Subgroups — M₁₆×Q₁₆×C₆₇ #30697289

Order 17152.

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	Q₈×M₁₆×C₆₇	C₂	1	2	yes	maximal
2	C₂×C₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
3	C₆₇×#1591	C₂	1	2	yes	maximal, complemented
4	C₆₇×#1584	C₂	1	4	yes	maximal, complemented
5	C₈×Q₁₆×C₆₇	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1567	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1487	C₂	1	2	yes	maximal
8	C₈×M₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
11	#446×C₆₇	—	2	4	no	—
12	#446×C₆₇	C₂²	1	4	yes	—
13	C₈×Q₈×C₆₇	C₂²	1	4	yes	—
14	C₄×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
15	C₄×Q₁₆×C₆₇	C₄	1	2	yes	—
16	C₄×Q₁₆×C₆₇	C₄	1	1	yes	characteristic
17	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
18	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
19	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
20	#431×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×#107×C₆₇	C₄	1	1	yes	characteristic
22	#423×C₆₇	—	2	2	no	—
23	#423×C₆₇	C₂²	1	2	yes	—
24	C₂×#103×C₆₇	C₄	1	2	yes	—
25	C₄×M₁₆×C₆₇	—	2	2	no	—
26	C₄×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
28	#335×C₆₇	C₂²	1	2	yes	—
29	#335×C₆₇	C₂²	1	2	yes	—
30	Q₈⋊C₈×C₆₇	C₂²	1	4	yes	—
31	Q₈⋊C₈×C₆₇	C₂²	1	4	yes	—
32	#322×C₆₇	C₂²	1	2	yes	—
33	C₈²×C₆₇	C₂²	1	2	yes	—
34	C₂²×Q₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₂×Q₁₆×C₆₇	—	2	4	no	—
36	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
37	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
38	C₂×Q₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
39	C₂×M₁₆×C₆₇	—	2	2	no	—
40	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
41	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
42	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₄×Q₈×C₆₇	—	2	2	no	—
45	C₄×Q₈×C₆₇	—	2	2	no	—
46	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
47	C₄×Q₈×C₆₇	C₂×C₄	1	2	yes	—
48	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
49	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
50	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
51	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₆₇	—	2	2	no	—
53	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
54	#107×C₆₇	C₂×C₄	1	2	yes	—
55	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
56	#107×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₄⋊C₈×C₆₇	—	2	4	no	—
58	C₄⋊C₈×C₆₇	—	2	4	no	—
59	C₄⋊C₈×C₆₇	—	2	2	no	—
60	C₄⋊C₈×C₆₇	—	2	1	no	—
61	C₄⋊C₈×C₆₇	C₂³	1	4	yes	—
62	#103×C₆₇	C₂×C₄	1	2	yes	—
63	#103×C₆₇	C₂×C₄	1	2	yes	—
64	#103×C₆₇	C₂×C₄	1	2	yes	—
65	#103×C₆₇	C₂×C₄	1	2	yes	—
66	C₂²⋊C₈×C₆₇	—	2	1	no	—
67	#97×C₆₇	—	2	4	no	—
68	#97×C₆₇	D₄	1	2	yes	—
69	C₄×C₈×C₆₇	—	2	4	no	—
70	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
71	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
72	C₄×C₈×C₆₇	C₂³	1	2	yes	—
73	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₆₇	C₂³	1	2	yes	—
76	C₂×Q₈×C₆₇	—	2	2	no	—
77	C₂×Q₈×C₆₇	—	2	4	no	—
78	C₂×Q₈×C₆₇	—	4	2	no	—
79	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
80	C₂×Q₈×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₆₇	—	2	2	no	—
82	C₂²×C₄×C₆₇	—	2	2	no	—
83	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
86	Q₁₆×C₆₇	—	2	4	no	—
87	Q₁₆×C₆₇	—	2	2	no	—
88	Q₁₆×C₆₇	M₁₆	1	4	yes	direct product factor
89	M₁₆×C₆₇	—	2	2	no	—
90	M₁₆×C₆₇	—	4	4	no	—
91	M₁₆×C₆₇	Q₁₆	1	4	yes	direct product factor
92	C₂×C₈×C₆₇	—	2	2	no	—
93	C₂×C₈×C₆₇	—	2	4	no	—
94	C₂×C₈×C₆₇	—	2	2	no	—
95	C₂×C₈×C₆₇	—	4	1	no	—
96	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
98	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
102	C₄⋊C₄×C₆₇	—	2	2	no	—
103	C₄⋊C₄×C₆₇	—	2	2	no	—
104	C₄⋊C₄×C₆₇	—	2	2	no	—
105	C₄⋊C₄×C₆₇	—	2	2	no	—
106	C₄⋊C₄×C₆₇	—	2	2	no	—
107	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
108	C₄⋊C₄×C₆₇	C₂²×C₄	1	2	yes	—
109	C₄²×C₆₇	—	2	2	no	—
110	C₄²×C₆₇	—	2	2	no	—
111	C₄²×C₆₇	—	2	2	no	—
112	C₄²×C₆₇	—	2	2	no	—
113	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
114	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
115	C₄²×C₆₇	#41	1	1	yes	characteristic
116	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
117	C₂³×C₆₇	C₄×D₄	1	1	yes	characteristic
118	Q₈×C₆₇	—	2	2	no	—
119	Q₈×C₆₇	—	2	2	no	—
120	Q₈×C₆₇	—	2	2	no	—
121	Q₈×C₆₇	—	4	4	no	—
122	Q₈×C₆₇	C₂×M₁₆	1	4	yes	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	1	no	—
125	C₂×C₄×C₆₇	—	2	1	no	—
126	C₂×C₄×C₆₇	—	2	2	no	—
127	C₂×C₄×C₆₇	—	2	2	no	—
128	C₂×C₄×C₆₇	—	2	2	no	—
129	C₂×C₄×C₆₇	—	4	1	no	—
130	C₂×C₄×C₆₇	—	4	4	no	—
131	C₂×C₄×C₆₇	—	8	2	no	—
132	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
136	C₂×C₄×C₆₇	#135	1	2	yes	—
137	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
140	C₈×C₆₇	—	2	1	no	—
141	C₈×C₆₇	—	2	1	no	—
142	C₈×C₆₇	—	2	1	no	—
143	C₈×C₆₇	—	2	2	no	—
144	C₈×C₆₇	—	4	2	no	—
145	C₈×C₆₇	—	4	4	no	—
146	C₈×C₆₇	C₂×M₁₆	1	2	yes	—
147	C₈×C₆₇	C₂×Q₁₆	1	4	yes	complemented
148	C₂²×C₆₇	—	2	1	no	—
149	C₂²×C₆₇	—	2	1	no	—
150	C₂²×C₆₇	C₄×Q₁₆	1	2	yes	—
151	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
152	C₄×C₆₇	—	2	1	no	—
153	C₄×C₆₇	—	2	4	no	—
154	C₄×C₆₇	—	2	1	no	—
155	C₄×C₆₇	—	2	1	no	—
156	C₄×C₆₇	—	4	2	no	—
157	C₄×C₆₇	—	4	2	no	—
158	C₄×C₆₇	—	4	2	no	—
159	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
160	C₄×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
161	C₄×C₆₇	#568	1	1	yes	characteristic
162	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
163	C₄×C₆₇	C₂×#135	1	1	yes	characteristic
164	M₁₆×Q₁₆	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
165	C₂×C₆₇	—	2	2	no	—
166	C₂×C₆₇	D₄×M₁₆	1	1	yes	characteristic
167	C₂×C₆₇	C₂×C₄×Q₁₆	1	1	yes	characteristic
168	C₂×C₆₇	#2957	1	1	yes	characteristic
169	Q₈×M₁₆	C₂×C₆₇	1	2	yes	—
170	C₂×C₄×Q₁₆	C₂×C₆₇	1	1	yes	characteristic
171	#1591	C₂×C₆₇	1	2	yes	complemented
172	#1584	C₂×C₆₇	1	4	yes	complemented
173	C₈×Q₁₆	C₂×C₆₇	1	2	yes	complemented
174	#1567	C₂×C₆₇	1	1	yes	characteristic
175	#1487	C₂×C₆₇	1	2	yes	—
176	C₈×M₁₆	C₂×C₆₇	1	1	yes	characteristic
177	C₆₇	M₁₆×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
178	C₂²×Q₁₆	C₄×C₆₇	1	1	yes	characteristic
179	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
180	#446	—	2	4	no	—
181	#446	C₂²×C₆₇	1	4	yes	—
182	C₈×Q₈	C₂²×C₆₇	1	4	yes	—
183	C₄×Q₁₆	C₄×C₆₇	1	2	yes	—
184	C₄×Q₁₆	C₄×C₆₇	1	1	yes	characteristic
185	C₄×Q₁₆	C₄×C₆₇	1	1	yes	characteristic
186	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
187	C₄×Q₁₆	C₂²×C₆₇	1	2	yes	—
188	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
189	#431	C₂²×C₆₇	1	1	yes	characteristic
190	C₂×#107	C₄×C₆₇	1	1	yes	characteristic
191	#423	—	2	2	no	—
192	#423	C₂²×C₆₇	1	2	yes	—
193	C₂×#103	C₄×C₆₇	1	2	yes	—
194	C₄×M₁₆	—	2	2	no	—
195	C₄×M₁₆	C₂²×C₆₇	1	1	yes	characteristic
196	C₂×C₄×C₈	C₂²×C₆₇	1	1	yes	characteristic
197	#335	C₂²×C₆₇	1	2	yes	—
198	#335	C₂²×C₆₇	1	2	yes	—
199	Q₈⋊C₈	C₂²×C₆₇	1	4	yes	—
200	Q₈⋊C₈	C₂²×C₆₇	1	4	yes	—
201	#322	C₂²×C₆₇	1	2	yes	—
202	C₈²	C₂²×C₆₇	1	2	yes	—
203	C₂²×Q₈	C₂×C₄×C₆₇	1	2	yes	—
204	C₂×Q₁₆	—	2	4	no	—
205	C₂×Q₁₆	C₂×C₄×C₆₇	1	2	yes	—
206	C₂×Q₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
207	C₂×Q₁₆	C₂×C₄×C₆₇	1	1	yes	characteristic
208	C₂×M₁₆	—	2	2	no	—
209	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
210	C₂×M₁₆	D₄×C₆₇	1	1	yes	characteristic
211	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
212	C₂²×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
213	C₄×Q₈	—	2	2	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
216	C₄×Q₈	C₂×C₄×C₆₇	1	2	yes	—
217	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
218	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	C₂×C₄×C₆₇	1	2	yes	—
221	C₂×C₄²	—	2	2	no	—
222	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
223	#107	C₂×C₄×C₆₇	1	2	yes	—
224	#107	C₂×C₄×C₆₇	1	1	yes	characteristic
225	#107	C₂×C₄×C₆₇	1	1	yes	characteristic
226	C₄⋊C₈	—	2	4	no	—
227	C₄⋊C₈	—	2	4	no	—
228	C₄⋊C₈	—	2	2	no	—
229	C₄⋊C₈	—	2	1	no	—
230	C₄⋊C₈	C₂³×C₆₇	1	4	yes	—
231	#103	C₂×C₄×C₆₇	1	2	yes	—
232	#103	C₂×C₄×C₆₇	1	2	yes	—
233	#103	C₂×C₄×C₆₇	1	2	yes	—
234	#103	C₂×C₄×C₆₇	1	2	yes	—
235	C₂²⋊C₈	—	2	1	no	—
236	#97	—	2	4	no	—
237	#97	D₄×C₆₇	1	2	yes	—
238	C₄×C₈	—	2	4	no	—
239	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
240	C₄×C₈	C₂×C₄×C₆₇	1	1	yes	characteristic
241	C₄×C₈	C₂³×C₆₇	1	2	yes	—
242	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
243	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
244	C₄×C₈	C₂³×C₆₇	1	2	yes	—
245	C₂×Q₈	—	2	2	no	—
246	C₂×Q₈	—	2	4	no	—
247	C₂×Q₈	—	4	2	no	—
248	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
249	C₂×Q₈	C₂²×C₄×C₆₇	1	2	yes	—
250	C₂²×C₄	—	2	2	no	—
251	C₂²×C₄	—	2	2	no	—
252	C₂²×C₄	C₂²×C₄×C₆₇	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
254	C₂²×C₄	#41×C₆₇	1	1	yes	characteristic
255	Q₁₆	—	2	4	no	—
256	Q₁₆	—	2	2	no	—
257	Q₁₆	M₁₆×C₆₇	1	4	yes	direct product factor
258	M₁₆	—	2	2	no	—
259	M₁₆	—	4	4	no	—
260	M₁₆	Q₁₆×C₆₇	1	4	yes	direct product factor
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	2	4	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	4	1	no	—
265	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
266	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
267	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
268	C₂×C₈	C₂²×C₄×C₆₇	1	1	yes	characteristic
269	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
270	C₂×C₈	C₂×D₄×C₆₇	1	2	yes	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
277	C₄⋊C₄	C₂²×C₄×C₆₇	1	2	yes	—
278	C₄²	—	2	2	no	—
279	C₄²	—	2	2	no	—
280	C₄²	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	C₂²×C₄×C₆₇	1	1	yes	characteristic
283	C₄²	C₂×D₄×C₆₇	1	1	yes	characteristic
284	C₄²	#41×C₆₇	1	1	yes	characteristic
285	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
286	C₂³	C₄×D₄×C₆₇	1	1	yes	characteristic
287	Q₈	—	2	2	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	2	2	no	—
290	Q₈	—	4	4	no	—
291	Q₈	C₂×M₁₆×C₆₇	1	4	yes	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	2	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
302	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
303	C₂×C₄	C₄×D₄×C₆₇	1	1	yes	characteristic
304	C₂×C₄	C₂×Q₁₆×C₆₇	1	2	yes	—
305	C₂×C₄	#135×C₆₇	1	2	yes	—
306	C₂×C₄	C₂³×C₄×C₆₇	1	1	yes	characteristic
307	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
308	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
309	C₈	—	2	1	no	—
310	C₈	—	2	1	no	—
311	C₈	—	2	1	no	—
312	C₈	—	2	2	no	—
313	C₈	—	4	2	no	—
314	C₈	—	4	4	no	—
315	C₈	C₂×M₁₆×C₆₇	1	2	yes	—
316	C₈	C₂×Q₁₆×C₆₇	1	4	yes	complemented
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	C₄×Q₁₆×C₆₇	1	2	yes	—
320	C₂²	C₂×C₄×D₄×C₆₇	1	1	yes	characteristic
321	C₄	—	2	1	no	—
322	C₄	—	2	4	no	—
323	C₄	—	2	1	no	—
324	C₄	—	2	1	no	—
325	C₄	—	4	2	no	—
326	C₄	—	4	2	no	—
327	C₄	—	4	2	no	—
328	C₄	C₄×Q₁₆×C₆₇	1	2	yes	—
329	C₄	C₂²×M₁₆×C₆₇	1	1	yes	characteristic
330	C₄	#568×C₆₇	1	1	yes	characteristic
331	C₄	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
332	C₄	C₂×#135×C₆₇	1	1	yes	characteristic
333	C₂	—	2	2	no	—
334	C₂	D₄×M₁₆×C₆₇	1	1	yes	characteristic
335	C₂	C₂×C₄×Q₁₆×C₆₇	1	1	yes	characteristic
336	C₂	C₆₇×#2957	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₆₇	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₆₇
Frattini, cosocle	C₄²	C₂⁴×C₆₇
socle	C₂²×C₆₇	C₂×C₄×D₄
prime radical	C₂³×C₆₇	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₆₇ ◁ C₄×M₁₆×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₆₇×#1591	C₂	semidirect
4. C₆₇×#1584	C₂	semidirect
5. C₈×Q₁₆×C₆₇	C₂	semidirect
88. Q₁₆×C₆₇	M₁₆	direct (N × H)
91. M₁₆×C₆₇	Q₁₆	direct (N × H)
147. C₈×C₆₇	C₂×Q₁₆	semidirect
164. M₁₆×Q₁₆	C₆₇	direct (N × H)
171. #1591	C₂×C₆₇	semidirect
172. #1584	C₂×C₆₇	semidirect
173. C₈×Q₁₆	C₂×C₆₇	semidirect
177. C₆₇	M₁₆×Q₁₆	direct (N × H)
257. Q₁₆	M₁₆×C₆₇	direct (N × H)
260. M₁₆	Q₁₆×C₆₇	direct (N × H)
316. C₈	C₂×Q₁₆×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
Q₈×M₁₆×C₆₇	2
C₂×C₄×Q₁₆×C₆₇	2
C₆₇×#1591	2
C₆₇×#1584	2
C₈×Q₁₆×C₆₇	2
C₆₇×#1567	2
C₆₇×#1487	2
C₈×M₁₆×C₆₇	2
M₁₆×Q₁₆	67

Subgroups — M16×Q16×C67 #30697289

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — M₁₆×Q₁₆×C₆₇ #30697289