Subgroups — SD₁₆×Q₁₆×C₆₇ #30699235

Order 17152.

622 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×SD₁₆×C₆₇	C₂	1	2	yes	maximal
3	D₄×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1707	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1675	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#1672	C₂	1	2	yes	maximal, complemented
7	C₆₇×#1670	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#1645	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1637	C₂	1	2	yes	maximal
10	C₈×Q₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈×SD₁₆×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₆₇	—	2	1	no	—
13	Q₈²×C₆₇	C₂²	1	2	yes	—
14	D₄×Q₈×C₆₇	C₂²	1	2	yes	—
15	#500×C₆₇	—	2	1	no	—
16	#500×C₆₇	C₂²	1	1	yes	characteristic
17	#498×C₆₇	—	2	2	no	—
18	#498×C₆₇	C₂²	1	2	yes	—
19	#494×C₆₇	C₂²	1	1	yes	characteristic
20	#487×C₆₇	C₂²	1	2	yes	—
21	#477×C₆₇	—	2	2	no	—
22	#477×C₆₇	C₂²	1	2	yes	—
23	#476×C₆₇	—	2	2	no	—
24	#476×C₆₇	C₂²	1	2	yes	—
25	#475×C₆₇	—	2	2	no	—
26	#475×C₆₇	C₂²	1	2	yes	—
27	#467×C₆₇	—	2	1	no	—
28	#462×C₆₇	C₂²	1	2	yes	—
29	#451×C₆₇	—	2	2	no	—
30	C₈×Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₈×Q₈×C₆₇	C₂²	1	2	yes	—
32	C₄×Q₁₆×C₆₇	—	2	1	no	—
33	C₄×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
34	C₄×SD₁₆×C₆₇	—	2	2	no	—
35	C₄×SD₁₆×C₆₇	C₂²	1	1	yes	characteristic
36	C₈×D₄×C₆₇	C₂²	1	1	yes	characteristic
37	#335×C₆₇	C₂²	1	1	yes	characteristic
38	#334×C₆₇	C₂²	1	1	yes	characteristic
39	#333×C₆₇	C₂²	1	2	yes	—
40	#332×C₆₇	C₂²	1	2	yes	—
41	Q₈⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
42	Q₈⋊C₈×C₆₇	C₂²	1	2	yes	—
43	D₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
44	C₈²×C₆₇	C₂²	1	1	yes	characteristic
45	C₂²×Q₈×C₆₇	—	2	2	no	—
46	C₂×Q₁₆×C₆₇	—	2	2	no	—
47	C₂×Q₁₆×C₆₇	—	2	1	no	—
48	C₂×Q₁₆×C₆₇	—	2	1	no	—
49	C₂×Q₁₆×C₆₇	—	4	4	no	—
50	C₂×Q₁₆×C₆₇	D₄	1	1	yes	characteristic
51	C₂×SD₁₆×C₆₇	D₄	1	1	yes	characteristic
52	C₂²×C₈×C₆₇	—	2	1	no	—
53	C₄⋊Q₈×C₆₇	—	2	2	no	—
54	C₄⋊Q₈×C₆₇	—	2	2	no	—
55	C₄⋊Q₈×C₆₇	—	2	2	no	—
56	C₄⋊Q₈×C₆₇	—	4	2	no	—
57	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
58	C₄⋊Q₈×C₆₇	C₂³	1	2	yes	—
59	C₂²⋊Q₈×C₆₇	—	2	2	no	—
60	C₂²⋊Q₈×C₆₇	—	4	2	no	—
61	C₄×Q₈×C₆₇	—	2	2	no	—
62	C₄×Q₈×C₆₇	—	2	2	no	—
63	C₄×Q₈×C₆₇	C₂³	1	2	yes	—
64	C₄×Q₈×C₆₇	C₂³	1	1	yes	characteristic
65	C₄×D₄×C₆₇	—	2	2	no	—
66	C₄×D₄×C₆₇	C₂³	1	1	yes	characteristic
67	#107×C₆₇	—	2	1	no	—
68	#107×C₆₇	—	2	1	no	—
69	#107×C₆₇	—	2	2	no	—
70	#107×C₆₇	D₄	1	1	yes	characteristic
71	#106×C₆₇	—	2	2	no	—
72	#106×C₆₇	—	2	2	no	—
73	#106×C₆₇	—	2	2	no	—
74	#106×C₆₇	—	2	2	no	—
75	#106×C₆₇	D₄	1	1	yes	characteristic
76	C₄⋊C₈×C₆₇	—	2	2	no	—
77	C₄⋊C₈×C₆₇	—	2	1	no	—
78	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
79	C₄⋊C₈×C₆₇	C₂³	1	2	yes	—
80	C₄⋊C₈×C₆₇	C₂³	1	1	yes	characteristic
81	#103×C₆₇	—	2	2	no	—
82	#103×C₆₇	—	2	2	no	—
83	#103×C₆₇	—	2	2	no	—
84	#103×C₆₇	—	4	2	no	—
85	#103×C₆₇	D₄	1	2	yes	—
86	#103×C₆₇	D₄	1	1	yes	characteristic
87	#102×C₆₇	—	2	2	no	—
88	#102×C₆₇	D₄	1	1	yes	characteristic
89	C₂²⋊C₈×C₆₇	—	2	1	no	—
90	C₄×C₈×C₆₇	—	2	2	no	—
91	C₄×C₈×C₆₇	—	2	1	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×Q₈×C₆₇	—	2	2	no	—
96	C₂×Q₈×C₆₇	—	2	2	no	—
97	C₂×Q₈×C₆₇	—	4	2	no	—
98	C₂×Q₈×C₆₇	—	4	2	no	—
99	C₂×Q₈×C₆₇	—	4	4	no	—
100	C₂×Q₈×C₆₇	—	8	2	no	—
101	C₂×Q₈×C₆₇	C₂×D₄	1	2	yes	—
102	C₂×Q₈×C₆₇	C₂×D₄	1	1	yes	characteristic
103	C₂×D₄×C₆₇	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	—	2	1	no	—
105	C₂²×C₄×C₆₇	—	4	2	no	—
106	Q₁₆×C₆₇	—	4	4	no	—
107	Q₁₆×C₆₇	—	4	2	no	—
108	Q₁₆×C₆₇	—	8	1	no	—
109	Q₁₆×C₆₇	SD₁₆	1	4	yes	direct product factor
110	SD₁₆×C₆₇	Q₁₆	1	4	yes	direct product factor
111	C₂×C₈×C₆₇	—	2	2	no	—
112	C₂×C₈×C₆₇	—	2	1	no	—
113	C₂×C₈×C₆₇	—	2	1	no	—
114	C₂×C₈×C₆₇	—	2	1	no	—
115	C₂×C₈×C₆₇	—	4	2	no	—
116	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
117	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
118	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
119	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
120	C₄⋊C₄×C₆₇	—	2	2	no	—
121	C₄⋊C₄×C₆₇	—	2	1	no	—
122	C₄⋊C₄×C₆₇	—	2	2	no	—
123	C₄⋊C₄×C₆₇	—	2	2	no	—
124	C₄⋊C₄×C₆₇	—	2	2	no	—
125	C₄⋊C₄×C₆₇	—	2	2	no	—
126	C₄⋊C₄×C₆₇	—	4	2	no	—
127	C₄⋊C₄×C₆₇	—	4	2	no	—
128	C₄⋊C₄×C₆₇	—	4	2	no	—
129	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
130	C₄⋊C₄×C₆₇	C₂×D₄	1	2	yes	—
131	C₄⋊C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
132	C₂²⋊C₄×C₆₇	—	2	1	no	—
133	C₂²⋊C₄×C₆₇	—	4	2	no	—
134	C₄²×C₆₇	—	2	2	no	—
135	C₄²×C₆₇	—	2	1	no	—
136	C₄²×C₆₇	—	4	2	no	—
137	C₄²×C₆₇	C₂⁴	1	1	yes	characteristic
138	C₂³×C₆₇	—	2	1	no	—
139	Q₈×C₆₇	—	2	2	no	—
140	Q₈×C₆₇	—	2	1	no	—
141	Q₈×C₆₇	—	4	2	no	—
142	Q₈×C₆₇	—	4	2	no	—
143	Q₈×C₆₇	—	8	4	no	—
144	Q₈×C₆₇	—	8	2	no	—
145	Q₈×C₆₇	—	16	2	no	—
146	Q₈×C₆₇	C₂×SD₁₆	1	4	yes	—
147	Q₈×C₆₇	C₂×Q₁₆	1	2	yes	complemented
148	D₄×C₆₇	—	2	1	no	—
149	D₄×C₆₇	C₂×Q₁₆	1	2	yes	—
150	C₂×C₄×C₆₇	—	2	1	no	—
151	C₂×C₄×C₆₇	—	2	2	no	—
152	C₂×C₄×C₆₇	—	2	2	no	—
153	C₂×C₄×C₆₇	—	2	1	no	—
154	C₂×C₄×C₆₇	—	2	1	no	—
155	C₂×C₄×C₆₇	—	4	1	no	—
156	C₂×C₄×C₆₇	—	4	1	no	—
157	C₂×C₄×C₆₇	—	4	2	no	—
158	C₂×C₄×C₆₇	—	4	2	no	—
159	C₂×C₄×C₆₇	—	8	4	no	—
160	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
161	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
162	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
163	C₈×C₆₇	—	2	1	no	—
164	C₈×C₆₇	—	2	1	no	—
165	C₈×C₆₇	—	4	1	no	—
166	C₈×C₆₇	—	4	1	no	—
167	C₈×C₆₇	—	4	1	no	—
168	C₈×C₆₇	—	4	2	no	—
169	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
170	C₈×C₆₇	C₂×Q₁₆	1	2	yes	complemented
171	C₂²×C₆₇	—	2	2	no	—
172	C₂²×C₆₇	—	4	1	no	—
173	C₂²×C₆₇	—	4	1	no	—
174	C₂²×C₆₇	D₄²	1	1	yes	characteristic
175	C₄×C₆₇	—	2	4	no	—
176	C₄×C₆₇	—	2	2	no	—
177	C₄×C₆₇	—	2	1	no	—
178	C₄×C₆₇	—	4	1	no	—
179	C₄×C₆₇	—	4	2	no	—
180	C₄×C₆₇	—	4	1	no	—
181	C₄×C₆₇	—	8	2	no	—
182	C₄×C₆₇	—	8	2	no	—
183	C₄×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
184	C₄×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
185	C₄×C₆₇	#577	1	1	yes	characteristic
186	C₄×C₆₇	#578	1	1	yes	characteristic
187	SD₁₆×Q₁₆	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
188	C₂×C₆₇	—	4	2	no	—
189	C₂×C₆₇	D₄×SD₁₆	1	1	yes	characteristic
190	C₂×C₆₇	D₄×Q₁₆	1	1	yes	characteristic
191	C₂×C₆₇	#3290	1	1	yes	characteristic
192	Q₈×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
193	Q₈×SD₁₆	C₂×C₆₇	1	2	yes	—
194	D₄×Q₁₆	C₂×C₆₇	1	1	yes	characteristic
195	#1707	C₂×C₆₇	1	1	yes	characteristic, complemented
196	#1675	C₂×C₆₇	1	1	yes	characteristic
197	#1672	C₂×C₆₇	1	2	yes	complemented
198	#1670	C₂×C₆₇	1	1	yes	characteristic, complemented
199	#1645	C₂×C₆₇	1	2	yes	complemented
200	#1637	C₂×C₆₇	1	2	yes	—
201	C₈×Q₁₆	C₂×C₆₇	1	1	yes	characteristic, complemented
202	C₈×SD₁₆	C₂×C₆₇	1	1	yes	characteristic
203	C₆₇	SD₁₆×Q₁₆	1	1	yes	characteristic, direct product factor, Sylow
204	C₂²×Q₁₆	—	2	1	no	—
205	Q₈²	C₂²×C₆₇	1	2	yes	—
206	D₄×Q₈	C₂²×C₆₇	1	2	yes	—
207	#500	—	2	1	no	—
208	#500	C₂²×C₆₇	1	1	yes	characteristic
209	#498	—	2	2	no	—
210	#498	C₂²×C₆₇	1	2	yes	—
211	#494	C₂²×C₆₇	1	1	yes	characteristic
212	#487	C₂²×C₆₇	1	2	yes	—
213	#477	—	2	2	no	—
214	#477	C₂²×C₆₇	1	2	yes	—
215	#476	—	2	2	no	—
216	#476	C₂²×C₆₇	1	2	yes	—
217	#475	—	2	2	no	—
218	#475	C₂²×C₆₇	1	2	yes	—
219	#467	—	2	1	no	—
220	#462	C₂²×C₆₇	1	2	yes	—
221	#451	—	2	2	no	—
222	C₈×Q₈	C₂²×C₆₇	1	1	yes	characteristic
223	C₈×Q₈	C₂²×C₆₇	1	2	yes	—
224	C₄×Q₁₆	—	2	1	no	—
225	C₄×Q₁₆	C₂²×C₆₇	1	1	yes	characteristic
226	C₄×SD₁₆	—	2	2	no	—
227	C₄×SD₁₆	C₂²×C₆₇	1	1	yes	characteristic
228	C₈×D₄	C₂²×C₆₇	1	1	yes	characteristic
229	#335	C₂²×C₆₇	1	1	yes	characteristic
230	#334	C₂²×C₆₇	1	1	yes	characteristic
231	#333	C₂²×C₆₇	1	2	yes	—
232	#332	C₂²×C₆₇	1	2	yes	—
233	Q₈⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
234	Q₈⋊C₈	C₂²×C₆₇	1	2	yes	—
235	D₄⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
236	C₈²	C₂²×C₆₇	1	1	yes	characteristic
237	C₂²×Q₈	—	2	2	no	—
238	C₂×Q₁₆	—	2	2	no	—
239	C₂×Q₁₆	—	2	1	no	—
240	C₂×Q₁₆	—	2	1	no	—
241	C₂×Q₁₆	—	4	4	no	—
242	C₂×Q₁₆	D₄×C₆₇	1	1	yes	characteristic
243	C₂×SD₁₆	D₄×C₆₇	1	1	yes	characteristic
244	C₂²×C₈	—	2	1	no	—
245	C₄⋊Q₈	—	2	2	no	—
246	C₄⋊Q₈	—	2	2	no	—
247	C₄⋊Q₈	—	2	2	no	—
248	C₄⋊Q₈	—	4	2	no	—
249	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
250	C₄⋊Q₈	C₂³×C₆₇	1	2	yes	—
251	C₂²⋊Q₈	—	2	2	no	—
252	C₂²⋊Q₈	—	4	2	no	—
253	C₄×Q₈	—	2	2	no	—
254	C₄×Q₈	—	2	2	no	—
255	C₄×Q₈	C₂³×C₆₇	1	2	yes	—
256	C₄×Q₈	C₂³×C₆₇	1	1	yes	characteristic
257	C₄×D₄	—	2	2	no	—
258	C₄×D₄	C₂³×C₆₇	1	1	yes	characteristic
259	#107	—	2	1	no	—
260	#107	—	2	1	no	—
261	#107	—	2	2	no	—
262	#107	D₄×C₆₇	1	1	yes	characteristic
263	#106	—	2	2	no	—
264	#106	—	2	2	no	—
265	#106	—	2	2	no	—
266	#106	—	2	2	no	—
267	#106	D₄×C₆₇	1	1	yes	characteristic
268	C₄⋊C₈	—	2	2	no	—
269	C₄⋊C₈	—	2	1	no	—
270	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
271	C₄⋊C₈	C₂³×C₆₇	1	2	yes	—
272	C₄⋊C₈	C₂³×C₆₇	1	1	yes	characteristic
273	#103	—	2	2	no	—
274	#103	—	2	2	no	—
275	#103	—	2	2	no	—
276	#103	—	4	2	no	—
277	#103	D₄×C₆₇	1	1	yes	characteristic
278	#103	D₄×C₆₇	1	2	yes	—
279	#102	—	2	2	no	—
280	#102	D₄×C₆₇	1	1	yes	characteristic
281	C₂²⋊C₈	—	2	1	no	—
282	C₄×C₈	—	2	2	no	—
283	C₄×C₈	—	2	1	no	—
284	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
285	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
286	C₄×C₈	C₂³×C₆₇	1	1	yes	characteristic
287	C₂×Q₈	—	2	2	no	—
288	C₂×Q₈	—	2	2	no	—
289	C₂×Q₈	—	4	2	no	—
290	C₂×Q₈	—	4	2	no	—
291	C₂×Q₈	—	4	4	no	—
292	C₂×Q₈	—	8	2	no	—
293	C₂×Q₈	C₂×D₄×C₆₇	1	2	yes	—
294	C₂×Q₈	C₂×D₄×C₆₇	1	1	yes	characteristic
295	C₂×D₄	C₂×D₄×C₆₇	1	1	yes	characteristic
296	C₂²×C₄	—	2	1	no	—
297	C₂²×C₄	—	4	2	no	—
298	Q₁₆	—	4	4	no	—
299	Q₁₆	—	4	2	no	—
300	Q₁₆	—	8	1	no	—
301	Q₁₆	SD₁₆×C₆₇	1	4	yes	direct product factor
302	SD₁₆	Q₁₆×C₆₇	1	4	yes	direct product factor
303	C₂×C₈	—	2	2	no	—
304	C₂×C₈	—	2	1	no	—
305	C₂×C₈	—	2	1	no	—
306	C₂×C₈	—	2	1	no	—
307	C₂×C₈	—	4	2	no	—
308	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
309	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
310	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
311	C₂×C₈	C₂×D₄×C₆₇	1	1	yes	characteristic
312	C₄⋊C₄	—	2	2	no	—
313	C₄⋊C₄	—	2	1	no	—
314	C₄⋊C₄	—	2	2	no	—
315	C₄⋊C₄	—	2	2	no	—
316	C₄⋊C₄	—	2	2	no	—
317	C₄⋊C₄	—	2	2	no	—
318	C₄⋊C₄	—	4	2	no	—
319	C₄⋊C₄	—	4	2	no	—
320	C₄⋊C₄	—	4	2	no	—
321	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
322	C₄⋊C₄	C₂×D₄×C₆₇	1	2	yes	—
323	C₄⋊C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
324	C₂²⋊C₄	—	2	1	no	—
325	C₂²⋊C₄	—	4	2	no	—
326	C₄²	—	2	2	no	—
327	C₄²	—	2	1	no	—
328	C₄²	—	4	2	no	—
329	C₄²	C₂⁴×C₆₇	1	1	yes	characteristic
330	C₂³	—	2	1	no	—
331	Q₈	—	2	2	no	—
332	Q₈	—	2	1	no	—
333	Q₈	—	4	2	no	—
334	Q₈	—	4	2	no	—
335	Q₈	—	8	4	no	—
336	Q₈	—	8	2	no	—
337	Q₈	—	16	2	no	—
338	Q₈	C₂×SD₁₆×C₆₇	1	4	yes	—
339	Q₈	C₂×Q₁₆×C₆₇	1	2	yes	complemented
340	D₄	—	2	1	no	—
341	D₄	C₂×Q₁₆×C₆₇	1	2	yes	—
342	C₂×C₄	—	2	1	no	—
343	C₂×C₄	—	2	2	no	—
344	C₂×C₄	—	2	2	no	—
345	C₂×C₄	—	2	1	no	—
346	C₂×C₄	—	2	1	no	—
347	C₂×C₄	—	4	1	no	—
348	C₂×C₄	—	4	1	no	—
349	C₂×C₄	—	4	2	no	—
350	C₂×C₄	—	4	2	no	—
351	C₂×C₄	—	8	4	no	—
352	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
353	C₂×C₄	C₂²×D₄×C₆₇	1	1	yes	characteristic
354	C₂×C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
355	C₈	—	2	1	no	—
356	C₈	—	2	1	no	—
357	C₈	—	4	1	no	—
358	C₈	—	4	1	no	—
359	C₈	—	4	1	no	—
360	C₈	—	4	2	no	—
361	C₈	C₂×SD₁₆×C₆₇	1	2	yes	—
362	C₈	C₂×Q₁₆×C₆₇	1	2	yes	complemented
363	C₂²	—	2	2	no	—
364	C₂²	—	4	1	no	—
365	C₂²	—	4	1	no	—
366	C₂²	D₄²×C₆₇	1	1	yes	characteristic
367	C₄	—	2	4	no	—
368	C₄	—	2	2	no	—
369	C₄	—	2	1	no	—
370	C₄	—	4	1	no	—
371	C₄	—	4	1	no	—
372	C₄	—	4	2	no	—
373	C₄	—	8	2	no	—
374	C₄	—	8	2	no	—
375	C₄	C₂²×SD₁₆×C₆₇	1	1	yes	characteristic
376	C₄	C₂²×Q₁₆×C₆₇	1	1	yes	characteristic
377	C₄	#577×C₆₇	1	1	yes	characteristic
378	C₄	#578×C₆₇	1	1	yes	characteristic
379	C₂	—	4	2	no	—
380	C₂	D₄×SD₁₆×C₆₇	1	1	yes	characteristic
381	C₂	D₄×Q₁₆×C₆₇	1	1	yes	characteristic
382	C₂	C₆₇×#3290	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₆₇	D₄²
commutator, Frattini, cosocle	C₄²	C₂⁴×C₆₇
prime radical	C₂×D₄×C₆₇	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. Q₈×Q₁₆×C₆₇	C₂	semidirect
4. C₆₇×#1707	C₂	semidirect
6. C₆₇×#1672	C₂	semidirect
7. C₆₇×#1670	C₂	semidirect
8. C₆₇×#1645	C₂	semidirect
10. C₈×Q₁₆×C₆₇	C₂	semidirect
109. Q₁₆×C₆₇	SD₁₆	direct (N × H)
110. SD₁₆×C₆₇	Q₁₆	direct (N × H)
147. Q₈×C₆₇	C₂×Q₁₆	semidirect
170. C₈×C₆₇	C₂×Q₁₆	semidirect
187. SD₁₆×Q₁₆	C₆₇	direct (N × H)
192. Q₈×Q₁₆	C₂×C₆₇	semidirect
195. #1707	C₂×C₆₇	semidirect
197. #1672	C₂×C₆₇	semidirect
198. #1670	C₂×C₆₇	semidirect
199. #1645	C₂×C₆₇	semidirect
201. C₈×Q₁₆	C₂×C₆₇	semidirect
203. C₆₇	SD₁₆×Q₁₆	direct (N × H)
301. Q₁₆	SD₁₆×C₆₇	direct (N × H)
302. SD₁₆	Q₁₆×C₆₇	direct (N × H)
339. Q₈	C₂×Q₁₆×C₆₇	semidirect
362. 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×Q₁₆×C₆₇	2
Q₈×SD₁₆×C₆₇	2
D₄×Q₁₆×C₆₇	2
C₆₇×#1707	2
C₆₇×#1675	2
C₆₇×#1672	2
C₆₇×#1670	2
C₆₇×#1645	2
C₆₇×#1637	2
C₈×Q₁₆×C₆₇	2
C₈×SD₁₆×C₆₇	2
SD₁₆×Q₁₆	67

Subgroups — SD16×Q16×C67 #30699235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — SD₁₆×Q₁₆×C₆₇ #30699235