Subgroups — C₃×#5648 #10595745

Order 576.

579 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃×#237	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃²×#123	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#935	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃×#929	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#896	C₂	1	2	yes	maximal, complemented
7	C₃×#888	C₂	1	2	yes	maximal, complemented
8	C₃×#886	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃×#885	C₂	1	2	yes	maximal, complemented
10	C₃×#878	C₂	1	2	yes	maximal, complemented
11	C₃×#877	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃×#536	—	3	1	no	maximal
13	#5648	C₃	1	1	yes	maximal, characteristic, direct product factor
14	C₂×C₃²×D₄	C₂²	1	1	yes	characteristic
15	C₂²×C₃²×C₄	C₂²	1	1	yes	characteristic
16	C₂×C₃×#67	C₂²	1	1	yes	characteristic
17	C₂×C₃×#67	C₂²	1	1	yes	characteristic
18	C₂²×C₃×Dic₃	C₂²	1	1	yes	characteristic
19	C₂²×C₃×Dic₃	C₂²	1	1	yes	characteristic
20	C₂²×C₃×Dic₃	C₂²	1	1	yes	characteristic
21	C₃×#237	—	2	4	no	—
22	C₂×C₃×C₄×D₃	C₂²	1	1	yes	characteristic
23	C₂×C₃×Dic₆	—	2	4	no	—
24	C₂×C₃×Dic₆	C₂²	1	2	yes	—
25	C₂×C₃×Dic₆	C₂²	1	2	yes	—
26	C₂×C₃×Dic₆	C₂²	1	1	yes	characteristic
27	C₃²×C₄⋊C₄	C₂²	1	2	yes	—
28	C₃²×C₂²⋊C₄	C₂²	1	1	yes	characteristic
29	C₃²×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₃×#217	C₂²	1	1	yes	characteristic
31	C₃×#217	C₂²	1	2	yes	—
32	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
33	C₃×(C₂×D₃)⋊C₄	C₂²	1	2	yes	—
34	C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
35	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
36	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
37	C₃×Dic₃⋊C₄	C₂²	1	2	yes	—
38	C₃×C₄×Dic₃	C₂²	1	2	yes	—
39	C₂×C₃×#41	—	3	1	no	—
40	C₂²×C₃×Q₈	—	3	1	no	—
41	C₂×#237	C₂×C₃	1	1	yes	characteristic
42	C₂²×Dic₆	C₂×C₃	1	1	yes	characteristic, complemented
43	C₃×C₄⋊Q₈	—	3	2	no	—
44	C₃×#124	—	3	2	no	—
45	C₃×#123	—	2	1	no	—
46	C₃×#123	—	3	1	no	—
47	C₃×#123	—	3	1	no	—
48	C₃×#123	—	3	1	no	—
49	C₃×#123	D₃	1	1	yes	characteristic, complemented
50	C₃×#123	C₂×C₃	1	1	yes	characteristic, complemented
51	C₃×C₂²⋊Q₈	—	3	2	no	—
52	C₃×C₂²⋊Q₈	—	3	2	no	—
53	C₃×#117	—	3	1	no	—
54	#935	C₂×C₃	1	1	yes	characteristic, complemented
55	#929	C₂×C₃	1	1	yes	characteristic, complemented
56	#896	C₂×C₃	1	2	yes	complemented
57	#888	C₂×C₃	1	2	yes	complemented
58	#886	C₂×C₃	1	1	yes	characteristic, complemented
59	#885	C₂×C₃	1	2	yes	complemented
60	#878	C₂×C₃	1	2	yes	complemented
61	#877	C₂×C₃	1	1	yes	characteristic, complemented
62	C₂³×C₃²	C₂³	1	1	yes	characteristic
63	C₂³×C₃²	C₂³	1	1	yes	characteristic
64	C₂²×C₃×D₃	C₂³	1	1	yes	characteristic
65	C₃²×D₄	—	2	2	no	—
66	C₂×C₃²×C₄	—	2	2	no	—
67	C₂×C₃²×C₄	C₂³	1	2	yes	—
68	C₂×C₃²×C₄	C₂³	1	1	yes	characteristic
69	C₂×C₃²×C₄	C₂³	1	2	yes	—
70	C₃×#67	—	2	2	no	—
71	C₃×#67	—	2	2	no	—
72	C₂×C₃×Dic₃	—	2	2	no	—
73	C₂×C₃×Dic₃	—	2	2	no	—
74	C₂×C₃×Dic₃	D₄	1	4	yes	—
75	C₂×C₃×Dic₃	C₂³	1	2	yes	—
76	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
77	C₂×C₃×Dic₃	C₂³	1	2	yes	—
78	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
79	C₂×C₃×Dic₃	C₂³	1	1	yes	characteristic
80	C₃×C₄×D₃	—	2	2	no	—
81	C₃×Dic₆	—	2	4	no	—
82	C₃×Dic₆	—	2	4	no	—
83	C₃×Dic₆	—	2	2	no	—
84	#536	—	3	1	no	Sylow
85	C₃×#41	—	6	4	no	—
86	C₂×C₃×Q₈	—	3	2	no	—
87	C₂×C₃×Q₈	—	3	2	no	—
88	C₂×C₃×Q₈	—	3	1	no	—
89	C₂×C₃×Q₈	—	6	4	no	—
90	C₂×C₃×D₄	—	2	1	no	—
91	C₂×C₃×D₄	—	3	1	no	—
92	C₂×C₃×D₄	—	3	1	no	—
93	C₂×C₃×D₄	C₂×D₃	1	1	yes	characteristic
94	C₂×C₃×D₄	C₂²×C₃	1	1	yes	characteristic
95	C₂²×C₃×C₄	—	2	1	no	—
96	C₂²×C₃×C₄	—	3	1	no	—
97	C₂²×C₃×C₄	—	3	1	no	—
98	C₂²×C₃×C₄	—	3	1	no	—
99	C₂²×C₃×C₄	—	3	1	no	—
100	C₂²×C₃×C₄	C₂×D₃	1	1	yes	characteristic
101	C₂²×C₃×C₄	C₂²×C₃	1	1	yes	characteristic
102	C₂×#67	C₂²×C₃	1	1	yes	characteristic
103	C₂×#67	C₂²×C₃	1	1	yes	characteristic
104	C₂²×Dic₃	C₂²×C₃	1	1	yes	characteristic
105	C₂²×Dic₃	C₂²×C₃	1	1	yes	characteristic
106	C₂²×Dic₃	C₂²×C₃	1	1	yes	characteristic
107	#237	—	2	4	no	—
108	C₂×C₄×D₃	C₂²×C₃	1	1	yes	characteristic
109	C₂×Dic₆	—	2	4	no	—
110	C₂×Dic₆	C₂²×C₃	1	2	yes	—
111	C₂×Dic₆	C₂²×C₃	1	2	yes	—
112	C₂×Dic₆	C₂²×C₃	1	1	yes	characteristic
113	C₃×C₄⋊C₄	—	2	2	no	—
114	C₃×C₄⋊C₄	—	3	2	no	—
115	C₃×C₄⋊C₄	—	3	2	no	—
116	C₃×C₄⋊C₄	—	3	2	no	—
117	C₃×C₄⋊C₄	—	3	2	no	—
118	C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
119	C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
120	C₃×C₂²⋊C₄	—	2	1	no	—
121	C₃×C₂²⋊C₄	—	2	2	no	—
122	C₃×C₂²⋊C₄	—	3	1	no	—
123	C₃×C₂²⋊C₄	—	3	2	no	—
124	C₃×C₂²⋊C₄	—	3	2	no	—
125	C₃×C₂²⋊C₄	—	3	2	no	—
126	C₃×C₂²⋊C₄	C₂×D₃	1	1	yes	characteristic
127	C₃×C₂²⋊C₄	C₂×D₃	1	2	yes	—
128	C₃×C₂²⋊C₄	C₂²×C₃	1	1	yes	characteristic
129	C₃×C₂²⋊C₄	C₂²×C₃	1	2	yes	—
130	C₃×C₄²	—	3	2	no	—
131	#217	C₂²×C₃	1	1	yes	characteristic
132	#217	C₂²×C₃	1	2	yes	—
133	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
134	(C₂×D₃)⋊C₄	C₂²×C₃	1	2	yes	—
135	C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
136	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
137	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
138	Dic₃⋊C₄	C₂²×C₃	1	2	yes	—
139	C₄×Dic₃	C₂²×C₃	1	2	yes	—
140	C₂²×C₃²	—	2	2	no	—
141	C₂²×C₃²	—	2	1	no	—
142	C₂²×C₃²	—	2	2	no	—
143	C₂²×C₃²	C₂×D₄	1	2	yes	—
144	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
145	C₂×C₃×D₃	—	2	1	no	—
146	C₂×C₃×D₃	—	2	2	no	—
147	C₃²×C₄	—	2	2	no	—
148	C₃²×C₄	—	2	1	no	—
149	C₃²×C₄	—	2	2	no	—
150	C₃×Dic₃	—	2	2	no	—
151	C₃×Dic₃	—	2	1	no	—
152	C₃×Dic₃	—	2	2	no	—
153	C₃×Dic₃	C₂×D₄	1	2	yes	—
154	C₃×Dic₃	C₂×D₄	1	2	yes	—
155	C₂×#41	—	3	1	no	—
156	C₂²×Q₈	—	3	1	no	—
157	C₄⋊Q₈	—	3	2	no	—
158	#124	—	3	2	no	—
159	#123	—	3	1	no	—
160	#123	—	3	1	no	—
161	#123	—	3	1	no	—
162	#123	C₃×D₃	1	1	yes	characteristic, complemented
163	C₂²⋊Q₈	—	3	2	no	—
164	C₂²⋊Q₈	—	3	2	no	—
165	#117	—	3	1	no	—
166	C₂³×C₃	—	2	1	no	—
167	C₂³×C₃	—	2	1	no	—
168	C₂³×C₃	—	3	1	no	—
169	C₂³×C₃	C₂²×D₃	1	1	yes	characteristic
170	C₂³×C₃	C₂²×D₃	1	1	yes	characteristic
171	C₂³×C₃	C₂³×C₃	1	1	yes	characteristic
172	C₂³×C₃	C₂³×C₃	1	1	yes	characteristic
173	C₂²×D₃	C₂³×C₃	1	1	yes	characteristic
174	C₃×Q₈	—	6	4	no	—
175	C₃×Q₈	—	6	4	no	—
176	C₃×Q₈	—	6	2	no	—
177	C₃×D₄	—	2	2	no	—
178	C₃×D₄	—	2	2	no	—
179	C₃×D₄	—	4	2	no	—
180	C₃×D₄	—	6	2	no	—
181	C₃×D₄	—	6	2	no	—
182	C₂×C₃×C₄	—	2	2	no	—
183	C₂×C₃×C₄	—	2	2	no	—
184	C₂×C₃×C₄	—	2	2	no	—
185	C₂×C₃×C₄	—	2	2	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	3	4	no	—
188	C₂×C₃×C₄	—	3	1	no	—
189	C₂×C₃×C₄	—	3	1	no	—
190	C₂×C₃×C₄	—	3	2	no	—
191	C₂×C₃×C₄	—	3	2	no	—
192	C₂×C₃×C₄	—	3	1	no	—
193	C₂×C₃×C₄	—	4	2	no	—
194	C₂×C₃×C₄	—	6	2	no	—
195	C₂×C₃×C₄	—	6	2	no	—
196	C₂×C₃×C₄	—	6	2	no	—
197	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
198	C₂×C₃×C₄	C₂²×D₃	1	2	yes	—
199	C₂×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
200	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
201	C₂×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂³×C₃	1	2	yes	—
203	#67	—	2	2	no	—
204	#67	—	2	2	no	—
205	C₂×Dic₃	—	2	2	no	—
206	C₂×Dic₃	—	2	2	no	—
207	C₂×Dic₃	C₃×D₄	1	4	yes	—
208	C₂×Dic₃	C₂³×C₃	1	2	yes	—
209	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
210	C₂×Dic₃	C₂³×C₃	1	2	yes	—
211	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
212	C₂×Dic₃	C₂³×C₃	1	1	yes	characteristic
213	C₄×D₃	—	2	2	no	—
214	Dic₆	—	2	4	no	—
215	Dic₆	—	2	4	no	—
216	Dic₆	—	2	2	no	—
217	C₂×C₃²	—	2	2	no	—
218	C₂×C₃²	—	4	1	no	—
219	C₂×C₃²	C₂²×D₄	1	1	yes	characteristic
220	C₂×C₃²	2₋¹⁺⁴	1	2	yes	—
221	C₃×D₃	—	4	1	no	—
222	#41	—	6	4	no	—
223	C₂×Q₈	—	3	2	no	—
224	C₂×Q₈	—	3	2	no	—
225	C₂×Q₈	—	3	1	no	—
226	C₂×Q₈	—	6	4	no	—
227	C₂×D₄	—	3	1	no	—
228	C₂×D₄	—	3	1	no	—
229	C₂×D₄	C₂×C₃×D₃	1	1	yes	characteristic
230	C₂²×C₄	—	3	1	no	—
231	C₂²×C₄	—	3	1	no	—
232	C₂²×C₄	—	3	1	no	—
233	C₂²×C₄	—	3	1	no	—
234	C₂²×C₄	C₂×C₃×D₃	1	1	yes	characteristic
235	C₄⋊C₄	—	3	2	no	—
236	C₄⋊C₄	—	3	2	no	—
237	C₄⋊C₄	—	3	2	no	—
238	C₄⋊C₄	—	3	2	no	—
239	C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
240	C₂²⋊C₄	—	3	1	no	—
241	C₂²⋊C₄	—	3	2	no	—
242	C₂²⋊C₄	—	3	2	no	—
243	C₂²⋊C₄	—	3	2	no	—
244	C₂²⋊C₄	C₂×C₃×D₃	1	1	yes	characteristic
245	C₂²⋊C₄	C₂×C₃×D₃	1	2	yes	—
246	C₄²	—	3	2	no	—
247	C₂²×C₃	—	2	2	no	—
248	C₂²×C₃	—	2	2	no	—
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	2	2	no	—
251	C₂²×C₃	—	2	1	no	—
252	C₂²×C₃	—	2	2	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	2	2	no	—
255	C₂²×C₃	—	4	2	no	—
256	C₂²×C₃	—	4	1	no	—
257	C₂²×C₃	—	4	2	no	—
258	C₂²×C₃	—	6	1	no	—
259	C₂²×C₃	—	6	2	no	—
260	C₂²×C₃	D₃×D₄	1	2	yes	—
261	C₂²×C₃	C₂×C₃×D₄	1	2	yes	—
262	C₂²×C₃	C₂³×D₃	1	1	yes	characteristic
263	C₂²×C₃	C₂⁴×C₃	1	1	yes	characteristic
264	C₂×D₃	—	2	1	no	—
265	C₂×D₃	—	2	2	no	—
266	C₃×C₄	—	2	2	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	2	2	no	—
269	C₃×C₄	—	2	2	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	2	2	no	—
272	C₃×C₄	—	3	2	no	—
273	C₃×C₄	—	3	2	no	—
274	C₃×C₄	—	4	2	no	—
275	C₃×C₄	—	4	1	no	—
276	C₃×C₄	—	4	2	no	—
277	C₃×C₄	—	6	1	no	—
278	C₃×C₄	—	6	2	no	—
279	C₃×C₄	—	6	2	no	—
280	Dic₃	—	2	2	no	—
281	Dic₃	—	2	1	no	—
282	Dic₃	—	2	2	no	—
283	Dic₃	C₂×C₃×D₄	1	2	yes	—
284	Dic₃	C₂×C₃×D₄	1	2	yes	—
285	C₃²	#536	1	1	yes	characteristic, complemented, Sylow
286	C₂³	—	3	1	no	—
287	C₂³	C₂²×C₃×D₃	1	1	yes	characteristic
288	C₂³	C₂²×C₃×D₃	1	1	yes	characteristic
289	Q₈	—	6	4	no	—
290	Q₈	—	6	4	no	—
291	Q₈	—	6	2	no	—
292	D₄	—	2	2	no	—
293	D₄	—	6	2	no	—
294	D₄	—	6	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	3	1	no	—
297	C₂×C₄	—	3	1	no	—
298	C₂×C₄	—	3	4	no	—
299	C₂×C₄	—	3	2	no	—
300	C₂×C₄	—	3	2	no	—
301	C₂×C₄	—	3	1	no	—
302	C₂×C₄	—	6	2	no	—
303	C₂×C₄	—	6	2	no	—
304	C₂×C₄	—	6	2	no	—
305	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
306	C₂×C₄	C₂²×C₃×D₃	1	2	yes	—
307	C₂×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
308	C₂×C₃	—	2	2	no	—
309	C₂×C₃	—	2	2	no	—
310	C₂×C₃	—	2	1	no	—
311	C₂×C₃	—	2	2	no	—
312	C₂×C₃	—	4	2	no	—
313	C₂×C₃	—	4	1	no	—
314	C₂×C₃	—	4	1	no	—
315	C₂×C₃	—	8	1	no	—
316	C₂×C₃	—	12	1	no	—
317	C₂×C₃	C₂×D₃×D₄	1	1	yes	characteristic
318	C₂×C₃	#1010	1	2	yes	—
319	C₂×C₃	C₂²×C₃×D₄	1	1	yes	characteristic
320	C₂×C₃	C₃×2₋¹⁺⁴	1	2	yes	—
321	D₃	—	4	1	no	—
322	C₂²	—	2	2	no	—
323	C₂²	—	2	1	no	—
324	C₂²	—	2	2	no	—
325	C₂²	—	6	1	no	—
326	C₂²	—	6	2	no	—
327	C₂²	C₃×D₃×D₄	1	2	yes	—
328	C₂²	C₂³×C₃×D₃	1	1	yes	characteristic
329	C₄	—	2	2	no	—
330	C₄	—	2	1	no	—
331	C₄	—	2	2	no	—
332	C₄	—	3	2	no	—
333	C₄	—	3	2	no	—
334	C₄	—	6	1	no	—
335	C₄	—	6	2	no	—
336	C₄	—	6	2	no	—
337	C₃	—	2	1	no	—
338	C₃	#5648	1	1	yes	characteristic, direct product factor
339	C₃	C₃×#536	1	1	yes	characteristic, complemented
340	C₂	—	2	2	no	—
341	C₂	—	4	1	no	—
342	C₂	—	12	1	no	—
343	C₂	C₂×C₃×D₃×D₄	1	1	yes	characteristic
344	C₂	C₃×#1010	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	C₂³×D₃
commutator, cosocle	C₂²×C₃	C₂⁴×C₃
Frattini	C₂²	C₂³×C₃×D₃
Fitting	C₃²×#123	C₂
socle	C₂²×C₃²	C₂⁴
prime radical	C₂×C₃×#237	C₂
hypercenter	C₃×#123	D₃
hypocenter	C₃	C₃×#536
LCS	G ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₃×#123

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃×Dic₆	C₂	semidirect
3. C₃²×#123	C₂	semidirect
4. C₃×#935	C₂	semidirect
5. C₃×#929	C₂	semidirect
6. C₃×#896	C₂	semidirect
7. C₃×#888	C₂	semidirect
8. C₃×#886	C₂	semidirect
9. C₃×#885	C₂	semidirect
10. C₃×#878	C₂	semidirect
11. C₃×#877	C₂	semidirect
13. #5648	C₃	direct (N × H)
42. C₂²×Dic₆	C₂×C₃	semidirect
49. C₃×#123	D₃	semidirect
50. C₃×#123	C₂×C₃	semidirect
54. #935	C₂×C₃	semidirect
55. #929	C₂×C₃	semidirect
56. #896	C₂×C₃	semidirect
57. #888	C₂×C₃	semidirect
58. #886	C₂×C₃	semidirect
59. #885	C₂×C₃	semidirect
60. #878	C₂×C₃	semidirect
61. #877	C₂×C₃	semidirect
162. #123	C₃×D₃	semidirect
285. C₃²	#536	semidirect
338. C₃	#5648	direct (N × H)
339. C₃	C₃×#536	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₃×#237	2
C₂²×C₃×Dic₆	2
C₃²×#123	2
C₃×#935	2
C₃×#929	2
C₃×#896	2
C₃×#888	2
C₃×#886	2
C₃×#885	2
C₃×#878	2
C₃×#877	2
C₃×#536	3
#5648	3

Subgroups — C3×#5648 #10595745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#5648 #10595745