Subgroups — C₈₉×#34127 #36411975

Order 22784.

2698 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#41×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2355	C₂	1	2	yes	maximal, complemented
3	C₈₉×#2352	C₂	1	4	yes	maximal, complemented
4	D₄×C₄⋊C₄×C₈₉	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2347	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2343	C₂	1	4	yes	maximal, complemented
7	C₈₉×#2339	C₂	1	2	yes	maximal
8	C₈₉×#2295	C₂	1	2	yes	maximal, complemented
9	C₈₉×#2290	C₂	1	4	yes	maximal, complemented
10	C₈₉×#2285	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2283	C₂	1	2	yes	maximal, complemented
12	C₄²×D₄×C₈₉	C₂	1	2	yes	maximal, complemented
13	C₄×#117×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×C₄×C₄⋊C₄×C₈₉	C₂	1	2	yes	maximal, complemented
15	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
16	C₄×#41×C₈₉	C₄	1	16	yes	complemented
17	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
18	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄×D₄×C₈₉	C₂²	1	4	yes	—
20	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
23	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
24	C₂×#117×C₈₉	C₂²	1	2	yes	—
25	C₂×#117×C₈₉	C₂²	1	2	yes	—
26	C₂×#117×C₈₉	C₂²	1	2	yes	—
27	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
28	C₂²×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
29	C₂²×C₄²×C₈₉	C₂²	1	4	yes	—
30	C₂²×C₄²×C₈₉	C₂²	1	2	yes	—
31	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
32	#391×C₈₉	C₂²	1	4	yes	—
33	#389×C₈₉	C₂²	1	4	yes	—
34	#388×C₈₉	C₂²	1	8	yes	—
35	#387×C₈₉	C₂²	1	8	yes	—
36	#387×C₈₉	C₂²	1	8	yes	—
37	#386×C₈₉	C₂²	1	8	yes	—
38	#385×C₈₉	C₂²	1	8	yes	—
39	#384×C₈₉	C₂²	1	2	yes	—
40	#383×C₈₉	C₂²	1	4	yes	—
41	#382×C₈₉	C₂²	1	2	yes	—
42	#382×C₈₉	C₂²	1	4	yes	—
43	#381×C₈₉	C₂²	1	4	yes	—
44	#380×C₈₉	C₂²	1	2	yes	—
45	#380×C₈₉	C₂²	1	4	yes	—
46	#380×C₈₉	C₂²	1	2	yes	—
47	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
48	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
49	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
50	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
51	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
52	C₄×C₄⋊C₄×C₈₉	C₂²	1	4	yes	—
53	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
54	C₄×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
55	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
56	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
57	C₄×C₂²⋊C₄×C₈₉	C₂²	1	8	yes	—
58	C₄×C₂²⋊C₄×C₈₉	C₂²	1	2	yes	—
59	C₄×C₂²⋊C₄×C₈₉	C₂²	1	4	yes	—
60	#376×C₈₉	C₂²	1	4	yes	—
61	#376×C₈₉	C₂²	1	2	yes	—
62	C₂×#95×C₈₉	C₂²	1	4	yes	—
63	C₄³×C₈₉	C₂²	1	2	yes	—
64	C₂×#41×C₈₉	—	2	8	no	—
65	C₂×#41×C₈₉	C₂×C₄	1	8	yes	—
66	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
67	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
68	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
69	C₂³×C₄×C₈₉	C₂³	1	8	yes	—
70	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
71	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
72	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
73	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
74	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
75	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
76	C₄×Q₈×C₈₉	C₂×C₄	1	8	yes	complemented
77	C₄×Q₈×C₈₉	C₂×C₄	1	8	yes	complemented
78	C₄×D₄×C₈₉	—	2	16	no	—
79	C₄×D₄×C₈₉	C₂×C₄	1	8	yes	—
80	C₄×D₄×C₈₉	C₂×C₄	1	8	yes	—
81	C₄×D₄×C₈₉	C₂×C₄	1	16	yes	complemented
82	C₄×D₄×C₈₉	C₂×C₄	1	16	yes	complemented
83	#117×C₈₉	—	2	8	no	—
84	#117×C₈₉	—	4	4	no	—
85	#117×C₈₉	C₂×C₄	1	16	yes	complemented
86	#117×C₈₉	C₂×C₄	1	8	yes	complemented
87	C₂×C₄⋊C₄×C₈₉	—	2	16	no	—
88	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
90	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
91	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
92	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
93	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
94	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
95	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
96	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
97	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
98	C₂×C₂²⋊C₄×C₈₉	C₂³	1	8	yes	—
99	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
100	C₂×C₂²⋊C₄×C₈₉	C₂³	1	4	yes	—
101	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
102	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
103	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
104	C₂×C₄²×C₈₉	—	4	8	no	—
105	C₂×C₄²×C₈₉	C₂×C₄	1	8	yes	complemented
106	C₂×C₄²×C₈₉	C₂×C₄	1	16	yes	complemented
107	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
108	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
109	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
110	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
111	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
112	C₂×C₄²×C₈₉	C₂³	1	8	yes	—
113	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
114	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
115	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
116	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
117	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
118	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
119	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
120	#95×C₈₉	C₂³	1	2	yes	—
121	#95×C₈₉	C₂³	1	4	yes	—
122	#95×C₈₉	C₂³	1	8	yes	—
123	#95×C₈₉	C₂³	1	2	yes	—
124	#95×C₈₉	C₂³	1	4	yes	—
125	#95×C₈₉	C₂³	1	8	yes	—
126	#95×C₈₉	C₂³	1	8	yes	—
127	#95×C₈₉	C₂³	1	4	yes	—
128	#95×C₈₉	C₂³	1	8	yes	—
129	C₂⁴×C₈₉	C₂⁴	1	1	yes	characteristic
130	C₂⁴×C₈₉	C₂⁴	1	2	yes	—
131	#41×C₈₉	—	2	32	no	—
132	C₂×Q₈×C₈₉	—	2	4	no	—
133	C₂×Q₈×C₈₉	C₂²×C₄	1	4	yes	—
134	C₂×D₄×C₈₉	—	2	8	no	—
135	C₂×D₄×C₈₉	C₂²×C₄	1	8	yes	—
136	C₂×D₄×C₈₉	C₂²×C₄	1	4	yes	—
137	C₂×D₄×C₈₉	#41	1	8	yes	—
138	C₂²×C₄×C₈₉	—	2	8	no	—
139	C₂²×C₄×C₈₉	—	2	8	no	—
140	C₂²×C₄×C₈₉	—	2	4	no	—
141	C₂²×C₄×C₈₉	—	2	4	no	—
142	C₂²×C₄×C₈₉	—	2	4	no	—
143	C₂²×C₄×C₈₉	—	2	2	no	—
144	C₂²×C₄×C₈₉	—	2	4	no	—
145	C₂²×C₄×C₈₉	—	2	2	no	—
146	C₂²×C₄×C₈₉	—	2	16	no	—
147	C₂²×C₄×C₈₉	—	2	16	no	—
148	C₂²×C₄×C₈₉	—	4	8	no	—
149	C₂²×C₄×C₈₉	C₂²×C₄	1	8	yes	—
150	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
151	C₂²×C₄×C₈₉	C₂²×C₄	1	8	yes	—
152	C₂²×C₄×C₈₉	C₂²×C₄	1	8	yes	—
153	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
154	C₂²×C₄×C₈₉	C₂²×C₄	1	4	yes	—
155	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
156	C₂²×C₄×C₈₉	C₂⁴	1	8	yes	—
157	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
158	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
159	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
160	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
161	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
162	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
163	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
164	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
165	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
166	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
167	C₄⋊C₄×C₈₉	—	2	16	no	—
168	C₄⋊C₄×C₈₉	—	2	8	no	—
169	C₄⋊C₄×C₈₉	—	2	4	no	—
170	C₄⋊C₄×C₈₉	—	2	8	no	—
171	C₄⋊C₄×C₈₉	—	4	4	no	—
172	C₄⋊C₄×C₈₉	C₂²×C₄	1	4	yes	—
173	C₄⋊C₄×C₈₉	C₂²×C₄	1	4	yes	—
174	C₄⋊C₄×C₈₉	C₂²×C₄	1	8	yes	—
175	C₄⋊C₄×C₈₉	C₂²×C₄	1	8	yes	—
176	C₄⋊C₄×C₈₉	#41	1	8	yes	complemented
177	C₂²⋊C₄×C₈₉	—	2	16	no	—
178	C₂²⋊C₄×C₈₉	—	2	8	no	—
179	C₂²⋊C₄×C₈₉	—	4	4	no	—
180	C₂²⋊C₄×C₈₉	C₂²×C₄	1	8	yes	—
181	C₂²⋊C₄×C₈₉	C₂²×C₄	1	8	yes	—
182	C₂²⋊C₄×C₈₉	C₂²×C₄	1	8	yes	—
183	C₄²×C₈₉	—	2	8	no	—
184	C₄²×C₈₉	—	2	8	no	—
185	C₄²×C₈₉	—	2	4	no	—
186	C₄²×C₈₉	—	2	8	no	—
187	C₄²×C₈₉	—	2	4	no	—
188	C₄²×C₈₉	—	4	8	no	—
189	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
190	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
191	C₄²×C₈₉	C₂²×C₄	1	8	yes	—
192	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
193	C₄²×C₈₉	C₂²×C₄	1	4	yes	—
194	C₄²×C₈₉	C₂²×C₄	1	8	yes	—
195	C₄²×C₈₉	C₂²×C₄	1	8	yes	—
196	C₂³×C₈₉	—	2	4	no	—
197	C₂³×C₈₉	—	2	2	no	—
198	C₂³×C₈₉	—	2	2	no	—
199	C₂³×C₈₉	—	2	2	no	—
200	C₂³×C₈₉	—	2	2	no	—
201	C₂³×C₈₉	—	2	1	no	—
202	C₂³×C₈₉	—	2	1	no	—
203	C₂³×C₈₉	C₂³×C₄	1	2	yes	—
204	C₂³×C₈₉	C₂³×C₄	1	4	yes	—
205	C₂³×C₈₉	C₂×#41	1	8	yes	—
206	C₂³×C₈₉	C₂⁵	1	1	yes	characteristic
207	Q₈×C₈₉	—	2	8	no	—
208	D₄×C₈₉	—	2	16	no	—
209	D₄×C₈₉	#117	1	16	yes	complemented
210	C₂×C₄×C₈₉	—	2	16	no	—
211	C₂×C₄×C₈₉	—	2	2	no	—
212	C₂×C₄×C₈₉	—	2	4	no	—
213	C₂×C₄×C₈₉	—	2	4	no	—
214	C₂×C₄×C₈₉	—	2	2	no	—
215	C₂×C₄×C₈₉	—	2	4	no	—
216	C₂×C₄×C₈₉	—	2	4	no	—
217	C₂×C₄×C₈₉	—	2	2	no	—
218	C₂×C₄×C₈₉	—	2	1	no	—
219	C₂×C₄×C₈₉	—	2	2	no	—
220	C₂×C₄×C₈₉	—	2	1	no	—
221	C₂×C₄×C₈₉	—	2	8	no	—
222	C₂×C₄×C₈₉	—	2	16	no	—
223	C₂×C₄×C₈₉	—	2	8	no	—
224	C₂×C₄×C₈₉	—	2	16	no	—
225	C₂×C₄×C₈₉	—	2	8	no	—
226	C₂×C₄×C₈₉	—	2	4	no	—
227	C₂×C₄×C₈₉	—	2	4	no	—
228	C₂×C₄×C₈₉	—	2	8	no	—
229	C₂×C₄×C₈₉	—	2	8	no	—
230	C₂×C₄×C₈₉	—	2	8	no	—
231	C₂×C₄×C₈₉	—	2	4	no	—
232	C₂×C₄×C₈₉	—	4	32	no	—
233	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
234	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
235	C₂×C₄×C₈₉	C₂³×C₄	1	4	yes	—
236	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
237	C₂×C₄×C₈₉	C₂³×C₄	1	4	yes	—
238	C₂×C₄×C₈₉	C₂³×C₄	1	4	yes	—
239	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
240	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
241	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
242	C₂×C₄×C₈₉	C₂×#41	1	8	yes	—
243	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
244	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
245	C₂×C₄×C₈₉	H(5,𝔽₂)	1	2	yes	—
246	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
247	C₂²×C₈₉	—	2	4	no	—
248	C₂²×C₈₉	—	2	4	no	—
249	C₂²×C₈₉	—	2	8	no	—
250	C₂²×C₈₉	—	2	8	no	—
251	C₂²×C₈₉	—	2	2	no	—
252	C₂²×C₈₉	—	2	2	no	—
253	C₂²×C₈₉	—	2	2	no	—
254	C₂²×C₈₉	—	4	2	no	—
255	C₂²×C₈₉	—	4	2	no	—
256	C₂²×C₈₉	C₂×#117	1	8	yes	—
257	C₂²×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
258	C₂²×C₈₉	C₂²×#41	1	2	yes	—
259	C₂²×C₈₉	C₂²×#41	1	2	yes	—
260	C₂²×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
261	C₂²×C₈₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
262	C₄×C₈₉	—	2	2	no	—
263	C₄×C₈₉	—	2	4	no	—
264	C₄×C₈₉	—	2	2	no	—
265	C₄×C₈₉	—	2	4	no	—
266	C₄×C₈₉	—	2	4	no	—
267	C₄×C₈₉	—	2	2	no	—
268	C₄×C₈₉	—	2	8	no	—
269	C₄×C₈₉	—	4	8	no	—
270	C₄×C₈₉	—	4	4	no	—
271	C₄×C₈₉	C₂×#117	1	4	yes	—
272	C₄×C₈₉	C₄×#41	1	4	yes	—
273	C₄×C₈₉	#520	1	4	yes	—
274	#34127	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
275	C₂×C₈₉	—	2	8	no	—
276	C₂×C₈₉	—	4	2	no	—
277	C₂×C₈₉	C₂²×#117	1	1	yes	characteristic
278	C₂×C₈₉	C₂×C₄×#41	1	1	yes	characteristic
279	C₂×C₈₉	C₂×#520	1	1	yes	characteristic
280	C₂×C₈₉	#3481	1	2	yes	—
281	C₂×C₈₉	#3482	1	2	yes	—
282	C₂×C₄×#41	C₂×C₈₉	1	1	yes	characteristic
283	#2355	C₂×C₈₉	1	2	yes	complemented
284	#2352	C₂×C₈₉	1	4	yes	complemented
285	D₄×C₄⋊C₄	C₂×C₈₉	1	2	yes	complemented
286	#2347	C₂×C₈₉	1	2	yes	complemented
287	#2343	C₂×C₈₉	1	4	yes	complemented
288	#2339	C₂×C₈₉	1	2	yes	—
289	#2295	C₂×C₈₉	1	2	yes	complemented
290	#2290	C₂×C₈₉	1	4	yes	complemented
291	#2285	C₂×C₈₉	1	1	yes	characteristic, complemented
292	#2283	C₂×C₈₉	1	2	yes	complemented
293	C₄²×D₄	C₂×C₈₉	1	2	yes	complemented
294	C₄×#117	C₂×C₈₉	1	1	yes	characteristic, complemented
295	C₂×C₄×C₄⋊C₄	C₂×C₈₉	1	2	yes	complemented
296	C₈₉	#34127	1	1	yes	characteristic, direct product factor, Sylow
297	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
298	C₄×#41	C₄×C₈₉	1	16	yes	complemented
299	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
300	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
301	C₂×C₄×D₄	C₂²×C₈₉	1	4	yes	—
302	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
303	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
304	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
305	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
306	C₂×#117	C₂²×C₈₉	1	2	yes	—
307	C₂×#117	C₂²×C₈₉	1	2	yes	—
308	C₂×#117	C₂²×C₈₉	1	2	yes	—
309	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
310	C₂²×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
311	C₂²×C₄²	C₂²×C₈₉	1	4	yes	—
312	C₂²×C₄²	C₂²×C₈₉	1	2	yes	—
313	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
314	#391	C₂²×C₈₉	1	4	yes	—
315	#389	C₂²×C₈₉	1	4	yes	—
316	#388	C₂²×C₈₉	1	8	yes	—
317	#387	C₂²×C₈₉	1	8	yes	—
318	#387	C₂²×C₈₉	1	8	yes	—
319	#386	C₂²×C₈₉	1	8	yes	—
320	#385	C₂²×C₈₉	1	8	yes	—
321	#384	C₂²×C₈₉	1	2	yes	—
322	#383	C₂²×C₈₉	1	4	yes	—
323	#382	C₂²×C₈₉	1	2	yes	—
324	#382	C₂²×C₈₉	1	4	yes	—
325	#381	C₂²×C₈₉	1	4	yes	—
326	#380	C₂²×C₈₉	1	2	yes	—
327	#380	C₂²×C₈₉	1	4	yes	—
328	#380	C₂²×C₈₉	1	2	yes	—
329	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
330	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
331	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
332	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
333	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
334	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
335	C₄×C₄⋊C₄	C₂²×C₈₉	1	4	yes	—
336	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
337	C₄×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
338	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
339	C₄×C₂²⋊C₄	C₂²×C₈₉	1	8	yes	—
340	C₄×C₂²⋊C₄	C₂²×C₈₉	1	2	yes	—
341	C₄×C₂²⋊C₄	C₂²×C₈₉	1	4	yes	—
342	#376	C₂²×C₈₉	1	2	yes	—
343	#376	C₂²×C₈₉	1	4	yes	—
344	C₂×#95	C₂²×C₈₉	1	4	yes	—
345	C₄³	C₂²×C₈₉	1	2	yes	—
346	C₂×#41	—	2	8	no	—
347	C₂×#41	C₂×C₄×C₈₉	1	8	yes	—
348	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
349	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
350	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
351	C₂³×C₄	C₂³×C₈₉	1	8	yes	—
352	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
353	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
354	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
355	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
356	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
357	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
358	C₄×Q₈	C₂×C₄×C₈₉	1	8	yes	complemented
359	C₄×Q₈	C₂×C₄×C₈₉	1	8	yes	complemented
360	C₄×D₄	—	2	16	no	—
361	C₄×D₄	C₂×C₄×C₈₉	1	8	yes	—
362	C₄×D₄	C₂×C₄×C₈₉	1	8	yes	—
363	C₄×D₄	C₂×C₄×C₈₉	1	16	yes	complemented
364	C₄×D₄	C₂×C₄×C₈₉	1	16	yes	complemented
365	#117	—	2	8	no	—
366	#117	—	4	4	no	—
367	#117	C₂×C₄×C₈₉	1	16	yes	complemented
368	#117	C₂×C₄×C₈₉	1	8	yes	complemented
369	C₂×C₄⋊C₄	—	2	16	no	—
370	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
371	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
372	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
373	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
374	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
375	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
376	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
377	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
378	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
379	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
380	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
381	C₂×C₂²⋊C₄	C₂³×C₈₉	1	8	yes	—
382	C₂×C₂²⋊C₄	C₂³×C₈₉	1	4	yes	—
383	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
384	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
385	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
386	C₂×C₄²	—	4	8	no	—
387	C₂×C₄²	C₂×C₄×C₈₉	1	8	yes	complemented
388	C₂×C₄²	C₂×C₄×C₈₉	1	16	yes	complemented
389	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
390	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
391	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
392	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
393	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
394	C₂×C₄²	C₂³×C₈₉	1	8	yes	—
395	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
396	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
397	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
398	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
399	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
400	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
401	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
402	#95	C₂³×C₈₉	1	2	yes	—
403	#95	C₂³×C₈₉	1	4	yes	—
404	#95	C₂³×C₈₉	1	8	yes	—
405	#95	C₂³×C₈₉	1	2	yes	—
406	#95	C₂³×C₈₉	1	4	yes	—
407	#95	C₂³×C₈₉	1	8	yes	—
408	#95	C₂³×C₈₉	1	8	yes	—
409	#95	C₂³×C₈₉	1	4	yes	—
410	#95	C₂³×C₈₉	1	8	yes	—
411	C₂⁴	C₂⁴×C₈₉	1	1	yes	characteristic
412	C₂⁴	C₂⁴×C₈₉	1	2	yes	—
413	#41	—	2	32	no	—
414	C₂×Q₈	—	2	4	no	—
415	C₂×Q₈	C₂²×C₄×C₈₉	1	4	yes	—
416	C₂×D₄	—	2	8	no	—
417	C₂×D₄	C₂²×C₄×C₈₉	1	8	yes	—
418	C₂×D₄	C₂²×C₄×C₈₉	1	4	yes	—
419	C₂×D₄	#41×C₈₉	1	8	yes	—
420	C₂²×C₄	—	2	8	no	—
421	C₂²×C₄	—	2	8	no	—
422	C₂²×C₄	—	2	4	no	—
423	C₂²×C₄	—	2	4	no	—
424	C₂²×C₄	—	2	4	no	—
425	C₂²×C₄	—	2	2	no	—
426	C₂²×C₄	—	2	4	no	—
427	C₂²×C₄	—	2	2	no	—
428	C₂²×C₄	—	2	16	no	—
429	C₂²×C₄	—	2	16	no	—
430	C₂²×C₄	—	4	8	no	—
431	C₂²×C₄	C₂²×C₄×C₈₉	1	8	yes	—
432	C₂²×C₄	C₂²×C₄×C₈₉	1	4	yes	—
433	C₂²×C₄	C₂²×C₄×C₈₉	1	8	yes	—
434	C₂²×C₄	C₂²×C₄×C₈₉	1	8	yes	—
435	C₂²×C₄	C₂²×C₄×C₈₉	1	4	yes	—
436	C₂²×C₄	C₂²×C₄×C₈₉	1	4	yes	—
437	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
438	C₂²×C₄	C₂⁴×C₈₉	1	8	yes	—
439	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
440	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
441	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
442	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
443	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
444	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
445	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
446	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
447	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
448	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
449	C₄⋊C₄	—	2	16	no	—
450	C₄⋊C₄	—	2	8	no	—
451	C₄⋊C₄	—	2	4	no	—
452	C₄⋊C₄	—	2	8	no	—
453	C₄⋊C₄	—	4	4	no	—
454	C₄⋊C₄	C₂²×C₄×C₈₉	1	4	yes	—
455	C₄⋊C₄	C₂²×C₄×C₈₉	1	4	yes	—
456	C₄⋊C₄	C₂²×C₄×C₈₉	1	8	yes	—
457	C₄⋊C₄	C₂²×C₄×C₈₉	1	8	yes	—
458	C₄⋊C₄	#41×C₈₉	1	8	yes	complemented
459	C₂²⋊C₄	—	2	16	no	—
460	C₂²⋊C₄	—	2	8	no	—
461	C₂²⋊C₄	—	4	4	no	—
462	C₂²⋊C₄	C₂²×C₄×C₈₉	1	8	yes	—
463	C₂²⋊C₄	C₂²×C₄×C₈₉	1	8	yes	—
464	C₂²⋊C₄	C₂²×C₄×C₈₉	1	8	yes	—
465	C₄²	—	2	8	no	—
466	C₄²	—	2	8	no	—
467	C₄²	—	2	4	no	—
468	C₄²	—	2	8	no	—
469	C₄²	—	2	4	no	—
470	C₄²	—	4	8	no	—
471	C₄²	C₂²×C₄×C₈₉	1	4	yes	—
472	C₄²	C₂²×C₄×C₈₉	1	4	yes	—
473	C₄²	C₂²×C₄×C₈₉	1	8	yes	—
474	C₄²	C₂²×C₄×C₈₉	1	4	yes	—
475	C₄²	C₂²×C₄×C₈₉	1	4	yes	—
476	C₄²	C₂²×C₄×C₈₉	1	8	yes	—
477	C₄²	C₂²×C₄×C₈₉	1	8	yes	—
478	C₂³	—	2	4	no	—
479	C₂³	—	2	2	no	—
480	C₂³	—	2	2	no	—
481	C₂³	—	2	2	no	—
482	C₂³	—	2	2	no	—
483	C₂³	—	2	1	no	—
484	C₂³	—	2	1	no	—
485	C₂³	C₂³×C₄×C₈₉	1	2	yes	—
486	C₂³	C₂³×C₄×C₈₉	1	4	yes	—
487	C₂³	C₂×#41×C₈₉	1	8	yes	—
488	C₂³	C₂⁵×C₈₉	1	1	yes	characteristic
489	Q₈	—	2	8	no	—
490	D₄	—	2	16	no	—
491	D₄	#117×C₈₉	1	16	yes	complemented
492	C₂×C₄	—	2	16	no	—
493	C₂×C₄	—	2	2	no	—
494	C₂×C₄	—	2	4	no	—
495	C₂×C₄	—	2	4	no	—
496	C₂×C₄	—	2	2	no	—
497	C₂×C₄	—	2	4	no	—
498	C₂×C₄	—	2	4	no	—
499	C₂×C₄	—	2	2	no	—
500	C₂×C₄	—	2	1	no	—
501	C₂×C₄	—	2	2	no	—
502	C₂×C₄	—	2	1	no	—
503	C₂×C₄	—	2	8	no	—
504	C₂×C₄	—	2	16	no	—
505	C₂×C₄	—	2	8	no	—
506	C₂×C₄	—	2	16	no	—
507	C₂×C₄	—	2	8	no	—
508	C₂×C₄	—	2	4	no	—
509	C₂×C₄	—	2	4	no	—
510	C₂×C₄	—	2	8	no	—
511	C₂×C₄	—	2	8	no	—
512	C₂×C₄	—	2	8	no	—
513	C₂×C₄	—	2	4	no	—
514	C₂×C₄	—	4	32	no	—
515	C₂×C₄	C₂³×C₄×C₈₉	1	2	yes	—
516	C₂×C₄	C₂³×C₄×C₈₉	1	2	yes	—
517	C₂×C₄	C₂³×C₄×C₈₉	1	4	yes	—
518	C₂×C₄	C₂³×C₄×C₈₉	1	2	yes	—
519	C₂×C₄	C₂³×C₄×C₈₉	1	4	yes	—
520	C₂×C₄	C₂³×C₄×C₈₉	1	4	yes	—
521	C₂×C₄	C₂³×C₄×C₈₉	1	2	yes	—
522	C₂×C₄	C₂³×C₄×C₈₉	1	2	yes	—
523	C₂×C₄	C₂³×C₄×C₈₉	1	2	yes	—
524	C₂×C₄	C₂×#41×C₈₉	1	8	yes	—
525	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
526	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
527	C₂×C₄	H(5,𝔽₂)×C₈₉	1	2	yes	—
528	C₂×C₄	2₋¹⁺⁴×C₈₉	1	2	yes	—
529	C₂²	—	2	4	no	—
530	C₂²	—	2	4	no	—
531	C₂²	—	2	8	no	—
532	C₂²	—	2	8	no	—
533	C₂²	—	2	2	no	—
534	C₂²	—	2	2	no	—
535	C₂²	—	2	2	no	—
536	C₂²	—	4	2	no	—
537	C₂²	—	4	2	no	—
538	C₂²	C₂×#117×C₈₉	1	8	yes	—
539	C₂²	C₂⁴×C₄×C₈₉	1	1	yes	characteristic
540	C₂²	C₂²×#41×C₈₉	1	2	yes	—
541	C₂²	C₂²×#41×C₈₉	1	2	yes	—
542	C₂²	C₂×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
543	C₂²	C₂×2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
544	C₄	—	2	2	no	—
545	C₄	—	2	4	no	—
546	C₄	—	2	2	no	—
547	C₄	—	2	4	no	—
548	C₄	—	2	4	no	—
549	C₄	—	2	2	no	—
550	C₄	—	2	8	no	—
551	C₄	—	4	8	no	—
552	C₄	—	4	4	no	—
553	C₄	C₂×#117×C₈₉	1	4	yes	—
554	C₄	C₄×#41×C₈₉	1	4	yes	—
555	C₄	#520×C₈₉	1	4	yes	—
556	C₂	—	2	8	no	—
557	C₂	—	4	2	no	—
558	C₂	C₂²×#117×C₈₉	1	1	yes	characteristic
559	C₂	C₂×C₄×#41×C₈₉	1	1	yes	characteristic
560	C₂	C₂×#520×C₈₉	1	1	yes	characteristic
561	C₂	C₈₉×#3481	1	2	yes	—
562	C₂	C₈₉×#3482	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₈₉	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₈₉
Frattini, cosocle	C₂³	C₂⁵×C₈₉
socle	C₂³×C₈₉	C₂⁵
prime radical	C₂²×#41×C₈₉	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#2355	C₂	semidirect
3. C₈₉×#2352	C₂	semidirect
4. D₄×C₄⋊C₄×C₈₉	C₂	semidirect
5. C₈₉×#2347	C₂	semidirect
6. C₈₉×#2343	C₂	semidirect
8. C₈₉×#2295	C₂	semidirect
9. C₈₉×#2290	C₂	semidirect
10. C₈₉×#2285	C₂	semidirect
11. C₈₉×#2283	C₂	semidirect
12. C₄²×D₄×C₈₉	C₂	semidirect
13. C₄×#117×C₈₉	C₂	semidirect
14. C₂×C₄×C₄⋊C₄×C₈₉	C₂	semidirect
16. C₄×#41×C₈₉	C₄	semidirect
76. C₄×Q₈×C₈₉	C₂×C₄	semidirect
77. C₄×Q₈×C₈₉	C₂×C₄	semidirect
81. C₄×D₄×C₈₉	C₂×C₄	semidirect
82. C₄×D₄×C₈₉	C₂×C₄	semidirect
85. #117×C₈₉	C₂×C₄	semidirect
86. #117×C₈₉	C₂×C₄	semidirect
105. C₂×C₄²×C₈₉	C₂×C₄	semidirect
106. C₂×C₄²×C₈₉	C₂×C₄	semidirect
176. C₄⋊C₄×C₈₉	#41	semidirect
209. D₄×C₈₉	#117	semidirect
274. #34127	C₈₉	direct (N × H)
283. #2355	C₂×C₈₉	semidirect
284. #2352	C₂×C₈₉	semidirect
285. D₄×C₄⋊C₄	C₂×C₈₉	semidirect
286. #2347	C₂×C₈₉	semidirect
287. #2343	C₂×C₈₉	semidirect
289. #2295	C₂×C₈₉	semidirect
290. #2290	C₂×C₈₉	semidirect
291. #2285	C₂×C₈₉	semidirect
292. #2283	C₂×C₈₉	semidirect
293. C₄²×D₄	C₂×C₈₉	semidirect
294. C₄×#117	C₂×C₈₉	semidirect
295. C₂×C₄×C₄⋊C₄	C₂×C₈₉	semidirect
296. C₈₉	#34127	direct (N × H)
298. C₄×#41	C₄×C₈₉	semidirect
358. C₄×Q₈	C₂×C₄×C₈₉	semidirect
359. C₄×Q₈	C₂×C₄×C₈₉	semidirect
363. C₄×D₄	C₂×C₄×C₈₉	semidirect
364. C₄×D₄	C₂×C₄×C₈₉	semidirect
367. #117	C₂×C₄×C₈₉	semidirect
368. #117	C₂×C₄×C₈₉	semidirect
387. C₂×C₄²	C₂×C₄×C₈₉	semidirect
388. C₂×C₄²	C₂×C₄×C₈₉	semidirect
458. C₄⋊C₄	#41×C₈₉	semidirect
491. D₄	#117×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#41×C₈₉	2
C₈₉×#2355	2
C₈₉×#2352	2
D₄×C₄⋊C₄×C₈₉	2
C₈₉×#2347	2
C₈₉×#2343	2
C₈₉×#2339	2
C₈₉×#2295	2
C₈₉×#2290	2
C₈₉×#2285	2
C₈₉×#2283	2
C₄²×D₄×C₈₉	2
C₄×#117×C₈₉	2
C₂×C₄×C₄⋊C₄×C₈₉	2
#34127	89

Subgroups — C89×#34127 #36411975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#34127 #36411975