Subgroups — C₂×C₉×#872 #13152289

Order 1728.

1098 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×D₃×C₉	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₄²×C₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₉×(C₂×D₃)⋊C₄	C₂	1	2	yes	maximal
4	C₂×C₉×Dic₃⋊C₄	C₂	1	2	yes	maximal, complemented
5	C₂×C₄×C₉×Dic₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₉×#872	C₂	1	8	yes	maximal, direct product factor
7	C₂×C₃×#872	C₃	1	1	yes	maximal, characteristic
8	C₂×C₉×#117	—	3	1	no	maximal
9	C₂³×D₃×C₉	C₂²	1	1	yes	characteristic
10	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
11	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
12	C₂²×C₉×Dic₃	C₂²	1	2	yes	—
13	C₂²×C₉×Dic₃	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₃×C₉	C₄	1	8	yes	complemented
15	C₂×C₄×D₃×C₉	C₂²	1	4	yes	—
16	C₃×C₄²×C₉	C₂²	1	4	yes	complemented
17	C₉×(C₂×D₃)⋊C₄	C₂²	1	8	yes	—
18	C₉×Dic₃⋊C₄	C₂²	1	8	yes	complemented
19	C₄×C₉×Dic₃	C₂²	1	4	yes	complemented
20	C₂²×C₃×C₄×D₃	C₂×C₃	1	1	yes	characteristic
21	C₂×C₃²×C₄²	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×(C₂×D₃)⋊C₄	C₂×C₃	1	2	yes	—
23	C₂×C₃×Dic₃⋊C₄	C₂×C₃	1	2	yes	—
24	C₂×C₃×C₄×Dic₃	C₂×C₃	1	1	yes	characteristic
25	C₃×#872	C₂×C₃	1	8	yes	—
26	C₂³×C₄×C₉	—	3	1	no	—
27	C₉×#117	—	3	8	no	—
28	C₂×C₉×C₄⋊C₄	—	3	2	no	—
29	C₂×C₉×C₂²⋊C₄	—	3	2	no	—
30	C₂×C₄²×C₉	—	2	1	no	—
31	C₂×C₄²×C₉	—	3	1	no	—
32	C₂×C₄²×C₉	D₃	1	1	yes	characteristic, complemented
33	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
34	C₂²×D₃×C₉	—	2	4	no	—
35	C₂²×D₃×C₉	C₂×C₄	1	4	yes	—
36	C₂²×D₃×C₉	C₂³	1	2	yes	—
37	C₂×C₃×C₄×C₉	—	2	4	no	—
38	C₂×C₃×C₄×C₉	C₂×C₄	1	4	yes	—
39	C₂×C₃×C₄×C₉	C₂³	1	4	yes	—
40	C₂×C₃×C₄×C₉	C₂³	1	2	yes	—
41	C₂×C₉×Dic₃	—	2	4	no	—
42	C₂×C₉×Dic₃	C₂×C₄	1	4	yes	—
43	C₂×C₉×Dic₃	C₂³	1	4	yes	—
44	C₂×C₉×Dic₃	C₂³	1	2	yes	—
45	C₄×D₃×C₉	C₂×C₄	1	16	yes	complemented
46	C₂×C₃×#117	—	3	1	no	—
47	C₂×#872	C₉	1	1	yes	characteristic, direct product factor
48	C₂³×C₃×D₃	C₂²×C₃	1	1	yes	characteristic
49	C₂²×C₃²×C₄	C₂²×C₃	1	2	yes	—
50	C₂²×C₃²×C₄	C₂²×C₃	1	1	yes	characteristic
51	C₂²×C₃×Dic₃	C₂²×C₃	1	2	yes	—
52	C₂²×C₃×Dic₃	C₂²×C₃	1	1	yes	characteristic
53	C₂×C₃×C₄×D₃	C₃×C₄	1	8	yes	—
54	C₂×C₃×C₄×D₃	C₂²×C₃	1	4	yes	—
55	C₂⁴×C₉	—	3	1	no	—
56	C₃²×C₄²	C₂²×C₃	1	4	yes	—
57	C₃×(C₂×D₃)⋊C₄	C₂²×C₃	1	8	yes	—
58	C₃×Dic₃⋊C₄	C₂²×C₃	1	8	yes	—
59	C₃×C₄×Dic₃	C₂²×C₃	1	4	yes	—
60	C₂²×C₄×C₉	—	2	2	no	—
61	C₂²×C₄×C₉	—	2	1	no	—
62	C₂²×C₄×C₉	—	3	1	no	—
63	C₂²×C₄×C₉	—	3	8	no	—
64	C₂²×C₄×C₉	—	3	4	no	—
65	C₂²×C₄×C₉	—	3	2	no	—
66	C₂²×C₄×C₉	C₂×D₃	1	2	yes	—
67	C₂²×C₄×C₉	C₂×D₃	1	1	yes	characteristic
68	C₉×C₄⋊C₄	—	3	8	no	—
69	C₉×C₂²⋊C₄	—	3	8	no	—
70	C₄²×C₉	—	2	4	no	—
71	C₄²×C₉	—	3	4	no	—
72	C₄²×C₉	C₂×D₃	1	4	yes	complemented
73	C₂²×C₃×C₉	C₂²×C₄	1	2	yes	—
74	C₂²×C₃×C₉	#41	1	4	yes	—
75	C₂²×C₃×C₉	C₂⁴	1	1	yes	characteristic
76	C₂×D₃×C₉	—	2	8	no	—
77	C₂×D₃×C₉	—	2	4	no	—
78	C₂×D₃×C₉	C₂²×C₄	1	4	yes	—
79	C₃×C₄×C₉	—	2	4	no	—
80	C₃×C₄×C₉	C₂²×C₄	1	4	yes	—
81	C₉×Dic₃	—	2	4	no	—
82	C₉×Dic₃	C₂²×C₄	1	4	yes	—
83	C₂³×C₃×C₄	—	3	1	no	—
84	C₂²×C₄×D₃	C₂×C₉	1	1	yes	characteristic
85	C₃×#117	—	3	8	no	—
86	C₂×C₃×C₄⋊C₄	—	3	2	no	—
87	C₂×C₃×C₂²⋊C₄	—	3	2	no	—
88	C₂×C₃×C₄²	—	2	1	no	—
89	C₂×C₃×C₄²	—	3	1	no	—
90	C₂×C₃×C₄²	C₂×C₉	1	1	yes	characteristic, complemented
91	C₂×C₃×C₄²	C₃×D₃	1	1	yes	characteristic
92	C₂×(C₂×D₃)⋊C₄	C₂×C₉	1	2	yes	—
93	C₂×Dic₃⋊C₄	C₂×C₉	1	2	yes	complemented
94	C₂×C₄×Dic₃	C₂×C₉	1	1	yes	characteristic, complemented
95	#872	C₂×C₉	1	8	yes	direct product factor
96	C₂³×C₃²	C₂³×C₃	1	1	yes	characteristic
97	C₂²×C₃×D₃	—	2	4	no	—
98	C₂²×C₃×D₃	C₂×C₃×C₄	1	4	yes	—
99	C₂²×C₃×D₃	C₂³×C₃	1	2	yes	—
100	C₂×C₃²×C₄	—	2	4	no	—
101	C₂×C₃²×C₄	C₂×C₃×C₄	1	4	yes	—
102	C₂×C₃²×C₄	C₂³×C₃	1	4	yes	—
103	C₂×C₃²×C₄	C₂³×C₃	1	2	yes	—
104	C₂×C₃×Dic₃	—	2	4	no	—
105	C₂×C₃×Dic₃	C₂×C₃×C₄	1	4	yes	—
106	C₂×C₃×Dic₃	C₂³×C₃	1	4	yes	—
107	C₂×C₃×Dic₃	C₂³×C₃	1	2	yes	—
108	C₃×C₄×D₃	C₂×C₃×C₄	1	16	yes	—
109	C₂³×C₉	—	2	1	no	—
110	C₂³×C₉	—	3	4	no	—
111	C₂³×C₉	—	3	2	no	—
112	C₂³×C₉	—	6	4	no	—
113	C₂³×C₉	C₂²×D₃	1	1	yes	characteristic
114	C₂×C₄×C₉	—	2	8	no	—
115	C₂×C₄×C₉	—	2	4	no	—
116	C₂×C₄×C₉	—	2	4	no	—
117	C₂×C₄×C₉	—	2	4	no	—
118	C₂×C₄×C₉	—	2	2	no	—
119	C₂×C₄×C₉	—	3	4	no	—
120	C₂×C₄×C₉	—	3	2	no	—
121	C₂×C₄×C₉	—	3	16	no	—
122	C₂×C₄×C₉	—	3	4	no	—
123	C₂×C₄×C₉	—	6	4	no	—
124	C₂×C₄×C₉	C₄×D₃	1	4	yes	—
125	C₂×C₄×C₉	C₂²×D₃	1	4	yes	—
126	C₂×C₄×C₉	C₂²×D₃	1	2	yes	—
127	C₂×#117	—	3	1	no	Sylow
128	C₂×C₃×C₉	#117	1	4	yes	complemented
129	C₂×C₃×C₉	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₃×C₉	C₂×#41	1	2	yes	—
131	D₃×C₉	—	2	4	no	—
132	C₂⁴×C₃	—	3	1	no	—
133	C₂³×D₃	C₂²×C₉	1	1	yes	characteristic
134	C₂²×C₃×C₄	—	2	2	no	—
135	C₂²×C₃×C₄	—	2	1	no	—
136	C₂²×C₃×C₄	—	3	1	no	—
137	C₂²×C₃×C₄	—	3	8	no	—
138	C₂²×C₃×C₄	—	3	4	no	—
139	C₂²×C₃×C₄	—	3	2	no	—
140	C₂²×C₃×C₄	C₂²×C₉	1	2	yes	—
141	C₂²×C₃×C₄	C₂²×C₉	1	1	yes	characteristic
142	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
143	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
144	C₂²×Dic₃	C₂²×C₉	1	2	yes	—
145	C₂²×Dic₃	C₂²×C₉	1	1	yes	characteristic
146	C₂×C₄×D₃	C₄×C₉	1	8	yes	complemented
147	C₂×C₄×D₃	C₂²×C₉	1	4	yes	—
148	C₃×C₄⋊C₄	—	3	8	no	—
149	C₃×C₂²⋊C₄	—	3	8	no	—
150	C₃×C₄²	—	2	4	no	—
151	C₃×C₄²	—	3	4	no	—
152	C₃×C₄²	C₂²×C₉	1	4	yes	complemented
153	C₃×C₄²	C₂×C₃×D₃	1	4	yes	—
154	(C₂×D₃)⋊C₄	C₂²×C₉	1	8	yes	—
155	Dic₃⋊C₄	C₂²×C₉	1	8	yes	complemented
156	C₄×Dic₃	C₂²×C₉	1	4	yes	complemented
157	C₂²×C₃²	C₂²×C₃×C₄	1	2	yes	—
158	C₂²×C₃²	C₃×#41	1	4	yes	—
159	C₂²×C₃²	C₂⁴×C₃	1	1	yes	characteristic
160	C₂×C₃×D₃	—	2	8	no	—
161	C₂×C₃×D₃	—	2	4	no	—
162	C₂×C₃×D₃	C₂²×C₃×C₄	1	4	yes	—
163	C₃²×C₄	—	2	4	no	—
164	C₃²×C₄	C₂²×C₃×C₄	1	4	yes	—
165	C₃×Dic₃	—	2	4	no	—
166	C₃×Dic₃	C₂²×C₃×C₄	1	4	yes	—
167	C₂²×C₉	—	2	1	no	—
168	C₂²×C₉	—	2	2	no	—
169	C₂²×C₉	—	2	4	no	—
170	C₂²×C₉	—	3	4	no	—
171	C₂²×C₉	—	6	8	no	—
172	C₂²×C₉	—	6	4	no	—
173	C₂²×C₉	C₂×C₄×D₃	1	2	yes	—
174	C₂²×C₉	#235	1	4	yes	—
175	C₂²×C₉	C₂³×D₃	1	1	yes	characteristic
176	C₄×C₉	—	2	8	no	—
177	C₄×C₉	—	2	4	no	—
178	C₄×C₉	—	2	4	no	—
179	C₄×C₉	—	3	4	no	—
180	C₄×C₉	—	6	4	no	—
181	C₄×C₉	C₂×C₄×D₃	1	4	yes	—
182	C₂³×C₄	—	3	1	no	—
183	#117	—	3	8	no	—
184	C₂×C₄⋊C₄	—	3	2	no	—
185	C₂×C₂²⋊C₄	—	3	2	no	—
186	C₂×C₄²	—	3	1	no	—
187	C₂×C₄²	D₃×C₉	1	1	yes	characteristic, complemented
188	C₃×C₉	C₂×#117	1	1	yes	characteristic, complemented, Sylow
189	C₂³×C₃	—	2	1	no	—
190	C₂³×C₃	—	3	2	no	—
191	C₂³×C₃	—	3	4	no	—
192	C₂³×C₃	—	6	4	no	—
193	C₂³×C₃	C₂³×C₉	1	1	yes	characteristic
194	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
195	C₂²×D₃	—	2	4	no	—
196	C₂²×D₃	C₂×C₄×C₉	1	4	yes	—
197	C₂²×D₃	C₂³×C₉	1	2	yes	—
198	C₂×C₃×C₄	—	2	8	no	—
199	C₂×C₃×C₄	—	2	4	no	—
200	C₂×C₃×C₄	—	2	4	no	—
201	C₂×C₃×C₄	—	2	4	no	—
202	C₂×C₃×C₄	—	2	4	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	3	2	no	—
205	C₂×C₃×C₄	—	3	4	no	—
206	C₂×C₃×C₄	—	3	16	no	—
207	C₂×C₃×C₄	—	3	4	no	—
208	C₂×C₃×C₄	—	6	4	no	—
209	C₂×C₃×C₄	C₂×C₄×C₉	1	4	yes	—
210	C₂×C₃×C₄	C₂³×C₉	1	4	yes	—
211	C₂×C₃×C₄	C₂³×C₉	1	2	yes	—
212	C₂×C₃×C₄	C₃×C₄×D₃	1	4	yes	—
213	C₂×C₃×C₄	C₂²×C₃×D₃	1	4	yes	—
214	C₂×C₃×C₄	C₂²×C₃×D₃	1	2	yes	—
215	C₂×Dic₃	—	2	4	no	—
216	C₂×Dic₃	C₂×C₄×C₉	1	4	yes	—
217	C₂×Dic₃	C₂³×C₉	1	4	yes	—
218	C₂×Dic₃	C₂³×C₉	1	2	yes	—
219	C₄×D₃	C₂×C₄×C₉	1	16	yes	complemented
220	C₂×C₃²	C₃×#117	1	4	yes	—
221	C₂×C₃²	C₂³×C₃×C₄	1	1	yes	characteristic
222	C₂×C₃²	C₂×C₃×#41	1	2	yes	—
223	C₃×D₃	—	2	4	no	—
224	C₂×C₉	—	2	1	no	—
225	C₂×C₉	—	2	2	no	—
226	C₂×C₉	—	2	4	no	—
227	C₂×C₉	—	6	4	no	—
228	C₂×C₉	#872	1	4	yes	direct product factor
229	C₂×C₉	C₂²×C₄×D₃	1	1	yes	characteristic
230	C₂×C₉	C₂×#235	1	2	yes	—
231	C₂⁴	—	3	1	no	—
232	C₂²×C₄	—	3	1	no	—
233	C₂²×C₄	—	3	8	no	—
234	C₂²×C₄	—	3	4	no	—
235	C₂²×C₄	—	3	2	no	—
236	C₂²×C₄	C₂×D₃×C₉	1	2	yes	—
237	C₂²×C₄	C₂×D₃×C₉	1	1	yes	characteristic
238	C₄⋊C₄	—	3	8	no	—
239	C₂²⋊C₄	—	3	8	no	—
240	C₄²	—	3	4	no	—
241	C₄²	C₂×D₃×C₉	1	4	yes	complemented
242	C₂²×C₃	—	2	1	no	—
243	C₂²×C₃	—	2	2	no	—
244	C₂²×C₃	—	2	4	no	—
245	C₂²×C₃	—	3	4	no	—
246	C₂²×C₃	—	6	4	no	—
247	C₂²×C₃	—	6	8	no	—
248	C₂²×C₃	C₂²×C₄×C₉	1	2	yes	—
249	C₂²×C₃	C₉×#41	1	4	yes	—
250	C₂²×C₃	C₂⁴×C₉	1	1	yes	characteristic
251	C₂²×C₃	C₂×C₃×C₄×D₃	1	2	yes	—
252	C₂²×C₃	C₃×#235	1	4	yes	—
253	C₂²×C₃	C₂³×C₃×D₃	1	1	yes	characteristic
254	C₂×D₃	—	2	8	no	—
255	C₂×D₃	—	2	4	no	—
256	C₂×D₃	C₂²×C₄×C₉	1	4	yes	—
257	C₃×C₄	—	2	8	no	—
258	C₃×C₄	—	2	4	no	—
259	C₃×C₄	—	2	4	no	—
260	C₃×C₄	—	2	4	no	—
261	C₃×C₄	—	3	4	no	—
262	C₃×C₄	—	6	4	no	—
263	C₃×C₄	C₂²×C₄×C₉	1	4	yes	—
264	C₃×C₄	C₂×C₃×C₄×D₃	1	4	yes	—
265	Dic₃	—	2	4	no	—
266	Dic₃	C₂²×C₄×C₉	1	4	yes	—
267	C₃²	C₂×C₃×#117	1	1	yes	characteristic
268	C₉	—	2	1	no	—
269	C₉	C₂×#872	1	1	yes	characteristic, direct product factor
270	C₂³	—	3	2	no	—
271	C₂³	—	3	4	no	—
272	C₂³	—	6	4	no	—
273	C₂³	C₂²×D₃×C₉	1	1	yes	characteristic
274	C₂×C₄	—	2	4	no	—
275	C₂×C₄	—	3	2	no	—
276	C₂×C₄	—	3	4	no	—
277	C₂×C₄	—	3	16	no	—
278	C₂×C₄	—	3	4	no	—
279	C₂×C₄	—	6	4	no	—
280	C₂×C₄	C₄×D₃×C₉	1	4	yes	—
281	C₂×C₄	C₂²×D₃×C₉	1	4	yes	—
282	C₂×C₄	C₂²×D₃×C₉	1	2	yes	—
283	C₂×C₃	—	2	1	no	—
284	C₂×C₃	—	2	2	no	—
285	C₂×C₃	—	2	4	no	—
286	C₂×C₃	—	6	4	no	—
287	C₂×C₃	C₉×#117	1	4	yes	complemented
288	C₂×C₃	C₂³×C₄×C₉	1	1	yes	characteristic
289	C₂×C₃	C₂×C₉×#41	1	2	yes	—
290	C₂×C₃	C₃×#872	1	4	yes	—
291	C₂×C₃	C₂²×C₃×C₄×D₃	1	1	yes	characteristic
292	C₂×C₃	C₂×C₃×#235	1	2	yes	—
293	D₃	—	2	4	no	—
294	C₂²	—	3	4	no	—
295	C₂²	—	6	4	no	—
296	C₂²	—	6	8	no	—
297	C₂²	C₂×C₄×D₃×C₉	1	2	yes	—
298	C₂²	C₉×#235	1	4	yes	—
299	C₂²	C₂³×D₃×C₉	1	1	yes	characteristic
300	C₄	—	2	4	no	—
301	C₄	—	3	4	no	—
302	C₄	—	6	4	no	—
303	C₄	C₂×C₄×D₃×C₉	1	4	yes	—
304	C₃	—	2	1	no	—
305	C₃	C₂×C₉×#117	1	1	yes	characteristic, complemented
306	C₃	C₂×C₃×#872	1	1	yes	characteristic
307	C₂	—	6	4	no	—
308	C₂	C₉×#872	1	4	yes	direct product factor
309	C₂	C₂²×C₄×D₃×C₉	1	1	yes	characteristic
310	C₂	C₂×C₉×#235	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₉	C₂×D₃
commutator	C₂×C₃	C₂³×C₄×C₉
Frattini	C₂²×C₃	C₂³×C₃×D₃
Fitting	C₂×C₃×C₄²×C₉	C₂
socle	C₂³×C₃²	C₂³×C₃
prime radical	C₂³×C₃×D₃	C₂²×C₃
cosocle	C₂²×C₃²	C₂⁴×C₃
hypercenter	C₂×C₄²×C₉	D₃
hypocenter	C₃	C₂×C₉×#117
LCS	G ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄×C₉ ◁ C₂×C₄²×C₉

Complements

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

N	H ≅ G/N	type
2. C₂×C₃×C₄²×C₉	C₂	semidirect
4. C₂×C₉×Dic₃⋊C₄	C₂	semidirect
5. C₂×C₄×C₉×Dic₃	C₂	semidirect
6. C₉×#872	C₂	direct (N × H)
14. C₂×C₄×D₃×C₉	C₄	semidirect
16. C₃×C₄²×C₉	C₂²	semidirect
18. C₉×Dic₃⋊C₄	C₂²	semidirect
19. C₄×C₉×Dic₃	C₂²	semidirect
32. C₂×C₄²×C₉	D₃	semidirect
45. C₄×D₃×C₉	C₂×C₄	semidirect
47. C₂×#872	C₉	direct (N × H)
72. C₄²×C₉	C₂×D₃	semidirect
90. C₂×C₃×C₄²	C₂×C₉	semidirect
93. C₂×Dic₃⋊C₄	C₂×C₉	semidirect
94. C₂×C₄×Dic₃	C₂×C₉	semidirect
95. #872	C₂×C₉	direct (N × H)
128. C₂×C₃×C₉	#117	semidirect
146. C₂×C₄×D₃	C₄×C₉	semidirect
152. C₃×C₄²	C₂²×C₉	semidirect
155. Dic₃⋊C₄	C₂²×C₉	semidirect
156. C₄×Dic₃	C₂²×C₉	semidirect
187. C₂×C₄²	D₃×C₉	semidirect
188. C₃×C₉	C₂×#117	semidirect
219. C₄×D₃	C₂×C₄×C₉	semidirect
228. C₂×C₉	#872	direct (N × H)
241. C₄²	C₂×D₃×C₉	semidirect
269. C₉	C₂×#872	direct (N × H)
287. C₂×C₃	C₉×#117	semidirect
305. C₃	C₂×C₉×#117	semidirect
308. C₂	C₉×#872	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×C₄×D₃×C₉	2
C₂×C₃×C₄²×C₉	2
C₂×C₉×(C₂×D₃)⋊C₄	2
C₂×C₉×Dic₃⋊C₄	2
C₂×C₄×C₉×Dic₃	2
C₉×#872	2
C₂×C₃×#872	3
C₂×C₉×#117	3

Subgroups — C2×C9×#872 #13152289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₉×#872 #13152289