Subgroups — C₂×C₈₉×#3106 #36438381

Order 22784.

2450 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#532×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×#41×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₉×#3106	C₂	1	16	yes	maximal, direct product factor
4	C₂×#482×C₈₉	C₂	1	2	yes	maximal, complemented
5	C₂×#477×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#476×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#460×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#459×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#449×C₈₉	C₂	1	2	yes	maximal, complemented
10	C₂×C₄×SD₁₆×C₈₉	C₂	1	2	yes	maximal, complemented
11	C₂×C₄×D₈×C₈₉	C₂	1	2	yes	maximal, complemented
12	C₂×#431×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂²×#41×C₈₉	—	2	1	no	—
14	C₂²×#41×C₈₉	C₂²	1	1	yes	characteristic
15	C₂²×SD₁₆×C₈₉	C₂²	1	2	yes	—
16	C₂²×D₈×C₈₉	C₂²	1	2	yes	—
17	#532×C₈₉	—	2	4	no	—
18	#532×C₈₉	C₂²	1	8	yes	complemented
19	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
20	C₂×#127×C₈₉	C₂²	1	1	yes	characteristic
21	C₂×#124×C₈₉	—	2	1	no	—
22	C₂×#121×C₈₉	—	2	1	no	—
23	C₂×#121×C₈₉	C₂²	1	2	yes	—
24	C₄×#41×C₈₉	—	2	4	no	—
25	C₄×#41×C₈₉	C₂²	1	8	yes	complemented
26	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
28	C₂×C₄×D₄×C₈₉	—	2	1	no	—
29	C₂×C₄×D₄×C₈₉	—	2	1	no	—
30	C₂×C₄×D₄×C₈₉	—	2	1	no	—
31	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
32	C₂×C₄×D₄×C₈₉	C₂²	1	1	yes	characteristic
33	C₂×C₄×D₄×C₈₉	C₂²	1	2	yes	—
34	C₂×#117×C₈₉	—	2	1	no	—
35	C₂×#117×C₈₉	C₂²	1	1	yes	characteristic
36	C₂²×C₄²×C₈₉	—	2	1	no	—
37	C₂²×C₄²×C₈₉	C₂²	1	1	yes	characteristic
38	#482×C₈₉	C₂²	1	16	yes	complemented
39	#477×C₈₉	C₂²	1	8	yes	complemented
40	#476×C₈₉	C₂²	1	8	yes	complemented
41	#460×C₈₉	C₂²	1	8	yes	complemented
42	#459×C₈₉	C₂²	1	8	yes	complemented
43	#449×C₈₉	C₂²	1	16	yes	complemented
44	C₄×SD₁₆×C₈₉	C₂²	1	16	yes	complemented
45	C₄×D₈×C₈₉	C₂²	1	16	yes	complemented
46	#431×C₈₉	C₂²	1	8	yes	complemented
47	C₂×#107×C₈₉	C₂²	1	2	yes	—
48	C₂×#106×C₈₉	C₂²	1	2	yes	—
49	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
50	C₂×C₄⋊C₈×C₈₉	C₂²	1	1	yes	characteristic
51	C₂×#103×C₈₉	C₂²	1	2	yes	—
52	C₂×#102×C₈₉	C₂²	1	2	yes	—
53	C₂×#102×C₈₉	C₂²	1	2	yes	—
54	C₂×#102×C₈₉	C₂²	1	2	yes	—
55	C₂×C₂²⋊C₈×C₈₉	C₂²	1	2	yes	—
56	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
57	C₂×#41×C₈₉	—	2	4	no	—
58	C₂×#41×C₈₉	—	2	4	no	—
59	C₂×#41×C₈₉	—	2	2	no	—
60	C₂×#41×C₈₉	—	2	8	no	—
61	C₂×#41×C₈₉	—	4	8	no	—
62	C₂×#41×C₈₉	C₂³	1	4	yes	—
63	C₂²×Q₈×C₈₉	—	2	1	no	—
64	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
65	C₂²×D₄×C₈₉	—	2	1	no	—
66	C₂²×D₄×C₈₉	—	2	2	no	—
67	C₂²×D₄×C₈₉	—	2	1	no	—
68	C₂²×D₄×C₈₉	—	2	1	no	—
69	C₂²×D₄×C₈₉	C₂³	1	2	yes	—
70	C₂²×D₄×C₈₉	C₂³	1	1	yes	characteristic
71	C₂³×C₄×C₈₉	—	2	1	no	—
72	C₂³×C₄×C₈₉	—	2	1	no	—
73	C₂³×C₄×C₈₉	—	2	1	no	—
74	C₂³×C₄×C₈₉	—	2	2	no	—
75	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
76	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
77	C₂³×C₄×C₈₉	C₂³	1	1	yes	characteristic
78	C₂×SD₁₆×C₈₉	—	2	8	no	—
79	C₂×SD₁₆×C₈₉	C₂³	1	8	yes	—
80	C₂×D₈×C₈₉	—	2	8	no	—
81	C₂×D₈×C₈₉	C₂³	1	8	yes	—
82	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
83	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
84	C₄⋊Q₈×C₈₉	—	2	2	no	—
85	C₄⋊Q₈×C₈₉	C₂³	1	4	yes	—
86	#127×C₈₉	—	2	2	no	—
87	#127×C₈₉	C₂³	1	4	yes	—
88	#124×C₈₉	—	2	8	no	—
89	#121×C₈₉	—	2	8	no	—
90	#121×C₈₉	—	2	4	no	—
91	#121×C₈₉	C₂³	1	8	yes	—
92	C₄×Q₈×C₈₉	—	2	2	no	—
93	C₄×Q₈×C₈₉	—	2	2	no	—
94	C₄×Q₈×C₈₉	C₂³	1	4	yes	—
95	C₄×Q₈×C₈₉	C₂³	1	4	yes	—
96	C₄×D₄×C₈₉	—	2	8	no	—
97	C₄×D₄×C₈₉	—	2	8	no	—
98	C₄×D₄×C₈₉	—	2	8	no	—
99	C₄×D₄×C₈₉	—	2	2	no	—
100	C₄×D₄×C₈₉	—	2	2	no	—
101	C₄×D₄×C₈₉	—	2	4	no	—
102	C₄×D₄×C₈₉	C₂³	1	4	yes	—
103	C₄×D₄×C₈₉	C₂³	1	8	yes	—
104	C₄×D₄×C₈₉	C₂³	1	4	yes	—
105	#117×C₈₉	—	2	2	no	—
106	#117×C₈₉	—	2	8	no	—
107	#117×C₈₉	C₂³	1	4	yes	—
108	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
109	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
110	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
111	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
112	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
113	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
114	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
115	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
116	C₂×C₂²⋊C₄×C₈₉	—	2	1	no	—
117	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
118	C₂×C₂²⋊C₄×C₈₉	—	2	2	no	—
119	C₂×C₄²×C₈₉	—	2	1	no	—
120	C₂×C₄²×C₈₉	—	2	1	no	—
121	C₂×C₄²×C₈₉	—	2	1	no	—
122	C₂×C₄²×C₈₉	—	2	8	no	—
123	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
124	C₂×C₄²×C₈₉	D₄	1	4	yes	—
125	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
126	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
127	C₂×C₄²×C₈₉	C₂³	1	4	yes	—
128	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
129	#107×C₈₉	C₂³	1	8	yes	—
130	#106×C₈₉	C₂³	1	8	yes	—
131	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
132	C₄⋊C₈×C₈₉	C₂³	1	4	yes	—
133	#103×C₈₉	C₂³	1	8	yes	—
134	#102×C₈₉	C₂³	1	8	yes	—
135	#102×C₈₉	C₂³	1	8	yes	—
136	#102×C₈₉	C₂³	1	8	yes	—
137	C₂²⋊C₈×C₈₉	C₂³	1	8	yes	—
138	C₄×C₈×C₈₉	C₂³	1	8	yes	—
139	C₂⁴×C₈₉	—	2	1	no	—
140	C₂⁴×C₈₉	—	2	2	no	—
141	C₂⁴×C₈₉	C₂×D₄	1	1	yes	characteristic
142	#41×C₈₉	—	2	16	no	—
143	#41×C₈₉	—	2	8	no	—
144	#41×C₈₉	—	2	8	no	—
145	#41×C₈₉	—	4	32	no	—
146	C₂×Q₈×C₈₉	—	2	1	no	—
147	C₂×Q₈×C₈₉	—	2	4	no	—
148	C₂×Q₈×C₈₉	—	2	2	no	—
149	C₂×Q₈×C₈₉	—	4	4	no	—
150	C₂×Q₈×C₈₉	#41	1	4	yes	complemented
151	C₂×Q₈×C₈₉	C₂⁴	1	2	yes	—
152	C₂×D₄×C₈₉	—	2	8	no	—
153	C₂×D₄×C₈₉	—	2	4	no	—
154	C₂×D₄×C₈₉	—	2	8	no	—
155	C₂×D₄×C₈₉	—	2	1	no	—
156	C₂×D₄×C₈₉	—	2	2	no	—
157	C₂×D₄×C₈₉	—	2	4	no	—
158	C₂×D₄×C₈₉	—	2	4	no	—
159	C₂×D₄×C₈₉	—	2	2	no	—
160	C₂×D₄×C₈₉	—	2	4	no	—
161	C₂×D₄×C₈₉	—	2	4	no	—
162	C₂×D₄×C₈₉	—	4	4	no	—
163	C₂×D₄×C₈₉	—	4	8	no	—
164	C₂×D₄×C₈₉	—	4	4	no	—
165	C₂×D₄×C₈₉	#41	1	4	yes	complemented
166	C₂×D₄×C₈₉	C₂⁴	1	4	yes	—
167	C₂×D₄×C₈₉	C₂⁴	1	2	yes	—
168	C₂²×C₄×C₈₉	—	2	8	no	—
169	C₂²×C₄×C₈₉	—	2	1	no	—
170	C₂²×C₄×C₈₉	—	2	4	no	—
171	C₂²×C₄×C₈₉	—	2	4	no	—
172	C₂²×C₄×C₈₉	—	2	4	no	—
173	C₂²×C₄×C₈₉	—	2	2	no	—
174	C₂²×C₄×C₈₉	—	2	2	no	—
175	C₂²×C₄×C₈₉	—	2	1	no	—
176	C₂²×C₄×C₈₉	—	2	1	no	—
177	C₂²×C₄×C₈₉	—	2	2	no	—
178	C₂²×C₄×C₈₉	—	2	4	no	—
179	C₂²×C₄×C₈₉	—	2	4	no	—
180	C₂²×C₄×C₈₉	—	2	8	no	—
181	C₂²×C₄×C₈₉	—	4	4	no	—
182	C₂²×C₄×C₈₉	—	4	4	no	—
183	C₂²×C₄×C₈₉	—	4	8	no	—
184	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
185	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
186	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
187	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
188	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
189	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
190	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
191	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
192	C₂²×C₄×C₈₉	C₂⁴	1	2	yes	—
193	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
194	SD₁₆×C₈₉	—	2	16	no	—
195	D₈×C₈₉	—	2	16	no	—
196	C₂×C₈×C₈₉	—	2	4	no	—
197	C₂×C₈×C₈₉	—	2	4	no	—
198	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
199	C₂×C₈×C₈₉	C₂⁴	1	4	yes	—
200	C₄⋊C₄×C₈₉	—	2	4	no	—
201	C₄⋊C₄×C₈₉	—	2	4	no	—
202	C₄⋊C₄×C₈₉	—	2	4	no	—
203	C₄⋊C₄×C₈₉	—	2	1	no	—
204	C₄⋊C₄×C₈₉	—	2	1	no	—
205	C₄⋊C₄×C₈₉	—	2	2	no	—
206	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
207	C₄⋊C₄×C₈₉	C₂⁴	1	2	yes	—
208	C₄⋊C₄×C₈₉	C₂⁴	1	4	yes	—
209	C₂²⋊C₄×C₈₉	—	2	4	no	—
210	C₂²⋊C₄×C₈₉	—	2	4	no	—
211	C₂²⋊C₄×C₈₉	—	2	4	no	—
212	C₂²⋊C₄×C₈₉	—	2	8	no	—
213	C₂²⋊C₄×C₈₉	—	2	8	no	—
214	C₄²×C₈₉	—	2	4	no	—
215	C₄²×C₈₉	—	2	4	no	—
216	C₄²×C₈₉	—	2	4	no	—
217	C₄²×C₈₉	C₂×D₄	1	4	yes	—
218	C₄²×C₈₉	C₂×D₄	1	4	yes	—
219	C₄²×C₈₉	C₂×D₄	1	2	yes	—
220	C₄²×C₈₉	C₂×D₄	1	2	yes	—
221	C₄²×C₈₉	C₂⁴	1	2	yes	—
222	C₄²×C₈₉	C₂⁴	1	2	yes	—
223	C₂³×C₈₉	—	2	2	no	—
224	C₂³×C₈₉	—	2	4	no	—
225	C₂³×C₈₉	—	2	2	no	—
226	C₂³×C₈₉	—	2	2	no	—
227	C₂³×C₈₉	—	2	2	no	—
228	C₂³×C₈₉	—	2	4	no	—
229	C₂³×C₈₉	—	4	2	no	—
230	C₂³×C₈₉	—	4	2	no	—
231	C₂³×C₈₉	—	4	4	no	—
232	C₂³×C₈₉	—	4	4	no	—
233	C₂³×C₈₉	—	4	4	no	—
234	C₂³×C₈₉	C₂²×D₄	1	2	yes	—
235	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
236	Q₈×C₈₉	—	2	4	no	—
237	Q₈×C₈₉	—	2	2	no	—
238	Q₈×C₈₉	—	4	8	no	—
239	Q₈×C₈₉	C₂×#41	1	4	yes	complemented
240	D₄×C₈₉	—	2	16	no	—
241	D₄×C₈₉	—	2	4	no	—
242	D₄×C₈₉	—	2	2	no	—
243	D₄×C₈₉	—	2	4	no	—
244	D₄×C₈₉	—	2	4	no	—
245	D₄×C₈₉	—	4	8	no	—
246	D₄×C₈₉	—	4	16	no	—
247	D₄×C₈₉	—	4	4	no	—
248	D₄×C₈₉	—	4	8	no	—
249	D₄×C₈₉	C₂×#41	1	4	yes	complemented
250	C₂×C₄×C₈₉	—	2	16	no	—
251	C₂×C₄×C₈₉	—	2	2	no	—
252	C₂×C₄×C₈₉	—	2	4	no	—
253	C₂×C₄×C₈₉	—	2	2	no	—
254	C₂×C₄×C₈₉	—	2	2	no	—
255	C₂×C₄×C₈₉	—	2	2	no	—
256	C₂×C₄×C₈₉	—	2	4	no	—
257	C₂×C₄×C₈₉	—	2	8	no	—
258	C₂×C₄×C₈₉	—	2	8	no	—
259	C₂×C₄×C₈₉	—	2	2	no	—
260	C₂×C₄×C₈₉	—	2	2	no	—
261	C₂×C₄×C₈₉	—	2	2	no	—
262	C₂×C₄×C₈₉	—	2	2	no	—
263	C₂×C₄×C₈₉	—	2	4	no	—
264	C₂×C₄×C₈₉	—	4	2	no	—
265	C₂×C₄×C₈₉	—	4	2	no	—
266	C₂×C₄×C₈₉	—	4	4	no	—
267	C₂×C₄×C₈₉	—	4	8	no	—
268	C₂×C₄×C₈₉	—	4	8	no	—
269	C₂×C₄×C₈₉	—	4	16	no	—
270	C₂×C₄×C₈₉	#135	1	4	yes	—
271	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	4	yes	—
272	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
273	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
274	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
275	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
276	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
277	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
278	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
279	C₂×C₄×C₈₉	C₂×#41	1	2	yes	—
280	C₂×C₄×C₈₉	C₂⁵	1	1	yes	characteristic
281	C₈×C₈₉	—	2	4	no	—
282	C₈×C₈₉	—	2	4	no	—
283	C₂²×C₈₉	—	2	2	no	—
284	C₂²×C₈₉	—	2	4	no	—
285	C₂²×C₈₉	—	2	2	no	—
286	C₂²×C₈₉	—	2	2	no	—
287	C₂²×C₈₉	—	4	4	no	—
288	C₂²×C₈₉	—	4	8	no	—
289	C₂²×C₈₉	—	4	2	no	—
290	C₂²×C₈₉	—	4	2	no	—
291	C₂²×C₈₉	—	4	4	no	—
292	C₂²×C₈₉	—	4	4	no	—
293	C₂²×C₈₉	—	4	4	no	—
294	C₂²×C₈₉	—	8	8	no	—
295	C₂²×C₈₉	#525	1	2	yes	—
296	C₂²×C₈₉	C₂×#135	1	2	yes	—
297	C₂²×C₈₉	C₂×AGL(1,ℤ/8)	1	2	yes	—
298	C₂²×C₈₉	C₂³×D₄	1	1	yes	characteristic
299	C₄×C₈₉	—	2	2	no	—
300	C₄×C₈₉	—	2	4	no	—
301	C₄×C₈₉	—	2	2	no	—
302	C₄×C₈₉	—	2	2	no	—
303	C₄×C₈₉	—	4	2	no	—
304	C₄×C₈₉	—	4	2	no	—
305	C₄×C₈₉	—	4	4	no	—
306	C₄×C₈₉	#525	1	2	yes	—
307	C₄×C₈₉	C₂×#135	1	4	yes	—
308	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	4	yes	—
309	C₄×C₈₉	C₂²×#41	1	1	yes	characteristic
310	C₄×C₈₉	C₂²×#41	1	1	yes	characteristic
311	C₂×#3106	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
312	C₂×C₈₉	—	4	2	no	—
313	C₂×C₈₉	—	4	4	no	—
314	C₂×C₈₉	—	8	4	no	—
315	C₂×C₈₉	#3106	1	4	yes	direct product factor
316	C₂×C₈₉	C₂×#525	1	1	yes	characteristic
317	C₂×C₈₉	C₂²×#135	1	1	yes	characteristic
318	C₂×C₈₉	C₂²×AGL(1,ℤ/8)	1	1	yes	characteristic
319	C₂×#532	C₂×C₈₉	1	1	yes	characteristic, complemented
320	C₂×C₄×#41	C₂×C₈₉	1	1	yes	characteristic, complemented
321	#3106	C₂×C₈₉	1	16	yes	direct product factor
322	C₂×#482	C₂×C₈₉	1	2	yes	complemented
323	C₂×#477	C₂×C₈₉	1	1	yes	characteristic, complemented
324	C₂×#476	C₂×C₈₉	1	1	yes	characteristic, complemented
325	C₂×#460	C₂×C₈₉	1	1	yes	characteristic, complemented
326	C₂×#459	C₂×C₈₉	1	1	yes	characteristic, complemented
327	C₂×#449	C₂×C₈₉	1	2	yes	complemented
328	C₂×C₄×SD₁₆	C₂×C₈₉	1	2	yes	complemented
329	C₂×C₄×D₈	C₂×C₈₉	1	2	yes	complemented
330	C₂×#431	C₂×C₈₉	1	1	yes	characteristic, complemented
331	C₈₉	C₂×#3106	1	1	yes	characteristic, direct product factor, Sylow
332	C₂²×#41	—	2	1	no	—
333	C₂²×#41	C₂²×C₈₉	1	1	yes	characteristic
334	C₂²×SD₁₆	C₂²×C₈₉	1	2	yes	—
335	C₂²×D₈	C₂²×C₈₉	1	2	yes	—
336	#532	—	2	4	no	—
337	#532	C₂²×C₈₉	1	8	yes	complemented
338	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
339	C₂×#127	C₂²×C₈₉	1	1	yes	characteristic
340	C₂×#124	—	2	1	no	—
341	C₂×#121	—	2	1	no	—
342	C₂×#121	C₂²×C₈₉	1	2	yes	—
343	C₄×#41	—	2	4	no	—
344	C₄×#41	C₂²×C₈₉	1	8	yes	complemented
345	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
346	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
347	C₂×C₄×D₄	—	2	1	no	—
348	C₂×C₄×D₄	—	2	1	no	—
349	C₂×C₄×D₄	—	2	1	no	—
350	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
351	C₂×C₄×D₄	C₂²×C₈₉	1	1	yes	characteristic
352	C₂×C₄×D₄	C₂²×C₈₉	1	2	yes	—
353	C₂×#117	—	2	1	no	—
354	C₂×#117	C₂²×C₈₉	1	1	yes	characteristic
355	C₂²×C₄²	—	2	1	no	—
356	C₂²×C₄²	C₂²×C₈₉	1	1	yes	characteristic
357	#482	C₂²×C₈₉	1	16	yes	complemented
358	#477	C₂²×C₈₉	1	8	yes	complemented
359	#476	C₂²×C₈₉	1	8	yes	complemented
360	#460	C₂²×C₈₉	1	8	yes	complemented
361	#459	C₂²×C₈₉	1	8	yes	complemented
362	#449	C₂²×C₈₉	1	16	yes	complemented
363	C₄×SD₁₆	C₂²×C₈₉	1	16	yes	complemented
364	C₄×D₈	C₂²×C₈₉	1	16	yes	complemented
365	#431	C₂²×C₈₉	1	8	yes	complemented
366	C₂×#107	C₂²×C₈₉	1	2	yes	—
367	C₂×#106	C₂²×C₈₉	1	2	yes	—
368	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
369	C₂×C₄⋊C₈	C₂²×C₈₉	1	1	yes	characteristic
370	C₂×#103	C₂²×C₈₉	1	2	yes	—
371	C₂×#102	C₂²×C₈₉	1	2	yes	—
372	C₂×#102	C₂²×C₈₉	1	2	yes	—
373	C₂×#102	C₂²×C₈₉	1	2	yes	—
374	C₂×C₂²⋊C₈	C₂²×C₈₉	1	2	yes	—
375	C₂×C₄×C₈	C₂²×C₈₉	1	2	yes	—
376	C₂×#41	—	2	4	no	—
377	C₂×#41	—	2	4	no	—
378	C₂×#41	—	2	2	no	—
379	C₂×#41	—	2	8	no	—
380	C₂×#41	—	4	8	no	—
381	C₂×#41	C₂³×C₈₉	1	4	yes	—
382	C₂²×Q₈	—	2	1	no	—
383	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
384	C₂²×D₄	—	2	1	no	—
385	C₂²×D₄	—	2	2	no	—
386	C₂²×D₄	—	2	1	no	—
387	C₂²×D₄	—	2	1	no	—
388	C₂²×D₄	C₂³×C₈₉	1	2	yes	—
389	C₂²×D₄	C₂³×C₈₉	1	1	yes	characteristic
390	C₂³×C₄	—	2	1	no	—
391	C₂³×C₄	—	2	2	no	—
392	C₂³×C₄	—	2	1	no	—
393	C₂³×C₄	—	2	1	no	—
394	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
395	C₂³×C₄	D₄×C₈₉	1	1	yes	characteristic
396	C₂³×C₄	C₂³×C₈₉	1	1	yes	characteristic
397	C₂×SD₁₆	—	2	8	no	—
398	C₂×SD₁₆	C₂³×C₈₉	1	8	yes	—
399	C₂×D₈	—	2	8	no	—
400	C₂×D₈	C₂³×C₈₉	1	8	yes	—
401	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
402	C₂²×C₈	C₂³×C₈₉	1	2	yes	—
403	C₄⋊Q₈	—	2	2	no	—
404	C₄⋊Q₈	C₂³×C₈₉	1	4	yes	—
405	#127	—	2	2	no	—
406	#127	C₂³×C₈₉	1	4	yes	—
407	#124	—	2	8	no	—
408	#121	—	2	8	no	—
409	#121	—	2	4	no	—
410	#121	C₂³×C₈₉	1	8	yes	—
411	C₄×Q₈	—	2	2	no	—
412	C₄×Q₈	—	2	2	no	—
413	C₄×Q₈	C₂³×C₈₉	1	4	yes	—
414	C₄×Q₈	C₂³×C₈₉	1	4	yes	—
415	C₄×D₄	—	2	8	no	—
416	C₄×D₄	—	2	8	no	—
417	C₄×D₄	—	2	8	no	—
418	C₄×D₄	—	2	2	no	—
419	C₄×D₄	—	2	2	no	—
420	C₄×D₄	—	2	4	no	—
421	C₄×D₄	C₂³×C₈₉	1	4	yes	—
422	C₄×D₄	C₂³×C₈₉	1	4	yes	—
423	C₄×D₄	C₂³×C₈₉	1	8	yes	—
424	#117	—	2	2	no	—
425	#117	—	2	8	no	—
426	#117	C₂³×C₈₉	1	4	yes	—
427	C₂×C₄⋊C₄	—	2	1	no	—
428	C₂×C₄⋊C₄	—	2	1	no	—
429	C₂×C₄⋊C₄	—	2	1	no	—
430	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
431	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
432	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
433	C₂×C₂²⋊C₄	—	2	1	no	—
434	C₂×C₂²⋊C₄	—	2	1	no	—
435	C₂×C₂²⋊C₄	—	2	1	no	—
436	C₂×C₂²⋊C₄	—	2	2	no	—
437	C₂×C₂²⋊C₄	—	2	2	no	—
438	C₂×C₄²	—	2	1	no	—
439	C₂×C₄²	—	2	1	no	—
440	C₂×C₄²	—	2	1	no	—
441	C₂×C₄²	—	2	8	no	—
442	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
443	C₂×C₄²	D₄×C₈₉	1	1	yes	characteristic
444	C₂×C₄²	D₄×C₈₉	1	4	yes	—
445	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
446	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
447	C₂×C₄²	C₂³×C₈₉	1	4	yes	—
448	#107	C₂³×C₈₉	1	8	yes	—
449	#106	C₂³×C₈₉	1	8	yes	—
450	C₄⋊C₈	C₂³×C₈₉	1	4	yes	—
451	C₄⋊C₈	C₂³×C₈₉	1	4	yes	—
452	#103	C₂³×C₈₉	1	8	yes	—
453	#102	C₂³×C₈₉	1	8	yes	—
454	#102	C₂³×C₈₉	1	8	yes	—
455	#102	C₂³×C₈₉	1	8	yes	—
456	C₂²⋊C₈	C₂³×C₈₉	1	8	yes	—
457	C₄×C₈	C₂³×C₈₉	1	8	yes	—
458	C₂⁴	—	2	1	no	—
459	C₂⁴	—	2	2	no	—
460	C₂⁴	C₂×D₄×C₈₉	1	1	yes	characteristic
461	#41	—	2	16	no	—
462	#41	—	2	8	no	—
463	#41	—	2	8	no	—
464	#41	—	4	32	no	—
465	C₂×Q₈	—	2	1	no	—
466	C₂×Q₈	—	2	4	no	—
467	C₂×Q₈	—	2	2	no	—
468	C₂×Q₈	—	4	4	no	—
469	C₂×Q₈	#41×C₈₉	1	4	yes	complemented
470	C₂×Q₈	C₂⁴×C₈₉	1	2	yes	—
471	C₂×D₄	—	2	8	no	—
472	C₂×D₄	—	2	4	no	—
473	C₂×D₄	—	2	8	no	—
474	C₂×D₄	—	2	1	no	—
475	C₂×D₄	—	2	2	no	—
476	C₂×D₄	—	2	4	no	—
477	C₂×D₄	—	2	4	no	—
478	C₂×D₄	—	2	2	no	—
479	C₂×D₄	—	2	4	no	—
480	C₂×D₄	—	2	4	no	—
481	C₂×D₄	—	4	4	no	—
482	C₂×D₄	—	4	8	no	—
483	C₂×D₄	—	4	4	no	—
484	C₂×D₄	#41×C₈₉	1	4	yes	complemented
485	C₂×D₄	C₂⁴×C₈₉	1	4	yes	—
486	C₂×D₄	C₂⁴×C₈₉	1	2	yes	—
487	C₂²×C₄	—	2	8	no	—
488	C₂²×C₄	—	2	1	no	—
489	C₂²×C₄	—	2	4	no	—
490	C₂²×C₄	—	2	4	no	—
491	C₂²×C₄	—	2	4	no	—
492	C₂²×C₄	—	2	2	no	—
493	C₂²×C₄	—	2	2	no	—
494	C₂²×C₄	—	2	1	no	—
495	C₂²×C₄	—	2	1	no	—
496	C₂²×C₄	—	2	2	no	—
497	C₂²×C₄	—	2	4	no	—
498	C₂²×C₄	—	2	4	no	—
499	C₂²×C₄	—	2	8	no	—
500	C₂²×C₄	—	4	4	no	—
501	C₂²×C₄	—	4	8	no	—
502	C₂²×C₄	—	4	4	no	—
503	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
504	C₂²×C₄	C₂×D₄×C₈₉	1	2	yes	—
505	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
506	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
507	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
508	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
509	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
510	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
511	C₂²×C₄	C₂⁴×C₈₉	1	2	yes	—
512	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
513	SD₁₆	—	2	16	no	—
514	D₈	—	2	16	no	—
515	C₂×C₈	—	2	4	no	—
516	C₂×C₈	—	2	4	no	—
517	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
518	C₂×C₈	C₂⁴×C₈₉	1	4	yes	—
519	C₄⋊C₄	—	2	4	no	—
520	C₄⋊C₄	—	2	4	no	—
521	C₄⋊C₄	—	2	4	no	—
522	C₄⋊C₄	—	2	1	no	—
523	C₄⋊C₄	—	2	1	no	—
524	C₄⋊C₄	—	2	2	no	—
525	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
526	C₄⋊C₄	C₂⁴×C₈₉	1	2	yes	—
527	C₄⋊C₄	C₂⁴×C₈₉	1	4	yes	—
528	C₂²⋊C₄	—	2	4	no	—
529	C₂²⋊C₄	—	2	4	no	—
530	C₂²⋊C₄	—	2	4	no	—
531	C₂²⋊C₄	—	2	8	no	—
532	C₂²⋊C₄	—	2	8	no	—
533	C₄²	—	2	4	no	—
534	C₄²	—	2	4	no	—
535	C₄²	—	2	4	no	—
536	C₄²	C₂×D₄×C₈₉	1	4	yes	—
537	C₄²	C₂×D₄×C₈₉	1	4	yes	—
538	C₄²	C₂×D₄×C₈₉	1	2	yes	—
539	C₄²	C₂×D₄×C₈₉	1	2	yes	—
540	C₄²	C₂⁴×C₈₉	1	2	yes	—
541	C₄²	C₂⁴×C₈₉	1	2	yes	—
542	C₂³	—	2	2	no	—
543	C₂³	—	2	4	no	—
544	C₂³	—	2	2	no	—
545	C₂³	—	2	2	no	—
546	C₂³	—	2	2	no	—
547	C₂³	—	2	4	no	—
548	C₂³	—	4	2	no	—
549	C₂³	—	4	2	no	—
550	C₂³	—	4	4	no	—
551	C₂³	—	4	4	no	—
552	C₂³	—	4	4	no	—
553	C₂³	C₂²×D₄×C₈₉	1	2	yes	—
554	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
555	Q₈	—	2	4	no	—
556	Q₈	—	2	2	no	—
557	Q₈	—	4	8	no	—
558	Q₈	C₂×#41×C₈₉	1	4	yes	complemented
559	D₄	—	2	16	no	—
560	D₄	—	2	4	no	—
561	D₄	—	2	2	no	—
562	D₄	—	2	4	no	—
563	D₄	—	2	4	no	—
564	D₄	—	4	8	no	—
565	D₄	—	4	16	no	—
566	D₄	—	4	4	no	—
567	D₄	—	4	8	no	—
568	D₄	C₂×#41×C₈₉	1	4	yes	complemented
569	C₂×C₄	—	2	16	no	—
570	C₂×C₄	—	2	2	no	—
571	C₂×C₄	—	2	4	no	—
572	C₂×C₄	—	2	2	no	—
573	C₂×C₄	—	2	2	no	—
574	C₂×C₄	—	2	2	no	—
575	C₂×C₄	—	2	4	no	—
576	C₂×C₄	—	2	8	no	—
577	C₂×C₄	—	2	8	no	—
578	C₂×C₄	—	2	2	no	—
579	C₂×C₄	—	2	2	no	—
580	C₂×C₄	—	2	2	no	—
581	C₂×C₄	—	2	2	no	—
582	C₂×C₄	—	2	4	no	—
583	C₂×C₄	—	4	2	no	—
584	C₂×C₄	—	4	2	no	—
585	C₂×C₄	—	4	4	no	—
586	C₂×C₄	—	4	8	no	—
587	C₂×C₄	—	4	8	no	—
588	C₂×C₄	—	4	16	no	—
589	C₂×C₄	#135×C₈₉	1	4	yes	—
590	C₂×C₄	AGL(1,ℤ/8)×C₈₉	1	4	yes	—
591	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
592	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
593	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
594	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
595	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
596	C₂×C₄	C₂²×D₄×C₈₉	1	2	yes	—
597	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
598	C₂×C₄	C₂×#41×C₈₉	1	2	yes	—
599	C₂×C₄	C₂⁵×C₈₉	1	1	yes	characteristic
600	C₈	—	2	4	no	—
601	C₈	—	2	4	no	—
602	C₂²	—	2	2	no	—
603	C₂²	—	2	2	no	—
604	C₂²	—	2	2	no	—
605	C₂²	—	2	4	no	—
606	C₂²	—	4	2	no	—
607	C₂²	—	4	2	no	—
608	C₂²	—	4	4	no	—
609	C₂²	—	4	8	no	—
610	C₂²	—	4	4	no	—
611	C₂²	—	4	4	no	—
612	C₂²	—	4	4	no	—
613	C₂²	—	8	8	no	—
614	C₂²	#525×C₈₉	1	2	yes	—
615	C₂²	C₂×#135×C₈₉	1	2	yes	—
616	C₂²	C₂×AGL(1,ℤ/8)×C₈₉	1	2	yes	—
617	C₂²	C₂³×D₄×C₈₉	1	1	yes	characteristic
618	C₄	—	2	2	no	—
619	C₄	—	2	2	no	—
620	C₄	—	2	2	no	—
621	C₄	—	2	4	no	—
622	C₄	—	4	2	no	—
623	C₄	—	4	4	no	—
624	C₄	—	4	2	no	—
625	C₄	#525×C₈₉	1	2	yes	—
626	C₄	C₂×#135×C₈₉	1	4	yes	—
627	C₄	C₂×AGL(1,ℤ/8)×C₈₉	1	4	yes	—
628	C₄	C₂²×#41×C₈₉	1	1	yes	characteristic
629	C₄	C₂²×#41×C₈₉	1	1	yes	characteristic
630	C₂	—	4	2	no	—
631	C₂	—	4	4	no	—
632	C₂	—	8	4	no	—
633	C₂	C₈₉×#3106	1	4	yes	direct product factor
634	C₂	C₂×#525×C₈₉	1	1	yes	characteristic
635	C₂	C₂²×#135×C₈₉	1	1	yes	characteristic
636	C₂	C₂²×AGL(1,ℤ/8)×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
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
1. C₂×#532×C₈₉	C₂	semidirect
2. C₂×C₄×#41×C₈₉	C₂	semidirect
3. C₈₉×#3106	C₂	direct (N × H)
4. C₂×#482×C₈₉	C₂	semidirect
5. C₂×#477×C₈₉	C₂	semidirect
6. C₂×#476×C₈₉	C₂	semidirect
7. C₂×#460×C₈₉	C₂	semidirect
8. C₂×#459×C₈₉	C₂	semidirect
9. C₂×#449×C₈₉	C₂	semidirect
10. C₂×C₄×SD₁₆×C₈₉	C₂	semidirect
11. C₂×C₄×D₈×C₈₉	C₂	semidirect
12. C₂×#431×C₈₉	C₂	semidirect
18. #532×C₈₉	C₂²	semidirect
25. C₄×#41×C₈₉	C₂²	semidirect
38. #482×C₈₉	C₂²	semidirect
39. #477×C₈₉	C₂²	semidirect
40. #476×C₈₉	C₂²	semidirect
41. #460×C₈₉	C₂²	semidirect
42. #459×C₈₉	C₂²	semidirect
43. #449×C₈₉	C₂²	semidirect
44. C₄×SD₁₆×C₈₉	C₂²	semidirect
45. C₄×D₈×C₈₉	C₂²	semidirect
46. #431×C₈₉	C₂²	semidirect
150. C₂×Q₈×C₈₉	#41	semidirect
165. C₂×D₄×C₈₉	#41	semidirect
239. Q₈×C₈₉	C₂×#41	semidirect
249. D₄×C₈₉	C₂×#41	semidirect
311. C₂×#3106	C₈₉	direct (N × H)
315. C₂×C₈₉	#3106	direct (N × H)
319. C₂×#532	C₂×C₈₉	semidirect
320. C₂×C₄×#41	C₂×C₈₉	semidirect
321. #3106	C₂×C₈₉	direct (N × H)
322. C₂×#482	C₂×C₈₉	semidirect
323. C₂×#477	C₂×C₈₉	semidirect
324. C₂×#476	C₂×C₈₉	semidirect
325. C₂×#460	C₂×C₈₉	semidirect
326. C₂×#459	C₂×C₈₉	semidirect
327. C₂×#449	C₂×C₈₉	semidirect
328. C₂×C₄×SD₁₆	C₂×C₈₉	semidirect
329. C₂×C₄×D₈	C₂×C₈₉	semidirect
330. C₂×#431	C₂×C₈₉	semidirect
331. C₈₉	C₂×#3106	direct (N × H)
337. #532	C₂²×C₈₉	semidirect
344. C₄×#41	C₂²×C₈₉	semidirect
357. #482	C₂²×C₈₉	semidirect
358. #477	C₂²×C₈₉	semidirect
359. #476	C₂²×C₈₉	semidirect
360. #460	C₂²×C₈₉	semidirect
361. #459	C₂²×C₈₉	semidirect
362. #449	C₂²×C₈₉	semidirect
363. C₄×SD₁₆	C₂²×C₈₉	semidirect
364. C₄×D₈	C₂²×C₈₉	semidirect
365. #431	C₂²×C₈₉	semidirect
469. C₂×Q₈	#41×C₈₉	semidirect
484. C₂×D₄	#41×C₈₉	semidirect
558. Q₈	C₂×#41×C₈₉	semidirect
568. D₄	C₂×#41×C₈₉	semidirect
633. C₂	C₈₉×#3106	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×#532×C₈₉	2
C₂×C₄×#41×C₈₉	2
C₈₉×#3106	2
C₂×#482×C₈₉	2
C₂×#477×C₈₉	2
C₂×#476×C₈₉	2
C₂×#460×C₈₉	2
C₂×#459×C₈₉	2
C₂×#449×C₈₉	2
C₂×C₄×SD₁₆×C₈₉	2
C₂×C₄×D₈×C₈₉	2
C₂×#431×C₈₉	2
C₂×#3106	89

Subgroups — C2×C89×#3106 #36438381

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₈₉×#3106 #36438381