	no	—
59	C₃²⋊C₉	—	9	2	no	—
60	C₃²⋊C₉	—	9	2	no	—
61	C₃²⋊C₉	3₋¹⁺²	1	3	yes	—
62	C₉²	—	9	3	no	—
63	C₃³	—	3	2	no	—
64	C₃³	—	3	2	no	—
65	C₃³	—	3	1	no	—
66	C₃³	—	3	1	no	—
67	C₃³	—	3	1	no	—
68	C₃³	—	3	1	no	—
69	C₃³	—	3	2	no	—
70	C₃³	—	3	2	no	—
71	C₃³	—	3	6	no	—
72	C₃³	—	9	2	no	—
73	C₃³	—	9	2	no	—
74	C₃³	—	9	2	no	—
75	C₃³	—	9	2	no	—
76	C₃³	—	9	2	no	—
77	C₃³	—	9	2	no	—
78	C₃³	—	9	2	no	—
79	C₃³	—	9	2	no	—
80	C₃³	—	9	2	no	—
81	C₃³	—	9	2	no	—
82	C₃³	—	9	2	no	—
83	C₃³	—	9	2	no	—
84	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
85	C₃³	C₃×3₋¹⁺²	1	1	yes	characteristic
86	C₃³	#737	1	2	yes	—
87	H(3,𝔽₃)	—	9	6	no	—
88	H(3,𝔽₃)	—	9	12	no	—
89	C₃×C₉	—	3	1	no	—
90	C₃×C₉	—	3	1	no	—
91	C₃×C₉	—	3	2	no	—
92	C₃×C₉	—	3	2	no	—
93	C₃×C₉	—	3	3	no	—
94	C₃×C₉	—	3	1	no	—
95	C₃×C₉	—	3	1	no	—
96	C₃×C₉	—	3	3	no	—
97	C₃×C₉	—	3	3	no	—
98	C₃×C₉	—	3	2	no	—
99	C₃×C₉	—	9	9	no	—
100	C₃×C₉	—	9	2	no	—
101	C₃×C₉	—	9	2	no	—
102	C₃×C₉	—	9	1	no	—
103	C₃×C₉	—	9	1	no	—
104	C₃×C₉	—	9	1	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	2	no	—
110	C₃×C₉	—	9	2	no	—
111	C₃×C₉	—	9	2	no	—
112	C₃×C₉	—	9	2	no	—
113	C₃×C₉	—	9	1	no	—
114	C₃×C₉	—	9	2	no	—
115	C₃×C₉	—	9	1	no	—
116	C₃×C₉	—	9	1	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	2	no	—
122	C₃×C₉	#732	1	3	yes	—
123	C₃×C₉	#732	1	3	yes	—
124	C₃²	—	3	1	no	—
125	C₃²	—	3	2	no	—
126	C₃²	—	3	1	no	—
127	C₃²	—	3	2	no	—
128	C₃²	—	3	2	no	—
129	C₃²	—	3	4	no	—
130	C₃²	—	9	1	no	—
131	C₃²	—	9	1	no	—
132	C₃²	—	9	1	no	—
133	C₃²	—	9	2	no	—
134	C₃²	—	9	2	no	—
135	C₃²	—	9	2	no	—
136	C₃²	—	9	6	no	—
137	C₃²	—	9	6	no	—
138	C₃²	—	9	6	no	—
139	C₃²	—	9	6	no	—
140	C₃²	—	27	2	no	—
141	C₃²	—	27	2	no	—
142	C₃²	—	27	2	no	—
143	C₃²	—	27	2	no	—
144	C₃²	—	27	2	no	—
145	C₃²	—	27	2	no	—
146	C₃²	—	27	2	no	—
147	C₃²	—	27	2	no	—
148	C₃²	—	27	2	no	—
149	C₃²	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
150	C₃²	C₃×#732	1	1	yes	characteristic
151	C₃²	C₃×#732	1	1	yes	characteristic
152	C₃²	#6908	1	1	yes	characteristic
153	C₃²	#6908	1	1	yes	characteristic
154	C₃²	#6908	1	2	yes	—
155	C₃²	#6909	1	2	yes	—
156	C₃²	#6913	1	2	yes	—
157	C₃²	#6913	1	2	yes	—
158	C₉	—	3	3	no	—
159	C₉	—	3	3	no	—
160	C₉	—	9	3	no	—
161	C₉	—	9	3	no	—
162	C₉	—	9	1	no	—
163	C₉	—	9	3	no	—
164	C₉	—	27	1	no	—
165	C₉	—	27	2	no	—
166	C₉	—	27	2	no	—
167	C₉	—	27	2	no	—
168	C₃	—	9	1	no	—
169	C₃	—	9	2	no	—
170	C₃	—	27	6	no	—
171	C₃	#10624585	1	1	yes	characteristic
172	C₃	#10624599	1	1	yes	characteristic
173	C₃	#10624601	1	1	yes	characteristic
174	C₃	#10624601	1	2	yes	—
175	C₃	#10624623	1	2	yes	—
176	C₃	#10624651	1	2	yes	—
177	C₃	#10624786	1	1	yes	characteristic
178	C₃	#10624792	1	1	yes	characteristic
179	C₃	#10624801	1	2	yes	—

	subgroup	G/H
center, socle	C₃³	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	C₃×#6856	C₃
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. #10624555	C₃	semidirect
3. #10624548	C₃	semidirect
4. C₉×C₃²⋊C₉	C₃	semidirect
5. #10624451	C₃	semidirect
6. #10624451	C₃	semidirect
7. #10624451	C₃	semidirect
8. #10624449	C₃	semidirect
9. #10624449	C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6856	3
#10624555	3
#10624548	3
C₉×C₃²⋊C₉	3
#10624451	3
#10624449	3