Finite Group Databasebeta

Subgroups — #14554419

Order 2187.

942 conjugacy classes of subgroups; only the 281 normal subgroups are shown.

normal subgroups only

This group has 942 conjugacy classes of subgroups — too many to draw. Only its 281 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×#6909	C₃	1	1	yes	maximal, characteristic
2	C₃×#6894	C₃	1	2	yes	maximal, complemented
3	C₃×C₉⋊H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×C₉×H(3,𝔽₃)	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624788	C₃	1	18	yes	maximal, complemented
6	#10624784	C₃	1	9	yes	maximal, complemented
7	#10624699	C₃	1	4	yes	maximal, complemented
8	#10624698	C₃	1	2	yes	maximal, complemented
9	#10624689	C₃	1	2	yes	maximal, complemented
10	C₃²×3₋¹⁺²	—	3	1	no	—
11	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
12	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
13	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
14	C₃³×C₉	C₃²	1	1	yes	characteristic
15	C₃³×C₉	C₃²	1	3	yes	complemented
16	#6909	—	3	9	no	—
17	C₃×AΣL(1,𝔽₂₇)	C₃²	1	9	yes	—
18	#6894	—	3	2	no	—
19	#6894	—	3	2	no	—
20	#6894	—	3	4	no	—
21	#6894	—	3	4	no	—
22	#6894	C₃²	1	6	yes	complemented
23	#6894	C₃²	1	12	yes	complemented
24	C₉⋊H(3,𝔽₃)	—	3	2	no	—
25	C₉⋊H(3,𝔽₃)	—	3	4	no	—
26	C₉⋊H(3,𝔽₃)	C₃²	1	3	yes	—
27	C₉⋊H(3,𝔽₃)	C₃²	1	6	yes	—
28	C₉×H(3,𝔽₃)	—	3	1	no	—
29	C₉×H(3,𝔽₃)	—	3	2	no	—
30	C₉×H(3,𝔽₃)	—	3	2	no	—
31	C₉×H(3,𝔽₃)	—	3	4	no	—
32	C₃×C₉⋊C₉	—	3	2	no	—
33	C₃×C₉⋊C₉	—	3	4	no	—
34	C₃×C₃²⋊C₉	—	3	2	no	—
35	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
36	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
37	C₃×C₃²⋊C₉	C₃²	1	4	yes	—
38	C₃×C₃²⋊C₉	C₃²	1	6	yes	complemented
39	3₋¹⁺²⋊C₉	C₃²	1	36	yes	complemented
40	H(3,𝔽₃)⋊C₉	C₃²	1	18	yes	—
41	#6866	C₃²	1	18	yes	complemented
42	C₃⁴	—	3	1	no	—
43	C₃⁴	C₃³	1	3	yes	—
44	C₃⁴	C₃³	1	1	yes	characteristic
45	C₃×3₋¹⁺²	—	3	2	no	—
46	C₃×3₋¹⁺²	—	3	2	no	—
47	C₃×3₋¹⁺²	—	3	6	no	—
48	C₃×3₋¹⁺²	—	3	3	no	—
49	C₃×3₋¹⁺²	—	3	6	no	—
50	C₃×3₋¹⁺²	—	3	12	no	—
51	C₃×3₋¹⁺²	—	9	9	no	—
52	C₃×3₋¹⁺²	C₃³	1	6	yes	—
53	C₃×H(3,𝔽₃)	—	3	2	no	—
54	C₃×H(3,𝔽₃)	—	3	1	no	—
55	C₃×H(3,𝔽₃)	—	3	6	no	—
56	C₃×H(3,𝔽₃)	—	3	3	no	—
57	C₃×H(3,𝔽₃)	—	3	6	no	—
58	C₃×H(3,𝔽₃)	—	3	2	no	—
59	C₃×H(3,𝔽₃)	3₋¹⁺²	1	9	yes	complemented
60	C₃×H(3,𝔽₃)	C₃³	1	3	yes	—
61	C₃²×C₉	—	3	2	no	—
62	C₃²×C₉	—	3	2	no	—
63	C₃²×C₉	—	3	4	no	—
64	C₃²×C₉	—	3	6	no	—
65	C₃²×C₉	—	3	3	no	—
66	C₃²×C₉	—	9	3	no	—
67	C₃²×C₉	—	9	9	no	—
68	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
69	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
70	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
71	C₃²×C₉	H(3,𝔽₃)	1	4	yes	—
72	C₃²×C₉	C₃³	1	1	yes	characteristic
73	C₃²×C₉	C₃³	1	6	yes	—
74	C₃²×C₉	C₃³	1	2	yes	—
75	AΣL(1,𝔽₂₇)	—	3	27	no	—
76	C₉⋊C₉	—	3	12	no	—
77	C₉⋊C₉	—	3	12	no	—
78	C₉⋊C₉	—	3	12	no	—
79	C₉⋊C₉	—	3	12	no	—
80	C₉⋊C₉	—	3	6	no	—
81	C₃²⋊C₉	—	3	4	no	—
82	C₃²⋊C₉	—	3	2	no	—
83	C₃²⋊C₉	—	3	4	no	—
84	C₃²⋊C₉	—	3	4	no	—
85	C₃²⋊C₉	—	3	4	no	—
86	C₃²⋊C₉	—	3	12	no	—
87	C₃²⋊C₉	—	3	6	no	—
88	C₃²⋊C₉	—	3	12	no	—
89	C₃²⋊C₉	—	3	6	no	—
90	C₃²⋊C₉	C₃³	1	12	yes	—
91	C₃²⋊C₉	C₃³	1	6	yes	—
92	C₃³	—	3	3	no	—
93	C₃³	—	3	6	no	—
94	C₃³	—	3	3	no	—
95	C₃³	—	9	1	no	—
96	C₃³	—	9	2	no	—
97	C₃³	—	9	3	no	—
98	C₃³	—	9	6	no	—
99	C₃³	—	9	3	no	—
100	C₃³	—	9	6	no	—
101	C₃³	—	9	3	no	—
102	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
103	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
104	C₃³	C₃×3₋¹⁺²	1	3	yes	—
105	C₃³	C₃×3₋¹⁺²	1	9	yes	complemented
106	C₃³	#737	1	6	yes	—
107	C₃³	C₃⁴	1	1	yes	characteristic
108	3₋¹⁺²	—	3	6	no	—
109	3₋¹⁺²	—	3	12	no	—
110	3₋¹⁺²	—	9	18	no	—
111	3₋¹⁺²	—	9	6	no	—
112	3₋¹⁺²	—	9	6	no	—
113	3₋¹⁺²	—	9	9	no	—
114	H(3,𝔽₃)	—	3	18	no	—
115	H(3,𝔽₃)	—	3	9	no	—
116	H(3,𝔽₃)	—	3	3	no	—
117	H(3,𝔽₃)	—	3	6	no	—
118	H(3,𝔽₃)	—	9	6	no	—
119	C₃×C₉	—	3	3	no	—
120	C₃×C₉	—	3	6	no	—
121	C₃×C₉	—	3	3	no	—
122	C₃×C₉	—	3	6	no	—
123	C₃×C₉	—	3	6	no	—
124	C₃×C₉	—	3	12	no	—
125	C₃×C₉	—	3	6	no	—
126	C₃×C₉	—	9	2	no	—
127	C₃×C₉	—	9	4	no	—
128	C₃×C₉	—	9	2	no	—
129	C₃×C₉	—	9	18	no	—
130	C₃×C₉	—	9	12	no	—
131	C₃×C₉	—	9	6	no	—
132	C₃×C₉	—	9	6	no	—
133	C₃×C₉	—	9	9	no	—
134	C₃×C₉	—	9	6	no	—
135	C₃×C₉	C₃×H(3,𝔽₃)	1	4	yes	—
136	C₃×C₉	C₃×H(3,𝔽₃)	1	2	yes	—
137	C₃×C₉	C₃×H(3,𝔽₃)	1	2	yes	—
138	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
139	C₃²	—	3	1	no	—
140	C₃²	—	3	1	no	—
141	C₃²	—	3	2	no	—
142	C₃²	—	3	4	no	—
143	C₃²	—	3	2	no	—
144	C₃²	—	3	2	no	—
145	C₃²	—	3	9	no	—
146	C₃²	—	9	3	no	—
147	C₃²	—	9	6	no	—
148	C₃²	—	9	9	no	—
149	C₃²	—	9	18	no	—
150	C₃²	—	9	3	no	—
151	C₃²	—	9	6	no	—
152	C₃²	—	9	3	no	—
153	C₃²	—	9	3	no	—
154	C₃²	—	9	6	no	—
155	C₃²	—	27	9	no	—
156	C₃²	C₉⋊H(3,𝔽₃)	1	6	yes	—
157	C₃²	C₉⋊H(3,𝔽₃)	1	3	yes	—
158	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
159	C₃²	C₃²×3₋¹⁺²	1	1	yes	characteristic
160	C₃²	C₃×#737	1	2	yes	—
161	C₉	—	3	2	no	—
162	C₉	—	3	4	no	—
163	C₉	—	3	2	no	—
164	C₉	—	9	6	no	—
165	C₉	—	9	6	no	—
166	C₉	—	9	6	no	—
167	C₉	—	9	12	no	—
168	C₉	#6909	1	3	yes	complemented
169	C₃	—	3	1	no	—
170	C₃	—	3	2	no	—
171	C₃	—	3	3	no	—
172	C₃	—	3	6	no	—
173	C₃	—	9	9	no	—
174	C₃	—	27	3	no	—
175	C₃	C₃×C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
176	C₃	C₃×#6909	1	1	yes	characteristic
177	C₃	#10624929	1	2	yes	—

	subgroup	G/H
center, socle	C₃²	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃⁴
prime radical	C₃×#6909	C₃
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₃×#6894	C₃	semidirect
3. C₃×C₉⋊H(3,𝔽₃)	C₃	semidirect
4. C₃×C₉×H(3,𝔽₃)	C₃	semidirect
5. #10624788	C₃	semidirect
6. #10624784	C₃	semidirect
7. #10624699	C₃	semidirect
8. #10624698	C₃	semidirect
9. #10624689	C₃	semidirect
15. C₃³×C₉	C₃²	semidirect
22. #6894	C₃²	semidirect
23. #6894	C₃²	semidirect
38. C₃×C₃²⋊C₉	C₃²	semidirect
39. 3₋¹⁺²⋊C₉	C₃²	semidirect
41. #6866	C₃²	semidirect
59. C₃×H(3,𝔽₃)	3₋¹⁺²	semidirect
105. C₃³	C₃×3₋¹⁺²	semidirect
168. C₉	#6909	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6909	3
C₃×#6894	3
C₃×C₉⋊H(3,𝔽₃)	3
C₃×C₉×H(3,𝔽₃)	3
#10624788	3
#10624784	3
#10624699	3
#10624698	3
#10624689	3