Finite Group Databasebeta

Subgroups — #14546959

Order 2187.

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

normal subgroups only

This group has 509 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	C₃×#6862	C₃	1	1	yes	maximal, characteristic, complemented
3	C₃×#6861	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×#6859	C₃	1	2	yes	maximal
5	C₃×#6858	C₃	1	2	yes	maximal, complemented
6	C₃×#6858	C₃	1	2	yes	maximal, complemented
7	C₃×#6857	C₃	1	1	yes	maximal, characteristic
8	#10624558	C₃	1	2	yes	maximal, 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	—	3	1	no	—
13	C₃³×C₉	C₃²	1	1	yes	characteristic
14	C₉×H(3,𝔽₃)	—	3	3	no	—
15	C₉×H(3,𝔽₃)	—	3	3	no	—
16	C₉×H(3,𝔽₃)	—	3	3	no	—
17	C₃×C₃²⋊C₉	—	3	2	no	—
18	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
19	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
20	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
21	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
22	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
23	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
24	#6862	—	3	3	no	—
25	#6861	—	3	3	no	—
26	#6859	—	3	6	no	—
27	#6858	—	3	6	no	—
28	#6858	—	3	6	no	—
29	#6857	—	3	3	no	—
30	#6855	—	3	2	no	—
31	#6855	—	3	1	no	—
32	#6855	—	3	4	no	—
33	#6855	—	3	2	no	—
34	C₃⁴	—	3	1	no	—
35	C₃⁴	C₃³	1	1	yes	characteristic
36	C₃×3₋¹⁺²	—	3	6	no	—
37	C₃×3₋¹⁺²	—	3	2	no	—
38	C₃×3₋¹⁺²	—	3	2	no	—
39	C₃×3₋¹⁺²	—	3	2	no	—
40	C₃×3₋¹⁺²	—	9	4	no	—
41	C₃×H(3,𝔽₃)	—	3	3	no	—
42	C₃×H(3,𝔽₃)	—	3	6	no	—
43	C₃×H(3,𝔽₃)	—	3	1	no	—
44	C₃×H(3,𝔽₃)	—	3	1	no	—
45	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
46	C₃²×C₉	—	3	9	no	—
47	C₃²×C₉	—	3	1	no	—
48	C₃²×C₉	—	3	1	no	—
49	C₃²×C₉	—	3	1	no	—
50	C₃²×C₉	—	3	1	no	—
51	C₃²×C₉	—	3	1	no	—
52	C₃²×C₉	—	3	2	no	—
53	C₃²×C₉	—	3	3	no	—
54	C₃²×C₉	—	3	2	no	—
55	C₃²×C₉	—	3	2	no	—
56	C₃²×C₉	—	3	1	no	—
57	C₃²×C₉	—	3	2	no	—
58	C₃²×C₉	—	9	2	no	—
59	C₃²×C₉	—	9	6	no	—
60	C₃²×C₉	—	9	3	no	—
61	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
62	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
63	C₃²⋊C₉	—	3	18	no	—
64	C₃²⋊C₉	—	3	1	no	—
65	C₃²⋊C₉	—	3	1	no	—
66	C₃²⋊C₉	—	3	1	no	—
67	C₃²⋊C₉	—	3	1	no	—
68	C₃²⋊C₉	—	3	1	no	—
69	C₃²⋊C₉	—	3	2	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	1	no	—
77	C₃²⋊C₉	—	3	2	no	—
78	C₃²⋊C₉	—	3	1	no	—
79	C₃²⋊C₉	—	3	2	no	—
80	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
81	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
82	C₃³	—	3	4	no	—
83	C₃³	—	3	3	no	—
84	C₃³	—	3	1	no	—
85	C₃³	—	3	1	no	—
86	C₃³	—	3	2	no	—
87	C₃³	—	3	3	no	—
88	C₃³	—	3	6	no	—
89	C₃³	—	3	2	no	—
90	C₃³	—	3	2	no	—
91	C₃³	—	9	4	no	—
92	C₃³	—	9	1	no	—
93	C₃³	—	9	2	no	—
94	C₃³	—	9	2	no	—
95	C₃³	C₃×H(3,𝔽₃)	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	3₋¹⁺²	—	9	6	no	—
100	3₋¹⁺²	—	9	6	no	—
101	3₋¹⁺²	—	9	6	no	—
102	H(3,𝔽₃)	—	3	9	no	—
103	H(3,𝔽₃)	—	3	9	no	—
104	H(3,𝔽₃)	—	3	9	no	—
105	C₃×C₉	—	3	1	no	—
106	C₃×C₉	—	3	2	no	—
107	C₃×C₉	—	3	2	no	—
108	C₃×C₉	—	3	3	no	—
109	C₃×C₉	—	3	3	no	—
110	C₃×C₉	—	3	2	no	—
111	C₃×C₉	—	3	6	no	—
112	C₃×C₉	—	3	4	no	—
113	C₃×C₉	—	9	3	no	—
114	C₃×C₉	—	9	3	no	—
115	C₃×C₉	—	9	3	no	—
116	C₃×C₉	—	9	27	no	—
117	C₃×C₉	—	9	4	no	—
118	C₃×C₉	—	9	2	no	—
119	C₃×C₉	—	9	1	no	—
120	C₃×C₉	—	9	1	no	—
121	C₃×C₉	—	9	2	no	—
122	C₃×C₉	—	9	2	no	—
123	C₃×C₉	—	9	4	no	—
124	C₃×C₉	—	9	4	no	—
125	C₃×C₉	—	9	1	no	—
126	C₃×C₉	—	9	2	no	—
127	C₃×C₉	—	9	2	no	—
128	C₃×C₉	—	9	2	no	—
129	C₃×C₉	—	9	1	no	—
130	C₃×C₉	—	9	2	no	—
131	C₃×C₉	—	9	2	no	—
132	C₃×C₉	—	9	4	no	—
133	C₃×C₉	—	9	2	no	—
134	C₃×C₉	—	9	2	no	—
135	C₃×C₉	#731	1	3	yes	—
136	C₃²	—	3	1	no	—
137	C₃²	—	3	1	no	—
138	C₃²	—	3	1	no	—
139	C₃²	—	3	9	no	—
140	C₃²	—	3	1	no	—
141	C₃²	—	3	1	no	—
142	C₃²	—	3	1	no	—
143	C₃²	—	3	2	no	—
144	C₃²	—	3	2	no	—
145	C₃²	—	3	2	no	—
146	C₃²	—	9	1	no	—
147	C₃²	—	9	3	no	—
148	C₃²	—	9	2	no	—
149	C₃²	—	9	6	no	—
150	C₃²	—	9	4	no	—
151	C₃²	—	9	12	no	—
152	C₃²	—	9	2	no	—
153	C₃²	—	9	6	no	—
154	C₃²	—	9	3	no	—
155	C₃²	—	9	6	no	—
156	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
157	C₃²	C₃×#731	1	1	yes	characteristic
158	C₃²	#6908	1	2	yes	—
159	C₃²	#6908	1	1	yes	characteristic
160	C₃²	#6908	1	2	yes	—
161	C₃²	#6910	1	2	yes	—
162	C₃²	#6913	1	4	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	1	no	—
169	C₉	—	27	2	no	—
170	C₉	—	27	2	no	—
171	C₉	—	27	2	no	—
172	C₃	—	9	1	no	—
173	C₃	—	9	1	no	—
174	C₃	—	9	1	no	—
175	C₃	—	9	9	no	—
176	C₃	#10624587	1	1	yes	characteristic
177	C₃	#10624606	1	2	yes	—
178	C₃	#10624624	1	4	yes	—
179	C₃	#10624652	1	2	yes	—
180	C₃	#10624734	1	1	yes	characteristic
181	C₃	#10624741	1	1	yes	characteristic
182	C₃	#10624746	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. C₃×#6862	C₃	semidirect
3. C₃×#6861	C₃	semidirect
5. C₃×#6858	C₃	semidirect
6. C₃×#6858	C₃	semidirect
8. #10624558	C₃	semidirect
9. #10624549	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
C₃×#6862	3
C₃×#6861	3
C₃×#6859	3
C₃×#6858	3
C₃×#6857	3
#10624558	3
#10624549	3