Subgroups — Q₈×#14523970 #30872530

Order 17152.

1096 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#11761612	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×Q₈×Dic₆₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#22561187	C₂	1	3	yes	maximal, complemented
4	#22561154	C₂	1	3	yes	maximal, complemented
5	#22560229	C₂	1	3	yes	maximal
6	C₄×#14523970	C₂	1	3	yes	maximal
7	Q₈×#41×C₆₇	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₆₇	—	2	1	no	—
9	#16863527	—	2	3	no	—
10	Q₈×C₆₇⋊C₈	—	2	1	no	—
11	#16863420	C₂²	1	3	yes	—
12	C₂×#14523970	C₂²	1	3	yes	—
13	C₂×#14523970	C₂²	1	1	yes	characteristic
14	C₄×#11761612	C₂²	1	3	yes	—
15	C₄⋊C₄×Dic₆₇	—	2	3	no	—
16	C₄×C₄⋊Dic₆₇	C₂²	1	3	yes	—
17	C₄²×Dic₆₇	C₂²	1	3	yes	—
18	#16863155	C₂²	1	3	yes	—
19	#16863155	C₂²	1	3	yes	—
20	#16863155	C₂²	1	6	yes	—
21	Q₈²×C₆₇	C₄	1	1	yes	characteristic, complemented
22	#557×C₆₇	C₄	1	3	yes	complemented
23	#554×C₆₇	—	2	3	no	—
24	#554×C₆₇	C₄	1	3	yes	complemented
25	D₄×Q₈×C₆₇	—	2	1	no	—
26	D₄×Q₈×C₆₇	C₄	1	1	yes	characteristic, complemented
27	#533×C₆₇	—	2	3	no	—
28	#533×C₆₇	C₂²	1	3	yes	—
29	C₄×#41×C₆₇	C₂²	1	3	yes	—
30	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
31	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×#11761612	C₂³	1	3	yes	—
33	C₂×#11761612	C₂³	1	1	yes	characteristic
34	Q₈×Dic₆₇	—	2	8	no	—
35	C₂×C₄⋊Dic₆₇	—	2	3	no	—
36	C₂×C₄×Dic₆₇	—	2	3	no	—
37	C₂×C₄×Dic₆₇	—	2	3	no	—
38	C₂×C₄×Dic₆₇	C₂³	1	3	yes	—
39	C₂×C₄×Dic₆₇	C₂³	1	1	yes	characteristic
40	C₄⋊(C₆₇⋊C₈)	—	2	3	no	—
41	C₄⋊(C₆₇⋊C₈)	—	2	3	no	—
42	#14523970	—	2	6	no	—
43	#14523970	Q₈	1	4	yes	direct product factor
44	#14523954	—	2	3	no	—
45	C₄×C₆₇⋊C₈	—	2	3	no	—
46	C₆₇⋊#95	—	2	3	no	—
47	C₂×#41×C₆₇	C₂³	1	1	yes	characteristic
48	C₂²×Q₈×C₆₇	—	2	1	no	—
49	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
50	C₄⋊Q₈×C₆₇	—	2	3	no	—
51	C₄⋊Q₈×C₆₇	—	2	3	no	—
52	C₄⋊Q₈×C₆₇	C₂×C₄	1	3	yes	—
53	C₄⋊Q₈×C₆₇	C₂×C₄	1	3	yes	—
54	#125×C₆₇	—	2	6	no	—
55	#125×C₆₇	C₂×C₄	1	6	yes	—
56	C₂²⋊Q₈×C₆₇	—	2	3	no	—
57	C₂²⋊Q₈×C₆₇	—	2	3	no	—
58	C₂²⋊Q₈×C₆₇	—	2	3	no	—
59	C₂²⋊Q₈×C₆₇	—	2	3	no	—
60	C₂²⋊Q₈×C₆₇	—	2	6	no	—
61	C₄×Q₈×C₆₇	—	2	1	no	—
62	C₄×Q₈×C₆₇	—	2	3	no	—
63	C₄×Q₈×C₆₇	—	2	3	no	—
64	C₄×Q₈×C₆₇	—	2	3	no	—
65	C₄×Q₈×C₆₇	C₂×C₄	1	3	yes	—
66	C₄×Q₈×C₆₇	C₂×C₄	1	3	yes	—
67	C₄×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₄×Q₈×C₆₇	C₂×C₄	1	3	yes	—
69	C₄×Q₈×C₆₇	D₄	1	1	yes	characteristic
70	C₄×Q₈×C₆₇	D₄	1	3	yes	—
71	C₄×D₄×C₆₇	—	2	3	no	—
72	C₄×D₄×C₆₇	—	2	3	no	—
73	C₄×D₄×C₆₇	C₂×C₄	1	3	yes	—
74	C₄×D₄×C₆₇	C₂×C₄	1	3	yes	—
75	#117×C₆₇	—	2	3	no	—
76	#117×C₆₇	C₂³	1	3	yes	—
77	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
78	C₂×C₄⋊C₄×C₆₇	D₄	1	3	yes	—
79	C₂×C₄²×C₆₇	—	2	3	no	—
80	C₂×C₄²×C₆₇	C₂³	1	3	yes	—
81	C₂²×Dic₆₇	—	2	1	no	—
82	C₂²×Dic₆₇	—	2	3	no	—
83	#11761612	—	2	1	no	—
84	#11761612	—	2	3	no	—
85	#11761612	—	2	3	no	—
86	#11761612	C₂×Q₈	1	2	yes	complemented
87	C₂×C₆₇⋊C₈	—	2	1	no	—
88	C₂×C₆₇⋊C₈	—	2	3	no	—
89	C₄⋊Dic₆₇	—	2	12	no	—
90	C₄×Dic₆₇	—	2	12	no	—
91	C₄×Dic₆₇	—	2	1	no	—
92	C₄×Dic₆₇	—	2	3	no	—
93	C₄×Dic₆₇	—	2	3	no	—
94	C₄×Dic₆₇	—	4	6	no	—
95	C₄×Dic₆₇	C₂×Q₈	1	2	yes	complemented
96	#41×C₆₇	—	2	1	no	—
97	#41×C₆₇	—	2	1	no	—
98	#41×C₆₇	C₂×Q₈	1	2	yes	—
99	#41×C₆₇	#41	1	2	yes	—
100	C₂×Q₈×C₆₇	—	2	4	no	—
101	C₂×Q₈×C₆₇	—	2	3	no	—
102	C₂×Q₈×C₆₇	—	2	3	no	—
103	C₂×Q₈×C₆₇	—	4	4	no	—
104	C₂×Q₈×C₆₇	C₂²⋊C₄	1	3	yes	—
105	C₂×Q₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
106	C₂×Q₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
107	C₂×D₄×C₆₇	—	2	1	no	—
108	C₂×D₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
109	C₂²×C₄×C₆₇	—	2	1	no	—
110	C₂²×C₄×C₆₇	—	2	3	no	—
111	C₂²×C₄×C₆₇	—	2	3	no	—
112	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
113	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
114	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
115	C₄⋊C₄×C₆₇	—	2	6	no	—
116	C₄⋊C₄×C₆₇	—	2	3	no	—
117	C₄⋊C₄×C₆₇	—	2	3	no	—
118	C₄⋊C₄×C₆₇	—	2	3	no	—
119	C₄⋊C₄×C₆₇	—	2	3	no	—
120	C₄⋊C₄×C₆₇	—	2	6	no	—
121	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	3	yes	—
122	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	6	yes	—
123	C₄⋊C₄×C₆₇	C₂²⋊C₄	1	3	yes	—
124	C₄⋊C₄×C₆₇	C₂²×C₄	1	3	yes	—
125	C₄⋊C₄×C₆₇	C₂²×C₄	1	3	yes	—
126	C₂²⋊C₄×C₆₇	—	2	3	no	—
127	C₂²⋊C₄×C₆₇	—	2	3	no	—
128	C₂²⋊C₄×C₆₇	—	2	3	no	—
129	C₄²×C₆₇	—	2	3	no	—
130	C₄²×C₆₇	—	2	3	no	—
131	C₄²×C₆₇	—	2	3	no	—
132	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
133	C₄²×C₆₇	C₂²×C₄	1	3	yes	—
134	C₄²×C₆₇	C₂×D₄	1	3	yes	—
135	C₄²×C₆₇	C₂×D₄	1	3	yes	—
136	C₂³×C₆₇	—	2	1	no	—
137	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
138	Q₈×C₆₇	—	2	1	no	—
139	Q₈×C₆₇	—	2	6	no	—
140	Q₈×C₆₇	—	4	6	no	—
141	Q₈×C₆₇	—	8	3	no	—
142	Q₈×C₆₇	#104	1	4	yes	complemented
143	Q₈×C₆₇	C₄×Q₈	1	2	yes	complemented
144	D₄×C₆₇	—	2	2	no	—
145	D₄×C₆₇	—	2	2	no	—
146	D₄×C₆₇	—	2	1	no	—
147	D₄×C₆₇	C₄×Q₈	1	2	yes	complemented
148	C₂×C₄×C₆₇	—	2	1	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	—	2	6	no	—
152	C₂×C₄×C₆₇	—	2	3	no	—
153	C₂×C₄×C₆₇	—	2	3	no	—
154	C₂×C₄×C₆₇	—	4	3	no	—
155	C₂×C₄×C₆₇	—	4	6	no	—
156	C₂×C₄×C₆₇	—	8	3	no	—
157	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
158	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
159	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
160	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
161	C₂×C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
162	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
163	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
164	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
165	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
166	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
167	C₂×Dic₆₇	—	2	2	no	—
168	C₂×Dic₆₇	—	2	2	no	—
169	C₂×Dic₆₇	—	2	2	no	—
170	C₂×Dic₆₇	—	2	6	no	—
171	C₂×Dic₆₇	—	4	3	no	—
172	C₂×Dic₆₇	—	4	3	no	—
173	C₆₇⋊C₈	—	2	2	no	—
174	C₆₇⋊C₈	—	4	3	no	—
175	C₂²×C₆₇	—	2	1	no	—
176	C₂²×C₆₇	—	2	1	no	—
177	C₂²×C₆₇	—	2	2	no	—
178	C₂²×C₆₇	—	4	1	no	—
179	C₂²×C₆₇	—	4	1	no	—
180	C₂²×C₆₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic
181	C₂²×C₆₇	#548	1	1	yes	characteristic
182	C₂²×C₆₇	D₄×Q₈	1	1	yes	characteristic
183	C₄×C₆₇	—	2	2	no	—
184	C₄×C₆₇	—	2	3	no	—
185	C₄×C₆₇	—	2	3	no	—
186	C₄×C₆₇	—	2	3	no	—
187	C₄×C₆₇	—	4	3	no	—
188	C₄×C₆₇	—	4	3	no	—
189	C₄×C₆₇	C₂×#104	1	6	yes	—
190	C₄×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
191	C₄×C₆₇	#519	1	1	yes	characteristic
192	C₄×C₆₇	#548	1	1	yes	characteristic
193	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
194	Dic₆₇	—	2	4	no	—
195	Dic₆₇	—	4	6	no	—
196	Q₈×#104	—	67	1	no	maximal, Sylow
197	C₂×C₆₇	—	2	1	no	—
198	C₂×C₆₇	—	2	1	no	—
199	C₂×C₆₇	—	4	2	no	—
200	C₂×C₆₇	Q₈×C₂²⋊C₄	1	1	yes	characteristic
201	C₂×C₆₇	C₂²×#104	1	1	yes	characteristic
202	C₂×C₆₇	#2901	1	1	yes	characteristic
203	Q₈×#41	D₆₇	1	1	yes	characteristic
204	Q₈×M₁₆	—	67	1	no	—
205	C₄²×Q₈	—	67	1	no	—
206	#1994	—	67	3	no	—
207	#1986	—	67	3	no	—
208	#1806	—	67	3	no	—
209	C₄×#104	—	67	3	no	—
210	C₆₇	Q₈×#104	1	1	yes	characteristic, complemented, Sylow
211	Q₈²	Dic₆₇	1	1	yes	characteristic, complemented
212	#557	Dic₆₇	1	3	yes	complemented
213	#554	—	2	3	no	—
214	#554	Dic₆₇	1	3	yes	complemented
215	D₄×Q₈	—	2	1	no	—
216	D₄×Q₈	Dic₆₇	1	1	yes	characteristic, complemented
217	#533	—	2	3	no	—
218	#533	C₂×D₆₇	1	3	yes	—
219	C₄×#41	C₂×D₆₇	1	3	yes	—
220	C₂×C₄×Q₈	—	2	1	no	—
221	C₂×C₄×Q₈	—	134	1	no	—
222	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
223	#446	—	134	3	no	—
224	C₈×Q₈	—	134	1	no	—
225	#423	—	67	3	no	—
226	C₂×#104	—	67	1	no	—
227	C₂×#104	—	67	3	no	—
228	C₄×M₁₆	—	67	3	no	—
229	C₄×C₄⋊C₄	—	67	3	no	—
230	C₄×C₄⋊C₄	—	134	3	no	—
231	C₄³	—	67	3	no	—
232	#339	—	67	6	no	—
233	#339	—	67	3	no	—
234	#339	—	67	3	no	—
235	C₂×#41	C₂²×D₆₇	1	1	yes	characteristic
236	C₂²×Q₈	—	2	1	no	—
237	C₂²×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
238	C₂×M₁₆	—	67	1	no	—
239	C₂×M₁₆	—	67	3	no	—
240	C₄⋊Q₈	—	2	3	no	—
241	C₄⋊Q₈	—	2	3	no	—
242	C₄⋊Q₈	C₂×Dic₆₇	1	3	yes	—
243	C₄⋊Q₈	C₂×Dic₆₇	1	3	yes	—
244	#125	—	2	6	no	—
245	#125	C₂×Dic₆₇	1	6	yes	—
246	C₂²⋊Q₈	—	2	3	no	—
247	C₂²⋊Q₈	—	2	3	no	—
248	C₂²⋊Q₈	—	2	3	no	—
249	C₂²⋊Q₈	—	2	3	no	—
250	C₂²⋊Q₈	—	2	6	no	—
251	C₄×Q₈	—	2	1	no	—
252	C₄×Q₈	—	2	3	no	—
253	C₄×Q₈	—	2	3	no	—
254	C₄×Q₈	—	2	3	no	—
255	C₄×Q₈	—	134	8	no	—
256	C₄×Q₈	C₂×Dic₆₇	1	3	yes	—
257	C₄×Q₈	C₂×Dic₆₇	1	3	yes	—
258	C₄×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
259	C₄×Q₈	C₂×Dic₆₇	1	3	yes	—
260	C₄×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
261	C₄×Q₈	C₂²⋊D₆₇	1	3	yes	—
262	C₄×D₄	—	2	3	no	—
263	C₄×D₄	—	2	3	no	—
264	C₄×D₄	C₂×Dic₆₇	1	3	yes	—
265	C₄×D₄	C₂×Dic₆₇	1	3	yes	—
266	#117	—	2	3	no	—
267	#117	C₂²×D₆₇	1	3	yes	—
268	C₂×C₄⋊C₄	—	2	3	no	—
269	C₂×C₄⋊C₄	—	134	3	no	—
270	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	3	yes	—
271	C₂×C₄²	—	2	3	no	—
272	C₂×C₄²	—	67	1	no	—
273	C₂×C₄²	—	67	3	no	—
274	C₂×C₄²	—	134	3	no	—
275	C₂×C₄²	—	134	3	no	—
276	C₂×C₄²	C₂²×D₆₇	1	3	yes	—
277	C₄⋊C₈	—	134	3	no	—
278	C₄⋊C₈	—	134	3	no	—
279	#104	—	67	4	no	—
280	#104	—	134	6	no	—
281	#97	—	134	3	no	—
282	C₄×C₈	—	134	3	no	—
283	#95	—	134	3	no	—
284	#41	—	2	1	no	—
285	#41	—	2	1	no	—
286	#41	Q₈×D₆₇	1	2	yes	—
287	#41	#11761629	1	2	yes	—
288	C₂×Q₈	—	2	4	no	—
289	C₂×Q₈	—	2	3	no	—
290	C₂×Q₈	—	2	3	no	—
291	C₂×Q₈	—	4	4	no	—
292	C₂×Q₈	C₂²⋊Dic₆₇	1	3	yes	—
293	C₂×Q₈	C₂²⋊Dic₆₇	1	1	yes	characteristic
294	C₂×Q₈	C₂²×Dic₆₇	1	1	yes	characteristic
295	C₂×D₄	—	2	1	no	—
296	C₂×D₄	C₂²×Dic₆₇	1	1	yes	characteristic
297	C₂²×C₄	—	2	1	no	—
298	C₂²×C₄	—	2	3	no	—
299	C₂²×C₄	—	2	3	no	—
300	C₂²×C₄	—	134	3	no	—
301	C₂²×C₄	—	134	1	no	—
302	C₂²×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
303	C₂²×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
304	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
305	M₁₆	—	67	2	no	—
306	M₁₆	—	134	3	no	—
307	M₁₆	—	134	3	no	—
308	M₁₆	—	134	1	no	—
309	C₂×C₈	—	134	3	no	—
310	C₂×C₈	—	134	1	no	—
311	C₄⋊C₄	—	2	6	no	—
312	C₄⋊C₄	—	2	3	no	—
313	C₄⋊C₄	—	2	3	no	—
314	C₄⋊C₄	—	2	3	no	—
315	C₄⋊C₄	—	2	3	no	—
316	C₄⋊C₄	—	2	6	no	—
317	C₄⋊C₄	—	134	12	no	—
318	C₄⋊C₄	C₂²⋊Dic₆₇	1	3	yes	—
319	C₄⋊C₄	C₂²⋊Dic₆₇	1	6	yes	—
320	C₄⋊C₄	C₂²⋊Dic₆₇	1	3	yes	—
321	C₄⋊C₄	C₂²×Dic₆₇	1	3	yes	—
322	C₄⋊C₄	C₂²×Dic₆₇	1	3	yes	—
323	C₂²⋊C₄	—	2	3	no	—
324	C₂²⋊C₄	—	2	3	no	—
325	C₂²⋊C₄	—	2	3	no	—
326	C₄²	—	2	3	no	—
327	C₄²	—	2	3	no	—
328	C₄²	—	2	3	no	—
329	C₄²	—	268	6	no	—
330	C₄²	—	67	2	no	—
331	C₄²	—	134	12	no	—
332	C₄²	—	134	3	no	—
333	C₄²	—	134	3	no	—
334	C₄²	—	134	1	no	—
335	C₄²	C₂²×Dic₆₇	1	3	yes	—
336	C₄²	C₂²×Dic₆₇	1	3	yes	—
337	C₄²	C₂×C₂²⋊D₆₇	1	3	yes	—
338	C₄²	C₂×C₂²⋊D₆₇	1	3	yes	—
339	C₂³	—	2	1	no	—
340	C₂³	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
341	Q₈	—	2	1	no	—
342	Q₈	—	2	6	no	—
343	Q₈	—	4	6	no	—
344	Q₈	—	8	3	no	—
345	Q₈	#14523970	1	4	yes	direct product factor
346	Q₈	Q₈×Dic₆₇	1	2	yes	complemented
347	D₄	—	2	2	no	—
348	D₄	—	2	2	no	—
349	D₄	—	2	1	no	—
350	D₄	Q₈×Dic₆₇	1	2	yes	complemented
351	C₂×C₄	—	2	1	no	—
352	C₂×C₄	—	2	2	no	—
353	C₂×C₄	—	2	2	no	—
354	C₂×C₄	—	2	6	no	—
355	C₂×C₄	—	2	3	no	—
356	C₂×C₄	—	2	3	no	—
357	C₂×C₄	—	4	3	no	—
358	C₂×C₄	—	4	6	no	—
359	C₂×C₄	—	8	3	no	—
360	C₂×C₄	—	268	3	no	—
361	C₂×C₄	—	268	3	no	—
362	C₂×C₄	—	134	6	no	—
363	C₂×C₄	—	134	2	no	—
364	C₂×C₄	—	134	2	no	—
365	C₂×C₄	—	134	2	no	—
366	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	3	yes	—
367	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	3	yes	—
368	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	3	yes	—
369	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	3	yes	—
370	C₂×C₄	C₂³×Dic₆₇	1	1	yes	characteristic
371	C₂×C₄	C₂²×C₂²⋊D₆₇	1	1	yes	characteristic
372	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
373	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
374	C₂×C₄	#14524091	1	1	yes	characteristic
375	C₂×C₄	#14524091	1	1	yes	characteristic
376	C₈	—	268	3	no	—
377	C₈	—	134	2	no	—
378	C₂²	—	2	1	no	—
379	C₂²	—	2	1	no	—
380	C₂²	—	2	2	no	—
381	C₂²	—	4	1	no	—
382	C₂²	—	4	1	no	—
383	C₂²	C₂²×C₂²⋊Dic₆₇	1	1	yes	characteristic
384	C₂²	#16864057	1	1	yes	characteristic
385	C₂²	Q₈×C₂²⋊D₆₇	1	1	yes	characteristic
386	C₄	—	2	3	no	—
387	C₄	—	2	3	no	—
388	C₄	—	2	3	no	—
389	C₄	—	2	2	no	—
390	C₄	—	4	3	no	—
391	C₄	—	4	3	no	—
392	C₄	—	268	6	no	—
393	C₄	—	134	4	no	—
394	C₄	C₂×#14523970	1	6	yes	—
395	C₄	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
396	C₄	#16863874	1	1	yes	characteristic
397	C₄	#16864057	1	1	yes	characteristic
398	C₄	Q₈×C₂²⋊D₆₇	1	1	yes	characteristic
399	C₂	—	2	1	no	—
400	C₂	—	2	1	no	—
401	C₂	—	4	2	no	—
402	C₂	Q₈×C₂²⋊Dic₆₇	1	1	yes	characteristic
403	C₂	C₂²×#14523970	1	1	yes	characteristic
404	C₂	#22568756	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₆₇
Fitting	Q₈×#41×C₆₇	C₂
socle	C₂²×C₆₇	C₂²×C₂²⋊C₄
prime radical	C₂×#41×C₆₇	C₂³
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	Q₈×#41	D₆₇
hypocenter	C₆₇	Q₈×#104
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×Q₈ ◁ Q₈×#41

Complements

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

N	H ≅ G/N	type
1. Q₈×#11761612	C₂	semidirect
2. C₄×Q₈×Dic₆₇	C₂	semidirect
3. #22561187	C₂	semidirect
4. #22561154	C₂	semidirect
21. Q₈²×C₆₇	C₄	semidirect
22. #557×C₆₇	C₄	semidirect
24. #554×C₆₇	C₄	semidirect
26. D₄×Q₈×C₆₇	C₄	semidirect
43. #14523970	Q₈	direct (N × H)
86. #11761612	C₂×Q₈	semidirect
95. C₄×Dic₆₇	C₂×Q₈	semidirect
142. Q₈×C₆₇	#104	semidirect
143. Q₈×C₆₇	C₄×Q₈	semidirect
147. D₄×C₆₇	C₄×Q₈	semidirect
210. C₆₇	Q₈×#104	semidirect
211. Q₈²	Dic₆₇	semidirect
212. #557	Dic₆₇	semidirect
214. #554	Dic₆₇	semidirect
216. D₄×Q₈	Dic₆₇	semidirect
345. Q₈	#14523970	direct (N × H)
346. Q₈	Q₈×Dic₆₇	semidirect
350. D₄	Q₈×Dic₆₇	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#11761612	2
C₄×Q₈×Dic₆₇	2
#22561187	2
#22561154	2
#22560229	2
C₄×#14523970	2
Q₈×#41×C₆₇	2
Q₈×#104	67

Subgroups — Q8×#14523970 #30872530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14523970 #30872530