Subgroups — C₈₉×#35279 #36413127

Order 22784.

2606 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#41×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2857	C₂	1	4	yes	maximal, complemented
4	C₈₉×#2628	C₂	1	4	yes	maximal, complemented
5	C₈₉×#2568	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2567	C₂	1	2	yes	maximal, complemented
7	C₈₉×#2563	C₂	1	2	yes	maximal, complemented
8	C₈₉×#2542	C₂	1	2	yes	maximal, complemented
9	C₈₉×#2487	C₂	1	4	yes	maximal, complemented
10	C₈₉×#2409	C₂	1	4	yes	maximal, complemented
11	C₈₉×#2317	C₂	1	2	yes	maximal, complemented
12	C₈₉×#2311	C₂	1	2	yes	maximal
13	C₈₉×#2285	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
15	C₂³×Q₈×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄⋊Q₈×C₈₉	—	2	8	no	—
17	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
19	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
21	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	8	yes	—
22	C₄×#41×C₈₉	—	2	8	no	—
23	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
24	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
28	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
29	C₂×#117×C₈₉	C₂²	1	2	yes	—
30	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×#117×C₈₉	C₂²	1	2	yes	—
32	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
33	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
34	C₂²×C₄²×C₈₉	C₂²	1	2	yes	—
35	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
36	#400×C₈₉	C₂²	1	8	yes	—
37	#399×C₈₉	C₂²	1	8	yes	—
38	#395×C₈₉	C₂²	1	8	yes	—
39	#393×C₈₉	C₂²	1	8	yes	—
40	#391×C₈₉	C₂²	1	4	yes	—
41	#391×C₈₉	C₂²	1	8	yes	—
42	#390×C₈₉	C₂²	1	4	yes	—
43	#389×C₈₉	C₂²	1	8	yes	—
44	#389×C₈₉	C₂²	1	8	yes	—
45	#386×C₈₉	C₂²	1	8	yes	—
46	#385×C₈₉	C₂²	1	8	yes	—
47	#384×C₈₉	C₂²	1	4	yes	—
48	#384×C₈₉	C₂²	1	2	yes	—
49	#382×C₈₉	C₂²	1	4	yes	—
50	#382×C₈₉	C₂²	1	2	yes	—
51	#380×C₈₉	C₂²	1	8	yes	—
52	#380×C₈₉	C₂²	1	8	yes	—
53	#380×C₈₉	C₂²	1	2	yes	—
54	#380×C₈₉	C₂²	1	2	yes	—
55	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
56	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
57	C₂×#41×C₈₉	—	2	4	no	—
58	C₂×#41×C₈₉	—	2	4	no	—
59	C₂×#41×C₈₉	Q₈	1	8	yes	complemented
60	C₂²×Q₈×C₈₉	—	2	8	no	—
61	C₂²×Q₈×C₈₉	—	4	8	no	—
62	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
63	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
64	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
65	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
66	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
67	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
68	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
69	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
70	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
72	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
73	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
74	C₄⋊Q₈×C₈₉	—	2	8	no	—
75	C₄⋊Q₈×C₈₉	—	2	8	no	—
76	C₄⋊Q₈×C₈₉	—	2	8	no	—
77	C₄⋊Q₈×C₈₉	—	2	8	no	—
78	C₂²⋊Q₈×C₈₉	—	4	16	no	—
79	C₄×Q₈×C₈₉	—	2	4	no	—
80	C₄×Q₈×C₈₉	—	2	4	no	—
81	C₄×D₄×C₈₉	—	2	8	no	—
82	C₄×D₄×C₈₉	—	2	8	no	—
83	C₄×D₄×C₈₉	—	2	4	no	—
84	C₄×D₄×C₈₉	—	2	4	no	—
85	#117×C₈₉	—	2	4	no	—
86	#117×C₈₉	—	2	8	no	—
87	#117×C₈₉	—	2	8	no	—
88	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
89	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
90	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
92	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
94	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
95	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
96	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
97	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
98	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
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	16	yes	—
102	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
103	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
104	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
105	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
106	C₂×C₂²⋊C₄×C₈₉	C₂³	1	16	yes	—
107	C₂×C₄²×C₈₉	—	2	8	no	—
108	C₂×C₄²×C₈₉	D₄	1	8	yes	—
109	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
110	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
111	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
112	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
113	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
115	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
116	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
117	#95×C₈₉	C₂³	1	4	yes	—
118	#95×C₈₉	C₂³	1	4	yes	—
119	#95×C₈₉	C₂³	1	16	yes	—
120	#95×C₈₉	C₂³	1	16	yes	—
121	C₂⁴×C₈₉	C₂⁴	1	2	yes	—
122	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
123	#41×C₈₉	—	2	32	no	—
124	C₂×Q₈×C₈₉	—	2	2	no	—
125	C₂×Q₈×C₈₉	—	2	8	no	—
126	C₂×Q₈×C₈₉	—	2	8	no	—
127	C₂×Q₈×C₈₉	—	2	2	no	—
128	C₂×Q₈×C₈₉	—	2	16	no	—
129	C₂×Q₈×C₈₉	—	4	8	no	—
130	C₂×Q₈×C₈₉	—	4	32	no	—
131	C₂×Q₈×C₈₉	C₂×Q₈	1	4	yes	complemented
132	C₂×D₄×C₈₉	—	2	4	no	—
133	C₂×D₄×C₈₉	—	2	4	no	—
134	C₂×D₄×C₈₉	—	2	2	no	—
135	C₂×D₄×C₈₉	—	2	2	no	—
136	C₂×D₄×C₈₉	C₂×Q₈	1	8	yes	—
137	C₂×D₄×C₈₉	C₂×Q₈	1	4	yes	complemented
138	C₂²×C₄×C₈₉	—	2	4	no	—
139	C₂²×C₄×C₈₉	—	2	4	no	—
140	C₂²×C₄×C₈₉	—	2	4	no	—
141	C₂²×C₄×C₈₉	—	2	4	no	—
142	C₂²×C₄×C₈₉	—	2	8	no	—
143	C₂²×C₄×C₈₉	—	2	4	no	—
144	C₂²×C₄×C₈₉	—	2	4	no	—
145	C₂²×C₄×C₈₉	—	2	8	no	—
146	C₂²×C₄×C₈₉	—	2	8	no	—
147	C₂²×C₄×C₈₉	—	4	8	no	—
148	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
149	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
150	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
151	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
152	C₂²×C₄×C₈₉	C₂×Q₈	1	8	yes	complemented
153	C₂²×C₄×C₈₉	#41	1	4	yes	—
154	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
155	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
156	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
157	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
158	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
159	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
160	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
162	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
163	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
164	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
165	C₄⋊C₄×C₈₉	—	2	8	no	—
166	C₄⋊C₄×C₈₉	—	2	8	no	—
167	C₄⋊C₄×C₈₉	—	2	8	no	—
168	C₄⋊C₄×C₈₉	—	2	8	no	—
169	C₄⋊C₄×C₈₉	—	2	4	no	—
170	C₄⋊C₄×C₈₉	—	2	4	no	—
171	C₄⋊C₄×C₈₉	—	2	2	no	—
172	C₄⋊C₄×C₈₉	—	2	2	no	—
173	C₄⋊C₄×C₈₉	—	2	8	no	—
174	C₄⋊C₄×C₈₉	—	2	8	no	—
175	C₄⋊C₄×C₈₉	—	2	8	no	—
176	C₄⋊C₄×C₈₉	—	4	16	no	—
177	C₂²⋊C₄×C₈₉	—	2	4	no	—
178	C₂²⋊C₄×C₈₉	—	2	8	no	—
179	C₂²⋊C₄×C₈₉	—	2	4	no	—
180	C₂²⋊C₄×C₈₉	—	2	4	no	—
181	C₂²⋊C₄×C₈₉	—	4	16	no	—
182	C₄²×C₈₉	—	2	4	no	—
183	C₄²×C₈₉	—	2	4	no	—
184	C₄²×C₈₉	—	2	4	no	—
185	C₄²×C₈₉	—	2	8	no	—
186	C₄²×C₈₉	C₂×D₄	1	4	yes	—
187	C₄²×C₈₉	C₂×D₄	1	4	yes	—
188	C₄²×C₈₉	C₂×D₄	1	4	yes	—
189	C₄²×C₈₉	C₂×D₄	1	4	yes	—
190	C₂³×C₈₉	—	2	2	no	—
191	C₂³×C₈₉	—	2	2	no	—
192	C₂³×C₈₉	—	2	2	no	—
193	C₂³×C₈₉	—	2	2	no	—
194	C₂³×C₈₉	—	2	4	no	—
195	C₂³×C₈₉	—	2	4	no	—
196	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
197	C₂³×C₈₉	C₂²×Q₈	1	4	yes	—
198	C₂³×C₈₉	C₂²×Q₈	1	2	yes	—
199	C₂³×C₈₉	C₂×#41	1	2	yes	—
200	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
201	Q₈×C₈₉	—	2	8	no	—
202	Q₈×C₈₉	—	4	16	no	—
203	Q₈×C₈₉	—	4	16	no	—
204	D₄×C₈₉	—	2	16	no	—
205	D₄×C₈₉	—	2	8	no	—
206	C₂×C₄×C₈₉	—	2	2	no	—
207	C₂×C₄×C₈₉	—	2	2	no	—
208	C₂×C₄×C₈₉	—	2	2	no	—
209	C₂×C₄×C₈₉	—	2	2	no	—
210	C₂×C₄×C₈₉	—	2	16	no	—
211	C₂×C₄×C₈₉	—	2	8	no	—
212	C₂×C₄×C₈₉	—	2	8	no	—
213	C₂×C₄×C₈₉	—	2	2	no	—
214	C₂×C₄×C₈₉	—	2	2	no	—
215	C₂×C₄×C₈₉	—	2	8	no	—
216	C₂×C₄×C₈₉	—	2	8	no	—
217	C₂×C₄×C₈₉	—	2	8	no	—
218	C₂×C₄×C₈₉	—	2	16	no	—
219	C₂×C₄×C₈₉	—	2	4	no	—
220	C₂×C₄×C₈₉	—	2	8	no	—
221	C₂×C₄×C₈₉	—	2	4	no	—
222	C₂×C₄×C₈₉	—	4	16	no	—
223	C₂×C₄×C₈₉	—	4	8	no	—
224	C₂×C₄×C₈₉	—	4	8	no	—
225	C₂×C₄×C₈₉	C₂²⋊Q₈	1	16	yes	complemented
226	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
227	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
228	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
229	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
230	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
231	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
232	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
233	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
234	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
235	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
236	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
237	C₂×C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
238	C₂×C₄×C₈₉	2₋¹⁺⁴	1	4	yes	—
239	C₂²×C₈₉	—	2	2	no	—
240	C₂²×C₈₉	—	2	2	no	—
241	C₂²×C₈₉	—	2	4	no	—
242	C₂²×C₈₉	—	2	4	no	—
243	C₂²×C₈₉	—	2	4	no	—
244	C₂²×C₈₉	—	2	8	no	—
245	C₂²×C₈₉	—	4	2	no	—
246	C₂²×C₈₉	—	4	2	no	—
247	C₂²×C₈₉	—	4	4	no	—
248	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	4	yes	—
249	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
250	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
251	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
252	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
253	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	2	yes	—
254	C₄×C₈₉	—	2	4	no	—
255	C₄×C₈₉	—	2	2	no	—
256	C₄×C₈₉	—	2	2	no	—
257	C₄×C₈₉	—	4	2	no	—
258	C₄×C₈₉	—	4	8	no	—
259	C₄×C₈₉	—	4	2	no	—
260	C₄×C₈₉	—	4	8	no	—
261	C₄×C₈₉	C₂×C₂²⋊Q₈	1	4	yes	—
262	C₄×C₈₉	C₂×C₂²⋊Q₈	1	4	yes	—
263	C₄×C₈₉	#535	1	4	yes	—
264	C₄×C₈₉	#536	1	4	yes	—
265	#35279	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
266	C₂×C₈₉	—	2	4	no	—
267	C₂×C₈₉	—	4	4	no	—
268	C₂×C₈₉	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
269	C₂×C₈₉	C₂×#535	1	1	yes	characteristic
270	C₂×C₈₉	C₂×#536	1	1	yes	characteristic
271	C₂×C₈₉	#3503	1	2	yes	—
272	C₂×C₈₉	#3511	1	2	yes	—
273	C₂²×C₄⋊Q₈	C₂×C₈₉	1	1	yes	characteristic, complemented
274	C₂×C₄×#41	C₂×C₈₉	1	1	yes	characteristic
275	#2857	C₂×C₈₉	1	4	yes	complemented
276	#2628	C₂×C₈₉	1	4	yes	complemented
277	#2568	C₂×C₈₉	1	2	yes	complemented
278	#2567	C₂×C₈₉	1	2	yes	complemented
279	#2563	C₂×C₈₉	1	2	yes	complemented
280	#2542	C₂×C₈₉	1	2	yes	complemented
281	#2487	C₂×C₈₉	1	4	yes	complemented
282	#2409	C₂×C₈₉	1	4	yes	complemented
283	#2317	C₂×C₈₉	1	2	yes	complemented
284	#2311	C₂×C₈₉	1	2	yes	—
285	#2285	C₂×C₈₉	1	1	yes	characteristic, complemented
286	C₈₉	#35279	1	1	yes	characteristic, direct product factor, Sylow
287	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
288	C₂³×Q₈	C₂²×C₈₉	1	2	yes	—
289	C₂×C₄⋊Q₈	—	2	8	no	—
290	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
291	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
292	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
293	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
294	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	8	yes	—
295	C₄×#41	—	2	8	no	—
296	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
297	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
298	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
299	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
300	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
301	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
302	C₂×#117	C₂²×C₈₉	1	2	yes	—
303	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
304	C₂×#117	C₂²×C₈₉	1	2	yes	—
305	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
306	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
307	C₂²×C₄²	C₂²×C₈₉	1	2	yes	—
308	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
309	#400	C₂²×C₈₉	1	8	yes	—
310	#399	C₂²×C₈₉	1	8	yes	—
311	#395	C₂²×C₈₉	1	8	yes	—
312	#393	C₂²×C₈₉	1	8	yes	—
313	#391	C₂²×C₈₉	1	4	yes	—
314	#391	C₂²×C₈₉	1	8	yes	—
315	#390	C₂²×C₈₉	1	4	yes	—
316	#389	C₂²×C₈₉	1	8	yes	—
317	#389	C₂²×C₈₉	1	8	yes	—
318	#386	C₂²×C₈₉	1	8	yes	—
319	#385	C₂²×C₈₉	1	8	yes	—
320	#384	C₂²×C₈₉	1	4	yes	—
321	#384	C₂²×C₈₉	1	2	yes	—
322	#382	C₂²×C₈₉	1	4	yes	—
323	#382	C₂²×C₈₉	1	2	yes	—
324	#380	C₂²×C₈₉	1	8	yes	—
325	#380	C₂²×C₈₉	1	8	yes	—
326	#380	C₂²×C₈₉	1	2	yes	—
327	#380	C₂²×C₈₉	1	2	yes	—
328	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
329	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
330	C₂×#41	—	2	4	no	—
331	C₂×#41	—	2	4	no	—
332	C₂×#41	Q₈×C₈₉	1	8	yes	complemented
333	C₂²×Q₈	—	2	8	no	—
334	C₂²×Q₈	—	4	8	no	—
335	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
336	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
337	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
338	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
339	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
340	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
341	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
342	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
343	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
344	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
345	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
346	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
347	C₄⋊Q₈	—	2	8	no	—
348	C₄⋊Q₈	—	2	8	no	—
349	C₄⋊Q₈	—	2	8	no	—
350	C₄⋊Q₈	—	2	8	no	—
351	C₂²⋊Q₈	—	4	16	no	—
352	C₄×Q₈	—	2	4	no	—
353	C₄×Q₈	—	2	4	no	—
354	C₄×D₄	—	2	8	no	—
355	C₄×D₄	—	2	8	no	—
356	C₄×D₄	—	2	4	no	—
357	C₄×D₄	—	2	4	no	—
358	#117	—	2	4	no	—
359	#117	—	2	8	no	—
360	#117	—	2	8	no	—
361	C₂×C₄⋊C₄	—	2	8	no	—
362	C₂×C₄⋊C₄	—	2	8	no	—
363	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
364	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
365	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
366	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
367	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
368	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
369	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
370	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
371	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
372	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
373	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
374	C₂×C₄⋊C₄	C₂³×C₈₉	1	16	yes	—
375	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
376	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
377	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
378	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
379	C₂×C₂²⋊C₄	C₂³×C₈₉	1	16	yes	—
380	C₂×C₄²	—	2	8	no	—
381	C₂×C₄²	D₄×C₈₉	1	8	yes	—
382	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
383	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
384	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
385	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
386	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
387	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
388	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
389	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
390	#95	C₂³×C₈₉	1	4	yes	—
391	#95	C₂³×C₈₉	1	4	yes	—
392	#95	C₂³×C₈₉	1	16	yes	—
393	#95	C₂³×C₈₉	1	16	yes	—
394	C₂⁴	C₂⁴×C₈₉	1	2	yes	—
395	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
396	#41	—	2	32	no	—
397	C₂×Q₈	—	2	2	no	—
398	C₂×Q₈	—	2	8	no	—
399	C₂×Q₈	—	2	8	no	—
400	C₂×Q₈	—	2	2	no	—
401	C₂×Q₈	—	2	16	no	—
402	C₂×Q₈	—	4	8	no	—
403	C₂×Q₈	—	4	32	no	—
404	C₂×Q₈	C₂×Q₈×C₈₉	1	4	yes	complemented
405	C₂×D₄	—	2	4	no	—
406	C₂×D₄	—	2	4	no	—
407	C₂×D₄	—	2	2	no	—
408	C₂×D₄	—	2	2	no	—
409	C₂×D₄	C₂×Q₈×C₈₉	1	8	yes	—
410	C₂×D₄	C₂×Q₈×C₈₉	1	4	yes	complemented
411	C₂²×C₄	—	2	4	no	—
412	C₂²×C₄	—	2	4	no	—
413	C₂²×C₄	—	2	4	no	—
414	C₂²×C₄	—	2	4	no	—
415	C₂²×C₄	—	2	8	no	—
416	C₂²×C₄	—	2	4	no	—
417	C₂²×C₄	—	2	4	no	—
418	C₂²×C₄	—	2	8	no	—
419	C₂²×C₄	—	2	8	no	—
420	C₂²×C₄	—	4	8	no	—
421	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
422	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
423	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
424	C₂²×C₄	C₂×Q₈×C₈₉	1	4	yes	—
425	C₂²×C₄	C₂×Q₈×C₈₉	1	8	yes	complemented
426	C₂²×C₄	#41×C₈₉	1	4	yes	—
427	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
428	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
429	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
430	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
431	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
432	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
433	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
434	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
435	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
436	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
437	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
438	C₄⋊C₄	—	2	8	no	—
439	C₄⋊C₄	—	2	8	no	—
440	C₄⋊C₄	—	2	8	no	—
441	C₄⋊C₄	—	2	8	no	—
442	C₄⋊C₄	—	2	4	no	—
443	C₄⋊C₄	—	2	4	no	—
444	C₄⋊C₄	—	2	2	no	—
445	C₄⋊C₄	—	2	2	no	—
446	C₄⋊C₄	—	2	8	no	—
447	C₄⋊C₄	—	2	8	no	—
448	C₄⋊C₄	—	2	8	no	—
449	C₄⋊C₄	—	4	16	no	—
450	C₂²⋊C₄	—	2	4	no	—
451	C₂²⋊C₄	—	2	8	no	—
452	C₂²⋊C₄	—	2	4	no	—
453	C₂²⋊C₄	—	2	4	no	—
454	C₂²⋊C₄	—	4	16	no	—
455	C₄²	—	2	4	no	—
456	C₄²	—	2	4	no	—
457	C₄²	—	2	4	no	—
458	C₄²	—	2	8	no	—
459	C₄²	C₂×D₄×C₈₉	1	4	yes	—
460	C₄²	C₂×D₄×C₈₉	1	4	yes	—
461	C₄²	C₂×D₄×C₈₉	1	4	yes	—
462	C₄²	C₂×D₄×C₈₉	1	4	yes	—
463	C₂³	—	2	2	no	—
464	C₂³	—	2	2	no	—
465	C₂³	—	2	2	no	—
466	C₂³	—	2	2	no	—
467	C₂³	—	2	4	no	—
468	C₂³	—	2	4	no	—
469	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
470	C₂³	C₂²×Q₈×C₈₉	1	4	yes	—
471	C₂³	C₂²×Q₈×C₈₉	1	2	yes	—
472	C₂³	C₂×#41×C₈₉	1	2	yes	—
473	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
474	Q₈	—	2	8	no	—
475	Q₈	—	4	16	no	—
476	Q₈	—	4	16	no	—
477	D₄	—	2	16	no	—
478	D₄	—	2	8	no	—
479	C₂×C₄	—	2	2	no	—
480	C₂×C₄	—	2	2	no	—
481	C₂×C₄	—	2	2	no	—
482	C₂×C₄	—	2	2	no	—
483	C₂×C₄	—	2	16	no	—
484	C₂×C₄	—	2	8	no	—
485	C₂×C₄	—	2	8	no	—
486	C₂×C₄	—	2	2	no	—
487	C₂×C₄	—	2	2	no	—
488	C₂×C₄	—	2	8	no	—
489	C₂×C₄	—	2	8	no	—
490	C₂×C₄	—	2	8	no	—
491	C₂×C₄	—	2	16	no	—
492	C₂×C₄	—	2	4	no	—
493	C₂×C₄	—	2	8	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	4	16	no	—
496	C₂×C₄	—	4	8	no	—
497	C₂×C₄	—	4	8	no	—
498	C₂×C₄	C₂²⋊Q₈×C₈₉	1	16	yes	complemented
499	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
500	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
501	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
502	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
503	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
504	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
505	C₂×C₄	C₂²×Q₈×C₈₉	1	4	yes	—
506	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
507	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
508	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
509	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
510	C₂×C₄	H(5,𝔽₂)×C₈₉	1	4	yes	—
511	C₂×C₄	2₋¹⁺⁴×C₈₉	1	4	yes	—
512	C₂²	—	2	2	no	—
513	C₂²	—	2	2	no	—
514	C₂²	—	2	4	no	—
515	C₂²	—	2	4	no	—
516	C₂²	—	2	4	no	—
517	C₂²	—	2	8	no	—
518	C₂²	—	4	2	no	—
519	C₂²	—	4	2	no	—
520	C₂²	—	4	4	no	—
521	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	4	yes	—
522	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
523	C₂²	C₂³×Q₈×C₈₉	1	1	yes	characteristic
524	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
525	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
526	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	2	yes	—
527	C₄	—	2	4	no	—
528	C₄	—	2	2	no	—
529	C₄	—	2	2	no	—
530	C₄	—	4	8	no	—
531	C₄	—	4	2	no	—
532	C₄	—	4	2	no	—
533	C₄	—	4	8	no	—
534	C₄	C₂×C₂²⋊Q₈×C₈₉	1	4	yes	—
535	C₄	C₂×C₂²⋊Q₈×C₈₉	1	4	yes	—
536	C₄	#535×C₈₉	1	4	yes	—
537	C₄	#536×C₈₉	1	4	yes	—
538	C₂	—	2	4	no	—
539	C₂	—	4	4	no	—
540	C₂	C₂²×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
541	C₂	C₂×#535×C₈₉	1	1	yes	characteristic
542	C₂	C₂×#536×C₈₉	1	1	yes	characteristic
543	C₂	C₈₉×#3503	1	2	yes	—
544	C₂	C₈₉×#3511	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂²×#41×C₈₉	C₂²
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. C₂²×C₄⋊Q₈×C₈₉	C₂	semidirect
3. C₈₉×#2857	C₂	semidirect
4. C₈₉×#2628	C₂	semidirect
5. C₈₉×#2568	C₂	semidirect
6. C₈₉×#2567	C₂	semidirect
7. C₈₉×#2563	C₂	semidirect
8. C₈₉×#2542	C₂	semidirect
9. C₈₉×#2487	C₂	semidirect
10. C₈₉×#2409	C₂	semidirect
11. C₈₉×#2317	C₂	semidirect
13. C₈₉×#2285	C₂	semidirect
59. C₂×#41×C₈₉	Q₈	semidirect
131. C₂×Q₈×C₈₉	C₂×Q₈	semidirect
137. C₂×D₄×C₈₉	C₂×Q₈	semidirect
152. C₂²×C₄×C₈₉	C₂×Q₈	semidirect
225. C₂×C₄×C₈₉	C₂²⋊Q₈	semidirect
265. #35279	C₈₉	direct (N × H)
273. C₂²×C₄⋊Q₈	C₂×C₈₉	semidirect
275. #2857	C₂×C₈₉	semidirect
276. #2628	C₂×C₈₉	semidirect
277. #2568	C₂×C₈₉	semidirect
278. #2567	C₂×C₈₉	semidirect
279. #2563	C₂×C₈₉	semidirect
280. #2542	C₂×C₈₉	semidirect
281. #2487	C₂×C₈₉	semidirect
282. #2409	C₂×C₈₉	semidirect
283. #2317	C₂×C₈₉	semidirect
285. #2285	C₂×C₈₉	semidirect
286. C₈₉	#35279	direct (N × H)
332. C₂×#41	Q₈×C₈₉	semidirect
404. C₂×Q₈	C₂×Q₈×C₈₉	semidirect
410. C₂×D₄	C₂×Q₈×C₈₉	semidirect
425. C₂²×C₄	C₂×Q₈×C₈₉	semidirect
498. C₂×C₄	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
C₂²×C₄⋊Q₈×C₈₉	2
C₂×C₄×#41×C₈₉	2
C₈₉×#2857	2
C₈₉×#2628	2
C₈₉×#2568	2
C₈₉×#2567	2
C₈₉×#2563	2
C₈₉×#2542	2
C₈₉×#2487	2
C₈₉×#2409	2
C₈₉×#2317	2
C₈₉×#2311	2
C₈₉×#2285	2
#35279	89

Subgroups — C89×#35279 #36413127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35279 #36413127