Finite Group Databasebeta

Subgroups — #14550667

Order 2187.

1568 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 1568 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	C₃³×3₋¹⁺²	C₃	1	1	yes	maximal, characteristic, complemented
2	C₃²×AΣL(1,𝔽₂₇)	C₃	1	1	yes	maximal, characteristic
3	C₃×3₋¹⁺²⋊C₉	C₃	1	4	yes	maximal, complemented
4	C₃×H(3,𝔽₃)⋊C₉	C₃	1	2	yes	maximal, complemented
5	C₃×#6866	C₃	1	2	yes	maximal, complemented
6	#10624558	C₃	1	2	yes	maximal, complemented
7	#10624555	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃⁵	C₃²	1	1	yes	characteristic
9	C₃²×3₋¹⁺²	—	3	3	no	—
10	C₃²×3₋¹⁺²	—	3	27	no	—
11	C₃²×3₋¹⁺²	—	3	6	no	—
12	C₃²×3₋¹⁺²	—	3	3	no	—
13	C₃²×3₋¹⁺²	C₃²	1	2	yes	—
14	C₃²×H(3,𝔽₃)	C₃²	1	1	yes	characteristic
15	C₃³×C₉	C₃²	1	2	yes	—
16	C₃³×C₉	C₃²	1	1	yes	characteristic
17	C₃×AΣL(1,𝔽₂₇)	—	3	3	no	—
18	C₃×AΣL(1,𝔽₂₇)	—	3	6	no	—
19	C₃×AΣL(1,𝔽₂₇)	—	3	3	no	—
20	C₃×C₃²⋊C₉	—	3	3	no	—
21	C₃×C₃²⋊C₉	C₃²	1	2	yes	—
22	C₃×C₃²⋊C₉	C₃²	1	4	yes	—
23	3₋¹⁺²⋊C₉	—	3	12	no	—
24	H(3,𝔽₃)⋊C₉	—	3	6	no	—
25	#6866	—	3	6	no	—
26	#6855	—	3	6	no	—
27	C₃⁴	—	3	9	no	—
28	C₃⁴	—	3	2	no	—
29	C₃⁴	—	3	1	no	—
30	C₃⁴	—	3	1	no	—
31	C₃⁴	—	3	2	no	—
32	C₃⁴	—	3	1	no	—
33	C₃⁴	—	3	2	no	—
34	C₃⁴	—	3	4	no	—
35	C₃⁴	—	9	6	no	—
36	C₃⁴	H(3,𝔽₃)	1	3	yes	—
37	C₃⁴	C₃³	1	1	yes	characteristic
38	C₃×3₋¹⁺²	—	3	81	no	—
39	C₃×3₋¹⁺²	—	3	162	no	—
40	C₃×3₋¹⁺²	—	3	27	no	—
41	C₃×3₋¹⁺²	—	3	54	no	—
42	C₃×3₋¹⁺²	—	3	4	no	—
43	C₃×3₋¹⁺²	—	3	27	no	—
44	C₃×3₋¹⁺²	—	3	4	no	—
45	C₃×3₋¹⁺²	—	3	2	no	—
46	C₃×3₋¹⁺²	—	3	2	no	—
47	C₃×3₋¹⁺²	—	3	2	no	—
48	C₃×3₋¹⁺²	—	3	4	no	—
49	C₃×3₋¹⁺²	—	3	4	no	—
50	C₃×3₋¹⁺²	3₋¹⁺²	1	6	yes	complemented
51	C₃×H(3,𝔽₃)	—	3	2	no	—
52	C₃×H(3,𝔽₃)	—	3	1	no	—
53	C₃×H(3,𝔽₃)	—	3	2	no	—
54	C₃×H(3,𝔽₃)	—	3	4	no	—
55	C₃×H(3,𝔽₃)	—	3	2	no	—
56	C₃×H(3,𝔽₃)	3₋¹⁺²	1	3	yes	complemented
57	C₃²×C₉	—	3	2	no	—
58	C₃²×C₉	—	3	2	no	—
59	C₃²×C₉	—	3	1	no	—
60	C₃²×C₉	—	3	2	no	—
61	C₃²×C₉	—	3	2	no	—
62	C₃²×C₉	—	3	9	no	—
63	C₃²×C₉	—	3	18	no	—
64	C₃²×C₉	—	3	2	no	—
65	C₃²×C₉	—	3	4	no	—
66	C₃²×C₉	—	3	2	no	—
67	C₃²×C₉	H(3,𝔽₃)	1	6	yes	—
68	C₃²×C₉	H(3,𝔽₃)	1	3	yes	—
69	AΣL(1,𝔽₂₇)	—	3	27	no	—
70	C₃²⋊C₉	—	3	2	no	—
71	C₃²⋊C₉	—	3	4	no	—
72	C₃²⋊C₉	—	3	4	no	—
73	C₃²⋊C₉	—	3	4	no	—
74	C₃²⋊C₉	—	3	4	no	—
75	C₃²⋊C₉	—	9	9	no	—
76	C₃³	—	3	6	no	—
77	C₃³	—	3	3	no	—
78	C₃³	—	3	18	no	—
79	C₃³	—	3	9	no	—
80	C₃³	—	3	3	no	—
81	C₃³	—	3	9	no	—
82	C₃³	—	3	3	no	—
83	C₃³	—	3	3	no	—
84	C₃³	—	3	6	no	—
85	C₃³	—	9	6	no	—
86	C₃³	—	9	9	no	—
87	C₃³	—	9	18	no	—
88	C₃³	—	9	18	no	—
89	C₃³	—	9	36	no	—
90	C₃³	—	9	12	no	—
91	C₃³	—	9	6	no	—
92	C₃³	—	9	2	no	—
93	C₃³	—	9	4	no	—
94	C₃³	—	9	1	no	—
95	C₃³	—	9	2	no	—
96	C₃³	—	9	6	no	—
97	C₃³	—	9	1	no	—
98	C₃³	—	9	2	no	—
99	C₃³	—	9	3	no	—
100	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
101	C₃³	C₃×H(3,𝔽₃)	1	2	yes	—
102	C₃³	C₃×H(3,𝔽₃)	1	1	yes	characteristic
103	C₃³	C₃×3₋¹⁺²	1	2	yes	—
104	C₃³	C₃×3₋¹⁺²	1	1	yes	characteristic
105	C₃³	#737	1	2	yes	—
106	C₃³	#737	1	4	yes	—
107	3₋¹⁺²	—	3	243	no	—
108	3₋¹⁺²	—	3	6	no	—
109	3₋¹⁺²	—	3	12	no	—
110	3₋¹⁺²	—	9	4	no	—
111	3₋¹⁺²	—	9	4	no	—
112	3₋¹⁺²	—	9	4	no	—
113	H(3,𝔽₃)	—	3	3	no	—
114	H(3,𝔽₃)	—	3	6	no	—
115	H(3,𝔽₃)	—	9	4	no	—
116	H(3,𝔽₃)	—	9	2	no	—
117	C₃×C₉	—	3	9	no	—
118	C₃×C₉	—	3	18	no	—
119	C₃×C₉	—	3	18	no	—
120	C₃×C₉	—	3	18	no	—
121	C₃×C₉	—	3	3	no	—
122	C₃×C₉	—	3	6	no	—
123	C₃×C₉	—	3	6	no	—
124	C₃×C₉	—	3	18	no	—
125	C₃×C₉	—	3	18	no	—
126	C₃×C₉	—	3	6	no	—
127	C₃×C₉	—	9	2	no	—
128	C₃×C₉	—	9	2	no	—
129	C₃×C₉	—	9	2	no	—
130	C₃×C₉	—	9	12	no	—
131	C₃×C₉	—	9	2	no	—
132	C₃×C₉	—	9	4	no	—
133	C₃×C₉	—	9	6	no	—
134	C₃×C₉	AΣL(1,𝔽₂₇)	1	9	yes	complemented
135	C₃²	—	3	3	no	—
136	C₃²	—	3	6	no	—
137	C₃²	—	3	6	no	—
138	C₃²	—	3	12	no	—
139	C₃²	—	3	1	no	—
140	C₃²	—	3	2	no	—
141	C₃²	—	3	2	no	—
142	C₃²	—	3	4	no	—
143	C₃²	—	3	9	no	—
144	C₃²	—	9	9	no	—
145	C₃²	—	9	18	no	—
146	C₃²	—	9	18	no	—
147	C₃²	—	9	36	no	—
148	C₃²	—	9	9	no	—
149	C₃²	—	9	18	no	—
150	C₃²	—	9	3	no	—
151	C₃²	—	9	6	no	—
152	C₃²	—	9	9	no	—
153	C₃²	—	9	3	no	—
154	C₃²	—	27	6	no	—
155	C₃²	—	27	1	no	—
156	C₃²	—	27	2	no	—
157	C₃²	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
158	C₃²	#6891	1	1	yes	characteristic
159	C₃²	C₉⋊H(3,𝔽₃)	1	2	yes	—
160	C₃²	C₉⋊H(3,𝔽₃)	1	4	yes	—
161	C₃²	C₉⋊H(3,𝔽₃)	1	1	yes	characteristic
162	C₃²	C₉⋊H(3,𝔽₃)	1	2	yes	—
163	C₃²	#6893	1	2	yes	—
164	C₃²	C₃×AΣL(1,𝔽₂₇)	1	3	yes	—
165	C₃²	#6913	1	6	yes	—
166	C₉	—	3	9	no	—
167	C₉	—	3	18	no	—
168	C₉	—	9	6	no	—
169	C₉	—	27	2	no	—
170	C₉	—	27	4	no	—
171	C₃	—	3	1	no	—
172	C₃	—	3	2	no	—
173	C₃	—	3	4	no	—
174	C₃	—	3	2	no	—
175	C₃	—	9	9	no	—
176	C₃	—	27	3	no	—
177	C₃	#10624563	1	1	yes	characteristic
178	C₃	#10624730	1	1	yes	characteristic
179	C₃	#10624739	1	2	yes	—
180	C₃	#10624794	1	3	yes	—
181	C₃	#10624798	1	6	yes	—

	subgroup	G/H
center, socle	C₃³	C₃×H(3,𝔽₃)
commutator, Frattini, cosocle	C₃⁴	C₃³
prime radical	C₃²×AΣL(1,𝔽₂₇)	C₃
LCS	G ▷ C₃⁴ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₃³ ◁ C₃³×C₉ ◁ G
LECS	G ▷ C₃⁴ ▷ 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
3. C₃×3₋¹⁺²⋊C₉	C₃	semidirect
4. C₃×H(3,𝔽₃)⋊C₉	C₃	semidirect
5. C₃×#6866	C₃	semidirect
6. #10624558	C₃	semidirect
7. #10624555	C₃	semidirect
50. C₃×3₋¹⁺²	3₋¹⁺²	semidirect
56. C₃×H(3,𝔽₃)	3₋¹⁺²	semidirect
134. C₃×C₉	AΣL(1,𝔽₂₇)	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
C₃²×AΣL(1,𝔽₂₇)	3
C₃×3₋¹⁺²⋊C₉	3
C₃×H(3,𝔽₃)⋊C₉	3
C₃×#6866	3
#10624558	3
#10624555	3