Subgroups — C₃×#10588842 #13143329

Order 1728.

428 conjugacy classes of subgroups.

The subgroup lattice has 368 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
2	C₃×#63506	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃×D₄×Dic₉	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#3672	C₂	1	1	yes	maximal, characteristic
5	C₃×#63395	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#63376	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×Dic₉⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#5162	—	3	1	no	maximal
9	#10588842	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	C₂²×C₃×Dic₉	—	2	1	no	—
13	C₂×C₃×Dic₁₈	C₂²	1	1	yes	characteristic
14	C₃×C₉×SD₁₆	—	2	2	no	—
15	C₂×C₃×C₈×C₉	C₂²	1	1	yes	characteristic
16	C₃×#3676	—	2	1	no	—
17	C₃×#3672	—	2	2	no	—
18	C₃×C₄⋊Dic₉	C₂²	1	1	yes	characteristic
19	C₃×Dic₉⋊C₄	—	2	1	no	—
20	C₃×Dic₉⋊C₄	—	2	1	no	—
21	C₃×C₄×Dic₉	C₂²	1	1	yes	characteristic
22	C₂×C₃×C₉⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₃²×SD₁₆	D₃	1	1	yes	characteristic
24	C₃×#944	—	3	1	no	—
25	C₃×D₄×Dic₃	—	3	1	no	—
26	C₂×C₃×#214	—	3	1	no	—
27	C₃×#835	—	3	1	no	—
28	C₃×#816	—	3	1	no	—
29	C₃×Dic₃⋊C₈	—	3	1	no	—
30	C₂×C₉×SD₁₆	—	2	1	no	—
31	C₂×C₉×SD₁₆	C₂×C₃	1	1	yes	characteristic
32	#63506	C₂×C₃	1	1	yes	characteristic, complemented
33	D₄×Dic₉	C₂×C₃	1	1	yes	characteristic
34	C₂×#3672	C₂×C₃	1	1	yes	characteristic
35	#63395	C₂×C₃	1	1	yes	characteristic, complemented
36	#63376	C₂×C₃	1	1	yes	characteristic, complemented
37	Dic₉⋊C₈	C₂×C₃	1	1	yes	characteristic, complemented
38	C₂³×C₃×C₉	—	2	1	no	—
39	C₃×Q₈×C₉	—	2	1	no	—
40	C₃×Q₈×C₉	—	2	1	no	—
41	C₃×D₄×C₉	—	2	1	no	—
42	C₃×D₄×C₉	D₄	1	2	yes	—
43	C₂×C₃×C₄×C₉	—	2	1	no	—
44	C₂×C₃×C₄×C₉	C₂³	1	1	yes	characteristic
45	C₂×C₃×Dic₉	—	2	1	no	—
46	C₂×C₃×Dic₉	—	2	1	no	—
47	C₂×C₃×Dic₉	—	4	2	no	—
48	C₂×C₃×Dic₉	D₄	1	1	yes	characteristic
49	C₂×C₃×Dic₉	D₄	1	1	yes	characteristic
50	C₃×Dic₁₈	—	2	1	no	—
51	C₃×Dic₁₈	—	2	1	no	—
52	C₃×C₈×C₉	—	2	1	no	—
53	C₃×C₉⋊C₈	—	2	1	no	—
54	C₃×#461	—	9	1	no	—
55	#5162	—	3	1	no	—
56	C₂×C₃²×Q₈	C₂×D₃	1	1	yes	characteristic
57	C₂×C₃²×D₄	C₂×D₃	1	1	yes	characteristic
58	C₂²×C₃×Dic₃	—	6	1	no	—
59	C₂×C₃×Dic₆	—	3	1	no	—
60	C₃²×SD₁₆	—	2	2	no	—
61	C₂×C₃²×C₈	C₂×D₃	1	1	yes	characteristic
62	C₃×#217	—	6	1	no	—
63	C₃×#214	—	6	2	no	—
64	C₃×C₄⋊Dic₃	—	3	1	no	—
65	C₃×Dic₃⋊C₄	—	6	1	no	—
66	C₃×Dic₃⋊C₄	—	6	1	no	—
67	C₃×C₄×Dic₃	—	3	1	no	—
68	C₂×C₃×C₃⋊C₈	—	3	1	no	—
69	C₂×Q₈×C₉	—	2	1	no	—
70	C₂×Q₈×C₉	C₂²×C₃	1	1	yes	characteristic
71	C₂×D₄×C₉	—	2	1	no	—
72	C₂×D₄×C₉	C₂²×C₃	1	1	yes	characteristic
73	C₂²×Dic₉	—	2	1	no	—
74	C₂×Dic₁₈	C₂²×C₃	1	1	yes	characteristic
75	C₉×SD₁₆	—	2	4	no	—
76	C₉×SD₁₆	—	2	2	no	—
77	C₂×C₈×C₉	—	2	1	no	—
78	C₂×C₈×C₉	C₂²×C₃	1	1	yes	characteristic
79	#3676	—	2	1	no	—
80	#3672	—	2	2	no	—
81	C₄⋊Dic₉	C₂²×C₃	1	1	yes	characteristic
82	Dic₉⋊C₄	—	2	1	no	—
83	Dic₉⋊C₄	—	2	1	no	—
84	C₄×Dic₉	C₂²×C₃	1	1	yes	characteristic
85	C₂×C₉⋊C₈	C₂²×C₃	1	1	yes	characteristic
86	C₂²×C₃×C₉	—	2	2	no	—
87	C₂²×C₃×C₉	—	4	1	no	—
88	C₂²×C₃×C₉	—	4	1	no	—
89	C₂²×C₃×C₉	C₂×D₄	1	1	yes	characteristic
90	C₃×C₄×C₉	—	4	1	no	—
91	C₃×C₄×C₉	C₂×D₄	1	1	yes	characteristic
92	C₃×C₄×C₉	#41	1	1	yes	characteristic
93	C₃×Dic₉	—	2	1	no	—
94	C₃×Dic₉	—	4	1	no	—
95	C₃×Dic₉	—	4	1	no	—
96	C₃×Dic₉	SD₁₆	1	2	yes	complemented
97	C₂×C₃×SD₁₆	—	2	1	no	—
98	C₂×C₃×SD₁₆	—	9	1	no	—
99	C₂×C₃×SD₁₆	D₉	1	1	yes	characteristic
100	C₂×C₃×SD₁₆	C₃×D₃	1	1	yes	characteristic
101	C₃×C₄⋊Q₈	—	9	1	no	—
102	C₃×C₄×D₄	—	9	1	no	—
103	#944	—	3	1	no	—
104	D₄×Dic₃	—	3	1	no	—
105	C₂×#214	—	3	1	no	—
106	C₃×C₄⋊C₈	—	9	1	no	—
107	C₃×#103	—	9	1	no	—
108	C₃×#103	—	9	1	no	—
109	#835	—	3	1	no	—
110	#816	—	3	1	no	—
111	Dic₃⋊C₈	—	3	1	no	—
112	C₂³×C₃²	—	2	1	no	—
113	C₃²×Q₈	—	2	1	no	—
114	C₃²×Q₈	—	2	1	no	—
115	C₃²×D₄	—	2	1	no	—
116	C₃²×D₄	#67	1	2	yes	—
117	C₂×C₃²×C₄	—	2	1	no	—
118	C₂×C₃²×C₄	C₂²×D₃	1	1	yes	characteristic
119	C₂×C₃×Dic₃	—	3	1	no	—
120	C₂×C₃×Dic₃	—	3	1	no	—
121	C₂×C₃×Dic₃	—	6	1	no	—
122	C₂×C₃×Dic₃	—	6	1	no	—
123	C₂×C₃×Dic₃	—	12	2	no	—
124	C₃×Dic₆	—	6	1	no	—
125	C₃×Dic₆	—	6	1	no	—
126	C₂³×C₉	—	2	1	no	—
127	C₂³×C₉	—	4	1	no	—
128	C₃²×C₈	—	2	1	no	—
129	C₃×C₃⋊C₈	—	6	1	no	—
130	Q₈×C₉	—	2	2	no	—
131	Q₈×C₉	—	2	1	no	—
132	Q₈×C₉	—	2	1	no	—
133	Q₈×C₉	—	4	1	no	—
134	D₄×C₉	—	2	1	no	—
135	D₄×C₉	—	2	2	no	—
136	D₄×C₉	—	4	1	no	—
137	D₄×C₉	C₃×D₄	1	2	yes	—
138	C₂×C₄×C₉	—	2	1	no	—
139	C₂×C₄×C₉	—	2	1	no	—
140	C₂×C₄×C₉	—	4	1	no	—
141	C₂×C₄×C₉	C₂³×C₃	1	1	yes	characteristic
142	C₂×Dic₉	—	2	1	no	—
143	C₂×Dic₉	—	2	1	no	—
144	C₂×Dic₉	—	4	2	no	—
145	C₂×Dic₉	C₃×D₄	1	1	yes	characteristic
146	C₂×Dic₉	C₃×D₄	1	1	yes	characteristic
147	Dic₁₈	—	2	1	no	—
148	Dic₁₈	—	2	1	no	—
149	C₈×C₉	—	2	2	no	—
150	C₈×C₉	—	2	1	no	—
151	C₉⋊C₈	—	2	1	no	—
152	#461	—	9	1	no	Sylow
153	C₂×C₃×C₉	—	4	2	no	—
154	C₂×C₃×C₉	#121	1	1	yes	characteristic
155	C₂×C₃×C₉	C₂×SD₁₆	1	1	yes	characteristic
156	C₂×C₃×C₉	#137	1	1	yes	characteristic
157	C₂×C₃×Q₈	—	2	1	no	—
158	C₂×C₃×Q₈	—	9	1	no	—
159	C₂×C₃×Q₈	C₂×D₉	1	1	yes	characteristic
160	C₂×C₃×Q₈	C₂×C₃×D₃	1	1	yes	characteristic
161	C₂×C₃×D₄	—	2	1	no	—
162	C₂×C₃×D₄	C₂×D₉	1	1	yes	characteristic
163	C₂×C₃×D₄	C₂×C₃×D₃	1	1	yes	characteristic
164	C₂²×C₃×C₄	—	18	1	no	—
165	C₂²×Dic₃	—	6	1	no	—
166	C₂×Dic₆	—	3	1	no	—
167	C₃×SD₁₆	—	2	4	no	—
168	C₃×SD₁₆	—	2	2	no	—
169	C₃×SD₁₆	—	2	2	no	—
170	C₃×SD₁₆	—	18	2	no	—
171	C₂×C₃×C₈	—	2	1	no	—
172	C₂×C₃×C₈	—	9	1	no	—
173	C₂×C₃×C₈	C₂×D₉	1	1	yes	characteristic
174	C₂×C₃×C₈	C₂×C₃×D₃	1	1	yes	characteristic
175	C₃×C₄⋊C₄	—	9	1	no	—
176	C₃×C₄⋊C₄	—	18	1	no	—
177	C₃×C₄⋊C₄	—	18	1	no	—
178	C₃×C₂²⋊C₄	—	18	1	no	—
179	C₃×C₄²	—	9	1	no	—
180	#217	—	6	1	no	—
181	#214	—	6	2	no	—
182	C₄⋊Dic₃	—	3	1	no	—
183	Dic₃⋊C₄	—	6	1	no	—
184	Dic₃⋊C₄	—	6	1	no	—
185	C₄×Dic₃	—	3	1	no	—
186	C₂×C₃⋊C₈	—	3	1	no	—
187	C₂²×C₃²	—	2	2	no	—
188	C₂²×C₃²	—	4	1	no	—
189	C₂²×C₃²	—	4	1	no	—
190	C₂²×C₃²	D₃×D₄	1	1	yes	characteristic
191	C₃²×C₄	—	4	1	no	—
192	C₃²×C₄	#237	1	1	yes	characteristic
193	C₃²×C₄	C₂×#67	1	1	yes	characteristic
194	C₃×Dic₃	—	3	2	no	—
195	C₃×Dic₃	—	6	1	no	—
196	C₃×Dic₃	—	12	1	no	—
197	C₃×Dic₃	—	12	1	no	—
198	C₂²×C₉	—	2	2	no	—
199	C₂²×C₉	—	2	1	no	—
200	C₂²×C₉	—	4	2	no	—
201	C₂²×C₉	—	4	1	no	—
202	C₂²×C₉	—	4	1	no	—
203	C₂²×C₉	—	8	1	no	—
204	C₂²×C₉	—	8	1	no	—
205	C₂²×C₉	C₂×C₃×D₄	1	1	yes	characteristic
206	C₄×C₉	—	2	1	no	—
207	C₄×C₉	—	2	1	no	—
208	C₄×C₉	—	4	2	no	—
209	C₄×C₉	—	4	1	no	—
210	C₄×C₉	C₂×C₃×D₄	1	1	yes	characteristic
211	C₄×C₉	C₃×#41	1	1	yes	characteristic
212	Dic₉	—	2	1	no	—
213	Dic₉	—	4	1	no	—
214	Dic₉	—	4	1	no	—
215	Dic₉	C₃×SD₁₆	1	2	yes	complemented
216	C₂×SD₁₆	—	9	1	no	—
217	C₂×SD₁₆	C₃×D₉	1	1	yes	characteristic
218	C₄⋊Q₈	—	9	1	no	—
219	C₄×D₄	—	9	1	no	—
220	C₄⋊C₈	—	9	1	no	—
221	#103	—	9	1	no	—
222	#103	—	9	1	no	—
223	C₃×C₉	#461	1	1	yes	characteristic, complemented, Sylow
224	C₂³×C₃	—	2	1	no	—
225	C₂³×C₃	—	2	1	no	—
226	C₂³×C₃	—	4	1	no	—
227	C₃×Q₈	—	2	2	no	—
228	C₃×Q₈	—	2	1	no	—
229	C₃×Q₈	—	2	1	no	—
230	C₃×Q₈	—	2	1	no	—
231	C₃×Q₈	—	2	1	no	—
232	C₃×Q₈	—	4	1	no	—
233	C₃×Q₈	—	18	1	no	—
234	C₃×Q₈	—	18	1	no	—
235	C₃×D₄	—	2	1	no	—
236	C₃×D₄	—	2	1	no	—
237	C₃×D₄	—	2	2	no	—
238	C₃×D₄	—	4	1	no	—
239	C₃×D₄	#611	1	2	yes	—
240	C₃×D₄	C₃×#67	1	2	yes	—
241	C₂×C₃×C₄	—	2	1	no	—
242	C₂×C₃×C₄	—	2	1	no	—
243	C₂×C₃×C₄	—	2	1	no	—
244	C₂×C₃×C₄	—	4	1	no	—
245	C₂×C₃×C₄	—	9	1	no	—
246	C₂×C₃×C₄	—	9	1	no	—
247	C₂×C₃×C₄	—	18	1	no	—
248	C₂×C₃×C₄	—	18	1	no	—
249	C₂×C₃×C₄	—	36	2	no	—
250	C₂×C₃×C₄	C₂²×D₉	1	1	yes	characteristic
251	C₂×C₃×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
252	C₂×Dic₃	—	3	1	no	—
253	C₂×Dic₃	—	3	1	no	—
254	C₂×Dic₃	—	6	1	no	—
255	C₂×Dic₃	—	6	1	no	—
256	C₂×Dic₃	—	12	2	no	—
257	Dic₆	—	6	1	no	—
258	Dic₆	—	6	1	no	—
259	C₃×C₈	—	2	2	no	—
260	C₃×C₈	—	2	1	no	—
261	C₃×C₈	—	2	1	no	—
262	C₃×C₈	—	18	1	no	—
263	C₃⋊C₈	—	6	1	no	—
264	C₂×C₃²	—	4	2	no	—
265	C₂×C₃²	D₃×SD₁₆	1	1	yes	characteristic
266	C₂×C₃²	#915	1	1	yes	characteristic
267	C₂×C₃²	#939	1	1	yes	characteristic
268	C₂×C₉	—	2	1	no	—
269	C₂×C₉	—	2	1	no	—
270	C₂×C₉	—	2	1	no	—
271	C₂×C₉	—	4	2	no	—
272	C₂×C₉	—	8	2	no	—
273	C₂×C₉	C₃×#121	1	1	yes	characteristic
274	C₂×C₉	C₂×C₃×SD₁₆	1	1	yes	characteristic
275	C₂×C₉	C₃×#137	1	1	yes	characteristic
276	C₂×Q₈	—	9	1	no	—
277	C₂×Q₈	C₂×C₃×D₉	1	1	yes	characteristic
278	C₂×D₄	C₂×C₃×D₉	1	1	yes	characteristic
279	C₂²×C₄	—	18	1	no	—
280	SD₁₆	—	2	2	no	—
281	SD₁₆	—	18	2	no	—
282	C₂×C₈	—	9	1	no	—
283	C₂×C₈	C₂×C₃×D₉	1	1	yes	characteristic
284	C₄⋊C₄	—	9	1	no	—
285	C₄⋊C₄	—	18	1	no	—
286	C₄⋊C₄	—	18	1	no	—
287	C₂²⋊C₄	—	18	1	no	—
288	C₄²	—	9	1	no	—
289	C₂²×C₃	—	2	2	no	—
290	C₂²×C₃	—	2	2	no	—
291	C₂²×C₃	—	2	1	no	—
292	C₂²×C₃	—	4	2	no	—
293	C₂²×C₃	—	4	1	no	—
294	C₂²×C₃	—	4	1	no	—
295	C₂²×C₃	—	4	1	no	—
296	C₂²×C₃	—	4	1	no	—
297	C₂²×C₃	—	8	1	no	—
298	C₂²×C₃	—	8	1	no	—
299	C₂²×C₃	D₄×D₉	1	1	yes	characteristic
300	C₂²×C₃	C₃×D₃×D₄	1	1	yes	characteristic
301	C₃×C₄	—	2	1	no	—
302	C₃×C₄	—	2	1	no	—
303	C₃×C₄	—	4	2	no	—
304	C₃×C₄	—	4	1	no	—
305	C₃×C₄	—	4	1	no	—
306	C₃×C₄	—	9	2	no	—
307	C₃×C₄	—	18	1	no	—
308	C₃×C₄	—	36	1	no	—
309	C₃×C₄	—	36	1	no	—
310	C₃×C₄	#3699	1	1	yes	characteristic
311	C₃×C₄	C₂×#611	1	1	yes	characteristic
312	C₃×C₄	C₃×#237	1	1	yes	characteristic
313	C₃×C₄	C₂×C₃×#67	1	1	yes	characteristic
314	Dic₃	—	3	2	no	—
315	Dic₃	—	6	1	no	—
316	Dic₃	—	12	1	no	—
317	Dic₃	—	12	1	no	—
318	C₃²	#5162	1	1	yes	characteristic
319	C₉	—	2	1	no	—
320	C₉	C₃×#461	1	1	yes	characteristic, complemented
321	C₂³	—	2	1	no	—
322	Q₈	—	2	1	no	—
323	Q₈	—	2	1	no	—
324	Q₈	—	18	1	no	—
325	Q₈	—	18	1	no	—
326	D₄	—	2	1	no	—
327	D₄	C₃×#611	1	2	yes	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	9	1	no	—
330	C₂×C₄	—	9	1	no	—
331	C₂×C₄	—	18	1	no	—
332	C₂×C₄	—	18	1	no	—
333	C₂×C₄	—	36	2	no	—
334	C₂×C₄	C₂²×C₃×D₉	1	1	yes	characteristic
335	C₈	—	2	1	no	—
336	C₈	—	18	1	no	—
337	C₂×C₃	—	2	1	no	—
338	C₂×C₃	—	2	1	no	—
339	C₂×C₃	—	2	1	no	—
340	C₂×C₃	—	4	2	no	—
341	C₂×C₃	—	4	2	no	—
342	C₂×C₃	—	8	2	no	—
343	C₂×C₃	SD₁₆×D₉	1	1	yes	characteristic
344	C₂×C₃	#63477	1	1	yes	characteristic
345	C₂×C₃	#63501	1	1	yes	characteristic
346	C₂×C₃	C₃×D₃×SD₁₆	1	1	yes	characteristic
347	C₂×C₃	C₃×#915	1	1	yes	characteristic
348	C₂×C₃	C₃×#939	1	1	yes	characteristic
349	C₂²	—	2	2	no	—
350	C₂²	—	4	1	no	—
351	C₂²	—	4	1	no	—
352	C₂²	C₃×D₄×D₉	1	1	yes	characteristic
353	C₄	—	4	1	no	—
354	C₄	—	9	2	no	—
355	C₄	—	18	1	no	—
356	C₄	—	36	1	no	—
357	C₄	—	36	1	no	—
358	C₄	C₃×#3699	1	1	yes	characteristic
359	C₄	C₂×C₃×#611	1	1	yes	characteristic
360	C₃	—	2	1	no	—
361	C₃	#10588842	1	1	yes	characteristic, direct product factor
362	C₃	C₃×#5162	1	1	yes	characteristic
363	C₂	—	4	2	no	—
364	C₂	C₃×SD₁₆×D₉	1	1	yes	characteristic
365	C₂	C₃×#63477	1	1	yes	characteristic
366	C₂	C₃×#63501	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	C₂×C₃²×D₄	C₂×D₃
hypercenter	C₂×C₃×SD₁₆	D₉
hypocenter	C₉	C₃×#461
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
2. C₃×#63506	C₂	semidirect
5. C₃×#63395	C₂	semidirect
6. C₃×#63376	C₂	semidirect
7. C₃×Dic₉⋊C₈	C₂	semidirect
9. #10588842	C₃	direct (N × H)
32. #63506	C₂×C₃	semidirect
35. #63395	C₂×C₃	semidirect
36. #63376	C₂×C₃	semidirect
37. Dic₉⋊C₈	C₂×C₃	semidirect
96. C₃×Dic₉	SD₁₆	semidirect
215. Dic₉	C₃×SD₁₆	semidirect
223. C₃×C₉	#461	semidirect
320. C₉	C₃×#461	semidirect
361. C₃	#10588842	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
C₃×D₄×Dic₉	2
C₂×C₃×#3672	2
C₃×#63395	2
C₃×#63376	2
C₃×Dic₉⋊C₈	2
C₃×#5162	3
#10588842	3

Subgroups — C3×#10588842 #13143329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#10588842 #13143329