Finite Group Databasebeta

Subgroups — #14554277

Order 2187.

2078 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 2078 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	C₃²×#737	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃×#6910	C₃	1	3	yes	maximal, complemented
3	C₃×#6909	C₃	1	3	yes	maximal, complemented
4	C₃²×#733	C₃	1	2	yes	maximal, complemented
5	C₃²×#732	C₃	1	1	yes	maximal, characteristic, 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	C₃×#737	—	3	3	no	—
10	C₃×#737	—	3	6	no	—
11	C₃×#737	—	3	9	no	—
12	C₃×#737	—	3	9	no	—
13	C₃×#737	—	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	6	yes	complemented
18	C₃²×H(3,𝔽₃)	C₃²	1	3	yes	—
19	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
20	C₃³×C₉	C₃²	1	1	yes	characteristic
21	C₃³×C₉	C₃²	1	3	yes	complemented
22	#6910	—	3	27	no	—
23	#6909	—	3	27	no	—
24	C₃×#733	—	3	18	no	—
25	C₃×#733	C₃²	1	6	yes	complemented
26	C₃×#733	C₃²	1	12	yes	complemented
27	C₃×#732	—	3	9	no	—
28	C₃×#732	C₃²	1	3	yes	—
29	C₃×#732	C₃²	1	6	yes	—
30	C₃×#731	C₃²	1	27	yes	complemented
31	C₃×AΣL(1,𝔽₂₇)	C₃²	1	27	yes	complemented
32	#6893	—	3	18	no	—
33	#6893	C₃²	1	27	yes	complemented
34	#6891	—	3	9	no	—
35	C₃⁴	—	3	3	no	—
36	C₃⁴	C₃³	1	3	yes	—
37	C₃⁴	C₃³	1	1	yes	characteristic
38	#737	—	3	162	no	—
39	#737	—	3	81	no	—
40	C₃×3₋¹⁺²	—	3	3	no	—
41	C₃×3₋¹⁺²	—	3	3	no	—
42	C₃×3₋¹⁺²	—	3	2	no	—
43	C₃×3₋¹⁺²	—	3	1	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	18	no	—
48	C₃×3₋¹⁺²	—	3	6	no	—
49	C₃×3₋¹⁺²	—	3	6	no	—
50	C₃×3₋¹⁺²	—	3	18	no	—
51	C₃×3₋¹⁺²	—	3	18	no	—
52	C₃×3₋¹⁺²	—	3	18	no	—
53	C₃×3₋¹⁺²	—	3	18	no	—
54	C₃×3₋¹⁺²	—	3	18	no	—
55	C₃×3₋¹⁺²	—	3	18	no	—
56	C₃×3₋¹⁺²	H(3,𝔽₃)	1	9	yes	—
57	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
58	C₃×3₋¹⁺²	H(3,𝔽₃)	1	6	yes	—
59	C₃×3₋¹⁺²	C₃³	1	18	yes	—
60	C₃×H(3,𝔽₃)	—	3	3	no	—
61	C₃×H(3,𝔽₃)	—	3	3	no	—
62	C₃×H(3,𝔽₃)	—	3	2	no	—
63	C₃×H(3,𝔽₃)	—	3	1	no	—
64	C₃×H(3,𝔽₃)	—	3	9	no	—
65	C₃×H(3,𝔽₃)	—	3	9	no	—
66	C₃×H(3,𝔽₃)	—	3	9	no	—
67	C₃×H(3,𝔽₃)	—	3	6	no	—
68	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	9	yes	—
69	C₃×H(3,𝔽₃)	C₃³	1	9	yes	—
70	C₃²×C₉	—	3	3	no	—
71	C₃²×C₉	—	3	3	no	—
72	C₃²×C₉	—	3	2	no	—
73	C₃²×C₉	—	3	1	no	—
74	C₃²×C₉	—	3	9	no	—
75	C₃²×C₉	—	3	9	no	—
76	C₃²×C₉	—	3	9	no	—
77	C₃²×C₉	—	3	9	no	—
78	C₃²×C₉	—	3	6	no	—
79	C₃²×C₉	H(3,𝔽₃)	1	9	yes	—
80	C₃²×C₉	C₃³	1	1	yes	characteristic
81	C₃²×C₉	C₃³	1	2	yes	—
82	C₃²×C₉	C₃³	1	3	yes	—
83	C₃²×C₉	C₃³	1	3	yes	—
84	#733	—	3	18	no	—
85	#733	—	3	36	no	—
86	#732	—	3	9	no	—
87	#732	—	3	18	no	—
88	#731	—	3	81	no	—
89	AΣL(1,𝔽₂₇)	—	3	81	no	—
90	C₃²⋊C₉	—	3	81	no	—
91	C₃³	—	3	3	no	—
92	C₃³	—	3	3	no	—
93	C₃³	—	3	3	no	—
94	C₃³	—	3	9	no	—
95	C₃³	—	9	3	no	—
96	C₃³	—	9	9	no	—
97	C₃³	—	9	9	no	—
98	C₃³	—	27	9	no	—
99	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
100	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
101	C₃³	C₃×H(3,𝔽₃)	1	9	yes	—
102	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
103	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
104	C₃³	C₃×H(3,𝔽₃)	1	3	yes	—
105	C₃³	C₃⁴	1	1	yes	characteristic
106	3₋¹⁺²	—	3	162	no	—
107	3₋¹⁺²	—	3	18	no	—
108	3₋¹⁺²	—	3	18	no	—
109	3₋¹⁺²	—	3	9	no	—
110	3₋¹⁺²	—	3	9	no	—
111	3₋¹⁺²	—	3	18	no	—
112	3₋¹⁺²	—	3	18	no	—
113	3₋¹⁺²	—	3	18	no	—
114	3₋¹⁺²	—	9	18	no	—
115	3₋¹⁺²	—	9	18	no	—
116	H(3,𝔽₃)	—	3	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)	—	3	9	no	—
121	H(3,𝔽₃)	—	9	18	no	—
122	C₃×C₉	—	3	81	no	—
123	C₃×C₉	—	3	1	no	—
124	C₃×C₉	—	3	1	no	—
125	C₃×C₉	—	3	2	no	—
126	C₃×C₉	—	3	3	no	—
127	C₃×C₉	—	3	3	no	—
128	C₃×C₉	—	3	2	no	—
129	C₃×C₉	—	3	3	no	—
130	C₃×C₉	—	3	3	no	—
131	C₃×C₉	—	3	3	no	—
132	C₃×C₉	—	3	1	no	—
133	C₃×C₉	—	3	2	no	—
134	C₃×C₉	—	3	3	no	—
135	C₃×C₉	—	3	18	no	—
136	C₃×C₉	—	3	9	no	—
137	C₃×C₉	—	3	18	no	—
138	C₃×C₉	—	9	18	no	—
139	C₃×C₉	C₃×H(3,𝔽₃)	1	9	yes	—
140	C₃×C₉	C₃×H(3,𝔽₃)	1	3	yes	—
141	C₃×C₉	C₃×H(3,𝔽₃)	1	6	yes	—
142	C₃×C₉	C₃×H(3,𝔽₃)	1	9	yes	—
143	C₃²	—	3	3	no	—
144	C₃²	—	3	3	no	—
145	C₃²	—	3	6	no	—
146	C₃²	—	3	9	no	—
147	C₃²	—	9	3	no	—
148	C₃²	—	9	3	no	—
149	C₃²	—	9	3	no	—
150	C₃²	—	9	9	no	—
151	C₃²	—	9	9	no	—
152	C₃²	—	9	9	no	—
153	C₃²	—	9	9	no	—
154	C₃²	—	9	9	no	—
155	C₃²	—	9	27	no	—
156	C₃²	—	27	27	no	—
157	C₃²	C₃²⋊H(3,𝔽₃)	1	6	yes	—
158	C₃²	C₃²⋊H(3,𝔽₃)	1	3	yes	—
159	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
160	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
161	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
162	C₃²	C₃²×H(3,𝔽₃)	1	1	yes	characteristic
163	C₉	—	3	3	no	—
164	C₉	—	3	9	no	—
165	C₉	—	3	9	no	—
166	C₉	—	3	6	no	—
167	C₉	—	9	18	no	—
168	C₃	—	3	1	no	—
169	C₃	—	3	1	no	—
170	C₃	—	3	1	no	—
171	C₃	—	3	3	no	—
172	C₃	—	3	6	no	—
173	C₃	—	9	9	no	—
174	C₃	—	27	9	no	—
175	C₃	C₃×C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
176	C₃	#10624936	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. C₃²×#737	C₃	semidirect
2. C₃×#6910	C₃	semidirect
3. C₃×#6909	C₃	semidirect
4. C₃²×#733	C₃	semidirect
5. C₃²×#732	C₃	semidirect
6. #10624881	C₃	semidirect
7. #10624739	C₃	semidirect
8. #10624734	C₃	semidirect
17. C₃²×3₋¹⁺²	C₃²	semidirect
21. C₃³×C₉	C₃²	semidirect
25. C₃×#733	C₃²	semidirect
26. C₃×#733	C₃²	semidirect
30. C₃×#731	C₃²	semidirect
31. C₃×AΣL(1,𝔽₂₇)	C₃²	semidirect
33. #6893	C₃²	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#737	3
C₃×#6910	3
C₃×#6909	3
C₃²×#733	3
C₃²×#732	3
#10624881	3
#10624739	3
#10624734	3