Subgroups — C₈₉×#36566 #36414414

Order 22784.

1790 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2864	C₂	1	2	yes	maximal, complemented
2	C₈₉×#2862	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2589	C₂	1	4	yes	maximal, complemented
4	C₈₉×#2581	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2580	C₂	1	1	yes	maximal, characteristic
6	C₈₉×#2574	C₂	1	4	yes	maximal, complemented
7	C₈₉×#2465	C₂	1	4	yes	maximal, complemented
8	C₈₉×#2452	C₂	1	2	yes	maximal
9	C₈₉×#2451	C₂	1	2	yes	maximal, complemented
10	C₈₉×#2368	C₂	1	4	yes	maximal, complemented
11	C₈₉×#2351	C₂	1	1	yes	maximal, characteristic, complemented
12	C₈₉×#2349	C₂	1	1	yes	maximal, characteristic
13	C₄×#124×C₈₉	C₂	1	1	yes	maximal, characteristic
14	C₄×#124×C₈₉	C₂	1	2	yes	maximal
15	C₄²×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
17	C₂×#124×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	4	yes	—
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	2	yes	—
22	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
25	#400×C₈₉	C₂²	1	2	yes	—
26	#399×C₈₉	C₂²	1	2	yes	—
27	#398×C₈₉	C₂²	1	4	yes	—
28	#396×C₈₉	C₂²	1	4	yes	—
29	#395×C₈₉	C₂²	1	4	yes	—
30	#395×C₈₉	C₂²	1	2	yes	—
31	#393×C₈₉	C₂²	1	4	yes	—
32	#393×C₈₉	C₂²	1	2	yes	—
33	#391×C₈₉	C₂²	1	2	yes	—
34	#391×C₈₉	C₂²	1	1	yes	characteristic
35	#391×C₈₉	C₂²	1	4	yes	—
36	#391×C₈₉	C₂²	1	4	yes	—
37	#391×C₈₉	C₂²	1	4	yes	—
38	#390×C₈₉	C₂²	1	2	yes	—
39	#390×C₈₉	C₂²	1	1	yes	characteristic
40	#389×C₈₉	C₂²	1	2	yes	—
41	#389×C₈₉	C₂²	1	2	yes	—
42	#389×C₈₉	C₂²	1	4	yes	—
43	#389×C₈₉	C₂²	1	2	yes	—
44	#389×C₈₉	C₂²	1	2	yes	—
45	#388×C₈₉	C₂²	1	4	yes	—
46	#388×C₈₉	C₂²	1	4	yes	—
47	#388×C₈₉	C₂²	1	4	yes	—
48	#387×C₈₉	C₂²	1	4	yes	—
49	#387×C₈₉	C₂²	1	4	yes	—
50	#387×C₈₉	C₂²	1	4	yes	—
51	#387×C₈₉	C₂²	1	4	yes	—
52	#387×C₈₉	C₂²	1	4	yes	—
53	#387×C₈₉	C₂²	1	4	yes	—
54	#385×C₈₉	C₂²	1	4	yes	—
55	#385×C₈₉	C₂²	1	4	yes	—
56	#385×C₈₉	C₂²	1	4	yes	—
57	#382×C₈₉	C₂²	1	2	yes	—
58	#382×C₈₉	C₂²	1	1	yes	characteristic
59	#382×C₈₉	C₂²	1	2	yes	—
60	#382×C₈₉	C₂²	1	1	yes	characteristic
61	#382×C₈₉	C₂²	1	1	yes	characteristic
62	#380×C₈₉	C₂²	1	2	yes	—
63	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
64	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
65	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
66	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
67	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
68	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
69	C₄×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
70	C₄×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
71	C₄×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
72	C₂×#95×C₈₉	C₂²	1	2	yes	—
73	C₄³×C₈₉	C₂²	1	1	yes	characteristic
74	C₄³×C₈₉	C₂²	1	2	yes	—
75	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
76	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
77	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
78	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
79	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
80	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
81	C₄⋊Q₈×C₈₉	—	4	4	no	—
82	#124×C₈₉	Q₈	1	8	yes	complemented
83	C₂²⋊Q₈×C₈₉	—	2	16	no	—
84	C₄×Q₈×C₈₉	—	2	8	no	—
85	C₄×Q₈×C₈₉	—	2	4	no	—
86	C₄×Q₈×C₈₉	—	2	4	no	—
87	C₄×Q₈×C₈₉	—	4	4	no	—
88	C₄×D₄×C₈₉	—	2	4	no	—
89	C₄×D₄×C₈₉	—	4	4	no	—
90	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
91	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
92	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
93	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
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	4	yes	—
100	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
101	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
102	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
103	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
104	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
105	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
106	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
107	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
108	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
109	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
110	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
111	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
112	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
113	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
114	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
116	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
117	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
118	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
119	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
120	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
121	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
122	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
123	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
124	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
125	#95×C₈₉	C₂³	1	1	yes	characteristic
126	#95×C₈₉	C₂³	1	2	yes	—
127	#95×C₈₉	C₂³	1	4	yes	—
128	#95×C₈₉	C₂³	1	4	yes	—
129	#95×C₈₉	C₂³	1	4	yes	—
130	#95×C₈₉	C₂³	1	4	yes	—
131	#95×C₈₉	C₂³	1	4	yes	—
132	#95×C₈₉	C₂³	1	4	yes	—
133	#95×C₈₉	C₂³	1	4	yes	—
134	#95×C₈₉	C₂³	1	1	yes	characteristic
135	#95×C₈₉	C₂³	1	2	yes	—
136	#95×C₈₉	C₂³	1	2	yes	—
137	#95×C₈₉	C₂³	1	2	yes	—
138	#95×C₈₉	C₂³	1	2	yes	—
139	#95×C₈₉	C₂³	1	4	yes	—
140	#95×C₈₉	C₂³	1	2	yes	—
141	#95×C₈₉	C₂³	1	4	yes	—
142	#95×C₈₉	C₂³	1	4	yes	—
143	C₂⁴×C₈₉	C₂⁴	1	2	yes	—
144	C₂×Q₈×C₈₉	—	2	4	no	—
145	C₂×Q₈×C₈₉	—	2	2	no	—
146	C₂×Q₈×C₈₉	—	2	2	no	—
147	C₂×Q₈×C₈₉	—	2	8	no	—
148	C₂×Q₈×C₈₉	—	2	2	no	—
149	C₂×Q₈×C₈₉	—	4	4	no	—
150	C₂×Q₈×C₈₉	—	4	4	no	—
151	C₂×Q₈×C₈₉	C₂×Q₈	1	4	yes	—
152	C₂×D₄×C₈₉	—	2	2	no	—
153	C₂×D₄×C₈₉	—	2	2	no	—
154	C₂×D₄×C₈₉	C₂×Q₈	1	4	yes	—
155	C₂²×C₄×C₈₉	—	2	4	no	—
156	C₂²×C₄×C₈₉	—	2	8	no	—
157	C₂²×C₄×C₈₉	—	4	4	no	—
158	C₂²×C₄×C₈₉	—	4	4	no	—
159	C₂²×C₄×C₈₉	—	4	4	no	—
160	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
162	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
163	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
164	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
165	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
166	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
167	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
169	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
170	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
171	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
172	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
173	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
174	C₄⋊C₄×C₈₉	—	2	2	no	—
175	C₄⋊C₄×C₈₉	—	2	4	no	—
176	C₄⋊C₄×C₈₉	—	2	4	no	—
177	C₄⋊C₄×C₈₉	—	2	4	no	—
178	C₄⋊C₄×C₈₉	—	2	4	no	—
179	C₄⋊C₄×C₈₉	—	2	2	no	—
180	C₄⋊C₄×C₈₉	—	2	2	no	—
181	C₄⋊C₄×C₈₉	—	2	8	no	—
182	C₄⋊C₄×C₈₉	—	2	8	no	—
183	C₄⋊C₄×C₈₉	—	2	4	no	—
184	C₄⋊C₄×C₈₉	—	2	4	no	—
185	C₄⋊C₄×C₈₉	—	4	4	no	—
186	C₄⋊C₄×C₈₉	—	4	4	no	—
187	C₄⋊C₄×C₈₉	—	4	4	no	—
188	C₄⋊C₄×C₈₉	—	4	4	no	—
189	C₄⋊C₄×C₈₉	#41	1	8	yes	—
190	C₂²⋊C₄×C₈₉	—	2	4	no	—
191	C₂²⋊C₄×C₈₉	—	2	8	no	—
192	C₂²⋊C₄×C₈₉	—	2	8	no	—
193	C₂²⋊C₄×C₈₉	—	4	4	no	—
194	C₂²⋊C₄×C₈₉	—	4	4	no	—
195	C₂²⋊C₄×C₈₉	C₂×Q₈	1	16	yes	—
196	C₄²×C₈₉	—	2	4	no	—
197	C₄²×C₈₉	—	2	4	no	—
198	C₄²×C₈₉	—	2	4	no	—
199	C₄²×C₈₉	—	2	4	no	—
200	C₄²×C₈₉	—	2	4	no	—
201	C₄²×C₈₉	—	2	4	no	—
202	C₄²×C₈₉	—	2	2	no	—
203	C₄²×C₈₉	—	4	2	no	—
204	C₄²×C₈₉	—	4	2	no	—
205	C₄²×C₈₉	—	4	2	no	—
206	C₄²×C₈₉	—	4	4	no	—
207	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
208	C₄²×C₈₉	#41	1	4	yes	—
209	C₂³×C₈₉	—	2	4	no	—
210	C₂³×C₈₉	—	2	2	no	—
211	C₂³×C₈₉	—	2	2	no	—
212	C₂³×C₈₉	—	2	4	no	—
213	C₂³×C₈₉	C₂²×Q₈	1	4	yes	—
214	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
215	Q₈×C₈₉	—	2	8	no	—
216	Q₈×C₈₉	—	2	8	no	—
217	Q₈×C₈₉	—	8	8	no	—
218	D₄×C₈₉	—	2	8	no	—
219	C₂×C₄×C₈₉	—	2	2	no	—
220	C₂×C₄×C₈₉	—	2	2	no	—
221	C₂×C₄×C₈₉	—	2	2	no	—
222	C₂×C₄×C₈₉	—	2	2	no	—
223	C₂×C₄×C₈₉	—	2	2	no	—
224	C₂×C₄×C₈₉	—	2	2	no	—
225	C₂×C₄×C₈₉	—	2	2	no	—
226	C₂×C₄×C₈₉	—	2	2	no	—
227	C₂×C₄×C₈₉	—	2	2	no	—
228	C₂×C₄×C₈₉	—	2	2	no	—
229	C₂×C₄×C₈₉	—	2	2	no	—
230	C₂×C₄×C₈₉	—	2	2	no	—
231	C₂×C₄×C₈₉	—	2	2	no	—
232	C₂×C₄×C₈₉	—	2	2	no	—
233	C₂×C₄×C₈₉	—	2	2	no	—
234	C₂×C₄×C₈₉	—	2	4	no	—
235	C₂×C₄×C₈₉	—	2	2	no	—
236	C₂×C₄×C₈₉	—	2	4	no	—
237	C₂×C₄×C₈₉	—	2	4	no	—
238	C₂×C₄×C₈₉	—	2	2	no	—
239	C₂×C₄×C₈₉	—	2	4	no	—
240	C₂×C₄×C₈₉	—	2	4	no	—
241	C₂×C₄×C₈₉	—	2	2	no	—
242	C₂×C₄×C₈₉	—	2	2	no	—
243	C₂×C₄×C₈₉	—	2	2	no	—
244	C₂×C₄×C₈₉	—	2	4	no	—
245	C₂×C₄×C₈₉	—	2	4	no	—
246	C₂×C₄×C₈₉	—	2	4	no	—
247	C₂×C₄×C₈₉	—	8	4	no	—
248	C₂×C₄×C₈₉	—	8	8	no	—
249	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
250	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
251	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
252	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
253	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
254	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
255	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
256	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
257	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
258	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
259	C₂²×C₈₉	—	2	4	no	—
260	C₂²×C₈₉	—	2	8	no	—
261	C₂²×C₈₉	—	4	2	no	—
262	C₂²×C₈₉	—	4	2	no	—
263	C₂²×C₈₉	—	4	4	no	—
264	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
265	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
266	C₂²×C₈₉	C₂²×#41	1	2	yes	—
267	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
268	C₂²×C₈₉	#585	1	2	yes	—
269	C₄×C₈₉	—	2	4	no	—
270	C₄×C₈₉	—	2	2	no	—
271	C₄×C₈₉	—	2	4	no	—
272	C₄×C₈₉	—	2	4	no	—
273	C₄×C₈₉	—	2	2	no	—
274	C₄×C₈₉	—	2	4	no	—
275	C₄×C₈₉	—	2	4	no	—
276	C₄×C₈₉	—	4	2	no	—
277	C₄×C₈₉	—	4	2	no	—
278	C₄×C₈₉	—	4	2	no	—
279	C₄×C₈₉	—	4	2	no	—
280	C₄×C₈₉	—	4	4	no	—
281	C₄×C₈₉	—	4	4	no	—
282	C₄×C₈₉	#554	1	4	yes	—
283	#36566	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
284	C₂×C₈₉	—	4	4	no	—
285	C₂×C₈₉	#3444	1	1	yes	characteristic
286	C₂×C₈₉	C₂×#554	1	1	yes	characteristic
287	C₂×C₈₉	Q₈×#41	1	2	yes	—
288	C₂×C₈₉	#3510	1	1	yes	characteristic
289	C₂×C₈₉	#3514	1	2	yes	—
290	#2864	C₂×C₈₉	1	2	yes	complemented
291	#2862	C₂×C₈₉	1	1	yes	characteristic, complemented
292	#2589	C₂×C₈₉	1	4	yes	complemented
293	#2581	C₂×C₈₉	1	1	yes	characteristic, complemented
294	#2580	C₂×C₈₉	1	1	yes	characteristic
295	#2574	C₂×C₈₉	1	4	yes	complemented
296	#2465	C₂×C₈₉	1	4	yes	complemented
297	#2452	C₂×C₈₉	1	2	yes	—
298	#2451	C₂×C₈₉	1	2	yes	complemented
299	#2368	C₂×C₈₉	1	4	yes	complemented
300	#2351	C₂×C₈₉	1	1	yes	characteristic, complemented
301	#2349	C₂×C₈₉	1	1	yes	characteristic
302	C₄×#124	C₂×C₈₉	1	1	yes	characteristic
303	C₄×#124	C₂×C₈₉	1	2	yes	—
304	C₄²×Q₈	C₂×C₈₉	1	1	yes	characteristic, complemented
305	C₈₉	#36566	1	1	yes	characteristic, direct product factor, Sylow
306	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
307	C₂×#124	C₂²×C₈₉	1	1	yes	characteristic
308	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	4	yes	—
309	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
310	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
311	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
312	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
313	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
314	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
315	#400	C₂²×C₈₉	1	2	yes	—
316	#399	C₂²×C₈₉	1	2	yes	—
317	#398	C₂²×C₈₉	1	4	yes	—
318	#396	C₂²×C₈₉	1	4	yes	—
319	#395	C₂²×C₈₉	1	4	yes	—
320	#395	C₂²×C₈₉	1	2	yes	—
321	#393	C₂²×C₈₉	1	4	yes	—
322	#393	C₂²×C₈₉	1	2	yes	—
323	#391	C₂²×C₈₉	1	4	yes	—
324	#391	C₂²×C₈₉	1	2	yes	—
325	#391	C₂²×C₈₉	1	4	yes	—
326	#391	C₂²×C₈₉	1	4	yes	—
327	#391	C₂²×C₈₉	1	1	yes	characteristic
328	#390	C₂²×C₈₉	1	2	yes	—
329	#390	C₂²×C₈₉	1	1	yes	characteristic
330	#389	C₂²×C₈₉	1	2	yes	—
331	#389	C₂²×C₈₉	1	2	yes	—
332	#389	C₂²×C₈₉	1	4	yes	—
333	#389	C₂²×C₈₉	1	2	yes	—
334	#389	C₂²×C₈₉	1	2	yes	—
335	#388	C₂²×C₈₉	1	4	yes	—
336	#388	C₂²×C₈₉	1	4	yes	—
337	#388	C₂²×C₈₉	1	4	yes	—
338	#387	C₂²×C₈₉	1	4	yes	—
339	#387	C₂²×C₈₉	1	4	yes	—
340	#387	C₂²×C₈₉	1	4	yes	—
341	#387	C₂²×C₈₉	1	4	yes	—
342	#387	C₂²×C₈₉	1	4	yes	—
343	#387	C₂²×C₈₉	1	4	yes	—
344	#385	C₂²×C₈₉	1	4	yes	—
345	#385	C₂²×C₈₉	1	4	yes	—
346	#385	C₂²×C₈₉	1	4	yes	—
347	#382	C₂²×C₈₉	1	2	yes	—
348	#382	C₂²×C₈₉	1	1	yes	characteristic
349	#382	C₂²×C₈₉	1	1	yes	characteristic
350	#382	C₂²×C₈₉	1	2	yes	—
351	#382	C₂²×C₈₉	1	1	yes	characteristic
352	#380	C₂²×C₈₉	1	2	yes	—
353	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
354	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
355	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
356	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
357	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
358	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
359	C₄×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
360	C₄×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
361	C₄×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
362	C₂×#95	C₂²×C₈₉	1	2	yes	—
363	C₄³	C₂²×C₈₉	1	1	yes	characteristic
364	C₄³	C₂²×C₈₉	1	2	yes	—
365	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
366	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
367	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
368	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
369	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
370	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
371	C₄⋊Q₈	—	4	4	no	—
372	#124	Q₈×C₈₉	1	8	yes	complemented
373	C₂²⋊Q₈	—	2	16	no	—
374	C₄×Q₈	—	2	8	no	—
375	C₄×Q₈	—	2	4	no	—
376	C₄×Q₈	—	2	4	no	—
377	C₄×Q₈	—	4	4	no	—
378	C₄×D₄	—	2	4	no	—
379	C₄×D₄	—	4	4	no	—
380	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
381	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
382	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
383	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
384	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
385	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
386	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
387	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
388	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
389	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
390	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
391	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
392	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
393	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
394	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
395	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
396	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
397	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
398	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
399	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
400	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
401	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
402	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
403	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
404	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
405	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
406	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
407	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
408	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
409	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
410	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
411	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
412	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
413	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
414	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
415	#95	C₂³×C₈₉	1	1	yes	characteristic
416	#95	C₂³×C₈₉	1	2	yes	—
417	#95	C₂³×C₈₉	1	4	yes	—
418	#95	C₂³×C₈₉	1	4	yes	—
419	#95	C₂³×C₈₉	1	4	yes	—
420	#95	C₂³×C₈₉	1	4	yes	—
421	#95	C₂³×C₈₉	1	4	yes	—
422	#95	C₂³×C₈₉	1	4	yes	—
423	#95	C₂³×C₈₉	1	4	yes	—
424	#95	C₂³×C₈₉	1	1	yes	characteristic
425	#95	C₂³×C₈₉	1	2	yes	—
426	#95	C₂³×C₈₉	1	2	yes	—
427	#95	C₂³×C₈₉	1	2	yes	—
428	#95	C₂³×C₈₉	1	2	yes	—
429	#95	C₂³×C₈₉	1	4	yes	—
430	#95	C₂³×C₈₉	1	2	yes	—
431	#95	C₂³×C₈₉	1	4	yes	—
432	#95	C₂³×C₈₉	1	4	yes	—
433	C₂⁴	C₂⁴×C₈₉	1	2	yes	—
434	C₂×Q₈	—	2	4	no	—
435	C₂×Q₈	—	2	2	no	—
436	C₂×Q₈	—	2	2	no	—
437	C₂×Q₈	—	2	8	no	—
438	C₂×Q₈	—	2	2	no	—
439	C₂×Q₈	—	4	4	no	—
440	C₂×Q₈	—	4	4	no	—
441	C₂×Q₈	C₂×Q₈×C₈₉	1	4	yes	—
442	C₂×D₄	—	2	2	no	—
443	C₂×D₄	—	2	2	no	—
444	C₂×D₄	C₂×Q₈×C₈₉	1	4	yes	—
445	C₂²×C₄	—	2	4	no	—
446	C₂²×C₄	—	2	8	no	—
447	C₂²×C₄	—	4	4	no	—
448	C₂²×C₄	—	4	4	no	—
449	C₂²×C₄	—	4	4	no	—
450	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
451	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
452	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
453	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
454	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
455	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
456	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
457	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
458	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
459	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
460	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
461	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
462	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
463	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
464	C₄⋊C₄	—	2	2	no	—
465	C₄⋊C₄	—	2	4	no	—
466	C₄⋊C₄	—	2	4	no	—
467	C₄⋊C₄	—	2	4	no	—
468	C₄⋊C₄	—	2	4	no	—
469	C₄⋊C₄	—	2	2	no	—
470	C₄⋊C₄	—	2	2	no	—
471	C₄⋊C₄	—	2	8	no	—
472	C₄⋊C₄	—	2	8	no	—
473	C₄⋊C₄	—	2	4	no	—
474	C₄⋊C₄	—	2	4	no	—
475	C₄⋊C₄	—	4	4	no	—
476	C₄⋊C₄	—	4	4	no	—
477	C₄⋊C₄	—	4	4	no	—
478	C₄⋊C₄	—	4	4	no	—
479	C₄⋊C₄	#41×C₈₉	1	8	yes	—
480	C₂²⋊C₄	—	2	4	no	—
481	C₂²⋊C₄	—	2	8	no	—
482	C₂²⋊C₄	—	2	8	no	—
483	C₂²⋊C₄	—	4	4	no	—
484	C₂²⋊C₄	—	4	4	no	—
485	C₂²⋊C₄	C₂×Q₈×C₈₉	1	16	yes	—
486	C₄²	—	2	4	no	—
487	C₄²	—	2	4	no	—
488	C₄²	—	2	4	no	—
489	C₄²	—	2	4	no	—
490	C₄²	—	2	4	no	—
491	C₄²	—	2	4	no	—
492	C₄²	—	2	2	no	—
493	C₄²	—	4	4	no	—
494	C₄²	—	4	2	no	—
495	C₄²	—	4	2	no	—
496	C₄²	—	4	2	no	—
497	C₄²	C₂×Q₈×C₈₉	1	4	yes	—
498	C₄²	#41×C₈₉	1	4	yes	—
499	C₂³	—	2	4	no	—
500	C₂³	—	2	2	no	—
501	C₂³	—	2	2	no	—
502	C₂³	—	2	4	no	—
503	C₂³	C₂²×Q₈×C₈₉	1	4	yes	—
504	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
505	Q₈	—	2	8	no	—
506	Q₈	—	2	8	no	—
507	Q₈	—	8	8	no	—
508	D₄	—	2	8	no	—
509	C₂×C₄	—	2	2	no	—
510	C₂×C₄	—	2	2	no	—
511	C₂×C₄	—	2	2	no	—
512	C₂×C₄	—	2	2	no	—
513	C₂×C₄	—	2	2	no	—
514	C₂×C₄	—	2	2	no	—
515	C₂×C₄	—	2	2	no	—
516	C₂×C₄	—	2	2	no	—
517	C₂×C₄	—	2	2	no	—
518	C₂×C₄	—	2	2	no	—
519	C₂×C₄	—	2	2	no	—
520	C₂×C₄	—	2	2	no	—
521	C₂×C₄	—	2	2	no	—
522	C₂×C₄	—	2	2	no	—
523	C₂×C₄	—	2	2	no	—
524	C₂×C₄	—	2	4	no	—
525	C₂×C₄	—	2	2	no	—
526	C₂×C₄	—	2	4	no	—
527	C₂×C₄	—	2	4	no	—
528	C₂×C₄	—	2	2	no	—
529	C₂×C₄	—	2	4	no	—
530	C₂×C₄	—	2	4	no	—
531	C₂×C₄	—	2	2	no	—
532	C₂×C₄	—	2	2	no	—
533	C₂×C₄	—	2	2	no	—
534	C₂×C₄	—	2	4	no	—
535	C₂×C₄	—	2	4	no	—
536	C₂×C₄	—	2	4	no	—
537	C₂×C₄	—	8	4	no	—
538	C₂×C₄	—	8	8	no	—
539	C₂×C₄	C₂²×Q₈×C₈₉	1	4	yes	—
540	C₂×C₄	C₂²×Q₈×C₈₉	1	4	yes	—
541	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
542	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
543	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
544	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
545	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
546	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
547	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
548	C₂×C₄	H(5,𝔽₂)×C₈₉	1	2	yes	—
549	C₂²	—	2	4	no	—
550	C₂²	—	2	8	no	—
551	C₂²	—	4	2	no	—
552	C₂²	—	4	2	no	—
553	C₂²	—	4	4	no	—
554	C₂²	C₂³×Q₈×C₈₉	1	1	yes	characteristic
555	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
556	C₂²	C₂²×#41×C₈₉	1	2	yes	—
557	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
558	C₂²	#585×C₈₉	1	2	yes	—
559	C₄	—	2	4	no	—
560	C₄	—	2	2	no	—
561	C₄	—	2	4	no	—
562	C₄	—	2	4	no	—
563	C₄	—	2	2	no	—
564	C₄	—	2	4	no	—
565	C₄	—	2	4	no	—
566	C₄	—	4	2	no	—
567	C₄	—	4	2	no	—
568	C₄	—	4	2	no	—
569	C₄	—	4	4	no	—
570	C₄	—	4	4	no	—
571	C₄	—	4	2	no	—
572	C₄	#554×C₈₉	1	4	yes	—
573	C₂	—	4	4	no	—
574	C₂	C₈₉×#3444	1	1	yes	characteristic
575	C₂	C₂×#554×C₈₉	1	1	yes	characteristic
576	C₂	Q₈×#41×C₈₉	1	2	yes	—
577	C₂	C₈₉×#3510	1	1	yes	characteristic
578	C₂	C₈₉×#3514	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂²×D₄×C₈₉	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₈₉×#2864	C₂	semidirect
2. C₈₉×#2862	C₂	semidirect
3. C₈₉×#2589	C₂	semidirect
4. C₈₉×#2581	C₂	semidirect
6. C₈₉×#2574	C₂	semidirect
7. C₈₉×#2465	C₂	semidirect
9. C₈₉×#2451	C₂	semidirect
10. C₈₉×#2368	C₂	semidirect
11. C₈₉×#2351	C₂	semidirect
15. C₄²×Q₈×C₈₉	C₂	semidirect
82. #124×C₈₉	Q₈	semidirect
283. #36566	C₈₉	direct (N × H)
290. #2864	C₂×C₈₉	semidirect
291. #2862	C₂×C₈₉	semidirect
292. #2589	C₂×C₈₉	semidirect
293. #2581	C₂×C₈₉	semidirect
295. #2574	C₂×C₈₉	semidirect
296. #2465	C₂×C₈₉	semidirect
298. #2451	C₂×C₈₉	semidirect
299. #2368	C₂×C₈₉	semidirect
300. #2351	C₂×C₈₉	semidirect
304. C₄²×Q₈	C₂×C₈₉	semidirect
305. C₈₉	#36566	direct (N × H)
372. #124	Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#2864	2
C₈₉×#2862	2
C₈₉×#2589	2
C₈₉×#2581	2
C₈₉×#2580	2
C₈₉×#2574	2
C₈₉×#2465	2
C₈₉×#2452	2
C₈₉×#2451	2
C₈₉×#2368	2
C₈₉×#2351	2
C₈₉×#2349	2
C₄×#124×C₈₉	2
C₄²×Q₈×C₈₉	2
#36566	89

Subgroups — C89×#36566 #36414414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#36566 #36414414