Subgroups — C₈₉×#35327 #36413175

Order 22784.

3134 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#536×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#535×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂²×#124×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄×#41×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2850	C₂	1	4	yes	maximal, complemented
6	C₈₉×#2620	C₂	1	4	yes	maximal, complemented
7	C₈₉×#2545	C₂	1	4	yes	maximal, complemented
8	C₈₉×#2544	C₂	1	4	yes	maximal, complemented
9	C₈₉×#2534	C₂	1	8	yes	maximal, complemented
10	C₈₉×#2310	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2309	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈₉×#2285	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
14	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic
16	#536×C₈₉	—	2	8	no	—
17	#535×C₈₉	—	2	8	no	—
18	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	complemented
19	C₂×#127×C₈₉	C₂²	1	2	yes	—
20	C₂×#124×C₈₉	—	2	8	no	—
21	C₂×#124×C₈₉	C₂²	1	2	yes	—
22	C₂×#124×C₈₉	C₂²	1	2	yes	—
23	C₂×#124×C₈₉	C₂²	1	4	yes	—
24	C₂×#124×C₈₉	C₂²	1	2	yes	—
25	C₂×#124×C₈₉	C₂²	1	2	yes	complemented
26	C₂×#123×C₈₉	C₂²	1	4	yes	—
27	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	4	yes	complemented
28	C₂×#121×C₈₉	C₂²	1	4	yes	—
29	C₂×#120×C₈₉	C₂²	1	4	yes	—
30	C₄×#41×C₈₉	—	2	8	no	—
31	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	complemented
32	C₂×C₄×D₄×C₈₉	C₂²	1	4	yes	—
33	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	complemented
34	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
36	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×#117×C₈₉	C₂²	1	2	yes	complemented
38	C₂²×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
39	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
40	C₂²×C₄²×C₈₉	C₂²	1	2	yes	complemented
41	#396×C₈₉	C₂²	1	16	yes	complemented
42	#394×C₈₉	C₂²	1	16	yes	complemented
43	#391×C₈₉	C₂²	1	4	yes	complemented
44	#391×C₈₉	C₂²	1	4	yes	complemented
45	#390×C₈₉	C₂²	1	4	yes	—
46	#390×C₈₉	C₂²	1	4	yes	—
47	#388×C₈₉	C₂²	1	16	yes	complemented
48	#386×C₈₉	C₂²	1	16	yes	complemented
49	#384×C₈₉	C₂²	1	2	yes	complemented
50	#382×C₈₉	C₂²	1	2	yes	complemented
51	#381×C₈₉	C₂²	1	4	yes	—
52	#380×C₈₉	C₂²	1	4	yes	—
53	#379×C₈₉	C₂²	1	4	yes	—
54	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	complemented
55	C₂⁵×C₈₉	C₂³	1	2	yes	—
56	C₂×#41×C₈₉	—	2	4	no	—
57	C₂×#41×C₈₉	—	2	4	no	—
58	C₂×#41×C₈₉	D₄	1	8	yes	—
59	C₂²×Q₈×C₈₉	—	2	4	no	—
60	C₂²×Q₈×C₈₉	—	2	4	no	—
61	C₂²×Q₈×C₈₉	—	2	4	no	—
62	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
63	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
64	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₈₉	—	2	4	no	—
66	C₂²×D₄×C₈₉	—	2	4	no	—
67	C₂²×D₄×C₈₉	—	2	4	no	—
68	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
69	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
70	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
71	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
72	C₂²×D₄×C₈₉	C₂³	1	8	yes	—
73	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
74	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
75	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
76	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
77	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
78	C₄⋊Q₈×C₈₉	—	2	8	no	—
79	#127×C₈₉	—	2	8	no	—
80	#124×C₈₉	—	2	8	no	—
81	#124×C₈₉	—	2	8	no	—
82	#124×C₈₉	—	2	8	no	—
83	#124×C₈₉	—	2	8	no	—
84	#124×C₈₉	—	2	16	no	—
85	#123×C₈₉	—	2	16	no	—
86	C₂²⋊Q₈×C₈₉	—	2	16	no	—
87	#121×C₈₉	—	2	16	no	—
88	#120×C₈₉	—	2	16	no	—
89	C₄×Q₈×C₈₉	—	2	8	no	—
90	C₄×D₄×C₈₉	—	2	16	no	—
91	C₄×D₄×C₈₉	—	2	8	no	—
92	#117×C₈₉	—	2	4	no	—
93	#117×C₈₉	—	2	4	no	—
94	#117×C₈₉	—	2	4	no	—
95	#117×C₈₉	—	2	8	no	—
96	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
97	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
98	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
99	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
100	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
101	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
102	C₂×C₂²⋊C₄×C₈₉	—	2	16	no	—
103	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
104	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
105	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
106	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
107	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
108	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
109	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
111	C₂×C₂²⋊C₄×C₈₉	C₂³	1	16	yes	—
112	C₂×C₄²×C₈₉	—	2	8	no	—
113	C₂×C₄²×C₈₉	D₄	1	8	yes	complemented
114	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
115	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
116	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
117	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
118	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
119	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
121	#95×C₈₉	C₂³	1	8	yes	—
122	#95×C₈₉	C₂³	1	16	yes	—
123	C₂⁴×C₈₉	—	2	4	no	—
124	C₂⁴×C₈₉	—	2	4	no	—
125	C₂⁴×C₈₉	—	2	4	no	—
126	C₂⁴×C₈₉	—	4	8	no	—
127	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
128	C₂⁴×C₈₉	C₂⁴	1	2	yes	—
129	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
130	#41×C₈₉	—	2	32	no	—
131	C₂×Q₈×C₈₉	—	2	2	no	—
132	C₂×Q₈×C₈₉	—	2	4	no	—
133	C₂×Q₈×C₈₉	—	2	4	no	—
134	C₂×Q₈×C₈₉	—	2	4	no	—
135	C₂×Q₈×C₈₉	—	2	4	no	—
136	C₂×Q₈×C₈₉	—	2	2	no	—
137	C₂×Q₈×C₈₉	—	2	4	no	—
138	C₂×Q₈×C₈₉	—	2	4	no	—
139	C₂×Q₈×C₈₉	—	4	16	no	—
140	C₂×Q₈×C₈₉	C₂×D₄	1	4	yes	—
141	C₂×D₄×C₈₉	—	2	4	no	—
142	C₂×D₄×C₈₉	—	2	4	no	—
143	C₂×D₄×C₈₉	—	2	16	no	—
144	C₂×D₄×C₈₉	—	2	2	no	—
145	C₂×D₄×C₈₉	—	2	4	no	—
146	C₂×D₄×C₈₉	—	2	4	no	—
147	C₂×D₄×C₈₉	—	2	4	no	—
148	C₂×D₄×C₈₉	—	2	4	no	—
149	C₂×D₄×C₈₉	—	2	2	no	—
150	C₂×D₄×C₈₉	—	2	4	no	—
151	C₂×D₄×C₈₉	—	2	4	no	—
152	C₂×D₄×C₈₉	—	4	16	no	—
153	C₂×D₄×C₈₉	—	4	16	no	—
154	C₂×D₄×C₈₉	C₂×D₄	1	8	yes	—
155	C₂×D₄×C₈₉	C₂×D₄	1	4	yes	—
156	C₂²×C₄×C₈₉	—	2	4	no	—
157	C₂²×C₄×C₈₉	—	2	4	no	—
158	C₂²×C₄×C₈₉	—	2	4	no	—
159	C₂²×C₄×C₈₉	—	2	4	no	—
160	C₂²×C₄×C₈₉	—	2	4	no	—
161	C₂²×C₄×C₈₉	—	2	8	no	—
162	C₂²×C₄×C₈₉	—	4	4	no	—
163	C₂²×C₄×C₈₉	—	4	8	no	—
164	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
165	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
166	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
167	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
168	C₂²×C₄×C₈₉	#41	1	4	yes	—
169	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
170	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
171	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
172	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
173	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
174	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
175	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
176	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
177	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
178	C₄⋊C₄×C₈₉	—	2	8	no	—
179	C₄⋊C₄×C₈₉	—	2	4	no	—
180	C₄⋊C₄×C₈₉	—	2	4	no	—
181	C₄⋊C₄×C₈₉	—	2	4	no	—
182	C₄⋊C₄×C₈₉	—	2	16	no	—
183	C₄⋊C₄×C₈₉	—	4	8	no	—
184	C₂²⋊C₄×C₈₉	—	2	4	no	—
185	C₂²⋊C₄×C₈₉	—	2	4	no	—
186	C₂²⋊C₄×C₈₉	—	2	4	no	—
187	C₂²⋊C₄×C₈₉	—	2	16	no	—
188	C₂²⋊C₄×C₈₉	—	2	16	no	—
189	C₂²⋊C₄×C₈₉	—	2	16	no	—
190	C₂²⋊C₄×C₈₉	—	2	8	no	—
191	C₂²⋊C₄×C₈₉	—	2	16	no	—
192	C₂²⋊C₄×C₈₉	—	4	16	no	—
193	C₄²×C₈₉	—	2	4	no	—
194	C₄²×C₈₉	—	2	8	no	—
195	C₄²×C₈₉	—	2	4	no	—
196	C₄²×C₈₉	—	2	4	no	—
197	C₄²×C₈₉	C₂×D₄	1	4	yes	—
198	C₄²×C₈₉	C₂×D₄	1	4	yes	—
199	C₄²×C₈₉	C₂×D₄	1	8	yes	—
200	C₂³×C₈₉	—	2	4	no	—
201	C₂³×C₈₉	—	2	2	no	—
202	C₂³×C₈₉	—	2	2	no	—
203	C₂³×C₈₉	—	2	8	no	—
204	C₂³×C₈₉	—	2	4	no	—
205	C₂³×C₈₉	—	2	4	no	—
206	C₂³×C₈₉	—	2	4	no	—
207	C₂³×C₈₉	—	2	16	no	—
208	C₂³×C₈₉	—	4	16	no	—
209	C₂³×C₈₉	—	4	8	no	—
210	C₂³×C₈₉	—	8	8	no	—
211	C₂³×C₈₉	—	8	8	no	—
212	C₂³×C₈₉	C₂²×D₄	1	4	yes	—
213	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
214	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
215	C₂³×C₈₉	C₂×#41	1	2	yes	—
216	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
217	Q₈×C₈₉	—	2	8	no	—
218	Q₈×C₈₉	—	4	8	no	—
219	Q₈×C₈₉	—	4	8	no	—
220	D₄×C₈₉	—	2	16	no	—
221	D₄×C₈₉	—	2	8	no	—
222	D₄×C₈₉	—	4	8	no	—
223	D₄×C₈₉	—	4	8	no	—
224	D₄×C₈₉	—	8	16	no	—
225	C₂×C₄×C₈₉	—	2	2	no	—
226	C₂×C₄×C₈₉	—	2	2	no	—
227	C₂×C₄×C₈₉	—	2	4	no	—
228	C₂×C₄×C₈₉	—	2	4	no	—
229	C₂×C₄×C₈₉	—	2	16	no	—
230	C₂×C₄×C₈₉	—	2	4	no	—
231	C₂×C₄×C₈₉	—	2	4	no	—
232	C₂×C₄×C₈₉	—	2	4	no	—
233	C₂×C₄×C₈₉	—	2	4	no	—
234	C₂×C₄×C₈₉	—	2	4	no	—
235	C₂×C₄×C₈₉	—	2	4	no	—
236	C₂×C₄×C₈₉	—	2	8	no	—
237	C₂×C₄×C₈₉	—	2	8	no	—
238	C₂×C₄×C₈₉	—	2	8	no	—
239	C₂×C₄×C₈₉	—	4	8	no	—
240	C₂×C₄×C₈₉	—	4	16	no	—
241	C₂×C₄×C₈₉	—	4	8	no	—
242	C₂×C₄×C₈₉	#121	1	16	yes	—
243	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
244	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
245	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
246	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
247	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
248	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
249	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
250	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
251	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
252	C₂²×C₈₉	—	2	4	no	—
253	C₂²×C₈₉	—	2	4	no	—
254	C₂²×C₈₉	—	2	4	no	—
255	C₂²×C₈₉	—	2	4	no	—
256	C₂²×C₈₉	—	2	8	no	—
257	C₂²×C₈₉	—	4	4	no	—
258	C₂²×C₈₉	—	4	4	no	—
259	C₂²×C₈₉	—	4	4	no	—
260	C₂²×C₈₉	—	4	8	no	—
261	C₂²×C₈₉	—	4	8	no	—
262	C₂²×C₈₉	—	4	8	no	—
263	C₂²×C₈₉	—	8	16	no	—
264	C₂²×C₈₉	C₂×#121	1	4	yes	—
265	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
266	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
267	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
268	C₂²×C₈₉	#585	1	4	yes	—
269	C₄×C₈₉	—	2	4	no	—
270	C₄×C₈₉	—	2	4	no	—
271	C₄×C₈₉	—	2	4	no	—
272	C₄×C₈₉	—	4	4	no	—
273	C₄×C₈₉	—	4	4	no	—
274	C₄×C₈₉	—	4	4	no	—
275	C₄×C₈₉	—	4	4	no	—
276	C₄×C₈₉	C₂×#121	1	4	yes	—
277	C₄×C₈₉	C₂×#121	1	4	yes	—
278	#35327	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
279	C₂×C₈₉	—	2	4	no	—
280	C₂×C₈₉	—	4	4	no	—
281	C₂×C₈₉	—	8	4	no	—
282	C₂×C₈₉	C₂²×#121	1	1	yes	characteristic
283	C₂×C₈₉	#3449	1	2	yes	—
284	C₂×C₈₉	#3500	1	2	yes	—
285	C₂×C₈₉	#3510	1	2	yes	—
286	C₂×#536	C₂×C₈₉	1	1	yes	characteristic, complemented
287	C₂×#535	C₂×C₈₉	1	1	yes	characteristic
288	C₂²×#124	C₂×C₈₉	1	1	yes	characteristic, complemented
289	C₂×C₄×#41	C₂×C₈₉	1	1	yes	characteristic, complemented
290	#2850	C₂×C₈₉	1	4	yes	complemented
291	#2620	C₂×C₈₉	1	4	yes	complemented
292	#2545	C₂×C₈₉	1	4	yes	complemented
293	#2544	C₂×C₈₉	1	4	yes	complemented
294	#2534	C₂×C₈₉	1	8	yes	complemented
295	#2310	C₂×C₈₉	1	1	yes	characteristic, complemented
296	#2309	C₂×C₈₉	1	1	yes	characteristic, complemented
297	#2285	C₂×C₈₉	1	1	yes	characteristic, complemented
298	C₈₉	#35327	1	1	yes	characteristic, direct product factor, Sylow
299	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
300	C₂³×Q₈	C₂²×C₈₉	1	1	yes	characteristic
301	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
302	#536	—	2	8	no	—
303	#535	—	2	8	no	—
304	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	complemented
305	C₂×#127	C₂²×C₈₉	1	2	yes	—
306	C₂×#124	—	2	8	no	—
307	C₂×#124	C₂²×C₈₉	1	2	yes	—
308	C₂×#124	C₂²×C₈₉	1	2	yes	—
309	C₂×#124	C₂²×C₈₉	1	4	yes	—
310	C₂×#124	C₂²×C₈₉	1	2	yes	—
311	C₂×#124	C₂²×C₈₉	1	2	yes	complemented
312	C₂×#123	C₂²×C₈₉	1	4	yes	—
313	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	4	yes	complemented
314	C₂×#121	C₂²×C₈₉	1	4	yes	—
315	C₂×#120	C₂²×C₈₉	1	4	yes	—
316	C₄×#41	—	2	8	no	—
317	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	complemented
318	C₂×C₄×D₄	C₂²×C₈₉	1	4	yes	—
319	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	complemented
320	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
321	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
322	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
323	C₂×#117	C₂²×C₈₉	1	2	yes	complemented
324	C₂²×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
325	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
326	C₂²×C₄²	C₂²×C₈₉	1	2	yes	complemented
327	#396	C₂²×C₈₉	1	16	yes	complemented
328	#394	C₂²×C₈₉	1	16	yes	complemented
329	#391	C₂²×C₈₉	1	4	yes	complemented
330	#391	C₂²×C₈₉	1	4	yes	complemented
331	#390	C₂²×C₈₉	1	4	yes	—
332	#390	C₂²×C₈₉	1	4	yes	—
333	#388	C₂²×C₈₉	1	16	yes	complemented
334	#386	C₂²×C₈₉	1	16	yes	complemented
335	#384	C₂²×C₈₉	1	2	yes	complemented
336	#382	C₂²×C₈₉	1	2	yes	complemented
337	#381	C₂²×C₈₉	1	4	yes	—
338	#380	C₂²×C₈₉	1	4	yes	—
339	#379	C₂²×C₈₉	1	4	yes	—
340	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	complemented
341	C₂⁵	C₂³×C₈₉	1	2	yes	—
342	C₂×#41	—	2	4	no	—
343	C₂×#41	—	2	4	no	—
344	C₂×#41	D₄×C₈₉	1	8	yes	—
345	C₂²×Q₈	—	2	4	no	—
346	C₂²×Q₈	—	2	4	no	—
347	C₂²×Q₈	—	2	4	no	—
348	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
349	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
350	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
351	C₂²×D₄	—	2	4	no	—
352	C₂²×D₄	—	2	4	no	—
353	C₂²×D₄	—	2	4	no	—
354	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
355	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
356	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
357	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
358	C₂²×D₄	C₂³×C₈₉	1	8	yes	—
359	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
360	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
361	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
362	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
363	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
364	C₄⋊Q₈	—	2	8	no	—
365	#127	—	2	8	no	—
366	#124	—	2	8	no	—
367	#124	—	2	8	no	—
368	#124	—	2	8	no	—
369	#124	—	2	8	no	—
370	#124	—	2	16	no	—
371	#123	—	2	16	no	—
372	C₂²⋊Q₈	—	2	16	no	—
373	#121	—	2	16	no	—
374	#120	—	2	16	no	—
375	C₄×Q₈	—	2	8	no	—
376	C₄×D₄	—	2	16	no	—
377	C₄×D₄	—	2	8	no	—
378	#117	—	2	4	no	—
379	#117	—	2	4	no	—
380	#117	—	2	4	no	—
381	#117	—	2	8	no	—
382	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
383	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
384	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
385	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
386	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
387	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
388	C₂×C₂²⋊C₄	—	2	16	no	—
389	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
390	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
391	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
392	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
393	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
394	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
395	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
396	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
397	C₂×C₂²⋊C₄	C₂³×C₈₉	1	16	yes	—
398	C₂×C₄²	—	2	8	no	—
399	C₂×C₄²	D₄×C₈₉	1	8	yes	complemented
400	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
401	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
402	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
403	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
404	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
405	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
406	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
407	#95	C₂³×C₈₉	1	8	yes	—
408	#95	C₂³×C₈₉	1	16	yes	—
409	C₂⁴	—	2	4	no	—
410	C₂⁴	—	2	4	no	—
411	C₂⁴	—	2	4	no	—
412	C₂⁴	—	4	8	no	—
413	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
414	C₂⁴	C₂⁴×C₈₉	1	2	yes	—
415	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
416	#41	—	2	32	no	—
417	C₂×Q₈	—	2	2	no	—
418	C₂×Q₈	—	2	4	no	—
419	C₂×Q₈	—	2	4	no	—
420	C₂×Q₈	—	2	4	no	—
421	C₂×Q₈	—	2	4	no	—
422	C₂×Q₈	—	2	2	no	—
423	C₂×Q₈	—	2	4	no	—
424	C₂×Q₈	—	2	4	no	—
425	C₂×Q₈	—	4	16	no	—
426	C₂×Q₈	C₂×D₄×C₈₉	1	4	yes	—
427	C₂×D₄	—	2	4	no	—
428	C₂×D₄	—	2	4	no	—
429	C₂×D₄	—	2	16	no	—
430	C₂×D₄	—	2	2	no	—
431	C₂×D₄	—	2	4	no	—
432	C₂×D₄	—	2	4	no	—
433	C₂×D₄	—	2	4	no	—
434	C₂×D₄	—	2	4	no	—
435	C₂×D₄	—	2	2	no	—
436	C₂×D₄	—	2	4	no	—
437	C₂×D₄	—	2	4	no	—
438	C₂×D₄	—	4	16	no	—
439	C₂×D₄	—	4	16	no	—
440	C₂×D₄	C₂×D₄×C₈₉	1	8	yes	—
441	C₂×D₄	C₂×D₄×C₈₉	1	4	yes	—
442	C₂²×C₄	—	2	4	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	2	4	no	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	2	4	no	—
447	C₂²×C₄	—	2	8	no	—
448	C₂²×C₄	—	4	4	no	—
449	C₂²×C₄	—	4	8	no	—
450	C₂²×C₄	C₂×D₄×C₈₉	1	8	yes	—
451	C₂²×C₄	C₂×D₄×C₈₉	1	8	yes	—
452	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
453	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
454	C₂²×C₄	#41×C₈₉	1	4	yes	—
455	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
456	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
457	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
458	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
459	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
460	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
461	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
462	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
463	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
464	C₄⋊C₄	—	2	8	no	—
465	C₄⋊C₄	—	2	4	no	—
466	C₄⋊C₄	—	2	4	no	—
467	C₄⋊C₄	—	2	4	no	—
468	C₄⋊C₄	—	2	16	no	—
469	C₄⋊C₄	—	4	8	no	—
470	C₂²⋊C₄	—	2	4	no	—
471	C₂²⋊C₄	—	2	4	no	—
472	C₂²⋊C₄	—	2	4	no	—
473	C₂²⋊C₄	—	2	16	no	—
474	C₂²⋊C₄	—	2	16	no	—
475	C₂²⋊C₄	—	2	16	no	—
476	C₂²⋊C₄	—	2	8	no	—
477	C₂²⋊C₄	—	2	16	no	—
478	C₂²⋊C₄	—	4	16	no	—
479	C₄²	—	2	4	no	—
480	C₄²	—	2	8	no	—
481	C₄²	—	2	4	no	—
482	C₄²	—	2	4	no	—
483	C₄²	C₂×D₄×C₈₉	1	4	yes	—
484	C₄²	C₂×D₄×C₈₉	1	4	yes	—
485	C₄²	C₂×D₄×C₈₉	1	8	yes	—
486	C₂³	—	2	4	no	—
487	C₂³	—	2	2	no	—
488	C₂³	—	2	2	no	—
489	C₂³	—	2	8	no	—
490	C₂³	—	2	4	no	—
491	C₂³	—	2	4	no	—
492	C₂³	—	2	4	no	—
493	C₂³	—	2	16	no	—
494	C₂³	—	4	16	no	—
495	C₂³	—	4	8	no	—
496	C₂³	—	8	8	no	—
497	C₂³	—	8	8	no	—
498	C₂³	C₂²×D₄×C₈₉	1	4	yes	—
499	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
500	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
501	C₂³	C₂×#41×C₈₉	1	2	yes	—
502	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
503	Q₈	—	2	8	no	—
504	Q₈	—	4	8	no	—
505	Q₈	—	4	8	no	—
506	D₄	—	2	16	no	—
507	D₄	—	2	8	no	—
508	D₄	—	4	8	no	—
509	D₄	—	4	8	no	—
510	D₄	—	8	16	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	16	no	—
516	C₂×C₄	—	2	4	no	—
517	C₂×C₄	—	2	4	no	—
518	C₂×C₄	—	2	4	no	—
519	C₂×C₄	—	2	4	no	—
520	C₂×C₄	—	2	4	no	—
521	C₂×C₄	—	2	4	no	—
522	C₂×C₄	—	2	8	no	—
523	C₂×C₄	—	2	8	no	—
524	C₂×C₄	—	2	8	no	—
525	C₂×C₄	—	4	8	no	—
526	C₂×C₄	—	4	16	no	—
527	C₂×C₄	—	4	8	no	—
528	C₂×C₄	#121×C₈₉	1	16	yes	—
529	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
530	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
531	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
532	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
533	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
534	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
535	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
536	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
537	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
538	C₂²	—	2	4	no	—
539	C₂²	—	2	4	no	—
540	C₂²	—	2	4	no	—
541	C₂²	—	2	4	no	—
542	C₂²	—	2	8	no	—
543	C₂²	—	4	4	no	—
544	C₂²	—	4	4	no	—
545	C₂²	—	4	4	no	—
546	C₂²	—	4	8	no	—
547	C₂²	—	4	8	no	—
548	C₂²	—	4	8	no	—
549	C₂²	—	8	16	no	—
550	C₂²	C₂×#121×C₈₉	1	4	yes	—
551	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
552	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
553	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
554	C₂²	#585×C₈₉	1	4	yes	—
555	C₄	—	2	4	no	—
556	C₄	—	2	4	no	—
557	C₄	—	2	4	no	—
558	C₄	—	4	4	no	—
559	C₄	—	4	4	no	—
560	C₄	—	4	4	no	—
561	C₄	—	4	4	no	—
562	C₄	C₂×#121×C₈₉	1	4	yes	—
563	C₄	C₂×#121×C₈₉	1	4	yes	—
564	C₂	—	2	4	no	—
565	C₂	—	4	4	no	—
566	C₂	—	8	4	no	—
567	C₂	C₂²×#121×C₈₉	1	1	yes	characteristic
568	C₂	C₈₉×#3449	1	2	yes	—
569	C₂	C₈₉×#3500	1	2	yes	—
570	C₂	C₈₉×#3510	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂×#535×C₈₉	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#536×C₈₉	C₂	semidirect
3. C₂²×#124×C₈₉	C₂	semidirect
4. C₂×C₄×#41×C₈₉	C₂	semidirect
5. C₈₉×#2850	C₂	semidirect
6. C₈₉×#2620	C₂	semidirect
7. C₈₉×#2545	C₂	semidirect
8. C₈₉×#2544	C₂	semidirect
9. C₈₉×#2534	C₂	semidirect
10. C₈₉×#2310	C₂	semidirect
11. C₈₉×#2309	C₂	semidirect
12. C₈₉×#2285	C₂	semidirect
18. C₂×C₄⋊Q₈×C₈₉	C₂²	semidirect
25. C₂×#124×C₈₉	C₂²	semidirect
27. C₂×C₂²⋊Q₈×C₈₉	C₂²	semidirect
31. C₂×C₄×Q₈×C₈₉	C₂²	semidirect
33. C₂×C₄×D₄×C₈₉	C₂²	semidirect
37. C₂×#117×C₈₉	C₂²	semidirect
40. C₂²×C₄²×C₈₉	C₂²	semidirect
41. #396×C₈₉	C₂²	semidirect
42. #394×C₈₉	C₂²	semidirect
43. #391×C₈₉	C₂²	semidirect
44. #391×C₈₉	C₂²	semidirect
47. #388×C₈₉	C₂²	semidirect
48. #386×C₈₉	C₂²	semidirect
49. #384×C₈₉	C₂²	semidirect
50. #382×C₈₉	C₂²	semidirect
54. C₄×C₄⋊C₄×C₈₉	C₂²	semidirect
113. C₂×C₄²×C₈₉	D₄	semidirect
278. #35327	C₈₉	direct (N × H)
286. C₂×#536	C₂×C₈₉	semidirect
288. C₂²×#124	C₂×C₈₉	semidirect
289. C₂×C₄×#41	C₂×C₈₉	semidirect
290. #2850	C₂×C₈₉	semidirect
291. #2620	C₂×C₈₉	semidirect
292. #2545	C₂×C₈₉	semidirect
293. #2544	C₂×C₈₉	semidirect
294. #2534	C₂×C₈₉	semidirect
295. #2310	C₂×C₈₉	semidirect
296. #2309	C₂×C₈₉	semidirect
297. #2285	C₂×C₈₉	semidirect
298. C₈₉	#35327	direct (N × H)
304. C₂×C₄⋊Q₈	C₂²×C₈₉	semidirect
311. C₂×#124	C₂²×C₈₉	semidirect
313. C₂×C₂²⋊Q₈	C₂²×C₈₉	semidirect
317. C₂×C₄×Q₈	C₂²×C₈₉	semidirect
319. C₂×C₄×D₄	C₂²×C₈₉	semidirect
323. C₂×#117	C₂²×C₈₉	semidirect
326. C₂²×C₄²	C₂²×C₈₉	semidirect
327. #396	C₂²×C₈₉	semidirect
328. #394	C₂²×C₈₉	semidirect
329. #391	C₂²×C₈₉	semidirect
330. #391	C₂²×C₈₉	semidirect
333. #388	C₂²×C₈₉	semidirect
334. #386	C₂²×C₈₉	semidirect
335. #384	C₂²×C₈₉	semidirect
336. #382	C₂²×C₈₉	semidirect
340. C₄×C₄⋊C₄	C₂²×C₈₉	semidirect
399. C₂×C₄²	D₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#536×C₈₉	2
C₂×#535×C₈₉	2
C₂²×#124×C₈₉	2
C₂×C₄×#41×C₈₉	2
C₈₉×#2850	2
C₈₉×#2620	2
C₈₉×#2545	2
C₈₉×#2544	2
C₈₉×#2534	2
C₈₉×#2310	2
C₈₉×#2309	2
C₈₉×#2285	2
#35327	89

Subgroups — C89×#35327 #36413175

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35327 #36413175