Subgroups — Q₁₆×#11761611 #30827286

Order 17152.

638 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11761611	C₂	1	2	yes	maximal
2	C₄×Q₁₆×D₆₇	C₂	1	1	yes	maximal, characteristic
3	#22559107	C₂	1	1	yes	maximal, characteristic, complemented
4	#22559105	C₂	1	1	yes	maximal, characteristic, complemented
5	#22559059	C₂	1	2	yes	maximal, complemented
6	#22559056	C₂	1	2	yes	maximal, complemented
7	Q₁₆×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
8	#22558966	C₂	1	1	yes	maximal, characteristic
9	#22558636	C₂	1	2	yes	maximal
10	C₈×#11761611	C₂	1	1	yes	maximal, characteristic
11	C₈×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₁₆×D₆₇	C₄	1	1	yes	characteristic
13	C₄×Q₈×D₆₇	C₂²	1	2	yes	—
14	#16863526	—	2	2	no	—
15	#16863526	C₂²	1	2	yes	—
16	#16863523	—	2	2	no	—
17	#16863523	C₂²	1	2	yes	—
18	Q₈×C₆₇⋊C₈	C₂²	1	2	yes	—
19	C₄×Dic₂₆₈	C₂²	1	1	yes	characteristic
20	Q₁₆×Dic₆₇	C₄	1	1	yes	characteristic
21	#16863490	C₄	1	1	yes	characteristic
22	C₄×#11761618	C₂²	1	2	yes	—
23	#16863488	C₄	1	2	yes	—
24	#16863447	C₂²	1	1	yes	characteristic
25	#107×D₆₇	C₄	1	1	yes	characteristic
26	#16863418	—	2	2	no	—
27	#16863418	C₂²	1	2	yes	—
28	#103×D₆₇	C₄	1	2	yes	—
29	C₄×#11761611	—	2	2	no	—
30	C₄×#11761611	C₂²	1	1	yes	characteristic
31	C₄×C₈×D₆₇	C₂²	1	1	yes	characteristic
32	#16863149	C₂²	1	1	yes	characteristic
33	#16863148	C₂²	1	2	yes	—
34	#16863128	C₂²	1	2	yes	—
35	Q₈⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
36	#16863126	C₂²	1	2	yes	—
37	#16863118	C₂²	1	1	yes	characteristic
38	#16863117	C₂²	1	1	yes	characteristic
39	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
40	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
41	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
42	#335×C₆₇	C₂²	1	1	yes	characteristic
43	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
44	C₈²×C₆₇	C₂²	1	1	yes	characteristic
45	C₂×Q₈×D₆₇	C₂×C₄	1	2	yes	—
46	C₂×Dic₂₆₈	C₂×C₄	1	1	yes	characteristic
47	C₂×#11761618	C₂×C₄	1	2	yes	—
48	Q₁₆×D₆₇	—	2	4	no	—
49	C₂×#11761611	—	2	2	no	—
50	C₂×#11761611	D₄	1	1	yes	characteristic
51	C₂×#11761611	D₄	1	1	yes	characteristic
52	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈×D₆₇	C₂×C₄	1	1	yes	characteristic
54	Q₈×Dic₆₇	C₂×C₄	1	2	yes	—
55	C₄×Dic₁₃₄	—	2	2	no	—
56	C₄×Dic₁₃₄	C₂³	1	2	yes	—
57	#14524008	—	2	2	no	—
58	#14524008	C₂×C₄	1	2	yes	—
59	C₄⋊C₄×D₆₇	—	2	2	no	—
60	C₄⋊C₄×D₆₇	C₂×C₄	1	2	yes	—
61	C₄²×D₆₇	—	2	2	no	—
62	C₄²×D₆₇	C₂³	1	1	yes	characteristic
63	#14523977	C₂×C₄	1	1	yes	characteristic
64	#14523976	C₂×C₄	1	2	yes	—
65	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
66	C₄⋊(C₆₇⋊C₈)	C₂³	1	2	yes	—
67	Dic₆₇⋊C₈	—	2	2	no	—
68	Dic₆₇⋊C₈	—	2	2	no	—
69	Dic₆₇⋊C₈	—	2	1	no	—
70	Dic₆₇⋊C₈	—	2	2	no	—
71	#14523968	C₂×C₄	1	2	yes	—
72	#14523967	C₂×C₄	1	2	yes	—
73	#14523966	C₂×C₄	1	2	yes	—
74	(C₂×D₆₇)⋊C₈	—	2	1	no	—
75	#14523953	—	2	2	no	—
76	#14523953	—	2	2	no	—
77	#14523953	D₄	1	1	yes	characteristic
78	#14523953	D₄	1	1	yes	characteristic
79	C₄×C₆₇⋊C₈	—	2	2	no	—
80	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
81	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
82	C₄×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
83	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
84	C₈×Dic₆₇	C₂×C₄	1	1	yes	characteristic
85	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
86	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
87	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
88	C₄⋊C₈×C₆₇	—	2	2	no	—
89	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
90	#103×C₆₇	C₂×C₄	1	2	yes	—
91	C₄×C₈×C₆₇	—	2	2	no	—
92	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
93	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
94	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
95	C₂×Dic₁₃₄	—	2	2	no	—
96	C₂×Dic₁₃₄	C₂²×C₄	1	2	yes	—
97	Q₈×D₆₇	—	2	4	no	—
98	Q₈×D₆₇	—	4	2	no	—
99	C₂×C₄×D₆₇	—	2	2	no	—
100	C₂×C₄×D₆₇	—	2	2	no	—
101	C₂×C₄×D₆₇	C₂²×C₄	1	1	yes	characteristic
102	C₂×C₄×D₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×D₆₇	#41	1	1	yes	characteristic
104	Dic₂₆₈	—	2	2	no	—
105	#11761618	—	2	4	no	—
106	#11761611	—	2	2	no	—
107	#11761611	—	4	4	no	—
108	#11761611	Q₁₆	1	4	yes	direct product factor
109	C₂×C₆₇⋊C₈	—	2	2	no	—
110	C₂×C₆₇⋊C₈	—	2	1	no	—
111	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₆₇⋊C₈	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
114	C₂×C₆₇⋊C₈	C₂×D₄	1	1	yes	characteristic
115	C₈×D₆₇	—	2	2	no	—
116	C₈×D₆₇	—	4	1	no	—
117	C₄⋊Dic₆₇	—	2	2	no	—
118	C₄⋊Dic₆₇	C₂²×C₄	1	2	yes	—
119	Dic₆₇⋊C₄	—	2	2	no	—
120	Dic₆₇⋊C₄	—	2	2	no	—
121	Dic₆₇⋊C₄	—	2	2	no	—
122	C₄×Dic₆₇	—	2	2	no	—
123	C₄×Dic₆₇	—	2	2	no	—
124	C₄×Dic₆₇	—	2	2	no	—
125	C₄×Dic₆₇	C₂²×C₄	1	1	yes	characteristic
126	C₄×Dic₆₇	C₂×D₄	1	1	yes	characteristic
127	C₄×Dic₆₇	#41	1	1	yes	characteristic
128	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
129	Q₁₆×C₆₇	M₁₆	1	4	yes	complemented
130	C₂×C₈×C₆₇	—	2	2	no	—
131	C₂×C₈×C₆₇	—	2	1	no	—
132	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
134	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
135	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
136	C₄⋊C₄×C₆₇	—	2	2	no	—
137	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
138	C₄²×C₆₇	—	2	2	no	—
139	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
140	C₂²×D₆₇	C₄×D₄	1	1	yes	characteristic
141	Q₈×C₆₇	—	2	2	no	—
142	Q₈×C₆₇	C₂×M₁₆	1	4	yes	—
143	C₂×C₄×C₆₇	—	2	2	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
147	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
148	C₂×Dic₆₇	—	2	2	no	—
149	C₂×Dic₆₇	—	2	2	no	—
150	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
151	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
152	C₂×Dic₆₇	C₄×D₄	1	1	yes	characteristic
153	C₄×D₆₇	—	2	1	no	—
154	C₄×D₆₇	—	2	1	no	—
155	C₄×D₆₇	—	4	1	no	—
156	C₄×D₆₇	—	4	4	no	—
157	C₄×D₆₇	—	8	2	no	—
158	C₄×D₆₇	C₂×Q₁₆	1	2	yes	—
159	C₄×D₆₇	#135	1	2	yes	—
160	Dic₁₃₄	—	2	2	no	—
161	Dic₁₃₄	—	2	2	no	—
162	Dic₁₃₄	—	4	4	no	—
163	C₈×C₆₇	—	2	1	no	—
164	C₈×C₆₇	—	2	1	no	—
165	C₈×C₆₇	—	4	1	no	—
166	C₈×C₆₇	—	4	2	no	—
167	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
168	C₈×C₆₇	C₂×Q₁₆	1	2	yes	complemented
169	C₆₇⋊C₈	—	2	1	no	—
170	C₆₇⋊C₈	—	2	1	no	—
171	C₆₇⋊C₈	—	2	1	no	—
172	C₆₇⋊C₈	—	4	1	no	—
173	C₆₇⋊C₈	—	4	2	no	—
174	C₆₇⋊C₈	C₂×Q₁₆	1	2	yes	complemented
175	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
176	C₂×D₆₇	—	2	1	no	—
177	C₂×D₆₇	—	2	1	no	—
178	C₂×D₆₇	C₄×Q₁₆	1	2	yes	—
179	C₄×C₆₇	—	2	4	no	—
180	C₄×C₆₇	—	2	1	no	—
181	C₄×C₆₇	—	4	2	no	—
182	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
183	C₄×C₆₇	#568	1	1	yes	characteristic
184	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
185	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
186	Dic₆₇	—	2	1	no	—
187	Dic₆₇	—	2	1	no	—
188	Dic₆₇	—	4	2	no	—
189	Dic₆₇	—	4	2	no	—
190	Dic₆₇	C₄×Q₁₆	1	2	yes	—
191	M₁₆×Q₁₆	—	67	1	no	maximal, Sylow
192	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
193	C₂×C₆₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
194	C₂×C₆₇	#2957	1	1	yes	characteristic
195	D₆₇	—	2	2	no	—
196	Q₈×M₁₆	—	67	2	no	—
197	C₂×C₄×Q₁₆	—	67	1	no	—
198	#1591	—	67	1	no	—
199	#1591	—	67	1	no	—
200	#1584	—	67	2	no	—
201	#1584	—	67	2	no	—
202	C₈×Q₁₆	—	67	1	no	—
203	C₈×Q₁₆	D₆₇	1	1	yes	characteristic, complemented
204	#1567	—	67	1	no	—
205	#1487	—	67	2	no	—
206	C₈×M₁₆	—	67	1	no	—
207	C₆₇	M₁₆×Q₁₆	1	1	yes	characteristic, complemented, Sylow
208	C₂²×Q₁₆	—	67	1	no	—
209	C₂×C₄×Q₈	—	67	2	no	—
210	#446	—	67	2	no	—
211	#446	—	67	2	no	—
212	#446	—	134	2	no	—
213	#446	—	134	2	no	—
214	C₈×Q₈	—	67	2	no	—
215	C₈×Q₈	C₂×D₆₇	1	2	yes	—
216	C₄×Q₁₆	—	67	1	no	—
217	C₄×Q₁₆	—	67	2	no	—
218	C₄×Q₁₆	—	67	1	no	—
219	C₄×Q₁₆	—	67	2	no	—
220	C₄×Q₁₆	—	67	1	no	—
221	C₄×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
222	#431	—	67	1	no	—
223	C₂×#107	—	67	1	no	—
224	#423	—	67	2	no	—
225	#423	—	134	2	no	—
226	C₂×#103	—	67	2	no	—
227	C₄×M₁₆	—	67	1	no	—
228	C₄×M₁₆	—	134	2	no	—
229	C₂×C₄×C₈	—	67	1	no	—
230	#335	—	67	2	no	—
231	#335	—	67	1	no	—
232	#335	C₂×D₆₇	1	1	yes	characteristic
233	Q₈⋊C₈	—	67	2	no	—
234	Q₈⋊C₈	—	67	2	no	—
235	Q₈⋊C₈	—	67	2	no	—
236	Q₈⋊C₈	C₂×D₆₇	1	2	yes	—
237	#322	—	67	1	no	—
238	#322	—	67	1	no	—
239	C₈²	—	67	1	no	—
240	C₈²	C₂×D₆₇	1	1	yes	characteristic
241	C₂²×Q₈	—	67	2	no	—
242	C₂×Q₁₆	—	67	2	no	—
243	C₂×Q₁₆	—	67	1	no	—
244	C₂×Q₁₆	—	134	4	no	—
245	C₂×Q₁₆	C₄×D₆₇	1	1	yes	characteristic
246	C₂×M₁₆	—	67	1	no	—
247	C₂×M₁₆	—	67	1	no	—
248	C₂×M₁₆	—	134	2	no	—
249	C₂²×C₈	—	67	1	no	—
250	C₂²×C₈	—	67	1	no	—
251	C₄×Q₈	—	67	2	no	—
252	C₄×Q₈	—	67	2	no	—
253	C₄×Q₈	—	67	2	no	—
254	C₄×Q₈	—	134	2	no	—
255	C₄×Q₈	—	134	2	no	—
256	C₄×Q₈	C₂²×D₆₇	1	2	yes	—
257	C₂×C₄⋊C₄	—	67	2	no	—
258	C₂×C₄⋊C₄	—	134	2	no	—
259	C₂×C₄²	—	67	1	no	—
260	C₂×C₄²	—	134	2	no	—
261	#107	—	67	1	no	—
262	#107	—	67	2	no	—
263	#107	C₄×D₆₇	1	1	yes	characteristic
264	C₄⋊C₈	—	2	2	no	—
265	C₄⋊C₈	—	67	2	no	—
266	C₄⋊C₈	—	134	2	no	—
267	C₄⋊C₈	—	134	2	no	—
268	C₄⋊C₈	—	134	1	no	—
269	C₄⋊C₈	—	134	2	no	—
270	C₄⋊C₈	—	134	2	no	—
271	C₄⋊C₈	C₂²×D₆₇	1	2	yes	—
272	#103	—	67	2	no	—
273	#103	—	67	2	no	—
274	#103	—	67	2	no	—
275	#103	C₄×D₆₇	1	2	yes	—
276	C₂²⋊C₈	—	134	1	no	—
277	#97	—	67	1	no	—
278	#97	—	67	1	no	—
279	#97	—	134	2	no	—
280	#97	—	134	2	no	—
281	C₄×C₈	—	2	2	no	—
282	C₄×C₈	—	67	1	no	—
283	C₄×C₈	—	67	1	no	—
284	C₄×C₈	—	67	1	no	—
285	C₄×C₈	—	67	1	no	—
286	C₄×C₈	—	67	1	no	—
287	C₄×C₈	—	134	2	no	—
288	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
289	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
290	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
291	C₂×Q₈	—	268	2	no	—
292	C₂×Q₈	—	67	2	no	—
293	C₂×Q₈	—	134	2	no	—
294	C₂×Q₈	—	134	4	no	—
295	C₂×Q₈	C₂×C₄×D₆₇	1	2	yes	—
296	C₂²×C₄	—	67	1	no	—
297	C₂²×C₄	—	67	1	no	—
298	C₂²×C₄	—	67	1	no	—
299	C₂²×C₄	—	134	2	no	—
300	C₂²×C₄	—	134	2	no	—
301	Q₁₆	—	134	4	no	—
302	Q₁₆	—	134	2	no	—
303	Q₁₆	#11761611	1	4	yes	direct product factor
304	M₁₆	—	268	4	no	—
305	M₁₆	—	67	4	no	—
306	M₁₆	—	134	2	no	—
307	C₂×C₈	—	2	2	no	—
308	C₂×C₈	—	2	1	no	—
309	C₂×C₈	—	268	1	no	—
310	C₂×C₈	—	67	1	no	—
311	C₂×C₈	—	67	1	no	—
312	C₂×C₈	—	67	1	no	—
313	C₂×C₈	—	67	1	no	—
314	C₂×C₈	—	134	2	no	—
315	C₂×C₈	—	134	1	no	—
316	C₂×C₈	—	134	2	no	—
317	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
318	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
319	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
320	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
321	C₄⋊C₄	—	2	2	no	—
322	C₄⋊C₄	—	67	2	no	—
323	C₄⋊C₄	—	134	2	no	—
324	C₄⋊C₄	—	134	2	no	—
325	C₄⋊C₄	—	134	2	no	—
326	C₄⋊C₄	—	134	2	no	—
327	C₄⋊C₄	C₂×C₄×D₆₇	1	2	yes	—
328	C₄²	—	2	2	no	—
329	C₄²	—	67	1	no	—
330	C₄²	—	67	1	no	—
331	C₄²	—	67	1	no	—
332	C₄²	—	134	2	no	—
333	C₄²	—	134	2	no	—
334	C₄²	—	134	2	no	—
335	C₄²	C₂³×D₆₇	1	1	yes	characteristic
336	C₂³	—	67	1	no	—
337	Q₈	—	2	2	no	—
338	Q₈	—	268	4	no	—
339	Q₈	—	134	2	no	—
340	Q₈	—	134	2	no	—
341	Q₈	C₂×#11761611	1	4	yes	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	268	4	no	—
345	C₂×C₄	—	268	1	no	—
346	C₂×C₄	—	536	2	no	—
347	C₂×C₄	—	67	1	no	—
348	C₂×C₄	—	67	2	no	—
349	C₂×C₄	—	67	2	no	—
350	C₂×C₄	—	67	1	no	—
351	C₂×C₄	—	67	1	no	—
352	C₂×C₄	—	134	2	no	—
353	C₂×C₄	—	134	2	no	—
354	C₂×C₄	—	134	1	no	—
355	C₂×C₄	—	134	1	no	—
356	C₂×C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
357	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
358	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
359	C₈	—	2	1	no	—
360	C₈	—	2	1	no	—
361	C₈	—	4	1	no	—
362	C₈	—	4	2	no	—
363	C₈	—	268	1	no	—
364	C₈	—	268	2	no	—
365	C₈	—	67	2	no	—
366	C₈	—	134	1	no	—
367	C₈	—	134	1	no	—
368	C₈	—	134	1	no	—
369	C₈	C₂×#11761611	1	2	yes	—
370	C₈	Q₁₆×D₆₇	1	2	yes	complemented
371	C₂²	—	67	2	no	—
372	C₂²	—	134	1	no	—
373	C₂²	—	134	1	no	—
374	C₂²	C₄×D₄×D₆₇	1	1	yes	characteristic
375	C₄	—	2	1	no	—
376	C₄	—	2	4	no	—
377	C₄	—	4	2	no	—
378	C₄	—	268	2	no	—
379	C₄	—	268	2	no	—
380	C₄	—	67	2	no	—
381	C₄	—	134	1	no	—
382	C₄	—	134	1	no	—
383	C₄	C₂²×#11761611	1	1	yes	characteristic
384	C₄	#16864161	1	1	yes	characteristic
385	C₄	C₂×Q₁₆×D₆₇	1	1	yes	characteristic
386	C₄	#135×D₆₇	1	1	yes	characteristic
387	C₂	—	134	2	no	—
388	C₂	D₄×#11761611	1	1	yes	characteristic
389	C₂	C₄×Q₁₆×D₆₇	1	1	yes	characteristic
390	C₂	#22569121	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×D₄×D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₄²	C₂³×D₆₇
Fitting	C₈×Q₁₆×C₆₇	C₂
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂²×D₆₇	C₄×D₄
cosocle	C₄²×C₆₇	C₂⁴
hypercenter	C₈×Q₁₆	D₆₇
hypocenter	C₆₇	M₁₆×Q₁₆
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×C₈ ◁ C₈×Q₁₆

Complements

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

N	H ≅ G/N	type
3. #22559107	C₂	semidirect
4. #22559105	C₂	semidirect
5. #22559059	C₂	semidirect
6. #22559056	C₂	semidirect
7. Q₁₆×C₆₇⋊C₈	C₂	semidirect
11. C₈×Q₁₆×C₆₇	C₂	semidirect
108. #11761611	Q₁₆	direct (N × H)
129. Q₁₆×C₆₇	M₁₆	semidirect
168. C₈×C₆₇	C₂×Q₁₆	semidirect
174. C₆₇⋊C₈	C₂×Q₁₆	semidirect
203. C₈×Q₁₆	D₆₇	semidirect
207. C₆₇	M₁₆×Q₁₆	semidirect
303. Q₁₆	#11761611	direct (N × H)
370. C₈	Q₁₆×D₆₇	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11761611	2
C₄×Q₁₆×D₆₇	2
#22559107	2
#22559105	2
#22559059	2
#22559056	2
Q₁₆×C₆₇⋊C₈	2
#22558966	2
#22558636	2
C₈×#11761611	2
C₈×Q₁₆×C₆₇	2
M₁₆×Q₁₆	67

Subgroups — Q16×#11761611 #30827286

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₁₆×#11761611 #30827286