Subgroups — C₈₉×#29392 #36407240

Order 22784.

774 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈₉×#2293	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2084	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2071	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#2058	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#2023	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1926	C₂	1	2	yes	maximal
8	C₈₉×#1920	C₂	1	2	yes	maximal, complemented
9	C₈₉×#1919	C₂	1	2	yes	maximal, complemented
10	C₈₉×#1864	C₂	1	2	yes	maximal
11	C₈₉×#1839	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
14	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#117×C₈₉	—	2	1	no	—
16	C₂×#106×C₈₉	C₂²	1	2	yes	—
17	C₂×#106×C₈₉	C₂²	1	2	yes	—
18	C₂×#102×C₈₉	C₂²	1	2	yes	—
19	C₂×#102×C₈₉	C₂²	1	2	yes	—
20	#391×C₈₉	—	2	1	no	—
21	#390×C₈₉	C₂²	1	1	yes	characteristic
22	#390×C₈₉	C₂²	1	1	yes	characteristic
23	#389×C₈₉	—	2	1	no	—
24	#389×C₈₉	C₂²	1	2	yes	—
25	#389×C₈₉	C₂²	1	1	yes	characteristic
26	#389×C₈₉	C₂²	1	1	yes	characteristic
27	#388×C₈₉	—	2	1	no	—
28	#388×C₈₉	—	2	1	no	—
29	#384×C₈₉	C₂²	1	1	yes	characteristic
30	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
31	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
32	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊C₄×C₈₉	—	2	1	no	—
34	C₄³×C₈₉	C₂²	1	1	yes	characteristic
35	#340×C₈₉	C₂²	1	2	yes	—
36	#340×C₈₉	C₂²	1	2	yes	—
37	#340×C₈₉	C₂²	1	2	yes	—
38	#340×C₈₉	C₂²	1	2	yes	—
39	#336×C₈₉	C₂²	1	1	yes	characteristic
40	#336×C₈₉	C₂²	1	1	yes	characteristic
41	#336×C₈₉	C₂²	1	2	yes	—
42	#336×C₈₉	C₂²	1	2	yes	—
43	C₂²×Q₈×C₈₉	—	2	1	no	—
44	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
45	C₂³×C₄×C₈₉	—	2	1	no	—
46	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
47	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
48	C₄⋊Q₈×C₈₉	—	2	4	no	—
49	C₄×Q₈×C₈₉	—	4	4	no	—
50	C₄×D₄×C₈₉	—	2	2	no	—
51	C₄×D₄×C₈₉	—	2	2	no	—
52	#117×C₈₉	—	4	4	no	—
53	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
54	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
55	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
56	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
57	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
58	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
59	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
60	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
63	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
65	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
66	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
67	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
69	C₂×C₄²×C₈₉	—	2	1	no	—
70	C₂×C₄²×C₈₉	—	2	2	no	—
71	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
72	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
73	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
74	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
75	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
76	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
77	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
78	#106×C₈₉	—	2	4	no	—
79	#106×C₈₉	—	2	4	no	—
80	#102×C₈₉	—	2	4	no	—
81	#102×C₈₉	—	2	4	no	—
82	#95×C₈₉	—	2	1	no	—
83	#95×C₈₉	—	2	1	no	—
84	#95×C₈₉	—	2	2	no	—
85	#95×C₈₉	—	2	1	no	—
86	#95×C₈₉	—	2	2	no	—
87	C₂⁴×C₈₉	—	2	1	no	—
88	C₂×Q₈×C₈₉	—	2	4	no	—
89	C₂×Q₈×C₈₉	—	4	2	no	—
90	C₂×Q₈×C₈₉	—	4	2	no	—
91	C₂×D₄×C₈₉	—	2	2	no	—
92	C₂×D₄×C₈₉	—	2	1	no	—
93	C₂×D₄×C₈₉	—	2	2	no	—
94	C₂×D₄×C₈₉	#41	1	2	yes	—
95	C₂²×C₄×C₈₉	—	2	4	no	—
96	C₂²×C₄×C₈₉	—	2	2	no	—
97	C₂²×C₄×C₈₉	—	2	1	no	—
98	C₂²×C₄×C₈₉	—	2	1	no	—
99	C₂²×C₄×C₈₉	—	2	1	no	—
100	C₂²×C₄×C₈₉	—	2	2	no	—
101	C₂²×C₄×C₈₉	—	4	2	no	—
102	C₂²×C₄×C₈₉	—	4	2	no	—
103	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
109	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
110	C₂×C₈×C₈₉	—	2	2	no	—
111	C₂×C₈×C₈₉	—	2	2	no	—
112	C₂×C₈×C₈₉	—	2	2	no	—
113	C₂×C₈×C₈₉	—	2	2	no	—
114	C₂×C₈×C₈₉	—	2	2	no	—
115	C₂×C₈×C₈₉	—	2	2	no	—
116	C₄⋊C₄×C₈₉	—	2	4	no	—
117	C₄⋊C₄×C₈₉	—	2	1	no	—
118	C₄⋊C₄×C₈₉	—	2	2	no	—
119	C₄⋊C₄×C₈₉	—	2	2	no	—
120	C₄⋊C₄×C₈₉	—	2	2	no	—
121	C₄⋊C₄×C₈₉	—	2	2	no	—
122	C₄⋊C₄×C₈₉	—	2	2	no	—
123	C₄⋊C₄×C₈₉	—	4	2	no	—
124	C₄⋊C₄×C₈₉	—	4	4	no	—
125	C₄⋊C₄×C₈₉	—	4	2	no	—
126	C₄⋊C₄×C₈₉	—	4	2	no	—
127	C₄⋊C₄×C₈₉	—	4	2	no	—
128	C₄⋊C₄×C₈₉	#41	1	4	yes	—
129	C₄⋊C₄×C₈₉	#41	1	2	yes	—
130	C₂²⋊C₄×C₈₉	—	2	4	no	—
131	C₂²⋊C₄×C₈₉	—	4	2	no	—
132	C₂²⋊C₄×C₈₉	—	4	2	no	—
133	C₂²⋊C₄×C₈₉	—	4	2	no	—
134	C₂²⋊C₄×C₈₉	—	4	2	no	—
135	C₄²×C₈₉	—	2	2	no	—
136	C₄²×C₈₉	—	2	1	no	—
137	C₄²×C₈₉	—	2	1	no	—
138	C₄²×C₈₉	—	2	2	no	—
139	C₄²×C₈₉	—	2	2	no	—
140	C₄²×C₈₉	—	4	1	no	—
141	C₄²×C₈₉	—	4	1	no	—
142	C₄²×C₈₉	—	4	4	no	—
143	C₄²×C₈₉	—	4	2	no	—
144	C₄²×C₈₉	SD₁₆	1	4	yes	—
145	C₂³×C₈₉	—	2	2	no	—
146	C₂³×C₈₉	—	4	1	no	—
147	C₂³×C₈₉	—	4	1	no	—
148	C₂³×C₈₉	—	4	1	no	—
149	C₂³×C₈₉	—	4	1	no	—
150	C₂³×C₈₉	—	4	1	no	—
151	C₂³×C₈₉	—	4	1	no	—
152	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
153	Q₈×C₈₉	—	4	8	no	—
154	D₄×C₈₉	—	2	4	no	—
155	D₄×C₈₉	—	4	2	no	—
156	C₂×C₄×C₈₉	—	2	1	no	—
157	C₂×C₄×C₈₉	—	2	1	no	—
158	C₂×C₄×C₈₉	—	2	1	no	—
159	C₂×C₄×C₈₉	—	2	1	no	—
160	C₂×C₄×C₈₉	—	2	1	no	—
161	C₂×C₄×C₈₉	—	2	1	no	—
162	C₂×C₄×C₈₉	—	2	1	no	—
163	C₂×C₄×C₈₉	—	2	1	no	—
164	C₂×C₄×C₈₉	—	2	1	no	—
165	C₂×C₄×C₈₉	—	2	2	no	—
166	C₂×C₄×C₈₉	—	2	4	no	—
167	C₂×C₄×C₈₉	—	2	1	no	—
168	C₂×C₄×C₈₉	—	2	1	no	—
169	C₂×C₄×C₈₉	—	2	1	no	—
170	C₂×C₄×C₈₉	—	2	1	no	—
171	C₂×C₄×C₈₉	—	2	1	no	—
172	C₂×C₄×C₈₉	—	4	8	no	—
173	C₂×C₄×C₈₉	—	4	1	no	—
174	C₂×C₄×C₈₉	—	4	2	no	—
175	C₂×C₄×C₈₉	—	4	1	no	—
176	C₂×C₄×C₈₉	—	4	2	no	—
177	C₂×C₄×C₈₉	—	4	1	no	—
178	C₂×C₄×C₈₉	—	4	2	no	—
179	C₂×C₄×C₈₉	—	4	1	no	—
180	C₂×C₄×C₈₉	—	4	1	no	—
181	C₂×C₄×C₈₉	—	4	1	no	—
182	C₂×C₄×C₈₉	—	4	2	no	—
183	C₂×C₄×C₈₉	—	4	1	no	—
184	C₂×C₄×C₈₉	—	4	1	no	—
185	C₂×C₄×C₈₉	—	4	2	no	—
186	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
187	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
188	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
189	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
190	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
191	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
192	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
193	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
194	C₈×C₈₉	—	4	2	no	—
195	C₈×C₈₉	—	4	2	no	—
196	C₂²×C₈₉	—	4	2	no	—
197	C₂²×C₈₉	—	4	2	no	—
198	C₂²×C₈₉	—	4	2	no	—
199	C₂²×C₈₉	—	8	1	no	—
200	C₂²×C₈₉	—	8	1	no	—
201	C₂²×C₈₉	—	8	1	no	—
202	C₂²×C₈₉	—	8	1	no	—
203	C₂²×C₈₉	#525	1	1	yes	characteristic
204	C₂²×C₈₉	#525	1	1	yes	characteristic
205	C₂²×C₈₉	#536	1	1	yes	characteristic
206	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
207	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
208	C₂²×C₈₉	#575	1	1	yes	characteristic
209	C₂²×C₈₉	#575	1	1	yes	characteristic
210	C₄×C₈₉	—	2	2	no	—
211	C₄×C₈₉	—	2	2	no	—
212	C₄×C₈₉	—	2	2	no	—
213	C₄×C₈₉	—	2	2	no	—
214	C₄×C₈₉	—	4	2	no	—
215	C₄×C₈₉	—	4	4	no	—
216	C₄×C₈₉	—	4	1	no	—
217	C₄×C₈₉	—	4	1	no	—
218	C₄×C₈₉	—	8	1	no	—
219	C₄×C₈₉	—	8	2	no	—
220	C₄×C₈₉	—	8	1	no	—
221	C₄×C₈₉	#552	1	2	yes	—
222	C₄×C₈₉	#556	1	2	yes	—
223	#29392	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
224	C₂×C₈₉	—	8	2	no	—
225	C₂×C₈₉	#2599	1	1	yes	characteristic
226	C₂×C₈₉	#3101	1	1	yes	characteristic
227	C₂×C₈₉	#3103	1	1	yes	characteristic
228	C₂×C₈₉	#3114	1	1	yes	characteristic
229	C₂×C₈₉	#3115	1	1	yes	characteristic
230	C₂×C₈₉	#3227	1	1	yes	characteristic
231	C₂×C₈₉	#3240	1	1	yes	characteristic
232	C₄×C₄⋊Q₈	C₂×C₈₉	1	1	yes	characteristic, complemented
233	#2293	C₂×C₈₉	1	1	yes	characteristic
234	#2084	C₂×C₈₉	1	1	yes	characteristic, complemented
235	#2071	C₂×C₈₉	1	1	yes	characteristic
236	#2058	C₂×C₈₉	1	1	yes	characteristic, complemented
237	#2023	C₂×C₈₉	1	1	yes	characteristic
238	#1926	C₂×C₈₉	1	2	yes	—
239	#1920	C₂×C₈₉	1	2	yes	complemented
240	#1919	C₂×C₈₉	1	2	yes	complemented
241	#1864	C₂×C₈₉	1	2	yes	—
242	#1839	C₂×C₈₉	1	1	yes	characteristic, complemented
243	C₈₉	#29392	1	1	yes	characteristic, direct product factor, Sylow
244	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
245	C₂×C₄×Q₈	—	2	1	no	—
246	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
247	C₂×#117	—	2	1	no	—
248	C₂×#106	C₂²×C₈₉	1	2	yes	—
249	C₂×#106	C₂²×C₈₉	1	2	yes	—
250	C₂×#102	C₂²×C₈₉	1	2	yes	—
251	C₂×#102	C₂²×C₈₉	1	2	yes	—
252	#391	—	2	1	no	—
253	#390	C₂²×C₈₉	1	1	yes	characteristic
254	#390	C₂²×C₈₉	1	1	yes	characteristic
255	#389	—	2	1	no	—
256	#389	C₂²×C₈₉	1	2	yes	—
257	#389	C₂²×C₈₉	1	1	yes	characteristic
258	#389	C₂²×C₈₉	1	1	yes	characteristic
259	#388	—	2	1	no	—
260	#388	—	2	1	no	—
261	#384	C₂²×C₈₉	1	1	yes	characteristic
262	C₄×C₄⋊C₄	—	2	1	no	—
263	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
264	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
265	C₄×C₂²⋊C₄	—	2	1	no	—
266	C₄³	C₂²×C₈₉	1	1	yes	characteristic
267	#340	C₂²×C₈₉	1	2	yes	—
268	#340	C₂²×C₈₉	1	2	yes	—
269	#340	C₂²×C₈₉	1	2	yes	—
270	#340	C₂²×C₈₉	1	2	yes	—
271	#336	C₂²×C₈₉	1	1	yes	characteristic
272	#336	C₂²×C₈₉	1	1	yes	characteristic
273	#336	C₂²×C₈₉	1	2	yes	—
274	#336	C₂²×C₈₉	1	2	yes	—
275	C₂²×Q₈	—	2	1	no	—
276	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
277	C₂³×C₄	—	2	1	no	—
278	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
279	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
280	C₄⋊Q₈	—	2	4	no	—
281	C₄×Q₈	—	4	4	no	—
282	C₄×D₄	—	2	2	no	—
283	C₄×D₄	—	2	2	no	—
284	#117	—	4	4	no	—
285	C₂×C₄⋊C₄	—	2	1	no	—
286	C₂×C₄⋊C₄	—	2	1	no	—
287	C₂×C₄⋊C₄	—	2	1	no	—
288	C₂×C₄⋊C₄	—	2	1	no	—
289	C₂×C₄⋊C₄	—	2	2	no	—
290	C₂×C₄⋊C₄	—	2	1	no	—
291	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
292	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
293	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
294	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
295	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
296	C₂×C₂²⋊C₄	—	2	1	no	—
297	C₂×C₂²⋊C₄	—	2	1	no	—
298	C₂×C₂²⋊C₄	—	2	1	no	—
299	C₂×C₂²⋊C₄	—	2	1	no	—
300	C₂×C₂²⋊C₄	—	2	1	no	—
301	C₂×C₄²	—	2	1	no	—
302	C₂×C₄²	—	2	2	no	—
303	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
304	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
305	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
306	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
307	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
308	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
309	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
310	#106	—	2	4	no	—
311	#106	—	2	4	no	—
312	#102	—	2	4	no	—
313	#102	—	2	4	no	—
314	#95	—	2	1	no	—
315	#95	—	2	1	no	—
316	#95	—	2	2	no	—
317	#95	—	2	1	no	—
318	#95	—	2	2	no	—
319	C₂⁴	—	2	1	no	—
320	C₂×Q₈	—	2	4	no	—
321	C₂×Q₈	—	4	2	no	—
322	C₂×Q₈	—	4	2	no	—
323	C₂×D₄	—	2	2	no	—
324	C₂×D₄	—	2	1	no	—
325	C₂×D₄	—	2	2	no	—
326	C₂×D₄	#41×C₈₉	1	2	yes	—
327	C₂²×C₄	—	2	4	no	—
328	C₂²×C₄	—	2	2	no	—
329	C₂²×C₄	—	2	1	no	—
330	C₂²×C₄	—	2	1	no	—
331	C₂²×C₄	—	2	1	no	—
332	C₂²×C₄	—	2	2	no	—
333	C₂²×C₄	—	4	2	no	—
334	C₂²×C₄	—	4	2	no	—
335	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
336	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
337	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
338	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
339	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
340	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
341	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
342	C₂×C₈	—	2	2	no	—
343	C₂×C₈	—	2	2	no	—
344	C₂×C₈	—	2	2	no	—
345	C₂×C₈	—	2	2	no	—
346	C₂×C₈	—	2	2	no	—
347	C₂×C₈	—	2	2	no	—
348	C₄⋊C₄	—	2	4	no	—
349	C₄⋊C₄	—	2	1	no	—
350	C₄⋊C₄	—	2	2	no	—
351	C₄⋊C₄	—	2	2	no	—
352	C₄⋊C₄	—	2	2	no	—
353	C₄⋊C₄	—	2	2	no	—
354	C₄⋊C₄	—	2	2	no	—
355	C₄⋊C₄	—	4	4	no	—
356	C₄⋊C₄	—	4	2	no	—
357	C₄⋊C₄	—	4	2	no	—
358	C₄⋊C₄	—	4	2	no	—
359	C₄⋊C₄	—	4	2	no	—
360	C₄⋊C₄	#41×C₈₉	1	4	yes	—
361	C₄⋊C₄	#41×C₈₉	1	2	yes	—
362	C₂²⋊C₄	—	2	4	no	—
363	C₂²⋊C₄	—	4	2	no	—
364	C₂²⋊C₄	—	4	2	no	—
365	C₂²⋊C₄	—	4	2	no	—
366	C₂²⋊C₄	—	4	2	no	—
367	C₄²	—	2	2	no	—
368	C₄²	—	2	1	no	—
369	C₄²	—	2	1	no	—
370	C₄²	—	2	2	no	—
371	C₄²	—	2	2	no	—
372	C₄²	—	4	1	no	—
373	C₄²	—	4	1	no	—
374	C₄²	—	4	4	no	—
375	C₄²	—	4	2	no	—
376	C₄²	SD₁₆×C₈₉	1	4	yes	—
377	C₂³	—	2	2	no	—
378	C₂³	—	4	1	no	—
379	C₂³	—	4	1	no	—
380	C₂³	—	4	1	no	—
381	C₂³	—	4	1	no	—
382	C₂³	—	4	1	no	—
383	C₂³	—	4	1	no	—
384	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
385	Q₈	—	4	8	no	—
386	D₄	—	2	4	no	—
387	D₄	—	4	2	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	1	no	—
391	C₂×C₄	—	2	1	no	—
392	C₂×C₄	—	2	1	no	—
393	C₂×C₄	—	2	1	no	—
394	C₂×C₄	—	2	1	no	—
395	C₂×C₄	—	2	1	no	—
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	2	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	2	1	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	2	1	no	—
403	C₂×C₄	—	2	1	no	—
404	C₂×C₄	—	4	8	no	—
405	C₂×C₄	—	4	1	no	—
406	C₂×C₄	—	4	2	no	—
407	C₂×C₄	—	4	1	no	—
408	C₂×C₄	—	4	2	no	—
409	C₂×C₄	—	4	1	no	—
410	C₂×C₄	—	4	2	no	—
411	C₂×C₄	—	4	1	no	—
412	C₂×C₄	—	4	1	no	—
413	C₂×C₄	—	4	1	no	—
414	C₂×C₄	—	4	2	no	—
415	C₂×C₄	—	4	1	no	—
416	C₂×C₄	—	4	1	no	—
417	C₂×C₄	—	4	2	no	—
418	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
419	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
420	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
421	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	2	yes	—
422	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
423	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
424	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
425	C₂×C₄	2₋¹⁺⁴×C₈₉	1	2	yes	—
426	C₈	—	4	2	no	—
427	C₈	—	4	2	no	—
428	C₂²	—	4	2	no	—
429	C₂²	—	4	2	no	—
430	C₂²	—	4	2	no	—
431	C₂²	—	8	1	no	—
432	C₂²	—	8	1	no	—
433	C₂²	—	8	1	no	—
434	C₂²	—	8	1	no	—
435	C₂²	#525×C₈₉	1	1	yes	characteristic
436	C₂²	#525×C₈₉	1	1	yes	characteristic
437	C₂²	#536×C₈₉	1	1	yes	characteristic
438	C₂²	C₂²×SD₁₆×C₈₉	1	1	yes	characteristic
439	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
440	C₂²	#575×C₈₉	1	1	yes	characteristic
441	C₂²	#575×C₈₉	1	1	yes	characteristic
442	C₄	—	2	2	no	—
443	C₄	—	2	2	no	—
444	C₄	—	2	2	no	—
445	C₄	—	2	2	no	—
446	C₄	—	4	2	no	—
447	C₄	—	4	4	no	—
448	C₄	—	4	1	no	—
449	C₄	—	4	1	no	—
450	C₄	—	8	1	no	—
451	C₄	—	8	2	no	—
452	C₄	—	8	1	no	—
453	C₄	#552×C₈₉	1	2	yes	—
454	C₄	#556×C₈₉	1	2	yes	—
455	C₂	—	8	2	no	—
456	C₂	C₈₉×#2599	1	1	yes	characteristic
457	C₂	C₈₉×#3101	1	1	yes	characteristic
458	C₂	C₈₉×#3103	1	1	yes	characteristic
459	C₂	C₈₉×#3114	1	1	yes	characteristic
460	C₂	C₈₉×#3115	1	1	yes	characteristic
461	C₂	C₈₉×#3227	1	1	yes	characteristic
462	C₂	C₈₉×#3240	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 ◁ C₄³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₄×C₄⋊Q₈×C₈₉	C₂	semidirect
3. C₈₉×#2084	C₂	semidirect
5. C₈₉×#2058	C₂	semidirect
8. C₈₉×#1920	C₂	semidirect
9. C₈₉×#1919	C₂	semidirect
11. C₈₉×#1839	C₂	semidirect
223. #29392	C₈₉	direct (N × H)
232. C₄×C₄⋊Q₈	C₂×C₈₉	semidirect
234. #2084	C₂×C₈₉	semidirect
236. #2058	C₂×C₈₉	semidirect
239. #1920	C₂×C₈₉	semidirect
240. #1919	C₂×C₈₉	semidirect
242. #1839	C₂×C₈₉	semidirect
243. C₈₉	#29392	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×C₄⋊Q₈×C₈₉	2
C₈₉×#2293	2
C₈₉×#2084	2
C₈₉×#2071	2
C₈₉×#2058	2
C₈₉×#2023	2
C₈₉×#1926	2
C₈₉×#1920	2
C₈₉×#1919	2
C₈₉×#1864	2
C₈₉×#1839	2
#29392	89

Subgroups — C89×#29392 #36407240

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#29392 #36407240