Subgroups — C₃×#70792 #11775082

Order 1152.

539 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×#5470	C₂	1	1	yes	maximal, characteristic
2	C₃²×#431	C₂	1	1	yes	maximal, characteristic
3	C₃×C₄×C₄⋊Dic₃	C₂	1	1	yes	maximal, characteristic
4	C₃×#4467	C₂	1	4	yes	maximal
5	C₃×#1503	—	3	1	no	maximal
6	#70792	C₃	1	1	yes	maximal, characteristic, direct product factor
7	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
8	C₃×#926	—	2	1	no	—
9	C₂×C₃×C₄⋊Dic₃	—	2	1	no	—
10	C₃×#924	—	2	1	no	—
11	C₃×#924	C₂²	1	2	yes	—
12	C₂×C₃×C₄×Dic₃	—	2	1	no	—
13	C₃×#870	—	2	1	no	—
14	C₃×#869	C₄	1	1	yes	characteristic, complemented
15	C₃×#869	C₄	1	1	yes	characteristic, complemented
16	C₃×C₄×Dic₆	—	2	1	no	—
17	C₃×C₄×Dic₆	C₄	1	2	yes	complemented
18	C₃²×C₄⋊C₈	—	2	1	no	—
19	C₃²×C₂²⋊C₈	C₂²	1	2	yes	—
20	C₃²×C₄×C₈	—	2	1	no	—
21	C₃×C₃⋊#95	C₂²	1	2	yes	—
22	C₃×#533	—	3	1	no	—
23	#5470	C₂×C₃	1	1	yes	characteristic
24	C₃×#431	—	2	1	no	—
25	C₃×#431	D₃	1	1	yes	characteristic
26	C₃×#431	C₂×C₃	1	1	yes	characteristic
27	C₃×C₄×C₄⋊C₄	—	3	1	no	—
28	C₄×C₄⋊Dic₃	C₂×C₃	1	1	yes	characteristic
29	C₃×#328	—	3	4	no	—
30	#4467	C₂×C₃	1	4	yes	—
31	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
32	C₂²×C₃²×C₄	D₄	1	1	yes	characteristic
33	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
34	C₂²×C₃×Dic₃	—	2	2	no	—
35	C₂×C₃×Dic₆	—	2	1	no	—
36	C₂×C₃×Dic₆	C₂×C₄	1	2	yes	—
37	C₂×C₃²×C₈	—	2	2	no	—
38	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
39	C₃²×C₄²	C₂×C₄	1	1	yes	characteristic
40	C₃²×C₄²	D₄	1	1	yes	characteristic
41	C₃²×C₄²	D₄	1	1	yes	characteristic
42	C₃×#217	—	2	2	no	—
43	C₃×C₄⋊Dic₃	—	2	4	no	—
44	C₃×C₄⋊Dic₃	—	2	1	no	—
45	C₃×C₄⋊Dic₃	—	2	1	no	—
46	C₃×C₄⋊Dic₃	—	2	1	no	—
47	C₃×C₄⋊Dic₃	C₂×C₄	1	2	yes	—
48	C₃×Dic₃⋊C₄	—	2	2	no	—
49	C₃×Dic₃⋊C₄	—	2	2	no	—
50	C₃×C₄×Dic₃	—	2	2	no	—
51	C₃×C₄×Dic₃	—	4	2	no	—
52	#1503	—	3	1	no	Sylow
53	C₃×C₄⋊Q₈	—	3	1	no	—
54	C₃×C₄⋊Q₈	—	3	1	no	—
55	C₃×#125	—	6	1	no	—
56	C₃×C₂²⋊Q₈	—	3	2	no	—
57	C₃×C₂²⋊Q₈	—	6	1	no	—
58	C₃×C₄×Q₈	—	3	2	no	—
59	C₃×C₄×Q₈	—	6	1	no	—
60	C₃×#117	—	6	1	no	—
61	C₂×C₃×C₄⋊C₄	—	6	1	no	—
62	C₂×C₃×C₄²	—	2	1	no	—
63	C₂×C₃×C₄²	—	6	1	no	—
64	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
65	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
66	#926	—	2	1	no	—
67	C₂×C₄⋊Dic₃	—	2	1	no	—
68	#924	—	2	1	no	—
69	#924	C₂²×C₃	1	2	yes	—
70	C₂×C₄×Dic₃	—	2	1	no	—
71	#870	—	2	1	no	—
72	#869	C₃×C₄	1	1	yes	characteristic, complemented
73	#869	C₃×C₄	1	1	yes	characteristic, complemented
74	C₄×Dic₆	—	2	1	no	—
75	C₄×Dic₆	C₃×C₄	1	2	yes	complemented
76	C₃×C₄⋊C₈	—	2	2	no	—
77	C₃×C₄⋊C₈	—	2	1	no	—
78	C₃×C₄⋊C₈	—	2	1	no	—
79	C₃×C₂²⋊C₈	—	2	2	no	—
80	C₃×C₂²⋊C₈	C₂×D₃	1	2	yes	—
81	C₃×C₂²⋊C₈	C₂²×C₃	1	2	yes	—
82	C₃×C₄×C₈	—	2	2	no	—
83	C₃×C₄×C₈	—	2	1	no	—
84	C₃×C₄×C₈	—	2	1	no	—
85	C₃×#95	—	3	2	no	—
86	C₃⋊#95	C₂²×C₃	1	2	yes	—
87	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃²×C₄	—	2	2	no	—
89	C₂×C₃²×C₄	—	4	1	no	—
90	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	SD₁₆	1	2	yes	—
95	C₂×C₃²×C₄	Q₁₆	1	2	yes	—
96	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
98	C₂×C₃×Dic₃	—	2	4	no	—
99	C₂×C₃×Dic₃	—	2	2	no	—
100	C₂×C₃×Dic₃	—	2	2	no	—
101	C₂×C₃×Dic₃	—	4	2	no	—
102	C₂×C₃×Dic₃	—	4	2	no	—
103	C₃×Dic₆	—	4	2	no	—
104	C₃×Dic₆	—	4	2	no	—
105	C₃²×C₈	—	4	2	no	—
106	#533	—	3	1	no	—
107	#431	C₃×D₃	1	1	yes	characteristic
108	C₄×C₄⋊C₄	—	3	1	no	—
109	#328	—	3	4	no	—
110	C₂×C₃×Q₈	—	3	2	no	—
111	C₂×C₃×Q₈	—	6	1	no	—
112	C₂²×C₃×C₄	—	2	1	no	—
113	C₂²×C₃×C₄	—	2	1	no	—
114	C₂²×C₃×C₄	—	2	1	no	—
115	C₂²×C₃×C₄	—	6	2	no	—
116	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
117	C₂²×C₃×C₄	D₁₂	1	1	yes	characteristic
118	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₃×D₄	1	1	yes	characteristic
120	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
121	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
122	C₂²×Dic₃	—	2	2	no	—
123	C₂×Dic₆	—	2	1	no	—
124	C₂×Dic₆	C₂×C₃×C₄	1	2	yes	—
125	C₂×C₃×C₈	—	2	4	no	—
126	C₂×C₃×C₈	—	2	2	no	—
127	C₂×C₃×C₈	—	2	2	no	—
128	C₃×C₄⋊C₄	—	3	2	no	—
129	C₃×C₄⋊C₄	—	6	4	no	—
130	C₃×C₄⋊C₄	—	6	2	no	—
131	C₃×C₄⋊C₄	—	6	1	no	—
132	C₃×C₄⋊C₄	—	6	1	no	—
133	C₃×C₄⋊C₄	—	6	1	no	—
134	C₃×C₄⋊C₄	—	6	2	no	—
135	C₃×C₂²⋊C₄	—	6	2	no	—
136	C₃×C₄²	—	2	1	no	—
137	C₃×C₄²	—	2	1	no	—
138	C₃×C₄²	—	2	1	no	—
139	C₃×C₄²	—	2	1	no	—
140	C₃×C₄²	—	6	2	no	—
141	C₃×C₄²	—	12	2	no	—
142	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
143	C₃×C₄²	C₄×D₃	1	1	yes	characteristic
144	C₃×C₄²	#67	1	1	yes	characteristic
145	C₃×C₄²	#67	1	1	yes	characteristic
146	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
147	C₃×C₄²	C₂×C₃×C₄	1	1	yes	characteristic
148	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
149	C₃×C₄²	C₃×D₄	1	1	yes	characteristic
150	#217	—	2	2	no	—
151	C₄⋊Dic₃	—	2	4	no	—
152	C₄⋊Dic₃	—	2	1	no	—
153	C₄⋊Dic₃	—	2	1	no	—
154	C₄⋊Dic₃	—	2	1	no	—
155	C₄⋊Dic₃	C₂×C₃×C₄	1	2	yes	—
156	Dic₃⋊C₄	—	2	2	no	—
157	Dic₃⋊C₄	—	2	2	no	—
158	C₄×Dic₃	—	2	2	no	—
159	C₄×Dic₃	—	4	2	no	—
160	C₂²×C₃²	—	2	1	no	—
161	C₂²×C₃²	—	2	1	no	—
162	C₂²×C₃²	C₂×C₂²⋊C₄	1	1	yes	characteristic
163	C₂²×C₃²	C₂×SD₁₆	1	1	yes	characteristic
164	C₂²×C₃²	C₂×Q₁₆	1	1	yes	characteristic
165	C₃²×C₄	—	2	1	no	—
166	C₃²×C₄	—	2	1	no	—
167	C₃²×C₄	—	2	1	no	—
168	C₃²×C₄	#103	1	2	yes	—
169	C₃²×C₄	#103	1	2	yes	—
170	C₃²×C₄	#104	1	2	yes	—
171	C₃×Dic₃	—	4	4	no	—
172	C₃×Dic₃	—	4	2	no	—
173	C₃×Dic₃	—	4	2	no	—
174	C₄⋊Q₈	—	3	1	no	—
175	C₄⋊Q₈	—	3	1	no	—
176	#125	—	6	1	no	—
177	C₂²⋊Q₈	—	3	2	no	—
178	C₂²⋊Q₈	—	6	1	no	—
179	C₄×Q₈	—	3	2	no	—
180	C₄×Q₈	—	6	1	no	—
181	#117	—	6	1	no	—
182	C₂×C₄⋊C₄	—	6	1	no	—
183	C₂×C₄²	—	6	1	no	—
184	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
185	C₄⋊C₈	—	2	1	no	—
186	C₂²⋊C₈	C₂×C₃×D₃	1	2	yes	—
187	C₄×C₈	—	2	1	no	—
188	#95	—	3	2	no	—
189	C₂³×C₃	—	2	1	no	—
190	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
191	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
192	C₃×Q₈	—	12	2	no	—
193	C₃×Q₈	—	12	2	no	—
194	C₂×C₃×C₄	—	2	2	no	—
195	C₂×C₃×C₄	—	2	2	no	—
196	C₂×C₃×C₄	—	2	2	no	—
197	C₂×C₃×C₄	—	2	2	no	—
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	2	1	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	2	1	no	—
203	C₂×C₃×C₄	—	2	1	no	—
204	C₂×C₃×C₄	—	4	2	no	—
205	C₂×C₃×C₄	—	4	1	no	—
206	C₂×C₃×C₄	—	4	1	no	—
207	C₂×C₃×C₄	—	4	2	no	—
208	C₂×C₃×C₄	—	6	4	no	—
209	C₂×C₃×C₄	—	6	2	no	—
210	C₂×C₃×C₄	—	6	2	no	—
211	C₂×C₃×C₄	—	12	2	no	—
212	C₂×C₃×C₄	—	12	2	no	—
213	C₂×C₃×C₄	#204	1	2	yes	—
214	C₂×C₃×C₄	Dic₁₂	1	2	yes	—
215	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
216	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
217	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
218	C₂×C₃×C₄	(C₂×D₃)⋊C₄	1	1	yes	characteristic
219	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
220	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
221	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
222	C₂×C₃×C₄	C₃×C₂²⋊C₄	1	1	yes	characteristic
223	C₂×C₃×C₄	C₃×SD₁₆	1	2	yes	—
224	C₂×C₃×C₄	C₃×Q₁₆	1	2	yes	—
225	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
228	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
229	C₂×Dic₃	—	2	4	no	—
230	C₂×Dic₃	—	2	2	no	—
231	C₂×Dic₃	—	2	2	no	—
232	C₂×Dic₃	—	4	2	no	—
233	C₂×Dic₃	—	4	2	no	—
234	Dic₆	—	4	2	no	—
235	Dic₆	—	4	2	no	—
236	C₃×C₈	—	4	4	no	—
237	C₃×C₈	—	4	2	no	—
238	C₃×C₈	—	4	2	no	—
239	C₂×C₃²	—	2	1	no	—
240	C₂×C₃²	—	2	1	no	—
241	C₂×C₃²	#410	1	1	yes	characteristic
242	C₂×C₃²	C₂×#103	1	1	yes	characteristic
243	C₂×C₃²	C₂×#104	1	1	yes	characteristic
244	C₂×Q₈	—	3	2	no	—
245	C₂×Q₈	—	6	1	no	—
246	C₂²×C₄	—	6	2	no	—
247	C₂²×C₄	C₃×D₁₂	1	1	yes	characteristic
248	C₂²×C₄	C₃×D₁₂	1	1	yes	characteristic
249	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
250	C₂×C₈	—	2	2	no	—
251	C₄⋊C₄	—	3	2	no	—
252	C₄⋊C₄	—	6	4	no	—
253	C₄⋊C₄	—	6	1	no	—
254	C₄⋊C₄	—	6	1	no	—
255	C₄⋊C₄	—	6	2	no	—
256	C₄⋊C₄	—	6	1	no	—
257	C₄⋊C₄	—	6	2	no	—
258	C₂²⋊C₄	—	6	2	no	—
259	C₄²	—	6	2	no	—
260	C₄²	—	12	2	no	—
261	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
262	C₄²	C₃×C₄×D₃	1	1	yes	characteristic
263	C₄²	C₃×#67	1	1	yes	characteristic
264	C₄²	C₃×#67	1	1	yes	characteristic
265	C₂²×C₃	—	2	1	no	—
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	—	2	1	no	—
269	C₂²×C₃	—	2	1	no	—
270	C₂²×C₃	—	2	1	no	—
271	C₂²×C₃	—	2	1	no	—
272	C₂²×C₃	—	4	1	no	—
273	C₂²×C₃	—	4	1	no	—
274	C₂²×C₃	C₂×#204	1	1	yes	characteristic
275	C₂²×C₃	C₂×Dic₁₂	1	1	yes	characteristic
276	C₂²×C₃	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
277	C₂²×C₃	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
278	C₂²×C₃	C₂×C₃×SD₁₆	1	1	yes	characteristic
279	C₂²×C₃	C₂×C₃×Q₁₆	1	1	yes	characteristic
280	C₃×C₄	—	2	2	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	—	2	1	no	—
286	C₃×C₄	—	2	1	no	—
287	C₃×C₄	—	2	2	no	—
288	C₃×C₄	—	2	2	no	—
289	C₃×C₄	—	2	2	no	—
290	C₃×C₄	—	4	1	no	—
291	C₃×C₄	—	4	1	no	—
292	C₃×C₄	—	12	4	no	—
293	C₃×C₄	—	12	2	no	—
294	C₃×C₄	—	12	2	no	—
295	C₃×C₄	#816	1	2	yes	—
296	C₃×C₄	#816	1	2	yes	—
297	C₃×C₄	#825	1	2	yes	—
298	C₃×C₄	C₃×#103	1	2	yes	—
299	C₃×C₄	C₃×#103	1	2	yes	—
300	C₃×C₄	C₃×#104	1	2	yes	—
301	Dic₃	—	4	4	no	—
302	Dic₃	—	4	2	no	—
303	Dic₃	—	4	2	no	—
304	C₃²	#1503	1	1	yes	characteristic, complemented, Sylow
305	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
306	Q₈	—	12	2	no	—
307	Q₈	—	12	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	6	4	no	—
311	C₂×C₄	—	6	2	no	—
312	C₂×C₄	—	6	2	no	—
313	C₂×C₄	—	12	2	no	—
314	C₂×C₄	—	12	2	no	—
315	C₂×C₄	C₃×#204	1	2	yes	—
316	C₂×C₄	C₃×Dic₁₂	1	2	yes	—
317	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
318	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
319	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
320	C₂×C₄	C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
321	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
322	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
323	C₈	—	4	2	no	—
324	C₂×C₃	—	2	1	no	—
325	C₂×C₃	—	2	1	no	—
326	C₂×C₃	—	2	1	no	—
327	C₂×C₃	—	2	1	no	—
328	C₂×C₃	—	2	1	no	—
329	C₂×C₃	—	2	1	no	—
330	C₂×C₃	—	2	1	no	—
331	C₂×C₃	—	4	1	no	—
332	C₂×C₃	—	4	1	no	—
333	C₂×C₃	#5006	1	1	yes	characteristic
334	C₂×C₃	C₂×#816	1	1	yes	characteristic
335	C₂×C₃	C₂×#825	1	1	yes	characteristic
336	C₂×C₃	C₃×#410	1	1	yes	characteristic
337	C₂×C₃	C₂×C₃×#103	1	1	yes	characteristic
338	C₂×C₃	C₂×C₃×#104	1	1	yes	characteristic
339	C₂²	—	2	1	no	—
340	C₂²	—	2	1	no	—
341	C₂²	C₂×C₃×#204	1	1	yes	characteristic
342	C₂²	C₂×C₃×Dic₁₂	1	1	yes	characteristic
343	C₂²	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	12	4	no	—
348	C₄	—	12	2	no	—
349	C₄	—	12	2	no	—
350	C₄	C₃×#816	1	2	yes	—
351	C₄	C₃×#816	1	2	yes	—
352	C₄	C₃×#825	1	2	yes	—
353	C₃	—	2	1	no	—
354	C₃	#70792	1	1	yes	characteristic, direct product factor
355	C₃	C₃×#1503	1	1	yes	characteristic, complemented
356	C₂	—	2	1	no	—
357	C₂	—	2	1	no	—
358	C₂	C₃×#5006	1	1	yes	characteristic
359	C₂	C₂×C₃×#816	1	1	yes	characteristic
360	C₂	C₂×C₃×#825	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	(C₂×D₃)⋊C₄
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×#431	C₂
socle	C₂²×C₃²	C₂×C₂²⋊C₄
prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₃×#431	D₃
hypocenter	C₃	C₃×#1503
LCS	G ▷ C₂×C₃×C₄ ▷ C₂²×C₃ ▷ C₃
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₂×C₃×C₄² ◁ C₃×#431

Complements

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

N	H ≅ G/N	type
6. #70792	C₃	direct (N × H)
14. C₃×#869	C₄	semidirect
15. C₃×#869	C₄	semidirect
17. C₃×C₄×Dic₆	C₄	semidirect
72. #869	C₃×C₄	semidirect
73. #869	C₃×C₄	semidirect
75. C₄×Dic₆	C₃×C₄	semidirect
304. C₃²	#1503	semidirect
354. C₃	#70792	direct (N × H)
355. C₃	C₃×#1503	semidirect

Maximal subgroups

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

subgroup	index
C₃×#5470	2
C₃²×#431	2
C₃×C₄×C₄⋊Dic₃	2
C₃×#4467	2
C₃×#1503	3
#70792	3

Subgroups — C3×#70792 #11775082

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70792 #11775082