Finite Group Databasebeta

Subgroups — #14554275

Order 2187.

2312 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 2312 conjugacy classes of subgroups — too many to draw. Only its 314 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	C₃³×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#6912	C₃	1	2	yes	maximal, complemented
3	C₃×#6911	C₃	1	1	yes	maximal, characteristic, complemented
4	C₃×#6910	C₃	1	3	yes	maximal, complemented
5	C₃²×AΣL(1,𝔽₂₇)	C₃	1	3	yes	maximal, complemented
6	#10624881	C₃	1	3	yes	maximal, complemented
7	#10624739	C₃	1	18	yes	maximal, complemented
8	#10624734	C₃	1	9	yes	maximal, complemented
9	C₃⁵	C₃²	1	1	yes	characteristic
10	C₃²×3₋¹⁺²	—	3	9	no	—
11	C₃²×3₋¹⁺²	—	3	9	no	—
12	C₃²×3₋¹⁺²	—	3	9	no	—
13	C₃²×3₋¹⁺²	—	3	9	no	—
14	C₃²×3₋¹⁺²	C₃²	1	6	yes	—
15	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
16	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
17	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
18	C₃²×3₋¹⁺²	C₃²	1	6	yes	complemented
19	C₃²×H(3,𝔽₃)	C₃²	1	3	yes	—
20	C₃³×C₉	C₃²	1	3	yes	complemented
21	#6912	—	3	18	no	—
22	#6911	—	3	9	no	—
23	#6910	—	3	27	no	—
24	C₃×#733	C₃²	1	18	yes	complemented
25	C₃×#732	C₃²	1	9	yes	complemented
26	C₃×#731	C₃²	1	27	yes	complemented
27	C₃×AΣL(1,𝔽₂₇)	—	3	27	no	—
28	C₃×AΣL(1,𝔽₂₇)	C₃²	1	9	yes	complemented
29	C₃×AΣL(1,𝔽₂₇)	C₃²	1	18	yes	complemented
30	#6893	—	3	18	no	—
31	#6893	C₃²	1	27	yes	complemented
32	#6891	—	3	9	no	—
33	C₃⁴	—	3	3	no	—
34	C₃⁴	—	3	3	no	—
35	C₃⁴	—	3	3	no	—
36	C₃⁴	—	3	3	no	—
37	C₃⁴	—	9	9	no	—
38	C₃⁴	H(3,𝔽₃)	1	9	yes	—
39	C₃⁴	C₃³	1	1	yes	characteristic
40	C₃⁴	C₃³	1	2	yes	—
41	C₃⁴	C₃³	1	1	yes	characteristic
42	C₃×3₋¹⁺²	—	3	243	no	—
43	C₃×3₋¹⁺²	—	3	3	no	—
44	C₃×3₋¹⁺²	—	3	3	no	—
45	C₃×3₋¹⁺²	—	3	3	no	—
46	C₃×3₋¹⁺²	—	3	3	no	—
47	C₃×3₋¹⁺²	—	3	3	no	—
48	C₃×3₋¹⁺²	—	3	3	no	—
49	C₃×3₋¹⁺²	—	3	3	no	—
50	C₃×3₋¹⁺²	—	3	3	no	—
51	C₃×3₋¹⁺²	—	3	3	no	—
52	C₃×3₋¹⁺²	—	3	18	no	—
53	C₃×3₋¹⁺²	—	3	6	no	—
54	C₃×3₋¹⁺²	—	3	6	no	—
55	C₃×3₋¹⁺²	—	3	18	no	—
56	C₃×3₋¹⁺²	—	3	18	no	—
57	C₃×3₋¹⁺²	—	3	18	no	—
58	C₃×3₋¹⁺²	—	3	18	no	—
59	C₃×3₋¹⁺²	—	3	18	no	—
60	C₃×3₋¹⁺²	—	3	18	no	—
61	C₃×3₋¹⁺²	H(3,𝔽₃)	1	9	yes	—
62	C₃×3₋¹⁺²	H(3,𝔽₃)	1	9	yes	—
63	C₃×3₋¹⁺²	H(3,𝔽₃)	1	9	yes	—
64	C₃×3₋¹⁺²	C₃³	1	18	yes	—
65	C₃×H(3,𝔽₃)	—	3	9	no	—
66	C₃×H(3,𝔽₃)	—	3	9	no	—
67	C₃×H(3,𝔽₃)	—	3	9	no	—
68	C₃×H(3,𝔽₃)	—	3	6	no	—
69	C₃×H(3,𝔽₃)	C₃³	1	9	yes	—
70	C₃²×C₉	—	3	9	no	—
71	C₃²×C₉	—	3	9	no	—
72	C₃²×C₉	—	3	9	no	—
73	C₃²×C₉	—	3	9	no	—
74	C₃²×C₉	—	3	6	no	—
75	C₃²×C₉	C₃³	1	3	yes	—
76	C₃²×C₉	C₃³	1	3	yes	—
77	C₃²×C₉	C₃³	1	3	yes	—
78	#733	—	3	54	no	—
79	#732	—	3	27	no	—
80	#731	—	3	81	no	—
81	AΣL(1,𝔽₂₇)	—	3	27	no	—
82	AΣL(1,𝔽₂₇)	—	3	54	no	—
83	C₃²⋊C₉	—	3	81	no	—
84	C₃³	—	3	9	no	—
85	C₃³	—	3	1	no	—
86	C₃³	—	3	1	no	—
87	C₃³	—	3	1	no	—
88	C₃³	—	3	2	no	—
89	C₃³	—	3	2	no	—
90	C₃³	—	3	2	no	—
91	C₃³	—	3	27	no	—
92	C₃³	—	3	9	no	—
93	C₃³	—	9	3	no	—
94	C₃³	—	9	27	no	—
95	C₃³	—	9	27	no	—
96	C₃³	—	9	6	no	—
97	C₃³	—	9	27	no	—
98	C₃³	—	9	27	no	—
99	C₃³	—	9	9	no	—
100	C₃³	—	27	9	no	—
101	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
102	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
103	C₃³	C₃×H(3,𝔽₃)	1	6	yes	—
104	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
105	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
106	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
107	C₃³	C₃⁴	1	1	yes	characteristic
108	3₋¹⁺²	—	3	162	no	—
109	3₋¹⁺²	—	3	27	no	—
110	3₋¹⁺²	—	3	27	no	—
111	3₋¹⁺²	—	3	27	no	—
112	3₋¹⁺²	—	3	18	no	—
113	3₋¹⁺²	—	3	18	no	—
114	3₋¹⁺²	—	3	18	no	—
115	3₋¹⁺²	—	9	18	no	—
116	3₋¹⁺²	—	9	18	no	—
117	H(3,𝔽₃)	—	3	9	no	—
118	H(3,𝔽₃)	—	3	9	no	—
119	H(3,𝔽₃)	—	3	9	no	—
120	H(3,𝔽₃)	—	9	18	no	—
121	C₃×C₉	—	3	81	no	—
122	C₃×C₉	—	3	3	no	—
123	C₃×C₉	—	3	3	no	—
124	C₃×C₉	—	3	3	no	—
125	C₃×C₉	—	3	3	no	—
126	C₃×C₉	—	3	3	no	—
127	C₃×C₉	—	3	3	no	—
128	C₃×C₉	—	3	3	no	—
129	C₃×C₉	—	3	3	no	—
130	C₃×C₉	—	3	3	no	—
131	C₃×C₉	—	3	18	no	—
132	C₃×C₉	—	9	18	no	—
133	C₃×C₉	C₃×H(3,𝔽₃)	1	9	yes	—
134	C₃×C₉	C₃×H(3,𝔽₃)	1	9	yes	—
135	C₃×C₉	C₃×H(3,𝔽₃)	1	9	yes	—
136	C₃²	—	3	9	no	—
137	C₃²	—	3	9	no	—
138	C₃²	—	3	9	no	—
139	C₃²	—	3	3	no	—
140	C₃²	—	3	9	no	—
141	C₃²	—	3	3	no	—
142	C₃²	—	3	9	no	—
143	C₃²	—	3	6	no	—
144	C₃²	—	9	3	no	—
145	C₃²	—	9	3	no	—
146	C₃²	—	9	3	no	—
147	C₃²	—	9	81	no	—
148	C₃²	—	9	6	no	—
149	C₃²	—	9	6	no	—
150	C₃²	—	9	6	no	—
151	C₃²	—	9	6	no	—
152	C₃²	—	9	9	no	—
153	C₃²	—	9	27	no	—
154	C₃²	—	27	27	no	—
155	C₃²	C₃²⋊H(3,𝔽₃)	1	9	yes	—
156	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
157	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
158	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
159	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
160	C₉	—	3	9	no	—
161	C₉	—	3	9	no	—
162	C₉	—	3	9	no	—
163	C₉	—	9	18	no	—
164	C₃	—	3	9	no	—
165	C₃	—	3	1	no	—
166	C₃	—	3	1	no	—
167	C₃	—	3	1	no	—
168	C₃	—	3	9	no	—
169	C₃	—	9	6	no	—
170	C₃	—	27	9	no	—
171	C₃	C₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
172	C₃	#10624934	1	3	yes	—

	subgroup	G/H
center, socle	C₃²	C₃²×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃³	C₃⁴
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃²×3₋¹⁺² ◁ G
LECS	G ▷ C₃³ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₃³×3₋¹⁺²	C₃	semidirect
2. C₃×#6912	C₃	semidirect
3. C₃×#6911	C₃	semidirect
4. C₃×#6910	C₃	semidirect
5. C₃²×AΣL(1,𝔽₂₇)	C₃	semidirect
6. #10624881	C₃	semidirect
7. #10624739	C₃	semidirect
8. #10624734	C₃	semidirect
18. C₃²×3₋¹⁺²	C₃²	semidirect
20. C₃³×C₉	C₃²	semidirect
24. C₃×#733	C₃²	semidirect
25. C₃×#732	C₃²	semidirect
26. C₃×#731	C₃²	semidirect
28. C₃×AΣL(1,𝔽₂₇)	C₃²	semidirect
29. C₃×AΣL(1,𝔽₂₇)	C₃²	semidirect
31. #6893	C₃²	semidirect

Maximal subgroups

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

subgroup	index
C₃³×3₋¹⁺²	3
C₃×#6912	3
C₃×#6911	3
C₃×#6910	3
C₃²×AΣL(1,𝔽₂₇)	3
#10624881	3
#10624739	3
#10624734	3