Subgroups — C₃×M₁₆×Q₁₆ #10636645

Order 768.

638 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃×Q₈×M₁₆	C₂	1	2	yes	maximal
2	C₂×C₃×C₄×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₃×#1591	C₂	1	2	yes	maximal, complemented
4	C₃×#1584	C₂	1	4	yes	maximal, complemented
5	C₃×C₈×Q₁₆	C₂	1	2	yes	maximal, complemented
6	C₃×#1567	C₂	1	1	yes	maximal, characteristic
7	C₃×#1487	C₂	1	2	yes	maximal
8	C₃×C₈×M₁₆	C₂	1	1	yes	maximal, characteristic
9	M₁₆×Q₁₆	C₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
10	C₂²×C₃×Q₁₆	C₄	1	1	yes	characteristic
11	C₂×C₃×C₄×Q₈	C₂²	1	2	yes	—
12	C₃×#446	—	2	4	no	—
13	C₃×#446	C₂²	1	4	yes	—
14	C₃×C₈×Q₈	C₂²	1	4	yes	—
15	C₃×C₄×Q₁₆	C₄	1	2	yes	—
16	C₃×C₄×Q₁₆	C₄	1	1	yes	characteristic
17	C₃×C₄×Q₁₆	C₄	1	1	yes	characteristic
18	C₃×C₄×Q₁₆	C₂²	1	1	yes	characteristic
19	C₃×C₄×Q₁₆	C₂²	1	2	yes	—
20	C₃×C₄×Q₁₆	C₂²	1	1	yes	characteristic
21	C₃×#431	C₂²	1	1	yes	characteristic
22	C₂×C₃×#107	C₄	1	1	yes	characteristic
23	C₃×#423	—	2	2	no	—
24	C₃×#423	C₂²	1	2	yes	—
25	C₂×C₃×#103	C₄	1	2	yes	—
26	C₃×C₄×M₁₆	—	2	2	no	—
27	C₃×C₄×M₁₆	C₂²	1	1	yes	characteristic
28	C₂×C₃×C₄×C₈	C₂²	1	1	yes	characteristic
29	C₃×#335	C₂²	1	2	yes	—
30	C₃×#335	C₂²	1	2	yes	—
31	C₃×Q₈⋊C₈	C₂²	1	4	yes	—
32	C₃×Q₈⋊C₈	C₂²	1	4	yes	—
33	C₃×#322	C₂²	1	2	yes	—
34	C₃×C₈²	C₂²	1	2	yes	—
35	Q₈×M₁₆	C₂×C₃	1	2	yes	—
36	C₂×C₄×Q₁₆	C₂×C₃	1	1	yes	characteristic
37	#1591	C₂×C₃	1	2	yes	complemented
38	#1584	C₂×C₃	1	4	yes	complemented
39	C₈×Q₁₆	C₂×C₃	1	2	yes	complemented
40	#1567	C₂×C₃	1	1	yes	characteristic
41	#1487	C₂×C₃	1	2	yes	—
42	C₈×M₁₆	C₂×C₃	1	1	yes	characteristic
43	C₂²×C₃×Q₈	C₂×C₄	1	2	yes	—
44	C₂×C₃×Q₁₆	—	2	4	no	—
45	C₂×C₃×Q₁₆	C₂×C₄	1	2	yes	—
46	C₂×C₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×Q₁₆	C₂×C₄	1	1	yes	characteristic
48	C₂×C₃×M₁₆	—	2	2	no	—
49	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
50	C₂×C₃×M₁₆	D₄	1	1	yes	characteristic
51	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₃×C₈	C₂×C₄	1	1	yes	characteristic
53	C₃×C₄×Q₈	—	2	2	no	—
54	C₃×C₄×Q₈	—	2	2	no	—
55	C₃×C₄×Q₈	C₂×C₄	1	2	yes	—
56	C₃×C₄×Q₈	C₂×C₄	1	2	yes	—
57	C₃×C₄×Q₈	C₂³	1	2	yes	—
58	C₃×C₄×Q₈	C₂³	1	2	yes	—
59	C₂×C₃×C₄⋊C₄	—	2	2	no	—
60	C₂×C₃×C₄⋊C₄	C₂×C₄	1	2	yes	—
61	C₂×C₃×C₄²	—	2	2	no	—
62	C₂×C₃×C₄²	C₂³	1	1	yes	characteristic
63	C₃×#107	C₂×C₄	1	2	yes	—
64	C₃×#107	C₂×C₄	1	1	yes	characteristic
65	C₃×#107	C₂×C₄	1	1	yes	characteristic
66	C₃×C₄⋊C₈	—	2	4	no	—
67	C₃×C₄⋊C₈	—	2	4	no	—
68	C₃×C₄⋊C₈	—	2	2	no	—
69	C₃×C₄⋊C₈	—	2	1	no	—
70	C₃×C₄⋊C₈	C₂³	1	4	yes	—
71	C₃×#103	C₂×C₄	1	2	yes	—
72	C₃×#103	C₂×C₄	1	2	yes	—
73	C₃×#103	C₂×C₄	1	2	yes	—
74	C₃×#103	C₂×C₄	1	2	yes	—
75	C₃×C₂²⋊C₈	—	2	1	no	—
76	C₃×#97	—	2	4	no	—
77	C₃×#97	D₄	1	2	yes	—
78	C₃×C₄×C₈	—	2	4	no	—
79	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
80	C₃×C₄×C₈	C₂×C₄	1	1	yes	characteristic
81	C₃×C₄×C₈	C₂³	1	2	yes	—
82	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
83	C₃×C₄×C₈	C₂³	1	1	yes	characteristic
84	C₃×C₄×C₈	C₂³	1	2	yes	—
85	C₂²×Q₁₆	C₃×C₄	1	1	yes	characteristic
86	C₂×C₄×Q₈	C₂²×C₃	1	2	yes	—
87	#446	—	2	4	no	—
88	#446	C₂²×C₃	1	4	yes	—
89	C₈×Q₈	C₂²×C₃	1	4	yes	—
90	C₄×Q₁₆	C₃×C₄	1	2	yes	—
91	C₄×Q₁₆	C₃×C₄	1	1	yes	characteristic
92	C₄×Q₁₆	C₃×C₄	1	1	yes	characteristic
93	C₄×Q₁₆	C₂²×C₃	1	1	yes	characteristic
94	C₄×Q₁₆	C₂²×C₃	1	2	yes	—
95	C₄×Q₁₆	C₂²×C₃	1	1	yes	characteristic
96	#431	C₂²×C₃	1	1	yes	characteristic
97	C₂×#107	C₃×C₄	1	1	yes	characteristic
98	#423	—	2	2	no	—
99	#423	C₂²×C₃	1	2	yes	—
100	C₂×#103	C₃×C₄	1	2	yes	—
101	C₄×M₁₆	—	2	2	no	—
102	C₄×M₁₆	C₂²×C₃	1	1	yes	characteristic
103	C₂×C₄×C₈	C₂²×C₃	1	1	yes	characteristic
104	#335	C₂²×C₃	1	2	yes	—
105	#335	C₂²×C₃	1	2	yes	—
106	Q₈⋊C₈	C₂²×C₃	1	4	yes	—
107	Q₈⋊C₈	C₂²×C₃	1	4	yes	—
108	#322	C₂²×C₃	1	2	yes	—
109	C₈²	C₂²×C₃	1	2	yes	—
110	C₂×C₃×Q₈	—	2	2	no	—
111	C₂×C₃×Q₈	—	2	4	no	—
112	C₂×C₃×Q₈	—	4	2	no	—
113	C₂×C₃×Q₈	C₂²×C₄	1	2	yes	—
114	C₂×C₃×Q₈	C₂²×C₄	1	2	yes	—
115	C₂²×C₃×C₄	—	2	2	no	—
116	C₂²×C₃×C₄	—	2	2	no	—
117	C₂²×C₃×C₄	C₂²×C₄	1	1	yes	characteristic
118	C₂²×C₃×C₄	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₃×C₄	#41	1	1	yes	characteristic
120	C₃×Q₁₆	—	2	4	no	—
121	C₃×Q₁₆	—	2	2	no	—
122	C₃×Q₁₆	M₁₆	1	4	yes	direct product factor
123	C₃×M₁₆	—	2	2	no	—
124	C₃×M₁₆	—	4	4	no	—
125	C₃×M₁₆	Q₁₆	1	4	yes	direct product factor
126	C₂×C₃×C₈	—	2	2	no	—
127	C₂×C₃×C₈	—	2	4	no	—
128	C₂×C₃×C₈	—	2	2	no	—
129	C₂×C₃×C₈	—	4	1	no	—
130	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
131	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
132	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₃×C₈	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
135	C₂×C₃×C₈	C₂×D₄	1	2	yes	—
136	C₃×C₄⋊C₄	—	2	2	no	—
137	C₃×C₄⋊C₄	—	2	2	no	—
138	C₃×C₄⋊C₄	—	2	2	no	—
139	C₃×C₄⋊C₄	—	2	2	no	—
140	C₃×C₄⋊C₄	—	2	2	no	—
141	C₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
142	C₃×C₄⋊C₄	C₂²×C₄	1	2	yes	—
143	C₃×C₄²	—	2	2	no	—
144	C₃×C₄²	—	2	2	no	—
145	C₃×C₄²	—	2	2	no	—
146	C₃×C₄²	—	2	2	no	—
147	C₃×C₄²	C₂²×C₄	1	1	yes	characteristic
148	C₃×C₄²	C₂×D₄	1	1	yes	characteristic
149	C₃×C₄²	#41	1	1	yes	characteristic
150	C₃×C₄²	C₂⁴	1	1	yes	characteristic
151	C₂²×Q₈	C₂×C₃×C₄	1	2	yes	—
152	C₂×Q₁₆	—	2	4	no	—
153	C₂×Q₁₆	C₂×C₃×C₄	1	2	yes	—
154	C₂×Q₁₆	C₂×C₃×C₄	1	1	yes	characteristic
155	C₂×Q₁₆	C₂×C₃×C₄	1	1	yes	characteristic
156	C₂×M₁₆	—	2	2	no	—
157	C₂×M₁₆	C₃×D₄	1	1	yes	characteristic
158	C₂×M₁₆	C₃×D₄	1	1	yes	characteristic
159	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
160	C₂²×C₈	C₂×C₃×C₄	1	1	yes	characteristic
161	C₄×Q₈	—	2	2	no	—
162	C₄×Q₈	—	2	2	no	—
163	C₄×Q₈	C₂×C₃×C₄	1	2	yes	—
164	C₄×Q₈	C₂×C₃×C₄	1	2	yes	—
165	C₄×Q₈	C₂³×C₃	1	2	yes	—
166	C₄×Q₈	C₂³×C₃	1	2	yes	—
167	C₂×C₄⋊C₄	—	2	2	no	—
168	C₂×C₄⋊C₄	C₂×C₃×C₄	1	2	yes	—
169	C₂×C₄²	—	2	2	no	—
170	C₂×C₄²	C₂³×C₃	1	1	yes	characteristic
171	#107	C₂×C₃×C₄	1	2	yes	—
172	#107	C₂×C₃×C₄	1	1	yes	characteristic
173	#107	C₂×C₃×C₄	1	1	yes	characteristic
174	C₄⋊C₈	—	2	4	no	—
175	C₄⋊C₈	—	2	4	no	—
176	C₄⋊C₈	—	2	2	no	—
177	C₄⋊C₈	—	2	1	no	—
178	C₄⋊C₈	C₂³×C₃	1	4	yes	—
179	#103	C₂×C₃×C₄	1	2	yes	—
180	#103	C₂×C₃×C₄	1	2	yes	—
181	#103	C₂×C₃×C₄	1	2	yes	—
182	#103	C₂×C₃×C₄	1	2	yes	—
183	C₂²⋊C₈	—	2	1	no	—
184	#97	—	2	4	no	—
185	#97	C₃×D₄	1	2	yes	—
186	C₄×C₈	—	2	4	no	—
187	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
188	C₄×C₈	C₂×C₃×C₄	1	1	yes	characteristic
189	C₄×C₈	C₂³×C₃	1	2	yes	—
190	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
191	C₄×C₈	C₂³×C₃	1	1	yes	characteristic
192	C₄×C₈	C₂³×C₃	1	2	yes	—
193	C₂³×C₃	C₄×D₄	1	1	yes	characteristic
194	C₃×Q₈	—	2	2	no	—
195	C₃×Q₈	—	2	2	no	—
196	C₃×Q₈	—	2	2	no	—
197	C₃×Q₈	—	4	4	no	—
198	C₃×Q₈	C₂×M₁₆	1	4	yes	—
199	C₂×C₃×C₄	—	2	2	no	—
200	C₂×C₃×C₄	—	2	1	no	—
201	C₂×C₃×C₄	—	2	1	no	—
202	C₂×C₃×C₄	—	2	2	no	—
203	C₂×C₃×C₄	—	2	2	no	—
204	C₂×C₃×C₄	—	2	2	no	—
205	C₂×C₃×C₄	—	4	1	no	—
206	C₂×C₃×C₄	—	4	4	no	—
207	C₂×C₃×C₄	—	8	2	no	—
208	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
209	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
210	C₂×C₃×C₄	C₄×D₄	1	1	yes	characteristic
211	C₂×C₃×C₄	C₂×Q₁₆	1	2	yes	—
212	C₂×C₃×C₄	#135	1	2	yes	—
213	C₂×C₃×C₄	C₂³×C₄	1	1	yes	characteristic
214	C₂×C₃×C₄	C₂²×D₄	1	1	yes	characteristic
215	C₂×C₃×C₄	C₂×#41	1	1	yes	characteristic
216	C₃×C₈	—	2	1	no	—
217	C₃×C₈	—	2	1	no	—
218	C₃×C₈	—	2	1	no	—
219	C₃×C₈	—	2	2	no	—
220	C₃×C₈	—	4	2	no	—
221	C₃×C₈	—	4	4	no	—
222	C₃×C₈	C₂×M₁₆	1	2	yes	—
223	C₃×C₈	C₂×Q₁₆	1	4	yes	complemented
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	—	4	2	no	—
227	C₂×Q₈	C₂²×C₃×C₄	1	2	yes	—
228	C₂×Q₈	C₂²×C₃×C₄	1	2	yes	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	2	no	—
231	C₂²×C₄	C₂²×C₃×C₄	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₃×D₄	1	1	yes	characteristic
233	C₂²×C₄	C₃×#41	1	1	yes	characteristic
234	Q₁₆	—	2	4	no	—
235	Q₁₆	—	2	2	no	—
236	Q₁₆	C₃×M₁₆	1	4	yes	direct product factor
237	M₁₆	—	2	2	no	—
238	M₁₆	—	4	4	no	—
239	M₁₆	C₃×Q₁₆	1	4	yes	direct product factor
240	C₂×C₈	—	2	2	no	—
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	4	1	no	—
244	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
245	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
246	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
247	C₂×C₈	C₂²×C₃×C₄	1	1	yes	characteristic
248	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
249	C₂×C₈	C₂×C₃×D₄	1	2	yes	—
250	C₄⋊C₄	—	2	2	no	—
251	C₄⋊C₄	—	2	2	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	C₂²×C₃×C₄	1	2	yes	—
256	C₄⋊C₄	C₂²×C₃×C₄	1	2	yes	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	2	2	no	—
261	C₄²	C₂²×C₃×C₄	1	1	yes	characteristic
262	C₄²	C₂×C₃×D₄	1	1	yes	characteristic
263	C₄²	C₃×#41	1	1	yes	characteristic
264	C₄²	C₂⁴×C₃	1	1	yes	characteristic
265	C₂²×C₃	—	2	1	no	—
266	C₂²×C₃	—	2	1	no	—
267	C₂²×C₃	C₄×Q₁₆	1	2	yes	—
268	C₂²×C₃	C₂×C₄×D₄	1	1	yes	characteristic
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	2	4	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	2	1	no	—
273	C₃×C₄	—	4	2	no	—
274	C₃×C₄	—	4	2	no	—
275	C₃×C₄	—	4	2	no	—
276	C₃×C₄	C₄×Q₁₆	1	2	yes	—
277	C₃×C₄	C₂²×M₁₆	1	1	yes	characteristic
278	C₃×C₄	#568	1	1	yes	characteristic
279	C₃×C₄	C₂²×Q₁₆	1	1	yes	characteristic
280	C₃×C₄	C₂×#135	1	1	yes	characteristic
281	C₂³	C₃×C₄×D₄	1	1	yes	characteristic
282	Q₈	—	2	2	no	—
283	Q₈	—	2	2	no	—
284	Q₈	—	2	2	no	—
285	Q₈	—	4	4	no	—
286	Q₈	C₂×C₃×M₁₆	1	4	yes	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	8	2	no	—
296	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
297	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
298	C₂×C₄	C₃×C₄×D₄	1	1	yes	characteristic
299	C₂×C₄	C₂×C₃×Q₁₆	1	2	yes	—
300	C₂×C₄	C₃×#135	1	2	yes	—
301	C₂×C₄	C₂³×C₃×C₄	1	1	yes	characteristic
302	C₂×C₄	C₂²×C₃×D₄	1	1	yes	characteristic
303	C₂×C₄	C₂×C₃×#41	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	—	2	2	no	—
308	C₈	—	4	2	no	—
309	C₈	—	4	4	no	—
310	C₈	C₂×C₃×M₁₆	1	2	yes	—
311	C₈	C₂×C₃×Q₁₆	1	4	yes	complemented
312	C₂×C₃	—	2	2	no	—
313	C₂×C₃	D₄×M₁₆	1	1	yes	characteristic
314	C₂×C₃	C₂×C₄×Q₁₆	1	1	yes	characteristic
315	C₂×C₃	#2957	1	1	yes	characteristic
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	C₃×C₄×Q₁₆	1	2	yes	—
319	C₂²	C₂×C₃×C₄×D₄	1	1	yes	characteristic
320	C₄	—	2	1	no	—
321	C₄	—	2	4	no	—
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	2	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	C₃×C₄×Q₁₆	1	2	yes	—
328	C₄	C₂²×C₃×M₁₆	1	1	yes	characteristic
329	C₄	C₃×#568	1	1	yes	characteristic
330	C₄	C₂²×C₃×Q₁₆	1	1	yes	characteristic
331	C₄	C₂×C₃×#135	1	1	yes	characteristic
332	C₃	M₁₆×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
333	C₂	—	2	2	no	—
334	C₂	C₃×D₄×M₁₆	1	1	yes	characteristic
335	C₂	C₂×C₃×C₄×Q₁₆	1	1	yes	characteristic
336	C₂	C₃×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₃×C₄	C₂²×D₄
commutator	C₂×C₄	C₂³×C₃×C₄
Frattini, cosocle	C₄²	C₂⁴×C₃
socle	C₂²×C₃	C₂×C₄×D₄
prime radical	C₂³×C₃	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₃×C₄ ◁ C₃×C₄×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₃×#1591	C₂	semidirect
4. C₃×#1584	C₂	semidirect
5. C₃×C₈×Q₁₆	C₂	semidirect
9. M₁₆×Q₁₆	C₃	direct (N × H)
37. #1591	C₂×C₃	semidirect
38. #1584	C₂×C₃	semidirect
39. C₈×Q₁₆	C₂×C₃	semidirect
122. C₃×Q₁₆	M₁₆	direct (N × H)
125. C₃×M₁₆	Q₁₆	direct (N × H)
223. C₃×C₈	C₂×Q₁₆	semidirect
236. Q₁₆	C₃×M₁₆	direct (N × H)
239. M₁₆	C₃×Q₁₆	direct (N × H)
311. C₈	C₂×C₃×Q₁₆	semidirect
332. C₃	M₁₆×Q₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₃×Q₈×M₁₆	2
C₂×C₃×C₄×Q₁₆	2
C₃×#1591	2
C₃×#1584	2
C₃×C₈×Q₁₆	2
C₃×#1567	2
C₃×#1487	2
C₃×C₈×M₁₆	2
M₁₆×Q₁₆	3

Subgroups — C3×M16×Q16 #10636645

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃×M₁₆×Q₁₆ #10636645