Finite Group Databasebeta

Subgroups — #14550435

Order 2187.

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

normal subgroups only

This group has 755 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	C₃²×AΣL(1,𝔽₂₇)	C₃	1	1	yes	maximal, characteristic
2	#10624863	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624699	C₃	1	2	yes	maximal, complemented
4	#10624699	C₃	1	2	yes	maximal, complemented
5	#10624698	C₃	1	2	yes	maximal, complemented
6	#10624691	C₃	1	1	yes	maximal, characteristic, complemented
7	#10624689	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃×#6866	C₃	1	2	yes	maximal, complemented
9	C₃×#6866	C₃	1	1	yes	maximal, characteristic, complemented
10	C₃⁵	C₃²	1	1	yes	characteristic
11	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
12	C₃²×3₋¹⁺²	C₃²	1	1	yes	characteristic
13	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
14	C₃×AΣL(1,𝔽₂₇)	—	3	3	no	—
15	C₃×AΣL(1,𝔽₂₇)	—	3	6	no	—
16	C₃×AΣL(1,𝔽₂₇)	C₉	1	9	yes	complemented
17	C₉×3₋¹⁺²	—	3	3	no	—
18	C₉×3₋¹⁺²	—	3	6	no	—
19	#6887	—	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	2	yes	—
23	C₃×C₃²⋊C₉	C₃²	1	1	yes	characteristic
24	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
25	3₋¹⁺²⋊C₉	—	3	6	no	—
26	3₋¹⁺²⋊C₉	—	3	6	no	—
27	H(3,𝔽₃)⋊C₉	—	3	6	no	—
28	#6868	—	3	3	no	—
29	#6866	—	3	3	no	—
30	#6866	—	3	6	no	—
31	#6866	—	3	3	no	—
32	C₃⁴	—	3	1	no	—
33	C₃⁴	—	3	2	no	—
34	C₃⁴	—	3	1	no	—
35	C₃⁴	—	3	2	no	—
36	C₃⁴	—	3	2	no	—
37	C₃⁴	—	3	4	no	—
38	C₃⁴	—	3	1	no	—
39	C₃⁴	—	9	3	no	—
40	C₃⁴	—	9	6	no	—
41	C₃⁴	C₃×C₉	1	3	yes	—
42	C₃⁴	C₃³	1	1	yes	characteristic
43	C₃×3₋¹⁺²	—	3	2	no	—
44	C₃×3₋¹⁺²	—	3	2	no	—
45	C₃×3₋¹⁺²	—	3	4	no	—
46	C₃×3₋¹⁺²	—	3	2	no	—
47	C₃×3₋¹⁺²	—	3	2	no	—
48	C₃×3₋¹⁺²	—	3	1	no	—
49	C₃×3₋¹⁺²	—	3	2	no	—
50	C₃×3₋¹⁺²	—	3	4	no	—
51	C₃×3₋¹⁺²	C₃×C₉	1	3	yes	—
52	C₃×3₋¹⁺²	C₃×C₉	1	3	yes	—
53	C₃×3₋¹⁺²	H(3,𝔽₃)	1	3	yes	—
54	C₃×3₋¹⁺²	3₋¹⁺²	1	6	yes	—
55	C₃×H(3,𝔽₃)	—	3	2	no	—
56	C₃×H(3,𝔽₃)	—	3	1	no	—
57	C₃×H(3,𝔽₃)	—	3	2	no	—
58	C₃×H(3,𝔽₃)	—	3	4	no	—
59	C₃×H(3,𝔽₃)	—	3	2	no	—
60	C₃×H(3,𝔽₃)	C₃×C₉	1	3	yes	—
61	C₃²×C₉	—	3	1	no	—
62	C₃²×C₉	—	3	1	no	—
63	C₃²×C₉	—	3	1	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	AΣL(1,𝔽₂₇)	—	3	27	no	—
69	C₉⋊C₉	—	3	18	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	2	no	—
79	C₃²⋊C₉	—	3	2	no	—
80	C₃²⋊C₉	—	3	2	no	—
81	C₃²⋊C₉	—	3	2	no	—
82	C₃²⋊C₉	H(3,𝔽₃)	1	3	yes	—
83	C₃²⋊C₉	H(3,𝔽₃)	1	6	yes	—
84	C₉²	—	3	9	no	—
85	C₃³	—	3	3	no	—
86	C₃³	—	3	3	no	—
87	C₃³	—	3	6	no	—
88	C₃³	—	3	3	no	—
89	C₃³	—	9	1	no	—
90	C₃³	—	9	2	no	—
91	C₃³	—	9	6	no	—
92	C₃³	—	9	6	no	—
93	C₃³	—	9	6	no	—
94	C₃³	—	9	3	no	—
95	C₃³	—	9	6	no	—
96	C₃³	—	9	3	no	—
97	C₃³	—	9	6	no	—
98	C₃³	—	9	9	no	—
99	C₃³	—	9	18	no	—
100	C₃³	—	9	18	no	—
101	C₃³	—	9	36	no	—
102	C₃³	—	9	1	no	—
103	C₃³	—	9	2	no	—
104	C₃³	—	9	4	no	—
105	C₃³	—	9	2	no	—
106	C₃³	—	9	1	no	—
107	C₃³	—	9	2	no	—
108	C₃³	—	27	1	no	—
109	C₃³	—	27	2	no	—
110	C₃³	C₃²×C₉	1	1	yes	characteristic
111	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
112	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
113	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
114	C₃³	C₃×3₋¹⁺²	1	2	yes	—
115	C₃³	#737	1	2	yes	—
116	C₃³	#737	1	4	yes	—
117	3₋¹⁺²	—	3	6	no	—
118	3₋¹⁺²	—	3	12	no	—
119	3₋¹⁺²	—	3	6	no	—
120	3₋¹⁺²	—	3	3	no	—
121	3₋¹⁺²	—	3	6	no	—
122	3₋¹⁺²	—	9	4	no	—
123	3₋¹⁺²	—	9	2	no	—
124	3₋¹⁺²	C₃²⋊C₉	1	9	yes	—
125	H(3,𝔽₃)	—	3	3	no	—
126	H(3,𝔽₃)	—	3	6	no	—
127	H(3,𝔽₃)	—	9	4	no	—
128	H(3,𝔽₃)	—	9	2	no	—
129	C₃×C₉	—	3	3	no	—
130	C₃×C₉	—	3	3	no	—
131	C₃×C₉	—	3	6	no	—
132	C₃×C₉	—	3	3	no	—
133	C₃×C₉	—	3	6	no	—
134	C₃×C₉	—	3	3	no	—
135	C₃×C₉	—	9	1	no	—
136	C₃×C₉	—	9	2	no	—
137	C₃×C₉	—	9	2	no	—
138	C₃×C₉	—	9	2	no	—
139	C₃×C₉	—	9	2	no	—
140	C₃×C₉	—	27	1	no	—
141	C₃×C₉	—	27	2	no	—
142	C₃×C₉	—	27	2	no	—
143	C₃×C₉	—	27	2	no	—
144	C₃×C₉	—	27	2	no	—
145	C₃²	—	3	1	no	—
146	C₃²	—	3	2	no	—
147	C₃²	—	3	2	no	—
148	C₃²	—	3	2	no	—
149	C₃²	—	3	2	no	—
150	C₃²	—	3	1	no	—
151	C₃²	—	3	2	no	—
152	C₃²	—	9	1	no	—
153	C₃²	—	9	2	no	—
154	C₃²	—	9	2	no	—
155	C₃²	—	9	4	no	—
156	C₃²	—	9	9	no	—
157	C₃²	—	9	9	no	—
158	C₃²	—	9	18	no	—
159	C₃²	—	9	18	no	—
160	C₃²	—	9	36	no	—
161	C₃²	—	9	18	no	—
162	C₃²	—	9	3	no	—
163	C₃²	—	9	3	no	—
164	C₃²	—	9	6	no	—
165	C₃²	—	9	3	no	—
166	C₃²	—	27	3	no	—
167	C₃²	—	27	6	no	—
168	C₃²	—	27	1	no	—
169	C₃²	—	27	2	no	—
170	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
171	C₃²	C₉×H(3,𝔽₃)	1	1	yes	characteristic
172	C₃²	C₉×H(3,𝔽₃)	1	2	yes	—
173	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
174	C₃²	C₉⋊H(3,𝔽₃)	1	2	yes	—
175	C₃²	C₉⋊H(3,𝔽₃)	1	4	yes	—
176	C₃²	#6893	1	2	yes	—
177	C₉	—	3	9	no	—
178	C₉	—	9	3	no	—
179	C₉	—	9	6	no	—
180	C₉	—	9	3	no	—
181	C₉	—	27	2	no	—
182	C₉	—	27	2	no	—
183	C₉	—	27	2	no	—
184	C₃	—	3	1	no	—
185	C₃	—	3	2	no	—
186	C₃	—	3	1	no	—
187	C₃	—	3	2	no	—
188	C₃	—	3	4	no	—
189	C₃	—	3	2	no	—
190	C₃	—	9	9	no	—
191	C₃	—	27	3	no	—
192	C₃	#10624555	1	1	yes	characteristic
193	C₃	#10624762	1	1	yes	characteristic
194	C₃	#10624762	1	2	yes	—

	subgroup	G/H
center, socle	C₃²	C₃²⋊H(3,𝔽₃)
commutator	C₃³	C₃²×C₉
Frattini, cosocle	C₃⁴	C₃³
prime radical	C₃²×AΣL(1,𝔽₂₇)	C₃
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
2. #10624863	C₃	semidirect
3. #10624699	C₃	semidirect
4. #10624699	C₃	semidirect
5. #10624698	C₃	semidirect
6. #10624691	C₃	semidirect
7. #10624689	C₃	semidirect
8. C₃×#6866	C₃	semidirect
9. C₃×#6866	C₃	semidirect
16. C₃×AΣL(1,𝔽₂₇)	C₉	semidirect

Maximal subgroups

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

subgroup	index
C₃²×AΣL(1,𝔽₂₇)	3
#10624863	3
#10624699	3
#10624698	3
#10624691	3
#10624689	3
C₃×#6866	3