Subgroups — C₃×C₉×#392 #13144934

Order 1728.

1610 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×D₄×C₉	C₂	1	3	yes	maximal, complemented
2	C₂×C₃×C₉×C₂²⋊C₄	C₂	1	3	yes	maximal, complemented
3	C₃×C₉×#95	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃²×#392	C₃	1	1	yes	maximal, characteristic
5	C₉×#392	C₃	1	3	yes	maximal, direct product factor
6	C₂⁴×C₃×C₉	C₂²	1	3	yes	—
7	C₂×C₃×D₄×C₉	—	2	12	no	—
8	C₂×C₃×D₄×C₉	—	2	6	no	—
9	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄×C₉	C₂²	1	3	yes	—
11	C₃×C₉×C₂²⋊C₄	—	2	6	no	—
12	C₂²×C₃²×D₄	C₂×C₃	1	3	yes	—
13	C₂×C₃²×C₂²⋊C₄	C₂×C₃	1	3	yes	—
14	C₂²×D₄×C₉	C₂×C₃	1	9	yes	complemented
15	C₃²×#95	C₂×C₃	1	1	yes	characteristic
16	C₂×C₉×C₂²⋊C₄	C₂×C₃	1	9	yes	complemented
17	C₉×#95	C₂×C₃	1	3	yes	complemented
18	C₂³×C₃×C₉	—	2	6	no	—
19	C₂³×C₃×C₉	—	2	6	no	—
20	C₂³×C₃×C₉	—	2	3	no	—
21	C₂³×C₃×C₉	—	2	3	no	—
22	C₂³×C₃×C₉	D₄	1	6	yes	complemented
23	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
24	C₃×D₄×C₉	—	2	24	no	—
25	C₂×C₃×C₄×C₉	—	2	6	no	—
26	C₂×C₃×C₄×C₉	—	2	3	no	—
27	C₂×C₃×C₄×C₉	D₄	1	6	yes	complemented
28	C₃×#392	C₉	1	3	yes	direct product factor
29	C₃×#392	C₃²	1	1	yes	characteristic
30	C₂⁴×C₃²	C₂²×C₃	1	3	yes	—
31	C₂×C₃²×D₄	—	2	12	no	—
32	C₂×C₃²×D₄	—	2	6	no	—
33	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
34	C₂²×C₃²×C₄	C₂²×C₃	1	3	yes	—
35	C₂⁴×C₉	C₂²×C₃	1	9	yes	—
36	C₃²×C₂²⋊C₄	—	2	6	no	—
37	C₂×D₄×C₉	—	2	36	no	—
38	C₂×D₄×C₉	—	2	18	no	—
39	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
40	C₂²×C₄×C₉	C₂²×C₃	1	9	yes	—
41	C₉×C₂²⋊C₄	—	2	18	no	—
42	C₂²×C₃×C₉	—	2	12	no	—
43	C₂²×C₃×C₉	—	2	6	no	—
44	C₂²×C₃×C₉	—	4	3	no	—
45	C₂²×C₃×C₉	—	4	6	no	—
46	C₂²×C₃×C₉	—	4	3	no	—
47	C₂²×C₃×C₉	C₂×D₄	1	3	yes	—
48	C₂²×C₃×C₉	C₂×D₄	1	3	yes	—
49	C₂²×C₃×C₉	#41	1	1	yes	characteristic
50	C₃×C₄×C₉	—	2	6	no	—
51	C₃×C₄×C₉	—	4	1	no	—
52	C₂²×C₃×D₄	C₂×C₉	1	9	yes	complemented
53	C₂²×C₃×D₄	C₂×C₃²	1	3	yes	—
54	C₂×C₃×C₂²⋊C₄	C₂×C₉	1	9	yes	complemented
55	C₂×C₃×C₂²⋊C₄	C₂×C₃²	1	3	yes	—
56	C₃×#95	C₂×C₉	1	3	yes	complemented
57	C₃×#95	C₂×C₃²	1	1	yes	characteristic
58	C₂³×C₃²	—	2	3	no	—
59	C₂³×C₃²	—	2	6	no	—
60	C₂³×C₃²	—	2	6	no	—
61	C₂³×C₃²	—	2	3	no	—
62	C₂³×C₃²	C₃×D₄	1	6	yes	—
63	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
64	C₃²×D₄	—	2	24	no	—
65	C₂×C₃²×C₄	—	2	6	no	—
66	C₂×C₃²×C₄	—	2	3	no	—
67	C₂×C₃²×C₄	C₃×D₄	1	6	yes	—
68	C₂³×C₉	—	2	9	no	—
69	C₂³×C₉	—	2	18	no	—
70	C₂³×C₉	—	2	18	no	—
71	C₂³×C₉	—	2	9	no	—
72	C₂³×C₉	C₃×D₄	1	18	yes	complemented
73	C₂³×C₉	C₂³×C₃	1	3	yes	—
74	D₄×C₉	—	2	72	no	—
75	C₂×C₄×C₉	—	2	18	no	—
76	C₂×C₄×C₉	—	2	9	no	—
77	C₂×C₄×C₉	C₃×D₄	1	18	yes	complemented
78	#392	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
79	C₂×C₃×C₉	—	4	6	no	—
80	C₂×C₃×C₉	#120	1	3	yes	—
81	C₂×C₃×C₉	#121	1	3	yes	—
82	C₂×C₃×C₉	#127	1	1	yes	characteristic
83	C₂⁴×C₃	C₂²×C₉	1	9	yes	—
84	C₂⁴×C₃	C₂²×C₃²	1	3	yes	—
85	C₂×C₃×D₄	—	2	36	no	—
86	C₂×C₃×D₄	—	2	12	no	—
87	C₂×C₃×D₄	—	2	18	no	—
88	C₂×C₃×D₄	—	2	6	no	—
89	C₂²×C₃×C₄	C₂²×C₉	1	3	yes	—
90	C₂²×C₃×C₄	C₂²×C₉	1	9	yes	—
91	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
92	C₂²×C₃×C₄	C₂²×C₃²	1	3	yes	—
93	C₃×C₂²⋊C₄	—	2	18	no	—
94	C₃×C₂²⋊C₄	—	2	6	no	—
95	C₂²×C₃²	—	2	12	no	—
96	C₂²×C₃²	—	2	6	no	—
97	C₂²×C₃²	—	4	3	no	—
98	C₂²×C₃²	—	4	6	no	—
99	C₂²×C₃²	—	4	3	no	—
100	C₂²×C₃²	C₂×C₃×D₄	1	3	yes	—
101	C₂²×C₃²	C₂×C₃×D₄	1	3	yes	—
102	C₂²×C₃²	C₃×#41	1	1	yes	characteristic
103	C₃²×C₄	—	2	6	no	—
104	C₃²×C₄	—	4	1	no	—
105	C₂²×C₉	—	2	36	no	—
106	C₂²×C₉	—	2	18	no	—
107	C₂²×C₉	—	4	9	no	—
108	C₂²×C₉	—	4	18	no	—
109	C₂²×C₉	—	4	9	no	—
110	C₂²×C₉	C₂×C₃×D₄	1	9	yes	—
111	C₂²×C₉	C₂×C₃×D₄	1	9	yes	—
112	C₂²×C₉	C₃×#41	1	3	yes	—
113	C₄×C₉	—	2	18	no	—
114	C₄×C₉	—	4	3	no	—
115	C₂²×D₄	C₂×C₃×C₉	1	3	yes	complemented
116	C₂×C₂²⋊C₄	C₂×C₃×C₉	1	3	yes	complemented
117	#95	C₂×C₃×C₉	1	1	yes	characteristic, complemented
118	C₃×C₉	#392	1	1	yes	characteristic, direct product factor, Sylow
119	C₂³×C₃	—	2	9	no	—
120	C₂³×C₃	—	2	18	no	—
121	C₂³×C₃	—	2	18	no	—
122	C₂³×C₃	—	2	9	no	—
123	C₂³×C₃	—	2	3	no	—
124	C₂³×C₃	—	2	6	no	—
125	C₂³×C₃	—	2	6	no	—
126	C₂³×C₃	—	2	3	no	—
127	C₂³×C₃	D₄×C₉	1	18	yes	complemented
128	C₂³×C₃	C₂³×C₉	1	3	yes	—
129	C₂³×C₃	C₃²×D₄	1	6	yes	—
130	C₂³×C₃	C₂³×C₃²	1	1	yes	characteristic
131	C₃×D₄	—	2	24	no	—
132	C₃×D₄	—	2	72	no	—
133	C₂×C₃×C₄	—	2	18	no	—
134	C₂×C₃×C₄	—	2	6	no	—
135	C₂×C₃×C₄	—	2	9	no	—
136	C₂×C₃×C₄	—	2	3	no	—
137	C₂×C₃×C₄	D₄×C₉	1	18	yes	complemented
138	C₂×C₃×C₄	C₃²×D₄	1	6	yes	—
139	C₂×C₃²	—	4	6	no	—
140	C₂×C₃²	C₃×#120	1	3	yes	—
141	C₂×C₃²	C₃×#121	1	3	yes	—
142	C₂×C₃²	C₃×#127	1	1	yes	characteristic
143	C₂×C₉	—	4	18	no	—
144	C₂×C₉	C₃×#120	1	9	yes	—
145	C₂×C₉	C₃×#121	1	9	yes	—
146	C₂×C₉	C₃×#127	1	3	yes	—
147	C₂⁴	C₂²×C₃×C₉	1	3	yes	—
148	C₂×D₄	—	2	12	no	—
149	C₂×D₄	—	2	6	no	—
150	C₂²×C₄	C₂²×C₃×C₉	1	1	yes	characteristic
151	C₂²×C₄	C₂²×C₃×C₉	1	3	yes	—
152	C₂²⋊C₄	—	2	6	no	—
153	C₂²×C₃	—	2	36	no	—
154	C₂²×C₃	—	2	18	no	—
155	C₂²×C₃	—	2	12	no	—
156	C₂²×C₃	—	2	6	no	—
157	C₂²×C₃	—	4	9	no	—
158	C₂²×C₃	—	4	18	no	—
159	C₂²×C₃	—	4	9	no	—
160	C₂²×C₃	—	4	3	no	—
161	C₂²×C₃	—	4	6	no	—
162	C₂²×C₃	—	4	3	no	—
163	C₂²×C₃	C₂×D₄×C₉	1	9	yes	—
164	C₂²×C₃	C₂×D₄×C₉	1	9	yes	—
165	C₂²×C₃	C₉×#41	1	3	yes	—
166	C₂²×C₃	C₂×C₃²×D₄	1	3	yes	—
167	C₂²×C₃	C₂×C₃²×D₄	1	3	yes	—
168	C₂²×C₃	C₃²×#41	1	1	yes	characteristic
169	C₃×C₄	—	2	18	no	—
170	C₃×C₄	—	2	6	no	—
171	C₃×C₄	—	4	3	no	—
172	C₃×C₄	—	4	1	no	—
173	C₃²	C₃×#392	1	1	yes	characteristic
174	C₉	C₃×#392	1	3	yes	direct product factor
175	C₂³	—	2	3	no	—
176	C₂³	—	2	6	no	—
177	C₂³	—	2	6	no	—
178	C₂³	—	2	3	no	—
179	C₂³	C₃×D₄×C₉	1	6	yes	complemented
180	C₂³	C₂³×C₃×C₉	1	1	yes	characteristic
181	D₄	—	2	24	no	—
182	C₂×C₄	—	2	6	no	—
183	C₂×C₄	—	2	3	no	—
184	C₂×C₄	C₃×D₄×C₉	1	6	yes	complemented
185	C₂×C₃	—	4	18	no	—
186	C₂×C₃	—	4	6	no	—
187	C₂×C₃	C₉×#120	1	9	yes	—
188	C₂×C₃	C₉×#121	1	9	yes	—
189	C₂×C₃	C₉×#127	1	3	yes	—
190	C₂×C₃	C₃²×#120	1	3	yes	—
191	C₂×C₃	C₃²×#121	1	3	yes	—
192	C₂×C₃	C₃²×#127	1	1	yes	characteristic
193	C₂²	—	2	12	no	—
194	C₂²	—	2	6	no	—
195	C₂²	—	4	3	no	—
196	C₂²	—	4	6	no	—
197	C₂²	—	4	3	no	—
198	C₂²	C₂×C₃×D₄×C₉	1	3	yes	—
199	C₂²	C₂×C₃×D₄×C₉	1	3	yes	—
200	C₂²	C₃×C₉×#41	1	1	yes	characteristic
201	C₄	—	2	6	no	—
202	C₄	—	4	1	no	—
203	C₃	C₉×#392	1	3	yes	direct product factor
204	C₃	C₃²×#392	1	1	yes	characteristic
205	C₂	—	4	6	no	—
206	C₂	C₃×C₉×#120	1	3	yes	—
207	C₂	C₃×C₉×#121	1	3	yes	—
208	C₂	C₃×C₉×#127	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃×C₉	C₂³
commutator	C₂³	C₂³×C₃×C₉
Frattini, cosocle	C₂³×C₃	C₂³×C₃²
socle	C₂³×C₃²	C₂³×C₃
prime radical	C₃²×#392	C₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₃×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃×D₄×C₉	C₂	semidirect
2. C₂×C₃×C₉×C₂²⋊C₄	C₂	semidirect
3. C₃×C₉×#95	C₂	semidirect
5. C₉×#392	C₃	direct (N × H)
14. C₂²×D₄×C₉	C₂×C₃	semidirect
16. C₂×C₉×C₂²⋊C₄	C₂×C₃	semidirect
17. C₉×#95	C₂×C₃	semidirect
22. C₂³×C₃×C₉	D₄	semidirect
27. C₂×C₃×C₄×C₉	D₄	semidirect
28. C₃×#392	C₉	direct (N × H)
52. C₂²×C₃×D₄	C₂×C₉	semidirect
54. C₂×C₃×C₂²⋊C₄	C₂×C₉	semidirect
56. C₃×#95	C₂×C₉	semidirect
72. C₂³×C₉	C₃×D₄	semidirect
77. C₂×C₄×C₉	C₃×D₄	semidirect
78. #392	C₃×C₉	direct (N × H)
115. C₂²×D₄	C₂×C₃×C₉	semidirect
116. C₂×C₂²⋊C₄	C₂×C₃×C₉	semidirect
117. #95	C₂×C₃×C₉	semidirect
118. C₃×C₉	#392	direct (N × H)
127. C₂³×C₃	D₄×C₉	semidirect
137. C₂×C₃×C₄	D₄×C₉	semidirect
174. C₉	C₃×#392	direct (N × H)
179. C₂³	C₃×D₄×C₉	semidirect
184. C₂×C₄	C₃×D₄×C₉	semidirect
203. C₃	C₉×#392	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃×D₄×C₉	2
C₂×C₃×C₉×C₂²⋊C₄	2
C₃×C₉×#95	2
C₃²×#392	3
C₉×#392	3

Subgroups — C3×C9×#392 #13144934

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#392 #13144934