Subgroups — Q₈×#14581715 #35421007

Order 21248.

1086 conjugacy classes of subgroups.

The subgroup lattice has 310 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈×Dic₁₆₆	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×C₈₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#24908423	C₂	1	3	yes	maximal
4	#24908411	C₂	1	3	yes	maximal
5	#24907465	C₂	1	3	yes	maximal
6	C₄×#14581715	C₂	1	3	yes	maximal
7	Q₈×C₄⋊C₄×C₈₃	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₁₆₆	—	2	4	no	—
9	Q₈×Dic₁₆₆	C₄	1	8	yes	complemented
10	C₂×#14581792	C₂²	1	3	yes	—
11	C₂×#14581791	—	2	3	no	—
12	C₂×Q₈×Dic₈₃	—	2	1	no	—
13	C₂×C₄×Dic₁₆₆	C₂²	1	3	yes	—
14	Q₈×C₈₃⋊C₈	—	2	4	no	—
15	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	3	yes	—
16	C₂×#14581715	C₂²	1	3	yes	—
17	C₂×#14581715	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈₃⋊C₈	C₂²	1	3	yes	—
19	#19127463	C₂²	1	3	yes	—
20	#19127463	C₂²	1	3	yes	—
21	#19127463	C₂²	1	6	yes	—
22	C₂×C₄×Q₈×C₈₃	—	2	1	no	—
23	C₂×C₄×Q₈×C₈₃	C₂²	1	1	yes	characteristic
24	#389×C₈₃	—	2	3	no	—
25	#384×C₈₃	C₂²	1	3	yes	—
26	C₄×C₄⋊C₄×C₈₃	C₂²	1	3	yes	—
27	C₂²×Dic₁₆₆	—	2	3	no	—
28	C₂²×Dic₁₆₆	C₂³	1	1	yes	characteristic
29	C₂²×C₈₃⋊C₈	C₂³	1	3	yes	—
30	C₂²×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
31	#14581792	—	2	6	no	—
32	#14581792	C₂×C₄	1	12	yes	—
33	#14581791	—	2	24	no	—
34	Q₈×Dic₈₃	—	2	8	no	—
35	C₄×Dic₁₆₆	—	2	6	no	—
36	C₄×Dic₁₆₆	C₂×C₄	1	12	yes	—
37	C₂×C₄⋊Dic₈₃	—	2	3	no	—
38	C₂×C₄⋊Dic₈₃	C₂³	1	3	yes	—
39	C₂×Dic₈₃⋊C₄	—	2	3	no	—
40	C₂×C₄×Dic₈₃	—	2	3	no	—
41	C₄⋊(C₈₃⋊C₈)	—	2	6	no	—
42	#14581715	—	2	6	no	—
43	#14581715	Q₈	1	4	yes	direct product factor
44	C₄×C₈₃⋊C₈	—	2	6	no	—
45	C₂²×Q₈×C₈₃	D₄	1	1	yes	characteristic
46	C₄×Q₈×C₈₃	—	4	4	no	—
47	C₄×Q₈×C₈₃	C₂×C₄	1	4	yes	—
48	C₄×Q₈×C₈₃	D₄	1	4	yes	—
49	C₂×C₄⋊C₄×C₈₃	—	2	3	no	—
50	C₂×C₄⋊C₄×C₈₃	—	2	3	no	—
51	C₂×C₄⋊C₄×C₈₃	D₄	1	3	yes	—
52	C₂×C₄⋊C₄×C₈₃	C₂³	1	3	yes	—
53	C₂×C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₈₃	—	2	3	no	—
55	C₂×C₄²×C₈₃	C₂³	1	3	yes	—
56	#95×C₈₃	—	2	3	no	—
57	C₂×Dic₁₆₆	—	2	1	no	—
58	C₂×Dic₁₆₆	—	2	1	no	—
59	C₂×Dic₁₆₆	—	2	1	no	—
60	C₂×Dic₁₆₆	—	2	12	no	—
61	C₂×Dic₁₆₆	—	8	3	no	—
62	C₂×Dic₁₆₆	—	8	3	no	—
63	C₂×Dic₁₆₆	C₂²×C₄	1	2	yes	—
64	C₂×Dic₁₆₆	C₂×Q₈	1	2	yes	—
65	C₂×Dic₁₆₆	#41	1	2	yes	—
66	C₂²×Dic₈₃	—	2	1	no	—
67	C₂²×Dic₈₃	—	2	3	no	—
68	C₂×C₈₃⋊C₈	—	2	1	no	—
69	C₂×C₈₃⋊C₈	—	2	1	no	—
70	C₂×C₈₃⋊C₈	—	2	3	no	—
71	C₂×C₈₃⋊C₈	—	2	3	no	—
72	C₂×C₈₃⋊C₈	—	2	3	no	—
73	C₂×C₈₃⋊C₈	C₂×Q₈	1	2	yes	—
74	C₄⋊Dic₈₃	—	2	12	no	—
75	C₄⋊Dic₈₃	—	2	3	no	—
76	C₄⋊Dic₈₃	C₂²×C₄	1	6	yes	—
77	Dic₈₃⋊C₄	—	2	12	no	—
78	C₄×Dic₈₃	—	2	12	no	—
79	C₂×Q₈×C₈₃	C₂²⋊C₄	1	4	yes	—
80	C₂×Q₈×C₈₃	SD₁₆	1	4	yes	—
81	C₂×Q₈×C₈₃	Q₁₆	1	4	yes	—
82	C₂²×C₄×C₈₃	—	2	1	no	—
83	C₂²×C₄×C₈₃	—	2	3	no	—
84	C₂²×C₄×C₈₃	C₂×D₄	1	3	yes	—
85	C₂²×C₄×C₈₃	C₂×D₄	1	3	yes	—
86	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
87	C₄⋊C₄×C₈₃	—	2	1	no	—
88	C₄⋊C₄×C₈₃	—	2	3	no	—
89	C₄⋊C₄×C₈₃	—	2	3	no	—
90	C₄⋊C₄×C₈₃	—	4	6	no	—
91	C₄⋊C₄×C₈₃	—	8	3	no	—
92	C₄⋊C₄×C₈₃	C₂²⋊C₄	1	12	yes	—
93	C₄⋊C₄×C₈₃	C₂×Q₈	1	2	yes	—
94	C₄²×C₈₃	—	4	6	no	—
95	C₄²×C₈₃	C₂²×C₄	1	6	yes	—
96	C₄²×C₈₃	C₂×D₄	1	6	yes	—
97	C₂³×C₈₃	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₈₃	#103	1	16	yes	complemented
99	C₂×C₄×C₈₃	—	2	3	no	—
100	C₂×C₄×C₈₃	—	2	3	no	—
101	C₂×C₄×C₈₃	—	2	2	no	—
102	C₂×C₄×C₈₃	—	4	1	no	—
103	C₂×C₄×C₈₃	—	4	1	no	—
104	C₂×C₄×C₈₃	—	4	3	no	—
105	C₂×C₄×C₈₃	—	4	3	no	—
106	C₂×C₄×C₈₃	—	4	3	no	—
107	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	6	yes	—
108	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	6	yes	—
109	C₂×C₄×C₈₃	C₂×SD₁₆	1	6	yes	—
110	C₂×C₄×C₈₃	C₂×Q₁₆	1	6	yes	—
111	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
112	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
113	C₂×C₄×C₈₃	C₂²×Q₈	1	1	yes	characteristic
114	C₂×C₄×C₈₃	C₂×#41	1	1	yes	characteristic
115	C₂×C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
116	C₂×C₄×C₈₃	2₋¹⁺⁴	1	1	yes	characteristic
117	C₂×Dic₈₃	—	2	2	no	—
118	C₂×Dic₈₃	—	2	2	no	—
119	C₂×Dic₈₃	—	2	2	no	—
120	C₂×Dic₈₃	—	2	6	no	—
121	C₂×Dic₈₃	—	4	3	no	—
122	C₂×Dic₈₃	—	4	3	no	—
123	Dic₁₆₆	—	2	2	no	—
124	Dic₁₆₆	—	2	2	no	—
125	Dic₁₆₆	—	2	2	no	—
126	Dic₁₆₆	—	8	12	no	—
127	Dic₁₆₆	C₄×Q₈	1	4	yes	complemented
128	C₈₃⋊C₈	—	2	2	no	—
129	C₈₃⋊C₈	—	4	3	no	—
130	C₂²×C₈₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
131	C₂²×C₈₃	#548	1	1	yes	characteristic
132	C₂²×C₈₃	D₄×Q₈	1	1	yes	characteristic
133	C₂²×C₈₃	C₂²×SD₁₆	1	1	yes	characteristic
134	C₂²×C₈₃	C₂²×Q₁₆	1	1	yes	characteristic
135	C₂²×C₈₃	#576	1	1	yes	characteristic
136	C₂²×C₈₃	#577	1	1	yes	characteristic
137	C₄×C₈₃	—	2	6	no	—
138	C₄×C₈₃	—	4	2	no	—
139	C₄×C₈₃	—	8	3	no	—
140	C₄×C₈₃	C₂×#103	1	12	yes	—
141	C₄×C₈₃	C₂×C₄×Q₈	1	1	yes	characteristic
142	C₄×C₈₃	#519	1	1	yes	characteristic
143	C₄×C₈₃	#548	1	1	yes	characteristic
144	C₄×C₈₃	D₄×Q₈	1	1	yes	characteristic
145	Dic₈₃	—	2	4	no	—
146	Dic₈₃	—	4	6	no	—
147	Q₈×#103	—	83	1	no	maximal, Sylow
148	C₂×C₈₃	Q₈×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₈₃	C₂²×#103	1	1	yes	characteristic
150	C₂×C₈₃	#2897	1	1	yes	characteristic
151	C₂×C₈₃	#3362	1	1	yes	characteristic
152	C₂×C₈₃	#3363	1	1	yes	characteristic
153	C₂×C₈₃	Q₈×SD₁₆	1	1	yes	characteristic
154	C₂×C₈₃	Q₈×Q₁₆	1	1	yes	characteristic
155	C₂×Q₈²	—	83	1	no	—
156	C₂×C₈×Q₈	—	83	1	no	—
157	Q₈×C₄⋊C₄	D₈₃	1	1	yes	characteristic
158	#1990	—	83	3	no	—
159	#1987	—	83	3	no	—
160	#1803	—	83	3	no	—
161	C₄×#103	—	83	3	no	—
162	C₈₃	Q₈×#103	1	1	yes	characteristic, complemented, Sylow
163	Q₈²	—	83	8	no	—
164	Q₈²	—	166	4	no	—
165	C₂×C₄⋊Q₈	—	83	3	no	—
166	C₂×C₄⋊Q₈	—	166	3	no	—
167	C₂×C₄×Q₈	—	2	1	no	—
168	C₂×C₄×Q₈	—	83	3	no	—
169	C₂×C₄×Q₈	—	166	1	no	—
170	C₂×C₄×Q₈	C₂×D₈₃	1	1	yes	characteristic
171	C₈×Q₈	—	166	4	no	—
172	C₂×C₄⋊C₈	—	83	3	no	—
173	C₂×#103	—	83	1	no	—
174	C₂×#103	—	83	3	no	—
175	C₂×C₄×C₈	—	83	3	no	—
176	#389	—	2	3	no	—
177	#384	C₂×D₈₃	1	3	yes	—
178	C₄×C₄⋊C₄	C₂×D₈₃	1	3	yes	—
179	#340	—	83	6	no	—
180	#340	—	83	3	no	—
181	#340	—	83	3	no	—
182	C₂²×Q₈	—	83	1	no	—
183	C₂²×Q₈	—	166	3	no	—
184	C₂²×Q₈	C₂²⋊D₈₃	1	1	yes	characteristic
185	C₂²×C₈	—	83	1	no	—
186	C₂²×C₈	—	83	3	no	—
187	C₄⋊Q₈	—	83	12	no	—
188	C₄⋊Q₈	—	166	24	no	—
189	C₄⋊Q₈	—	166	6	no	—
190	C₄×Q₈	—	4	4	no	—
191	C₄×Q₈	—	83	12	no	—
192	C₄×Q₈	—	166	8	no	—
193	C₄×Q₈	—	166	6	no	—
194	C₄×Q₈	C₄×D₈₃	1	4	yes	—
195	C₄×Q₈	D₃₃₂	1	4	yes	—
196	C₂×C₄⋊C₄	—	2	3	no	—
197	C₂×C₄⋊C₄	—	2	3	no	—
198	C₂×C₄⋊C₄	—	83	3	no	—
199	C₂×C₄⋊C₄	—	166	3	no	—
200	C₂×C₄⋊C₄	—	166	3	no	—
201	C₂×C₄⋊C₄	C₂²⋊D₈₃	1	3	yes	—
202	C₂×C₄⋊C₄	C₂²×D₈₃	1	3	yes	—
203	C₂×C₄⋊C₄	C₂²×D₈₃	1	1	yes	characteristic
204	C₂×C₄²	—	2	3	no	—
205	C₂×C₄²	—	166	3	no	—
206	C₂×C₄²	C₂²×D₈₃	1	3	yes	—
207	C₄⋊C₈	—	166	6	no	—
208	#103	—	83	4	no	—
209	#103	—	166	6	no	—
210	C₄×C₈	—	166	6	no	—
211	#95	—	2	3	no	—
212	C₂×Q₈	—	83	2	no	—
213	C₂×Q₈	—	83	2	no	—
214	C₂×Q₈	—	83	2	no	—
215	C₂×Q₈	—	664	3	no	—
216	C₂×Q₈	—	664	3	no	—
217	C₂×Q₈	—	166	1	no	—
218	C₂×Q₈	—	166	1	no	—
219	C₂×Q₈	—	166	1	no	—
220	C₂×Q₈	—	166	12	no	—
221	C₂×Q₈	(C₂×D₈₃)⋊C₄	1	4	yes	—
222	C₂×Q₈	#13050578	1	4	yes	—
223	C₂×Q₈	#13050580	1	4	yes	—
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	2	3	no	—
226	C₂²×C₄	—	166	3	no	—
227	C₂²×C₄	—	166	1	no	—
228	C₂²×C₄	C₂×C₂²⋊D₈₃	1	3	yes	—
229	C₂²×C₄	C₂×C₂²⋊D₈₃	1	3	yes	—
230	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
231	C₂×C₈	—	83	2	no	—
232	C₂×C₈	—	166	3	no	—
233	C₂×C₈	—	166	3	no	—
234	C₂×C₈	—	166	3	no	—
235	C₂×C₈	—	166	1	no	—
236	C₂×C₈	—	166	1	no	—
237	C₄⋊C₄	—	2	1	no	—
238	C₄⋊C₄	—	2	3	no	—
239	C₄⋊C₄	—	2	3	no	—
240	C₄⋊C₄	—	4	6	no	—
241	C₄⋊C₄	—	8	3	no	—
242	C₄⋊C₄	—	83	6	no	—
243	C₄⋊C₄	—	166	3	no	—
244	C₄⋊C₄	—	166	12	no	—
245	C₄⋊C₄	—	166	12	no	—
246	C₄⋊C₄	(C₂×D₈₃)⋊C₄	1	12	yes	—
247	C₄⋊C₄	Q₈×D₈₃	1	2	yes	—
248	C₄²	—	4	6	no	—
249	C₄²	—	166	12	no	—
250	C₄²	C₂×C₄×D₈₃	1	6	yes	—
251	C₄²	C₂×D₃₃₂	1	6	yes	—
252	C₂³	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
253	Q₈	—	83	4	no	—
254	Q₈	—	664	12	no	—
255	Q₈	—	166	2	no	—
256	Q₈	—	166	2	no	—
257	Q₈	—	166	2	no	—
258	Q₈	#14581715	1	16	yes	direct product factor
259	C₂×C₄	—	2	3	no	—
260	C₂×C₄	—	2	3	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	1	no	—
263	C₂×C₄	—	4	1	no	—
264	C₂×C₄	—	4	3	no	—
265	C₂×C₄	—	4	3	no	—
266	C₂×C₄	—	4	3	no	—
267	C₂×C₄	—	332	3	no	—
268	C₂×C₄	—	332	3	no	—
269	C₂×C₄	—	166	6	no	—
270	C₂×C₄	—	166	2	no	—
271	C₂×C₄	—	166	2	no	—
272	C₂×C₄	—	166	2	no	—
273	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	6	yes	—
274	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	6	yes	—
275	C₂×C₄	C₂×#13050578	1	6	yes	—
276	C₂×C₄	C₂×#13050580	1	6	yes	—
277	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
278	C₂×C₄	C₂²×D₃₃₂	1	1	yes	characteristic
279	C₂×C₄	C₂×Q₈×D₈₃	1	1	yes	characteristic
280	C₂×C₄	#41×D₈₃	1	1	yes	characteristic
281	C₂×C₄	#14581839	1	1	yes	characteristic
282	C₂×C₄	#14581840	1	1	yes	characteristic
283	C₈	—	332	3	no	—
284	C₈	—	166	2	no	—
285	C₂²	C₂²×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
286	C₂²	#19128358	1	1	yes	characteristic
287	C₂²	Q₈×C₂²⋊D₈₃	1	1	yes	characteristic
288	C₂²	C₂²×#13050578	1	1	yes	characteristic
289	C₂²	C₂²×#13050580	1	1	yes	characteristic
290	C₂²	#19128504	1	1	yes	characteristic
291	C₂²	#19128509	1	1	yes	characteristic
292	C₄	—	2	6	no	—
293	C₄	—	4	2	no	—
294	C₄	—	8	3	no	—
295	C₄	—	332	6	no	—
296	C₄	—	166	4	no	—
297	C₄	C₂×#14581715	1	12	yes	—
298	C₄	C₄×Q₈×D₈₃	1	1	yes	characteristic
299	C₄	#19128177	1	1	yes	characteristic
300	C₄	#19128363	1	1	yes	characteristic
301	C₄	Q₈×D₃₃₂	1	1	yes	characteristic
302	C₂	Q₈×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
303	C₂	C₂²×#14581715	1	1	yes	characteristic
304	C₂	#24915986	1	1	yes	characteristic
305	C₂	#24920130	1	1	yes	characteristic
306	C₂	#24920137	1	1	yes	characteristic
307	C₂	Q₈×#13050578	1	1	yes	characteristic
308	C₂	Q₈×#13050580	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×C₂²⋊D₈₃
commutator	C₂×C₄×C₈₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₃
Fitting	Q₈×C₄⋊C₄×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂²×D₄
cosocle	C₂²×C₄×C₈₃	C₂⁴
hypercenter	Q₈×C₄⋊C₄	D₈₃
hypocenter	C₈₃	Q₈×#103
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ Q₈×C₄⋊C₄

Complements

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

N	H ≅ G/N	type
9. Q₈×Dic₁₆₆	C₄	semidirect
43. #14581715	Q₈	direct (N × H)
98. Q₈×C₈₃	#103	semidirect
127. Dic₁₆₆	C₄×Q₈	semidirect
162. C₈₃	Q₈×#103	semidirect
258. Q₈	#14581715	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×Dic₁₆₆	2
C₂×Q₈×C₈₃⋊C₈	2
#24908423	2
#24908411	2
#24907465	2
C₄×#14581715	2
Q₈×C₄⋊C₄×C₈₃	2
Q₈×#103	83

Subgroups — Q8×#14581715 #35421007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#14581715 #35421007