Subgroups — C₃×#10588855 #13143331

Order 1728.

494 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×C₉×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×#63506	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃×#3675	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#63502	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃×#3672	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₃×#3664	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×C₈×Dic₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#5175	—	3	1	no	maximal
9	#10588855	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃×Q₈×C₉	C₂²	1	1	yes	characteristic
11	C₂×C₃×D₄×C₉	C₂²	1	1	yes	characteristic
12	C₂×C₃×#611	—	2	2	no	—
13	C₂×C₃×D₃₆	C₂²	1	1	yes	characteristic
14	C₂×C₃×Dic₁₈	C₂²	1	1	yes	characteristic
15	C₃×C₉×SD₁₆	—	2	2	no	—
16	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
17	C₃×#3675	—	2	2	no	—
18	C₃×#3672	—	2	2	no	—
19	C₃×Dic₉⋊C₄	—	2	2	no	—
20	C₃×C₄×Dic₉	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
22	C₃×#3664	—	2	2	no	—
23	C₂×C₃²×SD₁₆	D₃	1	1	yes	characteristic
24	C₃×#944	—	3	1	no	—
25	C₂×C₃×#215	—	3	1	no	—
26	C₃×#940	—	3	1	no	—
27	C₂×C₃×#214	—	3	1	no	—
28	C₂×C₃×#204	—	3	1	no	—
29	C₃×C₈×Dic₃	—	3	1	no	—
30	C₂×C₉×SD₁₆	—	2	1	no	—
31	C₂×C₉×SD₁₆	C₂×C₃	1	1	yes	characteristic, complemented
32	#63506	C₂×C₃	1	1	yes	characteristic, complemented
33	C₂×#3675	C₂×C₃	1	1	yes	characteristic, complemented
34	#63502	C₂×C₃	1	1	yes	characteristic
35	C₂×#3672	C₂×C₃	1	1	yes	characteristic, complemented
36	C₂×#3664	C₂×C₃	1	1	yes	characteristic, complemented
37	C₈×Dic₉	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂³×C₃×C₉	—	2	1	no	—
39	C₂²×C₃×D₉	—	2	1	no	—
40	C₃×Q₈×C₉	—	2	1	no	—
41	C₃×Q₈×C₉	—	2	1	no	—
42	C₃×D₄×C₉	—	2	1	no	—
43	C₃×D₄×C₉	—	2	1	no	—
44	C₂×C₃×C₄×C₉	—	2	1	no	—
45	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
46	C₃×#611	—	4	4	no	—
47	C₂×C₃×Dic₉	—	2	1	no	—
48	C₂×C₃×Dic₉	D₄	1	2	yes	—
49	C₃×D₃₆	—	2	1	no	—
50	C₃×D₃₆	—	2	1	no	—
51	C₃×Dic₁₈	—	2	1	no	—
52	C₃×Dic₁₈	—	2	1	no	—
53	C₃×C₈×C₉	D₄	1	2	yes	complemented
54	C₃×C₉⋊C₈	D₄	1	2	yes	complemented
55	C₃×#492	—	9	1	no	—
56	#5175	—	3	1	no	—
57	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
58	C₂×C₃²×D₄	C₂×D₃	1	1	yes	characteristic
59	C₂×C₃×#67	—	6	2	no	—
60	C₂×C₃×D₁₂	—	3	1	no	—
61	C₂×C₃×Dic₆	—	3	1	no	—
62	C₃²×SD₁₆	—	2	2	no	—
63	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
64	C₃×#215	—	6	2	no	—
65	C₃×#214	—	6	2	no	—
66	C₃×Dic₃⋊C₄	—	6	2	no	—
67	C₃×C₄×Dic₃	—	3	1	no	—
68	C₂×C₃×C₃⋊C₈	—	3	1	no	—
69	C₃×#204	—	6	2	no	—
70	C₂×Q₈×C₉	—	2	1	no	—
71	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
72	C₂×D₄×C₉	—	2	1	no	—
73	C₂×D₄×C₉	C₂²×C₃	1	1	yes	characteristic
74	C₂×#611	—	2	2	no	—
75	C₂×D₃₆	C₂²×C₃	1	1	yes	characteristic
76	C₂×Dic₁₈	C₂²×C₃	1	1	yes	characteristic
77	C₉×SD₁₆	—	2	4	no	—
78	C₉×SD₁₆	—	2	2	no	—
79	C₂×C₈×C₉	—	2	1	no	—
80	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
81	#3675	—	2	2	no	—
82	#3672	—	2	2	no	—
83	Dic₉⋊C₄	—	2	2	no	—
84	C₄×Dic₉	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
86	#3664	—	2	2	no	—
87	C₂²×C₃×C₉	—	4	1	no	—
88	C₂²×C₃×C₉	—	4	2	no	—
89	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
90	C₂×C₃×D₉	—	4	1	no	—
91	C₂×C₃×D₉	—	4	2	no	—
92	C₃×C₄×C₉	—	4	1	no	—
93	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
94	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
95	C₃×Dic₉	—	4	1	no	—
96	C₃×Dic₉	SD₁₆	1	4	yes	complemented
97	C₂×C₃×SD₁₆	—	2	1	no	—
98	C₂×C₃×SD₁₆	—	9	1	no	—
99	C₂×C₃×SD₁₆	—	9	1	no	—
100	C₂×C₃×SD₁₆	—	9	1	no	—
101	C₂×C₃×SD₁₆	D₉	1	1	yes	characteristic, complemented
102	C₂×C₃×SD₁₆	C₃×D₃	1	1	yes	characteristic
103	C₃×C₄⋊Q₈	—	9	1	no	—
104	C₃×#127	—	9	1	no	—
105	#944	—	3	1	no	—
106	C₂×#215	—	3	1	no	—
107	#940	—	3	1	no	—
108	C₂×#214	—	3	1	no	—
109	C₂×#204	—	3	1	no	—
110	C₃×C₄×C₈	—	9	1	no	—
111	C₈×Dic₃	—	3	1	no	—
112	C₂³×C₃²	—	2	1	no	—
113	C₂²×C₃×D₃	—	6	1	no	—
114	C₃²×Q₈	—	2	1	no	—
115	C₃²×Q₈	—	2	1	no	—
116	C₃²×D₄	—	2	1	no	—
117	C₃²×D₄	—	2	1	no	—
118	C₂×C₃²×C₄	—	2	1	no	—
119	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
120	C₃×#67	—	12	4	no	—
121	C₂×C₃×Dic₃	—	3	2	no	—
122	C₂×C₃×Dic₃	—	6	1	no	—
123	C₃×D₁₂	—	6	1	no	—
124	C₃×D₁₂	—	6	1	no	—
125	C₃×Dic₆	—	6	1	no	—
126	C₃×Dic₆	—	6	1	no	—
127	C₂³×C₉	—	2	1	no	—
128	C₂³×C₉	—	4	1	no	—
129	C₂²×D₉	—	2	1	no	—
130	C₃²×C₈	#67	1	2	yes	—
131	C₃×C₃⋊C₈	—	3	2	no	—
132	Q₈×C₉	—	2	2	no	—
133	Q₈×C₉	—	2	1	no	—
134	Q₈×C₉	—	2	1	no	—
135	Q₈×C₉	—	4	1	no	—
136	D₄×C₉	—	2	2	no	—
137	D₄×C₉	—	2	1	no	—
138	D₄×C₉	—	2	1	no	—
139	D₄×C₉	—	4	1	no	—
140	C₂×C₄×C₉	—	2	1	no	—
141	C₂×C₄×C₉	—	2	1	no	—
142	C₂×C₄×C₉	—	4	1	no	—
143	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
144	#611	—	4	4	no	—
145	C₂×Dic₉	—	2	1	no	—
146	C₂×Dic₉	C₃×D₄	1	2	yes	—
147	D₃₆	—	2	1	no	—
148	D₃₆	—	2	1	no	—
149	Dic₁₈	—	2	1	no	—
150	Dic₁₈	—	2	1	no	—
151	C₈×C₉	—	2	2	no	—
152	C₈×C₉	C₃×D₄	1	2	yes	complemented
153	C₉⋊C₈	C₃×D₄	1	2	yes	complemented
154	#492	—	9	1	no	Sylow
155	C₂×C₃×C₉	—	8	1	no	—
156	C₂×C₃×C₉	#127	1	1	yes	characteristic
157	C₂×C₃×C₉	C₂×SD₁₆	1	2	yes	—
158	C₃×D₉	—	8	1	no	—
159	C₂×C₃×Q₈	—	2	1	no	—
160	C₂×C₃×Q₈	—	9	1	no	—
161	C₂×C₃×Q₈	C₂×D₉	1	1	yes	characteristic
162	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
163	C₂×C₃×D₄	—	2	1	no	—
164	C₂×C₃×D₄	—	9	1	no	—
165	C₂×C₃×D₄	—	18	2	no	—
166	C₂×C₃×D₄	C₂×D₉	1	1	yes	characteristic
167	C₂×C₃×D₄	C₂×C₃×D₃	1	1	yes	characteristic
168	C₂×#67	—	6	2	no	—
169	C₂×D₁₂	—	3	1	no	—
170	C₂×Dic₆	—	3	1	no	—
171	C₃×SD₁₆	—	2	4	no	—
172	C₃×SD₁₆	—	2	2	no	—
173	C₃×SD₁₆	—	2	2	no	—
174	C₃×SD₁₆	—	18	2	no	—
175	C₃×SD₁₆	—	18	2	no	—
176	C₃×SD₁₆	—	18	2	no	—
177	C₂×C₃×C₈	—	2	1	no	—
178	C₂×C₃×C₈	—	9	1	no	—
179	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
180	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
181	C₃×C₄⋊C₄	—	18	2	no	—
182	C₃×C₄²	—	9	1	no	—
183	#215	—	6	2	no	—
184	#214	—	6	2	no	—
185	Dic₃⋊C₄	—	6	2	no	—
186	C₄×Dic₃	—	3	1	no	—
187	C₂×C₃⋊C₈	—	3	1	no	—
188	#204	—	6	2	no	—
189	C₂²×C₃²	—	4	1	no	—
190	C₂²×C₃²	—	4	2	no	—
191	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
192	C₂×C₃×D₃	—	12	2	no	—
193	C₂×C₃×D₃	—	12	1	no	—
194	C₃²×C₄	—	4	1	no	—
195	C₃²×C₄	D₃×D₄	1	1	yes	characteristic
196	C₃²×C₄	C₂×#67	1	1	yes	characteristic
197	C₃×Dic₃	—	3	4	no	—
198	C₃×Dic₃	—	12	1	no	—
199	C₂²×C₉	—	2	1	no	—
200	C₂²×C₉	—	4	2	no	—
201	C₂²×C₉	—	4	1	no	—
202	C₂²×C₉	—	4	2	no	—
203	C₂²×C₉	—	8	2	no	—
204	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
205	C₂×D₉	—	4	1	no	—
206	C₂×D₉	—	4	2	no	—
207	C₄×C₉	—	2	1	no	—
208	C₄×C₉	—	2	1	no	—
209	C₄×C₉	—	4	2	no	—
210	C₄×C₉	—	4	1	no	—
211	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
212	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
213	Dic₉	—	4	1	no	—
214	Dic₉	C₃×SD₁₆	1	4	yes	complemented
215	C₂×SD₁₆	—	9	1	no	—
216	C₂×SD₁₆	—	9	1	no	—
217	C₂×SD₁₆	—	9	1	no	—
218	C₂×SD₁₆	C₃×D₉	1	1	yes	characteristic, complemented
219	C₄⋊Q₈	—	9	1	no	—
220	#127	—	9	1	no	—
221	C₄×C₈	—	9	1	no	—
222	C₃×C₉	#492	1	1	yes	characteristic, complemented, Sylow
223	C₂³×C₃	—	2	1	no	—
224	C₂³×C₃	—	2	1	no	—
225	C₂³×C₃	—	4	1	no	—
226	C₂³×C₃	—	18	1	no	—
227	C₂²×D₃	—	6	1	no	—
228	C₃×Q₈	—	2	2	no	—
229	C₃×Q₈	—	2	1	no	—
230	C₃×Q₈	—	2	1	no	—
231	C₃×Q₈	—	2	1	no	—
232	C₃×Q₈	—	2	1	no	—
233	C₃×Q₈	—	4	1	no	—
234	C₃×Q₈	—	18	1	no	—
235	C₃×Q₈	—	18	1	no	—
236	C₃×D₄	—	2	2	no	—
237	C₃×D₄	—	2	1	no	—
238	C₃×D₄	—	2	1	no	—
239	C₃×D₄	—	2	1	no	—
240	C₃×D₄	—	2	1	no	—
241	C₃×D₄	—	4	1	no	—
242	C₃×D₄	—	18	1	no	—
243	C₃×D₄	—	18	1	no	—
244	C₃×D₄	—	36	4	no	—
245	C₂×C₃×C₄	—	2	1	no	—
246	C₂×C₃×C₄	—	2	1	no	—
247	C₂×C₃×C₄	—	2	1	no	—
248	C₂×C₃×C₄	—	4	1	no	—
249	C₂×C₃×C₄	—	9	2	no	—
250	C₂×C₃×C₄	—	18	1	no	—
251	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
252	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
253	#67	—	12	4	no	—
254	C₂×Dic₃	—	3	2	no	—
255	C₂×Dic₃	—	6	1	no	—
256	D₁₂	—	6	1	no	—
257	D₁₂	—	6	1	no	—
258	Dic₆	—	6	1	no	—
259	Dic₆	—	6	1	no	—
260	C₃×C₈	—	2	2	no	—
261	C₃×C₈	—	9	2	no	—
262	C₃×C₈	#611	1	2	yes	complemented
263	C₃×C₈	C₃×#67	1	2	yes	—
264	C₃⋊C₈	—	3	2	no	—
265	C₂×C₃²	—	8	1	no	—
266	C₂×C₃²	D₃×SD₁₆	1	2	yes	—
267	C₂×C₃²	#940	1	1	yes	characteristic
268	C₃×D₃	—	24	1	no	—
269	C₂×C₉	—	2	2	no	—
270	C₂×C₉	—	2	1	no	—
271	C₂×C₉	—	8	2	no	—
272	C₂×C₉	—	8	1	no	—
273	C₂×C₉	C₃×#127	1	1	yes	characteristic
274	C₂×C₉	C₂×C₃×SD₁₆	1	2	yes	—
275	D₉	—	8	1	no	—
276	C₂×Q₈	—	9	1	no	—
277	C₂×Q₈	C₂×C₃×D₉	1	1	yes	characteristic
278	C₂×D₄	—	9	1	no	—
279	C₂×D₄	—	18	2	no	—
280	C₂×D₄	C₂×C₃×D₉	1	1	yes	characteristic
281	SD₁₆	—	2	2	no	—
282	SD₁₆	—	18	2	no	—
283	SD₁₆	—	18	2	no	—
284	SD₁₆	—	18	2	no	—
285	C₂×C₈	—	9	1	no	—
286	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
287	C₄⋊C₄	—	18	2	no	—
288	C₄²	—	9	1	no	—
289	C₂²×C₃	—	2	1	no	—
290	C₂²×C₃	—	4	2	no	—
291	C₂²×C₃	—	4	1	no	—
292	C₂²×C₃	—	4	2	no	—
293	C₂²×C₃	—	4	1	no	—
294	C₂²×C₃	—	4	2	no	—
295	C₂²×C₃	—	8	2	no	—
296	C₂²×C₃	—	36	1	no	—
297	C₂²×C₃	—	36	2	no	—
298	C₂²×C₃	D₄×D₉	1	1	yes	characteristic
299	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
300	C₂×D₃	—	12	2	no	—
301	C₂×D₃	—	12	1	no	—
302	C₃×C₄	—	2	1	no	—
303	C₃×C₄	—	2	1	no	—
304	C₃×C₄	—	4	2	no	—
305	C₃×C₄	—	4	1	no	—
306	C₃×C₄	—	4	1	no	—
307	C₃×C₄	—	9	4	no	—
308	C₃×C₄	—	36	1	no	—
309	C₃×C₄	D₄×D₉	1	1	yes	characteristic
310	C₃×C₄	C₂×#611	1	1	yes	characteristic
311	C₃×C₄	C₃×D₃×D₄	1	1	yes	characteristic
312	C₃×C₄	C₂×C₃×#67	1	1	yes	characteristic
313	Dic₃	—	3	4	no	—
314	Dic₃	—	12	1	no	—
315	C₃²	#5175	1	1	yes	characteristic
316	C₉	—	2	1	no	—
317	C₉	C₃×#492	1	1	yes	characteristic, complemented
318	C₂³	—	2	1	no	—
319	C₂³	—	18	1	no	—
320	Q₈	—	2	1	no	—
321	Q₈	—	2	1	no	—
322	Q₈	—	18	1	no	—
323	Q₈	—	18	1	no	—
324	D₄	—	2	1	no	—
325	D₄	—	2	1	no	—
326	D₄	—	18	1	no	—
327	D₄	—	18	1	no	—
328	D₄	—	36	4	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	9	2	no	—
331	C₂×C₄	—	18	1	no	—
332	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
333	C₈	—	9	2	no	—
334	C₈	C₃×#611	1	2	yes	complemented
335	C₂×C₃	—	2	2	no	—
336	C₂×C₃	—	2	1	no	—
337	C₂×C₃	—	8	2	no	—
338	C₂×C₃	—	8	1	no	—
339	C₂×C₃	—	8	1	no	—
340	C₂×C₃	—	72	1	no	—
341	C₂×C₃	SD₁₆×D₉	1	2	yes	—
342	C₂×C₃	#63502	1	1	yes	characteristic
343	C₂×C₃	C₃×D₃×SD₁₆	1	2	yes	—
344	C₂×C₃	C₃×#940	1	1	yes	characteristic
345	D₃	—	24	1	no	—
346	C₂²	—	4	2	no	—
347	C₂²	—	4	1	no	—
348	C₂²	—	36	2	no	—
349	C₂²	—	36	1	no	—
350	C₂²	C₃×D₄×D₉	1	1	yes	characteristic
351	C₄	—	4	1	no	—
352	C₄	—	9	4	no	—
353	C₄	—	36	1	no	—
354	C₄	C₃×D₄×D₉	1	1	yes	characteristic
355	C₄	C₂×C₃×#611	1	1	yes	characteristic
356	C₃	—	2	1	no	—
357	C₃	#10588855	1	1	yes	characteristic, direct product factor
358	C₃	C₃×#5175	1	1	yes	characteristic
359	C₂	—	8	1	no	—
360	C₂	—	72	1	no	—
361	C₂	C₃×SD₁₆×D₉	1	2	yes	—
362	C₂	C₃×#63502	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	D₄×D₉
commutator, cosocle	C₂×C₄×C₉	C₂³×C₃
Frattini	C₂×C₃×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃×C₉×SD₁₆	C₂
socle	C₂²×C₃²	D₃×D₄
prime radical	C₃×#63502	C₂
hypercenter	C₂×C₃×SD₁₆	D₉
hypocenter	C₉	C₃×#492
LCS	G ▷ C₂×C₄×C₉ ▷ C₂×C₉ ▷ C₉
UCS	C₁ ◁ C₂²×C₃ ◁ C₂×C₃×C₄ ◁ C₂×C₃×SD₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₃×C₉×SD₁₆	C₂	semidirect
2. C₃×#63506	C₂	semidirect
3. C₂×C₃×#3675	C₂	semidirect
5. C₂×C₃×#3672	C₂	semidirect
6. C₂×C₃×#3664	C₂	semidirect
7. C₃×C₈×Dic₉	C₂	semidirect
9. #10588855	C₃	direct (N × H)
31. C₂×C₉×SD₁₆	C₂×C₃	semidirect
32. #63506	C₂×C₃	semidirect
33. C₂×#3675	C₂×C₃	semidirect
35. C₂×#3672	C₂×C₃	semidirect
36. C₂×#3664	C₂×C₃	semidirect
37. C₈×Dic₉	C₂×C₃	semidirect
53. C₃×C₈×C₉	D₄	semidirect
54. C₃×C₉⋊C₈	D₄	semidirect
96. C₃×Dic₉	SD₁₆	semidirect
101. C₂×C₃×SD₁₆	D₉	semidirect
152. C₈×C₉	C₃×D₄	semidirect
153. C₉⋊C₈	C₃×D₄	semidirect
214. Dic₉	C₃×SD₁₆	semidirect
218. C₂×SD₁₆	C₃×D₉	semidirect
222. C₃×C₉	#492	semidirect
262. C₃×C₈	#611	semidirect
317. C₉	C₃×#492	semidirect
334. C₈	C₃×#611	semidirect
357. C₃	#10588855	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×SD₁₆	2
C₃×#63506	2
C₂×C₃×#3675	2
C₃×#63502	2
C₂×C₃×#3672	2
C₂×C₃×#3664	2
C₃×C₈×Dic₉	2
C₃×#5175	3
#10588855	3

Subgroups — C3×#10588855 #13143331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588855 #13143331