Subgroups — C₈₉×#60564 #36438412

Order 22784.

2218 conjugacy classes of subgroups.

The subgroup lattice has 546 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×#518×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3123	C₂	1	4	yes	maximal, complemented
4	C₈₉×#3122	C₂	1	4	yes	maximal, complemented
5	C₈₉×#3118	C₂	1	8	yes	maximal, complemented
6	C₈₉×#3093	C₂	1	2	yes	maximal, complemented
7	C₈₉×#3077	C₂	1	2	yes	maximal, complemented
8	C₈₉×#3036	C₂	1	2	yes	maximal, complemented
9	C₈₉×#3003	C₂	1	2	yes	maximal, complemented
10	C₄×AGL(1,ℤ/8)×C₈₉	C₂	1	4	yes	maximal, complemented
11	C₈₉×#2921	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	4	yes	—
14	#542×C₈₉	C₂²	1	4	yes	complemented
15	#541×C₈₉	C₂²	1	2	yes	complemented
16	#540×C₈₉	C₂²	1	2	yes	complemented
17	#539×C₈₉	—	2	2	no	—
18	#538×C₈₉	—	2	2	no	—
19	#533×C₈₉	C₂²	1	1	yes	characteristic, complemented
20	#532×C₈₉	C₂²	1	1	yes	characteristic, complemented
21	#529×C₈₉	—	2	4	no	—
22	#525×C₈₉	C₂²	1	2	yes	—
23	#518×C₈₉	—	2	4	no	—
24	#518×C₈₉	C₂²	1	4	yes	—
25	#518×C₈₉	C₂²	1	4	yes	—
26	C₄×#41×C₈₉	C₂²	1	2	yes	complemented
27	C₂×C₄×D₄×C₈₉	—	2	2	no	—
28	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
29	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
30	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
32	C₂²×C₂²⋊C₄×C₈₉	—	2	2	no	—
33	#485×C₈₉	C₂²	1	8	yes	complemented
34	#482×C₈₉	C₂²	1	8	yes	complemented
35	#478×C₈₉	C₂²	1	8	yes	complemented
36	#476×C₈₉	C₂²	1	4	yes	complemented
37	#474×C₈₉	C₂²	1	4	yes	complemented
38	#463×C₈₉	C₂²	1	8	yes	complemented
39	#461×C₈₉	C₂²	1	4	yes	complemented
40	#459×C₈₉	C₂²	1	4	yes	complemented
41	#450×C₈₉	C₂²	1	8	yes	—
42	#447×C₈₉	C₂²	1	8	yes	—
43	#442×C₈₉	C₂²	1	8	yes	complemented
44	#440×C₈₉	C₂²	1	8	yes	complemented
45	C₄×SD₁₆×C₈₉	C₂²	1	8	yes	complemented
46	C₄×D₈×C₈₉	C₂²	1	8	yes	complemented
47	#433×C₈₉	C₂²	1	4	yes	complemented
48	#433×C₈₉	C₂²	1	4	yes	complemented
49	#428×C₈₉	C₂²	1	4	yes	complemented
50	#423×C₈₉	C₂²	1	2	yes	complemented
51	#418×C₈₉	C₂²	1	4	yes	—
52	#417×C₈₉	C₂²	1	4	yes	complemented
53	#407×C₈₉	C₂²	1	2	yes	—
54	C₄×M₁₆×C₈₉	C₂²	1	2	yes	complemented
55	C₂⁵×C₈₉	—	2	1	no	—
56	C₂×#41×C₈₉	C₂³	1	2	yes	—
57	C₂²×D₄×C₈₉	—	2	8	no	—
58	C₂²×D₄×C₈₉	—	2	1	no	—
59	C₂²×D₄×C₈₉	—	2	2	no	—
60	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
61	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
62	C₂³×C₄×C₈₉	—	2	2	no	—
63	C₂³×C₄×C₈₉	—	2	2	no	—
64	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
65	AGL(1,ℤ/8)×C₈₉	—	2	16	no	—
66	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	—
67	C₂×D₈×C₈₉	C₂³	1	8	yes	—
68	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
69	C₄⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic, complemented
70	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	complemented
71	#127×C₈₉	C₂³	1	1	yes	characteristic, complemented
72	#126×C₈₉	—	2	4	no	—
73	#126×C₈₉	—	2	4	no	—
74	#125×C₈₉	—	2	4	no	—
75	#125×C₈₉	C₂³	1	2	yes	complemented
76	#124×C₈₉	—	2	4	no	—
77	#124×C₈₉	C₂³	1	2	yes	complemented
78	#123×C₈₉	—	2	2	no	—
79	#123×C₈₉	—	2	4	no	—
80	#123×C₈₉	—	2	4	no	—
81	C₂²⋊Q₈×C₈₉	—	2	2	no	—
82	C₂²⋊Q₈×C₈₉	—	2	4	no	—
83	C₂²⋊Q₈×C₈₉	C₂³	1	4	yes	—
84	#121×C₈₉	—	2	2	no	—
85	#121×C₈₉	—	2	4	no	—
86	#121×C₈₉	C₂³	1	4	yes	—
87	#120×C₈₉	—	2	2	no	—
88	C₄×Q₈×C₈₉	—	2	2	no	—
89	C₄×Q₈×C₈₉	C₂³	1	4	yes	complemented
90	C₄×D₄×C₈₉	—	2	4	no	—
91	C₄×D₄×C₈₉	—	2	4	no	—
92	C₄×D₄×C₈₉	—	2	8	no	—
93	C₄×D₄×C₈₉	—	2	8	no	—
94	C₄×D₄×C₈₉	—	2	4	no	—
95	C₄×D₄×C₈₉	—	2	2	no	—
96	C₄×D₄×C₈₉	—	2	2	no	—
97	C₄×D₄×C₈₉	—	2	2	no	—
98	C₄×D₄×C₈₉	C₂³	1	8	yes	—
99	C₄×D₄×C₈₉	C₂³	1	4	yes	—
100	C₄×D₄×C₈₉	C₂³	1	4	yes	—
101	C₄×D₄×C₈₉	C₂³	1	4	yes	complemented
102	#117×C₈₉	—	2	2	no	—
103	#117×C₈₉	—	2	2	no	—
104	#117×C₈₉	D₄	1	4	yes	—
105	#117×C₈₉	C₂³	1	2	yes	—
106	#117×C₈₉	C₂³	1	4	yes	—
107	#117×C₈₉	C₂³	1	2	yes	—
108	#117×C₈₉	C₂³	1	2	yes	—
109	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
110	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
111	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
112	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
113	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
114	C₂×C₂²⋊C₄×C₈₉	—	2	8	no	—
115	C₂×C₂²⋊C₄×C₈₉	—	2	8	no	—
116	C₂×C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
117	C₂×C₄²×C₈₉	—	2	1	no	—
118	C₂×C₄²×C₈₉	—	2	2	no	—
119	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
121	#107×C₈₉	C₂³	1	8	yes	complemented
122	#106×C₈₉	C₂³	1	8	yes	complemented
123	C₄⋊C₈×C₈₉	C₂³	1	4	yes	complemented
124	C₄⋊C₈×C₈₉	C₂³	1	4	yes	complemented
125	#103×C₈₉	C₂³	1	8	yes	complemented
126	#102×C₈₉	C₂³	1	8	yes	—
127	#102×C₈₉	C₂³	1	8	yes	—
128	#102×C₈₉	C₂³	1	8	yes	complemented
129	C₂²⋊C₈×C₈₉	C₂³	1	8	yes	—
130	#97×C₈₉	C₂³	1	4	yes	complemented
131	C₄×C₈×C₈₉	C₂³	1	4	yes	complemented
132	C₂⁴×C₈₉	—	2	8	no	—
133	C₂⁴×C₈₉	—	2	2	no	—
134	C₂⁴×C₈₉	—	2	4	no	—
135	C₂⁴×C₈₉	—	8	4	no	—
136	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
137	#41×C₈₉	—	2	8	no	—
138	C₂×Q₈×C₈₉	—	2	2	no	—
139	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
140	C₂×D₄×C₈₉	—	2	8	no	—
141	C₂×D₄×C₈₉	—	2	16	no	—
142	C₂×D₄×C₈₉	—	2	16	no	—
143	C₂×D₄×C₈₉	—	2	2	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	4	no	—
149	C₂×D₄×C₈₉	#41	1	8	yes	—
150	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
151	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
152	C₂²×C₄×C₈₉	—	2	4	no	—
153	C₂²×C₄×C₈₉	—	2	2	no	—
154	C₂²×C₄×C₈₉	—	2	2	no	—
155	C₂²×C₄×C₈₉	—	2	4	no	—
156	C₂²×C₄×C₈₉	—	2	8	no	—
157	C₂²×C₄×C₈₉	—	2	4	no	—
158	C₂²×C₄×C₈₉	—	2	4	no	—
159	C₂²×C₄×C₈₉	—	8	4	no	—
160	C₂²×C₄×C₈₉	—	8	4	no	—
161	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
163	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
164	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
165	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
167	SD₁₆×C₈₉	—	2	16	no	—
168	D₈×C₈₉	—	2	16	no	—
169	M₁₆×C₈₉	—	2	8	no	—
170	C₂×C₈×C₈₉	C₂⁴	1	8	yes	—
171	C₄⋊C₄×C₈₉	—	2	2	no	—
172	C₄⋊C₄×C₈₉	—	2	4	no	—
173	C₄⋊C₄×C₈₉	—	2	2	no	—
174	C₄⋊C₄×C₈₉	—	2	4	no	—
175	C₄⋊C₄×C₈₉	—	2	4	no	—
176	C₄⋊C₄×C₈₉	—	2	2	no	—
177	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
178	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
179	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
180	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
181	C₂²⋊C₄×C₈₉	—	2	4	no	—
182	C₂²⋊C₄×C₈₉	—	2	2	no	—
183	C₂²⋊C₄×C₈₉	—	2	4	no	—
184	C₂²⋊C₄×C₈₉	—	2	8	no	—
185	C₂²⋊C₄×C₈₉	—	2	4	no	—
186	C₂²⋊C₄×C₈₉	—	2	4	no	—
187	C₂²⋊C₄×C₈₉	—	2	4	no	—
188	C₂²⋊C₄×C₈₉	—	2	4	no	—
189	C₂²⋊C₄×C₈₉	—	2	4	no	—
190	C₂²⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
191	C₂²⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
192	C₄²×C₈₉	—	2	4	no	—
193	C₄²×C₈₉	—	2	2	no	—
194	C₄²×C₈₉	—	2	4	no	—
195	C₄²×C₈₉	C₂×D₄	1	2	yes	—
196	C₄²×C₈₉	C₂×D₄	1	2	yes	—
197	C₄²×C₈₉	C₂⁴	1	2	yes	—
198	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
199	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
200	C₂³×C₈₉	—	2	16	no	—
201	C₂³×C₈₉	—	2	2	no	—
202	C₂³×C₈₉	—	2	8	no	—
203	C₂³×C₈₉	—	2	4	no	—
204	C₂³×C₈₉	—	2	2	no	—
205	C₂³×C₈₉	—	4	2	no	—
206	C₂³×C₈₉	—	4	4	no	—
207	C₂³×C₈₉	—	4	4	no	—
208	C₂³×C₈₉	—	8	8	no	—
209	C₂³×C₈₉	—	8	16	no	—
210	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
211	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
212	Q₈×C₈₉	—	2	4	no	—
213	Q₈×C₈₉	—	8	2	no	—
214	D₄×C₈₉	—	2	16	no	—
215	D₄×C₈₉	—	2	8	no	—
216	D₄×C₈₉	—	2	4	no	—
217	D₄×C₈₉	—	2	4	no	—
218	D₄×C₈₉	—	4	4	no	—
219	D₄×C₈₉	—	8	2	no	—
220	D₄×C₈₉	—	8	4	no	—
221	D₄×C₈₉	C₂×#41	1	8	yes	—
222	C₂×C₄×C₈₉	—	2	2	no	—
223	C₂×C₄×C₈₉	—	2	4	no	—
224	C₂×C₄×C₈₉	—	2	4	no	—
225	C₂×C₄×C₈₉	—	2	4	no	—
226	C₂×C₄×C₈₉	—	2	2	no	—
227	C₂×C₄×C₈₉	—	2	4	no	—
228	C₂×C₄×C₈₉	—	4	2	no	—
229	C₂×C₄×C₈₉	—	4	2	no	—
230	C₂×C₄×C₈₉	—	4	4	no	—
231	C₂×C₄×C₈₉	—	4	4	no	—
232	C₂×C₄×C₈₉	—	8	4	no	—
233	C₂×C₄×C₈₉	—	8	8	no	—
234	C₂×C₄×C₈₉	—	8	4	no	—
235	C₂×C₄×C₈₉	—	8	4	no	—
236	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
237	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
238	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
239	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
240	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
241	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
242	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
243	C₈×C₈₉	—	2	8	no	—
244	C₂²×C₈₉	—	2	2	no	—
245	C₂²×C₈₉	—	2	4	no	—
246	C₂²×C₈₉	—	2	2	no	—
247	C₂²×C₈₉	—	2	8	no	—
248	C₂²×C₈₉	—	4	16	no	—
249	C₂²×C₈₉	—	4	4	no	—
250	C₂²×C₈₉	—	4	4	no	—
251	C₂²×C₈₉	—	4	4	no	—
252	C₂²×C₈₉	—	4	4	no	—
253	C₂²×C₈₉	—	4	2	no	—
254	C₂²×C₈₉	—	8	16	no	—
255	C₂²×C₈₉	#525	1	2	yes	—
256	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
257	C₄×C₈₉	—	2	4	no	—
258	C₄×C₈₉	—	2	2	no	—
259	C₄×C₈₉	—	2	2	no	—
260	C₄×C₈₉	—	2	2	no	—
261	C₄×C₈₉	—	4	4	no	—
262	C₄×C₈₉	—	4	4	no	—
263	C₄×C₈₉	—	4	2	no	—
264	C₄×C₈₉	#525	1	2	yes	—
265	C₄×C₈₉	C₂²×#41	1	2	yes	—
266	#60564	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
267	C₂×C₈₉	—	2	2	no	—
268	C₂×C₈₉	—	4	2	no	—
269	C₂×C₈₉	—	4	8	no	—
270	C₂×C₈₉	—	8	2	no	—
271	C₂×C₈₉	C₂×#525	1	1	yes	characteristic
272	C₂×C₈₉	#3585	1	2	yes	—
273	#3455	C₂×C₈₉	1	1	yes	characteristic, complemented
274	C₂×#518	C₂×C₈₉	1	1	yes	characteristic, complemented
275	#3123	C₂×C₈₉	1	4	yes	complemented
276	#3122	C₂×C₈₉	1	4	yes	complemented
277	#3118	C₂×C₈₉	1	8	yes	complemented
278	#3093	C₂×C₈₉	1	2	yes	complemented
279	#3077	C₂×C₈₉	1	2	yes	complemented
280	#3036	C₂×C₈₉	1	2	yes	complemented
281	#3003	C₂×C₈₉	1	2	yes	complemented
282	C₄×AGL(1,ℤ/8)	C₂×C₈₉	1	4	yes	complemented
283	#2921	C₂×C₈₉	1	1	yes	characteristic, complemented
284	C₈₉	#60564	1	1	yes	characteristic, direct product factor, Sylow
285	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
286	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	4	yes	—
287	#542	C₂²×C₈₉	1	4	yes	complemented
288	#541	C₂²×C₈₉	1	2	yes	complemented
289	#540	C₂²×C₈₉	1	2	yes	complemented
290	#539	—	2	2	no	—
291	#538	—	2	2	no	—
292	#533	C₂²×C₈₉	1	1	yes	characteristic, complemented
293	#532	C₂²×C₈₉	1	1	yes	characteristic, complemented
294	#529	—	2	4	no	—
295	#525	C₂²×C₈₉	1	2	yes	—
296	#518	—	2	4	no	—
297	#518	C₂²×C₈₉	1	4	yes	—
298	#518	C₂²×C₈₉	1	4	yes	—
299	C₄×#41	C₂²×C₈₉	1	2	yes	complemented
300	C₂×C₄×D₄	—	2	2	no	—
301	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
302	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
303	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
304	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
305	C₂²×C₂²⋊C₄	—	2	2	no	—
306	#485	C₂²×C₈₉	1	8	yes	complemented
307	#482	C₂²×C₈₉	1	8	yes	complemented
308	#478	C₂²×C₈₉	1	8	yes	complemented
309	#476	C₂²×C₈₉	1	4	yes	complemented
310	#474	C₂²×C₈₉	1	4	yes	complemented
311	#463	C₂²×C₈₉	1	8	yes	complemented
312	#461	C₂²×C₈₉	1	4	yes	complemented
313	#459	C₂²×C₈₉	1	4	yes	complemented
314	#450	C₂²×C₈₉	1	8	yes	—
315	#447	C₂²×C₈₉	1	8	yes	—
316	#442	C₂²×C₈₉	1	8	yes	complemented
317	#440	C₂²×C₈₉	1	8	yes	complemented
318	C₄×SD₁₆	C₂²×C₈₉	1	8	yes	complemented
319	C₄×D₈	C₂²×C₈₉	1	8	yes	complemented
320	#433	C₂²×C₈₉	1	4	yes	complemented
321	#433	C₂²×C₈₉	1	4	yes	complemented
322	#428	C₂²×C₈₉	1	4	yes	complemented
323	#423	C₂²×C₈₉	1	2	yes	complemented
324	#418	C₂²×C₈₉	1	4	yes	—
325	#417	C₂²×C₈₉	1	4	yes	complemented
326	#407	C₂²×C₈₉	1	2	yes	—
327	C₄×M₁₆	C₂²×C₈₉	1	2	yes	complemented
328	C₂⁵	—	2	1	no	—
329	C₂×#41	C₂³×C₈₉	1	2	yes	—
330	C₂²×D₄	—	2	8	no	—
331	C₂²×D₄	—	2	1	no	—
332	C₂²×D₄	—	2	2	no	—
333	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
334	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
335	C₂³×C₄	—	2	2	no	—
336	C₂³×C₄	—	2	2	no	—
337	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
338	AGL(1,ℤ/8)	—	2	16	no	—
339	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	—
340	C₂×D₈	C₂³×C₈₉	1	8	yes	—
341	C₂×M₁₆	C₂³×C₈₉	1	4	yes	—
342	C₄⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic, complemented
343	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	complemented
344	#127	C₂³×C₈₉	1	1	yes	characteristic, complemented
345	#126	—	2	4	no	—
346	#126	—	2	4	no	—
347	#125	—	2	4	no	—
348	#125	C₂³×C₈₉	1	2	yes	complemented
349	#124	—	2	4	no	—
350	#124	C₂³×C₈₉	1	2	yes	complemented
351	#123	—	2	2	no	—
352	#123	—	2	4	no	—
353	#123	—	2	4	no	—
354	C₂²⋊Q₈	—	2	2	no	—
355	C₂²⋊Q₈	—	2	4	no	—
356	C₂²⋊Q₈	C₂³×C₈₉	1	4	yes	—
357	#121	—	2	2	no	—
358	#121	—	2	4	no	—
359	#121	C₂³×C₈₉	1	4	yes	—
360	#120	—	2	2	no	—
361	C₄×Q₈	—	2	2	no	—
362	C₄×Q₈	C₂³×C₈₉	1	4	yes	complemented
363	C₄×D₄	—	2	4	no	—
364	C₄×D₄	—	2	4	no	—
365	C₄×D₄	—	2	8	no	—
366	C₄×D₄	—	2	8	no	—
367	C₄×D₄	—	2	4	no	—
368	C₄×D₄	—	2	2	no	—
369	C₄×D₄	—	2	2	no	—
370	C₄×D₄	—	2	2	no	—
371	C₄×D₄	C₂³×C₈₉	1	8	yes	—
372	C₄×D₄	C₂³×C₈₉	1	4	yes	—
373	C₄×D₄	C₂³×C₈₉	1	4	yes	—
374	C₄×D₄	C₂³×C₈₉	1	4	yes	complemented
375	#117	—	2	2	no	—
376	#117	—	2	2	no	—
377	#117	D₄×C₈₉	1	4	yes	—
378	#117	C₂³×C₈₉	1	4	yes	—
379	#117	C₂³×C₈₉	1	2	yes	—
380	#117	C₂³×C₈₉	1	2	yes	—
381	#117	C₂³×C₈₉	1	2	yes	—
382	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
383	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
384	C₂×C₂²⋊C₄	—	2	1	no	—
385	C₂×C₂²⋊C₄	—	2	2	no	—
386	C₂×C₂²⋊C₄	—	2	2	no	—
387	C₂×C₂²⋊C₄	—	2	8	no	—
388	C₂×C₂²⋊C₄	—	2	8	no	—
389	C₂×C₂²⋊C₄	D₄×C₈₉	1	2	yes	—
390	C₂×C₄²	—	2	1	no	—
391	C₂×C₄²	—	2	2	no	—
392	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
393	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
394	#107	C₂³×C₈₉	1	8	yes	complemented
395	#106	C₂³×C₈₉	1	8	yes	complemented
396	C₄⋊C₈	C₂³×C₈₉	1	4	yes	complemented
397	C₄⋊C₈	C₂³×C₈₉	1	4	yes	complemented
398	#103	C₂³×C₈₉	1	8	yes	complemented
399	#102	C₂³×C₈₉	1	8	yes	—
400	#102	C₂³×C₈₉	1	8	yes	—
401	#102	C₂³×C₈₉	1	8	yes	complemented
402	C₂²⋊C₈	C₂³×C₈₉	1	8	yes	—
403	#97	C₂³×C₈₉	1	4	yes	complemented
404	C₄×C₈	C₂³×C₈₉	1	4	yes	complemented
405	C₂⁴	—	2	8	no	—
406	C₂⁴	—	2	2	no	—
407	C₂⁴	—	2	4	no	—
408	C₂⁴	—	8	4	no	—
409	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
410	#41	—	2	8	no	—
411	C₂×Q₈	—	2	2	no	—
412	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
413	C₂×D₄	—	2	8	no	—
414	C₂×D₄	—	2	16	no	—
415	C₂×D₄	—	2	16	no	—
416	C₂×D₄	—	2	2	no	—
417	C₂×D₄	—	2	4	no	—
418	C₂×D₄	—	2	4	no	—
419	C₂×D₄	—	2	2	no	—
420	C₂×D₄	—	2	2	no	—
421	C₂×D₄	—	2	4	no	—
422	C₂×D₄	#41×C₈₉	1	8	yes	—
423	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
424	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
425	C₂²×C₄	—	2	4	no	—
426	C₂²×C₄	—	2	2	no	—
427	C₂²×C₄	—	2	2	no	—
428	C₂²×C₄	—	2	4	no	—
429	C₂²×C₄	—	2	8	no	—
430	C₂²×C₄	—	2	4	no	—
431	C₂²×C₄	—	2	4	no	—
432	C₂²×C₄	—	8	4	no	—
433	C₂²×C₄	—	8	4	no	—
434	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
435	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
436	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
437	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
438	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
439	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
440	SD₁₆	—	2	16	no	—
441	D₈	—	2	16	no	—
442	M₁₆	—	2	8	no	—
443	C₂×C₈	C₂⁴×C₈₉	1	8	yes	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	4	no	—
446	C₄⋊C₄	—	2	2	no	—
447	C₄⋊C₄	—	2	4	no	—
448	C₄⋊C₄	—	2	4	no	—
449	C₄⋊C₄	—	2	2	no	—
450	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
451	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
452	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
453	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
454	C₂²⋊C₄	—	2	4	no	—
455	C₂²⋊C₄	—	2	2	no	—
456	C₂²⋊C₄	—	2	4	no	—
457	C₂²⋊C₄	—	2	8	no	—
458	C₂²⋊C₄	—	2	4	no	—
459	C₂²⋊C₄	—	2	4	no	—
460	C₂²⋊C₄	—	2	4	no	—
461	C₂²⋊C₄	—	2	4	no	—
462	C₂²⋊C₄	—	2	4	no	—
463	C₂²⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
464	C₂²⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
465	C₄²	—	2	4	no	—
466	C₄²	—	2	2	no	—
467	C₄²	—	2	4	no	—
468	C₄²	C₂×D₄×C₈₉	1	2	yes	—
469	C₄²	C₂×D₄×C₈₉	1	2	yes	—
470	C₄²	C₂⁴×C₈₉	1	2	yes	—
471	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
472	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
473	C₂³	—	2	16	no	—
474	C₂³	—	2	2	no	—
475	C₂³	—	2	8	no	—
476	C₂³	—	2	4	no	—
477	C₂³	—	2	2	no	—
478	C₂³	—	4	2	no	—
479	C₂³	—	4	4	no	—
480	C₂³	—	4	4	no	—
481	C₂³	—	8	8	no	—
482	C₂³	—	8	16	no	—
483	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
484	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
485	Q₈	—	2	4	no	—
486	Q₈	—	8	2	no	—
487	D₄	—	2	16	no	—
488	D₄	—	2	8	no	—
489	D₄	—	2	4	no	—
490	D₄	—	2	4	no	—
491	D₄	—	4	4	no	—
492	D₄	—	8	2	no	—
493	D₄	—	8	4	no	—
494	D₄	C₂×#41×C₈₉	1	8	yes	—
495	C₂×C₄	—	2	2	no	—
496	C₂×C₄	—	2	4	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	4	no	—
501	C₂×C₄	—	4	2	no	—
502	C₂×C₄	—	4	2	no	—
503	C₂×C₄	—	4	4	no	—
504	C₂×C₄	—	4	4	no	—
505	C₂×C₄	—	8	4	no	—
506	C₂×C₄	—	8	8	no	—
507	C₂×C₄	—	8	4	no	—
508	C₂×C₄	—	8	4	no	—
509	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
510	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
511	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
512	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
513	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
514	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
515	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
516	C₈	—	2	8	no	—
517	C₂²	—	2	2	no	—
518	C₂²	—	2	4	no	—
519	C₂²	—	2	2	no	—
520	C₂²	—	2	8	no	—
521	C₂²	—	4	4	no	—
522	C₂²	—	4	16	no	—
523	C₂²	—	4	4	no	—
524	C₂²	—	4	4	no	—
525	C₂²	—	4	4	no	—
526	C₂²	—	4	2	no	—
527	C₂²	—	8	16	no	—
528	C₂²	#525×C₈₉	1	2	yes	—
529	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
530	C₄	—	2	4	no	—
531	C₄	—	2	2	no	—
532	C₄	—	2	2	no	—
533	C₄	—	2	2	no	—
534	C₄	—	4	4	no	—
535	C₄	—	4	4	no	—
536	C₄	—	4	2	no	—
537	C₄	#525×C₈₉	1	2	yes	—
538	C₄	C₂²×#41×C₈₉	1	2	yes	—
539	C₂	—	2	2	no	—
540	C₂	—	4	2	no	—
541	C₂	—	4	8	no	—
542	C₂	—	8	2	no	—
543	C₂	C₂×#525×C₈₉	1	1	yes	characteristic
544	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. C₈₉×#3455	C₂	semidirect
2. C₂×#518×C₈₉	C₂	semidirect
3. C₈₉×#3123	C₂	semidirect
4. C₈₉×#3122	C₂	semidirect
5. C₈₉×#3118	C₂	semidirect
6. C₈₉×#3093	C₂	semidirect
7. C₈₉×#3077	C₂	semidirect
8. C₈₉×#3036	C₂	semidirect
9. C₈₉×#3003	C₂	semidirect
10. C₄×AGL(1,ℤ/8)×C₈₉	C₂	semidirect
11. C₈₉×#2921	C₂	semidirect
14. #542×C₈₉	C₂²	semidirect
15. #541×C₈₉	C₂²	semidirect
16. #540×C₈₉	C₂²	semidirect
19. #533×C₈₉	C₂²	semidirect
20. #532×C₈₉	C₂²	semidirect
26. C₄×#41×C₈₉	C₂²	semidirect
33. #485×C₈₉	C₂²	semidirect
34. #482×C₈₉	C₂²	semidirect
35. #478×C₈₉	C₂²	semidirect
36. #476×C₈₉	C₂²	semidirect
37. #474×C₈₉	C₂²	semidirect
38. #463×C₈₉	C₂²	semidirect
39. #461×C₈₉	C₂²	semidirect
40. #459×C₈₉	C₂²	semidirect
43. #442×C₈₉	C₂²	semidirect
44. #440×C₈₉	C₂²	semidirect
45. C₄×SD₁₆×C₈₉	C₂²	semidirect
46. C₄×D₈×C₈₉	C₂²	semidirect
47. #433×C₈₉	C₂²	semidirect
48. #433×C₈₉	C₂²	semidirect
49. #428×C₈₉	C₂²	semidirect
50. #423×C₈₉	C₂²	semidirect
52. #417×C₈₉	C₂²	semidirect
54. C₄×M₁₆×C₈₉	C₂²	semidirect
69. C₄⋊Q₈×C₈₉	C₂³	semidirect
70. C₄⋊Q₈×C₈₉	C₂³	semidirect
71. #127×C₈₉	C₂³	semidirect
75. #125×C₈₉	C₂³	semidirect
77. #124×C₈₉	C₂³	semidirect
89. C₄×Q₈×C₈₉	C₂³	semidirect
101. C₄×D₄×C₈₉	C₂³	semidirect
121. #107×C₈₉	C₂³	semidirect
122. #106×C₈₉	C₂³	semidirect
123. C₄⋊C₈×C₈₉	C₂³	semidirect
124. C₄⋊C₈×C₈₉	C₂³	semidirect
125. #103×C₈₉	C₂³	semidirect
128. #102×C₈₉	C₂³	semidirect
130. #97×C₈₉	C₂³	semidirect
131. C₄×C₈×C₈₉	C₂³	semidirect
266. #60564	C₈₉	direct (N × H)
273. #3455	C₂×C₈₉	semidirect
274. C₂×#518	C₂×C₈₉	semidirect
275. #3123	C₂×C₈₉	semidirect
276. #3122	C₂×C₈₉	semidirect
277. #3118	C₂×C₈₉	semidirect
278. #3093	C₂×C₈₉	semidirect
279. #3077	C₂×C₈₉	semidirect
280. #3036	C₂×C₈₉	semidirect
281. #3003	C₂×C₈₉	semidirect
282. C₄×AGL(1,ℤ/8)	C₂×C₈₉	semidirect
283. #2921	C₂×C₈₉	semidirect
284. C₈₉	#60564	direct (N × H)
287. #542	C₂²×C₈₉	semidirect
288. #541	C₂²×C₈₉	semidirect
289. #540	C₂²×C₈₉	semidirect
292. #533	C₂²×C₈₉	semidirect
293. #532	C₂²×C₈₉	semidirect
299. C₄×#41	C₂²×C₈₉	semidirect
306. #485	C₂²×C₈₉	semidirect
307. #482	C₂²×C₈₉	semidirect
308. #478	C₂²×C₈₉	semidirect
309. #476	C₂²×C₈₉	semidirect
310. #474	C₂²×C₈₉	semidirect
311. #463	C₂²×C₈₉	semidirect
312. #461	C₂²×C₈₉	semidirect
313. #459	C₂²×C₈₉	semidirect
316. #442	C₂²×C₈₉	semidirect
317. #440	C₂²×C₈₉	semidirect
318. C₄×SD₁₆	C₂²×C₈₉	semidirect
319. C₄×D₈	C₂²×C₈₉	semidirect
320. #433	C₂²×C₈₉	semidirect
321. #433	C₂²×C₈₉	semidirect
322. #428	C₂²×C₈₉	semidirect
323. #423	C₂²×C₈₉	semidirect
325. #417	C₂²×C₈₉	semidirect
327. C₄×M₁₆	C₂²×C₈₉	semidirect
342. C₄⋊Q₈	C₂³×C₈₉	semidirect
343. C₄⋊Q₈	C₂³×C₈₉	semidirect
344. #127	C₂³×C₈₉	semidirect
348. #125	C₂³×C₈₉	semidirect
350. #124	C₂³×C₈₉	semidirect
362. C₄×Q₈	C₂³×C₈₉	semidirect
374. C₄×D₄	C₂³×C₈₉	semidirect
394. #107	C₂³×C₈₉	semidirect
395. #106	C₂³×C₈₉	semidirect
396. C₄⋊C₈	C₂³×C₈₉	semidirect
397. C₄⋊C₈	C₂³×C₈₉	semidirect
398. #103	C₂³×C₈₉	semidirect
401. #102	C₂³×C₈₉	semidirect
403. #97	C₂³×C₈₉	semidirect
404. C₄×C₈	C₂³×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×#518×C₈₉	2
C₈₉×#3123	2
C₈₉×#3122	2
C₈₉×#3118	2
C₈₉×#3093	2
C₈₉×#3077	2
C₈₉×#3036	2
C₈₉×#3003	2
C₄×AGL(1,ℤ/8)×C₈₉	2
C₈₉×#2921	2
#60564	89

Subgroups — C89×#60564 #36438412

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#60564 #36438412