Subgroups — C₈₉×#60682 #36438530

Order 22784.

1878 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3455	C₂	1	2	yes	maximal, complemented
2	C₈₉×#3177	C₂	1	2	yes	maximal, complemented
3	C₈₉×#3176	C₂	1	2	yes	maximal, complemented
4	C₈₉×#3175	C₂	1	4	yes	maximal, complemented
5	C₈₉×#3174	C₂	1	4	yes	maximal, complemented
6	C₈₉×#3173	C₂	1	4	yes	maximal, complemented
7	C₈₉×#3160	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#3147	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#3135	C₂	1	2	yes	maximal, complemented
10	C₈₉×#3054	C₂	1	2	yes	maximal, complemented
11	C₈₉×#3050	C₂	1	2	yes	maximal, complemented
12	C₈₉×#2943	C₂	1	2	yes	maximal, complemented
13	C₄×#135×C₈₉	C₂	1	2	yes	maximal, complemented
14	C₂×#405×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×#135×C₈₉	C₂²	1	2	yes	—
16	C₂×#135×C₈₉	C₂²	1	2	yes	—
17	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
19	#542×C₈₉	C₂²	1	8	yes	—
20	#541×C₈₉	C₂²	1	4	yes	—
21	#540×C₈₉	C₂²	1	4	yes	—
22	#539×C₈₉	—	2	4	no	—
23	#538×C₈₉	—	2	4	no	—
24	#533×C₈₉	C₂²	1	2	yes	—
25	#532×C₈₉	C₂²	1	2	yes	—
26	#529×C₈₉	—	2	8	no	—
27	#525×C₈₉	C₂²	1	4	yes	—
28	C₄×#41×C₈₉	C₂²	1	4	yes	—
29	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
30	#500×C₈₉	C₂²	1	2	yes	complemented
31	#499×C₈₉	C₂²	1	2	yes	complemented
32	#499×C₈₉	C₂²	1	2	yes	complemented
33	#498×C₈₉	C₂²	1	2	yes	complemented
34	#495×C₈₉	C₂²	1	2	yes	complemented
35	#495×C₈₉	C₂²	1	2	yes	complemented
36	#494×C₈₉	C₂²	1	1	yes	characteristic, complemented
37	#493×C₈₉	C₂²	1	1	yes	characteristic, complemented
38	#492×C₈₉	C₂²	1	2	yes	complemented
39	#491×C₈₉	C₂²	1	4	yes	complemented
40	#490×C₈₉	C₂²	1	4	yes	complemented
41	#489×C₈₉	C₂²	1	8	yes	complemented
42	#470×C₈₉	C₂²	1	4	yes	—
43	#469×C₈₉	C₂²	1	4	yes	—
44	#468×C₈₉	C₂²	1	8	yes	—
45	#467×C₈₉	C₂²	1	4	yes	—
46	#466×C₈₉	C₂²	1	4	yes	—
47	#465×C₈₉	C₂²	1	8	yes	—
48	#442×C₈₉	C₂²	1	4	yes	complemented
49	#441×C₈₉	C₂²	1	4	yes	complemented
50	#440×C₈₉	C₂²	1	8	yes	complemented
51	C₄×Q₁₆×C₈₉	C₂²	1	4	yes	complemented
52	C₄×SD₁₆×C₈₉	C₂²	1	8	yes	complemented
53	C₄×D₈×C₈₉	C₂²	1	4	yes	complemented
54	#427×C₈₉	C₂²	1	2	yes	—
55	#427×C₈₉	C₂²	1	2	yes	—
56	#416×C₈₉	C₂²	1	4	yes	—
57	#416×C₈₉	C₂²	1	4	yes	—
58	#405×C₈₉	C₂²	1	2	yes	—
59	#405×C₈₉	C₂²	1	2	yes	—
60	#405×C₈₉	C₂²	1	2	yes	—
61	#405×C₈₉	C₂²	1	2	yes	—
62	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
63	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
64	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
65	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
66	C₂×#41×C₈₉	C₂³	1	4	yes	—
67	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
68	#135×C₈₉	—	2	8	no	—
69	#135×C₈₉	—	2	8	no	—
70	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
71	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
72	C₂×SD₁₆×C₈₉	C₂³	1	4	yes	—
73	C₂×SD₁₆×C₈₉	C₂³	1	4	yes	—
74	C₂×D₈×C₈₉	C₂³	1	2	yes	—
75	C₂×D₈×C₈₉	C₂³	1	2	yes	—
76	C₂×M₁₆×C₈₉	—	2	4	no	—
77	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
78	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
79	C₂²×C₈×C₈₉	—	2	4	no	—
80	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
83	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
84	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
85	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
86	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
87	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
88	#127×C₈₉	C₂³	1	2	yes	—
89	#126×C₈₉	—	2	8	no	—
90	#126×C₈₉	—	2	8	no	—
91	#125×C₈₉	—	2	8	no	—
92	#125×C₈₉	C₂³	1	4	yes	—
93	#124×C₈₉	—	2	8	no	—
94	#124×C₈₉	C₂³	1	4	yes	—
95	#123×C₈₉	—	2	4	no	—
96	#123×C₈₉	—	2	8	no	—
97	#123×C₈₉	—	2	8	no	—
98	C₂²⋊Q₈×C₈₉	—	2	4	no	—
99	C₂²⋊Q₈×C₈₉	—	2	8	no	—
100	C₂²⋊Q₈×C₈₉	C₂³	1	8	yes	—
101	#121×C₈₉	—	2	4	no	—
102	#121×C₈₉	—	2	8	no	—
103	#121×C₈₉	C₂³	1	8	yes	—
104	#120×C₈₉	—	2	4	no	—
105	C₄×Q₈×C₈₉	—	2	4	no	—
106	C₄×Q₈×C₈₉	C₂³	1	8	yes	—
107	C₄×D₄×C₈₉	—	2	8	no	—
108	C₄×D₄×C₈₉	—	2	8	no	—
109	C₄×D₄×C₈₉	—	2	4	no	—
110	C₄×D₄×C₈₉	C₂³	1	8	yes	—
111	#117×C₈₉	—	2	4	no	—
112	#117×C₈₉	D₄	1	4	yes	—
113	#117×C₈₉	C₂³	1	4	yes	—
114	#117×C₈₉	C₂³	1	2	yes	—
115	#117×C₈₉	C₂³	1	2	yes	—
116	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
117	C₂×C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
118	C₂×C₄²×C₈₉	—	2	4	no	—
119	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
121	#107×C₈₉	C₂³	1	4	yes	—
122	#107×C₈₉	C₂³	1	4	yes	—
123	#106×C₈₉	C₂³	1	4	yes	—
124	#106×C₈₉	C₂³	1	4	yes	—
125	#103×C₈₉	C₂³	1	8	yes	—
126	#103×C₈₉	C₂³	1	8	yes	—
127	#102×C₈₉	C₂³	1	8	yes	—
128	#102×C₈₉	C₂³	1	8	yes	—
129	#97×C₈₉	C₂³	1	2	yes	—
130	#97×C₈₉	C₂³	1	2	yes	—
131	#97×C₈₉	C₂³	1	2	yes	—
132	#97×C₈₉	C₂³	1	2	yes	—
133	C₄×C₈×C₈₉	C₂³	1	2	yes	—
134	C₄×C₈×C₈₉	C₂³	1	2	yes	—
135	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
136	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
137	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
138	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
139	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
140	#41×C₈₉	—	2	8	no	—
141	#41×C₈₉	—	2	8	no	—
142	C₂×Q₈×C₈₉	—	2	4	no	—
143	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
144	C₂×D₄×C₈₉	—	2	4	no	—
145	C₂×D₄×C₈₉	—	2	8	no	—
146	C₂×D₄×C₈₉	—	2	4	no	—
147	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
148	C₂²×C₄×C₈₉	—	2	2	no	—
149	C₂²×C₄×C₈₉	—	2	2	no	—
150	C₂²×C₄×C₈₉	—	2	4	no	—
151	C₂²×C₄×C₈₉	—	2	8	no	—
152	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
153	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
154	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
155	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
156	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
157	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
158	Q₁₆×C₈₉	—	2	4	no	—
159	Q₁₆×C₈₉	—	2	4	no	—
160	SD₁₆×C₈₉	—	2	8	no	—
161	SD₁₆×C₈₉	—	2	8	no	—
162	D₈×C₈₉	—	2	4	no	—
163	D₈×C₈₉	—	2	4	no	—
164	M₁₆×C₈₉	—	2	4	no	—
165	M₁₆×C₈₉	—	2	4	no	—
166	C₂×C₈×C₈₉	—	2	2	no	—
167	C₂×C₈×C₈₉	—	2	2	no	—
168	C₂×C₈×C₈₉	—	2	4	no	—
169	C₂×C₈×C₈₉	—	2	4	no	—
170	C₂×C₈×C₈₉	#41	1	4	yes	—
171	C₂×C₈×C₈₉	#41	1	4	yes	—
172	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
173	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
174	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
175	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
176	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
177	C₂×C₈×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	8	no	—
181	C₄⋊C₄×C₈₉	—	2	8	no	—
182	C₄⋊C₄×C₈₉	—	2	4	no	—
183	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
184	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
185	C₄⋊C₄×C₈₉	C₂⁴	1	8	yes	—
186	C₂²⋊C₄×C₈₉	—	2	4	no	—
187	C₂²⋊C₄×C₈₉	—	2	8	no	—
188	C₂²⋊C₄×C₈₉	—	2	8	no	—
189	C₂²⋊C₄×C₈₉	—	2	8	no	—
190	C₂²⋊C₄×C₈₉	—	2	8	no	—
191	C₂²⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
192	C₂²⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
193	C₄²×C₈₉	—	2	8	no	—
194	C₄²×C₈₉	—	2	4	no	—
195	C₄²×C₈₉	C₂×D₄	1	2	yes	—
196	C₄²×C₈₉	C₂×D₄	1	2	yes	—
197	C₄²×C₈₉	C₂⁴	1	2	yes	—
198	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
199	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
200	C₂³×C₈₉	—	2	2	no	—
201	C₂³×C₈₉	—	2	4	no	—
202	C₂³×C₈₉	—	4	2	no	—
203	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
204	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
205	Q₈×C₈₉	—	2	4	no	—
206	Q₈×C₈₉	—	2	4	no	—
207	Q₈×C₈₉	—	8	4	no	—
208	D₄×C₈₉	—	2	4	no	—
209	D₄×C₈₉	—	2	4	no	—
210	D₄×C₈₉	—	4	8	no	—
211	D₄×C₈₉	—	8	4	no	—
212	D₄×C₈₉	—	8	8	no	—
213	C₂×C₄×C₈₉	—	2	2	no	—
214	C₂×C₄×C₈₉	—	2	4	no	—
215	C₂×C₄×C₈₉	—	2	2	no	—
216	C₂×C₄×C₈₉	—	2	2	no	—
217	C₂×C₄×C₈₉	—	2	4	no	—
218	C₂×C₄×C₈₉	—	2	8	no	—
219	C₂×C₄×C₈₉	—	4	2	no	—
220	C₂×C₄×C₈₉	—	4	2	no	—
221	C₂×C₄×C₈₉	—	4	8	no	—
222	C₂×C₄×C₈₉	—	8	8	no	—
223	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
224	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
225	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
226	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
227	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
228	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
229	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
230	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
231	C₈×C₈₉	—	2	2	no	—
232	C₈×C₈₉	—	2	2	no	—
233	C₈×C₈₉	C₂×#41	1	2	yes	—
234	C₈×C₈₉	C₂×#41	1	2	yes	—
235	C₈×C₈₉	C₂×#41	1	2	yes	—
236	C₈×C₈₉	C₂×#41	1	2	yes	—
237	C₂²×C₈₉	—	2	2	no	—
238	C₂²×C₈₉	—	2	4	no	—
239	C₂²×C₈₉	—	2	2	no	—
240	C₂²×C₈₉	—	4	4	no	—
241	C₂²×C₈₉	—	4	8	no	—
242	C₂²×C₈₉	—	4	4	no	—
243	C₂²×C₈₉	#532	1	2	yes	—
244	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
245	C₄×C₈₉	—	2	4	no	—
246	C₄×C₈₉	—	2	2	no	—
247	C₄×C₈₉	—	2	2	no	—
248	C₄×C₈₉	—	2	2	no	—
249	C₄×C₈₉	—	4	8	no	—
250	C₄×C₈₉	—	4	4	no	—
251	C₄×C₈₉	#532	1	2	yes	—
252	C₄×C₈₉	C₂²×#41	1	1	yes	characteristic
253	C₄×C₈₉	C₂²×#41	1	1	yes	characteristic
254	#60682	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
255	C₂×C₈₉	—	2	2	no	—
256	C₂×C₈₉	—	4	2	no	—
257	C₂×C₈₉	—	8	4	no	—
258	C₂×C₈₉	C₂×#532	1	1	yes	characteristic
259	C₂×C₈₉	#3584	1	2	yes	—
260	#3455	C₂×C₈₉	1	2	yes	complemented
261	#3177	C₂×C₈₉	1	2	yes	complemented
262	#3176	C₂×C₈₉	1	2	yes	complemented
263	#3175	C₂×C₈₉	1	4	yes	complemented
264	#3174	C₂×C₈₉	1	4	yes	complemented
265	#3173	C₂×C₈₉	1	4	yes	complemented
266	#3160	C₂×C₈₉	1	1	yes	characteristic, complemented
267	#3147	C₂×C₈₉	1	1	yes	characteristic, complemented
268	#3135	C₂×C₈₉	1	2	yes	complemented
269	#3054	C₂×C₈₉	1	2	yes	complemented
270	#3050	C₂×C₈₉	1	2	yes	complemented
271	#2943	C₂×C₈₉	1	2	yes	complemented
272	C₄×#135	C₂×C₈₉	1	2	yes	complemented
273	C₂×#405	C₂×C₈₉	1	1	yes	characteristic, complemented
274	C₈₉	#60682	1	1	yes	characteristic, direct product factor, Sylow
275	C₂×#135	C₂²×C₈₉	1	2	yes	—
276	C₂×#135	C₂²×C₈₉	1	2	yes	—
277	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
278	C₂³×C₈	C₂²×C₈₉	1	1	yes	characteristic
279	#542	C₂²×C₈₉	1	8	yes	—
280	#541	C₂²×C₈₉	1	4	yes	—
281	#540	C₂²×C₈₉	1	4	yes	—
282	#539	—	2	4	no	—
283	#538	—	2	4	no	—
284	#533	C₂²×C₈₉	1	2	yes	—
285	#532	C₂²×C₈₉	1	2	yes	—
286	#529	—	2	8	no	—
287	#525	C₂²×C₈₉	1	4	yes	—
288	C₄×#41	C₂²×C₈₉	1	4	yes	—
289	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
290	#500	C₂²×C₈₉	1	2	yes	complemented
291	#499	C₂²×C₈₉	1	2	yes	complemented
292	#499	C₂²×C₈₉	1	2	yes	complemented
293	#498	C₂²×C₈₉	1	2	yes	complemented
294	#495	C₂²×C₈₉	1	2	yes	complemented
295	#495	C₂²×C₈₉	1	2	yes	complemented
296	#494	C₂²×C₈₉	1	1	yes	characteristic, complemented
297	#493	C₂²×C₈₉	1	1	yes	characteristic, complemented
298	#492	C₂²×C₈₉	1	2	yes	complemented
299	#491	C₂²×C₈₉	1	4	yes	complemented
300	#490	C₂²×C₈₉	1	4	yes	complemented
301	#489	C₂²×C₈₉	1	8	yes	complemented
302	#470	C₂²×C₈₉	1	4	yes	—
303	#469	C₂²×C₈₉	1	4	yes	—
304	#468	C₂²×C₈₉	1	8	yes	—
305	#467	C₂²×C₈₉	1	4	yes	—
306	#466	C₂²×C₈₉	1	4	yes	—
307	#465	C₂²×C₈₉	1	8	yes	—
308	#442	C₂²×C₈₉	1	4	yes	complemented
309	#441	C₂²×C₈₉	1	4	yes	complemented
310	#440	C₂²×C₈₉	1	8	yes	complemented
311	C₄×Q₁₆	C₂²×C₈₉	1	4	yes	complemented
312	C₄×SD₁₆	C₂²×C₈₉	1	8	yes	complemented
313	C₄×D₈	C₂²×C₈₉	1	4	yes	complemented
314	#427	C₂²×C₈₉	1	2	yes	—
315	#427	C₂²×C₈₉	1	2	yes	—
316	#416	C₂²×C₈₉	1	4	yes	—
317	#416	C₂²×C₈₉	1	4	yes	—
318	#405	C₂²×C₈₉	1	2	yes	—
319	#405	C₂²×C₈₉	1	2	yes	—
320	#405	C₂²×C₈₉	1	2	yes	—
321	#405	C₂²×C₈₉	1	2	yes	—
322	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
323	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
324	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
325	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
326	C₂×#41	C₂³×C₈₉	1	4	yes	—
327	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
328	#135	—	2	8	no	—
329	#135	—	2	8	no	—
330	C₂×Q₁₆	C₂³×C₈₉	1	2	yes	—
331	C₂×Q₁₆	C₂³×C₈₉	1	2	yes	—
332	C₂×SD₁₆	C₂³×C₈₉	1	4	yes	—
333	C₂×SD₁₆	C₂³×C₈₉	1	4	yes	—
334	C₂×D₈	C₂³×C₈₉	1	2	yes	—
335	C₂×D₈	C₂³×C₈₉	1	2	yes	—
336	C₂×M₁₆	—	2	4	no	—
337	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
338	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
339	C₂²×C₈	—	2	4	no	—
340	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
341	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
342	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
343	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
344	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
345	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
346	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
347	C₄⋊Q₈	C₂³×C₈₉	1	4	yes	—
348	#127	C₂³×C₈₉	1	2	yes	—
349	#126	—	2	8	no	—
350	#126	—	2	8	no	—
351	#125	—	2	8	no	—
352	#125	C₂³×C₈₉	1	4	yes	—
353	#124	—	2	8	no	—
354	#124	C₂³×C₈₉	1	4	yes	—
355	#123	—	2	4	no	—
356	#123	—	2	8	no	—
357	#123	—	2	8	no	—
358	C₂²⋊Q₈	—	2	4	no	—
359	C₂²⋊Q₈	—	2	8	no	—
360	C₂²⋊Q₈	C₂³×C₈₉	1	8	yes	—
361	#121	—	2	4	no	—
362	#121	—	2	8	no	—
363	#121	C₂³×C₈₉	1	8	yes	—
364	#120	—	2	4	no	—
365	C₄×Q₈	—	2	4	no	—
366	C₄×Q₈	C₂³×C₈₉	1	8	yes	—
367	C₄×D₄	—	2	8	no	—
368	C₄×D₄	—	2	8	no	—
369	C₄×D₄	—	2	4	no	—
370	C₄×D₄	C₂³×C₈₉	1	8	yes	—
371	#117	—	2	4	no	—
372	#117	D₄×C₈₉	1	4	yes	—
373	#117	C₂³×C₈₉	1	4	yes	—
374	#117	C₂³×C₈₉	1	2	yes	—
375	#117	C₂³×C₈₉	1	2	yes	—
376	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
377	C₂×C₂²⋊C₄	D₄×C₈₉	1	2	yes	—
378	C₂×C₄²	—	2	4	no	—
379	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
380	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
381	#107	C₂³×C₈₉	1	4	yes	—
382	#107	C₂³×C₈₉	1	4	yes	—
383	#106	C₂³×C₈₉	1	4	yes	—
384	#106	C₂³×C₈₉	1	4	yes	—
385	#103	C₂³×C₈₉	1	8	yes	—
386	#103	C₂³×C₈₉	1	8	yes	—
387	#102	C₂³×C₈₉	1	8	yes	—
388	#102	C₂³×C₈₉	1	8	yes	—
389	#97	C₂³×C₈₉	1	2	yes	—
390	#97	C₂³×C₈₉	1	2	yes	—
391	#97	C₂³×C₈₉	1	2	yes	—
392	#97	C₂³×C₈₉	1	2	yes	—
393	C₄×C₈	C₂³×C₈₉	1	2	yes	—
394	C₄×C₈	C₂³×C₈₉	1	2	yes	—
395	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
396	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
397	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
398	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
399	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
400	#41	—	2	8	no	—
401	#41	—	2	8	no	—
402	C₂×Q₈	—	2	4	no	—
403	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
404	C₂×D₄	—	2	4	no	—
405	C₂×D₄	—	2	8	no	—
406	C₂×D₄	—	2	4	no	—
407	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
408	C₂²×C₄	—	2	2	no	—
409	C₂²×C₄	—	2	2	no	—
410	C₂²×C₄	—	2	4	no	—
411	C₂²×C₄	—	2	8	no	—
412	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
413	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
414	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
415	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
416	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
417	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
418	Q₁₆	—	2	4	no	—
419	Q₁₆	—	2	4	no	—
420	SD₁₆	—	2	8	no	—
421	SD₁₆	—	2	8	no	—
422	D₈	—	2	4	no	—
423	D₈	—	2	4	no	—
424	M₁₆	—	2	4	no	—
425	M₁₆	—	2	4	no	—
426	C₂×C₈	—	2	2	no	—
427	C₂×C₈	—	2	2	no	—
428	C₂×C₈	—	2	4	no	—
429	C₂×C₈	—	2	4	no	—
430	C₂×C₈	#41×C₈₉	1	4	yes	—
431	C₂×C₈	#41×C₈₉	1	4	yes	—
432	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
433	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
434	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
435	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
436	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
437	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
438	C₄⋊C₄	—	2	8	no	—
439	C₄⋊C₄	—	2	4	no	—
440	C₄⋊C₄	—	2	8	no	—
441	C₄⋊C₄	—	2	8	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
444	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
445	C₄⋊C₄	C₂⁴×C₈₉	1	8	yes	—
446	C₂²⋊C₄	—	2	4	no	—
447	C₂²⋊C₄	—	2	8	no	—
448	C₂²⋊C₄	—	2	8	no	—
449	C₂²⋊C₄	—	2	8	no	—
450	C₂²⋊C₄	—	2	8	no	—
451	C₂²⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
452	C₂²⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
453	C₄²	—	2	4	no	—
454	C₄²	—	2	8	no	—
455	C₄²	C₂×D₄×C₈₉	1	2	yes	—
456	C₄²	C₂×D₄×C₈₉	1	2	yes	—
457	C₄²	C₂⁴×C₈₉	1	2	yes	—
458	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
459	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
460	C₂³	—	2	2	no	—
461	C₂³	—	2	4	no	—
462	C₂³	—	4	2	no	—
463	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
464	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
465	Q₈	—	2	4	no	—
466	Q₈	—	2	4	no	—
467	Q₈	—	8	4	no	—
468	D₄	—	2	4	no	—
469	D₄	—	2	4	no	—
470	D₄	—	4	8	no	—
471	D₄	—	8	4	no	—
472	D₄	—	8	8	no	—
473	C₂×C₄	—	2	2	no	—
474	C₂×C₄	—	2	4	no	—
475	C₂×C₄	—	2	2	no	—
476	C₂×C₄	—	2	2	no	—
477	C₂×C₄	—	2	4	no	—
478	C₂×C₄	—	2	8	no	—
479	C₂×C₄	—	4	2	no	—
480	C₂×C₄	—	4	2	no	—
481	C₂×C₄	—	4	8	no	—
482	C₂×C₄	—	8	8	no	—
483	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
484	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
485	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
486	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
487	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
488	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
489	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
490	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
491	C₈	—	2	2	no	—
492	C₈	—	2	2	no	—
493	C₈	C₂×#41×C₈₉	1	2	yes	—
494	C₈	C₂×#41×C₈₉	1	2	yes	—
495	C₈	C₂×#41×C₈₉	1	2	yes	—
496	C₈	C₂×#41×C₈₉	1	2	yes	—
497	C₂²	—	2	2	no	—
498	C₂²	—	2	4	no	—
499	C₂²	—	2	2	no	—
500	C₂²	—	4	4	no	—
501	C₂²	—	4	8	no	—
502	C₂²	—	4	4	no	—
503	C₂²	#532×C₈₉	1	2	yes	—
504	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
505	C₄	—	2	4	no	—
506	C₄	—	2	2	no	—
507	C₄	—	2	2	no	—
508	C₄	—	2	2	no	—
509	C₄	—	4	8	no	—
510	C₄	—	4	4	no	—
511	C₄	#532×C₈₉	1	2	yes	—
512	C₄	C₂²×#41×C₈₉	1	1	yes	characteristic
513	C₄	C₂²×#41×C₈₉	1	1	yes	characteristic
514	C₂	—	2	2	no	—
515	C₂	—	4	2	no	—
516	C₂	—	8	4	no	—
517	C₂	C₂×#532×C₈₉	1	1	yes	characteristic
518	C₂	C₈₉×#3584	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
socle	C₂²×C₈₉	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈₉ ◁ C₂×#117×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#3455	C₂	semidirect
2. C₈₉×#3177	C₂	semidirect
3. C₈₉×#3176	C₂	semidirect
4. C₈₉×#3175	C₂	semidirect
5. C₈₉×#3174	C₂	semidirect
6. C₈₉×#3173	C₂	semidirect
7. C₈₉×#3160	C₂	semidirect
8. C₈₉×#3147	C₂	semidirect
9. C₈₉×#3135	C₂	semidirect
10. C₈₉×#3054	C₂	semidirect
11. C₈₉×#3050	C₂	semidirect
12. C₈₉×#2943	C₂	semidirect
13. C₄×#135×C₈₉	C₂	semidirect
14. C₂×#405×C₈₉	C₂	semidirect
30. #500×C₈₉	C₂²	semidirect
31. #499×C₈₉	C₂²	semidirect
32. #499×C₈₉	C₂²	semidirect
33. #498×C₈₉	C₂²	semidirect
34. #495×C₈₉	C₂²	semidirect
35. #495×C₈₉	C₂²	semidirect
36. #494×C₈₉	C₂²	semidirect
37. #493×C₈₉	C₂²	semidirect
38. #492×C₈₉	C₂²	semidirect
39. #491×C₈₉	C₂²	semidirect
40. #490×C₈₉	C₂²	semidirect
41. #489×C₈₉	C₂²	semidirect
48. #442×C₈₉	C₂²	semidirect
49. #441×C₈₉	C₂²	semidirect
50. #440×C₈₉	C₂²	semidirect
51. C₄×Q₁₆×C₈₉	C₂²	semidirect
52. C₄×SD₁₆×C₈₉	C₂²	semidirect
53. C₄×D₈×C₈₉	C₂²	semidirect
254. #60682	C₈₉	direct (N × H)
260. #3455	C₂×C₈₉	semidirect
261. #3177	C₂×C₈₉	semidirect
262. #3176	C₂×C₈₉	semidirect
263. #3175	C₂×C₈₉	semidirect
264. #3174	C₂×C₈₉	semidirect
265. #3173	C₂×C₈₉	semidirect
266. #3160	C₂×C₈₉	semidirect
267. #3147	C₂×C₈₉	semidirect
268. #3135	C₂×C₈₉	semidirect
269. #3054	C₂×C₈₉	semidirect
270. #3050	C₂×C₈₉	semidirect
271. #2943	C₂×C₈₉	semidirect
272. C₄×#135	C₂×C₈₉	semidirect
273. C₂×#405	C₂×C₈₉	semidirect
274. C₈₉	#60682	direct (N × H)
290. #500	C₂²×C₈₉	semidirect
291. #499	C₂²×C₈₉	semidirect
292. #499	C₂²×C₈₉	semidirect
293. #498	C₂²×C₈₉	semidirect
294. #495	C₂²×C₈₉	semidirect
295. #495	C₂²×C₈₉	semidirect
296. #494	C₂²×C₈₉	semidirect
297. #493	C₂²×C₈₉	semidirect
298. #492	C₂²×C₈₉	semidirect
299. #491	C₂²×C₈₉	semidirect
300. #490	C₂²×C₈₉	semidirect
301. #489	C₂²×C₈₉	semidirect
308. #442	C₂²×C₈₉	semidirect
309. #441	C₂²×C₈₉	semidirect
310. #440	C₂²×C₈₉	semidirect
311. C₄×Q₁₆	C₂²×C₈₉	semidirect
312. C₄×SD₁₆	C₂²×C₈₉	semidirect
313. C₄×D₈	C₂²×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3455	2
C₈₉×#3177	2
C₈₉×#3176	2
C₈₉×#3175	2
C₈₉×#3174	2
C₈₉×#3173	2
C₈₉×#3160	2
C₈₉×#3147	2
C₈₉×#3135	2
C₈₉×#3054	2
C₈₉×#3050	2
C₈₉×#2943	2
C₄×#135×C₈₉	2
C₂×#405×C₈₉	2
#60682	89

Subgroups — C89×#60682 #36438530

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#60682 #36438530