Subgroups — C₃×#10588354 #13141318

Order 1728.

586 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×(C₂×D₉)⋊C₄	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×(C₂×D₉)⋊C₄	C₂	1	2	yes	maximal
3	C₂×C₃×C₄⋊Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×C₉×#95	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×C₉⋊#95	C₂	1	2	yes	maximal, complemented
6	C₃×#4674	—	3	1	no	maximal
7	#10588354	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂³×C₃×D₉	C₂²	1	1	yes	characteristic
9	C₂²×C₃×C₄×C₉	C₂²	1	2	yes	—
10	C₂²×C₃×C₄×C₉	C₂²	1	1	yes	characteristic
11	C₂²×C₃×Dic₉	C₂²	1	1	yes	characteristic
12	C₂²×C₃×Dic₉	C₂²	1	2	yes	—
13	C₃×(C₂×D₉)⋊C₄	—	2	4	no	—
14	C₃×(C₂×D₉)⋊C₄	—	2	2	no	—
15	C₃×C₄⋊Dic₉	—	2	2	no	—
16	C₂×C₃×(C₂×D₃)⋊C₄	—	3	2	no	—
17	C₂×C₃×(C₂×D₃)⋊C₄	—	3	1	no	—
18	C₂×C₃×C₄⋊Dic₃	—	3	1	no	—
19	C₃²×#95	D₃	1	1	yes	characteristic
20	C₃×C₃⋊#95	—	3	2	no	—
21	C₂×(C₂×D₉)⋊C₄	C₂×C₃	1	2	yes	—
22	C₂×(C₂×D₉)⋊C₄	C₂×C₃	1	1	yes	characteristic
23	C₂×C₄⋊Dic₉	C₂×C₃	1	1	yes	characteristic, complemented
24	C₉×#95	—	2	1	no	—
25	C₉×#95	C₂×C₃	1	1	yes	characteristic, complemented
26	C₉⋊#95	C₂×C₃	1	2	yes	complemented
27	C₂³×C₃×C₉	C₂³	1	1	yes	characteristic
28	C₂²×C₃×D₉	—	2	1	no	—
29	C₂²×C₃×D₉	—	2	1	no	—
30	C₂²×C₃×D₉	—	2	2	no	—
31	C₂²×C₃×D₉	—	2	2	no	—
32	C₂²×C₃×D₉	D₄	1	2	yes	—
33	C₂×C₃×C₄×C₉	—	2	2	no	—
34	C₂×C₃×C₄×C₉	—	2	2	no	—
35	C₂×C₃×C₄×C₉	—	2	2	no	—
36	C₂×C₃×C₄×C₉	—	2	2	no	—
37	C₂×C₃×C₄×C₉	D₄	1	2	yes	—
38	C₂×C₃×Dic₉	—	2	2	no	—
39	C₂×C₃×Dic₉	—	2	2	no	—
40	C₂×C₃×Dic₉	—	2	1	no	—
41	C₂×C₃×Dic₉	—	2	2	no	—
42	C₂×C₃×Dic₉	—	2	2	no	—
43	C₃×#397	—	9	1	no	—
44	#4674	—	3	1	no	—
45	C₂³×C₃×D₃	—	3	1	no	—
46	C₂²×C₃²×C₄	C₂×D₃	1	2	yes	—
47	C₂²×C₃²×C₄	C₂×D₃	1	1	yes	characteristic
48	C₂²×C₃×Dic₃	—	3	2	no	—
49	C₂²×C₃×Dic₃	—	3	1	no	—
50	C₂³×D₉	C₂²×C₃	1	1	yes	characteristic
51	C₃×(C₂×D₃)⋊C₄	—	6	4	no	—
52	C₃×(C₂×D₃)⋊C₄	—	6	2	no	—
53	C₃×C₄⋊Dic₃	—	6	2	no	—
54	C₂²×C₄×C₉	—	2	2	no	—
55	C₂²×C₄×C₉	—	2	1	no	—
56	C₂²×C₄×C₉	C₂²×C₃	1	2	yes	—
57	C₂²×C₄×C₉	C₂²×C₃	1	1	yes	characteristic
58	C₂²×Dic₉	C₂²×C₃	1	1	yes	characteristic
59	C₂²×Dic₉	C₂²×C₃	1	2	yes	—
60	(C₂×D₉)⋊C₄	—	2	4	no	—
61	(C₂×D₉)⋊C₄	—	2	2	no	—
62	C₄⋊Dic₉	—	2	2	no	—
63	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₃×C₉	#41	1	1	yes	characteristic
66	C₂²×C₃×C₉	#41	1	2	yes	—
67	C₂²×C₃×C₉	#41	1	2	yes	—
68	C₂×C₃×D₉	—	2	2	no	—
69	C₂×C₃×D₉	—	2	4	no	—
70	C₂×C₃×D₉	—	4	1	no	—
71	C₂×C₃×D₉	—	4	1	no	—
72	C₂×C₃×D₉	—	4	2	no	—
73	C₃×C₄×C₉	—	2	2	no	—
74	C₃×C₄×C₉	—	4	2	no	—
75	C₃×Dic₉	—	4	2	no	—
76	C₃×Dic₉	—	4	1	no	—
77	C₂×C₃×C₄⋊C₄	—	9	1	no	—
78	C₂×C₃×C₂²⋊C₄	—	9	2	no	—
79	C₂×C₃×C₂²⋊C₄	—	9	1	no	—
80	C₂×(C₂×D₃)⋊C₄	—	3	2	no	—
81	C₂×(C₂×D₃)⋊C₄	—	3	1	no	—
82	C₂×C₄⋊Dic₃	—	3	1	no	—
83	C₃×#95	—	2	1	no	—
84	C₃×#95	—	9	2	no	—
85	C₃×#95	D₉	1	1	yes	characteristic, complemented
86	C₃×#95	C₃×D₃	1	1	yes	characteristic
87	C₃⋊#95	—	3	2	no	—
88	C₂³×C₃²	C₂²×D₃	1	1	yes	characteristic
89	C₂²×C₃×D₃	—	3	2	no	—
90	C₂²×C₃×D₃	—	6	1	no	—
91	C₂²×C₃×D₃	—	6	2	no	—
92	C₂²×C₃×D₃	—	6	2	no	—
93	C₂²×C₃×D₃	—	6	1	no	—
94	C₂×C₃²×C₄	—	2	2	no	—
95	C₂×C₃²×C₄	—	2	2	no	—
96	C₂×C₃²×C₄	—	2	2	no	—
97	C₂×C₃²×C₄	—	2	2	no	—
98	C₂×C₃²×C₄	D₁₂	1	2	yes	—
99	C₂×C₃×Dic₃	—	6	2	no	—
100	C₂×C₃×Dic₃	—	6	1	no	—
101	C₂×C₃×Dic₃	—	6	2	no	—
102	C₂×C₃×Dic₃	—	6	2	no	—
103	C₂×C₃×Dic₃	—	6	2	no	—
104	C₂³×C₉	—	2	1	no	—
105	C₂³×C₉	C₂³×C₃	1	1	yes	characteristic
106	C₂²×D₉	—	2	1	no	—
107	C₂²×D₉	—	2	2	no	—
108	C₂²×D₉	—	2	2	no	—
109	C₂²×D₉	—	2	1	no	—
110	C₂²×D₉	C₃×D₄	1	2	yes	—
111	C₂×C₄×C₉	—	2	2	no	—
112	C₂×C₄×C₉	—	2	4	no	—
113	C₂×C₄×C₉	—	2	2	no	—
114	C₂×C₄×C₉	—	2	2	no	—
115	C₂×C₄×C₉	—	2	2	no	—
116	C₂×C₄×C₉	—	2	2	no	—
117	C₂×C₄×C₉	—	4	2	no	—
118	C₂×C₄×C₉	—	4	2	no	—
119	C₂×C₄×C₉	—	4	2	no	—
120	C₂×C₄×C₉	C₃×D₄	1	2	yes	—
121	C₂×Dic₉	—	2	2	no	—
122	C₂×Dic₉	—	2	1	no	—
123	C₂×Dic₉	—	2	2	no	—
124	C₂×Dic₉	—	2	2	no	—
125	C₂×Dic₉	—	2	2	no	—
126	#397	—	9	1	no	Sylow
127	C₂×C₃×C₉	#121	1	1	yes	characteristic
128	C₂×C₃×C₉	#123	1	2	yes	—
129	C₂×C₃×C₉	#123	1	1	yes	characteristic
130	C₂×C₃×C₉	#124	1	1	yes	characteristic
131	C₂×C₃×C₉	#126	1	2	yes	—
132	C₃×D₉	—	4	2	no	—
133	C₂⁴×C₃	—	9	1	no	—
134	C₂³×D₃	—	3	1	no	—
135	C₂²×C₃×C₄	—	2	2	no	—
136	C₂²×C₃×C₄	—	2	1	no	—
137	C₂²×C₃×C₄	—	9	1	no	—
138	C₂²×C₃×C₄	—	9	2	no	—
139	C₂²×C₃×C₄	C₂×D₉	1	2	yes	—
140	C₂²×C₃×C₄	C₂×D₉	1	1	yes	characteristic
141	C₂²×C₃×C₄	C₂×C₃×D₃	1	2	yes	—
142	C₂²×C₃×C₄	C₂×C₃×D₃	1	1	yes	characteristic
143	C₂²×Dic₃	—	3	1	no	—
144	C₂²×Dic₃	—	3	2	no	—
145	C₃×C₄⋊C₄	—	18	2	no	—
146	C₃×C₂²⋊C₄	—	18	4	no	—
147	C₃×C₂²⋊C₄	—	18	2	no	—
148	(C₂×D₃)⋊C₄	—	6	4	no	—
149	(C₂×D₃)⋊C₄	—	6	2	no	—
150	C₄⋊Dic₃	—	6	2	no	—
151	C₂²×C₃²	C₂×D₁₂	1	1	yes	characteristic
152	C₂²×C₃²	#235	1	2	yes	—
153	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
154	C₂²×C₃²	#237	1	2	yes	—
155	C₂²×C₃²	#239	1	1	yes	characteristic
156	C₂×C₃×D₃	—	6	4	no	—
157	C₂×C₃×D₃	—	6	2	no	—
158	C₂×C₃×D₃	—	12	1	no	—
159	C₂×C₃×D₃	—	12	1	no	—
160	C₂×C₃×D₃	—	12	2	no	—
161	C₃²×C₄	—	2	2	no	—
162	C₃²×C₄	—	4	2	no	—
163	C₃×Dic₃	—	12	1	no	—
164	C₃×Dic₃	—	12	2	no	—
165	C₂²×C₉	—	2	1	no	—
166	C₂²×C₉	—	2	2	no	—
167	C₂²×C₉	—	2	1	no	—
168	C₂²×C₉	—	2	2	no	—
169	C₂²×C₉	—	2	1	no	—
170	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
171	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
172	C₂²×C₉	C₃×#41	1	2	yes	—
173	C₂²×C₉	C₃×#41	1	2	yes	—
174	C₂²×C₉	C₃×#41	1	1	yes	characteristic
175	C₂×D₉	—	2	2	no	—
176	C₂×D₉	—	2	4	no	—
177	C₂×D₉	—	4	1	no	—
178	C₂×D₉	—	4	1	no	—
179	C₂×D₉	—	4	2	no	—
180	C₄×C₉	—	2	2	no	—
181	C₄×C₉	—	4	2	no	—
182	C₄×C₉	—	4	4	no	—
183	C₄×C₉	—	4	2	no	—
184	Dic₉	—	4	2	no	—
185	Dic₉	—	4	1	no	—
186	C₂×C₄⋊C₄	—	9	1	no	—
187	C₂×C₂²⋊C₄	—	9	2	no	—
188	C₂×C₂²⋊C₄	—	9	1	no	—
189	#95	—	9	2	no	—
190	#95	C₃×D₉	1	1	yes	characteristic, complemented
191	C₃×C₉	#397	1	1	yes	characteristic, complemented, Sylow
192	C₂³×C₃	—	2	1	no	—
193	C₂³×C₃	—	9	2	no	—
194	C₂³×C₃	—	18	1	no	—
195	C₂³×C₃	—	18	2	no	—
196	C₂³×C₃	—	18	2	no	—
197	C₂³×C₃	—	18	1	no	—
198	C₂³×C₃	C₂²×D₉	1	1	yes	characteristic
199	C₂³×C₃	C₂²×C₃×D₃	1	1	yes	characteristic
200	C₂²×D₃	—	3	2	no	—
201	C₂²×D₃	—	6	1	no	—
202	C₂²×D₃	—	6	2	no	—
203	C₂²×D₃	—	6	2	no	—
204	C₂²×D₃	—	6	1	no	—
205	C₂×C₃×C₄	—	2	2	no	—
206	C₂×C₃×C₄	—	2	2	no	—
207	C₂×C₃×C₄	—	2	4	no	—
208	C₂×C₃×C₄	—	2	2	no	—
209	C₂×C₃×C₄	—	2	2	no	—
210	C₂×C₃×C₄	—	2	2	no	—
211	C₂×C₃×C₄	—	2	2	no	—
212	C₂×C₃×C₄	—	2	2	no	—
213	C₂×C₃×C₄	—	2	2	no	—
214	C₂×C₃×C₄	—	2	2	no	—
215	C₂×C₃×C₄	—	4	2	no	—
216	C₂×C₃×C₄	—	4	2	no	—
217	C₂×C₃×C₄	—	4	2	no	—
218	C₂×C₃×C₄	—	18	1	no	—
219	C₂×C₃×C₄	—	18	2	no	—
220	C₂×C₃×C₄	—	18	2	no	—
221	C₂×C₃×C₄	—	18	2	no	—
222	C₂×C₃×C₄	—	18	2	no	—
223	C₂×C₃×C₄	D₃₆	1	2	yes	—
224	C₂×C₃×C₄	C₃×D₁₂	1	2	yes	—
225	C₂×Dic₃	—	6	2	no	—
226	C₂×Dic₃	—	6	1	no	—
227	C₂×Dic₃	—	6	2	no	—
228	C₂×Dic₃	—	6	2	no	—
229	C₂×Dic₃	—	6	2	no	—
230	C₂×C₃²	#875	1	1	yes	characteristic
231	C₂×C₃²	#883	1	2	yes	—
232	C₂×C₃²	#886	1	1	yes	characteristic
233	C₂×C₃²	#895	1	1	yes	characteristic
234	C₂×C₃²	#898	1	2	yes	—
235	C₃×D₃	—	12	2	no	—
236	C₂×C₉	—	2	2	no	—
237	C₂×C₉	—	2	1	no	—
238	C₂×C₉	—	2	1	no	—
239	C₂×C₉	—	2	2	no	—
240	C₂×C₉	—	2	1	no	—
241	C₂×C₉	C₃×#121	1	1	yes	characteristic
242	C₂×C₉	C₃×#123	1	2	yes	—
243	C₂×C₉	C₃×#123	1	1	yes	characteristic
244	C₂×C₉	C₃×#124	1	1	yes	characteristic
245	C₂×C₉	C₃×#126	1	2	yes	—
246	D₉	—	4	2	no	—
247	C₂⁴	—	9	1	no	—
248	C₂²×C₄	—	9	1	no	—
249	C₂²×C₄	—	9	2	no	—
250	C₂²×C₄	C₂×C₃×D₉	1	2	yes	—
251	C₂²×C₄	C₂×C₃×D₉	1	1	yes	characteristic
252	C₄⋊C₄	—	18	2	no	—
253	C₂²⋊C₄	—	18	4	no	—
254	C₂²⋊C₄	—	18	2	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	—	2	2	no	—
257	C₂²×C₃	—	2	1	no	—
258	C₂²×C₃	—	2	2	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	18	4	no	—
261	C₂²×C₃	—	18	2	no	—
262	C₂²×C₃	—	36	1	no	—
263	C₂²×C₃	—	36	1	no	—
264	C₂²×C₃	—	36	2	no	—
265	C₂²×C₃	C₂×D₃₆	1	1	yes	characteristic
266	C₂²×C₃	#3697	1	2	yes	—
267	C₂²×C₃	D₄×D₉	1	1	yes	characteristic
268	C₂²×C₃	#3699	1	2	yes	—
269	C₂²×C₃	#3701	1	1	yes	characteristic
270	C₂²×C₃	C₂×C₃×D₁₂	1	1	yes	characteristic
271	C₂²×C₃	C₃×#235	1	2	yes	—
272	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
273	C₂²×C₃	C₃×#237	1	2	yes	—
274	C₂²×C₃	C₃×#239	1	1	yes	characteristic
275	C₂×D₃	—	6	4	no	—
276	C₂×D₃	—	6	2	no	—
277	C₂×D₃	—	12	1	no	—
278	C₂×D₃	—	12	2	no	—
279	C₂×D₃	—	12	1	no	—
280	C₃×C₄	—	2	2	no	—
281	C₃×C₄	—	2	2	no	—
282	C₃×C₄	—	4	2	no	—
283	C₃×C₄	—	4	4	no	—
284	C₃×C₄	—	4	2	no	—
285	C₃×C₄	—	4	2	no	—
286	C₃×C₄	—	36	1	no	—
287	C₃×C₄	—	36	2	no	—
288	Dic₃	—	12	2	no	—
289	Dic₃	—	12	1	no	—
290	C₃²	#4674	1	1	yes	characteristic
291	C₉	—	2	1	no	—
292	C₉	C₃×#397	1	1	yes	characteristic, complemented
293	C₂³	—	9	2	no	—
294	C₂³	—	18	1	no	—
295	C₂³	—	18	2	no	—
296	C₂³	—	18	2	no	—
297	C₂³	—	18	1	no	—
298	C₂³	C₂²×C₃×D₉	1	1	yes	characteristic
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	2	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	18	1	no	—
304	C₂×C₄	—	18	2	no	—
305	C₂×C₄	—	18	2	no	—
306	C₂×C₄	—	18	2	no	—
307	C₂×C₄	—	18	2	no	—
308	C₂×C₄	C₃×D₃₆	1	2	yes	—
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	—	2	2	no	—
311	C₂×C₃	—	2	1	no	—
312	C₂×C₃	—	2	1	no	—
313	C₂×C₃	—	2	2	no	—
314	C₂×C₃	—	36	2	no	—
315	C₂×C₃	#63437	1	1	yes	characteristic
316	C₂×C₃	#63445	1	2	yes	—
317	C₂×C₃	#63448	1	1	yes	characteristic
318	C₂×C₃	#63457	1	1	yes	characteristic
319	C₂×C₃	#63460	1	2	yes	—
320	C₂×C₃	C₃×#875	1	1	yes	characteristic
321	C₂×C₃	C₃×#883	1	2	yes	—
322	C₂×C₃	C₃×#886	1	1	yes	characteristic
323	C₂×C₃	C₃×#895	1	1	yes	characteristic
324	C₂×C₃	C₃×#898	1	2	yes	—
325	D₃	—	12	2	no	—
326	C₂²	—	18	4	no	—
327	C₂²	—	18	2	no	—
328	C₂²	—	36	1	no	—
329	C₂²	—	36	1	no	—
330	C₂²	—	36	2	no	—
331	C₂²	C₂×C₃×D₃₆	1	1	yes	characteristic
332	C₂²	C₃×#3697	1	2	yes	—
333	C₂²	C₃×D₄×D₉	1	1	yes	characteristic
334	C₂²	C₃×#3699	1	2	yes	—
335	C₂²	C₃×#3701	1	1	yes	characteristic
336	C₄	—	2	2	no	—
337	C₄	—	4	2	no	—
338	C₄	—	36	1	no	—
339	C₄	—	36	2	no	—
340	C₃	—	2	1	no	—
341	C₃	#10588354	1	1	yes	characteristic, direct product factor
342	C₃	C₃×#4674	1	1	yes	characteristic
343	C₂	—	36	2	no	—
344	C₂	C₃×#63437	1	1	yes	characteristic
345	C₂	C₃×#63445	1	2	yes	—
346	C₂	C₃×#63448	1	1	yes	characteristic
347	C₂	C₃×#63457	1	1	yes	characteristic
348	C₂	C₃×#63460	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂²×D₉
commutator, cosocle	C₂³×C₉	C₂³×C₃
Frattini	C₂³×C₃	C₂²×C₃×D₃
Fitting	C₃×C₉×#95	C₂
socle	C₂³×C₃²	C₂²×D₃
prime radical	C₂³×C₃×D₉	C₂²
hypercenter	C₃×#95	D₉
hypocenter	C₉	C₃×#397
LCS	G ▷ C₂³×C₉ ▷ C₉
UCS	C₁ ◁ C₂³×C₃ ◁ C₃×#95

Complements

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

N	H ≅ G/N	type
3. C₂×C₃×C₄⋊Dic₉	C₂	semidirect
4. C₃×C₉×#95	C₂	semidirect
5. C₃×C₉⋊#95	C₂	semidirect
7. #10588354	C₃	direct (N × H)
23. C₂×C₄⋊Dic₉	C₂×C₃	semidirect
25. C₉×#95	C₂×C₃	semidirect
26. C₉⋊#95	C₂×C₃	semidirect
85. C₃×#95	D₉	semidirect
190. #95	C₃×D₉	semidirect
191. C₃×C₉	#397	semidirect
292. C₉	C₃×#397	semidirect
341. C₃	#10588354	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×(C₂×D₉)⋊C₄	2
C₂×C₃×C₄⋊Dic₉	2
C₃×C₉×#95	2
C₃×C₉⋊#95	2
C₃×#4674	3
#10588354	3

Subgroups — C3×#10588354 #13141318

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588354 #13141318