Subgroups — C₈₉×#18519 #36396367

Order 22784.

914 conjugacy classes of subgroups.

The subgroup lattice has 712 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×#41×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#3170	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3104	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#3076	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1649	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#1648	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#1645	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#1644	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1637	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1636	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1633	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈₉×#1632	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈₉×#1542	C₂	1	1	yes	maximal, characteristic, complemented
14	C₈₉×#1476	C₂	1	1	yes	maximal, characteristic
15	C₂×Q₈⋊C₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	Q₈²×C₈₉	C₂²	1	1	yes	characteristic
17	#557×C₈₉	—	2	1	no	—
18	#557×C₈₉	C₂²	1	1	yes	characteristic
19	#554×C₈₉	—	2	2	no	—
20	#554×C₈₉	—	2	1	no	—
21	#554×C₈₉	—	2	1	no	—
22	#554×C₈₉	C₂²	1	1	yes	characteristic
23	D₄×Q₈×C₈₉	—	2	1	no	—
24	D₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
25	#533×C₈₉	—	2	2	no	—
26	#533×C₈₉	—	2	1	no	—
27	#533×C₈₉	—	2	1	no	—
28	#533×C₈₉	C₂²	1	1	yes	characteristic
29	#533×C₈₉	C₂²	1	1	yes	characteristic
30	C₄×#41×C₈₉	—	2	1	no	—
31	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
33	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
34	#500×C₈₉	C₂²	1	1	yes	characteristic
35	#494×C₈₉	C₂²	1	1	yes	characteristic
36	#487×C₈₉	C₂²	1	1	yes	characteristic
37	#487×C₈₉	C₂²	1	1	yes	characteristic
38	#484×C₈₉	—	2	1	no	—
39	#479×C₈₉	—	2	1	no	—
40	#478×C₈₉	—	2	1	no	—
41	#477×C₈₉	C₂²	1	1	yes	characteristic
42	#477×C₈₉	C₂²	1	1	yes	characteristic
43	#477×C₈₉	C₂²	1	1	yes	characteristic
44	#476×C₈₉	C₂²	1	1	yes	characteristic
45	#475×C₈₉	C₂²	1	1	yes	characteristic
46	#474×C₈₉	C₂²	1	1	yes	characteristic
47	#467×C₈₉	—	2	2	no	—
48	#462×C₈₉	C₂²	1	1	yes	characteristic
49	#462×C₈₉	C₂²	1	1	yes	characteristic
50	#451×C₈₉	—	2	1	no	—
51	C₄×Q₁₆×C₈₉	—	2	1	no	—
52	C₄×Q₁₆×C₈₉	—	2	1	no	—
53	C₄×Q₁₆×C₈₉	—	2	2	no	—
54	#431×C₈₉	C₂²	1	1	yes	characteristic
55	#427×C₈₉	—	2	1	no	—
56	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
57	#416×C₈₉	—	2	1	no	—
58	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
59	#333×C₈₉	C₂²	1	1	yes	characteristic
60	#333×C₈₉	C₂²	1	1	yes	characteristic
61	#332×C₈₉	C₂²	1	1	yes	characteristic
62	#332×C₈₉	C₂²	1	1	yes	characteristic
63	Q₈⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
64	Q₈⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
65	Q₈⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
66	Q₈⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
67	Q₈⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
68	Q₈⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
69	D₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
70	D₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
71	C₂×#41×C₈₉	D₄	1	1	yes	characteristic
72	C₂²×Q₈×C₈₉	—	2	1	no	—
73	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
74	C₂×Q₁₆×C₈₉	—	2	1	no	—
75	C₂×Q₁₆×C₈₉	—	2	1	no	—
76	C₂×Q₁₆×C₈₉	—	2	2	no	—
77	C₂²×C₈×C₈₉	—	2	1	no	—
78	C₂²×C₈×C₈₉	—	2	1	no	—
79	C₄⋊Q₈×C₈₉	—	2	2	no	—
80	C₄⋊Q₈×C₈₉	—	2	1	no	—
81	C₄⋊Q₈×C₈₉	—	2	2	no	—
82	C₄⋊Q₈×C₈₉	—	2	1	no	—
83	C₄⋊Q₈×C₈₉	—	2	1	no	—
84	C₄⋊Q₈×C₈₉	—	2	1	no	—
85	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
86	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
87	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
88	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
89	#125×C₈₉	—	2	2	no	—
90	#125×C₈₉	—	2	2	no	—
91	#125×C₈₉	—	2	2	no	—
92	#125×C₈₉	—	2	1	no	—
93	#125×C₈₉	—	2	1	no	—
94	#125×C₈₉	—	2	1	no	—
95	#125×C₈₉	—	2	1	no	—
96	C₂²⋊Q₈×C₈₉	—	2	2	no	—
97	C₂²⋊Q₈×C₈₉	—	2	2	no	—
98	C₂²⋊Q₈×C₈₉	—	2	2	no	—
99	C₂²⋊Q₈×C₈₉	—	2	2	no	—
100	C₂²⋊Q₈×C₈₉	—	2	2	no	—
101	C₂²⋊Q₈×C₈₉	—	2	1	no	—
102	C₂²⋊Q₈×C₈₉	—	2	2	no	—
103	C₂²⋊Q₈×C₈₉	—	2	1	no	—
104	C₂²⋊Q₈×C₈₉	—	2	1	no	—
105	C₂²⋊Q₈×C₈₉	—	2	2	no	—
106	C₂²⋊Q₈×C₈₉	—	2	1	no	—
107	C₂²⋊Q₈×C₈₉	—	2	1	no	—
108	C₂²⋊Q₈×C₈₉	D₄	1	2	yes	—
109	C₄×Q₈×C₈₉	—	2	2	no	—
110	C₄×Q₈×C₈₉	—	2	2	no	—
111	C₄×Q₈×C₈₉	—	2	2	no	—
112	C₄×Q₈×C₈₉	—	2	1	no	—
113	C₄×Q₈×C₈₉	—	2	1	no	—
114	C₄×Q₈×C₈₉	—	2	1	no	—
115	C₄×Q₈×C₈₉	—	2	1	no	—
116	C₄×Q₈×C₈₉	—	2	1	no	—
117	C₄×Q₈×C₈₉	—	2	1	no	—
118	C₄×Q₈×C₈₉	—	2	1	no	—
119	C₄×Q₈×C₈₉	—	2	1	no	—
120	C₄×Q₈×C₈₉	—	2	1	no	—
121	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
122	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
123	C₄×Q₈×C₈₉	D₄	1	2	yes	—
124	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
125	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
126	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
127	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
128	C₄×D₄×C₈₉	—	2	2	no	—
129	C₄×D₄×C₈₉	—	2	2	no	—
130	C₄×D₄×C₈₉	—	2	1	no	—
131	C₄×D₄×C₈₉	—	2	1	no	—
132	C₄×D₄×C₈₉	—	2	1	no	—
133	C₄×D₄×C₈₉	—	2	1	no	—
134	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
135	C₄×D₄×C₈₉	C₂³	1	1	yes	characteristic
136	#117×C₈₉	—	2	2	no	—
137	#117×C₈₉	—	2	1	no	—
138	#117×C₈₉	—	2	1	no	—
139	#117×C₈₉	—	2	1	no	—
140	#117×C₈₉	D₄	1	1	yes	characteristic
141	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
142	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
143	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
144	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
145	C₂×C₄²×C₈₉	—	2	2	no	—
146	C₂×C₄²×C₈₉	—	2	1	no	—
147	C₂×C₄²×C₈₉	—	2	1	no	—
148	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
149	#107×C₈₉	—	2	1	no	—
150	#107×C₈₉	—	2	1	no	—
151	#107×C₈₉	—	2	2	no	—
152	#107×C₈₉	—	2	1	no	—
153	#107×C₈₉	—	2	1	no	—
154	#106×C₈₉	—	2	1	no	—
155	#106×C₈₉	—	2	1	no	—
156	C₄⋊C₈×C₈₉	—	2	1	no	—
157	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
158	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
159	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
160	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
161	#103×C₈₉	—	2	1	no	—
162	#103×C₈₉	—	2	1	no	—
163	#103×C₈₉	—	2	1	no	—
164	#103×C₈₉	—	2	1	no	—
165	#103×C₈₉	—	2	2	no	—
166	#103×C₈₉	—	2	1	no	—
167	#103×C₈₉	—	2	1	no	—
168	#103×C₈₉	—	2	2	no	—
169	#102×C₈₉	—	2	1	no	—
170	#102×C₈₉	—	2	1	no	—
171	C₂²⋊C₈×C₈₉	—	2	1	no	—
172	C₄×C₈×C₈₉	—	2	1	no	—
173	C₄×C₈×C₈₉	—	2	1	no	—
174	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
175	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
176	#41×C₈₉	—	2	2	no	—
177	#41×C₈₉	Q₁₆	1	4	yes	complemented
178	C₂×Q₈×C₈₉	—	2	2	no	—
179	C₂×Q₈×C₈₉	—	2	2	no	—
180	C₂×Q₈×C₈₉	—	2	1	no	—
181	C₂×Q₈×C₈₉	—	2	1	no	—
182	C₂×Q₈×C₈₉	—	2	2	no	—
183	C₂×Q₈×C₈₉	—	2	1	no	—
184	C₂×Q₈×C₈₉	—	2	1	no	—
185	C₂×Q₈×C₈₉	—	4	1	no	—
186	C₂×Q₈×C₈₉	—	4	2	no	—
187	C₂×Q₈×C₈₉	—	4	2	no	—
188	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
189	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
190	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
191	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
192	C₂×D₄×C₈₉	—	2	1	no	—
193	C₂×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
194	C₂²×C₄×C₈₉	—	2	1	no	—
195	C₂²×C₄×C₈₉	—	2	2	no	—
196	C₂²×C₄×C₈₉	—	2	1	no	—
197	C₂²×C₄×C₈₉	—	2	2	no	—
198	C₂²×C₄×C₈₉	—	2	1	no	—
199	C₂²×C₄×C₈₉	—	2	1	no	—
200	C₂²×C₄×C₈₉	—	2	1	no	—
201	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
202	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
203	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
204	Q₁₆×C₈₉	—	4	4	no	—
205	Q₁₆×C₈₉	—	4	2	no	—
206	Q₁₆×C₈₉	—	4	2	no	—
207	C₂×C₈×C₈₉	—	2	1	no	—
208	C₂×C₈×C₈₉	—	2	1	no	—
209	C₂×C₈×C₈₉	—	2	1	no	—
210	C₂×C₈×C₈₉	—	2	1	no	—
211	C₂×C₈×C₈₉	—	2	1	no	—
212	C₂×C₈×C₈₉	—	2	1	no	—
213	C₂×C₈×C₈₉	—	4	2	no	—
214	C₂×C₈×C₈₉	—	4	2	no	—
215	C₄⋊C₄×C₈₉	—	2	2	no	—
216	C₄⋊C₄×C₈₉	—	2	2	no	—
217	C₄⋊C₄×C₈₉	—	2	2	no	—
218	C₄⋊C₄×C₈₉	—	2	2	no	—
219	C₄⋊C₄×C₈₉	—	2	2	no	—
220	C₄⋊C₄×C₈₉	—	2	2	no	—
221	C₄⋊C₄×C₈₉	—	2	2	no	—
222	C₄⋊C₄×C₈₉	—	2	2	no	—
223	C₄⋊C₄×C₈₉	—	2	1	no	—
224	C₄⋊C₄×C₈₉	—	2	2	no	—
225	C₄⋊C₄×C₈₉	—	2	1	no	—
226	C₄⋊C₄×C₈₉	—	2	1	no	—
227	C₄⋊C₄×C₈₉	—	2	1	no	—
228	C₄⋊C₄×C₈₉	—	2	2	no	—
229	C₄⋊C₄×C₈₉	—	2	1	no	—
230	C₄⋊C₄×C₈₉	—	2	1	no	—
231	C₄⋊C₄×C₈₉	—	2	1	no	—
232	C₄⋊C₄×C₈₉	—	2	1	no	—
233	C₄⋊C₄×C₈₉	—	2	1	no	—
234	C₄⋊C₄×C₈₉	—	2	1	no	—
235	C₄⋊C₄×C₈₉	—	2	1	no	—
236	C₄⋊C₄×C₈₉	—	2	1	no	—
237	C₄⋊C₄×C₈₉	—	2	1	no	—
238	C₄⋊C₄×C₈₉	—	2	1	no	—
239	C₄⋊C₄×C₈₉	—	4	1	no	—
240	C₄⋊C₄×C₈₉	—	4	1	no	—
241	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
242	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
243	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
244	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
245	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
246	C₂²⋊C₄×C₈₉	—	2	2	no	—
247	C₂²⋊C₄×C₈₉	—	2	2	no	—
248	C₂²⋊C₄×C₈₉	—	2	2	no	—
249	C₂²⋊C₄×C₈₉	—	2	1	no	—
250	C₂²⋊C₄×C₈₉	—	2	1	no	—
251	C₂²⋊C₄×C₈₉	—	2	1	no	—
252	C₂²⋊C₄×C₈₉	—	4	1	no	—
253	C₄²×C₈₉	—	2	2	no	—
254	C₄²×C₈₉	—	2	2	no	—
255	C₄²×C₈₉	—	2	1	no	—
256	C₄²×C₈₉	—	2	1	no	—
257	C₄²×C₈₉	—	2	2	no	—
258	C₄²×C₈₉	—	2	1	no	—
259	C₄²×C₈₉	—	2	1	no	—
260	C₄²×C₈₉	—	2	1	no	—
261	C₄²×C₈₉	—	2	1	no	—
262	C₄²×C₈₉	—	2	1	no	—
263	C₄²×C₈₉	—	4	1	no	—
264	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
265	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
266	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
267	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
268	C₂³×C₈₉	—	2	1	no	—
269	C₂³×C₈₉	#120	1	1	yes	characteristic
270	Q₈×C₈₉	—	2	1	no	—
271	Q₈×C₈₉	—	2	1	no	—
272	Q₈×C₈₉	—	2	1	no	—
273	Q₈×C₈₉	—	2	1	no	—
274	Q₈×C₈₉	—	4	2	no	—
275	Q₈×C₈₉	—	4	2	no	—
276	Q₈×C₈₉	—	4	4	no	—
277	Q₈×C₈₉	—	4	2	no	—
278	Q₈×C₈₉	—	4	1	no	—
279	Q₈×C₈₉	—	4	1	no	—
280	Q₈×C₈₉	—	4	1	no	—
281	Q₈×C₈₉	—	4	1	no	—
282	Q₈×C₈₉	—	8	2	no	—
283	Q₈×C₈₉	C₂×Q₁₆	1	2	yes	—
284	Q₈×C₈₉	#135	1	2	yes	—
285	D₄×C₈₉	—	2	4	no	—
286	D₄×C₈₉	—	2	1	no	—
287	D₄×C₈₉	C₂×Q₁₆	1	2	yes	—
288	C₂×C₄×C₈₉	—	2	1	no	—
289	C₂×C₄×C₈₉	—	2	1	no	—
290	C₂×C₄×C₈₉	—	2	1	no	—
291	C₂×C₄×C₈₉	—	2	1	no	—
292	C₂×C₄×C₈₉	—	2	4	no	—
293	C₂×C₄×C₈₉	—	2	1	no	—
294	C₂×C₄×C₈₉	—	2	1	no	—
295	C₂×C₄×C₈₉	—	2	2	no	—
296	C₂×C₄×C₈₉	—	2	2	no	—
297	C₂×C₄×C₈₉	—	2	1	no	—
298	C₂×C₄×C₈₉	—	2	1	no	—
299	C₂×C₄×C₈₉	—	2	1	no	—
300	C₂×C₄×C₈₉	—	4	1	no	—
301	C₂×C₄×C₈₉	—	4	2	no	—
302	C₂×C₄×C₈₉	—	4	4	no	—
303	C₂×C₄×C₈₉	—	4	1	no	—
304	C₂×C₄×C₈₉	—	4	1	no	—
305	C₂×C₄×C₈₉	—	4	1	no	—
306	C₂×C₄×C₈₉	—	4	1	no	—
307	C₂×C₄×C₈₉	—	8	1	no	—
308	C₂×C₄×C₈₉	—	8	1	no	—
309	C₂×C₄×C₈₉	—	8	2	no	—
310	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
311	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
312	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
313	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
314	C₂×C₄×C₈₉	#137	1	2	yes	—
315	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
316	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
317	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
318	C₈×C₈₉	—	2	2	no	—
319	C₈×C₈₉	—	4	1	no	—
320	C₈×C₈₉	—	4	1	no	—
321	C₈×C₈₉	—	4	1	no	—
322	C₈×C₈₉	—	4	1	no	—
323	C₈×C₈₉	—	4	1	no	—
324	C₂²×C₈₉	—	2	1	no	—
325	C₂²×C₈₉	—	2	1	no	—
326	C₂²×C₈₉	—	2	2	no	—
327	C₂²×C₈₉	—	4	1	no	—
328	C₂²×C₈₉	—	4	1	no	—
329	C₂²×C₈₉	#451	1	2	yes	—
330	C₂²×C₈₉	C₂×#120	1	1	yes	characteristic
331	C₄×C₈₉	—	2	2	no	—
332	C₄×C₈₉	—	2	1	no	—
333	C₄×C₈₉	—	2	2	no	—
334	C₄×C₈₉	—	2	1	no	—
335	C₄×C₈₉	—	2	1	no	—
336	C₄×C₈₉	—	4	1	no	—
337	C₄×C₈₉	—	4	2	no	—
338	C₄×C₈₉	—	4	1	no	—
339	C₄×C₈₉	—	4	1	no	—
340	C₄×C₈₉	—	4	2	no	—
341	C₄×C₈₉	—	4	1	no	—
342	C₄×C₈₉	—	4	1	no	—
343	C₄×C₈₉	—	8	1	no	—
344	C₄×C₈₉	—	8	1	no	—
345	C₄×C₈₉	#451	1	2	yes	—
346	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
347	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
348	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
349	C₄×C₈₉	#575	1	1	yes	characteristic
350	#18519	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
351	C₂×C₈₉	—	2	2	no	—
352	C₂×C₈₉	—	4	2	no	—
353	C₂×C₈₉	C₂×#451	1	1	yes	characteristic
354	C₂×C₈₉	#3002	1	1	yes	characteristic
355	C₂×C₈₉	#3018	1	1	yes	characteristic
356	Q₈×#41	C₂×C₈₉	1	1	yes	characteristic
357	#3170	C₂×C₈₉	1	1	yes	characteristic, complemented
358	#3104	C₂×C₈₉	1	1	yes	characteristic, complemented
359	#3076	C₂×C₈₉	1	1	yes	characteristic
360	#1649	C₂×C₈₉	1	1	yes	characteristic, complemented
361	#1648	C₂×C₈₉	1	1	yes	characteristic, complemented
362	#1645	C₂×C₈₉	1	1	yes	characteristic, complemented
363	#1644	C₂×C₈₉	1	1	yes	characteristic, complemented
364	#1637	C₂×C₈₉	1	1	yes	characteristic, complemented
365	#1636	C₂×C₈₉	1	1	yes	characteristic, complemented
366	#1633	C₂×C₈₉	1	1	yes	characteristic, complemented
367	#1632	C₂×C₈₉	1	1	yes	characteristic, complemented
368	#1542	C₂×C₈₉	1	1	yes	characteristic, complemented
369	#1476	C₂×C₈₉	1	1	yes	characteristic
370	C₂×Q₈⋊C₈	C₂×C₈₉	1	1	yes	characteristic, complemented
371	C₈₉	#18519	1	1	yes	characteristic, direct product factor, Sylow
372	Q₈²	C₂²×C₈₉	1	1	yes	characteristic
373	#557	—	2	1	no	—
374	#557	C₂²×C₈₉	1	1	yes	characteristic
375	#554	—	2	2	no	—
376	#554	—	2	1	no	—
377	#554	—	2	1	no	—
378	#554	C₂²×C₈₉	1	1	yes	characteristic
379	D₄×Q₈	—	2	1	no	—
380	D₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
381	#533	—	2	2	no	—
382	#533	—	2	1	no	—
383	#533	—	2	1	no	—
384	#533	C₂²×C₈₉	1	1	yes	characteristic
385	#533	C₂²×C₈₉	1	1	yes	characteristic
386	C₄×#41	—	2	1	no	—
387	C₄×#41	C₂²×C₈₉	1	1	yes	characteristic
388	C₂×C₄×Q₈	—	2	1	no	—
389	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
390	#500	C₂²×C₈₉	1	1	yes	characteristic
391	#494	C₂²×C₈₉	1	1	yes	characteristic
392	#487	C₂²×C₈₉	1	1	yes	characteristic
393	#487	C₂²×C₈₉	1	1	yes	characteristic
394	#484	—	2	1	no	—
395	#479	—	2	1	no	—
396	#478	—	2	1	no	—
397	#477	C₂²×C₈₉	1	1	yes	characteristic
398	#477	C₂²×C₈₉	1	1	yes	characteristic
399	#477	C₂²×C₈₉	1	1	yes	characteristic
400	#476	C₂²×C₈₉	1	1	yes	characteristic
401	#475	C₂²×C₈₉	1	1	yes	characteristic
402	#474	C₂²×C₈₉	1	1	yes	characteristic
403	#467	—	2	2	no	—
404	#462	C₂²×C₈₉	1	1	yes	characteristic
405	#462	C₂²×C₈₉	1	1	yes	characteristic
406	#451	—	2	1	no	—
407	C₄×Q₁₆	—	2	1	no	—
408	C₄×Q₁₆	—	2	1	no	—
409	C₄×Q₁₆	—	2	2	no	—
410	#431	C₂²×C₈₉	1	1	yes	characteristic
411	#427	—	2	1	no	—
412	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
413	#416	—	2	1	no	—
414	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
415	#333	C₂²×C₈₉	1	1	yes	characteristic
416	#333	C₂²×C₈₉	1	1	yes	characteristic
417	#332	C₂²×C₈₉	1	1	yes	characteristic
418	#332	C₂²×C₈₉	1	1	yes	characteristic
419	Q₈⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
420	Q₈⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
421	Q₈⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
422	Q₈⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
423	Q₈⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
424	Q₈⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
425	D₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
426	D₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
427	C₂×#41	D₄×C₈₉	1	1	yes	characteristic
428	C₂²×Q₈	—	2	1	no	—
429	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
430	C₂×Q₁₆	—	2	1	no	—
431	C₂×Q₁₆	—	2	1	no	—
432	C₂×Q₁₆	—	2	2	no	—
433	C₂²×C₈	—	2	1	no	—
434	C₂²×C₈	—	2	1	no	—
435	C₄⋊Q₈	—	2	2	no	—
436	C₄⋊Q₈	—	2	1	no	—
437	C₄⋊Q₈	—	2	2	no	—
438	C₄⋊Q₈	—	2	1	no	—
439	C₄⋊Q₈	—	2	1	no	—
440	C₄⋊Q₈	—	2	1	no	—
441	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
442	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
443	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
444	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
445	#125	—	2	2	no	—
446	#125	—	2	2	no	—
447	#125	—	2	2	no	—
448	#125	—	2	1	no	—
449	#125	—	2	1	no	—
450	#125	—	2	1	no	—
451	#125	—	2	1	no	—
452	C₂²⋊Q₈	—	2	2	no	—
453	C₂²⋊Q₈	—	2	2	no	—
454	C₂²⋊Q₈	—	2	2	no	—
455	C₂²⋊Q₈	—	2	2	no	—
456	C₂²⋊Q₈	—	2	2	no	—
457	C₂²⋊Q₈	—	2	1	no	—
458	C₂²⋊Q₈	—	2	2	no	—
459	C₂²⋊Q₈	—	2	1	no	—
460	C₂²⋊Q₈	—	2	1	no	—
461	C₂²⋊Q₈	—	2	2	no	—
462	C₂²⋊Q₈	—	2	1	no	—
463	C₂²⋊Q₈	—	2	1	no	—
464	C₂²⋊Q₈	D₄×C₈₉	1	2	yes	—
465	C₄×Q₈	—	2	2	no	—
466	C₄×Q₈	—	2	2	no	—
467	C₄×Q₈	—	2	2	no	—
468	C₄×Q₈	—	2	1	no	—
469	C₄×Q₈	—	2	1	no	—
470	C₄×Q₈	—	2	1	no	—
471	C₄×Q₈	—	2	1	no	—
472	C₄×Q₈	—	2	1	no	—
473	C₄×Q₈	—	2	1	no	—
474	C₄×Q₈	—	2	1	no	—
475	C₄×Q₈	—	2	1	no	—
476	C₄×Q₈	—	2	1	no	—
477	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
478	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
479	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
480	C₄×Q₈	D₄×C₈₉	1	2	yes	—
481	C₄×Q₈	C₂³×C₈₉	1	1	yes	characteristic
482	C₄×Q₈	C₂³×C₈₉	1	1	yes	characteristic
483	C₄×Q₈	C₂³×C₈₉	1	1	yes	characteristic
484	C₄×D₄	—	2	2	no	—
485	C₄×D₄	—	2	2	no	—
486	C₄×D₄	—	2	1	no	—
487	C₄×D₄	—	2	1	no	—
488	C₄×D₄	—	2	1	no	—
489	C₄×D₄	—	2	1	no	—
490	C₄×D₄	D₄×C₈₉	1	1	yes	characteristic
491	C₄×D₄	C₂³×C₈₉	1	1	yes	characteristic
492	#117	—	2	2	no	—
493	#117	—	2	1	no	—
494	#117	—	2	1	no	—
495	#117	—	2	1	no	—
496	#117	D₄×C₈₉	1	1	yes	characteristic
497	C₂×C₄⋊C₄	—	2	2	no	—
498	C₂×C₄⋊C₄	—	2	1	no	—
499	C₂×C₄⋊C₄	—	2	1	no	—
500	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
501	C₂×C₄²	—	2	2	no	—
502	C₂×C₄²	—	2	1	no	—
503	C₂×C₄²	—	2	1	no	—
504	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
505	#107	—	2	1	no	—
506	#107	—	2	1	no	—
507	#107	—	2	2	no	—
508	#107	—	2	1	no	—
509	#107	—	2	1	no	—
510	#106	—	2	1	no	—
511	#106	—	2	1	no	—
512	C₄⋊C₈	—	2	1	no	—
513	C₄⋊C₈	C₂³×C₈₉	1	1	yes	characteristic
514	C₄⋊C₈	C₂³×C₈₉	1	1	yes	characteristic
515	C₄⋊C₈	C₂³×C₈₉	1	1	yes	characteristic
516	C₄⋊C₈	C₂³×C₈₉	1	1	yes	characteristic
517	#103	—	2	1	no	—
518	#103	—	2	1	no	—
519	#103	—	2	1	no	—
520	#103	—	2	1	no	—
521	#103	—	2	2	no	—
522	#103	—	2	1	no	—
523	#103	—	2	1	no	—
524	#103	—	2	2	no	—
525	#102	—	2	1	no	—
526	#102	—	2	1	no	—
527	C₂²⋊C₈	—	2	1	no	—
528	C₄×C₈	—	2	1	no	—
529	C₄×C₈	—	2	1	no	—
530	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
531	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
532	#41	—	2	2	no	—
533	#41	Q₁₆×C₈₉	1	4	yes	complemented
534	C₂×Q₈	—	2	2	no	—
535	C₂×Q₈	—	2	2	no	—
536	C₂×Q₈	—	2	1	no	—
537	C₂×Q₈	—	2	1	no	—
538	C₂×Q₈	—	2	2	no	—
539	C₂×Q₈	—	2	1	no	—
540	C₂×Q₈	—	2	1	no	—
541	C₂×Q₈	—	4	1	no	—
542	C₂×Q₈	—	4	2	no	—
543	C₂×Q₈	—	4	2	no	—
544	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
545	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
546	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
547	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
548	C₂×D₄	—	2	1	no	—
549	C₂×D₄	C₂×D₄×C₈₉	1	1	yes	characteristic
550	C₂²×C₄	—	2	1	no	—
551	C₂²×C₄	—	2	2	no	—
552	C₂²×C₄	—	2	1	no	—
553	C₂²×C₄	—	2	2	no	—
554	C₂²×C₄	—	2	1	no	—
555	C₂²×C₄	—	2	1	no	—
556	C₂²×C₄	—	2	1	no	—
557	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
558	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
559	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
560	Q₁₆	—	4	4	no	—
561	Q₁₆	—	4	2	no	—
562	Q₁₆	—	4	2	no	—
563	C₂×C₈	—	2	1	no	—
564	C₂×C₈	—	2	1	no	—
565	C₂×C₈	—	2	1	no	—
566	C₂×C₈	—	2	1	no	—
567	C₂×C₈	—	2	1	no	—
568	C₂×C₈	—	2	1	no	—
569	C₂×C₈	—	4	2	no	—
570	C₂×C₈	—	4	2	no	—
571	C₄⋊C₄	—	2	2	no	—
572	C₄⋊C₄	—	2	2	no	—
573	C₄⋊C₄	—	2	2	no	—
574	C₄⋊C₄	—	2	2	no	—
575	C₄⋊C₄	—	2	2	no	—
576	C₄⋊C₄	—	2	2	no	—
577	C₄⋊C₄	—	2	2	no	—
578	C₄⋊C₄	—	2	2	no	—
579	C₄⋊C₄	—	2	1	no	—
580	C₄⋊C₄	—	2	2	no	—
581	C₄⋊C₄	—	2	1	no	—
582	C₄⋊C₄	—	2	1	no	—
583	C₄⋊C₄	—	2	1	no	—
584	C₄⋊C₄	—	2	2	no	—
585	C₄⋊C₄	—	2	1	no	—
586	C₄⋊C₄	—	2	1	no	—
587	C₄⋊C₄	—	2	1	no	—
588	C₄⋊C₄	—	2	1	no	—
589	C₄⋊C₄	—	2	1	no	—
590	C₄⋊C₄	—	2	1	no	—
591	C₄⋊C₄	—	2	1	no	—
592	C₄⋊C₄	—	2	1	no	—
593	C₄⋊C₄	—	2	1	no	—
594	C₄⋊C₄	—	2	1	no	—
595	C₄⋊C₄	—	4	1	no	—
596	C₄⋊C₄	—	4	1	no	—
597	C₄⋊C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
598	C₄⋊C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
599	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
600	C₄⋊C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
601	C₄⋊C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
602	C₂²⋊C₄	—	2	2	no	—
603	C₂²⋊C₄	—	2	2	no	—
604	C₂²⋊C₄	—	2	2	no	—
605	C₂²⋊C₄	—	2	1	no	—
606	C₂²⋊C₄	—	2	1	no	—
607	C₂²⋊C₄	—	2	1	no	—
608	C₂²⋊C₄	—	4	1	no	—
609	C₄²	—	2	2	no	—
610	C₄²	—	2	2	no	—
611	C₄²	—	2	1	no	—
612	C₄²	—	2	2	no	—
613	C₄²	—	2	1	no	—
614	C₄²	—	2	1	no	—
615	C₄²	—	2	1	no	—
616	C₄²	—	2	1	no	—
617	C₄²	—	2	1	no	—
618	C₄²	—	2	1	no	—
619	C₄²	—	4	1	no	—
620	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
621	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
622	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
623	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
624	C₂³	—	2	1	no	—
625	C₂³	#120×C₈₉	1	1	yes	characteristic
626	Q₈	—	2	1	no	—
627	Q₈	—	2	1	no	—
628	Q₈	—	2	1	no	—
629	Q₈	—	2	1	no	—
630	Q₈	—	4	2	no	—
631	Q₈	—	4	2	no	—
632	Q₈	—	4	4	no	—
633	Q₈	—	4	2	no	—
634	Q₈	—	4	1	no	—
635	Q₈	—	4	1	no	—
636	Q₈	—	4	1	no	—
637	Q₈	—	4	1	no	—
638	Q₈	—	8	2	no	—
639	Q₈	C₂×Q₁₆×C₈₉	1	2	yes	—
640	Q₈	#135×C₈₉	1	2	yes	—
641	D₄	—	2	4	no	—
642	D₄	—	2	1	no	—
643	D₄	C₂×Q₁₆×C₈₉	1	2	yes	—
644	C₂×C₄	—	2	1	no	—
645	C₂×C₄	—	2	1	no	—
646	C₂×C₄	—	2	4	no	—
647	C₂×C₄	—	2	1	no	—
648	C₂×C₄	—	2	1	no	—
649	C₂×C₄	—	2	1	no	—
650	C₂×C₄	—	2	1	no	—
651	C₂×C₄	—	2	2	no	—
652	C₂×C₄	—	2	2	no	—
653	C₂×C₄	—	2	1	no	—
654	C₂×C₄	—	2	1	no	—
655	C₂×C₄	—	2	1	no	—
656	C₂×C₄	—	4	1	no	—
657	C₂×C₄	—	4	2	no	—
658	C₂×C₄	—	4	4	no	—
659	C₂×C₄	—	4	1	no	—
660	C₂×C₄	—	4	1	no	—
661	C₂×C₄	—	4	1	no	—
662	C₂×C₄	—	4	1	no	—
663	C₂×C₄	—	8	1	no	—
664	C₂×C₄	—	8	1	no	—
665	C₂×C₄	—	8	2	no	—
666	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
667	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
668	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
669	C₂×C₄	C₂×Q₁₆×C₈₉	1	2	yes	—
670	C₂×C₄	#137×C₈₉	1	2	yes	—
671	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
672	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
673	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
674	C₈	—	2	2	no	—
675	C₈	—	4	1	no	—
676	C₈	—	4	1	no	—
677	C₈	—	4	1	no	—
678	C₈	—	4	1	no	—
679	C₈	—	4	1	no	—
680	C₂²	—	2	1	no	—
681	C₂²	—	2	1	no	—
682	C₂²	—	2	2	no	—
683	C₂²	—	4	1	no	—
684	C₂²	—	4	1	no	—
685	C₂²	#451×C₈₉	1	2	yes	—
686	C₂²	C₂×#120×C₈₉	1	1	yes	characteristic
687	C₄	—	2	2	no	—
688	C₄	—	2	1	no	—
689	C₄	—	2	2	no	—
690	C₄	—	2	1	no	—
691	C₄	—	2	1	no	—
692	C₄	—	4	1	no	—
693	C₄	—	4	2	no	—
694	C₄	—	4	1	no	—
695	C₄	—	4	1	no	—
696	C₄	—	4	2	no	—
697	C₄	—	4	1	no	—
698	C₄	—	4	1	no	—
699	C₄	—	8	1	no	—
700	C₄	—	8	1	no	—
701	C₄	#451×C₈₉	1	2	yes	—
702	C₄	C₂²×Q₁₆×C₈₉	1	1	yes	characteristic
703	C₄	C₂×#135×C₈₉	1	1	yes	characteristic
704	C₄	C₂×#137×C₈₉	1	1	yes	characteristic
705	C₄	#575×C₈₉	1	1	yes	characteristic
706	C₂	—	2	2	no	—
707	C₂	—	4	2	no	—
708	C₂	C₂×#451×C₈₉	1	1	yes	characteristic
709	C₂	C₈₉×#3002	1	1	yes	characteristic
710	C₂	C₈₉×#3018	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈₉	#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₈₉
socle	C₂²×C₈₉	C₂×#120
prime radical	C₂×#41×C₈₉	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#3170	C₂	semidirect
3. C₈₉×#3104	C₂	semidirect
5. C₈₉×#1649	C₂	semidirect
6. C₈₉×#1648	C₂	semidirect
7. C₈₉×#1645	C₂	semidirect
8. C₈₉×#1644	C₂	semidirect
9. C₈₉×#1637	C₂	semidirect
10. C₈₉×#1636	C₂	semidirect
11. C₈₉×#1633	C₂	semidirect
12. C₈₉×#1632	C₂	semidirect
13. C₈₉×#1542	C₂	semidirect
15. C₂×Q₈⋊C₈×C₈₉	C₂	semidirect
177. #41×C₈₉	Q₁₆	semidirect
350. #18519	C₈₉	direct (N × H)
357. #3170	C₂×C₈₉	semidirect
358. #3104	C₂×C₈₉	semidirect
360. #1649	C₂×C₈₉	semidirect
361. #1648	C₂×C₈₉	semidirect
362. #1645	C₂×C₈₉	semidirect
363. #1644	C₂×C₈₉	semidirect
364. #1637	C₂×C₈₉	semidirect
365. #1636	C₂×C₈₉	semidirect
366. #1633	C₂×C₈₉	semidirect
367. #1632	C₂×C₈₉	semidirect
368. #1542	C₂×C₈₉	semidirect
370. C₂×Q₈⋊C₈	C₂×C₈₉	semidirect
371. C₈₉	#18519	direct (N × H)
533. #41	Q₁₆×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
Q₈×#41×C₈₉	2
C₈₉×#3170	2
C₈₉×#3104	2
C₈₉×#3076	2
C₈₉×#1649	2
C₈₉×#1648	2
C₈₉×#1645	2
C₈₉×#1644	2
C₈₉×#1637	2
C₈₉×#1636	2
C₈₉×#1633	2
C₈₉×#1632	2
C₈₉×#1542	2
C₈₉×#1476	2
C₂×Q₈⋊C₈×C₈₉	2
#18519	89

Subgroups — C89×#18519 #36396367

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#18519 #36396367