Subgroups — C₃×C₉×#459 #13145135

Order 1728.

700 conjugacy classes of subgroups.

The subgroup lattice has 288 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	2	yes	maximal, complemented
2	C₃×C₉×#127	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×C₄×D₄×C₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₉×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₉×#102	C₂	1	2	yes	maximal, complemented
6	C₃²×#459	C₃	1	1	yes	maximal, characteristic
7	C₉×#459	C₃	1	3	yes	maximal, direct product factor
8	C₂×C₃×D₄×C₉	—	2	1	no	—
9	C₂×C₃×D₄×C₉	—	2	1	no	—
10	C₂×C₃×D₄×C₉	C₂²	1	2	yes	—
11	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₄×C₉	—	2	1	no	—
13	C₃×C₉×D₈	—	2	4	no	—
14	C₂×C₃×C₈×C₉	C₂²	1	2	yes	—
15	C₃×C₉×C₄⋊C₄	C₂²	1	1	yes	characteristic
16	C₃×C₉×C₂²⋊C₄	—	2	1	no	—
17	C₃×C₄²×C₉	C₂²	1	1	yes	characteristic
18	C₂×C₃²×D₈	C₂×C₃	1	2	yes	—
19	C₃²×#127	C₂×C₃	1	1	yes	characteristic
20	C₃²×C₄×D₄	C₂×C₃	1	1	yes	characteristic
21	C₃²×C₄⋊C₈	C₂×C₃	1	1	yes	characteristic
22	C₃²×#102	C₂×C₃	1	2	yes	—
23	C₂×C₉×D₈	C₂×C₃	1	6	yes	complemented
24	C₉×#127	C₂×C₃	1	3	yes	complemented
25	C₄×D₄×C₉	C₂×C₃	1	3	yes	complemented
26	C₉×C₄⋊C₈	C₂×C₃	1	3	yes	complemented
27	C₉×#102	C₂×C₃	1	6	yes	complemented
28	C₂³×C₃×C₉	—	2	1	no	—
29	C₂³×C₃×C₉	—	2	2	no	—
30	C₃×D₄×C₉	—	2	2	no	—
31	C₃×D₄×C₉	—	2	1	no	—
32	C₃×D₄×C₉	—	2	2	no	—
33	C₃×D₄×C₉	—	4	2	no	—
34	C₃×D₄×C₉	—	4	2	no	—
35	C₃×D₄×C₉	D₄	1	2	yes	complemented
36	C₂×C₃×C₄×C₉	—	2	1	no	—
37	C₂×C₃×C₄×C₉	—	4	2	no	—
38	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
39	C₂×C₃×C₄×C₉	D₄	1	1	yes	characteristic
40	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
41	C₃×C₈×C₉	—	2	2	no	—
42	C₃×#459	C₉	1	3	yes	direct product factor
43	C₃×#459	C₃²	1	1	yes	characteristic
44	C₂×C₃²×D₄	—	2	1	no	—
45	C₂×C₃²×D₄	—	2	1	no	—
46	C₂×C₃²×D₄	C₂²×C₃	1	2	yes	—
47	C₂×C₃²×D₄	C₂²×C₃	1	1	yes	characteristic
48	C₂²×C₃²×C₄	—	2	1	no	—
49	C₃²×D₈	—	2	4	no	—
50	C₂×C₃²×C₈	C₂²×C₃	1	2	yes	—
51	C₃²×C₄⋊C₄	C₂²×C₃	1	1	yes	characteristic
52	C₃²×C₂²⋊C₄	—	2	1	no	—
53	C₃²×C₄²	C₂²×C₃	1	1	yes	characteristic
54	C₂×D₄×C₉	—	2	3	no	—
55	C₂×D₄×C₉	—	2	3	no	—
56	C₂×D₄×C₉	C₂²×C₃	1	6	yes	—
57	C₂×D₄×C₉	C₂²×C₃	1	3	yes	—
58	C₂²×C₄×C₉	—	2	3	no	—
59	C₉×D₈	—	2	12	no	—
60	C₂×C₈×C₉	C₂²×C₃	1	6	yes	—
61	C₉×C₄⋊C₄	C₂²×C₃	1	3	yes	—
62	C₉×C₂²⋊C₄	—	2	3	no	—
63	C₄²×C₉	C₂²×C₃	1	3	yes	—
64	C₂²×C₃×C₉	—	2	2	no	—
65	C₂²×C₃×C₉	—	4	1	no	—
66	C₂²×C₃×C₉	—	4	1	no	—
67	C₂²×C₃×C₉	—	4	2	no	—
68	C₂²×C₃×C₉	—	4	2	no	—
69	C₂²×C₃×C₉	—	4	2	no	—
70	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
71	C₃×C₄×C₉	—	2	1	no	—
72	C₃×C₄×C₉	—	4	1	no	—
73	C₃×C₄×C₉	D₈	1	2	yes	complemented
74	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
75	C₃×C₄×C₉	#41	1	1	yes	characteristic
76	C₂×C₃×D₈	C₂×C₉	1	6	yes	complemented
77	C₂×C₃×D₈	C₂×C₃²	1	2	yes	—
78	C₃×#127	C₂×C₉	1	3	yes	complemented
79	C₃×#127	C₂×C₃²	1	1	yes	characteristic
80	C₃×C₄×D₄	C₂×C₉	1	3	yes	complemented
81	C₃×C₄×D₄	C₂×C₃²	1	1	yes	characteristic
82	C₃×C₄⋊C₈	C₂×C₉	1	3	yes	complemented
83	C₃×C₄⋊C₈	C₂×C₃²	1	1	yes	characteristic
84	C₃×#102	C₂×C₉	1	6	yes	complemented
85	C₃×#102	C₂×C₃²	1	2	yes	—
86	C₂³×C₃²	—	2	1	no	—
87	C₂³×C₃²	—	2	2	no	—
88	C₃²×D₄	—	2	2	no	—
89	C₃²×D₄	—	2	1	no	—
90	C₃²×D₄	—	2	2	no	—
91	C₃²×D₄	—	4	2	no	—
92	C₃²×D₄	—	4	2	no	—
93	C₃²×D₄	C₃×D₄	1	2	yes	—
94	C₂×C₃²×C₄	—	2	1	no	—
95	C₂×C₃²×C₄	—	4	2	no	—
96	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
97	C₂×C₃²×C₄	C₃×D₄	1	1	yes	characteristic
98	C₂×C₃²×C₄	C₂³×C₃	1	1	yes	characteristic
99	C₂³×C₉	—	2	3	no	—
100	C₂³×C₉	—	2	6	no	—
101	C₃²×C₈	—	2	2	no	—
102	D₄×C₉	—	2	6	no	—
103	D₄×C₉	—	2	3	no	—
104	D₄×C₉	—	2	6	no	—
105	D₄×C₉	—	4	6	no	—
106	D₄×C₉	—	4	6	no	—
107	D₄×C₉	C₃×D₄	1	6	yes	complemented
108	C₂×C₄×C₉	—	2	3	no	—
109	C₂×C₄×C₉	—	4	6	no	—
110	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
111	C₂×C₄×C₉	C₃×D₄	1	3	yes	—
112	C₂×C₄×C₉	C₂³×C₃	1	3	yes	—
113	C₈×C₉	—	2	6	no	—
114	#459	C₃×C₉	1	1	yes	characteristic, direct product factor, Sylow
115	C₂×C₃×C₉	—	4	2	no	—
116	C₂×C₃×C₉	—	8	2	no	—
117	C₂×C₃×C₉	#121	1	1	yes	characteristic
118	C₂×C₃×C₉	C₂×D₈	1	1	yes	characteristic
119	C₂×C₃×C₉	AGL(1,ℤ/8)	1	1	yes	characteristic
120	C₂×C₃×D₄	—	2	3	no	—
121	C₂×C₃×D₄	—	2	3	no	—
122	C₂×C₃×D₄	—	2	1	no	—
123	C₂×C₃×D₄	—	2	1	no	—
124	C₂×C₃×D₄	C₂²×C₉	1	6	yes	—
125	C₂×C₃×D₄	C₂²×C₉	1	3	yes	—
126	C₂×C₃×D₄	C₂²×C₃²	1	2	yes	—
127	C₂×C₃×D₄	C₂²×C₃²	1	1	yes	characteristic
128	C₂²×C₃×C₄	—	2	3	no	—
129	C₂²×C₃×C₄	—	2	1	no	—
130	C₃×D₈	—	2	12	no	—
131	C₃×D₈	—	2	4	no	—
132	C₂×C₃×C₈	C₂²×C₉	1	6	yes	—
133	C₂×C₃×C₈	C₂²×C₃²	1	2	yes	—
134	C₃×C₄⋊C₄	C₂²×C₉	1	3	yes	—
135	C₃×C₄⋊C₄	C₂²×C₃²	1	1	yes	characteristic
136	C₃×C₂²⋊C₄	—	2	3	no	—
137	C₃×C₂²⋊C₄	—	2	1	no	—
138	C₃×C₄²	C₂²×C₉	1	3	yes	—
139	C₃×C₄²	C₂²×C₃²	1	1	yes	characteristic
140	C₂²×C₃²	—	2	2	no	—
141	C₂²×C₃²	—	4	1	no	—
142	C₂²×C₃²	—	4	1	no	—
143	C₂²×C₃²	—	4	2	no	—
144	C₂²×C₃²	—	4	2	no	—
145	C₂²×C₃²	—	4	2	no	—
146	C₂²×C₃²	C₂×C₃×D₄	1	1	yes	characteristic
147	C₃²×C₄	—	2	1	no	—
148	C₃²×C₄	—	4	1	no	—
149	C₃²×C₄	C₃×D₈	1	2	yes	—
150	C₃²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
151	C₃²×C₄	C₃×#41	1	1	yes	characteristic
152	C₂²×C₉	—	2	6	no	—
153	C₂²×C₉	—	4	3	no	—
154	C₂²×C₉	—	4	3	no	—
155	C₂²×C₉	—	4	6	no	—
156	C₂²×C₉	—	4	6	no	—
157	C₂²×C₉	—	4	6	no	—
158	C₂²×C₉	C₂×C₃×D₄	1	3	yes	—
159	C₄×C₉	—	2	3	no	—
160	C₄×C₉	—	4	3	no	—
161	C₄×C₉	C₃×D₈	1	6	yes	complemented
162	C₄×C₉	C₂×C₃×D₄	1	3	yes	—
163	C₄×C₉	C₃×#41	1	3	yes	—
164	C₂×D₈	C₂×C₃×C₉	1	2	yes	complemented
165	#127	C₂×C₃×C₉	1	1	yes	characteristic, complemented
166	C₄×D₄	C₂×C₃×C₉	1	1	yes	characteristic, complemented
167	C₄⋊C₈	C₂×C₃×C₉	1	1	yes	characteristic, complemented
168	#102	C₂×C₃×C₉	1	2	yes	complemented
169	C₃×C₉	#459	1	1	yes	characteristic, direct product factor, Sylow
170	C₂³×C₃	—	2	3	no	—
171	C₂³×C₃	—	2	1	no	—
172	C₂³×C₃	—	2	6	no	—
173	C₂³×C₃	—	2	2	no	—
174	C₃×D₄	—	2	6	no	—
175	C₃×D₄	—	2	3	no	—
176	C₃×D₄	—	2	6	no	—
177	C₃×D₄	—	2	2	no	—
178	C₃×D₄	—	2	1	no	—
179	C₃×D₄	—	2	2	no	—
180	C₃×D₄	—	4	2	no	—
181	C₃×D₄	—	4	6	no	—
182	C₃×D₄	—	4	6	no	—
183	C₃×D₄	—	4	2	no	—
184	C₃×D₄	D₄×C₉	1	6	yes	complemented
185	C₃×D₄	C₃²×D₄	1	2	yes	—
186	C₂×C₃×C₄	—	2	3	no	—
187	C₂×C₃×C₄	—	2	1	no	—
188	C₂×C₃×C₄	—	4	6	no	—
189	C₂×C₃×C₄	—	4	2	no	—
190	C₂×C₃×C₄	D₄×C₉	1	3	yes	—
191	C₂×C₃×C₄	D₄×C₉	1	3	yes	—
192	C₂×C₃×C₄	C₂³×C₉	1	3	yes	—
193	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
194	C₂×C₃×C₄	C₃²×D₄	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂³×C₃²	1	1	yes	characteristic
196	C₃×C₈	—	2	6	no	—
197	C₃×C₈	—	2	2	no	—
198	C₂×C₃²	—	4	2	no	—
199	C₂×C₃²	—	8	2	no	—
200	C₂×C₃²	C₃×#121	1	1	yes	characteristic
201	C₂×C₃²	C₂×C₃×D₈	1	1	yes	characteristic
202	C₂×C₃²	C₃×AGL(1,ℤ/8)	1	1	yes	characteristic
203	C₂×C₉	—	4	6	no	—
204	C₂×C₉	—	8	6	no	—
205	C₂×C₉	C₃×#121	1	3	yes	—
206	C₂×C₉	C₂×C₃×D₈	1	3	yes	—
207	C₂×C₉	C₃×AGL(1,ℤ/8)	1	3	yes	—
208	C₂×D₄	—	2	1	no	—
209	C₂×D₄	—	2	1	no	—
210	C₂×D₄	C₂²×C₃×C₉	1	2	yes	—
211	C₂×D₄	C₂²×C₃×C₉	1	1	yes	characteristic
212	C₂²×C₄	—	2	1	no	—
213	D₈	—	2	4	no	—
214	C₂×C₈	C₂²×C₃×C₉	1	2	yes	—
215	C₄⋊C₄	C₂²×C₃×C₉	1	1	yes	characteristic
216	C₂²⋊C₄	—	2	1	no	—
217	C₄²	C₂²×C₃×C₉	1	1	yes	characteristic
218	C₂²×C₃	—	2	6	no	—
219	C₂²×C₃	—	2	2	no	—
220	C₂²×C₃	—	4	3	no	—
221	C₂²×C₃	—	4	3	no	—
222	C₂²×C₃	—	4	1	no	—
223	C₂²×C₃	—	4	1	no	—
224	C₂²×C₃	—	4	6	no	—
225	C₂²×C₃	—	4	6	no	—
226	C₂²×C₃	—	4	6	no	—
227	C₂²×C₃	—	4	2	no	—
228	C₂²×C₃	—	4	2	no	—
229	C₂²×C₃	—	4	2	no	—
230	C₂²×C₃	C₂×D₄×C₉	1	3	yes	—
231	C₂²×C₃	C₂×C₃²×D₄	1	1	yes	characteristic
232	C₃×C₄	—	2	3	no	—
233	C₃×C₄	—	2	1	no	—
234	C₃×C₄	—	4	3	no	—
235	C₃×C₄	—	4	1	no	—
236	C₃×C₄	C₉×D₈	1	6	yes	complemented
237	C₃×C₄	C₂×D₄×C₉	1	3	yes	—
238	C₃×C₄	C₉×#41	1	3	yes	—
239	C₃×C₄	C₃²×D₈	1	2	yes	—
240	C₃×C₄	C₂×C₃²×D₄	1	1	yes	characteristic
241	C₃×C₄	C₃²×#41	1	1	yes	characteristic
242	C₃²	C₃×#459	1	1	yes	characteristic
243	C₉	C₃×#459	1	3	yes	direct product factor
244	C₂³	—	2	1	no	—
245	C₂³	—	2	2	no	—
246	D₄	—	2	2	no	—
247	D₄	—	2	1	no	—
248	D₄	—	2	2	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	D₄	C₃×D₄×C₉	1	2	yes	complemented
252	C₂×C₄	—	2	1	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	C₃×D₄×C₉	1	1	yes	characteristic
255	C₂×C₄	C₃×D₄×C₉	1	1	yes	characteristic
256	C₂×C₄	C₂³×C₃×C₉	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₂×C₃	—	4	2	no	—
259	C₂×C₃	—	4	6	no	—
260	C₂×C₃	—	8	2	no	—
261	C₂×C₃	—	8	6	no	—
262	C₂×C₃	C₉×#121	1	3	yes	—
263	C₂×C₃	C₂×C₉×D₈	1	3	yes	—
264	C₂×C₃	C₉×AGL(1,ℤ/8)	1	3	yes	—
265	C₂×C₃	C₃²×#121	1	1	yes	characteristic
266	C₂×C₃	C₂×C₃²×D₈	1	1	yes	characteristic
267	C₂×C₃	C₃²×AGL(1,ℤ/8)	1	1	yes	characteristic
268	C₂²	—	2	2	no	—
269	C₂²	—	4	1	no	—
270	C₂²	—	4	1	no	—
271	C₂²	—	4	2	no	—
272	C₂²	—	4	2	no	—
273	C₂²	—	4	2	no	—
274	C₂²	C₂×C₃×D₄×C₉	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	4	1	no	—
277	C₄	C₃×C₉×D₈	1	2	yes	complemented
278	C₄	C₂×C₃×D₄×C₉	1	1	yes	characteristic
279	C₄	C₃×C₉×#41	1	1	yes	characteristic
280	C₃	C₉×#459	1	3	yes	direct product factor
281	C₃	C₃²×#459	1	1	yes	characteristic
282	C₂	—	4	2	no	—
283	C₂	—	8	2	no	—
284	C₂	C₃×C₉×#121	1	1	yes	characteristic
285	C₂	C₂×C₃×C₉×D₈	1	1	yes	characteristic
286	C₂	C₃×C₉×AGL(1,ℤ/8)	1	1	yes	characteristic

	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×#459	C₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₉ ◁ C₃×C₄²×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. C₃×C₉×#127	C₂	semidirect
3. C₃×C₄×D₄×C₉	C₂	semidirect
4. C₃×C₉×C₄⋊C₈	C₂	semidirect
5. C₃×C₉×#102	C₂	semidirect
7. C₉×#459	C₃	direct (N × H)
23. C₂×C₉×D₈	C₂×C₃	semidirect
24. C₉×#127	C₂×C₃	semidirect
25. C₄×D₄×C₉	C₂×C₃	semidirect
26. C₉×C₄⋊C₈	C₂×C₃	semidirect
27. C₉×#102	C₂×C₃	semidirect
35. C₃×D₄×C₉	D₄	semidirect
42. C₃×#459	C₉	direct (N × H)
73. C₃×C₄×C₉	D₈	semidirect
76. C₂×C₃×D₈	C₂×C₉	semidirect
78. C₃×#127	C₂×C₉	semidirect
80. C₃×C₄×D₄	C₂×C₉	semidirect
82. C₃×C₄⋊C₈	C₂×C₉	semidirect
84. C₃×#102	C₂×C₉	semidirect
107. D₄×C₉	C₃×D₄	semidirect
114. #459	C₃×C₉	direct (N × H)
161. C₄×C₉	C₃×D₈	semidirect
164. C₂×D₈	C₂×C₃×C₉	semidirect
165. #127	C₂×C₃×C₉	semidirect
166. C₄×D₄	C₂×C₃×C₉	semidirect
167. C₄⋊C₈	C₂×C₃×C₉	semidirect
168. #102	C₂×C₃×C₉	semidirect
169. C₃×C₉	#459	direct (N × H)
184. C₃×D₄	D₄×C₉	semidirect
236. C₃×C₄	C₉×D₈	semidirect
243. C₉	C₃×#459	direct (N × H)
251. D₄	C₃×D₄×C₉	semidirect
277. C₄	C₃×C₉×D₈	semidirect
280. C₃	C₉×#459	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
C₃×C₉×#127	2
C₃×C₄×D₄×C₉	2
C₃×C₉×C₄⋊C₈	2
C₃×C₉×#102	2
C₃²×#459	3
C₉×#459	3

Subgroups — C3×C9×#459 #13145135

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×C₉×#459 #13145135