Subgroups — C₈₉×#60828 #36438676

Order 22784.

2018 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3510	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈₉×#3499	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3225	C₂	1	8	yes	maximal, complemented
4	C₈₉×#3219	C₂	1	4	yes	maximal, complemented
5	C₈₉×#3218	C₂	1	4	yes	maximal, complemented
6	C₈₉×#3153	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#497×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#3139	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#3135	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#3044	C₂	1	2	yes	maximal, complemented
11	C₈₉×#3042	C₂	1	2	yes	maximal, complemented
12	C₈₉×#3011	C₂	1	2	yes	maximal, complemented
13	C₈₉×#3010	C₂	1	2	yes	maximal, complemented
14	C₈₉×#2923	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×2₋¹⁺⁴×C₈₉	C₂²	1	2	yes	—
16	C₂×#137×C₈₉	C₂²	1	2	yes	—
17	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	2	yes	—
18	C₂²×Q₁₆×C₈₉	C₂²	1	2	yes	—
19	C₂²×SD₁₆×C₈₉	C₂²	1	2	yes	—
20	#559×C₈₉	—	2	2	no	—
21	#556×C₈₉	—	2	2	no	—
22	#551×C₈₉	—	2	4	no	—
23	D₄×Q₈×C₈₉	—	2	2	no	—
24	#547×C₈₉	—	2	4	no	—
25	#547×C₈₉	—	2	2	no	—
26	#542×C₈₉	C₂²	1	4	yes	—
27	#536×C₈₉	C₂²	1	2	yes	—
28	#536×C₈₉	C₂²	1	1	yes	characteristic
29	#536×C₈₉	C₂²	1	4	yes	complemented
30	#535×C₈₉	C₂²	1	1	yes	characteristic
31	#533×C₈₉	C₂²	1	1	yes	characteristic
32	#532×C₈₉	C₂²	1	1	yes	characteristic
33	#532×C₈₉	C₂²	1	2	yes	complemented
34	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	complemented
36	#529×C₈₉	C₂²	1	4	yes	—
37	C₂×#124×C₈₉	C₂²	1	1	yes	characteristic
38	#519×C₈₉	C₂²	1	1	yes	characteristic
39	#518×C₈₉	C₂²	1	1	yes	characteristic
40	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
41	#497×C₈₉	C₂²	1	4	yes	—
42	#497×C₈₉	C₂²	1	4	yes	—
43	#497×C₈₉	C₂²	1	2	yes	—
44	#496×C₈₉	C₂²	1	2	yes	—
45	#495×C₈₉	C₂²	1	4	yes	—
46	#491×C₈₉	C₂²	1	2	yes	complemented
47	#491×C₈₉	C₂²	1	4	yes	complemented
48	#490×C₈₉	C₂²	1	2	yes	complemented
49	#489×C₈₉	C₂²	1	4	yes	complemented
50	#487×C₈₉	C₂²	1	4	yes	complemented
51	#464×C₈₉	C₂²	1	8	yes	complemented
52	#463×C₈₉	C₂²	1	8	yes	—
53	#462×C₈₉	C₂²	1	8	yes	complemented
54	#461×C₈₉	C₂²	1	8	yes	—
55	#452×C₈₉	C₂²	1	8	yes	complemented
56	#451×C₈₉	C₂²	1	8	yes	complemented
57	#449×C₈₉	C₂²	1	8	yes	—
58	#448×C₈₉	C₂²	1	8	yes	—
59	#433×C₈₉	C₂²	1	4	yes	complemented
60	#432×C₈₉	C₂²	1	4	yes	complemented
61	#424×C₈₉	C₂²	1	2	yes	—
62	#419×C₈₉	C₂²	1	2	yes	—
63	#419×C₈₉	C₂²	1	2	yes	—
64	#416×C₈₉	C₂²	1	2	yes	—
65	#416×C₈₉	C₂²	1	2	yes	—
66	#408×C₈₉	C₂²	1	2	yes	—
67	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
68	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
69	2₋¹⁺⁴×C₈₉	—	4	4	no	—
70	2₋¹⁺⁴×C₈₉	—	4	4	no	—
71	C₂×#41×C₈₉	—	2	4	no	—
72	C₂×#41×C₈₉	—	2	2	no	—
73	C₂×#41×C₈₉	C₂³	1	2	yes	—
74	C₂×#41×C₈₉	C₂³	1	4	yes	—
75	C₂×#41×C₈₉	C₂³	1	1	yes	characteristic
76	C₂²×Q₈×C₈₉	—	2	2	no	—
77	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
78	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
80	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
81	#137×C₈₉	—	4	4	no	—
82	AGL(1,ℤ/8)×C₈₉	—	4	4	no	—
83	C₂×Q₁₆×C₈₉	—	4	4	no	—
84	C₂×Q₁₆×C₈₉	C₂³	1	4	yes	—
85	C₂×Q₁₆×C₈₉	C₂³	1	8	yes	—
86	C₂×SD₁₆×C₈₉	—	4	4	no	—
87	C₂×SD₁₆×C₈₉	C₂³	1	4	yes	—
88	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	—
89	C₂×SD₁₆×C₈₉	C₂³	1	4	yes	—
90	C₂×D₈×C₈₉	C₂³	1	4	yes	—
91	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
92	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
93	C₄⋊Q₈×C₈₉	—	2	1	no	—
94	C₄⋊Q₈×C₈₉	—	2	2	no	—
95	C₄⋊Q₈×C₈₉	—	2	4	no	—
96	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
97	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
98	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
99	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
100	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
101	#127×C₈₉	—	2	1	no	—
102	#127×C₈₉	C₂³	1	1	yes	characteristic
103	#126×C₈₉	—	2	4	no	—
104	#126×C₈₉	—	2	4	no	—
105	#125×C₈₉	C₂³	1	2	yes	—
106	#124×C₈₉	—	2	1	no	—
107	#124×C₈₉	—	2	1	no	—
108	#124×C₈₉	—	2	4	no	—
109	#124×C₈₉	—	2	2	no	—
110	#124×C₈₉	C₂³	1	4	yes	—
111	#124×C₈₉	C₂³	1	2	yes	—
112	#124×C₈₉	C₂³	1	2	yes	—
113	#124×C₈₉	C₂³	1	2	yes	—
114	#123×C₈₉	—	2	2	no	—
115	#123×C₈₉	—	2	4	no	—
116	#123×C₈₉	—	2	4	no	—
117	#123×C₈₉	—	2	8	no	—
118	#123×C₈₉	—	2	4	no	—
119	C₂²⋊Q₈×C₈₉	—	2	2	no	—
120	C₂²⋊Q₈×C₈₉	—	2	4	no	—
121	C₂²⋊Q₈×C₈₉	—	2	8	no	—
122	C₂²⋊Q₈×C₈₉	—	2	4	no	—
123	C₂²⋊Q₈×C₈₉	C₂³	1	4	yes	—
124	#121×C₈₉	—	2	2	no	—
125	#121×C₈₉	—	2	4	no	—
126	#121×C₈₉	C₂³	1	4	yes	—
127	#120×C₈₉	—	2	2	no	—
128	C₄×Q₈×C₈₉	—	2	2	no	—
129	C₄×Q₈×C₈₉	—	2	2	no	—
130	C₄×Q₈×C₈₉	D₄	1	2	yes	—
131	C₄×Q₈×C₈₉	C₂³	1	4	yes	—
132	C₄×D₄×C₈₉	—	2	4	no	—
133	C₄×D₄×C₈₉	—	2	2	no	—
134	C₄×D₄×C₈₉	—	2	2	no	—
135	C₄×D₄×C₈₉	—	2	4	no	—
136	C₄×D₄×C₈₉	D₄	1	4	yes	—
137	C₄×D₄×C₈₉	D₄	1	2	yes	—
138	C₄×D₄×C₈₉	C₂³	1	4	yes	—
139	#117×C₈₉	—	2	2	no	—
140	#117×C₈₉	—	2	4	no	—
141	#117×C₈₉	C₂³	1	1	yes	characteristic
142	#117×C₈₉	C₂³	1	1	yes	characteristic
143	#117×C₈₉	C₂³	1	1	yes	characteristic
144	#117×C₈₉	C₂³	1	2	yes	—
145	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
146	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
147	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
148	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
149	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
150	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
151	C₄⋊C₈×C₈₉	C₂³	1	8	yes	—
152	#103×C₈₉	C₂³	1	4	yes	—
153	#103×C₈₉	C₂³	1	4	yes	—
154	#103×C₈₉	C₂³	1	8	yes	—
155	#103×C₈₉	C₂³	1	8	yes	—
156	#102×C₈₉	C₂³	1	4	yes	—
157	#102×C₈₉	C₂³	1	4	yes	—
158	C₂²⋊C₈×C₈₉	C₂³	1	8	yes	—
159	#97×C₈₉	C₂³	1	2	yes	—
160	#97×C₈₉	C₂³	1	2	yes	—
161	#97×C₈₉	C₂³	1	2	yes	—
162	C₄×C₈×C₈₉	C₂³	1	2	yes	—
163	C₂⁴×C₈₉	—	2	1	no	—
164	#41×C₈₉	—	2	4	no	—
165	#41×C₈₉	—	2	4	no	—
166	#41×C₈₉	—	2	4	no	—
167	#41×C₈₉	—	4	4	no	—
168	#41×C₈₉	—	4	4	no	—
169	#41×C₈₉	—	4	16	no	—
170	#41×C₈₉	—	4	8	no	—
171	C₂×Q₈×C₈₉	—	2	1	no	—
172	C₂×Q₈×C₈₉	—	2	4	no	—
173	C₂×Q₈×C₈₉	—	2	2	no	—
174	C₂×Q₈×C₈₉	—	2	2	no	—
175	C₂×Q₈×C₈₉	—	2	4	no	—
176	C₂×Q₈×C₈₉	—	2	4	no	—
177	C₂×Q₈×C₈₉	—	2	4	no	—
178	C₂×Q₈×C₈₉	—	4	4	no	—
179	C₂×Q₈×C₈₉	—	4	4	no	—
180	C₂×Q₈×C₈₉	—	4	8	no	—
181	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
182	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
183	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
184	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
185	C₂×D₄×C₈₉	—	2	4	no	—
186	C₂×D₄×C₈₉	—	2	2	no	—
187	C₂×D₄×C₈₉	—	2	4	no	—
188	C₂×D₄×C₈₉	—	2	2	no	—
189	C₂×D₄×C₈₉	—	2	1	no	—
190	C₂×D₄×C₈₉	—	2	4	no	—
191	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
192	C₂×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
193	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
194	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
195	C₂²×C₄×C₈₉	—	2	1	no	—
196	C₂²×C₄×C₈₉	—	2	4	no	—
197	C₂²×C₄×C₈₉	—	2	2	no	—
198	C₂²×C₄×C₈₉	—	2	4	no	—
199	C₂²×C₄×C₈₉	—	2	2	no	—
200	C₂²×C₄×C₈₉	—	2	4	no	—
201	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
202	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
203	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
204	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
205	Q₁₆×C₈₉	—	4	8	no	—
206	Q₁₆×C₈₉	—	4	4	no	—
207	SD₁₆×C₈₉	—	4	8	no	—
208	SD₁₆×C₈₉	—	4	4	no	—
209	SD₁₆×C₈₉	—	4	4	no	—
210	D₈×C₈₉	—	4	4	no	—
211	M₁₆×C₈₉	—	2	4	no	—
212	C₂×C₈×C₈₉	—	2	4	no	—
213	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
214	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
215	C₄⋊C₄×C₈₉	—	2	4	no	—
216	C₄⋊C₄×C₈₉	—	2	2	no	—
217	C₄⋊C₄×C₈₉	—	2	8	no	—
218	C₄⋊C₄×C₈₉	—	2	4	no	—
219	C₄⋊C₄×C₈₉	—	2	2	no	—
220	C₄⋊C₄×C₈₉	—	2	4	no	—
221	C₄⋊C₄×C₈₉	—	2	4	no	—
222	C₄⋊C₄×C₈₉	—	2	2	no	—
223	C₄⋊C₄×C₈₉	—	2	2	no	—
224	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
225	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
226	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
227	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
228	C₂²⋊C₄×C₈₉	—	2	1	no	—
229	C₂²⋊C₄×C₈₉	—	2	1	no	—
230	C₂²⋊C₄×C₈₉	—	2	4	no	—
231	C₂²⋊C₄×C₈₉	—	2	4	no	—
232	C₂²⋊C₄×C₈₉	—	2	4	no	—
233	C₂²⋊C₄×C₈₉	—	2	4	no	—
234	C₂²⋊C₄×C₈₉	—	2	4	no	—
235	C₂²⋊C₄×C₈₉	—	2	8	no	—
236	C₂²⋊C₄×C₈₉	—	2	4	no	—
237	C₂²⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
238	C₂²⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
239	C₄²×C₈₉	—	2	4	no	—
240	C₄²×C₈₉	—	2	1	no	—
241	C₄²×C₈₉	—	2	1	no	—
242	C₄²×C₈₉	C₂×D₄	1	4	yes	—
243	C₄²×C₈₉	C₂×D₄	1	2	yes	—
244	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
245	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
246	C₄²×C₈₉	C₂⁴	1	2	yes	—
247	C₂³×C₈₉	—	2	2	no	—
248	C₂³×C₈₉	—	2	2	no	—
249	C₂³×C₈₉	—	4	2	no	—
250	C₂³×C₈₉	—	8	2	no	—
251	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
252	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
253	Q₈×C₈₉	—	2	2	no	—
254	Q₈×C₈₉	—	2	4	no	—
255	Q₈×C₈₉	—	2	2	no	—
256	Q₈×C₈₉	—	2	4	no	—
257	Q₈×C₈₉	—	2	4	no	—
258	Q₈×C₈₉	—	2	2	no	—
259	Q₈×C₈₉	—	4	8	no	—
260	Q₈×C₈₉	—	4	2	no	—
261	Q₈×C₈₉	—	4	4	no	—
262	Q₈×C₈₉	—	8	2	no	—
263	D₄×C₈₉	—	2	4	no	—
264	D₄×C₈₉	—	2	2	no	—
265	D₄×C₈₉	—	2	2	no	—
266	D₄×C₈₉	—	2	4	no	—
267	D₄×C₈₉	—	2	2	no	—
268	D₄×C₈₉	—	4	4	no	—
269	D₄×C₈₉	—	4	8	no	—
270	D₄×C₈₉	—	4	4	no	—
271	D₄×C₈₉	—	4	2	no	—
272	D₄×C₈₉	—	8	4	no	—
273	C₂×C₄×C₈₉	—	2	4	no	—
274	C₂×C₄×C₈₉	—	2	2	no	—
275	C₂×C₄×C₈₉	—	2	2	no	—
276	C₂×C₄×C₈₉	—	2	2	no	—
277	C₂×C₄×C₈₉	—	2	4	no	—
278	C₂×C₄×C₈₉	—	2	2	no	—
279	C₂×C₄×C₈₉	—	4	4	no	—
280	C₂×C₄×C₈₉	—	4	2	no	—
281	C₂×C₄×C₈₉	—	4	8	no	—
282	C₂×C₄×C₈₉	—	4	8	no	—
283	C₂×C₄×C₈₉	—	4	2	no	—
284	C₂×C₄×C₈₉	—	4	4	no	—
285	C₂×C₄×C₈₉	—	4	4	no	—
286	C₂×C₄×C₈₉	—	8	4	no	—
287	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
288	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
289	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
290	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
291	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
292	C₂×C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
293	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
294	C₈×C₈₉	—	2	4	no	—
295	C₈×C₈₉	—	2	4	no	—
296	C₂²×C₈₉	—	2	2	no	—
297	C₂²×C₈₉	—	2	4	no	—
298	C₂²×C₈₉	—	2	2	no	—
299	C₂²×C₈₉	—	4	4	no	—
300	C₂²×C₈₉	—	4	4	no	—
301	C₂²×C₈₉	—	4	2	no	—
302	C₂²×C₈₉	—	4	2	no	—
303	C₂²×C₈₉	—	16	2	no	—
304	C₂²×C₈₉	#535	1	2	yes	—
305	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
306	C₄×C₈₉	—	2	2	no	—
307	C₄×C₈₉	—	2	4	no	—
308	C₄×C₈₉	—	2	2	no	—
309	C₄×C₈₉	—	2	2	no	—
310	C₄×C₈₉	—	4	2	no	—
311	C₄×C₈₉	—	4	2	no	—
312	C₄×C₈₉	—	4	4	no	—
313	C₄×C₈₉	—	4	2	no	—
314	C₄×C₈₉	—	4	2	no	—
315	C₄×C₈₉	#535	1	2	yes	—
316	C₄×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
317	#60828	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
318	C₂×C₈₉	—	2	2	no	—
319	C₂×C₈₉	—	4	2	no	—
320	C₂×C₈₉	—	8	2	no	—
321	C₂×C₈₉	—	8	2	no	—
322	C₂×C₈₉	C₂×#535	1	1	yes	characteristic
323	C₂×C₈₉	#3586	1	2	yes	—
324	#3510	C₂×C₈₉	1	1	yes	characteristic, complemented
325	#3499	C₂×C₈₉	1	1	yes	characteristic, complemented
326	#3225	C₂×C₈₉	1	8	yes	complemented
327	#3219	C₂×C₈₉	1	4	yes	complemented
328	#3218	C₂×C₈₉	1	4	yes	complemented
329	#3153	C₂×C₈₉	1	1	yes	characteristic, complemented
330	C₂×#497	C₂×C₈₉	1	1	yes	characteristic, complemented
331	#3139	C₂×C₈₉	1	1	yes	characteristic, complemented
332	#3135	C₂×C₈₉	1	1	yes	characteristic, complemented
333	#3044	C₂×C₈₉	1	2	yes	complemented
334	#3042	C₂×C₈₉	1	2	yes	complemented
335	#3011	C₂×C₈₉	1	2	yes	complemented
336	#3010	C₂×C₈₉	1	2	yes	complemented
337	#2923	C₂×C₈₉	1	1	yes	characteristic, complemented
338	C₈₉	#60828	1	1	yes	characteristic, direct product factor, Sylow
339	C₂×2₋¹⁺⁴	C₂²×C₈₉	1	2	yes	—
340	C₂×#137	C₂²×C₈₉	1	2	yes	—
341	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	2	yes	—
342	C₂²×Q₁₆	C₂²×C₈₉	1	2	yes	—
343	C₂²×SD₁₆	C₂²×C₈₉	1	2	yes	—
344	#559	—	2	2	no	—
345	#556	—	2	2	no	—
346	#551	—	2	4	no	—
347	D₄×Q₈	—	2	2	no	—
348	#547	—	2	4	no	—
349	#547	—	2	2	no	—
350	#542	C₂²×C₈₉	1	4	yes	—
351	#536	C₂²×C₈₉	1	2	yes	—
352	#536	C₂²×C₈₉	1	1	yes	characteristic
353	#536	C₂²×C₈₉	1	4	yes	complemented
354	#535	C₂²×C₈₉	1	1	yes	characteristic
355	#533	C₂²×C₈₉	1	1	yes	characteristic
356	#532	C₂²×C₈₉	1	1	yes	characteristic
357	#532	C₂²×C₈₉	1	2	yes	complemented
358	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
359	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	complemented
360	#529	C₂²×C₈₉	1	4	yes	—
361	C₂×#124	C₂²×C₈₉	1	1	yes	characteristic
362	#519	C₂²×C₈₉	1	1	yes	characteristic
363	#518	C₂²×C₈₉	1	1	yes	characteristic
364	C₄×#41	C₂²×C₈₉	1	1	yes	characteristic
365	#497	C₂²×C₈₉	1	4	yes	—
366	#497	C₂²×C₈₉	1	4	yes	—
367	#497	C₂²×C₈₉	1	2	yes	—
368	#496	C₂²×C₈₉	1	2	yes	—
369	#495	C₂²×C₈₉	1	4	yes	—
370	#491	C₂²×C₈₉	1	2	yes	complemented
371	#491	C₂²×C₈₉	1	4	yes	complemented
372	#490	C₂²×C₈₉	1	2	yes	complemented
373	#489	C₂²×C₈₉	1	4	yes	complemented
374	#487	C₂²×C₈₉	1	4	yes	complemented
375	#464	C₂²×C₈₉	1	8	yes	complemented
376	#463	C₂²×C₈₉	1	8	yes	—
377	#462	C₂²×C₈₉	1	8	yes	complemented
378	#461	C₂²×C₈₉	1	8	yes	—
379	#452	C₂²×C₈₉	1	8	yes	complemented
380	#451	C₂²×C₈₉	1	8	yes	complemented
381	#449	C₂²×C₈₉	1	8	yes	—
382	#448	C₂²×C₈₉	1	8	yes	—
383	#433	C₂²×C₈₉	1	4	yes	complemented
384	#432	C₂²×C₈₉	1	4	yes	complemented
385	#424	C₂²×C₈₉	1	2	yes	—
386	#419	C₂²×C₈₉	1	2	yes	—
387	#419	C₂²×C₈₉	1	2	yes	—
388	#416	C₂²×C₈₉	1	2	yes	—
389	#416	C₂²×C₈₉	1	2	yes	—
390	#408	C₂²×C₈₉	1	2	yes	—
391	C₄×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
392	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
393	2₋¹⁺⁴	—	4	4	no	—
394	2₋¹⁺⁴	—	4	4	no	—
395	C₂×#41	—	2	4	no	—
396	C₂×#41	—	2	2	no	—
397	C₂×#41	C₂³×C₈₉	1	2	yes	—
398	C₂×#41	C₂³×C₈₉	1	4	yes	—
399	C₂×#41	C₂³×C₈₉	1	1	yes	characteristic
400	C₂²×Q₈	—	2	2	no	—
401	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
402	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
403	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
404	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
405	#137	—	4	4	no	—
406	AGL(1,ℤ/8)	—	4	4	no	—
407	C₂×Q₁₆	—	4	4	no	—
408	C₂×Q₁₆	C₂³×C₈₉	1	8	yes	—
409	C₂×Q₁₆	C₂³×C₈₉	1	4	yes	—
410	C₂×SD₁₆	—	4	4	no	—
411	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	—
412	C₂×SD₁₆	C₂³×C₈₉	1	4	yes	—
413	C₂×SD₁₆	C₂³×C₈₉	1	4	yes	—
414	C₂×D₈	C₂³×C₈₉	1	4	yes	—
415	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
416	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
417	C₄⋊Q₈	—	2	1	no	—
418	C₄⋊Q₈	—	2	2	no	—
419	C₄⋊Q₈	—	2	4	no	—
420	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
421	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
422	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
423	C₄⋊Q₈	C₂³×C₈₉	1	4	yes	—
424	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
425	#127	—	2	1	no	—
426	#127	C₂³×C₈₉	1	1	yes	characteristic
427	#126	—	2	4	no	—
428	#126	—	2	4	no	—
429	#125	C₂³×C₈₉	1	2	yes	—
430	#124	—	2	1	no	—
431	#124	—	2	1	no	—
432	#124	—	2	4	no	—
433	#124	—	2	2	no	—
434	#124	C₂³×C₈₉	1	2	yes	—
435	#124	C₂³×C₈₉	1	2	yes	—
436	#124	C₂³×C₈₉	1	2	yes	—
437	#124	C₂³×C₈₉	1	4	yes	—
438	#123	—	2	2	no	—
439	#123	—	2	4	no	—
440	#123	—	2	4	no	—
441	#123	—	2	8	no	—
442	#123	—	2	4	no	—
443	C₂²⋊Q₈	—	2	2	no	—
444	C₂²⋊Q₈	—	2	4	no	—
445	C₂²⋊Q₈	—	2	8	no	—
446	C₂²⋊Q₈	—	2	4	no	—
447	C₂²⋊Q₈	C₂³×C₈₉	1	4	yes	—
448	#121	—	2	2	no	—
449	#121	—	2	4	no	—
450	#121	C₂³×C₈₉	1	4	yes	—
451	#120	—	2	2	no	—
452	C₄×Q₈	—	2	2	no	—
453	C₄×Q₈	—	2	2	no	—
454	C₄×Q₈	D₄×C₈₉	1	2	yes	—
455	C₄×Q₈	C₂³×C₈₉	1	4	yes	—
456	C₄×D₄	—	2	4	no	—
457	C₄×D₄	—	2	2	no	—
458	C₄×D₄	—	2	2	no	—
459	C₄×D₄	—	2	4	no	—
460	C₄×D₄	D₄×C₈₉	1	4	yes	—
461	C₄×D₄	D₄×C₈₉	1	2	yes	—
462	C₄×D₄	C₂³×C₈₉	1	4	yes	—
463	#117	—	2	2	no	—
464	#117	—	2	4	no	—
465	#117	C₂³×C₈₉	1	1	yes	characteristic
466	#117	C₂³×C₈₉	1	1	yes	characteristic
467	#117	C₂³×C₈₉	1	1	yes	characteristic
468	#117	C₂³×C₈₉	1	2	yes	—
469	C₂×C₄⋊C₄	—	2	2	no	—
470	C₂×C₄⋊C₄	—	2	4	no	—
471	C₂×C₂²⋊C₄	—	2	2	no	—
472	C₂×C₂²⋊C₄	—	2	2	no	—
473	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
474	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
475	C₄⋊C₈	C₂³×C₈₉	1	8	yes	—
476	#103	C₂³×C₈₉	1	4	yes	—
477	#103	C₂³×C₈₉	1	4	yes	—
478	#103	C₂³×C₈₉	1	8	yes	—
479	#103	C₂³×C₈₉	1	8	yes	—
480	#102	C₂³×C₈₉	1	4	yes	—
481	#102	C₂³×C₈₉	1	4	yes	—
482	C₂²⋊C₈	C₂³×C₈₉	1	8	yes	—
483	#97	C₂³×C₈₉	1	2	yes	—
484	#97	C₂³×C₈₉	1	2	yes	—
485	#97	C₂³×C₈₉	1	2	yes	—
486	C₄×C₈	C₂³×C₈₉	1	2	yes	—
487	C₂⁴	—	2	1	no	—
488	#41	—	2	4	no	—
489	#41	—	2	4	no	—
490	#41	—	2	4	no	—
491	#41	—	4	4	no	—
492	#41	—	4	4	no	—
493	#41	—	4	16	no	—
494	#41	—	4	8	no	—
495	C₂×Q₈	—	2	1	no	—
496	C₂×Q₈	—	2	4	no	—
497	C₂×Q₈	—	2	2	no	—
498	C₂×Q₈	—	2	2	no	—
499	C₂×Q₈	—	2	4	no	—
500	C₂×Q₈	—	2	4	no	—
501	C₂×Q₈	—	2	4	no	—
502	C₂×Q₈	—	4	4	no	—
503	C₂×Q₈	—	4	4	no	—
504	C₂×Q₈	—	4	8	no	—
505	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
506	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
507	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
508	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
509	C₂×D₄	—	2	4	no	—
510	C₂×D₄	—	2	2	no	—
511	C₂×D₄	—	2	4	no	—
512	C₂×D₄	—	2	2	no	—
513	C₂×D₄	—	2	1	no	—
514	C₂×D₄	—	2	4	no	—
515	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
516	C₂×D₄	C₂×D₄×C₈₉	1	1	yes	characteristic
517	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
518	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
519	C₂²×C₄	—	2	1	no	—
520	C₂²×C₄	—	2	4	no	—
521	C₂²×C₄	—	2	2	no	—
522	C₂²×C₄	—	2	4	no	—
523	C₂²×C₄	—	2	2	no	—
524	C₂²×C₄	—	2	4	no	—
525	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
526	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
527	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
528	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
529	Q₁₆	—	4	8	no	—
530	Q₁₆	—	4	4	no	—
531	SD₁₆	—	4	8	no	—
532	SD₁₆	—	4	4	no	—
533	SD₁₆	—	4	4	no	—
534	D₈	—	4	4	no	—
535	M₁₆	—	2	4	no	—
536	C₂×C₈	—	2	4	no	—
537	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
538	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
539	C₄⋊C₄	—	2	4	no	—
540	C₄⋊C₄	—	2	2	no	—
541	C₄⋊C₄	—	2	8	no	—
542	C₄⋊C₄	—	2	4	no	—
543	C₄⋊C₄	—	2	2	no	—
544	C₄⋊C₄	—	2	4	no	—
545	C₄⋊C₄	—	2	4	no	—
546	C₄⋊C₄	—	2	2	no	—
547	C₄⋊C₄	—	2	2	no	—
548	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
549	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
550	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
551	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
552	C₂²⋊C₄	—	2	1	no	—
553	C₂²⋊C₄	—	2	1	no	—
554	C₂²⋊C₄	—	2	4	no	—
555	C₂²⋊C₄	—	2	4	no	—
556	C₂²⋊C₄	—	2	4	no	—
557	C₂²⋊C₄	—	2	4	no	—
558	C₂²⋊C₄	—	2	4	no	—
559	C₂²⋊C₄	—	2	8	no	—
560	C₂²⋊C₄	—	2	4	no	—
561	C₂²⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
562	C₂²⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
563	C₄²	—	2	4	no	—
564	C₄²	—	2	1	no	—
565	C₄²	—	2	1	no	—
566	C₄²	C₂×D₄×C₈₉	1	4	yes	—
567	C₄²	C₂×D₄×C₈₉	1	2	yes	—
568	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
569	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
570	C₄²	C₂⁴×C₈₉	1	2	yes	—
571	C₂³	—	2	2	no	—
572	C₂³	—	2	2	no	—
573	C₂³	—	4	2	no	—
574	C₂³	—	8	2	no	—
575	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
576	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
577	Q₈	—	2	2	no	—
578	Q₈	—	2	4	no	—
579	Q₈	—	2	2	no	—
580	Q₈	—	2	4	no	—
581	Q₈	—	2	4	no	—
582	Q₈	—	2	2	no	—
583	Q₈	—	4	8	no	—
584	Q₈	—	4	2	no	—
585	Q₈	—	4	4	no	—
586	Q₈	—	8	2	no	—
587	D₄	—	2	4	no	—
588	D₄	—	2	2	no	—
589	D₄	—	2	2	no	—
590	D₄	—	2	4	no	—
591	D₄	—	2	2	no	—
592	D₄	—	4	4	no	—
593	D₄	—	4	8	no	—
594	D₄	—	4	4	no	—
595	D₄	—	4	2	no	—
596	D₄	—	8	4	no	—
597	C₂×C₄	—	2	4	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	2	no	—
601	C₂×C₄	—	2	4	no	—
602	C₂×C₄	—	2	2	no	—
603	C₂×C₄	—	4	4	no	—
604	C₂×C₄	—	4	2	no	—
605	C₂×C₄	—	4	8	no	—
606	C₂×C₄	—	4	8	no	—
607	C₂×C₄	—	4	2	no	—
608	C₂×C₄	—	4	4	no	—
609	C₂×C₄	—	4	4	no	—
610	C₂×C₄	—	8	4	no	—
611	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
612	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
613	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
614	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
615	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
616	C₂×C₄	H(5,𝔽₂)×C₈₉	1	4	yes	—
617	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
618	C₈	—	2	4	no	—
619	C₈	—	2	4	no	—
620	C₂²	—	2	2	no	—
621	C₂²	—	2	4	no	—
622	C₂²	—	2	2	no	—
623	C₂²	—	4	4	no	—
624	C₂²	—	4	4	no	—
625	C₂²	—	4	2	no	—
626	C₂²	—	4	2	no	—
627	C₂²	—	16	2	no	—
628	C₂²	#535×C₈₉	1	2	yes	—
629	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
630	C₄	—	2	2	no	—
631	C₄	—	2	4	no	—
632	C₄	—	2	2	no	—
633	C₄	—	2	2	no	—
634	C₄	—	4	2	no	—
635	C₄	—	4	2	no	—
636	C₄	—	4	4	no	—
637	C₄	—	4	2	no	—
638	C₄	—	4	2	no	—
639	C₄	#535×C₈₉	1	2	yes	—
640	C₄	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
641	C₂	—	2	2	no	—
642	C₂	—	4	2	no	—
643	C₂	—	8	2	no	—
644	C₂	—	8	2	no	—
645	C₂	C₂×#535×C₈₉	1	1	yes	characteristic
646	C₂	C₈₉×#3586	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 ◁ C₄×#41×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#3510	C₂	semidirect
2. C₈₉×#3499	C₂	semidirect
3. C₈₉×#3225	C₂	semidirect
4. C₈₉×#3219	C₂	semidirect
5. C₈₉×#3218	C₂	semidirect
6. C₈₉×#3153	C₂	semidirect
7. C₂×#497×C₈₉	C₂	semidirect
8. C₈₉×#3139	C₂	semidirect
9. C₈₉×#3135	C₂	semidirect
10. C₈₉×#3044	C₂	semidirect
11. C₈₉×#3042	C₂	semidirect
12. C₈₉×#3011	C₂	semidirect
13. C₈₉×#3010	C₂	semidirect
14. C₈₉×#2923	C₂	semidirect
29. #536×C₈₉	C₂²	semidirect
33. #532×C₈₉	C₂²	semidirect
35. C₂×C₄⋊Q₈×C₈₉	C₂²	semidirect
46. #491×C₈₉	C₂²	semidirect
47. #491×C₈₉	C₂²	semidirect
48. #490×C₈₉	C₂²	semidirect
49. #489×C₈₉	C₂²	semidirect
50. #487×C₈₉	C₂²	semidirect
51. #464×C₈₉	C₂²	semidirect
53. #462×C₈₉	C₂²	semidirect
55. #452×C₈₉	C₂²	semidirect
56. #451×C₈₉	C₂²	semidirect
59. #433×C₈₉	C₂²	semidirect
60. #432×C₈₉	C₂²	semidirect
317. #60828	C₈₉	direct (N × H)
324. #3510	C₂×C₈₉	semidirect
325. #3499	C₂×C₈₉	semidirect
326. #3225	C₂×C₈₉	semidirect
327. #3219	C₂×C₈₉	semidirect
328. #3218	C₂×C₈₉	semidirect
329. #3153	C₂×C₈₉	semidirect
330. C₂×#497	C₂×C₈₉	semidirect
331. #3139	C₂×C₈₉	semidirect
332. #3135	C₂×C₈₉	semidirect
333. #3044	C₂×C₈₉	semidirect
334. #3042	C₂×C₈₉	semidirect
335. #3011	C₂×C₈₉	semidirect
336. #3010	C₂×C₈₉	semidirect
337. #2923	C₂×C₈₉	semidirect
338. C₈₉	#60828	direct (N × H)
353. #536	C₂²×C₈₉	semidirect
357. #532	C₂²×C₈₉	semidirect
359. C₂×C₄⋊Q₈	C₂²×C₈₉	semidirect
370. #491	C₂²×C₈₉	semidirect
371. #491	C₂²×C₈₉	semidirect
372. #490	C₂²×C₈₉	semidirect
373. #489	C₂²×C₈₉	semidirect
374. #487	C₂²×C₈₉	semidirect
375. #464	C₂²×C₈₉	semidirect
377. #462	C₂²×C₈₉	semidirect
379. #452	C₂²×C₈₉	semidirect
380. #451	C₂²×C₈₉	semidirect
383. #433	C₂²×C₈₉	semidirect
384. #432	C₂²×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3510	2
C₈₉×#3499	2
C₈₉×#3225	2
C₈₉×#3219	2
C₈₉×#3218	2
C₈₉×#3153	2
C₂×#497×C₈₉	2
C₈₉×#3139	2
C₈₉×#3135	2
C₈₉×#3044	2
C₈₉×#3042	2
C₈₉×#3011	2
C₈₉×#3010	2
C₈₉×#2923	2
#60828	89

Subgroups — C89×#60828 #36438676

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#60828 #36438676