Subgroups — C₈₉×#25512 #36403360

Order 22784.

1028 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#541×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#470×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×#467×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2625	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#2059	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#2043	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2026	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#2016	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1894	C₂	1	1	yes	maximal, characteristic
10	C₈₉×#1892	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#1878	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈₉×#1878	C₂	1	1	yes	maximal, characteristic, complemented
13	C₈₉×#1865	C₂	1	1	yes	maximal, characteristic, complemented
14	C₈₉×#1863	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈₉×#1729	C₂	1	1	yes	maximal, characteristic
16	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
19	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
20	#541×C₈₉	—	2	4	no	—
21	#541×C₈₉	—	2	4	no	—
22	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×#126×C₈₉	—	2	1	no	—
24	C₂×#126×C₈₉	—	2	1	no	—
25	C₂×#125×C₈₉	—	2	1	no	—
26	C₂×#125×C₈₉	—	2	1	no	—
27	C₂×#125×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
34	#470×C₈₉	—	2	4	no	—
35	#467×C₈₉	—	2	4	no	—
36	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
38	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
40	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
42	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
44	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
46	#400×C₈₉	—	2	1	no	—
47	#398×C₈₉	—	2	1	no	—
48	#397×C₈₉	—	2	1	no	—
49	#397×C₈₉	—	2	1	no	—
50	#391×C₈₉	C₂²	1	1	yes	characteristic
51	#391×C₈₉	C₂²	1	1	yes	characteristic
52	#389×C₈₉	C₂²	1	1	yes	characteristic
53	#389×C₈₉	C₂²	1	1	yes	characteristic
54	#380×C₈₉	C₂²	1	1	yes	characteristic
55	#340×C₈₉	C₂²	1	1	yes	characteristic
56	#340×C₈₉	C₂²	1	1	yes	characteristic
57	#340×C₈₉	C₂²	1	1	yes	characteristic
58	#340×C₈₉	C₂²	1	1	yes	characteristic
59	#336×C₈₉	C₂²	1	1	yes	characteristic
60	#336×C₈₉	C₂²	1	1	yes	characteristic
61	#336×C₈₉	C₂²	1	1	yes	characteristic
62	#336×C₈₉	C₂²	1	1	yes	characteristic
63	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
64	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
67	C₂×Q₁₆×C₈₉	—	2	2	no	—
68	C₂×Q₁₆×C₈₉	—	2	2	no	—
69	C₂×Q₁₆×C₈₉	—	2	2	no	—
70	C₂×Q₁₆×C₈₉	—	2	2	no	—
71	C₂×Q₁₆×C₈₉	—	4	1	no	—
72	C₂×Q₁₆×C₈₉	—	4	1	no	—
73	C₂×M₁₆×C₈₉	—	2	2	no	—
74	C₂×M₁₆×C₈₉	—	2	2	no	—
75	C₂×M₁₆×C₈₉	—	4	1	no	—
76	C₂²×C₈×C₈₉	—	2	2	no	—
77	C₂²×C₈×C₈₉	—	2	2	no	—
78	C₂²×C₈×C₈₉	—	4	1	no	—
79	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
80	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
81	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
82	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
83	C₄⋊Q₈×C₈₉	—	2	2	no	—
84	C₄⋊Q₈×C₈₉	—	2	2	no	—
85	#126×C₈₉	—	2	8	no	—
86	#126×C₈₉	—	2	8	no	—
87	#125×C₈₉	—	2	8	no	—
88	#125×C₈₉	—	2	8	no	—
89	#125×C₈₉	—	2	2	no	—
90	#125×C₈₉	—	2	2	no	—
91	C₂²⋊Q₈×C₈₉	—	2	2	no	—
92	C₂²⋊Q₈×C₈₉	—	2	2	no	—
93	C₂²⋊Q₈×C₈₉	—	2	2	no	—
94	C₂²⋊Q₈×C₈₉	—	2	2	no	—
95	C₂²⋊Q₈×C₈₉	D₄	1	4	yes	—
96	C₂²⋊Q₈×C₈₉	D₄	1	4	yes	—
97	C₄×Q₈×C₈₉	—	2	2	no	—
98	C₄×Q₈×C₈₉	—	2	2	no	—
99	C₄×Q₈×C₈₉	—	2	2	no	—
100	C₄×Q₈×C₈₉	—	2	2	no	—
101	#117×C₈₉	—	2	2	no	—
102	#117×C₈₉	—	2	2	no	—
103	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
104	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
105	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
106	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
107	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
108	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
109	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
110	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
111	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
112	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
113	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
114	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
115	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
116	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
117	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
118	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
120	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
121	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
122	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
123	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
124	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
125	C₂×C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
126	C₂×C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
127	C₂×C₄²×C₈₉	—	2	1	no	—
128	C₂×C₄²×C₈₉	—	2	1	no	—
129	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
130	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
131	#107×C₈₉	—	2	2	no	—
132	#106×C₈₉	—	2	2	no	—
133	#103×C₈₉	—	2	2	no	—
134	#103×C₈₉	—	2	2	no	—
135	#103×C₈₉	—	2	2	no	—
136	#103×C₈₉	—	2	2	no	—
137	#103×C₈₉	—	2	2	no	—
138	#103×C₈₉	—	2	2	no	—
139	#103×C₈₉	—	4	1	no	—
140	#103×C₈₉	—	4	1	no	—
141	#95×C₈₉	—	2	1	no	—
142	#95×C₈₉	—	2	1	no	—
143	#95×C₈₉	—	2	1	no	—
144	#95×C₈₉	—	2	1	no	—
145	#95×C₈₉	—	2	1	no	—
146	#95×C₈₉	—	2	1	no	—
147	#95×C₈₉	—	2	1	no	—
148	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
149	C₂×Q₈×C₈₉	—	2	1	no	—
150	C₂×Q₈×C₈₉	—	2	1	no	—
151	C₂×Q₈×C₈₉	—	2	1	no	—
152	C₂×Q₈×C₈₉	—	2	1	no	—
153	C₂×Q₈×C₈₉	—	2	1	no	—
154	C₂×Q₈×C₈₉	—	2	1	no	—
155	C₂×Q₈×C₈₉	—	2	1	no	—
156	C₂×Q₈×C₈₉	—	2	1	no	—
157	C₂×Q₈×C₈₉	—	2	1	no	—
158	C₂×Q₈×C₈₉	—	2	1	no	—
159	C₂×Q₈×C₈₉	—	2	1	no	—
160	C₂×Q₈×C₈₉	—	2	1	no	—
161	C₂×Q₈×C₈₉	—	2	1	no	—
162	C₂×Q₈×C₈₉	—	2	1	no	—
163	C₂×Q₈×C₈₉	—	2	1	no	—
164	C₂×Q₈×C₈₉	—	2	1	no	—
165	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
166	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
167	C₂²×C₄×C₈₉	—	2	1	no	—
168	C₂²×C₄×C₈₉	—	2	1	no	—
169	C₂²×C₄×C₈₉	—	2	1	no	—
170	C₂²×C₄×C₈₉	—	2	1	no	—
171	C₂²×C₄×C₈₉	—	2	1	no	—
172	C₂²×C₄×C₈₉	—	2	1	no	—
173	C₂²×C₄×C₈₉	—	2	1	no	—
174	C₂²×C₄×C₈₉	—	2	1	no	—
175	C₂²×C₄×C₈₉	—	2	1	no	—
176	C₂²×C₄×C₈₉	—	2	1	no	—
177	C₂²×C₄×C₈₉	—	2	1	no	—
178	C₂²×C₄×C₈₉	—	2	1	no	—
179	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
182	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
183	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
184	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
185	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
186	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
187	Q₁₆×C₈₉	—	4	4	no	—
188	Q₁₆×C₈₉	—	4	4	no	—
189	M₁₆×C₈₉	—	4	4	no	—
190	C₂×C₈×C₈₉	—	2	1	no	—
191	C₂×C₈×C₈₉	—	2	1	no	—
192	C₂×C₈×C₈₉	—	2	1	no	—
193	C₂×C₈×C₈₉	—	2	1	no	—
194	C₂×C₈×C₈₉	—	2	1	no	—
195	C₂×C₈×C₈₉	—	2	1	no	—
196	C₂×C₈×C₈₉	—	2	1	no	—
197	C₂×C₈×C₈₉	—	2	1	no	—
198	C₂×C₈×C₈₉	—	2	1	no	—
199	C₂×C₈×C₈₉	—	2	1	no	—
200	C₂×C₈×C₈₉	—	4	4	no	—
201	C₂×C₈×C₈₉	#41	1	2	yes	—
202	C₂×C₈×C₈₉	#41	1	2	yes	—
203	C₄⋊C₄×C₈₉	—	2	4	no	—
204	C₄⋊C₄×C₈₉	—	2	4	no	—
205	C₄⋊C₄×C₈₉	—	2	4	no	—
206	C₄⋊C₄×C₈₉	—	2	4	no	—
207	C₄⋊C₄×C₈₉	—	2	4	no	—
208	C₄⋊C₄×C₈₉	—	2	4	no	—
209	C₄⋊C₄×C₈₉	—	2	4	no	—
210	C₄⋊C₄×C₈₉	—	2	4	no	—
211	C₄⋊C₄×C₈₉	—	2	2	no	—
212	C₄⋊C₄×C₈₉	—	2	2	no	—
213	C₄⋊C₄×C₈₉	—	2	1	no	—
214	C₄⋊C₄×C₈₉	—	2	1	no	—
215	C₄⋊C₄×C₈₉	—	2	1	no	—
216	C₄⋊C₄×C₈₉	—	2	1	no	—
217	C₄⋊C₄×C₈₉	—	2	1	no	—
218	C₄⋊C₄×C₈₉	—	2	1	no	—
219	C₄⋊C₄×C₈₉	—	2	1	no	—
220	C₄⋊C₄×C₈₉	—	2	1	no	—
221	C₄⋊C₄×C₈₉	—	4	2	no	—
222	C₄⋊C₄×C₈₉	—	8	1	no	—
223	C₄⋊C₄×C₈₉	—	8	1	no	—
224	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
225	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
226	C₂²⋊C₄×C₈₉	—	2	4	no	—
227	C₂²⋊C₄×C₈₉	—	2	4	no	—
228	C₂²⋊C₄×C₈₉	—	2	2	no	—
229	C₂²⋊C₄×C₈₉	—	2	2	no	—
230	C₂²⋊C₄×C₈₉	—	4	2	no	—
231	C₂²⋊C₄×C₈₉	—	8	1	no	—
232	C₄²×C₈₉	—	2	4	no	—
233	C₄²×C₈₉	—	2	4	no	—
234	C₄²×C₈₉	—	2	1	no	—
235	C₄²×C₈₉	—	2	1	no	—
236	C₄²×C₈₉	—	2	1	no	—
237	C₄²×C₈₉	—	2	1	no	—
238	C₂³×C₈₉	—	2	1	no	—
239	C₂³×C₈₉	—	2	1	no	—
240	C₂³×C₈₉	—	2	1	no	—
241	C₂³×C₈₉	—	2	1	no	—
242	C₂³×C₈₉	—	2	1	no	—
243	C₂³×C₈₉	—	2	1	no	—
244	C₂³×C₈₉	#120	1	2	yes	—
245	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
246	Q₈×C₈₉	—	2	2	no	—
247	Q₈×C₈₉	—	2	2	no	—
248	Q₈×C₈₉	—	2	2	no	—
249	Q₈×C₈₉	—	2	2	no	—
250	Q₈×C₈₉	—	4	1	no	—
251	Q₈×C₈₉	—	4	1	no	—
252	Q₈×C₈₉	—	4	1	no	—
253	Q₈×C₈₉	—	4	1	no	—
254	Q₈×C₈₉	—	4	1	no	—
255	Q₈×C₈₉	—	4	1	no	—
256	Q₈×C₈₉	—	4	1	no	—
257	Q₈×C₈₉	—	4	1	no	—
258	C₂×C₄×C₈₉	—	2	1	no	—
259	C₂×C₄×C₈₉	—	2	1	no	—
260	C₂×C₄×C₈₉	—	2	2	no	—
261	C₂×C₄×C₈₉	—	2	2	no	—
262	C₂×C₄×C₈₉	—	2	2	no	—
263	C₂×C₄×C₈₉	—	2	2	no	—
264	C₂×C₄×C₈₉	—	2	1	no	—
265	C₂×C₄×C₈₉	—	2	1	no	—
266	C₂×C₄×C₈₉	—	2	1	no	—
267	C₂×C₄×C₈₉	—	2	1	no	—
268	C₂×C₄×C₈₉	—	2	2	no	—
269	C₂×C₄×C₈₉	—	2	2	no	—
270	C₂×C₄×C₈₉	—	2	2	no	—
271	C₂×C₄×C₈₉	—	2	2	no	—
272	C₂×C₄×C₈₉	—	2	1	no	—
273	C₂×C₄×C₈₉	—	2	1	no	—
274	C₂×C₄×C₈₉	—	2	1	no	—
275	C₂×C₄×C₈₉	—	2	1	no	—
276	C₂×C₄×C₈₉	—	2	1	no	—
277	C₂×C₄×C₈₉	—	2	1	no	—
278	C₂×C₄×C₈₉	—	2	1	no	—
279	C₂×C₄×C₈₉	—	2	1	no	—
280	C₂×C₄×C₈₉	—	4	1	no	—
281	C₂×C₄×C₈₉	—	4	1	no	—
282	C₂×C₄×C₈₉	—	4	1	no	—
283	C₂×C₄×C₈₉	—	4	1	no	—
284	C₂×C₄×C₈₉	—	4	1	no	—
285	C₂×C₄×C₈₉	—	4	1	no	—
286	C₂×C₄×C₈₉	—	4	1	no	—
287	C₂×C₄×C₈₉	—	4	1	no	—
288	C₂×C₄×C₈₉	—	4	1	no	—
289	C₂×C₄×C₈₉	—	4	1	no	—
290	C₂×C₄×C₈₉	—	4	1	no	—
291	C₂×C₄×C₈₉	—	4	1	no	—
292	C₂×C₄×C₈₉	—	4	1	no	—
293	C₂×C₄×C₈₉	—	4	1	no	—
294	C₂×C₄×C₈₉	—	4	1	no	—
295	C₂×C₄×C₈₉	—	4	1	no	—
296	C₂×C₄×C₈₉	—	4	1	no	—
297	C₂×C₄×C₈₉	—	4	1	no	—
298	C₂×C₄×C₈₉	—	4	1	no	—
299	C₂×C₄×C₈₉	—	4	1	no	—
300	C₂×C₄×C₈₉	#120	1	2	yes	—
301	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
302	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
303	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
304	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
305	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
306	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
307	C₈×C₈₉	—	2	2	no	—
308	C₈×C₈₉	—	2	2	no	—
309	C₈×C₈₉	—	4	1	no	—
310	C₈×C₈₉	—	4	1	no	—
311	C₂²×C₈₉	—	2	2	no	—
312	C₂²×C₈₉	—	2	2	no	—
313	C₂²×C₈₉	—	2	2	no	—
314	C₂²×C₈₉	—	4	1	no	—
315	C₂²×C₈₉	—	4	1	no	—
316	C₂²×C₈₉	—	4	1	no	—
317	C₂²×C₈₉	—	4	1	no	—
318	C₂²×C₈₉	C₂×#120	1	1	yes	characteristic
319	C₂²×C₈₉	#532	1	1	yes	characteristic
320	C₂²×C₈₉	#537	1	1	yes	characteristic
321	C₂²×C₈₉	#577	1	1	yes	characteristic
322	C₂²×C₈₉	#578	1	1	yes	characteristic
323	C₂²×C₈₉	#578	1	1	yes	characteristic
324	C₂²×C₈₉	#578	1	1	yes	characteristic
325	C₄×C₈₉	—	2	2	no	—
326	C₄×C₈₉	—	4	2	no	—
327	C₄×C₈₉	—	4	2	no	—
328	C₄×C₈₉	—	4	1	no	—
329	C₄×C₈₉	—	4	1	no	—
330	C₄×C₈₉	—	4	2	no	—
331	C₄×C₈₉	—	4	1	no	—
332	C₄×C₈₉	—	4	1	no	—
333	C₄×C₈₉	—	4	2	no	—
334	C₄×C₈₉	—	8	1	no	—
335	C₄×C₈₉	—	8	1	no	—
336	C₄×C₈₉	—	8	1	no	—
337	C₄×C₈₉	—	8	1	no	—
338	C₄×C₈₉	#546	1	1	yes	characteristic
339	C₄×C₈₉	#546	1	1	yes	characteristic
340	C₄×C₈₉	#547	1	1	yes	characteristic
341	C₄×C₈₉	#547	1	1	yes	characteristic
342	#25512	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
343	C₂×C₈₉	—	4	2	no	—
344	C₂×C₈₉	#2430	1	1	yes	characteristic
345	C₂×C₈₉	#3011	1	1	yes	characteristic
346	C₂×C₈₉	#3013	1	1	yes	characteristic
347	C₂×C₈₉	#3177	1	1	yes	characteristic
348	C₂×C₈₉	#3178	1	1	yes	characteristic
349	C₂×C₈₉	#3258	1	1	yes	characteristic
350	C₂×C₈₉	#3261	1	1	yes	characteristic
351	C₂×#541	C₂×C₈₉	1	1	yes	characteristic
352	C₂×#470	C₂×C₈₉	1	1	yes	characteristic
353	C₂×#467	C₂×C₈₉	1	1	yes	characteristic
354	#2625	C₂×C₈₉	1	1	yes	characteristic
355	#2059	C₂×C₈₉	1	1	yes	characteristic, complemented
356	#2043	C₂×C₈₉	1	1	yes	characteristic, complemented
357	#2026	C₂×C₈₉	1	1	yes	characteristic, complemented
358	#2016	C₂×C₈₉	1	1	yes	characteristic, complemented
359	#1894	C₂×C₈₉	1	1	yes	characteristic
360	#1892	C₂×C₈₉	1	1	yes	characteristic
361	#1878	C₂×C₈₉	1	1	yes	characteristic, complemented
362	#1878	C₂×C₈₉	1	1	yes	characteristic, complemented
363	#1865	C₂×C₈₉	1	1	yes	characteristic, complemented
364	#1863	C₂×C₈₉	1	1	yes	characteristic, complemented
365	#1729	C₂×C₈₉	1	1	yes	characteristic
366	C₈₉	#25512	1	1	yes	characteristic, direct product factor, Sylow
367	C₂²×Q₁₆	C₂²×C₈₉	1	1	yes	characteristic
368	C₂²×Q₁₆	C₂²×C₈₉	1	1	yes	characteristic
369	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
370	C₂³×C₈	C₂²×C₈₉	1	1	yes	characteristic
371	#541	—	2	4	no	—
372	#541	—	2	4	no	—
373	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
374	C₂×#126	—	2	1	no	—
375	C₂×#126	—	2	1	no	—
376	C₂×#125	—	2	1	no	—
377	C₂×#125	—	2	1	no	—
378	C₂×#125	C₂²×C₈₉	1	1	yes	characteristic
379	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
380	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
381	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
382	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
383	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
384	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
385	#470	—	2	4	no	—
386	#467	—	2	4	no	—
387	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
388	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
389	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
390	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
391	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
392	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
393	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
394	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
395	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
396	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
397	#400	—	2	1	no	—
398	#398	—	2	1	no	—
399	#397	—	2	1	no	—
400	#397	—	2	1	no	—
401	#391	C₂²×C₈₉	1	1	yes	characteristic
402	#391	C₂²×C₈₉	1	1	yes	characteristic
403	#389	C₂²×C₈₉	1	1	yes	characteristic
404	#389	C₂²×C₈₉	1	1	yes	characteristic
405	#380	C₂²×C₈₉	1	1	yes	characteristic
406	#340	C₂²×C₈₉	1	1	yes	characteristic
407	#340	C₂²×C₈₉	1	1	yes	characteristic
408	#340	C₂²×C₈₉	1	1	yes	characteristic
409	#340	C₂²×C₈₉	1	1	yes	characteristic
410	#336	C₂²×C₈₉	1	1	yes	characteristic
411	#336	C₂²×C₈₉	1	1	yes	characteristic
412	#336	C₂²×C₈₉	1	1	yes	characteristic
413	#336	C₂²×C₈₉	1	1	yes	characteristic
414	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
415	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
416	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
417	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
418	C₂×Q₁₆	—	2	2	no	—
419	C₂×Q₁₆	—	2	2	no	—
420	C₂×Q₁₆	—	2	2	no	—
421	C₂×Q₁₆	—	2	2	no	—
422	C₂×Q₁₆	—	4	1	no	—
423	C₂×Q₁₆	—	4	1	no	—
424	C₂×M₁₆	—	2	2	no	—
425	C₂×M₁₆	—	2	2	no	—
426	C₂×M₁₆	—	4	1	no	—
427	C₂²×C₈	—	2	2	no	—
428	C₂²×C₈	—	2	2	no	—
429	C₂²×C₈	—	4	1	no	—
430	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
431	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
432	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
433	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
434	C₄⋊Q₈	—	2	2	no	—
435	C₄⋊Q₈	—	2	2	no	—
436	#126	—	2	8	no	—
437	#126	—	2	8	no	—
438	#125	—	2	8	no	—
439	#125	—	2	8	no	—
440	#125	—	2	2	no	—
441	#125	—	2	2	no	—
442	C₂²⋊Q₈	—	2	2	no	—
443	C₂²⋊Q₈	—	2	2	no	—
444	C₂²⋊Q₈	—	2	2	no	—
445	C₂²⋊Q₈	—	2	2	no	—
446	C₂²⋊Q₈	D₄×C₈₉	1	4	yes	—
447	C₂²⋊Q₈	D₄×C₈₉	1	4	yes	—
448	C₄×Q₈	—	2	2	no	—
449	C₄×Q₈	—	2	2	no	—
450	C₄×Q₈	—	2	2	no	—
451	C₄×Q₈	—	2	2	no	—
452	#117	—	2	2	no	—
453	#117	—	2	2	no	—
454	C₂×C₄⋊C₄	—	2	1	no	—
455	C₂×C₄⋊C₄	—	2	1	no	—
456	C₂×C₄⋊C₄	—	2	1	no	—
457	C₂×C₄⋊C₄	—	2	1	no	—
458	C₂×C₄⋊C₄	—	2	1	no	—
459	C₂×C₄⋊C₄	—	2	1	no	—
460	C₂×C₄⋊C₄	—	2	1	no	—
461	C₂×C₄⋊C₄	—	2	1	no	—
462	C₂×C₄⋊C₄	—	2	1	no	—
463	C₂×C₄⋊C₄	—	2	1	no	—
464	C₂×C₄⋊C₄	—	2	1	no	—
465	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
466	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
467	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
468	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
469	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
470	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
471	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
472	C₂×C₂²⋊C₄	—	2	1	no	—
473	C₂×C₂²⋊C₄	—	2	1	no	—
474	C₂×C₂²⋊C₄	—	2	1	no	—
475	C₂×C₂²⋊C₄	—	2	1	no	—
476	C₂×C₂²⋊C₄	D₄×C₈₉	1	1	yes	characteristic
477	C₂×C₂²⋊C₄	D₄×C₈₉	1	1	yes	characteristic
478	C₂×C₄²	—	2	1	no	—
479	C₂×C₄²	—	2	1	no	—
480	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
481	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
482	#107	—	2	2	no	—
483	#106	—	2	2	no	—
484	#103	—	2	2	no	—
485	#103	—	2	2	no	—
486	#103	—	2	2	no	—
487	#103	—	2	2	no	—
488	#103	—	2	2	no	—
489	#103	—	2	2	no	—
490	#103	—	4	1	no	—
491	#103	—	4	1	no	—
492	#95	—	2	1	no	—
493	#95	—	2	1	no	—
494	#95	—	2	1	no	—
495	#95	—	2	1	no	—
496	#95	—	2	1	no	—
497	#95	—	2	1	no	—
498	#95	—	2	1	no	—
499	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
500	C₂×Q₈	—	2	1	no	—
501	C₂×Q₈	—	2	1	no	—
502	C₂×Q₈	—	2	1	no	—
503	C₂×Q₈	—	2	1	no	—
504	C₂×Q₈	—	2	1	no	—
505	C₂×Q₈	—	2	1	no	—
506	C₂×Q₈	—	2	1	no	—
507	C₂×Q₈	—	2	1	no	—
508	C₂×Q₈	—	2	1	no	—
509	C₂×Q₈	—	2	1	no	—
510	C₂×Q₈	—	2	1	no	—
511	C₂×Q₈	—	2	1	no	—
512	C₂×Q₈	—	2	1	no	—
513	C₂×Q₈	—	2	1	no	—
514	C₂×Q₈	—	2	1	no	—
515	C₂×Q₈	—	2	1	no	—
516	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
517	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
518	C₂²×C₄	—	2	1	no	—
519	C₂²×C₄	—	2	1	no	—
520	C₂²×C₄	—	2	1	no	—
521	C₂²×C₄	—	2	1	no	—
522	C₂²×C₄	—	2	1	no	—
523	C₂²×C₄	—	2	1	no	—
524	C₂²×C₄	—	2	1	no	—
525	C₂²×C₄	—	2	1	no	—
526	C₂²×C₄	—	2	1	no	—
527	C₂²×C₄	—	2	1	no	—
528	C₂²×C₄	—	2	1	no	—
529	C₂²×C₄	—	2	1	no	—
530	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
531	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
532	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
533	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
534	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
535	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
536	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
537	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
538	Q₁₆	—	4	4	no	—
539	Q₁₆	—	4	4	no	—
540	M₁₆	—	4	4	no	—
541	C₂×C₈	—	2	1	no	—
542	C₂×C₈	—	2	1	no	—
543	C₂×C₈	—	2	1	no	—
544	C₂×C₈	—	2	1	no	—
545	C₂×C₈	—	2	1	no	—
546	C₂×C₈	—	2	1	no	—
547	C₂×C₈	—	2	1	no	—
548	C₂×C₈	—	2	1	no	—
549	C₂×C₈	—	2	1	no	—
550	C₂×C₈	—	2	1	no	—
551	C₂×C₈	—	4	4	no	—
552	C₂×C₈	#41×C₈₉	1	2	yes	—
553	C₂×C₈	#41×C₈₉	1	2	yes	—
554	C₄⋊C₄	—	2	4	no	—
555	C₄⋊C₄	—	2	4	no	—
556	C₄⋊C₄	—	2	4	no	—
557	C₄⋊C₄	—	2	4	no	—
558	C₄⋊C₄	—	2	4	no	—
559	C₄⋊C₄	—	2	4	no	—
560	C₄⋊C₄	—	2	4	no	—
561	C₄⋊C₄	—	2	4	no	—
562	C₄⋊C₄	—	2	2	no	—
563	C₄⋊C₄	—	2	2	no	—
564	C₄⋊C₄	—	2	1	no	—
565	C₄⋊C₄	—	2	1	no	—
566	C₄⋊C₄	—	2	1	no	—
567	C₄⋊C₄	—	2	1	no	—
568	C₄⋊C₄	—	2	1	no	—
569	C₄⋊C₄	—	2	1	no	—
570	C₄⋊C₄	—	2	1	no	—
571	C₄⋊C₄	—	2	1	no	—
572	C₄⋊C₄	—	4	2	no	—
573	C₄⋊C₄	—	8	1	no	—
574	C₄⋊C₄	—	8	1	no	—
575	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
576	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
577	C₂²⋊C₄	—	2	4	no	—
578	C₂²⋊C₄	—	2	4	no	—
579	C₂²⋊C₄	—	2	2	no	—
580	C₂²⋊C₄	—	2	2	no	—
581	C₂²⋊C₄	—	4	2	no	—
582	C₂²⋊C₄	—	8	1	no	—
583	C₄²	—	2	4	no	—
584	C₄²	—	2	4	no	—
585	C₄²	—	2	1	no	—
586	C₄²	—	2	1	no	—
587	C₄²	—	2	1	no	—
588	C₄²	—	2	1	no	—
589	C₂³	—	2	1	no	—
590	C₂³	—	2	1	no	—
591	C₂³	—	2	1	no	—
592	C₂³	—	2	1	no	—
593	C₂³	—	2	1	no	—
594	C₂³	—	2	1	no	—
595	C₂³	#120×C₈₉	1	2	yes	—
596	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
597	Q₈	—	2	2	no	—
598	Q₈	—	2	2	no	—
599	Q₈	—	2	2	no	—
600	Q₈	—	2	2	no	—
601	Q₈	—	4	1	no	—
602	Q₈	—	4	1	no	—
603	Q₈	—	4	1	no	—
604	Q₈	—	4	1	no	—
605	Q₈	—	4	1	no	—
606	Q₈	—	4	1	no	—
607	Q₈	—	4	1	no	—
608	Q₈	—	4	1	no	—
609	C₂×C₄	—	2	1	no	—
610	C₂×C₄	—	2	1	no	—
611	C₂×C₄	—	2	2	no	—
612	C₂×C₄	—	2	2	no	—
613	C₂×C₄	—	2	2	no	—
614	C₂×C₄	—	2	2	no	—
615	C₂×C₄	—	2	1	no	—
616	C₂×C₄	—	2	1	no	—
617	C₂×C₄	—	2	1	no	—
618	C₂×C₄	—	2	1	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	2	2	no	—
622	C₂×C₄	—	2	2	no	—
623	C₂×C₄	—	2	1	no	—
624	C₂×C₄	—	2	1	no	—
625	C₂×C₄	—	2	1	no	—
626	C₂×C₄	—	2	1	no	—
627	C₂×C₄	—	2	1	no	—
628	C₂×C₄	—	2	1	no	—
629	C₂×C₄	—	2	1	no	—
630	C₂×C₄	—	2	1	no	—
631	C₂×C₄	—	4	1	no	—
632	C₂×C₄	—	4	1	no	—
633	C₂×C₄	—	4	1	no	—
634	C₂×C₄	—	4	1	no	—
635	C₂×C₄	—	4	1	no	—
636	C₂×C₄	—	4	1	no	—
637	C₂×C₄	—	4	1	no	—
638	C₂×C₄	—	4	1	no	—
639	C₂×C₄	—	4	1	no	—
640	C₂×C₄	—	4	1	no	—
641	C₂×C₄	—	4	1	no	—
642	C₂×C₄	—	4	1	no	—
643	C₂×C₄	—	4	1	no	—
644	C₂×C₄	—	4	1	no	—
645	C₂×C₄	—	4	1	no	—
646	C₂×C₄	—	4	1	no	—
647	C₂×C₄	—	4	1	no	—
648	C₂×C₄	—	4	1	no	—
649	C₂×C₄	—	4	1	no	—
650	C₂×C₄	—	4	1	no	—
651	C₂×C₄	#120×C₈₉	1	2	yes	—
652	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
653	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
654	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
655	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
656	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
657	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
658	C₈	—	2	2	no	—
659	C₈	—	2	2	no	—
660	C₈	—	4	1	no	—
661	C₈	—	4	1	no	—
662	C₂²	—	2	2	no	—
663	C₂²	—	2	2	no	—
664	C₂²	—	2	2	no	—
665	C₂²	—	4	1	no	—
666	C₂²	—	4	1	no	—
667	C₂²	—	4	1	no	—
668	C₂²	—	4	1	no	—
669	C₂²	C₂×#120×C₈₉	1	1	yes	characteristic
670	C₂²	#532×C₈₉	1	1	yes	characteristic
671	C₂²	#537×C₈₉	1	1	yes	characteristic
672	C₂²	#577×C₈₉	1	1	yes	characteristic
673	C₂²	#578×C₈₉	1	1	yes	characteristic
674	C₂²	#578×C₈₉	1	1	yes	characteristic
675	C₂²	#578×C₈₉	1	1	yes	characteristic
676	C₄	—	2	2	no	—
677	C₄	—	4	2	no	—
678	C₄	—	4	2	no	—
679	C₄	—	4	1	no	—
680	C₄	—	4	1	no	—
681	C₄	—	4	2	no	—
682	C₄	—	4	1	no	—
683	C₄	—	4	1	no	—
684	C₄	—	4	2	no	—
685	C₄	—	8	1	no	—
686	C₄	—	8	1	no	—
687	C₄	—	8	1	no	—
688	C₄	—	8	1	no	—
689	C₄	#546×C₈₉	1	1	yes	characteristic
690	C₄	#546×C₈₉	1	1	yes	characteristic
691	C₄	#547×C₈₉	1	1	yes	characteristic
692	C₄	#547×C₈₉	1	1	yes	characteristic
693	C₂	—	4	2	no	—
694	C₂	C₈₉×#2430	1	1	yes	characteristic
695	C₂	C₈₉×#3011	1	1	yes	characteristic
696	C₂	C₈₉×#3013	1	1	yes	characteristic
697	C₂	C₈₉×#3177	1	1	yes	characteristic
698	C₂	C₈₉×#3178	1	1	yes	characteristic
699	C₂	C₈₉×#3258	1	1	yes	characteristic
700	C₂	C₈₉×#3261	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂⁴×C₈₉	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂×#117×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₈₉×#2059	C₂	semidirect
6. C₈₉×#2043	C₂	semidirect
7. C₈₉×#2026	C₂	semidirect
8. C₈₉×#2016	C₂	semidirect
11. C₈₉×#1878	C₂	semidirect
12. C₈₉×#1878	C₂	semidirect
13. C₈₉×#1865	C₂	semidirect
14. C₈₉×#1863	C₂	semidirect
342. #25512	C₈₉	direct (N × H)
355. #2059	C₂×C₈₉	semidirect
356. #2043	C₂×C₈₉	semidirect
357. #2026	C₂×C₈₉	semidirect
358. #2016	C₂×C₈₉	semidirect
361. #1878	C₂×C₈₉	semidirect
362. #1878	C₂×C₈₉	semidirect
363. #1865	C₂×C₈₉	semidirect
364. #1863	C₂×C₈₉	semidirect
366. C₈₉	#25512	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×#541×C₈₉	2
C₂×#470×C₈₉	2
C₂×#467×C₈₉	2
C₈₉×#2625	2
C₈₉×#2059	2
C₈₉×#2043	2
C₈₉×#2026	2
C₈₉×#2016	2
C₈₉×#1894	2
C₈₉×#1892	2
C₈₉×#1878	2
C₈₉×#1865	2
C₈₉×#1863	2
C₈₉×#1729	2
#25512	89

Subgroups — C89×#25512 #36403360

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#25512 #36403360