Subgroups — C₈₉×#60444 #36438292

Order 22784.

2918 conjugacy classes of subgroups.

The subgroup lattice has 574 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	1	yes	maximal, characteristic, complemented
2	C₂³×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#525×C₈₉	C₂	1	2	yes	maximal, complemented
4	C₈₉×#3056	C₂	1	8	yes	maximal, complemented
5	C₈₉×#3055	C₂	1	8	yes	maximal, complemented
6	C₈₉×#3054	C₂	1	4	yes	maximal, complemented
7	C₈₉×#3050	C₂	1	4	yes	maximal, complemented
8	C₈₉×#2911	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#2896	C₂	1	2	yes	maximal, complemented
10	C₂²×#41×C₈₉	C₂²	1	2	yes	—
11	C₂⁴×C₄×C₈₉	C₂²	1	1	yes	characteristic
12	#575×C₈₉	—	2	8	no	—
13	C₂×#137×C₈₉	C₂²	1	4	yes	complemented
14	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	4	yes	complemented
15	C₂×#135×C₈₉	C₂²	1	4	yes	complemented
16	C₂²×M₁₆×C₈₉	—	2	8	no	—
17	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
18	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
19	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	complemented
20	C₂²×M₁₆×C₈₉	C₂²	1	8	yes	complemented
21	C₂³×C₈×C₈₉	C₂²	1	2	yes	complemented
22	#525×C₈₉	—	2	2	no	—
23	#525×C₈₉	—	2	2	no	—
24	#525×C₈₉	—	2	4	no	—
25	#525×C₈₉	C₂²	1	8	yes	—
26	#525×C₈₉	C₂²	1	4	yes	complemented
27	#525×C₈₉	C₂²	1	4	yes	complemented
28	C₂×#123×C₈₉	—	2	2	no	—
29	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	4	yes	complemented
30	C₂×#121×C₈₉	C₂²	1	4	yes	complemented
31	C₂×#120×C₈₉	—	2	2	no	—
32	C₂×C₄×D₄×C₈₉	—	2	4	no	—
33	C₂×#117×C₈₉	C₂²	1	2	yes	—
34	#470×C₈₉	C₂²	1	8	yes	complemented
35	#469×C₈₉	C₂²	1	8	yes	complemented
36	#468×C₈₉	C₂²	1	16	yes	complemented
37	#467×C₈₉	C₂²	1	8	yes	complemented
38	#466×C₈₉	C₂²	1	8	yes	complemented
39	#465×C₈₉	C₂²	1	16	yes	complemented
40	#428×C₈₉	C₂²	1	8	yes	complemented
41	#427×C₈₉	C₂²	1	4	yes	complemented
42	#419×C₈₉	C₂²	1	4	yes	complemented
43	#418×C₈₉	C₂²	1	4	yes	complemented
44	#417×C₈₉	C₂²	1	8	yes	complemented
45	#416×C₈₉	C₂²	1	8	yes	complemented
46	C₂⁵×C₈₉	D₄	1	1	yes	characteristic
47	C₂×#41×C₈₉	—	2	2	no	—
48	C₂×#41×C₈₉	—	2	2	no	—
49	C₂×#41×C₈₉	—	2	8	no	—
50	C₂×#41×C₈₉	—	2	8	no	—
51	C₂×#41×C₈₉	C₂³	1	4	yes	—
52	C₂×#41×C₈₉	C₂³	1	4	yes	—
53	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
54	C₂²×D₄×C₈₉	—	2	4	no	—
55	C₂²×D₄×C₈₉	—	2	2	no	—
56	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
57	C₂³×C₄×C₈₉	—	2	4	no	—
58	C₂³×C₄×C₈₉	—	2	4	no	—
59	C₂³×C₄×C₈₉	—	2	2	no	—
60	C₂³×C₄×C₈₉	D₄	1	4	yes	—
61	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
62	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
63	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
64	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
65	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
66	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
68	#137×C₈₉	—	2	16	no	—
69	AGL(1,ℤ/8)×C₈₉	—	2	16	no	—
70	#135×C₈₉	—	2	16	no	—
71	C₂×Q₁₆×C₈₉	C₂³	1	4	yes	complemented
72	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	complemented
73	C₂×D₈×C₈₉	C₂³	1	4	yes	complemented
74	C₂×M₁₆×C₈₉	—	2	4	no	—
75	C₂×M₁₆×C₈₉	—	2	32	no	—
76	C₂×M₁₆×C₈₉	—	2	8	no	—
77	C₂×M₁₆×C₈₉	D₄	1	8	yes	complemented
78	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
79	C₂×M₁₆×C₈₉	C₂³	1	8	yes	—
80	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
81	C₂²×C₈×C₈₉	—	2	8	no	—
82	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
83	C₂²×C₈×C₈₉	C₂³	1	8	yes	—
84	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
85	#123×C₈₉	—	2	8	no	—
86	#123×C₈₉	—	2	4	no	—
87	#123×C₈₉	—	2	4	no	—
88	C₂²⋊Q₈×C₈₉	—	2	4	no	—
89	C₂²⋊Q₈×C₈₉	—	2	4	no	—
90	C₂²⋊Q₈×C₈₉	C₂³	1	8	yes	—
91	C₂²⋊Q₈×C₈₉	C₂³	1	8	yes	—
92	#121×C₈₉	—	2	4	no	—
93	#121×C₈₉	—	2	4	no	—
94	#121×C₈₉	C₂³	1	8	yes	—
95	#121×C₈₉	C₂³	1	8	yes	—
96	#120×C₈₉	—	2	8	no	—
97	#120×C₈₉	—	2	4	no	—
98	#120×C₈₉	—	2	4	no	—
99	C₄×D₄×C₈₉	—	2	8	no	—
100	C₄×D₄×C₈₉	—	2	8	no	—
101	C₄×D₄×C₈₉	—	2	8	no	—
102	C₄×D₄×C₈₉	—	2	8	no	—
103	#117×C₈₉	—	2	2	no	—
104	#117×C₈₉	—	2	2	no	—
105	#117×C₈₉	C₂³	1	4	yes	—
106	#117×C₈₉	C₂³	1	4	yes	—
107	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
108	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
109	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
110	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
111	C₂×C₂²⋊C₄×C₈₉	—	2	4	no	—
112	C₂×C₄²×C₈₉	—	2	2	no	—
113	#107×C₈₉	C₂³	1	8	yes	complemented
114	#106×C₈₉	C₂³	1	8	yes	complemented
115	#103×C₈₉	C₂³	1	16	yes	complemented
116	#102×C₈₉	C₂³	1	16	yes	complemented
117	C₂⁴×C₈₉	—	2	4	no	—
118	C₂⁴×C₈₉	—	2	2	no	—
119	C₂⁴×C₈₉	—	4	4	no	—
120	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
121	C₂⁴×C₈₉	C₂×D₄	1	4	yes	—
122	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
123	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
124	#41×C₈₉	—	2	8	no	—
125	#41×C₈₉	—	2	8	no	—
126	#41×C₈₉	—	2	8	no	—
127	#41×C₈₉	—	2	8	no	—
128	#41×C₈₉	—	4	8	no	—
129	#41×C₈₉	—	4	8	no	—
130	C₂×Q₈×C₈₉	—	2	2	no	—
131	C₂×Q₈×C₈₉	—	2	4	no	—
132	C₂×Q₈×C₈₉	—	2	4	no	—
133	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
134	C₂×D₄×C₈₉	—	2	8	no	—
135	C₂×D₄×C₈₉	—	2	8	no	—
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	2	no	—
140	C₂×D₄×C₈₉	—	2	4	no	—
141	C₂×D₄×C₈₉	—	4	16	no	—
142	C₂×D₄×C₈₉	—	4	8	no	—
143	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
144	C₂²×C₄×C₈₉	—	2	4	no	—
145	C₂²×C₄×C₈₉	—	2	16	no	—
146	C₂²×C₄×C₈₉	—	2	4	no	—
147	C₂²×C₄×C₈₉	—	2	4	no	—
148	C₂²×C₄×C₈₉	—	2	2	no	—
149	C₂²×C₄×C₈₉	—	2	8	no	—
150	C₂²×C₄×C₈₉	—	2	2	no	—
151	C₂²×C₄×C₈₉	—	2	8	no	—
152	C₂²×C₄×C₈₉	—	2	2	no	—
153	C₂²×C₄×C₈₉	—	2	2	no	—
154	C₂²×C₄×C₈₉	—	2	4	no	—
155	C₂²×C₄×C₈₉	—	2	2	no	—
156	C₂²×C₄×C₈₉	—	2	4	no	—
157	C₂²×C₄×C₈₉	—	2	4	no	—
158	C₂²×C₄×C₈₉	—	4	8	no	—
159	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
160	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
161	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
162	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
166	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
167	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
168	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
169	C₂²×C₄×C₈₉	#41	1	4	yes	—
170	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
171	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
172	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
173	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
174	Q₁₆×C₈₉	—	2	8	no	—
175	SD₁₆×C₈₉	—	2	16	no	—
176	D₈×C₈₉	—	2	8	no	—
177	M₁₆×C₈₉	—	2	8	no	—
178	M₁₆×C₈₉	—	2	16	no	—
179	M₁₆×C₈₉	C₂×D₄	1	16	yes	complemented
180	C₂×C₈×C₈₉	—	2	4	no	—
181	C₂×C₈×C₈₉	—	2	16	no	—
182	C₂×C₈×C₈₉	C₂×D₄	1	8	yes	complemented
183	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
184	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
185	C₄⋊C₄×C₈₉	—	2	8	no	—
186	C₄⋊C₄×C₈₉	—	2	8	no	—
187	C₄⋊C₄×C₈₉	—	2	4	no	—
188	C₄⋊C₄×C₈₉	C₂⁴	1	8	yes	—
189	C₂²⋊C₄×C₈₉	—	2	4	no	—
190	C₂²⋊C₄×C₈₉	—	2	4	no	—
191	C₂²⋊C₄×C₈₉	—	2	8	no	—
192	C₂²⋊C₄×C₈₉	—	2	8	no	—
193	C₂²⋊C₄×C₈₉	—	2	8	no	—
194	C₂²⋊C₄×C₈₉	—	2	8	no	—
195	C₄²×C₈₉	—	2	4	no	—
196	C₄²×C₈₉	—	2	4	no	—
197	C₂³×C₈₉	—	2	2	no	—
198	C₂³×C₈₉	—	2	4	no	—
199	C₂³×C₈₉	—	2	8	no	—
200	C₂³×C₈₉	—	2	2	no	—
201	C₂³×C₈₉	—	2	8	no	—
202	C₂³×C₈₉	—	2	4	no	—
203	C₂³×C₈₉	—	4	2	no	—
204	C₂³×C₈₉	—	4	4	no	—
205	C₂³×C₈₉	—	4	2	no	—
206	C₂³×C₈₉	—	4	16	no	—
207	C₂³×C₈₉	—	4	8	no	—
208	C₂³×C₈₉	—	4	4	no	—
209	C₂³×C₈₉	#121	1	4	yes	—
210	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
211	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
212	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
213	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
214	Q₈×C₈₉	—	2	4	no	—
215	Q₈×C₈₉	—	2	4	no	—
216	Q₈×C₈₉	—	4	4	no	—
217	D₄×C₈₉	—	2	4	no	—
218	D₄×C₈₉	—	2	4	no	—
219	D₄×C₈₉	—	4	16	no	—
220	D₄×C₈₉	—	4	8	no	—
221	D₄×C₈₉	—	4	8	no	—
222	D₄×C₈₉	—	4	16	no	—
223	D₄×C₈₉	—	4	4	no	—
224	C₂×C₄×C₈₉	—	2	8	no	—
225	C₂×C₄×C₈₉	—	2	2	no	—
226	C₂×C₄×C₈₉	—	2	2	no	—
227	C₂×C₄×C₈₉	—	2	4	no	—
228	C₂×C₄×C₈₉	—	2	4	no	—
229	C₂×C₄×C₈₉	—	2	8	no	—
230	C₂×C₄×C₈₉	—	2	8	no	—
231	C₂×C₄×C₈₉	—	2	2	no	—
232	C₂×C₄×C₈₉	—	2	2	no	—
233	C₂×C₄×C₈₉	—	2	2	no	—
234	C₂×C₄×C₈₉	—	2	2	no	—
235	C₂×C₄×C₈₉	—	2	8	no	—
236	C₂×C₄×C₈₉	—	2	4	no	—
237	C₂×C₄×C₈₉	—	4	8	no	—
238	C₂×C₄×C₈₉	—	4	4	no	—
239	C₂×C₄×C₈₉	—	4	8	no	—
240	C₂×C₄×C₈₉	—	4	8	no	—
241	C₂×C₄×C₈₉	#121	1	8	yes	—
242	C₂×C₄×C₈₉	#121	1	4	yes	—
243	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
244	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
245	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
246	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
247	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
248	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
249	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
250	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
251	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
252	C₈×C₈₉	—	2	4	no	—
253	C₈×C₈₉	C₂²×D₄	1	8	yes	—
254	C₂²×C₈₉	—	2	8	no	—
255	C₂²×C₈₉	—	2	2	no	—
256	C₂²×C₈₉	—	2	4	no	—
257	C₂²×C₈₉	—	2	2	no	—
258	C₂²×C₈₉	—	2	2	no	—
259	C₂²×C₈₉	—	2	2	no	—
260	C₂²×C₈₉	—	2	4	no	—
261	C₂²×C₈₉	—	4	4	no	—
262	C₂²×C₈₉	—	4	16	no	—
263	C₂²×C₈₉	—	4	4	no	—
264	C₂²×C₈₉	—	4	4	no	—
265	C₂²×C₈₉	—	4	8	no	—
266	C₂²×C₈₉	—	8	8	no	—
267	C₂²×C₈₉	C₂×#121	1	4	yes	—
268	C₂²×C₈₉	C₂×#121	1	2	yes	—
269	C₂²×C₈₉	#575	1	4	yes	complemented
270	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
271	C₄×C₈₉	—	2	2	no	—
272	C₄×C₈₉	—	2	2	no	—
273	C₄×C₈₉	—	4	4	no	—
274	C₄×C₈₉	—	4	8	no	—
275	C₄×C₈₉	C₂×#121	1	2	yes	—
276	C₄×C₈₉	C₂×#121	1	4	yes	—
277	C₄×C₈₉	C₂³×D₄	1	1	yes	characteristic
278	C₄×C₈₉	C₂²×#41	1	1	yes	characteristic
279	#60444	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
280	C₂×C₈₉	—	2	2	no	—
281	C₂×C₈₉	—	2	4	no	—
282	C₂×C₈₉	—	2	4	no	—
283	C₂×C₈₉	—	4	2	no	—
284	C₂×C₈₉	—	8	4	no	—
285	C₂×C₈₉	C₂²×#121	1	1	yes	characteristic
286	C₂×C₈₉	C₂×#575	1	2	yes	—
287	C₂×#575	C₂×C₈₉	1	1	yes	characteristic, complemented
288	C₂³×M₁₆	C₂×C₈₉	1	1	yes	characteristic, complemented
289	C₂×#525	C₂×C₈₉	1	2	yes	complemented
290	#3056	C₂×C₈₉	1	8	yes	complemented
291	#3055	C₂×C₈₉	1	8	yes	complemented
292	#3054	C₂×C₈₉	1	4	yes	complemented
293	#3050	C₂×C₈₉	1	4	yes	complemented
294	#2911	C₂×C₈₉	1	1	yes	characteristic, complemented
295	#2896	C₂×C₈₉	1	2	yes	complemented
296	C₈₉	#60444	1	1	yes	characteristic, direct product factor, Sylow
297	C₂²×#41	C₂²×C₈₉	1	2	yes	—
298	C₂⁴×C₄	C₂²×C₈₉	1	1	yes	characteristic
299	#575	—	2	8	no	—
300	C₂×#137	C₂²×C₈₉	1	4	yes	complemented
301	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	4	yes	complemented
302	C₂×#135	C₂²×C₈₉	1	4	yes	complemented
303	C₂²×M₁₆	—	2	8	no	—
304	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
305	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
306	C₂²×M₁₆	C₂²×C₈₉	1	2	yes	complemented
307	C₂²×M₁₆	C₂²×C₈₉	1	8	yes	complemented
308	C₂³×C₈	C₂²×C₈₉	1	2	yes	complemented
309	#525	—	2	2	no	—
310	#525	—	2	2	no	—
311	#525	—	2	4	no	—
312	#525	C₂²×C₈₉	1	8	yes	—
313	#525	C₂²×C₈₉	1	4	yes	complemented
314	#525	C₂²×C₈₉	1	4	yes	complemented
315	C₂×#123	—	2	2	no	—
316	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	4	yes	complemented
317	C₂×#121	C₂²×C₈₉	1	4	yes	complemented
318	C₂×#120	—	2	2	no	—
319	C₂×C₄×D₄	—	2	4	no	—
320	C₂×#117	C₂²×C₈₉	1	2	yes	—
321	#470	C₂²×C₈₉	1	8	yes	complemented
322	#469	C₂²×C₈₉	1	8	yes	complemented
323	#468	C₂²×C₈₉	1	16	yes	complemented
324	#467	C₂²×C₈₉	1	8	yes	complemented
325	#466	C₂²×C₈₉	1	8	yes	complemented
326	#465	C₂²×C₈₉	1	16	yes	complemented
327	#428	C₂²×C₈₉	1	8	yes	complemented
328	#427	C₂²×C₈₉	1	4	yes	complemented
329	#419	C₂²×C₈₉	1	4	yes	complemented
330	#418	C₂²×C₈₉	1	4	yes	complemented
331	#417	C₂²×C₈₉	1	8	yes	complemented
332	#416	C₂²×C₈₉	1	8	yes	complemented
333	C₂⁵	D₄×C₈₉	1	1	yes	characteristic
334	C₂×#41	—	2	2	no	—
335	C₂×#41	—	2	2	no	—
336	C₂×#41	—	2	8	no	—
337	C₂×#41	—	2	8	no	—
338	C₂×#41	C₂³×C₈₉	1	4	yes	—
339	C₂×#41	C₂³×C₈₉	1	4	yes	—
340	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
341	C₂²×D₄	—	2	4	no	—
342	C₂²×D₄	—	2	2	no	—
343	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
344	C₂³×C₄	—	2	4	no	—
345	C₂³×C₄	—	2	4	no	—
346	C₂³×C₄	—	2	2	no	—
347	C₂³×C₄	D₄×C₈₉	1	4	yes	—
348	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
349	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
350	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
351	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
352	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
353	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
354	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
355	#137	—	2	16	no	—
356	AGL(1,ℤ/8)	—	2	16	no	—
357	#135	—	2	16	no	—
358	C₂×Q₁₆	C₂³×C₈₉	1	4	yes	complemented
359	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	complemented
360	C₂×D₈	C₂³×C₈₉	1	4	yes	complemented
361	C₂×M₁₆	—	2	4	no	—
362	C₂×M₁₆	—	2	32	no	—
363	C₂×M₁₆	—	2	8	no	—
364	C₂×M₁₆	D₄×C₈₉	1	8	yes	complemented
365	C₂×M₁₆	C₂³×C₈₉	1	4	yes	—
366	C₂×M₁₆	C₂³×C₈₉	1	8	yes	—
367	C₂×M₁₆	C₂³×C₈₉	1	4	yes	—
368	C₂²×C₈	—	2	8	no	—
369	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
370	C₂²×C₈	C₂³×C₈₉	1	8	yes	—
371	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
372	#123	—	2	8	no	—
373	#123	—	2	4	no	—
374	#123	—	2	4	no	—
375	C₂²⋊Q₈	—	2	4	no	—
376	C₂²⋊Q₈	—	2	4	no	—
377	C₂²⋊Q₈	C₂³×C₈₉	1	8	yes	—
378	C₂²⋊Q₈	C₂³×C₈₉	1	8	yes	—
379	#121	—	2	4	no	—
380	#121	—	2	4	no	—
381	#121	C₂³×C₈₉	1	8	yes	—
382	#121	C₂³×C₈₉	1	8	yes	—
383	#120	—	2	8	no	—
384	#120	—	2	4	no	—
385	#120	—	2	4	no	—
386	C₄×D₄	—	2	8	no	—
387	C₄×D₄	—	2	8	no	—
388	C₄×D₄	—	2	8	no	—
389	C₄×D₄	—	2	8	no	—
390	#117	—	2	2	no	—
391	#117	—	2	2	no	—
392	#117	C₂³×C₈₉	1	4	yes	—
393	#117	C₂³×C₈₉	1	4	yes	—
394	C₂×C₄⋊C₄	—	2	4	no	—
395	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
396	C₂×C₂²⋊C₄	—	2	2	no	—
397	C₂×C₂²⋊C₄	—	2	4	no	—
398	C₂×C₂²⋊C₄	—	2	4	no	—
399	C₂×C₄²	—	2	2	no	—
400	#107	C₂³×C₈₉	1	8	yes	complemented
401	#106	C₂³×C₈₉	1	8	yes	complemented
402	#103	C₂³×C₈₉	1	16	yes	complemented
403	#102	C₂³×C₈₉	1	16	yes	complemented
404	C₂⁴	—	2	4	no	—
405	C₂⁴	—	2	2	no	—
406	C₂⁴	—	4	4	no	—
407	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
408	C₂⁴	C₂×D₄×C₈₉	1	4	yes	—
409	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
410	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
411	#41	—	2	8	no	—
412	#41	—	2	8	no	—
413	#41	—	2	8	no	—
414	#41	—	2	8	no	—
415	#41	—	4	8	no	—
416	#41	—	4	8	no	—
417	C₂×Q₈	—	2	2	no	—
418	C₂×Q₈	—	2	4	no	—
419	C₂×Q₈	—	2	4	no	—
420	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
421	C₂×D₄	—	2	8	no	—
422	C₂×D₄	—	2	8	no	—
423	C₂×D₄	—	2	4	no	—
424	C₂×D₄	—	2	4	no	—
425	C₂×D₄	—	2	4	no	—
426	C₂×D₄	—	2	2	no	—
427	C₂×D₄	—	2	4	no	—
428	C₂×D₄	—	4	16	no	—
429	C₂×D₄	—	4	8	no	—
430	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
431	C₂²×C₄	—	2	4	no	—
432	C₂²×C₄	—	2	16	no	—
433	C₂²×C₄	—	2	4	no	—
434	C₂²×C₄	—	2	4	no	—
435	C₂²×C₄	—	2	2	no	—
436	C₂²×C₄	—	2	8	no	—
437	C₂²×C₄	—	2	2	no	—
438	C₂²×C₄	—	2	8	no	—
439	C₂²×C₄	—	2	2	no	—
440	C₂²×C₄	—	2	2	no	—
441	C₂²×C₄	—	2	4	no	—
442	C₂²×C₄	—	2	2	no	—
443	C₂²×C₄	—	2	4	no	—
444	C₂²×C₄	—	2	4	no	—
445	C₂²×C₄	—	4	8	no	—
446	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
447	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
448	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
449	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
450	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
451	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
452	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
453	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
454	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
455	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
456	C₂²×C₄	#41×C₈₉	1	4	yes	—
457	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
458	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
459	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
460	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
461	Q₁₆	—	2	8	no	—
462	SD₁₆	—	2	16	no	—
463	D₈	—	2	8	no	—
464	M₁₆	—	2	8	no	—
465	M₁₆	—	2	16	no	—
466	M₁₆	C₂×D₄×C₈₉	1	16	yes	complemented
467	C₂×C₈	—	2	4	no	—
468	C₂×C₈	—	2	16	no	—
469	C₂×C₈	C₂×D₄×C₈₉	1	8	yes	complemented
470	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
471	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
472	C₄⋊C₄	—	2	8	no	—
473	C₄⋊C₄	—	2	8	no	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	C₂⁴×C₈₉	1	8	yes	—
476	C₂²⋊C₄	—	2	4	no	—
477	C₂²⋊C₄	—	2	4	no	—
478	C₂²⋊C₄	—	2	8	no	—
479	C₂²⋊C₄	—	2	8	no	—
480	C₂²⋊C₄	—	2	8	no	—
481	C₂²⋊C₄	—	2	8	no	—
482	C₄²	—	2	4	no	—
483	C₄²	—	2	4	no	—
484	C₂³	—	2	2	no	—
485	C₂³	—	2	4	no	—
486	C₂³	—	2	8	no	—
487	C₂³	—	2	2	no	—
488	C₂³	—	2	8	no	—
489	C₂³	—	2	4	no	—
490	C₂³	—	4	2	no	—
491	C₂³	—	4	4	no	—
492	C₂³	—	4	2	no	—
493	C₂³	—	4	16	no	—
494	C₂³	—	4	8	no	—
495	C₂³	—	4	4	no	—
496	C₂³	#121×C₈₉	1	4	yes	—
497	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
498	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
499	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
500	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
501	Q₈	—	2	4	no	—
502	Q₈	—	2	4	no	—
503	Q₈	—	4	4	no	—
504	D₄	—	2	4	no	—
505	D₄	—	2	4	no	—
506	D₄	—	4	16	no	—
507	D₄	—	4	8	no	—
508	D₄	—	4	8	no	—
509	D₄	—	4	16	no	—
510	D₄	—	4	4	no	—
511	C₂×C₄	—	2	8	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	4	no	—
515	C₂×C₄	—	2	4	no	—
516	C₂×C₄	—	2	8	no	—
517	C₂×C₄	—	2	8	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	8	no	—
523	C₂×C₄	—	2	4	no	—
524	C₂×C₄	—	4	8	no	—
525	C₂×C₄	—	4	4	no	—
526	C₂×C₄	—	4	8	no	—
527	C₂×C₄	—	4	8	no	—
528	C₂×C₄	#121×C₈₉	1	8	yes	—
529	C₂×C₄	#121×C₈₉	1	4	yes	—
530	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
531	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
532	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
533	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
534	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
535	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
536	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
537	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
538	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
539	C₈	—	2	4	no	—
540	C₈	C₂²×D₄×C₈₉	1	8	yes	—
541	C₂²	—	2	8	no	—
542	C₂²	—	2	4	no	—
543	C₂²	—	2	2	no	—
544	C₂²	—	2	2	no	—
545	C₂²	—	2	2	no	—
546	C₂²	—	2	4	no	—
547	C₂²	—	2	2	no	—
548	C₂²	—	4	4	no	—
549	C₂²	—	4	16	no	—
550	C₂²	—	4	4	no	—
551	C₂²	—	4	8	no	—
552	C₂²	—	4	4	no	—
553	C₂²	—	8	8	no	—
554	C₂²	C₂×#121×C₈₉	1	4	yes	—
555	C₂²	C₂×#121×C₈₉	1	2	yes	—
556	C₂²	#575×C₈₉	1	4	yes	complemented
557	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
558	C₄	—	2	2	no	—
559	C₄	—	2	2	no	—
560	C₄	—	4	8	no	—
561	C₄	—	4	4	no	—
562	C₄	C₂×#121×C₈₉	1	2	yes	—
563	C₄	C₂×#121×C₈₉	1	4	yes	—
564	C₄	C₂³×D₄×C₈₉	1	1	yes	characteristic
565	C₄	C₂²×#41×C₈₉	1	1	yes	characteristic
566	C₂	—	2	2	no	—
567	C₂	—	2	4	no	—
568	C₂	—	2	4	no	—
569	C₂	—	4	2	no	—
570	C₂	—	8	4	no	—
571	C₂	C₂²×#121×C₈₉	1	1	yes	characteristic
572	C₂	C₂×#575×C₈₉	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
socle	C₂²×C₈₉	C₂³×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈₉ ◁ C₂⁴×C₄×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. C₂³×M₁₆×C₈₉	C₂	semidirect
3. C₂×#525×C₈₉	C₂	semidirect
4. C₈₉×#3056	C₂	semidirect
5. C₈₉×#3055	C₂	semidirect
6. C₈₉×#3054	C₂	semidirect
7. C₈₉×#3050	C₂	semidirect
8. C₈₉×#2911	C₂	semidirect
9. C₈₉×#2896	C₂	semidirect
13. C₂×#137×C₈₉	C₂²	semidirect
14. C₂×AGL(1,ℤ/8)×C₈₉	C₂²	semidirect
15. C₂×#135×C₈₉	C₂²	semidirect
19. C₂²×M₁₆×C₈₉	C₂²	semidirect
20. C₂²×M₁₆×C₈₉	C₂²	semidirect
21. C₂³×C₈×C₈₉	C₂²	semidirect
26. #525×C₈₉	C₂²	semidirect
27. #525×C₈₉	C₂²	semidirect
29. C₂×C₂²⋊Q₈×C₈₉	C₂²	semidirect
30. C₂×#121×C₈₉	C₂²	semidirect
34. #470×C₈₉	C₂²	semidirect
35. #469×C₈₉	C₂²	semidirect
36. #468×C₈₉	C₂²	semidirect
37. #467×C₈₉	C₂²	semidirect
38. #466×C₈₉	C₂²	semidirect
39. #465×C₈₉	C₂²	semidirect
40. #428×C₈₉	C₂²	semidirect
41. #427×C₈₉	C₂²	semidirect
42. #419×C₈₉	C₂²	semidirect
43. #418×C₈₉	C₂²	semidirect
44. #417×C₈₉	C₂²	semidirect
45. #416×C₈₉	C₂²	semidirect
71. C₂×Q₁₆×C₈₉	C₂³	semidirect
72. C₂×SD₁₆×C₈₉	C₂³	semidirect
73. C₂×D₈×C₈₉	C₂³	semidirect
77. C₂×M₁₆×C₈₉	D₄	semidirect
113. #107×C₈₉	C₂³	semidirect
114. #106×C₈₉	C₂³	semidirect
115. #103×C₈₉	C₂³	semidirect
116. #102×C₈₉	C₂³	semidirect
179. M₁₆×C₈₉	C₂×D₄	semidirect
182. C₂×C₈×C₈₉	C₂×D₄	semidirect
269. C₂²×C₈₉	#575	semidirect
279. #60444	C₈₉	direct (N × H)
287. C₂×#575	C₂×C₈₉	semidirect
288. C₂³×M₁₆	C₂×C₈₉	semidirect
289. C₂×#525	C₂×C₈₉	semidirect
290. #3056	C₂×C₈₉	semidirect
291. #3055	C₂×C₈₉	semidirect
292. #3054	C₂×C₈₉	semidirect
293. #3050	C₂×C₈₉	semidirect
294. #2911	C₂×C₈₉	semidirect
295. #2896	C₂×C₈₉	semidirect
296. C₈₉	#60444	direct (N × H)
300. C₂×#137	C₂²×C₈₉	semidirect
301. C₂×AGL(1,ℤ/8)	C₂²×C₈₉	semidirect
302. C₂×#135	C₂²×C₈₉	semidirect
306. C₂²×M₁₆	C₂²×C₈₉	semidirect
307. C₂²×M₁₆	C₂²×C₈₉	semidirect
308. C₂³×C₈	C₂²×C₈₉	semidirect
313. #525	C₂²×C₈₉	semidirect
314. #525	C₂²×C₈₉	semidirect
316. C₂×C₂²⋊Q₈	C₂²×C₈₉	semidirect
317. C₂×#121	C₂²×C₈₉	semidirect
321. #470	C₂²×C₈₉	semidirect
322. #469	C₂²×C₈₉	semidirect
323. #468	C₂²×C₈₉	semidirect
324. #467	C₂²×C₈₉	semidirect
325. #466	C₂²×C₈₉	semidirect
326. #465	C₂²×C₈₉	semidirect
327. #428	C₂²×C₈₉	semidirect
328. #427	C₂²×C₈₉	semidirect
329. #419	C₂²×C₈₉	semidirect
330. #418	C₂²×C₈₉	semidirect
331. #417	C₂²×C₈₉	semidirect
332. #416	C₂²×C₈₉	semidirect
358. C₂×Q₁₆	C₂³×C₈₉	semidirect
359. C₂×SD₁₆	C₂³×C₈₉	semidirect
360. C₂×D₈	C₂³×C₈₉	semidirect
364. C₂×M₁₆	D₄×C₈₉	semidirect
400. #107	C₂³×C₈₉	semidirect
401. #106	C₂³×C₈₉	semidirect
402. #103	C₂³×C₈₉	semidirect
403. #102	C₂³×C₈₉	semidirect
466. M₁₆	C₂×D₄×C₈₉	semidirect
469. C₂×C₈	C₂×D₄×C₈₉	semidirect
556. C₂²	#575×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
C₂³×M₁₆×C₈₉	2
C₂×#525×C₈₉	2
C₈₉×#3056	2
C₈₉×#3055	2
C₈₉×#3054	2
C₈₉×#3050	2
C₈₉×#2911	2
C₈₉×#2896	2
#60444	89

Subgroups — C89×#60444 #36438292

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#60444 #36438292