Subgroups — C₂×C₈₉×#3078 #36438329

Order 22784.

2098 conjugacy classes of subgroups.

The subgroup lattice has 474 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×#3078	C₂	1	16	yes	maximal, direct product factor
4	C₂×#479×C₈₉	C₂	1	4	yes	maximal, complemented
5	C₂×#477×C₈₉	C₂	1	2	yes	maximal, complemented
6	C₂×#475×C₈₉	C₂	1	2	yes	maximal, complemented
7	C₂×#428×C₈₉	C₂	1	2	yes	maximal
8	C₂×#423×C₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#419×C₈₉	C₂	1	2	yes	maximal
10	C₂³×Q₈×C₈₉	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₈₉	C₂²	1	2	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	C₂²	1	2	yes	—
17	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
18	C₂×C₂²⋊Q₈×C₈₉	—	2	1	no	—
19	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
20	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
21	C₂×C₄×Q₈×C₈₉	—	2	4	no	—
22	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
23	C₂×C₄×Q₈×C₈₉	—	2	1	no	—
24	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
25	C₂×C₄×Q₈×C₈₉	C₂²	1	2	yes	—
26	C₂×C₄×Q₈×C₈₉	C₂²	1	8	yes	—
27	C₂×#117×C₈₉	C₂²	1	2	yes	—
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	#479×C₈₉	C₂²	1	32	yes	complemented
33	#477×C₈₉	C₂²	1	16	yes	complemented
34	#475×C₈₉	C₂²	1	16	yes	complemented
35	#428×C₈₉	C₂²	1	16	yes	—
36	C₂×#107×C₈₉	C₂²	1	4	yes	—
37	C₂×#106×C₈₉	C₂²	1	4	yes	—
38	#423×C₈₉	C₂²	1	8	yes	—
39	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
40	C₂×C₄⋊C₈×C₈₉	C₂²	1	2	yes	—
41	#419×C₈₉	C₂²	1	16	yes	—
42	C₂×#103×C₈₉	C₂²	1	4	yes	—
43	C₂×#103×C₈₉	C₂²	1	4	yes	—
44	C₂²×Q₈×C₈₉	—	2	2	no	—
45	C₂²×Q₈×C₈₉	—	2	1	no	—
46	C₂²×Q₈×C₈₉	—	2	1	no	—
47	C₂²×Q₈×C₈₉	—	2	1	no	—
48	C₂²×Q₈×C₈₉	—	2	4	no	—
49	C₂²×Q₈×C₈₉	Q₈	1	8	yes	complemented
50	C₂²×Q₈×C₈₉	C₂³	1	2	yes	—
51	C₂²×Q₈×C₈₉	C₂³	1	4	yes	—
52	C₂³×C₄×C₈₉	—	2	1	no	—
53	C₂³×C₄×C₈₉	—	2	1	no	—
54	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
55	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
56	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
57	C₂×M₁₆×C₈₉	—	2	8	no	—
58	C₂×M₁₆×C₈₉	C₂³	1	8	yes	—
59	C₂²×C₈×C₈₉	C₂³	1	4	yes	—
60	C₄⋊Q₈×C₈₉	—	2	2	no	—
61	C₄⋊Q₈×C₈₉	—	2	2	no	—
62	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
63	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
64	#125×C₈₉	—	2	4	no	—
65	#125×C₈₉	C₂³	1	8	yes	—
66	C₂²⋊Q₈×C₈₉	—	2	8	no	—
67	C₂²⋊Q₈×C₈₉	—	2	8	no	—
68	C₄×Q₈×C₈₉	—	2	8	no	—
69	C₄×Q₈×C₈₉	—	2	8	no	—
70	C₄×Q₈×C₈₉	—	2	4	no	—
71	C₄×Q₈×C₈₉	—	2	4	no	—
72	C₄×Q₈×C₈₉	—	2	8	no	—
73	C₄×Q₈×C₈₉	—	2	8	no	—
74	C₄×Q₈×C₈₉	C₂³	1	8	yes	—
75	C₄×Q₈×C₈₉	C₂³	1	8	yes	—
76	#117×C₈₉	—	2	4	no	—
77	#117×C₈₉	C₂³	1	8	yes	—
78	C₂×C₄⋊C₄×C₈₉	—	2	8	no	—
79	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
80	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
81	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
82	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
83	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
84	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
85	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
86	C₂×C₄⋊C₄×C₈₉	—	2	1	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	2	yes	—
90	C₂×C₄⋊C₄×C₈₉	C₂³	1	4	yes	—
91	C₂×C₂²⋊C₄×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	—	2	2	no	—
96	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
97	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
98	C₂×C₄²×C₈₉	D₄	1	4	yes	—
99	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
100	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
101	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
102	#107×C₈₉	C₂³	1	16	yes	—
103	#106×C₈₉	C₂³	1	16	yes	—
104	C₄⋊C₈×C₈₉	C₂³	1	8	yes	—
105	C₄⋊C₈×C₈₉	C₂³	1	8	yes	—
106	#103×C₈₉	C₂³	1	16	yes	—
107	#103×C₈₉	C₂³	1	16	yes	—
108	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
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	4	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	—	4	4	no	—
119	C₂×Q₈×C₈₉	C₂×Q₈	1	8	yes	—
120	C₂×Q₈×C₈₉	C₂×Q₈	1	16	yes	complemented
121	C₂×Q₈×C₈₉	C₂⁴	1	4	yes	—
122	C₂²×C₄×C₈₉	—	2	8	no	—
123	C₂²×C₄×C₈₉	—	2	4	no	—
124	C₂²×C₄×C₈₉	—	2	2	no	—
125	C₂²×C₄×C₈₉	—	2	4	no	—
126	C₂²×C₄×C₈₉	—	2	4	no	—
127	C₂²×C₄×C₈₉	—	2	2	no	—
128	C₂²×C₄×C₈₉	—	2	2	no	—
129	C₂²×C₄×C₈₉	—	2	2	no	—
130	C₂²×C₄×C₈₉	—	4	2	no	—
131	C₂²×C₄×C₈₉	—	4	4	no	—
132	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
133	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
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	2	yes	—
138	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₈₉	C₂×Q₈	1	4	yes	—
141	C₂²×C₄×C₈₉	#41	1	4	yes	—
142	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
143	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
144	M₁₆×C₈₉	—	2	16	no	—
145	C₂×C₈×C₈₉	—	2	8	no	—
146	C₂×C₈×C₈₉	C₂⁴	1	8	yes	—
147	C₄⋊C₄×C₈₉	—	2	8	no	—
148	C₄⋊C₄×C₈₉	—	2	8	no	—
149	C₄⋊C₄×C₈₉	—	2	4	no	—
150	C₄⋊C₄×C₈₉	—	2	2	no	—
151	C₄⋊C₄×C₈₉	—	2	8	no	—
152	C₄⋊C₄×C₈₉	—	2	8	no	—
153	C₄⋊C₄×C₈₉	—	2	4	no	—
154	C₄⋊C₄×C₈₉	—	2	4	no	—
155	C₄⋊C₄×C₈₉	—	2	2	no	—
156	C₄⋊C₄×C₈₉	—	2	2	no	—
157	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
158	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
159	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
160	C₂²⋊C₄×C₈₉	—	2	8	no	—
161	C₄²×C₈₉	—	2	8	no	—
162	C₄²×C₈₉	—	2	8	no	—
163	C₄²×C₈₉	—	2	8	no	—
164	C₄²×C₈₉	C₂×D₄	1	4	yes	—
165	C₄²×C₈₉	C₂×D₄	1	4	yes	—
166	C₄²×C₈₉	C₂×D₄	1	2	yes	—
167	C₄²×C₈₉	C₂×D₄	1	2	yes	—
168	C₄²×C₈₉	C₂⁴	1	2	yes	—
169	C₄²×C₈₉	C₂⁴	1	2	yes	—
170	C₂³×C₈₉	—	2	2	no	—
171	C₂³×C₈₉	—	2	2	no	—
172	C₂³×C₈₉	C₂²⋊Q₈	1	4	yes	—
173	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
174	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
175	Q₈×C₈₉	—	2	8	no	—
176	Q₈×C₈₉	—	2	16	no	—
177	Q₈×C₈₉	—	2	4	no	—
178	Q₈×C₈₉	—	4	8	no	—
179	Q₈×C₈₉	—	4	8	no	—
180	Q₈×C₈₉	C₂²×Q₈	1	8	yes	—
181	C₂×C₄×C₈₉	—	2	8	no	—
182	C₂×C₄×C₈₉	—	2	16	no	—
183	C₂×C₄×C₈₉	—	2	2	no	—
184	C₂×C₄×C₈₉	—	2	2	no	—
185	C₂×C₄×C₈₉	—	2	2	no	—
186	C₂×C₄×C₈₉	—	2	4	no	—
187	C₂×C₄×C₈₉	—	2	8	no	—
188	C₂×C₄×C₈₉	—	2	4	no	—
189	C₂×C₄×C₈₉	—	2	4	no	—
190	C₂×C₄×C₈₉	—	2	4	no	—
191	C₂×C₄×C₈₉	—	4	4	no	—
192	C₂×C₄×C₈₉	—	4	8	no	—
193	C₂×C₄×C₈₉	—	4	4	no	—
194	C₂×C₄×C₈₉	—	4	4	no	—
195	C₂×C₄×C₈₉	—	4	4	no	—
196	C₂×C₄×C₈₉	C₂²⋊Q₈	1	8	yes	—
197	C₂×C₄×C₈₉	C₂²⋊Q₈	1	4	yes	—
198	C₂×C₄×C₈₉	#137	1	4	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	2	yes	—
202	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
203	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
205	C₂×C₄×C₈₉	C₂²×Q₈	1	2	yes	—
206	C₂×C₄×C₈₉	C₂²×Q₈	1	4	yes	—
207	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
208	C₂×C₄×C₈₉	C₂×#41	1	4	yes	—
209	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
210	C₈×C₈₉	—	2	8	no	—
211	C₂²×C₈₉	—	2	8	no	—
212	C₂²×C₈₉	—	2	2	no	—
213	C₂²×C₈₉	—	2	2	no	—
214	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	4	yes	—
215	C₂²×C₈₉	C₂×C₂²⋊Q₈	1	2	yes	—
216	C₂²×C₈₉	C₂×#137	1	2	yes	—
217	C₂²×C₈₉	#575	1	2	yes	—
218	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
219	C₄×C₈₉	—	2	8	no	—
220	C₄×C₈₉	—	2	2	no	—
221	C₄×C₈₉	—	2	2	no	—
222	C₄×C₈₉	—	2	2	no	—
223	C₄×C₈₉	—	4	4	no	—
224	C₄×C₈₉	—	4	4	no	—
225	C₄×C₈₉	—	4	4	no	—
226	C₄×C₈₉	C₂×C₂²⋊Q₈	1	4	yes	—
227	C₄×C₈₉	C₂×C₂²⋊Q₈	1	2	yes	—
228	C₄×C₈₉	C₂×#137	1	4	yes	—
229	C₄×C₈₉	C₂³×Q₈	1	1	yes	characteristic
230	C₄×C₈₉	C₂²×#41	1	1	yes	characteristic
231	C₂×#3078	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
232	C₂×C₈₉	—	2	4	no	—
233	C₂×C₈₉	#3078	1	4	yes	direct product factor
234	C₂×C₈₉	C₂²×C₂²⋊Q₈	1	1	yes	characteristic
235	C₂×C₈₉	C₂²×#137	1	1	yes	characteristic
236	C₂×C₈₉	C₂×#575	1	1	yes	characteristic
237	C₂×#533	C₂×C₈₉	1	1	yes	characteristic
238	C₂²×C₄×Q₈	C₂×C₈₉	1	1	yes	characteristic
239	#3078	C₂×C₈₉	1	16	yes	direct product factor
240	C₂×#479	C₂×C₈₉	1	4	yes	complemented
241	C₂×#477	C₂×C₈₉	1	2	yes	complemented
242	C₂×#475	C₂×C₈₉	1	2	yes	complemented
243	C₂×#428	C₂×C₈₉	1	2	yes	—
244	C₂×#423	C₂×C₈₉	1	1	yes	characteristic
245	C₂×#419	C₂×C₈₉	1	2	yes	—
246	C₈₉	C₂×#3078	1	1	yes	characteristic, direct product factor, Sylow
247	C₂³×Q₈	C₂²×C₈₉	1	1	yes	characteristic
248	C₂²×M₁₆	C₂²×C₈₉	1	2	yes	—
249	#533	—	2	4	no	—
250	#533	C₂²×C₈₉	1	8	yes	—
251	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
252	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
253	C₂×#125	C₂²×C₈₉	1	2	yes	—
254	C₂×C₂²⋊Q₈	—	2	1	no	—
255	C₂×C₂²⋊Q₈	—	2	1	no	—
256	C₂×C₄×Q₈	—	2	1	no	—
257	C₂×C₄×Q₈	—	2	1	no	—
258	C₂×C₄×Q₈	—	2	4	no	—
259	C₂×C₄×Q₈	—	2	1	no	—
260	C₂×C₄×Q₈	—	2	1	no	—
261	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
262	C₂×C₄×Q₈	C₂²×C₈₉	1	2	yes	—
263	C₂×C₄×Q₈	C₂²×C₈₉	1	8	yes	—
264	C₂×#117	C₂²×C₈₉	1	2	yes	—
265	C₂²×C₄⋊C₄	—	2	1	no	—
266	C₂²×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
267	C₂²×C₄²	—	2	1	no	—
268	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
269	#479	C₂²×C₈₉	1	32	yes	complemented
270	#477	C₂²×C₈₉	1	16	yes	complemented
271	#475	C₂²×C₈₉	1	16	yes	complemented
272	#428	C₂²×C₈₉	1	16	yes	—
273	C₂×#107	C₂²×C₈₉	1	4	yes	—
274	C₂×#106	C₂²×C₈₉	1	4	yes	—
275	#423	C₂²×C₈₉	1	8	yes	—
276	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
277	C₂×C₄⋊C₈	C₂²×C₈₉	1	2	yes	—
278	#419	C₂²×C₈₉	1	16	yes	—
279	C₂×#103	C₂²×C₈₉	1	4	yes	—
280	C₂×#103	C₂²×C₈₉	1	4	yes	—
281	C₂²×Q₈	—	2	1	no	—
282	C₂²×Q₈	—	2	2	no	—
283	C₂²×Q₈	—	2	1	no	—
284	C₂²×Q₈	—	2	1	no	—
285	C₂²×Q₈	—	2	4	no	—
286	C₂²×Q₈	Q₈×C₈₉	1	8	yes	complemented
287	C₂²×Q₈	C₂³×C₈₉	1	2	yes	—
288	C₂²×Q₈	C₂³×C₈₉	1	4	yes	—
289	C₂³×C₄	—	2	1	no	—
290	C₂³×C₄	—	2	1	no	—
291	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
292	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
293	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
294	C₂×M₁₆	—	2	8	no	—
295	C₂×M₁₆	C₂³×C₈₉	1	8	yes	—
296	C₂²×C₈	C₂³×C₈₉	1	4	yes	—
297	C₄⋊Q₈	—	2	2	no	—
298	C₄⋊Q₈	—	2	2	no	—
299	C₄⋊Q₈	C₂³×C₈₉	1	4	yes	—
300	C₄⋊Q₈	C₂³×C₈₉	1	4	yes	—
301	#125	—	2	4	no	—
302	#125	C₂³×C₈₉	1	8	yes	—
303	C₂²⋊Q₈	—	2	8	no	—
304	C₂²⋊Q₈	—	2	8	no	—
305	C₄×Q₈	—	2	8	no	—
306	C₄×Q₈	—	2	8	no	—
307	C₄×Q₈	—	2	4	no	—
308	C₄×Q₈	—	2	4	no	—
309	C₄×Q₈	—	2	8	no	—
310	C₄×Q₈	—	2	8	no	—
311	C₄×Q₈	C₂³×C₈₉	1	8	yes	—
312	C₄×Q₈	C₂³×C₈₉	1	8	yes	—
313	#117	—	2	4	no	—
314	#117	C₂³×C₈₉	1	8	yes	—
315	C₂×C₄⋊C₄	—	2	2	no	—
316	C₂×C₄⋊C₄	—	2	2	no	—
317	C₂×C₄⋊C₄	—	2	8	no	—
318	C₂×C₄⋊C₄	—	2	1	no	—
319	C₂×C₄⋊C₄	—	2	2	no	—
320	C₂×C₄⋊C₄	—	2	2	no	—
321	C₂×C₄⋊C₄	—	2	2	no	—
322	C₂×C₄⋊C₄	—	2	1	no	—
323	C₂×C₄⋊C₄	—	2	1	no	—
324	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
325	C₂×C₄⋊C₄	C₂³×C₈₉	1	4	yes	—
326	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
327	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
328	C₂×C₂²⋊C₄	—	2	2	no	—
329	C₂×C₄²	—	2	2	no	—
330	C₂×C₄²	—	2	2	no	—
331	C₂×C₄²	—	2	8	no	—
332	C₂×C₄²	—	2	2	no	—
333	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
334	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
335	C₂×C₄²	D₄×C₈₉	1	4	yes	—
336	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
337	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
338	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
339	#107	C₂³×C₈₉	1	16	yes	—
340	#106	C₂³×C₈₉	1	16	yes	—
341	C₄⋊C₈	C₂³×C₈₉	1	8	yes	—
342	C₄⋊C₈	C₂³×C₈₉	1	8	yes	—
343	#103	C₂³×C₈₉	1	16	yes	—
344	#103	C₂³×C₈₉	1	16	yes	—
345	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
346	C₂×Q₈	—	2	8	no	—
347	C₂×Q₈	—	2	16	no	—
348	C₂×Q₈	—	2	4	no	—
349	C₂×Q₈	—	2	2	no	—
350	C₂×Q₈	—	2	4	no	—
351	C₂×Q₈	—	2	4	no	—
352	C₂×Q₈	—	2	4	no	—
353	C₂×Q₈	—	2	8	no	—
354	C₂×Q₈	—	4	4	no	—
355	C₂×Q₈	—	4	4	no	—
356	C₂×Q₈	C₂×Q₈×C₈₉	1	8	yes	—
357	C₂×Q₈	C₂×Q₈×C₈₉	1	16	yes	complemented
358	C₂×Q₈	C₂⁴×C₈₉	1	4	yes	—
359	C₂²×C₄	—	2	8	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	4	no	—
362	C₂²×C₄	—	2	4	no	—
363	C₂²×C₄	—	2	4	no	—
364	C₂²×C₄	—	2	2	no	—
365	C₂²×C₄	—	2	2	no	—
366	C₂²×C₄	—	2	2	no	—
367	C₂²×C₄	—	4	2	no	—
368	C₂²×C₄	—	4	4	no	—
369	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
370	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
371	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
372	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
373	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
374	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
375	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
376	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
377	C₂²×C₄	C₂×Q₈×C₈₉	1	4	yes	—
378	C₂²×C₄	#41×C₈₉	1	4	yes	—
379	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
380	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
381	M₁₆	—	2	16	no	—
382	C₂×C₈	—	2	8	no	—
383	C₂×C₈	C₂⁴×C₈₉	1	8	yes	—
384	C₄⋊C₄	—	2	8	no	—
385	C₄⋊C₄	—	2	8	no	—
386	C₄⋊C₄	—	2	4	no	—
387	C₄⋊C₄	—	2	2	no	—
388	C₄⋊C₄	—	2	8	no	—
389	C₄⋊C₄	—	2	8	no	—
390	C₄⋊C₄	—	2	4	no	—
391	C₄⋊C₄	—	2	4	no	—
392	C₄⋊C₄	—	2	2	no	—
393	C₄⋊C₄	—	2	2	no	—
394	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
395	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
396	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
397	C₂²⋊C₄	—	2	8	no	—
398	C₄²	—	2	8	no	—
399	C₄²	—	2	8	no	—
400	C₄²	—	2	8	no	—
401	C₄²	C₂×D₄×C₈₉	1	4	yes	—
402	C₄²	C₂×D₄×C₈₉	1	4	yes	—
403	C₄²	C₂×D₄×C₈₉	1	2	yes	—
404	C₄²	C₂×D₄×C₈₉	1	2	yes	—
405	C₄²	C₂⁴×C₈₉	1	2	yes	—
406	C₄²	C₂⁴×C₈₉	1	2	yes	—
407	C₂³	—	2	2	no	—
408	C₂³	—	2	2	no	—
409	C₂³	C₂²⋊Q₈×C₈₉	1	4	yes	—
410	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
411	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
412	Q₈	—	2	8	no	—
413	Q₈	—	2	16	no	—
414	Q₈	—	2	4	no	—
415	Q₈	—	4	8	no	—
416	Q₈	—	4	8	no	—
417	Q₈	C₂²×Q₈×C₈₉	1	8	yes	—
418	C₂×C₄	—	2	8	no	—
419	C₂×C₄	—	2	16	no	—
420	C₂×C₄	—	2	2	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	2	8	no	—
425	C₂×C₄	—	2	4	no	—
426	C₂×C₄	—	2	4	no	—
427	C₂×C₄	—	2	4	no	—
428	C₂×C₄	—	4	4	no	—
429	C₂×C₄	—	4	8	no	—
430	C₂×C₄	—	4	4	no	—
431	C₂×C₄	—	4	4	no	—
432	C₂×C₄	—	4	4	no	—
433	C₂×C₄	C₂²⋊Q₈×C₈₉	1	8	yes	—
434	C₂×C₄	C₂²⋊Q₈×C₈₉	1	4	yes	—
435	C₂×C₄	#137×C₈₉	1	4	yes	—
436	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
437	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
438	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
439	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
440	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
441	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
442	C₂×C₄	C₂²×Q₈×C₈₉	1	2	yes	—
443	C₂×C₄	C₂²×Q₈×C₈₉	1	4	yes	—
444	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
445	C₂×C₄	C₂×#41×C₈₉	1	4	yes	—
446	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
447	C₈	—	2	8	no	—
448	C₂²	—	2	2	no	—
449	C₂²	—	2	2	no	—
450	C₂²	—	2	8	no	—
451	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	4	yes	—
452	C₂²	C₂×C₂²⋊Q₈×C₈₉	1	2	yes	—
453	C₂²	C₂×#137×C₈₉	1	2	yes	—
454	C₂²	#575×C₈₉	1	2	yes	—
455	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
456	C₄	—	2	2	no	—
457	C₄	—	2	2	no	—
458	C₄	—	2	2	no	—
459	C₄	—	2	8	no	—
460	C₄	—	4	4	no	—
461	C₄	—	4	4	no	—
462	C₄	—	4	4	no	—
463	C₄	C₂×C₂²⋊Q₈×C₈₉	1	4	yes	—
464	C₄	C₂×C₂²⋊Q₈×C₈₉	1	2	yes	—
465	C₄	C₂×#137×C₈₉	1	4	yes	—
466	C₄	C₂³×Q₈×C₈₉	1	1	yes	characteristic
467	C₄	C₂²×#41×C₈₉	1	1	yes	characteristic
468	C₂	—	2	4	no	—
469	C₂	C₈₉×#3078	1	4	yes	direct product factor
470	C₂	C₂²×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
471	C₂	C₂²×#137×C₈₉	1	1	yes	characteristic
472	C₂	C₂×#575×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×#3078	C₂	direct (N × H)
4. C₂×#479×C₈₉	C₂	semidirect
5. C₂×#477×C₈₉	C₂	semidirect
6. C₂×#475×C₈₉	C₂	semidirect
32. #479×C₈₉	C₂²	semidirect
33. #477×C₈₉	C₂²	semidirect
34. #475×C₈₉	C₂²	semidirect
49. C₂²×Q₈×C₈₉	Q₈	semidirect
120. C₂×Q₈×C₈₉	C₂×Q₈	semidirect
231. C₂×#3078	C₈₉	direct (N × H)
233. C₂×C₈₉	#3078	direct (N × H)
239. #3078	C₂×C₈₉	direct (N × H)
240. C₂×#479	C₂×C₈₉	semidirect
241. C₂×#477	C₂×C₈₉	semidirect
242. C₂×#475	C₂×C₈₉	semidirect
246. C₈₉	C₂×#3078	direct (N × H)
269. #479	C₂²×C₈₉	semidirect
270. #477	C₂²×C₈₉	semidirect
271. #475	C₂²×C₈₉	semidirect
286. C₂²×Q₈	Q₈×C₈₉	semidirect
357. C₂×Q₈	C₂×Q₈×C₈₉	semidirect
469. C₂	C₈₉×#3078	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×#3078	2
C₂×#479×C₈₉	2
C₂×#477×C₈₉	2
C₂×#475×C₈₉	2
C₂×#428×C₈₉	2
C₂×#423×C₈₉	2
C₂×#419×C₈₉	2
C₂×#3078	89

Subgroups — C2×C89×#3078 #36438329

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3078 #36438329