Finite Group Databasebeta

Subgroups — #14550451

Order 2187.

689 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 689 conjugacy classes of subgroups — too many to draw. Only its 128 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	C₃²×#731	C₃	1	1	yes	maximal, characteristic
2	C₃×C₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic
3	C₃×3₋¹⁺²⋊C₉	C₃	1	4	yes	maximal, complemented
4	C₃×H(3,𝔽₃)⋊C₉	C₃	1	2	yes	maximal
5	C₃×#6867	C₃	1	2	yes	maximal, complemented
6	#10624558	C₃	1	2	yes	maximal, complemented
7	C₃×#6855	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
9	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
10	C₃³×C₉	—	3	1	no	—
11	C₃³×C₉	C₃²	1	1	yes	characteristic
12	C₃³×C₉	C₃²	1	2	yes	—
13	C₃³×C₉	C₃²	1	1	yes	characteristic
14	C₃×#731	—	3	3	no	—
15	C₃×#731	—	3	6	no	—
16	C₃×#731	C₉	1	9	yes	complemented
17	C₉×H(3,𝔽₃)	—	3	3	no	—
18	C₉×H(3,𝔽₃)	—	3	6	no	—
19	C₃×C₃²⋊C₉	—	3	2	no	—
20	C₃×C₃²⋊C₉	C₃²	1	4	yes	—
21	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
22	3₋¹⁺²⋊C₉	—	3	12	no	—
23	H(3,𝔽₃)⋊C₉	—	3	6	no	—
24	#6867	—	3	6	no	—
25	#6855	—	3	2	no	—
26	#6855	—	3	1	no	—
27	#6855	—	3	4	no	—
28	#6855	—	3	2	no	—
29	C₃⁴	—	3	1	no	—
30	C₃⁴	C₃³	1	1	yes	characteristic
31	C₃×3₋¹⁺²	—	3	4	no	—
32	C₃×3₋¹⁺²	—	3	4	no	—
33	C₃×3₋¹⁺²	—	3	2	no	—
34	C₃×3₋¹⁺²	—	3	4	no	—
35	C₃×3₋¹⁺²	—	3	8	no	—
36	C₃×3₋¹⁺²	C₃×C₉	1	6	yes	complemented
37	C₃×H(3,𝔽₃)	—	3	2	no	—
38	C₃×H(3,𝔽₃)	—	3	4	no	—
39	C₃×H(3,𝔽₃)	—	3	2	no	—
40	C₃×H(3,𝔽₃)	C₃×C₉	1	3	yes	—
41	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
42	C₃×H(3,𝔽₃)	3₋¹⁺²	1	6	yes	—
43	C₃²×C₉	—	3	9	no	—
44	C₃²×C₉	—	3	2	no	—
45	C₃²×C₉	—	3	1	no	—
46	C₃²×C₉	—	3	1	no	—
47	C₃²×C₉	—	3	2	no	—
48	C₃²×C₉	—	3	2	no	—
49	C₃²×C₉	—	3	1	no	—
50	C₃²×C₉	—	3	2	no	—
51	C₃²×C₉	—	3	2	no	—
52	C₃²×C₉	—	3	2	no	—
53	C₃²×C₉	—	3	4	no	—
54	C₃²×C₉	—	3	2	no	—
55	C₃²×C₉	—	3	3	no	—
56	C₃²×C₉	—	3	6	no	—
57	C₃²×C₉	—	3	2	no	—
58	C₃²×C₉	—	3	4	no	—
59	C₃²×C₉	—	3	4	no	—
60	C₃²×C₉	—	9	2	no	—
61	C₃²×C₉	—	9	6	no	—
62	C₃²×C₉	—	9	3	no	—
63	C₃²×C₉	—	9	4	no	—
64	C₃²×C₉	C₃×C₉	1	3	yes	complemented
65	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
66	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
67	#731	—	3	27	no	—
68	C₃²⋊C₉	—	3	18	no	—
69	C₃²⋊C₉	—	3	2	no	—
70	C₃²⋊C₉	—	3	2	no	—
71	C₃²⋊C₉	—	3	4	no	—
72	C₃²⋊C₉	—	3	4	no	—
73	C₃²⋊C₉	—	3	4	no	—
74	C₃²⋊C₉	H(3,𝔽₃)	1	6	yes	—
75	C₃³	—	3	4	no	—
76	C₃³	—	3	1	no	—
77	C₃³	—	3	2	no	—
78	C₃³	—	3	2	no	—
79	C₃³	—	3	3	no	—
80	C₃³	—	3	3	no	—
81	C₃³	—	3	6	no	—
82	C₃³	—	9	4	no	—
83	C₃³	—	9	1	no	—
84	C₃³	—	9	2	no	—
85	C₃³	—	9	2	no	—
86	C₃³	C₃²×C₉	1	1	yes	characteristic
87	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
88	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
89	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
90	C₃³	C₃×3₋¹⁺²	1	2	yes	—
91	C₃³	#737	1	4	yes	—
92	C₃³	#737	1	2	yes	—
93	3₋¹⁺²	—	3	6	no	—
94	3₋¹⁺²	—	3	12	no	—
95	3₋¹⁺²	—	9	8	no	—
96	3₋¹⁺²	—	9	4	no	—
97	H(3,𝔽₃)	—	3	6	no	—
98	H(3,𝔽₃)	—	3	12	no	—
99	H(3,𝔽₃)	—	3	6	no	—
100	H(3,𝔽₃)	C₃²⋊C₉	1	9	yes	—
101	C₃×C₉	—	3	1	no	—
102	C₃×C₉	—	3	2	no	—
103	C₃×C₉	—	3	3	no	—
104	C₃×C₉	—	3	6	no	—
105	C₃×C₉	—	3	3	no	—
106	C₃×C₉	—	3	6	no	—
107	C₃×C₉	—	3	2	no	—
108	C₃×C₉	—	3	6	no	—
109	C₃×C₉	—	3	2	no	—
110	C₃×C₉	—	3	6	no	—
111	C₃×C₉	—	3	4	no	—
112	C₃×C₉	—	9	3	no	—
113	C₃×C₉	—	9	6	no	—
114	C₃×C₉	—	9	27	no	—
115	C₃×C₉	—	9	4	no	—
116	C₃×C₉	—	9	1	no	—
117	C₃×C₉	—	9	2	no	—
118	C₃×C₉	—	9	2	no	—
119	C₃×C₉	—	9	4	no	—
120	C₃×C₉	—	9	2	no	—
121	C₃×C₉	—	9	4	no	—
122	C₃×C₉	—	9	2	no	—
123	C₃×C₉	—	9	2	no	—
124	C₃×C₉	—	9	4	no	—
125	C₃×C₉	—	9	6	no	—
126	C₃×C₉	—	9	12	no	—
127	C₃×C₉	—	9	8	no	—
128	C₃×C₉	—	9	4	no	—
129	C₃×C₉	—	9	4	no	—
130	C₃×C₉	#731	1	3	yes	complemented
131	C₃²	—	3	2	no	—
132	C₃²	—	3	4	no	—
133	C₃²	—	3	4	no	—
134	C₃²	—	3	8	no	—
135	C₃²	—	3	1	no	—
136	C₃²	—	3	2	no	—
137	C₃²	—	3	2	no	—
138	C₃²	—	3	4	no	—
139	C₃²	—	3	1	no	—
140	C₃²	—	3	2	no	—
141	C₃²	—	3	2	no	—
142	C₃²	—	3	4	no	—
143	C₃²	—	3	9	no	—
144	C₃²	—	9	6	no	—
145	C₃²	—	9	12	no	—
146	C₃²	—	9	3	no	—
147	C₃²	—	9	6	no	—
148	C₃²	—	9	3	no	—
149	C₃²	—	9	6	no	—
150	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
151	C₃²	C₉×H(3,𝔽₃)	1	2	yes	—
152	C₃²	C₉×H(3,𝔽₃)	1	1	yes	characteristic
153	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
154	C₃²	C₉⋊H(3,𝔽₃)	1	4	yes	—
155	C₃²	C₉⋊H(3,𝔽₃)	1	2	yes	—
156	C₃²	#6893	1	2	yes	—
157	C₃²	C₃×#731	1	1	yes	characteristic
158	C₃²	#6908	1	2	yes	—
159	C₃²	#6910	1	2	yes	—
160	C₃²	#6913	1	4	yes	—
161	C₉	—	3	3	no	—
162	C₉	—	9	3	no	—
163	C₉	—	9	6	no	—
164	C₉	—	9	2	no	—
165	C₉	—	9	6	no	—
166	C₉	—	9	6	no	—
167	C₉	—	27	4	no	—
168	C₃	—	3	1	no	—
169	C₃	—	3	2	no	—
170	C₃	—	3	4	no	—
171	C₃	—	3	2	no	—
172	C₃	—	9	9	no	—
173	C₃	#10624555	1	1	yes	characteristic
174	C₃	C₉×#731	1	1	yes	characteristic
175	C₃	#731⋊C₉	1	2	yes	—
176	C₃	#10624741	1	1	yes	characteristic
177	C₃	#10624746	1	2	yes	—
178	C₃	#10624793	1	2	yes	—
179	C₃	#10624796	1	4	yes	—

	subgroup	G/H
center, socle	C₃³	C₃×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 ◁ C₃³ ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. C₃×3₋¹⁺²⋊C₉	C₃	semidirect
5. C₃×#6867	C₃	semidirect
6. #10624558	C₃	semidirect
7. C₃×#6855	C₃	semidirect
16. C₃×#731	C₉	semidirect
36. C₃×3₋¹⁺²	C₃×C₉	semidirect
64. C₃²×C₉	C₃×C₉	semidirect
130. C₃×C₉	#731	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#731	3
C₃×C₉×H(3,𝔽₃)	3
C₃×3₋¹⁺²⋊C₉	3
C₃×H(3,𝔽₃)⋊C₉	3
C₃×#6867	3
#10624558	3
C₃×#6855	3