Subgroups — #67⋊(C₃⋊C₁₆) #11797447

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	C₂²×C₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃×C₈×#67	C₂	1	1	yes	maximal, characteristic
3	C₂×D₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃×#4549	C₂	1	1	yes	maximal, characteristic, complemented
5	Dic₃⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
6	(C₂×D₃)⋊(C₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
7	Dic₃×C₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	D₄×C₃⋊C₁₆	—	3	1	no	maximal
9	#67⋊C₁₆	—	3	1	no	maximal
10	C₂²×C₃²×C₈	C₂²	1	1	yes	characteristic
11	C₃×C₄×#67	C₄	1	1	yes	characteristic
12	C₂×C₃×D₃×C₈	C₂²	1	1	yes	characteristic
13	C₂×C₃⋊(C₃⋊C₁₆)	—	2	2	no	—
14	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
15	C₂×C₃⋊(C₃⋊C₁₆)	C₂²	1	1	yes	characteristic
16	C₃×#830	C₄	1	1	yes	characteristic
17	C₃×(C₂×D₃)⋊C₈	C₄	1	1	yes	characteristic
18	C₃×Dic₃⋊C₈	C₄	1	1	yes	characteristic
19	C₃×C₈×Dic₃	C₂²	1	1	yes	characteristic
20	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₂×C₃×C₃⋊C₁₆	C₂²	1	1	yes	characteristic
22	D₃×C₃⋊C₁₆	—	2	2	no	—
23	C₃×C₈×D₄	—	3	1	no	—
24	C₈×#67	D₃	1	1	yes	characteristic
25	C₂²×C₃⋊C₁₆	—	3	1	no	—
26	C₂²×C₃⋊C₁₆	—	3	1	no	—
27	C₂²×C₃⋊C₁₆	—	3	1	no	—
28	C₂²×C₃⋊C₁₆	—	6	1	no	—
29	C₂×D₃×C₁₆	—	3	1	no	—
30	C₃×C₂²⋊C₁₆	—	3	1	no	—
31	#4549	—	3	1	no	—
32	#4549	D₃	1	1	yes	characteristic, complemented
33	(C₂×D₃)⋊C₁₆	—	3	1	no	—
34	Dic₃⋊C₁₆	—	3	1	no	—
35	Dic₃×C₁₆	—	3	1	no	—
36	C₄⋊(C₃⋊C₁₆)	—	3	1	no	—
37	C₄×C₃⋊C₁₆	—	3	1	no	—
38	C₂²×C₃²×C₄	C₂×C₄	1	1	yes	characteristic
39	C₂×C₃×#67	C₈	1	1	yes	characteristic
40	C₂×C₃×C₄×D₃	C₂×C₄	1	1	yes	characteristic
41	C₂×C₃²×C₈	—	2	1	no	—
42	C₂×C₃²×C₈	—	2	1	no	—
43	C₂×C₃²×C₈	C₂×C₄	1	1	yes	characteristic
44	C₂×C₃²×C₈	C₂³	1	1	yes	characteristic
45	C₃×#217	C₈	1	1	yes	characteristic
46	C₃×(C₂×D₃)⋊C₄	C₈	1	1	yes	characteristic
47	C₃×Dic₃⋊C₄	C₈	1	1	yes	characteristic
48	C₃×C₄×Dic₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
50	C₂×C₃×C₃⋊C₈	C₂×C₄	1	1	yes	characteristic
51	C₃×D₃×C₈	—	2	1	no	—
52	C₃×D₃×C₈	—	2	1	no	—
53	C₃⋊(C₃⋊C₁₆)	—	2	1	no	—
54	C₃⋊(C₃⋊C₁₆)	D₄	1	2	yes	complemented
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	—	2	1	no	—
59	C₂²×C₃×C₈	—	3	1	no	—
60	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
61	C₂²×C₃×C₈	C₂×D₃	1	1	yes	characteristic
62	C₃×C₄×D₄	—	3	1	no	—
63	C₄×#67	Dic₃	1	1	yes	characteristic
64	C₂×D₃×C₈	C₂×D₃	1	1	yes	characteristic
65	C₂×C₃×C₁₆	—	3	1	no	—
66	C₂×C₃×C₁₆	—	3	1	no	—
67	C₃×C₄⋊C₈	—	3	1	no	—
68	C₃×C₂²⋊C₈	—	3	1	no	—
69	C₃×C₂²⋊C₈	—	3	1	no	—
70	C₃×C₄×C₈	—	3	1	no	—
71	#830	Dic₃	1	1	yes	characteristic
72	(C₂×D₃)⋊C₈	Dic₃	1	1	yes	characteristic
73	Dic₃⋊C₈	Dic₃	1	1	yes	characteristic
74	C₈×Dic₃	C₂×D₃	1	1	yes	characteristic
75	C₂×C₃⋊C₁₆	—	3	1	no	—
76	C₂×C₃⋊C₁₆	—	3	1	no	—
77	C₂×C₃⋊C₁₆	—	3	1	no	—
78	C₂×C₃⋊C₁₆	—	3	1	no	—
79	C₂×C₃⋊C₁₆	—	6	1	no	—
80	C₂×C₃⋊C₁₆	—	6	1	no	—
81	C₂×C₃⋊C₁₆	—	6	2	no	—
82	C₂×C₃⋊C₁₆	—	6	2	no	—
83	C₂×C₃⋊C₁₆	—	6	2	no	—
84	C₂×C₃⋊C₁₆	—	6	4	no	—
85	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
86	C₂×C₃⋊C₁₆	C₂×D₃	1	1	yes	characteristic
87	D₃×C₁₆	—	6	2	no	—
88	C₂³×C₃²	C₂×C₈	1	1	yes	characteristic
89	C₂²×C₃×D₃	C₂×C₈	1	1	yes	characteristic
90	C₂×C₃²×C₄	—	2	1	no	—
91	C₂×C₃²×C₄	—	2	1	no	—
92	C₂×C₃²×C₄	C₂×C₈	1	1	yes	characteristic
93	C₂×C₃²×C₄	C₂²×C₄	1	1	yes	characteristic
94	C₃×#67	C₁₆	1	4	yes	complemented
95	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
96	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
97	C₂×C₃×Dic₃	C₂×C₈	1	1	yes	characteristic
98	C₃×C₄×D₃	—	2	1	no	—
99	C₃×C₄×D₃	—	2	1	no	—
100	C₃²×C₈	—	2	1	no	—
101	C₃²×C₈	C₂×D₄	1	1	yes	characteristic
102	C₃²×C₈	#41	1	1	yes	characteristic
103	C₃×C₃⋊C₈	—	2	1	no	—
104	C₃×C₃⋊C₈	—	2	1	no	—
105	C₂²×C₁₆	—	9	1	no	—
106	C₂²×C₁₆	—	9	1	no	—
107	C₈×D₄	—	3	1	no	—
108	C₄⋊C₁₆	—	9	1	no	—
109	C₂²⋊C₁₆	—	3	1	no	—
110	C₂²⋊C₁₆	—	9	1	no	—
111	C₄×C₁₆	—	9	1	no	—
112	C₂×C₃×D₄	—	3	1	no	—
113	C₂²×C₃×C₄	—	2	1	no	—
114	C₂²×C₃×C₄	—	3	1	no	—
115	C₂²×C₃×C₄	C₄×D₃	1	1	yes	characteristic
116	C₂²×C₃×C₄	C₂×Dic₃	1	1	yes	characteristic
117	C₂×#67	C₃⋊C₈	1	1	yes	characteristic
118	C₂×C₄×D₃	C₂×Dic₃	1	1	yes	characteristic
119	C₂×C₃×C₈	—	2	2	no	—
120	C₂×C₃×C₈	—	2	2	no	—
121	C₂×C₃×C₈	—	2	1	no	—
122	C₂×C₃×C₈	—	2	1	no	—
123	C₂×C₃×C₈	—	2	1	no	—
124	C₂×C₃×C₈	—	2	1	no	—
125	C₂×C₃×C₈	—	2	1	no	—
126	C₂×C₃×C₈	—	2	1	no	—
127	C₂×C₃×C₈	—	3	1	no	—
128	C₂×C₃×C₈	—	3	1	no	—
129	C₂×C₃×C₈	—	6	1	no	—
130	C₂×C₃×C₈	—	6	1	no	—
131	C₂×C₃×C₈	C₄×D₃	1	1	yes	characteristic
132	C₂×C₃×C₈	C₂×Dic₃	1	1	yes	characteristic
133	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
134	C₂×C₃×C₈	C₂²×D₃	1	1	yes	characteristic
135	C₃×C₄⋊C₄	—	3	1	no	—
136	C₃×C₂²⋊C₄	—	3	1	no	—
137	C₃×C₂²⋊C₄	—	3	1	no	—
138	C₃×C₄²	—	3	1	no	—
139	#217	C₃⋊C₈	1	1	yes	characteristic
140	(C₂×D₃)⋊C₄	C₃⋊C₈	1	1	yes	characteristic
141	Dic₃⋊C₄	C₃⋊C₈	1	1	yes	characteristic
142	C₄×Dic₃	C₂×Dic₃	1	1	yes	characteristic
143	C₂×C₃⋊C₈	C₂×Dic₃	1	1	yes	characteristic
144	C₂×C₃⋊C₈	C₂×Dic₃	1	1	yes	characteristic
145	D₃×C₈	—	2	1	no	—
146	D₃×C₈	—	2	1	no	—
147	C₃×C₁₆	—	6	1	no	—
148	C₃×C₁₆	—	6	1	no	—
149	C₃⋊C₁₆	—	2	1	no	—
150	C₃⋊C₁₆	—	2	1	no	—
151	C₃⋊C₁₆	—	3	2	no	—
152	C₃⋊C₁₆	—	3	2	no	—
153	C₃⋊C₁₆	—	6	2	no	—
154	C₃⋊C₁₆	—	6	2	no	—
155	C₃⋊C₁₆	—	6	1	no	—
156	C₃⋊C₁₆	—	6	1	no	—
157	C₂²×C₃²	—	2	1	no	—
158	C₂²×C₃²	—	2	1	no	—
159	C₂²×C₃²	C₂×C₁₆	1	2	yes	complemented
160	C₂²×C₃²	C₂²×C₈	1	1	yes	characteristic
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	—	2	1	no	—
165	C₃²×C₄	C₄×D₄	1	1	yes	characteristic
166	C₃²×C₄	#131	1	1	yes	characteristic
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	D₃²	1	1	yes	characteristic
172	C₄×D₄	—	3	1	no	—
173	C₂×C₁₆	—	3	1	no	—
174	C₂×C₁₆	—	3	1	no	—
175	C₂×C₁₆	—	9	1	no	—
176	C₂×C₁₆	—	9	1	no	—
177	C₂×C₁₆	—	18	2	no	—
178	C₂×C₁₆	—	18	2	no	—
179	C₄⋊C₈	—	3	1	no	—
180	C₂²⋊C₈	—	3	1	no	—
181	C₂²⋊C₈	—	3	1	no	—
182	C₄×C₈	—	3	1	no	—
183	C₂³×C₃	—	2	1	no	—
184	C₂³×C₃	—	3	1	no	—
185	C₂³×C₃	D₃×C₈	1	1	yes	characteristic
186	C₂³×C₃	C₂×C₃⋊C₈	1	1	yes	characteristic
187	C₂²×D₃	C₂×C₃⋊C₈	1	1	yes	characteristic
188	C₃×D₄	—	3	4	no	—
189	C₂×C₃×C₄	—	2	2	no	—
190	C₂×C₃×C₄	—	2	2	no	—
191	C₂×C₃×C₄	—	2	1	no	—
192	C₂×C₃×C₄	—	2	1	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	—	3	1	no	—
198	C₂×C₃×C₄	—	3	1	no	—
199	C₂×C₃×C₄	—	3	1	no	—
200	C₂×C₃×C₄	—	6	1	no	—
201	C₂×C₃×C₄	—	6	1	no	—
202	C₂×C₃×C₄	D₃×C₈	1	1	yes	characteristic
203	C₂×C₃×C₄	C₂×C₃⋊C₈	1	1	yes	characteristic
204	C₂×C₃×C₄	C₂×C₄×D₃	1	1	yes	characteristic
205	C₂×C₃×C₄	C₂²×Dic₃	1	1	yes	characteristic
206	#67	C₃⋊C₁₆	1	4	yes	complemented
207	C₂×Dic₃	C₂×C₃⋊C₈	1	1	yes	characteristic
208	C₂×Dic₃	C₂×C₃⋊C₈	1	1	yes	characteristic
209	C₂×Dic₃	C₂×C₃⋊C₈	1	1	yes	characteristic
210	C₄×D₃	—	2	1	no	—
211	C₄×D₃	—	2	1	no	—
212	C₃×C₈	—	2	2	no	—
213	C₃×C₈	—	2	1	no	—
214	C₃×C₈	—	2	1	no	—
215	C₃×C₈	—	2	1	no	—
216	C₃×C₈	—	2	1	no	—
217	C₃×C₈	—	6	1	no	—
218	C₃×C₈	—	6	1	no	—
219	C₃×C₈	D₃×D₄	1	1	yes	characteristic
220	C₃×C₈	D₃×D₄	1	1	yes	characteristic
221	C₃×C₈	#237	1	1	yes	characteristic
222	C₃×C₈	#237	1	1	yes	characteristic
223	C₃⋊C₈	—	2	1	no	—
224	C₃⋊C₈	—	2	1	no	—
225	C₂×C₃²	—	2	2	no	—
226	C₂×C₃²	C₈×D₄	1	1	yes	characteristic
227	C₂×C₃²	C₂²×C₁₆	1	1	yes	characteristic
228	C₂×C₃²	#504	1	1	yes	characteristic
229	C₃×D₃	—	2	2	no	—
230	C₂×D₄	—	3	1	no	—
231	C₂²×C₄	—	3	1	no	—
232	C₂²×C₄	D₃×Dic₃	1	1	yes	characteristic
233	C₂×C₈	—	2	1	no	—
234	C₂×C₈	—	2	1	no	—
235	C₂×C₈	—	3	1	no	—
236	C₂×C₈	—	3	1	no	—
237	C₂×C₈	—	6	1	no	—
238	C₂×C₈	—	6	1	no	—
239	C₂×C₈	D₃×Dic₃	1	1	yes	characteristic
240	C₂×C₈	C₂×D₃²	1	1	yes	characteristic
241	C₄⋊C₄	—	3	1	no	—
242	C₂²⋊C₄	—	3	1	no	—
243	C₂²⋊C₄	—	3	1	no	—
244	C₄²	—	3	1	no	—
245	C₁₆	—	6	1	no	—
246	C₁₆	—	6	1	no	—
247	C₁₆	—	9	2	no	—
248	C₁₆	—	18	1	no	—
249	C₂²×C₃	—	2	2	no	—
250	C₂²×C₃	—	2	2	no	—
251	C₂²×C₃	—	2	1	no	—
252	C₂²×C₃	—	2	1	no	—
253	C₂²×C₃	—	2	1	no	—
254	C₂²×C₃	—	2	1	no	—
255	C₂²×C₃	—	2	2	no	—
256	C₂²×C₃	—	2	1	no	—
257	C₂²×C₃	—	3	2	no	—
258	C₂²×C₃	—	6	1	no	—
259	C₂²×C₃	—	6	1	no	—
260	C₂²×C₃	D₃×C₁₆	1	2	yes	complemented
261	C₂²×C₃	C₂×C₃⋊C₁₆	1	2	yes	complemented
262	C₂²×C₃	C₂×D₃×C₈	1	1	yes	characteristic
263	C₂²×C₃	C₂²×C₃⋊C₈	1	1	yes	characteristic
264	C₂×D₃	—	2	1	no	—
265	C₂×D₃	—	2	1	no	—
266	C₂×D₃	C₂×C₃⋊C₁₆	1	2	yes	complemented
267	C₃×C₄	—	2	2	no	—
268	C₃×C₄	—	2	1	no	—
269	C₃×C₄	—	2	1	no	—
270	C₃×C₄	—	2	1	no	—
271	C₃×C₄	—	2	1	no	—
272	C₃×C₄	—	3	2	no	—
273	C₃×C₄	—	6	1	no	—
274	C₃×C₄	—	6	1	no	—
275	C₃×C₄	#67⋊C₄	1	1	yes	characteristic
276	C₃×C₄	#907	1	1	yes	characteristic
277	C₃×C₄	D₄×Dic₃	1	1	yes	characteristic
278	C₃×C₄	#948	1	1	yes	characteristic
279	Dic₃	—	2	1	no	—
280	Dic₃	—	2	1	no	—
281	Dic₃	C₂×C₃⋊C₁₆	1	2	yes	complemented
282	C₃²	D₄×C₁₆	1	1	yes	characteristic, complemented, Sylow
283	C₂³	—	3	1	no	—
284	C₂³	D₃×C₃⋊C₈	1	1	yes	characteristic
285	D₄	—	3	4	no	—
286	C₂×C₄	—	2	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	3	1	no	—
289	C₂×C₄	—	3	1	no	—
290	C₂×C₄	—	3	1	no	—
291	C₂×C₄	—	6	1	no	—
292	C₂×C₄	—	6	1	no	—
293	C₂×C₄	D₃×C₃⋊C₈	1	1	yes	characteristic
294	C₂×C₄	C₂×D₃×Dic₃	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	6	1	no	—
297	C₈	—	6	1	no	—
298	C₈	#3805	1	1	yes	characteristic
299	C₈	#3811	1	1	yes	characteristic
300	C₂×C₃	—	2	4	no	—
301	C₂×C₃	—	2	2	no	—
302	C₂×C₃	—	2	2	no	—
303	C₂×C₃	—	2	1	no	—
304	C₂×C₃	—	2	1	no	—
305	C₂×C₃	—	2	1	no	—
306	C₂×C₃	—	6	2	no	—
307	C₂×C₃	#67⋊C₈	1	1	yes	characteristic
308	C₂×C₃	C₂×D₃×C₁₆	1	1	yes	characteristic
309	C₂×C₃	#4907	1	1	yes	characteristic
310	C₂×C₃	D₄×C₃⋊C₈	1	1	yes	characteristic
311	C₂×C₃	C₂²×C₃⋊C₁₆	1	1	yes	characteristic
312	C₂×C₃	#5140	1	1	yes	characteristic
313	D₃	—	2	2	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	2	1	no	—
316	C₂²	—	3	2	no	—
317	C₂²	—	6	1	no	—
318	C₂²	—	6	1	no	—
319	C₂²	D₃×C₃⋊C₁₆	1	2	yes	complemented
320	C₂²	C₂×D₃×C₃⋊C₈	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	3	2	no	—
323	C₄	—	6	1	no	—
324	C₄	—	6	1	no	—
325	C₄	#63815	1	1	yes	characteristic
326	C₄	#67⋊Dic₃	1	1	yes	characteristic
327	C₃	—	2	1	no	—
328	C₃	#67⋊C₁₆	1	1	yes	characteristic, complemented
329	C₃	D₄×C₃⋊C₁₆	1	1	yes	characteristic, complemented
330	C₂	—	2	2	no	—
331	C₂	—	6	2	no	—
332	C₂	C₂×D₃×C₃⋊C₁₆	1	1	yes	characteristic
333	C₂	#10590622	1	1	yes	characteristic
334	C₂	#67⋊(C₃⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂×D₃²
commutator	C₂×C₃²	C₂²×C₁₆
Fitting	C₂²×C₃²×C₈	C₂²
socle	C₂²×C₃²	C₂²×C₈
prime radical	C₂×C₃×#67	C₈
cosocle	C₂×C₃²×C₈	C₂³
hypercenter	C₂²×C₈	D₃²
hypocenter	C₃²	D₄×C₁₆
LCS	G ▷ C₂×C₃² ▷ C₃²
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃⋊(C₃⋊C₁₆)	C₂	semidirect
3. C₂×D₃×C₃⋊C₁₆	C₂	semidirect
4. C₃×#4549	C₂	semidirect
5. Dic₃⋊(C₃⋊C₁₆)	C₂	semidirect
6. (C₂×D₃)⋊(C₃⋊C₁₆)	C₂	semidirect
7. Dic₃×C₃⋊C₁₆	C₂	semidirect
32. #4549	D₃	semidirect
54. C₃⋊(C₃⋊C₁₆)	D₄	semidirect
94. C₃×#67	C₁₆	semidirect
159. C₂²×C₃²	C₂×C₁₆	semidirect
163. C₂×C₃×D₃	C₂×C₁₆	semidirect
169. C₃×Dic₃	C₂×C₁₆	semidirect
206. #67	C₃⋊C₁₆	semidirect
260. C₂²×C₃	D₃×C₁₆	semidirect
261. C₂²×C₃	C₂×C₃⋊C₁₆	semidirect
266. C₂×D₃	C₂×C₃⋊C₁₆	semidirect
281. Dic₃	C₂×C₃⋊C₁₆	semidirect
282. C₃²	D₄×C₁₆	semidirect
319. C₂²	D₃×C₃⋊C₁₆	semidirect
328. C₃	#67⋊C₁₆	semidirect
329. C₃	D₄×C₃⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₃⋊(C₃⋊C₁₆)	2
C₃×C₈×#67	2
C₂×D₃×C₃⋊C₁₆	2
C₃×#4549	2
Dic₃⋊(C₃⋊C₁₆)	2
(C₂×D₃)⋊(C₃⋊C₁₆)	2
Dic₃×C₃⋊C₁₆	2
D₄×C₃⋊C₁₆	3
#67⋊C₁₆	3

Subgroups — #67⋊(C3⋊C16) #11797447

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #67⋊(C₃⋊C₁₆) #11797447