Subgroups — C₈₉×#60807 #36438655

Order 22784.

2030 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3455	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#536×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3225	C₂	1	8	yes	maximal, complemented
4	C₈₉×#3223	C₂	1	4	yes	maximal, complemented
5	C₈₉×#3221	C₂	1	4	yes	maximal, complemented
6	C₈₉×#3153	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#3152	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#3140	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#3139	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#3040	C₂	1	2	yes	maximal, complemented
11	C₈₉×#3039	C₂	1	2	yes	maximal, complemented
12	C₈₉×#3005	C₂	1	2	yes	maximal, complemented
13	C₈₉×#3004	C₂	1	2	yes	maximal, complemented
14	C₈₉×#2921	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
16	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×#137×C₈₉	C₂²	1	2	yes	—
18	C₂×#137×C₈₉	C₂²	1	2	yes	—
19	C₂×#137×C₈₉	C₂²	1	2	yes	—
20	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	2	yes	—
21	#542×C₈₉	C₂²	1	4	yes	complemented
22	#541×C₈₉	C₂²	1	2	yes	complemented
23	#540×C₈₉	C₂²	1	2	yes	complemented
24	#539×C₈₉	—	2	2	no	—
25	#538×C₈₉	—	2	2	no	—
26	#536×C₈₉	—	2	4	no	—
27	#536×C₈₉	C₂²	1	4	yes	—
28	#536×C₈₉	C₂²	1	4	yes	—
29	#533×C₈₉	C₂²	1	1	yes	characteristic
30	#532×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
33	#529×C₈₉	—	2	4	no	—
34	C₂×#124×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×#124×C₈₉	C₂²	1	1	yes	characteristic
36	#525×C₈₉	C₂²	1	2	yes	—
37	C₂×#123×C₈₉	—	2	2	no	—
38	C₂×C₂²⋊Q₈×C₈₉	—	2	2	no	—
39	C₄×#41×C₈₉	C₂²	1	2	yes	—
40	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
41	#497×C₈₉	C₂²	1	4	yes	complemented
42	#497×C₈₉	C₂²	1	2	yes	complemented
43	#496×C₈₉	C₂²	1	2	yes	complemented
44	#495×C₈₉	C₂²	1	4	yes	complemented
45	#494×C₈₉	C₂²	1	2	yes	complemented
46	#492×C₈₉	C₂²	1	2	yes	complemented
47	#491×C₈₉	C₂²	1	4	yes	complemented
48	#489×C₈₉	C₂²	1	4	yes	complemented
49	#488×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	complemented
53	#462×C₈₉	C₂²	1	8	yes	complemented
54	#461×C₈₉	C₂²	1	8	yes	complemented
55	#452×C₈₉	C₂²	1	8	yes	—
56	#451×C₈₉	C₂²	1	8	yes	complemented
57	#449×C₈₉	C₂²	1	8	yes	—
58	#448×C₈₉	C₂²	1	8	yes	complemented
59	#433×C₈₉	C₂²	1	4	yes	complemented
60	#433×C₈₉	C₂²	1	4	yes	complemented
61	#423×C₈₉	C₂²	1	2	yes	—
62	#419×C₈₉	C₂²	1	2	yes	—
63	#419×C₈₉	C₂²	1	2	yes	—
64	#419×C₈₉	C₂²	1	2	yes	—
65	#418×C₈₉	C₂²	1	2	yes	—
66	#407×C₈₉	C₂²	1	2	yes	—
67	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
68	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
69	C₂×#41×C₈₉	—	2	2	no	—
70	C₂×#41×C₈₉	—	4	4	no	—
71	C₂×#41×C₈₉	C₂³	1	2	yes	—
72	C₂×#41×C₈₉	C₂³	1	4	yes	—
73	C₂²×Q₈×C₈₉	—	2	1	no	—
74	C₂²×Q₈×C₈₉	—	2	2	no	—
75	C₂²×Q₈×C₈₉	—	4	4	no	—
76	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
77	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
80	C₂²×D₄×C₈₉	—	2	1	no	—
81	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
82	C₂³×C₄×C₈₉	—	2	1	no	—
83	C₂³×C₄×C₈₉	—	2	1	no	—
84	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
85	#137×C₈₉	—	4	4	no	—
86	#137×C₈₉	—	4	4	no	—
87	#137×C₈₉	—	4	4	no	—
88	AGL(1,ℤ/8)×C₈₉	—	4	4	no	—
89	C₂×Q₁₆×C₈₉	C₂³	1	4	yes	—
90	C₂×Q₁₆×C₈₉	C₂³	1	4	yes	—
91	C₂×Q₁₆×C₈₉	C₂³	1	4	yes	—
92	C₂×SD₁₆×C₈₉	C₂³	1	4	yes	—
93	C₂×SD₁₆×C₈₉	C₂³	1	4	yes	—
94	C₂×SD₁₆×C₈₉	C₂³	1	4	yes	—
95	C₂×SD₁₆×C₈₉	C₂³	1	4	yes	—
96	C₂×D₈×C₈₉	C₂³	1	4	yes	—
97	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
98	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
99	C₄⋊Q₈×C₈₉	—	2	2	no	—
100	C₄⋊Q₈×C₈₉	—	2	2	no	—
101	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
102	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
103	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
104	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
105	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
106	#127×C₈₉	C₂³	1	1	yes	characteristic
107	#126×C₈₉	—	2	4	no	—
108	#126×C₈₉	—	2	4	no	—
109	#125×C₈₉	—	2	4	no	—
110	#125×C₈₉	C₂³	1	2	yes	—
111	#124×C₈₉	—	2	4	no	—
112	#124×C₈₉	—	2	2	no	—
113	#124×C₈₉	—	2	2	no	—
114	#124×C₈₉	C₂³	1	4	yes	—
115	#124×C₈₉	C₂³	1	2	yes	—
116	#124×C₈₉	C₂³	1	2	yes	—
117	#124×C₈₉	C₂³	1	2	yes	—
118	#123×C₈₉	—	2	2	no	—
119	#123×C₈₉	—	2	4	no	—
120	#123×C₈₉	—	2	8	no	—
121	#123×C₈₉	—	2	8	no	—
122	#123×C₈₉	—	2	4	no	—
123	C₂²⋊Q₈×C₈₉	—	2	2	no	—
124	C₂²⋊Q₈×C₈₉	—	2	8	no	—
125	C₂²⋊Q₈×C₈₉	—	2	8	no	—
126	C₂²⋊Q₈×C₈₉	—	2	4	no	—
127	C₂²⋊Q₈×C₈₉	C₂³	1	4	yes	—
128	#121×C₈₉	—	2	2	no	—
129	#121×C₈₉	—	2	4	no	—
130	#121×C₈₉	C₂³	1	4	yes	—
131	#120×C₈₉	—	2	2	no	—
132	C₄×Q₈×C₈₉	—	2	2	no	—
133	C₄×Q₈×C₈₉	C₂³	1	4	yes	—
134	C₄×D₄×C₈₉	—	2	4	no	—
135	C₄×D₄×C₈₉	—	2	4	no	—
136	C₄×D₄×C₈₉	—	2	2	no	—
137	C₄×D₄×C₈₉	C₂³	1	4	yes	—
138	#117×C₈₉	—	2	2	no	—
139	#117×C₈₉	D₄	1	4	yes	—
140	#117×C₈₉	C₂³	1	2	yes	—
141	#117×C₈₉	C₂³	1	2	yes	—
142	#117×C₈₉	C₂³	1	2	yes	—
143	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
144	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
145	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
146	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
147	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
148	C₂×C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
149	C₂×C₄²×C₈₉	—	2	2	no	—
150	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
151	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
152	C₄⋊C₈×C₈₉	C₂³	1	8	yes	—
153	#103×C₈₉	C₂³	1	8	yes	—
154	#103×C₈₉	C₂³	1	8	yes	—
155	#103×C₈₉	C₂³	1	8	yes	—
156	#102×C₈₉	C₂³	1	8	yes	—
157	C₂²⋊C₈×C₈₉	C₂³	1	8	yes	—
158	#97×C₈₉	C₂³	1	2	yes	—
159	#97×C₈₉	C₂³	1	2	yes	—
160	C₄×C₈×C₈₉	C₂³	1	2	yes	—
161	C₄×C₈×C₈₉	C₂³	1	2	yes	—
162	C₂⁴×C₈₉	—	2	1	no	—
163	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
164	#41×C₈₉	—	2	4	no	—
165	#41×C₈₉	—	2	4	no	—
166	#41×C₈₉	—	4	8	no	—
167	#41×C₈₉	—	4	8	no	—
168	C₂×Q₈×C₈₉	—	2	4	no	—
169	C₂×Q₈×C₈₉	—	2	1	no	—
170	C₂×Q₈×C₈₉	—	2	1	no	—
171	C₂×Q₈×C₈₉	—	2	1	no	—
172	C₂×Q₈×C₈₉	—	2	2	no	—
173	C₂×Q₈×C₈₉	—	2	4	no	—
174	C₂×Q₈×C₈₉	—	2	4	no	—
175	C₂×Q₈×C₈₉	—	2	4	no	—
176	C₂×Q₈×C₈₉	—	4	8	no	—
177	C₂×Q₈×C₈₉	—	4	4	no	—
178	C₂×Q₈×C₈₉	—	4	8	no	—
179	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
180	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
181	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
182	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
183	C₂×D₄×C₈₉	—	2	2	no	—
184	C₂×D₄×C₈₉	—	2	4	no	—
185	C₂×D₄×C₈₉	—	2	4	no	—
186	C₂×D₄×C₈₉	—	2	1	no	—
187	C₂×D₄×C₈₉	—	2	2	no	—
188	C₂×D₄×C₈₉	—	2	4	no	—
189	C₂×D₄×C₈₉	—	4	4	no	—
190	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
191	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
192	C₂²×C₄×C₈₉	—	2	4	no	—
193	C₂²×C₄×C₈₉	—	2	1	no	—
194	C₂²×C₄×C₈₉	—	2	1	no	—
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	4	no	—
200	C₂²×C₄×C₈₉	—	4	4	no	—
201	C₂²×C₄×C₈₉	—	4	4	no	—
202	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
203	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
204	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
205	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
206	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
207	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
208	Q₁₆×C₈₉	—	4	4	no	—
209	Q₁₆×C₈₉	—	4	4	no	—
210	Q₁₆×C₈₉	—	4	4	no	—
211	SD₁₆×C₈₉	—	4	4	no	—
212	SD₁₆×C₈₉	—	4	4	no	—
213	SD₁₆×C₈₉	—	4	4	no	—
214	SD₁₆×C₈₉	—	4	4	no	—
215	D₈×C₈₉	—	4	4	no	—
216	M₁₆×C₈₉	—	2	4	no	—
217	M₁₆×C₈₉	—	2	4	no	—
218	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
219	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
220	C₄⋊C₄×C₈₉	—	2	8	no	—
221	C₄⋊C₄×C₈₉	—	2	4	no	—
222	C₄⋊C₄×C₈₉	—	2	4	no	—
223	C₄⋊C₄×C₈₉	—	2	4	no	—
224	C₄⋊C₄×C₈₉	—	2	2	no	—
225	C₄⋊C₄×C₈₉	—	2	4	no	—
226	C₄⋊C₄×C₈₉	—	2	4	no	—
227	C₄⋊C₄×C₈₉	—	2	2	no	—
228	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
229	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
230	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
231	C₂²⋊C₄×C₈₉	—	2	2	no	—
232	C₂²⋊C₄×C₈₉	—	2	4	no	—
233	C₂²⋊C₄×C₈₉	—	2	8	no	—
234	C₂²⋊C₄×C₈₉	—	2	4	no	—
235	C₂²⋊C₄×C₈₉	—	2	4	no	—
236	C₂²⋊C₄×C₈₉	—	2	4	no	—
237	C₂²⋊C₄×C₈₉	—	2	4	no	—
238	C₂²⋊C₄×C₈₉	—	2	4	no	—
239	C₂²⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
240	C₂²⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
241	C₄²×C₈₉	—	2	2	no	—
242	C₄²×C₈₉	—	2	4	no	—
243	C₄²×C₈₉	C₂×D₄	1	2	yes	—
244	C₄²×C₈₉	C₂×D₄	1	2	yes	—
245	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
246	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
247	C₄²×C₈₉	C₂⁴	1	2	yes	—
248	C₂³×C₈₉	—	2	2	no	—
249	C₂³×C₈₉	—	2	2	no	—
250	C₂³×C₈₉	—	2	2	no	—
251	C₂³×C₈₉	—	4	2	no	—
252	C₂³×C₈₉	—	4	4	no	—
253	C₂³×C₈₉	—	4	2	no	—
254	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
255	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
256	Q₈×C₈₉	—	2	2	no	—
257	Q₈×C₈₉	—	2	4	no	—
258	Q₈×C₈₉	—	2	4	no	—
259	Q₈×C₈₉	—	2	4	no	—
260	Q₈×C₈₉	—	2	2	no	—
261	Q₈×C₈₉	—	4	8	no	—
262	Q₈×C₈₉	—	4	2	no	—
263	Q₈×C₈₉	—	4	2	no	—
264	Q₈×C₈₉	—	4	2	no	—
265	Q₈×C₈₉	—	8	2	no	—
266	D₄×C₈₉	—	2	2	no	—
267	D₄×C₈₉	—	2	4	no	—
268	D₄×C₈₉	—	2	2	no	—
269	D₄×C₈₉	—	4	8	no	—
270	D₄×C₈₉	—	4	4	no	—
271	D₄×C₈₉	—	4	2	no	—
272	D₄×C₈₉	—	8	2	no	—
273	D₄×C₈₉	—	8	4	no	—
274	C₂×C₄×C₈₉	—	2	2	no	—
275	C₂×C₄×C₈₉	—	2	4	no	—
276	C₂×C₄×C₈₉	—	2	4	no	—
277	C₂×C₄×C₈₉	—	2	2	no	—
278	C₂×C₄×C₈₉	—	2	4	no	—
279	C₂×C₄×C₈₉	—	2	2	no	—
280	C₂×C₄×C₈₉	—	2	2	no	—
281	C₂×C₄×C₈₉	—	2	2	no	—
282	C₂×C₄×C₈₉	—	4	2	no	—
283	C₂×C₄×C₈₉	—	4	2	no	—
284	C₂×C₄×C₈₉	—	4	8	no	—
285	C₂×C₄×C₈₉	—	4	8	no	—
286	C₂×C₄×C₈₉	—	4	2	no	—
287	C₂×C₄×C₈₉	—	4	2	no	—
288	C₂×C₄×C₈₉	—	4	2	no	—
289	C₂×C₄×C₈₉	—	4	4	no	—
290	C₂×C₄×C₈₉	—	8	4	no	—
291	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
292	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
293	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
294	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
295	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
296	C₂×C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
297	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
298	C₈×C₈₉	—	2	4	no	—
299	C₈×C₈₉	—	2	4	no	—
300	C₂²×C₈₉	—	2	2	no	—
301	C₂²×C₈₉	—	2	4	no	—
302	C₂²×C₈₉	—	2	2	no	—
303	C₂²×C₈₉	—	4	4	no	—
304	C₂²×C₈₉	—	4	4	no	—
305	C₂²×C₈₉	—	4	4	no	—
306	C₂²×C₈₉	—	4	2	no	—
307	C₂²×C₈₉	—	4	2	no	—
308	C₂²×C₈₉	—	8	4	no	—
309	C₂²×C₈₉	#535	1	2	yes	—
310	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
311	C₄×C₈₉	—	2	4	no	—
312	C₄×C₈₉	—	2	2	no	—
313	C₄×C₈₉	—	2	2	no	—
314	C₄×C₈₉	—	2	2	no	—
315	C₄×C₈₉	—	4	4	no	—
316	C₄×C₈₉	—	4	2	no	—
317	C₄×C₈₉	—	4	2	no	—
318	C₄×C₈₉	—	4	2	no	—
319	C₄×C₈₉	—	4	2	no	—
320	C₄×C₈₉	#535	1	2	yes	—
321	C₄×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
322	#60807	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
323	C₂×C₈₉	—	2	2	no	—
324	C₂×C₈₉	—	4	2	no	—
325	C₂×C₈₉	—	8	2	no	—
326	C₂×C₈₉	—	8	2	no	—
327	C₂×C₈₉	C₂×#535	1	1	yes	characteristic
328	C₂×C₈₉	#3586	1	2	yes	—
329	#3455	C₂×C₈₉	1	1	yes	characteristic, complemented
330	C₂×#536	C₂×C₈₉	1	1	yes	characteristic, complemented
331	#3225	C₂×C₈₉	1	8	yes	complemented
332	#3223	C₂×C₈₉	1	4	yes	complemented
333	#3221	C₂×C₈₉	1	4	yes	complemented
334	#3153	C₂×C₈₉	1	1	yes	characteristic, complemented
335	#3152	C₂×C₈₉	1	1	yes	characteristic, complemented
336	#3140	C₂×C₈₉	1	1	yes	characteristic, complemented
337	#3139	C₂×C₈₉	1	1	yes	characteristic, complemented
338	#3040	C₂×C₈₉	1	2	yes	complemented
339	#3039	C₂×C₈₉	1	2	yes	complemented
340	#3005	C₂×C₈₉	1	2	yes	complemented
341	#3004	C₂×C₈₉	1	2	yes	complemented
342	#2921	C₂×C₈₉	1	1	yes	characteristic, complemented
343	C₈₉	#60807	1	1	yes	characteristic, direct product factor, Sylow
344	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
345	C₂³×Q₈	C₂²×C₈₉	1	1	yes	characteristic
346	C₂×#137	C₂²×C₈₉	1	2	yes	—
347	C₂×#137	C₂²×C₈₉	1	2	yes	—
348	C₂×#137	C₂²×C₈₉	1	2	yes	—
349	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	2	yes	—
350	#542	C₂²×C₈₉	1	4	yes	complemented
351	#541	C₂²×C₈₉	1	2	yes	complemented
352	#540	C₂²×C₈₉	1	2	yes	complemented
353	#539	—	2	2	no	—
354	#538	—	2	2	no	—
355	#536	—	2	4	no	—
356	#536	C₂²×C₈₉	1	4	yes	—
357	#536	C₂²×C₈₉	1	4	yes	—
358	#533	C₂²×C₈₉	1	1	yes	characteristic
359	#532	C₂²×C₈₉	1	1	yes	characteristic
360	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
361	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
362	#529	—	2	4	no	—
363	C₂×#124	C₂²×C₈₉	1	1	yes	characteristic
364	C₂×#124	C₂²×C₈₉	1	1	yes	characteristic
365	#525	C₂²×C₈₉	1	2	yes	—
366	C₂×#123	—	2	2	no	—
367	C₂×C₂²⋊Q₈	—	2	2	no	—
368	C₄×#41	C₂²×C₈₉	1	2	yes	—
369	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
370	#497	C₂²×C₈₉	1	4	yes	complemented
371	#497	C₂²×C₈₉	1	2	yes	complemented
372	#496	C₂²×C₈₉	1	2	yes	complemented
373	#495	C₂²×C₈₉	1	4	yes	complemented
374	#494	C₂²×C₈₉	1	2	yes	complemented
375	#492	C₂²×C₈₉	1	2	yes	complemented
376	#491	C₂²×C₈₉	1	4	yes	complemented
377	#489	C₂²×C₈₉	1	4	yes	complemented
378	#488	C₂²×C₈₉	1	4	yes	complemented
379	#487	C₂²×C₈₉	1	4	yes	complemented
380	#464	C₂²×C₈₉	1	8	yes	complemented
381	#463	C₂²×C₈₉	1	8	yes	complemented
382	#462	C₂²×C₈₉	1	8	yes	complemented
383	#461	C₂²×C₈₉	1	8	yes	complemented
384	#452	C₂²×C₈₉	1	8	yes	—
385	#451	C₂²×C₈₉	1	8	yes	complemented
386	#449	C₂²×C₈₉	1	8	yes	—
387	#448	C₂²×C₈₉	1	8	yes	complemented
388	#433	C₂²×C₈₉	1	4	yes	complemented
389	#433	C₂²×C₈₉	1	4	yes	complemented
390	#423	C₂²×C₈₉	1	2	yes	—
391	#419	C₂²×C₈₉	1	2	yes	—
392	#419	C₂²×C₈₉	1	2	yes	—
393	#419	C₂²×C₈₉	1	2	yes	—
394	#418	C₂²×C₈₉	1	2	yes	—
395	#407	C₂²×C₈₉	1	2	yes	—
396	C₄×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
397	C₄×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
398	C₂×#41	—	2	2	no	—
399	C₂×#41	—	4	4	no	—
400	C₂×#41	C₂³×C₈₉	1	2	yes	—
401	C₂×#41	C₂³×C₈₉	1	4	yes	—
402	C₂²×Q₈	—	2	1	no	—
403	C₂²×Q₈	—	2	2	no	—
404	C₂²×Q₈	—	4	4	no	—
405	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
406	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
407	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
408	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
409	C₂²×D₄	—	2	1	no	—
410	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
411	C₂³×C₄	—	2	1	no	—
412	C₂³×C₄	—	2	1	no	—
413	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
414	#137	—	4	4	no	—
415	#137	—	4	4	no	—
416	#137	—	4	4	no	—
417	AGL(1,ℤ/8)	—	4	4	no	—
418	C₂×Q₁₆	C₂³×C₈₉	1	4	yes	—
419	C₂×Q₁₆	C₂³×C₈₉	1	4	yes	—
420	C₂×Q₁₆	C₂³×C₈₉	1	4	yes	—
421	C₂×SD₁₆	C₂³×C₈₉	1	4	yes	—
422	C₂×SD₁₆	C₂³×C₈₉	1	4	yes	—
423	C₂×SD₁₆	C₂³×C₈₉	1	4	yes	—
424	C₂×SD₁₆	C₂³×C₈₉	1	4	yes	—
425	C₂×D₈	C₂³×C₈₉	1	4	yes	—
426	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
427	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
428	C₄⋊Q₈	—	2	2	no	—
429	C₄⋊Q₈	—	2	2	no	—
430	C₄⋊Q₈	C₂³×C₈₉	1	4	yes	—
431	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
432	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
433	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
434	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
435	#127	C₂³×C₈₉	1	1	yes	characteristic
436	#126	—	2	4	no	—
437	#126	—	2	4	no	—
438	#125	—	2	4	no	—
439	#125	C₂³×C₈₉	1	2	yes	—
440	#124	—	2	4	no	—
441	#124	—	2	2	no	—
442	#124	—	2	2	no	—
443	#124	C₂³×C₈₉	1	4	yes	—
444	#124	C₂³×C₈₉	1	2	yes	—
445	#124	C₂³×C₈₉	1	2	yes	—
446	#124	C₂³×C₈₉	1	2	yes	—
447	#123	—	2	2	no	—
448	#123	—	2	4	no	—
449	#123	—	2	8	no	—
450	#123	—	2	8	no	—
451	#123	—	2	4	no	—
452	C₂²⋊Q₈	—	2	2	no	—
453	C₂²⋊Q₈	—	2	8	no	—
454	C₂²⋊Q₈	—	2	8	no	—
455	C₂²⋊Q₈	—	2	4	no	—
456	C₂²⋊Q₈	C₂³×C₈₉	1	4	yes	—
457	#121	—	2	2	no	—
458	#121	—	2	4	no	—
459	#121	C₂³×C₈₉	1	4	yes	—
460	#120	—	2	2	no	—
461	C₄×Q₈	—	2	2	no	—
462	C₄×Q₈	C₂³×C₈₉	1	4	yes	—
463	C₄×D₄	—	2	4	no	—
464	C₄×D₄	—	2	4	no	—
465	C₄×D₄	—	2	2	no	—
466	C₄×D₄	C₂³×C₈₉	1	4	yes	—
467	#117	—	2	2	no	—
468	#117	D₄×C₈₉	1	4	yes	—
469	#117	C₂³×C₈₉	1	2	yes	—
470	#117	C₂³×C₈₉	1	2	yes	—
471	#117	C₂³×C₈₉	1	2	yes	—
472	C₂×C₄⋊C₄	—	2	2	no	—
473	C₂×C₄⋊C₄	—	2	2	no	—
474	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
475	C₂×C₂²⋊C₄	—	2	2	no	—
476	C₂×C₂²⋊C₄	—	2	2	no	—
477	C₂×C₂²⋊C₄	D₄×C₈₉	1	2	yes	—
478	C₂×C₄²	—	2	2	no	—
479	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
480	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
481	C₄⋊C₈	C₂³×C₈₉	1	8	yes	—
482	#103	C₂³×C₈₉	1	8	yes	—
483	#103	C₂³×C₈₉	1	8	yes	—
484	#103	C₂³×C₈₉	1	8	yes	—
485	#102	C₂³×C₈₉	1	8	yes	—
486	C₂²⋊C₈	C₂³×C₈₉	1	8	yes	—
487	#97	C₂³×C₈₉	1	2	yes	—
488	#97	C₂³×C₈₉	1	2	yes	—
489	C₄×C₈	C₂³×C₈₉	1	2	yes	—
490	C₄×C₈	C₂³×C₈₉	1	2	yes	—
491	C₂⁴	—	2	1	no	—
492	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
493	#41	—	2	4	no	—
494	#41	—	2	4	no	—
495	#41	—	4	8	no	—
496	#41	—	4	8	no	—
497	C₂×Q₈	—	2	4	no	—
498	C₂×Q₈	—	2	1	no	—
499	C₂×Q₈	—	2	1	no	—
500	C₂×Q₈	—	2	1	no	—
501	C₂×Q₈	—	2	2	no	—
502	C₂×Q₈	—	2	4	no	—
503	C₂×Q₈	—	2	4	no	—
504	C₂×Q₈	—	2	4	no	—
505	C₂×Q₈	—	4	8	no	—
506	C₂×Q₈	—	4	4	no	—
507	C₂×Q₈	—	4	8	no	—
508	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
509	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
510	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
511	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
512	C₂×D₄	—	2	2	no	—
513	C₂×D₄	—	2	4	no	—
514	C₂×D₄	—	2	4	no	—
515	C₂×D₄	—	2	1	no	—
516	C₂×D₄	—	2	2	no	—
517	C₂×D₄	—	2	4	no	—
518	C₂×D₄	—	4	4	no	—
519	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
520	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
521	C₂²×C₄	—	2	4	no	—
522	C₂²×C₄	—	2	1	no	—
523	C₂²×C₄	—	2	1	no	—
524	C₂²×C₄	—	2	1	no	—
525	C₂²×C₄	—	2	4	no	—
526	C₂²×C₄	—	2	2	no	—
527	C₂²×C₄	—	2	4	no	—
528	C₂²×C₄	—	2	4	no	—
529	C₂²×C₄	—	4	4	no	—
530	C₂²×C₄	—	4	4	no	—
531	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
532	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
533	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
534	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
535	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
536	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
537	Q₁₆	—	4	4	no	—
538	Q₁₆	—	4	4	no	—
539	Q₁₆	—	4	4	no	—
540	SD₁₆	—	4	4	no	—
541	SD₁₆	—	4	4	no	—
542	SD₁₆	—	4	4	no	—
543	SD₁₆	—	4	4	no	—
544	D₈	—	4	4	no	—
545	M₁₆	—	2	4	no	—
546	M₁₆	—	2	4	no	—
547	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
548	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
549	C₄⋊C₄	—	2	8	no	—
550	C₄⋊C₄	—	2	4	no	—
551	C₄⋊C₄	—	2	4	no	—
552	C₄⋊C₄	—	2	4	no	—
553	C₄⋊C₄	—	2	2	no	—
554	C₄⋊C₄	—	2	4	no	—
555	C₄⋊C₄	—	2	4	no	—
556	C₄⋊C₄	—	2	2	no	—
557	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
558	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
559	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
560	C₂²⋊C₄	—	2	2	no	—
561	C₂²⋊C₄	—	2	4	no	—
562	C₂²⋊C₄	—	2	8	no	—
563	C₂²⋊C₄	—	2	4	no	—
564	C₂²⋊C₄	—	2	4	no	—
565	C₂²⋊C₄	—	2	4	no	—
566	C₂²⋊C₄	—	2	4	no	—
567	C₂²⋊C₄	—	2	4	no	—
568	C₂²⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
569	C₂²⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
570	C₄²	—	2	2	no	—
571	C₄²	—	2	4	no	—
572	C₄²	C₂×D₄×C₈₉	1	2	yes	—
573	C₄²	C₂×D₄×C₈₉	1	2	yes	—
574	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
575	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
576	C₄²	C₂⁴×C₈₉	1	2	yes	—
577	C₂³	—	2	2	no	—
578	C₂³	—	2	2	no	—
579	C₂³	—	2	2	no	—
580	C₂³	—	4	2	no	—
581	C₂³	—	4	4	no	—
582	C₂³	—	4	2	no	—
583	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
584	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
585	Q₈	—	2	2	no	—
586	Q₈	—	2	4	no	—
587	Q₈	—	2	4	no	—
588	Q₈	—	2	4	no	—
589	Q₈	—	2	2	no	—
590	Q₈	—	4	8	no	—
591	Q₈	—	4	2	no	—
592	Q₈	—	4	2	no	—
593	Q₈	—	4	2	no	—
594	Q₈	—	8	2	no	—
595	D₄	—	2	2	no	—
596	D₄	—	2	4	no	—
597	D₄	—	2	2	no	—
598	D₄	—	4	8	no	—
599	D₄	—	4	4	no	—
600	D₄	—	4	2	no	—
601	D₄	—	8	2	no	—
602	D₄	—	8	4	no	—
603	C₂×C₄	—	2	2	no	—
604	C₂×C₄	—	2	4	no	—
605	C₂×C₄	—	2	4	no	—
606	C₂×C₄	—	2	2	no	—
607	C₂×C₄	—	2	4	no	—
608	C₂×C₄	—	2	2	no	—
609	C₂×C₄	—	2	2	no	—
610	C₂×C₄	—	2	2	no	—
611	C₂×C₄	—	4	2	no	—
612	C₂×C₄	—	4	2	no	—
613	C₂×C₄	—	4	8	no	—
614	C₂×C₄	—	4	8	no	—
615	C₂×C₄	—	4	2	no	—
616	C₂×C₄	—	4	2	no	—
617	C₂×C₄	—	4	2	no	—
618	C₂×C₄	—	4	4	no	—
619	C₂×C₄	—	8	4	no	—
620	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
621	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
622	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
623	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
624	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
625	C₂×C₄	H(5,𝔽₂)×C₈₉	1	4	yes	—
626	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
627	C₈	—	2	4	no	—
628	C₈	—	2	4	no	—
629	C₂²	—	2	2	no	—
630	C₂²	—	2	4	no	—
631	C₂²	—	2	2	no	—
632	C₂²	—	4	4	no	—
633	C₂²	—	4	4	no	—
634	C₂²	—	4	4	no	—
635	C₂²	—	4	2	no	—
636	C₂²	—	4	2	no	—
637	C₂²	—	8	4	no	—
638	C₂²	#535×C₈₉	1	2	yes	—
639	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
640	C₄	—	2	4	no	—
641	C₄	—	2	2	no	—
642	C₄	—	2	2	no	—
643	C₄	—	2	2	no	—
644	C₄	—	4	4	no	—
645	C₄	—	4	2	no	—
646	C₄	—	4	2	no	—
647	C₄	—	4	2	no	—
648	C₄	—	4	2	no	—
649	C₄	#535×C₈₉	1	2	yes	—
650	C₄	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
651	C₂	—	2	2	no	—
652	C₂	—	4	2	no	—
653	C₂	—	8	2	no	—
654	C₂	—	8	2	no	—
655	C₂	C₂×#535×C₈₉	1	1	yes	characteristic
656	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 ◁ C₂×#117×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#3455	C₂	semidirect
2. C₂×#536×C₈₉	C₂	semidirect
3. C₈₉×#3225	C₂	semidirect
4. C₈₉×#3223	C₂	semidirect
5. C₈₉×#3221	C₂	semidirect
6. C₈₉×#3153	C₂	semidirect
7. C₈₉×#3152	C₂	semidirect
8. C₈₉×#3140	C₂	semidirect
9. C₈₉×#3139	C₂	semidirect
10. C₈₉×#3040	C₂	semidirect
11. C₈₉×#3039	C₂	semidirect
12. C₈₉×#3005	C₂	semidirect
13. C₈₉×#3004	C₂	semidirect
14. C₈₉×#2921	C₂	semidirect
21. #542×C₈₉	C₂²	semidirect
22. #541×C₈₉	C₂²	semidirect
23. #540×C₈₉	C₂²	semidirect
41. #497×C₈₉	C₂²	semidirect
42. #497×C₈₉	C₂²	semidirect
43. #496×C₈₉	C₂²	semidirect
44. #495×C₈₉	C₂²	semidirect
45. #494×C₈₉	C₂²	semidirect
46. #492×C₈₉	C₂²	semidirect
47. #491×C₈₉	C₂²	semidirect
48. #489×C₈₉	C₂²	semidirect
49. #488×C₈₉	C₂²	semidirect
50. #487×C₈₉	C₂²	semidirect
51. #464×C₈₉	C₂²	semidirect
52. #463×C₈₉	C₂²	semidirect
53. #462×C₈₉	C₂²	semidirect
54. #461×C₈₉	C₂²	semidirect
56. #451×C₈₉	C₂²	semidirect
58. #448×C₈₉	C₂²	semidirect
59. #433×C₈₉	C₂²	semidirect
60. #433×C₈₉	C₂²	semidirect
322. #60807	C₈₉	direct (N × H)
329. #3455	C₂×C₈₉	semidirect
330. C₂×#536	C₂×C₈₉	semidirect
331. #3225	C₂×C₈₉	semidirect
332. #3223	C₂×C₈₉	semidirect
333. #3221	C₂×C₈₉	semidirect
334. #3153	C₂×C₈₉	semidirect
335. #3152	C₂×C₈₉	semidirect
336. #3140	C₂×C₈₉	semidirect
337. #3139	C₂×C₈₉	semidirect
338. #3040	C₂×C₈₉	semidirect
339. #3039	C₂×C₈₉	semidirect
340. #3005	C₂×C₈₉	semidirect
341. #3004	C₂×C₈₉	semidirect
342. #2921	C₂×C₈₉	semidirect
343. C₈₉	#60807	direct (N × H)
350. #542	C₂²×C₈₉	semidirect
351. #541	C₂²×C₈₉	semidirect
352. #540	C₂²×C₈₉	semidirect
370. #497	C₂²×C₈₉	semidirect
371. #497	C₂²×C₈₉	semidirect
372. #496	C₂²×C₈₉	semidirect
373. #495	C₂²×C₈₉	semidirect
374. #494	C₂²×C₈₉	semidirect
375. #492	C₂²×C₈₉	semidirect
376. #491	C₂²×C₈₉	semidirect
377. #489	C₂²×C₈₉	semidirect
378. #488	C₂²×C₈₉	semidirect
379. #487	C₂²×C₈₉	semidirect
380. #464	C₂²×C₈₉	semidirect
381. #463	C₂²×C₈₉	semidirect
382. #462	C₂²×C₈₉	semidirect
383. #461	C₂²×C₈₉	semidirect
385. #451	C₂²×C₈₉	semidirect
387. #448	C₂²×C₈₉	semidirect
388. #433	C₂²×C₈₉	semidirect
389. #433	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×#3455	2
C₂×#536×C₈₉	2
C₈₉×#3225	2
C₈₉×#3223	2
C₈₉×#3221	2
C₈₉×#3153	2
C₈₉×#3152	2
C₈₉×#3140	2
C₈₉×#3139	2
C₈₉×#3040	2
C₈₉×#3039	2
C₈₉×#3005	2
C₈₉×#3004	2
C₈₉×#2921	2
#60807	89

Subgroups — C89×#60807 #36438655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#60807 #36438655