Subgroups — C₃×#72067 #11791066

Order 1152.

737 conjugacy classes of subgroups.

The subgroup lattice has 347 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₃×#5229	C₂	1	2	yes	maximal
3	C₂×C₃×#835	C₂	1	2	yes	maximal
4	C₂×C₃×#817	C₂	1	1	yes	maximal, characteristic
5	C₂×C₃×#815	C₂	1	1	yes	maximal, characteristic
6	C₃×#1971	—	3	1	no	maximal
7	#72067	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×Q₈	C₂²	1	2	yes	—
9	C₂×C₃²×Q₁₆	—	2	2	no	—
10	C₂×C₃²×Q₁₆	—	2	2	no	—
11	C₂×C₃²×Q₁₆	C₄	1	4	yes	complemented
12	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃×C₄⋊Dic₃	C₂²	1	2	yes	—
14	C₂×C₃×C₄×Dic₃	C₂²	1	1	yes	characteristic
15	C₂²×C₃×C₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₃×#835	—	2	4	no	—
17	C₃×C₃⋊#95	—	2	4	no	—
18	C₃×#817	—	2	2	no	—
19	C₃×#815	—	2	2	no	—
20	C₂²×C₃×Q₁₆	—	2	1	no	—
21	C₂²×C₃×Q₁₆	D₃	1	1	yes	characteristic
22	C₂²×C₃×Q₁₆	C₂×C₃	1	1	yes	characteristic
23	C₂×C₃×#106	—	3	1	no	—
24	C₂×C₃×#103	—	3	2	no	—
25	C₂×C₃×#97	—	3	1	no	—
26	C₃×#391	—	3	2	no	—
27	#5229	C₂×C₃	1	2	yes	—
28	C₂×#835	C₂×C₃	1	2	yes	—
29	C₂×#817	C₂×C₃	1	1	yes	characteristic
30	C₂×#815	C₂×C₃	1	1	yes	characteristic
31	C₂×C₃²×Q₈	—	2	2	no	—
32	C₂×C₃²×Q₈	—	2	2	no	—
33	C₂×C₃²×Q₈	—	2	2	no	—
34	C₂×C₃²×Q₈	—	2	2	no	—
35	C₂×C₃²×Q₈	—	2	2	no	—
36	C₂×C₃²×Q₈	C₂×C₄	1	4	yes	—
37	C₂²×C₃²×C₄	—	2	2	no	—
38	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
39	C₂²×C₃×Dic₃	—	2	2	no	—
40	C₂²×C₃×Dic₃	D₄	1	2	yes	—
41	C₃²×Q₁₆	—	2	8	no	—
42	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
43	C₂×C₃²×C₈	D₄	1	2	yes	—
44	C₂×C₃²×C₈	D₄	1	2	yes	—
45	C₃×C₄⋊Dic₃	—	2	2	no	—
46	C₃×C₄⋊Dic₃	—	2	2	no	—
47	C₃×C₄×Dic₃	—	2	1	no	—
48	C₃×C₄×Dic₃	—	2	1	no	—
49	C₂×C₃×C₃⋊C₈	—	2	1	no	—
50	C₂×C₃×C₃⋊C₈	—	2	1	no	—
51	C₂×C₃×C₃⋊C₈	D₄	1	2	yes	—
52	#1971	—	3	1	no	Sylow
53	C₂²×C₃×Q₈	—	2	2	no	—
54	C₂²×C₃×Q₈	C₂×D₃	1	2	yes	—
55	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
56	C₂×C₃×Q₁₆	—	2	4	no	—
57	C₂×C₃×Q₁₆	—	2	4	no	—
58	C₂×C₃×Q₁₆	—	2	2	no	—
59	C₂×C₃×Q₁₆	—	2	2	no	—
60	C₂×C₃×Q₁₆	—	2	2	no	—
61	C₂×C₃×Q₁₆	—	2	2	no	—
62	C₂×C₃×Q₁₆	—	2	4	no	—
63	C₂×C₃×Q₁₆	Dic₃	1	4	yes	complemented
64	C₂×C₃×Q₁₆	C₃×C₄	1	4	yes	complemented
65	C₂²×C₃×C₈	—	2	1	no	—
66	C₂²×C₃×C₈	—	3	1	no	—
67	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
68	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
69	C₂×C₃×C₄⋊C₄	—	3	2	no	—
70	C₂×C₃×C₄²	—	3	1	no	—
71	C₂×C₄⋊Dic₃	C₂²×C₃	1	2	yes	—
72	C₂×C₄×Dic₃	C₂²×C₃	1	1	yes	characteristic
73	C₂²×C₃⋊C₈	C₂²×C₃	1	1	yes	characteristic
74	C₃×#106	—	6	2	no	—
75	C₃×#103	—	6	4	no	—
76	C₃×#97	—	6	2	no	—
77	C₃×#95	—	6	4	no	—
78	#835	—	2	4	no	—
79	C₃⋊#95	—	2	4	no	—
80	#817	—	2	2	no	—
81	#815	—	2	2	no	—
82	C₂³×C₃²	C₂×D₄	1	1	yes	characteristic
83	C₃²×Q₈	—	2	4	no	—
84	C₃²×Q₈	—	2	4	no	—
85	C₃²×Q₈	—	4	2	no	—
86	C₃²×Q₈	—	4	2	no	—
87	C₂×C₃²×C₄	—	2	4	no	—
88	C₂×C₃²×C₄	—	4	2	no	—
89	C₂×C₃²×C₄	—	4	2	no	—
90	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
92	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
94	C₂×C₃²×C₄	#41	1	1	yes	characteristic
95	C₂×C₃²×C₄	#41	1	1	yes	characteristic
96	C₂×C₃×Dic₃	—	2	2	no	—
97	C₂×C₃×Dic₃	—	2	4	no	—
98	C₂×C₃×Dic₃	—	4	2	no	—
99	C₂×C₃×Dic₃	—	4	4	no	—
100	C₃²×C₈	C₂²⋊C₄	1	4	yes	—
101	C₃×C₃⋊C₈	—	2	2	no	—
102	C₂²×Q₁₆	C₃×D₃	1	1	yes	characteristic
103	C₂×#106	—	3	1	no	—
104	C₂×#103	—	3	2	no	—
105	C₂×#97	—	3	1	no	—
106	#391	—	3	2	no	—
107	C₂×C₃×Q₈	—	2	2	no	—
108	C₂×C₃×Q₈	—	2	2	no	—
109	C₂×C₃×Q₈	—	2	4	no	—
110	C₂×C₃×Q₈	—	2	4	no	—
111	C₂×C₃×Q₈	—	2	2	no	—
112	C₂×C₃×Q₈	—	2	2	no	—
113	C₂×C₃×Q₈	—	2	2	no	—
114	C₂×C₃×Q₈	—	2	2	no	—
115	C₂×C₃×Q₈	—	2	2	no	—
116	C₂×C₃×Q₈	—	2	2	no	—
117	C₂×C₃×Q₈	—	2	2	no	—
118	C₂×C₃×Q₈	—	2	2	no	—
119	C₂×C₃×Q₈	—	2	4	no	—
120	C₂×C₃×Q₈	—	4	2	no	—
121	C₂×C₃×Q₈	—	4	2	no	—
122	C₂×C₃×Q₈	—	4	2	no	—
123	C₂×C₃×Q₈	C₂×Dic₃	1	4	yes	—
124	C₂×C₃×Q₈	C₂×C₃×C₄	1	4	yes	—
125	C₂²×C₃×C₄	—	2	2	no	—
126	C₂²×C₃×C₄	—	2	2	no	—
127	C₂²×C₃×C₄	—	2	1	no	—
128	C₂²×C₃×C₄	—	3	2	no	—
129	C₂²×C₃×C₄	—	4	2	no	—
130	C₂²×C₃×C₄	—	6	2	no	—
131	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
132	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
133	C₂²×Dic₃	—	2	2	no	—
134	C₂²×Dic₃	C₃×D₄	1	2	yes	—
135	C₃×Q₁₆	—	2	16	no	—
136	C₃×Q₁₆	—	2	8	no	—
137	C₃×Q₁₆	—	2	8	no	—
138	C₂×C₃×C₈	—	2	2	no	—
139	C₂×C₃×C₈	—	2	2	no	—
140	C₂×C₃×C₈	—	2	2	no	—
141	C₂×C₃×C₈	—	3	2	no	—
142	C₂×C₃×C₈	—	6	1	no	—
143	C₂×C₃×C₈	—	6	1	no	—
144	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
145	C₂×C₃×C₈	#67	1	2	yes	—
146	C₂×C₃×C₈	#67	1	2	yes	—
147	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
148	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
149	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
150	C₃×C₄⋊C₄	—	6	2	no	—
151	C₃×C₄⋊C₄	—	6	2	no	—
152	C₃×C₄²	—	6	1	no	—
153	C₃×C₄²	—	6	1	no	—
154	C₄⋊Dic₃	—	2	2	no	—
155	C₄⋊Dic₃	—	2	2	no	—
156	C₄×Dic₃	—	2	1	no	—
157	C₄×Dic₃	—	2	1	no	—
158	C₂×C₃⋊C₈	—	2	1	no	—
159	C₂×C₃⋊C₈	—	2	1	no	—
160	C₂×C₃⋊C₈	C₃×D₄	1	2	yes	—
161	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
162	C₂²×C₃²	#121	1	1	yes	characteristic
163	C₂²×C₃²	#127	1	1	yes	characteristic
164	C₂²×C₃²	#137	1	4	yes	—
165	C₃²×C₄	—	4	4	no	—
166	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
167	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
168	C₃²×C₄	#121	1	1	yes	characteristic
169	C₃²×C₄	#124	1	1	yes	characteristic
170	C₃×Dic₃	—	4	2	no	—
171	C₃×Dic₃	—	8	2	no	—
172	C₂²×Q₈	C₂×C₃×D₃	1	2	yes	—
173	C₂×Q₁₆	—	2	2	no	—
174	C₂×Q₁₆	—	2	2	no	—
175	C₂×Q₁₆	C₃×Dic₃	1	4	yes	complemented
176	C₂²×C₈	—	3	1	no	—
177	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
178	C₂×C₄⋊C₄	—	3	2	no	—
179	C₂×C₄²	—	3	1	no	—
180	#106	—	6	2	no	—
181	#103	—	6	4	no	—
182	#97	—	6	2	no	—
183	#95	—	6	4	no	—
184	C₂³×C₃	—	2	1	no	—
185	C₂³×C₃	D₃×D₄	1	1	yes	characteristic
186	C₂³×C₃	C₂×C₃×D₄	1	1	yes	characteristic
187	C₃×Q₈	—	2	8	no	—
188	C₃×Q₈	—	2	4	no	—
189	C₃×Q₈	—	2	4	no	—
190	C₃×Q₈	—	2	4	no	—
191	C₃×Q₈	—	2	4	no	—
192	C₃×Q₈	—	4	4	no	—
193	C₃×Q₈	—	4	4	no	—
194	C₃×Q₈	—	4	4	no	—
195	C₃×Q₈	—	4	2	no	—
196	C₃×Q₈	—	4	2	no	—
197	C₃×Q₈	—	4	2	no	—
198	C₃×Q₈	—	4	2	no	—
199	C₂×C₃×C₄	—	2	4	no	—
200	C₂×C₃×C₄	—	2	4	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	—	2	1	no	—
205	C₂×C₃×C₄	—	2	1	no	—
206	C₂×C₃×C₄	—	2	1	no	—
207	C₂×C₃×C₄	—	4	4	no	—
208	C₂×C₃×C₄	—	4	4	no	—
209	C₂×C₃×C₄	—	4	4	no	—
210	C₂×C₃×C₄	—	4	2	no	—
211	C₂×C₃×C₄	—	4	2	no	—
212	C₂×C₃×C₄	—	4	2	no	—
213	C₂×C₃×C₄	—	4	2	no	—
214	C₂×C₃×C₄	—	6	4	no	—
215	C₂×C₃×C₄	—	6	2	no	—
216	C₂×C₃×C₄	—	12	4	no	—
217	C₂×C₃×C₄	—	12	2	no	—
218	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
219	C₂×C₃×C₄	#237	1	1	yes	characteristic
220	C₂×C₃×C₄	#237	1	1	yes	characteristic
221	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
222	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
223	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
224	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
225	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
226	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
227	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
228	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
229	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
230	C₂×Dic₃	—	2	2	no	—
231	C₂×Dic₃	—	2	4	no	—
232	C₂×Dic₃	—	4	2	no	—
233	C₂×Dic₃	—	4	4	no	—
234	C₃×C₈	—	2	4	no	—
235	C₃×C₈	—	6	2	no	—
236	C₃×C₈	#217	1	4	yes	—
237	C₃×C₈	C₃×C₂²⋊C₄	1	4	yes	—
238	C₃⋊C₈	—	2	2	no	—
239	C₂×C₃²	#390	1	1	yes	characteristic
240	C₂×C₃²	#441	1	2	yes	—
241	C₂×C₃²	#470	1	2	yes	—
242	C₂×C₃²	#497	1	2	yes	—
243	C₂×Q₈	—	2	2	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	2	no	—
247	C₂×Q₈	—	2	2	no	—
248	C₂×Q₈	C₂×C₃×Dic₃	1	4	yes	—
249	C₂²×C₄	—	2	2	no	—
250	C₂²×C₄	—	3	2	no	—
251	C₂²×C₄	—	6	2	no	—
252	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
253	Q₁₆	—	2	8	no	—
254	C₂×C₈	—	3	2	no	—
255	C₂×C₈	—	6	1	no	—
256	C₂×C₈	—	6	1	no	—
257	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
258	C₂×C₈	C₃×#67	1	2	yes	—
259	C₂×C₈	C₃×#67	1	2	yes	—
260	C₄⋊C₄	—	6	2	no	—
261	C₄⋊C₄	—	6	2	no	—
262	C₄²	—	6	1	no	—
263	C₄²	—	6	1	no	—
264	C₂²×C₃	—	2	4	no	—
265	C₂²×C₃	—	2	1	no	—
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	—	2	1	no	—
268	C₂²×C₃	#918	1	4	yes	—
269	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
270	C₂²×C₃	#938	1	1	yes	characteristic
271	C₂²×C₃	#940	1	1	yes	characteristic
272	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
273	C₂²×C₃	C₃×#121	1	1	yes	characteristic
274	C₂²×C₃	C₃×#127	1	1	yes	characteristic
275	C₂²×C₃	C₃×#137	1	4	yes	—
276	C₃×C₄	—	2	1	no	—
277	C₃×C₄	—	2	1	no	—
278	C₃×C₄	—	2	1	no	—
279	C₃×C₄	—	2	1	no	—
280	C₃×C₄	—	4	8	no	—
281	C₃×C₄	—	4	4	no	—
282	C₃×C₄	—	4	4	no	—
283	C₃×C₄	—	12	2	no	—
284	C₃×C₄	—	24	2	no	—
285	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
286	C₃×C₄	#936	1	1	yes	characteristic
287	C₃×C₄	#938	1	1	yes	characteristic
288	C₃×C₄	C₂×#217	1	1	yes	characteristic
289	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
290	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
291	C₃×C₄	C₃×#121	1	1	yes	characteristic
292	C₃×C₄	C₃×#124	1	1	yes	characteristic
293	Dic₃	—	4	2	no	—
294	Dic₃	—	8	2	no	—
295	C₃²	#1971	1	1	yes	characteristic, complemented, Sylow
296	C₂³	C₃×D₃×D₄	1	1	yes	characteristic
297	Q₈	—	2	4	no	—
298	Q₈	—	2	4	no	—
299	Q₈	—	4	2	no	—
300	Q₈	—	4	2	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	6	4	no	—
305	C₂×C₄	—	6	2	no	—
306	C₂×C₄	—	12	4	no	—
307	C₂×C₄	—	12	2	no	—
308	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
309	C₂×C₄	C₃×#237	1	1	yes	characteristic
310	C₂×C₄	C₃×#237	1	1	yes	characteristic
311	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
312	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
313	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
314	C₈	—	6	2	no	—
315	C₈	C₃×#217	1	4	yes	—
316	C₂×C₃	—	2	2	no	—
317	C₂×C₃	—	2	2	no	—
318	C₂×C₃	—	2	2	no	—
319	C₂×C₃	—	2	1	no	—
320	C₂×C₃	#5184	1	2	yes	—
321	C₂×C₃	#5189	1	2	yes	—
322	C₂×C₃	#5190	1	2	yes	—
323	C₂×C₃	#5221	1	1	yes	characteristic
324	C₂×C₃	C₃×#390	1	1	yes	characteristic
325	C₂×C₃	C₃×#441	1	2	yes	—
326	C₂×C₃	C₃×#470	1	2	yes	—
327	C₂×C₃	C₃×#497	1	2	yes	—
328	C₂²	C₃×#918	1	4	yes	—
329	C₂²	C₃×D₄×Dic₃	1	1	yes	characteristic
330	C₂²	C₃×#938	1	1	yes	characteristic
331	C₂²	C₃×#940	1	1	yes	characteristic
332	C₄	—	4	4	no	—
333	C₄	—	12	2	no	—
334	C₄	—	24	2	no	—
335	C₄	C₃×D₄×Dic₃	1	1	yes	characteristic
336	C₄	C₃×#936	1	1	yes	characteristic
337	C₄	C₃×#938	1	1	yes	characteristic
338	C₄	C₂×C₃×#217	1	1	yes	characteristic
339	C₃	—	2	1	no	—
340	C₃	#72067	1	1	yes	characteristic, direct product factor
341	C₃	C₃×#1971	1	1	yes	characteristic, complemented
342	C₂	C₃×#5184	1	2	yes	—
343	C₂	C₃×#5189	1	2	yes	—
344	C₂	C₃×#5190	1	2	yes	—
345	C₂	C₃×#5221	1	1	yes	characteristic

	subgroup	G/H
center	C₂³×C₃	D₃×D₄
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂²×C₃²×Q₁₆	C₂
socle, prime radical	C₂³×C₃²	C₂×D₄
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂²×C₃×Q₁₆	D₃
hypocenter	C₃	C₃×#1971
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂³×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×Q₁₆

Complements

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

N	H ≅ G/N	type
7. #72067	C₃	direct (N × H)
11. C₂×C₃²×Q₁₆	C₄	semidirect
63. C₂×C₃×Q₁₆	Dic₃	semidirect
64. C₂×C₃×Q₁₆	C₃×C₄	semidirect
175. C₂×Q₁₆	C₃×Dic₃	semidirect
295. C₃²	#1971	semidirect
340. C₃	#72067	direct (N × H)
341. 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×C₃²×Q₁₆	2
C₃×#5229	2
C₂×C₃×#835	2
C₂×C₃×#817	2
C₂×C₃×#815	2
C₃×#1971	3
#72067	3

Subgroups — C3×#72067 #11791066

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#72067 #11791066