Subgroups — C₈₉×#35281 #36413129

Order 22784.

2374 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	C₂×#533×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄⋊Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#2868	C₂	1	2	yes	maximal, complemented
4	C₈₉×#2862	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2660	C₂	1	4	yes	maximal, complemented
6	C₈₉×#2567	C₂	1	4	yes	maximal, complemented
7	C₈₉×#2546	C₂	1	4	yes	maximal, complemented
8	C₈₉×#2502	C₂	1	4	yes	maximal
9	C₈₉×#2434	C₂	1	4	yes	maximal
10	C₈₉×#2317	C₂	1	4	yes	maximal
11	C₄×#117×C₈₉	C₂	1	1	yes	maximal, characteristic
12	C₂³×Q₈×C₈₉	C₂²	1	2	yes	—
13	#533×C₈₉	—	2	8	no	—
14	C₂×C₄⋊Q₈×C₈₉	—	2	8	no	—
15	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
16	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
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×C₄⋊Q₈×C₈₉	C₂²	1	8	yes	—
22	C₂×#125×C₈₉	C₂²	1	2	yes	—
23	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	2	yes	—
24	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	2	yes	—
25	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
27	C₂×#117×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	#398×C₈₉	C₂²	1	16	yes	—
33	#395×C₈₉	C₂²	1	16	yes	—
34	#391×C₈₉	C₂²	1	8	yes	—
35	#391×C₈₉	C₂²	1	8	yes	—
36	#389×C₈₉	C₂²	1	16	yes	—
37	#385×C₈₉	C₂²	1	16	yes	—
38	#384×C₈₉	C₂²	1	8	yes	—
39	#384×C₈₉	C₂²	1	8	yes	—
40	#382×C₈₉	C₂²	1	8	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	C₂²×Q₈×C₈₉	—	2	8	no	—
48	C₂²×Q₈×C₈₉	—	4	8	no	—
49	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
50	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
51	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
52	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
53	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
54	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
55	C₂³×C₄×C₈₉	C₂³	1	4	yes	—
56	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
57	C₄⋊Q₈×C₈₉	—	2	8	no	—
58	C₄⋊Q₈×C₈₉	—	2	8	no	—
59	C₄⋊Q₈×C₈₉	—	2	8	no	—
60	C₄⋊Q₈×C₈₉	—	2	8	no	—
61	C₄⋊Q₈×C₈₉	—	2	4	no	—
62	C₄⋊Q₈×C₈₉	—	2	4	no	—
63	C₄⋊Q₈×C₈₉	—	4	16	no	—
64	#125×C₈₉	—	2	8	no	—
65	C₂²⋊Q₈×C₈₉	—	2	8	no	—
66	C₂²⋊Q₈×C₈₉	—	2	8	no	—
67	C₄×Q₈×C₈₉	—	2	8	no	—
68	C₄×Q₈×C₈₉	—	2	8	no	—
69	#117×C₈₉	—	2	8	no	—
70	#117×C₈₉	Q₈	1	16	yes	complemented
71	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
72	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
73	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
75	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
76	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
77	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
78	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
79	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	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	32	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×C₄²×C₈₉	D₄	1	8	yes	—
88	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
89	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
90	C₂×C₄²×C₈₉	C₂³	1	4	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	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
96	#95×C₈₉	C₂³	1	4	yes	—
97	#95×C₈₉	C₂³	1	4	yes	—
98	#95×C₈₉	C₂³	1	32	yes	—
99	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
100	C₂×Q₈×C₈₉	—	2	4	no	—
101	C₂×Q₈×C₈₉	—	2	4	no	—
102	C₂×Q₈×C₈₉	—	2	8	no	—
103	C₂×Q₈×C₈₉	—	2	8	no	—
104	C₂×Q₈×C₈₉	—	2	8	no	—
105	C₂×Q₈×C₈₉	—	2	16	no	—
106	C₂×Q₈×C₈₉	—	2	8	no	—
107	C₂×Q₈×C₈₉	—	4	8	no	—
108	C₂×Q₈×C₈₉	—	4	32	no	—
109	C₂²×C₄×C₈₉	—	2	4	no	—
110	C₂²×C₄×C₈₉	—	2	4	no	—
111	C₂²×C₄×C₈₉	—	2	8	no	—
112	C₂²×C₄×C₈₉	—	4	8	no	—
113	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
114	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
115	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
116	C₂²×C₄×C₈₉	C₂×Q₈	1	8	yes	—
117	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
118	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
119	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
120	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
121	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
122	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
123	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
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	8	no	—
128	C₄⋊C₄×C₈₉	—	2	8	no	—
129	C₄⋊C₄×C₈₉	—	2	8	no	—
130	C₄⋊C₄×C₈₉	—	2	8	no	—
131	C₄⋊C₄×C₈₉	—	2	4	no	—
132	C₄⋊C₄×C₈₉	—	2	4	no	—
133	C₄⋊C₄×C₈₉	—	2	8	no	—
134	C₄⋊C₄×C₈₉	—	2	8	no	—
135	C₄⋊C₄×C₈₉	—	2	4	no	—
136	C₄⋊C₄×C₈₉	—	2	4	no	—
137	C₄⋊C₄×C₈₉	—	4	32	no	—
138	C₄⋊C₄×C₈₉	C₂×Q₈	1	16	yes	complemented
139	C₂²⋊C₄×C₈₉	—	2	8	no	—
140	C₂²⋊C₄×C₈₉	C₂×Q₈	1	16	yes	—
141	C₄²×C₈₉	—	2	8	no	—
142	C₄²×C₈₉	—	2	8	no	—
143	C₄²×C₈₉	—	2	8	no	—
144	C₄²×C₈₉	C₂×D₄	1	4	yes	—
145	C₄²×C₈₉	C₂×D₄	1	4	yes	—
146	C₄²×C₈₉	C₂×D₄	1	4	yes	—
147	C₄²×C₈₉	C₂×D₄	1	4	yes	—
148	C₄²×C₈₉	C₂×Q₈	1	16	yes	complemented
149	C₂³×C₈₉	—	2	2	no	—
150	C₂³×C₈₉	—	2	2	no	—
151	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
152	C₂³×C₈₉	C₂²×Q₈	1	4	yes	—
153	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
154	Q₈×C₈₉	—	4	16	no	—
155	Q₈×C₈₉	—	4	8	no	—
156	Q₈×C₈₉	—	4	16	no	—
157	Q₈×C₈₉	—	4	8	no	—
158	C₂×C₄×C₈₉	—	2	8	no	—
159	C₂×C₄×C₈₉	—	2	2	no	—
160	C₂×C₄×C₈₉	—	2	8	no	—
161	C₂×C₄×C₈₉	—	2	2	no	—
162	C₂×C₄×C₈₉	—	2	8	no	—
163	C₂×C₄×C₈₉	—	2	8	no	—
164	C₂×C₄×C₈₉	—	2	4	no	—
165	C₂×C₄×C₈₉	—	2	4	no	—
166	C₂×C₄×C₈₉	—	2	8	no	—
167	C₂×C₄×C₈₉	—	2	8	no	—
168	C₂×C₄×C₈₉	—	2	16	no	—
169	C₂×C₄×C₈₉	—	2	8	no	—
170	C₂×C₄×C₈₉	—	4	16	no	—
171	C₂×C₄×C₈₉	C₄⋊Q₈	1	16	yes	complemented
172	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
173	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
174	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
175	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
176	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
177	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
178	C₂×C₄×C₈₉	C₂²×Q₈	1	16	yes	—
179	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
180	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
181	C₂×C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
182	C₂×C₄×C₈₉	2₋¹⁺⁴	1	4	yes	—
183	C₂²×C₈₉	—	2	2	no	—
184	C₂²×C₈₉	—	2	2	no	—
185	C₂²×C₈₉	—	2	8	no	—
186	C₂²×C₈₉	C₂×C₄⋊Q₈	1	4	yes	—
187	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
188	C₂²×C₈₉	C₂³×Q₈	1	2	yes	—
189	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
190	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	2	yes	—
191	C₄×C₈₉	—	2	2	no	—
192	C₄×C₈₉	—	2	2	no	—
193	C₄×C₈₉	—	2	16	no	—
194	C₄×C₈₉	—	4	8	no	—
195	C₄×C₈₉	—	4	4	no	—
196	C₄×C₈₉	—	4	4	no	—
197	C₄×C₈₉	C₂×C₄⋊Q₈	1	4	yes	—
198	C₄×C₈₉	C₂×C₄⋊Q₈	1	4	yes	—
199	C₄×C₈₉	#535	1	4	yes	—
200	C₄×C₈₉	#536	1	4	yes	—
201	#35281	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
202	C₂×C₈₉	—	2	4	no	—
203	C₂×C₈₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
204	C₂×C₈₉	C₂×#535	1	1	yes	characteristic
205	C₂×C₈₉	C₂×#536	1	1	yes	characteristic
206	C₂×C₈₉	#3503	1	4	yes	—
207	C₂×#533	C₂×C₈₉	1	1	yes	characteristic
208	C₂²×C₄⋊Q₈	C₂×C₈₉	1	1	yes	characteristic
209	#2868	C₂×C₈₉	1	2	yes	complemented
210	#2862	C₂×C₈₉	1	2	yes	complemented
211	#2660	C₂×C₈₉	1	4	yes	complemented
212	#2567	C₂×C₈₉	1	4	yes	complemented
213	#2546	C₂×C₈₉	1	4	yes	complemented
214	#2502	C₂×C₈₉	1	4	yes	—
215	#2434	C₂×C₈₉	1	4	yes	—
216	#2317	C₂×C₈₉	1	4	yes	—
217	C₄×#117	C₂×C₈₉	1	1	yes	characteristic
218	C₈₉	#35281	1	1	yes	characteristic, direct product factor, Sylow
219	C₂³×Q₈	C₂²×C₈₉	1	2	yes	—
220	#533	—	2	8	no	—
221	C₂×C₄⋊Q₈	—	2	8	no	—
222	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
223	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
224	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
225	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
226	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
227	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
228	C₂×C₄⋊Q₈	C₂²×C₈₉	1	8	yes	—
229	C₂×#125	C₂²×C₈₉	1	2	yes	—
230	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	2	yes	—
231	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	2	yes	—
232	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
233	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
234	C₂×#117	C₂²×C₈₉	1	2	yes	—
235	C₂×#117	C₂²×C₈₉	1	2	yes	—
236	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
237	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
238	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
239	#398	C₂²×C₈₉	1	16	yes	—
240	#395	C₂²×C₈₉	1	16	yes	—
241	#391	C₂²×C₈₉	1	8	yes	—
242	#391	C₂²×C₈₉	1	8	yes	—
243	#389	C₂²×C₈₉	1	16	yes	—
244	#385	C₂²×C₈₉	1	16	yes	—
245	#384	C₂²×C₈₉	1	8	yes	—
246	#384	C₂²×C₈₉	1	8	yes	—
247	#382	C₂²×C₈₉	1	8	yes	—
248	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
249	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
250	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
251	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
252	#376	C₂²×C₈₉	1	2	yes	—
253	C₄³	C₂²×C₈₉	1	2	yes	—
254	C₂²×Q₈	—	2	8	no	—
255	C₂²×Q₈	—	4	8	no	—
256	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
257	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
258	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
259	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
260	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
261	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
262	C₂³×C₄	C₂³×C₈₉	1	4	yes	—
263	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
264	C₄⋊Q₈	—	2	8	no	—
265	C₄⋊Q₈	—	2	8	no	—
266	C₄⋊Q₈	—	2	8	no	—
267	C₄⋊Q₈	—	2	8	no	—
268	C₄⋊Q₈	—	2	4	no	—
269	C₄⋊Q₈	—	2	4	no	—
270	C₄⋊Q₈	—	4	16	no	—
271	#125	—	2	8	no	—
272	C₂²⋊Q₈	—	2	8	no	—
273	C₂²⋊Q₈	—	2	8	no	—
274	C₄×Q₈	—	2	8	no	—
275	C₄×Q₈	—	2	8	no	—
276	#117	—	2	8	no	—
277	#117	Q₈×C₈₉	1	16	yes	complemented
278	C₂×C₄⋊C₄	—	2	8	no	—
279	C₂×C₄⋊C₄	—	2	8	no	—
280	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
281	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
282	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
283	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
284	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
285	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
286	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
287	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
288	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
289	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
290	C₂×C₄⋊C₄	C₂³×C₈₉	1	32	yes	—
291	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
292	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
293	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
294	C₂×C₄²	D₄×C₈₉	1	8	yes	—
295	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
296	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
297	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
298	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
299	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
300	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
301	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
302	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
303	#95	C₂³×C₈₉	1	4	yes	—
304	#95	C₂³×C₈₉	1	4	yes	—
305	#95	C₂³×C₈₉	1	32	yes	—
306	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
307	C₂×Q₈	—	2	4	no	—
308	C₂×Q₈	—	2	4	no	—
309	C₂×Q₈	—	2	8	no	—
310	C₂×Q₈	—	2	8	no	—
311	C₂×Q₈	—	2	8	no	—
312	C₂×Q₈	—	2	16	no	—
313	C₂×Q₈	—	2	8	no	—
314	C₂×Q₈	—	4	8	no	—
315	C₂×Q₈	—	4	32	no	—
316	C₂²×C₄	—	2	4	no	—
317	C₂²×C₄	—	2	4	no	—
318	C₂²×C₄	—	2	8	no	—
319	C₂²×C₄	—	4	8	no	—
320	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
321	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
322	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
323	C₂²×C₄	C₂×Q₈×C₈₉	1	8	yes	—
324	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
325	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
326	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
327	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
328	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
329	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
330	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
331	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
332	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
333	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
334	C₄⋊C₄	—	2	8	no	—
335	C₄⋊C₄	—	2	8	no	—
336	C₄⋊C₄	—	2	8	no	—
337	C₄⋊C₄	—	2	8	no	—
338	C₄⋊C₄	—	2	4	no	—
339	C₄⋊C₄	—	2	4	no	—
340	C₄⋊C₄	—	2	8	no	—
341	C₄⋊C₄	—	2	8	no	—
342	C₄⋊C₄	—	2	4	no	—
343	C₄⋊C₄	—	2	4	no	—
344	C₄⋊C₄	—	4	32	no	—
345	C₄⋊C₄	C₂×Q₈×C₈₉	1	16	yes	complemented
346	C₂²⋊C₄	—	2	8	no	—
347	C₂²⋊C₄	C₂×Q₈×C₈₉	1	16	yes	—
348	C₄²	—	2	8	no	—
349	C₄²	—	2	8	no	—
350	C₄²	—	2	8	no	—
351	C₄²	C₂×D₄×C₈₉	1	4	yes	—
352	C₄²	C₂×D₄×C₈₉	1	4	yes	—
353	C₄²	C₂×D₄×C₈₉	1	4	yes	—
354	C₄²	C₂×D₄×C₈₉	1	4	yes	—
355	C₄²	C₂×Q₈×C₈₉	1	16	yes	complemented
356	C₂³	—	2	2	no	—
357	C₂³	—	2	2	no	—
358	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
359	C₂³	C₂²×Q₈×C₈₉	1	4	yes	—
360	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
361	Q₈	—	4	16	no	—
362	Q₈	—	4	8	no	—
363	Q₈	—	4	16	no	—
364	Q₈	—	4	8	no	—
365	C₂×C₄	—	2	8	no	—
366	C₂×C₄	—	2	2	no	—
367	C₂×C₄	—	2	8	no	—
368	C₂×C₄	—	2	2	no	—
369	C₂×C₄	—	2	8	no	—
370	C₂×C₄	—	2	8	no	—
371	C₂×C₄	—	2	4	no	—
372	C₂×C₄	—	2	4	no	—
373	C₂×C₄	—	2	8	no	—
374	C₂×C₄	—	2	8	no	—
375	C₂×C₄	—	2	16	no	—
376	C₂×C₄	—	2	8	no	—
377	C₂×C₄	—	4	16	no	—
378	C₂×C₄	C₄⋊Q₈×C₈₉	1	16	yes	complemented
379	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
380	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
381	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
382	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
383	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
384	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
385	C₂×C₄	C₂²×Q₈×C₈₉	1	16	yes	—
386	C₂×C₄	C₂²×Q₈×C₈₉	1	4	yes	—
387	C₂×C₄	C₂²×Q₈×C₈₉	1	4	yes	—
388	C₂×C₄	H(5,𝔽₂)×C₈₉	1	4	yes	—
389	C₂×C₄	2₋¹⁺⁴×C₈₉	1	4	yes	—
390	C₂²	—	2	2	no	—
391	C₂²	—	2	2	no	—
392	C₂²	—	2	8	no	—
393	C₂²	C₂×C₄⋊Q₈×C₈₉	1	4	yes	—
394	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
395	C₂²	C₂³×Q₈×C₈₉	1	2	yes	—
396	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
397	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	2	yes	—
398	C₄	—	2	2	no	—
399	C₄	—	2	2	no	—
400	C₄	—	2	16	no	—
401	C₄	—	4	8	no	—
402	C₄	—	4	4	no	—
403	C₄	—	4	4	no	—
404	C₄	C₂×C₄⋊Q₈×C₈₉	1	4	yes	—
405	C₄	C₂×C₄⋊Q₈×C₈₉	1	4	yes	—
406	C₄	#535×C₈₉	1	4	yes	—
407	C₄	#536×C₈₉	1	4	yes	—
408	C₂	—	2	4	no	—
409	C₂	C₂²×C₄⋊Q₈×C₈₉	1	1	yes	characteristic
410	C₂	C₂×#535×C₈₉	1	1	yes	characteristic
411	C₂	C₂×#536×C₈₉	1	1	yes	characteristic
412	C₂	C₈₉×#3503	1	4	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
3. C₈₉×#2868	C₂	semidirect
4. C₈₉×#2862	C₂	semidirect
5. C₈₉×#2660	C₂	semidirect
6. C₈₉×#2567	C₂	semidirect
7. C₈₉×#2546	C₂	semidirect
70. #117×C₈₉	Q₈	semidirect
138. C₄⋊C₄×C₈₉	C₂×Q₈	semidirect
148. C₄²×C₈₉	C₂×Q₈	semidirect
171. C₂×C₄×C₈₉	C₄⋊Q₈	semidirect
201. #35281	C₈₉	direct (N × H)
209. #2868	C₂×C₈₉	semidirect
210. #2862	C₂×C₈₉	semidirect
211. #2660	C₂×C₈₉	semidirect
212. #2567	C₂×C₈₉	semidirect
213. #2546	C₂×C₈₉	semidirect
218. C₈₉	#35281	direct (N × H)
277. #117	Q₈×C₈₉	semidirect
345. C₄⋊C₄	C₂×Q₈×C₈₉	semidirect
355. C₄²	C₂×Q₈×C₈₉	semidirect
378. C₂×C₄	C₄⋊Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#533×C₈₉	2
C₂²×C₄⋊Q₈×C₈₉	2
C₈₉×#2868	2
C₈₉×#2862	2
C₈₉×#2660	2
C₈₉×#2567	2
C₈₉×#2546	2
C₈₉×#2502	2
C₈₉×#2434	2
C₈₉×#2317	2
C₄×#117×C₈₉	2
#35281	89

Subgroups — C89×#35281 #36413129

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#35281 #36413129