Subgroups — C₈₉×#35284 #36413132

Order 22784.

1722 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#2868	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈₉×#2859	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2857	C₂	1	2	yes	maximal
4	C₈₉×#2647	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2629	C₂	1	2	yes	maximal
6	C₈₉×#2507	C₂	1	4	yes	maximal, complemented
7	C₈₉×#2497	C₂	1	4	yes	maximal
8	C₈₉×#2446	C₂	1	4	yes	maximal, complemented
9	C₈₉×#2429	C₂	1	4	yes	maximal
10	C₈₉×#2330	C₂	1	2	yes	maximal, complemented
11	C₈₉×#2314	C₂	1	2	yes	maximal
12	C₄×C₄⋊Q₈×C₈₉	C₂	1	2	yes	maximal, complemented
13	C₄×#117×C₈₉	C₂	1	1	yes	maximal, characteristic
14	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
15	C₂×C₄⋊Q₈×C₈₉	C₂²	1	4	yes	—
16	C₂×#125×C₈₉	C₂²	1	2	yes	—
17	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	4	yes	—
18	C₂×C₄×Q₈×C₈₉	C₂²	1	4	yes	—
19	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
22	#400×C₈₉	C₂²	1	8	yes	—
23	#398×C₈₉	C₂²	1	8	yes	—
24	#398×C₈₉	C₂²	1	8	yes	—
25	#396×C₈₉	C₂²	1	8	yes	—
26	#391×C₈₉	C₂²	1	4	yes	—
27	#391×C₈₉	C₂²	1	4	yes	—
28	#389×C₈₉	C₂²	1	4	yes	—
29	#389×C₈₉	C₂²	1	4	yes	—
30	#389×C₈₉	C₂²	1	4	yes	—
31	#389×C₈₉	C₂²	1	8	yes	—
32	#389×C₈₉	C₂²	1	4	yes	—
33	#389×C₈₉	C₂²	1	4	yes	—
34	#389×C₈₉	C₂²	1	4	yes	—
35	#388×C₈₉	C₂²	1	8	yes	—
36	#387×C₈₉	C₂²	1	8	yes	—
37	#387×C₈₉	C₂²	1	8	yes	—
38	#384×C₈₉	C₂²	1	2	yes	—
39	#384×C₈₉	C₂²	1	2	yes	—
40	#384×C₈₉	C₂²	1	2	yes	—
41	#384×C₈₉	C₂²	1	4	yes	—
42	#382×C₈₉	C₂²	1	2	yes	—
43	#380×C₈₉	C₂²	1	4	yes	—
44	#380×C₈₉	C₂²	1	4	yes	—
45	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
46	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
47	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
48	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
49	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
50	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
51	#376×C₈₉	C₂²	1	1	yes	characteristic
52	#376×C₈₉	C₂²	1	1	yes	characteristic
53	C₄³×C₈₉	C₂²	1	1	yes	characteristic
54	C₄³×C₈₉	C₂²	1	1	yes	characteristic
55	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
56	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
57	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
59	C₄⋊Q₈×C₈₉	—	2	8	no	—
60	C₄⋊Q₈×C₈₉	—	4	8	no	—
61	#125×C₈₉	—	2	8	no	—
62	C₂²⋊Q₈×C₈₉	—	4	8	no	—
63	C₄×Q₈×C₈₉	—	4	8	no	—
64	#117×C₈₉	—	2	4	no	—
65	#117×C₈₉	Q₈	1	8	yes	complemented
66	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
67	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
68	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
69	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
70	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
71	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
72	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
73	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
74	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
75	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
76	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
77	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
78	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
79	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
80	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
81	C₂×C₄²×C₈₉	—	2	4	no	—
82	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
83	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
84	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
85	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
86	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
87	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₈₉	C₂³	1	8	yes	—
91	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
92	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
93	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
94	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
95	#95×C₈₉	C₂³	1	2	yes	—
96	#95×C₈₉	C₂³	1	2	yes	—
97	#95×C₈₉	C₂³	1	2	yes	—
98	#95×C₈₉	C₂³	1	2	yes	—
99	#95×C₈₉	C₂³	1	8	yes	—
100	#95×C₈₉	C₂³	1	8	yes	—
101	#95×C₈₉	C₂³	1	8	yes	—
102	#95×C₈₉	C₂³	1	8	yes	—
103	#95×C₈₉	C₂³	1	8	yes	—
104	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
105	C₂×Q₈×C₈₉	—	2	8	no	—
106	C₂×Q₈×C₈₉	—	2	8	no	—
107	C₂×Q₈×C₈₉	—	2	8	no	—
108	C₂²×C₄×C₈₉	—	2	2	no	—
109	C₂²×C₄×C₈₉	—	2	2	no	—
110	C₂²×C₄×C₈₉	—	2	2	no	—
111	C₂²×C₄×C₈₉	—	2	2	no	—
112	C₂²×C₄×C₈₉	—	2	4	no	—
113	C₂²×C₄×C₈₉	—	2	4	no	—
114	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
115	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
116	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
117	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
118	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
119	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
120	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
121	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
122	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
123	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
124	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
125	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
126	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
127	C₄⋊C₄×C₈₉	—	2	4	no	—
128	C₄⋊C₄×C₈₉	—	2	16	no	—
129	C₄⋊C₄×C₈₉	—	2	8	no	—
130	C₄⋊C₄×C₈₉	—	2	8	no	—
131	C₄⋊C₄×C₈₉	—	2	8	no	—
132	C₄⋊C₄×C₈₉	—	2	8	no	—
133	C₄⋊C₄×C₈₉	—	2	8	no	—
134	C₄⋊C₄×C₈₉	—	4	8	no	—
135	C₄⋊C₄×C₈₉	—	4	8	no	—
136	C₄⋊C₄×C₈₉	—	4	8	no	—
137	C₄⋊C₄×C₈₉	—	4	8	no	—
138	C₄⋊C₄×C₈₉	C₂×Q₈	1	8	yes	—
139	C₂²⋊C₄×C₈₉	—	2	4	no	—
140	C₂²⋊C₄×C₈₉	—	4	8	no	—
141	C₂²⋊C₄×C₈₉	C₂×Q₈	1	8	yes	—
142	C₄²×C₈₉	—	2	2	no	—
143	C₄²×C₈₉	—	2	2	no	—
144	C₄²×C₈₉	—	2	2	no	—
145	C₄²×C₈₉	—	2	2	no	—
146	C₄²×C₈₉	—	2	4	no	—
147	C₄²×C₈₉	—	2	4	no	—
148	C₄²×C₈₉	—	2	4	no	—
149	C₄²×C₈₉	—	2	4	no	—
150	C₄²×C₈₉	—	4	8	no	—
151	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
152	C₄²×C₈₉	C₂×Q₈	1	4	yes	—
153	C₄²×C₈₉	#41	1	4	yes	—
154	C₄²×C₈₉	#41	1	4	yes	—
155	C₂³×C₈₉	—	2	1	no	—
156	C₂³×C₈₉	—	2	1	no	—
157	C₂³×C₈₉	—	2	2	no	—
158	C₂³×C₈₉	—	2	2	no	—
159	C₂³×C₈₉	C₂²×Q₈	1	2	yes	—
160	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
161	Q₈×C₈₉	—	4	16	no	—
162	C₂×C₄×C₈₉	—	2	1	no	—
163	C₂×C₄×C₈₉	—	2	1	no	—
164	C₂×C₄×C₈₉	—	2	2	no	—
165	C₂×C₄×C₈₉	—	2	2	no	—
166	C₂×C₄×C₈₉	—	2	2	no	—
167	C₂×C₄×C₈₉	—	2	2	no	—
168	C₂×C₄×C₈₉	—	2	2	no	—
169	C₂×C₄×C₈₉	—	2	2	no	—
170	C₂×C₄×C₈₉	—	2	8	no	—
171	C₂×C₄×C₈₉	—	2	1	no	—
172	C₂×C₄×C₈₉	—	2	1	no	—
173	C₂×C₄×C₈₉	—	2	2	no	—
174	C₂×C₄×C₈₉	—	2	8	no	—
175	C₂×C₄×C₈₉	—	2	8	no	—
176	C₂×C₄×C₈₉	—	2	2	no	—
177	C₂×C₄×C₈₉	—	2	8	no	—
178	C₂×C₄×C₈₉	—	2	8	no	—
179	C₂×C₄×C₈₉	—	2	8	no	—
180	C₂×C₄×C₈₉	—	2	2	no	—
181	C₂×C₄×C₈₉	—	2	2	no	—
182	C₂×C₄×C₈₉	—	2	8	no	—
183	C₂×C₄×C₈₉	—	2	2	no	—
184	C₂×C₄×C₈₉	—	4	4	no	—
185	C₂×C₄×C₈₉	—	4	4	no	—
186	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
187	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
188	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
189	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
190	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
191	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
192	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
193	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
194	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
195	C₂×C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
196	C₂×C₄×C₈₉	2₋¹⁺⁴	1	4	yes	—
197	C₂²×C₈₉	—	2	2	no	—
198	C₂²×C₈₉	—	2	4	no	—
199	C₂²×C₈₉	—	4	1	no	—
200	C₂²×C₈₉	—	4	1	no	—
201	C₂²×C₈₉	—	4	2	no	—
202	C₂²×C₈₉	C₂³×Q₈	1	1	yes	characteristic
203	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
204	C₂²×C₈₉	C₂²×#41	1	1	yes	characteristic
205	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
206	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	2	yes	—
207	C₄×C₈₉	—	2	2	no	—
208	C₄×C₈₉	—	2	4	no	—
209	C₄×C₈₉	—	2	4	no	—
210	C₄×C₈₉	—	2	4	no	—
211	C₄×C₈₉	—	4	1	no	—
212	C₄×C₈₉	—	4	8	no	—
213	C₄×C₈₉	—	4	1	no	—
214	C₄×C₈₉	—	4	2	no	—
215	C₄×C₈₉	—	4	8	no	—
216	C₄×C₈₉	#533	1	4	yes	—
217	C₄×C₈₉	#540	1	4	yes	—
218	C₄×C₈₉	#541	1	4	yes	—
219	#35284	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
220	C₂×C₈₉	—	4	2	no	—
221	C₂×C₈₉	C₂×#533	1	1	yes	characteristic
222	C₂×C₈₉	C₂×#540	1	1	yes	characteristic
223	C₂×C₈₉	C₂×#541	1	1	yes	characteristic
224	C₂×C₈₉	#3503	1	2	yes	—
225	C₂×C₈₉	#3511	1	2	yes	—
226	#2868	C₂×C₈₉	1	1	yes	characteristic, complemented
227	#2859	C₂×C₈₉	1	1	yes	characteristic, complemented
228	#2857	C₂×C₈₉	1	2	yes	—
229	#2647	C₂×C₈₉	1	2	yes	complemented
230	#2629	C₂×C₈₉	1	2	yes	—
231	#2507	C₂×C₈₉	1	4	yes	complemented
232	#2497	C₂×C₈₉	1	4	yes	—
233	#2446	C₂×C₈₉	1	4	yes	complemented
234	#2429	C₂×C₈₉	1	4	yes	—
235	#2330	C₂×C₈₉	1	2	yes	complemented
236	#2314	C₂×C₈₉	1	2	yes	—
237	C₄×C₄⋊Q₈	C₂×C₈₉	1	2	yes	complemented
238	C₄×#117	C₂×C₈₉	1	1	yes	characteristic
239	C₈₉	#35284	1	1	yes	characteristic, direct product factor, Sylow
240	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
241	C₂×C₄⋊Q₈	C₂²×C₈₉	1	4	yes	—
242	C₂×#125	C₂²×C₈₉	1	2	yes	—
243	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	4	yes	—
244	C₂×C₄×Q₈	C₂²×C₈₉	1	4	yes	—
245	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
246	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
247	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
248	#400	C₂²×C₈₉	1	8	yes	—
249	#398	C₂²×C₈₉	1	8	yes	—
250	#398	C₂²×C₈₉	1	8	yes	—
251	#396	C₂²×C₈₉	1	8	yes	—
252	#391	C₂²×C₈₉	1	4	yes	—
253	#391	C₂²×C₈₉	1	4	yes	—
254	#389	C₂²×C₈₉	1	4	yes	—
255	#389	C₂²×C₈₉	1	4	yes	—
256	#389	C₂²×C₈₉	1	4	yes	—
257	#389	C₂²×C₈₉	1	8	yes	—
258	#389	C₂²×C₈₉	1	4	yes	—
259	#389	C₂²×C₈₉	1	4	yes	—
260	#389	C₂²×C₈₉	1	4	yes	—
261	#388	C₂²×C₈₉	1	8	yes	—
262	#387	C₂²×C₈₉	1	8	yes	—
263	#387	C₂²×C₈₉	1	8	yes	—
264	#384	C₂²×C₈₉	1	2	yes	—
265	#384	C₂²×C₈₉	1	2	yes	—
266	#384	C₂²×C₈₉	1	2	yes	—
267	#384	C₂²×C₈₉	1	4	yes	—
268	#382	C₂²×C₈₉	1	2	yes	—
269	#380	C₂²×C₈₉	1	4	yes	—
270	#380	C₂²×C₈₉	1	4	yes	—
271	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
272	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
273	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
274	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
275	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
276	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
277	#376	C₂²×C₈₉	1	1	yes	characteristic
278	#376	C₂²×C₈₉	1	1	yes	characteristic
279	C₄³	C₂²×C₈₉	1	1	yes	characteristic
280	C₄³	C₂²×C₈₉	1	1	yes	characteristic
281	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
282	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
283	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
284	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
285	C₄⋊Q₈	—	2	8	no	—
286	C₄⋊Q₈	—	4	8	no	—
287	#125	—	2	8	no	—
288	C₂²⋊Q₈	—	4	8	no	—
289	C₄×Q₈	—	4	8	no	—
290	#117	—	2	4	no	—
291	#117	Q₈×C₈₉	1	8	yes	complemented
292	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
293	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
294	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
295	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
296	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
297	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
298	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
299	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
300	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
301	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
302	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
303	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
304	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
305	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
306	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
307	C₂×C₄²	—	2	4	no	—
308	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
309	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
310	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
311	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
312	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
313	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
314	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
315	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
316	C₂×C₄²	C₂³×C₈₉	1	8	yes	—
317	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
318	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
319	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
320	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
321	#95	C₂³×C₈₉	1	2	yes	—
322	#95	C₂³×C₈₉	1	2	yes	—
323	#95	C₂³×C₈₉	1	2	yes	—
324	#95	C₂³×C₈₉	1	2	yes	—
325	#95	C₂³×C₈₉	1	8	yes	—
326	#95	C₂³×C₈₉	1	8	yes	—
327	#95	C₂³×C₈₉	1	8	yes	—
328	#95	C₂³×C₈₉	1	8	yes	—
329	#95	C₂³×C₈₉	1	8	yes	—
330	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
331	C₂×Q₈	—	2	8	no	—
332	C₂×Q₈	—	2	8	no	—
333	C₂×Q₈	—	2	8	no	—
334	C₂²×C₄	—	2	2	no	—
335	C₂²×C₄	—	2	2	no	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	2	no	—
338	C₂²×C₄	—	2	4	no	—
339	C₂²×C₄	—	2	4	no	—
340	C₂²×C₄	C₂×Q₈×C₈₉	1	4	yes	—
341	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
342	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
343	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
344	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
345	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
346	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
347	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
348	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
349	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
350	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
351	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
352	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
353	C₄⋊C₄	—	2	4	no	—
354	C₄⋊C₄	—	2	16	no	—
355	C₄⋊C₄	—	2	8	no	—
356	C₄⋊C₄	—	2	8	no	—
357	C₄⋊C₄	—	2	8	no	—
358	C₄⋊C₄	—	2	8	no	—
359	C₄⋊C₄	—	2	8	no	—
360	C₄⋊C₄	—	4	8	no	—
361	C₄⋊C₄	—	4	8	no	—
362	C₄⋊C₄	—	4	8	no	—
363	C₄⋊C₄	—	4	8	no	—
364	C₄⋊C₄	C₂×Q₈×C₈₉	1	8	yes	—
365	C₂²⋊C₄	—	2	4	no	—
366	C₂²⋊C₄	—	4	8	no	—
367	C₂²⋊C₄	C₂×Q₈×C₈₉	1	8	yes	—
368	C₄²	—	2	2	no	—
369	C₄²	—	2	2	no	—
370	C₄²	—	2	2	no	—
371	C₄²	—	2	2	no	—
372	C₄²	—	2	4	no	—
373	C₄²	—	2	4	no	—
374	C₄²	—	2	4	no	—
375	C₄²	—	2	4	no	—
376	C₄²	—	4	8	no	—
377	C₄²	C₂×Q₈×C₈₉	1	4	yes	—
378	C₄²	C₂×Q₈×C₈₉	1	4	yes	—
379	C₄²	#41×C₈₉	1	4	yes	—
380	C₄²	#41×C₈₉	1	4	yes	—
381	C₂³	—	2	1	no	—
382	C₂³	—	2	1	no	—
383	C₂³	—	2	2	no	—
384	C₂³	—	2	2	no	—
385	C₂³	C₂²×Q₈×C₈₉	1	2	yes	—
386	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
387	Q₈	—	4	16	no	—
388	C₂×C₄	—	2	1	no	—
389	C₂×C₄	—	2	1	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	2	no	—
392	C₂×C₄	—	2	2	no	—
393	C₂×C₄	—	2	2	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	8	no	—
397	C₂×C₄	—	2	1	no	—
398	C₂×C₄	—	2	1	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	8	no	—
401	C₂×C₄	—	2	8	no	—
402	C₂×C₄	—	2	2	no	—
403	C₂×C₄	—	2	8	no	—
404	C₂×C₄	—	2	8	no	—
405	C₂×C₄	—	2	8	no	—
406	C₂×C₄	—	2	2	no	—
407	C₂×C₄	—	2	2	no	—
408	C₂×C₄	—	2	8	no	—
409	C₂×C₄	—	2	2	no	—
410	C₂×C₄	—	4	4	no	—
411	C₂×C₄	—	4	4	no	—
412	C₂×C₄	C₂²×Q₈×C₈₉	1	4	yes	—
413	C₂×C₄	C₂²×Q₈×C₈₉	1	4	yes	—
414	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
415	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
416	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
417	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
418	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
419	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
420	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
421	C₂×C₄	H(5,𝔽₂)×C₈₉	1	4	yes	—
422	C₂×C₄	2₋¹⁺⁴×C₈₉	1	4	yes	—
423	C₂²	—	2	2	no	—
424	C₂²	—	2	4	no	—
425	C₂²	—	4	1	no	—
426	C₂²	—	4	1	no	—
427	C₂²	—	4	2	no	—
428	C₂²	C₂³×Q₈×C₈₉	1	1	yes	characteristic
429	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
430	C₂²	C₂²×#41×C₈₉	1	1	yes	characteristic
431	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
432	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	2	yes	—
433	C₄	—	2	2	no	—
434	C₄	—	2	4	no	—
435	C₄	—	2	4	no	—
436	C₄	—	2	4	no	—
437	C₄	—	4	1	no	—
438	C₄	—	4	8	no	—
439	C₄	—	4	1	no	—
440	C₄	—	4	2	no	—
441	C₄	—	4	8	no	—
442	C₄	#533×C₈₉	1	4	yes	—
443	C₄	#540×C₈₉	1	4	yes	—
444	C₄	#541×C₈₉	1	4	yes	—
445	C₂	—	4	2	no	—
446	C₂	C₂×#533×C₈₉	1	1	yes	characteristic
447	C₂	C₂×#540×C₈₉	1	1	yes	characteristic
448	C₂	C₂×#541×C₈₉	1	1	yes	characteristic
449	C₂	C₈₉×#3503	1	2	yes	—
450	C₂	C₈₉×#3511	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂⁴×C₈₉	C₂⁴
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×#2868	C₂	semidirect
2. C₈₉×#2859	C₂	semidirect
4. C₈₉×#2647	C₂	semidirect
6. C₈₉×#2507	C₂	semidirect
8. C₈₉×#2446	C₂	semidirect
10. C₈₉×#2330	C₂	semidirect
12. C₄×C₄⋊Q₈×C₈₉	C₂	semidirect
65. #117×C₈₉	Q₈	semidirect
219. #35284	C₈₉	direct (N × H)
226. #2868	C₂×C₈₉	semidirect
227. #2859	C₂×C₈₉	semidirect
229. #2647	C₂×C₈₉	semidirect
231. #2507	C₂×C₈₉	semidirect
233. #2446	C₂×C₈₉	semidirect
235. #2330	C₂×C₈₉	semidirect
237. C₄×C₄⋊Q₈	C₂×C₈₉	semidirect
239. C₈₉	#35284	direct (N × H)
291. #117	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×#2868	2
C₈₉×#2859	2
C₈₉×#2857	2
C₈₉×#2647	2
C₈₉×#2629	2
C₈₉×#2507	2
C₈₉×#2497	2
C₈₉×#2446	2
C₈₉×#2429	2
C₈₉×#2330	2
C₈₉×#2314	2
C₄×C₄⋊Q₈×C₈₉	2
C₄×#117×C₈₉	2
#35284	89

Subgroups — C89×#35284 #36413132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35284 #36413132