Subgroups — C₈₉×#35289 #36413137

Order 22784.

2654 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2862	C₂	1	2	yes	maximal, complemented
4	C₈₉×#2858	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2640	C₂	1	4	yes	maximal, complemented
6	C₈₉×#2563	C₂	1	4	yes	maximal, complemented
7	C₈₉×#2547	C₂	1	4	yes	maximal, complemented
8	C₈₉×#2504	C₂	1	4	yes	maximal, complemented
9	C₈₉×#2436	C₂	1	4	yes	maximal, complemented
10	C₈₉×#2311	C₂	1	4	yes	maximal, complemented
11	C₄×#117×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₈₉	C₂²	1	2	yes	—
13	C₂³×Q₈×C₈₉	C₂²	1	2	yes	—
14	#532×C₈₉	—	2	8	no	—
15	C₂×C₄⋊Q₈×C₈₉	—	2	8	no	—
16	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
17	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
21	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×#124×C₈₉	C₂²	1	2	yes	—
23	C₂×#124×C₈₉	C₂²	1	8	yes	—
24	C₂×#121×C₈₉	C₂²	1	2	yes	—
25	C₂×#121×C₈₉	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
27	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
28	C₂×#117×C₈₉	C₂²	1	2	yes	—
29	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
30	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
31	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
32	#397×C₈₉	C₂²	1	16	yes	—
33	#393×C₈₉	C₂²	1	16	yes	—
34	#391×C₈₉	C₂²	1	8	yes	—
35	#391×C₈₉	C₂²	1	8	yes	—
36	#390×C₈₉	C₂²	1	8	yes	—
37	#390×C₈₉	C₂²	1	8	yes	—
38	#386×C₈₉	C₂²	1	16	yes	—
39	#382×C₈₉	C₂²	1	8	yes	—
40	#380×C₈₉	C₂²	1	16	yes	—
41	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
42	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
43	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
44	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
45	#376×C₈₉	C₂²	1	2	yes	—
46	C₄³×C₈₉	C₂²	1	2	yes	—
47	C₂×#41×C₈₉	—	2	8	no	—
48	C₂×#41×C₈₉	—	2	16	no	—
49	C₂²×Q₈×C₈₉	—	2	8	no	—
50	C₂²×Q₈×C₈₉	—	4	8	no	—
51	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
52	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
53	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
54	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
55	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
56	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
57	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
58	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
59	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
60	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
62	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
63	C₄⋊Q₈×C₈₉	—	2	8	no	—
64	C₄⋊Q₈×C₈₉	—	2	8	no	—
65	C₄⋊Q₈×C₈₉	—	2	8	no	—
66	C₄⋊Q₈×C₈₉	—	2	8	no	—
67	C₄⋊Q₈×C₈₉	—	2	4	no	—
68	#127×C₈₉	—	2	4	no	—
69	#124×C₈₉	—	2	8	no	—
70	#124×C₈₉	—	4	16	no	—
71	#121×C₈₉	—	2	8	no	—
72	#121×C₈₉	—	2	8	no	—
73	C₄×D₄×C₈₉	—	2	8	no	—
74	C₄×D₄×C₈₉	—	2	8	no	—
75	#117×C₈₉	—	2	8	no	—
76	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
77	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
78	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
79	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
80	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
81	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
82	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
83	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
84	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
85	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
86	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
87	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
88	C₂×C₂²⋊C₄×C₈₉	C₂³	1	32	yes	—
89	C₂×C₄²×C₈₉	D₄	1	8	yes	—
90	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
91	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
92	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
93	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
94	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
95	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
97	#95×C₈₉	C₂³	1	4	yes	—
98	#95×C₈₉	C₂³	1	4	yes	—
99	#95×C₈₉	C₂³	1	32	yes	—
100	C₂⁴×C₈₉	C₂⁴	1	4	yes	—
101	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
102	#41×C₈₉	—	4	32	no	—
103	C₂×Q₈×C₈₉	—	2	4	no	—
104	C₂×Q₈×C₈₉	—	2	8	no	—
105	C₂×Q₈×C₈₉	—	2	8	no	—
106	C₂×Q₈×C₈₉	—	2	8	no	—
107	C₂×Q₈×C₈₉	—	2	16	no	—
108	C₂×Q₈×C₈₉	—	4	8	no	—
109	C₂×Q₈×C₈₉	—	4	32	no	—
110	C₂×D₄×C₈₉	—	2	8	no	—
111	C₂×D₄×C₈₉	—	2	16	no	—
112	C₂×D₄×C₈₉	—	2	4	no	—
113	C₂×D₄×C₈₉	—	2	4	no	—
114	C₂×D₄×C₈₉	—	2	4	no	—
115	C₂×D₄×C₈₉	—	2	8	no	—
116	C₂×D₄×C₈₉	—	2	8	no	—
117	C₂×D₄×C₈₉	—	2	8	no	—
118	C₂²×C₄×C₈₉	—	2	4	no	—
119	C₂²×C₄×C₈₉	—	2	4	no	—
120	C₂²×C₄×C₈₉	—	2	8	no	—
121	C₂²×C₄×C₈₉	—	2	8	no	—
122	C₂²×C₄×C₈₉	—	2	16	no	—
123	C₂²×C₄×C₈₉	—	4	8	no	—
124	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
125	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
126	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
127	C₂²×C₄×C₈₉	#41	1	8	yes	—
128	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
129	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
130	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
131	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
132	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
133	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
134	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
135	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
136	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
137	C₄⋊C₄×C₈₉	—	2	8	no	—
138	C₄⋊C₄×C₈₉	—	2	8	no	—
139	C₄⋊C₄×C₈₉	—	2	8	no	—
140	C₄⋊C₄×C₈₉	—	2	8	no	—
141	C₄⋊C₄×C₈₉	—	2	4	no	—
142	C₄⋊C₄×C₈₉	—	2	4	no	—
143	C₄⋊C₄×C₈₉	—	2	8	no	—
144	C₂²⋊C₄×C₈₉	—	2	8	no	—
145	C₂²⋊C₄×C₈₉	—	2	8	no	—
146	C₂²⋊C₄×C₈₉	—	2	8	no	—
147	C₂²⋊C₄×C₈₉	—	4	32	no	—
148	C₄²×C₈₉	—	2	8	no	—
149	C₄²×C₈₉	—	2	8	no	—
150	C₄²×C₈₉	—	2	8	no	—
151	C₄²×C₈₉	C₂×D₄	1	4	yes	—
152	C₄²×C₈₉	C₂×D₄	1	4	yes	—
153	C₄²×C₈₉	C₂×D₄	1	4	yes	—
154	C₄²×C₈₉	C₂×D₄	1	4	yes	—
155	C₂³×C₈₉	—	2	2	no	—
156	C₂³×C₈₉	—	2	2	no	—
157	C₂³×C₈₉	—	2	4	no	—
158	C₂³×C₈₉	—	2	8	no	—
159	C₂³×C₈₉	—	2	8	no	—
160	C₂³×C₈₉	—	2	8	no	—
161	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
162	C₂³×C₈₉	C₂×#41	1	4	yes	—
163	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
164	Q₈×C₈₉	—	4	16	no	—
165	Q₈×C₈₉	—	4	8	no	—
166	Q₈×C₈₉	—	4	16	no	—
167	D₄×C₈₉	—	4	16	no	—
168	D₄×C₈₉	—	4	8	no	—
169	D₄×C₈₉	—	4	8	no	—
170	D₄×C₈₉	—	4	8	no	—
171	C₂×C₄×C₈₉	—	2	2	no	—
172	C₂×C₄×C₈₉	—	2	2	no	—
173	C₂×C₄×C₈₉	—	2	8	no	—
174	C₂×C₄×C₈₉	—	2	8	no	—
175	C₂×C₄×C₈₉	—	2	4	no	—
176	C₂×C₄×C₈₉	—	2	8	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	16	no	—
181	C₂×C₄×C₈₉	—	2	8	no	—
182	C₂×C₄×C₈₉	—	4	16	no	—
183	C₂×C₄×C₈₉	—	4	16	no	—
184	C₂×C₄×C₈₉	#124	1	16	yes	complemented
185	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
186	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
187	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
188	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
189	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
190	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
191	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
192	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
193	C₂×C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
194	C₂×C₄×C₈₉	2₋¹⁺⁴	1	4	yes	—
195	C₂²×C₈₉	—	2	2	no	—
196	C₂²×C₈₉	—	2	2	no	—
197	C₂²×C₈₉	—	2	8	no	—
198	C₂²×C₈₉	—	4	4	no	—
199	C₂²×C₈₉	—	4	4	no	—
200	C₂²×C₈₉	—	4	4	no	—
201	C₂²×C₈₉	—	4	16	no	—
202	C₂²×C₈₉	C₂×#124	1	4	yes	—
203	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
204	C₂²×C₈₉	C₂²×#41	1	2	yes	—
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	2	no	—
209	C₄×C₈₉	—	4	4	no	—
210	C₄×C₈₉	—	4	8	no	—
211	C₄×C₈₉	—	4	8	no	—
212	C₄×C₈₉	C₂×#124	1	4	yes	—
213	C₄×C₈₉	C₂×#124	1	4	yes	—
214	C₄×C₈₉	#535	1	4	yes	—
215	C₄×C₈₉	#536	1	4	yes	—
216	#35289	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
217	C₂×C₈₉	—	2	4	no	—
218	C₂×C₈₉	—	8	4	no	—
219	C₂×C₈₉	C₂²×#124	1	1	yes	characteristic
220	C₂×C₈₉	C₂×#535	1	1	yes	characteristic
221	C₂×C₈₉	C₂×#536	1	1	yes	characteristic
222	C₂×C₈₉	#3511	1	4	yes	—
223	C₂×#532	C₂×C₈₉	1	1	yes	characteristic
224	C₂²×C₄⋊Q₈	C₂×C₈₉	1	1	yes	characteristic, complemented
225	#2862	C₂×C₈₉	1	2	yes	complemented
226	#2858	C₂×C₈₉	1	2	yes	complemented
227	#2640	C₂×C₈₉	1	4	yes	complemented
228	#2563	C₂×C₈₉	1	4	yes	complemented
229	#2547	C₂×C₈₉	1	4	yes	complemented
230	#2504	C₂×C₈₉	1	4	yes	complemented
231	#2436	C₂×C₈₉	1	4	yes	complemented
232	#2311	C₂×C₈₉	1	4	yes	complemented
233	C₄×#117	C₂×C₈₉	1	1	yes	characteristic, complemented
234	C₈₉	#35289	1	1	yes	characteristic, direct product factor, Sylow
235	C₂²×#41	C₂²×C₈₉	1	2	yes	—
236	C₂³×Q₈	C₂²×C₈₉	1	2	yes	—
237	#532	—	2	8	no	—
238	C₂×C₄⋊Q₈	—	2	8	no	—
239	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
240	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
241	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
242	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
243	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
244	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
245	C₂×#124	C₂²×C₈₉	1	2	yes	—
246	C₂×#124	C₂²×C₈₉	1	8	yes	—
247	C₂×#121	C₂²×C₈₉	1	2	yes	—
248	C₂×#121	C₂²×C₈₉	1	2	yes	—
249	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
250	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
251	C₂×#117	C₂²×C₈₉	1	2	yes	—
252	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
253	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
254	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
255	#397	C₂²×C₈₉	1	16	yes	—
256	#393	C₂²×C₈₉	1	16	yes	—
257	#391	C₂²×C₈₉	1	8	yes	—
258	#391	C₂²×C₈₉	1	8	yes	—
259	#390	C₂²×C₈₉	1	8	yes	—
260	#390	C₂²×C₈₉	1	8	yes	—
261	#386	C₂²×C₈₉	1	16	yes	—
262	#382	C₂²×C₈₉	1	8	yes	—
263	#380	C₂²×C₈₉	1	16	yes	—
264	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
265	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
266	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
267	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
268	#376	C₂²×C₈₉	1	2	yes	—
269	C₄³	C₂²×C₈₉	1	2	yes	—
270	C₂×#41	—	2	8	no	—
271	C₂×#41	—	2	16	no	—
272	C₂²×Q₈	—	2	8	no	—
273	C₂²×Q₈	—	4	8	no	—
274	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
275	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
276	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
277	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
278	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
279	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
280	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
281	C₂³×C₄	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	4	yes	—
285	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
286	C₄⋊Q₈	—	2	8	no	—
287	C₄⋊Q₈	—	2	8	no	—
288	C₄⋊Q₈	—	2	8	no	—
289	C₄⋊Q₈	—	2	8	no	—
290	C₄⋊Q₈	—	2	4	no	—
291	#127	—	2	4	no	—
292	#124	—	2	8	no	—
293	#124	—	4	16	no	—
294	#121	—	2	8	no	—
295	#121	—	2	8	no	—
296	C₄×D₄	—	2	8	no	—
297	C₄×D₄	—	2	8	no	—
298	#117	—	2	8	no	—
299	C₂×C₄⋊C₄	—	2	8	no	—
300	C₂×C₄⋊C₄	—	2	8	no	—
301	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
302	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
303	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
304	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
305	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
306	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
307	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
308	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
309	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
310	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
311	C₂×C₂²⋊C₄	C₂³×C₈₉	1	32	yes	—
312	C₂×C₄²	D₄×C₈₉	1	8	yes	—
313	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
314	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
315	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
316	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
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	#95	C₂³×C₈₉	1	4	yes	—
321	#95	C₂³×C₈₉	1	4	yes	—
322	#95	C₂³×C₈₉	1	32	yes	—
323	C₂⁴	C₂⁴×C₈₉	1	4	yes	—
324	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
325	#41	—	4	32	no	—
326	C₂×Q₈	—	2	4	no	—
327	C₂×Q₈	—	2	8	no	—
328	C₂×Q₈	—	2	8	no	—
329	C₂×Q₈	—	2	8	no	—
330	C₂×Q₈	—	2	16	no	—
331	C₂×Q₈	—	4	8	no	—
332	C₂×Q₈	—	4	32	no	—
333	C₂×D₄	—	2	8	no	—
334	C₂×D₄	—	2	16	no	—
335	C₂×D₄	—	2	4	no	—
336	C₂×D₄	—	2	4	no	—
337	C₂×D₄	—	2	4	no	—
338	C₂×D₄	—	2	8	no	—
339	C₂×D₄	—	2	8	no	—
340	C₂×D₄	—	2	8	no	—
341	C₂²×C₄	—	2	4	no	—
342	C₂²×C₄	—	2	4	no	—
343	C₂²×C₄	—	2	8	no	—
344	C₂²×C₄	—	2	8	no	—
345	C₂²×C₄	—	2	16	no	—
346	C₂²×C₄	—	4	8	no	—
347	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
348	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
349	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
350	C₂²×C₄	#41×C₈₉	1	8	yes	—
351	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
352	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
353	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
354	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
355	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
356	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
357	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
358	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
359	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
360	C₄⋊C₄	—	2	8	no	—
361	C₄⋊C₄	—	2	8	no	—
362	C₄⋊C₄	—	2	8	no	—
363	C₄⋊C₄	—	2	8	no	—
364	C₄⋊C₄	—	2	4	no	—
365	C₄⋊C₄	—	2	4	no	—
366	C₄⋊C₄	—	2	8	no	—
367	C₂²⋊C₄	—	2	8	no	—
368	C₂²⋊C₄	—	2	8	no	—
369	C₂²⋊C₄	—	2	8	no	—
370	C₂²⋊C₄	—	4	32	no	—
371	C₄²	—	2	8	no	—
372	C₄²	—	2	8	no	—
373	C₄²	—	2	8	no	—
374	C₄²	C₂×D₄×C₈₉	1	4	yes	—
375	C₄²	C₂×D₄×C₈₉	1	4	yes	—
376	C₄²	C₂×D₄×C₈₉	1	4	yes	—
377	C₄²	C₂×D₄×C₈₉	1	4	yes	—
378	C₂³	—	2	2	no	—
379	C₂³	—	2	2	no	—
380	C₂³	—	2	4	no	—
381	C₂³	—	2	8	no	—
382	C₂³	—	2	8	no	—
383	C₂³	—	2	8	no	—
384	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
385	C₂³	C₂×#41×C₈₉	1	4	yes	—
386	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
387	Q₈	—	4	16	no	—
388	Q₈	—	4	8	no	—
389	Q₈	—	4	16	no	—
390	D₄	—	4	16	no	—
391	D₄	—	4	8	no	—
392	D₄	—	4	8	no	—
393	D₄	—	4	8	no	—
394	C₂×C₄	—	2	2	no	—
395	C₂×C₄	—	2	2	no	—
396	C₂×C₄	—	2	8	no	—
397	C₂×C₄	—	2	8	no	—
398	C₂×C₄	—	2	4	no	—
399	C₂×C₄	—	2	8	no	—
400	C₂×C₄	—	2	8	no	—
401	C₂×C₄	—	2	8	no	—
402	C₂×C₄	—	2	8	no	—
403	C₂×C₄	—	2	16	no	—
404	C₂×C₄	—	2	8	no	—
405	C₂×C₄	—	4	16	no	—
406	C₂×C₄	—	4	16	no	—
407	C₂×C₄	#124×C₈₉	1	16	yes	complemented
408	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
409	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
410	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
411	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
412	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
413	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
414	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
415	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
416	C₂×C₄	H(5,𝔽₂)×C₈₉	1	4	yes	—
417	C₂×C₄	2₋¹⁺⁴×C₈₉	1	4	yes	—
418	C₂²	—	2	2	no	—
419	C₂²	—	2	2	no	—
420	C₂²	—	2	8	no	—
421	C₂²	—	4	4	no	—
422	C₂²	—	4	4	no	—
423	C₂²	—	4	4	no	—
424	C₂²	—	4	16	no	—
425	C₂²	C₂×#124×C₈₉	1	4	yes	—
426	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
427	C₂²	C₂²×#41×C₈₉	1	2	yes	—
428	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
429	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	2	yes	—
430	C₄	—	2	2	no	—
431	C₄	—	2	2	no	—
432	C₄	—	4	4	no	—
433	C₄	—	4	8	no	—
434	C₄	—	4	8	no	—
435	C₄	C₂×#124×C₈₉	1	4	yes	—
436	C₄	C₂×#124×C₈₉	1	4	yes	—
437	C₄	#535×C₈₉	1	4	yes	—
438	C₄	#536×C₈₉	1	4	yes	—
439	C₂	—	2	4	no	—
440	C₂	—	8	4	no	—
441	C₂	C₂²×#124×C₈₉	1	1	yes	characteristic
442	C₂	C₂×#535×C₈₉	1	1	yes	characteristic
443	C₂	C₂×#536×C₈₉	1	1	yes	characteristic
444	C₂	C₈₉×#3511	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×C₈₉
prime radical	C₂×#532×C₈₉	C₂
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
2. C₂²×C₄⋊Q₈×C₈₉	C₂	semidirect
3. C₈₉×#2862	C₂	semidirect
4. C₈₉×#2858	C₂	semidirect
5. C₈₉×#2640	C₂	semidirect
6. C₈₉×#2563	C₂	semidirect
7. C₈₉×#2547	C₂	semidirect
8. C₈₉×#2504	C₂	semidirect
9. C₈₉×#2436	C₂	semidirect
10. C₈₉×#2311	C₂	semidirect
11. C₄×#117×C₈₉	C₂	semidirect
184. C₂×C₄×C₈₉	#124	semidirect
216. #35289	C₈₉	direct (N × H)
224. C₂²×C₄⋊Q₈	C₂×C₈₉	semidirect
225. #2862	C₂×C₈₉	semidirect
226. #2858	C₂×C₈₉	semidirect
227. #2640	C₂×C₈₉	semidirect
228. #2563	C₂×C₈₉	semidirect
229. #2547	C₂×C₈₉	semidirect
230. #2504	C₂×C₈₉	semidirect
231. #2436	C₂×C₈₉	semidirect
232. #2311	C₂×C₈₉	semidirect
233. C₄×#117	C₂×C₈₉	semidirect
234. C₈₉	#35289	direct (N × H)
407. C₂×C₄	#124×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#532×C₈₉	2
C₂²×C₄⋊Q₈×C₈₉	2
C₈₉×#2862	2
C₈₉×#2858	2
C₈₉×#2640	2
C₈₉×#2563	2
C₈₉×#2547	2
C₈₉×#2504	2
C₈₉×#2436	2
C₈₉×#2311	2
C₄×#117×C₈₉	2
#35289	89

Subgroups — C89×#35289 #36413137

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35289 #36413137