Subgroups — C₈₉×#24326 #36402174

Order 22784.

1330 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₄×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#474×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×D₈×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2538	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2057	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2023	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1990	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#1984	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1879	C₂	1	2	yes	maximal
10	C₈₉×#1862	C₂	1	2	yes	maximal, complemented
11	C₈₉×#1802	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×Q₈×C₈₉	—	2	1	no	—
13	C₂²×D₈×C₈₉	C₂²	1	1	yes	characteristic
14	D₄×Q₈×C₈₉	—	2	4	no	—
15	D₄×Q₈×C₈₉	—	2	4	no	—
16	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
19	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
20	C₂×C₂²⋊Q₈×C₈₉	—	2	2	no	—
21	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
24	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
25	#474×C₈₉	—	2	4	no	—
26	C₄×D₈×C₈₉	—	2	4	no	—
27	C₂×#107×C₈₉	C₂²	1	2	yes	—
28	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×#102×C₈₉	C₂²	1	2	yes	—
31	C₂×#102×C₈₉	C₂²	1	2	yes	—
32	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
35	#396×C₈₉	—	2	2	no	—
36	#390×C₈₉	C₂²	1	2	yes	—
37	#389×C₈₉	C₂²	1	2	yes	—
38	#384×C₈₉	C₂²	1	1	yes	characteristic
39	#380×C₈₉	—	2	1	no	—
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	8	no	—
46	C₂²×Q₈×C₈₉	—	2	2	no	—
47	C₂²×Q₈×C₈₉	—	2	1	no	—
48	C₂²×Q₈×C₈₉	—	4	4	no	—
49	C₂²×Q₈×C₈₉	—	4	4	no	—
50	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
51	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
52	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
53	C₂³×C₄×C₈₉	—	2	2	no	—
54	C₂³×C₄×C₈₉	—	2	1	no	—
55	C₂³×C₄×C₈₉	—	2	1	no	—
56	C₂×D₈×C₈₉	—	2	2	no	—
57	C₂×D₈×C₈₉	—	2	2	no	—
58	C₂×D₈×C₈₉	Q₈	1	4	yes	complemented
59	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
61	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₄⋊Q₈×C₈₉	—	2	4	no	—
63	C₄⋊Q₈×C₈₉	—	2	4	no	—
64	C₄⋊Q₈×C₈₉	—	2	2	no	—
65	C₄⋊Q₈×C₈₉	D₄	1	4	yes	—
66	C₂²⋊Q₈×C₈₉	—	2	8	no	—
67	C₂²⋊Q₈×C₈₉	—	2	16	no	—
68	C₂²⋊Q₈×C₈₉	—	4	4	no	—
69	C₄×Q₈×C₈₉	—	2	2	no	—
70	C₄×Q₈×C₈₉	—	2	2	no	—
71	C₄×D₄×C₈₉	—	2	4	no	—
72	C₄×D₄×C₈₉	—	2	4	no	—
73	C₄×D₄×C₈₉	—	2	2	no	—
74	C₄×D₄×C₈₉	—	2	2	no	—
75	C₄×D₄×C₈₉	—	2	2	no	—
76	C₄×D₄×C₈₉	D₄	1	4	yes	—
77	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
78	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
79	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
80	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
81	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
82	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
83	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
84	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
85	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
86	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
87	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
88	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
89	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
90	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
91	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
92	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
93	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
94	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
95	#107×C₈₉	—	2	2	no	—
96	#107×C₈₉	—	2	4	no	—
97	C₄⋊C₈×C₈₉	—	2	2	no	—
98	#102×C₈₉	—	2	2	no	—
99	#102×C₈₉	—	2	4	no	—
100	#102×C₈₉	—	2	2	no	—
101	#102×C₈₉	—	2	4	no	—
102	C₄×C₈×C₈₉	—	2	2	no	—
103	#95×C₈₉	—	2	2	no	—
104	C₂⁴×C₈₉	—	2	1	no	—
105	C₂⁴×C₈₉	—	2	1	no	—
106	C₂×Q₈×C₈₉	—	2	8	no	—
107	C₂×Q₈×C₈₉	—	2	2	no	—
108	C₂×Q₈×C₈₉	—	2	4	no	—
109	C₂×Q₈×C₈₉	—	2	2	no	—
110	C₂×Q₈×C₈₉	—	4	4	no	—
111	C₂×Q₈×C₈₉	—	4	32	no	—
112	C₂×Q₈×C₈₉	—	4	4	no	—
113	C₂×Q₈×C₈₉	—	4	2	no	—
114	C₂×Q₈×C₈₉	—	4	2	no	—
115	C₂×Q₈×C₈₉	D₈	1	4	yes	complemented
116	C₂×D₄×C₈₉	—	2	1	no	—
117	C₂×D₄×C₈₉	—	2	1	no	—
118	C₂×D₄×C₈₉	—	2	1	no	—
119	C₂×D₄×C₈₉	—	2	1	no	—
120	C₂×D₄×C₈₉	—	2	1	no	—
121	C₂×D₄×C₈₉	—	2	1	no	—
122	C₂×D₄×C₈₉	—	2	1	no	—
123	C₂×D₄×C₈₉	—	2	1	no	—
124	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
125	C₂×D₄×C₈₉	C₂×Q₈	1	2	yes	—
126	C₂×D₄×C₈₉	C₂×Q₈	1	2	yes	complemented
127	C₂×D₄×C₈₉	#41	1	2	yes	—
128	C₂²×C₄×C₈₉	—	2	8	no	—
129	C₂²×C₄×C₈₉	—	2	2	no	—
130	C₂²×C₄×C₈₉	—	2	1	no	—
131	C₂²×C₄×C₈₉	—	2	4	no	—
132	C₂²×C₄×C₈₉	—	2	1	no	—
133	C₂²×C₄×C₈₉	—	2	2	no	—
134	C₂²×C₄×C₈₉	—	4	4	no	—
135	C₂²×C₄×C₈₉	—	4	4	no	—
136	C₂²×C₄×C₈₉	—	4	2	no	—
137	C₂²×C₄×C₈₉	—	4	2	no	—
138	C₂²×C₄×C₈₉	—	4	2	no	—
139	C₂²×C₄×C₈₉	—	4	2	no	—
140	C₂²×C₄×C₈₉	—	4	2	no	—
141	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
142	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
143	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
144	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
145	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
146	D₈×C₈₉	—	2	8	no	—
147	C₂×C₈×C₈₉	—	2	1	no	—
148	C₂×C₈×C₈₉	—	2	1	no	—
149	C₂×C₈×C₈₉	—	2	1	no	—
150	C₂×C₈×C₈₉	—	2	2	no	—
151	C₂×C₈×C₈₉	—	2	2	no	—
152	C₂×C₈×C₈₉	—	2	1	no	—
153	C₂×C₈×C₈₉	—	2	1	no	—
154	C₂×C₈×C₈₉	—	2	2	no	—
155	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	complemented
156	C₄⋊C₄×C₈₉	—	2	8	no	—
157	C₄⋊C₄×C₈₉	—	2	4	no	—
158	C₄⋊C₄×C₈₉	—	2	4	no	—
159	C₄⋊C₄×C₈₉	—	2	2	no	—
160	C₄⋊C₄×C₈₉	—	2	2	no	—
161	C₄⋊C₄×C₈₉	—	2	1	no	—
162	C₄⋊C₄×C₈₉	—	2	1	no	—
163	C₄⋊C₄×C₈₉	—	2	1	no	—
164	C₄⋊C₄×C₈₉	—	2	2	no	—
165	C₄⋊C₄×C₈₉	—	2	2	no	—
166	C₄⋊C₄×C₈₉	—	4	2	no	—
167	C₄⋊C₄×C₈₉	—	4	4	no	—
168	C₄⋊C₄×C₈₉	—	8	1	no	—
169	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
170	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
171	C₂²⋊C₄×C₈₉	—	2	8	no	—
172	C₂²⋊C₄×C₈₉	—	2	4	no	—
173	C₂²⋊C₄×C₈₉	—	4	2	no	—
174	C₂²⋊C₄×C₈₉	—	4	4	no	—
175	C₂²⋊C₄×C₈₉	—	8	2	no	—
176	C₄²×C₈₉	—	2	2	no	—
177	C₄²×C₈₉	—	2	2	no	—
178	C₄²×C₈₉	C₂×D₄	1	2	yes	—
179	C₄²×C₈₉	C₂×D₄	1	2	yes	—
180	C₂³×C₈₉	—	2	2	no	—
181	C₂³×C₈₉	—	2	2	no	—
182	C₂³×C₈₉	—	2	2	no	—
183	C₂³×C₈₉	—	2	2	no	—
184	C₂³×C₈₉	—	4	1	no	—
185	C₂³×C₈₉	—	4	1	no	—
186	C₂³×C₈₉	—	4	1	no	—
187	C₂³×C₈₉	—	4	1	no	—
188	C₂³×C₈₉	—	4	1	no	—
189	C₂³×C₈₉	—	4	1	no	—
190	C₂³×C₈₉	—	4	1	no	—
191	C₂³×C₈₉	—	4	1	no	—
192	C₂³×C₈₉	—	4	1	no	—
193	C₂³×C₈₉	—	4	1	no	—
194	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
195	Q₈×C₈₉	—	2	8	no	—
196	Q₈×C₈₉	—	4	16	no	—
197	Q₈×C₈₉	—	4	8	no	—
198	D₄×C₈₉	—	2	2	no	—
199	D₄×C₈₉	—	2	2	no	—
200	D₄×C₈₉	—	2	2	no	—
201	D₄×C₈₉	—	2	2	no	—
202	D₄×C₈₉	—	2	2	no	—
203	D₄×C₈₉	—	4	1	no	—
204	D₄×C₈₉	—	4	1	no	—
205	D₄×C₈₉	C₂²⋊Q₈	1	4	yes	complemented
206	C₂×C₄×C₈₉	—	2	2	no	—
207	C₂×C₄×C₈₉	—	2	2	no	—
208	C₂×C₄×C₈₉	—	2	1	no	—
209	C₂×C₄×C₈₉	—	2	1	no	—
210	C₂×C₄×C₈₉	—	2	2	no	—
211	C₂×C₄×C₈₉	—	2	4	no	—
212	C₂×C₄×C₈₉	—	2	2	no	—
213	C₂×C₄×C₈₉	—	2	2	no	—
214	C₂×C₄×C₈₉	—	2	2	no	—
215	C₂×C₄×C₈₉	—	4	16	no	—
216	C₂×C₄×C₈₉	—	4	2	no	—
217	C₂×C₄×C₈₉	—	4	1	no	—
218	C₂×C₄×C₈₉	—	4	2	no	—
219	C₂×C₄×C₈₉	—	4	1	no	—
220	C₂×C₄×C₈₉	—	4	8	no	—
221	C₂×C₄×C₈₉	—	4	1	no	—
222	C₂×C₄×C₈₉	—	4	2	no	—
223	C₂×C₄×C₈₉	—	4	2	no	—
224	C₂×C₄×C₈₉	—	4	1	no	—
225	C₂×C₄×C₈₉	—	4	1	no	—
226	C₂×C₄×C₈₉	—	4	2	no	—
227	C₂×C₄×C₈₉	—	8	4	no	—
228	C₂×C₄×C₈₉	#120	1	2	yes	—
229	C₂×C₄×C₈₉	#120	1	2	yes	—
230	C₂×C₄×C₈₉	C₂×D₈	1	4	yes	—
231	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
232	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
233	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
234	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
236	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
237	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
238	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
239	C₈×C₈₉	—	2	2	no	—
240	C₈×C₈₉	—	4	1	no	—
241	C₈×C₈₉	—	4	2	no	—
242	C₂²×C₈₉	—	2	4	no	—
243	C₂²×C₈₉	—	4	2	no	—
244	C₂²×C₈₉	—	4	2	no	—
245	C₂²×C₈₉	—	4	2	no	—
246	C₂²×C₈₉	—	4	2	no	—
247	C₂²×C₈₉	—	4	2	no	—
248	C₂²×C₈₉	—	4	2	no	—
249	C₂²×C₈₉	—	4	2	no	—
250	C₂²×C₈₉	—	4	2	no	—
251	C₂²×C₈₉	—	4	2	no	—
252	C₂²×C₈₉	—	8	1	no	—
253	C₂²×C₈₉	—	8	1	no	—
254	C₂²×C₈₉	—	8	1	no	—
255	C₂²×C₈₉	—	8	1	no	—
256	C₂²×C₈₉	C₂×#120	1	1	yes	characteristic
257	C₂²×C₈₉	#548	1	1	yes	characteristic
258	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
259	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
260	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
261	C₂²×C₈₉	#577	1	1	yes	characteristic
262	C₂²×C₈₉	#578	1	1	yes	characteristic
263	C₄×C₈₉	—	2	4	no	—
264	C₄×C₈₉	—	2	2	no	—
265	C₄×C₈₉	—	4	4	no	—
266	C₄×C₈₉	—	4	2	no	—
267	C₄×C₈₉	—	4	2	no	—
268	C₄×C₈₉	—	8	2	no	—
269	C₄×C₈₉	—	8	1	no	—
270	C₄×C₈₉	#447	1	4	yes	—
271	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
272	C₄×C₈₉	#536	1	1	yes	characteristic
273	C₄×C₈₉	#548	1	1	yes	characteristic
274	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
275	#24326	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
276	C₂×C₈₉	—	4	4	no	—
277	C₂×C₈₉	—	8	2	no	—
278	C₂×C₈₉	#2409	1	1	yes	characteristic
279	C₂×C₈₉	C₂×#447	1	1	yes	characteristic
280	C₂×C₈₉	#3013	1	1	yes	characteristic
281	C₂×C₈₉	#3358	1	1	yes	characteristic
282	C₂×C₈₉	#3367	1	1	yes	characteristic
283	C₂×C₈₉	Q₈×D₈	1	1	yes	characteristic
284	C₂×C₈₉	#3384	1	1	yes	characteristic
285	C₂×D₄×Q₈	C₂×C₈₉	1	1	yes	characteristic, complemented
286	C₂×#474	C₂×C₈₉	1	1	yes	characteristic, complemented
287	C₂×C₄×D₈	C₂×C₈₉	1	1	yes	characteristic
288	#2538	C₂×C₈₉	1	1	yes	characteristic, complemented
289	#2057	C₂×C₈₉	1	2	yes	complemented
290	#2023	C₂×C₈₉	1	2	yes	complemented
291	#1990	C₂×C₈₉	1	1	yes	characteristic, complemented
292	#1984	C₂×C₈₉	1	1	yes	characteristic, complemented
293	#1879	C₂×C₈₉	1	2	yes	—
294	#1862	C₂×C₈₉	1	2	yes	complemented
295	#1802	C₂×C₈₉	1	1	yes	characteristic, complemented
296	C₈₉	#24326	1	1	yes	characteristic, direct product factor, Sylow
297	C₂³×Q₈	—	2	1	no	—
298	C₂²×D₈	C₂²×C₈₉	1	1	yes	characteristic
299	D₄×Q₈	—	2	4	no	—
300	D₄×Q₈	—	2	4	no	—
301	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
302	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
303	C₂×C₂²⋊Q₈	—	2	1	no	—
304	C₂×C₂²⋊Q₈	—	2	1	no	—
305	C₂×C₂²⋊Q₈	—	2	2	no	—
306	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
307	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
308	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
309	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
310	#474	—	2	4	no	—
311	C₄×D₈	—	2	4	no	—
312	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
313	C₂×#107	C₂²×C₈₉	1	2	yes	—
314	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
315	C₂×#102	C₂²×C₈₉	1	2	yes	—
316	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
317	C₂×#102	C₂²×C₈₉	1	2	yes	—
318	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
319	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
320	#396	—	2	2	no	—
321	#390	C₂²×C₈₉	1	2	yes	—
322	#389	C₂²×C₈₉	1	2	yes	—
323	#384	C₂²×C₈₉	1	1	yes	characteristic
324	#380	—	2	1	no	—
325	#340	C₂²×C₈₉	1	2	yes	—
326	#340	C₂²×C₈₉	1	2	yes	—
327	#340	C₂²×C₈₉	1	2	yes	—
328	#336	C₂²×C₈₉	1	2	yes	—
329	#336	C₂²×C₈₉	1	2	yes	—
330	C₂²×Q₈	—	2	8	no	—
331	C₂²×Q₈	—	2	2	no	—
332	C₂²×Q₈	—	2	1	no	—
333	C₂²×Q₈	—	4	4	no	—
334	C₂²×Q₈	—	4	4	no	—
335	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
336	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
337	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
338	C₂³×C₄	—	2	2	no	—
339	C₂³×C₄	—	2	1	no	—
340	C₂³×C₄	—	2	1	no	—
341	C₂×D₈	—	2	2	no	—
342	C₂×D₈	—	2	2	no	—
343	C₂×D₈	Q₈×C₈₉	1	4	yes	complemented
344	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
345	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
346	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
347	C₄⋊Q₈	—	2	4	no	—
348	C₄⋊Q₈	—	2	4	no	—
349	C₄⋊Q₈	—	2	2	no	—
350	C₄⋊Q₈	D₄×C₈₉	1	4	yes	—
351	C₂²⋊Q₈	—	2	8	no	—
352	C₂²⋊Q₈	—	2	16	no	—
353	C₂²⋊Q₈	—	4	4	no	—
354	C₄×Q₈	—	2	2	no	—
355	C₄×Q₈	—	2	2	no	—
356	C₄×D₄	—	2	4	no	—
357	C₄×D₄	—	2	4	no	—
358	C₄×D₄	—	2	2	no	—
359	C₄×D₄	—	2	2	no	—
360	C₄×D₄	—	2	2	no	—
361	C₄×D₄	D₄×C₈₉	1	4	yes	—
362	C₂×C₄⋊C₄	—	2	2	no	—
363	C₂×C₄⋊C₄	—	2	1	no	—
364	C₂×C₄⋊C₄	—	2	1	no	—
365	C₂×C₄⋊C₄	—	2	2	no	—
366	C₂×C₄⋊C₄	—	2	1	no	—
367	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
368	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
369	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
370	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
371	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
372	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
373	C₂×C₂²⋊C₄	—	2	2	no	—
374	C₂×C₂²⋊C₄	—	2	1	no	—
375	C₂×C₂²⋊C₄	—	2	1	no	—
376	C₂×C₂²⋊C₄	—	2	2	no	—
377	C₂×C₂²⋊C₄	—	2	2	no	—
378	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
379	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
380	#107	—	2	2	no	—
381	#107	—	2	4	no	—
382	C₄⋊C₈	—	2	2	no	—
383	#102	—	2	2	no	—
384	#102	—	2	4	no	—
385	#102	—	2	2	no	—
386	#102	—	2	4	no	—
387	C₄×C₈	—	2	2	no	—
388	#95	—	2	2	no	—
389	C₂⁴	—	2	1	no	—
390	C₂⁴	—	2	1	no	—
391	C₂×Q₈	—	2	8	no	—
392	C₂×Q₈	—	2	2	no	—
393	C₂×Q₈	—	2	4	no	—
394	C₂×Q₈	—	2	2	no	—
395	C₂×Q₈	—	4	4	no	—
396	C₂×Q₈	—	4	32	no	—
397	C₂×Q₈	—	4	4	no	—
398	C₂×Q₈	—	4	2	no	—
399	C₂×Q₈	—	4	2	no	—
400	C₂×Q₈	D₈×C₈₉	1	4	yes	complemented
401	C₂×D₄	—	2	1	no	—
402	C₂×D₄	—	2	1	no	—
403	C₂×D₄	—	2	1	no	—
404	C₂×D₄	—	2	1	no	—
405	C₂×D₄	—	2	1	no	—
406	C₂×D₄	—	2	1	no	—
407	C₂×D₄	—	2	1	no	—
408	C₂×D₄	—	2	1	no	—
409	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
410	C₂×D₄	C₂×Q₈×C₈₉	1	2	yes	—
411	C₂×D₄	C₂×Q₈×C₈₉	1	2	yes	complemented
412	C₂×D₄	#41×C₈₉	1	2	yes	—
413	C₂²×C₄	—	2	8	no	—
414	C₂²×C₄	—	2	2	no	—
415	C₂²×C₄	—	2	1	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	1	no	—
418	C₂²×C₄	—	2	2	no	—
419	C₂²×C₄	—	4	4	no	—
420	C₂²×C₄	—	4	4	no	—
421	C₂²×C₄	—	4	2	no	—
422	C₂²×C₄	—	4	2	no	—
423	C₂²×C₄	—	4	2	no	—
424	C₂²×C₄	—	4	2	no	—
425	C₂²×C₄	—	4	2	no	—
426	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
427	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
428	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
429	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
430	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
431	D₈	—	2	8	no	—
432	C₂×C₈	—	2	1	no	—
433	C₂×C₈	—	2	1	no	—
434	C₂×C₈	—	2	1	no	—
435	C₂×C₈	—	2	2	no	—
436	C₂×C₈	—	2	2	no	—
437	C₂×C₈	—	2	1	no	—
438	C₂×C₈	—	2	1	no	—
439	C₂×C₈	—	2	2	no	—
440	C₂×C₈	C₂×Q₈×C₈₉	1	2	yes	complemented
441	C₄⋊C₄	—	2	8	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	—	2	4	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	—	2	1	no	—
447	C₄⋊C₄	—	2	1	no	—
448	C₄⋊C₄	—	2	1	no	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	—	2	2	no	—
451	C₄⋊C₄	—	4	2	no	—
452	C₄⋊C₄	—	4	4	no	—
453	C₄⋊C₄	—	8	1	no	—
454	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
455	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
456	C₂²⋊C₄	—	2	8	no	—
457	C₂²⋊C₄	—	2	4	no	—
458	C₂²⋊C₄	—	4	2	no	—
459	C₂²⋊C₄	—	4	4	no	—
460	C₂²⋊C₄	—	8	2	no	—
461	C₄²	—	2	2	no	—
462	C₄²	—	2	2	no	—
463	C₄²	C₂×D₄×C₈₉	1	2	yes	—
464	C₄²	C₂×D₄×C₈₉	1	2	yes	—
465	C₂³	—	2	2	no	—
466	C₂³	—	2	2	no	—
467	C₂³	—	2	2	no	—
468	C₂³	—	2	2	no	—
469	C₂³	—	4	1	no	—
470	C₂³	—	4	1	no	—
471	C₂³	—	4	1	no	—
472	C₂³	—	4	1	no	—
473	C₂³	—	4	1	no	—
474	C₂³	—	4	1	no	—
475	C₂³	—	4	1	no	—
476	C₂³	—	4	1	no	—
477	C₂³	—	4	1	no	—
478	C₂³	—	4	1	no	—
479	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
480	Q₈	—	2	8	no	—
481	Q₈	—	4	16	no	—
482	Q₈	—	4	8	no	—
483	D₄	—	2	2	no	—
484	D₄	—	2	2	no	—
485	D₄	—	2	2	no	—
486	D₄	—	2	2	no	—
487	D₄	—	2	2	no	—
488	D₄	—	4	1	no	—
489	D₄	—	4	1	no	—
490	D₄	C₂²⋊Q₈×C₈₉	1	4	yes	complemented
491	C₂×C₄	—	2	2	no	—
492	C₂×C₄	—	2	2	no	—
493	C₂×C₄	—	2	1	no	—
494	C₂×C₄	—	2	1	no	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	2	no	—
498	C₂×C₄	—	2	2	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	4	16	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	—	4	1	no	—
503	C₂×C₄	—	4	2	no	—
504	C₂×C₄	—	4	1	no	—
505	C₂×C₄	—	4	8	no	—
506	C₂×C₄	—	4	1	no	—
507	C₂×C₄	—	4	2	no	—
508	C₂×C₄	—	4	2	no	—
509	C₂×C₄	—	4	1	no	—
510	C₂×C₄	—	4	1	no	—
511	C₂×C₄	—	4	2	no	—
512	C₂×C₄	—	8	4	no	—
513	C₂×C₄	#120×C₈₉	1	2	yes	—
514	C₂×C₄	#120×C₈₉	1	2	yes	—
515	C₂×C₄	C₂×D₈×C₈₉	1	4	yes	—
516	C₂×C₄	C₂×D₈×C₈₉	1	2	yes	—
517	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
518	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
519	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
520	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
521	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
522	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
523	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
524	C₈	—	2	2	no	—
525	C₈	—	4	1	no	—
526	C₈	—	4	2	no	—
527	C₂²	—	2	4	no	—
528	C₂²	—	4	2	no	—
529	C₂²	—	4	2	no	—
530	C₂²	—	4	2	no	—
531	C₂²	—	4	2	no	—
532	C₂²	—	4	2	no	—
533	C₂²	—	4	2	no	—
534	C₂²	—	4	2	no	—
535	C₂²	—	4	2	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	8	1	no	—
538	C₂²	—	8	1	no	—
539	C₂²	—	8	1	no	—
540	C₂²	—	8	1	no	—
541	C₂²	C₂×#120×C₈₉	1	1	yes	characteristic
542	C₂²	#548×C₈₉	1	1	yes	characteristic
543	C₂²	D₄×Q₈×C₈₉	1	1	yes	characteristic
544	C₂²	C₂²×D₈×C₈₉	1	1	yes	characteristic
545	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
546	C₂²	#577×C₈₉	1	1	yes	characteristic
547	C₂²	#578×C₈₉	1	1	yes	characteristic
548	C₄	—	2	4	no	—
549	C₄	—	2	2	no	—
550	C₄	—	4	4	no	—
551	C₄	—	4	2	no	—
552	C₄	—	4	2	no	—
553	C₄	—	8	2	no	—
554	C₄	—	8	1	no	—
555	C₄	#447×C₈₉	1	4	yes	—
556	C₄	C₂×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
557	C₄	#536×C₈₉	1	1	yes	characteristic
558	C₄	#548×C₈₉	1	1	yes	characteristic
559	C₄	D₄×Q₈×C₈₉	1	1	yes	characteristic
560	C₂	—	4	4	no	—
561	C₂	—	8	2	no	—
562	C₂	C₈₉×#2409	1	1	yes	characteristic
563	C₂	C₂×#447×C₈₉	1	1	yes	characteristic
564	C₂	C₈₉×#3013	1	1	yes	characteristic
565	C₂	C₈₉×#3358	1	1	yes	characteristic
566	C₂	C₈₉×#3367	1	1	yes	characteristic
567	C₂	Q₈×D₈×C₈₉	1	1	yes	characteristic
568	C₂	C₈₉×#3384	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂²×D₈×C₈₉	C₂²
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
1. C₂×D₄×Q₈×C₈₉	C₂	semidirect
2. C₂×#474×C₈₉	C₂	semidirect
4. C₈₉×#2538	C₂	semidirect
5. C₈₉×#2057	C₂	semidirect
6. C₈₉×#2023	C₂	semidirect
7. C₈₉×#1990	C₂	semidirect
8. C₈₉×#1984	C₂	semidirect
10. C₈₉×#1862	C₂	semidirect
11. C₈₉×#1802	C₂	semidirect
58. C₂×D₈×C₈₉	Q₈	semidirect
115. C₂×Q₈×C₈₉	D₈	semidirect
126. C₂×D₄×C₈₉	C₂×Q₈	semidirect
155. C₂×C₈×C₈₉	C₂×Q₈	semidirect
205. D₄×C₈₉	C₂²⋊Q₈	semidirect
275. #24326	C₈₉	direct (N × H)
285. C₂×D₄×Q₈	C₂×C₈₉	semidirect
286. C₂×#474	C₂×C₈₉	semidirect
288. #2538	C₂×C₈₉	semidirect
289. #2057	C₂×C₈₉	semidirect
290. #2023	C₂×C₈₉	semidirect
291. #1990	C₂×C₈₉	semidirect
292. #1984	C₂×C₈₉	semidirect
294. #1862	C₂×C₈₉	semidirect
295. #1802	C₂×C₈₉	semidirect
296. C₈₉	#24326	direct (N × H)
343. C₂×D₈	Q₈×C₈₉	semidirect
400. C₂×Q₈	D₈×C₈₉	semidirect
411. C₂×D₄	C₂×Q₈×C₈₉	semidirect
440. C₂×C₈	C₂×Q₈×C₈₉	semidirect
490. D₄	C₂²⋊Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₄×Q₈×C₈₉	2
C₂×#474×C₈₉	2
C₂×C₄×D₈×C₈₉	2
C₈₉×#2538	2
C₈₉×#2057	2
C₈₉×#2023	2
C₈₉×#1990	2
C₈₉×#1984	2
C₈₉×#1879	2
C₈₉×#1862	2
C₈₉×#1802	2
#24326	89

Subgroups — C89×#24326 #36402174

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#24326 #36402174