Subgroups — C₁₆×#626 #11797449

Order 1152.

397 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₆×C₃⋊D₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃×D₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×#626	C₂	1	1	yes	maximal, characteristic
4	C₃×Dic₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	#10589108	C₂	1	1	yes	maximal, characteristic, complemented
7	(C₂×D₃)⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₆×#67	—	3	1	no	maximal
9	C₁₆×D₁₂	—	3	1	no	maximal
10	C₂×C₈×C₃⋊D₃	C₂²	1	1	yes	characteristic
11	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
12	C₄×#626	C₄	1	1	yes	characteristic
13	C₂×C₃²×C₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
15	C₁₆×C₃⋊D₃	—	2	2	no	—
16	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
17	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
18	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
19	C₃×D₃×C₁₆	—	2	2	no	—
20	Dic₃⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
21	#63557	C₄	1	1	yes	characteristic
22	(C₂×D₃)⋊(C₃⋊C₈)	C₄	1	1	yes	characteristic
23	C₂²×C₃×C₁₆	—	3	1	no	—
24	C₈×#67	—	3	1	no	—
25	C₂×D₃×C₁₆	—	3	1	no	—
26	C₂×D₃×C₁₆	—	3	1	no	—
27	C₂×D₃×C₁₆	—	6	1	no	—
28	C₂×D₃×C₁₆	D₃	1	1	yes	characteristic, complemented
29	C₈×D₁₂	—	3	1	no	—
30	C₃×C₄×C₁₆	—	3	1	no	—
31	#4549	—	3	1	no	—
32	(C₂×D₃)⋊C₁₆	—	3	1	no	—
33	(C₂×D₃)⋊C₁₆	—	3	1	no	—
34	(C₂×D₃)⋊C₁₆	—	3	1	no	—
35	Dic₃⋊C₁₆	—	3	1	no	—
36	Dic₃×C₁₆	D₃	1	1	yes	characteristic, complemented
37	C₄⋊(C₃⋊C₁₆)	—	3	1	no	—
38	C₂×C₄×C₃⋊D₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
40	C₂×#626	C₈	1	1	yes	characteristic
41	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
42	C₂×C₃⋊(C₃⋊C₈)	C₂×C₄	1	1	yes	characteristic
43	C₈×C₃⋊D₃	—	2	1	no	—
44	C₈×C₃⋊D₃	—	2	1	no	—
45	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
47	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
48	C₃×D₃×C₈	—	2	1	no	—
49	C₃×D₃×C₈	—	2	1	no	—
50	Dic₃⋊Dic₃	C₈	1	1	yes	characteristic
51	#3722	C₈	1	1	yes	characteristic
52	(C₂×D₃)⋊Dic₃	C₈	1	1	yes	characteristic
53	C₃²×C₁₆	D₄	1	2	yes	complemented
54	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
55	C₃×C₃⋊C₁₆	—	2	1	no	—
56	C₃×C₃⋊C₁₆	—	2	1	no	—
57	D₄×C₁₆	—	9	1	no	Sylow
58	C₂²×C₃×C₈	—	3	1	no	—
59	C₄×#67	—	3	1	no	—
60	C₂×D₃×C₈	—	3	1	no	—
61	C₂×D₃×C₈	—	3	1	no	—
62	C₂×D₃×C₈	—	6	1	no	—
63	C₂×D₃×C₈	C₂×D₃	1	1	yes	characteristic
64	C₄×D₁₂	—	3	1	no	—
65	C₂×C₃×C₁₆	—	2	1	no	—
66	C₂×C₃×C₁₆	—	3	1	no	—
67	C₂×C₃×C₁₆	—	3	1	no	—
68	C₂×C₃×C₁₆	—	6	2	no	—
69	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
70	C₂×C₃×C₁₆	C₂×D₃	1	1	yes	characteristic
71	C₃×C₄×C₈	—	3	1	no	—
72	#830	—	3	1	no	—
73	(C₂×D₃)⋊C₈	—	3	1	no	—
74	(C₂×D₃)⋊C₈	—	3	1	no	—
75	(C₂×D₃)⋊C₈	—	3	1	no	—
76	Dic₃⋊C₈	—	3	1	no	—
77	C₈×Dic₃	C₂×D₃	1	1	yes	characteristic
78	C₂×C₃⋊C₁₆	—	3	1	no	—
79	C₂×C₃⋊C₁₆	—	3	1	no	—
80	C₂×C₃⋊C₁₆	—	6	1	no	—
81	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
82	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
83	C₄⋊(C₃⋊C₈)	—	3	1	no	—
84	D₃×C₁₆	—	2	2	no	—
85	D₃×C₁₆	—	6	4	no	—
86	D₃×C₁₆	—	6	2	no	—
87	D₃×C₁₆	—	6	2	no	—
88	C₂²×C₃⋊D₃	C₂×C₈	1	1	yes	characteristic
89	C₂²×C₃×D₃	C₂×C₈	1	1	yes	characteristic
90	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₃⋊Dic₃	C₂×C₈	1	1	yes	characteristic
92	C₄×C₃⋊D₃	—	2	1	no	—
93	C₄×C₃⋊D₃	—	2	1	no	—
94	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
95	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
96	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
97	C₃×C₄×D₃	—	2	1	no	—
98	C₃×C₄×D₃	—	2	1	no	—
99	#626	C₁₆	1	4	yes	direct product factor
100	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
101	C₃²×C₈	#41	1	1	yes	characteristic
102	C₃⋊(C₃⋊C₈)	—	2	1	no	—
103	C₃×C₃⋊C₈	—	2	1	no	—
104	C₃×C₃⋊C₈	—	2	1	no	—
105	C₂²×C₁₆	—	3	1	no	—
106	C₂²×C₁₆	—	9	1	no	—
107	C₈×D₄	—	9	1	no	—
108	C₄⋊C₁₆	—	9	1	no	—
109	C₂²⋊C₁₆	—	9	1	no	—
110	C₂²⋊C₁₆	—	9	1	no	—
111	C₄×C₁₆	—	3	1	no	—
112	C₂²×C₃×C₄	—	3	1	no	—
113	C₂×#67	—	3	1	no	—
114	C₂×D₁₂	—	3	1	no	—
115	C₂×C₄×D₃	—	3	1	no	—
116	C₂×C₄×D₃	—	3	1	no	—
117	C₂×C₄×D₃	—	6	1	no	—
118	C₂×C₄×D₃	C₄×D₃	1	1	yes	characteristic
119	C₂×C₃×C₈	—	2	1	no	—
120	C₂×C₃×C₈	—	3	1	no	—
121	C₂×C₃×C₈	—	3	1	no	—
122	C₂×C₃×C₈	—	6	1	no	—
123	C₂×C₃×C₈	—	6	1	no	—
124	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
125	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
126	C₃×C₄²	—	3	1	no	—
127	#217	—	3	1	no	—
128	(C₂×D₃)⋊C₄	—	3	1	no	—
129	(C₂×D₃)⋊C₄	—	3	1	no	—
130	(C₂×D₃)⋊C₄	—	3	1	no	—
131	C₄⋊Dic₃	—	3	1	no	—
132	Dic₃⋊C₄	—	3	1	no	—
133	C₄×Dic₃	C₄×D₃	1	1	yes	characteristic
134	C₂×C₃⋊C₈	—	3	1	no	—
135	C₂×C₃⋊C₈	—	3	1	no	—
136	C₂×C₃⋊C₈	—	6	1	no	—
137	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
138	C₂×C₃⋊C₈	C₄×D₃	1	1	yes	characteristic
139	D₃×C₈	—	2	1	no	—
140	D₃×C₈	—	2	1	no	—
141	D₃×C₈	—	6	2	no	—
142	D₃×C₈	—	6	2	no	—
143	D₃×C₈	—	6	1	no	—
144	D₃×C₈	—	6	1	no	—
145	D₃×C₈	—	6	1	no	—
146	D₃×C₈	—	6	1	no	—
147	C₃×C₁₆	—	2	2	no	—
148	C₃×C₁₆	—	6	1	no	—
149	C₃×C₁₆	—	6	1	no	—
150	C₃×C₁₆	D₁₂	1	2	yes	complemented
151	C₃×C₁₆	#67	1	2	yes	complemented
152	C₃⋊C₁₆	—	2	1	no	—
153	C₃⋊C₁₆	—	2	1	no	—
154	C₃⋊C₁₆	—	6	2	no	—
155	C₃⋊C₁₆	—	6	1	no	—
156	C₃⋊C₁₆	—	6	1	no	—
157	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
158	C₂×C₃⋊D₃	—	2	1	no	—
159	C₂×C₃⋊D₃	—	2	1	no	—
160	C₂×C₃⋊D₃	C₂×C₁₆	1	2	yes	complemented
161	C₂×C₃×D₃	—	2	1	no	—
162	C₂×C₃×D₃	—	2	1	no	—
163	C₂×C₃×D₃	C₂×C₁₆	1	2	yes	complemented
164	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
165	C₃²×C₄	#131	1	1	yes	characteristic
166	C₃⋊Dic₃	—	2	1	no	—
167	C₃×Dic₃	—	2	1	no	—
168	C₃×Dic₃	—	2	1	no	—
169	C₃×Dic₃	C₂×C₁₆	1	2	yes	complemented
170	C₂²×C₈	—	3	1	no	—
171	C₂²×C₈	—	9	1	no	—
172	C₄×D₄	—	9	1	no	—
173	C₂×C₁₆	—	3	1	no	—
174	C₂×C₁₆	—	3	1	no	—
175	C₂×C₁₆	—	6	2	no	—
176	C₂×C₁₆	—	9	1	no	—
177	C₂×C₁₆	—	18	2	no	—
178	C₂×C₁₆	D₃²	1	1	yes	characteristic
179	C₄⋊C₈	—	9	1	no	—
180	C₂²⋊C₈	—	9	1	no	—
181	C₂²⋊C₈	—	9	1	no	—
182	C₄×C₈	—	3	1	no	—
183	C₂³×C₃	—	3	1	no	—
184	C₂²×D₃	—	3	1	no	—
185	C₂²×D₃	—	3	1	no	—
186	C₂²×D₃	—	6	1	no	—
187	C₂²×D₃	D₃×C₈	1	1	yes	characteristic
188	C₂×C₃×C₄	—	2	1	no	—
189	C₂×C₃×C₄	—	3	1	no	—
190	C₂×C₃×C₄	—	3	1	no	—
191	C₂×C₃×C₄	—	3	1	no	—
192	C₂×C₃×C₄	—	6	1	no	—
193	C₂×C₃×C₄	—	6	1	no	—
194	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
195	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
196	#67	—	3	4	no	—
197	C₂×Dic₃	—	3	1	no	—
198	C₂×Dic₃	—	3	1	no	—
199	C₂×Dic₃	—	6	1	no	—
200	C₂×Dic₃	D₃×C₈	1	1	yes	characteristic
201	C₂×Dic₃	D₃×C₈	1	1	yes	characteristic
202	C₂×Dic₃	D₃×C₈	1	1	yes	characteristic
203	D₁₂	—	3	4	no	—
204	C₄×D₃	—	2	1	no	—
205	C₄×D₃	—	2	1	no	—
206	C₄×D₃	—	6	2	no	—
207	C₄×D₃	—	6	2	no	—
208	C₄×D₃	—	6	1	no	—
209	C₄×D₃	—	6	1	no	—
210	C₄×D₃	—	6	1	no	—
211	C₄×D₃	—	6	1	no	—
212	C₃×C₈	—	2	1	no	—
213	C₃×C₈	—	2	1	no	—
214	C₃×C₈	—	6	1	no	—
215	C₃×C₈	—	6	1	no	—
216	C₃×C₈	C₂×D₁₂	1	1	yes	characteristic
217	C₃×C₈	#235	1	1	yes	characteristic
218	C₃×C₈	#235	1	1	yes	characteristic
219	C₃×C₈	C₂×#67	1	1	yes	characteristic
220	C₃⋊C₈	—	2	1	no	—
221	C₃⋊C₈	—	2	1	no	—
222	C₃⋊C₈	—	6	2	no	—
223	C₃⋊C₈	—	6	1	no	—
224	C₃⋊C₈	—	6	1	no	—
225	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
226	C₂×C₃²	C₂²×C₁₆	1	1	yes	characteristic
227	C₂×C₃²	#504	1	1	yes	characteristic
228	C₃⋊D₃	—	2	2	no	—
229	C₃×D₃	—	2	2	no	—
230	C₂×D₄	—	9	1	no	—
231	C₂²×C₄	—	3	1	no	—
232	C₂²×C₄	—	9	1	no	—
233	C₂×C₈	—	3	1	no	—
234	C₂×C₈	—	3	1	no	—
235	C₂×C₈	—	6	1	no	—
236	C₂×C₈	—	6	1	no	—
237	C₂×C₈	—	9	1	no	—
238	C₂×C₈	—	18	1	no	—
239	C₂×C₈	—	18	1	no	—
240	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
241	C₄⋊C₄	—	9	1	no	—
242	C₂²⋊C₄	—	9	1	no	—
243	C₂²⋊C₄	—	9	1	no	—
244	C₄²	—	3	1	no	—
245	C₁₆	—	6	1	no	—
246	C₁₆	—	6	1	no	—
247	C₁₆	—	18	1	no	—
248	C₁₆	#626	1	2	yes	direct product factor
249	C₂²×C₃	—	2	1	no	—
250	C₂²×C₃	—	3	2	no	—
251	C₂²×C₃	—	6	1	no	—
252	C₂²×C₃	—	6	1	no	—
253	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
254	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
255	C₂×D₃	—	2	1	no	—
256	C₂×D₃	—	2	1	no	—
257	C₂×D₃	—	3	2	no	—
258	C₂×D₃	—	3	2	no	—
259	C₂×D₃	—	6	2	no	—
260	C₂×D₃	—	6	2	no	—
261	C₂×D₃	—	6	2	no	—
262	C₂×D₃	—	6	1	no	—
263	C₂×D₃	—	6	1	no	—
264	C₂×D₃	—	6	1	no	—
265	C₂×D₃	—	6	1	no	—
266	C₂×D₃	D₃×C₁₆	1	2	yes	complemented
267	C₃×C₄	—	2	1	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	3	2	no	—
270	C₃×C₄	—	6	1	no	—
271	C₃×C₄	—	6	1	no	—
272	C₃×C₄	C₄×D₁₂	1	1	yes	characteristic
273	C₃×C₄	#901	1	1	yes	characteristic
274	C₃×C₄	#901	1	1	yes	characteristic
275	C₃×C₄	C₄×#67	1	1	yes	characteristic
276	Dic₃	—	2	1	no	—
277	Dic₃	—	2	1	no	—
278	Dic₃	—	6	2	no	—
279	Dic₃	—	6	1	no	—
280	Dic₃	—	6	1	no	—
281	Dic₃	D₃×C₁₆	1	2	yes	complemented
282	C₃²	D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
283	C₂³	—	3	1	no	—
284	C₂³	—	9	1	no	—
285	D₄	—	9	4	no	—
286	C₂×C₄	—	3	1	no	—
287	C₂×C₄	—	3	1	no	—
288	C₂×C₄	—	3	1	no	—
289	C₂×C₄	—	6	1	no	—
290	C₂×C₄	—	6	1	no	—
291	C₂×C₄	—	9	1	no	—
292	C₂×C₄	—	18	1	no	—
293	C₂×C₄	—	18	1	no	—
294	C₂×C₄	C₄×D₃²	1	1	yes	characteristic
295	C₈	—	6	1	no	—
296	C₈	—	6	1	no	—
297	C₈	—	18	1	no	—
298	C₈	#3798	1	1	yes	characteristic
299	C₈	C₂×#626	1	1	yes	characteristic
300	C₂×C₃	—	2	1	no	—
301	C₂×C₃	—	2	1	no	—
302	C₂×C₃	—	2	1	no	—
303	C₂×C₃	—	6	2	no	—
304	C₂×C₃	C₈×D₁₂	1	1	yes	characteristic
305	C₂×C₃	C₂×D₃×C₁₆	1	1	yes	characteristic
306	C₂×C₃	C₂×D₃×C₁₆	1	1	yes	characteristic
307	C₂×C₃	#4901	1	1	yes	characteristic
308	C₂×C₃	#4901	1	1	yes	characteristic
309	C₂×C₃	C₈×#67	1	1	yes	characteristic
310	D₃	—	2	2	no	—
311	D₃	—	6	4	no	—
312	D₃	—	6	2	no	—
313	D₃	—	6	2	no	—
314	C₂²	—	3	2	no	—
315	C₂²	—	6	1	no	—
316	C₂²	—	6	1	no	—
317	C₂²	—	9	2	no	—
318	C₂²	—	18	1	no	—
319	C₂²	—	18	1	no	—
320	C₂²	D₃²×C₈	1	1	yes	characteristic
321	C₄	—	3	2	no	—
322	C₄	—	6	1	no	—
323	C₄	—	6	1	no	—
324	C₄	—	18	1	no	—
325	C₄	#63803	1	1	yes	characteristic
326	C₄	C₄×#626	1	1	yes	characteristic
327	C₃	—	2	1	no	—
328	C₃	C₁₆×D₁₂	1	1	yes	characteristic, complemented
329	C₃	C₁₆×#67	1	1	yes	characteristic, complemented
330	C₂	—	6	2	no	—
331	C₂	—	18	2	no	—
332	C₂	D₃²×C₁₆	1	1	yes	characteristic
333	C₂	#10590227	1	1	yes	characteristic
334	C₂	C₈×#626	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂×C₁₆	D₃²
commutator	C₂×C₃²	C₂²×C₁₆
Frattini	C₂×C₈	C₂×D₃²
Fitting	C₂×C₃²×C₁₆	C₂²
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂×#626	C₈
cosocle	C₂×C₃²×C₈	C₂³
hypocenter	C₃²	D₄×C₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₆×C₃⋊D₃	C₂	semidirect
2. C₂×C₃×D₃×C₁₆	C₂	semidirect
4. C₃×Dic₃×C₁₆	C₂	semidirect
5. Dic₃⋊(C₃⋊C₁₆)	C₂	semidirect
6. #10589108	C₂	semidirect
7. (C₂×D₃)⋊(C₃⋊C₁₆)	C₂	semidirect
28. C₂×D₃×C₁₆	D₃	semidirect
36. Dic₃×C₁₆	D₃	semidirect
53. C₃²×C₁₆	D₄	semidirect
99. #626	C₁₆	direct (N × H)
150. C₃×C₁₆	D₁₂	semidirect
151. C₃×C₁₆	#67	semidirect
160. C₂×C₃⋊D₃	C₂×C₁₆	semidirect
163. C₂×C₃×D₃	C₂×C₁₆	semidirect
169. C₃×Dic₃	C₂×C₁₆	semidirect
248. C₁₆	#626	direct (N × H)
266. C₂×D₃	D₃×C₁₆	semidirect
281. Dic₃	D₃×C₁₆	semidirect
282. C₃²	D₄×C₁₆	semidirect
328. C₃	C₁₆×D₁₂	semidirect
329. C₃	C₁₆×#67	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₁₆×C₃⋊D₃	2
C₂×C₃×D₃×C₁₆	2
C₈×#626	2
C₃×Dic₃×C₁₆	2
Dic₃⋊(C₃⋊C₁₆)	2
#10589108	2
(C₂×D₃)⋊(C₃⋊C₁₆)	2
C₁₆×#67	3
C₁₆×D₁₂	3

Subgroups — C16×#626 #11797449

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₆×#626 #11797449