Subgroups — C₈₉×#60644 #36438492

Order 22784.

3350 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#575×C₈₉	C₂	1	2	yes	maximal, complemented
2	C₂²×AGL(1,ℤ/8)×C₈₉	C₂	1	2	yes	maximal, complemented
3	C₈₉×#3449	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#535×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#3155	C₂	1	8	yes	maximal, complemented
6	C₈₉×#3154	C₂	1	8	yes	maximal, complemented
7	C₈₉×#3153	C₂	1	4	yes	maximal, complemented
8	C₈₉×#3151	C₂	1	4	yes	maximal, complemented
9	C₈₉×#2873	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₈₉	C₂²	1	2	yes	—
11	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
12	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic
13	#575×C₈₉	—	2	16	no	—
14	C₂×#137×C₈₉	C₂²	1	4	yes	complemented
15	C₂×#137×C₈₉	C₂²	1	4	yes	complemented
16	C₂×AGL(1,ℤ/8)×C₈₉	—	2	16	no	—
17	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	8	yes	complemented
18	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	8	yes	complemented
19	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	4	yes	complemented
20	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	4	yes	complemented
21	C₂×#135×C₈₉	C₂²	1	8	yes	complemented
22	C₂²×SD₁₆×C₈₉	C₂²	1	4	yes	complemented
23	C₂²×D₈×C₈₉	C₂²	1	4	yes	complemented
24	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
25	#537×C₈₉	—	2	2	no	—
26	#536×C₈₉	C₂²	1	4	yes	complemented
27	#535×C₈₉	—	2	4	no	—
28	#535×C₈₉	C₂²	1	8	yes	complemented
29	#535×C₈₉	C₂²	1	4	yes	complemented
30	#534×C₈₉	—	2	2	no	—
31	#532×C₈₉	C₂²	1	4	yes	complemented
32	C₂×#127×C₈₉	C₂²	1	2	yes	complemented
33	C₂×#124×C₈₉	C₂²	1	2	yes	complemented
34	#525×C₈₉	—	2	4	no	—
35	C₂×#121×C₈₉	—	2	2	no	—
36	C₂×#120×C₈₉	—	2	2	no	—
37	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
38	#497×C₈₉	C₂²	1	8	yes	complemented
39	#496×C₈₉	C₂²	1	8	yes	complemented
40	#496×C₈₉	C₂²	1	16	yes	complemented
41	#495×C₈₉	C₂²	1	16	yes	complemented
42	#493×C₈₉	C₂²	1	8	yes	complemented
43	#492×C₈₉	C₂²	1	8	yes	complemented
44	#405×C₈₉	C₂²	1	8	yes	complemented
45	C₄×M₁₆×C₈₉	C₂²	1	4	yes	complemented
46	C₂⁵×C₈₉	—	2	1	no	—
47	C₂×#41×C₈₉	—	2	2	no	—
48	C₂×#41×C₈₉	—	2	2	no	—
49	C₂×#41×C₈₉	—	2	2	no	—
50	C₂×#41×C₈₉	—	2	8	no	—
51	C₂×#41×C₈₉	—	2	8	no	—
52	C₂×#41×C₈₉	—	2	8	no	—
53	C₂×#41×C₈₉	C₂³	1	4	yes	—
54	C₂×#41×C₈₉	C₂³	1	4	yes	—
55	C₂×#41×C₈₉	C₂³	1	4	yes	—
56	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
57	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
58	C₂²×D₄×C₈₉	—	2	1	no	—
59	C₂²×D₄×C₈₉	—	2	2	no	—
60	C₂²×D₄×C₈₉	—	2	8	no	—
61	C₂²×D₄×C₈₉	—	2	1	no	—
62	C₂²×D₄×C₈₉	—	2	2	no	—
63	C₂²×D₄×C₈₉	—	2	4	no	—
64	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
65	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
67	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
68	C₂³×C₄×C₈₉	—	2	1	no	—
69	C₂³×C₄×C₈₉	—	2	2	no	—
70	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
71	#137×C₈₉	—	2	16	no	—
72	#137×C₈₉	—	2	16	no	—
73	AGL(1,ℤ/8)×C₈₉	—	2	32	no	—
74	AGL(1,ℤ/8)×C₈₉	—	2	16	no	—
75	AGL(1,ℤ/8)×C₈₉	—	2	32	no	—
76	AGL(1,ℤ/8)×C₈₉	—	2	16	no	—
77	#135×C₈₉	—	2	32	no	—
78	C₂×Q₁₆×C₈₉	C₂³	1	8	yes	complemented
79	C₂×SD₁₆×C₈₉	—	2	16	no	—
80	C₂×SD₁₆×C₈₉	C₂³	1	16	yes	—
81	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	complemented
82	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	complemented
83	C₂×D₈×C₈₉	—	2	16	no	—
84	C₂×D₈×C₈₉	C₂³	1	16	yes	—
85	C₂×D₈×C₈₉	C₂³	1	8	yes	complemented
86	C₂×M₁₆×C₈₉	D₄	1	16	yes	complemented
87	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
88	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
89	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
90	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	complemented
91	#127×C₈₉	—	2	4	no	—
92	#127×C₈₉	C₂³	1	8	yes	—
93	#127×C₈₉	C₂³	1	4	yes	complemented
94	#124×C₈₉	—	2	4	no	—
95	#124×C₈₉	C₂³	1	8	yes	—
96	#124×C₈₉	C₂³	1	8	yes	complemented
97	#123×C₈₉	—	2	8	no	—
98	C₂²⋊Q₈×C₈₉	—	2	8	no	—
99	#121×C₈₉	—	2	8	no	—
100	#121×C₈₉	—	2	16	no	—
101	#121×C₈₉	—	2	8	no	—
102	#120×C₈₉	—	2	16	no	—
103	#120×C₈₉	—	2	8	no	—
104	C₄×D₄×C₈₉	—	2	8	no	—
105	#117×C₈₉	D₄	1	4	yes	—
106	#117×C₈₉	C₂³	1	4	yes	—
107	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
108	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
109	C₂×C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
110	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
111	#97×C₈₉	C₂³	1	8	yes	complemented
112	C₄×C₈×C₈₉	C₂³	1	8	yes	complemented
113	C₂⁴×C₈₉	—	2	2	no	—
114	C₂⁴×C₈₉	—	2	1	no	—
115	C₂⁴×C₈₉	—	2	2	no	—
116	C₂⁴×C₈₉	—	2	4	no	—
117	C₂⁴×C₈₉	—	4	4	no	—
118	C₂⁴×C₈₉	—	8	4	no	—
119	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
120	#41×C₈₉	—	2	8	no	—
121	#41×C₈₉	—	2	8	no	—
122	#41×C₈₉	—	2	8	no	—
123	#41×C₈₉	—	2	8	no	—
124	#41×C₈₉	—	2	8	no	—
125	#41×C₈₉	—	2	8	no	—
126	#41×C₈₉	—	4	8	no	—
127	#41×C₈₉	—	4	8	no	—
128	#41×C₈₉	—	4	8	no	—
129	C₂×Q₈×C₈₉	—	2	2	no	—
130	C₂×Q₈×C₈₉	—	2	1	no	—
131	C₂×Q₈×C₈₉	—	2	4	no	—
132	C₂×Q₈×C₈₉	—	2	4	no	—
133	C₂×Q₈×C₈₉	—	2	4	no	—
134	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
135	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
136	C₂×D₄×C₈₉	—	2	4	no	—
137	C₂×D₄×C₈₉	—	2	4	no	—
138	C₂×D₄×C₈₉	—	2	4	no	—
139	C₂×D₄×C₈₉	—	2	4	no	—
140	C₂×D₄×C₈₉	—	2	8	no	—
141	C₂×D₄×C₈₉	—	2	8	no	—
142	C₂×D₄×C₈₉	—	2	8	no	—
143	C₂×D₄×C₈₉	—	2	4	no	—
144	C₂×D₄×C₈₉	—	2	4	no	—
145	C₂×D₄×C₈₉	—	2	4	no	—
146	C₂×D₄×C₈₉	—	2	2	no	—
147	C₂×D₄×C₈₉	—	2	2	no	—
148	C₂×D₄×C₈₉	—	2	1	no	—
149	C₂×D₄×C₈₉	—	2	8	no	—
150	C₂×D₄×C₈₉	—	2	16	no	—
151	C₂×D₄×C₈₉	—	4	4	no	—
152	C₂×D₄×C₈₉	—	4	8	no	—
153	C₂×D₄×C₈₉	—	4	4	no	—
154	C₂×D₄×C₈₉	—	4	8	no	—
155	C₂×D₄×C₈₉	—	8	8	no	—
156	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
157	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
158	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₈₉	—	2	1	no	—
160	C₂²×C₄×C₈₉	—	2	2	no	—
161	C₂²×C₄×C₈₉	—	2	4	no	—
162	C₂²×C₄×C₈₉	—	2	4	no	—
163	C₂²×C₄×C₈₉	—	2	4	no	—
164	C₂²×C₄×C₈₉	—	4	4	no	—
165	C₂²×C₄×C₈₉	—	4	8	no	—
166	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
167	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
168	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
169	C₂²×C₄×C₈₉	C₂×D₄	1	8	yes	—
170	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
171	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
172	Q₁₆×C₈₉	—	2	16	no	—
173	SD₁₆×C₈₉	—	2	32	no	—
174	SD₁₆×C₈₉	—	2	16	no	—
175	SD₁₆×C₈₉	—	2	16	no	—
176	D₈×C₈₉	—	2	32	no	—
177	D₈×C₈₉	—	2	16	no	—
178	M₁₆×C₈₉	C₂×D₄	1	16	yes	—
179	M₁₆×C₈₉	C₂×D₄	1	16	yes	complemented
180	C₂×C₈×C₈₉	C₂×D₄	1	16	yes	complemented
181	C₂×C₈×C₈₉	C₂⁴	1	8	yes	—
182	C₄⋊C₄×C₈₉	—	2	8	no	—
183	C₄⋊C₄×C₈₉	C₂×D₄	1	8	yes	—
184	C₂²⋊C₄×C₈₉	—	2	16	no	—
185	C₂²⋊C₄×C₈₉	—	2	16	no	—
186	C₂²⋊C₄×C₈₉	C₂×D₄	1	8	yes	—
187	C₄²×C₈₉	C₂×D₄	1	4	yes	—
188	C₄²×C₈₉	C₂⁴	1	4	yes	—
189	C₂³×C₈₉	—	2	4	no	—
190	C₂³×C₈₉	—	2	4	no	—
191	C₂³×C₈₉	—	2	2	no	—
192	C₂³×C₈₉	—	4	2	no	—
193	C₂³×C₈₉	—	4	4	no	—
194	C₂³×C₈₉	—	4	8	no	—
195	C₂³×C₈₉	—	4	2	no	—
196	C₂³×C₈₉	—	4	4	no	—
197	C₂³×C₈₉	—	4	4	no	—
198	C₂³×C₈₉	—	4	8	no	—
199	C₂³×C₈₉	—	8	4	no	—
200	C₂³×C₈₉	—	8	8	no	—
201	C₂³×C₈₉	—	8	16	no	—
202	C₂³×C₈₉	#127	1	4	yes	—
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	—	2	4	no	—
208	Q₈×C₈₉	—	4	4	no	—
209	Q₈×C₈₉	—	4	2	no	—
210	D₄×C₈₉	—	2	4	no	—
211	D₄×C₈₉	—	2	8	no	—
212	D₄×C₈₉	—	2	4	no	—
213	D₄×C₈₉	—	2	4	no	—
214	D₄×C₈₉	—	4	8	no	—
215	D₄×C₈₉	—	4	8	no	—
216	D₄×C₈₉	—	4	8	no	—
217	D₄×C₈₉	—	4	8	no	—
218	D₄×C₈₉	—	4	4	no	—
219	D₄×C₈₉	—	4	4	no	—
220	D₄×C₈₉	—	4	2	no	—
221	D₄×C₈₉	—	8	8	no	—
222	D₄×C₈₉	—	8	16	no	—
223	C₂×C₄×C₈₉	—	2	2	no	—
224	C₂×C₄×C₈₉	—	2	4	no	—
225	C₂×C₄×C₈₉	—	4	4	no	—
226	C₂×C₄×C₈₉	—	4	2	no	—
227	C₂×C₄×C₈₉	—	4	4	no	—
228	C₂×C₄×C₈₉	—	4	8	no	—
229	C₂×C₄×C₈₉	—	4	8	no	—
230	C₂×C₄×C₈₉	—	4	8	no	—
231	C₂×C₄×C₈₉	#127	1	4	yes	—
232	C₂×C₄×C₈₉	#127	1	8	yes	—
233	C₂×C₄×C₈₉	C₂²×D₄	1	8	yes	—
234	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
235	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
236	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
237	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
238	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
239	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
240	C₈×C₈₉	C₂²×D₄	1	16	yes	—
241	C₂²×C₈₉	—	2	2	no	—
242	C₂²×C₈₉	—	2	4	no	—
243	C₂²×C₈₉	—	4	4	no	—
244	C₂²×C₈₉	—	4	4	no	—
245	C₂²×C₈₉	—	4	8	no	—
246	C₂²×C₈₉	—	4	8	no	—
247	C₂²×C₈₉	—	4	4	no	—
248	C₂²×C₈₉	—	4	2	no	—
249	C₂²×C₈₉	—	4	4	no	—
250	C₂²×C₈₉	—	8	4	no	—
251	C₂²×C₈₉	—	8	8	no	—
252	C₂²×C₈₉	—	8	8	no	—
253	C₂²×C₈₉	—	8	16	no	—
254	C₂²×C₈₉	C₂×#127	1	4	yes	—
255	C₂²×C₈₉	C₂×#127	1	2	yes	—
256	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
257	C₄×C₈₉	—	2	8	no	—
258	C₄×C₈₉	—	4	2	no	—
259	C₄×C₈₉	—	4	4	no	—
260	C₄×C₈₉	C₂×#127	1	2	yes	—
261	C₄×C₈₉	C₂×#127	1	4	yes	—
262	C₄×C₈₉	C₂³×D₄	1	2	yes	—
263	#60644	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
264	C₂×C₈₉	—	2	2	no	—
265	C₂×C₈₉	—	2	4	no	—
266	C₂×C₈₉	—	8	2	no	—
267	C₂×C₈₉	—	8	4	no	—
268	C₂×C₈₉	—	8	4	no	—
269	C₂×C₈₉	C₂²×#127	1	1	yes	characteristic
270	C₂×C₈₉	#3585	1	2	yes	—
271	C₂×#575	C₂×C₈₉	1	2	yes	complemented
272	C₂²×AGL(1,ℤ/8)	C₂×C₈₉	1	2	yes	complemented
273	#3449	C₂×C₈₉	1	1	yes	characteristic, complemented
274	C₂×#535	C₂×C₈₉	1	1	yes	characteristic, complemented
275	#3155	C₂×C₈₉	1	8	yes	complemented
276	#3154	C₂×C₈₉	1	8	yes	complemented
277	#3153	C₂×C₈₉	1	4	yes	complemented
278	#3151	C₂×C₈₉	1	4	yes	complemented
279	#2873	C₂×C₈₉	1	1	yes	characteristic, complemented
280	C₈₉	#60644	1	1	yes	characteristic, direct product factor, Sylow
281	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
282	C₂²×#41	C₂²×C₈₉	1	2	yes	—
283	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
284	#575	—	2	16	no	—
285	C₂×#137	C₂²×C₈₉	1	4	yes	complemented
286	C₂×#137	C₂²×C₈₉	1	4	yes	complemented
287	C₂×AGL(1,ℤ/8)	—	2	16	no	—
288	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	8	yes	complemented
289	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	8	yes	complemented
290	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	4	yes	complemented
291	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	4	yes	complemented
292	C₂×#135	C₂²×C₈₉	1	8	yes	complemented
293	C₂²×SD₁₆	C₂²×C₈₉	1	4	yes	complemented
294	C₂²×D₈	C₂²×C₈₉	1	4	yes	complemented
295	C₂²×M₁₆	C₂²×C₈₉	1	2	yes	—
296	#537	—	2	2	no	—
297	#536	C₂²×C₈₉	1	4	yes	complemented
298	#535	—	2	4	no	—
299	#535	C₂²×C₈₉	1	8	yes	complemented
300	#535	C₂²×C₈₉	1	4	yes	complemented
301	#534	—	2	2	no	—
302	#532	C₂²×C₈₉	1	4	yes	complemented
303	C₂×#127	C₂²×C₈₉	1	2	yes	complemented
304	C₂×#124	C₂²×C₈₉	1	2	yes	complemented
305	#525	—	2	4	no	—
306	C₂×#121	—	2	2	no	—
307	C₂×#120	—	2	2	no	—
308	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
309	#497	C₂²×C₈₉	1	8	yes	complemented
310	#496	C₂²×C₈₉	1	8	yes	complemented
311	#496	C₂²×C₈₉	1	16	yes	complemented
312	#495	C₂²×C₈₉	1	16	yes	complemented
313	#493	C₂²×C₈₉	1	8	yes	complemented
314	#492	C₂²×C₈₉	1	8	yes	complemented
315	#405	C₂²×C₈₉	1	8	yes	complemented
316	C₄×M₁₆	C₂²×C₈₉	1	4	yes	complemented
317	C₂⁵	—	2	1	no	—
318	C₂×#41	—	2	2	no	—
319	C₂×#41	—	2	2	no	—
320	C₂×#41	—	2	2	no	—
321	C₂×#41	—	2	8	no	—
322	C₂×#41	—	2	8	no	—
323	C₂×#41	—	2	8	no	—
324	C₂×#41	C₂³×C₈₉	1	4	yes	—
325	C₂×#41	C₂³×C₈₉	1	4	yes	—
326	C₂×#41	C₂³×C₈₉	1	4	yes	—
327	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
328	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
329	C₂²×D₄	—	2	1	no	—
330	C₂²×D₄	—	2	2	no	—
331	C₂²×D₄	—	2	8	no	—
332	C₂²×D₄	—	2	1	no	—
333	C₂²×D₄	—	2	2	no	—
334	C₂²×D₄	—	2	4	no	—
335	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
336	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
337	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
338	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
339	C₂³×C₄	—	2	1	no	—
340	C₂³×C₄	—	2	2	no	—
341	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
342	#137	—	2	16	no	—
343	#137	—	2	16	no	—
344	AGL(1,ℤ/8)	—	2	32	no	—
345	AGL(1,ℤ/8)	—	2	16	no	—
346	AGL(1,ℤ/8)	—	2	32	no	—
347	AGL(1,ℤ/8)	—	2	16	no	—
348	#135	—	2	32	no	—
349	C₂×Q₁₆	C₂³×C₈₉	1	8	yes	complemented
350	C₂×SD₁₆	—	2	16	no	—
351	C₂×SD₁₆	C₂³×C₈₉	1	16	yes	—
352	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	complemented
353	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	complemented
354	C₂×D₈	—	2	16	no	—
355	C₂×D₈	C₂³×C₈₉	1	16	yes	—
356	C₂×D₈	C₂³×C₈₉	1	8	yes	complemented
357	C₂×M₁₆	D₄×C₈₉	1	16	yes	complemented
358	C₂×M₁₆	C₂³×C₈₉	1	4	yes	—
359	C₂×M₁₆	C₂³×C₈₉	1	4	yes	—
360	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
361	C₄⋊Q₈	C₂³×C₈₉	1	4	yes	complemented
362	#127	—	2	4	no	—
363	#127	C₂³×C₈₉	1	8	yes	—
364	#127	C₂³×C₈₉	1	4	yes	complemented
365	#124	—	2	4	no	—
366	#124	C₂³×C₈₉	1	8	yes	—
367	#124	C₂³×C₈₉	1	8	yes	complemented
368	#123	—	2	8	no	—
369	C₂²⋊Q₈	—	2	8	no	—
370	#121	—	2	8	no	—
371	#121	—	2	16	no	—
372	#121	—	2	8	no	—
373	#120	—	2	16	no	—
374	#120	—	2	8	no	—
375	C₄×D₄	—	2	8	no	—
376	#117	D₄×C₈₉	1	4	yes	—
377	#117	C₂³×C₈₉	1	4	yes	—
378	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
379	C₂×C₂²⋊C₄	—	2	4	no	—
380	C₂×C₂²⋊C₄	D₄×C₈₉	1	2	yes	—
381	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
382	#97	C₂³×C₈₉	1	8	yes	complemented
383	C₄×C₈	C₂³×C₈₉	1	8	yes	complemented
384	C₂⁴	—	2	2	no	—
385	C₂⁴	—	2	1	no	—
386	C₂⁴	—	2	2	no	—
387	C₂⁴	—	2	4	no	—
388	C₂⁴	—	4	4	no	—
389	C₂⁴	—	8	4	no	—
390	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
391	#41	—	2	8	no	—
392	#41	—	2	8	no	—
393	#41	—	2	8	no	—
394	#41	—	2	8	no	—
395	#41	—	2	8	no	—
396	#41	—	2	8	no	—
397	#41	—	4	8	no	—
398	#41	—	4	8	no	—
399	#41	—	4	8	no	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	2	1	no	—
402	C₂×Q₈	—	2	4	no	—
403	C₂×Q₈	—	2	4	no	—
404	C₂×Q₈	—	2	4	no	—
405	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
406	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
407	C₂×D₄	—	2	4	no	—
408	C₂×D₄	—	2	4	no	—
409	C₂×D₄	—	2	4	no	—
410	C₂×D₄	—	2	4	no	—
411	C₂×D₄	—	2	8	no	—
412	C₂×D₄	—	2	8	no	—
413	C₂×D₄	—	2	8	no	—
414	C₂×D₄	—	2	4	no	—
415	C₂×D₄	—	2	4	no	—
416	C₂×D₄	—	2	4	no	—
417	C₂×D₄	—	2	2	no	—
418	C₂×D₄	—	2	2	no	—
419	C₂×D₄	—	2	1	no	—
420	C₂×D₄	—	2	8	no	—
421	C₂×D₄	—	2	16	no	—
422	C₂×D₄	—	4	4	no	—
423	C₂×D₄	—	4	8	no	—
424	C₂×D₄	—	4	4	no	—
425	C₂×D₄	—	4	8	no	—
426	C₂×D₄	—	8	8	no	—
427	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
428	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
429	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
430	C₂²×C₄	—	2	1	no	—
431	C₂²×C₄	—	2	2	no	—
432	C₂²×C₄	—	2	4	no	—
433	C₂²×C₄	—	2	4	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	4	4	no	—
436	C₂²×C₄	—	4	8	no	—
437	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
438	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
439	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
440	C₂²×C₄	C₂×D₄×C₈₉	1	8	yes	—
441	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
442	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
443	Q₁₆	—	2	16	no	—
444	SD₁₆	—	2	32	no	—
445	SD₁₆	—	2	16	no	—
446	SD₁₆	—	2	16	no	—
447	D₈	—	2	32	no	—
448	D₈	—	2	16	no	—
449	M₁₆	C₂×D₄×C₈₉	1	16	yes	—
450	M₁₆	C₂×D₄×C₈₉	1	16	yes	complemented
451	C₂×C₈	C₂×D₄×C₈₉	1	16	yes	complemented
452	C₂×C₈	C₂⁴×C₈₉	1	8	yes	—
453	C₄⋊C₄	—	2	8	no	—
454	C₄⋊C₄	C₂×D₄×C₈₉	1	8	yes	—
455	C₂²⋊C₄	—	2	16	no	—
456	C₂²⋊C₄	—	2	16	no	—
457	C₂²⋊C₄	C₂×D₄×C₈₉	1	8	yes	—
458	C₄²	C₂×D₄×C₈₉	1	4	yes	—
459	C₄²	C₂⁴×C₈₉	1	4	yes	—
460	C₂³	—	2	4	no	—
461	C₂³	—	2	4	no	—
462	C₂³	—	2	2	no	—
463	C₂³	—	4	2	no	—
464	C₂³	—	4	4	no	—
465	C₂³	—	4	8	no	—
466	C₂³	—	4	2	no	—
467	C₂³	—	4	4	no	—
468	C₂³	—	4	4	no	—
469	C₂³	—	4	8	no	—
470	C₂³	—	8	4	no	—
471	C₂³	—	8	8	no	—
472	C₂³	—	8	16	no	—
473	C₂³	#127×C₈₉	1	4	yes	—
474	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
475	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
476	Q₈	—	2	4	no	—
477	Q₈	—	2	4	no	—
478	Q₈	—	2	4	no	—
479	Q₈	—	4	4	no	—
480	Q₈	—	4	2	no	—
481	D₄	—	2	4	no	—
482	D₄	—	2	8	no	—
483	D₄	—	2	4	no	—
484	D₄	—	2	4	no	—
485	D₄	—	4	8	no	—
486	D₄	—	4	8	no	—
487	D₄	—	4	8	no	—
488	D₄	—	4	8	no	—
489	D₄	—	4	4	no	—
490	D₄	—	4	4	no	—
491	D₄	—	4	2	no	—
492	D₄	—	8	8	no	—
493	D₄	—	8	16	no	—
494	C₂×C₄	—	2	2	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	4	4	no	—
497	C₂×C₄	—	4	2	no	—
498	C₂×C₄	—	4	4	no	—
499	C₂×C₄	—	4	8	no	—
500	C₂×C₄	—	4	8	no	—
501	C₂×C₄	—	4	8	no	—
502	C₂×C₄	#127×C₈₉	1	4	yes	—
503	C₂×C₄	#127×C₈₉	1	8	yes	—
504	C₂×C₄	C₂²×D₄×C₈₉	1	8	yes	—
505	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
506	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
507	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
508	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
509	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
510	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
511	C₈	C₂²×D₄×C₈₉	1	16	yes	—
512	C₂²	—	2	2	no	—
513	C₂²	—	2	4	no	—
514	C₂²	—	4	4	no	—
515	C₂²	—	4	4	no	—
516	C₂²	—	4	8	no	—
517	C₂²	—	4	8	no	—
518	C₂²	—	4	4	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	4	no	—
521	C₂²	—	8	8	no	—
522	C₂²	—	8	4	no	—
523	C₂²	—	8	16	no	—
524	C₂²	—	8	8	no	—
525	C₂²	C₂×#127×C₈₉	1	4	yes	—
526	C₂²	C₂×#127×C₈₉	1	2	yes	—
527	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
528	C₄	—	2	8	no	—
529	C₄	—	4	2	no	—
530	C₄	—	4	4	no	—
531	C₄	C₂×#127×C₈₉	1	2	yes	—
532	C₄	C₂×#127×C₈₉	1	4	yes	—
533	C₄	C₂³×D₄×C₈₉	1	2	yes	—
534	C₂	—	2	2	no	—
535	C₂	—	2	4	no	—
536	C₂	—	8	2	no	—
537	C₂	—	8	4	no	—
538	C₂	—	8	4	no	—
539	C₂	C₂²×#127×C₈₉	1	1	yes	characteristic
540	C₂	C₈₉×#3585	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂×#575×C₈₉	C₂	semidirect
2. C₂²×AGL(1,ℤ/8)×C₈₉	C₂	semidirect
3. C₈₉×#3449	C₂	semidirect
4. C₂×#535×C₈₉	C₂	semidirect
5. C₈₉×#3155	C₂	semidirect
6. C₈₉×#3154	C₂	semidirect
7. C₈₉×#3153	C₂	semidirect
8. C₈₉×#3151	C₂	semidirect
9. C₈₉×#2873	C₂	semidirect
14. C₂×#137×C₈₉	C₂²	semidirect
15. C₂×#137×C₈₉	C₂²	semidirect
17. C₂×AGL(1,ℤ/8)×C₈₉	C₂²	semidirect
18. C₂×AGL(1,ℤ/8)×C₈₉	C₂²	semidirect
19. C₂×AGL(1,ℤ/8)×C₈₉	C₂²	semidirect
20. C₂×AGL(1,ℤ/8)×C₈₉	C₂²	semidirect
21. C₂×#135×C₈₉	C₂²	semidirect
22. C₂²×SD₁₆×C₈₉	C₂²	semidirect
23. C₂²×D₈×C₈₉	C₂²	semidirect
26. #536×C₈₉	C₂²	semidirect
28. #535×C₈₉	C₂²	semidirect
29. #535×C₈₉	C₂²	semidirect
31. #532×C₈₉	C₂²	semidirect
32. C₂×#127×C₈₉	C₂²	semidirect
33. C₂×#124×C₈₉	C₂²	semidirect
38. #497×C₈₉	C₂²	semidirect
39. #496×C₈₉	C₂²	semidirect
40. #496×C₈₉	C₂²	semidirect
41. #495×C₈₉	C₂²	semidirect
42. #493×C₈₉	C₂²	semidirect
43. #492×C₈₉	C₂²	semidirect
44. #405×C₈₉	C₂²	semidirect
45. C₄×M₁₆×C₈₉	C₂²	semidirect
78. C₂×Q₁₆×C₈₉	C₂³	semidirect
81. C₂×SD₁₆×C₈₉	C₂³	semidirect
82. C₂×SD₁₆×C₈₉	C₂³	semidirect
85. C₂×D₈×C₈₉	C₂³	semidirect
86. C₂×M₁₆×C₈₉	D₄	semidirect
90. C₄⋊Q₈×C₈₉	C₂³	semidirect
93. #127×C₈₉	C₂³	semidirect
96. #124×C₈₉	C₂³	semidirect
111. #97×C₈₉	C₂³	semidirect
112. C₄×C₈×C₈₉	C₂³	semidirect
179. M₁₆×C₈₉	C₂×D₄	semidirect
180. C₂×C₈×C₈₉	C₂×D₄	semidirect
263. #60644	C₈₉	direct (N × H)
271. C₂×#575	C₂×C₈₉	semidirect
272. C₂²×AGL(1,ℤ/8)	C₂×C₈₉	semidirect
273. #3449	C₂×C₈₉	semidirect
274. C₂×#535	C₂×C₈₉	semidirect
275. #3155	C₂×C₈₉	semidirect
276. #3154	C₂×C₈₉	semidirect
277. #3153	C₂×C₈₉	semidirect
278. #3151	C₂×C₈₉	semidirect
279. #2873	C₂×C₈₉	semidirect
280. C₈₉	#60644	direct (N × H)
285. C₂×#137	C₂²×C₈₉	semidirect
286. C₂×#137	C₂²×C₈₉	semidirect
288. C₂×AGL(1,ℤ/8)	C₂²×C₈₉	semidirect
289. C₂×AGL(1,ℤ/8)	C₂²×C₈₉	semidirect
290. C₂×AGL(1,ℤ/8)	C₂²×C₈₉	semidirect
291. C₂×AGL(1,ℤ/8)	C₂²×C₈₉	semidirect
292. C₂×#135	C₂²×C₈₉	semidirect
293. C₂²×SD₁₆	C₂²×C₈₉	semidirect
294. C₂²×D₈	C₂²×C₈₉	semidirect
297. #536	C₂²×C₈₉	semidirect
299. #535	C₂²×C₈₉	semidirect
300. #535	C₂²×C₈₉	semidirect
302. #532	C₂²×C₈₉	semidirect
303. C₂×#127	C₂²×C₈₉	semidirect
304. C₂×#124	C₂²×C₈₉	semidirect
309. #497	C₂²×C₈₉	semidirect
310. #496	C₂²×C₈₉	semidirect
311. #496	C₂²×C₈₉	semidirect
312. #495	C₂²×C₈₉	semidirect
313. #493	C₂²×C₈₉	semidirect
314. #492	C₂²×C₈₉	semidirect
315. #405	C₂²×C₈₉	semidirect
316. C₄×M₁₆	C₂²×C₈₉	semidirect
349. C₂×Q₁₆	C₂³×C₈₉	semidirect
352. C₂×SD₁₆	C₂³×C₈₉	semidirect
353. C₂×SD₁₆	C₂³×C₈₉	semidirect
356. C₂×D₈	C₂³×C₈₉	semidirect
357. C₂×M₁₆	D₄×C₈₉	semidirect
361. C₄⋊Q₈	C₂³×C₈₉	semidirect
364. #127	C₂³×C₈₉	semidirect
367. #124	C₂³×C₈₉	semidirect
382. #97	C₂³×C₈₉	semidirect
383. C₄×C₈	C₂³×C₈₉	semidirect
450. M₁₆	C₂×D₄×C₈₉	semidirect
451. C₂×C₈	C₂×D₄×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#575×C₈₉	2
C₂²×AGL(1,ℤ/8)×C₈₉	2
C₈₉×#3449	2
C₂×#535×C₈₉	2
C₈₉×#3155	2
C₈₉×#3154	2
C₈₉×#3153	2
C₈₉×#3151	2
C₈₉×#2873	2
#60644	89

Subgroups — C89×#60644 #36438492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#60644 #36438492