Subgroups — C₃×#69904 #11791365

Order 1152.

617 conjugacy classes of subgroups.

The subgroup lattice has 350 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	C₃²×#405	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#824	C₂	1	2	yes	maximal
4	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic
5	C₃×#5004	C₂	1	2	yes	maximal
6	C₃×#1979	—	3	1	no	maximal
7	#69904	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃×Dic₆	C₂²	1	2	yes	—
9	C₂×C₃²×M₁₆	C₂²	1	1	yes	characteristic
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₃²×#117	C₂²	1	1	yes	characteristic
12	C₂×C₃×#208	C₂²	1	2	yes	—
13	C₂×C₃×Dic₁₂	—	2	2	no	—
14	C₂×C₃×Dic₁₂	—	2	1	no	—
15	C₂×C₃×Dic₁₂	—	2	1	no	—
16	C₂×C₃×Dic₁₂	C₄	1	4	yes	complemented
17	C₃²×#97	—	2	1	no	—
18	C₃²×C₄×C₈	—	2	1	no	—
19	C₃×#824	—	2	4	no	—
20	C₃×#819	—	2	2	no	—
21	C₃×#810	—	2	4	no	—
22	C₂²×C₃×Q₁₆	—	3	1	no	—
23	C₂²×Dic₁₂	C₂×C₃	1	1	yes	characteristic
24	C₂×C₃×#108	—	3	1	no	—
25	C₃×#419	—	3	2	no	—
26	C₂×C₃×#101	—	3	2	no	—
27	C₃×#405	—	2	1	no	—
28	C₃×#405	D₃	1	1	yes	characteristic
29	C₃×#405	C₂×C₃	1	1	yes	characteristic
30	C₂×#824	C₂×C₃	1	2	yes	—
31	C₂×#819	C₂×C₃	1	1	yes	characteristic
32	#5004	C₂×C₃	1	2	yes	—
33	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
34	C₂²×C₃×Dic₃	—	2	2	no	—
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	C₃²×M₁₆	—	2	1	no	—
42	C₃²×M₁₆	D₄	1	2	yes	—
43	C₂×C₃²×C₈	—	2	1	no	—
44	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
45	C₂×C₃²×C₈	D₄	1	2	yes	—
46	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
47	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
48	C₃²×C₄⋊C₄	D₄	1	2	yes	—
49	C₃²×C₂²⋊C₄	—	2	1	no	—
50	C₃²×C₄²	—	2	1	no	—
51	C₃×#208	—	2	2	no	—
52	C₃×#208	—	2	2	no	—
53	C₂×C₃×C₃⋊C₈	—	2	2	no	—
54	C₃×Dic₁₂	—	2	4	no	—
55	C₃×Dic₁₂	—	2	4	no	—
56	#1979	—	3	1	no	Sylow
57	C₂²×C₃×Q₈	—	3	2	no	—
58	C₂²×Dic₆	C₂²×C₃	1	2	yes	—
59	C₂×C₃×Q₁₆	—	3	4	no	—
60	C₂×C₃×Q₁₆	—	6	2	no	—
61	C₂×C₃×Q₁₆	—	6	1	no	—
62	C₂×C₃×Q₁₆	—	6	1	no	—
63	C₂×C₃×M₁₆	—	2	1	no	—
64	C₂×C₃×M₁₆	—	3	2	no	—
65	C₂×C₃×M₁₆	C₂×D₃	1	1	yes	characteristic
66	C₂×C₃×M₁₆	C₂²×C₃	1	1	yes	characteristic
67	C₂²×C₃×C₈	—	2	1	no	—
68	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
69	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
70	C₃×#117	—	2	1	no	—
71	C₃×#117	C₂×D₃	1	1	yes	characteristic
72	C₃×#117	C₂²×C₃	1	1	yes	characteristic
73	C₂×#208	C₂²×C₃	1	2	yes	—
74	C₂×Dic₁₂	—	2	2	no	—
75	C₂×Dic₁₂	—	2	1	no	—
76	C₂×Dic₁₂	—	2	1	no	—
77	C₂×Dic₁₂	C₃×C₄	1	4	yes	complemented
78	C₃×#108	—	6	2	no	—
79	C₃×#103	—	6	4	no	—
80	C₃×#101	—	6	4	no	—
81	C₃×#97	—	2	2	no	—
82	C₃×#97	—	2	1	no	—
83	C₃×#97	—	2	1	no	—
84	C₃×C₄×C₈	—	2	2	no	—
85	C₃×C₄×C₈	—	2	1	no	—
86	C₃×C₄×C₈	—	2	1	no	—
87	#824	—	2	4	no	—
88	#819	—	2	2	no	—
89	#810	—	2	4	no	—
90	C₂³×C₃²	#41	1	1	yes	characteristic
91	C₂×C₃²×C₄	—	2	2	no	—
92	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
95	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
96	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
97	C₂×C₃²×C₄	#41	1	1	yes	characteristic
98	C₂×C₃×Dic₃	—	2	4	no	—
99	C₂×C₃×Dic₃	—	4	2	no	—
100	C₂×C₃×Dic₃	—	4	2	no	—
101	C₃×Dic₆	—	2	4	no	—
102	C₃×Dic₆	—	2	4	no	—
103	C₃×Dic₆	—	4	2	no	—
104	C₃×Dic₆	—	4	2	no	—
105	C₃²×C₈	—	2	2	no	—
106	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
107	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
108	C₃×C₃⋊C₈	—	4	2	no	—
109	C₂²×Q₁₆	—	3	1	no	—
110	C₂×#108	—	3	1	no	—
111	#419	—	3	2	no	—
112	C₂×#101	—	3	2	no	—
113	#405	C₃×D₃	1	1	yes	characteristic
114	C₂×C₃×Q₈	—	3	4	no	—
115	C₂×C₃×Q₈	—	6	2	no	—
116	C₂×C₃×Q₈	—	6	2	no	—
117	C₂×C₃×Q₈	—	6	2	no	—
118	C₂×C₃×Q₈	—	6	2	no	—
119	C₂×C₃×Q₈	—	6	2	no	—
120	C₂²×C₃×C₄	—	2	1	no	—
121	C₂²×C₃×C₄	—	6	2	no	—
122	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
123	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
124	C₂²×Dic₃	—	2	2	no	—
125	C₂×Dic₆	—	2	2	no	—
126	C₂×Dic₆	—	2	2	no	—
127	C₂×Dic₆	—	2	2	no	—
128	C₂×Dic₆	—	2	2	no	—
129	C₂×Dic₆	—	2	2	no	—
130	C₂×Dic₆	C₂×C₃×C₄	1	4	yes	—
131	C₃×Q₁₆	—	6	4	no	—
132	C₃×Q₁₆	—	6	4	no	—
133	C₃×M₁₆	—	2	2	no	—
134	C₃×M₁₆	—	2	1	no	—
135	C₃×M₁₆	—	2	1	no	—
136	C₃×M₁₆	—	2	2	no	—
137	C₃×M₁₆	—	6	2	no	—
138	C₃×M₁₆	—	6	2	no	—
139	C₃×M₁₆	D₁₂	1	2	yes	—
140	C₃×M₁₆	C₃×D₄	1	2	yes	—
141	C₂×C₃×C₈	—	2	2	no	—
142	C₂×C₃×C₈	—	2	1	no	—
143	C₂×C₃×C₈	—	2	1	no	—
144	C₂×C₃×C₈	—	2	2	no	—
145	C₂×C₃×C₈	—	2	2	no	—
146	C₂×C₃×C₈	—	2	1	no	—
147	C₂×C₃×C₈	—	2	1	no	—
148	C₂×C₃×C₈	—	6	2	no	—
149	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
150	C₂×C₃×C₈	D₁₂	1	2	yes	—
151	C₂×C₃×C₈	#67	1	1	yes	characteristic
152	C₂×C₃×C₈	#67	1	1	yes	characteristic
153	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
154	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
155	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
156	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
157	C₃×C₄⋊C₄	—	2	2	no	—
158	C₃×C₄⋊C₄	D₁₂	1	2	yes	—
159	C₃×C₄⋊C₄	C₃×D₄	1	2	yes	—
160	C₃×C₂²⋊C₄	—	2	2	no	—
161	C₃×C₂²⋊C₄	—	2	1	no	—
162	C₃×C₂²⋊C₄	—	2	1	no	—
163	C₃×C₄²	—	2	2	no	—
164	C₃×C₄²	—	2	1	no	—
165	C₃×C₄²	—	2	1	no	—
166	#208	—	2	2	no	—
167	#208	—	2	2	no	—
168	C₂×C₃⋊C₈	—	2	2	no	—
169	Dic₁₂	—	2	4	no	—
170	Dic₁₂	—	2	4	no	—
171	C₂²×C₃²	—	2	2	no	—
172	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
173	C₂²×C₃²	#121	1	1	yes	characteristic
174	C₂²×C₃²	#124	1	1	yes	characteristic
175	C₃²×C₄	—	4	2	no	—
176	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
177	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
178	C₃²×C₄	#121	1	1	yes	characteristic
179	C₃²×C₄	#127	1	1	yes	characteristic
180	C₃×Dic₃	—	4	4	no	—
181	C₂²×Q₈	—	3	2	no	—
182	C₂×Q₁₆	—	3	4	no	—
183	C₂×Q₁₆	—	6	2	no	—
184	C₂×Q₁₆	—	6	1	no	—
185	C₂×Q₁₆	—	6	1	no	—
186	C₂×M₁₆	—	3	2	no	—
187	C₂×M₁₆	C₂×C₃×D₃	1	1	yes	characteristic
188	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
189	#117	C₂×C₃×D₃	1	1	yes	characteristic
190	#108	—	6	2	no	—
191	#103	—	6	4	no	—
192	#101	—	6	4	no	—
193	#97	—	2	1	no	—
194	C₄×C₈	—	2	1	no	—
195	C₂³×C₃	—	2	1	no	—
196	C₂³×C₃	#235	1	1	yes	characteristic
197	C₂³×C₃	C₃×#41	1	1	yes	characteristic
198	C₃×Q₈	—	6	4	no	—
199	C₃×Q₈	—	6	4	no	—
200	C₃×Q₈	—	12	2	no	—
201	C₃×Q₈	—	12	2	no	—
202	C₂×C₃×C₄	—	2	4	no	—
203	C₂×C₃×C₄	—	2	2	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	—	2	1	no	—
211	C₂×C₃×C₄	—	6	4	no	—
212	C₂×C₃×C₄	—	12	2	no	—
213	C₂×C₃×C₄	—	12	2	no	—
214	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
216	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
217	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
218	C₂×C₃×C₄	#235	1	1	yes	characteristic
219	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
220	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
225	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
226	C₂×Dic₃	—	2	4	no	—
227	C₂×Dic₃	—	4	2	no	—
228	C₂×Dic₃	—	4	2	no	—
229	Dic₆	—	2	4	no	—
230	Dic₆	—	2	4	no	—
231	Dic₆	—	4	2	no	—
232	Dic₆	—	4	2	no	—
233	C₃×C₈	—	2	4	no	—
234	C₃×C₈	—	2	2	no	—
235	C₃×C₈	—	2	2	no	—
236	C₃×C₈	—	2	2	no	—
237	C₃×C₈	—	2	2	no	—
238	C₃×C₈	—	12	2	no	—
239	C₃×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
240	C₃×C₈	(C₂×D₃)⋊C₄	1	2	yes	—
241	C₃×C₈	C₃×C₂²⋊C₄	1	2	yes	—
242	C₃×C₈	C₃×C₂²⋊C₄	1	2	yes	—
243	C₃⋊C₈	—	4	2	no	—
244	C₂×C₃²	—	2	1	no	—
245	C₂×C₃²	—	2	1	no	—
246	C₂×C₃²	#390	1	1	yes	characteristic
247	C₂×C₃²	#473	1	2	yes	—
248	C₂×Q₈	—	3	4	no	—
249	C₂×Q₈	—	6	2	no	—
250	C₂×Q₈	—	6	2	no	—
251	C₂×Q₈	—	6	2	no	—
252	C₂×Q₈	—	6	2	no	—
253	C₂×Q₈	—	6	2	no	—
254	C₂²×C₄	—	6	2	no	—
255	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
256	Q₁₆	—	6	4	no	—
257	Q₁₆	—	6	4	no	—
258	M₁₆	—	2	1	no	—
259	M₁₆	—	6	2	no	—
260	M₁₆	—	6	2	no	—
261	M₁₆	C₃×D₁₂	1	2	yes	—
262	C₂×C₈	—	2	1	no	—
263	C₂×C₈	—	6	2	no	—
264	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
265	C₂×C₈	C₃×D₁₂	1	2	yes	—
266	C₂×C₈	C₃×#67	1	1	yes	characteristic
267	C₂×C₈	C₃×#67	1	1	yes	characteristic
268	C₄⋊C₄	C₃×D₁₂	1	2	yes	—
269	C₂²⋊C₄	—	2	1	no	—
270	C₄²	—	2	1	no	—
271	C₂²×C₃	—	2	2	no	—
272	C₂²×C₃	—	2	2	no	—
273	C₂²×C₃	—	2	1	no	—
274	C₂²×C₃	—	2	1	no	—
275	C₂²×C₃	—	2	1	no	—
276	C₂²×C₃	—	4	2	no	—
277	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
278	C₂²×C₃	#875	1	1	yes	characteristic
279	C₂²×C₃	#930	1	1	yes	characteristic
280	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
281	C₂²×C₃	C₃×#121	1	1	yes	characteristic
282	C₂²×C₃	C₃×#124	1	1	yes	characteristic
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	—	4	4	no	—
288	C₃×C₄	—	4	2	no	—
289	C₃×C₄	—	4	2	no	—
290	C₃×C₄	—	12	4	no	—
291	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
292	C₃×C₄	#874	1	1	yes	characteristic
293	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
294	C₃×C₄	#930	1	1	yes	characteristic
295	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
296	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
297	C₃×C₄	C₃×#121	1	1	yes	characteristic
298	C₃×C₄	C₃×#127	1	1	yes	characteristic
299	Dic₃	—	4	4	no	—
300	C₃²	#1979	1	1	yes	characteristic, complemented, Sylow
301	C₂³	C₃×#235	1	1	yes	characteristic
302	Q₈	—	6	4	no	—
303	Q₈	—	6	4	no	—
304	Q₈	—	12	2	no	—
305	Q₈	—	12	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	6	4	no	—
308	C₂×C₄	—	12	2	no	—
309	C₂×C₄	—	12	2	no	—
310	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
311	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
312	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
313	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
314	C₂×C₄	C₃×#235	1	1	yes	characteristic
315	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
316	C₈	—	2	2	no	—
317	C₈	—	12	2	no	—
318	C₈	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
319	C₈	C₃×(C₂×D₃)⋊C₄	1	2	yes	—
320	C₂×C₃	—	2	1	no	—
321	C₂×C₃	—	2	1	no	—
322	C₂×C₃	—	2	1	no	—
323	C₂×C₃	—	2	1	no	—
324	C₂×C₃	—	2	1	no	—
325	C₂×C₃	—	2	2	no	—
326	C₂×C₃	—	4	1	no	—
327	C₂×C₃	—	4	1	no	—
328	C₂×C₃	#4939	1	1	yes	characteristic
329	C₂×C₃	#5143	1	2	yes	—
330	C₂×C₃	C₃×#390	1	1	yes	characteristic
331	C₂×C₃	C₃×#473	1	2	yes	—
332	C₂²	—	2	2	no	—
333	C₂²	C₃×C₄×D₁₂	1	1	yes	characteristic
334	C₂²	C₃×#875	1	1	yes	characteristic
335	C₂²	C₃×#930	1	1	yes	characteristic
336	C₄	—	4	2	no	—
337	C₄	—	12	4	no	—
338	C₄	C₃×C₄×D₁₂	1	1	yes	characteristic
339	C₄	C₃×#874	1	1	yes	characteristic
340	C₄	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
341	C₄	C₃×#930	1	1	yes	characteristic
342	C₃	—	2	1	no	—
343	C₃	#69904	1	1	yes	characteristic, direct product factor
344	C₃	C₃×#1979	1	1	yes	characteristic, complemented
345	C₂	—	2	1	no	—
346	C₂	—	2	1	no	—
347	C₂	C₃×#4939	1	1	yes	characteristic
348	C₂	C₃×#5143	1	2	yes	—

	subgroup	G/H
center	C₂²×C₃	#930
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₃²×#405	C₂
socle	C₂²×C₃²	#121
prime radical	C₂³×C₃²	#41
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₃×#405	D₃
hypocenter	C₃	C₃×#1979
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₃×#405

Complements

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

N	H ≅ G/N	type
7. #69904	C₃	direct (N × H)
16. C₂×C₃×Dic₁₂	C₄	semidirect
77. C₂×Dic₁₂	C₃×C₄	semidirect
300. C₃²	#1979	semidirect
343. C₃	#69904	direct (N × H)
344. C₃	C₃×#1979	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
C₃²×#405	2
C₂×C₃×#824	2
C₂×C₃×#819	2
C₃×#5004	2
C₃×#1979	3
#69904	3

Subgroups — C3×#69904 #11791365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69904 #11791365