Subgroups — C₃×#70575 #11791057

Order 1152.

673 conjugacy classes of subgroups.

The subgroup lattice has 335 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃×Dic₁₂	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃²×#106	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#815	C₂	1	1	yes	maximal, characteristic
4	C₃×#4973	C₂	1	2	yes	maximal
5	C₂×C₃×#810	C₂	1	2	yes	maximal
6	C₃×#1971	—	3	1	no	maximal
7	#70575	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×Dic₆	C₂²	1	2	yes	—
9	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
10	C₂×C₃²×C₄⋊C₄	C₂²	1	2	yes	—
11	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
12	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×Dic₁₂	—	2	2	no	—
14	C₂×C₃×Dic₁₂	—	2	2	no	—
15	C₂×C₃×Dic₁₂	C₄	1	4	yes	complemented
16	C₃²×#106	—	2	2	no	—
17	C₃×C₃⋊#95	—	2	4	no	—
18	C₃×#815	—	2	2	no	—
19	C₃×#810	—	2	4	no	—
20	C₂²×C₃×Q₁₆	—	3	1	no	—
21	C₂²×Dic₁₂	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×#106	—	2	1	no	—
23	C₂×C₃×#106	D₃	1	1	yes	characteristic
24	C₂×C₃×#106	C₂×C₃	1	1	yes	characteristic
25	C₂×C₃×#103	—	3	2	no	—
26	C₂×C₃×#97	—	3	1	no	—
27	C₃×#391	—	3	2	no	—
28	C₂×#815	C₂×C₃	1	1	yes	characteristic
29	#4973	C₂×C₃	1	2	yes	—
30	C₂×#810	C₂×C₃	1	2	yes	—
31	C₂²×C₃²×C₄	—	2	2	no	—
32	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
33	C₂²×C₃×Dic₃	—	2	2	no	—
34	C₂²×C₃×Dic₃	D₄	1	2	yes	—
35	C₂×C₃×Dic₆	—	2	2	no	—
36	C₂×C₃×Dic₆	—	2	2	no	—
37	C₂×C₃×Dic₆	—	2	2	no	—
38	C₂×C₃×Dic₆	—	2	2	no	—
39	C₂×C₃×Dic₆	—	2	2	no	—
40	C₂×C₃×Dic₆	C₂×C₄	1	4	yes	—
41	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
42	C₂×C₃²×C₈	D₄	1	2	yes	—
43	C₂×C₃²×C₈	D₄	1	2	yes	—
44	C₃²×C₄⋊C₄	—	2	2	no	—
45	C₃²×C₄⋊C₄	—	2	2	no	—
46	C₃×C₄×Dic₃	—	2	1	no	—
47	C₃×C₄×Dic₃	—	2	1	no	—
48	C₂×C₃×C₃⋊C₈	—	2	1	no	—
49	C₂×C₃×C₃⋊C₈	—	2	1	no	—
50	C₂×C₃×C₃⋊C₈	D₄	1	2	yes	—
51	C₃×Dic₁₂	—	2	8	no	—
52	#1971	—	3	1	no	Sylow
53	C₂²×C₃×Q₈	—	3	2	no	—
54	C₂²×Dic₆	C₂²×C₃	1	2	yes	—
55	C₂×C₃×Q₁₆	—	3	4	no	—
56	C₂×C₃×Q₁₆	—	6	2	no	—
57	C₂×C₃×Q₁₆	—	6	2	no	—
58	C₂²×C₃×C₈	—	2	1	no	—
59	C₂²×C₃×C₈	—	3	1	no	—
60	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
61	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
62	C₂×C₃×C₄⋊C₄	—	2	2	no	—
63	C₂×C₃×C₄⋊C₄	C₂×D₃	1	2	yes	—
64	C₂×C₃×C₄⋊C₄	C₂²×C₃	1	2	yes	—
65	C₂×C₃×C₄²	—	3	1	no	—
66	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
67	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
68	C₂×Dic₁₂	—	2	2	no	—
69	C₂×Dic₁₂	—	2	2	no	—
70	C₂×Dic₁₂	C₃×C₄	1	4	yes	complemented
71	C₃×#106	—	2	4	no	—
72	C₃×#106	—	2	2	no	—
73	C₃×#106	—	2	2	no	—
74	C₃×#103	—	6	4	no	—
75	C₃×#97	—	6	2	no	—
76	C₃×#95	—	6	4	no	—
77	C₃⋊#95	—	2	4	no	—
78	#815	—	2	2	no	—
79	#810	—	2	4	no	—
80	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
81	C₂×C₃²×C₄	—	4	2	no	—
82	C₂×C₃²×C₄	—	4	4	no	—
83	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
84	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
85	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
86	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
87	C₂×C₃²×C₄	#41	1	1	yes	characteristic
88	C₂×C₃²×C₄	#41	1	1	yes	characteristic
89	C₂×C₃×Dic₃	—	2	4	no	—
90	C₂×C₃×Dic₃	—	2	2	no	—
91	C₂×C₃×Dic₃	—	2	4	no	—
92	C₂×C₃×Dic₃	—	4	2	no	—
93	C₂×C₃×Dic₃	—	4	2	no	—
94	C₃×Dic₆	—	2	4	no	—
95	C₃×Dic₆	—	2	4	no	—
96	C₃×Dic₆	—	4	2	no	—
97	C₃×Dic₆	—	4	2	no	—
98	C₃²×C₈	C₂²⋊C₄	1	4	yes	—
99	C₃×C₃⋊C₈	—	2	2	no	—
100	C₂²×Q₁₆	—	3	1	no	—
101	C₂×#106	C₃×D₃	1	1	yes	characteristic
102	C₂×#103	—	3	2	no	—
103	C₂×#97	—	3	1	no	—
104	#391	—	3	2	no	—
105	C₂×C₃×Q₈	—	3	4	no	—
106	C₂×C₃×Q₈	—	6	2	no	—
107	C₂×C₃×Q₈	—	6	2	no	—
108	C₂×C₃×Q₈	—	6	2	no	—
109	C₂×C₃×Q₈	—	6	2	no	—
110	C₂×C₃×Q₈	—	6	2	no	—
111	C₂²×C₃×C₄	—	2	2	no	—
112	C₂²×C₃×C₄	—	2	2	no	—
113	C₂²×C₃×C₄	—	2	1	no	—
114	C₂²×C₃×C₄	—	3	2	no	—
115	C₂²×C₃×C₄	—	4	2	no	—
116	C₂²×C₃×C₄	—	6	2	no	—
117	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
118	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
119	C₂²×Dic₃	—	2	2	no	—
120	C₂²×Dic₃	C₃×D₄	1	2	yes	—
121	C₂×Dic₆	—	2	2	no	—
122	C₂×Dic₆	—	2	2	no	—
123	C₂×Dic₆	—	2	2	no	—
124	C₂×Dic₆	—	2	2	no	—
125	C₂×Dic₆	—	2	2	no	—
126	C₂×Dic₆	C₂×C₃×C₄	1	4	yes	—
127	C₃×Q₁₆	—	6	8	no	—
128	C₂×C₃×C₈	—	2	2	no	—
129	C₂×C₃×C₈	—	2	2	no	—
130	C₂×C₃×C₈	—	2	2	no	—
131	C₂×C₃×C₈	—	3	2	no	—
132	C₂×C₃×C₈	—	6	1	no	—
133	C₂×C₃×C₈	—	6	1	no	—
134	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
135	C₂×C₃×C₈	D₁₂	1	2	yes	—
136	C₂×C₃×C₈	#67	1	2	yes	—
137	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
138	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
139	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
140	C₃×C₄⋊C₄	—	2	4	no	—
141	C₃×C₄⋊C₄	—	2	2	no	—
142	C₃×C₄⋊C₄	—	2	2	no	—
143	C₃×C₄⋊C₄	—	2	2	no	—
144	C₃×C₄⋊C₄	—	2	2	no	—
145	C₃×C₄⋊C₄	—	4	2	no	—
146	C₃×C₄²	—	6	1	no	—
147	C₃×C₄²	—	6	1	no	—
148	C₄×Dic₃	—	2	1	no	—
149	C₄×Dic₃	—	2	1	no	—
150	C₂×C₃⋊C₈	—	2	1	no	—
151	C₂×C₃⋊C₈	—	2	1	no	—
152	C₂×C₃⋊C₈	C₃×D₄	1	2	yes	—
153	Dic₁₂	—	2	8	no	—
154	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
155	C₂²×C₃²	#121	1	1	yes	characteristic
156	C₂²×C₃²	#127	1	1	yes	characteristic
157	C₂²×C₃²	#137	1	4	yes	—
158	C₃²×C₄	—	8	2	no	—
159	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
161	C₃²×C₄	#121	1	1	yes	characteristic
162	C₃²×C₄	#124	1	1	yes	characteristic
163	C₃×Dic₃	—	4	4	no	—
164	C₃×Dic₃	—	4	2	no	—
165	C₂²×Q₈	—	3	2	no	—
166	C₂×Q₁₆	—	3	4	no	—
167	C₂×Q₁₆	—	6	2	no	—
168	C₂×Q₁₆	—	6	2	no	—
169	C₂²×C₈	—	3	1	no	—
170	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
171	C₂×C₄⋊C₄	C₂×C₃×D₃	1	2	yes	—
172	C₂×C₄²	—	3	1	no	—
173	#106	—	2	2	no	—
174	#103	—	6	4	no	—
175	#97	—	6	2	no	—
176	#95	—	6	4	no	—
177	C₂³×C₃	—	2	1	no	—
178	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
179	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
180	C₃×Q₈	—	6	4	no	—
181	C₃×Q₈	—	6	4	no	—
182	C₃×Q₈	—	12	2	no	—
183	C₃×Q₈	—	12	2	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	1	no	—
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	2	1	no	—
190	C₂×C₃×C₄	—	4	4	no	—
191	C₂×C₃×C₄	—	4	4	no	—
192	C₂×C₃×C₄	—	4	2	no	—
193	C₂×C₃×C₄	—	4	2	no	—
194	C₂×C₃×C₄	—	4	4	no	—
195	C₂×C₃×C₄	—	6	4	no	—
196	C₂×C₃×C₄	—	6	2	no	—
197	C₂×C₃×C₄	—	6	4	no	—
198	C₂×C₃×C₄	—	8	4	no	—
199	C₂×C₃×C₄	—	12	2	no	—
200	C₂×C₃×C₄	—	12	2	no	—
201	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
202	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
203	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
204	C₂×C₃×C₄	#239	1	1	yes	characteristic
205	C₂×C₃×C₄	#239	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
208	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
209	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
211	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
212	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
213	C₂×Dic₃	—	2	4	no	—
214	C₂×Dic₃	—	2	2	no	—
215	C₂×Dic₃	—	2	4	no	—
216	C₂×Dic₃	—	4	2	no	—
217	C₂×Dic₃	—	4	2	no	—
218	Dic₆	—	2	4	no	—
219	Dic₆	—	2	4	no	—
220	Dic₆	—	4	2	no	—
221	Dic₆	—	4	2	no	—
222	C₃×C₈	—	2	4	no	—
223	C₃×C₈	—	6	2	no	—
224	C₃×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
225	C₃×C₈	C₃×C₂²⋊C₄	1	4	yes	—
226	C₃⋊C₈	—	2	2	no	—
227	C₂×C₃²	#390	1	1	yes	characteristic
228	C₂×C₃²	#441	1	2	yes	—
229	C₂×C₃²	#470	1	2	yes	—
230	C₂×C₃²	#497	1	2	yes	—
231	C₂×Q₈	—	3	4	no	—
232	C₂×Q₈	—	6	2	no	—
233	C₂×Q₈	—	6	2	no	—
234	C₂×Q₈	—	6	2	no	—
235	C₂×Q₈	—	6	2	no	—
236	C₂×Q₈	—	6	2	no	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	3	2	no	—
239	C₂²×C₄	—	6	2	no	—
240	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
241	Q₁₆	—	6	8	no	—
242	C₂×C₈	—	3	2	no	—
243	C₂×C₈	—	6	1	no	—
244	C₂×C₈	—	6	1	no	—
245	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
246	C₂×C₈	C₃×D₁₂	1	2	yes	—
247	C₂×C₈	C₃×#67	1	2	yes	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄²	—	6	1	no	—
251	C₄²	—	6	1	no	—
252	C₂²×C₃	—	2	1	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	2	4	no	—
255	C₂²×C₃	—	2	1	no	—
256	C₂²×C₃	#893	1	1	yes	characteristic
257	C₂²×C₃	#895	1	1	yes	characteristic
258	C₂²×C₃	#915	1	4	yes	—
259	C₂²×C₃	#940	1	1	yes	characteristic
260	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
261	C₂²×C₃	C₃×#121	1	1	yes	characteristic
262	C₂²×C₃	C₃×#127	1	1	yes	characteristic
263	C₂²×C₃	C₃×#137	1	4	yes	—
264	C₃×C₄	—	2	1	no	—
265	C₃×C₄	—	2	1	no	—
266	C₃×C₄	—	2	1	no	—
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	8	4	no	—
269	C₃×C₄	—	8	2	no	—
270	C₃×C₄	—	8	2	no	—
271	C₃×C₄	—	12	2	no	—
272	C₃×C₄	—	12	4	no	—
273	C₃×C₄	#893	1	1	yes	characteristic
274	C₃×C₄	#895	1	1	yes	characteristic
275	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
276	C₃×C₄	#947	1	1	yes	characteristic
277	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
278	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
279	C₃×C₄	C₃×#121	1	1	yes	characteristic
280	C₃×C₄	C₃×#124	1	1	yes	characteristic
281	Dic₃	—	4	4	no	—
282	Dic₃	—	4	2	no	—
283	C₃²	#1971	1	1	yes	characteristic, complemented, Sylow
284	C₂³	C₃×D₃×D₄	1	1	yes	characteristic
285	Q₈	—	6	4	no	—
286	Q₈	—	6	4	no	—
287	Q₈	—	12	2	no	—
288	Q₈	—	12	2	no	—
289	C₂×C₄	—	4	4	no	—
290	C₂×C₄	—	4	2	no	—
291	C₂×C₄	—	6	4	no	—
292	C₂×C₄	—	6	2	no	—
293	C₂×C₄	—	6	4	no	—
294	C₂×C₄	—	12	2	no	—
295	C₂×C₄	—	12	2	no	—
296	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
297	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
298	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
299	C₂×C₄	C₃×#239	1	1	yes	characteristic
300	C₂×C₄	C₃×#239	1	1	yes	characteristic
301	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
302	C₈	—	6	2	no	—
303	C₈	C₃×(C₂×D₃)⋊C₄	1	4	yes	—
304	C₂×C₃	—	2	2	no	—
305	C₂×C₃	—	2	1	no	—
306	C₂×C₃	—	2	2	no	—
307	C₂×C₃	—	2	2	no	—
308	C₂×C₃	#4853	1	2	yes	—
309	C₂×C₃	#4867	1	2	yes	—
310	C₂×C₃	#4988	1	1	yes	characteristic
311	C₂×C₃	#5167	1	2	yes	—
312	C₂×C₃	C₃×#390	1	1	yes	characteristic
313	C₂×C₃	C₃×#441	1	2	yes	—
314	C₂×C₃	C₃×#470	1	2	yes	—
315	C₂×C₃	C₃×#497	1	2	yes	—
316	C₂²	C₃×#893	1	1	yes	characteristic
317	C₂²	C₃×#895	1	1	yes	characteristic
318	C₂²	C₃×#915	1	4	yes	—
319	C₂²	C₃×#940	1	1	yes	characteristic
320	C₄	—	8	2	no	—
321	C₄	—	12	4	no	—
322	C₄	—	12	2	no	—
323	C₄	C₃×#893	1	1	yes	characteristic
324	C₄	C₃×#895	1	1	yes	characteristic
325	C₄	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
326	C₄	C₃×#947	1	1	yes	characteristic
327	C₃	—	2	1	no	—
328	C₃	#70575	1	1	yes	characteristic, direct product factor
329	C₃	C₃×#1971	1	1	yes	characteristic, complemented
330	C₂	C₃×#4853	1	2	yes	—
331	C₂	C₃×#4867	1	2	yes	—
332	C₂	C₃×#4988	1	1	yes	characteristic
333	C₂	C₃×#5167	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×#106	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×#106	D₃
hypocenter	C₃	C₃×#1971
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₂×C₃×#106

Complements

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

N	H ≅ G/N	type
7. #70575	C₃	direct (N × H)
15. C₂×C₃×Dic₁₂	C₄	semidirect
70. C₂×Dic₁₂	C₃×C₄	semidirect
283. C₃²	#1971	semidirect
328. C₃	#70575	direct (N × H)
329. C₃	C₃×#1971	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃×Dic₁₂	2
C₂×C₃²×#106	2
C₂×C₃×#815	2
C₃×#4973	2
C₂×C₃×#810	2
C₃×#1971	3
#70575	3

Subgroups — C3×#70575 #11791057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#70575 #11791057