Finite Group Databasebeta

Subgroups — #14550441

Order 2187.

552 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 107 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×#6910	C₃	1	1	yes	maximal, characteristic
2	#10624862	C₃	1	1	yes	maximal, characteristic
3	#10624699	C₃	1	2	yes	maximal, complemented
4	#10624699	C₃	1	2	yes	maximal, complemented
5	#10624699	C₃	1	2	yes	maximal, complemented
6	#10624698	C₃	1	1	yes	maximal, characteristic
7	C₃×3₋¹⁺²⋊C₉	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃×H(3,𝔽₃)⋊C₉	C₃	1	1	yes	maximal, characteristic
9	#10624558	C₃	1	2	yes	maximal, complemented
10	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
11	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
12	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
13	C₃³×C₉	C₃²	1	1	yes	characteristic
14	#6910	—	3	6	no	—
15	#6910	C₉	1	9	yes	complemented
16	C₃×#731	—	3	3	no	—
17	C₉×H(3,𝔽₃)	—	3	3	no	—
18	C₉×H(3,𝔽₃)	—	3	6	no	—
19	C₃×C₃²⋊C₉	—	3	3	no	—
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	2	yes	—
24	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
25	3₋¹⁺²⋊C₉	—	3	6	no	—
26	3₋¹⁺²⋊C₉	—	3	6	no	—
27	3₋¹⁺²⋊C₉	—	3	3	no	—
28	3₋¹⁺²⋊C₉	—	3	6	no	—
29	H(3,𝔽₃)⋊C₉	—	3	3	no	—
30	H(3,𝔽₃)⋊C₉	—	3	3	no	—
31	#6855	—	3	6	no	—
32	C₃⁴	—	3	1	no	—
33	C₃⁴	C₃³	1	1	yes	characteristic
34	C₃×3₋¹⁺²	—	3	2	no	—
35	C₃×3₋¹⁺²	—	3	4	no	—
36	C₃×3₋¹⁺²	—	3	2	no	—
37	C₃×3₋¹⁺²	—	3	2	no	—
38	C₃×3₋¹⁺²	—	3	2	no	—
39	C₃×3₋¹⁺²	—	3	1	no	—
40	C₃×3₋¹⁺²	—	3	2	no	—
41	C₃×3₋¹⁺²	—	3	4	no	—
42	C₃×3₋¹⁺²	—	3	4	no	—
43	C₃×3₋¹⁺²	—	3	4	no	—
44	C₃×3₋¹⁺²	—	3	4	no	—
45	C₃×3₋¹⁺²	—	3	2	no	—
46	C₃×3₋¹⁺²	C₃×C₉	1	3	yes	complemented
47	C₃×3₋¹⁺²	C₃×C₉	1	6	yes	complemented
48	C₃×H(3,𝔽₃)	—	3	2	no	—
49	C₃×H(3,𝔽₃)	—	3	4	no	—
50	C₃×H(3,𝔽₃)	—	3	2	no	—
51	C₃×H(3,𝔽₃)	C₃×C₉	1	3	yes	—
52	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
53	C₃×H(3,𝔽₃)	3₋¹⁺²	1	6	yes	—
54	C₃²×C₉	—	3	2	no	—
55	C₃²×C₉	—	3	2	no	—
56	C₃²×C₉	—	3	1	no	—
57	C₃²×C₉	—	3	9	no	—
58	C₃²×C₉	—	3	1	no	—
59	C₃²×C₉	—	3	1	no	—
60	C₃²×C₉	—	3	1	no	—
61	C₃²×C₉	—	3	2	no	—
62	C₃²×C₉	—	3	1	no	—
63	C₃²×C₉	—	3	2	no	—
64	C₃²×C₉	—	9	3	no	—
65	C₃²×C₉	H(3,𝔽₃)	1	6	yes	—
66	#731	—	3	27	no	—
67	C₃²⋊C₉	—	3	18	no	—
68	C₃²⋊C₉	—	3	2	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	2	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	C₃³	—	3	3	no	—
82	C₃³	—	3	3	no	—
83	C₃³	—	3	6	no	—
84	C₃³	—	9	1	no	—
85	C₃³	—	9	1	no	—
86	C₃³	—	9	1	no	—
87	C₃³	—	27	1	no	—
88	C₃³	—	27	1	no	—
89	C₃³	—	27	1	no	—
90	C₃³	C₃²×C₉	1	1	yes	characteristic
91	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
92	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
93	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
94	C₃³	C₃×3₋¹⁺²	1	2	yes	—
95	C₃³	#737	1	4	yes	—
96	C₃³	#737	1	2	yes	—
97	3₋¹⁺²	—	3	6	no	—
98	3₋¹⁺²	—	3	6	no	—
99	3₋¹⁺²	—	3	6	no	—
100	3₋¹⁺²	—	3	3	no	—
101	3₋¹⁺²	—	3	6	no	—
102	3₋¹⁺²	—	9	4	no	—
103	3₋¹⁺²	—	9	4	no	—
104	3₋¹⁺²	—	9	2	no	—
105	3₋¹⁺²	—	9	4	no	—
106	3₋¹⁺²	—	9	4	no	—
107	H(3,𝔽₃)	—	3	6	no	—
108	H(3,𝔽₃)	—	3	12	no	—
109	H(3,𝔽₃)	—	3	6	no	—
110	H(3,𝔽₃)	C₃²⋊C₉	1	9	yes	—
111	C₃×C₉	—	3	6	no	—
112	C₃×C₉	—	3	3	no	—
113	C₃×C₉	—	3	6	no	—
114	C₃×C₉	—	3	6	no	—
115	C₃×C₉	—	3	3	no	—
116	C₃×C₉	—	9	4	no	—
117	C₃×C₉	—	9	1	no	—
118	C₃×C₉	—	9	2	no	—
119	C₃×C₉	—	9	1	no	—
120	C₃×C₉	—	9	4	no	—
121	C₃×C₉	—	9	2	no	—
122	C₃×C₉	—	9	3	no	—
123	C₃×C₉	—	9	3	no	—
124	C₃×C₉	—	9	6	no	—
125	C₃×C₉	—	9	6	no	—
126	C₃×C₉	—	9	2	no	—
127	C₃×C₉	—	9	2	no	—
128	C₃×C₉	—	9	1	no	—
129	C₃×C₉	—	27	1	no	—
130	C₃×C₉	—	27	9	no	—
131	C₃×C₉	—	27	2	no	—
132	C₃×C₉	—	27	1	no	—
133	C₃²	—	3	2	no	—
134	C₃²	—	3	1	no	—
135	C₃²	—	3	1	no	—
136	C₃²	—	3	2	no	—
137	C₃²	—	3	1	no	—
138	C₃²	—	3	1	no	—
139	C₃²	—	3	2	no	—
140	C₃²	—	3	1	no	—
141	C₃²	—	3	1	no	—
142	C₃²	—	3	9	no	—
143	C₃²	—	9	1	no	—
144	C₃²	—	9	2	no	—
145	C₃²	—	9	2	no	—
146	C₃²	—	9	4	no	—
147	C₃²	—	9	3	no	—
148	C₃²	—	9	3	no	—
149	C₃²	—	9	3	no	—
150	C₃²	—	27	3	no	—
151	C₃²	—	27	6	no	—
152	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
153	C₃²	C₉×H(3,𝔽₃)	1	2	yes	—
154	C₃²	C₉×H(3,𝔽₃)	1	1	yes	characteristic
155	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
156	C₃²	C₉⋊H(3,𝔽₃)	1	4	yes	—
157	C₃²	C₉⋊H(3,𝔽₃)	1	2	yes	—
158	C₃²	#6893	1	2	yes	—
159	C₉	—	9	3	no	—
160	C₉	—	9	6	no	—
161	C₉	—	9	6	no	—
162	C₉	—	9	3	no	—
163	C₉	—	9	3	no	—
164	C₉	—	27	2	no	—
165	C₉	—	27	2	no	—
166	C₉	—	27	1	no	—
167	C₃	—	3	1	no	—
168	C₃	—	3	2	no	—
169	C₃	—	3	1	no	—
170	C₃	—	3	2	no	—
171	C₃	—	3	4	no	—
172	C₃	—	3	2	no	—
173	C₃	—	9	9	no	—
174	C₃	#10624555	1	1	yes	characteristic
175	C₃	#10624748	1	1	yes	characteristic
176	C₃	#10624751	1	1	yes	characteristic
177	C₃	#10624752	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₃²	C₃²⋊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 ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
3. #10624699	C₃	semidirect
4. #10624699	C₃	semidirect
5. #10624699	C₃	semidirect
7. C₃×3₋¹⁺²⋊C₉	C₃	semidirect
9. #10624558	C₃	semidirect
15. #6910	C₉	semidirect
46. C₃×3₋¹⁺²	C₃×C₉	semidirect
47. C₃×3₋¹⁺²	C₃×C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6910	3
#10624862	3
#10624699	3
#10624698	3
C₃×3₋¹⁺²⋊C₉	3
C₃×H(3,𝔽₃)⋊C₉	3
#10624558	3