Subgroups — C₈₉×#24337 #36402185

Order 22784.

1070 conjugacy classes of subgroups.

The subgroup lattice has 676 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#554×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂×#479×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₄×SD₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2581	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2059	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#2058	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2026	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#2024	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1991	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1984	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1877	C₂	1	1	yes	maximal, characteristic
12	C₈₉×#1877	C₂	1	1	yes	maximal, characteristic
13	C₈₉×#1864	C₂	1	1	yes	maximal, characteristic
14	C₈₉×#1862	C₂	1	1	yes	maximal, characteristic
15	C₈₉×#1804	C₂	1	1	yes	maximal, characteristic
16	C₂²×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	#554×C₈₉	—	2	4	no	—
18	#554×C₈₉	—	2	4	no	—
19	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#125×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×#125×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
23	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
24	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
25	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
30	C₂²×C₄⋊C₄×C₈₉	—	2	1	no	—
31	#479×C₈₉	—	2	4	no	—
32	C₄×SD₁₆×C₈₉	—	2	4	no	—
33	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
36	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
38	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
40	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
42	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
44	#400×C₈₉	—	2	1	no	—
45	#395×C₈₉	—	2	1	no	—
46	#391×C₈₉	C₂²	1	1	yes	characteristic
47	#391×C₈₉	C₂²	1	1	yes	characteristic
48	#389×C₈₉	C₂²	1	1	yes	characteristic
49	#389×C₈₉	C₂²	1	1	yes	characteristic
50	#387×C₈₉	—	2	1	no	—
51	#387×C₈₉	—	2	1	no	—
52	#382×C₈₉	C₂²	1	1	yes	characteristic
53	#340×C₈₉	C₂²	1	1	yes	characteristic
54	#340×C₈₉	C₂²	1	1	yes	characteristic
55	#340×C₈₉	C₂²	1	1	yes	characteristic
56	#340×C₈₉	C₂²	1	1	yes	characteristic
57	#340×C₈₉	C₂²	1	1	yes	characteristic
58	#340×C₈₉	C₂²	1	1	yes	characteristic
59	#336×C₈₉	C₂²	1	1	yes	characteristic
60	#336×C₈₉	C₂²	1	1	yes	characteristic
61	#336×C₈₉	C₂²	1	1	yes	characteristic
62	#336×C₈₉	C₂²	1	1	yes	characteristic
63	C₂²×Q₈×C₈₉	—	2	1	no	—
64	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
65	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
66	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₈₉	—	2	1	no	—
68	C₂³×C₄×C₈₉	—	2	1	no	—
69	C₂³×C₄×C₈₉	—	2	1	no	—
70	C₂×SD₁₆×C₈₉	—	2	2	no	—
71	C₂×SD₁₆×C₈₉	—	2	2	no	—
72	C₂×SD₁₆×C₈₉	Q₈	1	4	yes	complemented
73	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
74	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
76	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
77	C₄⋊Q₈×C₈₉	—	2	2	no	—
78	C₄⋊Q₈×C₈₉	—	2	2	no	—
79	#125×C₈₉	—	2	2	no	—
80	#125×C₈₉	—	2	2	no	—
81	#125×C₈₉	—	2	2	no	—
82	#125×C₈₉	D₄	1	4	yes	—
83	C₂²⋊Q₈×C₈₉	—	2	8	no	—
84	C₂²⋊Q₈×C₈₉	—	2	8	no	—
85	C₂²⋊Q₈×C₈₉	—	2	8	no	—
86	C₄×Q₈×C₈₉	—	2	2	no	—
87	C₄×Q₈×C₈₉	—	2	2	no	—
88	C₄×Q₈×C₈₉	—	2	2	no	—
89	C₄×Q₈×C₈₉	—	2	2	no	—
90	C₄×D₄×C₈₉	—	2	2	no	—
91	C₄×D₄×C₈₉	—	2	2	no	—
92	C₄×D₄×C₈₉	—	2	2	no	—
93	C₄×D₄×C₈₉	—	2	2	no	—
94	C₄×D₄×C₈₉	—	2	2	no	—
95	C₄×D₄×C₈₉	D₄	1	4	yes	—
96	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
97	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
98	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
99	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
100	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
101	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
102	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
103	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
104	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
105	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
106	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
107	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
108	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
109	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
110	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
111	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
114	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
116	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
117	C₂×C₄²×C₈₉	—	2	1	no	—
118	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
119	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
120	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
121	#107×C₈₉	—	2	2	no	—
122	#106×C₈₉	—	2	2	no	—
123	#106×C₈₉	—	2	2	no	—
124	C₄⋊C₈×C₈₉	—	2	2	no	—
125	#103×C₈₉	—	2	2	no	—
126	#103×C₈₉	—	2	2	no	—
127	#103×C₈₉	—	2	2	no	—
128	#102×C₈₉	—	2	2	no	—
129	#102×C₈₉	—	2	2	no	—
130	#102×C₈₉	—	2	2	no	—
131	C₄×C₈×C₈₉	—	2	2	no	—
132	#95×C₈₉	—	2	1	no	—
133	#95×C₈₉	—	2	1	no	—
134	#95×C₈₉	—	2	1	no	—
135	#95×C₈₉	—	2	1	no	—
136	#95×C₈₉	—	2	1	no	—
137	#95×C₈₉	—	2	1	no	—
138	#95×C₈₉	—	2	1	no	—
139	#95×C₈₉	—	2	1	no	—
140	C₂⁴×C₈₉	—	2	1	no	—
141	C₂×Q₈×C₈₉	—	2	4	no	—
142	C₂×Q₈×C₈₉	—	2	2	no	—
143	C₂×Q₈×C₈₉	—	2	2	no	—
144	C₂×Q₈×C₈₉	—	2	1	no	—
145	C₂×Q₈×C₈₉	—	2	1	no	—
146	C₂×Q₈×C₈₉	—	2	1	no	—
147	C₂×Q₈×C₈₉	—	2	1	no	—
148	C₂×Q₈×C₈₉	—	2	1	no	—
149	C₂×Q₈×C₈₉	—	4	1	no	—
150	C₂×Q₈×C₈₉	—	8	1	no	—
151	C₂×Q₈×C₈₉	—	8	1	no	—
152	C₂×Q₈×C₈₉	C₂×Q₈	1	2	yes	—
153	C₂×D₄×C₈₉	—	2	1	no	—
154	C₂×D₄×C₈₉	—	2	1	no	—
155	C₂×D₄×C₈₉	—	2	1	no	—
156	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
157	C₂×D₄×C₈₉	C₂×Q₈	1	2	yes	—
158	C₂×D₄×C₈₉	#41	1	2	yes	—
159	C₂²×C₄×C₈₉	—	2	4	no	—
160	C₂²×C₄×C₈₉	—	2	1	no	—
161	C₂²×C₄×C₈₉	—	2	1	no	—
162	C₂²×C₄×C₈₉	—	2	1	no	—
163	C₂²×C₄×C₈₉	—	2	1	no	—
164	C₂²×C₄×C₈₉	—	2	4	no	—
165	C₂²×C₄×C₈₉	—	2	4	no	—
166	C₂²×C₄×C₈₉	—	2	1	no	—
167	C₂²×C₄×C₈₉	—	2	1	no	—
168	C₂²×C₄×C₈₉	—	2	1	no	—
169	C₂²×C₄×C₈₉	—	4	2	no	—
170	C₂²×C₄×C₈₉	—	4	2	no	—
171	C₂²×C₄×C₈₉	—	8	1	no	—
172	C₂²×C₄×C₈₉	—	8	1	no	—
173	C₂²×C₄×C₈₉	—	8	1	no	—
174	C₂²×C₄×C₈₉	—	8	1	no	—
175	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
176	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
177	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
178	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
179	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
180	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
181	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
182	SD₁₆×C₈₉	—	2	8	no	—
183	C₂×C₈×C₈₉	—	2	1	no	—
184	C₂×C₈×C₈₉	—	2	1	no	—
185	C₂×C₈×C₈₉	—	2	1	no	—
186	C₂×C₈×C₈₉	—	2	1	no	—
187	C₂×C₈×C₈₉	—	2	1	no	—
188	C₂×C₈×C₈₉	—	2	1	no	—
189	C₂×C₈×C₈₉	—	2	1	no	—
190	C₂×C₈×C₈₉	—	2	1	no	—
191	C₂×C₈×C₈₉	—	2	1	no	—
192	C₂×C₈×C₈₉	—	2	1	no	—
193	C₂×C₈×C₈₉	—	2	1	no	—
194	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
195	C₄⋊C₄×C₈₉	—	2	4	no	—
196	C₄⋊C₄×C₈₉	—	2	4	no	—
197	C₄⋊C₄×C₈₉	—	2	4	no	—
198	C₄⋊C₄×C₈₉	—	2	2	no	—
199	C₄⋊C₄×C₈₉	—	2	4	no	—
200	C₄⋊C₄×C₈₉	—	2	4	no	—
201	C₄⋊C₄×C₈₉	—	2	2	no	—
202	C₄⋊C₄×C₈₉	—	2	2	no	—
203	C₄⋊C₄×C₈₉	—	2	1	no	—
204	C₄⋊C₄×C₈₉	—	2	1	no	—
205	C₄⋊C₄×C₈₉	—	2	1	no	—
206	C₄⋊C₄×C₈₉	—	2	1	no	—
207	C₄⋊C₄×C₈₉	—	2	1	no	—
208	C₄⋊C₄×C₈₉	—	2	1	no	—
209	C₄⋊C₄×C₈₉	—	2	1	no	—
210	C₄⋊C₄×C₈₉	—	2	1	no	—
211	C₄⋊C₄×C₈₉	—	2	1	no	—
212	C₄⋊C₄×C₈₉	—	4	2	no	—
213	C₄⋊C₄×C₈₉	—	4	2	no	—
214	C₄⋊C₄×C₈₉	—	4	2	no	—
215	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
216	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
217	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
218	C₂²⋊C₄×C₈₉	—	2	4	no	—
219	C₂²⋊C₄×C₈₉	—	2	4	no	—
220	C₂²⋊C₄×C₈₉	—	2	4	no	—
221	C₄²×C₈₉	—	2	1	no	—
222	C₄²×C₈₉	—	2	1	no	—
223	C₄²×C₈₉	—	2	1	no	—
224	C₄²×C₈₉	—	2	1	no	—
225	C₄²×C₈₉	—	4	2	no	—
226	C₄²×C₈₉	C₂×D₄	1	2	yes	—
227	C₄²×C₈₉	C₂×D₄	1	2	yes	—
228	C₂³×C₈₉	—	2	2	no	—
229	C₂³×C₈₉	—	2	2	no	—
230	C₂³×C₈₉	—	2	2	no	—
231	C₂³×C₈₉	—	4	1	no	—
232	C₂³×C₈₉	—	4	1	no	—
233	C₂³×C₈₉	—	4	1	no	—
234	C₂³×C₈₉	—	4	1	no	—
235	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
236	Q₈×C₈₉	—	2	2	no	—
237	Q₈×C₈₉	—	2	2	no	—
238	Q₈×C₈₉	—	4	4	no	—
239	Q₈×C₈₉	—	4	1	no	—
240	Q₈×C₈₉	—	4	1	no	—
241	Q₈×C₈₉	—	8	4	no	—
242	D₄×C₈₉	—	2	2	no	—
243	D₄×C₈₉	—	2	2	no	—
244	D₄×C₈₉	—	2	2	no	—
245	D₄×C₈₉	C₂²⋊Q₈	1	4	yes	—
246	C₂×C₄×C₈₉	—	2	1	no	—
247	C₂×C₄×C₈₉	—	2	1	no	—
248	C₂×C₄×C₈₉	—	2	1	no	—
249	C₂×C₄×C₈₉	—	2	1	no	—
250	C₂×C₄×C₈₉	—	2	1	no	—
251	C₂×C₄×C₈₉	—	2	1	no	—
252	C₂×C₄×C₈₉	—	2	1	no	—
253	C₂×C₄×C₈₉	—	2	1	no	—
254	C₂×C₄×C₈₉	—	2	1	no	—
255	C₂×C₄×C₈₉	—	2	2	no	—
256	C₂×C₄×C₈₉	—	2	2	no	—
257	C₂×C₄×C₈₉	—	2	2	no	—
258	C₂×C₄×C₈₉	—	2	2	no	—
259	C₂×C₄×C₈₉	—	2	1	no	—
260	C₂×C₄×C₈₉	—	2	1	no	—
261	C₂×C₄×C₈₉	—	2	1	no	—
262	C₂×C₄×C₈₉	—	2	1	no	—
263	C₂×C₄×C₈₉	—	2	1	no	—
264	C₂×C₄×C₈₉	—	2	1	no	—
265	C₂×C₄×C₈₉	—	4	8	no	—
266	C₂×C₄×C₈₉	—	4	1	no	—
267	C₂×C₄×C₈₉	—	4	1	no	—
268	C₂×C₄×C₈₉	—	4	1	no	—
269	C₂×C₄×C₈₉	—	4	1	no	—
270	C₂×C₄×C₈₉	—	4	1	no	—
271	C₂×C₄×C₈₉	—	4	1	no	—
272	C₂×C₄×C₈₉	—	4	1	no	—
273	C₂×C₄×C₈₉	—	4	1	no	—
274	C₂×C₄×C₈₉	—	4	1	no	—
275	C₂×C₄×C₈₉	—	4	1	no	—
276	C₂×C₄×C₈₉	—	4	1	no	—
277	C₂×C₄×C₈₉	—	4	1	no	—
278	C₂×C₄×C₈₉	—	4	1	no	—
279	C₂×C₄×C₈₉	—	4	1	no	—
280	C₂×C₄×C₈₉	—	4	1	no	—
281	C₂×C₄×C₈₉	—	4	1	no	—
282	C₂×C₄×C₈₉	—	4	1	no	—
283	C₂×C₄×C₈₉	—	8	4	no	—
284	C₂×C₄×C₈₉	—	8	4	no	—
285	C₂×C₄×C₈₉	#120	1	2	yes	—
286	C₂×C₄×C₈₉	#120	1	2	yes	—
287	C₂×C₄×C₈₉	#135	1	2	yes	—
288	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
289	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
290	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
291	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
292	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
293	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
294	C₈×C₈₉	—	2	2	no	—
295	C₈×C₈₉	—	4	1	no	—
296	C₈×C₈₉	—	4	1	no	—
297	C₈×C₈₉	—	4	1	no	—
298	C₂²×C₈₉	—	2	4	no	—
299	C₂²×C₈₉	—	4	2	no	—
300	C₂²×C₈₉	—	4	2	no	—
301	C₂²×C₈₉	—	4	2	no	—
302	C₂²×C₈₉	—	4	2	no	—
303	C₂²×C₈₉	—	4	2	no	—
304	C₂²×C₈₉	—	4	2	no	—
305	C₂²×C₈₉	C₂×#120	1	1	yes	characteristic
306	C₂²×C₈₉	#548	1	1	yes	characteristic
307	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
308	C₂²×C₈₉	C₂×#135	1	1	yes	characteristic
309	C₂²×C₈₉	#575	1	1	yes	characteristic
310	C₂²×C₈₉	#577	1	1	yes	characteristic
311	C₂²×C₈₉	#577	1	1	yes	characteristic
312	C₄×C₈₉	—	2	2	no	—
313	C₄×C₈₉	—	2	2	no	—
314	C₄×C₈₉	—	2	2	no	—
315	C₄×C₈₉	—	4	2	no	—
316	C₄×C₈₉	—	4	1	no	—
317	C₄×C₈₉	—	4	2	no	—
318	C₄×C₈₉	—	4	1	no	—
319	C₄×C₈₉	—	4	1	no	—
320	C₄×C₈₉	—	4	2	no	—
321	C₄×C₈₉	—	4	2	no	—
322	C₄×C₈₉	—	8	1	no	—
323	C₄×C₈₉	—	8	1	no	—
324	C₄×C₈₉	—	8	1	no	—
325	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
326	C₄×C₈₉	#536	1	1	yes	characteristic
327	C₄×C₈₉	#548	1	1	yes	characteristic
328	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
329	#24337	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
330	C₂×C₈₉	—	4	4	no	—
331	C₂×C₈₉	#2409	1	1	yes	characteristic
332	C₂×C₈₉	#3002	1	1	yes	characteristic
333	C₂×C₈₉	#3009	1	1	yes	characteristic
334	C₂×C₈₉	#3360	1	1	yes	characteristic
335	C₂×C₈₉	#3376	1	1	yes	characteristic
336	C₂×C₈₉	#3381	1	1	yes	characteristic
337	C₂×C₈₉	#3388	1	1	yes	characteristic
338	C₂×#554	C₂×C₈₉	1	1	yes	characteristic
339	C₂×#479	C₂×C₈₉	1	1	yes	characteristic, complemented
340	C₂×C₄×SD₁₆	C₂×C₈₉	1	1	yes	characteristic
341	#2581	C₂×C₈₉	1	1	yes	characteristic, complemented
342	#2059	C₂×C₈₉	1	1	yes	characteristic, complemented
343	#2058	C₂×C₈₉	1	1	yes	characteristic, complemented
344	#2026	C₂×C₈₉	1	1	yes	characteristic, complemented
345	#2024	C₂×C₈₉	1	1	yes	characteristic, complemented
346	#1991	C₂×C₈₉	1	1	yes	characteristic, complemented
347	#1984	C₂×C₈₉	1	1	yes	characteristic, complemented
348	#1877	C₂×C₈₉	1	1	yes	characteristic
349	#1877	C₂×C₈₉	1	1	yes	characteristic
350	#1864	C₂×C₈₉	1	1	yes	characteristic
351	#1862	C₂×C₈₉	1	1	yes	characteristic
352	#1804	C₂×C₈₉	1	1	yes	characteristic
353	C₈₉	#24337	1	1	yes	characteristic, direct product factor, Sylow
354	C₂²×SD₁₆	C₂²×C₈₉	1	1	yes	characteristic
355	#554	—	2	4	no	—
356	#554	—	2	4	no	—
357	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
358	C₂×#125	C₂²×C₈₉	1	1	yes	characteristic
359	C₂×#125	C₂²×C₈₉	1	1	yes	characteristic
360	C₂×C₂²⋊Q₈	—	2	1	no	—
361	C₂×C₂²⋊Q₈	—	2	1	no	—
362	C₂×C₂²⋊Q₈	—	2	1	no	—
363	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
364	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
365	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
366	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
367	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
368	C₂²×C₄⋊C₄	—	2	1	no	—
369	#479	—	2	4	no	—
370	C₄×SD₁₆	—	2	4	no	—
371	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
372	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
373	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
374	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
375	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
376	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
377	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
378	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
379	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
380	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
381	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
382	#400	—	2	1	no	—
383	#395	—	2	1	no	—
384	#391	C₂²×C₈₉	1	1	yes	characteristic
385	#391	C₂²×C₈₉	1	1	yes	characteristic
386	#389	C₂²×C₈₉	1	1	yes	characteristic
387	#389	C₂²×C₈₉	1	1	yes	characteristic
388	#387	—	2	1	no	—
389	#387	—	2	1	no	—
390	#382	C₂²×C₈₉	1	1	yes	characteristic
391	#340	C₂²×C₈₉	1	1	yes	characteristic
392	#340	C₂²×C₈₉	1	1	yes	characteristic
393	#340	C₂²×C₈₉	1	1	yes	characteristic
394	#340	C₂²×C₈₉	1	1	yes	characteristic
395	#340	C₂²×C₈₉	1	1	yes	characteristic
396	#340	C₂²×C₈₉	1	1	yes	characteristic
397	#336	C₂²×C₈₉	1	1	yes	characteristic
398	#336	C₂²×C₈₉	1	1	yes	characteristic
399	#336	C₂²×C₈₉	1	1	yes	characteristic
400	#336	C₂²×C₈₉	1	1	yes	characteristic
401	C₂²×Q₈	—	2	1	no	—
402	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
403	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
404	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
405	C₂³×C₄	—	2	1	no	—
406	C₂³×C₄	—	2	1	no	—
407	C₂³×C₄	—	2	1	no	—
408	C₂×SD₁₆	—	2	2	no	—
409	C₂×SD₁₆	—	2	2	no	—
410	C₂×SD₁₆	Q₈×C₈₉	1	4	yes	complemented
411	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
412	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
413	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
414	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
415	C₄⋊Q₈	—	2	2	no	—
416	C₄⋊Q₈	—	2	2	no	—
417	#125	—	2	2	no	—
418	#125	—	2	2	no	—
419	#125	—	2	2	no	—
420	#125	D₄×C₈₉	1	4	yes	—
421	C₂²⋊Q₈	—	2	8	no	—
422	C₂²⋊Q₈	—	2	8	no	—
423	C₂²⋊Q₈	—	2	8	no	—
424	C₄×Q₈	—	2	2	no	—
425	C₄×Q₈	—	2	2	no	—
426	C₄×Q₈	—	2	2	no	—
427	C₄×Q₈	—	2	2	no	—
428	C₄×D₄	—	2	2	no	—
429	C₄×D₄	—	2	2	no	—
430	C₄×D₄	—	2	2	no	—
431	C₄×D₄	—	2	2	no	—
432	C₄×D₄	—	2	2	no	—
433	C₄×D₄	D₄×C₈₉	1	4	yes	—
434	C₂×C₄⋊C₄	—	2	8	no	—
435	C₂×C₄⋊C₄	—	2	1	no	—
436	C₂×C₄⋊C₄	—	2	1	no	—
437	C₂×C₄⋊C₄	—	2	1	no	—
438	C₂×C₄⋊C₄	—	2	1	no	—
439	C₂×C₄⋊C₄	—	2	1	no	—
440	C₂×C₄⋊C₄	—	2	1	no	—
441	C₂×C₄⋊C₄	—	2	1	no	—
442	C₂×C₄⋊C₄	—	2	1	no	—
443	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
444	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
445	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
446	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
447	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
448	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
449	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
450	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
451	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
452	C₂×C₂²⋊C₄	—	2	1	no	—
453	C₂×C₂²⋊C₄	—	2	1	no	—
454	C₂×C₂²⋊C₄	—	2	1	no	—
455	C₂×C₄²	—	2	1	no	—
456	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
457	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
458	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
459	#107	—	2	2	no	—
460	#106	—	2	2	no	—
461	#106	—	2	2	no	—
462	C₄⋊C₈	—	2	2	no	—
463	#103	—	2	2	no	—
464	#103	—	2	2	no	—
465	#103	—	2	2	no	—
466	#102	—	2	2	no	—
467	#102	—	2	2	no	—
468	#102	—	2	2	no	—
469	C₄×C₈	—	2	2	no	—
470	#95	—	2	1	no	—
471	#95	—	2	1	no	—
472	#95	—	2	1	no	—
473	#95	—	2	1	no	—
474	#95	—	2	1	no	—
475	#95	—	2	1	no	—
476	#95	—	2	1	no	—
477	#95	—	2	1	no	—
478	C₂⁴	—	2	1	no	—
479	C₂×Q₈	—	2	4	no	—
480	C₂×Q₈	—	2	2	no	—
481	C₂×Q₈	—	2	2	no	—
482	C₂×Q₈	—	2	1	no	—
483	C₂×Q₈	—	2	1	no	—
484	C₂×Q₈	—	2	1	no	—
485	C₂×Q₈	—	2	1	no	—
486	C₂×Q₈	—	2	1	no	—
487	C₂×Q₈	—	4	1	no	—
488	C₂×Q₈	—	8	1	no	—
489	C₂×Q₈	—	8	1	no	—
490	C₂×Q₈	C₂×Q₈×C₈₉	1	2	yes	—
491	C₂×D₄	—	2	1	no	—
492	C₂×D₄	—	2	1	no	—
493	C₂×D₄	—	2	1	no	—
494	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
495	C₂×D₄	C₂×Q₈×C₈₉	1	2	yes	—
496	C₂×D₄	#41×C₈₉	1	2	yes	—
497	C₂²×C₄	—	2	4	no	—
498	C₂²×C₄	—	2	1	no	—
499	C₂²×C₄	—	2	1	no	—
500	C₂²×C₄	—	2	1	no	—
501	C₂²×C₄	—	2	1	no	—
502	C₂²×C₄	—	2	4	no	—
503	C₂²×C₄	—	2	4	no	—
504	C₂²×C₄	—	2	1	no	—
505	C₂²×C₄	—	2	1	no	—
506	C₂²×C₄	—	2	1	no	—
507	C₂²×C₄	—	4	2	no	—
508	C₂²×C₄	—	4	2	no	—
509	C₂²×C₄	—	8	1	no	—
510	C₂²×C₄	—	8	1	no	—
511	C₂²×C₄	—	8	1	no	—
512	C₂²×C₄	—	8	1	no	—
513	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
514	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
515	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
516	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
517	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
518	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
519	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
520	SD₁₆	—	2	8	no	—
521	C₂×C₈	—	2	1	no	—
522	C₂×C₈	—	2	1	no	—
523	C₂×C₈	—	2	1	no	—
524	C₂×C₈	—	2	1	no	—
525	C₂×C₈	—	2	1	no	—
526	C₂×C₈	—	2	1	no	—
527	C₂×C₈	—	2	1	no	—
528	C₂×C₈	—	2	1	no	—
529	C₂×C₈	—	2	1	no	—
530	C₂×C₈	—	2	1	no	—
531	C₂×C₈	—	2	1	no	—
532	C₂×C₈	C₂×Q₈×C₈₉	1	2	yes	—
533	C₄⋊C₄	—	2	4	no	—
534	C₄⋊C₄	—	2	4	no	—
535	C₄⋊C₄	—	2	4	no	—
536	C₄⋊C₄	—	2	2	no	—
537	C₄⋊C₄	—	2	4	no	—
538	C₄⋊C₄	—	2	4	no	—
539	C₄⋊C₄	—	2	2	no	—
540	C₄⋊C₄	—	2	2	no	—
541	C₄⋊C₄	—	2	1	no	—
542	C₄⋊C₄	—	2	1	no	—
543	C₄⋊C₄	—	2	1	no	—
544	C₄⋊C₄	—	2	1	no	—
545	C₄⋊C₄	—	2	1	no	—
546	C₄⋊C₄	—	2	1	no	—
547	C₄⋊C₄	—	2	1	no	—
548	C₄⋊C₄	—	2	1	no	—
549	C₄⋊C₄	—	2	1	no	—
550	C₄⋊C₄	—	4	2	no	—
551	C₄⋊C₄	—	4	2	no	—
552	C₄⋊C₄	—	4	2	no	—
553	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
554	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
555	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
556	C₂²⋊C₄	—	2	4	no	—
557	C₂²⋊C₄	—	2	4	no	—
558	C₂²⋊C₄	—	2	4	no	—
559	C₄²	—	2	1	no	—
560	C₄²	—	2	1	no	—
561	C₄²	—	2	1	no	—
562	C₄²	—	2	1	no	—
563	C₄²	—	4	2	no	—
564	C₄²	C₂×D₄×C₈₉	1	2	yes	—
565	C₄²	C₂×D₄×C₈₉	1	2	yes	—
566	C₂³	—	2	2	no	—
567	C₂³	—	2	2	no	—
568	C₂³	—	2	2	no	—
569	C₂³	—	4	1	no	—
570	C₂³	—	4	1	no	—
571	C₂³	—	4	1	no	—
572	C₂³	—	4	1	no	—
573	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
574	Q₈	—	2	2	no	—
575	Q₈	—	2	2	no	—
576	Q₈	—	4	4	no	—
577	Q₈	—	4	1	no	—
578	Q₈	—	4	1	no	—
579	Q₈	—	8	4	no	—
580	D₄	—	2	2	no	—
581	D₄	—	2	2	no	—
582	D₄	—	2	2	no	—
583	D₄	C₂²⋊Q₈×C₈₉	1	4	yes	—
584	C₂×C₄	—	2	1	no	—
585	C₂×C₄	—	2	1	no	—
586	C₂×C₄	—	2	1	no	—
587	C₂×C₄	—	2	1	no	—
588	C₂×C₄	—	2	1	no	—
589	C₂×C₄	—	2	1	no	—
590	C₂×C₄	—	2	1	no	—
591	C₂×C₄	—	2	1	no	—
592	C₂×C₄	—	2	1	no	—
593	C₂×C₄	—	2	2	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	2	no	—
596	C₂×C₄	—	2	2	no	—
597	C₂×C₄	—	2	1	no	—
598	C₂×C₄	—	2	1	no	—
599	C₂×C₄	—	2	1	no	—
600	C₂×C₄	—	2	1	no	—
601	C₂×C₄	—	2	1	no	—
602	C₂×C₄	—	2	1	no	—
603	C₂×C₄	—	4	8	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	1	no	—
606	C₂×C₄	—	4	1	no	—
607	C₂×C₄	—	4	1	no	—
608	C₂×C₄	—	4	1	no	—
609	C₂×C₄	—	4	1	no	—
610	C₂×C₄	—	4	1	no	—
611	C₂×C₄	—	4	1	no	—
612	C₂×C₄	—	4	1	no	—
613	C₂×C₄	—	4	1	no	—
614	C₂×C₄	—	4	1	no	—
615	C₂×C₄	—	4	1	no	—
616	C₂×C₄	—	4	1	no	—
617	C₂×C₄	—	4	1	no	—
618	C₂×C₄	—	4	1	no	—
619	C₂×C₄	—	4	1	no	—
620	C₂×C₄	—	4	1	no	—
621	C₂×C₄	—	8	4	no	—
622	C₂×C₄	—	8	4	no	—
623	C₂×C₄	#120×C₈₉	1	2	yes	—
624	C₂×C₄	#120×C₈₉	1	2	yes	—
625	C₂×C₄	#135×C₈₉	1	2	yes	—
626	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
627	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
628	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
629	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
630	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
631	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
632	C₈	—	2	2	no	—
633	C₈	—	4	1	no	—
634	C₈	—	4	1	no	—
635	C₈	—	4	1	no	—
636	C₂²	—	2	4	no	—
637	C₂²	—	4	2	no	—
638	C₂²	—	4	2	no	—
639	C₂²	—	4	2	no	—
640	C₂²	—	4	2	no	—
641	C₂²	—	4	2	no	—
642	C₂²	—	4	2	no	—
643	C₂²	C₂×#120×C₈₉	1	1	yes	characteristic
644	C₂²	#548×C₈₉	1	1	yes	characteristic
645	C₂²	D₄×Q₈×C₈₉	1	1	yes	characteristic
646	C₂²	C₂×#135×C₈₉	1	1	yes	characteristic
647	C₂²	#575×C₈₉	1	1	yes	characteristic
648	C₂²	#577×C₈₉	1	1	yes	characteristic
649	C₂²	#577×C₈₉	1	1	yes	characteristic
650	C₄	—	2	2	no	—
651	C₄	—	2	2	no	—
652	C₄	—	2	2	no	—
653	C₄	—	4	2	no	—
654	C₄	—	4	1	no	—
655	C₄	—	4	2	no	—
656	C₄	—	4	1	no	—
657	C₄	—	4	1	no	—
658	C₄	—	4	2	no	—
659	C₄	—	4	2	no	—
660	C₄	—	8	1	no	—
661	C₄	—	8	1	no	—
662	C₄	—	8	1	no	—
663	C₄	C₂×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
664	C₄	#536×C₈₉	1	1	yes	characteristic
665	C₄	#548×C₈₉	1	1	yes	characteristic
666	C₄	D₄×Q₈×C₈₉	1	1	yes	characteristic
667	C₂	—	4	4	no	—
668	C₂	C₈₉×#2409	1	1	yes	characteristic
669	C₂	C₈₉×#3002	1	1	yes	characteristic
670	C₂	C₈₉×#3009	1	1	yes	characteristic
671	C₂	C₈₉×#3360	1	1	yes	characteristic
672	C₂	C₈₉×#3376	1	1	yes	characteristic
673	C₂	C₈₉×#3381	1	1	yes	characteristic
674	C₂	C₈₉×#3388	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂²×D₄×C₈₉	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂×C₄×Q₈×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#479×C₈₉	C₂	semidirect
4. C₈₉×#2581	C₂	semidirect
5. C₈₉×#2059	C₂	semidirect
6. C₈₉×#2058	C₂	semidirect
7. C₈₉×#2026	C₂	semidirect
8. C₈₉×#2024	C₂	semidirect
9. C₈₉×#1991	C₂	semidirect
10. C₈₉×#1984	C₂	semidirect
72. C₂×SD₁₆×C₈₉	Q₈	semidirect
329. #24337	C₈₉	direct (N × H)
339. C₂×#479	C₂×C₈₉	semidirect
341. #2581	C₂×C₈₉	semidirect
342. #2059	C₂×C₈₉	semidirect
343. #2058	C₂×C₈₉	semidirect
344. #2026	C₂×C₈₉	semidirect
345. #2024	C₂×C₈₉	semidirect
346. #1991	C₂×C₈₉	semidirect
347. #1984	C₂×C₈₉	semidirect
353. C₈₉	#24337	direct (N × H)
410. C₂×SD₁₆	Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#554×C₈₉	2
C₂×#479×C₈₉	2
C₂×C₄×SD₁₆×C₈₉	2
C₈₉×#2581	2
C₈₉×#2059	2
C₈₉×#2058	2
C₈₉×#2026	2
C₈₉×#2024	2
C₈₉×#1991	2
C₈₉×#1984	2
C₈₉×#1877	2
C₈₉×#1864	2
C₈₉×#1862	2
C₈₉×#1804	2
#24337	89

Subgroups — C89×#24337 #36402185

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#24337 #36402185