Subgroups — C₃×#69824 #11791063

Order 1152.

681 conjugacy classes of subgroups.

The subgroup lattice has 340 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×#97	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃×#817	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃×#816	C₂	1	2	yes	maximal
5	C₃×#4929	C₂	1	2	yes	maximal
6	C₃×#1971	—	3	1	no	maximal
7	#69824	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	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₂×C₃²×C₄²	C₂²	1	1	yes	characteristic
12	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
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×#97	—	2	2	no	—
17	C₃×C₃⋊#95	—	2	4	no	—
18	C₃×#817	—	2	2	no	—
19	C₃×#816	—	2	4	no	—
20	C₂²×C₃×Q₁₆	—	3	1	no	—
21	C₂²×Dic₁₂	C₂×C₃	1	1	yes	characteristic
22	C₂×C₃×#106	—	3	1	no	—
23	C₂×C₃×#103	—	3	2	no	—
24	C₂×C₃×#97	—	2	1	no	—
25	C₂×C₃×#97	D₃	1	1	yes	characteristic
26	C₂×C₃×#97	C₂×C₃	1	1	yes	characteristic
27	C₃×#391	—	3	2	no	—
28	C₂×#817	C₂×C₃	1	1	yes	characteristic
29	C₂×#816	C₂×C₃	1	2	yes	—
30	#4929	C₂×C₃	1	2	yes	—
31	C₂²×C₃²×C₄	D₄	1	2	yes	—
32	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
33	C₂²×C₃×Dic₃	—	2	2	no	—
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	C₂×C₃²×C₈	—	2	1	no	—
42	C₂×C₃²×C₈	—	2	1	no	—
43	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
44	C₂×C₃²×C₈	D₄	1	2	yes	—
45	C₂×C₃²×C₈	D₄	1	2	yes	—
46	C₂×C₃²×C₈	D₄	1	2	yes	—
47	C₃²×C₄²	—	2	1	no	—
48	C₃²×C₄²	—	2	1	no	—
49	C₃×C₄⋊Dic₃	—	2	2	no	—
50	C₃×C₄⋊Dic₃	—	2	2	no	—
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	—	2	1	no	—
60	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
61	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
62	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
63	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
64	C₂×C₃×C₄⋊C₄	—	3	2	no	—
65	C₂×C₃×C₄²	—	2	1	no	—
66	C₂×C₃×C₄²	C₂×D₃	1	1	yes	characteristic
67	C₂×C₃×C₄²	C₂²×C₃	1	1	yes	characteristic
68	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
69	C₂×Dic₁₂	—	2	2	no	—
70	C₂×Dic₁₂	—	2	2	no	—
71	C₂×Dic₁₂	C₃×C₄	1	4	yes	complemented
72	C₃×#106	—	6	2	no	—
73	C₃×#103	—	6	4	no	—
74	C₃×#97	—	2	4	no	—
75	C₃×#97	—	2	2	no	—
76	C₃×#97	—	2	2	no	—
77	C₃×#95	—	6	4	no	—
78	C₃⋊#95	—	2	4	no	—
79	#817	—	2	2	no	—
80	#816	—	2	4	no	—
81	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
82	C₂×C₃²×C₄	—	2	2	no	—
83	C₂×C₃²×C₄	—	2	4	no	—
84	C₂×C₃²×C₄	C₂²×C₄	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	C₂×D₄	1	1	yes	characteristic
88	C₂×C₃²×C₄	#41	1	1	yes	characteristic
89	C₂×C₃²×C₄	#41	1	1	yes	characteristic
90	C₂×C₃×Dic₃	—	2	4	no	—
91	C₂×C₃×Dic₃	—	4	2	no	—
92	C₂×C₃×Dic₃	—	4	2	no	—
93	C₂×C₃×Dic₃	—	4	2	no	—
94	C₂×C₃×Dic₃	—	4	4	no	—
95	C₃×Dic₆	—	2	4	no	—
96	C₃×Dic₆	—	2	4	no	—
97	C₃×Dic₆	—	4	2	no	—
98	C₃×Dic₆	—	4	2	no	—
99	C₃²×C₈	—	2	2	no	—
100	C₃²×C₈	C₂²⋊C₄	1	4	yes	—
101	C₂²×Q₁₆	—	3	1	no	—
102	C₂×#106	—	3	1	no	—
103	C₂×#103	—	3	2	no	—
104	C₂×#97	C₃×D₃	1	1	yes	characteristic
105	#391	—	3	2	no	—
106	C₂×C₃×Q₈	—	3	4	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	C₂×C₃×Q₈	—	6	2	no	—
112	C₂²×C₃×C₄	—	2	2	no	—
113	C₂²×C₃×C₄	—	2	1	no	—
114	C₂²×C₃×C₄	—	6	2	no	—
115	C₂²×C₃×C₄	—	6	2	no	—
116	C₂²×C₃×C₄	D₁₂	1	2	yes	—
117	C₂²×C₃×C₄	C₃×D₄	1	2	yes	—
118	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
119	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
120	C₂²×Dic₃	—	2	2	no	—
121	C₂²×Dic₃	—	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	—	2	2	no	—
127	C₂×Dic₆	C₂×C₃×C₄	1	4	yes	—
128	C₃×Q₁₆	—	6	8	no	—
129	C₂×C₃×C₈	—	2	2	no	—
130	C₂×C₃×C₈	—	2	2	no	—
131	C₂×C₃×C₈	—	2	1	no	—
132	C₂×C₃×C₈	—	2	1	no	—
133	C₂×C₃×C₈	—	2	1	no	—
134	C₂×C₃×C₈	—	2	1	no	—
135	C₂×C₃×C₈	—	2	2	no	—
136	C₂×C₃×C₈	—	2	2	no	—
137	C₂×C₃×C₈	—	2	2	no	—
138	C₂×C₃×C₈	—	2	2	no	—
139	C₂×C₃×C₈	C₄×D₃	1	2	yes	—
140	C₂×C₃×C₈	D₁₂	1	2	yes	—
141	C₂×C₃×C₈	D₁₂	1	2	yes	—
142	C₂×C₃×C₈	#67	1	2	yes	—
143	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
144	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
145	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
146	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
147	C₃×C₄⋊C₄	—	6	2	no	—
148	C₃×C₄⋊C₄	—	6	2	no	—
149	C₃×C₄²	—	2	2	no	—
150	C₃×C₄²	—	2	2	no	—
151	C₃×C₄²	—	2	1	no	—
152	C₃×C₄²	—	2	1	no	—
153	C₃×C₄²	—	2	1	no	—
154	C₃×C₄²	—	2	1	no	—
155	C₄⋊Dic₃	—	2	2	no	—
156	C₄⋊Dic₃	—	2	2	no	—
157	Dic₁₂	—	2	8	no	—
158	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
159	C₂²×C₃²	#121	1	1	yes	characteristic
160	C₂²×C₃²	#127	1	1	yes	characteristic
161	C₂²×C₃²	#137	1	4	yes	—
162	C₃²×C₄	—	4	2	no	—
163	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
164	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
165	C₃²×C₄	#121	1	1	yes	characteristic
166	C₃²×C₄	#124	1	1	yes	characteristic
167	C₃×Dic₃	—	4	4	no	—
168	C₃×Dic₃	—	8	2	no	—
169	C₂²×Q₈	—	3	2	no	—
170	C₂×Q₁₆	—	3	4	no	—
171	C₂×Q₁₆	—	6	2	no	—
172	C₂×Q₁₆	—	6	2	no	—
173	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
174	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
175	C₂×C₄⋊C₄	—	3	2	no	—
176	C₂×C₄²	C₂×C₃×D₃	1	1	yes	characteristic
177	#106	—	6	2	no	—
178	#103	—	6	4	no	—
179	#97	—	2	2	no	—
180	#95	—	6	4	no	—
181	C₂³×C₃	—	2	1	no	—
182	C₂³×C₃	C₂×D₁₂	1	1	yes	characteristic
183	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
184	C₃×Q₈	—	6	4	no	—
185	C₃×Q₈	—	6	4	no	—
186	C₃×Q₈	—	12	2	no	—
187	C₃×Q₈	—	12	2	no	—
188	C₂×C₃×C₄	—	2	4	no	—
189	C₂×C₃×C₄	—	2	4	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	2	no	—
192	C₂×C₃×C₄	—	2	4	no	—
193	C₂×C₃×C₄	—	2	1	no	—
194	C₂×C₃×C₄	—	2	1	no	—
195	C₂×C₃×C₄	—	2	1	no	—
196	C₂×C₃×C₄	—	2	1	no	—
197	C₂×C₃×C₄	—	2	1	no	—
198	C₂×C₃×C₄	—	2	1	no	—
199	C₂×C₃×C₄	—	4	4	no	—
200	C₂×C₃×C₄	—	6	4	no	—
201	C₂×C₃×C₄	—	12	2	no	—
202	C₂×C₃×C₄	—	12	2	no	—
203	C₂×C₃×C₄	—	12	2	no	—
204	C₂×C₃×C₄	—	12	4	no	—
205	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
206	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
207	C₂×C₃×C₄	C₂×D₁₂	1	1	yes	characteristic
208	C₂×C₃×C₄	#235	1	1	yes	characteristic
209	C₂×C₃×C₄	#235	1	1	yes	characteristic
210	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
212	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
213	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
215	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
216	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
217	C₂×Dic₃	—	2	4	no	—
218	C₂×Dic₃	—	4	2	no	—
219	C₂×Dic₃	—	4	2	no	—
220	C₂×Dic₃	—	4	2	no	—
221	C₂×Dic₃	—	4	4	no	—
222	Dic₆	—	2	4	no	—
223	Dic₆	—	2	4	no	—
224	Dic₆	—	4	2	no	—
225	Dic₆	—	4	2	no	—
226	C₃×C₈	—	2	4	no	—
227	C₃×C₈	—	2	2	no	—
228	C₃×C₈	—	2	2	no	—
229	C₃×C₈	—	2	4	no	—
230	C₃×C₈	(C₂×D₃)⋊C₄	1	4	yes	—
231	C₃×C₈	C₃×C₂²⋊C₄	1	4	yes	—
232	C₂×C₃²	#390	1	1	yes	characteristic
233	C₂×C₃²	#441	1	2	yes	—
234	C₂×C₃²	#470	1	2	yes	—
235	C₂×C₃²	#497	1	2	yes	—
236	C₂×Q₈	—	3	4	no	—
237	C₂×Q₈	—	6	2	no	—
238	C₂×Q₈	—	6	2	no	—
239	C₂×Q₈	—	6	2	no	—
240	C₂×Q₈	—	6	2	no	—
241	C₂×Q₈	—	6	2	no	—
242	C₂²×C₄	—	6	2	no	—
243	C₂²×C₄	—	6	2	no	—
244	C₂²×C₄	C₃×D₁₂	1	2	yes	—
245	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
246	Q₁₆	—	6	8	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	C₃×C₄×D₃	1	2	yes	—
250	C₂×C₈	C₃×D₁₂	1	2	yes	—
251	C₂×C₈	C₃×D₁₂	1	2	yes	—
252	C₂×C₈	C₃×#67	1	2	yes	—
253	C₄⋊C₄	—	6	2	no	—
254	C₄⋊C₄	—	6	2	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	1	no	—
257	C₂²×C₃	—	2	4	no	—
258	C₂²×C₃	—	2	1	no	—
259	C₂²×C₃	—	2	1	no	—
260	C₂²×C₃	—	2	1	no	—
261	C₂²×C₃	C₄×D₁₂	1	1	yes	characteristic
262	C₂²×C₃	#874	1	1	yes	characteristic
263	C₂²×C₃	#909	1	4	yes	—
264	C₂²×C₃	#930	1	1	yes	characteristic
265	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
266	C₂²×C₃	C₃×#121	1	1	yes	characteristic
267	C₂²×C₃	C₃×#127	1	1	yes	characteristic
268	C₂²×C₃	C₃×#137	1	4	yes	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	2	1	no	—
273	C₃×C₄	—	4	4	no	—
274	C₃×C₄	—	4	2	no	—
275	C₃×C₄	—	4	2	no	—
276	C₃×C₄	—	12	4	no	—
277	C₃×C₄	—	24	2	no	—
278	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
279	C₃×C₄	#875	1	1	yes	characteristic
280	C₃×C₄	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
281	C₃×C₄	#930	1	1	yes	characteristic
282	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
283	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
284	C₃×C₄	C₃×#121	1	1	yes	characteristic
285	C₃×C₄	C₃×#124	1	1	yes	characteristic
286	Dic₃	—	4	4	no	—
287	Dic₃	—	8	2	no	—
288	C₃²	#1971	1	1	yes	characteristic, complemented, Sylow
289	C₂³	C₂×C₃×D₁₂	1	1	yes	characteristic
290	Q₈	—	6	4	no	—
291	Q₈	—	6	4	no	—
292	Q₈	—	12	2	no	—
293	Q₈	—	12	2	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	6	4	no	—
297	C₂×C₄	—	12	2	no	—
298	C₂×C₄	—	12	2	no	—
299	C₂×C₄	—	12	4	no	—
300	C₂×C₄	—	12	2	no	—
301	C₂×C₄	C₂×C₃×C₄×D₃	1	1	yes	characteristic
302	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
303	C₂×C₄	C₂×C₃×D₁₂	1	1	yes	characteristic
304	C₂×C₄	C₃×#235	1	1	yes	characteristic
305	C₂×C₄	C₃×#235	1	1	yes	characteristic
306	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
307	C₈	—	2	2	no	—
308	C₈	C₃×(C₂×D₃)⋊C₄	1	4	yes	—
309	C₂×C₃	—	2	1	no	—
310	C₂×C₃	—	2	2	no	—
311	C₂×C₃	—	2	2	no	—
312	C₂×C₃	—	2	2	no	—
313	C₂×C₃	#4715	1	2	yes	—
314	C₂×C₃	#4716	1	2	yes	—
315	C₂×C₃	#4939	1	1	yes	characteristic
316	C₂×C₃	#5137	1	2	yes	—
317	C₂×C₃	C₃×#390	1	1	yes	characteristic
318	C₂×C₃	C₃×#441	1	2	yes	—
319	C₂×C₃	C₃×#470	1	2	yes	—
320	C₂×C₃	C₃×#497	1	2	yes	—
321	C₂²	C₃×C₄×D₁₂	1	1	yes	characteristic
322	C₂²	C₃×#874	1	1	yes	characteristic
323	C₂²	C₃×#909	1	4	yes	—
324	C₂²	C₃×#930	1	1	yes	characteristic
325	C₄	—	4	2	no	—
326	C₄	—	12	4	no	—
327	C₄	—	24	2	no	—
328	C₄	C₃×C₄×D₁₂	1	1	yes	characteristic
329	C₄	C₃×#875	1	1	yes	characteristic
330	C₄	C₂×C₃×(C₂×D₃)⋊C₄	1	1	yes	characteristic
331	C₄	C₃×#930	1	1	yes	characteristic
332	C₃	—	2	1	no	—
333	C₃	#69824	1	1	yes	characteristic, direct product factor
334	C₃	C₃×#1971	1	1	yes	characteristic, complemented
335	C₂	C₃×#4715	1	2	yes	—
336	C₂	C₃×#4716	1	2	yes	—
337	C₂	C₃×#4939	1	1	yes	characteristic
338	C₂	C₃×#5137	1	2	yes	—

	subgroup	G/H
center	C₂³×C₃	C₂×D₁₂
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂×C₃²×#97	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂×C₃×#97	D₃
hypocenter	C₃	C₃×#1971
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂×C₃×C₄² ◁ C₂×C₃×#97

Complements

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

N	H ≅ G/N	type
7. #69824	C₃	direct (N × H)
15. C₂×C₃×Dic₁₂	C₄	semidirect
71. C₂×Dic₁₂	C₃×C₄	semidirect
288. C₃²	#1971	semidirect
333. C₃	#69824	direct (N × H)
334. 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×#97	2
C₂×C₃×#817	2
C₂×C₃×#816	2
C₃×#4929	2
C₃×#1971	3
#69824	3

Subgroups — C3×#69824 #11791063

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#69824 #11791063