Subgroups — C₃×C₉×#469 #13145165

Order 1728.

650 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×D₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×C₉×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₉×#121	C₂	1	2	yes	maximal, complemented
4	C₃×C₉×#106	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₉×#102	C₂	1	2	yes	maximal, complemented
6	C₃²×#469	C₃	1	1	yes	maximal, characteristic
7	C₉×#469	C₃	1	3	yes	maximal, direct product factor
8	C₂×C₃×D₄×C₉	—	2	2	no	—
9	C₂×C₃×D₄×C₉	C₂²	1	2	yes	—
10	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₃×C₉×D₈	—	2	2	no	—
12	C₃×C₉×M₁₆	—	2	2	no	—
13	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
15	C₃×C₉×C₄⋊C₄	C₂²	1	2	yes	—
16	C₃×C₉×C₂²⋊C₄	—	2	2	no	—
17	C₂×C₃²×D₈	C₂×C₃	1	1	yes	characteristic
18	C₂×C₃²×M₁₆	C₂×C₃	1	1	yes	characteristic
19	C₃²×#121	C₂×C₃	1	2	yes	—
20	C₃²×#106	C₂×C₃	1	1	yes	characteristic
21	C₃²×#102	C₂×C₃	1	2	yes	—
22	C₂×C₉×D₈	C₂×C₃	1	3	yes	complemented
23	C₂×C₉×M₁₆	C₂×C₃	1	3	yes	complemented
24	C₉×#121	C₂×C₃	1	6	yes	complemented
25	C₉×#106	C₂×C₃	1	3	yes	complemented
26	C₉×#102	C₂×C₃	1	6	yes	complemented
27	C₂³×C₃×C₉	—	2	2	no	—
28	C₂³×C₃×C₉	D₄	1	1	yes	characteristic
29	C₃×D₄×C₉	—	2	2	no	—
30	C₃×D₄×C₉	—	2	2	no	—
31	C₃×D₄×C₉	—	4	4	no	—
32	C₂×C₃×C₄×C₉	—	2	1	no	—
33	C₂×C₃×C₄×C₉	—	2	1	no	—
34	C₂×C₃×C₄×C₉	—	2	2	no	—
35	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
36	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
37	C₃×C₈×C₉	—	2	1	no	—
38	C₃×C₈×C₉	D₄	1	2	yes	complemented
39	C₃×#469	C₉	1	3	yes	direct product factor
40	C₃×#469	C₃²	1	1	yes	characteristic
41	C₂×C₃²×D₄	—	2	2	no	—
42	C₂×C₃²×D₄	C₂²×C₃	1	2	yes	—
43	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
44	C₃²×D₈	—	2	2	no	—
45	C₃²×M₁₆	—	2	2	no	—
46	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
47	C₂×C₃²×C₈	C₂²×C₃	1	1	yes	characteristic
48	C₃²×C₄⋊C₄	C₂²×C₃	1	2	yes	—
49	C₃²×C₂²⋊C₄	—	2	2	no	—
50	C₂×D₄×C₉	—	2	6	no	—
51	C₂×D₄×C₉	C₂²×C₃	1	6	yes	—
52	C₂²×C₄×C₉	C₂²×C₃	1	3	yes	—
53	C₉×D₈	—	2	6	no	—
54	C₉×M₁₆	—	2	6	no	—
55	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
56	C₂×C₈×C₉	C₂²×C₃	1	3	yes	—
57	C₉×C₄⋊C₄	C₂²×C₃	1	6	yes	—
58	C₉×C₂²⋊C₄	—	2	6	no	—
59	C₂²×C₃×C₉	—	2	1	no	—
60	C₂²×C₃×C₉	—	2	2	no	—
61	C₂²×C₃×C₉	—	4	2	no	—
62	C₂²×C₃×C₉	—	4	2	no	—
63	C₂²×C₃×C₉	—	4	2	no	—
64	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
65	C₃×C₄×C₉	—	2	1	no	—
66	C₃×C₄×C₉	—	4	2	no	—
67	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
68	C₃×C₄×C₉	#41	1	1	yes	characteristic
69	C₂×C₃×D₈	C₂×C₉	1	3	yes	complemented
70	C₂×C₃×D₈	C₂×C₃²	1	1	yes	characteristic
71	C₂×C₃×M₁₆	C₂×C₉	1	3	yes	complemented
72	C₂×C₃×M₁₆	C₂×C₃²	1	1	yes	characteristic
73	C₃×#121	C₂×C₉	1	6	yes	complemented
74	C₃×#121	C₂×C₃²	1	2	yes	—
75	C₃×#106	C₂×C₉	1	3	yes	complemented
76	C₃×#106	C₂×C₃²	1	1	yes	characteristic
77	C₃×#102	C₂×C₉	1	6	yes	complemented
78	C₃×#102	C₂×C₃²	1	2	yes	—
79	C₂³×C₃²	—	2	2	no	—
80	C₂³×C₃²	C₃×D₄	1	1	yes	characteristic
81	C₃²×D₄	—	2	2	no	—
82	C₃²×D₄	—	2	2	no	—
83	C₃²×D₄	—	4	4	no	—
84	C₂×C₃²×C₄	—	2	1	no	—
85	C₂×C₃²×C₄	—	2	1	no	—
86	C₂×C₃²×C₄	—	2	2	no	—
87	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
88	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
89	C₂³×C₉	—	2	6	no	—
90	C₂³×C₉	C₃×D₄	1	3	yes	—
91	C₃²×C₈	—	2	1	no	—
92	C₃²×C₈	C₃×D₄	1	2	yes	—
93	D₄×C₉	—	2	6	no	—
94	D₄×C₉	—	2	6	no	—
95	D₄×C₉	—	4	12	no	—
96	C₂×C₄×C₉	—	2	3	no	—
97	C₂×C₄×C₉	—	2	3	no	—
98	C₂×C₄×C₉	—	2	6	no	—
99	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
100	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
101	C₈×C₉	—	2	3	no	—
102	C₈×C₉	C₃×D₄	1	6	yes	complemented
103	#469	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
104	C₂×C₃×C₉	—	4	1	no	—
105	C₂×C₃×C₉	—	8	2	no	—
106	C₂×C₃×C₉	#121	1	1	yes	characteristic
107	C₂×C₃×C₉	AGL(1,ℤ/8)	1	2	yes	—
108	C₂×C₃×D₄	—	2	6	no	—
109	C₂×C₃×D₄	—	2	2	no	—
110	C₂×C₃×D₄	C₂²×C₉	1	6	yes	—
111	C₂×C₃×D₄	C₂²×C₃²	1	2	yes	—
112	C₂²×C₃×C₄	C₂²×C₉	1	3	yes	—
113	C₂²×C₃×C₄	C₂²×C₃²	1	1	yes	characteristic
114	C₃×D₈	—	2	6	no	—
115	C₃×D₈	—	2	2	no	—
116	C₃×M₁₆	—	2	6	no	—
117	C₃×M₁₆	—	2	2	no	—
118	C₂×C₃×C₈	C₂²×C₉	1	3	yes	—
119	C₂×C₃×C₈	C₂²×C₉	1	3	yes	—
120	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
121	C₂×C₃×C₈	C₂²×C₃²	1	1	yes	characteristic
122	C₃×C₄⋊C₄	C₂²×C₉	1	6	yes	—
123	C₃×C₄⋊C₄	C₂²×C₃²	1	2	yes	—
124	C₃×C₂²⋊C₄	—	2	6	no	—
125	C₃×C₂²⋊C₄	—	2	2	no	—
126	C₂²×C₃²	—	2	1	no	—
127	C₂²×C₃²	—	2	2	no	—
128	C₂²×C₃²	—	4	2	no	—
129	C₂²×C₃²	—	4	2	no	—
130	C₂²×C₃²	—	4	2	no	—
131	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
132	C₃²×C₄	—	2	1	no	—
133	C₃²×C₄	—	4	2	no	—
134	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
135	C₃²×C₄	C₃×#41	1	1	yes	characteristic
136	C₂²×C₉	—	2	3	no	—
137	C₂²×C₉	—	2	6	no	—
138	C₂²×C₉	—	4	6	no	—
139	C₂²×C₉	—	4	6	no	—
140	C₂²×C₉	—	4	6	no	—
141	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
142	C₄×C₉	—	2	3	no	—
143	C₄×C₉	—	4	6	no	—
144	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
145	C₄×C₉	C₃×#41	1	3	yes	—
146	C₂×D₈	C₂×C₃×C₉	1	1	yes	characteristic, complemented
147	C₂×M₁₆	C₂×C₃×C₉	1	1	yes	characteristic, complemented
148	#121	C₂×C₃×C₉	1	2	yes	complemented
149	#106	C₂×C₃×C₉	1	1	yes	characteristic, complemented
150	#102	C₂×C₃×C₉	1	2	yes	complemented
151	C₃×C₉	#469	1	1	yes	characteristic, direct product factor, Sylow
152	C₂³×C₃	—	2	6	no	—
153	C₂³×C₃	—	2	2	no	—
154	C₂³×C₃	D₄×C₉	1	3	yes	—
155	C₂³×C₃	C₃²×D₄	1	1	yes	characteristic
156	C₃×D₄	—	2	6	no	—
157	C₃×D₄	—	2	6	no	—
158	C₃×D₄	—	2	2	no	—
159	C₃×D₄	—	2	2	no	—
160	C₃×D₄	—	4	4	no	—
161	C₃×D₄	—	4	12	no	—
162	C₂×C₃×C₄	—	2	3	no	—
163	C₂×C₃×C₄	—	2	3	no	—
164	C₂×C₃×C₄	—	2	1	no	—
165	C₂×C₃×C₄	—	2	1	no	—
166	C₂×C₃×C₄	—	2	6	no	—
167	C₂×C₃×C₄	—	2	2	no	—
168	C₂×C₃×C₄	D₄×C₉	1	3	yes	—
169	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
170	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
171	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
172	C₃×C₈	—	2	3	no	—
173	C₃×C₈	—	2	1	no	—
174	C₃×C₈	D₄×C₉	1	6	yes	complemented
175	C₃×C₈	C₃²×D₄	1	2	yes	—
176	C₂×C₃²	—	4	1	no	—
177	C₂×C₃²	—	8	2	no	—
178	C₂×C₃²	C₃×#121	1	1	yes	characteristic
179	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	2	yes	—
180	C₂×C₉	—	4	3	no	—
181	C₂×C₉	—	8	6	no	—
182	C₂×C₉	C₃×#121	1	3	yes	—
183	C₂×C₉	C₃×AGL(1,ℤ/8)	1	6	yes	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	C₂²×C₃×C₉	1	2	yes	—
186	C₂²×C₄	C₂²×C₃×C₉	1	1	yes	characteristic
187	D₈	—	2	2	no	—
188	M₁₆	—	2	2	no	—
189	C₂×C₈	C₂²×C₃×C₉	1	1	yes	characteristic
190	C₂×C₈	C₂²×C₃×C₉	1	1	yes	characteristic
191	C₄⋊C₄	C₂²×C₃×C₉	1	2	yes	—
192	C₂²⋊C₄	—	2	2	no	—
193	C₂²×C₃	—	2	3	no	—
194	C₂²×C₃	—	2	6	no	—
195	C₂²×C₃	—	2	1	no	—
196	C₂²×C₃	—	2	2	no	—
197	C₂²×C₃	—	4	6	no	—
198	C₂²×C₃	—	4	6	no	—
199	C₂²×C₃	—	4	6	no	—
200	C₂²×C₃	—	4	2	no	—
201	C₂²×C₃	—	4	2	no	—
202	C₂²×C₃	—	4	2	no	—
203	C₂²×C₃	C₂×D₄×C₉	1	3	yes	—
204	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
205	C₃×C₄	—	2	3	no	—
206	C₃×C₄	—	2	1	no	—
207	C₃×C₄	—	4	6	no	—
208	C₃×C₄	—	4	2	no	—
209	C₃×C₄	C₂×D₄×C₉	1	3	yes	—
210	C₃×C₄	C₉×#41	1	3	yes	—
211	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
212	C₃×C₄	C₃²×#41	1	1	yes	characteristic
213	C₃²	C₃×#469	1	1	yes	characteristic
214	C₉	C₃×#469	1	3	yes	direct product factor
215	C₂³	—	2	2	no	—
216	C₂³	C₃×D₄×C₉	1	1	yes	characteristic
217	D₄	—	2	2	no	—
218	D₄	—	2	2	no	—
219	D₄	—	4	4	no	—
220	C₂×C₄	—	2	1	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	C₃×D₄×C₉	1	1	yes	characteristic
224	C₂×C₄	C₂³×C₃×C₉	1	1	yes	characteristic
225	C₈	—	2	1	no	—
226	C₈	C₃×D₄×C₉	1	2	yes	complemented
227	C₂×C₃	—	4	1	no	—
228	C₂×C₃	—	4	3	no	—
229	C₂×C₃	—	8	2	no	—
230	C₂×C₃	—	8	6	no	—
231	C₂×C₃	C₉×#121	1	3	yes	—
232	C₂×C₃	C₉×AGL(1,ℤ/8)	1	6	yes	—
233	C₂×C₃	C₃²×#121	1	1	yes	characteristic
234	C₂×C₃	C₃²×AGL(1,ℤ/8)	1	2	yes	—
235	C₂²	—	2	1	no	—
236	C₂²	—	2	2	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	C₂×C₃×D₄×C₉	1	1	yes	characteristic
241	C₄	—	2	1	no	—
242	C₄	—	4	2	no	—
243	C₄	C₂×C₃×D₄×C₉	1	1	yes	characteristic
244	C₄	C₃×C₉×#41	1	1	yes	characteristic
245	C₃	C₉×#469	1	3	yes	direct product factor
246	C₃	C₃²×#469	1	1	yes	characteristic
247	C₂	—	4	1	no	—
248	C₂	—	8	2	no	—
249	C₂	C₃×C₉×#121	1	1	yes	characteristic
250	C₂	C₃×C₉×AGL(1,ℤ/8)	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₉	C₂×D₄
commutator	C₂×C₄	C₂³×C₃×C₉
Frattini, cosocle	C₂×C₃×C₄	C₂³×C₃²
socle	C₂²×C₃²	C₂×C₃×D₄
prime radical	C₃²×#469	C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₉ ◁ C₂²×C₃×C₄×C₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₉×D₈	C₂	semidirect
2. C₂×C₃×C₉×M₁₆	C₂	semidirect
3. C₃×C₉×#121	C₂	semidirect
4. C₃×C₉×#106	C₂	semidirect
5. C₃×C₉×#102	C₂	semidirect
7. C₉×#469	C₃	direct (N × H)
22. C₂×C₉×D₈	C₂×C₃	semidirect
23. C₂×C₉×M₁₆	C₂×C₃	semidirect
24. C₉×#121	C₂×C₃	semidirect
25. C₉×#106	C₂×C₃	semidirect
26. C₉×#102	C₂×C₃	semidirect
38. C₃×C₈×C₉	D₄	semidirect
39. C₃×#469	C₉	direct (N × H)
69. C₂×C₃×D₈	C₂×C₉	semidirect
71. C₂×C₃×M₁₆	C₂×C₉	semidirect
73. C₃×#121	C₂×C₉	semidirect
75. C₃×#106	C₂×C₉	semidirect
77. C₃×#102	C₂×C₉	semidirect
102. C₈×C₉	C₃×D₄	semidirect
103. #469	C₃×C₉	direct (N × H)
146. C₂×D₈	C₂×C₃×C₉	semidirect
147. C₂×M₁₆	C₂×C₃×C₉	semidirect
148. #121	C₂×C₃×C₉	semidirect
149. #106	C₂×C₃×C₉	semidirect
150. #102	C₂×C₃×C₉	semidirect
151. C₃×C₉	#469	direct (N × H)
174. C₃×C₈	D₄×C₉	semidirect
214. C₉	C₃×#469	direct (N × H)
226. C₈	C₃×D₄×C₉	semidirect
245. C₃	C₉×#469	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃×C₉×D₈	2
C₂×C₃×C₉×M₁₆	2
C₃×C₉×#121	2
C₃×C₉×#106	2
C₃×C₉×#102	2
C₃²×#469	3
C₉×#469	3

Subgroups — C3×C9×#469 #13145165

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#469 #13145165