Subgroups — C₈₉×#22964 #36400812

Order 22784.

822 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#475×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2365	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#1990	C₂	1	1	yes	maximal, characteristic
5	C₈₉×#1948	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#1941	C₂	1	2	yes	maximal
7	C₈₉×#1920	C₂	1	2	yes	maximal
8	C₈₉×#1865	C₂	1	2	yes	maximal
9	C₈×C₄⋊C₄×C₈₉	C₂	1	1	yes	maximal, characteristic
10	C₄×#103×C₈₉	C₂	1	2	yes	maximal
11	C₄×#103×C₈₉	C₂	1	1	yes	maximal, characteristic
12	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
14	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
16	#475×C₈₉	C₄	1	8	yes	complemented
17	C₂×#106×C₈₉	C₂²	1	2	yes	—
18	C₂×#106×C₈₉	C₂²	1	2	yes	—
19	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×#103×C₈₉	C₂²	1	2	yes	—
22	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
25	#391×C₈₉	—	2	1	no	—
26	#389×C₈₉	—	2	2	no	—
27	#389×C₈₉	C₂²	1	2	yes	—
28	#389×C₈₉	C₂²	1	2	yes	—
29	#387×C₈₉	—	2	2	no	—
30	#384×C₈₉	—	2	1	no	—
31	#384×C₈₉	C₂²	1	1	yes	characteristic
32	C₄×C₄⋊C₄×C₈₉	—	2	1	no	—
33	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
34	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
35	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
36	#340×C₈₉	C₂²	1	2	yes	—
37	#340×C₈₉	C₂²	1	1	yes	characteristic
38	#340×C₈₉	C₂²	1	2	yes	—
39	#340×C₈₉	C₂²	1	1	yes	characteristic
40	#340×C₈₉	C₂²	1	2	yes	—
41	#336×C₈₉	C₂²	1	2	yes	—
42	C₂²×Q₈×C₈₉	—	2	1	no	—
43	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
46	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
47	C₄⋊Q₈×C₈₉	—	2	2	no	—
48	C₄⋊Q₈×C₈₉	C₂×C₄	1	4	yes	—
49	C₄×Q₈×C₈₉	—	2	2	no	—
50	C₄×Q₈×C₈₉	—	2	4	no	—
51	C₄×Q₈×C₈₉	—	4	2	no	—
52	C₄×Q₈×C₈₉	—	4	4	no	—
53	C₄×Q₈×C₈₉	C₂×C₄	1	4	yes	—
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	2	no	—
58	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
59	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
60	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
61	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
63	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
64	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
66	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
67	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
68	C₂×C₄²×C₈₉	—	2	1	no	—
69	C₂×C₄²×C₈₉	—	2	2	no	—
70	C₂×C₄²×C₈₉	—	2	2	no	—
71	C₂×C₄²×C₈₉	—	2	1	no	—
72	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
73	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
74	#106×C₈₉	—	4	2	no	—
75	#106×C₈₉	C₂×C₄	1	8	yes	—
76	C₄⋊C₈×C₈₉	C₂×C₄	1	4	yes	—
77	#103×C₈₉	—	2	2	no	—
78	#103×C₈₉	C₂×C₄	1	8	yes	—
79	#103×C₈₉	Q₈	1	4	yes	complemented
80	C₄×C₈×C₈₉	—	2	2	no	—
81	C₄×C₈×C₈₉	—	4	2	no	—
82	#95×C₈₉	—	2	2	no	—
83	#95×C₈₉	—	2	1	no	—
84	#95×C₈₉	—	2	2	no	—
85	#95×C₈₉	—	2	2	no	—
86	#95×C₈₉	D₄	1	2	yes	—
87	C₂×Q₈×C₈₉	—	2	4	no	—
88	C₂×Q₈×C₈₉	—	2	1	no	—
89	C₂×Q₈×C₈₉	—	2	1	no	—
90	C₂×Q₈×C₈₉	—	2	1	no	—
91	C₂×Q₈×C₈₉	—	4	2	no	—
92	C₂×Q₈×C₈₉	—	4	2	no	—
93	C₂×Q₈×C₈₉	C₂²×C₄	1	2	yes	—
94	C₂×Q₈×C₈₉	C₂×Q₈	1	2	yes	—
95	C₂×Q₈×C₈₉	#41	1	2	yes	—
96	C₂²×C₄×C₈₉	—	2	1	no	—
97	C₂²×C₄×C₈₉	—	2	1	no	—
98	C₂²×C₄×C₈₉	—	2	1	no	—
99	C₂²×C₄×C₈₉	—	2	2	no	—
100	C₂²×C₄×C₈₉	—	2	1	no	—
101	C₂²×C₄×C₈₉	—	2	2	no	—
102	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
103	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
105	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
107	C₂×C₈×C₈₉	—	2	1	no	—
108	C₂×C₈×C₈₉	—	2	1	no	—
109	C₂×C₈×C₈₉	—	2	2	no	—
110	C₂×C₈×C₈₉	—	2	2	no	—
111	C₂×C₈×C₈₉	—	2	1	no	—
112	C₂×C₈×C₈₉	—	2	1	no	—
113	C₂×C₈×C₈₉	—	2	1	no	—
114	C₂×C₈×C₈₉	C₂²×C₄	1	4	yes	—
115	C₂×C₈×C₈₉	C₂×Q₈	1	2	yes	—
116	C₄⋊C₄×C₈₉	—	2	4	no	—
117	C₄⋊C₄×C₈₉	—	2	4	no	—
118	C₄⋊C₄×C₈₉	—	2	2	no	—
119	C₄⋊C₄×C₈₉	—	2	1	no	—
120	C₄⋊C₄×C₈₉	—	2	2	no	—
121	C₄⋊C₄×C₈₉	—	2	1	no	—
122	C₄⋊C₄×C₈₉	—	2	1	no	—
123	C₄⋊C₄×C₈₉	—	2	2	no	—
124	C₄⋊C₄×C₈₉	—	2	1	no	—
125	C₄⋊C₄×C₈₉	—	2	2	no	—
126	C₄⋊C₄×C₈₉	—	4	4	no	—
127	C₄⋊C₄×C₈₉	—	4	4	no	—
128	C₄⋊C₄×C₈₉	—	4	2	no	—
129	C₄⋊C₄×C₈₉	—	8	1	no	—
130	C₄⋊C₄×C₈₉	—	8	2	no	—
131	C₄⋊C₄×C₈₉	SD₁₆	1	4	yes	—
132	C₄⋊C₄×C₈₉	C₂²×C₄	1	4	yes	—
133	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
134	C₄⋊C₄×C₈₉	C₂×Q₈	1	2	yes	—
135	C₄²×C₈₉	—	2	4	no	—
136	C₄²×C₈₉	—	2	2	no	—
137	C₄²×C₈₉	—	2	2	no	—
138	C₄²×C₈₉	—	4	4	no	—
139	C₄²×C₈₉	—	4	4	no	—
140	C₄²×C₈₉	—	4	2	no	—
141	C₄²×C₈₉	C₂²×C₄	1	2	yes	—
142	C₄²×C₈₉	#41	1	2	yes	—
143	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
144	Q₈×C₈₉	—	2	2	no	—
145	Q₈×C₈₉	—	2	2	no	—
146	Q₈×C₈₉	—	2	2	no	—
147	Q₈×C₈₉	—	4	8	no	—
148	Q₈×C₈₉	C₄×Q₈	1	4	yes	complemented
149	C₂×C₄×C₈₉	—	2	2	no	—
150	C₂×C₄×C₈₉	—	2	2	no	—
151	C₂×C₄×C₈₉	—	2	2	no	—
152	C₂×C₄×C₈₉	—	2	2	no	—
153	C₂×C₄×C₈₉	—	2	1	no	—
154	C₂×C₄×C₈₉	—	2	1	no	—
155	C₂×C₄×C₈₉	—	2	2	no	—
156	C₂×C₄×C₈₉	—	2	2	no	—
157	C₂×C₄×C₈₉	—	2	4	no	—
158	C₂×C₄×C₈₉	—	2	2	no	—
159	C₂×C₄×C₈₉	—	2	2	no	—
160	C₂×C₄×C₈₉	—	2	2	no	—
161	C₂×C₄×C₈₉	—	2	2	no	—
162	C₂×C₄×C₈₉	—	4	1	no	—
163	C₂×C₄×C₈₉	—	4	2	no	—
164	C₂×C₄×C₈₉	—	4	2	no	—
165	C₂×C₄×C₈₉	—	4	1	no	—
166	C₂×C₄×C₈₉	—	4	1	no	—
167	C₂×C₄×C₈₉	—	4	1	no	—
168	C₂×C₄×C₈₉	—	4	2	no	—
169	C₂×C₄×C₈₉	—	4	1	no	—
170	C₂×C₄×C₈₉	—	4	1	no	—
171	C₂×C₄×C₈₉	—	4	2	no	—
172	C₂×C₄×C₈₉	—	4	2	no	—
173	C₂×C₄×C₈₉	—	4	1	no	—
174	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
175	C₂×C₄×C₈₉	C₄×D₄	1	2	yes	—
176	C₂×C₄×C₈₉	C₂×SD₁₆	1	4	yes	—
177	C₂×C₄×C₈₉	C₂×SD₁₆	1	2	yes	—
178	C₂×C₄×C₈₉	#135	1	2	yes	—
179	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
180	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
181	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
182	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
183	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
184	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
185	C₈×C₈₉	—	2	4	no	—
186	C₈×C₈₉	—	2	2	no	—
187	C₈×C₈₉	—	4	1	no	—
188	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
189	C₂²×C₈₉	#546	1	1	yes	characteristic
190	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
191	C₂²×C₈₉	C₂²×SD₁₆	1	1	yes	characteristic
192	C₂²×C₈₉	C₂×#135	1	1	yes	characteristic
193	C₂²×C₈₉	#577	1	1	yes	characteristic
194	C₂²×C₈₉	#578	1	1	yes	characteristic
195	C₄×C₈₉	—	2	4	no	—
196	C₄×C₈₉	—	2	2	no	—
197	C₄×C₈₉	—	2	4	no	—
198	C₄×C₈₉	—	4	2	no	—
199	C₄×C₈₉	—	4	4	no	—
200	C₄×C₈₉	—	4	2	no	—
201	C₄×C₈₉	—	4	2	no	—
202	C₄×C₈₉	—	8	1	no	—
203	C₄×C₈₉	—	8	2	no	—
204	C₄×C₈₉	C₄×SD₁₆	1	4	yes	—
205	C₄×C₈₉	C₂×C₄×Q₈	1	1	yes	characteristic
206	C₄×C₈₉	#520	1	1	yes	characteristic
207	C₄×C₈₉	#552	1	1	yes	characteristic
208	C₄×C₈₉	#554	1	1	yes	characteristic
209	#22964	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
210	C₂×C₈₉	#2360	1	1	yes	characteristic
211	C₂×C₈₉	C₂×C₄×SD₁₆	1	1	yes	characteristic
212	C₂×C₈₉	#2947	1	1	yes	characteristic
213	C₂×C₈₉	#3298	1	1	yes	characteristic
214	C₂×C₈₉	#3304	1	1	yes	characteristic
215	C₂×C₈₉	Q₈×SD₁₆	1	1	yes	characteristic
216	C₂×C₈₉	#3383	1	1	yes	characteristic
217	C₂×#475	C₂×C₈₉	1	1	yes	characteristic
218	#2365	C₂×C₈₉	1	1	yes	characteristic
219	Q₈×C₄⋊C₄	C₂×C₈₉	1	1	yes	characteristic
220	#1990	C₂×C₈₉	1	1	yes	characteristic
221	#1948	C₂×C₈₉	1	1	yes	characteristic
222	#1941	C₂×C₈₉	1	2	yes	—
223	#1920	C₂×C₈₉	1	2	yes	—
224	#1865	C₂×C₈₉	1	2	yes	—
225	C₈×C₄⋊C₄	C₂×C₈₉	1	1	yes	characteristic
226	C₄×#103	C₂×C₈₉	1	2	yes	—
227	C₄×#103	C₂×C₈₉	1	1	yes	characteristic
228	C₈₉	#22964	1	1	yes	characteristic, direct product factor, Sylow
229	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
230	C₂×C₄×Q₈	—	2	1	no	—
231	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
232	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
233	#475	C₄×C₈₉	1	8	yes	complemented
234	C₂×#106	C₂²×C₈₉	1	2	yes	—
235	C₂×#106	C₂²×C₈₉	1	2	yes	—
236	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
237	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
238	C₂×#103	C₂²×C₈₉	1	2	yes	—
239	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
240	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
241	C₂×C₄×C₈	C₂²×C₈₉	1	2	yes	—
242	#391	—	2	1	no	—
243	#389	—	2	2	no	—
244	#389	C₂²×C₈₉	1	2	yes	—
245	#389	C₂²×C₈₉	1	2	yes	—
246	#387	—	2	2	no	—
247	#384	—	2	1	no	—
248	#384	C₂²×C₈₉	1	1	yes	characteristic
249	C₄×C₄⋊C₄	—	2	1	no	—
250	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
251	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
252	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
253	#340	C₂²×C₈₉	1	2	yes	—
254	#340	C₂²×C₈₉	1	1	yes	characteristic
255	#340	C₂²×C₈₉	1	2	yes	—
256	#340	C₂²×C₈₉	1	1	yes	characteristic
257	#340	C₂²×C₈₉	1	2	yes	—
258	#336	C₂²×C₈₉	1	2	yes	—
259	C₂²×Q₈	—	2	1	no	—
260	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
261	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
262	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
263	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
264	C₄⋊Q₈	—	2	2	no	—
265	C₄⋊Q₈	C₂×C₄×C₈₉	1	4	yes	—
266	C₄×Q₈	—	2	2	no	—
267	C₄×Q₈	—	2	4	no	—
268	C₄×Q₈	—	4	2	no	—
269	C₄×Q₈	—	4	4	no	—
270	C₄×Q₈	C₂×C₄×C₈₉	1	4	yes	—
271	C₂×C₄⋊C₄	—	2	1	no	—
272	C₂×C₄⋊C₄	—	2	1	no	—
273	C₂×C₄⋊C₄	—	2	2	no	—
274	C₂×C₄⋊C₄	—	2	2	no	—
275	C₂×C₄⋊C₄	—	2	1	no	—
276	C₂×C₄⋊C₄	—	2	2	no	—
277	C₂×C₄⋊C₄	—	2	1	no	—
278	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
279	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
280	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
281	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
282	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
283	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
284	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
285	C₂×C₄²	—	2	1	no	—
286	C₂×C₄²	—	2	2	no	—
287	C₂×C₄²	—	2	1	no	—
288	C₂×C₄²	—	2	2	no	—
289	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
290	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
291	#106	—	4	2	no	—
292	#106	C₂×C₄×C₈₉	1	8	yes	—
293	C₄⋊C₈	C₂×C₄×C₈₉	1	4	yes	—
294	#103	—	2	2	no	—
295	#103	C₂×C₄×C₈₉	1	8	yes	—
296	#103	Q₈×C₈₉	1	4	yes	complemented
297	C₄×C₈	—	2	2	no	—
298	C₄×C₈	—	4	2	no	—
299	#95	—	2	2	no	—
300	#95	—	2	1	no	—
301	#95	—	2	2	no	—
302	#95	—	2	2	no	—
303	#95	D₄×C₈₉	1	2	yes	—
304	C₂×Q₈	—	2	4	no	—
305	C₂×Q₈	—	2	1	no	—
306	C₂×Q₈	—	2	1	no	—
307	C₂×Q₈	—	2	1	no	—
308	C₂×Q₈	—	4	2	no	—
309	C₂×Q₈	—	4	2	no	—
310	C₂×Q₈	C₂²×C₄×C₈₉	1	2	yes	—
311	C₂×Q₈	C₂×Q₈×C₈₉	1	2	yes	—
312	C₂×Q₈	#41×C₈₉	1	2	yes	—
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	2	1	no	—
315	C₂²×C₄	—	2	1	no	—
316	C₂²×C₄	—	2	2	no	—
317	C₂²×C₄	—	2	1	no	—
318	C₂²×C₄	—	2	2	no	—
319	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
320	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
321	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
322	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
323	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
324	C₂×C₈	—	2	1	no	—
325	C₂×C₈	—	2	1	no	—
326	C₂×C₈	—	2	2	no	—
327	C₂×C₈	—	2	2	no	—
328	C₂×C₈	—	2	1	no	—
329	C₂×C₈	—	2	1	no	—
330	C₂×C₈	—	2	1	no	—
331	C₂×C₈	C₂²×C₄×C₈₉	1	4	yes	—
332	C₂×C₈	C₂×Q₈×C₈₉	1	2	yes	—
333	C₄⋊C₄	—	2	4	no	—
334	C₄⋊C₄	—	2	4	no	—
335	C₄⋊C₄	—	2	2	no	—
336	C₄⋊C₄	—	2	1	no	—
337	C₄⋊C₄	—	2	2	no	—
338	C₄⋊C₄	—	2	1	no	—
339	C₄⋊C₄	—	2	1	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	—	2	1	no	—
342	C₄⋊C₄	—	2	2	no	—
343	C₄⋊C₄	—	4	4	no	—
344	C₄⋊C₄	—	4	4	no	—
345	C₄⋊C₄	—	4	2	no	—
346	C₄⋊C₄	—	8	1	no	—
347	C₄⋊C₄	—	8	2	no	—
348	C₄⋊C₄	SD₁₆×C₈₉	1	4	yes	—
349	C₄⋊C₄	C₂²×C₄×C₈₉	1	4	yes	—
350	C₄⋊C₄	C₂²×C₄×C₈₉	1	2	yes	—
351	C₄⋊C₄	C₂×Q₈×C₈₉	1	2	yes	—
352	C₄²	—	2	4	no	—
353	C₄²	—	2	2	no	—
354	C₄²	—	2	2	no	—
355	C₄²	—	4	4	no	—
356	C₄²	—	4	4	no	—
357	C₄²	—	4	2	no	—
358	C₄²	C₂²×C₄×C₈₉	1	2	yes	—
359	C₄²	#41×C₈₉	1	2	yes	—
360	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
361	Q₈	—	2	2	no	—
362	Q₈	—	2	2	no	—
363	Q₈	—	2	2	no	—
364	Q₈	—	4	8	no	—
365	Q₈	C₄×Q₈×C₈₉	1	4	yes	complemented
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	2	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	2	no	—
370	C₂×C₄	—	2	1	no	—
371	C₂×C₄	—	2	1	no	—
372	C₂×C₄	—	2	2	no	—
373	C₂×C₄	—	2	2	no	—
374	C₂×C₄	—	2	4	no	—
375	C₂×C₄	—	2	2	no	—
376	C₂×C₄	—	2	2	no	—
377	C₂×C₄	—	2	2	no	—
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	4	1	no	—
380	C₂×C₄	—	4	2	no	—
381	C₂×C₄	—	4	2	no	—
382	C₂×C₄	—	4	1	no	—
383	C₂×C₄	—	4	1	no	—
384	C₂×C₄	—	4	1	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	1	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	2	no	—
389	C₂×C₄	—	4	2	no	—
390	C₂×C₄	—	4	1	no	—
391	C₂×C₄	C₄×D₄×C₈₉	1	2	yes	—
392	C₂×C₄	C₄×D₄×C₈₉	1	2	yes	—
393	C₂×C₄	C₂×SD₁₆×C₈₉	1	4	yes	—
394	C₂×C₄	C₂×SD₁₆×C₈₉	1	2	yes	—
395	C₂×C₄	#135×C₈₉	1	2	yes	—
396	C₂×C₄	C₂³×C₄×C₈₉	1	1	yes	characteristic
397	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
398	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
399	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
400	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
401	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
402	C₈	—	2	4	no	—
403	C₈	—	2	2	no	—
404	C₈	—	4	1	no	—
405	C₂²	C₂×C₄×D₄×C₈₉	1	1	yes	characteristic
406	C₂²	#546×C₈₉	1	1	yes	characteristic
407	C₂²	D₄×Q₈×C₈₉	1	1	yes	characteristic
408	C₂²	C₂²×SD₁₆×C₈₉	1	1	yes	characteristic
409	C₂²	C₂×#135×C₈₉	1	1	yes	characteristic
410	C₂²	#577×C₈₉	1	1	yes	characteristic
411	C₂²	#578×C₈₉	1	1	yes	characteristic
412	C₄	—	2	4	no	—
413	C₄	—	2	2	no	—
414	C₄	—	2	4	no	—
415	C₄	—	4	2	no	—
416	C₄	—	4	4	no	—
417	C₄	—	4	2	no	—
418	C₄	—	4	2	no	—
419	C₄	—	8	1	no	—
420	C₄	—	8	2	no	—
421	C₄	C₄×SD₁₆×C₈₉	1	4	yes	—
422	C₄	C₂×C₄×Q₈×C₈₉	1	1	yes	characteristic
423	C₄	#520×C₈₉	1	1	yes	characteristic
424	C₄	#552×C₈₉	1	1	yes	characteristic
425	C₄	#554×C₈₉	1	1	yes	characteristic
426	C₂	C₈₉×#2360	1	1	yes	characteristic
427	C₂	C₂×C₄×SD₁₆×C₈₉	1	1	yes	characteristic
428	C₂	C₈₉×#2947	1	1	yes	characteristic
429	C₂	C₈₉×#3298	1	1	yes	characteristic
430	C₂	C₈₉×#3304	1	1	yes	characteristic
431	C₂	Q₈×SD₁₆×C₈₉	1	1	yes	characteristic
432	C₂	C₈₉×#3383	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₈₉	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₈₉
Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
LCS	G ▷ C₂×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
16. #475×C₈₉	C₄	semidirect
79. #103×C₈₉	Q₈	semidirect
148. Q₈×C₈₉	C₄×Q₈	semidirect
209. #22964	C₈₉	direct (N × H)
228. C₈₉	#22964	direct (N × H)
233. #475	C₄×C₈₉	semidirect
296. #103	Q₈×C₈₉	semidirect
365. Q₈	C₄×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×#475×C₈₉	2
C₈₉×#2365	2
Q₈×C₄⋊C₄×C₈₉	2
C₈₉×#1990	2
C₈₉×#1948	2
C₈₉×#1941	2
C₈₉×#1920	2
C₈₉×#1865	2
C₈×C₄⋊C₄×C₈₉	2
C₄×#103×C₈₉	2
#22964	89

Subgroups — C89×#22964 #36400812

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#22964 #36400812