Finite Group Databasebeta

Subgroups — #14550410

Order 2187.

455 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 455 conjugacy classes of subgroups — too many to draw. Only its 104 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	C₃×#6908	C₃	1	1	yes	maximal, characteristic
2	C₃²×C₂₇⋊C₃	C₃	1	1	yes	maximal, characteristic, complemented
3	#10624704	C₃	1	3	yes	maximal, complemented
4	#10624702	C₃	1	3	yes	maximal, complemented
5	#10624694	C₃	1	3	yes	maximal, complemented
6	C₃×#6869	C₃	1	1	yes	maximal, characteristic, complemented
7	C₃×#6869	C₃	1	1	yes	maximal, characteristic, complemented
8	C₃×#737	C₃²	1	3	yes	—
9	C₃³×C₉	C₃²	1	1	yes	characteristic
10	#6908	—	3	1	no	—
11	#6908	—	3	2	no	—
12	#6908	—	3	2	no	—
13	#6908	—	3	2	no	—
14	#6908	—	3	2	no	—
15	C₃×#733	C₉	1	2	yes	—
16	C₃×#732	C₉	1	1	yes	characteristic
17	C₃×#731	C₉	1	3	yes	—
18	C₃×AΣL(1,𝔽₂₇)	C₉	1	3	yes	—
19	C₃×C₂₇⋊C₃	—	3	2	no	—
20	C₃×C₂₇⋊C₃	—	3	3	no	—
21	C₃×C₂₇⋊C₃	—	3	3	no	—
22	C₃×C₂₇⋊C₃	—	3	3	no	—
23	C₃×C₂₇⋊C₃	C₃²	1	3	yes	—
24	C₃×C₂₇⋊C₃	C₃²	1	3	yes	—
25	C₃×C₂₇⋊C₃	C₃²	1	1	yes	characteristic
26	C₃×C₂₇⋊C₃	C₃²	1	1	yes	characteristic
27	C₃×C₂₇⋊C₃	C₃²	1	1	yes	characteristic
28	C₃²×C₂₇	—	3	1	no	—
29	#6873	—	3	9	no	—
30	#6872	—	3	9	no	—
31	#6869	—	3	1	no	—
32	#6869	—	3	1	no	—
33	#6869	—	3	2	no	—
34	#6869	—	3	2	no	—
35	#6869	—	3	9	no	—
36	C₃⁴	C₃×C₉	1	1	yes	characteristic
37	#737	—	3	3	no	—
38	#737	—	3	3	no	—
39	#737	—	3	3	no	—
40	#737	—	3	6	no	—
41	#737	—	3	6	no	—
42	#737	—	3	6	no	—
43	C₃×3₋¹⁺²	—	3	6	no	—
44	C₃×3₋¹⁺²	C₃×C₉	1	6	yes	—
45	C₃×H(3,𝔽₃)	C₃×C₉	1	3	yes	—
46	C₃²×C₉	—	3	1	no	—
47	C₃²×C₉	—	3	1	no	—
48	C₃²×C₉	—	3	1	no	—
49	C₃²×C₉	—	3	2	no	—
50	C₃²×C₉	—	3	2	no	—
51	C₃²×C₉	—	3	2	no	—
52	C₃²×C₉	—	3	3	no	—
53	C₃²×C₉	C₃×C₉	1	1	yes	characteristic
54	C₃²×C₉	C₃×C₉	1	1	yes	characteristic
55	C₃²×C₉	H(3,𝔽₃)	1	1	yes	characteristic
56	C₃²×C₉	H(3,𝔽₃)	1	2	yes	—
57	C₃²×C₉	3₋¹⁺²	1	2	yes	—
58	C₃²×C₉	3₋¹⁺²	1	2	yes	—
59	C₃²×C₉	3₋¹⁺²	1	2	yes	—
60	C₃²×C₉	C₃³	1	1	yes	characteristic
61	#733	—	3	2	no	—
62	#733	—	3	4	no	—
63	#732	—	3	1	no	—
64	#732	—	3	2	no	—
65	#731	—	3	3	no	—
66	#731	—	3	6	no	—
67	AΣL(1,𝔽₂₇)	—	3	3	no	—
68	AΣL(1,𝔽₂₇)	—	3	6	no	—
69	C₂₇⋊C₃	—	3	18	no	—
70	C₂₇⋊C₃	—	3	1	no	—
71	C₂₇⋊C₃	—	3	1	no	—
72	C₂₇⋊C₃	—	3	1	no	—
73	C₂₇⋊C₃	—	3	1	no	—
74	C₂₇⋊C₃	—	3	1	no	—
75	C₂₇⋊C₃	—	3	1	no	—
76	C₂₇⋊C₃	—	3	3	no	—
77	C₂₇⋊C₃	—	3	3	no	—
78	C₂₇⋊C₃	—	3	3	no	—
79	C₂₇⋊C₃	—	3	3	no	—
80	C₂₇⋊C₃	—	3	3	no	—
81	C₂₇⋊C₃	—	3	3	no	—
82	C₂₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
83	C₂₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
84	C₂₇⋊C₃	H(3,𝔽₃)	1	3	yes	—
85	C₃×C₂₇	—	3	9	no	—
86	C₃×C₂₇	—	3	1	no	—
87	C₃×C₂₇	—	3	1	no	—
88	C₃×C₂₇	—	3	1	no	—
89	C₃×C₂₇	—	3	3	no	—
90	C₃×C₂₇	—	3	3	no	—
91	C₃³	—	3	1	no	—
92	C₃³	—	3	1	no	—
93	C₃³	—	3	1	no	—
94	C₃³	—	3	3	no	—
95	C₃³	—	9	3	no	—
96	C₃³	C₃²⋊C₉	1	1	yes	characteristic
97	C₃³	C₃²⋊C₉	1	2	yes	—
98	C₃³	C₃²×C₉	1	1	yes	characteristic
99	3₋¹⁺²	—	3	6	no	—
100	3₋¹⁺²	—	3	6	no	—
101	3₋¹⁺²	—	3	6	no	—
102	3₋¹⁺²	—	9	6	no	—
103	3₋¹⁺²	—	9	6	no	—
104	3₋¹⁺²	—	9	6	no	—
105	H(3,𝔽₃)	—	3	3	no	—
106	H(3,𝔽₃)	—	3	3	no	—
107	H(3,𝔽₃)	—	3	3	no	—
108	C₃×C₉	—	3	1	no	—
109	C₃×C₉	—	3	1	no	—
110	C₃×C₉	—	3	1	no	—
111	C₃×C₉	—	3	1	no	—
112	C₃×C₉	—	3	1	no	—
113	C₃×C₉	—	3	1	no	—
114	C₃×C₉	—	3	1	no	—
115	C₃×C₉	—	3	2	no	—
116	C₃×C₉	—	3	2	no	—
117	C₃×C₉	—	3	2	no	—
118	C₃×C₉	—	3	2	no	—
119	C₃×C₉	—	3	2	no	—
120	C₃×C₉	—	3	2	no	—
121	C₃×C₉	—	3	2	no	—
122	C₃×C₉	—	3	2	no	—
123	C₃×C₉	—	3	2	no	—
124	C₃×C₉	—	3	2	no	—
125	C₃×C₉	—	3	2	no	—
126	C₃×C₉	—	3	4	no	—
127	C₃×C₉	—	3	6	no	—
128	C₃×C₉	—	9	3	no	—
129	C₃×C₉	—	9	6	no	—
130	C₃×C₉	C₃²⋊C₉	1	1	yes	characteristic
131	C₃×C₉	C₃²⋊C₉	1	2	yes	—
132	C₃×C₉	C₃²⋊C₉	1	1	yes	characteristic
133	C₃×C₉	C₃²⋊C₉	1	2	yes	—
134	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
135	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
136	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
137	C₃×C₉	C₃×H(3,𝔽₃)	1	1	yes	characteristic
138	C₃×C₉	C₃×3₋¹⁺²	1	2	yes	—
139	C₃×C₉	#737	1	2	yes	—
140	C₃×C₉	#737	1	2	yes	—
141	C₃×C₉	#737	1	2	yes	—
142	C₂₇	—	3	3	no	—
143	C₂₇	—	3	3	no	—
144	C₂₇	—	3	3	no	—
145	C₂₇	—	9	3	no	—
146	C₂₇	—	9	3	no	—
147	C₃²	—	3	3	no	—
148	C₃²	—	3	1	no	—
149	C₃²	—	3	2	no	—
150	C₃²	—	9	1	no	—
151	C₃²	—	9	3	no	—
152	C₃²	—	9	3	no	—
153	C₃²	—	9	3	no	—
154	C₃²	—	9	2	no	—
155	C₃²	—	9	3	no	—
156	C₃²	—	27	3	no	—
157	C₃²	C₃×C₃²⋊C₉	1	1	yes	characteristic
158	C₃²	C₉×H(3,𝔽₃)	1	1	yes	characteristic
159	C₃²	C₉×H(3,𝔽₃)	1	1	yes	characteristic
160	C₃²	C₉×H(3,𝔽₃)	1	1	yes	characteristic
161	C₉	—	3	1	no	—
162	C₉	—	3	1	no	—
163	C₉	—	3	2	no	—
164	C₉	—	3	2	no	—
165	C₉	—	9	6	no	—
166	C₉	C₃²⋊H(3,𝔽₃)	1	1	yes	characteristic
167	C₉	C₉⋊H(3,𝔽₃)	1	2	yes	—
168	C₉	C₉⋊H(3,𝔽₃)	1	2	yes	—
169	C₉	C₉⋊H(3,𝔽₃)	1	2	yes	—
170	C₉	#6893	1	2	yes	—
171	C₃	—	3	3	no	—
172	C₃	—	3	1	no	—
173	C₃	—	3	1	no	—
174	C₃	—	3	1	no	—
175	C₃	—	3	1	no	—
176	C₃	—	9	2	no	—
177	C₃	—	27	3	no	—
178	C₃	#10624555	1	1	yes	characteristic

	subgroup	G/H
center	C₉	C₃²⋊H(3,𝔽₃)
commutator	C₃³	C₃²×C₉
Frattini, cosocle	C₃²×C₉	C₃³
socle	C₃	#10624555
prime radical	C₃×#6908	C₃
LCS	G ▷ C₃³ ▷ C₃ ▷ C₁
UCS	C₁ ◁ C₉ ◁ C₃²×C₉ ◁ G
LECS	G ▷ C₃²×C₉ ▷ C₃ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. C₃²×C₂₇⋊C₃	C₃	semidirect
3. #10624704	C₃	semidirect
4. #10624702	C₃	semidirect
5. #10624694	C₃	semidirect
6. C₃×#6869	C₃	semidirect
7. C₃×#6869	C₃	semidirect

Maximal subgroups

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

subgroup	index
C₃×#6908	3
C₃²×C₂₇⋊C₃	3
#10624704	3
#10624702	3
#10624694	3
C₃×#6869	3