Subgroups — C₈₉×#60810 #36438658

Order 22784.

2098 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#542×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₈₉×#3449	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3225	C₂	1	8	yes	maximal, complemented
4	C₈₉×#3224	C₂	1	8	yes	maximal, complemented
5	C₈₉×#3153	C₂	1	2	yes	maximal, complemented
6	C₈₉×#3141	C₂	1	2	yes	maximal, complemented
7	C₈₉×#3037	C₂	1	2	yes	maximal, complemented
8	C₈₉×#3036	C₂	1	2	yes	maximal, complemented
9	C₈₉×#3007	C₂	1	2	yes	maximal, complemented
10	C₈₉×#3006	C₂	1	2	yes	maximal, complemented
11	C₈₉×#2921	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#41×C₈₉	C₂²	1	2	yes	—
13	C₂×#137×C₈₉	C₂²	1	4	yes	complemented
14	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	4	yes	complemented
15	#542×C₈₉	—	2	4	no	—
16	#542×C₈₉	C₂²	1	8	yes	complemented
17	#537×C₈₉	—	2	1	no	—
18	#537×C₈₉	—	2	1	no	—
19	#536×C₈₉	C₂²	1	4	yes	complemented
20	#535×C₈₉	C₂²	1	4	yes	complemented
21	#534×C₈₉	—	2	1	no	—
22	#534×C₈₉	—	2	1	no	—
23	#532×C₈₉	C₂²	1	2	yes	—
24	#532×C₈₉	C₂²	1	2	yes	—
25	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
26	C₂×#126×C₈₉	—	2	1	no	—
27	C₂×#124×C₈₉	—	2	1	no	—
28	C₂×#124×C₈₉	C₂²	1	1	yes	characteristic, complemented
29	#525×C₈₉	—	2	4	no	—
30	C₂×#123×C₈₉	—	2	1	no	—
31	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
32	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×#121×C₈₉	C₂²	1	1	yes	characteristic
34	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic, complemented
35	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic, complemented
36	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
37	#497×C₈₉	C₂²	1	4	yes	complemented
38	#496×C₈₉	C₂²	1	4	yes	complemented
39	#495×C₈₉	C₂²	1	8	yes	complemented
40	#489×C₈₉	C₂²	1	8	yes	complemented
41	#487×C₈₉	C₂²	1	4	yes	complemented
42	#486×C₈₉	C₂²	1	4	yes	complemented
43	#464×C₈₉	C₂²	1	8	yes	complemented
44	#463×C₈₉	C₂²	1	8	yes	complemented
45	#462×C₈₉	C₂²	1	4	yes	complemented
46	#461×C₈₉	C₂²	1	4	yes	complemented
47	#460×C₈₉	C₂²	1	4	yes	complemented
48	#459×C₈₉	C₂²	1	4	yes	complemented
49	#452×C₈₉	C₂²	1	8	yes	complemented
50	#451×C₈₉	C₂²	1	4	yes	complemented
51	#450×C₈₉	C₂²	1	4	yes	complemented
52	#449×C₈₉	C₂²	1	8	yes	—
53	#448×C₈₉	C₂²	1	4	yes	complemented
54	#447×C₈₉	C₂²	1	4	yes	complemented
55	#433×C₈₉	C₂²	1	8	yes	complemented
56	#423×C₈₉	C₂²	1	2	yes	complemented
57	#417×C₈₉	C₂²	1	4	yes	complemented
58	#417×C₈₉	C₂²	1	4	yes	complemented
59	#407×C₈₉	C₂²	1	2	yes	—
60	C₄×M₁₆×C₈₉	C₂²	1	2	yes	complemented
61	C₂×#41×C₈₉	—	2	2	no	—
62	C₂×#41×C₈₉	—	2	2	no	—
63	C₂×#41×C₈₉	—	4	8	no	—
64	C₂×#41×C₈₉	C₂³	1	4	yes	—
65	C₂×#41×C₈₉	C₂³	1	4	yes	—
66	C₂²×Q₈×C₈₉	—	2	1	no	—
67	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
68	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
69	C₂²×D₄×C₈₉	—	2	2	no	—
70	C₂²×D₄×C₈₉	—	2	1	no	—
71	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
72	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
73	C₂³×C₄×C₈₉	—	2	2	no	—
74	C₂³×C₄×C₈₉	—	2	1	no	—
75	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
76	#137×C₈₉	—	4	8	no	—
77	AGL(1,ℤ/8)×C₈₉	—	4	8	no	—
78	C₂×Q₁₆×C₈₉	C₂³	1	8	yes	—
79	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	—
80	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	—
81	C₂×D₈×C₈₉	C₂³	1	8	yes	—
82	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
83	C₄⋊Q₈×C₈₉	—	2	2	no	—
84	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
85	C₄⋊Q₈×C₈₉	C₂³	1	2	yes	—
86	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
87	#127×C₈₉	C₂³	1	2	yes	—
88	#127×C₈₉	C₂³	1	2	yes	—
89	#126×C₈₉	—	2	8	no	—
90	#124×C₈₉	—	2	8	no	—
91	#124×C₈₉	—	2	2	no	—
92	#124×C₈₉	C₂³	1	4	yes	—
93	#124×C₈₉	C₂³	1	4	yes	—
94	#124×C₈₉	C₂³	1	4	yes	—
95	#123×C₈₉	—	2	8	no	—
96	#123×C₈₉	—	2	4	no	—
97	#123×C₈₉	—	2	4	no	—
98	C₂²⋊Q₈×C₈₉	—	2	2	no	—
99	C₂²⋊Q₈×C₈₉	—	2	8	no	—
100	C₂²⋊Q₈×C₈₉	—	2	4	no	—
101	C₂²⋊Q₈×C₈₉	—	2	4	no	—
102	C₂²⋊Q₈×C₈₉	C₂³	1	4	yes	—
103	#121×C₈₉	—	2	2	no	—
104	#121×C₈₉	—	2	2	no	—
105	#121×C₈₉	—	2	2	no	—
106	#121×C₈₉	—	2	2	no	—
107	#121×C₈₉	—	2	2	no	—
108	#121×C₈₉	—	2	4	no	—
109	#121×C₈₉	—	2	4	no	—
110	#121×C₈₉	C₂³	1	4	yes	—
111	#120×C₈₉	—	2	4	no	—
112	#120×C₈₉	—	2	4	no	—
113	C₄×Q₈×C₈₉	—	2	2	no	—
114	C₄×Q₈×C₈₉	C₂³	1	4	yes	—
115	C₄×D₄×C₈₉	—	2	2	no	—
116	C₄×D₄×C₈₉	—	2	2	no	—
117	C₄×D₄×C₈₉	—	2	2	no	—
118	C₄×D₄×C₈₉	—	2	2	no	—
119	C₄×D₄×C₈₉	—	2	2	no	—
120	C₄×D₄×C₈₉	C₂³	1	4	yes	—
121	#117×C₈₉	D₄	1	4	yes	—
122	#117×C₈₉	C₂³	1	4	yes	—
123	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
124	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
125	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
126	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
127	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
128	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
129	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
130	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
131	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
132	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
133	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
134	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
135	C₂×C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
136	C₂×C₂²⋊C₄×C₈₉	D₄	1	1	yes	characteristic
137	C₂×C₄²×C₈₉	—	2	1	no	—
138	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
139	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
140	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
141	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
142	#103×C₈₉	C₂³	1	8	yes	—
143	#103×C₈₉	C₂³	1	8	yes	—
144	#102×C₈₉	C₂³	1	8	yes	—
145	#102×C₈₉	C₂³	1	8	yes	—
146	C₂²⋊C₈×C₈₉	C₂³	1	4	yes	—
147	C₂²⋊C₈×C₈₉	C₂³	1	4	yes	—
148	#97×C₈₉	C₂³	1	4	yes	—
149	C₄×C₈×C₈₉	C₂³	1	4	yes	—
150	C₂⁴×C₈₉	—	2	1	no	—
151	C₂⁴×C₈₉	—	2	2	no	—
152	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
153	#41×C₈₉	—	2	8	no	—
154	#41×C₈₉	—	2	8	no	—
155	#41×C₈₉	—	4	4	no	—
156	#41×C₈₉	—	4	4	no	—
157	#41×C₈₉	—	4	8	no	—
158	#41×C₈₉	—	4	8	no	—
159	C₂×Q₈×C₈₉	—	2	1	no	—
160	C₂×Q₈×C₈₉	—	2	2	no	—
161	C₂×Q₈×C₈₉	—	2	4	no	—
162	C₂×Q₈×C₈₉	—	2	4	no	—
163	C₂×Q₈×C₈₉	—	4	4	no	—
164	C₂×Q₈×C₈₉	—	8	2	no	—
165	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
166	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
167	C₂×D₄×C₈₉	—	2	4	no	—
168	C₂×D₄×C₈₉	—	2	4	no	—
169	C₂×D₄×C₈₉	—	2	4	no	—
170	C₂×D₄×C₈₉	—	2	1	no	—
171	C₂×D₄×C₈₉	—	2	2	no	—
172	C₂×D₄×C₈₉	—	2	2	no	—
173	C₂×D₄×C₈₉	—	2	2	no	—
174	C₂×D₄×C₈₉	—	2	2	no	—
175	C₂×D₄×C₈₉	—	2	2	no	—
176	C₂×D₄×C₈₉	—	2	4	no	—
177	C₂×D₄×C₈₉	—	4	4	no	—
178	C₂×D₄×C₈₉	—	4	8	no	—
179	C₂×D₄×C₈₉	—	8	2	no	—
180	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
181	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
182	C₂²×C₄×C₈₉	—	2	4	no	—
183	C₂²×C₄×C₈₉	—	2	1	no	—
184	C₂²×C₄×C₈₉	—	2	1	no	—
185	C₂²×C₄×C₈₉	—	2	1	no	—
186	C₂²×C₄×C₈₉	—	2	2	no	—
187	C₂²×C₄×C₈₉	—	2	4	no	—
188	C₂²×C₄×C₈₉	—	2	4	no	—
189	C₂²×C₄×C₈₉	—	2	4	no	—
190	C₂²×C₄×C₈₉	—	4	8	no	—
191	C₂²×C₄×C₈₉	—	8	2	no	—
192	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
193	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
194	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
195	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
196	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
197	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
198	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
199	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
200	Q₁₆×C₈₉	—	4	8	no	—
201	SD₁₆×C₈₉	—	4	8	no	—
202	SD₁₆×C₈₉	—	4	8	no	—
203	D₈×C₈₉	—	4	8	no	—
204	M₁₆×C₈₉	—	2	8	no	—
205	C₂×C₈×C₈₉	C₂⁴	1	8	yes	—
206	C₄⋊C₄×C₈₉	—	2	4	no	—
207	C₄⋊C₄×C₈₉	—	2	4	no	—
208	C₄⋊C₄×C₈₉	—	2	4	no	—
209	C₄⋊C₄×C₈₉	—	2	1	no	—
210	C₄⋊C₄×C₈₉	—	2	1	no	—
211	C₄⋊C₄×C₈₉	—	2	2	no	—
212	C₄⋊C₄×C₈₉	—	2	2	no	—
213	C₄⋊C₄×C₈₉	—	2	2	no	—
214	C₄⋊C₄×C₈₉	—	2	2	no	—
215	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
216	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
217	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
218	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
219	C₂²⋊C₄×C₈₉	—	2	4	no	—
220	C₂²⋊C₄×C₈₉	—	2	4	no	—
221	C₂²⋊C₄×C₈₉	—	2	4	no	—
222	C₂²⋊C₄×C₈₉	—	2	4	no	—
223	C₂²⋊C₄×C₈₉	—	2	4	no	—
224	C₂²⋊C₄×C₈₉	—	2	4	no	—
225	C₂²⋊C₄×C₈₉	—	2	4	no	—
226	C₂²⋊C₄×C₈₉	—	2	4	no	—
227	C₂²⋊C₄×C₈₉	—	2	4	no	—
228	C₂²⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
229	C₂²⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
230	C₄²×C₈₉	—	2	4	no	—
231	C₄²×C₈₉	C₂×D₄	1	2	yes	—
232	C₄²×C₈₉	C₂×D₄	1	2	yes	—
233	C₄²×C₈₉	C₂⁴	1	2	yes	—
234	C₄²×C₈₉	C₂⁴	1	2	yes	—
235	C₂³×C₈₉	—	2	2	no	—
236	C₂³×C₈₉	—	2	2	no	—
237	C₂³×C₈₉	—	2	4	no	—
238	C₂³×C₈₉	—	4	1	no	—
239	C₂³×C₈₉	—	4	1	no	—
240	C₂³×C₈₉	—	4	4	no	—
241	C₂³×C₈₉	—	4	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	C₂²×D₄	1	2	yes	—
247	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
248	Q₈×C₈₉	—	2	4	no	—
249	Q₈×C₈₉	—	2	4	no	—
250	Q₈×C₈₉	—	2	4	no	—
251	Q₈×C₈₉	—	4	4	no	—
252	Q₈×C₈₉	—	4	2	no	—
253	Q₈×C₈₉	—	8	4	no	—
254	D₄×C₈₉	—	2	4	no	—
255	D₄×C₈₉	—	2	4	no	—
256	D₄×C₈₉	—	2	4	no	—
257	D₄×C₈₉	—	4	4	no	—
258	D₄×C₈₉	—	4	8	no	—
259	D₄×C₈₉	—	4	8	no	—
260	D₄×C₈₉	—	4	2	no	—
261	D₄×C₈₉	—	8	2	no	—
262	D₄×C₈₉	—	8	2	no	—
263	D₄×C₈₉	—	8	2	no	—
264	D₄×C₈₉	—	8	2	no	—
265	D₄×C₈₉	—	8	4	no	—
266	C₂×C₄×C₈₉	—	2	2	no	—
267	C₂×C₄×C₈₉	—	2	4	no	—
268	C₂×C₄×C₈₉	—	2	4	no	—
269	C₂×C₄×C₈₉	—	2	2	no	—
270	C₂×C₄×C₈₉	—	2	2	no	—
271	C₂×C₄×C₈₉	—	2	2	no	—
272	C₂×C₄×C₈₉	—	2	4	no	—
273	C₂×C₄×C₈₉	—	4	1	no	—
274	C₂×C₄×C₈₉	—	4	1	no	—
275	C₂×C₄×C₈₉	—	4	1	no	—
276	C₂×C₄×C₈₉	—	4	1	no	—
277	C₂×C₄×C₈₉	—	4	8	no	—
278	C₂×C₄×C₈₉	—	4	4	no	—
279	C₂×C₄×C₈₉	—	4	2	no	—
280	C₂×C₄×C₈₉	—	4	2	no	—
281	C₂×C₄×C₈₉	—	4	2	no	—
282	C₂×C₄×C₈₉	—	4	8	no	—
283	C₂×C₄×C₈₉	—	8	4	no	—
284	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
285	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
286	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
287	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
288	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
289	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
290	C₂×C₄×C₈₉	H(5,𝔽₂)	1	4	yes	—
291	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
292	C₈×C₈₉	—	2	8	no	—
293	C₂²×C₈₉	—	2	1	no	—
294	C₂²×C₈₉	—	2	1	no	—
295	C₂²×C₈₉	—	2	2	no	—
296	C₂²×C₈₉	—	2	2	no	—
297	C₂²×C₈₉	—	2	2	no	—
298	C₂²×C₈₉	—	4	2	no	—
299	C₂²×C₈₉	—	4	4	no	—
300	C₂²×C₈₉	—	4	4	no	—
301	C₂²×C₈₉	—	4	2	no	—
302	C₂²×C₈₉	—	4	4	no	—
303	C₂²×C₈₉	—	4	2	no	—
304	C₂²×C₈₉	—	8	2	no	—
305	C₂²×C₈₉	—	8	4	no	—
306	C₂²×C₈₉	—	8	8	no	—
307	C₂²×C₈₉	#535	1	2	yes	—
308	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
309	C₄×C₈₉	—	2	2	no	—
310	C₄×C₈₉	—	2	2	no	—
311	C₄×C₈₉	—	2	2	no	—
312	C₄×C₈₉	—	2	2	no	—
313	C₄×C₈₉	—	2	2	no	—
314	C₄×C₈₉	—	4	2	no	—
315	C₄×C₈₉	—	4	2	no	—
316	C₄×C₈₉	—	4	4	no	—
317	C₄×C₈₉	—	4	2	no	—
318	C₄×C₈₉	#535	1	2	yes	—
319	C₄×C₈₉	C₂×H(5,𝔽₂)	1	2	yes	—
320	#60810	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
321	C₂×C₈₉	—	2	2	no	—
322	C₂×C₈₉	—	4	2	no	—
323	C₂×C₈₉	—	8	2	no	—
324	C₂×C₈₉	—	8	4	no	—
325	C₂×C₈₉	C₂×#535	1	1	yes	characteristic
326	C₂×C₈₉	#3585	1	1	yes	characteristic
327	C₂×C₈₉	#3586	1	1	yes	characteristic
328	C₂×#542	C₂×C₈₉	1	1	yes	characteristic, complemented
329	#3449	C₂×C₈₉	1	1	yes	characteristic, complemented
330	#3225	C₂×C₈₉	1	8	yes	complemented
331	#3224	C₂×C₈₉	1	8	yes	complemented
332	#3153	C₂×C₈₉	1	2	yes	complemented
333	#3141	C₂×C₈₉	1	2	yes	complemented
334	#3037	C₂×C₈₉	1	2	yes	complemented
335	#3036	C₂×C₈₉	1	2	yes	complemented
336	#3007	C₂×C₈₉	1	2	yes	complemented
337	#3006	C₂×C₈₉	1	2	yes	complemented
338	#2921	C₂×C₈₉	1	1	yes	characteristic, complemented
339	C₈₉	#60810	1	1	yes	characteristic, direct product factor, Sylow
340	C₂²×#41	C₂²×C₈₉	1	2	yes	—
341	C₂×#137	C₂²×C₈₉	1	4	yes	complemented
342	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	4	yes	complemented
343	#542	—	2	4	no	—
344	#542	C₂²×C₈₉	1	8	yes	complemented
345	#537	—	2	1	no	—
346	#537	—	2	1	no	—
347	#536	C₂²×C₈₉	1	4	yes	complemented
348	#535	C₂²×C₈₉	1	4	yes	complemented
349	#534	—	2	1	no	—
350	#534	—	2	1	no	—
351	#532	C₂²×C₈₉	1	2	yes	—
352	#532	C₂²×C₈₉	1	2	yes	—
353	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
354	C₂×#126	—	2	1	no	—
355	C₂×#124	—	2	1	no	—
356	C₂×#124	C₂²×C₈₉	1	1	yes	characteristic, complemented
357	#525	—	2	4	no	—
358	C₂×#123	—	2	1	no	—
359	C₂×C₂²⋊Q₈	—	2	1	no	—
360	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
361	C₂×#121	C₂²×C₈₉	1	1	yes	characteristic
362	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic, complemented
363	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic, complemented
364	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
365	#497	C₂²×C₈₉	1	4	yes	complemented
366	#496	C₂²×C₈₉	1	4	yes	complemented
367	#495	C₂²×C₈₉	1	8	yes	complemented
368	#489	C₂²×C₈₉	1	8	yes	complemented
369	#487	C₂²×C₈₉	1	4	yes	complemented
370	#486	C₂²×C₈₉	1	4	yes	complemented
371	#464	C₂²×C₈₉	1	8	yes	complemented
372	#463	C₂²×C₈₉	1	8	yes	complemented
373	#462	C₂²×C₈₉	1	4	yes	complemented
374	#461	C₂²×C₈₉	1	4	yes	complemented
375	#460	C₂²×C₈₉	1	4	yes	complemented
376	#459	C₂²×C₈₉	1	4	yes	complemented
377	#452	C₂²×C₈₉	1	8	yes	complemented
378	#451	C₂²×C₈₉	1	4	yes	complemented
379	#450	C₂²×C₈₉	1	4	yes	complemented
380	#449	C₂²×C₈₉	1	8	yes	—
381	#448	C₂²×C₈₉	1	4	yes	complemented
382	#447	C₂²×C₈₉	1	4	yes	complemented
383	#433	C₂²×C₈₉	1	8	yes	complemented
384	#423	C₂²×C₈₉	1	2	yes	complemented
385	#417	C₂²×C₈₉	1	4	yes	complemented
386	#417	C₂²×C₈₉	1	4	yes	complemented
387	#407	C₂²×C₈₉	1	2	yes	—
388	C₄×M₁₆	C₂²×C₈₉	1	2	yes	complemented
389	C₂×#41	—	2	2	no	—
390	C₂×#41	—	2	2	no	—
391	C₂×#41	—	4	8	no	—
392	C₂×#41	C₂³×C₈₉	1	4	yes	—
393	C₂×#41	C₂³×C₈₉	1	4	yes	—
394	C₂²×Q₈	—	2	1	no	—
395	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
396	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
397	C₂²×D₄	—	2	2	no	—
398	C₂²×D₄	—	2	1	no	—
399	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
400	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
401	C₂³×C₄	—	2	2	no	—
402	C₂³×C₄	—	2	1	no	—
403	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
404	#137	—	4	8	no	—
405	AGL(1,ℤ/8)	—	4	8	no	—
406	C₂×Q₁₆	C₂³×C₈₉	1	8	yes	—
407	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	—
408	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	—
409	C₂×D₈	C₂³×C₈₉	1	8	yes	—
410	C₂×M₁₆	C₂³×C₈₉	1	4	yes	—
411	C₄⋊Q₈	—	2	2	no	—
412	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
413	C₄⋊Q₈	C₂³×C₈₉	1	2	yes	—
414	C₄⋊Q₈	C₂³×C₈₉	1	4	yes	—
415	#127	C₂³×C₈₉	1	2	yes	—
416	#127	C₂³×C₈₉	1	2	yes	—
417	#126	—	2	8	no	—
418	#124	—	2	8	no	—
419	#124	—	2	2	no	—
420	#124	C₂³×C₈₉	1	4	yes	—
421	#124	C₂³×C₈₉	1	4	yes	—
422	#124	C₂³×C₈₉	1	4	yes	—
423	#123	—	2	8	no	—
424	#123	—	2	4	no	—
425	#123	—	2	4	no	—
426	C₂²⋊Q₈	—	2	2	no	—
427	C₂²⋊Q₈	—	2	8	no	—
428	C₂²⋊Q₈	—	2	4	no	—
429	C₂²⋊Q₈	—	2	4	no	—
430	C₂²⋊Q₈	C₂³×C₈₉	1	4	yes	—
431	#121	—	2	2	no	—
432	#121	—	2	2	no	—
433	#121	—	2	2	no	—
434	#121	—	2	2	no	—
435	#121	—	2	2	no	—
436	#121	—	2	4	no	—
437	#121	—	2	4	no	—
438	#121	C₂³×C₈₉	1	4	yes	—
439	#120	—	2	4	no	—
440	#120	—	2	4	no	—
441	C₄×Q₈	—	2	2	no	—
442	C₄×Q₈	C₂³×C₈₉	1	4	yes	—
443	C₄×D₄	—	2	2	no	—
444	C₄×D₄	—	2	2	no	—
445	C₄×D₄	—	2	2	no	—
446	C₄×D₄	—	2	2	no	—
447	C₄×D₄	—	2	2	no	—
448	C₄×D₄	C₂³×C₈₉	1	4	yes	—
449	#117	D₄×C₈₉	1	4	yes	—
450	#117	C₂³×C₈₉	1	4	yes	—
451	C₂×C₄⋊C₄	—	2	1	no	—
452	C₂×C₄⋊C₄	—	2	1	no	—
453	C₂×C₄⋊C₄	—	2	1	no	—
454	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
455	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
456	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
457	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
458	C₂×C₂²⋊C₄	—	2	1	no	—
459	C₂×C₂²⋊C₄	—	2	1	no	—
460	C₂×C₂²⋊C₄	—	2	1	no	—
461	C₂×C₂²⋊C₄	—	2	1	no	—
462	C₂×C₂²⋊C₄	—	2	1	no	—
463	C₂×C₂²⋊C₄	D₄×C₈₉	1	1	yes	characteristic
464	C₂×C₂²⋊C₄	D₄×C₈₉	1	1	yes	characteristic
465	C₂×C₄²	—	2	1	no	—
466	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
467	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
468	C₄⋊C₈	C₂³×C₈₉	1	4	yes	—
469	C₄⋊C₈	C₂³×C₈₉	1	4	yes	—
470	#103	C₂³×C₈₉	1	8	yes	—
471	#103	C₂³×C₈₉	1	8	yes	—
472	#102	C₂³×C₈₉	1	8	yes	—
473	#102	C₂³×C₈₉	1	8	yes	—
474	C₂²⋊C₈	C₂³×C₈₉	1	4	yes	—
475	C₂²⋊C₈	C₂³×C₈₉	1	4	yes	—
476	#97	C₂³×C₈₉	1	4	yes	—
477	C₄×C₈	C₂³×C₈₉	1	4	yes	—
478	C₂⁴	—	2	1	no	—
479	C₂⁴	—	2	2	no	—
480	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
481	#41	—	2	8	no	—
482	#41	—	2	8	no	—
483	#41	—	4	4	no	—
484	#41	—	4	4	no	—
485	#41	—	4	8	no	—
486	#41	—	4	8	no	—
487	C₂×Q₈	—	2	1	no	—
488	C₂×Q₈	—	2	2	no	—
489	C₂×Q₈	—	2	4	no	—
490	C₂×Q₈	—	2	4	no	—
491	C₂×Q₈	—	4	4	no	—
492	C₂×Q₈	—	8	2	no	—
493	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
494	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
495	C₂×D₄	—	2	4	no	—
496	C₂×D₄	—	2	4	no	—
497	C₂×D₄	—	2	4	no	—
498	C₂×D₄	—	2	1	no	—
499	C₂×D₄	—	2	2	no	—
500	C₂×D₄	—	2	2	no	—
501	C₂×D₄	—	2	2	no	—
502	C₂×D₄	—	2	2	no	—
503	C₂×D₄	—	2	2	no	—
504	C₂×D₄	—	2	4	no	—
505	C₂×D₄	—	4	4	no	—
506	C₂×D₄	—	4	8	no	—
507	C₂×D₄	—	8	2	no	—
508	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
509	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
510	C₂²×C₄	—	2	4	no	—
511	C₂²×C₄	—	2	1	no	—
512	C₂²×C₄	—	2	1	no	—
513	C₂²×C₄	—	2	1	no	—
514	C₂²×C₄	—	2	2	no	—
515	C₂²×C₄	—	2	4	no	—
516	C₂²×C₄	—	2	4	no	—
517	C₂²×C₄	—	2	4	no	—
518	C₂²×C₄	—	4	8	no	—
519	C₂²×C₄	—	8	2	no	—
520	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
521	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
522	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
523	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
524	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
525	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
526	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
527	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
528	Q₁₆	—	4	8	no	—
529	SD₁₆	—	4	8	no	—
530	SD₁₆	—	4	8	no	—
531	D₈	—	4	8	no	—
532	M₁₆	—	2	8	no	—
533	C₂×C₈	C₂⁴×C₈₉	1	8	yes	—
534	C₄⋊C₄	—	2	4	no	—
535	C₄⋊C₄	—	2	4	no	—
536	C₄⋊C₄	—	2	4	no	—
537	C₄⋊C₄	—	2	1	no	—
538	C₄⋊C₄	—	2	1	no	—
539	C₄⋊C₄	—	2	2	no	—
540	C₄⋊C₄	—	2	2	no	—
541	C₄⋊C₄	—	2	2	no	—
542	C₄⋊C₄	—	2	2	no	—
543	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
544	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
545	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
546	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
547	C₂²⋊C₄	—	2	4	no	—
548	C₂²⋊C₄	—	2	4	no	—
549	C₂²⋊C₄	—	2	4	no	—
550	C₂²⋊C₄	—	2	4	no	—
551	C₂²⋊C₄	—	2	4	no	—
552	C₂²⋊C₄	—	2	4	no	—
553	C₂²⋊C₄	—	2	4	no	—
554	C₂²⋊C₄	—	2	4	no	—
555	C₂²⋊C₄	—	2	4	no	—
556	C₂²⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
557	C₂²⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
558	C₄²	—	2	4	no	—
559	C₄²	C₂×D₄×C₈₉	1	2	yes	—
560	C₄²	C₂×D₄×C₈₉	1	2	yes	—
561	C₄²	C₂⁴×C₈₉	1	2	yes	—
562	C₄²	C₂⁴×C₈₉	1	2	yes	—
563	C₂³	—	2	2	no	—
564	C₂³	—	2	2	no	—
565	C₂³	—	2	4	no	—
566	C₂³	—	4	1	no	—
567	C₂³	—	4	1	no	—
568	C₂³	—	4	4	no	—
569	C₂³	—	4	2	no	—
570	C₂³	—	4	2	no	—
571	C₂³	—	4	2	no	—
572	C₂³	—	8	2	no	—
573	C₂³	—	8	2	no	—
574	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
575	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
576	Q₈	—	2	4	no	—
577	Q₈	—	2	4	no	—
578	Q₈	—	2	4	no	—
579	Q₈	—	4	4	no	—
580	Q₈	—	4	2	no	—
581	Q₈	—	8	4	no	—
582	D₄	—	2	4	no	—
583	D₄	—	2	4	no	—
584	D₄	—	2	4	no	—
585	D₄	—	4	4	no	—
586	D₄	—	4	8	no	—
587	D₄	—	4	8	no	—
588	D₄	—	4	2	no	—
589	D₄	—	8	2	no	—
590	D₄	—	8	2	no	—
591	D₄	—	8	2	no	—
592	D₄	—	8	2	no	—
593	D₄	—	8	4	no	—
594	C₂×C₄	—	2	2	no	—
595	C₂×C₄	—	2	4	no	—
596	C₂×C₄	—	2	4	no	—
597	C₂×C₄	—	2	2	no	—
598	C₂×C₄	—	2	2	no	—
599	C₂×C₄	—	2	2	no	—
600	C₂×C₄	—	2	4	no	—
601	C₂×C₄	—	4	1	no	—
602	C₂×C₄	—	4	1	no	—
603	C₂×C₄	—	4	1	no	—
604	C₂×C₄	—	4	1	no	—
605	C₂×C₄	—	4	8	no	—
606	C₂×C₄	—	4	4	no	—
607	C₂×C₄	—	4	2	no	—
608	C₂×C₄	—	4	2	no	—
609	C₂×C₄	—	4	2	no	—
610	C₂×C₄	—	4	8	no	—
611	C₂×C₄	—	8	4	no	—
612	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
613	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
614	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
615	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
616	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
617	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
618	C₂×C₄	H(5,𝔽₂)×C₈₉	1	4	yes	—
619	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
620	C₈	—	2	8	no	—
621	C₂²	—	2	1	no	—
622	C₂²	—	2	1	no	—
623	C₂²	—	2	2	no	—
624	C₂²	—	2	2	no	—
625	C₂²	—	2	2	no	—
626	C₂²	—	4	2	no	—
627	C₂²	—	4	4	no	—
628	C₂²	—	4	4	no	—
629	C₂²	—	4	2	no	—
630	C₂²	—	4	4	no	—
631	C₂²	—	4	2	no	—
632	C₂²	—	8	2	no	—
633	C₂²	—	8	4	no	—
634	C₂²	—	8	8	no	—
635	C₂²	#535×C₈₉	1	2	yes	—
636	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
637	C₄	—	2	2	no	—
638	C₄	—	2	2	no	—
639	C₄	—	2	2	no	—
640	C₄	—	2	2	no	—
641	C₄	—	2	2	no	—
642	C₄	—	4	2	no	—
643	C₄	—	4	2	no	—
644	C₄	—	4	4	no	—
645	C₄	—	4	2	no	—
646	C₄	#535×C₈₉	1	2	yes	—
647	C₄	C₂×H(5,𝔽₂)×C₈₉	1	2	yes	—
648	C₂	—	2	2	no	—
649	C₂	—	4	2	no	—
650	C₂	—	8	2	no	—
651	C₂	—	8	4	no	—
652	C₂	C₂×#535×C₈₉	1	1	yes	characteristic
653	C₂	C₈₉×#3585	1	1	yes	characteristic
654	C₂	C₈₉×#3586	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×#117×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×#542×C₈₉	C₂	semidirect
2. C₈₉×#3449	C₂	semidirect
3. C₈₉×#3225	C₂	semidirect
4. C₈₉×#3224	C₂	semidirect
5. C₈₉×#3153	C₂	semidirect
6. C₈₉×#3141	C₂	semidirect
7. C₈₉×#3037	C₂	semidirect
8. C₈₉×#3036	C₂	semidirect
9. C₈₉×#3007	C₂	semidirect
10. C₈₉×#3006	C₂	semidirect
11. C₈₉×#2921	C₂	semidirect
13. C₂×#137×C₈₉	C₂²	semidirect
14. C₂×AGL(1,ℤ/8)×C₈₉	C₂²	semidirect
16. #542×C₈₉	C₂²	semidirect
19. #536×C₈₉	C₂²	semidirect
20. #535×C₈₉	C₂²	semidirect
25. C₂×C₄⋊Q₈×C₈₉	C₂²	semidirect
28. C₂×#124×C₈₉	C₂²	semidirect
34. C₂×C₄×Q₈×C₈₉	C₂²	semidirect
35. C₂×C₄×D₄×C₈₉	C₂²	semidirect
37. #497×C₈₉	C₂²	semidirect
38. #496×C₈₉	C₂²	semidirect
39. #495×C₈₉	C₂²	semidirect
40. #489×C₈₉	C₂²	semidirect
41. #487×C₈₉	C₂²	semidirect
42. #486×C₈₉	C₂²	semidirect
43. #464×C₈₉	C₂²	semidirect
44. #463×C₈₉	C₂²	semidirect
45. #462×C₈₉	C₂²	semidirect
46. #461×C₈₉	C₂²	semidirect
47. #460×C₈₉	C₂²	semidirect
48. #459×C₈₉	C₂²	semidirect
49. #452×C₈₉	C₂²	semidirect
50. #451×C₈₉	C₂²	semidirect
51. #450×C₈₉	C₂²	semidirect
53. #448×C₈₉	C₂²	semidirect
54. #447×C₈₉	C₂²	semidirect
55. #433×C₈₉	C₂²	semidirect
56. #423×C₈₉	C₂²	semidirect
57. #417×C₈₉	C₂²	semidirect
58. #417×C₈₉	C₂²	semidirect
60. C₄×M₁₆×C₈₉	C₂²	semidirect
320. #60810	C₈₉	direct (N × H)
328. C₂×#542	C₂×C₈₉	semidirect
329. #3449	C₂×C₈₉	semidirect
330. #3225	C₂×C₈₉	semidirect
331. #3224	C₂×C₈₉	semidirect
332. #3153	C₂×C₈₉	semidirect
333. #3141	C₂×C₈₉	semidirect
334. #3037	C₂×C₈₉	semidirect
335. #3036	C₂×C₈₉	semidirect
336. #3007	C₂×C₈₉	semidirect
337. #3006	C₂×C₈₉	semidirect
338. #2921	C₂×C₈₉	semidirect
339. C₈₉	#60810	direct (N × H)
341. C₂×#137	C₂²×C₈₉	semidirect
342. C₂×AGL(1,ℤ/8)	C₂²×C₈₉	semidirect
344. #542	C₂²×C₈₉	semidirect
347. #536	C₂²×C₈₉	semidirect
348. #535	C₂²×C₈₉	semidirect
353. C₂×C₄⋊Q₈	C₂²×C₈₉	semidirect
356. C₂×#124	C₂²×C₈₉	semidirect
362. C₂×C₄×Q₈	C₂²×C₈₉	semidirect
363. C₂×C₄×D₄	C₂²×C₈₉	semidirect
365. #497	C₂²×C₈₉	semidirect
366. #496	C₂²×C₈₉	semidirect
367. #495	C₂²×C₈₉	semidirect
368. #489	C₂²×C₈₉	semidirect
369. #487	C₂²×C₈₉	semidirect
370. #486	C₂²×C₈₉	semidirect
371. #464	C₂²×C₈₉	semidirect
372. #463	C₂²×C₈₉	semidirect
373. #462	C₂²×C₈₉	semidirect
374. #461	C₂²×C₈₉	semidirect
375. #460	C₂²×C₈₉	semidirect
376. #459	C₂²×C₈₉	semidirect
377. #452	C₂²×C₈₉	semidirect
378. #451	C₂²×C₈₉	semidirect
379. #450	C₂²×C₈₉	semidirect
381. #448	C₂²×C₈₉	semidirect
382. #447	C₂²×C₈₉	semidirect
383. #433	C₂²×C₈₉	semidirect
384. #423	C₂²×C₈₉	semidirect
385. #417	C₂²×C₈₉	semidirect
386. #417	C₂²×C₈₉	semidirect
388. C₄×M₁₆	C₂²×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×#542×C₈₉	2
C₈₉×#3449	2
C₈₉×#3225	2
C₈₉×#3224	2
C₈₉×#3153	2
C₈₉×#3141	2
C₈₉×#3037	2
C₈₉×#3036	2
C₈₉×#3007	2
C₈₉×#3006	2
C₈₉×#2921	2
#60810	89

Subgroups — C89×#60810 #36438658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#60810 #36438658