Subgroups — C₈₉×#25060 #36402908

Order 22784.

698 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2582	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2096	C₂	1	2	yes	maximal
4	C₈₉×#2084	C₂	1	2	yes	maximal
5	C₈₉×#1984	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#1948	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1927	C₂	1	2	yes	maximal
8	C₈₉×#1920	C₂	1	2	yes	maximal
9	C₈₉×#1844	C₂	1	1	yes	maximal, characteristic
10	C₈₉×#1803	C₂	1	1	yes	maximal, characteristic
11	C₄×#103×C₈₉	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
13	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#106×C₈₉	C₂²	1	2	yes	—
16	C₂×#106×C₈₉	C₂²	1	2	yes	—
17	C₂×#106×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#103×C₈₉	C₂²	1	2	yes	—
21	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
23	#398×C₈₉	—	2	2	no	—
24	#391×C₈₉	—	2	1	no	—
25	#391×C₈₉	C₂²	1	2	yes	—
26	#389×C₈₉	—	2	1	no	—
27	#389×C₈₉	—	2	1	no	—
28	#389×C₈₉	—	2	2	no	—
29	#389×C₈₉	C₂²	1	2	yes	—
30	#384×C₈₉	C₂²	1	1	yes	characteristic
31	#384×C₈₉	C₂²	1	1	yes	characteristic
32	#384×C₈₉	C₂²	1	2	yes	—
33	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
35	#340×C₈₉	C₂²	1	2	yes	—
36	#340×C₈₉	C₂²	1	1	yes	characteristic
37	#340×C₈₉	C₂²	1	1	yes	characteristic
38	#340×C₈₉	C₂²	1	2	yes	—
39	#336×C₈₉	C₂²	1	2	yes	—
40	#336×C₈₉	C₂²	1	2	yes	—
41	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
42	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
44	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
46	C₄×Q₈×C₈₉	—	2	2	no	—
47	C₄×Q₈×C₈₉	—	2	2	no	—
48	C₄×Q₈×C₈₉	—	2	2	no	—
49	C₄×Q₈×C₈₉	—	2	2	no	—
50	C₄×Q₈×C₈₉	—	4	4	no	—
51	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
52	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
53	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
55	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
56	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
57	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
58	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
59	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
60	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
61	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
65	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₈₉	—	2	2	no	—
67	C₂×C₄²×C₈₉	—	2	1	no	—
68	C₂×C₄²×C₈₉	—	2	1	no	—
69	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
70	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
71	#106×C₈₉	—	2	4	no	—
72	#106×C₈₉	—	4	2	no	—
73	#106×C₈₉	—	4	2	no	—
74	C₄⋊C₈×C₈₉	—	2	2	no	—
75	#103×C₈₉	—	2	2	no	—
76	#103×C₈₉	—	4	2	no	—
77	#103×C₈₉	Q₈	1	4	yes	complemented
78	C₄×C₈×C₈₉	—	2	2	no	—
79	#95×C₈₉	—	2	2	no	—
80	#95×C₈₉	—	2	1	no	—
81	#95×C₈₉	—	2	2	no	—
82	#95×C₈₉	—	2	2	no	—
83	#95×C₈₉	—	2	2	no	—
84	C₂×Q₈×C₈₉	—	2	2	no	—
85	C₂×Q₈×C₈₉	—	2	2	no	—
86	C₂×Q₈×C₈₉	—	2	1	no	—
87	C₂×Q₈×C₈₉	—	2	1	no	—
88	C₂×Q₈×C₈₉	—	2	1	no	—
89	C₂×Q₈×C₈₉	—	2	1	no	—
90	C₂×Q₈×C₈₉	—	2	1	no	—
91	C₂×Q₈×C₈₉	SD₁₆	1	4	yes	—
92	C₂×Q₈×C₈₉	C₂×Q₈	1	2	yes	—
93	C₂²×C₄×C₈₉	—	2	1	no	—
94	C₂²×C₄×C₈₉	—	2	1	no	—
95	C₂²×C₄×C₈₉	—	2	2	no	—
96	C₂²×C₄×C₈₉	—	2	1	no	—
97	C₂²×C₄×C₈₉	—	2	1	no	—
98	C₂²×C₄×C₈₉	—	2	2	no	—
99	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
104	C₂×C₈×C₈₉	—	2	1	no	—
105	C₂×C₈×C₈₉	—	2	1	no	—
106	C₂×C₈×C₈₉	—	2	1	no	—
107	C₂×C₈×C₈₉	—	2	1	no	—
108	C₂×C₈×C₈₉	—	2	2	no	—
109	C₂×C₈×C₈₉	—	2	2	no	—
110	C₂×C₈×C₈₉	—	2	2	no	—
111	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
112	C₂×C₈×C₈₉	#41	1	2	yes	—
113	C₄⋊C₄×C₈₉	—	2	4	no	—
114	C₄⋊C₄×C₈₉	—	2	2	no	—
115	C₄⋊C₄×C₈₉	—	2	1	no	—
116	C₄⋊C₄×C₈₉	—	2	1	no	—
117	C₄⋊C₄×C₈₉	—	2	2	no	—
118	C₄⋊C₄×C₈₉	—	2	2	no	—
119	C₄⋊C₄×C₈₉	—	2	1	no	—
120	C₄⋊C₄×C₈₉	—	2	1	no	—
121	C₄⋊C₄×C₈₉	—	2	2	no	—
122	C₄⋊C₄×C₈₉	—	2	1	no	—
123	C₄⋊C₄×C₈₉	—	2	2	no	—
124	C₄⋊C₄×C₈₉	—	4	2	no	—
125	C₄⋊C₄×C₈₉	—	4	4	no	—
126	C₄⋊C₄×C₈₉	—	4	2	no	—
127	C₄⋊C₄×C₈₉	—	4	2	no	—
128	C₄⋊C₄×C₈₉	—	8	1	no	—
129	C₄⋊C₄×C₈₉	—	8	1	no	—
130	C₄⋊C₄×C₈₉	—	8	2	no	—
131	C₄⋊C₄×C₈₉	—	8	2	no	—
132	C₄⋊C₄×C₈₉	C₂×Q₈	1	2	yes	—
133	C₄²×C₈₉	—	2	2	no	—
134	C₄²×C₈₉	—	2	2	no	—
135	C₄²×C₈₉	—	4	2	no	—
136	C₄²×C₈₉	—	4	4	no	—
137	C₄²×C₈₉	—	4	2	no	—
138	C₄²×C₈₉	#41	1	2	yes	—
139	C₄²×C₈₉	#41	1	2	yes	—
140	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
141	Q₈×C₈₉	—	2	8	no	—
142	Q₈×C₈₉	—	2	2	no	—
143	Q₈×C₈₉	—	2	2	no	—
144	Q₈×C₈₉	—	4	1	no	—
145	Q₈×C₈₉	—	4	1	no	—
146	C₂×C₄×C₈₉	—	2	2	no	—
147	C₂×C₄×C₈₉	—	2	2	no	—
148	C₂×C₄×C₈₉	—	2	1	no	—
149	C₂×C₄×C₈₉	—	2	1	no	—
150	C₂×C₄×C₈₉	—	2	2	no	—
151	C₂×C₄×C₈₉	—	2	2	no	—
152	C₂×C₄×C₈₉	—	2	2	no	—
153	C₂×C₄×C₈₉	—	2	2	no	—
154	C₂×C₄×C₈₉	—	2	2	no	—
155	C₂×C₄×C₈₉	—	4	1	no	—
156	C₂×C₄×C₈₉	—	4	1	no	—
157	C₂×C₄×C₈₉	—	4	1	no	—
158	C₂×C₄×C₈₉	—	4	1	no	—
159	C₂×C₄×C₈₉	—	4	1	no	—
160	C₂×C₄×C₈₉	—	4	1	no	—
161	C₂×C₄×C₈₉	—	4	2	no	—
162	C₂×C₄×C₈₉	—	4	2	no	—
163	C₂×C₄×C₈₉	—	4	2	no	—
164	C₂×C₄×C₈₉	—	4	1	no	—
165	C₂×C₄×C₈₉	—	4	2	no	—
166	C₂×C₄×C₈₉	—	4	1	no	—
167	C₂×C₄×C₈₉	—	4	2	no	—
168	C₂×C₄×C₈₉	—	4	1	no	—
169	C₂×C₄×C₈₉	—	4	2	no	—
170	C₂×C₄×C₈₉	—	4	1	no	—
171	C₂×C₄×C₈₉	#123	1	2	yes	—
172	C₂×C₄×C₈₉	#123	1	2	yes	—
173	C₂×C₄×C₈₉	C₂×SD₁₆	1	4	yes	—
174	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
175	C₂×C₄×C₈₉	#137	1	2	yes	—
176	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
177	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
178	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
179	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
180	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
181	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
182	C₈×C₈₉	—	2	2	no	—
183	C₈×C₈₉	—	2	2	no	—
184	C₈×C₈₉	—	4	2	no	—
185	C₂²×C₈₉	C₂×#123	1	1	yes	characteristic
186	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
187	C₂²×C₈₉	#550	1	1	yes	characteristic
188	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
189	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
190	C₂²×C₈₉	#577	1	1	yes	characteristic
191	C₂²×C₈₉	#577	1	1	yes	characteristic
192	C₄×C₈₉	—	2	4	no	—
193	C₄×C₈₉	—	2	2	no	—
194	C₄×C₈₉	—	4	2	no	—
195	C₄×C₈₉	—	4	2	no	—
196	C₄×C₈₉	—	4	2	no	—
197	C₄×C₈₉	—	8	1	no	—
198	C₄×C₈₉	—	8	1	no	—
199	C₄×C₈₉	—	8	2	no	—
200	C₄×C₈₉	—	8	2	no	—
201	C₄×C₈₉	#483	1	4	yes	—
202	C₄×C₈₉	#533	1	1	yes	characteristic
203	C₄×C₈₉	#542	1	1	yes	characteristic
204	C₄×C₈₉	#552	1	1	yes	characteristic
205	C₄×C₈₉	#554	1	1	yes	characteristic
206	#25060	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
207	C₂×C₈₉	#2423	1	1	yes	characteristic
208	C₂×C₈₉	C₂×#483	1	1	yes	characteristic
209	C₂×C₈₉	#3099	1	1	yes	characteristic
210	C₂×C₈₉	Q₈×SD₁₆	1	1	yes	characteristic
211	C₂×C₈₉	#3387	1	1	yes	characteristic
212	C₂×C₈₉	#3392	1	1	yes	characteristic
213	C₂×C₈₉	#3400	1	1	yes	characteristic
214	#2582	C₂×C₈₉	1	1	yes	characteristic
215	Q₈×C₄⋊C₄	C₂×C₈₉	1	1	yes	characteristic
216	#2096	C₂×C₈₉	1	2	yes	—
217	#2084	C₂×C₈₉	1	2	yes	—
218	#1984	C₂×C₈₉	1	1	yes	characteristic
219	#1948	C₂×C₈₉	1	1	yes	characteristic
220	#1927	C₂×C₈₉	1	2	yes	—
221	#1920	C₂×C₈₉	1	2	yes	—
222	#1844	C₂×C₈₉	1	1	yes	characteristic
223	#1803	C₂×C₈₉	1	1	yes	characteristic
224	C₄×#103	C₂×C₈₉	1	1	yes	characteristic
225	C₈₉	#25060	1	1	yes	characteristic, direct product factor, Sylow
226	C₂×C₄×Q₈	—	2	1	no	—
227	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
228	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
229	C₂×#106	C₂²×C₈₉	1	2	yes	—
230	C₂×#106	C₂²×C₈₉	1	2	yes	—
231	C₂×#106	C₂²×C₈₉	1	2	yes	—
232	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
233	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
234	C₂×#103	C₂²×C₈₉	1	2	yes	—
235	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
236	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
237	#398	—	2	2	no	—
238	#391	—	2	1	no	—
239	#391	C₂²×C₈₉	1	2	yes	—
240	#389	—	2	1	no	—
241	#389	—	2	1	no	—
242	#389	—	2	2	no	—
243	#389	C₂²×C₈₉	1	2	yes	—
244	#384	C₂²×C₈₉	1	1	yes	characteristic
245	#384	C₂²×C₈₉	1	1	yes	characteristic
246	#384	C₂²×C₈₉	1	2	yes	—
247	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
248	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
249	#340	C₂²×C₈₉	1	2	yes	—
250	#340	C₂²×C₈₉	1	1	yes	characteristic
251	#340	C₂²×C₈₉	1	1	yes	characteristic
252	#340	C₂²×C₈₉	1	2	yes	—
253	#336	C₂²×C₈₉	1	2	yes	—
254	#336	C₂²×C₈₉	1	2	yes	—
255	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
256	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
257	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
258	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
259	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
260	C₄×Q₈	—	2	2	no	—
261	C₄×Q₈	—	2	2	no	—
262	C₄×Q₈	—	2	2	no	—
263	C₄×Q₈	—	2	2	no	—
264	C₄×Q₈	—	4	4	no	—
265	C₂×C₄⋊C₄	—	2	2	no	—
266	C₂×C₄⋊C₄	—	2	1	no	—
267	C₂×C₄⋊C₄	—	2	1	no	—
268	C₂×C₄⋊C₄	—	2	1	no	—
269	C₂×C₄⋊C₄	—	2	2	no	—
270	C₂×C₄⋊C₄	—	2	1	no	—
271	C₂×C₄⋊C₄	—	2	1	no	—
272	C₂×C₄⋊C₄	—	2	2	no	—
273	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
274	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
275	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
276	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
277	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
278	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
279	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
280	C₂×C₄²	—	2	2	no	—
281	C₂×C₄²	—	2	1	no	—
282	C₂×C₄²	—	2	1	no	—
283	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
284	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
285	#106	—	2	4	no	—
286	#106	—	4	2	no	—
287	#106	—	4	2	no	—
288	C₄⋊C₈	—	2	2	no	—
289	#103	—	2	2	no	—
290	#103	—	4	2	no	—
291	#103	Q₈×C₈₉	1	4	yes	complemented
292	C₄×C₈	—	2	2	no	—
293	#95	—	2	2	no	—
294	#95	—	2	1	no	—
295	#95	—	2	2	no	—
296	#95	—	2	2	no	—
297	#95	—	2	2	no	—
298	C₂×Q₈	—	2	2	no	—
299	C₂×Q₈	—	2	2	no	—
300	C₂×Q₈	—	2	1	no	—
301	C₂×Q₈	—	2	1	no	—
302	C₂×Q₈	—	2	1	no	—
303	C₂×Q₈	—	2	1	no	—
304	C₂×Q₈	—	2	1	no	—
305	C₂×Q₈	SD₁₆×C₈₉	1	4	yes	—
306	C₂×Q₈	C₂×Q₈×C₈₉	1	2	yes	—
307	C₂²×C₄	—	2	1	no	—
308	C₂²×C₄	—	2	1	no	—
309	C₂²×C₄	—	2	2	no	—
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
314	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
315	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
316	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
317	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
318	C₂×C₈	—	2	1	no	—
319	C₂×C₈	—	2	1	no	—
320	C₂×C₈	—	2	1	no	—
321	C₂×C₈	—	2	1	no	—
322	C₂×C₈	—	2	2	no	—
323	C₂×C₈	—	2	2	no	—
324	C₂×C₈	—	2	2	no	—
325	C₂×C₈	C₂×Q₈×C₈₉	1	2	yes	—
326	C₂×C₈	#41×C₈₉	1	2	yes	—
327	C₄⋊C₄	—	2	4	no	—
328	C₄⋊C₄	—	2	2	no	—
329	C₄⋊C₄	—	2	1	no	—
330	C₄⋊C₄	—	2	1	no	—
331	C₄⋊C₄	—	2	2	no	—
332	C₄⋊C₄	—	2	2	no	—
333	C₄⋊C₄	—	2	1	no	—
334	C₄⋊C₄	—	2	1	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	1	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	4	2	no	—
339	C₄⋊C₄	—	4	4	no	—
340	C₄⋊C₄	—	4	2	no	—
341	C₄⋊C₄	—	4	2	no	—
342	C₄⋊C₄	—	8	1	no	—
343	C₄⋊C₄	—	8	1	no	—
344	C₄⋊C₄	—	8	2	no	—
345	C₄⋊C₄	—	8	2	no	—
346	C₄⋊C₄	C₂×Q₈×C₈₉	1	2	yes	—
347	C₄²	—	2	2	no	—
348	C₄²	—	2	2	no	—
349	C₄²	—	4	2	no	—
350	C₄²	—	4	4	no	—
351	C₄²	—	4	2	no	—
352	C₄²	#41×C₈₉	1	2	yes	—
353	C₄²	#41×C₈₉	1	2	yes	—
354	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
355	Q₈	—	2	8	no	—
356	Q₈	—	2	2	no	—
357	Q₈	—	2	2	no	—
358	Q₈	—	4	1	no	—
359	Q₈	—	4	1	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	1	no	—
363	C₂×C₄	—	2	1	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	4	1	no	—
370	C₂×C₄	—	4	1	no	—
371	C₂×C₄	—	4	1	no	—
372	C₂×C₄	—	4	1	no	—
373	C₂×C₄	—	4	1	no	—
374	C₂×C₄	—	4	1	no	—
375	C₂×C₄	—	4	2	no	—
376	C₂×C₄	—	4	2	no	—
377	C₂×C₄	—	4	2	no	—
378	C₂×C₄	—	4	1	no	—
379	C₂×C₄	—	4	2	no	—
380	C₂×C₄	—	4	1	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	#123×C₈₉	1	2	yes	—
386	C₂×C₄	#123×C₈₉	1	2	yes	—
387	C₂×C₄	C₂×SD₁₆×C₈₉	1	4	yes	—
388	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
389	C₂×C₄	#137×C₈₉	1	2	yes	—
390	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
391	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
392	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
393	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
394	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
395	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
396	C₈	—	2	2	no	—
397	C₈	—	2	2	no	—
398	C₈	—	4	2	no	—
399	C₂²	C₂×#123×C₈₉	1	1	yes	characteristic
400	C₂²	D₄×Q₈×C₈₉	1	1	yes	characteristic
401	C₂²	#550×C₈₉	1	1	yes	characteristic
402	C₂²	C₂²×SD₁₆×C₈₉	1	1	yes	characteristic
403	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
404	C₂²	#577×C₈₉	1	1	yes	characteristic
405	C₂²	#577×C₈₉	1	1	yes	characteristic
406	C₄	—	2	4	no	—
407	C₄	—	2	2	no	—
408	C₄	—	4	2	no	—
409	C₄	—	4	2	no	—
410	C₄	—	4	2	no	—
411	C₄	—	8	1	no	—
412	C₄	—	8	1	no	—
413	C₄	—	8	2	no	—
414	C₄	—	8	2	no	—
415	C₄	#483×C₈₉	1	4	yes	—
416	C₄	#533×C₈₉	1	1	yes	characteristic
417	C₄	#542×C₈₉	1	1	yes	characteristic
418	C₄	#552×C₈₉	1	1	yes	characteristic
419	C₄	#554×C₈₉	1	1	yes	characteristic
420	C₂	C₈₉×#2423	1	1	yes	characteristic
421	C₂	C₂×#483×C₈₉	1	1	yes	characteristic
422	C₂	C₈₉×#3099	1	1	yes	characteristic
423	C₂	Q₈×SD₁₆×C₈₉	1	1	yes	characteristic
424	C₂	C₈₉×#3387	1	1	yes	characteristic
425	C₂	C₈₉×#3392	1	1	yes	characteristic
426	C₂	C₈₉×#3400	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
77. #103×C₈₉	Q₈	semidirect
206. #25060	C₈₉	direct (N × H)
225. C₈₉	#25060	direct (N × H)
291. #103	Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#2582	2
Q₈×C₄⋊C₄×C₈₉	2
C₈₉×#2096	2
C₈₉×#2084	2
C₈₉×#1984	2
C₈₉×#1948	2
C₈₉×#1927	2
C₈₉×#1920	2
C₈₉×#1844	2
C₈₉×#1803	2
C₄×#103×C₈₉	2
#25060	89

Subgroups — C89×#25060 #36402908

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#25060 #36402908