Subgroups — C₃×#70864 #11777465

Order 1152.

393 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃²×#436	C₂	1	1	yes	maximal, characteristic
2	C₃×C₄×#214	C₂	1	1	yes	maximal, characteristic
3	C₃×C₈×Dic₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×D₄⋊(C₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
5	C₃×#4484	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃×#4480	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃×#4455	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃×#1583	—	3	1	no	maximal
9	#70864	C₃	1	1	yes	maximal, characteristic, direct product factor
10	C₂×C₃²×M₁₆	—	2	1	no	—
11	C₃²×C₄×D₄	C₂²	1	1	yes	characteristic
12	C₂×C₃×#214	C₄	1	1	yes	characteristic
13	C₃×C₄×Dic₆	C₂²	1	1	yes	characteristic
14	C₃²×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₃²×C₂²⋊C₈	—	2	1	no	—
16	C₃²×C₄×C₈	C₂²	1	1	yes	characteristic
17	C₃×#832	C₄	1	1	yes	characteristic
18	C₃×Dic₃⋊C₈	—	2	1	no	—
19	C₃×C₈×Dic₃	—	2	1	no	—
20	C₃×#810	C₄	1	1	yes	characteristic
21	C₃×#808	C₄	1	1	yes	characteristic
22	C₃×C₄⋊(C₃⋊C₈)	C₂²	1	1	yes	characteristic
23	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
24	C₃×C₄×C₃⋊C₈	C₂²	1	1	yes	characteristic
25	C₃×C₈×Q₈	—	3	1	no	—
26	C₃×C₄×SD₁₆	—	3	1	no	—
27	C₃×#436	—	2	1	no	—
28	C₃×#436	D₃	1	1	yes	characteristic
29	C₃×#436	C₂×C₃	1	1	yes	characteristic
30	C₄×#214	C₂×C₃	1	1	yes	characteristic
31	C₈×Dic₆	C₂×C₃	1	1	yes	characteristic, complemented
32	C₃×#334	—	3	1	no	—
33	C₃×Q₈⋊C₈	—	3	1	no	—
34	C₃×D₄⋊C₈	—	3	1	no	—
35	C₃×#322	—	3	1	no	—
36	D₄⋊(C₃⋊C₈)	C₂×C₃	1	1	yes	characteristic
37	#4484	C₂×C₃	1	1	yes	characteristic, complemented
38	#4480	C₂×C₃	1	1	yes	characteristic, complemented
39	#4455	C₂×C₃	1	1	yes	characteristic, complemented
40	C₂×C₃²×D₄	C₂×C₄	1	1	yes	characteristic
41	C₂²×C₃²×C₄	—	2	1	no	—
42	C₂×C₃×Dic₆	C₂×C₄	1	1	yes	characteristic
43	C₃²×M₁₆	—	4	2	no	—
44	C₂×C₃²×C₈	—	2	1	no	—
45	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
46	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
47	C₃²×C₄⋊C₄	C₂×C₄	1	1	yes	characteristic
48	C₃²×C₂²⋊C₄	—	2	1	no	—
49	C₃²×C₄²	C₂³	1	1	yes	characteristic
50	C₃×#214	—	2	2	no	—
51	C₃×C₄⋊Dic₃	C₂×C₄	1	1	yes	characteristic
52	C₃×Dic₃⋊C₄	—	2	1	no	—
53	C₃×C₄×Dic₃	—	2	1	no	—
54	C₂×C₃×C₃⋊C₈	—	2	1	no	—
55	C₂×C₃×C₃⋊C₈	—	2	1	no	—
56	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
57	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
58	#1583	—	3	1	no	Sylow
59	C₂×C₃×SD₁₆	—	3	1	no	—
60	C₂×C₃×M₁₆	—	2	2	no	—
61	C₂×C₃×M₁₆	—	2	1	no	—
62	C₂×C₃×M₁₆	—	2	1	no	—
63	C₃×C₄×Q₈	—	3	1	no	—
64	C₃×C₄×D₄	—	2	1	no	—
65	C₃×C₄×D₄	C₂×D₃	1	1	yes	characteristic
66	C₃×C₄×D₄	C₂²×C₃	1	1	yes	characteristic
67	C₂×#214	C₃×C₄	1	1	yes	characteristic
68	C₄×Dic₆	C₂²×C₃	1	1	yes	characteristic
69	C₃×#106	—	3	1	no	—
70	C₃×C₄⋊C₈	—	2	1	no	—
71	C₃×C₄⋊C₈	—	3	1	no	—
72	C₃×C₄⋊C₈	—	6	1	no	—
73	C₃×C₄⋊C₈	C₂×D₃	1	1	yes	characteristic
74	C₃×C₄⋊C₈	C₂²×C₃	1	1	yes	characteristic
75	C₃×#103	—	3	1	no	—
76	C₃×#102	—	3	1	no	—
77	C₃×C₂²⋊C₈	—	2	2	no	—
78	C₃×C₂²⋊C₈	—	2	1	no	—
79	C₃×C₂²⋊C₈	—	2	1	no	—
80	C₃×C₄×C₈	—	2	1	no	—
81	C₃×C₄×C₈	—	3	1	no	—
82	C₃×C₄×C₈	—	3	1	no	—
83	C₃×C₄×C₈	—	6	1	no	—
84	C₃×C₄×C₈	C₂×D₃	1	1	yes	characteristic
85	C₃×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
86	#832	C₃×C₄	1	1	yes	characteristic
87	Dic₃⋊C₈	—	2	1	no	—
88	C₈×Dic₃	—	2	1	no	—
89	#810	C₃×C₄	1	1	yes	characteristic
90	#808	C₃×C₄	1	1	yes	characteristic
91	C₄⋊(C₃⋊C₈)	C₂²×C₃	1	1	yes	characteristic
92	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
93	C₄×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
94	C₂³×C₃²	—	2	1	no	—
95	C₃²×D₄	—	2	1	no	—
96	C₃²×D₄	—	2	1	no	—
97	C₂×C₃²×C₄	—	2	1	no	—
98	C₂×C₃²×C₄	—	4	1	no	—
99	C₂×C₃²×C₄	—	4	1	no	—
100	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
101	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
102	C₂×C₃²×C₄	#41	1	1	yes	characteristic
103	C₂×C₃×Dic₃	—	2	1	no	—
104	C₂×C₃×Dic₃	—	2	1	no	—
105	C₃×Dic₆	—	2	1	no	—
106	C₃×Dic₆	M₁₆	1	2	yes	complemented
107	C₃²×C₈	—	2	1	no	—
108	C₃²×C₈	—	4	1	no	—
109	C₃²×C₈	SD₁₆	1	2	yes	complemented
110	C₃×C₃⋊C₈	—	2	1	no	—
111	C₃×C₃⋊C₈	—	2	1	no	—
112	C₃×C₃⋊C₈	—	4	1	no	—
113	C₃×C₃⋊C₈	—	4	1	no	—
114	C₈×Q₈	—	3	1	no	—
115	C₄×SD₁₆	—	3	1	no	—
116	#436	C₃×D₃	1	1	yes	characteristic
117	#334	—	3	1	no	—
118	Q₈⋊C₈	—	3	1	no	—
119	D₄⋊C₈	—	3	1	no	—
120	#322	—	3	1	no	—
121	C₂×C₃×Q₈	—	3	1	no	—
122	C₂×C₃×D₄	—	2	1	no	—
123	C₂×C₃×D₄	C₄×D₃	1	1	yes	characteristic
124	C₂×C₃×D₄	C₂×C₃×C₄	1	1	yes	characteristic
125	C₂²×C₃×C₄	—	2	2	no	—
126	C₂²×C₃×C₄	—	2	1	no	—
127	C₂²×C₃×C₄	—	2	1	no	—
128	C₂×Dic₆	C₂×C₃×C₄	1	1	yes	characteristic
129	C₃×SD₁₆	—	6	2	no	—
130	C₃×M₁₆	—	4	4	no	—
131	C₃×M₁₆	—	4	2	no	—
132	C₃×M₁₆	—	4	2	no	—
133	C₂×C₃×C₈	—	2	2	no	—
134	C₂×C₃×C₈	—	2	1	no	—
135	C₂×C₃×C₈	—	2	1	no	—
136	C₂×C₃×C₈	—	2	1	no	—
137	C₂×C₃×C₈	—	2	1	no	—
138	C₂×C₃×C₈	—	3	1	no	—
139	C₂×C₃×C₈	—	3	1	no	—
140	C₂×C₃×C₈	—	6	1	no	—
141	C₂×C₃×C₈	—	6	1	no	—
142	C₂×C₃×C₈	#67	1	1	yes	characteristic
143	C₂×C₃×C₈	#67	1	1	yes	characteristic
144	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
145	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
146	C₃×C₄⋊C₄	—	2	1	no	—
147	C₃×C₄⋊C₄	—	3	1	no	—
148	C₃×C₄⋊C₄	—	6	1	no	—
149	C₃×C₄⋊C₄	C₄×D₃	1	1	yes	characteristic
150	C₃×C₄⋊C₄	C₂×C₃×C₄	1	1	yes	characteristic
151	C₃×C₂²⋊C₄	—	2	2	no	—
152	C₃×C₂²⋊C₄	—	2	1	no	—
153	C₃×C₂²⋊C₄	—	2	1	no	—
154	C₃×C₄²	—	2	1	no	—
155	C₃×C₄²	—	6	1	no	—
156	C₃×C₄²	C₂²×D₃	1	1	yes	characteristic
157	C₃×C₄²	C₂³×C₃	1	1	yes	characteristic
158	#214	—	2	2	no	—
159	C₄⋊Dic₃	C₂×C₃×C₄	1	1	yes	characteristic
160	Dic₃⋊C₄	—	2	1	no	—
161	C₄×Dic₃	—	2	1	no	—
162	C₂×C₃⋊C₈	—	2	1	no	—
163	C₂×C₃⋊C₈	—	2	1	no	—
164	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
165	C₂×C₃⋊C₈	C₂×C₃×C₄	1	1	yes	characteristic
166	C₂²×C₃²	—	4	1	no	—
167	C₂²×C₃²	—	4	1	no	—
168	C₂²×C₃²	—	4	1	no	—
169	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
170	C₃²×C₄	—	2	1	no	—
171	C₃²×C₄	—	4	1	no	—
172	C₃²×C₄	C₂×M₁₆	1	1	yes	characteristic
173	C₃²×C₄	#131	1	1	yes	characteristic
174	C₃²×C₄	C₂×SD₁₆	1	1	yes	characteristic
175	C₃²×C₄	#135	1	1	yes	characteristic
176	C₃×Dic₃	—	2	2	no	—
177	C₃×Dic₃	—	4	1	no	—
178	C₂×SD₁₆	—	3	1	no	—
179	C₂×M₁₆	—	2	1	no	—
180	C₄×Q₈	—	3	1	no	—
181	C₄×D₄	C₂×C₃×D₃	1	1	yes	characteristic
182	#106	—	3	1	no	—
183	C₄⋊C₈	—	3	1	no	—
184	C₄⋊C₈	—	6	1	no	—
185	C₄⋊C₈	C₂×C₃×D₃	1	1	yes	characteristic
186	#103	—	3	1	no	—
187	#102	—	3	1	no	—
188	C₂²⋊C₈	—	2	1	no	—
189	C₄×C₈	—	3	1	no	—
190	C₄×C₈	—	3	1	no	—
191	C₄×C₈	—	6	1	no	—
192	C₄×C₈	C₂×C₃×D₃	1	1	yes	characteristic
193	C₂³×C₃	—	2	2	no	—
194	C₂³×C₃	—	2	1	no	—
195	C₂³×C₃	—	2	1	no	—
196	C₃×Q₈	—	3	2	no	—
197	C₃×Q₈	—	6	1	no	—
198	C₃×D₄	—	2	1	no	—
199	C₃×D₄	—	2	1	no	—
200	C₃×D₄	—	2	1	no	—
201	C₃×D₄	—	2	1	no	—
202	C₃×D₄	—	4	1	no	—
203	C₃×D₄	—	4	1	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	—	2	1	no	—
208	C₂×C₃×C₄	—	2	1	no	—
209	C₂×C₃×C₄	—	2	1	no	—
210	C₂×C₃×C₄	—	4	2	no	—
211	C₂×C₃×C₄	—	4	2	no	—
212	C₂×C₃×C₄	—	4	1	no	—
213	C₂×C₃×C₄	—	4	1	no	—
214	C₂×C₃×C₄	—	4	1	no	—
215	C₂×C₃×C₄	—	4	1	no	—
216	C₂×C₃×C₄	—	6	1	no	—
217	C₂×C₃×C₄	—	6	1	no	—
218	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
219	C₂×C₃×C₄	#235	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
223	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
224	C₂×Dic₃	—	2	1	no	—
225	C₂×Dic₃	—	2	1	no	—
226	Dic₆	—	2	1	no	—
227	Dic₆	C₃×M₁₆	1	2	yes	complemented
228	C₃×C₈	—	2	1	no	—
229	C₃×C₈	—	2	1	no	—
230	C₃×C₈	—	2	2	no	—
231	C₃×C₈	—	4	1	no	—
232	C₃×C₈	—	4	2	no	—
233	C₃×C₈	—	4	1	no	—
234	C₃×C₈	—	4	1	no	—
235	C₃×C₈	—	6	1	no	—
236	C₃×C₈	—	6	1	no	—
237	C₃×C₈	—	12	1	no	—
238	C₃×C₈	—	12	1	no	—
239	C₃×C₈	#214	1	2	yes	complemented
240	C₃×C₈	C₃×SD₁₆	1	2	yes	complemented
241	C₃⋊C₈	—	2	1	no	—
242	C₃⋊C₈	—	2	1	no	—
243	C₃⋊C₈	—	4	1	no	—
244	C₃⋊C₈	—	4	1	no	—
245	C₂×C₃²	—	8	1	no	—
246	C₂×C₃²	#435	1	1	yes	characteristic
247	C₂×C₃²	C₄×SD₁₆	1	1	yes	characteristic
248	C₂×C₃²	#444	1	1	yes	characteristic
249	C₂×Q₈	—	3	1	no	—
250	C₂×D₄	C₃×C₄×D₃	1	1	yes	characteristic
251	C₂²×C₄	—	2	1	no	—
252	SD₁₆	—	6	2	no	—
253	M₁₆	—	4	2	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	3	1	no	—
256	C₂×C₈	—	3	1	no	—
257	C₂×C₈	—	6	1	no	—
258	C₂×C₈	—	6	1	no	—
259	C₂×C₈	C₃×#67	1	1	yes	characteristic
260	C₂×C₈	C₃×#67	1	1	yes	characteristic
261	C₄⋊C₄	—	3	1	no	—
262	C₄⋊C₄	—	6	1	no	—
263	C₄⋊C₄	C₃×C₄×D₃	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₄²	—	6	1	no	—
266	C₄²	C₂²×C₃×D₃	1	1	yes	characteristic
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	—	4	2	no	—
269	C₂²×C₃	—	4	2	no	—
270	C₂²×C₃	—	4	2	no	—
271	C₂²×C₃	—	4	1	no	—
272	C₂²×C₃	—	4	1	no	—
273	C₂²×C₃	—	4	1	no	—
274	C₂²×C₃	—	4	1	no	—
275	C₂²×C₃	—	4	1	no	—
276	C₂²×C₃	—	4	1	no	—
277	C₂²×C₃	C₄×#67	1	1	yes	characteristic
278	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	2	1	no	—
281	C₃×C₄	—	2	1	no	—
282	C₃×C₄	—	2	1	no	—
283	C₃×C₄	—	2	1	no	—
284	C₃×C₄	—	2	1	no	—
285	C₃×C₄	—	4	1	no	—
286	C₃×C₄	—	4	2	no	—
287	C₃×C₄	—	4	1	no	—
288	C₃×C₄	—	4	1	no	—
289	C₃×C₄	—	6	2	no	—
290	C₃×C₄	—	12	1	no	—
291	C₃×C₄	D₃×M₁₆	1	1	yes	characteristic
292	C₃×C₄	#907	1	1	yes	characteristic
293	C₃×C₄	C₂×#214	1	1	yes	characteristic
294	C₃×C₄	#950	1	1	yes	characteristic
295	C₃×C₄	C₂×C₃×M₁₆	1	1	yes	characteristic
296	C₃×C₄	C₃×#131	1	1	yes	characteristic
297	C₃×C₄	C₂×C₃×SD₁₆	1	1	yes	characteristic
298	C₃×C₄	C₃×#135	1	1	yes	characteristic
299	Dic₃	—	2	2	no	—
300	Dic₃	—	4	1	no	—
301	C₃²	#1583	1	1	yes	characteristic, complemented, Sylow
302	C₂³	—	2	1	no	—
303	Q₈	—	3	2	no	—
304	Q₈	—	6	1	no	—
305	D₄	—	2	1	no	—
306	D₄	—	2	1	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	6	1	no	—
311	C₂×C₄	—	6	1	no	—
312	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
313	C₂×C₄	C₃×#235	1	1	yes	characteristic
314	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
315	C₈	—	2	1	no	—
316	C₈	—	4	1	no	—
317	C₈	—	6	1	no	—
318	C₈	—	6	1	no	—
319	C₈	—	12	1	no	—
320	C₈	—	12	1	no	—
321	C₈	C₃×#214	1	2	yes	complemented
322	C₂×C₃	—	2	1	no	—
323	C₂×C₃	—	2	1	no	—
324	C₂×C₃	—	2	1	no	—
325	C₂×C₃	—	8	2	no	—
326	C₂×C₃	—	8	1	no	—
327	C₂×C₃	—	8	1	no	—
328	C₂×C₃	C₄×#214	1	1	yes	characteristic
329	C₂×C₃	#5127	1	1	yes	characteristic
330	C₂×C₃	#5145	1	1	yes	characteristic
331	C₂×C₃	C₃×#435	1	1	yes	characteristic
332	C₂×C₃	C₃×C₄×SD₁₆	1	1	yes	characteristic
333	C₂×C₃	C₃×#444	1	1	yes	characteristic
334	C₂²	—	4	1	no	—
335	C₂²	—	4	1	no	—
336	C₂²	—	4	1	no	—
337	C₂²	C₃×C₄×#67	1	1	yes	characteristic
338	C₄	—	2	1	no	—
339	C₄	—	4	1	no	—
340	C₄	—	6	2	no	—
341	C₄	—	12	1	no	—
342	C₄	C₃×D₃×M₁₆	1	1	yes	characteristic
343	C₄	C₃×#907	1	1	yes	characteristic
344	C₄	C₂×C₃×#214	1	1	yes	characteristic
345	C₄	C₃×#950	1	1	yes	characteristic
346	C₃	—	2	1	no	—
347	C₃	#70864	1	1	yes	characteristic, direct product factor
348	C₃	C₃×#1583	1	1	yes	characteristic, complemented
349	C₂	—	8	1	no	—
350	C₂	C₃×C₄×#214	1	1	yes	characteristic
351	C₂	C₃×#5127	1	1	yes	characteristic
352	C₂	C₃×#5145	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂×#67
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₄²	C₂²×C₃×D₃
Fitting	C₃²×#436	C₂
socle	C₂²×C₃²	C₄×D₄
prime radical	C₂×C₃²×D₄	C₂×C₄
cosocle	C₃×C₄²	C₂³×C₃
hypercenter	C₃×#436	D₃
hypocenter	C₃	C₃×#1583
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×C₈ ◁ C₃×#436

Complements

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

N	H ≅ G/N	type
3. C₃×C₈×Dic₆	C₂	semidirect
5. C₃×#4484	C₂	semidirect
6. C₃×#4480	C₂	semidirect
7. C₃×#4455	C₂	semidirect
9. #70864	C₃	direct (N × H)
31. C₈×Dic₆	C₂×C₃	semidirect
37. #4484	C₂×C₃	semidirect
38. #4480	C₂×C₃	semidirect
39. #4455	C₂×C₃	semidirect
106. C₃×Dic₆	M₁₆	semidirect
109. C₃²×C₈	SD₁₆	semidirect
227. Dic₆	C₃×M₁₆	semidirect
239. C₃×C₈	#214	semidirect
240. C₃×C₈	C₃×SD₁₆	semidirect
301. C₃²	#1583	semidirect
321. C₈	C₃×#214	semidirect
347. C₃	#70864	direct (N × H)
348. C₃	C₃×#1583	semidirect

Maximal subgroups

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

subgroup	index
C₃²×#436	2
C₃×C₄×#214	2
C₃×C₈×Dic₆	2
C₃×D₄⋊(C₃⋊C₈)	2
C₃×#4484	2
C₃×#4480	2
C₃×#4455	2
C₃×#1583	3
#70864	3

Subgroups — C3×#70864 #11777465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70864 #11777465