Subgroups — Q₈×C₃₁×C₂²⋊C₈ #21479267

Order 7936.

1356 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×C₈×Q₈×C₃₁	C₂	1	2	yes	maximal, complemented
3	C₃₁×#1984	C₂	1	6	yes	maximal, complemented
4	C₃₁×#1797	C₂	1	3	yes	maximal
5	C₄×C₃₁×C₂²⋊C₈	C₂	1	3	yes	maximal
6	C₂³×Q₈×C₃₁	C₄	1	1	yes	characteristic
7	C₂×C₄×Q₈×C₃₁	—	2	8	no	—
8	C₂×C₄×Q₈×C₃₁	C₄	1	1	yes	characteristic
9	C₂×C₄×Q₈×C₃₁	C₄	1	3	yes	—
10	C₂×C₄×Q₈×C₃₁	C₂²	1	3	yes	—
11	C₂×C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×C₄⋊C₄×C₃₁	C₄	1	3	yes	—
13	C₂²×C₄²×C₃₁	C₂²	1	3	yes	—
14	C₈×Q₈×C₃₁	—	2	8	no	—
15	C₂×C₃₁×C₄⋊C₈	C₂²	1	6	yes	—
16	C₂×C₃₁×C₂²⋊C₈	C₂²	1	3	yes	—
17	C₂×C₃₁×C₂²⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₃₁	C₂²	1	6	yes	—
19	C₃₁×#336	C₂²	1	6	yes	—
20	C₃₁×#336	C₂²	1	6	yes	—
21	C₂²×Q₈×C₃₁	—	2	4	no	—
22	C₂²×Q₈×C₃₁	—	2	4	no	—
23	C₂²×Q₈×C₃₁	C₈	1	8	yes	complemented
24	C₂²×Q₈×C₃₁	C₂×C₄	1	3	yes	—
25	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic
26	C₂³×C₄×C₃₁	C₂×C₄	1	3	yes	—
27	C₂³×C₄×C₃₁	C₂×C₄	1	3	yes	—
28	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
29	C₂²×C₈×C₃₁	C₂³	1	6	yes	—
30	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
31	C₄×Q₈×C₃₁	—	2	4	no	—
32	C₄×Q₈×C₃₁	—	2	12	no	—
33	C₄×Q₈×C₃₁	—	2	12	no	—
34	C₄×Q₈×C₃₁	D₄	1	8	yes	—
35	C₂×C₄⋊C₄×C₃₁	—	2	12	no	—
36	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	3	yes	—
37	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	6	yes	—
38	C₂×C₄⋊C₄×C₃₁	C₂×C₄	1	3	yes	—
39	C₂×C₄²×C₃₁	—	2	12	no	—
40	C₂×C₄²×C₃₁	C₂×C₄	1	3	yes	—
41	C₂×C₄²×C₃₁	C₂×C₄	1	3	yes	—
42	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
43	C₂×C₄²×C₃₁	C₂³	1	3	yes	—
44	C₃₁×C₄⋊C₈	—	2	12	no	—
45	C₃₁×C₂²⋊C₈	—	2	6	no	—
46	C₃₁×C₂²⋊C₈	Q₈	1	4	yes	direct product factor
47	C₄×C₈×C₃₁	—	2	12	no	—
48	C₂⁴×C₃₁	C₂²×C₄	1	1	yes	characteristic
49	C₂×Q₈×C₃₁	—	2	32	no	—
50	C₂×Q₈×C₃₁	—	2	6	no	—
51	C₂×Q₈×C₃₁	—	2	6	no	—
52	C₂×Q₈×C₃₁	C₂²⋊C₄	1	4	yes	—
53	C₂×Q₈×C₃₁	C₂×C₈	1	12	yes	—
54	C₂×Q₈×C₃₁	C₂×C₈	1	4	yes	—
55	C₂×Q₈×C₃₁	M₁₆	1	4	yes	—
56	C₂²×C₄×C₃₁	—	2	6	no	—
57	C₂²×C₄×C₃₁	—	2	6	no	—
58	C₂²×C₄×C₃₁	—	2	2	no	—
59	C₂²×C₄×C₃₁	—	2	2	no	—
60	C₂²×C₄×C₃₁	—	2	6	no	—
61	C₂²×C₄×C₃₁	—	2	6	no	—
62	C₂²×C₄×C₃₁	—	4	3	no	—
63	C₂²×C₄×C₃₁	C₂×C₈	1	12	yes	—
64	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
65	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
66	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
67	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
68	C₂²×C₄×C₃₁	C₂²×C₄	1	3	yes	—
69	C₂²×C₄×C₃₁	C₂×Q₈	1	2	yes	—
70	C₂²×C₄×C₃₁	#41	1	2	yes	—
71	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
72	C₂×C₈×C₃₁	—	2	2	no	—
73	C₂×C₈×C₃₁	—	2	2	no	—
74	C₂×C₈×C₃₁	—	2	6	no	—
75	C₂×C₈×C₃₁	—	2	6	no	—
76	C₂×C₈×C₃₁	—	2	6	no	—
77	C₂×C₈×C₃₁	C₂×Q₈	1	4	yes	complemented
78	C₄⋊C₄×C₃₁	—	2	12	no	—
79	C₄⋊C₄×C₃₁	—	2	6	no	—
80	C₄⋊C₄×C₃₁	C₂²⋊C₄	1	12	yes	—
81	C₄²×C₃₁	—	2	6	no	—
82	C₄²×C₃₁	—	2	6	no	—
83	C₄²×C₃₁	—	2	6	no	—
84	C₄²×C₃₁	C₂×D₄	1	12	yes	—
85	Q₈×C₂²⋊C₈	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂³×C₃₁	—	2	1	no	—
87	C₂³×C₃₁	—	2	1	no	—
88	C₂³×C₃₁	—	2	1	no	—
89	C₂³×C₃₁	—	2	1	no	—
90	C₂³×C₃₁	C₄×Q₈	1	2	yes	—
91	C₂³×C₃₁	C₂²×C₈	1	2	yes	—
92	C₂³×C₃₁	#131	1	2	yes	—
93	C₂³×C₃₁	C₂³×C₄	1	1	yes	characteristic
94	Q₈×C₃₁	—	2	24	no	—
95	Q₈×C₃₁	C₂²⋊C₈	1	16	yes	direct product factor
96	C₂×C₄×C₃₁	—	2	1	no	—
97	C₂×C₄×C₃₁	—	2	3	no	—
98	C₂×C₄×C₃₁	—	2	1	no	—
99	C₂×C₄×C₃₁	—	2	3	no	—
100	C₂×C₄×C₃₁	—	2	3	no	—
101	C₂×C₄×C₃₁	—	2	3	no	—
102	C₂×C₄×C₃₁	—	2	4	no	—
103	C₂×C₄×C₃₁	—	2	4	no	—
104	C₂×C₄×C₃₁	—	2	24	no	—
105	C₂×C₄×C₃₁	—	2	3	no	—
106	C₂×C₄×C₃₁	—	2	3	no	—
107	C₂×C₄×C₃₁	—	2	3	no	—
108	C₂×C₄×C₃₁	—	4	12	no	—
109	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
110	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	6	yes	—
111	C₂×C₄×C₃₁	C₄×Q₈	1	2	yes	—
112	C₂×C₄×C₃₁	C₂²×C₈	1	6	yes	—
113	C₂×C₄×C₃₁	C₂²×C₈	1	6	yes	—
114	C₂×C₄×C₃₁	C₂×M₁₆	1	6	yes	—
115	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
116	C₂×C₄×C₃₁	C₂²×Q₈	1	1	yes	characteristic
117	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
118	C₂×C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
119	C₈×C₃₁	—	2	4	no	—
120	C₈×C₃₁	—	4	6	no	—
121	C₂²×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic
122	C₂×C₈×Q₈	C₂×C₃₁	1	2	yes	complemented
123	#1984	C₂×C₃₁	1	6	yes	complemented
124	#1797	C₂×C₃₁	1	3	yes	—
125	C₄×C₂²⋊C₈	C₂×C₃₁	1	3	yes	—
126	C₂²×C₃₁	—	2	2	no	—
127	C₂²×C₃₁	—	2	2	no	—
128	C₂²×C₃₁	—	2	2	no	—
129	C₂²×C₃₁	—	2	2	no	—
130	C₂²×C₃₁	—	2	2	no	—
131	C₂²×C₃₁	—	2	2	no	—
132	C₂²×C₃₁	C₈×Q₈	1	4	yes	complemented
133	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
134	C₂²×C₃₁	C₂×C₄×Q₈	1	1	yes	characteristic
135	C₂²×C₃₁	#519	1	1	yes	characteristic
136	C₂²×C₃₁	C₂³×C₈	1	1	yes	characteristic
137	C₂²×C₃₁	C₂²×M₁₆	1	1	yes	characteristic
138	C₂²×C₃₁	C₂×#131	1	1	yes	characteristic
139	C₂²×C₃₁	#568	1	1	yes	characteristic
140	C₄×C₃₁	—	2	2	no	—
141	C₄×C₃₁	—	2	6	no	—
142	C₄×C₃₁	—	2	6	no	—
143	C₄×C₃₁	—	4	3	no	—
144	C₄×C₃₁	C₂×C₂²⋊C₈	1	12	yes	—
145	C₄×C₃₁	#548	1	2	yes	—
146	C₄×C₃₁	D₄×Q₈	1	2	yes	—
147	C₂³×Q₈	C₄×C₃₁	1	1	yes	characteristic
148	C₂×C₄×Q₈	—	2	8	no	—
149	C₂×C₄×Q₈	C₄×C₃₁	1	1	yes	characteristic
150	C₂×C₄×Q₈	C₄×C₃₁	1	3	yes	—
151	C₂×C₄×Q₈	C₂²×C₃₁	1	3	yes	—
152	C₂×C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
153	C₂²×C₄⋊C₄	C₄×C₃₁	1	3	yes	—
154	C₂²×C₄²	C₂²×C₃₁	1	3	yes	—
155	C₈×Q₈	—	2	8	no	—
156	C₂×C₄⋊C₈	C₂²×C₃₁	1	6	yes	—
157	C₂×C₂²⋊C₈	C₂²×C₃₁	1	3	yes	—
158	C₂×C₂²⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂²×C₃₁	1	6	yes	—
160	#336	C₂²×C₃₁	1	6	yes	—
161	#336	C₂²×C₃₁	1	6	yes	—
162	C₂×C₃₁	—	2	4	no	—
163	C₂×C₃₁	Q₈×C₂²⋊C₄	1	1	yes	characteristic
164	C₂×C₃₁	C₂²×C₂²⋊C₈	1	1	yes	characteristic
165	C₂×C₃₁	#2880	1	1	yes	characteristic
166	C₂×C₃₁	C₂×C₈×Q₈	1	1	yes	characteristic
167	C₂×C₃₁	Q₈×M₁₆	1	1	yes	characteristic
168	C₂×C₃₁	#2982	1	1	yes	characteristic
169	C₂×C₃₁	#2984	1	1	yes	characteristic
170	C₂²×Q₈	—	2	4	no	—
171	C₂²×Q₈	—	2	4	no	—
172	C₂²×Q₈	C₈×C₃₁	1	8	yes	complemented
173	C₂²×Q₈	C₂×C₄×C₃₁	1	3	yes	—
174	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic
175	C₂³×C₄	C₂×C₄×C₃₁	1	3	yes	—
176	C₂³×C₄	C₂×C₄×C₃₁	1	3	yes	—
177	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
178	C₂²×C₈	C₂³×C₃₁	1	6	yes	—
179	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
180	C₄×Q₈	—	2	4	no	—
181	C₄×Q₈	—	2	12	no	—
182	C₄×Q₈	—	2	12	no	—
183	C₄×Q₈	D₄×C₃₁	1	8	yes	—
184	C₂×C₄⋊C₄	—	2	12	no	—
185	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	3	yes	—
186	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	6	yes	—
187	C₂×C₄⋊C₄	C₂×C₄×C₃₁	1	3	yes	—
188	C₂×C₄²	—	2	12	no	—
189	C₂×C₄²	C₂×C₄×C₃₁	1	3	yes	—
190	C₂×C₄²	C₂×C₄×C₃₁	1	3	yes	—
191	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
192	C₂×C₄²	C₂³×C₃₁	1	3	yes	—
193	C₄⋊C₈	—	2	12	no	—
194	C₂²⋊C₈	—	2	6	no	—
195	C₂²⋊C₈	Q₈×C₃₁	1	4	yes	direct product factor
196	C₄×C₈	—	2	12	no	—
197	C₃₁	Q₈×C₂²⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
198	C₂⁴	C₂²×C₄×C₃₁	1	1	yes	characteristic
199	C₂×Q₈	—	2	32	no	—
200	C₂×Q₈	—	2	6	no	—
201	C₂×Q₈	—	2	6	no	—
202	C₂×Q₈	C₂²⋊C₄×C₃₁	1	4	yes	—
203	C₂×Q₈	C₂×C₈×C₃₁	1	12	yes	—
204	C₂×Q₈	C₂×C₈×C₃₁	1	4	yes	—
205	C₂×Q₈	M₁₆×C₃₁	1	4	yes	—
206	C₂²×C₄	—	2	6	no	—
207	C₂²×C₄	—	2	6	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	6	no	—
211	C₂²×C₄	—	2	6	no	—
212	C₂²×C₄	—	4	3	no	—
213	C₂²×C₄	C₂×C₈×C₃₁	1	12	yes	—
214	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
215	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
216	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
217	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
218	C₂²×C₄	C₂²×C₄×C₃₁	1	3	yes	—
219	C₂²×C₄	C₂×Q₈×C₃₁	1	2	yes	—
220	C₂²×C₄	#41×C₃₁	1	2	yes	—
221	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
222	C₂×C₈	—	2	2	no	—
223	C₂×C₈	—	2	2	no	—
224	C₂×C₈	—	2	6	no	—
225	C₂×C₈	—	2	6	no	—
226	C₂×C₈	—	2	6	no	—
227	C₂×C₈	C₂×Q₈×C₃₁	1	4	yes	complemented
228	C₄⋊C₄	—	2	12	no	—
229	C₄⋊C₄	—	2	6	no	—
230	C₄⋊C₄	C₂²⋊C₄×C₃₁	1	12	yes	—
231	C₄²	—	2	6	no	—
232	C₄²	—	2	6	no	—
233	C₄²	—	2	6	no	—
234	C₄²	C₂×D₄×C₃₁	1	12	yes	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	1	no	—
238	C₂³	—	2	1	no	—
239	C₂³	C₄×Q₈×C₃₁	1	2	yes	—
240	C₂³	C₂²×C₈×C₃₁	1	2	yes	—
241	C₂³	C₃₁×#131	1	2	yes	—
242	C₂³	C₂³×C₄×C₃₁	1	1	yes	characteristic
243	Q₈	—	2	24	no	—
244	Q₈	C₃₁×C₂²⋊C₈	1	16	yes	direct product factor
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	3	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	3	no	—
249	C₂×C₄	—	2	3	no	—
250	C₂×C₄	—	2	3	no	—
251	C₂×C₄	—	2	24	no	—
252	C₂×C₄	—	2	4	no	—
253	C₂×C₄	—	2	4	no	—
254	C₂×C₄	—	2	3	no	—
255	C₂×C₄	—	2	3	no	—
256	C₂×C₄	—	2	3	no	—
257	C₂×C₄	—	4	12	no	—
258	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
259	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	6	yes	—
260	C₂×C₄	C₄×Q₈×C₃₁	1	2	yes	—
261	C₂×C₄	C₂²×C₈×C₃₁	1	6	yes	—
262	C₂×C₄	C₂²×C₈×C₃₁	1	6	yes	—
263	C₂×C₄	C₂×M₁₆×C₃₁	1	6	yes	—
264	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
265	C₂×C₄	C₂²×Q₈×C₃₁	1	1	yes	characteristic
266	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
267	C₂×C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
268	C₈	—	2	4	no	—
269	C₈	—	4	6	no	—
270	C₂²	—	2	2	no	—
271	C₂²	—	2	2	no	—
272	C₂²	—	2	2	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	C₈×Q₈×C₃₁	1	4	yes	complemented
277	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
278	C₂²	C₂×C₄×Q₈×C₃₁	1	1	yes	characteristic
279	C₂²	C₃₁×#519	1	1	yes	characteristic
280	C₂²	C₂³×C₈×C₃₁	1	1	yes	characteristic
281	C₂²	C₂²×M₁₆×C₃₁	1	1	yes	characteristic
282	C₂²	C₂×C₃₁×#131	1	1	yes	characteristic
283	C₂²	C₃₁×#568	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	2	6	no	—
286	C₄	—	2	6	no	—
287	C₄	—	4	3	no	—
288	C₄	C₂×C₃₁×C₂²⋊C₈	1	12	yes	—
289	C₄	C₃₁×#548	1	2	yes	—
290	C₄	D₄×Q₈×C₃₁	1	2	yes	—
291	C₂	—	2	4	no	—
292	C₂	Q₈×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
293	C₂	C₂²×C₃₁×C₂²⋊C₈	1	1	yes	characteristic
294	C₂	C₃₁×#2880	1	1	yes	characteristic
295	C₂	C₂×C₈×Q₈×C₃₁	1	1	yes	characteristic
296	C₂	Q₈×M₁₆×C₃₁	1	1	yes	characteristic
297	C₂	C₃₁×#2982	1	1	yes	characteristic
298	C₂	C₃₁×#2984	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂³×C₈×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂³×C₃₁	C₂³×C₄
prime radical	C₂⁴×C₃₁	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₈×Q₈×C₃₁	C₂	semidirect
3. C₃₁×#1984	C₂	semidirect
23. C₂²×Q₈×C₃₁	C₈	semidirect
46. C₃₁×C₂²⋊C₈	Q₈	direct (N × H)
77. C₂×C₈×C₃₁	C₂×Q₈	semidirect
85. Q₈×C₂²⋊C₈	C₃₁	direct (N × H)
95. Q₈×C₃₁	C₂²⋊C₈	direct (N × H)
122. C₂×C₈×Q₈	C₂×C₃₁	semidirect
123. #1984	C₂×C₃₁	semidirect
132. C₂²×C₃₁	C₈×Q₈	semidirect
172. C₂²×Q₈	C₈×C₃₁	semidirect
195. C₂²⋊C₈	Q₈×C₃₁	direct (N × H)
197. C₃₁	Q₈×C₂²⋊C₈	direct (N × H)
227. C₂×C₈	C₂×Q₈×C₃₁	semidirect
244. Q₈	C₃₁×C₂²⋊C₈	direct (N × H)
276. C₂²	C₈×Q₈×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₃₁	2
C₂×C₈×Q₈×C₃₁	2
C₃₁×#1984	2
C₃₁×#1797	2
C₄×C₃₁×C₂²⋊C₈	2
Q₈×C₂²⋊C₈	31

Subgroups — Q8×C31×C22⋊C8 #21479267

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×C₃₁×C₂²⋊C₈ #21479267