Subgroups — C₂²⋊Q₈×C₆₇⋊C₈ #30863246

Order 17152.

990 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#22561146	C₂	1	1	yes	maximal, characteristic, complemented
4	#22560928	C₂	1	1	yes	maximal, characteristic, complemented
5	#22560768	C₂	1	2	yes	maximal, complemented
6	#22560389	C₂	1	2	yes	maximal
7	#22560189	C₂	1	1	yes	maximal, characteristic
8	C₄⋊C₄×C₆₇⋊C₈	C₂	1	2	yes	maximal, complemented
9	C₄⋊C₄×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²⋊C₄×C₆₇⋊C₈	C₂	1	2	yes	maximal
11	C₄×C₂²⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
12	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	C₂³×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
14	Q₈×C₆₇⋊C₈	—	2	4	no	—
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
18	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
19	C₂×C₂²⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
20	C₂×C₄×C₆₇⋊C₈	—	2	4	no	—
21	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
24	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
25	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
27	#16863151	C₂²	1	2	yes	—
28	#16863151	C₂²	1	2	yes	—
29	#16863151	C₂²	1	2	yes	—
30	#16863151	C₂²	1	2	yes	—
31	#16863151	C₂²	1	2	yes	—
32	C₂×C₂²⋊Q₈×C₆₇	C₄	1	1	yes	characteristic
33	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
35	#391×C₆₇	C₄	1	1	yes	characteristic
36	#389×C₆₇	C₄	1	1	yes	characteristic
37	#387×C₆₇	C₄	1	2	yes	—
38	#385×C₆₇	C₄	1	2	yes	—
39	#380×C₆₇	C₄	1	1	yes	characteristic
40	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
41	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
42	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
43	C₂²×C₆₇⋊C₈	—	2	2	no	—
44	C₂²×C₆₇⋊C₈	—	2	2	no	—
45	C₂²×C₆₇⋊C₈	—	2	2	no	—
46	C₂²×C₆₇⋊C₈	—	2	2	no	—
47	C₂²×C₆₇⋊C₈	—	2	2	no	—
48	C₂²×C₆₇⋊C₈	Q₈	1	4	yes	complemented
49	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
50	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
51	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
52	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
53	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
54	C₂²×C₆₇⋊C₈	C₂³	1	1	yes	characteristic
55	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
56	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
57	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
58	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
59	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
60	C₄×C₆₇⋊C₈	—	2	4	no	—
61	C₄×C₆₇⋊C₈	—	2	2	no	—
62	C₄×C₆₇⋊C₈	—	2	2	no	—
63	C₄×C₆₇⋊C₈	—	2	4	no	—
64	C₄×C₆₇⋊C₈	—	2	2	no	—
65	C₄×C₆₇⋊C₈	D₄	1	4	yes	—
66	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂³×C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
69	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
70	C₂²⋊Q₈×C₆₇	C₈	1	8	yes	complemented
71	C₄×Q₈×C₆₇	—	2	2	no	—
72	C₄×Q₈×C₆₇	—	2	2	no	—
73	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
74	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
75	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
77	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
78	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
79	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
80	C₂×C₂²⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
81	C₂×C₄²×C₆₇	—	2	2	no	—
82	C₂×C₄²×C₆₇	—	2	2	no	—
83	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
84	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
85	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
86	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
87	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
88	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
89	#95×C₆₇	C₂×C₄	1	2	yes	—
90	#95×C₆₇	C₂×C₄	1	2	yes	—
91	#95×C₆₇	C₂×C₄	1	2	yes	—
92	#95×C₆₇	C₂×C₄	1	2	yes	—
93	#95×C₆₇	C₂×C₄	1	2	yes	—
94	C₂×C₆₇⋊C₈	—	2	8	no	—
95	C₂×C₆₇⋊C₈	—	2	1	no	—
96	C₂×C₆₇⋊C₈	—	2	1	no	—
97	C₂×C₆₇⋊C₈	—	2	1	no	—
98	C₂×C₆₇⋊C₈	—	2	1	no	—
99	C₂×C₆₇⋊C₈	—	2	1	no	—
100	C₂×C₆₇⋊C₈	—	2	1	no	—
101	C₂×C₆₇⋊C₈	—	2	8	no	—
102	C₂×C₆₇⋊C₈	—	2	2	no	—
103	C₂×C₆₇⋊C₈	—	2	2	no	—
104	C₂×C₆₇⋊C₈	—	2	2	no	—
105	C₂×C₆₇⋊C₈	—	2	2	no	—
106	C₂×C₆₇⋊C₈	—	2	2	no	—
107	C₂×C₆₇⋊C₈	—	2	2	no	—
108	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
109	C₂×C₆₇⋊C₈	C₂×D₄	1	2	yes	—
110	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
111	C₂×C₆₇⋊C₈	C₂×Q₈	1	2	yes	—
112	C₂×C₆₇⋊C₈	#41	1	2	yes	—
113	C₂×C₆₇⋊C₈	#41	1	2	yes	—
114	C₂⁴×C₆₇	C₂²×C₄	1	1	yes	characteristic
115	C₂×Q₈×C₆₇	—	2	2	no	—
116	C₂×Q₈×C₆₇	—	2	2	no	—
117	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	complemented
118	C₂²×C₄×C₆₇	—	2	2	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	2	2	no	—
121	C₂²×C₄×C₆₇	—	2	2	no	—
122	C₂²×C₄×C₆₇	—	2	2	no	—
123	C₂²×C₄×C₆₇	—	2	1	no	—
124	C₂²×C₄×C₆₇	—	2	1	no	—
125	C₂²×C₄×C₆₇	—	2	1	no	—
126	C₂²×C₄×C₆₇	—	2	1	no	—
127	C₂²×C₄×C₆₇	C₂×C₈	1	4	yes	—
128	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
129	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
130	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
131	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
132	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
133	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
134	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
135	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
136	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
138	C₂²×C₄×C₆₇	#41	1	2	yes	—
139	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
140	C₄⋊C₄×C₆₇	—	2	4	no	—
141	C₄⋊C₄×C₆₇	—	2	2	no	—
142	C₄⋊C₄×C₆₇	—	2	4	no	—
143	C₄⋊C₄×C₆₇	—	2	2	no	—
144	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	complemented
145	C₄⋊C₄×C₆₇	C₂×C₈	1	8	yes	complemented
146	C₂²⋊C₄×C₆₇	—	2	4	no	—
147	C₂²⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
148	C₄²×C₆₇	—	2	2	no	—
149	C₄²×C₆₇	—	2	2	no	—
150	C₄²×C₆₇	—	2	1	no	—
151	C₄²×C₆₇	—	2	2	no	—
152	C₄²×C₆₇	—	2	1	no	—
153	C₄²×C₆₇	—	2	2	no	—
154	C₄²×C₆₇	—	2	2	no	—
155	C₄²×C₆₇	—	2	2	no	—
156	C₄²×C₆₇	C₂×D₄	1	2	yes	—
157	C₄²×C₆₇	#41	1	2	yes	—
158	C₂³×C₆₇	—	2	1	no	—
159	C₂³×C₆₇	—	2	1	no	—
160	C₂³×C₆₇	—	2	1	no	—
161	C₂³×C₆₇	—	2	1	no	—
162	C₂³×C₆₇	C₄×Q₈	1	2	yes	—
163	C₂³×C₆₇	C₂²×C₈	1	2	yes	—
164	C₂³×C₆₇	#131	1	2	yes	—
165	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
166	Q₈×C₆₇	—	2	8	no	—
167	C₂×C₄×C₆₇	—	2	8	no	—
168	C₂×C₄×C₆₇	—	2	1	no	—
169	C₂×C₄×C₆₇	—	2	1	no	—
170	C₂×C₄×C₆₇	—	2	1	no	—
171	C₂×C₄×C₆₇	—	2	2	no	—
172	C₂×C₄×C₆₇	—	2	2	no	—
173	C₂×C₄×C₆₇	—	2	1	no	—
174	C₂×C₄×C₆₇	—	2	2	no	—
175	C₂×C₄×C₆₇	—	2	1	no	—
176	C₂×C₄×C₆₇	—	2	1	no	—
177	C₂×C₄×C₆₇	—	2	1	no	—
178	C₂×C₄×C₆₇	—	2	1	no	—
179	C₂×C₄×C₆₇	—	2	2	no	—
180	C₂×C₄×C₆₇	—	2	2	no	—
181	C₂×C₄×C₆₇	—	2	2	no	—
182	C₂×C₄×C₆₇	—	2	2	no	—
183	C₂×C₄×C₆₇	—	2	2	no	—
184	C₂×C₄×C₆₇	—	2	8	no	—
185	C₂×C₄×C₆₇	—	2	2	no	—
186	C₂×C₄×C₆₇	—	2	2	no	—
187	C₂×C₄×C₆₇	—	2	2	no	—
188	C₂×C₄×C₆₇	—	2	2	no	—
189	C₂×C₄×C₆₇	—	2	2	no	—
190	C₂×C₄×C₆₇	—	2	2	no	—
191	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
192	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
193	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
194	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
195	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
196	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
197	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
198	C₂×C₄×C₆₇	#131	1	2	yes	—
199	C₂×C₄×C₆₇	#131	1	2	yes	—
200	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
202	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
203	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
204	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
205	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
206	C₆₇⋊C₈	—	2	2	no	—
207	C₆₇⋊C₈	—	2	2	no	—
208	C₆₇⋊C₈	—	2	2	no	—
209	C₆₇⋊C₈	—	4	2	no	—
210	C₆₇⋊C₈	—	4	2	no	—
211	C₆₇⋊C₈	C₂²⋊Q₈	1	4	yes	direct product factor
212	C₂²×C₆₇	—	2	2	no	—
213	C₂²×C₆₇	—	2	2	no	—
214	C₂²×C₆₇	—	2	2	no	—
215	C₂²×C₆₇	—	2	2	no	—
216	C₂²×C₆₇	—	2	2	no	—
217	C₂²×C₆₇	—	2	2	no	—
218	C₂²×C₆₇	C₈×Q₈	1	4	yes	complemented
219	C₂²×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
220	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
221	C₂²×C₆₇	C₄×#41	1	1	yes	characteristic
222	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
223	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
224	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
225	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
226	C₄×C₆₇	—	2	2	no	—
227	C₄×C₆₇	—	2	4	no	—
228	C₄×C₆₇	—	2	2	no	—
229	C₄×C₆₇	—	2	2	no	—
230	C₄×C₆₇	—	2	4	no	—
231	C₄×C₆₇	—	2	2	no	—
232	C₄×C₆₇	—	4	2	no	—
233	C₄×C₆₇	—	4	2	no	—
234	C₄×C₆₇	C₈×D₄	1	4	yes	—
235	C₄×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
236	C₄×C₆₇	#525	1	1	yes	characteristic
237	C₄×C₆₇	#529	1	1	yes	characteristic
238	C₄×C₆₇	#533	1	1	yes	characteristic
239	C₈×C₂²⋊Q₈	—	67	1	no	maximal, Sylow
240	C₂×C₆₇	—	2	4	no	—
241	C₂×C₆₇	C₄×C₂²⋊Q₈	1	1	yes	characteristic
242	C₂×C₆₇	C₂×C₈×D₄	1	1	yes	characteristic
243	C₂×C₆₇	#2929	1	1	yes	characteristic
244	C₂×C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic
245	C₂×C₆₇	#2960	1	1	yes	characteristic
246	C₂×C₆₇	C₈×#41	1	1	yes	characteristic
247	C₂×C₆₇	#2966	1	1	yes	characteristic
248	C₂×C₈×Q₈	—	67	1	no	—
249	C₂²×C₄×C₈	—	67	1	no	—
250	C₄×C₂²⋊Q₈	D₆₇	1	1	yes	characteristic
251	#1984	—	67	1	no	—
252	#1939	—	67	1	no	—
253	#1916	—	67	2	no	—
254	#1850	—	67	2	no	—
255	#1797	—	67	1	no	—
256	C₈×C₄⋊C₄	—	67	2	no	—
257	C₈×C₄⋊C₄	—	67	1	no	—
258	C₈×C₂²⋊C₄	—	67	2	no	—
259	C₆₇	C₈×C₂²⋊Q₈	1	1	yes	characteristic, complemented, Sylow
260	C₂³×C₈	—	67	1	no	—
261	C₂³×C₈	—	67	1	no	—
262	C₂×C₂²⋊Q₈	Dic₆₇	1	1	yes	characteristic
263	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
264	C₂²×C₄²	C₂×D₆₇	1	1	yes	characteristic
265	C₈×Q₈	—	134	4	no	—
266	C₂×C₄⋊C₈	—	67	1	no	—
267	C₂×C₄⋊C₈	—	67	1	no	—
268	C₂×C₄⋊C₈	—	67	2	no	—
269	C₂×C₄⋊C₈	—	67	2	no	—
270	C₂×C₂²⋊C₈	—	67	2	no	—
271	C₂×C₄×C₈	—	67	1	no	—
272	C₂×C₄×C₈	—	67	1	no	—
273	C₂×C₄×C₈	—	67	1	no	—
274	C₂×C₄×C₈	—	67	1	no	—
275	C₂×C₄×C₈	—	67	2	no	—
276	C₂×C₄×C₈	—	67	2	no	—
277	C₂×C₄×C₈	—	134	4	no	—
278	#391	Dic₆₇	1	1	yes	characteristic
279	#389	Dic₆₇	1	1	yes	characteristic
280	#387	Dic₆₇	1	2	yes	—
281	#385	Dic₆₇	1	2	yes	—
282	#380	Dic₆₇	1	1	yes	characteristic
283	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
284	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
285	C₄×C₂²⋊C₄	C₂×D₆₇	1	2	yes	—
286	#336	—	67	2	no	—
287	#336	—	67	2	no	—
288	#336	—	67	2	no	—
289	#336	—	67	2	no	—
290	#336	—	67	2	no	—
291	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
292	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
293	C₂³×C₄	C₂×Dic₆₇	1	1	yes	characteristic
294	C₂³×C₄	C₂²×D₆₇	1	1	yes	characteristic
295	C₂²×C₈	—	67	1	no	—
296	C₂²×C₈	—	67	4	no	—
297	C₂²×C₈	—	67	1	no	—
298	C₂²×C₈	—	67	1	no	—
299	C₂²×C₈	—	67	1	no	—
300	C₂²×C₈	—	67	2	no	—
301	C₂²×C₈	—	67	2	no	—
302	C₂²×C₈	—	134	2	no	—
303	C₂²×C₈	—	134	2	no	—
304	C₂²×C₈	—	134	2	no	—
305	C₂²×C₈	—	134	2	no	—
306	C₂²×C₈	—	134	2	no	—
307	C₂²⋊Q₈	C₆₇⋊C₈	1	8	yes	direct product factor
308	C₄×Q₈	—	2	2	no	—
309	C₄×Q₈	—	2	2	no	—
310	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
311	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
312	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
313	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
314	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
315	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
316	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	2	yes	—
317	C₂×C₂²⋊C₄	C₂×Dic₆₇	1	2	yes	—
318	C₂×C₄²	—	2	2	no	—
319	C₂×C₄²	—	2	2	no	—
320	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
321	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
322	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
323	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
324	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
325	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
326	C₄⋊C₈	—	134	4	no	—
327	C₄⋊C₈	—	134	4	no	—
328	C₄⋊C₈	—	134	2	no	—
329	C₄⋊C₈	—	134	2	no	—
330	C₂²⋊C₈	—	134	4	no	—
331	C₄×C₈	—	67	4	no	—
332	C₄×C₈	—	134	4	no	—
333	C₄×C₈	—	134	2	no	—
334	C₄×C₈	—	134	2	no	—
335	C₄×C₈	—	134	4	no	—
336	C₄×C₈	—	134	2	no	—
337	#95	C₂×Dic₆₇	1	2	yes	—
338	#95	C₂×Dic₆₇	1	2	yes	—
339	#95	C₂×Dic₆₇	1	2	yes	—
340	#95	C₂×Dic₆₇	1	2	yes	—
341	#95	C₂×Dic₆₇	1	2	yes	—
342	C₂⁴	C₂²×Dic₆₇	1	1	yes	characteristic
343	C₂×Q₈	—	2	2	no	—
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	C₂×C₆₇⋊C₈	1	4	yes	complemented
346	C₂²×C₄	—	2	2	no	—
347	C₂²×C₄	—	2	2	no	—
348	C₂²×C₄	—	2	2	no	—
349	C₂²×C₄	—	2	2	no	—
350	C₂²×C₄	—	2	2	no	—
351	C₂²×C₄	—	2	1	no	—
352	C₂²×C₄	—	2	1	no	—
353	C₂²×C₄	—	2	1	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	C₂×C₆₇⋊C₈	1	4	yes	—
356	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
357	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
358	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
359	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
360	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
361	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
362	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
363	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
364	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
365	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
366	C₂²×C₄	#11761629	1	2	yes	—
367	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
368	C₂×C₈	—	67	2	no	—
369	C₂×C₈	—	67	2	no	—
370	C₂×C₈	—	67	2	no	—
371	C₂×C₈	—	67	2	no	—
372	C₂×C₈	—	67	2	no	—
373	C₂×C₈	—	67	2	no	—
374	C₂×C₈	—	134	2	no	—
375	C₂×C₈	—	134	2	no	—
376	C₂×C₈	—	134	2	no	—
377	C₂×C₈	—	134	2	no	—
378	C₂×C₈	—	134	2	no	—
379	C₂×C₈	—	134	2	no	—
380	C₂×C₈	—	134	8	no	—
381	C₂×C₈	—	134	8	no	—
382	C₂×C₈	—	134	1	no	—
383	C₂×C₈	—	134	1	no	—
384	C₂×C₈	—	134	1	no	—
385	C₂×C₈	—	134	1	no	—
386	C₂×C₈	—	134	1	no	—
387	C₂×C₈	—	134	1	no	—
388	C₄⋊C₄	—	2	4	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	4	no	—
391	C₄⋊C₄	—	2	2	no	—
392	C₄⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	complemented
393	C₄⋊C₄	C₂×C₆₇⋊C₈	1	8	yes	complemented
394	C₂²⋊C₄	—	2	4	no	—
395	C₂²⋊C₄	C₂×C₆₇⋊C₈	1	8	yes	—
396	C₄²	—	2	2	no	—
397	C₄²	—	2	2	no	—
398	C₄²	—	2	1	no	—
399	C₄²	—	2	2	no	—
400	C₄²	—	2	2	no	—
401	C₄²	—	2	2	no	—
402	C₄²	—	2	1	no	—
403	C₄²	—	2	2	no	—
404	C₄²	D₄×D₆₇	1	2	yes	—
405	C₄²	#11761627	1	2	yes	—
406	C₂³	—	2	1	no	—
407	C₂³	—	2	1	no	—
408	C₂³	—	2	1	no	—
409	C₂³	—	2	1	no	—
410	C₂³	Q₈×Dic₆₇	1	2	yes	—
411	C₂³	C₂²×C₆₇⋊C₈	1	2	yes	—
412	C₂³	#14524050	1	2	yes	—
413	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
414	Q₈	—	2	8	no	—
415	C₂×C₄	—	2	8	no	—
416	C₂×C₄	—	2	1	no	—
417	C₂×C₄	—	2	1	no	—
418	C₂×C₄	—	2	1	no	—
419	C₂×C₄	—	2	2	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	1	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	1	no	—
424	C₂×C₄	—	2	1	no	—
425	C₂×C₄	—	2	1	no	—
426	C₂×C₄	—	2	1	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	8	no	—
433	C₂×C₄	—	2	2	no	—
434	C₂×C₄	—	2	2	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	2	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
440	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
441	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
442	C₂×C₄	C₂²×C₆₇⋊C₈	1	4	yes	—
443	C₂×C₄	C₂²×C₆₇⋊C₈	1	4	yes	—
444	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
445	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
446	C₂×C₄	#14524050	1	2	yes	—
447	C₂×C₄	#14524050	1	2	yes	—
448	C₂×C₄	C₂×D₄×D₆₇	1	1	yes	characteristic
449	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
450	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
451	C₂×C₄	#41×D₆₇	1	1	yes	characteristic
452	C₂×C₄	C₂×#11761627	1	1	yes	characteristic
453	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
454	C₈	—	268	2	no	—
455	C₈	—	268	2	no	—
456	C₈	—	67	4	no	—
457	C₈	—	134	2	no	—
458	C₈	—	134	2	no	—
459	C₈	—	134	2	no	—
460	C₂²	—	2	2	no	—
461	C₂²	—	2	2	no	—
462	C₂²	—	2	2	no	—
463	C₂²	—	2	2	no	—
464	C₂²	—	2	2	no	—
465	C₂²	—	2	2	no	—
466	C₂²	Q₈×C₆₇⋊C₈	1	4	yes	complemented
467	C₂²	C₂×D₄×Dic₆₇	1	1	yes	characteristic
468	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
469	C₂²	#41×Dic₆₇	1	1	yes	characteristic
470	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
471	C₂²	C₂×#14524050	1	1	yes	characteristic
472	C₂²	C₂×#14524050	1	1	yes	characteristic
473	C₂²	C₂×#14524050	1	1	yes	characteristic
474	C₄	—	2	2	no	—
475	C₄	—	2	4	no	—
476	C₄	—	2	2	no	—
477	C₄	—	2	2	no	—
478	C₄	—	2	2	no	—
479	C₄	—	2	4	no	—
480	C₄	—	4	2	no	—
481	C₄	—	4	2	no	—
482	C₄	D₄×C₆₇⋊C₈	1	4	yes	—
483	C₄	C₂²⋊Q₈×D₆₇	1	1	yes	characteristic
484	C₄	#16863909	1	1	yes	characteristic
485	C₄	#16863932	1	1	yes	characteristic
486	C₄	#16863948	1	1	yes	characteristic
487	C₂	—	2	4	no	—
488	C₂	C₂²⋊Q₈×Dic₆₇	1	1	yes	characteristic
489	C₂	C₂×D₄×C₆₇⋊C₈	1	1	yes	characteristic
490	C₂	#22568904	1	1	yes	characteristic
491	C₂	C₂×Q₈×C₆₇⋊C₈	1	1	yes	characteristic
492	C₂	#22569144	1	1	yes	characteristic
493	C₂	#41×C₆₇⋊C₈	1	1	yes	characteristic
494	C₂	#22569185	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂³×C₈
Fitting	C₄×C₂²⋊Q₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂³×C₄
prime radical	C₂⁴×C₆₇	C₂²×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄×C₂²⋊Q₈	D₆₇
hypocenter	C₆₇	C₈×C₂²⋊Q₈
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊Q₈

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈×C₆₇⋊C₈	C₂	semidirect
3. #22561146	C₂	semidirect
4. #22560928	C₂	semidirect
5. #22560768	C₂	semidirect
8. C₄⋊C₄×C₆₇⋊C₈	C₂	semidirect
9. C₄⋊C₄×C₆₇⋊C₈	C₂	semidirect
48. C₂²×C₆₇⋊C₈	Q₈	semidirect
70. C₂²⋊Q₈×C₆₇	C₈	semidirect
117. C₂×Q₈×C₆₇	C₂×C₈	semidirect
144. C₄⋊C₄×C₆₇	C₂×C₈	semidirect
145. C₄⋊C₄×C₆₇	C₂×C₈	semidirect
211. C₆₇⋊C₈	C₂²⋊Q₈	direct (N × H)
218. C₂²×C₆₇	C₈×Q₈	semidirect
259. C₆₇	C₈×C₂²⋊Q₈	semidirect
307. C₂²⋊Q₈	C₆₇⋊C₈	direct (N × H)
345. C₂×Q₈	C₂×C₆₇⋊C₈	semidirect
392. C₄⋊C₄	C₂×C₆₇⋊C₈	semidirect
393. C₄⋊C₄	C₂×C₆₇⋊C₈	semidirect
466. C₂²	Q₈×C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₆₇⋊C₈	2
C₂²×C₄×C₆₇⋊C₈	2
#22561146	2
#22560928	2
#22560768	2
#22560389	2
#22560189	2
C₄⋊C₄×C₆₇⋊C₈	2
C₂²⋊C₄×C₆₇⋊C₈	2
C₄×C₂²⋊Q₈×C₆₇	2
C₈×C₂²⋊Q₈	67

Subgroups — C22⋊Q8×C67⋊C8 #30863246

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²⋊Q₈×C₆₇⋊C₈ #30863246