Subgroups — C₂×C₈₉×#3104 #36438379

Order 22784.

2138 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#533×C₈₉	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×Q₈×C₈₉	C₂	1	1	yes	maximal, characteristic
3	C₈₉×#3104	C₂	1	16	yes	maximal, direct product factor
4	C₂×#484×C₈₉	C₂	1	2	yes	maximal
5	C₂×#477×C₈₉	C₂	1	2	yes	maximal, complemented
6	C₂×#462×C₈₉	C₂	1	2	yes	maximal, complemented
7	C₂×#451×C₈₉	C₂	1	2	yes	maximal
8	C₂×C₄×Q₁₆×C₈₉	C₂	1	4	yes	maximal, complemented
9	C₂×#431×C₈₉	C₂	1	1	yes	maximal, characteristic
10	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic
11	C₂²×Q₁₆×C₈₉	C₂²	1	4	yes	—
12	#533×C₈₉	—	2	4	no	—
13	#533×C₈₉	C₂²	1	8	yes	—
14	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
16	C₂×#125×C₈₉	—	2	1	no	—
17	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
18	C₂×C₂²⋊Q₈×C₈₉	C₂²	1	2	yes	—
19	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
20	C₂×C₄×Q₈×C₈₉	—	2	4	no	—
21	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
22	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
23	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
24	C₂×C₄×Q₈×C₈₉	C₂²	1	8	yes	—
25	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
27	C₂×#117×C₈₉	—	2	1	no	—
28	C₂²×C₄⋊C₄×C₈₉	—	2	1	no	—
29	C₂²×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
30	C₂²×C₄²×C₈₉	—	2	1	no	—
31	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
32	#484×C₈₉	C₂²	1	16	yes	—
33	#477×C₈₉	C₂²	1	16	yes	complemented
34	#462×C₈₉	C₂²	1	16	yes	complemented
35	#451×C₈₉	C₂²	1	16	yes	—
36	C₄×Q₁₆×C₈₉	C₂²	1	32	yes	complemented
37	#431×C₈₉	C₂²	1	8	yes	—
38	C₂×#107×C₈₉	C₂²	1	4	yes	—
39	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
40	C₂×#103×C₈₉	C₂²	1	4	yes	—
41	C₂×#103×C₈₉	C₂²	1	4	yes	—
42	C₂×C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
43	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
44	C₂²×Q₈×C₈₉	—	2	8	no	—
45	C₂²×Q₈×C₈₉	—	2	1	no	—
46	C₂²×Q₈×C₈₉	—	2	2	no	—
47	C₂²×Q₈×C₈₉	—	2	1	no	—
48	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
49	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
50	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
51	C₂³×C₄×C₈₉	—	2	1	no	—
52	C₂³×C₄×C₈₉	—	2	1	no	—
53	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
55	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
56	C₂×Q₁₆×C₈₉	—	2	16	no	—
57	C₂×Q₁₆×C₈₉	C₂³	1	16	yes	—
58	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
59	C₄⋊Q₈×C₈₉	—	2	2	no	—
60	C₄⋊Q₈×C₈₉	—	2	2	no	—
61	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
62	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
63	#125×C₈₉	—	2	8	no	—
64	C₂²⋊Q₈×C₈₉	—	2	8	no	—
65	C₂²⋊Q₈×C₈₉	—	2	4	no	—
66	C₂²⋊Q₈×C₈₉	C₂³	1	8	yes	—
67	C₄×Q₈×C₈₉	—	2	8	no	—
68	C₄×Q₈×C₈₉	—	2	4	no	—
69	C₄×Q₈×C₈₉	—	2	8	no	—
70	C₄×Q₈×C₈₉	—	2	4	no	—
71	C₄×Q₈×C₈₉	—	2	8	no	—
72	C₄×Q₈×C₈₉	—	2	4	no	—
73	C₄×Q₈×C₈₉	C₂³	1	8	yes	—
74	C₄×Q₈×C₈₉	C₂³	1	8	yes	—
75	C₄×Q₈×C₈₉	C₂³	1	8	yes	—
76	#117×C₈₉	—	2	8	no	—
77	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
78	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
79	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
80	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
81	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
82	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
83	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
84	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
85	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
86	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
87	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
88	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
89	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
90	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
91	C₂×C₄²×C₈₉	—	2	2	no	—
92	C₂×C₄²×C₈₉	—	2	8	no	—
93	C₂×C₄²×C₈₉	—	2	2	no	—
94	C₂×C₄²×C₈₉	—	2	2	no	—
95	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
96	C₂×C₄²×C₈₉	D₄	1	4	yes	—
97	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
99	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
100	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
101	#107×C₈₉	C₂³	1	16	yes	—
102	C₄⋊C₈×C₈₉	C₂³	1	8	yes	—
103	#103×C₈₉	C₂³	1	16	yes	—
104	#103×C₈₉	C₂³	1	16	yes	—
105	C₂²⋊C₈×C₈₉	C₂³	1	8	yes	—
106	C₄×C₈×C₈₉	C₂³	1	8	yes	—
107	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₂×Q₈×C₈₉	—	2	16	no	—
109	C₂×Q₈×C₈₉	—	2	8	no	—
110	C₂×Q₈×C₈₉	—	2	16	no	—
111	C₂×Q₈×C₈₉	—	2	4	no	—
112	C₂×Q₈×C₈₉	—	2	2	no	—
113	C₂×Q₈×C₈₉	—	2	2	no	—
114	C₂×Q₈×C₈₉	—	2	4	no	—
115	C₂×Q₈×C₈₉	—	2	4	no	—
116	C₂×Q₈×C₈₉	—	2	8	no	—
117	C₂×Q₈×C₈₉	—	4	4	no	—
118	C₂×Q₈×C₈₉	#41	1	8	yes	—
119	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
120	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
121	C₂²×C₄×C₈₉	—	2	8	no	—
122	C₂²×C₄×C₈₉	—	2	4	no	—
123	C₂²×C₄×C₈₉	—	2	2	no	—
124	C₂²×C₄×C₈₉	—	2	4	no	—
125	C₂²×C₄×C₈₉	—	2	4	no	—
126	C₂²×C₄×C₈₉	—	2	2	no	—
127	C₂²×C₄×C₈₉	—	2	2	no	—
128	C₂²×C₄×C₈₉	—	2	2	no	—
129	C₂²×C₄×C₈₉	—	4	2	no	—
130	C₂²×C₄×C₈₉	—	4	4	no	—
131	C₂²×C₄×C₈₉	Q₁₆	1	8	yes	complemented
132	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
133	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
134	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
135	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
136	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
137	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
139	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
141	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
142	Q₁₆×C₈₉	—	2	32	no	—
143	C₂×C₈×C₈₉	—	2	8	no	—
144	C₂×C₈×C₈₉	C₂⁴	1	8	yes	—
145	C₄⋊C₄×C₈₉	—	2	8	no	—
146	C₄⋊C₄×C₈₉	—	2	8	no	—
147	C₄⋊C₄×C₈₉	—	2	4	no	—
148	C₄⋊C₄×C₈₉	—	2	2	no	—
149	C₄⋊C₄×C₈₉	—	2	4	no	—
150	C₄⋊C₄×C₈₉	—	2	4	no	—
151	C₄⋊C₄×C₈₉	—	2	4	no	—
152	C₄⋊C₄×C₈₉	—	2	8	no	—
153	C₄⋊C₄×C₈₉	—	2	8	no	—
154	C₄⋊C₄×C₈₉	—	2	2	no	—
155	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
156	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
157	C₂²⋊C₄×C₈₉	—	2	8	no	—
158	C₄²×C₈₉	—	2	8	no	—
159	C₄²×C₈₉	—	2	8	no	—
160	C₄²×C₈₉	—	2	8	no	—
161	C₄²×C₈₉	C₂×D₄	1	4	yes	—
162	C₄²×C₈₉	C₂×D₄	1	2	yes	—
163	C₄²×C₈₉	C₂×D₄	1	4	yes	—
164	C₄²×C₈₉	C₂×D₄	1	2	yes	—
165	C₄²×C₈₉	C₂⁴	1	2	yes	—
166	C₄²×C₈₉	C₂⁴	1	2	yes	—
167	C₂³×C₈₉	—	2	2	no	—
168	C₂³×C₈₉	—	2	2	no	—
169	C₂³×C₈₉	C₂×Q₁₆	1	4	yes	—
170	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
171	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
172	Q₈×C₈₉	—	2	16	no	—
173	Q₈×C₈₉	—	2	8	no	—
174	Q₈×C₈₉	—	2	4	no	—
175	Q₈×C₈₉	—	2	8	no	—
176	Q₈×C₈₉	—	4	8	no	—
177	Q₈×C₈₉	—	4	4	no	—
178	Q₈×C₈₉	C₂×#41	1	8	yes	—
179	C₂×C₄×C₈₉	—	2	16	no	—
180	C₂×C₄×C₈₉	—	2	8	no	—
181	C₂×C₄×C₈₉	—	2	2	no	—
182	C₂×C₄×C₈₉	—	2	2	no	—
183	C₂×C₄×C₈₉	—	2	2	no	—
184	C₂×C₄×C₈₉	—	2	4	no	—
185	C₂×C₄×C₈₉	—	2	4	no	—
186	C₂×C₄×C₈₉	—	2	4	no	—
187	C₂×C₄×C₈₉	—	2	4	no	—
188	C₂×C₄×C₈₉	—	2	4	no	—
189	C₂×C₄×C₈₉	—	2	4	no	—
190	C₂×C₄×C₈₉	—	4	4	no	—
191	C₂×C₄×C₈₉	—	4	8	no	—
192	C₂×C₄×C₈₉	—	4	4	no	—
193	C₂×C₄×C₈₉	—	4	4	no	—
194	C₂×C₄×C₈₉	—	4	4	no	—
195	C₂×C₄×C₈₉	C₂×Q₁₆	1	4	yes	—
196	C₂×C₄×C₈₉	C₂×Q₁₆	1	4	yes	—
197	C₂×C₄×C₈₉	C₂×Q₁₆	1	16	yes	complemented
198	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
199	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
200	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
201	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
203	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
204	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
205	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
206	C₈×C₈₉	—	2	8	no	—
207	C₂²×C₈₉	—	2	8	no	—
208	C₂²×C₈₉	—	2	2	no	—
209	C₂²×C₈₉	—	2	2	no	—
210	C₂²×C₈₉	#525	1	2	yes	—
211	C₂²×C₈₉	C₂²×Q₁₆	1	4	yes	—
212	C₂²×C₈₉	C₂²×Q₁₆	1	2	yes	—
213	C₂²×C₈₉	#575	1	2	yes	—
214	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
215	C₄×C₈₉	—	2	8	no	—
216	C₄×C₈₉	—	2	2	no	—
217	C₄×C₈₉	—	2	2	no	—
218	C₄×C₈₉	—	2	2	no	—
219	C₄×C₈₉	—	4	4	no	—
220	C₄×C₈₉	—	4	4	no	—
221	C₄×C₈₉	—	4	4	no	—
222	C₄×C₈₉	#525	1	2	yes	—
223	C₄×C₈₉	C₂²×Q₁₆	1	4	yes	—
224	C₄×C₈₉	C₂²×Q₁₆	1	4	yes	—
225	C₄×C₈₉	C₂²×#41	1	2	yes	—
226	C₂×#3104	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
227	C₂×C₈₉	—	2	4	no	—
228	C₂×C₈₉	#3104	1	4	yes	direct product factor
229	C₂×C₈₉	C₂×#525	1	1	yes	characteristic
230	C₂×C₈₉	C₂³×Q₁₆	1	1	yes	characteristic
231	C₂×C₈₉	C₂×#575	1	1	yes	characteristic
232	C₂×#533	C₂×C₈₉	1	1	yes	characteristic
233	C₂²×C₄×Q₈	C₂×C₈₉	1	1	yes	characteristic
234	#3104	C₂×C₈₉	1	16	yes	direct product factor
235	C₂×#484	C₂×C₈₉	1	2	yes	—
236	C₂×#477	C₂×C₈₉	1	2	yes	complemented
237	C₂×#462	C₂×C₈₉	1	2	yes	complemented
238	C₂×#451	C₂×C₈₉	1	2	yes	—
239	C₂×C₄×Q₁₆	C₂×C₈₉	1	4	yes	complemented
240	C₂×#431	C₂×C₈₉	1	1	yes	characteristic
241	C₈₉	C₂×#3104	1	1	yes	characteristic, direct product factor, Sylow
242	C₂³×Q₈	C₂²×C₈₉	1	1	yes	characteristic
243	C₂²×Q₁₆	C₂²×C₈₉	1	4	yes	—
244	#533	—	2	4	no	—
245	#533	C₂²×C₈₉	1	8	yes	—
246	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
247	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
248	C₂×#125	—	2	1	no	—
249	C₂×C₂²⋊Q₈	—	2	1	no	—
250	C₂×C₂²⋊Q₈	C₂²×C₈₉	1	2	yes	—
251	C₂×C₄×Q₈	—	2	1	no	—
252	C₂×C₄×Q₈	—	2	1	no	—
253	C₂×C₄×Q₈	—	2	4	no	—
254	C₂×C₄×Q₈	—	2	1	no	—
255	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
256	C₂×C₄×Q₈	C₂²×C₈₉	1	8	yes	—
257	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
258	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
259	C₂×#117	—	2	1	no	—
260	C₂²×C₄⋊C₄	—	2	1	no	—
261	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
262	C₂²×C₄²	—	2	1	no	—
263	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
264	#484	C₂²×C₈₉	1	16	yes	—
265	#477	C₂²×C₈₉	1	16	yes	complemented
266	#462	C₂²×C₈₉	1	16	yes	complemented
267	#451	C₂²×C₈₉	1	16	yes	—
268	C₄×Q₁₆	C₂²×C₈₉	1	32	yes	complemented
269	#431	C₂²×C₈₉	1	8	yes	—
270	C₂×#107	C₂²×C₈₉	1	4	yes	—
271	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
272	C₂×#103	C₂²×C₈₉	1	4	yes	—
273	C₂×#103	C₂²×C₈₉	1	4	yes	—
274	C₂×C₂²⋊C₈	C₂²×C₈₉	1	2	yes	—
275	C₂×C₄×C₈	C₂²×C₈₉	1	2	yes	—
276	C₂²×Q₈	—	2	8	no	—
277	C₂²×Q₈	—	2	1	no	—
278	C₂²×Q₈	—	2	2	no	—
279	C₂²×Q₈	—	2	1	no	—
280	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
281	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
282	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
283	C₂³×C₄	—	2	1	no	—
284	C₂³×C₄	—	2	1	no	—
285	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
286	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
287	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
288	C₂×Q₁₆	—	2	16	no	—
289	C₂×Q₁₆	C₂³×C₈₉	1	16	yes	—
290	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
291	C₄⋊Q₈	—	2	2	no	—
292	C₄⋊Q₈	—	2	2	no	—
293	C₄⋊Q₈	C₂³×C₈₉	1	4	yes	—
294	C₄⋊Q₈	C₂³×C₈₉	1	4	yes	—
295	#125	—	2	8	no	—
296	C₂²⋊Q₈	—	2	8	no	—
297	C₂²⋊Q₈	—	2	4	no	—
298	C₂²⋊Q₈	C₂³×C₈₉	1	8	yes	—
299	C₄×Q₈	—	2	8	no	—
300	C₄×Q₈	—	2	4	no	—
301	C₄×Q₈	—	2	8	no	—
302	C₄×Q₈	—	2	4	no	—
303	C₄×Q₈	—	2	8	no	—
304	C₄×Q₈	—	2	4	no	—
305	C₄×Q₈	C₂³×C₈₉	1	8	yes	—
306	C₄×Q₈	C₂³×C₈₉	1	8	yes	—
307	C₄×Q₈	C₂³×C₈₉	1	8	yes	—
308	#117	—	2	8	no	—
309	C₂×C₄⋊C₄	—	2	2	no	—
310	C₂×C₄⋊C₄	—	2	2	no	—
311	C₂×C₄⋊C₄	—	2	8	no	—
312	C₂×C₄⋊C₄	—	2	1	no	—
313	C₂×C₄⋊C₄	—	2	2	no	—
314	C₂×C₄⋊C₄	—	2	1	no	—
315	C₂×C₄⋊C₄	—	2	1	no	—
316	C₂×C₄⋊C₄	—	2	1	no	—
317	C₂×C₄⋊C₄	—	2	2	no	—
318	C₂×C₄⋊C₄	—	2	2	no	—
319	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
320	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
321	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
322	C₂×C₂²⋊C₄	—	2	2	no	—
323	C₂×C₄²	—	2	2	no	—
324	C₂×C₄²	—	2	2	no	—
325	C₂×C₄²	—	2	2	no	—
326	C₂×C₄²	—	2	8	no	—
327	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
328	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
329	C₂×C₄²	D₄×C₈₉	1	4	yes	—
330	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
331	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
332	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
333	#107	C₂³×C₈₉	1	16	yes	—
334	C₄⋊C₈	C₂³×C₈₉	1	8	yes	—
335	#103	C₂³×C₈₉	1	16	yes	—
336	#103	C₂³×C₈₉	1	16	yes	—
337	C₂²⋊C₈	C₂³×C₈₉	1	8	yes	—
338	C₄×C₈	C₂³×C₈₉	1	8	yes	—
339	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
340	C₂×Q₈	—	2	16	no	—
341	C₂×Q₈	—	2	8	no	—
342	C₂×Q₈	—	2	16	no	—
343	C₂×Q₈	—	2	4	no	—
344	C₂×Q₈	—	2	2	no	—
345	C₂×Q₈	—	2	2	no	—
346	C₂×Q₈	—	2	4	no	—
347	C₂×Q₈	—	2	4	no	—
348	C₂×Q₈	—	2	8	no	—
349	C₂×Q₈	—	4	4	no	—
350	C₂×Q₈	#41×C₈₉	1	8	yes	—
351	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
352	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
353	C₂²×C₄	—	2	8	no	—
354	C₂²×C₄	—	2	4	no	—
355	C₂²×C₄	—	2	2	no	—
356	C₂²×C₄	—	2	4	no	—
357	C₂²×C₄	—	2	4	no	—
358	C₂²×C₄	—	2	2	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	4	2	no	—
362	C₂²×C₄	—	4	4	no	—
363	C₂²×C₄	Q₁₆×C₈₉	1	8	yes	complemented
364	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
365	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
366	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
367	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
368	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
369	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
370	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
371	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
372	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
373	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
374	Q₁₆	—	2	32	no	—
375	C₂×C₈	—	2	8	no	—
376	C₂×C₈	C₂⁴×C₈₉	1	8	yes	—
377	C₄⋊C₄	—	2	8	no	—
378	C₄⋊C₄	—	2	8	no	—
379	C₄⋊C₄	—	2	4	no	—
380	C₄⋊C₄	—	2	2	no	—
381	C₄⋊C₄	—	2	4	no	—
382	C₄⋊C₄	—	2	4	no	—
383	C₄⋊C₄	—	2	4	no	—
384	C₄⋊C₄	—	2	8	no	—
385	C₄⋊C₄	—	2	8	no	—
386	C₄⋊C₄	—	2	2	no	—
387	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
388	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
389	C₂²⋊C₄	—	2	8	no	—
390	C₄²	—	2	8	no	—
391	C₄²	—	2	8	no	—
392	C₄²	—	2	8	no	—
393	C₄²	C₂×D₄×C₈₉	1	4	yes	—
394	C₄²	C₂×D₄×C₈₉	1	2	yes	—
395	C₄²	C₂×D₄×C₈₉	1	4	yes	—
396	C₄²	C₂×D₄×C₈₉	1	2	yes	—
397	C₄²	C₂⁴×C₈₉	1	2	yes	—
398	C₄²	C₂⁴×C₈₉	1	2	yes	—
399	C₂³	—	2	2	no	—
400	C₂³	—	2	2	no	—
401	C₂³	C₂×Q₁₆×C₈₉	1	4	yes	—
402	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
403	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
404	Q₈	—	2	16	no	—
405	Q₈	—	2	8	no	—
406	Q₈	—	2	4	no	—
407	Q₈	—	2	8	no	—
408	Q₈	—	4	8	no	—
409	Q₈	—	4	4	no	—
410	Q₈	C₂×#41×C₈₉	1	8	yes	—
411	C₂×C₄	—	2	16	no	—
412	C₂×C₄	—	2	8	no	—
413	C₂×C₄	—	2	2	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	2	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	4	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	2	4	no	—
420	C₂×C₄	—	2	4	no	—
421	C₂×C₄	—	2	4	no	—
422	C₂×C₄	—	4	4	no	—
423	C₂×C₄	—	4	8	no	—
424	C₂×C₄	—	4	4	no	—
425	C₂×C₄	—	4	4	no	—
426	C₂×C₄	—	4	4	no	—
427	C₂×C₄	C₂×Q₁₆×C₈₉	1	4	yes	—
428	C₂×C₄	C₂×Q₁₆×C₈₉	1	4	yes	—
429	C₂×C₄	C₂×Q₁₆×C₈₉	1	16	yes	complemented
430	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
431	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
432	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
433	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
434	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
435	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
436	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
437	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
438	C₈	—	2	8	no	—
439	C₂²	—	2	2	no	—
440	C₂²	—	2	8	no	—
441	C₂²	—	2	2	no	—
442	C₂²	#525×C₈₉	1	2	yes	—
443	C₂²	C₂²×Q₁₆×C₈₉	1	4	yes	—
444	C₂²	C₂²×Q₁₆×C₈₉	1	2	yes	—
445	C₂²	#575×C₈₉	1	2	yes	—
446	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
447	C₄	—	2	2	no	—
448	C₄	—	2	2	no	—
449	C₄	—	2	8	no	—
450	C₄	—	2	2	no	—
451	C₄	—	4	4	no	—
452	C₄	—	4	4	no	—
453	C₄	—	4	4	no	—
454	C₄	#525×C₈₉	1	2	yes	—
455	C₄	C₂²×Q₁₆×C₈₉	1	4	yes	—
456	C₄	C₂²×Q₁₆×C₈₉	1	4	yes	—
457	C₄	C₂²×#41×C₈₉	1	2	yes	—
458	C₂	—	2	4	no	—
459	C₂	C₈₉×#3104	1	4	yes	direct product factor
460	C₂	C₂×#525×C₈₉	1	1	yes	characteristic
461	C₂	C₂³×Q₁₆×C₈₉	1	1	yes	characteristic
462	C₂	C₂×#575×C₈₉	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₈₉	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₈₉
socle	C₂³×C₈₉	C₂²×D₄
prime radical	C₂⁴×C₈₉	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₈₉ ◁ C₂²×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₈₉×#3104	C₂	direct (N × H)
5. C₂×#477×C₈₉	C₂	semidirect
6. C₂×#462×C₈₉	C₂	semidirect
8. C₂×C₄×Q₁₆×C₈₉	C₂	semidirect
33. #477×C₈₉	C₂²	semidirect
34. #462×C₈₉	C₂²	semidirect
36. C₄×Q₁₆×C₈₉	C₂²	semidirect
131. C₂²×C₄×C₈₉	Q₁₆	semidirect
197. C₂×C₄×C₈₉	C₂×Q₁₆	semidirect
226. C₂×#3104	C₈₉	direct (N × H)
228. C₂×C₈₉	#3104	direct (N × H)
234. #3104	C₂×C₈₉	direct (N × H)
236. C₂×#477	C₂×C₈₉	semidirect
237. C₂×#462	C₂×C₈₉	semidirect
239. C₂×C₄×Q₁₆	C₂×C₈₉	semidirect
241. C₈₉	C₂×#3104	direct (N × H)
265. #477	C₂²×C₈₉	semidirect
266. #462	C₂²×C₈₉	semidirect
268. C₄×Q₁₆	C₂²×C₈₉	semidirect
363. C₂²×C₄	Q₁₆×C₈₉	semidirect
429. C₂×C₄	C₂×Q₁₆×C₈₉	semidirect
459. C₂	C₈₉×#3104	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×#533×C₈₉	2
C₂²×C₄×Q₈×C₈₉	2
C₈₉×#3104	2
C₂×#484×C₈₉	2
C₂×#477×C₈₉	2
C₂×#462×C₈₉	2
C₂×#451×C₈₉	2
C₂×C₄×Q₁₆×C₈₉	2
C₂×#431×C₈₉	2
C₂×#3104	89

Subgroups — C2×C89×#3104 #36438379

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3104 #36438379