Subgroups — C₃×#72069 #11791368

Order 1152.

681 conjugacy classes of subgroups.

The subgroup lattice has 359 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	C₂×C₃×#836	C₂	1	2	yes	maximal
3	C₃×#5225	C₂	1	2	yes	maximal
4	C₂×C₃×#819	C₂	1	1	yes	maximal, characteristic
5	C₃×#5101	C₂	1	1	yes	maximal, characteristic
6	C₃×#1979	—	3	1	no	maximal
7	#72069	C₃	1	1	yes	maximal, characteristic, direct product factor
8	C₂²×C₃²×Q₈	C₂²	1	2	yes	—
9	C₂×C₃²×Q₁₆	—	2	1	no	—
10	C₂×C₃²×Q₁₆	—	2	1	no	—
11	C₂×C₃²×Q₁₆	—	2	2	no	—
12	C₂×C₃²×Q₁₆	C₄	1	4	yes	complemented
13	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
14	C₃×#926	C₂²	1	1	yes	characteristic
15	C₂×C₃×#208	C₂²	1	2	yes	—
16	C₂×C₃×#208	C₂²	1	1	yes	characteristic
17	C₃×#836	—	2	4	no	—
18	C₃×#835	—	2	4	no	—
19	C₃×#819	—	2	2	no	—
20	C₃×#815	—	2	1	no	—
21	C₃×C₈×Dic₃	—	2	1	no	—
22	C₂²×C₃×Q₁₆	—	2	1	no	—
23	C₂²×C₃×Q₁₆	D₃	1	1	yes	characteristic
24	C₂²×C₃×Q₁₆	C₂×C₃	1	1	yes	characteristic
25	C₂×C₃×#108	—	3	1	no	—
26	C₃×#419	—	3	2	no	—
27	C₂×C₃×#101	—	3	2	no	—
28	C₃×#405	—	3	1	no	—
29	C₂×#836	C₂×C₃	1	2	yes	—
30	#5225	C₂×C₃	1	2	yes	—
31	C₂×#819	C₂×C₃	1	1	yes	characteristic
32	#5101	C₂×C₃	1	1	yes	characteristic
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	—	2	2	no	—
37	C₂×C₃²×Q₈	—	2	2	no	—
38	C₂×C₃²×Q₈	C₂×C₄	1	4	yes	—
39	C₂²×C₃²×C₄	—	2	2	no	—
40	C₂²×C₃²×C₄	C₂³	1	1	yes	characteristic
41	C₃²×Q₁₆	—	2	4	no	—
42	C₃²×Q₁₆	—	2	4	no	—
43	C₂×C₃²×C₈	C₂×C₄	1	2	yes	—
44	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
45	C₂×C₃²×C₈	D₄	1	1	yes	characteristic
46	C₂×C₃²×C₈	D₄	1	2	yes	—
47	C₃×#217	—	2	1	no	—
48	C₃×C₄⋊Dic₃	D₄	1	2	yes	—
49	C₃×C₄×Dic₃	—	2	1	no	—
50	C₃×#208	—	2	1	no	—
51	C₃×#208	—	2	2	no	—
52	C₃×#208	—	2	2	no	—
53	C₃×#208	D₄	1	2	yes	—
54	C₂×C₃×C₃⋊C₈	—	2	1	no	—
55	C₂×C₃×C₃⋊C₈	—	2	2	no	—
56	#1979	—	3	1	no	Sylow
57	C₂²×C₃×Q₈	—	2	2	no	—
58	C₂²×C₃×Q₈	C₂×D₃	1	2	yes	—
59	C₂²×C₃×Q₈	C₂²×C₃	1	2	yes	—
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	—	2	1	no	—
64	C₂×C₃×Q₁₆	—	2	1	no	—
65	C₂×C₃×Q₁₆	—	2	2	no	—
66	C₂×C₃×Q₁₆	—	2	1	no	—
67	C₂×C₃×Q₁₆	—	2	1	no	—
68	C₂×C₃×Q₁₆	—	2	2	no	—
69	C₂×C₃×Q₁₆	—	2	4	no	—
70	C₂×C₃×Q₁₆	Dic₃	1	4	yes	complemented
71	C₂×C₃×Q₁₆	C₃×C₄	1	4	yes	complemented
72	C₂×C₃×M₁₆	—	3	1	no	—
73	C₂×C₃×M₁₆	—	3	2	no	—
74	C₂²×C₃×C₈	—	2	1	no	—
75	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
76	C₂²×C₃×C₈	C₂²×C₃	1	1	yes	characteristic
77	C₃×#117	—	3	1	no	—
78	#926	C₂²×C₃	1	1	yes	characteristic
79	C₂×#208	C₂²×C₃	1	2	yes	—
80	C₂×#208	C₂²×C₃	1	1	yes	characteristic
81	C₃×#108	—	6	2	no	—
82	C₃×#103	—	6	4	no	—
83	C₃×#101	—	6	4	no	—
84	C₃×#97	—	6	1	no	—
85	C₃×C₄×C₈	—	6	1	no	—
86	#836	—	2	4	no	—
87	#835	—	2	4	no	—
88	#819	—	2	2	no	—
89	#815	—	2	1	no	—
90	C₈×Dic₃	—	2	1	no	—
91	C₂³×C₃²	#41	1	1	yes	characteristic
92	C₃²×Q₈	—	2	4	no	—
93	C₃²×Q₈	—	2	4	no	—
94	C₃²×Q₈	—	4	2	no	—
95	C₃²×Q₈	—	4	2	no	—
96	C₂×C₃²×C₄	—	2	4	no	—
97	C₂×C₃²×C₄	—	4	2	no	—
98	C₂×C₃²×C₄	—	4	2	no	—
99	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
101	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
102	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
103	C₂×C₃²×C₄	C₂×D₄	1	1	yes	characteristic
104	C₂×C₃²×C₄	#41	1	1	yes	characteristic
105	C₂×C₃×Dic₃	—	2	2	no	—
106	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
107	C₃²×C₈	C₂²⋊C₄	1	2	yes	—
108	C₃×C₃⋊C₈	—	2	2	no	—
109	C₃×C₃⋊C₈	—	4	2	no	—
110	C₂²×Q₁₆	C₃×D₃	1	1	yes	characteristic
111	C₂×#108	—	3	1	no	—
112	#419	—	3	2	no	—
113	C₂×#101	—	3	2	no	—
114	#405	—	3	1	no	—
115	C₂×C₃×Q₈	—	2	2	no	—
116	C₂×C₃×Q₈	—	2	2	no	—
117	C₂×C₃×Q₈	—	2	4	no	—
118	C₂×C₃×Q₈	—	2	4	no	—
119	C₂×C₃×Q₈	—	2	2	no	—
120	C₂×C₃×Q₈	—	2	2	no	—
121	C₂×C₃×Q₈	—	2	2	no	—
122	C₂×C₃×Q₈	—	2	2	no	—
123	C₂×C₃×Q₈	—	2	2	no	—
124	C₂×C₃×Q₈	—	2	2	no	—
125	C₂×C₃×Q₈	—	2	2	no	—
126	C₂×C₃×Q₈	—	2	2	no	—
127	C₂×C₃×Q₈	—	2	4	no	—
128	C₂×C₃×Q₈	—	4	2	no	—
129	C₂×C₃×Q₈	—	4	2	no	—
130	C₂×C₃×Q₈	—	4	2	no	—
131	C₂×C₃×Q₈	C₂×Dic₃	1	4	yes	—
132	C₂×C₃×Q₈	C₂×C₃×C₄	1	4	yes	—
133	C₂²×C₃×C₄	—	2	2	no	—
134	C₂²×C₃×C₄	—	2	2	no	—
135	C₂²×C₃×C₄	—	2	1	no	—
136	C₂²×C₃×C₄	—	4	2	no	—
137	C₂²×C₃×C₄	C₂²×D₃	1	1	yes	characteristic
138	C₂²×C₃×C₄	C₂³×C₃	1	1	yes	characteristic
139	C₃×Q₁₆	—	2	8	no	—
140	C₃×Q₁₆	—	2	8	no	—
141	C₃×Q₁₆	—	2	4	no	—
142	C₃×Q₁₆	—	2	4	no	—
143	C₃×Q₁₆	—	2	4	no	—
144	C₃×Q₁₆	—	2	4	no	—
145	C₃×M₁₆	—	3	2	no	—
146	C₃×M₁₆	—	6	1	no	—
147	C₃×M₁₆	—	6	2	no	—
148	C₃×M₁₆	—	6	2	no	—
149	C₂×C₃×C₈	—	2	2	no	—
150	C₂×C₃×C₈	—	2	1	no	—
151	C₂×C₃×C₈	—	2	1	no	—
152	C₂×C₃×C₈	—	2	2	no	—
153	C₂×C₃×C₈	—	6	1	no	—
154	C₂×C₃×C₈	—	6	2	no	—
155	C₂×C₃×C₈	C₂×Dic₃	1	2	yes	—
156	C₂×C₃×C₈	#67	1	1	yes	characteristic
157	C₂×C₃×C₈	#67	1	1	yes	characteristic
158	C₂×C₃×C₈	#67	1	2	yes	—
159	C₂×C₃×C₈	C₂×C₃×C₄	1	2	yes	—
160	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
161	C₂×C₃×C₈	C₃×D₄	1	1	yes	characteristic
162	C₂×C₃×C₈	C₃×D₄	1	2	yes	—
163	C₃×C₄⋊C₄	—	3	2	no	—
164	C₃×C₂²⋊C₄	—	6	1	no	—
165	C₃×C₄²	—	6	1	no	—
166	#217	—	2	1	no	—
167	C₄⋊Dic₃	C₃×D₄	1	2	yes	—
168	C₄×Dic₃	—	2	1	no	—
169	#208	—	2	1	no	—
170	#208	—	2	2	no	—
171	#208	—	2	2	no	—
172	#208	C₃×D₄	1	2	yes	—
173	C₂×C₃⋊C₈	—	2	1	no	—
174	C₂×C₃⋊C₈	—	2	2	no	—
175	C₂²×C₃²	—	2	2	no	—
176	C₂²×C₃²	C₄×D₄	1	1	yes	characteristic
177	C₂²×C₃²	#121	1	1	yes	characteristic
178	C₂²×C₃²	#124	1	1	yes	characteristic
179	C₃²×C₄	—	4	4	no	—
180	C₃²×C₄	C₂×C₂²⋊C₄	1	1	yes	characteristic
181	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
182	C₃²×C₄	#121	1	1	yes	characteristic
183	C₃²×C₄	#127	1	1	yes	characteristic
184	C₃×Dic₃	—	4	2	no	—
185	C₂²×Q₈	C₂×C₃×D₃	1	2	yes	—
186	C₂×Q₁₆	—	2	1	no	—
187	C₂×Q₁₆	—	2	1	no	—
188	C₂×Q₁₆	—	2	2	no	—
189	C₂×Q₁₆	C₃×Dic₃	1	4	yes	complemented
190	C₂×M₁₆	—	3	1	no	—
191	C₂×M₁₆	—	3	2	no	—
192	C₂²×C₈	C₂×C₃×D₃	1	1	yes	characteristic
193	#117	—	3	1	no	—
194	#108	—	6	2	no	—
195	#103	—	6	4	no	—
196	#101	—	6	4	no	—
197	#97	—	6	1	no	—
198	C₄×C₈	—	6	1	no	—
199	C₂³×C₃	—	2	1	no	—
200	C₂³×C₃	#237	1	1	yes	characteristic
201	C₂³×C₃	C₃×#41	1	1	yes	characteristic
202	C₃×Q₈	—	2	8	no	—
203	C₃×Q₈	—	2	4	no	—
204	C₃×Q₈	—	2	4	no	—
205	C₃×Q₈	—	2	4	no	—
206	C₃×Q₈	—	2	4	no	—
207	C₃×Q₈	—	4	4	no	—
208	C₃×Q₈	—	4	4	no	—
209	C₃×Q₈	—	4	4	no	—
210	C₃×Q₈	—	4	2	no	—
211	C₃×Q₈	—	4	2	no	—
212	C₃×Q₈	—	4	2	no	—
213	C₃×Q₈	—	4	2	no	—
214	C₂×C₃×C₄	—	2	4	no	—
215	C₂×C₃×C₄	—	2	4	no	—
216	C₂×C₃×C₄	—	2	1	no	—
217	C₂×C₃×C₄	—	2	1	no	—
218	C₂×C₃×C₄	—	2	1	no	—
219	C₂×C₃×C₄	—	2	1	no	—
220	C₂×C₃×C₄	—	2	1	no	—
221	C₂×C₃×C₄	—	2	1	no	—
222	C₂×C₃×C₄	—	4	4	no	—
223	C₂×C₃×C₄	—	4	4	no	—
224	C₂×C₃×C₄	—	4	4	no	—
225	C₂×C₃×C₄	—	4	2	no	—
226	C₂×C₃×C₄	—	4	2	no	—
227	C₂×C₃×C₄	—	4	2	no	—
228	C₂×C₃×C₄	—	4	2	no	—
229	C₂×C₃×C₄	—	6	2	no	—
230	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
231	C₂×C₃×C₄	D₃×D₄	1	1	yes	characteristic
232	C₂×C₃×C₄	#237	1	1	yes	characteristic
233	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
234	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
235	C₂×C₃×C₄	C₂×#67	1	1	yes	characteristic
236	C₂×C₃×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
237	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
238	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
239	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
240	C₂×C₃×C₄	C₂×C₃×D₄	1	1	yes	characteristic
241	C₂×C₃×C₄	C₃×#41	1	1	yes	characteristic
242	C₂×Dic₃	—	2	2	no	—
243	C₃×C₈	—	2	2	no	—
244	C₃×C₈	—	2	2	no	—
245	C₃×C₈	—	6	2	no	—
246	C₃×C₈	—	12	2	no	—
247	C₃×C₈	#217	1	2	yes	—
248	C₃×C₈	#217	1	2	yes	—
249	C₃×C₈	C₃×C₂²⋊C₄	1	2	yes	—
250	C₃×C₈	C₃×C₂²⋊C₄	1	2	yes	—
251	C₃⋊C₈	—	2	2	no	—
252	C₃⋊C₈	—	4	2	no	—
253	C₂×C₃²	—	2	1	no	—
254	C₂×C₃²	—	2	1	no	—
255	C₂×C₃²	#390	1	1	yes	characteristic
256	C₂×C₃²	#473	1	2	yes	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	2	2	no	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	C₂×C₃×Dic₃	1	4	yes	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	C₂²×C₃×D₃	1	1	yes	characteristic
265	Q₁₆	—	2	4	no	—
266	Q₁₆	—	2	4	no	—
267	M₁₆	—	3	2	no	—
268	M₁₆	—	6	1	no	—
269	M₁₆	—	6	2	no	—
270	M₁₆	—	6	2	no	—
271	C₂×C₈	—	6	1	no	—
272	C₂×C₈	—	6	2	no	—
273	C₂×C₈	C₂×C₃×Dic₃	1	2	yes	—
274	C₂×C₈	C₃×#67	1	1	yes	characteristic
275	C₂×C₈	C₃×#67	1	1	yes	characteristic
276	C₂×C₈	C₃×#67	1	2	yes	—
277	C₄⋊C₄	—	3	2	no	—
278	C₂²⋊C₄	—	6	1	no	—
279	C₄²	—	6	1	no	—
280	C₂²×C₃	—	2	4	no	—
281	C₂²×C₃	—	2	2	no	—
282	C₂²×C₃	—	2	2	no	—
283	C₂²×C₃	—	2	1	no	—
284	C₂²×C₃	—	2	1	no	—
285	C₂²×C₃	—	2	1	no	—
286	C₂²×C₃	D₄×Dic₃	1	1	yes	characteristic
287	C₂²×C₃	#936	1	1	yes	characteristic
288	C₂²×C₃	#938	1	1	yes	characteristic
289	C₂²×C₃	C₃×C₄×D₄	1	1	yes	characteristic
290	C₂²×C₃	C₃×#121	1	1	yes	characteristic
291	C₂²×C₃	C₃×#124	1	1	yes	characteristic
292	C₃×C₄	—	2	1	no	—
293	C₃×C₄	—	2	1	no	—
294	C₃×C₄	—	2	1	no	—
295	C₃×C₄	—	2	1	no	—
296	C₃×C₄	—	4	8	no	—
297	C₃×C₄	—	4	4	no	—
298	C₃×C₄	—	4	4	no	—
299	C₃×C₄	—	12	2	no	—
300	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
301	C₃×C₄	#938	1	1	yes	characteristic
302	C₃×C₄	#940	1	1	yes	characteristic
303	C₃×C₄	C₂×#217	1	1	yes	characteristic
304	C₃×C₄	C₂×C₃×C₂²⋊C₄	1	1	yes	characteristic
305	C₃×C₄	C₃×C₄×D₄	1	1	yes	characteristic
306	C₃×C₄	C₃×#121	1	1	yes	characteristic
307	C₃×C₄	C₃×#127	1	1	yes	characteristic
308	Dic₃	—	4	2	no	—
309	C₃²	#1979	1	1	yes	characteristic, complemented, Sylow
310	C₂³	C₃×#237	1	1	yes	characteristic
311	Q₈	—	2	4	no	—
312	Q₈	—	2	4	no	—
313	Q₈	—	4	2	no	—
314	Q₈	—	4	2	no	—
315	C₂×C₄	—	2	4	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	6	2	no	—
319	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
320	C₂×C₄	C₃×D₃×D₄	1	1	yes	characteristic
321	C₂×C₄	C₃×#237	1	1	yes	characteristic
322	C₂×C₄	C₂²×C₃×Dic₃	1	1	yes	characteristic
323	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
324	C₂×C₄	C₂×C₃×#67	1	1	yes	characteristic
325	C₈	—	6	2	no	—
326	C₈	—	12	2	no	—
327	C₈	C₃×#217	1	2	yes	—
328	C₈	C₃×#217	1	2	yes	—
329	C₂×C₃	—	2	2	no	—
330	C₂×C₃	—	2	2	no	—
331	C₂×C₃	—	2	1	no	—
332	C₂×C₃	—	2	1	no	—
333	C₂×C₃	—	2	1	no	—
334	C₂×C₃	—	2	1	no	—
335	C₂×C₃	—	2	2	no	—
336	C₂×C₃	—	2	1	no	—
337	C₂×C₃	#5192	1	2	yes	—
338	C₂×C₃	#5221	1	1	yes	characteristic
339	C₂×C₃	C₃×#390	1	1	yes	characteristic
340	C₂×C₃	C₃×#473	1	2	yes	—
341	C₂²	—	2	2	no	—
342	C₂²	C₃×D₄×Dic₃	1	1	yes	characteristic
343	C₂²	C₃×#936	1	1	yes	characteristic
344	C₂²	C₃×#938	1	1	yes	characteristic
345	C₄	—	4	4	no	—
346	C₄	—	12	2	no	—
347	C₄	C₃×D₄×Dic₃	1	1	yes	characteristic
348	C₄	C₃×#938	1	1	yes	characteristic
349	C₄	C₃×#940	1	1	yes	characteristic
350	C₄	C₂×C₃×#217	1	1	yes	characteristic
351	C₃	—	2	1	no	—
352	C₃	#72069	1	1	yes	characteristic, direct product factor
353	C₃	C₃×#1979	1	1	yes	characteristic, complemented
354	C₂	—	2	1	no	—
355	C₂	—	2	1	no	—
356	C₂	C₃×#5192	1	2	yes	—
357	C₂	C₃×#5221	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₃	#938
commutator	C₂×C₃×C₄	C₂²×C₃×C₄
Frattini	C₂²×C₄	C₂²×C₃×D₃
Fitting	C₂²×C₃²×Q₁₆	C₂
socle	C₂²×C₃²	#121
prime radical	C₂³×C₃²	#41
cosocle	C₂²×C₃×C₄	C₂³×C₃
hypercenter	C₂²×C₃×Q₁₆	D₃
hypocenter	C₃	C₃×#1979
LCS	G ▷ C₂×C₃×C₄ ▷ C₂×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₃ ◁ C₂²×C₃×C₄ ◁ C₂²×C₃×Q₁₆

Complements

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

N	H ≅ G/N	type
7. #72069	C₃	direct (N × H)
12. C₂×C₃²×Q₁₆	C₄	semidirect
70. C₂×C₃×Q₁₆	Dic₃	semidirect
71. C₂×C₃×Q₁₆	C₃×C₄	semidirect
189. C₂×Q₁₆	C₃×Dic₃	semidirect
309. C₃²	#1979	semidirect
352. C₃	#72069	direct (N × H)
353. 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×C₃²×Q₁₆	2
C₂×C₃×#836	2
C₃×#5225	2
C₂×C₃×#819	2
C₃×#5101	2
C₃×#1979	3
#72069	3

Subgroups — C3×#72069 #11791368

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×#72069 #11791368