Subgroups — C₂×C₈₉×#3369 #36439158

Order 22784.

1842 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈²×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#556×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#3369	C₂	1	16	yes	maximal, direct product factor
4	C₂×#489×C₈₉	C₂	1	2	yes	maximal
5	C₂×#487×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#475×C₈₉	C₂	1	2	yes	maximal, complemented
7	C₂×#476×C₈₉	C₂	1	1	yes	maximal, characteristic
8	C₂×#464×C₈₉	C₂	1	2	yes	maximal
9	C₂×#462×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#446×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#441×C₈₉	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×SD₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
13	C₂²×Q₁₆×C₈₉	C₂²	1	2	yes	—
14	C₂²×SD₁₆×C₈₉	C₂²	1	1	yes	characteristic
15	Q₈²×C₈₉	—	2	4	no	—
16	Q₈²×C₈₉	C₂²	1	8	yes	complemented
17	#556×C₈₉	—	2	4	no	—
18	#556×C₈₉	C₂²	1	8	yes	—
19	C₂×C₄⋊Q₈×C₈₉	—	2	1	no	—
20	C₂×C₄⋊Q₈×C₈₉	—	2	2	no	—
21	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
24	C₂×#126×C₈₉	—	2	2	no	—
25	C₂×#124×C₈₉	C₂²	1	2	yes	—
26	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
27	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
28	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
29	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
31	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×#117×C₈₉	—	2	1	no	—
34	#489×C₈₉	C₂²	1	16	yes	—
35	#487×C₈₉	C₂²	1	8	yes	complemented
36	#476×C₈₉	C₂²	1	8	yes	—
37	#475×C₈₉	C₂²	1	16	yes	complemented
38	#464×C₈₉	C₂²	1	16	yes	—
39	#462×C₈₉	C₂²	1	8	yes	complemented
40	#446×C₈₉	C₂²	1	8	yes	complemented
41	#441×C₈₉	C₂²	1	16	yes	complemented
42	C₄×SD₁₆×C₈₉	C₂²	1	8	yes	—
43	C₂×#106×C₈₉	C₂²	1	2	yes	—
44	C₂×#106×C₈₉	C₂²	1	1	yes	characteristic
45	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
46	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
47	C₂×#103×C₈₉	C₂²	1	2	yes	—
48	C₂×#103×C₈₉	C₂²	1	2	yes	—
49	C₂×#103×C₈₉	C₂²	1	2	yes	—
50	C₂×#103×C₈₉	C₂²	1	1	yes	characteristic
51	C₂×#102×C₈₉	C₂²	1	2	yes	—
52	C₂×#102×C₈₉	C₂²	1	1	yes	characteristic
53	C₂×#97×C₈₉	C₂²	1	2	yes	—
54	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
55	C₂²×Q₈×C₈₉	—	2	2	no	—
56	C₂²×Q₈×C₈₉	—	2	1	no	—
57	C₂²×Q₈×C₈₉	D₄	1	1	yes	characteristic
58	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
59	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₈₉	—	2	1	no	—
62	C₂×Q₁₆×C₈₉	—	2	8	no	—
63	C₂×Q₁₆×C₈₉	C₂³	1	8	yes	—
64	C₂×SD₁₆×C₈₉	—	2	4	no	—
65	C₂×SD₁₆×C₈₉	C₂³	1	4	yes	—
66	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
68	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
69	C₄⋊Q₈×C₈₉	—	2	8	no	—
70	C₄⋊Q₈×C₈₉	—	2	16	no	—
71	C₄⋊Q₈×C₈₉	—	2	2	no	—
72	C₄⋊Q₈×C₈₉	—	2	2	no	—
73	C₄⋊Q₈×C₈₉	—	2	4	no	—
74	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
75	C₄⋊Q₈×C₈₉	C₂³	1	8	yes	—
76	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
77	#126×C₈₉	—	2	16	no	—
78	#124×C₈₉	—	2	4	no	—
79	#124×C₈₉	C₂³	1	8	yes	—
80	C₂²⋊Q₈×C₈₉	—	2	8	no	—
81	C₄×Q₈×C₈₉	—	2	8	no	—
82	C₄×Q₈×C₈₉	—	2	4	no	—
83	C₄×Q₈×C₈₉	—	2	2	no	—
84	C₄×Q₈×C₈₉	—	2	4	no	—
85	C₄×Q₈×C₈₉	D₄	1	4	yes	—
86	C₄×Q₈×C₈₉	C₂³	1	4	yes	—
87	C₄×Q₈×C₈₉	C₂³	1	8	yes	—
88	C₄×Q₈×C₈₉	C₂³	1	8	yes	—
89	C₄×Q₈×C₈₉	C₂³	1	4	yes	—
90	C₄×D₄×C₈₉	—	2	2	no	—
91	C₄×D₄×C₈₉	C₂³	1	4	yes	—
92	#117×C₈₉	—	2	8	no	—
93	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
94	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
95	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
96	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
97	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
98	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
99	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
100	C₂×C₄⋊C₄×C₈₉	D₄	1	1	yes	characteristic
101	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
102	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
103	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
104	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
105	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
106	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
107	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
108	C₂×C₄²×C₈₉	—	2	2	no	—
109	C₂×C₄²×C₈₉	—	2	2	no	—
110	C₂×C₄²×C₈₉	—	2	1	no	—
111	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
112	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
113	#106×C₈₉	C₂³	1	4	yes	—
114	#106×C₈₉	C₂³	1	8	yes	—
115	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
116	C₄⋊C₈×C₈₉	C₂³	1	8	yes	—
117	#103×C₈₉	C₂³	1	4	yes	—
118	#103×C₈₉	C₂³	1	8	yes	—
119	#103×C₈₉	C₂³	1	8	yes	—
120	#103×C₈₉	C₂³	1	8	yes	—
121	#102×C₈₉	C₂³	1	4	yes	—
122	#102×C₈₉	C₂³	1	8	yes	—
123	#97×C₈₉	C₂³	1	8	yes	—
124	C₄×C₈×C₈₉	C₂³	1	4	yes	—
125	C₂⁴×C₈₉	—	2	1	no	—
126	C₂×Q₈×C₈₉	—	2	4	no	—
127	C₂×Q₈×C₈₉	—	2	1	no	—
128	C₂×Q₈×C₈₉	—	2	2	no	—
129	C₂×Q₈×C₈₉	—	2	8	no	—
130	C₂×Q₈×C₈₉	—	2	4	no	—
131	C₂×Q₈×C₈₉	—	2	2	no	—
132	C₂×Q₈×C₈₉	—	2	4	no	—
133	C₂×Q₈×C₈₉	—	2	4	no	—
134	C₂×Q₈×C₈₉	—	8	4	no	—
135	C₂×Q₈×C₈₉	—	8	2	no	—
136	C₂×Q₈×C₈₉	C₂×D₄	1	4	yes	—
137	C₂×Q₈×C₈₉	#41	1	4	yes	—
138	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
139	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
140	C₂×D₄×C₈₉	—	2	1	no	—
141	C₂×D₄×C₈₉	—	2	2	no	—
142	C₂×D₄×C₈₉	—	2	2	no	—
143	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
144	C₂²×C₄×C₈₉	—	2	1	no	—
145	C₂²×C₄×C₈₉	—	2	1	no	—
146	C₂²×C₄×C₈₉	—	2	2	no	—
147	C₂²×C₄×C₈₉	—	2	2	no	—
148	C₂²×C₄×C₈₉	—	2	1	no	—
149	C₂²×C₄×C₈₉	—	2	4	no	—
150	C₂²×C₄×C₈₉	—	4	4	no	—
151	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
152	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
153	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
154	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
155	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
156	Q₁₆×C₈₉	—	2	16	no	—
157	SD₁₆×C₈₉	—	2	8	no	—
158	C₂×C₈×C₈₉	—	2	2	no	—
159	C₂×C₈×C₈₉	—	2	4	no	—
160	C₂×C₈×C₈₉	—	2	2	no	—
161	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
162	C₂×C₈×C₈₉	C₂⁴	1	2	yes	—
163	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
164	C₄⋊C₄×C₈₉	—	2	4	no	—
165	C₄⋊C₄×C₈₉	—	2	8	no	—
166	C₄⋊C₄×C₈₉	—	2	4	no	—
167	C₄⋊C₄×C₈₉	—	2	8	no	—
168	C₄⋊C₄×C₈₉	—	2	8	no	—
169	C₄⋊C₄×C₈₉	—	2	4	no	—
170	C₄⋊C₄×C₈₉	—	2	4	no	—
171	C₄⋊C₄×C₈₉	—	2	1	no	—
172	C₄⋊C₄×C₈₉	—	2	2	no	—
173	C₄⋊C₄×C₈₉	—	2	1	no	—
174	C₄⋊C₄×C₈₉	C₂×D₄	1	4	yes	—
175	C₄⋊C₄×C₈₉	C₂×D₄	1	8	yes	—
176	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
177	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
178	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
179	C₂²⋊C₄×C₈₉	—	2	8	no	—
180	C₂²⋊C₄×C₈₉	—	2	4	no	—
181	C₂²⋊C₄×C₈₉	—	2	8	no	—
182	C₄²×C₈₉	—	2	8	no	—
183	C₄²×C₈₉	—	2	8	no	—
184	C₄²×C₈₉	—	2	4	no	—
185	C₄²×C₈₉	C₂×D₄	1	2	yes	—
186	C₄²×C₈₉	C₂×D₄	1	4	yes	—
187	C₄²×C₈₉	C₂⁴	1	2	yes	—
188	C₄²×C₈₉	C₂⁴	1	4	yes	—
189	C₂³×C₈₉	—	2	2	no	—
190	C₂³×C₈₉	—	4	2	no	—
191	C₂³×C₈₉	—	4	2	no	—
192	C₂³×C₈₉	—	4	2	no	—
193	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
194	Q₈×C₈₉	—	2	4	no	—
195	Q₈×C₈₉	—	2	8	no	—
196	Q₈×C₈₉	—	2	2	no	—
197	Q₈×C₈₉	—	2	4	no	—
198	Q₈×C₈₉	—	2	4	no	—
199	Q₈×C₈₉	—	4	4	no	—
200	Q₈×C₈₉	—	8	8	no	—
201	Q₈×C₈₉	—	8	4	no	—
202	Q₈×C₈₉	C₂×#41	1	4	yes	—
203	D₄×C₈₉	—	2	2	no	—
204	D₄×C₈₉	—	2	2	no	—
205	D₄×C₈₉	—	4	2	no	—
206	C₂×C₄×C₈₉	—	2	4	no	—
207	C₂×C₄×C₈₉	—	2	4	no	—
208	C₂×C₄×C₈₉	—	2	4	no	—
209	C₂×C₄×C₈₉	—	2	2	no	—
210	C₂×C₄×C₈₉	—	2	2	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	—	4	4	no	—
216	C₂×C₄×C₈₉	—	4	2	no	—
217	C₂×C₄×C₈₉	—	4	4	no	—
218	C₂×C₄×C₈₉	—	4	2	no	—
219	C₂×C₄×C₈₉	—	4	8	no	—
220	C₂×C₄×C₈₉	#137	1	4	yes	—
221	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
222	C₂×C₄×C₈₉	C₂²×D₄	1	4	yes	—
223	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
224	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
225	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
226	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
227	C₂×C₄×C₈₉	2₋¹⁺⁴	1	2	yes	—
228	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
229	C₈×C₈₉	—	2	2	no	—
230	C₈×C₈₉	—	2	4	no	—
231	C₈×C₈₉	—	2	2	no	—
232	C₂²×C₈₉	—	4	2	no	—
233	C₂²×C₈₉	—	4	2	no	—
234	C₂²×C₈₉	—	4	2	no	—
235	C₂²×C₈₉	—	8	4	no	—
236	C₂²×C₈₉	#548	1	2	yes	—
237	C₂²×C₈₉	C₂×#137	1	2	yes	—
238	C₂²×C₈₉	#577	1	2	yes	—
239	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
240	C₄×C₈₉	—	2	4	no	—
241	C₄×C₈₉	—	2	4	no	—
242	C₄×C₈₉	—	2	4	no	—
243	C₄×C₈₉	—	2	2	no	—
244	C₄×C₈₉	—	4	4	no	—
245	C₄×C₈₉	—	4	2	no	—
246	C₄×C₈₉	—	4	4	no	—
247	C₄×C₈₉	—	4	2	no	—
248	C₄×C₈₉	#548	1	2	yes	—
249	C₄×C₈₉	C₂×#137	1	4	yes	—
250	C₄×C₈₉	C₂²×#41	1	1	yes	characteristic
251	C₄×C₈₉	C₂×2₋¹⁺⁴	1	1	yes	characteristic
252	C₂×#3369	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
253	C₂×C₈₉	—	8	2	no	—
254	C₂×C₈₉	#3369	1	4	yes	direct product factor
255	C₂×C₈₉	C₂×#548	1	1	yes	characteristic
256	C₂×C₈₉	C₂²×#137	1	1	yes	characteristic
257	C₂×C₈₉	C₂×#577	1	1	yes	characteristic
258	C₂×Q₈²	C₂×C₈₉	1	1	yes	characteristic, complemented
259	C₂×#556	C₂×C₈₉	1	1	yes	characteristic
260	#3369	C₂×C₈₉	1	16	yes	direct product factor
261	C₂×#489	C₂×C₈₉	1	2	yes	—
262	C₂×#487	C₂×C₈₉	1	1	yes	characteristic, complemented
263	C₂×#475	C₂×C₈₉	1	2	yes	complemented
264	C₂×#476	C₂×C₈₉	1	1	yes	characteristic
265	C₂×#464	C₂×C₈₉	1	2	yes	—
266	C₂×#462	C₂×C₈₉	1	1	yes	characteristic, complemented
267	C₂×#446	C₂×C₈₉	1	1	yes	characteristic, complemented
268	C₂×#441	C₂×C₈₉	1	2	yes	complemented
269	C₂×C₄×SD₁₆	C₂×C₈₉	1	1	yes	characteristic
270	C₈₉	C₂×#3369	1	1	yes	characteristic, direct product factor, Sylow
271	C₂²×Q₁₆	C₂²×C₈₉	1	2	yes	—
272	C₂²×SD₁₆	C₂²×C₈₉	1	1	yes	characteristic
273	Q₈²	—	2	4	no	—
274	Q₈²	C₂²×C₈₉	1	8	yes	complemented
275	#556	—	2	4	no	—
276	#556	C₂²×C₈₉	1	8	yes	—
277	C₂×C₄⋊Q₈	—	2	1	no	—
278	C₂×C₄⋊Q₈	—	2	2	no	—
279	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
280	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
281	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
282	C₂×#126	—	2	2	no	—
283	C₂×#124	C₂²×C₈₉	1	2	yes	—
284	C₂×C₂²⋊Q₈	—	2	1	no	—
285	C₂×C₄×Q₈	—	2	1	no	—
286	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
287	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
288	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
289	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
290	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
291	C₂×#117	—	2	1	no	—
292	#489	C₂²×C₈₉	1	16	yes	—
293	#487	C₂²×C₈₉	1	8	yes	complemented
294	#476	C₂²×C₈₉	1	8	yes	—
295	#475	C₂²×C₈₉	1	16	yes	complemented
296	#464	C₂²×C₈₉	1	16	yes	—
297	#462	C₂²×C₈₉	1	8	yes	complemented
298	#446	C₂²×C₈₉	1	8	yes	complemented
299	#441	C₂²×C₈₉	1	16	yes	complemented
300	C₄×SD₁₆	C₂²×C₈₉	1	8	yes	—
301	C₂×#106	C₂²×C₈₉	1	1	yes	characteristic
302	C₂×#106	C₂²×C₈₉	1	2	yes	—
303	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
304	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
305	C₂×#103	C₂²×C₈₉	1	2	yes	—
306	C₂×#103	C₂²×C₈₉	1	2	yes	—
307	C₂×#103	C₂²×C₈₉	1	2	yes	—
308	C₂×#103	C₂²×C₈₉	1	1	yes	characteristic
309	C₂×#102	C₂²×C₈₉	1	1	yes	characteristic
310	C₂×#102	C₂²×C₈₉	1	2	yes	—
311	C₂×#97	C₂²×C₈₉	1	2	yes	—
312	C₂×C₄×C₈	C₂²×C₈₉	1	1	yes	characteristic
313	C₂²×Q₈	—	2	2	no	—
314	C₂²×Q₈	—	2	1	no	—
315	C₂²×Q₈	D₄×C₈₉	1	1	yes	characteristic
316	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
317	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
318	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
319	C₂³×C₄	—	2	1	no	—
320	C₂×Q₁₆	—	2	8	no	—
321	C₂×Q₁₆	C₂³×C₈₉	1	8	yes	—
322	C₂×SD₁₆	—	2	4	no	—
323	C₂×SD₁₆	C₂³×C₈₉	1	4	yes	—
324	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
325	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
326	C₂²×C₈	C₂³×C₈₉	1	1	yes	characteristic
327	C₄⋊Q₈	—	2	8	no	—
328	C₄⋊Q₈	—	2	16	no	—
329	C₄⋊Q₈	—	2	2	no	—
330	C₄⋊Q₈	—	2	2	no	—
331	C₄⋊Q₈	—	2	4	no	—
332	C₄⋊Q₈	C₂³×C₈₉	1	4	yes	—
333	C₄⋊Q₈	C₂³×C₈₉	1	8	yes	—
334	C₄⋊Q₈	C₂³×C₈₉	1	4	yes	—
335	#126	—	2	16	no	—
336	#124	—	2	4	no	—
337	#124	C₂³×C₈₉	1	8	yes	—
338	C₂²⋊Q₈	—	2	8	no	—
339	C₄×Q₈	—	2	8	no	—
340	C₄×Q₈	—	2	4	no	—
341	C₄×Q₈	—	2	2	no	—
342	C₄×Q₈	—	2	4	no	—
343	C₄×Q₈	D₄×C₈₉	1	4	yes	—
344	C₄×Q₈	C₂³×C₈₉	1	8	yes	—
345	C₄×Q₈	C₂³×C₈₉	1	4	yes	—
346	C₄×Q₈	C₂³×C₈₉	1	8	yes	—
347	C₄×Q₈	C₂³×C₈₉	1	4	yes	—
348	C₄×D₄	—	2	2	no	—
349	C₄×D₄	C₂³×C₈₉	1	4	yes	—
350	#117	—	2	8	no	—
351	C₂×C₄⋊C₄	—	2	1	no	—
352	C₂×C₄⋊C₄	—	2	2	no	—
353	C₂×C₄⋊C₄	—	2	1	no	—
354	C₂×C₄⋊C₄	—	2	2	no	—
355	C₂×C₄⋊C₄	—	2	2	no	—
356	C₂×C₄⋊C₄	—	2	1	no	—
357	C₂×C₄⋊C₄	—	2	1	no	—
358	C₂×C₄⋊C₄	D₄×C₈₉	1	1	yes	characteristic
359	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
360	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
361	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
362	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
363	C₂×C₂²⋊C₄	—	2	2	no	—
364	C₂×C₂²⋊C₄	—	2	1	no	—
365	C₂×C₂²⋊C₄	—	2	2	no	—
366	C₂×C₄²	—	2	2	no	—
367	C₂×C₄²	—	2	2	no	—
368	C₂×C₄²	—	2	1	no	—
369	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
370	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
371	#106	C₂³×C₈₉	1	4	yes	—
372	#106	C₂³×C₈₉	1	8	yes	—
373	C₄⋊C₈	C₂³×C₈₉	1	4	yes	—
374	C₄⋊C₈	C₂³×C₈₉	1	8	yes	—
375	#103	C₂³×C₈₉	1	4	yes	—
376	#103	C₂³×C₈₉	1	8	yes	—
377	#103	C₂³×C₈₉	1	8	yes	—
378	#103	C₂³×C₈₉	1	8	yes	—
379	#102	C₂³×C₈₉	1	4	yes	—
380	#102	C₂³×C₈₉	1	8	yes	—
381	#97	C₂³×C₈₉	1	8	yes	—
382	C₄×C₈	C₂³×C₈₉	1	4	yes	—
383	C₂⁴	—	2	1	no	—
384	C₂×Q₈	—	2	4	no	—
385	C₂×Q₈	—	2	1	no	—
386	C₂×Q₈	—	2	2	no	—
387	C₂×Q₈	—	2	8	no	—
388	C₂×Q₈	—	2	4	no	—
389	C₂×Q₈	—	2	2	no	—
390	C₂×Q₈	—	2	4	no	—
391	C₂×Q₈	—	2	4	no	—
392	C₂×Q₈	—	8	4	no	—
393	C₂×Q₈	—	8	2	no	—
394	C₂×Q₈	C₂×D₄×C₈₉	1	4	yes	—
395	C₂×Q₈	#41×C₈₉	1	4	yes	—
396	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
397	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
398	C₂×D₄	—	2	1	no	—
399	C₂×D₄	—	2	2	no	—
400	C₂×D₄	—	2	2	no	—
401	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
402	C₂²×C₄	—	2	1	no	—
403	C₂²×C₄	—	2	1	no	—
404	C₂²×C₄	—	2	2	no	—
405	C₂²×C₄	—	2	2	no	—
406	C₂²×C₄	—	2	1	no	—
407	C₂²×C₄	—	2	4	no	—
408	C₂²×C₄	—	4	4	no	—
409	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
410	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
411	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
412	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
413	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
414	Q₁₆	—	2	16	no	—
415	SD₁₆	—	2	8	no	—
416	C₂×C₈	—	2	2	no	—
417	C₂×C₈	—	2	4	no	—
418	C₂×C₈	—	2	2	no	—
419	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
420	C₂×C₈	C₂⁴×C₈₉	1	2	yes	—
421	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
422	C₄⋊C₄	—	2	4	no	—
423	C₄⋊C₄	—	2	8	no	—
424	C₄⋊C₄	—	2	4	no	—
425	C₄⋊C₄	—	2	8	no	—
426	C₄⋊C₄	—	2	8	no	—
427	C₄⋊C₄	—	2	4	no	—
428	C₄⋊C₄	—	2	4	no	—
429	C₄⋊C₄	—	2	1	no	—
430	C₄⋊C₄	—	2	1	no	—
431	C₄⋊C₄	—	2	2	no	—
432	C₄⋊C₄	C₂×D₄×C₈₉	1	4	yes	—
433	C₄⋊C₄	C₂×D₄×C₈₉	1	8	yes	—
434	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
435	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
436	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
437	C₂²⋊C₄	—	2	8	no	—
438	C₂²⋊C₄	—	2	4	no	—
439	C₂²⋊C₄	—	2	8	no	—
440	C₄²	—	2	8	no	—
441	C₄²	—	2	8	no	—
442	C₄²	—	2	4	no	—
443	C₄²	C₂×D₄×C₈₉	1	2	yes	—
444	C₄²	C₂×D₄×C₈₉	1	4	yes	—
445	C₄²	C₂⁴×C₈₉	1	2	yes	—
446	C₄²	C₂⁴×C₈₉	1	4	yes	—
447	C₂³	—	2	2	no	—
448	C₂³	—	4	2	no	—
449	C₂³	—	4	2	no	—
450	C₂³	—	4	2	no	—
451	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
452	Q₈	—	2	4	no	—
453	Q₈	—	2	8	no	—
454	Q₈	—	2	2	no	—
455	Q₈	—	2	4	no	—
456	Q₈	—	2	4	no	—
457	Q₈	—	4	4	no	—
458	Q₈	—	8	8	no	—
459	Q₈	—	8	4	no	—
460	Q₈	C₂×#41×C₈₉	1	4	yes	—
461	D₄	—	2	2	no	—
462	D₄	—	2	2	no	—
463	D₄	—	4	2	no	—
464	C₂×C₄	—	2	4	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	4	no	—
467	C₂×C₄	—	2	2	no	—
468	C₂×C₄	—	2	2	no	—
469	C₂×C₄	—	2	2	no	—
470	C₂×C₄	—	2	4	no	—
471	C₂×C₄	—	2	4	no	—
472	C₂×C₄	—	2	2	no	—
473	C₂×C₄	—	4	4	no	—
474	C₂×C₄	—	4	2	no	—
475	C₂×C₄	—	4	4	no	—
476	C₂×C₄	—	4	2	no	—
477	C₂×C₄	—	4	8	no	—
478	C₂×C₄	#137×C₈₉	1	4	yes	—
479	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
480	C₂×C₄	C₂²×D₄×C₈₉	1	4	yes	—
481	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
482	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
483	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
484	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
485	C₂×C₄	2₋¹⁺⁴×C₈₉	1	2	yes	—
486	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
487	C₈	—	2	2	no	—
488	C₈	—	2	4	no	—
489	C₈	—	2	2	no	—
490	C₂²	—	4	2	no	—
491	C₂²	—	4	2	no	—
492	C₂²	—	4	2	no	—
493	C₂²	—	8	4	no	—
494	C₂²	#548×C₈₉	1	2	yes	—
495	C₂²	C₂×#137×C₈₉	1	2	yes	—
496	C₂²	#577×C₈₉	1	2	yes	—
497	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
498	C₄	—	2	4	no	—
499	C₄	—	2	4	no	—
500	C₄	—	2	2	no	—
501	C₄	—	2	4	no	—
502	C₄	—	4	2	no	—
503	C₄	—	4	4	no	—
504	C₄	—	4	2	no	—
505	C₄	—	4	4	no	—
506	C₄	#548×C₈₉	1	2	yes	—
507	C₄	C₂×#137×C₈₉	1	4	yes	—
508	C₄	C₂²×#41×C₈₉	1	1	yes	characteristic
509	C₄	C₂×2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
510	C₂	—	8	2	no	—
511	C₂	C₈₉×#3369	1	4	yes	direct product factor
512	C₂	C₂×#548×C₈₉	1	1	yes	characteristic
513	C₂	C₂²×#137×C₈₉	1	1	yes	characteristic
514	C₂	C₂×#577×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₉	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
prime radical	C₂²×D₄×C₈₉	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₉ ◁ C₂×C₄×Q₈×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×Q₈²×C₈₉	C₂	semidirect
3. C₈₉×#3369	C₂	direct (N × H)
5. C₂×#487×C₈₉	C₂	semidirect
6. C₂×#475×C₈₉	C₂	semidirect
9. C₂×#462×C₈₉	C₂	semidirect
10. C₂×#446×C₈₉	C₂	semidirect
11. C₂×#441×C₈₉	C₂	semidirect
16. Q₈²×C₈₉	C₂²	semidirect
35. #487×C₈₉	C₂²	semidirect
37. #475×C₈₉	C₂²	semidirect
39. #462×C₈₉	C₂²	semidirect
40. #446×C₈₉	C₂²	semidirect
41. #441×C₈₉	C₂²	semidirect
252. C₂×#3369	C₈₉	direct (N × H)
254. C₂×C₈₉	#3369	direct (N × H)
258. C₂×Q₈²	C₂×C₈₉	semidirect
260. #3369	C₂×C₈₉	direct (N × H)
262. C₂×#487	C₂×C₈₉	semidirect
263. C₂×#475	C₂×C₈₉	semidirect
266. C₂×#462	C₂×C₈₉	semidirect
267. C₂×#446	C₂×C₈₉	semidirect
268. C₂×#441	C₂×C₈₉	semidirect
270. C₈₉	C₂×#3369	direct (N × H)
274. Q₈²	C₂²×C₈₉	semidirect
293. #487	C₂²×C₈₉	semidirect
295. #475	C₂²×C₈₉	semidirect
297. #462	C₂²×C₈₉	semidirect
298. #446	C₂²×C₈₉	semidirect
299. #441	C₂²×C₈₉	semidirect
511. C₂	C₈₉×#3369	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×Q₈²×C₈₉	2
C₂×#556×C₈₉	2
C₈₉×#3369	2
C₂×#489×C₈₉	2
C₂×#487×C₈₉	2
C₂×#475×C₈₉	2
C₂×#476×C₈₉	2
C₂×#464×C₈₉	2
C₂×#462×C₈₉	2
C₂×#446×C₈₉	2
C₂×#441×C₈₉	2
C₂×C₄×SD₁₆×C₈₉	2
C₂×#3369	89

Subgroups — C2×C89×#3369 #36439158

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3369 #36439158