Subgroups — Q₈×Dic₇₁⋊C₈ #32016243

Order 18176.

896 conjugacy classes of subgroups.

The subgroup lattice has 286 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₄×Q₈×Dic₇₁	C₂	1	1	yes	maximal, characteristic
3	#22589058	C₂	1	3	yes	maximal
4	#22589056	C₂	1	3	yes	maximal
5	#22588491	C₂	1	3	yes	maximal
6	C₄×Dic₇₁⋊C₈	C₂	1	3	yes	maximal
7	C₂×C₈×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂×Q₈×Dic₇₁	C₄	1	1	yes	characteristic
9	C₂×Q₈×Dic₇₁	C₄	1	1	yes	characteristic
10	Q₈×C₇₁⋊C₈	—	2	4	no	—
11	C₂×C₄⋊(C₇₁⋊C₈)	C₂²	1	3	yes	—
12	C₂×Dic₇₁⋊C₈	C₂²	1	3	yes	—
13	C₂×Dic₇₁⋊C₈	C₂²	1	6	yes	—
14	C₂×Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₇₁⋊C₈	C₂²	1	3	yes	—
16	C₄⋊C₄×Dic₇₁	C₄	1	3	yes	—
17	C₄⋊C₄×Dic₇₁	C₄	1	3	yes	—
18	C₄×C₄⋊Dic₇₁	C₂²	1	3	yes	—
19	C₄²×Dic₇₁	C₂²	1	3	yes	—
20	#17985753	C₂²	1	3	yes	—
21	#17985753	C₂²	1	3	yes	—
22	C₂×C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
23	C₈×Q₈×C₇₁	—	2	4	no	—
24	C₂×C₄⋊C₈×C₇₁	C₂²	1	3	yes	—
25	C₂×C₄×C₈×C₇₁	C₂²	1	3	yes	—
26	C₂²×C₇₁⋊C₈	C₂³	1	3	yes	—
27	C₂²×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
28	Q₈×Dic₇₁	—	2	4	no	—
29	Q₈×Dic₇₁	C₈	1	8	yes	complemented
30	C₂×C₄⋊Dic₇₁	C₂×C₄	1	3	yes	—
31	C₂×C₄⋊Dic₇₁	C₂×C₄	1	3	yes	—
32	C₂×C₄×Dic₇₁	C₂×C₄	1	3	yes	—
33	C₂×C₄×Dic₇₁	C₂×C₄	1	3	yes	—
34	C₂×C₄×Dic₇₁	C₂×C₄	1	3	yes	—
35	C₂×C₄×Dic₇₁	C₂×C₄	1	3	yes	—
36	C₂×C₄×Dic₇₁	C₂³	1	3	yes	—
37	C₂×C₄×Dic₇₁	C₂³	1	1	yes	characteristic
38	C₄⋊(C₇₁⋊C₈)	—	2	6	no	—
39	Dic₇₁⋊C₈	—	2	6	no	—
40	Dic₇₁⋊C₈	—	4	6	no	—
41	Dic₇₁⋊C₈	Q₈	1	4	yes	direct product factor
42	C₄×C₇₁⋊C₈	—	2	6	no	—
43	C₇₁⋊#95	C₂×C₄	1	6	yes	—
44	C₂²×Q₈×C₇₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₇₁	C₂³	1	3	yes	—
46	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
47	C₄×Q₈×C₇₁	D₄	1	4	yes	—
48	C₄×Q₈×C₇₁	Q₈	1	4	yes	—
49	C₂×C₄⋊C₄×C₇₁	C₂×C₄	1	3	yes	—
50	C₂×C₄²×C₇₁	C₂³	1	3	yes	—
51	C₄⋊C₈×C₇₁	—	2	6	no	—
52	C₄×C₈×C₇₁	—	2	6	no	—
53	C₂²×Dic₇₁	C₂²×C₄	1	3	yes	—
54	C₂²×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
55	C₂²×Dic₇₁	C₂²×C₄	1	3	yes	—
56	C₂²×Dic₇₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₇₁⋊C₈	—	2	1	no	—
58	C₂×C₇₁⋊C₈	—	2	1	no	—
59	C₂×C₇₁⋊C₈	—	2	3	no	—
60	C₂×C₇₁⋊C₈	—	2	3	no	—
61	C₂×C₇₁⋊C₈	—	2	3	no	—
62	C₂×C₇₁⋊C₈	C₂×Q₈	1	2	yes	—
63	C₄⋊Dic₇₁	—	2	6	no	—
64	C₄⋊Dic₇₁	C₂×C₈	1	12	yes	—
65	C₄×Dic₇₁	—	2	6	no	—
66	C₄×Dic₇₁	—	2	3	no	—
67	C₄×Dic₇₁	—	2	3	no	—
68	C₄×Dic₇₁	—	2	6	no	—
69	C₄×Dic₇₁	—	4	3	no	—
70	C₄×Dic₇₁	C₂×C₈	1	12	yes	—
71	C₄×Dic₇₁	C₂×Q₈	1	2	yes	—
72	C₄×Dic₇₁	#41	1	2	yes	—
73	C₂×Q₈×C₇₁	C₄⋊C₄	1	4	yes	—
74	C₂×Q₈×C₇₁	C₂×C₈	1	4	yes	—
75	C₂×Q₈×C₇₁	M₁₆	1	4	yes	—
76	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
77	C₂²×C₄×C₇₁	C₂²×C₄	1	3	yes	—
78	C₂²×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₇₁	—	2	1	no	—
80	C₂×C₈×C₇₁	—	2	1	no	—
81	C₂×C₈×C₇₁	—	2	3	no	—
82	C₂×C₈×C₇₁	—	2	3	no	—
83	C₂×C₈×C₇₁	—	2	3	no	—
84	C₂×C₈×C₇₁	C₂×Q₈	1	2	yes	—
85	C₄⋊C₄×C₇₁	C₄⋊C₄	1	12	yes	—
86	C₄²×C₇₁	C₂×D₄	1	6	yes	—
87	C₄²×C₇₁	C₂×Q₈	1	6	yes	—
88	C₂³×C₇₁	C₂³×C₄	1	1	yes	characteristic
89	Q₈×C₇₁	C₄⋊C₈	1	16	yes	complemented
90	C₂×C₄×C₇₁	—	2	3	no	—
91	C₂×C₄×C₇₁	—	2	3	no	—
92	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
93	C₂×C₄×C₇₁	C₂×C₄⋊C₄	1	6	yes	—
94	C₂×C₄×C₇₁	C₂²×C₈	1	6	yes	—
95	C₂×C₄×C₇₁	C₂×M₁₆	1	6	yes	—
96	C₂×C₄×C₇₁	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
98	C₂×C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
99	C₂×C₄×C₇₁	C₂×#41	1	1	yes	characteristic
100	C₂×C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
101	C₂×C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
102	C₂×Dic₇₁	—	2	3	no	—
103	C₂×Dic₇₁	—	2	1	no	—
104	C₂×Dic₇₁	—	2	1	no	—
105	C₂×Dic₇₁	—	2	3	no	—
106	C₂×Dic₇₁	—	2	3	no	—
107	C₂×Dic₇₁	—	2	3	no	—
108	C₂×Dic₇₁	—	2	3	no	—
109	C₂×Dic₇₁	C₄×Q₈	1	2	yes	—
110	C₂×Dic₇₁	C₄×Q₈	1	2	yes	—
111	C₂×Dic₇₁	C₂²×C₈	1	6	yes	—
112	C₂×Dic₇₁	C₂²×C₈	1	2	yes	—
113	C₂×Dic₇₁	#131	1	2	yes	—
114	C₈×C₇₁	—	2	2	no	—
115	C₈×C₇₁	—	4	3	no	—
116	C₇₁⋊C₈	—	2	2	no	—
117	C₇₁⋊C₈	—	4	3	no	—
118	C₂²×C₇₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
119	C₂²×C₇₁	C₂×C₄×Q₈	1	1	yes	characteristic
120	C₂²×C₇₁	#520	1	1	yes	characteristic
121	C₂²×C₇₁	C₂³×C₈	1	1	yes	characteristic
122	C₂²×C₇₁	C₂²×M₁₆	1	1	yes	characteristic
123	C₂²×C₇₁	C₂×#131	1	1	yes	characteristic
124	C₂²×C₇₁	#568	1	1	yes	characteristic
125	C₄×C₇₁	—	2	6	no	—
126	C₄×C₇₁	C₂×C₄⋊C₈	1	12	yes	—
127	C₄×C₇₁	D₄×Q₈	1	1	yes	characteristic
128	C₄×C₇₁	#550	1	1	yes	characteristic
129	C₄×C₇₁	#557	1	1	yes	characteristic
130	C₄×C₇₁	Q₈²	1	1	yes	characteristic
131	Dic₇₁	—	2	2	no	—
132	Dic₇₁	—	2	6	no	—
133	Dic₇₁	—	4	3	no	—
134	Dic₇₁	C₈×Q₈	1	4	yes	complemented
135	Q₈×C₄⋊C₈	—	71	1	no	maximal, Sylow
136	C₂×C₇₁	Q₈×C₄⋊C₄	1	1	yes	characteristic
137	C₂×C₇₁	C₂²×C₄⋊C₈	1	1	yes	characteristic
138	C₂×C₇₁	#2906	1	1	yes	characteristic
139	C₂×C₇₁	C₂×C₈×Q₈	1	1	yes	characteristic
140	C₂×C₇₁	Q₈×M₁₆	1	1	yes	characteristic
141	C₂×C₇₁	#2988	1	1	yes	characteristic
142	C₂×C₇₁	#2991	1	1	yes	characteristic
143	C₂×C₈×Q₈	—	71	1	no	—
144	C₂×C₈×Q₈	D₇₁	1	1	yes	characteristic
145	C₄²×Q₈	—	71	1	no	—
146	#1939	—	71	3	no	—
147	#1939	—	71	3	no	—
148	#1842	—	71	3	no	—
149	C₄×C₄⋊C₈	—	71	3	no	—
150	C₇₁	Q₈×C₄⋊C₈	1	1	yes	characteristic, complemented, Sylow
151	C₂×C₄×Q₈	—	71	1	no	—
152	C₂×C₄×Q₈	—	71	1	no	—
153	C₂×C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
154	C₈×Q₈	—	2	4	no	—
155	C₈×Q₈	—	142	4	no	—
156	C₂×C₄⋊C₈	—	71	6	no	—
157	C₂×C₄⋊C₈	—	71	1	no	—
158	C₂×C₄⋊C₈	—	71	3	no	—
159	C₂×C₄⋊C₈	—	71	3	no	—
160	C₂×C₄⋊C₈	C₂×D₇₁	1	3	yes	—
161	C₂×C₄×C₈	—	71	3	no	—
162	C₂×C₄×C₈	C₂×D₇₁	1	3	yes	—
163	C₄×C₄⋊C₄	—	71	3	no	—
164	C₄×C₄⋊C₄	—	71	3	no	—
165	C₄×C₄⋊C₄	—	71	3	no	—
166	C₄³	—	71	3	no	—
167	#336	—	71	3	no	—
168	#336	—	71	3	no	—
169	C₂²×Q₈	C₄×D₇₁	1	1	yes	characteristic
170	C₂²×C₈	—	71	1	no	—
171	C₂²×C₈	—	71	3	no	—
172	C₂²×C₈	C₂²×D₇₁	1	3	yes	—
173	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
174	C₄×Q₈	—	71	8	no	—
175	C₄×Q₈	—	142	4	no	—
176	C₄×Q₈	Dic₁₄₂	1	4	yes	—
177	C₄×Q₈	C₂²⋊D₇₁	1	4	yes	—
178	C₂×C₄⋊C₄	—	71	3	no	—
179	C₂×C₄⋊C₄	—	71	3	no	—
180	C₂×C₄⋊C₄	C₄×D₇₁	1	3	yes	—
181	C₂×C₄²	—	71	3	no	—
182	C₂×C₄²	—	71	3	no	—
183	C₂×C₄²	—	71	1	no	—
184	C₂×C₄²	—	71	3	no	—
185	C₂×C₄²	—	71	3	no	—
186	C₂×C₄²	—	71	3	no	—
187	C₂×C₄²	C₂²×D₇₁	1	3	yes	—
188	C₄⋊C₈	—	2	6	no	—
189	C₄⋊C₈	—	284	6	no	—
190	C₄⋊C₈	—	71	4	no	—
191	C₄⋊C₈	—	142	6	no	—
192	C₄⋊C₈	—	142	6	no	—
193	C₄×C₈	—	2	6	no	—
194	C₄×C₈	—	142	6	no	—
195	#95	—	71	6	no	—
196	C₂×Q₈	Dic₇₁⋊C₄	1	4	yes	—
197	C₂×Q₈	C₈×D₇₁	1	4	yes	—
198	C₂×Q₈	#11765848	1	4	yes	—
199	C₂²×C₄	—	71	1	no	—
200	C₂²×C₄	—	71	3	no	—
201	C₂²×C₄	—	71	1	no	—
202	C₂²×C₄	—	71	3	no	—
203	C₂²×C₄	C₂×C₄×D₇₁	1	3	yes	—
204	C₂²×C₄	C₂×C₄×D₇₁	1	3	yes	—
205	C₂²×C₄	C₂³×D₇₁	1	1	yes	characteristic
206	C₂×C₈	—	2	1	no	—
207	C₂×C₈	—	2	1	no	—
208	C₂×C₈	—	2	3	no	—
209	C₂×C₈	—	2	3	no	—
210	C₂×C₈	—	2	3	no	—
211	C₂×C₈	—	71	2	no	—
212	C₂×C₈	—	142	3	no	—
213	C₂×C₈	—	142	3	no	—
214	C₂×C₈	—	142	3	no	—
215	C₂×C₈	—	142	1	no	—
216	C₂×C₈	—	142	1	no	—
217	C₂×C₈	Q₈×D₇₁	1	2	yes	—
218	C₄⋊C₄	—	71	12	no	—
219	C₄⋊C₄	—	142	6	no	—
220	C₄⋊C₄	Dic₇₁⋊C₄	1	12	yes	—
221	C₄²	—	284	3	no	—
222	C₄²	—	71	2	no	—
223	C₄²	—	71	2	no	—
224	C₄²	—	71	12	no	—
225	C₄²	—	142	6	no	—
226	C₄²	—	142	3	no	—
227	C₄²	—	142	3	no	—
228	C₄²	—	142	6	no	—
229	C₄²	C₂×C₂²⋊D₇₁	1	6	yes	—
230	C₄²	C₂×Dic₁₄₂	1	6	yes	—
231	C₂³	C₂²×C₄×D₇₁	1	1	yes	characteristic
232	Q₈	Dic₇₁⋊C₈	1	16	yes	direct product factor
233	C₂×C₄	—	2	3	no	—
234	C₂×C₄	—	2	3	no	—
235	C₂×C₄	—	71	2	no	—
236	C₂×C₄	—	71	2	no	—
237	C₂×C₄	—	71	2	no	—
238	C₂×C₄	—	71	2	no	—
239	C₂×C₄	—	71	6	no	—
240	C₂×C₄	—	142	3	no	—
241	C₂×C₄	—	142	3	no	—
242	C₂×C₄	—	142	3	no	—
243	C₂×C₄	—	142	3	no	—
244	C₂×C₄	—	142	1	no	—
245	C₂×C₄	—	142	1	no	—
246	C₂×C₄	—	142	3	no	—
247	C₂×C₄	C₂×Dic₇₁⋊C₄	1	6	yes	—
248	C₂×C₄	C₂×Dic₇₁⋊C₄	1	6	yes	—
249	C₂×C₄	C₂×C₈×D₇₁	1	6	yes	—
250	C₂×C₄	C₂×#11765848	1	6	yes	—
251	C₂×C₄	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
252	C₂×C₄	C₂×Q₈×D₇₁	1	1	yes	characteristic
253	C₂×C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
254	C₂×C₄	#41×D₇₁	1	1	yes	characteristic
255	C₂×C₄	#14556174	1	1	yes	characteristic
256	C₂×C₄	#14556176	1	1	yes	characteristic
257	C₈	—	2	2	no	—
258	C₈	—	4	3	no	—
259	C₈	—	284	3	no	—
260	C₈	—	142	2	no	—
261	C₂²	C₂²×Dic₇₁⋊C₄	1	1	yes	characteristic
262	C₂²	C₄×Q₈×D₇₁	1	1	yes	characteristic
263	C₂²	#17986481	1	1	yes	characteristic
264	C₂²	C₂²×C₈×D₇₁	1	1	yes	characteristic
265	C₂²	C₂²×#11765848	1	1	yes	characteristic
266	C₂²	#131×D₇₁	1	1	yes	characteristic
267	C₂²	#17986764	1	1	yes	characteristic
268	C₄	—	2	6	no	—
269	C₄	—	284	3	no	—
270	C₄	—	71	4	no	—
271	C₄	—	142	6	no	—
272	C₄	—	142	2	no	—
273	C₄	C₂×Dic₇₁⋊C₈	1	12	yes	—
274	C₄	Q₈×C₂²⋊D₇₁	1	1	yes	characteristic
275	C₄	#17986668	1	1	yes	characteristic
276	C₄	#17986725	1	1	yes	characteristic
277	C₄	Q₈×Dic₁₄₂	1	1	yes	characteristic
278	C₂	Q₈×Dic₇₁⋊C₄	1	1	yes	characteristic
279	C₂	C₂²×Dic₇₁⋊C₈	1	1	yes	characteristic
280	C₂	#22596906	1	1	yes	characteristic
281	C₂	C₈×Q₈×D₇₁	1	1	yes	characteristic
282	C₂	Q₈×#11765848	1	1	yes	characteristic
283	C₂	#22597450	1	1	yes	characteristic
284	C₂	#22597471	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₇₁
commutator	C₂²×C₇₁	C₂³×C₈
Fitting	C₂×C₈×Q₈×C₇₁	C₂
socle, prime radical	C₂³×C₇₁	C₂³×C₄
cosocle	C₂²×C₄×C₇₁	C₂⁴
hypercenter	C₂×C₈×Q₈	D₇₁
hypocenter	C₇₁	Q₈×C₄⋊C₈
LCS	G ▷ C₂²×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
29. Q₈×Dic₇₁	C₈	semidirect
41. Dic₇₁⋊C₈	Q₈	direct (N × H)
89. Q₈×C₇₁	C₄⋊C₈	semidirect
134. Dic₇₁	C₈×Q₈	semidirect
150. C₇₁	Q₈×C₄⋊C₈	semidirect
232. Q₈	Dic₇₁⋊C₈	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈×C₇₁⋊C₈	2
C₄×Q₈×Dic₇₁	2
#22589058	2
#22589056	2
#22588491	2
C₄×Dic₇₁⋊C₈	2
C₂×C₈×Q₈×C₇₁	2
Q₈×C₄⋊C₈	71

Subgroups — Q8×Dic71⋊C8 #32016243

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×Dic₇₁⋊C₈ #32016243