Subgroups — C₂×C₈₉×#3062 #36438299

Order 22784.

2934 conjugacy classes of subgroups.

The subgroup lattice has 666 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×AGL(1,ℤ/8)×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#131×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#537×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#535×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#3062	C₂	1	16	yes	maximal, direct product factor
6	C₂×#469×C₈₉	C₂	1	2	yes	maximal, complemented
7	C₂×#468×C₈₉	C₂	1	2	yes	maximal, complemented
8	C₂×#466×C₈₉	C₂	1	2	yes	maximal, complemented
9	C₂×#465×C₈₉	C₂	1	2	yes	maximal, complemented
10	C₂×#428×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#416×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#102×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
14	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
15	C₂³×D₄×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×AGL(1,ℤ/8)×C₈₉	—	2	8	no	—
17	C₂×AGL(1,ℤ/8)×C₈₉	C₂²	1	8	yes	complemented
18	C₂²×SD₁₆×C₈₉	C₂²	1	2	yes	—
19	C₂²×D₈×C₈₉	C₂²	1	2	yes	—
20	C₂×#131×C₈₉	—	2	8	no	—
21	C₂×#131×C₈₉	C₂²	1	8	yes	complemented
22	C₂²×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
23	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	complemented
24	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂³×C₈×C₈₉	C₂²	1	2	yes	complemented
26	#537×C₈₉	—	2	4	no	—
27	#537×C₈₉	C₂²	1	8	yes	complemented
28	#535×C₈₉	—	2	4	no	—
29	#535×C₈₉	C₂²	1	8	yes	complemented
30	C₂×#127×C₈₉	—	2	1	no	—
31	C₂×#124×C₈₉	—	2	1	no	—
32	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
33	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	2	yes	complemented
34	C₂×#121×C₈₉	—	2	1	no	—
35	C₂×#121×C₈₉	—	2	2	no	—
36	C₂×#121×C₈₉	C₂²	1	2	yes	—
37	C₂×#121×C₈₉	C₂²	1	2	yes	—
38	C₂×#121×C₈₉	C₂²	1	2	yes	complemented
39	C₂×#120×C₈₉	—	2	2	no	—
40	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
41	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
42	#469×C₈₉	C₂²	1	16	yes	complemented
43	#468×C₈₉	C₂²	1	16	yes	complemented
44	#466×C₈₉	C₂²	1	16	yes	complemented
45	#465×C₈₉	C₂²	1	16	yes	complemented
46	#428×C₈₉	C₂²	1	8	yes	complemented
47	C₂×#107×C₈₉	C₂²	1	2	yes	complemented
48	C₂×#106×C₈₉	C₂²	1	2	yes	complemented
49	#416×C₈₉	C₂²	1	8	yes	complemented
50	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic, complemented
51	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic, complemented
52	C₂×#102×C₈₉	C₂²	1	2	yes	—
53	C₂×#102×C₈₉	C₂²	1	2	yes	—
54	C₂×#102×C₈₉	C₂²	1	8	yes	complemented
55	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic, complemented
56	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic, complemented
57	C₂⁵×C₈₉	—	2	1	no	—
58	C₂×#41×C₈₉	—	2	4	no	—
59	C₂×#41×C₈₉	—	2	2	no	—
60	C₂×#41×C₈₉	—	2	4	no	—
61	C₂×#41×C₈₉	—	2	4	no	—
62	C₂×#41×C₈₉	—	2	4	no	—
63	C₂×#41×C₈₉	D₄	1	4	yes	—
64	C₂×#41×C₈₉	C₂³	1	4	yes	—
65	C₂×#41×C₈₉	C₂³	1	4	yes	—
66	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
67	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
68	C₂²×D₄×C₈₉	—	2	1	no	—
69	C₂²×D₄×C₈₉	—	2	2	no	—
70	C₂²×D₄×C₈₉	—	2	2	no	—
71	C₂²×D₄×C₈₉	—	2	2	no	—
72	C₂²×D₄×C₈₉	—	2	4	no	—
73	C₂²×D₄×C₈₉	—	2	1	no	—
74	C₂²×D₄×C₈₉	—	2	2	no	—
75	C₂²×D₄×C₈₉	—	2	4	no	—
76	C₂²×D₄×C₈₉	D₄	1	2	yes	—
77	C₂²×D₄×C₈₉	D₄	1	1	yes	characteristic
78	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
79	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
80	C₂²×D₄×C₈₉	C₂³	1	4	yes	—
81	C₂³×C₄×C₈₉	—	2	1	no	—
82	C₂³×C₄×C₈₉	—	2	1	no	—
83	C₂³×C₄×C₈₉	C₂³	1	2	yes	—
84	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
85	AGL(1,ℤ/8)×C₈₉	—	2	32	no	—
86	C₂×SD₁₆×C₈₉	—	2	8	no	—
87	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	—
88	C₂×D₈×C₈₉	—	2	8	no	—
89	C₂×D₈×C₈₉	C₂³	1	8	yes	—
90	#131×C₈₉	—	2	32	no	—
91	C₂×M₁₆×C₈₉	—	2	8	no	—
92	C₂×M₁₆×C₈₉	D₄	1	8	yes	complemented
93	C₂×M₁₆×C₈₉	C₂³	1	4	yes	—
94	C₂×M₁₆×C₈₉	C₂³	1	8	yes	complemented
95	C₂²×C₈×C₈₉	—	2	4	no	—
96	C₂²×C₈×C₈₉	—	2	8	no	—
97	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
98	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
99	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
100	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
101	C₂²×C₈×C₈₉	C₂³	1	8	yes	complemented
102	#127×C₈₉	—	2	8	no	—
103	#124×C₈₉	—	2	8	no	—
104	C₂²⋊Q₈×C₈₉	—	2	8	no	—
105	C₂²⋊Q₈×C₈₉	—	2	4	no	—
106	C₂²⋊Q₈×C₈₉	C₂³	1	8	yes	complemented
107	#121×C₈₉	—	2	8	no	—
108	#121×C₈₉	—	2	16	no	—
109	#121×C₈₉	—	2	4	no	—
110	#121×C₈₉	—	2	4	no	—
111	#121×C₈₉	—	2	4	no	—
112	#121×C₈₉	C₂³	1	8	yes	—
113	#121×C₈₉	C₂³	1	8	yes	—
114	#121×C₈₉	C₂³	1	8	yes	complemented
115	#120×C₈₉	—	2	16	no	—
116	#117×C₈₉	—	2	2	no	—
117	#117×C₈₉	C₂³	1	4	yes	—
118	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
119	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
120	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
121	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
122	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
123	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
124	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
125	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
126	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
127	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
128	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
129	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
130	C₂×C₄²×C₈₉	—	2	1	no	—
131	#107×C₈₉	C₂³	1	8	yes	complemented
132	#106×C₈₉	C₂³	1	8	yes	complemented
133	#103×C₈₉	C₂³	1	4	yes	complemented
134	#103×C₈₉	C₂³	1	4	yes	complemented
135	#102×C₈₉	C₂³	1	8	yes	—
136	#102×C₈₉	C₂³	1	8	yes	—
137	#102×C₈₉	C₂³	1	4	yes	complemented
138	#102×C₈₉	C₂³	1	4	yes	complemented
139	C₂⁴×C₈₉	—	2	1	no	—
140	C₂⁴×C₈₉	—	2	2	no	—
141	C₂⁴×C₈₉	—	2	4	no	—
142	C₂⁴×C₈₉	—	4	4	no	—
143	C₂⁴×C₈₉	—	8	2	no	—
144	C₂⁴×C₈₉	—	8	2	no	—
145	C₂⁴×C₈₉	C₂×D₄	1	2	yes	—
146	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
147	#41×C₈₉	—	2	16	no	—
148	#41×C₈₉	—	2	8	no	—
149	#41×C₈₉	—	2	8	no	—
150	#41×C₈₉	—	2	8	no	—
151	#41×C₈₉	—	2	8	no	—
152	#41×C₈₉	—	4	8	no	—
153	C₂×Q₈×C₈₉	—	2	4	no	—
154	C₂×Q₈×C₈₉	—	2	1	no	—
155	C₂×Q₈×C₈₉	—	2	2	no	—
156	C₂×Q₈×C₈₉	—	2	2	no	—
157	C₂×Q₈×C₈₉	C₂×D₄	1	4	yes	—
158	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
159	C₂×D₄×C₈₉	—	2	8	no	—
160	C₂×D₄×C₈₉	—	2	4	no	—
161	C₂×D₄×C₈₉	—	2	8	no	—
162	C₂×D₄×C₈₉	—	2	8	no	—
163	C₂×D₄×C₈₉	—	2	8	no	—
164	C₂×D₄×C₈₉	—	2	4	no	—
165	C₂×D₄×C₈₉	—	2	2	no	—
166	C₂×D₄×C₈₉	—	2	4	no	—
167	C₂×D₄×C₈₉	—	2	8	no	—
168	C₂×D₄×C₈₉	—	2	2	no	—
169	C₂×D₄×C₈₉	—	2	1	no	—
170	C₂×D₄×C₈₉	—	2	4	no	—
171	C₂×D₄×C₈₉	—	2	4	no	—
172	C₂×D₄×C₈₉	—	2	8	no	—
173	C₂×D₄×C₈₉	—	2	2	no	—
174	C₂×D₄×C₈₉	—	4	4	no	—
175	C₂×D₄×C₈₉	—	4	4	no	—
176	C₂×D₄×C₈₉	—	4	8	no	—
177	C₂×D₄×C₈₉	—	8	4	no	—
178	C₂×D₄×C₈₉	—	8	4	no	—
179	C₂×D₄×C₈₉	—	8	4	no	—
180	C₂×D₄×C₈₉	C₂×D₄	1	8	yes	—
181	C₂×D₄×C₈₉	C₂×D₄	1	4	yes	—
182	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
183	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
184	C₂²×C₄×C₈₉	—	2	4	no	—
185	C₂²×C₄×C₈₉	—	2	4	no	—
186	C₂²×C₄×C₈₉	—	2	1	no	—
187	C₂²×C₄×C₈₉	—	2	1	no	—
188	C₂²×C₄×C₈₉	—	2	2	no	—
189	C₂²×C₄×C₈₉	—	2	1	no	—
190	C₂²×C₄×C₈₉	—	2	4	no	—
191	C₂²×C₄×C₈₉	—	2	4	no	—
192	C₂²×C₄×C₈₉	—	4	4	no	—
193	C₂²×C₄×C₈₉	—	8	2	no	—
194	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
195	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
196	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
197	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
198	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
199	C₂²×C₄×C₈₉	#41	1	4	yes	—
200	C₂²×C₄×C₈₉	C₂⁴	1	4	yes	—
201	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
202	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
203	SD₁₆×C₈₉	—	2	16	no	—
204	D₈×C₈₉	—	2	16	no	—
205	M₁₆×C₈₉	—	2	16	no	—
206	M₁₆×C₈₉	C₂×D₄	1	16	yes	complemented
207	C₂×C₈×C₈₉	—	2	2	no	—
208	C₂×C₈×C₈₉	—	2	2	no	—
209	C₂×C₈×C₈₉	—	2	8	no	—
210	C₂×C₈×C₈₉	—	2	16	no	—
211	C₂×C₈×C₈₉	C₂×D₄	1	8	yes	—
212	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
213	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
214	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
215	C₄⋊C₄×C₈₉	—	2	4	no	—
216	C₄⋊C₄×C₈₉	—	2	2	no	—
217	C₄⋊C₄×C₈₉	—	2	8	no	—
218	C₄⋊C₄×C₈₉	—	2	1	no	—
219	C₄⋊C₄×C₈₉	—	2	1	no	—
220	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
221	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
222	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
223	C₂²⋊C₄×C₈₉	—	2	4	no	—
224	C₂²⋊C₄×C₈₉	—	2	8	no	—
225	C₂²⋊C₄×C₈₉	—	2	8	no	—
226	C₂²⋊C₄×C₈₉	—	2	8	no	—
227	C₂²⋊C₄×C₈₉	—	2	8	no	—
228	C₄²×C₈₉	—	2	4	no	—
229	C₂³×C₈₉	—	2	2	no	—
230	C₂³×C₈₉	—	2	2	no	—
231	C₂³×C₈₉	—	2	4	no	—
232	C₂³×C₈₉	—	2	4	no	—
233	C₂³×C₈₉	—	2	4	no	—
234	C₂³×C₈₉	—	2	2	no	—
235	C₂³×C₈₉	—	2	4	no	—
236	C₂³×C₈₉	—	4	2	no	—
237	C₂³×C₈₉	—	4	8	no	—
238	C₂³×C₈₉	—	4	4	no	—
239	C₂³×C₈₉	—	4	2	no	—
240	C₂³×C₈₉	—	4	4	no	—
241	C₂³×C₈₉	—	4	4	no	—
242	C₂³×C₈₉	—	4	2	no	—
243	C₂³×C₈₉	—	8	4	no	—
244	C₂³×C₈₉	—	8	4	no	—
245	C₂³×C₈₉	—	8	16	no	—
246	C₂³×C₈₉	#121	1	4	yes	—
247	C₂³×C₈₉	C₂²×D₄	1	4	yes	—
248	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
249	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
250	Q₈×C₈₉	—	2	8	no	—
251	Q₈×C₈₉	—	2	2	no	—
252	Q₈×C₈₉	—	2	2	no	—
253	Q₈×C₈₉	—	4	2	no	—
254	D₄×C₈₉	—	2	16	no	—
255	D₄×C₈₉	—	2	8	no	—
256	D₄×C₈₉	—	2	2	no	—
257	D₄×C₈₉	—	2	4	no	—
258	D₄×C₈₉	—	2	4	no	—
259	D₄×C₈₉	—	2	2	no	—
260	D₄×C₈₉	—	4	8	no	—
261	D₄×C₈₉	—	4	4	no	—
262	D₄×C₈₉	—	4	8	no	—
263	D₄×C₈₉	—	4	2	no	—
264	D₄×C₈₉	—	8	8	no	—
265	D₄×C₈₉	—	8	8	no	—
266	D₄×C₈₉	—	8	8	no	—
267	C₂×C₄×C₈₉	—	2	4	no	—
268	C₂×C₄×C₈₉	—	2	8	no	—
269	C₂×C₄×C₈₉	—	2	4	no	—
270	C₂×C₄×C₈₉	—	2	4	no	—
271	C₂×C₄×C₈₉	—	2	2	no	—
272	C₂×C₄×C₈₉	—	2	2	no	—
273	C₂×C₄×C₈₉	—	2	4	no	—
274	C₂×C₄×C₈₉	—	2	2	no	—
275	C₂×C₄×C₈₉	—	4	2	no	—
276	C₂×C₄×C₈₉	—	4	4	no	—
277	C₂×C₄×C₈₉	—	4	2	no	—
278	C₂×C₄×C₈₉	—	4	2	no	—
279	C₂×C₄×C₈₉	—	4	8	no	—
280	C₂×C₄×C₈₉	—	8	4	no	—
281	C₂×C₄×C₈₉	#121	1	4	yes	—
282	C₂×C₄×C₈₉	#121	1	8	yes	—
283	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
284	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
285	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
286	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
287	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
288	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
289	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
290	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
291	C₈×C₈₉	—	2	2	no	—
292	C₈×C₈₉	—	2	2	no	—
293	C₈×C₈₉	C₂²×D₄	1	8	yes	—
294	C₂²×C₈₉	—	2	8	no	—
295	C₂²×C₈₉	—	2	2	no	—
296	C₂²×C₈₉	—	2	2	no	—
297	C₂²×C₈₉	—	2	4	no	—
298	C₂²×C₈₉	—	2	4	no	—
299	C₂²×C₈₉	—	2	4	no	—
300	C₂²×C₈₉	—	4	8	no	—
301	C₂²×C₈₉	—	4	2	no	—
302	C₂²×C₈₉	—	4	4	no	—
303	C₂²×C₈₉	—	4	2	no	—
304	C₂²×C₈₉	—	4	4	no	—
305	C₂²×C₈₉	—	4	4	no	—
306	C₂²×C₈₉	—	4	2	no	—
307	C₂²×C₈₉	—	8	16	no	—
308	C₂²×C₈₉	—	8	4	no	—
309	C₂²×C₈₉	—	8	8	no	—
310	C₂²×C₈₉	C₂×#121	1	4	yes	—
311	C₂²×C₈₉	C₂×#121	1	2	yes	—
312	C₂²×C₈₉	#576	1	2	yes	—
313	C₂²×C₈₉	#577	1	2	yes	—
314	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
315	C₄×C₈₉	—	2	4	no	—
316	C₄×C₈₉	—	4	2	no	—
317	C₄×C₈₉	—	4	4	no	—
318	C₄×C₈₉	—	4	2	no	—
319	C₄×C₈₉	—	4	2	no	—
320	C₄×C₈₉	C₂×#121	1	2	yes	—
321	C₄×C₈₉	C₂×#121	1	4	yes	—
322	C₄×C₈₉	C₂³×D₄	1	1	yes	characteristic
323	C₄×C₈₉	C₂²×#41	1	1	yes	characteristic
324	C₂×#3062	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
325	C₂×C₈₉	—	2	4	no	—
326	C₂×C₈₉	—	4	4	no	—
327	C₂×C₈₉	—	8	2	no	—
328	C₂×C₈₉	—	8	4	no	—
329	C₂×C₈₉	#3062	1	4	yes	direct product factor
330	C₂×C₈₉	C₂²×#121	1	1	yes	characteristic
331	C₂×C₈₉	C₂×#576	1	1	yes	characteristic
332	C₂×C₈₉	C₂×#577	1	1	yes	characteristic
333	C₂²×AGL(1,ℤ/8)	C₂×C₈₉	1	1	yes	characteristic, complemented
334	C₂²×#131	C₂×C₈₉	1	1	yes	characteristic, complemented
335	C₂×#537	C₂×C₈₉	1	1	yes	characteristic, complemented
336	C₂×#535	C₂×C₈₉	1	1	yes	characteristic, complemented
337	#3062	C₂×C₈₉	1	16	yes	direct product factor
338	C₂×#469	C₂×C₈₉	1	2	yes	complemented
339	C₂×#468	C₂×C₈₉	1	2	yes	complemented
340	C₂×#466	C₂×C₈₉	1	2	yes	complemented
341	C₂×#465	C₂×C₈₉	1	2	yes	complemented
342	C₂×#428	C₂×C₈₉	1	1	yes	characteristic, complemented
343	C₂×#416	C₂×C₈₉	1	1	yes	characteristic, complemented
344	C₂²×#102	C₂×C₈₉	1	1	yes	characteristic, complemented
345	C₈₉	C₂×#3062	1	1	yes	characteristic, direct product factor, Sylow
346	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
347	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
348	C₂³×D₄	C₂²×C₈₉	1	1	yes	characteristic
349	C₂×AGL(1,ℤ/8)	—	2	8	no	—
350	C₂×AGL(1,ℤ/8)	C₂²×C₈₉	1	8	yes	complemented
351	C₂²×SD₁₆	C₂²×C₈₉	1	2	yes	—
352	C₂²×D₈	C₂²×C₈₉	1	2	yes	—
353	C₂×#131	—	2	8	no	—
354	C₂×#131	C₂²×C₈₉	1	8	yes	complemented
355	C₂²×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
356	C₂²×M₁₆	C₂²×C₈₉	1	2	yes	complemented
357	C₂³×C₈	C₂²×C₈₉	1	1	yes	characteristic
358	C₂³×C₈	C₂²×C₈₉	1	2	yes	complemented
359	#537	—	2	4	no	—
360	#537	C₂²×C₈₉	1	8	yes	complemented
361	#535	—	2	4	no	—
362	#535	C₂²×C₈₉	1	8	yes	complemented
363	C₂×#127	—	2	1	no	—
364	C₂×#124	—	2	1	no	—
365	C₂×C₂²⋊Q₈	—	2	1	no	—
366	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	2	yes	complemented
367	C₂×#121	—	2	1	no	—
368	C₂×#121	—	2	2	no	—
369	C₂×#121	C₂²×C₈₉	1	2	yes	—
370	C₂×#121	C₂²×C₈₉	1	2	yes	—
371	C₂×#121	C₂²×C₈₉	1	2	yes	complemented
372	C₂×#120	—	2	2	no	—
373	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
374	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
375	#469	C₂²×C₈₉	1	16	yes	complemented
376	#468	C₂²×C₈₉	1	16	yes	complemented
377	#466	C₂²×C₈₉	1	16	yes	complemented
378	#465	C₂²×C₈₉	1	16	yes	complemented
379	#428	C₂²×C₈₉	1	8	yes	complemented
380	C₂×#107	C₂²×C₈₉	1	2	yes	complemented
381	C₂×#106	C₂²×C₈₉	1	2	yes	complemented
382	#416	C₂²×C₈₉	1	8	yes	complemented
383	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic, complemented
384	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic, complemented
385	C₂×#102	C₂²×C₈₉	1	2	yes	—
386	C₂×#102	C₂²×C₈₉	1	2	yes	—
387	C₂×#102	C₂²×C₈₉	1	8	yes	complemented
388	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic, complemented
389	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic, complemented
390	C₂⁵	—	2	1	no	—
391	C₂×#41	—	2	4	no	—
392	C₂×#41	—	2	2	no	—
393	C₂×#41	—	2	4	no	—
394	C₂×#41	—	2	4	no	—
395	C₂×#41	—	2	4	no	—
396	C₂×#41	D₄×C₈₉	1	4	yes	—
397	C₂×#41	C₂³×C₈₉	1	4	yes	—
398	C₂×#41	C₂³×C₈₉	1	4	yes	—
399	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
400	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
401	C₂²×D₄	—	2	1	no	—
402	C₂²×D₄	—	2	2	no	—
403	C₂²×D₄	—	2	2	no	—
404	C₂²×D₄	—	2	2	no	—
405	C₂²×D₄	—	2	4	no	—
406	C₂²×D₄	—	2	1	no	—
407	C₂²×D₄	—	2	2	no	—
408	C₂²×D₄	—	2	4	no	—
409	C₂²×D₄	D₄×C₈₉	1	2	yes	—
410	C₂²×D₄	D₄×C₈₉	1	1	yes	characteristic
411	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
412	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
413	C₂²×D₄	C₂³×C₈₉	1	4	yes	—
414	C₂³×C₄	—	2	1	no	—
415	C₂³×C₄	—	2	1	no	—
416	C₂³×C₄	C₂³×C₈₉	1	2	yes	—
417	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
418	AGL(1,ℤ/8)	—	2	32	no	—
419	C₂×SD₁₆	—	2	8	no	—
420	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	—
421	C₂×D₈	—	2	8	no	—
422	C₂×D₈	C₂³×C₈₉	1	8	yes	—
423	#131	—	2	32	no	—
424	C₂×M₁₆	—	2	8	no	—
425	C₂×M₁₆	D₄×C₈₉	1	8	yes	complemented
426	C₂×M₁₆	C₂³×C₈₉	1	4	yes	—
427	C₂×M₁₆	C₂³×C₈₉	1	8	yes	complemented
428	C₂²×C₈	—	2	4	no	—
429	C₂²×C₈	—	2	8	no	—
430	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
431	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
432	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
433	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
434	C₂²×C₈	C₂³×C₈₉	1	8	yes	complemented
435	#127	—	2	8	no	—
436	#124	—	2	8	no	—
437	C₂²⋊Q₈	—	2	8	no	—
438	C₂²⋊Q₈	—	2	4	no	—
439	C₂²⋊Q₈	C₂³×C₈₉	1	8	yes	complemented
440	#121	—	2	8	no	—
441	#121	—	2	16	no	—
442	#121	—	2	4	no	—
443	#121	—	2	4	no	—
444	#121	—	2	4	no	—
445	#121	C₂³×C₈₉	1	8	yes	—
446	#121	C₂³×C₈₉	1	8	yes	—
447	#121	C₂³×C₈₉	1	8	yes	complemented
448	#120	—	2	16	no	—
449	#117	—	2	2	no	—
450	#117	C₂³×C₈₉	1	4	yes	—
451	C₂×C₄⋊C₄	—	2	1	no	—
452	C₂×C₄⋊C₄	—	2	2	no	—
453	C₂×C₄⋊C₄	—	2	2	no	—
454	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
455	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
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	2	no	—
460	C₂×C₂²⋊C₄	—	2	2	no	—
461	C₂×C₂²⋊C₄	—	2	2	no	—
462	C₂×C₂²⋊C₄	—	2	2	no	—
463	C₂×C₄²	—	2	1	no	—
464	#107	C₂³×C₈₉	1	8	yes	complemented
465	#106	C₂³×C₈₉	1	8	yes	complemented
466	#103	C₂³×C₈₉	1	4	yes	complemented
467	#103	C₂³×C₈₉	1	4	yes	complemented
468	#102	C₂³×C₈₉	1	8	yes	—
469	#102	C₂³×C₈₉	1	8	yes	—
470	#102	C₂³×C₈₉	1	4	yes	complemented
471	#102	C₂³×C₈₉	1	4	yes	complemented
472	C₂⁴	—	2	1	no	—
473	C₂⁴	—	2	2	no	—
474	C₂⁴	—	2	4	no	—
475	C₂⁴	—	4	4	no	—
476	C₂⁴	—	8	2	no	—
477	C₂⁴	—	8	2	no	—
478	C₂⁴	C₂×D₄×C₈₉	1	2	yes	—
479	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
480	#41	—	2	16	no	—
481	#41	—	2	8	no	—
482	#41	—	2	8	no	—
483	#41	—	2	8	no	—
484	#41	—	2	8	no	—
485	#41	—	4	8	no	—
486	C₂×Q₈	—	2	4	no	—
487	C₂×Q₈	—	2	1	no	—
488	C₂×Q₈	—	2	2	no	—
489	C₂×Q₈	—	2	2	no	—
490	C₂×Q₈	C₂×D₄×C₈₉	1	4	yes	—
491	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
492	C₂×D₄	—	2	8	no	—
493	C₂×D₄	—	2	4	no	—
494	C₂×D₄	—	2	8	no	—
495	C₂×D₄	—	2	8	no	—
496	C₂×D₄	—	2	8	no	—
497	C₂×D₄	—	2	4	no	—
498	C₂×D₄	—	2	2	no	—
499	C₂×D₄	—	2	4	no	—
500	C₂×D₄	—	2	8	no	—
501	C₂×D₄	—	2	2	no	—
502	C₂×D₄	—	2	1	no	—
503	C₂×D₄	—	2	4	no	—
504	C₂×D₄	—	2	4	no	—
505	C₂×D₄	—	2	8	no	—
506	C₂×D₄	—	2	2	no	—
507	C₂×D₄	—	4	4	no	—
508	C₂×D₄	—	4	4	no	—
509	C₂×D₄	—	4	8	no	—
510	C₂×D₄	—	8	4	no	—
511	C₂×D₄	—	8	4	no	—
512	C₂×D₄	—	8	4	no	—
513	C₂×D₄	C₂×D₄×C₈₉	1	8	yes	—
514	C₂×D₄	C₂×D₄×C₈₉	1	4	yes	—
515	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
516	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
517	C₂²×C₄	—	2	4	no	—
518	C₂²×C₄	—	2	4	no	—
519	C₂²×C₄	—	2	1	no	—
520	C₂²×C₄	—	2	1	no	—
521	C₂²×C₄	—	2	2	no	—
522	C₂²×C₄	—	2	1	no	—
523	C₂²×C₄	—	2	4	no	—
524	C₂²×C₄	—	2	4	no	—
525	C₂²×C₄	—	4	4	no	—
526	C₂²×C₄	—	8	2	no	—
527	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
528	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
529	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
530	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
531	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
532	C₂²×C₄	#41×C₈₉	1	4	yes	—
533	C₂²×C₄	C₂⁴×C₈₉	1	4	yes	—
534	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
535	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
536	SD₁₆	—	2	16	no	—
537	D₈	—	2	16	no	—
538	M₁₆	—	2	16	no	—
539	M₁₆	C₂×D₄×C₈₉	1	16	yes	complemented
540	C₂×C₈	—	2	2	no	—
541	C₂×C₈	—	2	2	no	—
542	C₂×C₈	—	2	8	no	—
543	C₂×C₈	—	2	16	no	—
544	C₂×C₈	C₂×D₄×C₈₉	1	8	yes	—
545	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
546	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
547	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
548	C₄⋊C₄	—	2	4	no	—
549	C₄⋊C₄	—	2	2	no	—
550	C₄⋊C₄	—	2	8	no	—
551	C₄⋊C₄	—	2	1	no	—
552	C₄⋊C₄	—	2	1	no	—
553	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
554	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
555	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
556	C₂²⋊C₄	—	2	4	no	—
557	C₂²⋊C₄	—	2	8	no	—
558	C₂²⋊C₄	—	2	8	no	—
559	C₂²⋊C₄	—	2	8	no	—
560	C₂²⋊C₄	—	2	8	no	—
561	C₄²	—	2	4	no	—
562	C₂³	—	2	2	no	—
563	C₂³	—	2	2	no	—
564	C₂³	—	2	4	no	—
565	C₂³	—	2	4	no	—
566	C₂³	—	2	4	no	—
567	C₂³	—	2	2	no	—
568	C₂³	—	2	4	no	—
569	C₂³	—	4	2	no	—
570	C₂³	—	4	8	no	—
571	C₂³	—	4	4	no	—
572	C₂³	—	4	2	no	—
573	C₂³	—	4	4	no	—
574	C₂³	—	4	4	no	—
575	C₂³	—	4	2	no	—
576	C₂³	—	8	4	no	—
577	C₂³	—	8	4	no	—
578	C₂³	—	8	16	no	—
579	C₂³	#121×C₈₉	1	4	yes	—
580	C₂³	C₂²×D₄×C₈₉	1	4	yes	—
581	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
582	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
583	Q₈	—	2	8	no	—
584	Q₈	—	2	2	no	—
585	Q₈	—	2	2	no	—
586	Q₈	—	4	2	no	—
587	D₄	—	2	16	no	—
588	D₄	—	2	8	no	—
589	D₄	—	2	2	no	—
590	D₄	—	2	4	no	—
591	D₄	—	2	4	no	—
592	D₄	—	2	2	no	—
593	D₄	—	4	8	no	—
594	D₄	—	4	4	no	—
595	D₄	—	4	8	no	—
596	D₄	—	4	2	no	—
597	D₄	—	8	8	no	—
598	D₄	—	8	8	no	—
599	D₄	—	8	8	no	—
600	C₂×C₄	—	2	4	no	—
601	C₂×C₄	—	2	8	no	—
602	C₂×C₄	—	2	4	no	—
603	C₂×C₄	—	2	4	no	—
604	C₂×C₄	—	2	2	no	—
605	C₂×C₄	—	2	2	no	—
606	C₂×C₄	—	2	4	no	—
607	C₂×C₄	—	2	2	no	—
608	C₂×C₄	—	4	2	no	—
609	C₂×C₄	—	4	4	no	—
610	C₂×C₄	—	4	2	no	—
611	C₂×C₄	—	4	2	no	—
612	C₂×C₄	—	4	8	no	—
613	C₂×C₄	—	8	4	no	—
614	C₂×C₄	#121×C₈₉	1	4	yes	—
615	C₂×C₄	#121×C₈₉	1	8	yes	—
616	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
617	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
618	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
619	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
620	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
621	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
622	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
623	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
624	C₈	—	2	2	no	—
625	C₈	—	2	2	no	—
626	C₈	C₂²×D₄×C₈₉	1	8	yes	—
627	C₂²	—	2	8	no	—
628	C₂²	—	2	2	no	—
629	C₂²	—	2	2	no	—
630	C₂²	—	2	4	no	—
631	C₂²	—	2	4	no	—
632	C₂²	—	2	4	no	—
633	C₂²	—	4	8	no	—
634	C₂²	—	4	2	no	—
635	C₂²	—	4	4	no	—
636	C₂²	—	4	4	no	—
637	C₂²	—	4	2	no	—
638	C₂²	—	4	2	no	—
639	C₂²	—	4	4	no	—
640	C₂²	—	8	16	no	—
641	C₂²	—	8	8	no	—
642	C₂²	—	8	4	no	—
643	C₂²	C₂×#121×C₈₉	1	4	yes	—
644	C₂²	C₂×#121×C₈₉	1	2	yes	—
645	C₂²	#576×C₈₉	1	2	yes	—
646	C₂²	#577×C₈₉	1	2	yes	—
647	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
648	C₄	—	2	4	no	—
649	C₄	—	4	4	no	—
650	C₄	—	4	2	no	—
651	C₄	—	4	2	no	—
652	C₄	—	4	2	no	—
653	C₄	C₂×#121×C₈₉	1	2	yes	—
654	C₄	C₂×#121×C₈₉	1	4	yes	—
655	C₄	C₂³×D₄×C₈₉	1	1	yes	characteristic
656	C₄	C₂²×#41×C₈₉	1	1	yes	characteristic
657	C₂	—	2	4	no	—
658	C₂	—	4	4	no	—
659	C₂	—	8	2	no	—
660	C₂	—	8	4	no	—
661	C₂	C₈₉×#3062	1	4	yes	direct product factor
662	C₂	C₂²×#121×C₈₉	1	1	yes	characteristic
663	C₂	C₂×#576×C₈₉	1	1	yes	characteristic
664	C₂	C₂×#577×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂²×#41×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×AGL(1,ℤ/8)×C₈₉	C₂	semidirect
2. C₂²×#131×C₈₉	C₂	semidirect
3. C₂×#537×C₈₉	C₂	semidirect
4. C₂×#535×C₈₉	C₂	semidirect
5. C₈₉×#3062	C₂	direct (N × H)
6. C₂×#469×C₈₉	C₂	semidirect
7. C₂×#468×C₈₉	C₂	semidirect
8. C₂×#466×C₈₉	C₂	semidirect
9. C₂×#465×C₈₉	C₂	semidirect
10. C₂×#428×C₈₉	C₂	semidirect
11. C₂×#416×C₈₉	C₂	semidirect
12. C₂²×#102×C₈₉	C₂	semidirect
17. C₂×AGL(1,ℤ/8)×C₈₉	C₂²	semidirect
21. C₂×#131×C₈₉	C₂²	semidirect
23. C₂²×M₁₆×C₈₉	C₂²	semidirect
25. C₂³×C₈×C₈₉	C₂²	semidirect
27. #537×C₈₉	C₂²	semidirect
29. #535×C₈₉	C₂²	semidirect
33. C₂×C₂²⋊Q₈×C₈₉	C₂²	semidirect
38. C₂×#121×C₈₉	C₂²	semidirect
42. #469×C₈₉	C₂²	semidirect
43. #468×C₈₉	C₂²	semidirect
44. #466×C₈₉	C₂²	semidirect
45. #465×C₈₉	C₂²	semidirect
46. #428×C₈₉	C₂²	semidirect
47. C₂×#107×C₈₉	C₂²	semidirect
48. C₂×#106×C₈₉	C₂²	semidirect
49. #416×C₈₉	C₂²	semidirect
50. C₂×#103×C₈₉	C₂²	semidirect
51. C₂×#103×C₈₉	C₂²	semidirect
54. C₂×#102×C₈₉	C₂²	semidirect
55. C₂×#102×C₈₉	C₂²	semidirect
56. C₂×#102×C₈₉	C₂²	semidirect
92. C₂×M₁₆×C₈₉	D₄	semidirect
94. C₂×M₁₆×C₈₉	C₂³	semidirect
101. C₂²×C₈×C₈₉	C₂³	semidirect
106. C₂²⋊Q₈×C₈₉	C₂³	semidirect
114. #121×C₈₉	C₂³	semidirect
131. #107×C₈₉	C₂³	semidirect
132. #106×C₈₉	C₂³	semidirect
133. #103×C₈₉	C₂³	semidirect
134. #103×C₈₉	C₂³	semidirect
137. #102×C₈₉	C₂³	semidirect
138. #102×C₈₉	C₂³	semidirect
206. M₁₆×C₈₉	C₂×D₄	semidirect
324. C₂×#3062	C₈₉	direct (N × H)
329. C₂×C₈₉	#3062	direct (N × H)
333. C₂²×AGL(1,ℤ/8)	C₂×C₈₉	semidirect
334. C₂²×#131	C₂×C₈₉	semidirect
335. C₂×#537	C₂×C₈₉	semidirect
336. C₂×#535	C₂×C₈₉	semidirect
337. #3062	C₂×C₈₉	direct (N × H)
338. C₂×#469	C₂×C₈₉	semidirect
339. C₂×#468	C₂×C₈₉	semidirect
340. C₂×#466	C₂×C₈₉	semidirect
341. C₂×#465	C₂×C₈₉	semidirect
342. C₂×#428	C₂×C₈₉	semidirect
343. C₂×#416	C₂×C₈₉	semidirect
344. C₂²×#102	C₂×C₈₉	semidirect
345. C₈₉	C₂×#3062	direct (N × H)
350. C₂×AGL(1,ℤ/8)	C₂²×C₈₉	semidirect
354. C₂×#131	C₂²×C₈₉	semidirect
356. C₂²×M₁₆	C₂²×C₈₉	semidirect
358. C₂³×C₈	C₂²×C₈₉	semidirect
360. #537	C₂²×C₈₉	semidirect
362. #535	C₂²×C₈₉	semidirect
366. C₂×C₂²⋊Q₈	C₂²×C₈₉	semidirect
371. C₂×#121	C₂²×C₈₉	semidirect
375. #469	C₂²×C₈₉	semidirect
376. #468	C₂²×C₈₉	semidirect
377. #466	C₂²×C₈₉	semidirect
378. #465	C₂²×C₈₉	semidirect
379. #428	C₂²×C₈₉	semidirect
380. C₂×#107	C₂²×C₈₉	semidirect
381. C₂×#106	C₂²×C₈₉	semidirect
382. #416	C₂²×C₈₉	semidirect
383. C₂×#103	C₂²×C₈₉	semidirect
384. C₂×#103	C₂²×C₈₉	semidirect
387. C₂×#102	C₂²×C₈₉	semidirect
388. C₂×#102	C₂²×C₈₉	semidirect
389. C₂×#102	C₂²×C₈₉	semidirect
425. C₂×M₁₆	D₄×C₈₉	semidirect
427. C₂×M₁₆	C₂³×C₈₉	semidirect
434. C₂²×C₈	C₂³×C₈₉	semidirect
439. C₂²⋊Q₈	C₂³×C₈₉	semidirect
447. #121	C₂³×C₈₉	semidirect
464. #107	C₂³×C₈₉	semidirect
465. #106	C₂³×C₈₉	semidirect
466. #103	C₂³×C₈₉	semidirect
467. #103	C₂³×C₈₉	semidirect
470. #102	C₂³×C₈₉	semidirect
471. #102	C₂³×C₈₉	semidirect
539. M₁₆	C₂×D₄×C₈₉	semidirect
661. C₂	C₈₉×#3062	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×AGL(1,ℤ/8)×C₈₉	2
C₂²×#131×C₈₉	2
C₂×#537×C₈₉	2
C₂×#535×C₈₉	2
C₈₉×#3062	2
C₂×#469×C₈₉	2
C₂×#468×C₈₉	2
C₂×#466×C₈₉	2
C₂×#465×C₈₉	2
C₂×#428×C₈₉	2
C₂×#416×C₈₉	2
C₂²×#102×C₈₉	2
C₂×#3062	89

Subgroups — C2×C89×#3062 #36438299

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3062 #36438299