Subgroups — C₃×#20784 #10639391

Order 768.

758 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×C₄×#117	C₂	1	1	yes	maximal, characteristic
2	C₃×#1839	C₂	1	2	yes	maximal, complemented
3	C₃×#1831	C₂	1	2	yes	maximal, complemented
4	C₃×#1818	C₂	1	2	yes	maximal
5	C₃×#1798	C₂	1	2	yes	maximal
6	C₃×#1768	C₂	1	2	yes	maximal, complemented
7	C₃×#1749	C₂	1	1	yes	maximal, characteristic
8	C₃×C₄×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
9	C₃×#1746	C₂	1	2	yes	maximal, complemented
10	#20784	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
11	C₂×C₃×#117	C₂²	1	1	yes	characteristic
12	C₂×C₃×#117	C₂²	1	1	yes	characteristic
13	C₂²×C₃×C₄²	C₂²	1	1	yes	characteristic
14	C₂×C₃×C₄⋊C₈	C₂²	1	4	yes	—
15	C₂×C₃×C₂²⋊C₈	C₂²	1	2	yes	—
16	C₂×C₃×C₂²⋊C₈	C₂²	1	2	yes	—
17	C₂×C₃×#97	C₂²	1	2	yes	—
18	C₂×C₃×C₄×C₈	C₂²	1	2	yes	—
19	C₃×C₄×C₄⋊C₄	C₄	1	2	yes	—
20	C₃×C₄×C₄⋊C₄	C₄	1	2	yes	—
21	C₃×C₄×C₂²⋊C₄	C₄	1	2	yes	—
22	C₃×C₄×C₂²⋊C₄	C₄	1	2	yes	—
23	C₃×#376	C₂²	1	1	yes	characteristic
24	C₃×#376	C₂²	1	1	yes	characteristic
25	C₃×C₄³	C₂²	1	1	yes	characteristic
26	C₃×C₄³	C₂²	1	1	yes	characteristic
27	C₃×#336	C₂²	1	4	yes	—
28	C₃×#336	C₂²	1	4	yes	—
29	C₃×#336	C₂²	1	2	yes	—
30	C₃×#336	C₂²	1	2	yes	—
31	C₃×#336	C₂²	1	4	yes	—
32	C₄×#117	C₂×C₃	1	1	yes	characteristic
33	#1839	C₂×C₃	1	2	yes	complemented
34	#1831	C₂×C₃	1	2	yes	complemented
35	#1818	C₂×C₃	1	2	yes	—
36	#1798	C₂×C₃	1	2	yes	—
37	#1768	C₂×C₃	1	2	yes	complemented
38	#1749	C₂×C₃	1	1	yes	characteristic
39	C₄×C₂²⋊C₈	C₂×C₃	1	1	yes	characteristic
40	#1746	C₂×C₃	1	2	yes	complemented
41	C₂³×C₃×C₄	C₂×C₄	1	2	yes	—
42	C₂³×C₃×C₄	C₂³	1	1	yes	characteristic
43	C₂²×C₃×C₈	C₂³	1	4	yes	—
44	C₂²×C₃×C₈	C₂³	1	4	yes	—
45	C₃×#117	—	2	4	no	—
46	C₃×#117	—	2	4	no	—
47	C₂×C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
49	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
50	C₂×C₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
51	C₂×C₃×C₂²⋊C₄	C₂×C₄	1	1	yes	characteristic
52	C₂×C₃×C₂²⋊C₄	C₂×C₄	1	2	yes	—
53	C₂×C₃×C₄²	—	2	4	no	—
54	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
55	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
56	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
57	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
58	C₂×C₃×C₄²	C₂×C₄	1	2	yes	—
59	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
60	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
61	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
62	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
63	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
64	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
65	C₃×C₄⋊C₈	—	2	8	no	—
66	C₃×C₂²⋊C₈	—	2	4	no	—
67	C₃×C₂²⋊C₈	—	2	4	no	—
68	C₃×#97	—	2	4	no	—
69	C₃×C₄×C₈	—	2	4	no	—
70	C₃×#95	C₂×C₄	1	2	yes	—
71	C₃×#95	C₂×C₄	1	2	yes	—
72	C₃×#95	C₂×C₄	1	2	yes	—
73	C₃×#95	C₂×C₄	1	2	yes	—
74	C₂×#117	C₂²×C₃	1	1	yes	characteristic
75	C₂×#117	C₂²×C₃	1	1	yes	characteristic
76	C₂²×C₄²	C₂²×C₃	1	1	yes	characteristic
77	C₂×C₄⋊C₈	C₂²×C₃	1	4	yes	—
78	C₂×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
79	C₂×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
80	C₂×#97	C₂²×C₃	1	2	yes	—
81	C₂×C₄×C₈	C₂²×C₃	1	2	yes	—
82	C₄×C₄⋊C₄	C₃×C₄	1	2	yes	—
83	C₄×C₄⋊C₄	C₃×C₄	1	2	yes	—
84	C₄×C₂²⋊C₄	C₃×C₄	1	2	yes	—
85	C₄×C₂²⋊C₄	C₃×C₄	1	2	yes	—
86	#376	C₂²×C₃	1	1	yes	characteristic
87	#376	C₂²×C₃	1	1	yes	characteristic
88	C₄³	C₂²×C₃	1	1	yes	characteristic
89	C₄³	C₂²×C₃	1	1	yes	characteristic
90	#336	C₂²×C₃	1	4	yes	—
91	#336	C₂²×C₃	1	4	yes	—
92	#336	C₂²×C₃	1	4	yes	—
93	#336	C₂²×C₃	1	2	yes	—
94	#336	C₂²×C₃	1	2	yes	—
95	C₂⁴×C₃	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₃×C₄	—	2	2	no	—
97	C₂²×C₃×C₄	—	2	2	no	—
98	C₂²×C₃×C₄	—	2	4	no	—
99	C₂²×C₃×C₄	—	2	4	no	—
100	C₂²×C₃×C₄	—	4	2	no	—
101	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
102	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
104	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
106	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
107	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
108	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₃×C₄	C₂²×C₄	1	2	yes	—
110	C₂²×C₃×C₄	#41	1	4	yes	—
111	C₂²×C₃×C₄	C₂⁴	1	1	yes	characteristic
112	C₂×C₃×C₈	—	2	4	no	—
113	C₂×C₃×C₈	—	2	4	no	—
114	C₂×C₃×C₈	—	2	4	no	—
115	C₂×C₃×C₈	—	2	4	no	—
116	C₂×C₃×C₈	—	2	4	no	—
117	C₂×C₃×C₈	—	2	4	no	—
118	C₃×C₄⋊C₄	—	2	4	no	—
119	C₃×C₄⋊C₄	—	2	2	no	—
120	C₃×C₄⋊C₄	—	2	2	no	—
121	C₃×C₂²⋊C₄	—	2	2	no	—
122	C₃×C₂²⋊C₄	—	2	2	no	—
123	C₃×C₂²⋊C₄	—	2	4	no	—
124	C₃×C₄²	—	2	4	no	—
125	C₃×C₄²	—	2	4	no	—
126	C₃×C₄²	—	2	4	no	—
127	C₃×C₄²	—	2	2	no	—
128	C₃×C₄²	—	2	2	no	—
129	C₃×C₄²	—	2	2	no	—
130	C₃×C₄²	—	2	2	no	—
131	C₃×C₄²	—	4	2	no	—
132	C₃×C₄²	—	4	2	no	—
133	C₃×C₄²	#41	1	4	yes	—
134	C₃×C₄²	#41	1	4	yes	—
135	C₂³×C₄	C₂×C₃×C₄	1	2	yes	—
136	C₂³×C₄	C₂³×C₃	1	1	yes	characteristic
137	C₂²×C₈	C₂³×C₃	1	4	yes	—
138	C₂²×C₈	C₂³×C₃	1	4	yes	—
139	#117	—	2	4	no	—
140	#117	—	2	4	no	—
141	C₂×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
142	C₂×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
143	C₂×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
144	C₂×C₂²⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
145	C₂×C₂²⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
146	C₂×C₂²⋊C₄	C₂×C₃×C₄	1	2	yes	—
147	C₂×C₄²	—	2	4	no	—
148	C₂×C₄²	C₂×C₃×C₄	1	2	yes	—
149	C₂×C₄²	C₂×C₃×C₄	1	2	yes	—
150	C₂×C₄²	C₂×C₃×C₄	1	2	yes	—
151	C₂×C₄²	C₂×C₃×C₄	1	2	yes	—
152	C₂×C₄²	C₂×C₃×C₄	1	2	yes	—
153	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
154	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
155	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
156	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
157	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
158	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
159	C₄⋊C₈	—	2	8	no	—
160	C₂²⋊C₈	—	2	4	no	—
161	C₂²⋊C₈	—	2	4	no	—
162	#97	—	2	4	no	—
163	C₄×C₈	—	2	4	no	—
164	#95	C₂×C₃×C₄	1	2	yes	—
165	#95	C₂×C₃×C₄	1	2	yes	—
166	#95	C₂×C₃×C₄	1	2	yes	—
167	#95	C₂×C₃×C₄	1	2	yes	—
168	C₂³×C₃	—	2	1	no	—
169	C₂³×C₃	—	2	1	no	—
170	C₂³×C₃	—	2	2	no	—
171	C₂³×C₃	—	2	2	no	—
172	C₂³×C₃	#117	1	2	yes	—
173	C₂³×C₃	C₂³×C₄	1	1	yes	characteristic
174	C₂×C₃×C₄	—	2	2	no	—
175	C₂×C₃×C₄	—	2	1	no	—
176	C₂×C₃×C₄	—	2	1	no	—
177	C₂×C₃×C₄	—	2	2	no	—
178	C₂×C₃×C₄	—	2	2	no	—
179	C₂×C₃×C₄	—	2	2	no	—
180	C₂×C₃×C₄	—	2	2	no	—
181	C₂×C₃×C₄	—	2	2	no	—
182	C₂×C₃×C₄	—	2	8	no	—
183	C₂×C₃×C₄	—	2	1	no	—
184	C₂×C₃×C₄	—	2	1	no	—
185	C₂×C₃×C₄	—	2	1	no	—
186	C₂×C₃×C₄	—	2	1	no	—
187	C₂×C₃×C₄	—	2	2	no	—
188	C₂×C₃×C₄	—	2	2	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	2	no	—
193	C₂×C₃×C₄	—	2	2	no	—
194	C₂×C₃×C₄	—	4	8	no	—
195	C₂×C₃×C₄	#117	1	4	yes	—
196	C₂×C₃×C₄	#117	1	4	yes	—
197	C₂×C₃×C₄	#117	1	2	yes	—
198	C₂×C₃×C₄	#131	1	4	yes	—
199	C₂×C₃×C₄	C₂×#41	1	2	yes	—
200	C₂×C₃×C₄	C₂×#41	1	2	yes	—
201	C₂×C₃×C₄	C₂×#41	1	2	yes	—
202	C₃×C₈	—	4	4	no	—
203	C₃×C₈	—	4	4	no	—
204	C₂⁴	C₂²×C₃×C₄	1	1	yes	characteristic
205	C₂²×C₄	—	2	2	no	—
206	C₂²×C₄	—	2	2	no	—
207	C₂²×C₄	—	2	4	no	—
208	C₂²×C₄	—	2	4	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
211	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
212	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
213	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
214	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
215	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
216	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
217	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
218	C₂²×C₄	C₂²×C₃×C₄	1	2	yes	—
219	C₂²×C₄	C₃×#41	1	4	yes	—
220	C₂²×C₄	C₂⁴×C₃	1	1	yes	characteristic
221	C₂×C₈	—	2	4	no	—
222	C₂×C₈	—	2	4	no	—
223	C₂×C₈	—	2	4	no	—
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	2	2	no	—
229	C₄⋊C₄	—	2	2	no	—
230	C₂²⋊C₄	—	2	2	no	—
231	C₂²⋊C₄	—	2	2	no	—
232	C₂²⋊C₄	—	2	4	no	—
233	C₄²	—	2	4	no	—
234	C₄²	—	2	4	no	—
235	C₄²	—	2	4	no	—
236	C₄²	—	2	2	no	—
237	C₄²	—	2	2	no	—
238	C₄²	—	2	2	no	—
239	C₄²	—	2	2	no	—
240	C₄²	—	4	2	no	—
241	C₄²	—	4	2	no	—
242	C₄²	C₃×#41	1	4	yes	—
243	C₄²	C₃×#41	1	4	yes	—
244	C₂²×C₃	—	2	2	no	—
245	C₂²×C₃	—	2	2	no	—
246	C₂²×C₃	—	2	2	no	—
247	C₂²×C₃	—	4	2	no	—
248	C₂²×C₃	—	4	2	no	—
249	C₂²×C₃	C₂×#117	1	1	yes	characteristic
250	C₂²×C₃	C₂×#117	1	1	yes	characteristic
251	C₂²×C₃	C₂×#117	1	1	yes	characteristic
252	C₂²×C₃	C₂×#131	1	2	yes	—
253	C₂²×C₃	#568	1	2	yes	—
254	C₃×C₄	—	2	4	no	—
255	C₃×C₄	—	2	4	no	—
256	C₃×C₄	—	2	4	no	—
257	C₃×C₄	—	2	2	no	—
258	C₃×C₄	—	4	1	no	—
259	C₃×C₄	—	4	1	no	—
260	C₃×C₄	—	4	1	no	—
261	C₃×C₄	—	4	1	no	—
262	C₃×C₄	—	4	2	no	—
263	C₃×C₄	#529	1	4	yes	—
264	C₂³	—	2	2	no	—
265	C₂³	—	2	2	no	—
266	C₂³	—	2	1	no	—
267	C₂³	—	2	1	no	—
268	C₂³	C₃×#117	1	2	yes	—
269	C₂³	C₂³×C₃×C₄	1	1	yes	characteristic
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	8	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	4	8	no	—
291	C₂×C₄	C₃×#117	1	4	yes	—
292	C₂×C₄	C₃×#117	1	4	yes	—
293	C₂×C₄	C₃×#117	1	2	yes	—
294	C₂×C₄	C₃×#131	1	4	yes	—
295	C₂×C₄	C₂×C₃×#41	1	2	yes	—
296	C₂×C₄	C₂×C₃×#41	1	2	yes	—
297	C₂×C₄	C₂×C₃×#41	1	2	yes	—
298	C₈	—	4	4	no	—
299	C₈	—	4	4	no	—
300	C₂×C₃	—	4	2	no	—
301	C₂×C₃	#2283	1	1	yes	characteristic
302	C₂×C₃	#2922	1	1	yes	characteristic
303	C₂×C₃	#2923	1	1	yes	characteristic
304	C₂×C₃	#2924	1	2	yes	—
305	C₂×C₃	#2925	1	2	yes	—
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	—	4	2	no	—
310	C₂²	—	4	2	no	—
311	C₂²	C₂×C₃×#117	1	1	yes	characteristic
312	C₂²	C₂×C₃×#117	1	1	yes	characteristic
313	C₂²	C₂×C₃×#117	1	1	yes	characteristic
314	C₂²	C₂×C₃×#131	1	2	yes	—
315	C₂²	C₃×#568	1	2	yes	—
316	C₄	—	2	4	no	—
317	C₄	—	2	4	no	—
318	C₄	—	2	4	no	—
319	C₄	—	2	2	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	2	no	—
325	C₄	C₃×#529	1	4	yes	—
326	C₃	#20784	1	1	yes	characteristic, direct product factor, Sylow
327	C₂	—	4	2	no	—
328	C₂	C₃×#2283	1	1	yes	characteristic
329	C₂	C₃×#2922	1	1	yes	characteristic
330	C₂	C₃×#2923	1	1	yes	characteristic
331	C₂	C₃×#2924	1	2	yes	—
332	C₂	C₃×#2925	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃×C₄	C₂⁴
commutator	C₂³	C₂³×C₃×C₄
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃
socle	C₂³×C₃	C₂³×C₄
prime radical	C₂⁴×C₃	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₃×C₄ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃×#1839	C₂	semidirect
3. C₃×#1831	C₂	semidirect
6. C₃×#1768	C₂	semidirect
9. C₃×#1746	C₂	semidirect
10. #20784	C₃	direct (N × H)
33. #1839	C₂×C₃	semidirect
34. #1831	C₂×C₃	semidirect
37. #1768	C₂×C₃	semidirect
40. #1746	C₂×C₃	semidirect
326. C₃	#20784	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×C₄×#117	2
C₃×#1839	2
C₃×#1831	2
C₃×#1818	2
C₃×#1798	2
C₃×#1768	2
C₃×#1749	2
C₃×C₄×C₂²⋊C₈	2
C₃×#1746	2
#20784	3

Subgroups — C3×#20784 #10639391

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#20784 #10639391