Finite Group Databasebeta

Subgroups — #14547285

Order 2187.

518 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 56 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	#10624880	C₃	1	1	yes	maximal, characteristic
2	#10624820	C₃	1	2	yes	maximal, complemented
3	#10624769	C₃	1	1	yes	maximal, characteristic, complemented
4	#10624745	C₃	1	1	yes	maximal, characteristic, complemented
5	#10624689	C₃	1	1	yes	maximal, characteristic, complemented
6	#10624620	C₃	1	2	yes	maximal, complemented
7	#10624609	C₃	1	2	yes	maximal, complemented
8	#10624598	C₃	1	2	yes	maximal, complemented
9	#10624563	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic, complemented
11	C₃²×H(3,𝔽₃)	—	3	1	no	—
12	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
13	C₃×#731	—	3	3	no	—
14	#6891	C₃²	1	1	yes	characteristic, complemented
15	#6891	C₃²	1	2	yes	complemented
16	C₃²⋊H(3,𝔽₃)	—	3	2	no	—
17	C₃²⋊H(3,𝔽₃)	—	3	6	no	—
18	C₃²⋊H(3,𝔽₃)	—	3	3	no	—
19	C₃²⋊H(3,𝔽₃)	C₃²	1	2	yes	—
20	C₃²⋊H(3,𝔽₃)	C₃²	1	1	yes	characteristic
21	C₃×C₃²⋊C₉	—	3	2	no	—
22	C₃×C₃²⋊C₉	—	3	1	no	—
23	C₃×C₃²⋊C₉	C₃²	1	2	yes	complemented
24	C₃×C₃²⋊C₉	C₃²	1	2	yes	complemented
25	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic, complemented
26	H(3,𝔽₃)⋊C₉	—	3	6	no	—
27	#6866	—	3	2	no	—
28	#6866	—	3	1	no	—
29	#6866	—	3	2	no	—
30	#6866	—	3	1	no	—
31	#6859	—	3	6	no	—
32	#6859	—	3	6	no	—
33	#6857	—	3	6	no	—
34	C₃⁴	—	3	3	no	—
35	C₃⁴	H(3,𝔽₃)	1	2	yes	—
36	C₃⁴	H(3,𝔽₃)	1	1	yes	characteristic
37	C₃⁴	C₃³	1	1	yes	characteristic
38	C₃×3₋¹⁺²	—	3	1	no	—
39	C₃×3₋¹⁺²	—	3	2	no	—
40	C₃×3₋¹⁺²	—	3	1	no	—
41	C₃×3₋¹⁺²	—	3	2	no	—
42	C₃×3₋¹⁺²	—	9	3	no	—
43	C₃×H(3,𝔽₃)	—	3	6	no	—
44	C₃×H(3,𝔽₃)	—	3	3	no	—
45	C₃×H(3,𝔽₃)	—	3	2	no	—
46	C₃×H(3,𝔽₃)	—	3	1	no	—
47	C₃×H(3,𝔽₃)	—	3	1	no	—
48	C₃×H(3,𝔽₃)	—	3	2	no	—
49	C₃×H(3,𝔽₃)	—	3	6	no	—
50	C₃×H(3,𝔽₃)	—	3	12	no	—
51	C₃×H(3,𝔽₃)	—	3	2	no	—
52	C₃×H(3,𝔽₃)	—	3	2	no	—
53	C₃×H(3,𝔽₃)	—	3	2	no	—
54	C₃×H(3,𝔽₃)	—	9	2	no	—
55	C₃×H(3,𝔽₃)	—	9	6	no	—
56	C₃×H(3,𝔽₃)	—	9	3	no	—
57	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	6	yes	—
58	C₃×H(3,𝔽₃)	H(3,𝔽₃)	1	3	yes	—
59	C₃²×C₉	—	3	1	no	—
60	C₃²×C₉	—	3	2	no	—
61	C₃²×C₉	—	3	1	no	—
62	C₃²×C₉	—	3	2	no	—
63	#731	—	9	9	no	—
64	C₃²⋊C₉	—	3	2	no	—
65	C₃²⋊C₉	—	3	2	no	—
66	C₃²⋊C₉	—	3	2	no	—
67	C₃²⋊C₉	—	3	2	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	1	no	—
73	C₃²⋊C₉	—	3	1	no	—
74	C₃²⋊C₉	—	3	2	no	—
75	C₃²⋊C₉	—	3	2	no	—
76	C₃²⋊C₉	—	3	2	no	—
77	C₃²⋊C₉	—	9	3	no	—
78	C₃²⋊C₉	—	9	6	no	—
79	C₃³	—	3	1	no	—
80	C₃³	—	3	1	no	—
81	C₃³	—	3	2	no	—
82	C₃³	—	3	2	no	—
83	C₃³	—	3	2	no	—
84	C₃³	—	3	3	no	—
85	C₃³	—	3	6	no	—
86	C₃³	—	3	2	no	—
87	C₃³	—	3	2	no	—
88	C₃³	—	3	2	no	—
89	C₃³	—	9	1	no	—
90	C₃³	—	9	2	no	—
91	C₃³	—	9	1	no	—
92	C₃³	—	9	2	no	—
93	C₃³	—	9	2	no	—
94	C₃³	—	9	4	no	—
95	C₃³	—	9	9	no	—
96	C₃³	—	27	3	no	—
97	C₃³	—	27	6	no	—
98	C₃³	#731	1	3	yes	complemented
99	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
100	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
101	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
102	3₋¹⁺²	—	3	3	no	—
103	3₋¹⁺²	—	9	2	no	—
104	3₋¹⁺²	—	9	6	no	—
105	3₋¹⁺²	—	9	3	no	—
106	3₋¹⁺²	—	27	2	no	—
107	H(3,𝔽₃)	—	9	3	no	—
108	H(3,𝔽₃)	—	9	6	no	—
109	H(3,𝔽₃)	—	9	3	no	—
110	H(3,𝔽₃)	—	9	6	no	—
111	H(3,𝔽₃)	—	9	18	no	—
112	H(3,𝔽₃)	—	9	2	no	—
113	H(3,𝔽₃)	—	9	1	no	—
114	H(3,𝔽₃)	—	9	4	no	—
115	H(3,𝔽₃)	—	9	2	no	—
116	H(3,𝔽₃)	—	27	2	no	—
117	H(3,𝔽₃)	—	27	2	no	—
118	H(3,𝔽₃)	—	27	2	no	—
119	H(3,𝔽₃)	—	27	6	no	—
120	H(3,𝔽₃)	—	27	12	no	—
121	C₃×C₉	—	3	3	no	—
122	C₃×C₉	—	9	2	no	—
123	C₃×C₉	—	9	1	no	—
124	C₃×C₉	—	9	2	no	—
125	C₃×C₉	—	9	4	no	—
126	C₃×C₉	—	9	3	no	—
127	C₃×C₉	—	9	1	no	—
128	C₃×C₉	—	9	2	no	—
129	C₃×C₉	—	9	2	no	—
130	C₃×C₉	—	9	1	no	—
131	C₃×C₉	—	9	2	no	—
132	C₃×C₉	—	27	2	no	—
133	C₃²	—	3	1	no	—
134	C₃²	—	3	2	no	—
135	C₃²	—	3	3	no	—
136	C₃²	—	3	1	no	—
137	C₃²	—	3	2	no	—
138	C₃²	—	3	1	no	—
139	C₃²	—	3	2	no	—
140	C₃²	—	3	2	no	—
141	C₃²	—	3	2	no	—
142	C₃²	—	3	2	no	—
143	C₃²	—	9	3	no	—
144	C₃²	—	9	6	no	—
145	C₃²	—	9	1	no	—
146	C₃²	—	9	2	no	—
147	C₃²	—	9	2	no	—
148	C₃²	—	9	3	no	—
149	C₃²	—	9	6	no	—
150	C₃²	—	9	3	no	—
151	C₃²	—	9	2	no	—
152	C₃²	—	9	6	no	—
153	C₃²	—	9	6	no	—
154	C₃²	—	9	4	no	—
155	C₃²	—	9	6	no	—
156	C₃²	—	9	6	no	—
157	C₃²	—	27	1	no	—
158	C₃²	—	27	2	no	—
159	C₃²	—	27	2	no	—
160	C₃²	—	27	2	no	—
161	C₃²	—	27	2	no	—
162	C₃²	—	27	27	no	—
163	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
164	C₃²	C₃×#731	1	1	yes	characteristic
165	C₃²	#6910	1	2	yes	—
166	C₉	—	9	3	no	—
167	C₉	—	27	2	no	—
168	C₉	—	27	1	no	—
169	C₉	—	27	1	no	—
170	C₉	—	27	2	no	—
171	C₉	—	27	2	no	—
172	C₃	—	3	1	no	—
173	C₃	—	3	2	no	—
174	C₃	—	9	1	no	—
175	C₃	—	9	2	no	—
176	C₃	—	9	3	no	—
177	C₃	—	27	2	no	—
178	C₃	—	27	3	no	—
179	C₃	—	27	6	no	—
180	C₃	#10624741	1	1	yes	characteristic
181	C₃	UT(4,𝔽₃)	1	1	yes	characteristic
182	C₃	#10624748	1	2	yes	—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	#10624880	C₃
LCS	G ▷ C₃⁴ ▷ C₃² ▷ C₁
UCS	C₁ ◁ C₃² ◁ C₃⁴ ◁ G
LECS	G ▷ C₃⁴ ▷ C₃² ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #10624820	C₃	semidirect
3. #10624769	C₃	semidirect
4. #10624745	C₃	semidirect
5. #10624689	C₃	semidirect
6. #10624620	C₃	semidirect
7. #10624609	C₃	semidirect
8. #10624598	C₃	semidirect
9. #10624563	C₃	semidirect
10. C₃²×3₋¹⁺²	C₃²	semidirect
14. #6891	C₃²	semidirect
15. #6891	C₃²	semidirect
23. C₃×C₃²⋊C₉	C₃²	semidirect
24. C₃×C₃²⋊C₉	C₃²	semidirect
25. C₃×C₃²⋊C₉	C₃²	semidirect
98. C₃³	#731	semidirect

Maximal subgroups

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

subgroup	index
#10624880	3
#10624820	3
#10624769	3
#10624745	3
#10624689	3
#10624620	3
#10624609	3
#10624598	3
#10624563	3