Subgroups — C₈₉×#24333 #36402181

Order 22784.

1034 conjugacy classes of subgroups.

The subgroup lattice has 496 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×#477×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄×Q₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2582	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#2057	C₂	1	2	yes	maximal
6	C₈₉×#2024	C₂	1	2	yes	maximal
7	C₈₉×#1990	C₂	1	1	yes	maximal, characteristic
8	C₈₉×#1984	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#1878	C₂	1	2	yes	maximal
10	C₈₉×#1863	C₂	1	2	yes	maximal
11	C₈₉×#1803	C₂	1	1	yes	maximal, characteristic
12	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
13	Q₈²×C₈₉	—	2	4	no	—
14	Q₈²×C₈₉	—	2	4	no	—
15	C₂×C₄⋊Q₈×C₈₉	—	2	1	no	—
16	C₂×C₄⋊Q₈×C₈₉	—	2	2	no	—
17	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
20	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
22	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
24	#477×C₈₉	—	2	4	no	—
25	C₄×Q₁₆×C₈₉	—	2	4	no	—
26	C₂×#107×C₈₉	C₂²	1	2	yes	—
27	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×#103×C₈₉	C₂²	1	2	yes	—
30	C₂×#103×C₈₉	C₂²	1	2	yes	—
31	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
34	#398×C₈₉	—	2	2	no	—
35	#391×C₈₉	—	2	1	no	—
36	#391×C₈₉	C₂²	1	2	yes	—
37	#389×C₈₉	—	2	1	no	—
38	#389×C₈₉	C₂²	1	2	yes	—
39	#384×C₈₉	C₂²	1	1	yes	characteristic
40	#340×C₈₉	C₂²	1	2	yes	—
41	#340×C₈₉	C₂²	1	2	yes	—
42	#340×C₈₉	C₂²	1	2	yes	—
43	#336×C₈₉	C₂²	1	2	yes	—
44	#336×C₈₉	C₂²	1	2	yes	—
45	C₂²×Q₈×C₈₉	—	2	2	no	—
46	C₂²×Q₈×C₈₉	—	2	1	no	—
47	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
48	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
49	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
50	C₂×Q₁₆×C₈₉	—	2	2	no	—
51	C₂×Q₁₆×C₈₉	—	2	2	no	—
52	C₂×Q₁₆×C₈₉	Q₈	1	4	yes	complemented
53	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
54	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
55	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
56	C₄⋊Q₈×C₈₉	—	2	16	no	—
57	C₄⋊Q₈×C₈₉	—	2	8	no	—
58	C₄⋊Q₈×C₈₉	—	2	4	no	—
59	C₄⋊Q₈×C₈₉	—	2	4	no	—
60	C₄⋊Q₈×C₈₉	—	2	2	no	—
61	C₄⋊Q₈×C₈₉	D₄	1	4	yes	—
62	C₄×Q₈×C₈₉	—	2	8	no	—
63	C₄×Q₈×C₈₉	—	2	4	no	—
64	C₄×Q₈×C₈₉	—	2	4	no	—
65	C₄×Q₈×C₈₉	—	2	2	no	—
66	C₄×Q₈×C₈₉	—	2	2	no	—
67	C₄×Q₈×C₈₉	—	2	2	no	—
68	C₄×Q₈×C₈₉	—	2	2	no	—
69	C₄×Q₈×C₈₉	—	2	2	no	—
70	C₄×Q₈×C₈₉	D₄	1	4	yes	—
71	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
72	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
73	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
74	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
75	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
76	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
77	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
78	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
79	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
81	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
85	C₂×C₄²×C₈₉	—	2	2	no	—
86	C₂×C₄²×C₈₉	—	2	1	no	—
87	C₂×C₄²×C₈₉	—	2	1	no	—
88	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
89	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
90	#107×C₈₉	—	2	2	no	—
91	#107×C₈₉	—	2	4	no	—
92	C₄⋊C₈×C₈₉	—	2	2	no	—
93	#103×C₈₉	—	2	2	no	—
94	#103×C₈₉	—	2	4	no	—
95	#103×C₈₉	—	2	2	no	—
96	#103×C₈₉	—	2	4	no	—
97	C₄×C₈×C₈₉	—	2	2	no	—
98	#95×C₈₉	—	2	2	no	—
99	#95×C₈₉	—	2	2	no	—
100	#95×C₈₉	—	2	2	no	—
101	C₂×Q₈×C₈₉	—	2	8	no	—
102	C₂×Q₈×C₈₉	—	2	2	no	—
103	C₂×Q₈×C₈₉	—	2	4	no	—
104	C₂×Q₈×C₈₉	—	2	2	no	—
105	C₂×Q₈×C₈₉	—	2	1	no	—
106	C₂×Q₈×C₈₉	—	2	1	no	—
107	C₂×Q₈×C₈₉	—	2	1	no	—
108	C₂×Q₈×C₈₉	—	2	1	no	—
109	C₂×Q₈×C₈₉	—	2	1	no	—
110	C₂×Q₈×C₈₉	—	2	1	no	—
111	C₂×Q₈×C₈₉	—	2	1	no	—
112	C₂×Q₈×C₈₉	—	2	1	no	—
113	C₂×Q₈×C₈₉	—	8	1	no	—
114	C₂×Q₈×C₈₉	—	8	1	no	—
115	C₂×Q₈×C₈₉	—	8	2	no	—
116	C₂×Q₈×C₈₉	—	8	2	no	—
117	C₂×Q₈×C₈₉	Q₁₆	1	4	yes	complemented
118	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
119	C₂×Q₈×C₈₉	C₂×Q₈	1	2	yes	—
120	C₂×Q₈×C₈₉	C₂×Q₈	1	2	yes	—
121	C₂×Q₈×C₈₉	#41	1	2	yes	—
122	C₂²×C₄×C₈₉	—	2	1	no	—
123	C₂²×C₄×C₈₉	—	2	1	no	—
124	C₂²×C₄×C₈₉	—	2	2	no	—
125	C₂²×C₄×C₈₉	—	2	1	no	—
126	C₂²×C₄×C₈₉	—	2	2	no	—
127	C₂²×C₄×C₈₉	—	2	1	no	—
128	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
129	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
130	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
131	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
133	Q₁₆×C₈₉	—	2	8	no	—
134	C₂×C₈×C₈₉	—	2	1	no	—
135	C₂×C₈×C₈₉	—	2	1	no	—
136	C₂×C₈×C₈₉	—	2	1	no	—
137	C₂×C₈×C₈₉	—	2	2	no	—
138	C₂×C₈×C₈₉	—	2	2	no	—
139	C₂×C₈×C₈₉	—	2	1	no	—
140	C₂×C₈×C₈₉	—	2	1	no	—
141	C₂×C₈×C₈₉	—	2	2	no	—
142	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
143	C₄⋊C₄×C₈₉	—	2	8	no	—
144	C₄⋊C₄×C₈₉	—	2	8	no	—
145	C₄⋊C₄×C₈₉	—	2	4	no	—
146	C₄⋊C₄×C₈₉	—	2	4	no	—
147	C₄⋊C₄×C₈₉	—	2	4	no	—
148	C₄⋊C₄×C₈₉	—	2	2	no	—
149	C₄⋊C₄×C₈₉	—	2	2	no	—
150	C₄⋊C₄×C₈₉	—	2	1	no	—
151	C₄⋊C₄×C₈₉	—	2	1	no	—
152	C₄⋊C₄×C₈₉	—	2	1	no	—
153	C₄⋊C₄×C₈₉	—	2	2	no	—
154	C₄⋊C₄×C₈₉	—	2	2	no	—
155	C₄⋊C₄×C₈₉	—	4	2	no	—
156	C₄⋊C₄×C₈₉	—	4	2	no	—
157	C₄⋊C₄×C₈₉	—	8	1	no	—
158	C₄⋊C₄×C₈₉	—	8	2	no	—
159	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
160	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
161	C₄²×C₈₉	—	2	8	no	—
162	C₄²×C₈₉	—	2	4	no	—
163	C₄²×C₈₉	—	2	2	no	—
164	C₄²×C₈₉	—	2	2	no	—
165	C₄²×C₈₉	—	4	2	no	—
166	C₄²×C₈₉	C₂×D₄	1	2	yes	—
167	C₄²×C₈₉	C₂×D₄	1	2	yes	—
168	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
169	Q₈×C₈₉	—	2	2	no	—
170	Q₈×C₈₉	—	2	2	no	—
171	Q₈×C₈₉	—	2	2	no	—
172	Q₈×C₈₉	—	2	8	no	—
173	Q₈×C₈₉	—	2	2	no	—
174	Q₈×C₈₉	—	2	2	no	—
175	Q₈×C₈₉	—	4	1	no	—
176	Q₈×C₈₉	—	4	1	no	—
177	Q₈×C₈₉	—	8	4	no	—
178	Q₈×C₈₉	—	8	8	no	—
179	Q₈×C₈₉	C₂²⋊Q₈	1	4	yes	—
180	C₂×C₄×C₈₉	—	2	2	no	—
181	C₂×C₄×C₈₉	—	2	2	no	—
182	C₂×C₄×C₈₉	—	2	2	no	—
183	C₂×C₄×C₈₉	—	2	2	no	—
184	C₂×C₄×C₈₉	—	2	2	no	—
185	C₂×C₄×C₈₉	—	2	1	no	—
186	C₂×C₄×C₈₉	—	2	1	no	—
187	C₂×C₄×C₈₉	—	2	2	no	—
188	C₂×C₄×C₈₉	—	2	4	no	—
189	C₂×C₄×C₈₉	—	2	2	no	—
190	C₂×C₄×C₈₉	—	2	2	no	—
191	C₂×C₄×C₈₉	—	2	2	no	—
192	C₂×C₄×C₈₉	—	2	2	no	—
193	C₂×C₄×C₈₉	—	4	1	no	—
194	C₂×C₄×C₈₉	—	4	1	no	—
195	C₂×C₄×C₈₉	—	4	2	no	—
196	C₂×C₄×C₈₉	—	4	1	no	—
197	C₂×C₄×C₈₉	—	4	2	no	—
198	C₂×C₄×C₈₉	—	4	1	no	—
199	C₂×C₄×C₈₉	—	4	1	no	—
200	C₂×C₄×C₈₉	—	4	2	no	—
201	C₂×C₄×C₈₉	—	4	2	no	—
202	C₂×C₄×C₈₉	—	4	2	no	—
203	C₂×C₄×C₈₉	—	4	1	no	—
204	C₂×C₄×C₈₉	—	4	1	no	—
205	C₂×C₄×C₈₉	#120	1	2	yes	—
206	C₂×C₄×C₈₉	#120	1	2	yes	—
207	C₂×C₄×C₈₉	C₂×Q₁₆	1	4	yes	—
208	C₂×C₄×C₈₉	C₂×Q₁₆	1	2	yes	—
209	C₂×C₄×C₈₉	#137	1	2	yes	—
210	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
211	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
212	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
213	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
214	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
215	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
216	C₈×C₈₉	—	2	2	no	—
217	C₈×C₈₉	—	4	1	no	—
218	C₈×C₈₉	—	4	2	no	—
219	C₂²×C₈₉	C₂×#120	1	1	yes	characteristic
220	C₂²×C₈₉	#548	1	1	yes	characteristic
221	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
222	C₂²×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
223	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
224	C₂²×C₈₉	#576	1	1	yes	characteristic
225	C₂²×C₈₉	#577	1	1	yes	characteristic
226	C₄×C₈₉	—	2	4	no	—
227	C₄×C₈₉	—	2	4	no	—
228	C₄×C₈₉	—	2	2	no	—
229	C₄×C₈₉	—	4	2	no	—
230	C₄×C₈₉	—	4	2	no	—
231	C₄×C₈₉	—	4	4	no	—
232	C₄×C₈₉	—	4	2	no	—
233	C₄×C₈₉	—	8	2	no	—
234	C₄×C₈₉	—	8	1	no	—
235	C₄×C₈₉	#451	1	4	yes	—
236	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
237	C₄×C₈₉	#536	1	1	yes	characteristic
238	C₄×C₈₉	#548	1	1	yes	characteristic
239	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
240	#24333	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
241	C₂×C₈₉	#2409	1	1	yes	characteristic
242	C₂×C₈₉	C₂×#451	1	1	yes	characteristic
243	C₂×C₈₉	#3012	1	1	yes	characteristic
244	C₂×C₈₉	#3363	1	1	yes	characteristic
245	C₂×C₈₉	#3375	1	1	yes	characteristic
246	C₂×C₈₉	Q₈×Q₁₆	1	1	yes	characteristic
247	C₂×C₈₉	#3387	1	1	yes	characteristic
248	C₂×Q₈²	C₂×C₈₉	1	1	yes	characteristic
249	C₂×#477	C₂×C₈₉	1	1	yes	characteristic
250	C₂×C₄×Q₁₆	C₂×C₈₉	1	1	yes	characteristic
251	#2582	C₂×C₈₉	1	1	yes	characteristic
252	#2057	C₂×C₈₉	1	2	yes	—
253	#2024	C₂×C₈₉	1	2	yes	—
254	#1990	C₂×C₈₉	1	1	yes	characteristic
255	#1984	C₂×C₈₉	1	1	yes	characteristic
256	#1878	C₂×C₈₉	1	2	yes	—
257	#1863	C₂×C₈₉	1	2	yes	—
258	#1803	C₂×C₈₉	1	1	yes	characteristic
259	C₈₉	#24333	1	1	yes	characteristic, direct product factor, Sylow
260	C₂²×Q₁₆	C₂²×C₈₉	1	1	yes	characteristic
261	Q₈²	—	2	4	no	—
262	Q₈²	—	2	4	no	—
263	C₂×C₄⋊Q₈	—	2	1	no	—
264	C₂×C₄⋊Q₈	—	2	2	no	—
265	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
266	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
267	C₂×C₄×Q₈	—	2	1	no	—
268	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
269	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
270	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
271	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
272	#477	—	2	4	no	—
273	C₄×Q₁₆	—	2	4	no	—
274	C₂×#107	C₂²×C₈₉	1	2	yes	—
275	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
276	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
277	C₂×#103	C₂²×C₈₉	1	2	yes	—
278	C₂×#103	C₂²×C₈₉	1	2	yes	—
279	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
280	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
281	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
282	#398	—	2	2	no	—
283	#391	—	2	1	no	—
284	#391	C₂²×C₈₉	1	2	yes	—
285	#389	—	2	1	no	—
286	#389	C₂²×C₈₉	1	2	yes	—
287	#384	C₂²×C₈₉	1	1	yes	characteristic
288	#340	C₂²×C₈₉	1	2	yes	—
289	#340	C₂²×C₈₉	1	2	yes	—
290	#340	C₂²×C₈₉	1	2	yes	—
291	#336	C₂²×C₈₉	1	2	yes	—
292	#336	C₂²×C₈₉	1	2	yes	—
293	C₂²×Q₈	—	2	2	no	—
294	C₂²×Q₈	—	2	1	no	—
295	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
296	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
297	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
298	C₂×Q₁₆	—	2	2	no	—
299	C₂×Q₁₆	—	2	2	no	—
300	C₂×Q₁₆	Q₈×C₈₉	1	4	yes	complemented
301	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
302	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
303	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
304	C₄⋊Q₈	—	2	16	no	—
305	C₄⋊Q₈	—	2	8	no	—
306	C₄⋊Q₈	—	2	4	no	—
307	C₄⋊Q₈	—	2	4	no	—
308	C₄⋊Q₈	—	2	2	no	—
309	C₄⋊Q₈	D₄×C₈₉	1	4	yes	—
310	C₄×Q₈	—	2	8	no	—
311	C₄×Q₈	—	2	4	no	—
312	C₄×Q₈	—	2	4	no	—
313	C₄×Q₈	—	2	2	no	—
314	C₄×Q₈	—	2	2	no	—
315	C₄×Q₈	—	2	2	no	—
316	C₄×Q₈	—	2	2	no	—
317	C₄×Q₈	—	2	2	no	—
318	C₄×Q₈	D₄×C₈₉	1	4	yes	—
319	C₂×C₄⋊C₄	—	2	2	no	—
320	C₂×C₄⋊C₄	—	2	1	no	—
321	C₂×C₄⋊C₄	—	2	2	no	—
322	C₂×C₄⋊C₄	—	2	1	no	—
323	C₂×C₄⋊C₄	—	2	1	no	—
324	C₂×C₄⋊C₄	—	2	1	no	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₄⋊C₄	—	2	2	no	—
327	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
328	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
329	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
330	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
331	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
332	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
333	C₂×C₄²	—	2	2	no	—
334	C₂×C₄²	—	2	1	no	—
335	C₂×C₄²	—	2	1	no	—
336	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
337	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
338	#107	—	2	2	no	—
339	#107	—	2	4	no	—
340	C₄⋊C₈	—	2	2	no	—
341	#103	—	2	2	no	—
342	#103	—	2	4	no	—
343	#103	—	2	2	no	—
344	#103	—	2	4	no	—
345	C₄×C₈	—	2	2	no	—
346	#95	—	2	2	no	—
347	#95	—	2	2	no	—
348	#95	—	2	2	no	—
349	C₂×Q₈	—	2	8	no	—
350	C₂×Q₈	—	2	2	no	—
351	C₂×Q₈	—	2	4	no	—
352	C₂×Q₈	—	2	2	no	—
353	C₂×Q₈	—	2	1	no	—
354	C₂×Q₈	—	2	1	no	—
355	C₂×Q₈	—	2	1	no	—
356	C₂×Q₈	—	2	1	no	—
357	C₂×Q₈	—	2	1	no	—
358	C₂×Q₈	—	2	1	no	—
359	C₂×Q₈	—	2	1	no	—
360	C₂×Q₈	—	2	1	no	—
361	C₂×Q₈	—	8	1	no	—
362	C₂×Q₈	—	8	1	no	—
363	C₂×Q₈	—	8	2	no	—
364	C₂×Q₈	—	8	2	no	—
365	C₂×Q₈	Q₁₆×C₈₉	1	4	yes	complemented
366	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
367	C₂×Q₈	C₂×Q₈×C₈₉	1	2	yes	—
368	C₂×Q₈	C₂×Q₈×C₈₉	1	2	yes	—
369	C₂×Q₈	#41×C₈₉	1	2	yes	—
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	2	no	—
373	C₂²×C₄	—	2	1	no	—
374	C₂²×C₄	—	2	2	no	—
375	C₂²×C₄	—	2	1	no	—
376	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
377	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
378	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
379	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
380	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
381	Q₁₆	—	2	8	no	—
382	C₂×C₈	—	2	1	no	—
383	C₂×C₈	—	2	1	no	—
384	C₂×C₈	—	2	1	no	—
385	C₂×C₈	—	2	2	no	—
386	C₂×C₈	—	2	2	no	—
387	C₂×C₈	—	2	1	no	—
388	C₂×C₈	—	2	1	no	—
389	C₂×C₈	—	2	2	no	—
390	C₂×C₈	C₂×Q₈×C₈₉	1	2	yes	—
391	C₄⋊C₄	—	2	8	no	—
392	C₄⋊C₄	—	2	8	no	—
393	C₄⋊C₄	—	2	4	no	—
394	C₄⋊C₄	—	2	4	no	—
395	C₄⋊C₄	—	2	4	no	—
396	C₄⋊C₄	—	2	2	no	—
397	C₄⋊C₄	—	2	2	no	—
398	C₄⋊C₄	—	2	1	no	—
399	C₄⋊C₄	—	2	1	no	—
400	C₄⋊C₄	—	2	1	no	—
401	C₄⋊C₄	—	2	2	no	—
402	C₄⋊C₄	—	2	2	no	—
403	C₄⋊C₄	—	4	2	no	—
404	C₄⋊C₄	—	4	2	no	—
405	C₄⋊C₄	—	8	1	no	—
406	C₄⋊C₄	—	8	2	no	—
407	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
408	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
409	C₄²	—	2	8	no	—
410	C₄²	—	2	4	no	—
411	C₄²	—	2	2	no	—
412	C₄²	—	2	2	no	—
413	C₄²	—	4	2	no	—
414	C₄²	C₂×D₄×C₈₉	1	2	yes	—
415	C₄²	C₂×D₄×C₈₉	1	2	yes	—
416	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
417	Q₈	—	2	8	no	—
418	Q₈	—	2	2	no	—
419	Q₈	—	2	2	no	—
420	Q₈	—	2	2	no	—
421	Q₈	—	2	2	no	—
422	Q₈	—	2	2	no	—
423	Q₈	—	4	1	no	—
424	Q₈	—	4	1	no	—
425	Q₈	—	8	4	no	—
426	Q₈	—	8	8	no	—
427	Q₈	C₂²⋊Q₈×C₈₉	1	4	yes	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	2	2	no	—
432	C₂×C₄	—	2	2	no	—
433	C₂×C₄	—	2	1	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	2	no	—
436	C₂×C₄	—	2	4	no	—
437	C₂×C₄	—	2	2	no	—
438	C₂×C₄	—	2	2	no	—
439	C₂×C₄	—	2	2	no	—
440	C₂×C₄	—	2	2	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	1	no	—
443	C₂×C₄	—	4	2	no	—
444	C₂×C₄	—	4	1	no	—
445	C₂×C₄	—	4	2	no	—
446	C₂×C₄	—	4	1	no	—
447	C₂×C₄	—	4	1	no	—
448	C₂×C₄	—	4	2	no	—
449	C₂×C₄	—	4	2	no	—
450	C₂×C₄	—	4	2	no	—
451	C₂×C₄	—	4	1	no	—
452	C₂×C₄	—	4	1	no	—
453	C₂×C₄	#120×C₈₉	1	2	yes	—
454	C₂×C₄	#120×C₈₉	1	2	yes	—
455	C₂×C₄	C₂×Q₁₆×C₈₉	1	4	yes	—
456	C₂×C₄	C₂×Q₁₆×C₈₉	1	2	yes	—
457	C₂×C₄	#137×C₈₉	1	2	yes	—
458	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
459	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
460	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
461	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
462	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
463	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
464	C₈	—	2	2	no	—
465	C₈	—	4	1	no	—
466	C₈	—	4	2	no	—
467	C₂²	C₂×#120×C₈₉	1	1	yes	characteristic
468	C₂²	#548×C₈₉	1	1	yes	characteristic
469	C₂²	D₄×Q₈×C₈₉	1	1	yes	characteristic
470	C₂²	C₂²×Q₁₆×C₈₉	1	1	yes	characteristic
471	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
472	C₂²	#576×C₈₉	1	1	yes	characteristic
473	C₂²	#577×C₈₉	1	1	yes	characteristic
474	C₄	—	2	4	no	—
475	C₄	—	2	4	no	—
476	C₄	—	2	2	no	—
477	C₄	—	4	2	no	—
478	C₄	—	4	2	no	—
479	C₄	—	4	4	no	—
480	C₄	—	4	2	no	—
481	C₄	—	8	2	no	—
482	C₄	—	8	1	no	—
483	C₄	#451×C₈₉	1	4	yes	—
484	C₄	C₂×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
485	C₄	#536×C₈₉	1	1	yes	characteristic
486	C₄	#548×C₈₉	1	1	yes	characteristic
487	C₄	D₄×Q₈×C₈₉	1	1	yes	characteristic
488	C₂	C₈₉×#2409	1	1	yes	characteristic
489	C₂	C₂×#451×C₈₉	1	1	yes	characteristic
490	C₂	C₈₉×#3012	1	1	yes	characteristic
491	C₂	C₈₉×#3363	1	1	yes	characteristic
492	C₂	C₈₉×#3375	1	1	yes	characteristic
493	C₂	Q₈×Q₁₆×C₈₉	1	1	yes	characteristic
494	C₂	C₈₉×#3387	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
LCS	G ▷ C₂²×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
52. C₂×Q₁₆×C₈₉	Q₈	semidirect
117. C₂×Q₈×C₈₉	Q₁₆	semidirect
240. #24333	C₈₉	direct (N × H)
259. C₈₉	#24333	direct (N × H)
300. C₂×Q₁₆	Q₈×C₈₉	semidirect
365. C₂×Q₈	Q₁₆×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×#477×C₈₉	2
C₂×C₄×Q₁₆×C₈₉	2
C₈₉×#2582	2
C₈₉×#2057	2
C₈₉×#2024	2
C₈₉×#1990	2
C₈₉×#1984	2
C₈₉×#1878	2
C₈₉×#1863	2
C₈₉×#1803	2
#24333	89

Subgroups — C89×#24333 #36402181

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#24333 #36402181