Subgroups — #31890140

Order 18176.

588 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22588492	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₇₁⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊(C₇₁⋊C₈)	C₂	1	2	yes	maximal
4	#22586181	C₂	1	2	yes	maximal
5	C₄²×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
8	C₂×C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
9	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	2	yes	—
10	C₂×C₄×C₇₁⋊C₈	C₂²	1	2	yes	—
11	#17985754	—	2	1	no	—
12	Q₈⋊(C₇₁⋊C₈)	—	2	4	no	—
13	C₂×C₄×Q₈×C₇₁	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₇₁	C₄	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₇₁	—	2	1	no	—
17	C₄×C₄⋊C₄×C₇₁	—	2	1	no	—
18	C₄×C₄⋊C₄×C₇₁	—	2	1	no	—
19	C₄×C₄⋊C₄×C₇₁	C₄	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₇₁	C₄	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
22	C₄³×C₇₁	—	2	1	no	—
23	C₄³×C₇₁	C₂²	1	1	yes	characteristic
24	C₂²×C₇₁⋊C₈	—	2	2	no	—
25	C₂²×C₇₁⋊C₈	—	2	2	no	—
26	C₄⋊(C₇₁⋊C₈)	—	2	4	no	—
27	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
28	C₄⋊(C₇₁⋊C₈)	—	2	2	no	—
29	C₄⋊(C₇₁⋊C₈)	—	4	2	no	—
30	C₄⋊(C₇₁⋊C₈)	—	4	2	no	—
31	C₄×C₇₁⋊C₈	—	2	2	no	—
32	C₄×C₇₁⋊C₈	—	2	2	no	—
33	C₂²×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
34	C₄×Q₈×C₇₁	—	2	2	no	—
35	C₄×Q₈×C₇₁	—	2	2	no	—
36	C₄×Q₈×C₇₁	—	2	2	no	—
37	C₄×Q₈×C₇₁	—	2	1	no	—
38	C₄×Q₈×C₇₁	—	2	1	no	—
39	C₄×Q₈×C₇₁	C₈	1	4	yes	complemented
40	C₄×Q₈×C₇₁	D₄	1	2	yes	—
41	C₄×Q₈×C₇₁	Q₈	1	2	yes	—
42	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
43	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
44	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
45	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₇₁	—	2	1	no	—
49	C₂×C₄²×C₇₁	—	2	1	no	—
50	C₂×C₄²×C₇₁	—	2	1	no	—
51	C₂×C₄²×C₇₁	—	2	1	no	—
52	C₂×C₄²×C₇₁	—	2	1	no	—
53	C₂×C₄²×C₇₁	—	2	1	no	—
54	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
56	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₇₁	D₄	1	1	yes	characteristic
60	C₂×C₄²×C₇₁	C₂³	1	1	yes	characteristic
61	#95×C₇₁	—	2	1	no	—
62	#95×C₇₁	—	2	1	no	—
63	#95×C₇₁	—	2	1	no	—
64	C₂×C₇₁⋊C₈	—	2	4	no	—
65	C₂×C₇₁⋊C₈	—	4	2	no	—
66	C₂×C₇₁⋊C₈	—	4	2	no	—
67	C₂×C₇₁⋊C₈	—	4	2	no	—
68	C₂×C₇₁⋊C₈	—	4	2	no	—
69	C₂×C₇₁⋊C₈	—	4	2	no	—
70	C₂×Q₈×C₇₁	—	2	1	no	—
71	C₂×Q₈×C₇₁	—	2	1	no	—
72	C₂×Q₈×C₇₁	—	2	1	no	—
73	C₂×Q₈×C₇₁	C₄⋊C₄	1	2	yes	—
74	C₂×Q₈×C₇₁	C₂×C₈	1	2	yes	—
75	C₂×Q₈×C₇₁	M₁₆	1	2	yes	—
76	C₂²×C₄×C₇₁	—	2	1	no	—
77	C₂²×C₄×C₇₁	—	2	1	no	—
78	C₂²×C₄×C₇₁	—	2	1	no	—
79	C₂²×C₄×C₇₁	—	2	1	no	—
80	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
83	C₂²×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
84	C₂²×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
87	C₄⋊C₄×C₇₁	—	2	4	no	—
88	C₄⋊C₄×C₇₁	—	2	4	no	—
89	C₄⋊C₄×C₇₁	—	2	1	no	—
90	C₄⋊C₄×C₇₁	—	2	2	no	—
91	C₄⋊C₄×C₇₁	—	2	2	no	—
92	C₄⋊C₄×C₇₁	—	2	1	no	—
93	C₄⋊C₄×C₇₁	—	2	1	no	—
94	C₄⋊C₄×C₇₁	C₄⋊C₄	1	2	yes	—
95	C₄⋊C₄×C₇₁	C₂×C₈	1	2	yes	—
96	C₄⋊C₄×C₇₁	M₁₆	1	2	yes	—
97	C₄²×C₇₁	—	2	4	no	—
98	C₄²×C₇₁	—	2	4	no	—
99	C₄²×C₇₁	—	2	2	no	—
100	C₄²×C₇₁	—	2	2	no	—
101	C₄²×C₇₁	—	2	2	no	—
102	C₄²×C₇₁	—	2	1	no	—
103	C₄²×C₇₁	—	2	1	no	—
104	C₄²×C₇₁	—	2	1	no	—
105	C₄²×C₇₁	—	2	1	no	—
106	C₄²×C₇₁	—	4	1	no	—
107	C₄²×C₇₁	—	4	2	no	—
108	C₄²×C₇₁	—	4	2	no	—
109	C₄²×C₇₁	—	4	1	no	—
110	C₄²×C₇₁	—	4	1	no	—
111	C₄²×C₇₁	C₂×C₈	1	2	yes	—
112	C₄²×C₇₁	M₁₆	1	2	yes	—
113	C₄²×C₇₁	SD₁₆	1	2	yes	—
114	C₄²×C₇₁	Q₁₆	1	2	yes	—
115	C₄²×C₇₁	C₂×D₄	1	1	yes	characteristic
116	C₄²×C₇₁	C₂×Q₈	1	1	yes	characteristic
117	C₄²×C₇₁	#41	1	1	yes	characteristic
118	C₄²×C₇₁	#41	1	1	yes	characteristic
119	C₂³×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
120	Q₈×C₇₁	—	2	2	no	—
121	Q₈×C₇₁	—	2	2	no	—
122	Q₈×C₇₁	—	2	2	no	—
123	Q₈×C₇₁	C₄⋊C₈	1	4	yes	complemented
124	C₂×C₄×C₇₁	—	2	2	no	—
125	C₂×C₄×C₇₁	—	2	2	no	—
126	C₂×C₄×C₇₁	—	2	1	no	—
127	C₂×C₄×C₇₁	—	2	2	no	—
128	C₂×C₄×C₇₁	—	2	1	no	—
129	C₂×C₄×C₇₁	—	2	1	no	—
130	C₂×C₄×C₇₁	—	2	1	no	—
131	C₂×C₄×C₇₁	—	2	1	no	—
132	C₂×C₄×C₇₁	—	2	1	no	—
133	C₂×C₄×C₇₁	—	2	2	no	—
134	C₂×C₄×C₇₁	—	2	2	no	—
135	C₂×C₄×C₇₁	—	2	1	no	—
136	C₂×C₄×C₇₁	—	2	2	no	—
137	C₂×C₄×C₇₁	—	2	1	no	—
138	C₂×C₄×C₇₁	—	2	1	no	—
139	C₂×C₄×C₇₁	—	4	1	no	—
140	C₂×C₄×C₇₁	—	4	1	no	—
141	C₂×C₄×C₇₁	—	4	1	no	—
142	C₂×C₄×C₇₁	—	4	1	no	—
143	C₂×C₄×C₇₁	—	4	1	no	—
144	C₂×C₄×C₇₁	—	4	1	no	—
145	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
146	C₂×C₄×C₇₁	C₂²⋊C₈	1	2	yes	—
147	C₂×C₄×C₇₁	#103	1	2	yes	—
148	C₂×C₄×C₇₁	#103	1	2	yes	—
149	C₂×C₄×C₇₁	#104	1	2	yes	—
150	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
151	C₂×C₄×C₇₁	#117	1	1	yes	characteristic
152	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
153	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
154	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
155	C₂×C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
156	C₂×C₄×C₇₁	C₂²×C₈	1	1	yes	characteristic
157	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
158	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
159	C₂×C₄×C₇₁	C₂×M₁₆	1	1	yes	characteristic
160	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
161	C₂×C₄×C₇₁	C₂×Q₁₆	1	1	yes	characteristic
162	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
163	C₂×C₄×C₇₁	#137	1	1	yes	characteristic
164	C₇₁⋊C₈	—	4	4	no	—
165	C₇₁⋊C₈	—	8	2	no	—
166	C₂²×C₇₁	#380	1	1	yes	characteristic
167	C₂²×C₇₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
168	C₂²×C₇₁	#407	1	1	yes	characteristic
169	C₂²×C₇₁	C₂×#103	1	1	yes	characteristic
170	C₂²×C₇₁	#417	1	1	yes	characteristic
171	C₂²×C₇₁	C₂×#104	1	1	yes	characteristic
172	C₂²×C₇₁	#421	1	1	yes	characteristic
173	C₄×C₇₁	—	2	4	no	—
174	C₄×C₇₁	—	2	1	no	—
175	C₄×C₇₁	—	2	1	no	—
176	C₄×C₇₁	—	2	2	no	—
177	C₄×C₇₁	—	2	2	no	—
178	C₄×C₇₁	—	4	2	no	—
179	C₄×C₇₁	—	4	2	no	—
180	C₄×C₇₁	—	4	1	no	—
181	C₄×C₇₁	—	4	2	no	—
182	C₄×C₇₁	Q₈⋊C₈	1	4	yes	complemented
183	C₄×C₇₁	C₂×C₄⋊C₈	1	1	yes	characteristic
184	C₄×C₇₁	#423	1	1	yes	characteristic
185	C₄×C₇₁	#431	1	1	yes	characteristic
186	C₄×C₇₁	#432	1	1	yes	characteristic
187	C₄×C₇₁	#460	1	1	yes	characteristic
188	C₄×C₇₁	#462	1	1	yes	characteristic
189	C₄×C₇₁	#475	1	1	yes	characteristic
190	C₄×C₇₁	#477	1	1	yes	characteristic
191	#8564	—	71	1	no	maximal, Sylow
192	C₂×C₇₁	C₂×Q₈⋊C₈	1	1	yes	characteristic
193	C₂×C₇₁	#1477	1	1	yes	characteristic
194	C₂×C₇₁	#1487	1	1	yes	characteristic
195	C₂×C₇₁	#1489	1	1	yes	characteristic
196	C₂×C₇₁	#1797	1	1	yes	characteristic
197	C₂×C₇₁	#1803	1	1	yes	characteristic
198	C₂×C₇₁	#1806	1	1	yes	characteristic
199	C₄²×Q₈	D₇₁	1	1	yes	characteristic
200	#1842	—	71	1	no	—
201	C₄×C₄⋊C₈	—	71	1	no	—
202	C₂×Q₈⋊C₈	—	71	2	no	—
203	#1279	—	71	2	no	—
204	C₇₁	#8564	1	1	yes	characteristic, complemented, Sylow
205	C₂×C₄×Q₈	Dic₇₁	1	1	yes	characteristic
206	C₂×C₄×Q₈	Dic₇₁	1	1	yes	characteristic
207	C₂×C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
208	C₂×C₄⋊C₈	—	71	2	no	—
209	C₂×C₄⋊C₈	—	142	1	no	—
210	C₂×C₄⋊C₈	—	142	1	no	—
211	C₂×C₄⋊C₈	—	142	1	no	—
212	C₂×C₄×C₈	—	71	2	no	—
213	C₄×C₄⋊C₄	—	2	1	no	—
214	C₄×C₄⋊C₄	—	2	1	no	—
215	C₄×C₄⋊C₄	—	2	1	no	—
216	C₄×C₄⋊C₄	Dic₇₁	1	1	yes	characteristic
217	C₄×C₄⋊C₄	Dic₇₁	1	1	yes	characteristic
218	C₄×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
219	C₄³	—	2	1	no	—
220	C₄³	C₂×D₇₁	1	1	yes	characteristic
221	#336	—	142	1	no	—
222	Q₈⋊C₈	—	142	4	no	—
223	C₂²×Q₈	C₂×Dic₇₁	1	1	yes	characteristic
224	C₂²×C₈	—	142	2	no	—
225	C₂²×C₈	—	142	2	no	—
226	C₄×Q₈	—	2	2	no	—
227	C₄×Q₈	—	2	2	no	—
228	C₄×Q₈	—	2	2	no	—
229	C₄×Q₈	—	2	1	no	—
230	C₄×Q₈	—	2	1	no	—
231	C₄×Q₈	C₇₁⋊C₈	1	4	yes	complemented
232	C₄×Q₈	Dic₁₄₂	1	2	yes	—
233	C₄×Q₈	D₂₈₄	1	2	yes	—
234	C₂×C₄⋊C₄	—	2	1	no	—
235	C₂×C₄⋊C₄	—	2	1	no	—
236	C₂×C₄⋊C₄	—	2	1	no	—
237	C₂×C₄⋊C₄	C₂×Dic₇₁	1	1	yes	characteristic
238	C₂×C₄⋊C₄	C₂×Dic₇₁	1	1	yes	characteristic
239	C₂×C₄⋊C₄	C₂×Dic₇₁	1	1	yes	characteristic
240	C₂×C₄²	—	2	1	no	—
241	C₂×C₄²	—	2	1	no	—
242	C₂×C₄²	—	2	1	no	—
243	C₂×C₄²	—	2	1	no	—
244	C₂×C₄²	—	2	1	no	—
245	C₂×C₄²	—	2	1	no	—
246	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
247	C₂×C₄²	C₂×Dic₇₁	1	1	yes	characteristic
248	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
249	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
250	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
251	C₂×C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
252	C₂×C₄²	C₂²×D₇₁	1	1	yes	characteristic
253	C₄⋊C₈	—	284	2	no	—
254	C₄⋊C₈	—	284	2	no	—
255	C₄⋊C₈	—	142	4	no	—
256	C₄⋊C₈	—	142	2	no	—
257	C₄⋊C₈	—	142	2	no	—
258	C₄×C₈	—	142	2	no	—
259	C₄×C₈	—	142	2	no	—
260	#95	—	2	1	no	—
261	#95	—	2	1	no	—
262	#95	—	2	1	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	2	1	no	—
266	C₂×Q₈	C₄⋊Dic₇₁	1	2	yes	—
267	C₂×Q₈	C₂×C₇₁⋊C₈	1	2	yes	—
268	C₂×Q₈	#11765849	1	2	yes	—
269	C₂²×C₄	—	2	1	no	—
270	C₂²×C₄	—	2	1	no	—
271	C₂²×C₄	—	2	1	no	—
272	C₂²×C₄	—	2	1	no	—
273	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
274	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
275	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
276	C₂²×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
277	C₂²×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
278	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
279	C₂²×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
280	C₂×C₈	—	284	2	no	—
281	C₂×C₈	—	284	2	no	—
282	C₂×C₈	—	284	2	no	—
283	C₂×C₈	—	284	2	no	—
284	C₂×C₈	—	284	2	no	—
285	C₂×C₈	—	142	4	no	—
286	C₄⋊C₄	—	2	4	no	—
287	C₄⋊C₄	—	2	4	no	—
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	2	2	no	—
291	C₄⋊C₄	—	2	1	no	—
292	C₄⋊C₄	—	2	1	no	—
293	C₄⋊C₄	C₄⋊Dic₇₁	1	2	yes	—
294	C₄⋊C₄	C₂×C₇₁⋊C₈	1	2	yes	—
295	C₄⋊C₄	#11765849	1	2	yes	—
296	C₄²	—	2	4	no	—
297	C₄²	—	2	4	no	—
298	C₄²	—	2	2	no	—
299	C₄²	—	2	2	no	—
300	C₄²	—	2	2	no	—
301	C₄²	—	2	1	no	—
302	C₄²	—	2	1	no	—
303	C₄²	—	2	1	no	—
304	C₄²	—	2	1	no	—
305	C₄²	—	4	1	no	—
306	C₄²	—	4	2	no	—
307	C₄²	—	4	2	no	—
308	C₄²	—	4	1	no	—
309	C₄²	—	4	1	no	—
310	C₄²	C₂×C₇₁⋊C₈	1	2	yes	—
311	C₄²	#11765849	1	2	yes	—
312	C₄²	#11765852	1	2	yes	—
313	C₄²	#11765855	1	2	yes	—
314	C₄²	C₂×D₂₈₄	1	1	yes	characteristic
315	C₄²	C₂×Dic₁₄₂	1	1	yes	characteristic
316	C₄²	#11765865	1	1	yes	characteristic
317	C₄²	#11765865	1	1	yes	characteristic
318	C₂³	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
319	Q₈	—	2	2	no	—
320	Q₈	—	2	2	no	—
321	Q₈	—	2	2	no	—
322	Q₈	C₄⋊(C₇₁⋊C₈)	1	4	yes	complemented
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	2	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	2	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	1	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	2	no	—
333	C₂×C₄	—	2	2	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	2	no	—
336	C₂×C₄	—	2	1	no	—
337	C₂×C₄	—	2	1	no	—
338	C₂×C₄	—	4	1	no	—
339	C₂×C₄	—	4	1	no	—
340	C₂×C₄	—	4	1	no	—
341	C₂×C₄	—	4	1	no	—
342	C₂×C₄	—	4	1	no	—
343	C₂×C₄	—	4	1	no	—
344	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
345	C₂×C₄	C₂²⋊(C₇₁⋊C₈)	1	2	yes	—
346	C₂×C₄	#14556052	1	2	yes	—
347	C₂×C₄	#14556052	1	2	yes	—
348	C₂×C₄	#14556055	1	2	yes	—
349	C₂×C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
350	C₂×C₄	#14556087	1	1	yes	characteristic
351	C₂×C₄	#14556102	1	1	yes	characteristic
352	C₂×C₄	#14556102	1	1	yes	characteristic
353	C₂×C₄	#14556107	1	1	yes	characteristic
354	C₂×C₄	#14556107	1	1	yes	characteristic
355	C₂×C₄	C₂²×C₇₁⋊C₈	1	1	yes	characteristic
356	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
357	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
358	C₂×C₄	C₂×#11765849	1	1	yes	characteristic
359	C₂×C₄	C₂×#11765852	1	1	yes	characteristic
360	C₂×C₄	C₂×#11765855	1	1	yes	characteristic
361	C₂×C₄	#14556153	1	1	yes	characteristic
362	C₂×C₄	#14556158	1	1	yes	characteristic
363	C₈	—	284	4	no	—
364	C₈	—	568	2	no	—
365	C₂²	#17985859	1	1	yes	characteristic
366	C₂²	C₂×C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
367	C₂²	#17985963	1	1	yes	characteristic
368	C₂²	C₂×#14556052	1	1	yes	characteristic
369	C₂²	#17986000	1	1	yes	characteristic
370	C₂²	C₂×#14556055	1	1	yes	characteristic
371	C₂²	#17986016	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	2	2	no	—
374	C₄	—	2	4	no	—
375	C₄	—	2	1	no	—
376	C₄	—	2	1	no	—
377	C₄	—	4	2	no	—
378	C₄	—	4	1	no	—
379	C₄	—	4	2	no	—
380	C₄	—	4	2	no	—
381	C₄	Q₈⋊(C₇₁⋊C₈)	1	4	yes	complemented
382	C₄	C₂×C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
383	C₄	#17986023	1	1	yes	characteristic
384	C₄	#17986052	1	1	yes	characteristic
385	C₄	#17986056	1	1	yes	characteristic
386	C₄	#17986199	1	1	yes	characteristic
387	C₄	#17986209	1	1	yes	characteristic
388	C₄	#17986284	1	1	yes	characteristic
389	C₄	#17986294	1	1	yes	characteristic
390	C₂	C₂×Q₈⋊(C₇₁⋊C₈)	1	1	yes	characteristic
391	C₂	#22586727	1	1	yes	characteristic
392	C₂	#22586769	1	1	yes	characteristic
393	C₂	#22586779	1	1	yes	characteristic
394	C₂	#22588320	1	1	yes	characteristic
395	C₂	#22588345	1	1	yes	characteristic
396	C₂	#22588360	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₇₁
Fitting	C₄²×Q₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₇₁	C₂³
hypercenter	C₄²×Q₈	D₇₁
hypocenter	C₇₁	#8564
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
39. C₄×Q₈×C₇₁	C₈	semidirect
123. Q₈×C₇₁	C₄⋊C₈	semidirect
182. C₄×C₇₁	Q₈⋊C₈	semidirect
204. C₇₁	#8564	semidirect
231. C₄×Q₈	C₇₁⋊C₈	semidirect
322. Q₈	C₄⋊(C₇₁⋊C₈)	semidirect
381. C₄	Q₈⋊(C₇₁⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
#22588492	2
C₄×C₄⋊(C₇₁⋊C₈)	2
C₂×Q₈⋊(C₇₁⋊C₈)	2
#22586181	2
C₄²×Q₈×C₇₁	2
#8564	71