Subgroups — C₈₉×#26397 #36404245

Order 22784.

858 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#477×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#440×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2452	C₂	1	1	yes	maximal, characteristic
4	Q₈×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2093	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#2083	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈₉×#2065	C₂	1	1	yes	maximal, characteristic
8	C₈₉×#2057	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1944	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1917	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1880	C₂	1	1	yes	maximal, characteristic
12	C₈₉×#1880	C₂	1	1	yes	maximal, characteristic
13	C₈₉×#1865	C₂	1	1	yes	maximal, characteristic, complemented
14	C₈₉×#1863	C₂	1	1	yes	maximal, characteristic, complemented
15	C₈₉×#1862	C₂	1	1	yes	maximal, characteristic
16	C₂²×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
19	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
23	#477×C₈₉	—	2	4	no	—
24	#440×C₈₉	—	2	4	no	—
25	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×#107×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
36	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
37	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
38	#396×C₈₉	—	2	1	no	—
39	#393×C₈₉	—	2	1	no	—
40	#390×C₈₉	C₂²	1	1	yes	characteristic
41	#390×C₈₉	C₂²	1	1	yes	characteristic
42	#389×C₈₉	—	2	1	no	—
43	#389×C₈₉	—	2	1	no	—
44	#389×C₈₉	C₂²	1	1	yes	characteristic
45	#389×C₈₉	C₂²	1	1	yes	characteristic
46	#389×C₈₉	C₂²	1	1	yes	characteristic
47	#389×C₈₉	C₂²	1	1	yes	characteristic
48	#385×C₈₉	—	2	1	no	—
49	#384×C₈₉	—	2	1	no	—
50	#384×C₈₉	C₂²	1	1	yes	characteristic
51	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
52	C₄×C₄⋊C₄×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	#336×C₈₉	C₂²	1	1	yes	characteristic
58	#336×C₈₉	C₂²	1	1	yes	characteristic
59	#336×C₈₉	C₂²	1	1	yes	characteristic
60	#336×C₈₉	C₂²	1	1	yes	characteristic
61	C₂²×Q₈×C₈₉	—	2	1	no	—
62	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
63	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
64	C₂³×C₄×C₈₉	—	2	1	no	—
65	C₂×SD₁₆×C₈₉	—	2	2	no	—
66	C₂×SD₁₆×C₈₉	—	2	2	no	—
67	C₂×SD₁₆×C₈₉	Q₈	1	4	yes	complemented
68	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
69	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
70	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
71	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
72	C₄⋊Q₈×C₈₉	—	2	2	no	—
73	C₄⋊Q₈×C₈₉	—	4	1	no	—
74	C₂²⋊Q₈×C₈₉	—	4	4	no	—
75	C₄×Q₈×C₈₉	—	2	2	no	—
76	C₄×Q₈×C₈₉	—	2	2	no	—
77	C₄×Q₈×C₈₉	—	2	2	no	—
78	C₄×Q₈×C₈₉	—	2	2	no	—
79	C₄×Q₈×C₈₉	—	4	1	no	—
80	C₄×Q₈×C₈₉	—	4	1	no	—
81	C₄×D₄×C₈₉	—	2	2	no	—
82	C₄×D₄×C₈₉	—	4	1	no	—
83	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
84	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
85	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
86	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
87	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
88	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
89	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
90	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
91	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
92	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
93	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
95	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
99	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
100	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
102	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
103	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
104	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
105	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
106	C₂×C₄²×C₈₉	—	2	1	no	—
107	C₂×C₄²×C₈₉	—	2	1	no	—
108	C₂×C₄²×C₈₉	—	2	1	no	—
109	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
110	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
111	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
112	#107×C₈₉	—	2	2	no	—
113	#107×C₈₉	—	2	2	no	—
114	#107×C₈₉	—	2	2	no	—
115	#107×C₈₉	—	4	1	no	—
116	#106×C₈₉	—	2	2	no	—
117	C₄⋊C₈×C₈₉	—	2	2	no	—
118	#103×C₈₉	—	2	2	no	—
119	#103×C₈₉	—	2	2	no	—
120	#103×C₈₉	D₄	1	4	yes	—
121	#102×C₈₉	—	2	2	no	—
122	#102×C₈₉	—	2	2	no	—
123	#102×C₈₉	—	4	1	no	—
124	#97×C₈₉	—	2	2	no	—
125	#95×C₈₉	—	2	1	no	—
126	#95×C₈₉	—	2	1	no	—
127	#95×C₈₉	—	2	1	no	—
128	#95×C₈₉	—	2	1	no	—
129	#95×C₈₉	D₄	1	1	yes	characteristic
130	#95×C₈₉	D₄	1	1	yes	characteristic
131	C₂⁴×C₈₉	—	2	1	no	—
132	C₂×Q₈×C₈₉	—	2	1	no	—
133	C₂×Q₈×C₈₉	—	2	1	no	—
134	C₂×Q₈×C₈₉	—	2	1	no	—
135	C₂×Q₈×C₈₉	—	4	2	no	—
136	C₂×Q₈×C₈₉	—	4	2	no	—
137	C₂×Q₈×C₈₉	—	8	1	no	—
138	C₂×Q₈×C₈₉	C₂×D₄	1	2	yes	—
139	C₂×Q₈×C₈₉	C₂×Q₈	1	2	yes	—
140	C₂×Q₈×C₈₉	#41	1	2	yes	—
141	C₂×D₄×C₈₉	—	2	1	no	—
142	C₂×D₄×C₈₉	—	2	1	no	—
143	C₂×D₄×C₈₉	—	2	1	no	—
144	C₂×D₄×C₈₉	—	2	1	no	—
145	C₂×D₄×C₈₉	—	2	1	no	—
146	C₂×D₄×C₈₉	C₂×Q₈	1	2	yes	—
147	C₂²×C₄×C₈₉	—	2	1	no	—
148	C₂²×C₄×C₈₉	—	2	1	no	—
149	C₂²×C₄×C₈₉	—	2	1	no	—
150	C₂²×C₄×C₈₉	—	2	1	no	—
151	C₂²×C₄×C₈₉	—	2	1	no	—
152	C₂²×C₄×C₈₉	—	2	1	no	—
153	C₂²×C₄×C₈₉	—	2	1	no	—
154	C₂²×C₄×C₈₉	—	4	2	no	—
155	C₂²×C₄×C₈₉	—	4	2	no	—
156	C₂²×C₄×C₈₉	—	8	1	no	—
157	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
158	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
160	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
161	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
162	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
163	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
164	SD₁₆×C₈₉	—	2	8	no	—
165	C₂×C₈×C₈₉	—	2	1	no	—
166	C₂×C₈×C₈₉	—	2	1	no	—
167	C₂×C₈×C₈₉	—	2	1	no	—
168	C₂×C₈×C₈₉	—	2	1	no	—
169	C₂×C₈×C₈₉	—	2	1	no	—
170	C₂×C₈×C₈₉	—	2	1	no	—
171	C₂×C₈×C₈₉	—	2	1	no	—
172	C₂×C₈×C₈₉	—	2	1	no	—
173	C₂×C₈×C₈₉	—	2	1	no	—
174	C₂×C₈×C₈₉	—	2	1	no	—
175	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
176	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
177	C₄⋊C₄×C₈₉	—	2	4	no	—
178	C₄⋊C₄×C₈₉	—	2	2	no	—
179	C₄⋊C₄×C₈₉	—	2	2	no	—
180	C₄⋊C₄×C₈₉	—	2	1	no	—
181	C₄⋊C₄×C₈₉	—	2	1	no	—
182	C₄⋊C₄×C₈₉	—	2	1	no	—
183	C₄⋊C₄×C₈₉	—	2	1	no	—
184	C₄⋊C₄×C₈₉	—	2	1	no	—
185	C₄⋊C₄×C₈₉	—	2	1	no	—
186	C₄⋊C₄×C₈₉	—	2	1	no	—
187	C₄⋊C₄×C₈₉	—	2	1	no	—
188	C₄⋊C₄×C₈₉	—	2	1	no	—
189	C₄⋊C₄×C₈₉	—	2	1	no	—
190	C₄⋊C₄×C₈₉	—	4	2	no	—
191	C₄⋊C₄×C₈₉	—	4	2	no	—
192	C₄⋊C₄×C₈₉	—	4	2	no	—
193	C₄⋊C₄×C₈₉	—	4	2	no	—
194	C₄⋊C₄×C₈₉	—	4	2	no	—
195	C₄⋊C₄×C₈₉	—	4	2	no	—
196	C₄⋊C₄×C₈₉	—	8	1	no	—
197	C₄⋊C₄×C₈₉	—	8	1	no	—
198	C₄⋊C₄×C₈₉	—	8	1	no	—
199	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
200	C₄⋊C₄×C₈₉	#41	1	2	yes	—
201	C₂²⋊C₄×C₈₉	—	4	2	no	—
202	C₂²⋊C₄×C₈₉	—	4	2	no	—
203	C₂²⋊C₄×C₈₉	—	4	2	no	—
204	C₂²⋊C₄×C₈₉	—	4	2	no	—
205	C₂²⋊C₄×C₈₉	—	8	1	no	—
206	C₄²×C₈₉	—	2	4	no	—
207	C₄²×C₈₉	—	2	1	no	—
208	C₄²×C₈₉	—	2	1	no	—
209	C₄²×C₈₉	—	2	1	no	—
210	C₄²×C₈₉	—	2	1	no	—
211	C₄²×C₈₉	—	2	1	no	—
212	C₄²×C₈₉	—	2	1	no	—
213	C₄²×C₈₉	—	4	2	no	—
214	C₄²×C₈₉	—	4	2	no	—
215	C₂³×C₈₉	—	2	2	no	—
216	C₂³×C₈₉	—	4	1	no	—
217	C₂³×C₈₉	—	4	1	no	—
218	C₂³×C₈₉	—	4	1	no	—
219	C₂³×C₈₉	—	4	1	no	—
220	C₂³×C₈₉	—	4	1	no	—
221	C₂³×C₈₉	—	4	1	no	—
222	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
223	Q₈×C₈₉	—	2	2	no	—
224	Q₈×C₈₉	—	2	2	no	—
225	Q₈×C₈₉	—	2	2	no	—
226	Q₈×C₈₉	—	8	4	no	—
227	Q₈×C₈₉	C₂²⋊Q₈	1	4	yes	complemented
228	D₄×C₈₉	—	2	2	no	—
229	D₄×C₈₉	—	2	2	no	—
230	D₄×C₈₉	—	4	1	no	—
231	D₄×C₈₉	—	4	1	no	—
232	C₂×C₄×C₈₉	—	2	1	no	—
233	C₂×C₄×C₈₉	—	2	2	no	—
234	C₂×C₄×C₈₉	—	2	1	no	—
235	C₂×C₄×C₈₉	—	2	2	no	—
236	C₂×C₄×C₈₉	—	2	2	no	—
237	C₂×C₄×C₈₉	—	2	1	no	—
238	C₂×C₄×C₈₉	—	2	1	no	—
239	C₂×C₄×C₈₉	—	2	1	no	—
240	C₂×C₄×C₈₉	—	2	2	no	—
241	C₂×C₄×C₈₉	—	2	1	no	—
242	C₂×C₄×C₈₉	—	2	1	no	—
243	C₂×C₄×C₈₉	—	2	1	no	—
244	C₂×C₄×C₈₉	—	2	2	no	—
245	C₂×C₄×C₈₉	—	2	1	no	—
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	—	4	1	no	—
254	C₂×C₄×C₈₉	—	4	1	no	—
255	C₂×C₄×C₈₉	—	4	1	no	—
256	C₂×C₄×C₈₉	—	4	1	no	—
257	C₂×C₄×C₈₉	—	4	1	no	—
258	C₂×C₄×C₈₉	—	4	1	no	—
259	C₂×C₄×C₈₉	—	4	1	no	—
260	C₂×C₄×C₈₉	—	4	1	no	—
261	C₂×C₄×C₈₉	—	4	1	no	—
262	C₂×C₄×C₈₉	—	4	1	no	—
263	C₂×C₄×C₈₉	—	4	1	no	—
264	C₂×C₄×C₈₉	—	4	1	no	—
265	C₂×C₄×C₈₉	—	4	1	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	—	8	4	no	—
270	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
271	C₂×C₄×C₈₉	#137	1	2	yes	—
272	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
273	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
274	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
275	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
276	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
277	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
278	C₈×C₈₉	—	2	2	no	—
279	C₈×C₈₉	—	2	2	no	—
280	C₈×C₈₉	—	4	1	no	—
281	C₈×C₈₉	—	4	1	no	—
282	C₂²×C₈₉	—	4	2	no	—
283	C₂²×C₈₉	—	4	2	no	—
284	C₂²×C₈₉	—	4	2	no	—
285	C₂²×C₈₉	—	8	1	no	—
286	C₂²×C₈₉	—	8	1	no	—
287	C₂²×C₈₉	—	8	1	no	—
288	C₂²×C₈₉	—	8	1	no	—
289	C₂²×C₈₉	#525	1	1	yes	characteristic
290	C₂²×C₈₉	D₄²	1	1	yes	characteristic
291	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
292	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
293	C₂²×C₈₉	C₂×#137	1	1	yes	characteristic
294	C₂²×C₈₉	#576	1	1	yes	characteristic
295	C₂²×C₈₉	#578	1	1	yes	characteristic
296	C₄×C₈₉	—	2	2	no	—
297	C₄×C₈₉	—	2	4	no	—
298	C₄×C₈₉	—	2	2	no	—
299	C₄×C₈₉	—	4	1	no	—
300	C₄×C₈₉	—	4	2	no	—
301	C₄×C₈₉	—	4	1	no	—
302	C₄×C₈₉	—	4	1	no	—
303	C₄×C₈₉	—	4	1	no	—
304	C₄×C₈₉	—	4	2	no	—
305	C₄×C₈₉	—	8	1	no	—
306	C₄×C₈₉	—	8	1	no	—
307	C₄×C₈₉	—	8	1	no	—
308	C₄×C₈₉	—	8	1	no	—
309	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
310	C₄×C₈₉	#539	1	1	yes	characteristic
311	C₄×C₈₉	#547	1	1	yes	characteristic
312	C₄×C₈₉	#554	1	1	yes	characteristic
313	#26397	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
314	C₂×C₈₉	—	8	2	no	—
315	C₂×C₈₉	#2449	1	1	yes	characteristic
316	C₂×C₈₉	#3112	1	1	yes	characteristic
317	C₂×C₈₉	#3124	1	1	yes	characteristic
318	C₂×C₈₉	#3270	1	1	yes	characteristic
319	C₂×C₈₉	#3274	1	1	yes	characteristic
320	C₂×C₈₉	#3385	1	1	yes	characteristic
321	C₂×C₈₉	#3386	1	1	yes	characteristic
322	C₂×#477	C₂×C₈₉	1	1	yes	characteristic, complemented
323	C₂×#440	C₂×C₈₉	1	1	yes	characteristic
324	#2452	C₂×C₈₉	1	1	yes	characteristic
325	Q₈×C₄⋊C₄	C₂×C₈₉	1	1	yes	characteristic, complemented
326	#2093	C₂×C₈₉	1	1	yes	characteristic
327	#2083	C₂×C₈₉	1	1	yes	characteristic, complemented
328	#2065	C₂×C₈₉	1	1	yes	characteristic
329	#2057	C₂×C₈₉	1	1	yes	characteristic, complemented
330	#1944	C₂×C₈₉	1	1	yes	characteristic, complemented
331	#1917	C₂×C₈₉	1	1	yes	characteristic, complemented
332	#1880	C₂×C₈₉	1	1	yes	characteristic
333	#1880	C₂×C₈₉	1	1	yes	characteristic
334	#1865	C₂×C₈₉	1	1	yes	characteristic, complemented
335	#1863	C₂×C₈₉	1	1	yes	characteristic, complemented
336	#1862	C₂×C₈₉	1	1	yes	characteristic
337	C₈₉	#26397	1	1	yes	characteristic, direct product factor, Sylow
338	C₂²×SD₁₆	C₂²×C₈₉	1	1	yes	characteristic
339	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
340	C₂×C₂²⋊Q₈	—	2	1	no	—
341	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
342	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
343	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
344	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
345	#477	—	2	4	no	—
346	#440	—	2	4	no	—
347	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
348	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
349	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
350	C₂×#107	C₂²×C₈₉	1	1	yes	characteristic
351	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
352	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
353	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
354	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
355	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
356	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
357	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
358	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
359	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
360	#396	—	2	1	no	—
361	#393	—	2	1	no	—
362	#390	C₂²×C₈₉	1	1	yes	characteristic
363	#390	C₂²×C₈₉	1	1	yes	characteristic
364	#389	—	2	1	no	—
365	#389	—	2	1	no	—
366	#389	C₂²×C₈₉	1	1	yes	characteristic
367	#389	C₂²×C₈₉	1	1	yes	characteristic
368	#389	C₂²×C₈₉	1	1	yes	characteristic
369	#389	C₂²×C₈₉	1	1	yes	characteristic
370	#385	—	2	1	no	—
371	#384	—	2	1	no	—
372	#384	C₂²×C₈₉	1	1	yes	characteristic
373	C₄×C₄⋊C₄	—	2	1	no	—
374	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
375	#340	C₂²×C₈₉	1	1	yes	characteristic
376	#340	C₂²×C₈₉	1	1	yes	characteristic
377	#340	C₂²×C₈₉	1	1	yes	characteristic
378	#340	C₂²×C₈₉	1	1	yes	characteristic
379	#336	C₂²×C₈₉	1	1	yes	characteristic
380	#336	C₂²×C₈₉	1	1	yes	characteristic
381	#336	C₂²×C₈₉	1	1	yes	characteristic
382	#336	C₂²×C₈₉	1	1	yes	characteristic
383	C₂²×Q₈	—	2	1	no	—
384	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
385	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
386	C₂³×C₄	—	2	1	no	—
387	C₂×SD₁₆	—	2	2	no	—
388	C₂×SD₁₆	—	2	2	no	—
389	C₂×SD₁₆	Q₈×C₈₉	1	4	yes	complemented
390	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
391	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
392	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
393	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
394	C₄⋊Q₈	—	2	2	no	—
395	C₄⋊Q₈	—	4	1	no	—
396	C₂²⋊Q₈	—	4	4	no	—
397	C₄×Q₈	—	2	2	no	—
398	C₄×Q₈	—	2	2	no	—
399	C₄×Q₈	—	2	2	no	—
400	C₄×Q₈	—	2	2	no	—
401	C₄×Q₈	—	4	1	no	—
402	C₄×Q₈	—	4	1	no	—
403	C₄×D₄	—	2	2	no	—
404	C₄×D₄	—	4	1	no	—
405	C₂×C₄⋊C₄	—	2	1	no	—
406	C₂×C₄⋊C₄	—	2	1	no	—
407	C₂×C₄⋊C₄	—	2	1	no	—
408	C₂×C₄⋊C₄	—	2	1	no	—
409	C₂×C₄⋊C₄	—	2	1	no	—
410	C₂×C₄⋊C₄	—	2	1	no	—
411	C₂×C₄⋊C₄	—	2	1	no	—
412	C₂×C₄⋊C₄	—	2	1	no	—
413	C₂×C₄⋊C₄	—	2	1	no	—
414	C₂×C₄⋊C₄	—	2	1	no	—
415	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
416	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
417	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
418	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
419	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
420	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
421	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
422	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
423	C₂×C₂²⋊C₄	—	2	1	no	—
424	C₂×C₂²⋊C₄	—	2	1	no	—
425	C₂×C₂²⋊C₄	—	2	1	no	—
426	C₂×C₂²⋊C₄	—	2	1	no	—
427	C₂×C₂²⋊C₄	—	2	1	no	—
428	C₂×C₄²	—	2	1	no	—
429	C₂×C₄²	—	2	1	no	—
430	C₂×C₄²	—	2	1	no	—
431	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
432	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
433	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
434	#107	—	2	2	no	—
435	#107	—	2	2	no	—
436	#107	—	2	2	no	—
437	#107	—	4	1	no	—
438	#106	—	2	2	no	—
439	C₄⋊C₈	—	2	2	no	—
440	#103	—	2	2	no	—
441	#103	—	2	2	no	—
442	#103	D₄×C₈₉	1	4	yes	—
443	#102	—	2	2	no	—
444	#102	—	2	2	no	—
445	#102	—	4	1	no	—
446	#97	—	2	2	no	—
447	#95	—	2	1	no	—
448	#95	—	2	1	no	—
449	#95	—	2	1	no	—
450	#95	—	2	1	no	—
451	#95	D₄×C₈₉	1	1	yes	characteristic
452	#95	D₄×C₈₉	1	1	yes	characteristic
453	C₂⁴	—	2	1	no	—
454	C₂×Q₈	—	2	1	no	—
455	C₂×Q₈	—	2	1	no	—
456	C₂×Q₈	—	2	1	no	—
457	C₂×Q₈	—	4	2	no	—
458	C₂×Q₈	—	4	2	no	—
459	C₂×Q₈	—	8	1	no	—
460	C₂×Q₈	C₂×D₄×C₈₉	1	2	yes	—
461	C₂×Q₈	C₂×Q₈×C₈₉	1	2	yes	—
462	C₂×Q₈	#41×C₈₉	1	2	yes	—
463	C₂×D₄	—	2	1	no	—
464	C₂×D₄	—	2	1	no	—
465	C₂×D₄	—	2	1	no	—
466	C₂×D₄	—	2	1	no	—
467	C₂×D₄	—	2	1	no	—
468	C₂×D₄	C₂×Q₈×C₈₉	1	2	yes	—
469	C₂²×C₄	—	2	1	no	—
470	C₂²×C₄	—	2	1	no	—
471	C₂²×C₄	—	2	1	no	—
472	C₂²×C₄	—	2	1	no	—
473	C₂²×C₄	—	2	1	no	—
474	C₂²×C₄	—	2	1	no	—
475	C₂²×C₄	—	2	1	no	—
476	C₂²×C₄	—	4	2	no	—
477	C₂²×C₄	—	4	2	no	—
478	C₂²×C₄	—	8	1	no	—
479	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
480	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
481	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
482	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
483	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
484	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
485	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
486	SD₁₆	—	2	8	no	—
487	C₂×C₈	—	2	1	no	—
488	C₂×C₈	—	2	1	no	—
489	C₂×C₈	—	2	1	no	—
490	C₂×C₈	—	2	1	no	—
491	C₂×C₈	—	2	1	no	—
492	C₂×C₈	—	2	1	no	—
493	C₂×C₈	—	2	1	no	—
494	C₂×C₈	—	2	1	no	—
495	C₂×C₈	—	2	1	no	—
496	C₂×C₈	—	2	1	no	—
497	C₂×C₈	C₂×D₄×C₈₉	1	2	yes	—
498	C₂×C₈	C₂×Q₈×C₈₉	1	2	yes	—
499	C₄⋊C₄	—	2	4	no	—
500	C₄⋊C₄	—	2	2	no	—
501	C₄⋊C₄	—	2	2	no	—
502	C₄⋊C₄	—	2	1	no	—
503	C₄⋊C₄	—	2	1	no	—
504	C₄⋊C₄	—	2	1	no	—
505	C₄⋊C₄	—	2	1	no	—
506	C₄⋊C₄	—	2	1	no	—
507	C₄⋊C₄	—	2	1	no	—
508	C₄⋊C₄	—	2	1	no	—
509	C₄⋊C₄	—	2	1	no	—
510	C₄⋊C₄	—	2	1	no	—
511	C₄⋊C₄	—	2	1	no	—
512	C₄⋊C₄	—	4	2	no	—
513	C₄⋊C₄	—	4	2	no	—
514	C₄⋊C₄	—	4	2	no	—
515	C₄⋊C₄	—	4	2	no	—
516	C₄⋊C₄	—	4	2	no	—
517	C₄⋊C₄	—	4	2	no	—
518	C₄⋊C₄	—	8	1	no	—
519	C₄⋊C₄	—	8	1	no	—
520	C₄⋊C₄	—	8	1	no	—
521	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
522	C₄⋊C₄	#41×C₈₉	1	2	yes	—
523	C₂²⋊C₄	—	4	2	no	—
524	C₂²⋊C₄	—	4	2	no	—
525	C₂²⋊C₄	—	4	2	no	—
526	C₂²⋊C₄	—	4	2	no	—
527	C₂²⋊C₄	—	8	1	no	—
528	C₄²	—	2	4	no	—
529	C₄²	—	2	1	no	—
530	C₄²	—	2	1	no	—
531	C₄²	—	2	1	no	—
532	C₄²	—	2	1	no	—
533	C₄²	—	2	1	no	—
534	C₄²	—	2	1	no	—
535	C₄²	—	4	2	no	—
536	C₄²	—	4	2	no	—
537	C₂³	—	2	2	no	—
538	C₂³	—	4	1	no	—
539	C₂³	—	4	1	no	—
540	C₂³	—	4	1	no	—
541	C₂³	—	4	1	no	—
542	C₂³	—	4	1	no	—
543	C₂³	—	4	1	no	—
544	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
545	Q₈	—	2	2	no	—
546	Q₈	—	2	2	no	—
547	Q₈	—	2	2	no	—
548	Q₈	—	8	4	no	—
549	Q₈	C₂²⋊Q₈×C₈₉	1	4	yes	complemented
550	D₄	—	2	2	no	—
551	D₄	—	2	2	no	—
552	D₄	—	4	1	no	—
553	D₄	—	4	1	no	—
554	C₂×C₄	—	2	1	no	—
555	C₂×C₄	—	2	2	no	—
556	C₂×C₄	—	2	1	no	—
557	C₂×C₄	—	2	2	no	—
558	C₂×C₄	—	2	2	no	—
559	C₂×C₄	—	2	1	no	—
560	C₂×C₄	—	2	1	no	—
561	C₂×C₄	—	2	1	no	—
562	C₂×C₄	—	2	2	no	—
563	C₂×C₄	—	2	1	no	—
564	C₂×C₄	—	2	1	no	—
565	C₂×C₄	—	2	1	no	—
566	C₂×C₄	—	2	2	no	—
567	C₂×C₄	—	2	1	no	—
568	C₂×C₄	—	2	1	no	—
569	C₂×C₄	—	2	1	no	—
570	C₂×C₄	—	2	1	no	—
571	C₂×C₄	—	2	1	no	—
572	C₂×C₄	—	2	1	no	—
573	C₂×C₄	—	2	1	no	—
574	C₂×C₄	—	2	1	no	—
575	C₂×C₄	—	4	1	no	—
576	C₂×C₄	—	4	1	no	—
577	C₂×C₄	—	4	1	no	—
578	C₂×C₄	—	4	1	no	—
579	C₂×C₄	—	4	1	no	—
580	C₂×C₄	—	4	1	no	—
581	C₂×C₄	—	4	1	no	—
582	C₂×C₄	—	4	1	no	—
583	C₂×C₄	—	4	1	no	—
584	C₂×C₄	—	4	1	no	—
585	C₂×C₄	—	4	1	no	—
586	C₂×C₄	—	4	1	no	—
587	C₂×C₄	—	4	1	no	—
588	C₂×C₄	—	4	1	no	—
589	C₂×C₄	—	4	1	no	—
590	C₂×C₄	—	4	1	no	—
591	C₂×C₄	—	8	4	no	—
592	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
593	C₂×C₄	#137×C₈₉	1	2	yes	—
594	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
595	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
596	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
597	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
598	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
599	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
600	C₈	—	2	2	no	—
601	C₈	—	2	2	no	—
602	C₈	—	4	1	no	—
603	C₈	—	4	1	no	—
604	C₂²	—	4	2	no	—
605	C₂²	—	4	2	no	—
606	C₂²	—	4	2	no	—
607	C₂²	—	8	1	no	—
608	C₂²	—	8	1	no	—
609	C₂²	—	8	1	no	—
610	C₂²	—	8	1	no	—
611	C₂²	#525×C₈₉	1	1	yes	characteristic
612	C₂²	D₄²×C₈₉	1	1	yes	characteristic
613	C₂²	D₄×Q₈×C₈₉	1	1	yes	characteristic
614	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
615	C₂²	C₂×#137×C₈₉	1	1	yes	characteristic
616	C₂²	#576×C₈₉	1	1	yes	characteristic
617	C₂²	#578×C₈₉	1	1	yes	characteristic
618	C₄	—	2	2	no	—
619	C₄	—	2	4	no	—
620	C₄	—	2	2	no	—
621	C₄	—	4	1	no	—
622	C₄	—	4	2	no	—
623	C₄	—	4	1	no	—
624	C₄	—	4	1	no	—
625	C₄	—	4	1	no	—
626	C₄	—	4	2	no	—
627	C₄	—	8	1	no	—
628	C₄	—	8	1	no	—
629	C₄	—	8	1	no	—
630	C₄	—	8	1	no	—
631	C₄	C₂×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
632	C₄	#539×C₈₉	1	1	yes	characteristic
633	C₄	#547×C₈₉	1	1	yes	characteristic
634	C₄	#554×C₈₉	1	1	yes	characteristic
635	C₂	—	8	2	no	—
636	C₂	C₈₉×#2449	1	1	yes	characteristic
637	C₂	C₈₉×#3112	1	1	yes	characteristic
638	C₂	C₈₉×#3124	1	1	yes	characteristic
639	C₂	C₈₉×#3270	1	1	yes	characteristic
640	C₂	C₈₉×#3274	1	1	yes	characteristic
641	C₂	C₈₉×#3385	1	1	yes	characteristic
642	C₂	C₈₉×#3386	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 ◁ C₄×C₄⋊C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#477×C₈₉	C₂	semidirect
4. Q₈×C₄⋊C₄×C₈₉	C₂	semidirect
6. C₈₉×#2083	C₂	semidirect
8. C₈₉×#2057	C₂	semidirect
9. C₈₉×#1944	C₂	semidirect
10. C₈₉×#1917	C₂	semidirect
13. C₈₉×#1865	C₂	semidirect
14. C₈₉×#1863	C₂	semidirect
67. C₂×SD₁₆×C₈₉	Q₈	semidirect
227. Q₈×C₈₉	C₂²⋊Q₈	semidirect
313. #26397	C₈₉	direct (N × H)
322. C₂×#477	C₂×C₈₉	semidirect
325. Q₈×C₄⋊C₄	C₂×C₈₉	semidirect
327. #2083	C₂×C₈₉	semidirect
329. #2057	C₂×C₈₉	semidirect
330. #1944	C₂×C₈₉	semidirect
331. #1917	C₂×C₈₉	semidirect
334. #1865	C₂×C₈₉	semidirect
335. #1863	C₂×C₈₉	semidirect
337. C₈₉	#26397	direct (N × H)
389. C₂×SD₁₆	Q₈×C₈₉	semidirect
549. Q₈	C₂²⋊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×#477×C₈₉	2
C₂×#440×C₈₉	2
C₈₉×#2452	2
Q₈×C₄⋊C₄×C₈₉	2
C₈₉×#2093	2
C₈₉×#2083	2
C₈₉×#2065	2
C₈₉×#2057	2
C₈₉×#1944	2
C₈₉×#1917	2
C₈₉×#1880	2
C₈₉×#1865	2
C₈₉×#1863	2
C₈₉×#1862	2
#26397	89

Subgroups — C89×#26397 #36404245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#26397 #36404245