Subgroups — Q₈×#15667937 #36571036

Order 22784.

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×Dic₁₇₈	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊Dic₈₉	C₂	1	1	yes	maximal, characteristic
3	#26021213	C₂	1	3	yes	maximal
4	#26021199	C₂	1	3	yes	maximal
5	#26020258	C₂	1	3	yes	maximal
6	C₄×#15667937	C₂	1	3	yes	maximal
7	C₂×C₈×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₁₇₈	—	2	4	no	—
9	Q₈×Dic₁₇₈	C₄	1	8	yes	complemented
10	C₂×#15668012	C₂²	1	3	yes	—
11	C₂×#15668011	—	2	3	no	—
12	C₂×Q₈×Dic₈₉	—	2	1	no	—
13	C₂×Q₈×Dic₈₉	—	2	1	no	—
14	C₂×C₄×Dic₁₇₈	C₂²	1	3	yes	—
15	C₂×#15667937	C₂²	1	3	yes	—
16	C₂×#15667937	C₂²	1	1	yes	characteristic
17	#19151497	—	2	3	no	—
18	#19151468	C₂²	1	3	yes	—
19	C₄×C₄⋊Dic₈₉	C₂²	1	3	yes	—
20	#19151361	C₂²	1	3	yes	—
21	#19151361	C₂²	1	6	yes	—
22	#19151361	C₂²	1	3	yes	—
23	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₈×Q₈×C₈₉	—	2	4	no	—
25	C₂×C₄⋊C₈×C₈₉	C₂²	1	3	yes	—
26	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
27	C₂²×Dic₁₇₈	—	2	3	no	—
28	C₂²×Dic₁₇₈	C₂³	1	1	yes	characteristic
29	#15668012	—	2	6	no	—
30	#15668012	C₂×C₄	1	12	yes	—
31	#15668011	—	2	24	no	—
32	Q₈×Dic₈₉	—	2	8	no	—
33	Q₈×Dic₈₉	—	4	4	no	—
34	C₄×Dic₁₇₈	—	2	6	no	—
35	C₄×Dic₁₇₈	C₂×C₄	1	12	yes	—
36	C₂×C₄⋊Dic₈₉	—	2	3	no	—
37	C₂×C₄⋊Dic₈₉	—	2	3	no	—
38	C₂×C₄⋊Dic₈₉	—	2	3	no	—
39	C₂×C₄⋊Dic₈₉	C₂³	1	3	yes	—
40	C₂×C₄⋊Dic₈₉	C₂³	1	3	yes	—
41	C₂×C₄⋊Dic₈₉	C₂³	1	1	yes	characteristic
42	C₂×#13083553	—	2	3	no	—
43	C₂×C₄×Dic₈₉	—	2	3	no	—
44	C₂×C₄×Dic₈₉	—	2	3	no	—
45	#15667937	—	2	6	no	—
46	#15667937	Q₈	1	4	yes	direct product factor
47	#15667919	—	2	3	no	—
48	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
49	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
50	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
51	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
52	C₄×Q₈×C₈₉	D₄	1	4	yes	—
53	C₂×C₄⋊C₄×C₈₉	D₄	1	3	yes	—
54	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
55	C₄⋊C₈×C₈₉	—	2	6	no	—
56	C₄×C₈×C₈₉	—	2	6	no	—
57	C₂×Dic₁₇₈	—	2	1	no	—
58	C₂×Dic₁₇₈	—	2	1	no	—
59	C₂×Dic₁₇₈	—	2	1	no	—
60	C₂×Dic₁₇₈	—	2	12	no	—
61	C₂×Dic₁₇₈	—	8	3	no	—
62	C₂×Dic₁₇₈	—	8	3	no	—
63	C₂×Dic₁₇₈	C₂²×C₄	1	2	yes	—
64	C₂×Dic₁₇₈	C₂×Q₈	1	2	yes	—
65	C₂×Dic₁₇₈	#41	1	2	yes	—
66	C₂²×Dic₈₉	—	2	1	no	—
67	C₂²×Dic₈₉	—	2	3	no	—
68	C₂²×Dic₈₉	—	2	1	no	—
69	C₂²×Dic₈₉	—	2	3	no	—
70	C₄⋊Dic₈₉	—	2	12	no	—
71	C₄⋊Dic₈₉	—	2	3	no	—
72	C₄⋊Dic₈₉	—	2	1	no	—
73	C₄⋊Dic₈₉	—	2	3	no	—
74	C₄⋊Dic₈₉	—	2	3	no	—
75	C₄⋊Dic₈₉	—	4	6	no	—
76	C₄⋊Dic₈₉	—	8	3	no	—
77	C₄⋊Dic₈₉	C₂²×C₄	1	6	yes	—
78	C₄⋊Dic₈₉	C₂×Q₈	1	2	yes	—
79	#13083553	—	2	12	no	—
80	C₄×Dic₈₉	—	2	12	no	—
81	C₄×Dic₈₉	—	4	6	no	—
82	C₂×Q₈×C₈₉	C₂²⋊C₄	1	4	yes	—
83	C₂×Q₈×C₈₉	SD₁₆	1	4	yes	—
84	C₂×Q₈×C₈₉	Q₁₆	1	4	yes	—
85	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
86	C₂²×C₄×C₈₉	C₂×D₄	1	3	yes	—
87	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
88	C₂×C₈×C₈₉	—	2	1	no	—
89	C₂×C₈×C₈₉	—	2	1	no	—
90	C₂×C₈×C₈₉	—	2	3	no	—
91	C₂×C₈×C₈₉	—	2	3	no	—
92	C₂×C₈×C₈₉	—	2	3	no	—
93	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
94	C₄⋊C₄×C₈₉	C₂²⋊C₄	1	12	yes	—
95	C₄²×C₈₉	C₂²×C₄	1	6	yes	—
96	C₄²×C₈₉	C₂×D₄	1	6	yes	—
97	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
98	Q₈×C₈₉	#103	1	16	yes	complemented
99	C₂×C₄×C₈₉	—	2	3	no	—
100	C₂×C₄×C₈₉	—	2	3	no	—
101	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	6	yes	—
102	C₂×C₄×C₈₉	C₂×C₂²⋊C₄	1	6	yes	—
103	C₂×C₄×C₈₉	C₂×SD₁₆	1	6	yes	—
104	C₂×C₄×C₈₉	C₂×Q₁₆	1	6	yes	—
105	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
106	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
107	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
108	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
109	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
110	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
111	C₂×Dic₈₉	—	2	2	no	—
112	C₂×Dic₈₉	—	2	2	no	—
113	C₂×Dic₈₉	—	2	2	no	—
114	C₂×Dic₈₉	—	2	6	no	—
115	C₂×Dic₈₉	—	2	2	no	—
116	C₂×Dic₈₉	—	4	1	no	—
117	C₂×Dic₈₉	—	4	3	no	—
118	C₂×Dic₈₉	—	4	1	no	—
119	C₂×Dic₈₉	—	4	3	no	—
120	C₂×Dic₈₉	—	4	3	no	—
121	C₂×Dic₈₉	—	4	3	no	—
122	C₂×Dic₈₉	—	4	3	no	—
123	Dic₁₇₈	—	2	2	no	—
124	Dic₁₇₈	—	2	2	no	—
125	Dic₁₇₈	—	2	2	no	—
126	Dic₁₇₈	—	8	12	no	—
127	Dic₁₇₈	C₄×Q₈	1	4	yes	complemented
128	C₈×C₈₉	—	2	2	no	—
129	C₈×C₈₉	—	4	3	no	—
130	C₂²×C₈₉	C₂²×C₂²⋊C₄	1	1	yes	characteristic
131	C₂²×C₈₉	#548	1	1	yes	characteristic
132	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
133	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
134	C₂²×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
135	C₂²×C₈₉	#576	1	1	yes	characteristic
136	C₂²×C₈₉	#577	1	1	yes	characteristic
137	C₄×C₈₉	—	2	6	no	—
138	C₄×C₈₉	C₂×#103	1	12	yes	—
139	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
140	C₄×C₈₉	#519	1	1	yes	characteristic
141	C₄×C₈₉	#548	1	1	yes	characteristic
142	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
143	Dic₈₉	—	2	4	no	—
144	Dic₈₉	—	4	2	no	—
145	Dic₈₉	—	4	6	no	—
146	Dic₈₉	—	8	3	no	—
147	Q₈×#103	—	89	1	no	maximal, Sylow
148	C₂×C₈₉	Q₈×C₂²⋊C₄	1	1	yes	characteristic
149	C₂×C₈₉	C₂²×#103	1	1	yes	characteristic
150	C₂×C₈₉	#2897	1	1	yes	characteristic
151	C₂×C₈₉	#3362	1	1	yes	characteristic
152	C₂×C₈₉	#3363	1	1	yes	characteristic
153	C₂×C₈₉	Q₈×SD₁₆	1	1	yes	characteristic
154	C₂×C₈₉	Q₈×Q₁₆	1	1	yes	characteristic
155	C₂×Q₈²	—	89	1	no	—
156	C₂×C₈×Q₈	D₈₉	1	1	yes	characteristic
157	Q₈×C₄⋊C₄	—	89	1	no	—
158	#1990	—	89	3	no	—
159	#1987	—	89	3	no	—
160	#1803	—	89	3	no	—
161	C₄×#103	—	89	3	no	—
162	C₈₉	Q₈×#103	1	1	yes	characteristic, complemented, Sylow
163	Q₈²	—	89	8	no	—
164	Q₈²	—	178	4	no	—
165	C₂×C₄⋊Q₈	—	89	3	no	—
166	C₂×C₄⋊Q₈	—	178	3	no	—
167	C₂×C₄×Q₈	—	89	3	no	—
168	C₂×C₄×Q₈	—	178	1	no	—
169	C₂×C₄×Q₈	—	178	1	no	—
170	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
171	C₈×Q₈	—	2	4	no	—
172	C₂×C₄⋊C₈	C₂×D₈₉	1	3	yes	—
173	C₂×#103	—	89	1	no	—
174	C₂×#103	—	89	3	no	—
175	C₂×C₄×C₈	C₂×D₈₉	1	3	yes	—
176	#389	—	178	3	no	—
177	#384	—	89	3	no	—
178	C₄×C₄⋊C₄	—	89	3	no	—
179	#340	—	89	6	no	—
180	#340	—	89	3	no	—
181	#340	—	89	3	no	—
182	C₂²×Q₈	—	89	1	no	—
183	C₂²×Q₈	—	178	3	no	—
184	C₂²×Q₈	D₃₅₆	1	1	yes	characteristic
185	C₂²×C₈	C₂²×D₈₉	1	3	yes	—
186	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
187	C₄⋊Q₈	—	89	12	no	—
188	C₄⋊Q₈	—	178	24	no	—
189	C₄⋊Q₈	—	178	6	no	—
190	C₄×Q₈	—	89	12	no	—
191	C₄×Q₈	—	356	4	no	—
192	C₄×Q₈	—	178	8	no	—
193	C₄×Q₈	—	178	6	no	—
194	C₄×Q₈	C₄×D₈₉	1	4	yes	—
195	C₄×Q₈	C₂²⋊D₈₉	1	4	yes	—
196	C₂×C₄⋊C₄	—	89	3	no	—
197	C₂×C₄⋊C₄	—	89	1	no	—
198	C₂×C₄⋊C₄	—	89	3	no	—
199	C₂×C₄⋊C₄	—	178	3	no	—
200	C₂×C₄⋊C₄	—	178	3	no	—
201	C₂×C₄⋊C₄	—	178	3	no	—
202	C₂×C₄⋊C₄	—	178	3	no	—
203	C₂×C₄⋊C₄	D₃₅₆	1	3	yes	—
204	C₂×C₄²	—	178	3	no	—
205	C₂×C₄²	—	178	3	no	—
206	C₂×C₄²	C₂²×D₈₉	1	3	yes	—
207	C₄⋊C₈	—	2	6	no	—
208	#103	—	89	4	no	—
209	#103	—	178	6	no	—
210	C₄×C₈	—	2	6	no	—
211	#95	—	178	3	no	—
212	C₂×Q₈	—	89	2	no	—
213	C₂×Q₈	—	89	2	no	—
214	C₂×Q₈	—	89	2	no	—
215	C₂×Q₈	—	178	1	no	—
216	C₂×Q₈	—	178	1	no	—
217	C₂×Q₈	—	178	1	no	—
218	C₂×Q₈	—	178	12	no	—
219	C₂×Q₈	—	712	3	no	—
220	C₂×Q₈	—	712	3	no	—
221	C₂×Q₈	#13083551	1	4	yes	—
222	C₂×Q₈	#13083563	1	4	yes	—
223	C₂×Q₈	Dic₃₅₆	1	4	yes	—
224	C₂²×C₄	—	178	3	no	—
225	C₂²×C₄	—	178	1	no	—
226	C₂²×C₄	—	178	3	no	—
227	C₂²×C₄	—	178	1	no	—
228	C₂²×C₄	C₂×D₃₅₆	1	3	yes	—
229	C₂²×C₄	C₂×D₃₅₆	1	3	yes	—
230	C₂²×C₄	C₂³×D₈₉	1	1	yes	characteristic
231	C₂×C₈	—	2	1	no	—
232	C₂×C₈	—	2	1	no	—
233	C₂×C₈	—	2	3	no	—
234	C₂×C₈	—	2	3	no	—
235	C₂×C₈	—	2	3	no	—
236	C₂×C₈	Q₈×D₈₉	1	2	yes	—
237	C₄⋊C₄	—	89	6	no	—
238	C₄⋊C₄	—	89	2	no	—
239	C₄⋊C₄	—	356	6	no	—
240	C₄⋊C₄	—	178	3	no	—
241	C₄⋊C₄	—	178	3	no	—
242	C₄⋊C₄	—	178	3	no	—
243	C₄⋊C₄	—	178	1	no	—
244	C₄⋊C₄	—	178	12	no	—
245	C₄⋊C₄	—	178	12	no	—
246	C₄⋊C₄	—	712	3	no	—
247	C₄⋊C₄	#13083551	1	12	yes	—
248	C₄²	—	356	6	no	—
249	C₄²	—	178	12	no	—
250	C₄²	C₂×C₄×D₈₉	1	6	yes	—
251	C₄²	C₂×C₂²⋊D₈₉	1	6	yes	—
252	C₂³	C₂²×D₃₅₆	1	1	yes	characteristic
253	Q₈	—	89	4	no	—
254	Q₈	—	178	2	no	—
255	Q₈	—	178	2	no	—
256	Q₈	—	178	2	no	—
257	Q₈	—	712	12	no	—
258	Q₈	#15667937	1	16	yes	direct product factor
259	C₂×C₄	—	2	3	no	—
260	C₂×C₄	—	2	3	no	—
261	C₂×C₄	—	356	3	no	—
262	C₂×C₄	—	356	1	no	—
263	C₂×C₄	—	356	3	no	—
264	C₂×C₄	—	356	1	no	—
265	C₂×C₄	—	356	3	no	—
266	C₂×C₄	—	356	3	no	—
267	C₂×C₄	—	356	3	no	—
268	C₂×C₄	—	178	2	no	—
269	C₂×C₄	—	178	6	no	—
270	C₂×C₄	—	178	2	no	—
271	C₂×C₄	—	178	2	no	—
272	C₂×C₄	—	178	2	no	—
273	C₂×C₄	C₂×#13083551	1	6	yes	—
274	C₂×C₄	C₂×#13083551	1	6	yes	—
275	C₂×C₄	C₂×#13083563	1	6	yes	—
276	C₂×C₄	C₂×Dic₃₅₆	1	6	yes	—
277	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
278	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
279	C₂×C₄	C₂×Q₈×D₈₉	1	1	yes	characteristic
280	C₂×C₄	#41×D₈₉	1	1	yes	characteristic
281	C₂×C₄	#15668059	1	1	yes	characteristic
282	C₂×C₄	#15668060	1	1	yes	characteristic
283	C₈	—	2	2	no	—
284	C₈	—	4	3	no	—
285	C₂²	C₂²×#13083551	1	1	yes	characteristic
286	C₂²	#19152260	1	1	yes	characteristic
287	C₂²	Q₈×D₃₅₆	1	1	yes	characteristic
288	C₂²	C₂²×#13083563	1	1	yes	characteristic
289	C₂²	C₂²×Dic₃₅₆	1	1	yes	characteristic
290	C₂²	#19152403	1	1	yes	characteristic
291	C₂²	#19152410	1	1	yes	characteristic
292	C₄	—	2	6	no	—
293	C₄	—	356	2	no	—
294	C₄	—	356	6	no	—
295	C₄	—	178	4	no	—
296	C₄	—	712	3	no	—
297	C₄	C₂×#15667937	1	12	yes	—
298	C₄	C₄×Q₈×D₈₉	1	1	yes	characteristic
299	C₄	#19152074	1	1	yes	characteristic
300	C₄	#19152255	1	1	yes	characteristic
301	C₄	Q₈×C₂²⋊D₈₉	1	1	yes	characteristic
302	C₂	Q₈×#13083551	1	1	yes	characteristic
303	C₂	C₂²×#15667937	1	1	yes	characteristic
304	C₂	#26028779	1	1	yes	characteristic
305	C₂	#26032924	1	1	yes	characteristic
306	C₂	#26032931	1	1	yes	characteristic
307	C₂	Q₈×#13083563	1	1	yes	characteristic
308	C₂	Q₈×Dic₃₅₆	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²×D₃₅₆
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₈₉
Fitting	C₂×C₈×Q₈×C₈₉	C₂
socle, prime radical	C₂³×C₈₉	C₂²×D₄
cosocle	C₂²×C₄×C₈₉	C₂⁴
hypercenter	C₂×C₈×Q₈	D₈₉
hypocenter	C₈₉	Q₈×#103
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈ ◁ C₂×C₈×Q₈

Complements

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

N	H ≅ G/N	type
9. Q₈×Dic₁₇₈	C₄	semidirect
46. #15667937	Q₈	direct (N × H)
98. Q₈×C₈₉	#103	semidirect
127. Dic₁₇₈	C₄×Q₈	semidirect
162. C₈₉	Q₈×#103	semidirect
258. Q₈	#15667937	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×Dic₁₇₈	2
Q₈×C₄⋊Dic₈₉	2
#26021213	2
#26021199	2
#26020258	2
C₄×#15667937	2
C₂×C₈×Q₈×C₈₉	2
Q₈×#103	89

Subgroups — Q8×#15667937 #36571036

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#15667937 #36571036