Subgroups — C₈₉×#18273 #36396121

Order 22784.

1002 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3558	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#3112	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3108	C₂	1	1	yes	maximal, characteristic, complemented
4	C₈₉×#2592	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#1612	C₂	1	2	yes	maximal, complemented
6	C₈₉×#1607	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1606	C₂	1	2	yes	maximal, complemented
8	C₈₉×#1601	C₂	1	2	yes	maximal, complemented
9	C₈₉×#1509	C₂	1	1	yes	maximal, characteristic, complemented
10	C₈₉×#1502	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#1469	C₂	1	1	yes	maximal, characteristic, complemented
12	#562×C₈₉	—	2	1	no	—
13	H(3,𝔽₄)×C₈₉	—	2	1	no	—
14	#559×C₈₉	—	2	1	no	—
15	#548×C₈₉	—	2	1	no	—
16	#546×C₈₉	—	2	1	no	—
17	#546×C₈₉	—	2	2	no	—
18	D₄²×C₈₉	—	2	1	no	—
19	#542×C₈₉	—	2	1	no	—
20	#538×C₈₉	—	2	2	no	—
21	#538×C₈₉	C₂²	1	2	yes	—
22	#535×C₈₉	—	2	1	no	—
23	#534×C₈₉	C₂²	1	2	yes	—
24	#532×C₈₉	C₂²	1	1	yes	characteristic
25	#529×C₈₉	C₂²	1	1	yes	characteristic
26	C₂×#124×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
28	#483×C₈₉	C₂²	1	1	yes	characteristic, complemented
29	#480×C₈₉	C₂²	1	1	yes	characteristic
30	#480×C₈₉	C₂²	1	2	yes	complemented
31	#479×C₈₉	—	2	1	no	—
32	#477×C₈₉	—	2	1	no	—
33	#464×C₈₉	—	2	1	no	—
34	#460×C₈₉	—	2	1	no	—
35	#451×C₈₉	C₂²	1	1	yes	characteristic, complemented
36	#448×C₈₉	C₂²	1	1	yes	characteristic
37	#448×C₈₉	C₂²	1	2	yes	complemented
38	#440×C₈₉	—	2	2	no	—
39	C₄×Q₁₆×C₈₉	—	2	1	no	—
40	C₄×SD₁₆×C₈₉	—	2	1	no	—
41	#432×C₈₉	C₂²	1	1	yes	characteristic, complemented
42	#431×C₈₉	C₂²	1	1	yes	characteristic, complemented
43	#397×C₈₉	C₂²	1	2	yes	complemented
44	#393×C₈₉	C₂²	1	2	yes	complemented
45	#391×C₈₉	—	2	1	no	—
46	#390×C₈₉	—	2	1	no	—
47	#388×C₈₉	—	2	2	no	—
48	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic, complemented
49	#328×C₈₉	C₂²	1	2	yes	complemented
50	#327×C₈₉	C₂²	1	2	yes	—
51	#327×C₈₉	C₂²	1	2	yes	complemented
52	#327×C₈₉	C₂²	1	2	yes	complemented
53	#323×C₈₉	C₂²	1	2	yes	complemented
54	#323×C₈₉	C₂²	1	2	yes	complemented
55	C₂×#41×C₈₉	—	2	1	no	—
56	C₂×#41×C₈₉	—	2	1	no	—
57	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
58	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
59	C₂²×D₄×C₈₉	—	2	1	no	—
60	C₂²×D₄×C₈₉	—	2	2	no	—
61	C₂²×D₄×C₈₉	D₄	1	1	yes	characteristic
62	C₂×Q₁₆×C₈₉	—	2	1	no	—
63	C₂×SD₁₆×C₈₉	—	2	1	no	—
64	C₂×SD₁₆×C₈₉	—	2	2	no	—
65	C₄⋊Q₈×C₈₉	D₄	1	1	yes	characteristic
66	#127×C₈₉	—	2	1	no	—
67	#127×C₈₉	D₄	1	1	yes	characteristic
68	#126×C₈₉	—	2	2	no	—
69	#126×C₈₉	—	2	1	no	—
70	#125×C₈₉	D₄	1	1	yes	characteristic
71	#124×C₈₉	—	2	1	no	—
72	#124×C₈₉	—	2	2	no	—
73	#124×C₈₉	—	2	2	no	—
74	#124×C₈₉	—	2	2	no	—
75	#124×C₈₉	—	2	1	no	—
76	#124×C₈₉	—	2	1	no	—
77	#124×C₈₉	—	2	1	no	—
78	#124×C₈₉	—	2	1	no	—
79	#124×C₈₉	—	2	1	no	—
80	#124×C₈₉	D₄	1	1	yes	characteristic
81	#123×C₈₉	—	2	2	no	—
82	#123×C₈₉	—	2	1	no	—
83	#123×C₈₉	—	2	2	no	—
84	#123×C₈₉	—	2	2	no	—
85	#123×C₈₉	—	2	2	no	—
86	C₂²⋊Q₈×C₈₉	—	2	2	no	—
87	C₂²⋊Q₈×C₈₉	—	2	1	no	—
88	C₂²⋊Q₈×C₈₉	—	2	2	no	—
89	C₂²⋊Q₈×C₈₉	C₂³	1	1	yes	characteristic
90	#121×C₈₉	—	2	2	no	—
91	#121×C₈₉	—	2	2	no	—
92	#121×C₈₉	—	2	2	no	—
93	#121×C₈₉	—	2	1	no	—
94	#121×C₈₉	—	2	2	no	—
95	#121×C₈₉	—	2	1	no	—
96	#121×C₈₉	—	2	2	no	—
97	#121×C₈₉	C₂³	1	1	yes	characteristic
98	#121×C₈₉	C₂³	1	2	yes	—
99	#120×C₈₉	—	2	2	no	—
100	#120×C₈₉	—	2	2	no	—
101	#120×C₈₉	—	2	2	no	—
102	#120×C₈₉	—	2	2	no	—
103	#120×C₈₉	—	2	2	no	—
104	C₄×Q₈×C₈₉	—	2	1	no	—
105	C₄×Q₈×C₈₉	—	2	1	no	—
106	C₄×Q₈×C₈₉	—	2	1	no	—
107	C₄×Q₈×C₈₉	—	2	1	no	—
108	C₄×Q₈×C₈₉	—	2	1	no	—
109	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
110	C₄×D₄×C₈₉	—	2	2	no	—
111	C₄×D₄×C₈₉	—	2	1	no	—
112	C₄×D₄×C₈₉	—	2	1	no	—
113	C₄×D₄×C₈₉	—	2	2	no	—
114	C₄×D₄×C₈₉	—	2	1	no	—
115	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
116	C₄×D₄×C₈₉	D₄	1	2	yes	—
117	#117×C₈₉	—	2	1	no	—
118	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
119	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
120	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
121	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
122	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
123	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
124	C₂×C₂²⋊C₄×C₈₉	D₄	1	2	yes	—
125	C₂×C₂²⋊C₄×C₈₉	C₂³	1	2	yes	—
126	C₂×C₄²×C₈₉	—	2	1	no	—
127	C₂×C₄²×C₈₉	—	2	1	no	—
128	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
129	#107×C₈₉	—	2	1	no	—
130	#107×C₈₉	—	2	2	no	—
131	#106×C₈₉	—	2	1	no	—
132	C₄⋊C₈×C₈₉	—	2	1	no	—
133	C₄⋊C₈×C₈₉	—	2	1	no	—
134	#103×C₈₉	—	2	1	no	—
135	#103×C₈₉	—	2	1	no	—
136	#103×C₈₉	—	2	1	no	—
137	#103×C₈₉	—	2	2	no	—
138	#102×C₈₉	—	2	1	no	—
139	#102×C₈₉	—	2	2	no	—
140	C₂²⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
141	C₂²⋊C₈×C₈₉	C₂³	1	1	yes	characteristic
142	C₂²⋊C₈×C₈₉	C₂³	1	2	yes	—
143	#97×C₈₉	—	2	1	no	—
144	C₄×C₈×C₈₉	—	2	1	no	—
145	#95×C₈₉	—	2	2	no	—
146	#95×C₈₉	—	2	2	no	—
147	#95×C₈₉	C₂³	1	2	yes	—
148	C₂⁴×C₈₉	—	2	2	no	—
149	C₂⁴×C₈₉	C₂×D₄	1	2	yes	—
150	#41×C₈₉	—	8	1	no	—
151	#41×C₈₉	—	8	1	no	—
152	#41×C₈₉	—	8	2	no	—
153	C₂×Q₈×C₈₉	—	2	1	no	—
154	C₂×Q₈×C₈₉	—	2	1	no	—
155	C₂×Q₈×C₈₉	—	2	1	no	—
156	C₂×Q₈×C₈₉	—	2	1	no	—
157	C₂×Q₈×C₈₉	—	2	1	no	—
158	C₂×Q₈×C₈₉	—	4	2	no	—
159	C₂×Q₈×C₈₉	—	4	2	no	—
160	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
161	C₂×D₄×C₈₉	—	2	2	no	—
162	C₂×D₄×C₈₉	—	2	1	no	—
163	C₂×D₄×C₈₉	—	2	2	no	—
164	C₂×D₄×C₈₉	—	2	2	no	—
165	C₂×D₄×C₈₉	—	2	2	no	—
166	C₂×D₄×C₈₉	—	2	2	no	—
167	C₂×D₄×C₈₉	—	2	2	no	—
168	C₂×D₄×C₈₉	—	2	1	no	—
169	C₂×D₄×C₈₉	—	2	1	no	—
170	C₂×D₄×C₈₉	—	2	1	no	—
171	C₂×D₄×C₈₉	—	2	1	no	—
172	C₂×D₄×C₈₉	—	4	2	no	—
173	C₂×D₄×C₈₉	—	8	2	no	—
174	C₂×D₄×C₈₉	—	8	2	no	—
175	C₂×D₄×C₈₉	—	8	2	no	—
176	C₂×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
177	C₂×D₄×C₈₉	C₂×D₄	1	2	yes	—
178	C₂²×C₄×C₈₉	—	2	1	no	—
179	C₂²×C₄×C₈₉	—	2	2	no	—
180	C₂²×C₄×C₈₉	—	2	2	no	—
181	C₂²×C₄×C₈₉	—	2	1	no	—
182	C₂²×C₄×C₈₉	—	2	2	no	—
183	C₂²×C₄×C₈₉	—	2	2	no	—
184	C₂²×C₄×C₈₉	—	2	1	no	—
185	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
186	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
188	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
189	Q₁₆×C₈₉	—	4	2	no	—
190	SD₁₆×C₈₉	—	4	4	no	—
191	SD₁₆×C₈₉	—	4	2	no	—
192	C₂×C₈×C₈₉	—	2	1	no	—
193	C₂×C₈×C₈₉	—	2	1	no	—
194	C₂×C₈×C₈₉	—	2	2	no	—
195	C₄⋊C₄×C₈₉	—	2	2	no	—
196	C₄⋊C₄×C₈₉	—	2	2	no	—
197	C₄⋊C₄×C₈₉	—	2	1	no	—
198	C₄⋊C₄×C₈₉	—	2	1	no	—
199	C₄⋊C₄×C₈₉	—	2	1	no	—
200	C₄⋊C₄×C₈₉	—	2	1	no	—
201	C₄⋊C₄×C₈₉	—	2	1	no	—
202	C₄⋊C₄×C₈₉	—	2	1	no	—
203	C₄⋊C₄×C₈₉	—	2	1	no	—
204	C₄⋊C₄×C₈₉	—	4	2	no	—
205	C₄⋊C₄×C₈₉	—	4	1	no	—
206	C₄⋊C₄×C₈₉	—	4	1	no	—
207	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
208	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
209	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
210	C₂²⋊C₄×C₈₉	—	2	2	no	—
211	C₂²⋊C₄×C₈₉	—	2	2	no	—
212	C₂²⋊C₄×C₈₉	—	2	2	no	—
213	C₂²⋊C₄×C₈₉	—	2	2	no	—
214	C₂²⋊C₄×C₈₉	—	2	2	no	—
215	C₂²⋊C₄×C₈₉	—	2	1	no	—
216	C₂²⋊C₄×C₈₉	—	2	1	no	—
217	C₂²⋊C₄×C₈₉	—	2	2	no	—
218	C₂²⋊C₄×C₈₉	—	2	2	no	—
219	C₂²⋊C₄×C₈₉	—	2	1	no	—
220	C₂²⋊C₄×C₈₉	—	2	1	no	—
221	C₂²⋊C₄×C₈₉	—	2	2	no	—
222	C₂²⋊C₄×C₈₉	—	2	2	no	—
223	C₂²⋊C₄×C₈₉	—	2	2	no	—
224	C₂²⋊C₄×C₈₉	—	2	1	no	—
225	C₂²⋊C₄×C₈₉	—	2	2	no	—
226	C₂²⋊C₄×C₈₉	—	4	4	no	—
227	C₂²⋊C₄×C₈₉	—	8	2	no	—
228	C₄²×C₈₉	—	2	2	no	—
229	C₄²×C₈₉	—	2	1	no	—
230	C₄²×C₈₉	—	4	1	no	—
231	C₄²×C₈₉	—	4	1	no	—
232	C₄²×C₈₉	—	8	1	no	—
233	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
234	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
235	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
236	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
237	C₂³×C₈₉	—	2	2	no	—
238	C₂³×C₈₉	—	2	1	no	—
239	C₂³×C₈₉	—	2	2	no	—
240	C₂³×C₈₉	—	2	2	no	—
241	C₂³×C₈₉	—	2	2	no	—
242	C₂³×C₈₉	—	4	2	no	—
243	C₂³×C₈₉	—	4	2	no	—
244	C₂³×C₈₉	—	8	2	no	—
245	C₂³×C₈₉	—	8	2	no	—
246	C₂³×C₈₉	—	8	2	no	—
247	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
248	Q₈×C₈₉	—	2	2	no	—
249	Q₈×C₈₉	—	2	2	no	—
250	Q₈×C₈₉	—	4	1	no	—
251	Q₈×C₈₉	—	4	1	no	—
252	Q₈×C₈₉	—	4	1	no	—
253	Q₈×C₈₉	—	4	2	no	—
254	Q₈×C₈₉	—	8	1	no	—
255	Q₈×C₈₉	—	8	1	no	—
256	D₄×C₈₉	—	4	2	no	—
257	D₄×C₈₉	—	4	1	no	—
258	D₄×C₈₉	—	4	1	no	—
259	D₄×C₈₉	—	4	2	no	—
260	D₄×C₈₉	—	4	1	no	—
261	D₄×C₈₉	—	4	2	no	—
262	D₄×C₈₉	—	4	2	no	—
263	D₄×C₈₉	—	4	2	no	—
264	D₄×C₈₉	—	8	1	no	—
265	D₄×C₈₉	—	8	1	no	—
266	D₄×C₈₉	—	8	2	no	—
267	D₄×C₈₉	—	8	2	no	—
268	D₄×C₈₉	—	8	2	no	—
269	D₄×C₈₉	—	8	2	no	—
270	D₄×C₈₉	—	8	2	no	—
271	C₂×C₄×C₈₉	—	2	1	no	—
272	C₂×C₄×C₈₉	—	2	1	no	—
273	C₂×C₄×C₈₉	—	2	2	no	—
274	C₂×C₄×C₈₉	—	2	2	no	—
275	C₂×C₄×C₈₉	—	2	1	no	—
276	C₂×C₄×C₈₉	—	2	2	no	—
277	C₂×C₄×C₈₉	—	2	2	no	—
278	C₂×C₄×C₈₉	—	2	1	no	—
279	C₂×C₄×C₈₉	—	2	1	no	—
280	C₂×C₄×C₈₉	—	4	1	no	—
281	C₂×C₄×C₈₉	—	4	1	no	—
282	C₂×C₄×C₈₉	—	4	1	no	—
283	C₂×C₄×C₈₉	—	4	2	no	—
284	C₂×C₄×C₈₉	—	4	2	no	—
285	C₂×C₄×C₈₉	—	4	2	no	—
286	C₂×C₄×C₈₉	—	4	2	no	—
287	C₂×C₄×C₈₉	—	4	1	no	—
288	C₂×C₄×C₈₉	—	4	2	no	—
289	C₂×C₄×C₈₉	—	4	1	no	—
290	C₂×C₄×C₈₉	—	4	1	no	—
291	C₂×C₄×C₈₉	—	8	2	no	—
292	C₂×C₄×C₈₉	—	8	2	no	—
293	C₂×C₄×C₈₉	—	8	2	no	—
294	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
295	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
296	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
297	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
298	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
299	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
300	C₈×C₈₉	—	4	2	no	—
301	C₈×C₈₉	—	4	1	no	—
302	C₈×C₈₉	—	4	1	no	—
303	C₂²×C₈₉	—	2	1	no	—
304	C₂²×C₈₉	—	2	1	no	—
305	C₂²×C₈₉	—	4	2	no	—
306	C₂²×C₈₉	—	4	2	no	—
307	C₂²×C₈₉	—	4	1	no	—
308	C₂²×C₈₉	—	4	1	no	—
309	C₂²×C₈₉	—	4	2	no	—
310	C₂²×C₈₉	—	4	2	no	—
311	C₂²×C₈₉	—	4	2	no	—
312	C₂²×C₈₉	—	4	1	no	—
313	C₂²×C₈₉	—	4	2	no	—
314	C₂²×C₈₉	—	8	2	no	—
315	C₂²×C₈₉	—	8	2	no	—
316	C₂²×C₈₉	—	16	4	no	—
317	C₂²×C₈₉	C₂×#120	1	1	yes	characteristic
318	C₂²×C₈₉	#576	1	1	yes	characteristic
319	C₂²×C₈₉	#577	1	1	yes	characteristic
320	C₄×C₈₉	—	2	1	no	—
321	C₄×C₈₉	—	2	1	no	—
322	C₄×C₈₉	—	2	1	no	—
323	C₄×C₈₉	—	2	1	no	—
324	C₄×C₈₉	—	2	1	no	—
325	C₄×C₈₉	—	4	1	no	—
326	C₄×C₈₉	—	4	2	no	—
327	C₄×C₈₉	—	4	1	no	—
328	C₄×C₈₉	—	4	2	no	—
329	C₄×C₈₉	—	4	2	no	—
330	C₄×C₈₉	—	4	1	no	—
331	C₄×C₈₉	—	8	2	no	—
332	C₄×C₈₉	—	8	1	no	—
333	C₄×C₈₉	#455	1	2	yes	—
334	#18273	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
335	C₂×C₈₉	—	2	1	no	—
336	C₂×C₈₉	—	2	1	no	—
337	C₂×C₈₉	—	8	2	no	—
338	C₂×C₈₉	—	8	2	no	—
339	C₂×C₈₉	—	8	1	no	—
340	C₂×C₈₉	#3012	1	1	yes	characteristic
341	C₂×C₈₉	C₂×#455	1	1	yes	characteristic
342	C₂×C₈₉	#3025	1	1	yes	characteristic
343	#3558	C₂×C₈₉	1	1	yes	characteristic
344	#3112	C₂×C₈₉	1	1	yes	characteristic, complemented
345	#3108	C₂×C₈₉	1	1	yes	characteristic, complemented
346	#2592	C₂×C₈₉	1	1	yes	characteristic, complemented
347	#1612	C₂×C₈₉	1	2	yes	complemented
348	#1607	C₂×C₈₉	1	2	yes	complemented
349	#1606	C₂×C₈₉	1	2	yes	complemented
350	#1601	C₂×C₈₉	1	2	yes	complemented
351	#1509	C₂×C₈₉	1	1	yes	characteristic, complemented
352	#1502	C₂×C₈₉	1	1	yes	characteristic, complemented
353	#1469	C₂×C₈₉	1	1	yes	characteristic, complemented
354	C₈₉	#18273	1	1	yes	characteristic, direct product factor, Sylow
355	#562	—	2	1	no	—
356	H(3,𝔽₄)	—	2	1	no	—
357	#559	—	2	1	no	—
358	#548	—	2	1	no	—
359	#546	—	2	1	no	—
360	#546	—	2	2	no	—
361	D₄²	—	2	1	no	—
362	#542	—	2	1	no	—
363	#538	—	2	2	no	—
364	#538	C₂²×C₈₉	1	2	yes	—
365	#535	—	2	1	no	—
366	#534	C₂²×C₈₉	1	2	yes	—
367	#532	C₂²×C₈₉	1	1	yes	characteristic
368	#529	C₂²×C₈₉	1	1	yes	characteristic
369	C₂×#124	C₂²×C₈₉	1	1	yes	characteristic
370	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
371	#483	C₂²×C₈₉	1	1	yes	characteristic, complemented
372	#480	C₂²×C₈₉	1	1	yes	characteristic
373	#480	C₂²×C₈₉	1	2	yes	complemented
374	#479	—	2	1	no	—
375	#477	—	2	1	no	—
376	#464	—	2	1	no	—
377	#460	—	2	1	no	—
378	#451	C₂²×C₈₉	1	1	yes	characteristic, complemented
379	#448	C₂²×C₈₉	1	1	yes	characteristic
380	#448	C₂²×C₈₉	1	2	yes	complemented
381	#440	—	2	2	no	—
382	C₄×Q₁₆	—	2	1	no	—
383	C₄×SD₁₆	—	2	1	no	—
384	#432	C₂²×C₈₉	1	1	yes	characteristic, complemented
385	#431	C₂²×C₈₉	1	1	yes	characteristic, complemented
386	#397	C₂²×C₈₉	1	2	yes	complemented
387	#393	C₂²×C₈₉	1	2	yes	complemented
388	#391	—	2	1	no	—
389	#390	—	2	1	no	—
390	#388	—	2	2	no	—
391	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic, complemented
392	#328	C₂²×C₈₉	1	2	yes	complemented
393	#327	C₂²×C₈₉	1	2	yes	—
394	#327	C₂²×C₈₉	1	2	yes	complemented
395	#327	C₂²×C₈₉	1	2	yes	complemented
396	#323	C₂²×C₈₉	1	2	yes	complemented
397	#323	C₂²×C₈₉	1	2	yes	complemented
398	C₂×#41	—	2	1	no	—
399	C₂×#41	—	2	1	no	—
400	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
401	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
402	C₂²×D₄	—	2	1	no	—
403	C₂²×D₄	—	2	2	no	—
404	C₂²×D₄	D₄×C₈₉	1	1	yes	characteristic
405	C₂×Q₁₆	—	2	1	no	—
406	C₂×SD₁₆	—	2	1	no	—
407	C₂×SD₁₆	—	2	2	no	—
408	C₄⋊Q₈	D₄×C₈₉	1	1	yes	characteristic
409	#127	—	2	1	no	—
410	#127	D₄×C₈₉	1	1	yes	characteristic
411	#126	—	2	2	no	—
412	#126	—	2	1	no	—
413	#125	D₄×C₈₉	1	1	yes	characteristic
414	#124	—	2	1	no	—
415	#124	—	2	2	no	—
416	#124	—	2	2	no	—
417	#124	—	2	2	no	—
418	#124	—	2	1	no	—
419	#124	—	2	1	no	—
420	#124	—	2	1	no	—
421	#124	—	2	1	no	—
422	#124	—	2	1	no	—
423	#124	D₄×C₈₉	1	1	yes	characteristic
424	#123	—	2	2	no	—
425	#123	—	2	1	no	—
426	#123	—	2	2	no	—
427	#123	—	2	2	no	—
428	#123	—	2	2	no	—
429	C₂²⋊Q₈	—	2	2	no	—
430	C₂²⋊Q₈	—	2	1	no	—
431	C₂²⋊Q₈	—	2	2	no	—
432	C₂²⋊Q₈	C₂³×C₈₉	1	1	yes	characteristic
433	#121	—	2	2	no	—
434	#121	—	2	2	no	—
435	#121	—	2	2	no	—
436	#121	—	2	1	no	—
437	#121	—	2	2	no	—
438	#121	—	2	1	no	—
439	#121	—	2	2	no	—
440	#121	C₂³×C₈₉	1	1	yes	characteristic
441	#121	C₂³×C₈₉	1	2	yes	—
442	#120	—	2	2	no	—
443	#120	—	2	2	no	—
444	#120	—	2	2	no	—
445	#120	—	2	2	no	—
446	#120	—	2	2	no	—
447	C₄×Q₈	—	2	1	no	—
448	C₄×Q₈	—	2	1	no	—
449	C₄×Q₈	—	2	1	no	—
450	C₄×Q₈	—	2	1	no	—
451	C₄×Q₈	—	2	1	no	—
452	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
453	C₄×D₄	—	2	2	no	—
454	C₄×D₄	—	2	1	no	—
455	C₄×D₄	—	2	1	no	—
456	C₄×D₄	—	2	2	no	—
457	C₄×D₄	—	2	1	no	—
458	C₄×D₄	D₄×C₈₉	1	1	yes	characteristic
459	C₄×D₄	D₄×C₈₉	1	2	yes	—
460	#117	—	2	1	no	—
461	C₂×C₄⋊C₄	—	2	1	no	—
462	C₂×C₄⋊C₄	—	2	1	no	—
463	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
464	C₂×C₂²⋊C₄	—	2	2	no	—
465	C₂×C₂²⋊C₄	—	2	2	no	—
466	C₂×C₂²⋊C₄	—	2	2	no	—
467	C₂×C₂²⋊C₄	D₄×C₈₉	1	2	yes	—
468	C₂×C₂²⋊C₄	C₂³×C₈₉	1	2	yes	—
469	C₂×C₄²	—	2	1	no	—
470	C₂×C₄²	—	2	1	no	—
471	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
472	#107	—	2	1	no	—
473	#107	—	2	2	no	—
474	#106	—	2	1	no	—
475	C₄⋊C₈	—	2	1	no	—
476	C₄⋊C₈	—	2	1	no	—
477	#103	—	2	1	no	—
478	#103	—	2	1	no	—
479	#103	—	2	1	no	—
480	#103	—	2	2	no	—
481	#102	—	2	1	no	—
482	#102	—	2	2	no	—
483	C₂²⋊C₈	C₂³×C₈₉	1	2	yes	—
484	C₂²⋊C₈	C₂³×C₈₉	1	1	yes	characteristic
485	C₂²⋊C₈	C₂³×C₈₉	1	1	yes	characteristic
486	#97	—	2	1	no	—
487	C₄×C₈	—	2	1	no	—
488	#95	—	2	2	no	—
489	#95	—	2	2	no	—
490	#95	C₂³×C₈₉	1	2	yes	—
491	C₂⁴	—	2	2	no	—
492	C₂⁴	C₂×D₄×C₈₉	1	2	yes	—
493	#41	—	8	1	no	—
494	#41	—	8	1	no	—
495	#41	—	8	2	no	—
496	C₂×Q₈	—	2	1	no	—
497	C₂×Q₈	—	2	1	no	—
498	C₂×Q₈	—	2	1	no	—
499	C₂×Q₈	—	2	1	no	—
500	C₂×Q₈	—	2	1	no	—
501	C₂×Q₈	—	4	2	no	—
502	C₂×Q₈	—	4	2	no	—
503	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
504	C₂×D₄	—	2	2	no	—
505	C₂×D₄	—	2	1	no	—
506	C₂×D₄	—	2	2	no	—
507	C₂×D₄	—	2	2	no	—
508	C₂×D₄	—	2	2	no	—
509	C₂×D₄	—	2	2	no	—
510	C₂×D₄	—	2	2	no	—
511	C₂×D₄	—	2	1	no	—
512	C₂×D₄	—	2	1	no	—
513	C₂×D₄	—	2	1	no	—
514	C₂×D₄	—	2	1	no	—
515	C₂×D₄	—	4	2	no	—
516	C₂×D₄	—	8	2	no	—
517	C₂×D₄	—	8	2	no	—
518	C₂×D₄	—	8	2	no	—
519	C₂×D₄	C₂×D₄×C₈₉	1	1	yes	characteristic
520	C₂×D₄	C₂×D₄×C₈₉	1	2	yes	—
521	C₂²×C₄	—	2	1	no	—
522	C₂²×C₄	—	2	2	no	—
523	C₂²×C₄	—	2	2	no	—
524	C₂²×C₄	—	2	1	no	—
525	C₂²×C₄	—	2	2	no	—
526	C₂²×C₄	—	2	2	no	—
527	C₂²×C₄	—	2	1	no	—
528	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
529	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
530	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
531	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
532	Q₁₆	—	4	2	no	—
533	SD₁₆	—	4	4	no	—
534	SD₁₆	—	4	2	no	—
535	C₂×C₈	—	2	1	no	—
536	C₂×C₈	—	2	1	no	—
537	C₂×C₈	—	2	2	no	—
538	C₄⋊C₄	—	2	2	no	—
539	C₄⋊C₄	—	2	2	no	—
540	C₄⋊C₄	—	2	1	no	—
541	C₄⋊C₄	—	2	1	no	—
542	C₄⋊C₄	—	2	1	no	—
543	C₄⋊C₄	—	2	1	no	—
544	C₄⋊C₄	—	2	1	no	—
545	C₄⋊C₄	—	2	1	no	—
546	C₄⋊C₄	—	2	1	no	—
547	C₄⋊C₄	—	4	2	no	—
548	C₄⋊C₄	—	4	1	no	—
549	C₄⋊C₄	—	4	1	no	—
550	C₄⋊C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
551	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
552	C₄⋊C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
553	C₂²⋊C₄	—	2	2	no	—
554	C₂²⋊C₄	—	2	2	no	—
555	C₂²⋊C₄	—	2	2	no	—
556	C₂²⋊C₄	—	2	2	no	—
557	C₂²⋊C₄	—	2	2	no	—
558	C₂²⋊C₄	—	2	1	no	—
559	C₂²⋊C₄	—	2	1	no	—
560	C₂²⋊C₄	—	2	2	no	—
561	C₂²⋊C₄	—	2	2	no	—
562	C₂²⋊C₄	—	2	1	no	—
563	C₂²⋊C₄	—	2	1	no	—
564	C₂²⋊C₄	—	2	2	no	—
565	C₂²⋊C₄	—	2	2	no	—
566	C₂²⋊C₄	—	2	2	no	—
567	C₂²⋊C₄	—	2	1	no	—
568	C₂²⋊C₄	—	2	2	no	—
569	C₂²⋊C₄	—	4	4	no	—
570	C₂²⋊C₄	—	8	2	no	—
571	C₄²	—	2	2	no	—
572	C₄²	—	2	1	no	—
573	C₄²	—	4	1	no	—
574	C₄²	—	4	1	no	—
575	C₄²	—	8	1	no	—
576	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
577	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
578	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
579	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
580	C₂³	—	2	2	no	—
581	C₂³	—	2	1	no	—
582	C₂³	—	2	2	no	—
583	C₂³	—	2	2	no	—
584	C₂³	—	2	2	no	—
585	C₂³	—	4	2	no	—
586	C₂³	—	4	2	no	—
587	C₂³	—	8	2	no	—
588	C₂³	—	8	2	no	—
589	C₂³	—	8	2	no	—
590	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
591	Q₈	—	2	2	no	—
592	Q₈	—	2	2	no	—
593	Q₈	—	4	1	no	—
594	Q₈	—	4	1	no	—
595	Q₈	—	4	1	no	—
596	Q₈	—	4	2	no	—
597	Q₈	—	8	1	no	—
598	Q₈	—	8	1	no	—
599	D₄	—	4	2	no	—
600	D₄	—	4	1	no	—
601	D₄	—	4	1	no	—
602	D₄	—	4	2	no	—
603	D₄	—	4	1	no	—
604	D₄	—	4	2	no	—
605	D₄	—	4	2	no	—
606	D₄	—	4	2	no	—
607	D₄	—	8	1	no	—
608	D₄	—	8	1	no	—
609	D₄	—	8	2	no	—
610	D₄	—	8	2	no	—
611	D₄	—	8	2	no	—
612	D₄	—	8	2	no	—
613	D₄	—	8	2	no	—
614	C₂×C₄	—	2	1	no	—
615	C₂×C₄	—	2	1	no	—
616	C₂×C₄	—	2	2	no	—
617	C₂×C₄	—	2	2	no	—
618	C₂×C₄	—	2	1	no	—
619	C₂×C₄	—	2	2	no	—
620	C₂×C₄	—	2	2	no	—
621	C₂×C₄	—	2	1	no	—
622	C₂×C₄	—	2	1	no	—
623	C₂×C₄	—	4	1	no	—
624	C₂×C₄	—	4	1	no	—
625	C₂×C₄	—	4	1	no	—
626	C₂×C₄	—	4	2	no	—
627	C₂×C₄	—	4	2	no	—
628	C₂×C₄	—	4	2	no	—
629	C₂×C₄	—	4	2	no	—
630	C₂×C₄	—	4	1	no	—
631	C₂×C₄	—	4	2	no	—
632	C₂×C₄	—	4	1	no	—
633	C₂×C₄	—	4	1	no	—
634	C₂×C₄	—	8	2	no	—
635	C₂×C₄	—	8	2	no	—
636	C₂×C₄	—	8	2	no	—
637	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
638	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
639	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
640	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
641	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
642	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
643	C₈	—	4	2	no	—
644	C₈	—	4	1	no	—
645	C₈	—	4	1	no	—
646	C₂²	—	2	1	no	—
647	C₂²	—	2	1	no	—
648	C₂²	—	4	2	no	—
649	C₂²	—	4	2	no	—
650	C₂²	—	4	1	no	—
651	C₂²	—	4	1	no	—
652	C₂²	—	4	2	no	—
653	C₂²	—	4	2	no	—
654	C₂²	—	4	2	no	—
655	C₂²	—	4	1	no	—
656	C₂²	—	4	2	no	—
657	C₂²	—	8	2	no	—
658	C₂²	—	8	2	no	—
659	C₂²	—	16	4	no	—
660	C₂²	C₂×#120×C₈₉	1	1	yes	characteristic
661	C₂²	#576×C₈₉	1	1	yes	characteristic
662	C₂²	#577×C₈₉	1	1	yes	characteristic
663	C₄	—	2	1	no	—
664	C₄	—	2	1	no	—
665	C₄	—	2	1	no	—
666	C₄	—	2	1	no	—
667	C₄	—	2	1	no	—
668	C₄	—	4	1	no	—
669	C₄	—	4	2	no	—
670	C₄	—	4	1	no	—
671	C₄	—	4	2	no	—
672	C₄	—	4	2	no	—
673	C₄	—	4	1	no	—
674	C₄	—	8	2	no	—
675	C₄	—	8	1	no	—
676	C₄	#455×C₈₉	1	2	yes	—
677	C₂	—	2	1	no	—
678	C₂	—	2	1	no	—
679	C₂	—	8	2	no	—
680	C₂	—	8	2	no	—
681	C₂	—	8	1	no	—
682	C₂	C₈₉×#3012	1	1	yes	characteristic
683	C₂	C₂×#455×C₈₉	1	1	yes	characteristic
684	C₂	C₈₉×#3025	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₈₉×#3558	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#3112	C₂	semidirect
3. C₈₉×#3108	C₂	semidirect
4. C₈₉×#2592	C₂	semidirect
5. C₈₉×#1612	C₂	semidirect
6. C₈₉×#1607	C₂	semidirect
7. C₈₉×#1606	C₂	semidirect
8. C₈₉×#1601	C₂	semidirect
9. C₈₉×#1509	C₂	semidirect
10. C₈₉×#1502	C₂	semidirect
11. C₈₉×#1469	C₂	semidirect
27. C₂×C₄×Q₈×C₈₉	C₂²	semidirect
28. #483×C₈₉	C₂²	semidirect
30. #480×C₈₉	C₂²	semidirect
35. #451×C₈₉	C₂²	semidirect
37. #448×C₈₉	C₂²	semidirect
41. #432×C₈₉	C₂²	semidirect
42. #431×C₈₉	C₂²	semidirect
43. #397×C₈₉	C₂²	semidirect
44. #393×C₈₉	C₂²	semidirect
48. C₄×C₄⋊C₄×C₈₉	C₂²	semidirect
49. #328×C₈₉	C₂²	semidirect
51. #327×C₈₉	C₂²	semidirect
52. #327×C₈₉	C₂²	semidirect
53. #323×C₈₉	C₂²	semidirect
54. #323×C₈₉	C₂²	semidirect
334. #18273	C₈₉	direct (N × H)
344. #3112	C₂×C₈₉	semidirect
345. #3108	C₂×C₈₉	semidirect
346. #2592	C₂×C₈₉	semidirect
347. #1612	C₂×C₈₉	semidirect
348. #1607	C₂×C₈₉	semidirect
349. #1606	C₂×C₈₉	semidirect
350. #1601	C₂×C₈₉	semidirect
351. #1509	C₂×C₈₉	semidirect
352. #1502	C₂×C₈₉	semidirect
353. #1469	C₂×C₈₉	semidirect
354. C₈₉	#18273	direct (N × H)
370. C₂×C₄×Q₈	C₂²×C₈₉	semidirect
371. #483	C₂²×C₈₉	semidirect
373. #480	C₂²×C₈₉	semidirect
378. #451	C₂²×C₈₉	semidirect
380. #448	C₂²×C₈₉	semidirect
384. #432	C₂²×C₈₉	semidirect
385. #431	C₂²×C₈₉	semidirect
386. #397	C₂²×C₈₉	semidirect
387. #393	C₂²×C₈₉	semidirect
391. C₄×C₄⋊C₄	C₂²×C₈₉	semidirect
392. #328	C₂²×C₈₉	semidirect
394. #327	C₂²×C₈₉	semidirect
395. #327	C₂²×C₈₉	semidirect
396. #323	C₂²×C₈₉	semidirect
397. #323	C₂²×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3558	2
C₈₉×#3112	2
C₈₉×#3108	2
C₈₉×#2592	2
C₈₉×#1612	2
C₈₉×#1607	2
C₈₉×#1606	2
C₈₉×#1601	2
C₈₉×#1509	2
C₈₉×#1502	2
C₈₉×#1469	2
#18273	89

Subgroups — C89×#18273 #36396121

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#18273 #36396121