Subgroups — C₈₉×#60328 #36438176

Order 22784.

1920 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3483	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#2991	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#2990	C₂	1	3	yes	maximal, complemented
4	C₈₉×#2988	C₂	1	2	yes	maximal, complemented
5	C₈₉×#2981	C₂	1	6	yes	maximal, complemented
6	C₈₉×#2979	C₂	1	3	yes	maximal, complemented
7	C₈₉×#2971	C₂	1	2	yes	maximal, complemented
8	C₈₉×#2965	C₂	1	3	yes	maximal, complemented
9	C₈×#41×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#2935	C₂	1	6	yes	maximal, complemented
11	C₈₉×#2931	C₂	1	3	yes	maximal, complemented
12	C₂×#131×C₈₉	C₂²	1	6	yes	—
13	Q₈²×C₈₉	C₄	1	1	yes	characteristic
14	#555×C₈₉	C₄	1	6	yes	—
15	#555×C₈₉	C₄	1	3	yes	—
16	#550×C₈₉	C₄	1	1	yes	characteristic
17	#550×C₈₉	C₄	1	2	yes	—
18	#550×C₈₉	C₄	1	3	yes	—
19	#532×C₈₉	C₂²	1	3	yes	—
20	#532×C₈₉	C₂²	1	6	yes	—
21	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
22	C₄×#41×C₈₉	C₂²	1	3	yes	—
23	C₄×#41×C₈₉	C₂²	1	2	yes	—
24	#446×C₈₉	C₂²	1	1	yes	characteristic
25	#446×C₈₉	C₂²	1	3	yes	—
26	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₈×Q₈×C₈₉	C₂²	1	2	yes	—
29	#436×C₈₉	C₂²	1	3	yes	—
30	#436×C₈₉	C₂²	1	6	yes	—
31	#436×C₈₉	C₂²	1	6	yes	—
32	#436×C₈₉	C₂²	1	3	yes	—
33	#436×C₈₉	C₂²	1	6	yes	—
34	#435×C₈₉	C₂²	1	6	yes	—
35	#435×C₈₉	C₂²	1	12	yes	—
36	#435×C₈₉	C₂²	1	6	yes	—
37	C₈×D₄×C₈₉	C₂²	1	6	yes	—
38	C₈×D₄×C₈₉	C₂²	1	6	yes	—
39	C₈×D₄×C₈₉	C₂²	1	3	yes	—
40	C₈×D₄×C₈₉	C₂²	1	6	yes	—
41	C₈×D₄×C₈₉	C₂²	1	3	yes	—
42	#432×C₈₉	C₂²	1	6	yes	—
43	#431×C₈₉	C₂²	1	3	yes	—
44	#431×C₈₉	C₂²	1	3	yes	—
45	#431×C₈₉	C₂²	1	6	yes	—
46	#423×C₈₉	C₂²	1	3	yes	—
47	C₂×C₄⋊C₈×C₈₉	C₂²	1	6	yes	—
48	#408×C₈₉	C₂²	1	6	yes	—
49	#408×C₈₉	C₂²	1	12	yes	—
50	C₄×M₁₆×C₈₉	C₂²	1	6	yes	—
51	C₂×C₄×C₈×C₈₉	C₂²	1	3	yes	—
52	C₂×#41×C₈₉	C₂×C₄	1	1	yes	characteristic
53	C₂×#41×C₈₉	C₂×C₄	1	2	yes	—
54	C₂×#41×C₈₉	C₂×C₄	1	3	yes	—
55	C₂×#41×C₈₉	C₂³	1	6	yes	—
56	#131×C₈₉	—	4	12	no	—
57	C₂×M₁₆×C₈₉	C₂³	1	6	yes	—
58	C₂×M₁₆×C₈₉	C₂³	1	6	yes	—
59	C₂×M₁₆×C₈₉	C₂³	1	6	yes	—
60	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
61	C₂²×C₈×C₈₉	C₂³	1	3	yes	—
62	C₂²×C₈×C₈₉	C₂³	1	12	yes	—
63	C₄⋊Q₈×C₈₉	C₂×C₄	1	3	yes	—
64	C₄⋊Q₈×C₈₉	C₂×C₄	1	6	yes	—
65	#127×C₈₉	C₂×C₄	1	3	yes	—
66	#127×C₈₉	C₂×C₄	1	6	yes	—
67	#124×C₈₉	C₂×C₄	1	6	yes	—
68	#124×C₈₉	C₂×C₄	1	12	yes	—
69	#121×C₈₉	C₂×C₄	1	6	yes	—
70	#121×C₈₉	C₂×C₄	1	6	yes	—
71	#121×C₈₉	C₂×C₄	1	6	yes	—
72	#121×C₈₉	C₂×C₄	1	6	yes	—
73	#121×C₈₉	C₂×C₄	1	12	yes	—
74	C₄×Q₈×C₈₉	C₂×C₄	1	3	yes	—
75	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
76	C₄×Q₈×C₈₉	C₂×C₄	1	2	yes	—
77	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
78	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
79	C₄×D₄×C₈₉	C₂×C₄	1	3	yes	—
80	C₄×D₄×C₈₉	C₂×C₄	1	3	yes	—
81	C₄×D₄×C₈₉	C₂×C₄	1	6	yes	—
82	C₄×D₄×C₈₉	C₂×C₄	1	6	yes	—
83	C₄×D₄×C₈₉	C₂³	1	3	yes	—
84	C₄×D₄×C₈₉	C₂³	1	6	yes	—
85	C₄×D₄×C₈₉	C₂³	1	3	yes	—
86	C₄×D₄×C₈₉	C₂³	1	6	yes	—
87	#117×C₈₉	C₂×C₄	1	3	yes	—
88	#117×C₈₉	C₂×C₄	1	3	yes	—
89	#117×C₈₉	C₂×C₄	1	6	yes	—
90	#117×C₈₉	C₂×C₄	1	6	yes	—
91	C₂×C₄²×C₈₉	C₂³	1	3	yes	—
92	C₂×C₄²×C₈₉	C₂³	1	6	yes	—
93	C₄⋊C₈×C₈₉	C₂³	1	3	yes	—
94	C₄⋊C₈×C₈₉	C₂³	1	6	yes	—
95	C₄⋊C₈×C₈₉	C₂³	1	6	yes	—
96	C₄⋊C₈×C₈₉	C₂³	1	3	yes	—
97	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
98	C₄⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
99	C₄⋊C₈×C₈₉	C₂³	1	3	yes	—
100	C₂²⋊C₈×C₈₉	C₂³	1	6	yes	—
101	C₂²⋊C₈×C₈₉	C₂³	1	6	yes	—
102	C₂²⋊C₈×C₈₉	C₂³	1	6	yes	—
103	C₂²⋊C₈×C₈₉	C₂³	1	6	yes	—
104	C₂²⋊C₈×C₈₉	C₂³	1	12	yes	—
105	#97×C₈₉	C₂³	1	2	yes	—
106	#97×C₈₉	C₂³	1	6	yes	—
107	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
108	C₄×C₈×C₈₉	C₂³	1	3	yes	—
109	C₄×C₈×C₈₉	C₂³	1	3	yes	—
110	C₄×C₈×C₈₉	C₂³	1	3	yes	—
111	C₄×C₈×C₈₉	C₂³	1	6	yes	—
112	#41×C₈₉	—	2	4	no	—
113	#41×C₈₉	—	2	8	no	—
114	#41×C₈₉	—	4	6	no	—
115	#41×C₈₉	—	4	6	no	—
116	#41×C₈₉	—	4	6	no	—
117	C₂×Q₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
118	C₂×Q₈×C₈₉	C₂²×C₄	1	6	yes	—
119	C₂×Q₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
120	C₂×D₄×C₈₉	C₂²×C₄	1	3	yes	—
121	C₂×D₄×C₈₉	C₂²×C₄	1	6	yes	—
122	C₂×D₄×C₈₉	C₂²×C₄	1	3	yes	—
123	C₂×D₄×C₈₉	C₂²×C₄	1	6	yes	—
124	C₂×D₄×C₈₉	C₂²×C₄	1	12	yes	—
125	C₂×D₄×C₈₉	C₂²×C₄	1	6	yes	—
126	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
127	C₂²×C₄×C₈₉	C₂²×C₄	1	6	yes	—
128	C₂²×C₄×C₈₉	C₂²×C₄	1	3	yes	—
129	C₂²×C₄×C₈₉	C₂²×C₄	1	6	yes	—
130	C₂²×C₄×C₈₉	C₂⁴	1	3	yes	—
131	C₂²×C₄×C₈₉	C₂⁴	1	6	yes	—
132	M₁₆×C₈₉	—	2	12	no	—
133	M₁₆×C₈₉	—	4	6	no	—
134	M₁₆×C₈₉	—	4	6	no	—
135	C₂×C₈×C₈₉	—	2	6	no	—
136	C₂×C₈×C₈₉	—	4	3	no	—
137	C₂×C₈×C₈₉	—	4	12	no	—
138	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
139	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
140	C₂×C₈×C₈₉	C₂⁴	1	3	yes	—
141	C₂×C₈×C₈₉	C₂⁴	1	3	yes	—
142	C₂×C₈×C₈₉	C₂⁴	1	6	yes	—
143	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
144	C₄⋊C₄×C₈₉	C₂²×C₄	1	2	yes	—
145	C₄⋊C₄×C₈₉	C₂²×C₄	1	3	yes	—
146	C₄⋊C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
147	C₄⋊C₄×C₈₉	C₂²×C₄	1	6	yes	—
148	C₄⋊C₄×C₈₉	C₂²×C₄	1	3	yes	—
149	C₄⋊C₄×C₈₉	C₂²×C₄	1	6	yes	—
150	C₄⋊C₄×C₈₉	C₂²×C₄	1	3	yes	—
151	C₂²⋊C₄×C₈₉	C₂²×C₄	1	6	yes	—
152	C₂²⋊C₄×C₈₉	C₂²×C₄	1	6	yes	—
153	C₂²⋊C₄×C₈₉	C₂²×C₄	1	12	yes	—
154	C₂²⋊C₄×C₈₉	C₂²×C₄	1	6	yes	—
155	C₂²⋊C₄×C₈₉	C₂²×C₄	1	6	yes	—
156	C₄²×C₈₉	C₂²×C₄	1	3	yes	—
157	C₄²×C₈₉	C₂²×C₄	1	6	yes	—
158	C₄²×C₈₉	C₂²×C₄	1	6	yes	—
159	C₄²×C₈₉	C₂²×C₄	1	3	yes	—
160	C₄²×C₈₉	C₂⁴	1	3	yes	—
161	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
162	C₄²×C₈₉	C₂⁴	1	2	yes	—
163	C₂³×C₈₉	C₂³×C₄	1	3	yes	—
164	C₂³×C₈₉	C₂³×C₄	1	6	yes	—
165	Q₈×C₈₉	—	2	2	no	—
166	Q₈×C₈₉	—	4	6	no	—
167	Q₈×C₈₉	#131	1	4	yes	complemented
168	D₄×C₈₉	—	2	6	no	—
169	D₄×C₈₉	—	2	12	no	—
170	D₄×C₈₉	—	2	6	no	—
171	D₄×C₈₉	—	4	6	no	—
172	D₄×C₈₉	—	4	12	no	—
173	D₄×C₈₉	—	4	6	no	—
174	C₂×C₄×C₈₉	—	2	6	no	—
175	C₂×C₄×C₈₉	—	2	12	no	—
176	C₂×C₄×C₈₉	—	2	6	no	—
177	C₂×C₄×C₈₉	—	2	3	no	—
178	C₂×C₄×C₈₉	—	2	6	no	—
179	C₂×C₄×C₈₉	—	2	3	no	—
180	C₂×C₄×C₈₉	—	2	6	no	—
181	C₂×C₄×C₈₉	—	4	6	no	—
182	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
183	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
184	C₂×C₄×C₈₉	C₂³×C₄	1	3	yes	—
185	C₂×C₄×C₈₉	C₂³×C₄	1	2	yes	—
186	C₂×C₄×C₈₉	C₂³×C₄	1	3	yes	—
187	C₂×C₄×C₈₉	C₂³×C₄	1	6	yes	—
188	C₂×C₄×C₈₉	C₂³×C₄	1	3	yes	—
189	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
190	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
191	C₈×C₈₉	—	2	3	no	—
192	C₈×C₈₉	—	2	6	no	—
193	C₈×C₈₉	—	2	1	no	—
194	C₈×C₈₉	—	2	3	no	—
195	C₈×C₈₉	—	2	2	no	—
196	C₈×C₈₉	H(5,𝔽₂)	1	2	yes	—
197	C₂²×C₈₉	—	2	3	no	—
198	C₂²×C₈₉	—	2	6	no	—
199	C₂²×C₈₉	—	2	3	no	—
200	C₂²×C₈₉	—	2	6	no	—
201	C₂²×C₈₉	—	2	3	no	—
202	C₂²×C₈₉	—	2	6	no	—
203	C₂²×C₈₉	C₂⁴×C₄	1	1	yes	characteristic
204	C₄×C₈₉	—	2	3	no	—
205	C₄×C₈₉	—	2	6	no	—
206	C₄×C₈₉	—	2	1	no	—
207	C₄×C₈₉	—	2	2	no	—
208	C₄×C₈₉	—	2	3	no	—
209	C₄×C₈₉	—	2	2	no	—
210	C₄×C₈₉	C₂×#131	1	6	yes	—
211	C₄×C₈₉	#568	1	2	yes	—
212	C₄×C₈₉	C₂×H(5,𝔽₂)	1	1	yes	characteristic
213	C₄×C₈₉	#585	1	1	yes	characteristic
214	#60328	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
215	C₂×C₈₉	—	4	3	no	—
216	C₂×C₈₉	—	4	6	no	—
217	C₂×C₈₉	C₄×H(5,𝔽₂)	1	1	yes	characteristic
218	C₂×C₈₉	C₂²×#131	1	1	yes	characteristic
219	C₂×C₈₉	C₂×#568	1	1	yes	characteristic
220	#3483	C₂×C₈₉	1	1	yes	characteristic
221	#2991	C₂×C₈₉	1	1	yes	characteristic, complemented
222	#2990	C₂×C₈₉	1	3	yes	complemented
223	#2988	C₂×C₈₉	1	2	yes	complemented
224	#2981	C₂×C₈₉	1	6	yes	complemented
225	#2979	C₂×C₈₉	1	3	yes	complemented
226	#2971	C₂×C₈₉	1	2	yes	complemented
227	#2965	C₂×C₈₉	1	3	yes	complemented
228	C₈×#41	C₂×C₈₉	1	1	yes	characteristic, complemented
229	#2935	C₂×C₈₉	1	6	yes	complemented
230	#2931	C₂×C₈₉	1	3	yes	complemented
231	C₈₉	#60328	1	1	yes	characteristic, direct product factor, Sylow
232	C₂×#131	C₂²×C₈₉	1	6	yes	—
233	Q₈²	C₄×C₈₉	1	1	yes	characteristic
234	#555	C₄×C₈₉	1	6	yes	—
235	#555	C₄×C₈₉	1	3	yes	—
236	#550	C₄×C₈₉	1	1	yes	characteristic
237	#550	C₄×C₈₉	1	2	yes	—
238	#550	C₄×C₈₉	1	3	yes	—
239	#532	C₂²×C₈₉	1	3	yes	—
240	#532	C₂²×C₈₉	1	6	yes	—
241	C₄×#41	C₂²×C₈₉	1	1	yes	characteristic
242	C₄×#41	C₂²×C₈₉	1	3	yes	—
243	C₄×#41	C₂²×C₈₉	1	2	yes	—
244	#446	C₂²×C₈₉	1	1	yes	characteristic
245	#446	C₂²×C₈₉	1	3	yes	—
246	C₈×Q₈	C₂²×C₈₉	1	1	yes	characteristic
247	C₈×Q₈	C₂²×C₈₉	1	1	yes	characteristic
248	C₈×Q₈	C₂²×C₈₉	1	2	yes	—
249	#436	C₂²×C₈₉	1	3	yes	—
250	#436	C₂²×C₈₉	1	6	yes	—
251	#436	C₂²×C₈₉	1	6	yes	—
252	#436	C₂²×C₈₉	1	3	yes	—
253	#436	C₂²×C₈₉	1	6	yes	—
254	#435	C₂²×C₈₉	1	6	yes	—
255	#435	C₂²×C₈₉	1	12	yes	—
256	#435	C₂²×C₈₉	1	6	yes	—
257	C₈×D₄	C₂²×C₈₉	1	6	yes	—
258	C₈×D₄	C₂²×C₈₉	1	6	yes	—
259	C₈×D₄	C₂²×C₈₉	1	3	yes	—
260	C₈×D₄	C₂²×C₈₉	1	6	yes	—
261	C₈×D₄	C₂²×C₈₉	1	3	yes	—
262	#432	C₂²×C₈₉	1	6	yes	—
263	#431	C₂²×C₈₉	1	3	yes	—
264	#431	C₂²×C₈₉	1	3	yes	—
265	#431	C₂²×C₈₉	1	6	yes	—
266	#423	C₂²×C₈₉	1	3	yes	—
267	C₂×C₄⋊C₈	C₂²×C₈₉	1	6	yes	—
268	#408	C₂²×C₈₉	1	6	yes	—
269	#408	C₂²×C₈₉	1	12	yes	—
270	C₄×M₁₆	C₂²×C₈₉	1	6	yes	—
271	C₂×C₄×C₈	C₂²×C₈₉	1	3	yes	—
272	C₂×#41	C₂×C₄×C₈₉	1	1	yes	characteristic
273	C₂×#41	C₂×C₄×C₈₉	1	2	yes	—
274	C₂×#41	C₂×C₄×C₈₉	1	3	yes	—
275	C₂×#41	C₂³×C₈₉	1	6	yes	—
276	#131	—	4	12	no	—
277	C₂×M₁₆	C₂³×C₈₉	1	6	yes	—
278	C₂×M₁₆	C₂³×C₈₉	1	6	yes	—
279	C₂×M₁₆	C₂³×C₈₉	1	6	yes	—
280	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
281	C₂²×C₈	C₂³×C₈₉	1	12	yes	—
282	C₂²×C₈	C₂³×C₈₉	1	3	yes	—
283	C₄⋊Q₈	C₂×C₄×C₈₉	1	3	yes	—
284	C₄⋊Q₈	C₂×C₄×C₈₉	1	6	yes	—
285	#127	C₂×C₄×C₈₉	1	3	yes	—
286	#127	C₂×C₄×C₈₉	1	6	yes	—
287	#124	C₂×C₄×C₈₉	1	6	yes	—
288	#124	C₂×C₄×C₈₉	1	12	yes	—
289	#121	C₂×C₄×C₈₉	1	6	yes	—
290	#121	C₂×C₄×C₈₉	1	6	yes	—
291	#121	C₂×C₄×C₈₉	1	6	yes	—
292	#121	C₂×C₄×C₈₉	1	6	yes	—
293	#121	C₂×C₄×C₈₉	1	12	yes	—
294	C₄×Q₈	C₂×C₄×C₈₉	1	3	yes	—
295	C₄×Q₈	C₂×C₄×C₈₉	1	1	yes	characteristic
296	C₄×Q₈	C₂×C₄×C₈₉	1	2	yes	—
297	C₄×Q₈	C₂³×C₈₉	1	1	yes	characteristic
298	C₄×Q₈	C₂³×C₈₉	1	1	yes	characteristic
299	C₄×D₄	C₂×C₄×C₈₉	1	3	yes	—
300	C₄×D₄	C₂×C₄×C₈₉	1	3	yes	—
301	C₄×D₄	C₂×C₄×C₈₉	1	6	yes	—
302	C₄×D₄	C₂×C₄×C₈₉	1	6	yes	—
303	C₄×D₄	C₂³×C₈₉	1	3	yes	—
304	C₄×D₄	C₂³×C₈₉	1	3	yes	—
305	C₄×D₄	C₂³×C₈₉	1	6	yes	—
306	C₄×D₄	C₂³×C₈₉	1	6	yes	—
307	#117	C₂×C₄×C₈₉	1	3	yes	—
308	#117	C₂×C₄×C₈₉	1	3	yes	—
309	#117	C₂×C₄×C₈₉	1	6	yes	—
310	#117	C₂×C₄×C₈₉	1	6	yes	—
311	C₂×C₄²	C₂³×C₈₉	1	3	yes	—
312	C₂×C₄²	C₂³×C₈₉	1	6	yes	—
313	C₄⋊C₈	C₂³×C₈₉	1	3	yes	—
314	C₄⋊C₈	C₂³×C₈₉	1	6	yes	—
315	C₄⋊C₈	C₂³×C₈₉	1	6	yes	—
316	C₄⋊C₈	C₂³×C₈₉	1	3	yes	—
317	C₄⋊C₈	C₂³×C₈₉	1	2	yes	—
318	C₄⋊C₈	C₂³×C₈₉	1	1	yes	characteristic
319	C₄⋊C₈	C₂³×C₈₉	1	3	yes	—
320	C₂²⋊C₈	C₂³×C₈₉	1	6	yes	—
321	C₂²⋊C₈	C₂³×C₈₉	1	6	yes	—
322	C₂²⋊C₈	C₂³×C₈₉	1	6	yes	—
323	C₂²⋊C₈	C₂³×C₈₉	1	6	yes	—
324	C₂²⋊C₈	C₂³×C₈₉	1	12	yes	—
325	#97	C₂³×C₈₉	1	2	yes	—
326	#97	C₂³×C₈₉	1	6	yes	—
327	C₄×C₈	C₂³×C₈₉	1	1	yes	characteristic
328	C₄×C₈	C₂³×C₈₉	1	3	yes	—
329	C₄×C₈	C₂³×C₈₉	1	3	yes	—
330	C₄×C₈	C₂³×C₈₉	1	3	yes	—
331	C₄×C₈	C₂³×C₈₉	1	6	yes	—
332	#41	—	2	4	no	—
333	#41	—	2	8	no	—
334	#41	—	4	6	no	—
335	#41	—	4	6	no	—
336	#41	—	4	6	no	—
337	C₂×Q₈	C₂²×C₄×C₈₉	1	1	yes	characteristic
338	C₂×Q₈	C₂²×C₄×C₈₉	1	6	yes	—
339	C₂×Q₈	C₂²×C₄×C₈₉	1	1	yes	characteristic
340	C₂×D₄	C₂²×C₄×C₈₉	1	3	yes	—
341	C₂×D₄	C₂²×C₄×C₈₉	1	6	yes	—
342	C₂×D₄	C₂²×C₄×C₈₉	1	3	yes	—
343	C₂×D₄	C₂²×C₄×C₈₉	1	6	yes	—
344	C₂×D₄	C₂²×C₄×C₈₉	1	12	yes	—
345	C₂×D₄	C₂²×C₄×C₈₉	1	6	yes	—
346	C₂²×C₄	C₂²×C₄×C₈₉	1	3	yes	—
347	C₂²×C₄	C₂²×C₄×C₈₉	1	6	yes	—
348	C₂²×C₄	C₂²×C₄×C₈₉	1	3	yes	—
349	C₂²×C₄	C₂²×C₄×C₈₉	1	6	yes	—
350	C₂²×C₄	C₂⁴×C₈₉	1	3	yes	—
351	C₂²×C₄	C₂⁴×C₈₉	1	6	yes	—
352	M₁₆	—	2	12	no	—
353	M₁₆	—	4	6	no	—
354	M₁₆	—	4	6	no	—
355	C₂×C₈	—	2	6	no	—
356	C₂×C₈	—	4	12	no	—
357	C₂×C₈	—	4	3	no	—
358	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
359	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
360	C₂×C₈	C₂⁴×C₈₉	1	3	yes	—
361	C₂×C₈	C₂⁴×C₈₉	1	3	yes	—
362	C₂×C₈	C₂⁴×C₈₉	1	6	yes	—
363	C₂×C₈	C₂⁴×C₈₉	1	1	yes	characteristic
364	C₄⋊C₄	C₂²×C₄×C₈₉	1	2	yes	—
365	C₄⋊C₄	C₂²×C₄×C₈₉	1	3	yes	—
366	C₄⋊C₄	C₂²×C₄×C₈₉	1	1	yes	characteristic
367	C₄⋊C₄	C₂²×C₄×C₈₉	1	6	yes	—
368	C₄⋊C₄	C₂²×C₄×C₈₉	1	3	yes	—
369	C₄⋊C₄	C₂²×C₄×C₈₉	1	6	yes	—
370	C₄⋊C₄	C₂²×C₄×C₈₉	1	3	yes	—
371	C₂²⋊C₄	C₂²×C₄×C₈₉	1	6	yes	—
372	C₂²⋊C₄	C₂²×C₄×C₈₉	1	6	yes	—
373	C₂²⋊C₄	C₂²×C₄×C₈₉	1	12	yes	—
374	C₂²⋊C₄	C₂²×C₄×C₈₉	1	6	yes	—
375	C₂²⋊C₄	C₂²×C₄×C₈₉	1	6	yes	—
376	C₄²	C₂²×C₄×C₈₉	1	3	yes	—
377	C₄²	C₂²×C₄×C₈₉	1	6	yes	—
378	C₄²	C₂²×C₄×C₈₉	1	6	yes	—
379	C₄²	C₂²×C₄×C₈₉	1	3	yes	—
380	C₄²	C₂⁴×C₈₉	1	3	yes	—
381	C₄²	C₂⁴×C₈₉	1	1	yes	characteristic
382	C₄²	C₂⁴×C₈₉	1	2	yes	—
383	C₂³	C₂³×C₄×C₈₉	1	3	yes	—
384	C₂³	C₂³×C₄×C₈₉	1	6	yes	—
385	Q₈	—	2	2	no	—
386	Q₈	—	4	6	no	—
387	Q₈	#131×C₈₉	1	4	yes	complemented
388	D₄	—	2	6	no	—
389	D₄	—	2	12	no	—
390	D₄	—	2	6	no	—
391	D₄	—	4	6	no	—
392	D₄	—	4	12	no	—
393	D₄	—	4	6	no	—
394	C₂×C₄	—	2	6	no	—
395	C₂×C₄	—	2	12	no	—
396	C₂×C₄	—	2	6	no	—
397	C₂×C₄	—	2	3	no	—
398	C₂×C₄	—	2	6	no	—
399	C₂×C₄	—	2	3	no	—
400	C₂×C₄	—	2	6	no	—
401	C₂×C₄	—	4	6	no	—
402	C₂×C₄	C₂³×C₄×C₈₉	1	1	yes	characteristic
403	C₂×C₄	C₂³×C₄×C₈₉	1	2	yes	—
404	C₂×C₄	C₂³×C₄×C₈₉	1	3	yes	—
405	C₂×C₄	C₂³×C₄×C₈₉	1	2	yes	—
406	C₂×C₄	C₂³×C₄×C₈₉	1	3	yes	—
407	C₂×C₄	C₂³×C₄×C₈₉	1	6	yes	—
408	C₂×C₄	C₂³×C₄×C₈₉	1	3	yes	—
409	C₂×C₄	C₂³×C₄×C₈₉	1	1	yes	characteristic
410	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
411	C₈	—	2	3	no	—
412	C₈	—	2	6	no	—
413	C₈	—	2	1	no	—
414	C₈	—	2	3	no	—
415	C₈	—	2	2	no	—
416	C₈	H(5,𝔽₂)×C₈₉	1	2	yes	—
417	C₂²	—	2	3	no	—
418	C₂²	—	2	6	no	—
419	C₂²	—	2	3	no	—
420	C₂²	—	2	6	no	—
421	C₂²	—	2	3	no	—
422	C₂²	—	2	6	no	—
423	C₂²	C₂⁴×C₄×C₈₉	1	1	yes	characteristic
424	C₄	—	2	3	no	—
425	C₄	—	2	6	no	—
426	C₄	—	2	1	no	—
427	C₄	—	2	2	no	—
428	C₄	—	2	3	no	—
429	C₄	—	2	2	no	—
430	C₄	C₂×#131×C₈₉	1	6	yes	—
431	C₄	#568×C₈₉	1	2	yes	—
432	C₄	C₂×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
433	C₄	#585×C₈₉	1	1	yes	characteristic
434	C₂	—	4	3	no	—
435	C₂	—	4	6	no	—
436	C₂	C₄×H(5,𝔽₂)×C₈₉	1	1	yes	characteristic
437	C₂	C₂²×#131×C₈₉	1	1	yes	characteristic
438	C₂	C₂×#568×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₈₉	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₈₉
Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
socle	C₂²×C₈₉	C₂⁴×C₄
prime radical	C₈₉×#3483	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#2991	C₂	semidirect
3. C₈₉×#2990	C₂	semidirect
4. C₈₉×#2988	C₂	semidirect
5. C₈₉×#2981	C₂	semidirect
6. C₈₉×#2979	C₂	semidirect
7. C₈₉×#2971	C₂	semidirect
8. C₈₉×#2965	C₂	semidirect
9. C₈×#41×C₈₉	C₂	semidirect
10. C₈₉×#2935	C₂	semidirect
11. C₈₉×#2931	C₂	semidirect
167. Q₈×C₈₉	#131	semidirect
214. #60328	C₈₉	direct (N × H)
221. #2991	C₂×C₈₉	semidirect
222. #2990	C₂×C₈₉	semidirect
223. #2988	C₂×C₈₉	semidirect
224. #2981	C₂×C₈₉	semidirect
225. #2979	C₂×C₈₉	semidirect
226. #2971	C₂×C₈₉	semidirect
227. #2965	C₂×C₈₉	semidirect
228. C₈×#41	C₂×C₈₉	semidirect
229. #2935	C₂×C₈₉	semidirect
230. #2931	C₂×C₈₉	semidirect
231. C₈₉	#60328	direct (N × H)
387. Q₈	#131×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3483	2
C₈₉×#2991	2
C₈₉×#2990	2
C₈₉×#2988	2
C₈₉×#2981	2
C₈₉×#2979	2
C₈₉×#2971	2
C₈₉×#2965	2
C₈×#41×C₈₉	2
C₈₉×#2935	2
C₈₉×#2931	2
#60328	89

Subgroups — C89×#60328 #36438176

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#60328 #36438176