Subgroups — C₂×C₈₉×#3165 #36438509

Order 22784.

2118 conjugacy classes of subgroups.

The subgroup lattice has 410 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×#3165	C₂	1	16	yes	maximal, direct product factor
4	C₂×#501×C₈₉	C₂	1	4	yes	maximal, complemented
5	C₂×#500×C₈₉	C₂	1	2	yes	maximal, complemented
6	C₂×#498×C₈₉	C₂	1	2	yes	maximal, complemented
7	C₂×#428×C₈₉	C₂	1	4	yes	maximal
8	C₂×C₄×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₈₉	—	2	1	no	—
10	C₂²×M₁₆×C₈₉	C₂²	1	2	yes	—
11	#533×C₈₉	—	2	4	no	—
12	#533×C₈₉	C₂²	1	8	yes	—
13	C₂×C₄⋊Q₈×C₈₉	—	2	1	no	—
14	C₂×C₄⋊Q₈×C₈₉	—	2	1	no	—
15	C₂×C₄⋊Q₈×C₈₉	—	2	4	no	—
16	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
18	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊Q₈×C₈₉	C₂²	1	2	yes	—
20	C₂×C₄⋊Q₈×C₈₉	C₂²	1	8	yes	—
21	C₂×#125×C₈₉	C₂²	1	2	yes	—
22	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
23	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
24	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
25	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
26	C₂×#117×C₈₉	C₂²	1	2	yes	—
27	C₂²×C₄⋊C₄×C₈₉	—	2	1	no	—
28	C₂²×C₄⋊C₄×C₈₉	C₂²	1	2	yes	—
29	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
30	#501×C₈₉	C₂²	1	32	yes	complemented
31	#500×C₈₉	C₂²	1	16	yes	complemented
32	#498×C₈₉	C₂²	1	16	yes	complemented
33	#428×C₈₉	C₂²	1	32	yes	—
34	C₂×#107×C₈₉	C₂²	1	8	yes	—
35	C₂×#106×C₈₉	C₂²	1	8	yes	—
36	C₄×M₁₆×C₈₉	C₂²	1	8	yes	—
37	C₂×#97×C₈₉	C₂²	1	2	yes	—
38	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
39	C₂²×Q₈×C₈₉	—	2	1	no	—
40	C₂²×Q₈×C₈₉	—	2	1	no	—
41	C₂²×Q₈×C₈₉	—	2	8	no	—
42	C₂²×Q₈×C₈₉	—	2	2	no	—
43	C₂²×Q₈×C₈₉	—	2	2	no	—
44	C₂²×Q₈×C₈₉	—	4	8	no	—
45	C₂³×C₄×C₈₉	—	2	2	no	—
46	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
47	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
48	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
49	C₂×M₁₆×C₈₉	Q₈	1	16	yes	complemented
50	C₂×M₁₆×C₈₉	C₂³	1	8	yes	—
51	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
52	C₄⋊Q₈×C₈₉	—	2	8	no	—
53	C₄⋊Q₈×C₈₉	—	2	8	no	—
54	C₄⋊Q₈×C₈₉	—	2	4	no	—
55	C₄⋊Q₈×C₈₉	—	2	4	no	—
56	C₄⋊Q₈×C₈₉	—	2	2	no	—
57	C₄⋊Q₈×C₈₉	—	2	2	no	—
58	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
59	C₄⋊Q₈×C₈₉	C₂³	1	8	yes	—
60	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
61	C₄⋊Q₈×C₈₉	C₂³	1	8	yes	—
62	#125×C₈₉	—	2	4	no	—
63	#125×C₈₉	C₂³	1	8	yes	—
64	C₂²⋊Q₈×C₈₉	—	2	8	no	—
65	C₂²⋊Q₈×C₈₉	—	2	8	no	—
66	C₄×Q₈×C₈₉	—	2	8	no	—
67	C₄×Q₈×C₈₉	—	2	8	no	—
68	#117×C₈₉	—	2	4	no	—
69	#117×C₈₉	C₂³	1	8	yes	—
70	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
71	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
72	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
73	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
74	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
75	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
76	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
77	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
78	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
79	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
80	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
81	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
82	C₂×C₄⋊C₄×C₈₉	C₂³	1	8	yes	—
83	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
84	C₂×C₄²×C₈₉	—	2	2	no	—
85	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
86	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
87	C₂×C₄²×C₈₉	D₄	1	4	yes	—
88	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
89	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
90	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
91	#107×C₈₉	C₂³	1	32	yes	—
92	#106×C₈₉	C₂³	1	32	yes	—
93	#97×C₈₉	C₂³	1	8	yes	—
94	C₄×C₈×C₈₉	C₂³	1	8	yes	—
95	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
96	C₂×Q₈×C₈₉	—	2	4	no	—
97	C₂×Q₈×C₈₉	—	2	4	no	—
98	C₂×Q₈×C₈₉	—	2	8	no	—
99	C₂×Q₈×C₈₉	—	2	8	no	—
100	C₂×Q₈×C₈₉	—	4	8	no	—
101	C₂×Q₈×C₈₉	—	4	32	no	—
102	C₂×Q₈×C₈₉	—	4	4	no	—
103	C₂×Q₈×C₈₉	—	4	8	no	—
104	C₂×Q₈×C₈₉	—	4	4	no	—
105	C₂²×C₄×C₈₉	—	2	4	no	—
106	C₂²×C₄×C₈₉	—	2	4	no	—
107	C₂²×C₄×C₈₉	—	2	8	no	—
108	C₂²×C₄×C₈₉	—	2	2	no	—
109	C₂²×C₄×C₈₉	—	2	2	no	—
110	C₂²×C₄×C₈₉	—	2	4	no	—
111	C₂²×C₄×C₈₉	—	4	8	no	—
112	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
113	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
114	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
115	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
117	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
118	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₈₉	C₂×Q₈	1	8	yes	—
121	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
122	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
123	M₁₆×C₈₉	C₂×Q₈	1	32	yes	complemented
124	C₂×C₈×C₈₉	C₂×Q₈	1	16	yes	complemented
125	C₂×C₈×C₈₉	C₂⁴	1	8	yes	—
126	C₄⋊C₄×C₈₉	—	2	8	no	—
127	C₄⋊C₄×C₈₉	—	2	8	no	—
128	C₄⋊C₄×C₈₉	—	2	8	no	—
129	C₄⋊C₄×C₈₉	—	2	4	no	—
130	C₄⋊C₄×C₈₉	—	2	8	no	—
131	C₄⋊C₄×C₈₉	—	2	8	no	—
132	C₄⋊C₄×C₈₉	—	2	4	no	—
133	C₄⋊C₄×C₈₉	—	2	4	no	—
134	C₄⋊C₄×C₈₉	—	2	2	no	—
135	C₄⋊C₄×C₈₉	—	2	2	no	—
136	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
137	C₄⋊C₄×C₈₉	C₂⁴	1	8	yes	—
138	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
139	C₂²⋊C₄×C₈₉	—	2	8	no	—
140	C₄²×C₈₉	—	2	8	no	—
141	C₄²×C₈₉	C₂×D₄	1	4	yes	—
142	C₄²×C₈₉	C₂×D₄	1	4	yes	—
143	C₄²×C₈₉	C₂×D₄	1	2	yes	—
144	C₄²×C₈₉	C₂×D₄	1	2	yes	—
145	C₄²×C₈₉	C₂⁴	1	2	yes	—
146	C₄²×C₈₉	C₂⁴	1	2	yes	—
147	C₂³×C₈₉	—	2	2	no	—
148	C₂³×C₈₉	—	2	2	no	—
149	C₂³×C₈₉	C₄⋊Q₈	1	4	yes	—
150	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
151	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
152	Q₈×C₈₉	—	4	16	no	—
153	Q₈×C₈₉	—	4	8	no	—
154	Q₈×C₈₉	—	4	16	no	—
155	Q₈×C₈₉	—	4	8	no	—
156	C₂×C₄×C₈₉	—	2	2	no	—
157	C₂×C₄×C₈₉	—	2	2	no	—
158	C₂×C₄×C₈₉	—	2	8	no	—
159	C₂×C₄×C₈₉	—	2	8	no	—
160	C₂×C₄×C₈₉	—	2	4	no	—
161	C₂×C₄×C₈₉	—	2	4	no	—
162	C₂×C₄×C₈₉	—	2	8	no	—
163	C₂×C₄×C₈₉	—	4	16	no	—
164	C₂×C₄×C₈₉	—	4	8	no	—
165	C₂×C₄×C₈₉	—	4	4	no	—
166	C₂×C₄×C₈₉	—	4	4	no	—
167	C₂×C₄×C₈₉	C₄⋊Q₈	1	8	yes	—
168	C₂×C₄×C₈₉	C₄⋊Q₈	1	4	yes	—
169	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
170	C₂×C₄×C₈₉	#137	1	4	yes	—
171	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
172	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
173	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
174	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
175	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
176	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
177	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
178	C₂×C₄×C₈₉	C₂²×Q₈	1	8	yes	—
179	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
180	C₈×C₈₉	C₂²×Q₈	1	16	yes	complemented
181	C₂²×C₈₉	—	2	8	no	—
182	C₂²×C₈₉	—	2	2	no	—
183	C₂²×C₈₉	—	2	2	no	—
184	C₂²×C₈₉	C₂×C₄⋊Q₈	1	4	yes	—
185	C₂²×C₈₉	C₂×C₄⋊Q₈	1	2	yes	—
186	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
187	C₂²×C₈₉	C₂×#137	1	2	yes	—
188	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
189	C₄×C₈₉	—	2	2	no	—
190	C₄×C₈₉	—	2	2	no	—
191	C₄×C₈₉	—	4	8	no	—
192	C₄×C₈₉	—	4	4	no	—
193	C₄×C₈₉	—	4	4	no	—
194	C₄×C₈₉	C₂×C₄⋊Q₈	1	4	yes	—
195	C₄×C₈₉	C₂×C₄⋊Q₈	1	2	yes	—
196	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	4	yes	—
197	C₄×C₈₉	C₂×#137	1	4	yes	—
198	C₄×C₈₉	C₂³×Q₈	1	2	yes	—
199	C₂×#3165	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
200	C₂×C₈₉	—	2	4	no	—
201	C₂×C₈₉	#3165	1	4	yes	direct product factor
202	C₂×C₈₉	C₂²×C₄⋊Q₈	1	1	yes	characteristic
203	C₂×C₈₉	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
204	C₂×C₈₉	C₂²×#137	1	1	yes	characteristic
205	C₂×#533	C₂×C₈₉	1	1	yes	characteristic
206	C₂²×C₄⋊Q₈	C₂×C₈₉	1	1	yes	characteristic
207	#3165	C₂×C₈₉	1	16	yes	direct product factor
208	C₂×#501	C₂×C₈₉	1	4	yes	complemented
209	C₂×#500	C₂×C₈₉	1	2	yes	complemented
210	C₂×#498	C₂×C₈₉	1	2	yes	complemented
211	C₂×#428	C₂×C₈₉	1	4	yes	—
212	C₂×C₄×M₁₆	C₂×C₈₉	1	1	yes	characteristic
213	C₈₉	C₂×#3165	1	1	yes	characteristic, direct product factor, Sylow
214	C₂³×Q₈	—	2	1	no	—
215	C₂²×M₁₆	C₂²×C₈₉	1	2	yes	—
216	#533	—	2	4	no	—
217	#533	C₂²×C₈₉	1	8	yes	—
218	C₂×C₄⋊Q₈	—	2	1	no	—
219	C₂×C₄⋊Q₈	—	2	1	no	—
220	C₂×C₄⋊Q₈	—	2	4	no	—
221	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
222	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
223	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
224	C₂×C₄⋊Q₈	C₂²×C₈₉	1	2	yes	—
225	C₂×C₄⋊Q₈	C₂²×C₈₉	1	8	yes	—
226	C₂×#125	C₂²×C₈₉	1	2	yes	—
227	C₂×C₂²⋊Q₈	—	2	1	no	—
228	C₂×C₂²⋊Q₈	—	2	1	no	—
229	C₂×C₄×Q₈	—	2	1	no	—
230	C₂×C₄×Q₈	—	2	1	no	—
231	C₂×#117	C₂²×C₈₉	1	2	yes	—
232	C₂²×C₄⋊C₄	—	2	1	no	—
233	C₂²×C₄⋊C₄	C₂²×C₈₉	1	2	yes	—
234	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
235	#501	C₂²×C₈₉	1	32	yes	complemented
236	#500	C₂²×C₈₉	1	16	yes	complemented
237	#498	C₂²×C₈₉	1	16	yes	complemented
238	#428	C₂²×C₈₉	1	32	yes	—
239	C₂×#107	C₂²×C₈₉	1	8	yes	—
240	C₂×#106	C₂²×C₈₉	1	8	yes	—
241	C₄×M₁₆	C₂²×C₈₉	1	8	yes	—
242	C₂×#97	C₂²×C₈₉	1	2	yes	—
243	C₂×C₄×C₈	C₂²×C₈₉	1	2	yes	—
244	C₂²×Q₈	—	2	1	no	—
245	C₂²×Q₈	—	2	1	no	—
246	C₂²×Q₈	—	2	2	no	—
247	C₂²×Q₈	—	2	8	no	—
248	C₂²×Q₈	—	2	2	no	—
249	C₂²×Q₈	—	4	8	no	—
250	C₂³×C₄	—	2	2	no	—
251	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
252	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
253	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
254	C₂×M₁₆	Q₈×C₈₉	1	16	yes	complemented
255	C₂×M₁₆	C₂³×C₈₉	1	8	yes	—
256	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
257	C₄⋊Q₈	—	2	8	no	—
258	C₄⋊Q₈	—	2	8	no	—
259	C₄⋊Q₈	—	2	4	no	—
260	C₄⋊Q₈	—	2	4	no	—
261	C₄⋊Q₈	—	2	2	no	—
262	C₄⋊Q₈	—	2	2	no	—
263	C₄⋊Q₈	C₂³×C₈₉	1	8	yes	—
264	C₄⋊Q₈	C₂³×C₈₉	1	8	yes	—
265	C₄⋊Q₈	C₂³×C₈₉	1	4	yes	—
266	C₄⋊Q₈	C₂³×C₈₉	1	4	yes	—
267	#125	—	2	4	no	—
268	#125	C₂³×C₈₉	1	8	yes	—
269	C₂²⋊Q₈	—	2	8	no	—
270	C₂²⋊Q₈	—	2	8	no	—
271	C₄×Q₈	—	2	8	no	—
272	C₄×Q₈	—	2	8	no	—
273	#117	—	2	4	no	—
274	#117	C₂³×C₈₉	1	8	yes	—
275	C₂×C₄⋊C₄	—	2	2	no	—
276	C₂×C₄⋊C₄	—	2	8	no	—
277	C₂×C₄⋊C₄	—	2	2	no	—
278	C₂×C₄⋊C₄	—	2	2	no	—
279	C₂×C₄⋊C₄	—	2	4	no	—
280	C₂×C₄⋊C₄	—	2	2	no	—
281	C₂×C₄⋊C₄	—	2	2	no	—
282	C₂×C₄⋊C₄	—	2	1	no	—
283	C₂×C₄⋊C₄	—	2	1	no	—
284	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
285	C₂×C₄⋊C₄	C₂³×C₈₉	1	8	yes	—
286	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
287	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
288	C₂×C₂²⋊C₄	—	2	2	no	—
289	C₂×C₄²	—	2	2	no	—
290	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
291	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
292	C₂×C₄²	D₄×C₈₉	1	4	yes	—
293	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
294	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
295	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
296	#107	C₂³×C₈₉	1	32	yes	—
297	#106	C₂³×C₈₉	1	32	yes	—
298	#97	C₂³×C₈₉	1	8	yes	—
299	C₄×C₈	C₂³×C₈₉	1	8	yes	—
300	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
301	C₂×Q₈	—	2	4	no	—
302	C₂×Q₈	—	2	4	no	—
303	C₂×Q₈	—	2	8	no	—
304	C₂×Q₈	—	2	8	no	—
305	C₂×Q₈	—	4	8	no	—
306	C₂×Q₈	—	4	32	no	—
307	C₂×Q₈	—	4	8	no	—
308	C₂×Q₈	—	4	4	no	—
309	C₂×Q₈	—	4	4	no	—
310	C₂²×C₄	—	2	4	no	—
311	C₂²×C₄	—	2	4	no	—
312	C₂²×C₄	—	2	2	no	—
313	C₂²×C₄	—	2	2	no	—
314	C₂²×C₄	—	2	8	no	—
315	C₂²×C₄	—	2	4	no	—
316	C₂²×C₄	—	4	8	no	—
317	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
318	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
319	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
320	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
321	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
322	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
323	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
324	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
325	C₂²×C₄	C₂×Q₈×C₈₉	1	8	yes	—
326	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
327	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
328	M₁₆	C₂×Q₈×C₈₉	1	32	yes	complemented
329	C₂×C₈	C₂×Q₈×C₈₉	1	16	yes	complemented
330	C₂×C₈	C₂⁴×C₈₉	1	8	yes	—
331	C₄⋊C₄	—	2	8	no	—
332	C₄⋊C₄	—	2	8	no	—
333	C₄⋊C₄	—	2	8	no	—
334	C₄⋊C₄	—	2	4	no	—
335	C₄⋊C₄	—	2	8	no	—
336	C₄⋊C₄	—	2	8	no	—
337	C₄⋊C₄	—	2	4	no	—
338	C₄⋊C₄	—	2	4	no	—
339	C₄⋊C₄	—	2	2	no	—
340	C₄⋊C₄	—	2	2	no	—
341	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
342	C₄⋊C₄	C₂⁴×C₈₉	1	8	yes	—
343	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
344	C₂²⋊C₄	—	2	8	no	—
345	C₄²	—	2	8	no	—
346	C₄²	C₂×D₄×C₈₉	1	4	yes	—
347	C₄²	C₂×D₄×C₈₉	1	4	yes	—
348	C₄²	C₂×D₄×C₈₉	1	2	yes	—
349	C₄²	C₂×D₄×C₈₉	1	2	yes	—
350	C₄²	C₂⁴×C₈₉	1	2	yes	—
351	C₄²	C₂⁴×C₈₉	1	2	yes	—
352	C₂³	—	2	2	no	—
353	C₂³	—	2	2	no	—
354	C₂³	C₄⋊Q₈×C₈₉	1	4	yes	—
355	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
356	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
357	Q₈	—	4	16	no	—
358	Q₈	—	4	8	no	—
359	Q₈	—	4	16	no	—
360	Q₈	—	4	8	no	—
361	C₂×C₄	—	2	2	no	—
362	C₂×C₄	—	2	2	no	—
363	C₂×C₄	—	2	8	no	—
364	C₂×C₄	—	2	8	no	—
365	C₂×C₄	—	2	4	no	—
366	C₂×C₄	—	2	4	no	—
367	C₂×C₄	—	2	8	no	—
368	C₂×C₄	—	4	16	no	—
369	C₂×C₄	—	4	8	no	—
370	C₂×C₄	—	4	4	no	—
371	C₂×C₄	—	4	4	no	—
372	C₂×C₄	C₄⋊Q₈×C₈₉	1	8	yes	—
373	C₂×C₄	C₄⋊Q₈×C₈₉	1	4	yes	—
374	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	4	yes	—
375	C₂×C₄	#137×C₈₉	1	4	yes	—
376	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
377	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
378	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
379	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
380	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
381	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
382	C₂×C₄	C₂²×Q₈×C₈₉	1	4	yes	—
383	C₂×C₄	C₂²×Q₈×C₈₉	1	8	yes	—
384	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
385	C₈	C₂²×Q₈×C₈₉	1	16	yes	complemented
386	C₂²	—	2	2	no	—
387	C₂²	—	2	2	no	—
388	C₂²	—	2	8	no	—
389	C₂²	C₂×C₄⋊Q₈×C₈₉	1	4	yes	—
390	C₂²	C₂×C₄⋊Q₈×C₈₉	1	2	yes	—
391	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	2	yes	—
392	C₂²	C₂×#137×C₈₉	1	2	yes	—
393	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
394	C₄	—	2	2	no	—
395	C₄	—	2	2	no	—
396	C₄	—	4	8	no	—
397	C₄	—	4	4	no	—
398	C₄	—	4	4	no	—
399	C₄	C₂×C₄⋊Q₈×C₈₉	1	4	yes	—
400	C₄	C₂×C₄⋊Q₈×C₈₉	1	2	yes	—
401	C₄	C₂×AGL(1,ℤ/8)×C₈₉	1	4	yes	—
402	C₄	C₂×#137×C₈₉	1	4	yes	—
403	C₄	C₂³×Q₈×C₈₉	1	2	yes	—
404	C₂	—	2	4	no	—
405	C₂	C₈₉×#3165	1	4	yes	direct product factor
406	C₂	C₂²×C₄⋊Q₈×C₈₉	1	1	yes	characteristic
407	C₂	C₂²×AGL(1,ℤ/8)×C₈₉	1	1	yes	characteristic
408	C₂	C₂²×#137×C₈₉	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
3. C₈₉×#3165	C₂	direct (N × H)
4. C₂×#501×C₈₉	C₂	semidirect
5. C₂×#500×C₈₉	C₂	semidirect
6. C₂×#498×C₈₉	C₂	semidirect
30. #501×C₈₉	C₂²	semidirect
31. #500×C₈₉	C₂²	semidirect
32. #498×C₈₉	C₂²	semidirect
49. C₂×M₁₆×C₈₉	Q₈	semidirect
123. M₁₆×C₈₉	C₂×Q₈	semidirect
124. C₂×C₈×C₈₉	C₂×Q₈	semidirect
180. C₈×C₈₉	C₂²×Q₈	semidirect
199. C₂×#3165	C₈₉	direct (N × H)
201. C₂×C₈₉	#3165	direct (N × H)
207. #3165	C₂×C₈₉	direct (N × H)
208. C₂×#501	C₂×C₈₉	semidirect
209. C₂×#500	C₂×C₈₉	semidirect
210. C₂×#498	C₂×C₈₉	semidirect
213. C₈₉	C₂×#3165	direct (N × H)
235. #501	C₂²×C₈₉	semidirect
236. #500	C₂²×C₈₉	semidirect
237. #498	C₂²×C₈₉	semidirect
254. C₂×M₁₆	Q₈×C₈₉	semidirect
328. M₁₆	C₂×Q₈×C₈₉	semidirect
329. C₂×C₈	C₂×Q₈×C₈₉	semidirect
385. C₈	C₂²×Q₈×C₈₉	semidirect
405. C₂	C₈₉×#3165	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×#3165	2
C₂×#501×C₈₉	2
C₂×#500×C₈₉	2
C₂×#498×C₈₉	2
C₂×#428×C₈₉	2
C₂×C₄×M₁₆×C₈₉	2
C₂×#3165	89

Subgroups — C2×C89×#3165 #36438509

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3165 #36438509