Subgroups — C₈₉×#32678 #36410526

Order 22784.

758 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2592	C₂	1	2	yes	maximal, complemented
2	C₈₉×#2055	C₂	1	2	yes	maximal, complemented
3	C₈₉×#2015	C₂	1	2	yes	maximal, complemented
4	C₈₉×#2012	C₂	1	2	yes	maximal
5	C₈₉×#1985	C₂	1	1	yes	maximal, characteristic, complemented
6	C₈₉×#1972	C₂	1	1	yes	maximal, characteristic
7	C₈₉×#1971	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#1927	C₂	1	2	yes	maximal, complemented
9	C₈₉×#1926	C₂	1	2	yes	maximal, complemented
10	C₂²×Q₁₆×C₈₉	C₂²	1	1	yes	characteristic
11	C₂²×SD₁₆×C₈₉	C₂²	1	2	yes	—
12	C₂²×D₈×C₈₉	C₂²	1	1	yes	characteristic
13	C₂×#124×C₈₉	C₂²	1	2	yes	—
14	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×#103×C₈₉	C₂²	1	2	yes	—
19	C₂×#103×C₈₉	C₂²	1	2	yes	—
20	C₂×#102×C₈₉	C₂²	1	2	yes	—
21	C₂×#102×C₈₉	C₂²	1	2	yes	—
22	C₂×#97×C₈₉	C₂²	1	1	yes	characteristic
23	#397×C₈₉	—	2	2	no	—
24	#393×C₈₉	—	2	2	no	—
25	#391×C₈₉	C₂²	1	2	yes	—
26	#391×C₈₉	C₂²	1	2	yes	—
27	#390×C₈₉	C₂²	1	2	yes	—
28	#390×C₈₉	C₂²	1	2	yes	—
29	#388×C₈₉	—	2	2	no	—
30	#388×C₈₉	—	2	2	no	—
31	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
32	#340×C₈₉	C₂²	1	2	yes	—
33	#336×C₈₉	C₂²	1	2	yes	—
34	#336×C₈₉	C₂²	1	2	yes	—
35	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
36	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
37	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
38	C₂×Q₁₆×C₈₉	—	2	2	no	—
39	C₂×Q₁₆×C₈₉	—	4	1	no	—
40	C₂×Q₁₆×C₈₉	—	4	1	no	—
41	C₂×SD₁₆×C₈₉	—	4	2	no	—
42	C₂×SD₁₆×C₈₉	—	4	2	no	—
43	C₂×SD₁₆×C₈₉	—	4	2	no	—
44	C₂×D₈×C₈₉	—	2	2	no	—
45	C₂×D₈×C₈₉	—	4	1	no	—
46	C₂×D₈×C₈₉	—	4	1	no	—
47	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
48	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
50	#124×C₈₉	—	4	2	no	—
51	#124×C₈₉	—	4	2	no	—
52	C₄×Q₈×C₈₉	—	4	1	no	—
53	C₄×Q₈×C₈₉	—	4	1	no	—
54	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
55	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
56	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
57	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
58	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
59	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
60	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
61	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
62	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
63	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
64	C₂×C₄²×C₈₉	—	2	2	no	—
65	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
66	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
67	#106×C₈₉	—	4	1	no	—
68	#106×C₈₉	—	4	1	no	—
69	C₄⋊C₈×C₈₉	—	4	1	no	—
70	#103×C₈₉	—	2	4	no	—
71	#103×C₈₉	—	4	2	no	—
72	#102×C₈₉	—	2	4	no	—
73	#102×C₈₉	—	4	2	no	—
74	#97×C₈₉	—	4	1	no	—
75	#95×C₈₉	—	2	2	no	—
76	#95×C₈₉	—	2	2	no	—
77	#95×C₈₉	—	2	2	no	—
78	#95×C₈₉	—	2	2	no	—
79	#95×C₈₉	—	2	2	no	—
80	C₂⁴×C₈₉	—	2	2	no	—
81	C₂×Q₈×C₈₉	—	2	2	no	—
82	C₂×Q₈×C₈₉	—	2	2	no	—
83	C₂×Q₈×C₈₉	—	2	2	no	—
84	C₂×Q₈×C₈₉	—	2	2	no	—
85	C₂×Q₈×C₈₉	—	2	2	no	—
86	C₂×Q₈×C₈₉	—	2	2	no	—
87	C₂×Q₈×C₈₉	—	4	1	no	—
88	C₂×Q₈×C₈₉	—	4	2	no	—
89	C₂×D₄×C₈₉	—	2	2	no	—
90	C₂×D₄×C₈₉	—	2	2	no	—
91	C₂×D₄×C₈₉	—	2	2	no	—
92	C₂×D₄×C₈₉	—	2	2	no	—
93	C₂×D₄×C₈₉	—	2	2	no	—
94	C₂×D₄×C₈₉	—	2	2	no	—
95	C₂²×C₄×C₈₉	—	2	2	no	—
96	C₂²×C₄×C₈₉	—	2	2	no	—
97	C₂²×C₄×C₈₉	—	2	2	no	—
98	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
100	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
102	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
103	Q₁₆×C₈₉	—	4	4	no	—
104	SD₁₆×C₈₉	—	4	8	no	—
105	D₈×C₈₉	—	4	4	no	—
106	C₂×C₈×C₈₉	—	2	1	no	—
107	C₂×C₈×C₈₉	—	2	1	no	—
108	C₂×C₈×C₈₉	—	2	1	no	—
109	C₂×C₈×C₈₉	—	2	1	no	—
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	#41	1	2	yes	—
114	C₂×C₈×C₈₉	#41	1	2	yes	—
115	C₄⋊C₄×C₈₉	—	2	2	no	—
116	C₄⋊C₄×C₈₉	—	2	2	no	—
117	C₄⋊C₄×C₈₉	—	2	2	no	—
118	C₄⋊C₄×C₈₉	—	2	2	no	—
119	C₄⋊C₄×C₈₉	—	4	1	no	—
120	C₄⋊C₄×C₈₉	—	4	2	no	—
121	C₂²⋊C₄×C₈₉	—	4	4	no	—
122	C₂²⋊C₄×C₈₉	—	4	4	no	—
123	C₂²⋊C₄×C₈₉	—	4	4	no	—
124	C₂²⋊C₄×C₈₉	—	4	4	no	—
125	C₂²⋊C₄×C₈₉	—	8	2	no	—
126	C₂²⋊C₄×C₈₉	—	8	2	no	—
127	C₄²×C₈₉	—	2	1	no	—
128	C₄²×C₈₉	—	2	1	no	—
129	C₄²×C₈₉	—	2	2	no	—
130	C₄²×C₈₉	—	2	2	no	—
131	C₄²×C₈₉	—	8	2	no	—
132	C₂³×C₈₉	—	4	2	no	—
133	C₂³×C₈₉	—	4	2	no	—
134	C₂³×C₈₉	—	4	2	no	—
135	C₂³×C₈₉	—	4	2	no	—
136	C₂³×C₈₉	—	4	2	no	—
137	C₂³×C₈₉	—	4	2	no	—
138	C₂³×C₈₉	—	4	2	no	—
139	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
140	Q₈×C₈₉	—	4	4	no	—
141	Q₈×C₈₉	—	4	2	no	—
142	Q₈×C₈₉	—	4	2	no	—
143	Q₈×C₈₉	—	4	2	no	—
144	Q₈×C₈₉	—	4	2	no	—
145	D₄×C₈₉	—	4	2	no	—
146	D₄×C₈₉	—	4	2	no	—
147	D₄×C₈₉	—	4	2	no	—
148	D₄×C₈₉	—	4	2	no	—
149	C₂×C₄×C₈₉	—	2	2	no	—
150	C₂×C₄×C₈₉	—	2	2	no	—
151	C₂×C₄×C₈₉	—	2	1	no	—
152	C₂×C₄×C₈₉	—	2	1	no	—
153	C₂×C₄×C₈₉	—	2	2	no	—
154	C₂×C₄×C₈₉	—	2	2	no	—
155	C₂×C₄×C₈₉	—	2	2	no	—
156	C₂×C₄×C₈₉	—	2	2	no	—
157	C₂×C₄×C₈₉	—	2	2	no	—
158	C₂×C₄×C₈₉	—	2	2	no	—
159	C₂×C₄×C₈₉	—	4	2	no	—
160	C₂×C₄×C₈₉	—	4	2	no	—
161	C₂×C₄×C₈₉	—	4	2	no	—
162	C₂×C₄×C₈₉	—	4	2	no	—
163	C₂×C₄×C₈₉	—	4	2	no	—
164	C₂×C₄×C₈₉	—	4	2	no	—
165	C₂×C₄×C₈₉	—	4	2	no	—
166	C₂×C₄×C₈₉	—	4	2	no	—
167	C₂×C₄×C₈₉	—	4	2	no	—
168	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
169	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
170	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
171	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
172	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
173	C₂×C₄×C₈₉	H(5,𝔽₂)	1	1	yes	characteristic
174	C₈×C₈₉	—	2	2	no	—
175	C₈×C₈₉	—	2	2	no	—
176	C₈×C₈₉	—	4	2	no	—
177	C₂²×C₈₉	—	8	2	no	—
178	C₂²×C₈₉	—	8	2	no	—
179	C₂²×C₈₉	—	8	2	no	—
180	C₂²×C₈₉	—	8	2	no	—
181	C₂²×C₈₉	—	8	2	no	—
182	C₂²×C₈₉	—	8	2	no	—
183	C₂²×C₈₉	—	8	2	no	—
184	C₂²×C₈₉	#534	1	1	yes	characteristic
185	C₂²×C₈₉	#550	1	1	yes	characteristic
186	C₂²×C₈₉	#550	1	1	yes	characteristic
187	C₂²×C₈₉	#575	1	2	yes	—
188	C₂²×C₈₉	#577	1	2	yes	—
189	C₄×C₈₉	—	4	2	no	—
190	C₄×C₈₉	—	4	1	no	—
191	C₄×C₈₉	—	4	2	no	—
192	C₄×C₈₉	—	4	1	no	—
193	C₄×C₈₉	—	8	2	no	—
194	C₄×C₈₉	—	8	2	no	—
195	C₄×C₈₉	—	8	2	no	—
196	C₄×C₈₉	#538	1	1	yes	characteristic
197	C₄×C₈₉	#538	1	1	yes	characteristic
198	C₄×C₈₉	#555	1	1	yes	characteristic
199	C₄×C₈₉	H(3,𝔽₄)	1	1	yes	characteristic
200	#32678	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
201	C₂×C₈₉	—	16	2	no	—
202	C₂×C₈₉	#2733	1	1	yes	characteristic
203	C₂×C₈₉	#3197	1	1	yes	characteristic
204	C₂×C₈₉	#3204	1	1	yes	characteristic
205	C₂×C₈₉	#3401	1	2	yes	—
206	C₂×C₈₉	#3401	1	2	yes	—
207	#2592	C₂×C₈₉	1	2	yes	complemented
208	#2055	C₂×C₈₉	1	2	yes	complemented
209	#2015	C₂×C₈₉	1	2	yes	complemented
210	#2012	C₂×C₈₉	1	2	yes	—
211	#1985	C₂×C₈₉	1	1	yes	characteristic, complemented
212	#1972	C₂×C₈₉	1	1	yes	characteristic
213	#1971	C₂×C₈₉	1	1	yes	characteristic, complemented
214	#1927	C₂×C₈₉	1	2	yes	complemented
215	#1926	C₂×C₈₉	1	2	yes	complemented
216	C₈₉	#32678	1	1	yes	characteristic, direct product factor, Sylow
217	C₂²×Q₁₆	C₂²×C₈₉	1	1	yes	characteristic
218	C₂²×SD₁₆	C₂²×C₈₉	1	2	yes	—
219	C₂²×D₈	C₂²×C₈₉	1	1	yes	characteristic
220	C₂×#124	C₂²×C₈₉	1	2	yes	—
221	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
222	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
223	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
224	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
225	C₂×#103	C₂²×C₈₉	1	2	yes	—
226	C₂×#103	C₂²×C₈₉	1	2	yes	—
227	C₂×#102	C₂²×C₈₉	1	2	yes	—
228	C₂×#102	C₂²×C₈₉	1	2	yes	—
229	C₂×#97	C₂²×C₈₉	1	1	yes	characteristic
230	#397	—	2	2	no	—
231	#393	—	2	2	no	—
232	#391	C₂²×C₈₉	1	2	yes	—
233	#391	C₂²×C₈₉	1	2	yes	—
234	#390	C₂²×C₈₉	1	2	yes	—
235	#390	C₂²×C₈₉	1	2	yes	—
236	#388	—	2	2	no	—
237	#388	—	2	2	no	—
238	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
239	#340	C₂²×C₈₉	1	2	yes	—
240	#336	C₂²×C₈₉	1	2	yes	—
241	#336	C₂²×C₈₉	1	2	yes	—
242	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
243	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
244	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
245	C₂×Q₁₆	—	2	2	no	—
246	C₂×Q₁₆	—	4	1	no	—
247	C₂×Q₁₆	—	4	1	no	—
248	C₂×SD₁₆	—	4	2	no	—
249	C₂×SD₁₆	—	4	2	no	—
250	C₂×SD₁₆	—	4	2	no	—
251	C₂×D₈	—	2	2	no	—
252	C₂×D₈	—	4	1	no	—
253	C₂×D₈	—	4	1	no	—
254	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
255	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
256	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
257	#124	—	4	2	no	—
258	#124	—	4	2	no	—
259	C₄×Q₈	—	4	1	no	—
260	C₄×Q₈	—	4	1	no	—
261	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
262	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
263	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
264	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
265	C₂×C₂²⋊C₄	—	2	2	no	—
266	C₂×C₂²⋊C₄	—	2	2	no	—
267	C₂×C₂²⋊C₄	—	2	2	no	—
268	C₂×C₂²⋊C₄	—	2	2	no	—
269	C₂×C₂²⋊C₄	—	2	2	no	—
270	C₂×C₂²⋊C₄	—	2	2	no	—
271	C₂×C₄²	—	2	2	no	—
272	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
273	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
274	#106	—	4	1	no	—
275	#106	—	4	1	no	—
276	C₄⋊C₈	—	4	1	no	—
277	#103	—	2	4	no	—
278	#103	—	4	2	no	—
279	#102	—	2	4	no	—
280	#102	—	4	2	no	—
281	#97	—	4	1	no	—
282	#95	—	2	2	no	—
283	#95	—	2	2	no	—
284	#95	—	2	2	no	—
285	#95	—	2	2	no	—
286	#95	—	2	2	no	—
287	C₂⁴	—	2	2	no	—
288	C₂×Q₈	—	2	2	no	—
289	C₂×Q₈	—	2	2	no	—
290	C₂×Q₈	—	2	2	no	—
291	C₂×Q₈	—	2	2	no	—
292	C₂×Q₈	—	2	2	no	—
293	C₂×Q₈	—	2	2	no	—
294	C₂×Q₈	—	4	1	no	—
295	C₂×Q₈	—	4	2	no	—
296	C₂×D₄	—	2	2	no	—
297	C₂×D₄	—	2	2	no	—
298	C₂×D₄	—	2	2	no	—
299	C₂×D₄	—	2	2	no	—
300	C₂×D₄	—	2	2	no	—
301	C₂×D₄	—	2	2	no	—
302	C₂²×C₄	—	2	2	no	—
303	C₂²×C₄	—	2	2	no	—
304	C₂²×C₄	—	2	2	no	—
305	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
306	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
307	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
308	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
309	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
310	Q₁₆	—	4	4	no	—
311	SD₁₆	—	4	8	no	—
312	D₈	—	4	4	no	—
313	C₂×C₈	—	2	1	no	—
314	C₂×C₈	—	2	1	no	—
315	C₂×C₈	—	2	1	no	—
316	C₂×C₈	—	2	1	no	—
317	C₂×C₈	—	2	2	no	—
318	C₂×C₈	—	2	2	no	—
319	C₂×C₈	—	2	2	no	—
320	C₂×C₈	#41×C₈₉	1	2	yes	—
321	C₂×C₈	#41×C₈₉	1	2	yes	—
322	C₄⋊C₄	—	2	2	no	—
323	C₄⋊C₄	—	2	2	no	—
324	C₄⋊C₄	—	2	2	no	—
325	C₄⋊C₄	—	2	2	no	—
326	C₄⋊C₄	—	4	1	no	—
327	C₄⋊C₄	—	4	2	no	—
328	C₂²⋊C₄	—	4	4	no	—
329	C₂²⋊C₄	—	4	4	no	—
330	C₂²⋊C₄	—	4	4	no	—
331	C₂²⋊C₄	—	4	4	no	—
332	C₂²⋊C₄	—	8	2	no	—
333	C₂²⋊C₄	—	8	2	no	—
334	C₄²	—	2	1	no	—
335	C₄²	—	2	1	no	—
336	C₄²	—	2	2	no	—
337	C₄²	—	2	2	no	—
338	C₄²	—	8	2	no	—
339	C₂³	—	4	2	no	—
340	C₂³	—	4	2	no	—
341	C₂³	—	4	2	no	—
342	C₂³	—	4	2	no	—
343	C₂³	—	4	2	no	—
344	C₂³	—	4	2	no	—
345	C₂³	—	4	2	no	—
346	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
347	Q₈	—	4	4	no	—
348	Q₈	—	4	2	no	—
349	Q₈	—	4	2	no	—
350	Q₈	—	4	2	no	—
351	Q₈	—	4	2	no	—
352	D₄	—	4	2	no	—
353	D₄	—	4	2	no	—
354	D₄	—	4	2	no	—
355	D₄	—	4	2	no	—
356	C₂×C₄	—	2	2	no	—
357	C₂×C₄	—	2	2	no	—
358	C₂×C₄	—	2	1	no	—
359	C₂×C₄	—	2	1	no	—
360	C₂×C₄	—	2	2	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	2	no	—
364	C₂×C₄	—	2	2	no	—
365	C₂×C₄	—	2	2	no	—
366	C₂×C₄	—	4	2	no	—
367	C₂×C₄	—	4	2	no	—
368	C₂×C₄	—	4	2	no	—
369	C₂×C₄	—	4	2	no	—
370	C₂×C₄	—	4	2	no	—
371	C₂×C₄	—	4	2	no	—
372	C₂×C₄	—	4	2	no	—
373	C₂×C₄	—	4	2	no	—
374	C₂×C₄	—	4	2	no	—
375	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
376	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
377	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
378	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
379	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
380	C₂×C₄	H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
381	C₈	—	2	2	no	—
382	C₈	—	2	2	no	—
383	C₈	—	4	2	no	—
384	C₂²	—	8	2	no	—
385	C₂²	—	8	2	no	—
386	C₂²	—	8	2	no	—
387	C₂²	—	8	2	no	—
388	C₂²	—	8	2	no	—
389	C₂²	—	8	2	no	—
390	C₂²	—	8	2	no	—
391	C₂²	#534×C₈₉	1	1	yes	characteristic
392	C₂²	#550×C₈₉	1	1	yes	characteristic
393	C₂²	#550×C₈₉	1	1	yes	characteristic
394	C₂²	#575×C₈₉	1	2	yes	—
395	C₂²	#577×C₈₉	1	2	yes	—
396	C₄	—	4	2	no	—
397	C₄	—	4	1	no	—
398	C₄	—	4	2	no	—
399	C₄	—	4	1	no	—
400	C₄	—	8	2	no	—
401	C₄	—	8	2	no	—
402	C₄	—	8	2	no	—
403	C₄	#538×C₈₉	1	1	yes	characteristic
404	C₄	#538×C₈₉	1	1	yes	characteristic
405	C₄	#555×C₈₉	1	1	yes	characteristic
406	C₄	H(3,𝔽₄)×C₈₉	1	1	yes	characteristic
407	C₂	—	16	2	no	—
408	C₂	C₈₉×#2733	1	1	yes	characteristic
409	C₂	C₈₉×#3197	1	1	yes	characteristic
410	C₂	C₈₉×#3204	1	1	yes	characteristic
411	C₂	C₈₉×#3401	1	2	yes	—
412	C₂	C₈₉×#3401	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂²×D₈×C₈₉	C₂²
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
1. C₈₉×#2592	C₂	semidirect
2. C₈₉×#2055	C₂	semidirect
3. C₈₉×#2015	C₂	semidirect
5. C₈₉×#1985	C₂	semidirect
7. C₈₉×#1971	C₂	semidirect
8. C₈₉×#1927	C₂	semidirect
9. C₈₉×#1926	C₂	semidirect
200. #32678	C₈₉	direct (N × H)
207. #2592	C₂×C₈₉	semidirect
208. #2055	C₂×C₈₉	semidirect
209. #2015	C₂×C₈₉	semidirect
211. #1985	C₂×C₈₉	semidirect
213. #1971	C₂×C₈₉	semidirect
214. #1927	C₂×C₈₉	semidirect
215. #1926	C₂×C₈₉	semidirect
216. C₈₉	#32678	direct (N × H)

Maximal subgroups

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

subgroup	index
C₈₉×#2592	2
C₈₉×#2055	2
C₈₉×#2015	2
C₈₉×#2012	2
C₈₉×#1985	2
C₈₉×#1972	2
C₈₉×#1971	2
C₈₉×#1927	2
C₈₉×#1926	2
#32678	89

Subgroups — C89×#32678 #36410526

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#32678 #36410526