Subgroups — Q₈×#103×C₈₉ #36400368

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

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₈₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂×C₄×Q₈×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
9. Q₈²×C₈₉	C₄	semidirect
54. #103×C₈₉	Q₈	direct (N × H)
102. Q₈×C₈₉	#103	direct (N × H)
103. Q₈×C₈₉	C₄×Q₈	semidirect
147. Q₈×#103	C₈₉	direct (N × H)
162. C₈₉	Q₈×#103	direct (N × H)
164. Q₈²	C₄×C₈₉	semidirect
209. #103	Q₈×C₈₉	direct (N × H)
257. Q₈	#103×C₈₉	direct (N × H)
258. Q₈	C₄×Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×Q₈²×C₈₉	2
C₂×C₈×Q₈×C₈₉	2
Q₈×C₄⋊C₄×C₈₉	2
C₈₉×#1990	2
C₈₉×#1987	2
C₈₉×#1803	2
C₄×#103×C₈₉	2
Q₈×#103	89

Subgroups — Q8×#103×C89 #36400368

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — Q₈×#103×C₈₉ #36400368