Subgroups — C₈₉×#24323 #36402171

Order 22784.

1004 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₈₉×#3503	C₂	1	1	yes	maximal, characteristic
2	C₈₉×#3078	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×#137×C₈₉	C₂	1	1	yes	maximal, characteristic
4	C₈₉×#2567	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₉×#2060	C₂	1	2	yes	maximal, complemented
6	C₈₉×#2028	C₂	1	2	yes	maximal, complemented
7	C₈₉×#1997	C₂	1	1	yes	maximal, characteristic, complemented
8	C₈₉×#1986	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#1882	C₂	1	2	yes	maximal
10	C₈₉×#1859	C₂	1	2	yes	maximal
11	C₈₉×#1806	C₂	1	1	yes	maximal, characteristic
12	C₂×#137×C₈₉	C₂²	1	1	yes	characteristic
13	Q₈²×C₈₉	—	2	1	no	—
14	#557×C₈₉	—	2	1	no	—
15	#554×C₈₉	—	2	1	no	—
16	#554×C₈₉	—	2	2	no	—
17	D₄×Q₈×C₈₉	—	2	2	no	—
18	D₄×Q₈×C₈₉	—	2	1	no	—
19	#544×C₈₉	—	2	2	no	—
20	#544×C₈₉	C₂²	1	2	yes	—
21	#533×C₈₉	—	2	1	no	—
22	#533×C₈₉	C₂²	1	1	yes	characteristic
23	C₂×C₄⋊Q₈×C₈₉	—	2	1	no	—
24	C₂×C₄⋊Q₈×C₈₉	C₂²	1	1	yes	characteristic
25	#520×C₈₉	C₂²	1	2	yes	—
26	C₄×#41×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
28	#479×C₈₉	—	2	2	no	—
29	#477×C₈₉	—	2	1	no	—
30	#475×C₈₉	—	2	1	no	—
31	#441×C₈₉	—	2	1	no	—
32	#440×C₈₉	—	2	1	no	—
33	C₄×Q₁₆×C₈₉	—	2	1	no	—
34	C₄×SD₁₆×C₈₉	—	2	1	no	—
35	#428×C₈₉	C₂²	1	2	yes	—
36	#428×C₈₉	C₂²	1	1	yes	characteristic
37	#423×C₈₉	C₂²	1	1	yes	characteristic
38	C₂×#104×C₈₉	C₂²	1	2	yes	—
39	#419×C₈₉	C₂²	1	2	yes	—
40	#419×C₈₉	C₂²	1	1	yes	characteristic
41	#417×C₈₉	C₂²	1	1	yes	characteristic
42	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
43	#395×C₈₉	—	2	2	no	—
44	#391×C₈₉	C₂²	1	2	yes	—
45	#389×C₈₉	—	2	2	no	—
46	#384×C₈₉	C₂²	1	2	yes	—
47	C₄×C₄⋊C₄×C₈₉	C₂²	1	1	yes	characteristic
48	#343×C₈₉	C₂²	1	2	yes	—
49	#343×C₈₉	C₂²	1	2	yes	—
50	#339×C₈₉	C₂²	1	2	yes	—
51	#339×C₈₉	C₂²	1	2	yes	—
52	#339×C₈₉	C₂²	1	2	yes	—
53	C₂×#41×C₈₉	C₂³	1	1	yes	characteristic
54	C₂²×Q₈×C₈₉	—	2	2	no	—
55	C₂²×Q₈×C₈₉	D₄	1	2	yes	—
56	C₂²×Q₈×C₈₉	C₂³	1	1	yes	characteristic
57	#137×C₈₉	—	2	2	no	—
58	#137×C₈₉	Q₈	1	4	yes	complemented
59	C₂×Q₁₆×C₈₉	—	2	1	no	—
60	C₂×SD₁₆×C₈₉	—	2	1	no	—
61	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
62	C₂×M₁₆×C₈₉	C₂³	1	2	yes	—
63	C₂×M₁₆×C₈₉	C₂³	1	1	yes	characteristic
64	C₄⋊Q₈×C₈₉	—	2	2	no	—
65	C₄⋊Q₈×C₈₉	—	2	2	no	—
66	C₄⋊Q₈×C₈₉	—	2	4	no	—
67	C₄⋊Q₈×C₈₉	—	2	2	no	—
68	C₄⋊Q₈×C₈₉	—	2	2	no	—
69	C₄⋊Q₈×C₈₉	—	2	1	no	—
70	C₄⋊Q₈×C₈₉	—	2	1	no	—
71	C₄⋊Q₈×C₈₉	—	2	1	no	—
72	C₄⋊Q₈×C₈₉	—	2	1	no	—
73	C₄⋊Q₈×C₈₉	D₄	1	1	yes	characteristic
74	C₄⋊Q₈×C₈₉	D₄	1	1	yes	characteristic
75	#125×C₈₉	—	2	4	no	—
76	#125×C₈₉	—	2	2	no	—
77	#125×C₈₉	—	2	2	no	—
78	#125×C₈₉	D₄	1	2	yes	—
79	C₂²⋊Q₈×C₈₉	—	2	2	no	—
80	C₂²⋊Q₈×C₈₉	—	2	2	no	—
81	C₂²⋊Q₈×C₈₉	—	2	4	no	—
82	C₂²⋊Q₈×C₈₉	—	2	4	no	—
83	C₂²⋊Q₈×C₈₉	—	2	1	no	—
84	C₂²⋊Q₈×C₈₉	—	2	1	no	—
85	C₂²⋊Q₈×C₈₉	—	2	4	no	—
86	C₄×Q₈×C₈₉	—	2	2	no	—
87	C₄×Q₈×C₈₉	—	2	2	no	—
88	C₄×Q₈×C₈₉	—	2	2	no	—
89	C₄×Q₈×C₈₉	—	2	1	no	—
90	C₄×Q₈×C₈₉	—	2	1	no	—
91	C₄×Q₈×C₈₉	—	2	1	no	—
92	C₄×Q₈×C₈₉	—	2	1	no	—
93	C₄×Q₈×C₈₉	—	2	1	no	—
94	C₄×Q₈×C₈₉	—	2	1	no	—
95	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
96	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
97	C₄×D₄×C₈₉	—	2	2	no	—
98	C₄×D₄×C₈₉	—	2	1	no	—
99	C₄×D₄×C₈₉	—	2	1	no	—
100	C₄×D₄×C₈₉	—	2	4	no	—
101	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
102	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
103	#117×C₈₉	—	2	2	no	—
104	#117×C₈₉	—	2	1	no	—
105	#117×C₈₉	C₂³	1	1	yes	characteristic
106	#117×C₈₉	C₂³	1	2	yes	—
107	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
108	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
109	C₂×C₄⋊C₄×C₈₉	—	2	2	no	—
110	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
111	C₂×C₄⋊C₄×C₈₉	—	2	4	no	—
112	C₂×C₄⋊C₄×C₈₉	D₄	1	2	yes	—
113	C₂×C₄⋊C₄×C₈₉	C₂³	1	1	yes	characteristic
114	C₂×C₄⋊C₄×C₈₉	C₂³	1	2	yes	—
115	C₂×C₄²×C₈₉	—	2	1	no	—
116	C₂×C₄²×C₈₉	—	2	1	no	—
117	C₂×C₄²×C₈₉	C₂³	1	2	yes	—
118	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
119	#107×C₈₉	—	2	1	no	—
120	#107×C₈₉	—	2	2	no	—
121	#106×C₈₉	—	2	1	no	—
122	#106×C₈₉	—	2	2	no	—
123	C₄⋊C₈×C₈₉	—	2	1	no	—
124	C₄⋊C₈×C₈₉	—	2	1	no	—
125	#104×C₈₉	—	2	4	no	—
126	#103×C₈₉	—	2	1	no	—
127	#103×C₈₉	—	2	1	no	—
128	#103×C₈₉	—	2	1	no	—
129	#103×C₈₉	—	2	2	no	—
130	#103×C₈₉	—	2	2	no	—
131	#102×C₈₉	—	2	1	no	—
132	#97×C₈₉	—	2	1	no	—
133	C₄×C₈×C₈₉	—	2	1	no	—
134	#95×C₈₉	—	2	1	no	—
135	#95×C₈₉	—	2	4	no	—
136	#41×C₈₉	—	2	1	no	—
137	#41×C₈₉	—	2	1	no	—
138	#41×C₈₉	C₂×Q₈	1	2	yes	—
139	#41×C₈₉	#41	1	2	yes	—
140	C₂×Q₈×C₈₉	—	2	4	no	—
141	C₂×Q₈×C₈₉	—	2	1	no	—
142	C₂×Q₈×C₈₉	—	2	2	no	—
143	C₂×Q₈×C₈₉	—	2	1	no	—
144	C₂×Q₈×C₈₉	—	2	2	no	—
145	C₂×Q₈×C₈₉	—	2	1	no	—
146	C₂×Q₈×C₈₉	—	2	1	no	—
147	C₂×Q₈×C₈₉	—	2	1	no	—
148	C₂×Q₈×C₈₉	—	2	1	no	—
149	C₂×Q₈×C₈₉	—	2	1	no	—
150	C₂×Q₈×C₈₉	—	8	2	no	—
151	C₂×Q₈×C₈₉	—	8	4	no	—
152	C₂×Q₈×C₈₉	C₂×D₄	1	1	yes	characteristic
153	C₂×Q₈×C₈₉	C₂×Q₈	1	2	yes	complemented
154	C₂×D₄×C₈₉	—	2	1	no	—
155	C₂×D₄×C₈₉	C₂×D₄	1	1	yes	characteristic
156	C₂²×C₄×C₈₉	—	2	1	no	—
157	C₂²×C₄×C₈₉	—	2	1	no	—
158	C₂²×C₄×C₈₉	—	2	2	no	—
159	C₂²×C₄×C₈₉	—	2	4	no	—
160	C₂²×C₄×C₈₉	—	2	1	no	—
161	C₂²×C₄×C₈₉	—	2	1	no	—
162	C₂²×C₄×C₈₉	—	2	1	no	—
163	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
164	C₂²×C₄×C₈₉	C₂×D₄	1	4	yes	—
165	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
166	C₂²×C₄×C₈₉	C₂⁴	1	1	yes	characteristic
167	Q₁₆×C₈₉	—	2	4	no	—
168	SD₁₆×C₈₉	—	2	4	no	—
169	M₁₆×C₈₉	—	2	1	no	—
170	M₁₆×C₈₉	—	2	2	no	—
171	M₁₆×C₈₉	—	2	1	no	—
172	M₁₆×C₈₉	—	2	1	no	—
173	M₁₆×C₈₉	—	2	2	no	—
174	M₁₆×C₈₉	C₂×Q₈	1	2	yes	complemented
175	C₂×C₈×C₈₉	—	2	1	no	—
176	C₂×C₈×C₈₉	—	2	1	no	—
177	C₂×C₈×C₈₉	—	2	2	no	—
178	C₄⋊C₄×C₈₉	—	2	4	no	—
179	C₄⋊C₄×C₈₉	—	2	4	no	—
180	C₄⋊C₄×C₈₉	—	2	4	no	—
181	C₄⋊C₄×C₈₉	—	2	4	no	—
182	C₄⋊C₄×C₈₉	—	2	4	no	—
183	C₄⋊C₄×C₈₉	—	2	1	no	—
184	C₄⋊C₄×C₈₉	—	2	2	no	—
185	C₄⋊C₄×C₈₉	—	2	2	no	—
186	C₄⋊C₄×C₈₉	—	2	1	no	—
187	C₄⋊C₄×C₈₉	—	2	1	no	—
188	C₄⋊C₄×C₈₉	—	2	1	no	—
189	C₄⋊C₄×C₈₉	—	2	2	no	—
190	C₄⋊C₄×C₈₉	—	2	2	no	—
191	C₄⋊C₄×C₈₉	—	2	2	no	—
192	C₄⋊C₄×C₈₉	—	2	2	no	—
193	C₄⋊C₄×C₈₉	—	2	1	no	—
194	C₄⋊C₄×C₈₉	—	2	1	no	—
195	C₄⋊C₄×C₈₉	—	4	1	no	—
196	C₄⋊C₄×C₈₉	—	4	1	no	—
197	C₄⋊C₄×C₈₉	—	8	4	no	—
198	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
199	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
200	C₄⋊C₄×C₈₉	C₂×D₄	1	2	yes	—
201	C₄⋊C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
202	C₂²⋊C₄×C₈₉	—	2	4	no	—
203	C₂²⋊C₄×C₈₉	—	2	2	no	—
204	C₂²⋊C₄×C₈₉	—	2	1	no	—
205	C₂²⋊C₄×C₈₉	—	2	1	no	—
206	C₂²⋊C₄×C₈₉	—	2	1	no	—
207	C₄²×C₈₉	—	2	4	no	—
208	C₄²×C₈₉	—	2	1	no	—
209	C₄²×C₈₉	—	2	1	no	—
210	C₄²×C₈₉	—	2	1	no	—
211	C₄²×C₈₉	—	2	2	no	—
212	C₄²×C₈₉	—	2	2	no	—
213	C₄²×C₈₉	—	2	2	no	—
214	C₄²×C₈₉	—	4	1	no	—
215	C₄²×C₈₉	—	4	1	no	—
216	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
217	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
218	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
219	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
220	C₂³×C₈₉	—	2	1	no	—
221	C₂³×C₈₉	C₂²×D₄	1	1	yes	characteristic
222	Q₈×C₈₉	—	2	1	no	—
223	Q₈×C₈₉	—	2	2	no	—
224	Q₈×C₈₉	—	2	2	no	—
225	Q₈×C₈₉	—	4	4	no	—
226	Q₈×C₈₉	—	4	2	no	—
227	Q₈×C₈₉	—	4	1	no	—
228	Q₈×C₈₉	—	4	1	no	—
229	Q₈×C₈₉	—	4	2	no	—
230	Q₈×C₈₉	—	8	2	no	—
231	Q₈×C₈₉	—	8	4	no	—
232	Q₈×C₈₉	C₂²⋊Q₈	1	2	yes	—
233	D₄×C₈₉	—	2	2	no	—
234	D₄×C₈₉	—	2	2	no	—
235	D₄×C₈₉	—	2	1	no	—
236	D₄×C₈₉	C₂²⋊Q₈	1	2	yes	—
237	C₂×C₄×C₈₉	—	2	1	no	—
238	C₂×C₄×C₈₉	—	2	2	no	—
239	C₂×C₄×C₈₉	—	2	2	no	—
240	C₂×C₄×C₈₉	—	2	2	no	—
241	C₂×C₄×C₈₉	—	2	4	no	—
242	C₂×C₄×C₈₉	—	2	2	no	—
243	C₂×C₄×C₈₉	—	2	1	no	—
244	C₂×C₄×C₈₉	—	2	1	no	—
245	C₂×C₄×C₈₉	—	2	4	no	—
246	C₂×C₄×C₈₉	—	2	2	no	—
247	C₂×C₄×C₈₉	—	2	2	no	—
248	C₂×C₄×C₈₉	—	4	1	no	—
249	C₂×C₄×C₈₉	—	4	2	no	—
250	C₂×C₄×C₈₉	—	4	2	no	—
251	C₂×C₄×C₈₉	—	4	1	no	—
252	C₂×C₄×C₈₉	—	4	2	no	—
253	C₂×C₄×C₈₉	—	4	1	no	—
254	C₂×C₄×C₈₉	—	4	2	no	—
255	C₂×C₄×C₈₉	—	4	4	no	—
256	C₂×C₄×C₈₉	—	4	2	no	—
257	C₂×C₄×C₈₉	—	4	1	no	—
258	C₂×C₄×C₈₉	—	8	4	no	—
259	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
260	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
261	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
262	C₂×C₄×C₈₉	#120	1	1	yes	characteristic
263	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
264	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
265	C₂×C₄×C₈₉	C₂²×Q₈	1	1	yes	characteristic
266	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
267	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
268	C₂×C₄×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
269	C₈×C₈₉	—	2	2	no	—
270	C₈×C₈₉	—	4	1	no	—
271	C₈×C₈₉	—	4	2	no	—
272	C₂²×C₈₉	—	2	1	no	—
273	C₂²×C₈₉	—	2	1	no	—
274	C₂²×C₈₉	—	2	2	no	—
275	C₂²×C₈₉	—	4	1	no	—
276	C₂²×C₈₉	—	4	1	no	—
277	C₂²×C₈₉	C₂×#120	1	1	yes	characteristic
278	C₂²×C₈₉	#548	1	1	yes	characteristic
279	C₂²×C₈₉	D₄×Q₈	1	1	yes	characteristic
280	C₄×C₈₉	—	2	2	no	—
281	C₄×C₈₉	—	2	1	no	—
282	C₄×C₈₉	—	2	1	no	—
283	C₄×C₈₉	—	2	1	no	—
284	C₄×C₈₉	—	4	4	no	—
285	C₄×C₈₉	—	4	2	no	—
286	C₄×C₈₉	—	4	1	no	—
287	C₄×C₈₉	—	4	2	no	—
288	C₄×C₈₉	—	4	1	no	—
289	C₄×C₈₉	—	4	4	no	—
290	C₄×C₈₉	—	4	2	no	—
291	C₄×C₈₉	—	8	1	no	—
292	C₄×C₈₉	#456	1	2	yes	—
293	C₄×C₈₉	C₂×C₂²⋊Q₈	1	1	yes	characteristic
294	C₄×C₈₉	#536	1	1	yes	characteristic
295	C₄×C₈₉	#548	1	1	yes	characteristic
296	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
297	#24323	C₈₉	1	1	yes	maximal, characteristic, direct product factor, Sylow
298	C₂×C₈₉	—	2	1	no	—
299	C₂×C₈₉	—	2	1	no	—
300	C₂×C₈₉	—	4	2	no	—
301	C₂×C₈₉	#2409	1	1	yes	characteristic
302	C₂×C₈₉	C₂×#456	1	1	yes	characteristic
303	C₂×C₈₉	#3019	1	1	yes	characteristic
304	#3503	C₂×C₈₉	1	1	yes	characteristic
305	#3078	C₂×C₈₉	1	1	yes	characteristic, complemented
306	C₄×#137	C₂×C₈₉	1	1	yes	characteristic
307	#2567	C₂×C₈₉	1	1	yes	characteristic, complemented
308	#2060	C₂×C₈₉	1	2	yes	complemented
309	#2028	C₂×C₈₉	1	2	yes	complemented
310	#1997	C₂×C₈₉	1	1	yes	characteristic, complemented
311	#1986	C₂×C₈₉	1	1	yes	characteristic, complemented
312	#1882	C₂×C₈₉	1	2	yes	—
313	#1859	C₂×C₈₉	1	2	yes	—
314	#1806	C₂×C₈₉	1	1	yes	characteristic
315	C₈₉	#24323	1	1	yes	characteristic, direct product factor, Sylow
316	C₂×#137	C₂²×C₈₉	1	1	yes	characteristic
317	Q₈²	—	2	1	no	—
318	#557	—	2	1	no	—
319	#554	—	2	1	no	—
320	#554	—	2	2	no	—
321	D₄×Q₈	—	2	2	no	—
322	D₄×Q₈	—	2	1	no	—
323	#544	—	2	2	no	—
324	#544	C₂²×C₈₉	1	2	yes	—
325	#533	—	2	1	no	—
326	#533	C₂²×C₈₉	1	1	yes	characteristic
327	C₂×C₄⋊Q₈	—	2	1	no	—
328	C₂×C₄⋊Q₈	C₂²×C₈₉	1	1	yes	characteristic
329	#520	C₂²×C₈₉	1	2	yes	—
330	C₄×#41	C₂²×C₈₉	1	1	yes	characteristic
331	C₂×C₄×Q₈	C₂²×C₈₉	1	1	yes	characteristic
332	#479	—	2	2	no	—
333	#477	—	2	1	no	—
334	#475	—	2	1	no	—
335	#441	—	2	1	no	—
336	#440	—	2	1	no	—
337	C₄×Q₁₆	—	2	1	no	—
338	C₄×SD₁₆	—	2	1	no	—
339	#428	C₂²×C₈₉	1	2	yes	—
340	#428	C₂²×C₈₉	1	1	yes	characteristic
341	#423	C₂²×C₈₉	1	1	yes	characteristic
342	C₂×#104	C₂²×C₈₉	1	2	yes	—
343	#419	C₂²×C₈₉	1	2	yes	—
344	#419	C₂²×C₈₉	1	1	yes	characteristic
345	#417	C₂²×C₈₉	1	1	yes	characteristic
346	C₄×M₁₆	C₂²×C₈₉	1	1	yes	characteristic
347	#395	—	2	2	no	—
348	#391	C₂²×C₈₉	1	2	yes	—
349	#389	—	2	2	no	—
350	#384	C₂²×C₈₉	1	2	yes	—
351	C₄×C₄⋊C₄	C₂²×C₈₉	1	1	yes	characteristic
352	#343	C₂²×C₈₉	1	2	yes	—
353	#343	C₂²×C₈₉	1	2	yes	—
354	#339	C₂²×C₈₉	1	2	yes	—
355	#339	C₂²×C₈₉	1	2	yes	—
356	#339	C₂²×C₈₉	1	2	yes	—
357	C₂×#41	C₂³×C₈₉	1	1	yes	characteristic
358	C₂²×Q₈	—	2	2	no	—
359	C₂²×Q₈	D₄×C₈₉	1	2	yes	—
360	C₂²×Q₈	C₂³×C₈₉	1	1	yes	characteristic
361	#137	—	2	2	no	—
362	#137	Q₈×C₈₉	1	4	yes	complemented
363	C₂×Q₁₆	—	2	1	no	—
364	C₂×SD₁₆	—	2	1	no	—
365	C₂×M₁₆	C₂³×C₈₉	1	1	yes	characteristic
366	C₂×M₁₆	C₂³×C₈₉	1	2	yes	—
367	C₂×M₁₆	C₂³×C₈₉	1	1	yes	characteristic
368	C₄⋊Q₈	—	2	2	no	—
369	C₄⋊Q₈	—	2	2	no	—
370	C₄⋊Q₈	—	2	4	no	—
371	C₄⋊Q₈	—	2	2	no	—
372	C₄⋊Q₈	—	2	2	no	—
373	C₄⋊Q₈	—	2	1	no	—
374	C₄⋊Q₈	—	2	1	no	—
375	C₄⋊Q₈	—	2	1	no	—
376	C₄⋊Q₈	—	2	1	no	—
377	C₄⋊Q₈	D₄×C₈₉	1	1	yes	characteristic
378	C₄⋊Q₈	D₄×C₈₉	1	1	yes	characteristic
379	#125	—	2	4	no	—
380	#125	—	2	2	no	—
381	#125	—	2	2	no	—
382	#125	D₄×C₈₉	1	2	yes	—
383	C₂²⋊Q₈	—	2	2	no	—
384	C₂²⋊Q₈	—	2	2	no	—
385	C₂²⋊Q₈	—	2	4	no	—
386	C₂²⋊Q₈	—	2	4	no	—
387	C₂²⋊Q₈	—	2	1	no	—
388	C₂²⋊Q₈	—	2	1	no	—
389	C₂²⋊Q₈	—	2	4	no	—
390	C₄×Q₈	—	2	2	no	—
391	C₄×Q₈	—	2	2	no	—
392	C₄×Q₈	—	2	2	no	—
393	C₄×Q₈	—	2	1	no	—
394	C₄×Q₈	—	2	1	no	—
395	C₄×Q₈	—	2	1	no	—
396	C₄×Q₈	—	2	1	no	—
397	C₄×Q₈	—	2	1	no	—
398	C₄×Q₈	—	2	1	no	—
399	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
400	C₄×Q₈	D₄×C₈₉	1	1	yes	characteristic
401	C₄×D₄	—	2	2	no	—
402	C₄×D₄	—	2	1	no	—
403	C₄×D₄	—	2	1	no	—
404	C₄×D₄	—	2	4	no	—
405	C₄×D₄	D₄×C₈₉	1	1	yes	characteristic
406	C₄×D₄	D₄×C₈₉	1	1	yes	characteristic
407	#117	—	2	2	no	—
408	#117	—	2	1	no	—
409	#117	C₂³×C₈₉	1	1	yes	characteristic
410	#117	C₂³×C₈₉	1	2	yes	—
411	C₂×C₄⋊C₄	—	2	2	no	—
412	C₂×C₄⋊C₄	—	2	1	no	—
413	C₂×C₄⋊C₄	—	2	2	no	—
414	C₂×C₄⋊C₄	—	2	1	no	—
415	C₂×C₄⋊C₄	—	2	4	no	—
416	C₂×C₄⋊C₄	D₄×C₈₉	1	2	yes	—
417	C₂×C₄⋊C₄	C₂³×C₈₉	1	1	yes	characteristic
418	C₂×C₄⋊C₄	C₂³×C₈₉	1	2	yes	—
419	C₂×C₄²	—	2	1	no	—
420	C₂×C₄²	—	2	1	no	—
421	C₂×C₄²	C₂³×C₈₉	1	2	yes	—
422	C₂×C₄²	C₂³×C₈₉	1	1	yes	characteristic
423	#107	—	2	1	no	—
424	#107	—	2	2	no	—
425	#106	—	2	1	no	—
426	#106	—	2	2	no	—
427	C₄⋊C₈	—	2	1	no	—
428	C₄⋊C₈	—	2	1	no	—
429	#104	—	2	4	no	—
430	#103	—	2	1	no	—
431	#103	—	2	1	no	—
432	#103	—	2	1	no	—
433	#103	—	2	2	no	—
434	#103	—	2	2	no	—
435	#102	—	2	1	no	—
436	#97	—	2	1	no	—
437	C₄×C₈	—	2	1	no	—
438	#95	—	2	1	no	—
439	#95	—	2	4	no	—
440	#41	—	2	1	no	—
441	#41	—	2	1	no	—
442	#41	C₂×Q₈×C₈₉	1	2	yes	—
443	#41	#41×C₈₉	1	2	yes	—
444	C₂×Q₈	—	2	4	no	—
445	C₂×Q₈	—	2	1	no	—
446	C₂×Q₈	—	2	2	no	—
447	C₂×Q₈	—	2	1	no	—
448	C₂×Q₈	—	2	2	no	—
449	C₂×Q₈	—	2	1	no	—
450	C₂×Q₈	—	2	1	no	—
451	C₂×Q₈	—	2	1	no	—
452	C₂×Q₈	—	2	1	no	—
453	C₂×Q₈	—	2	1	no	—
454	C₂×Q₈	—	8	2	no	—
455	C₂×Q₈	—	8	4	no	—
456	C₂×Q₈	C₂×D₄×C₈₉	1	1	yes	characteristic
457	C₂×Q₈	C₂×Q₈×C₈₉	1	2	yes	complemented
458	C₂×D₄	—	2	1	no	—
459	C₂×D₄	C₂×D₄×C₈₉	1	1	yes	characteristic
460	C₂²×C₄	—	2	1	no	—
461	C₂²×C₄	—	2	1	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	4	no	—
464	C₂²×C₄	—	2	1	no	—
465	C₂²×C₄	—	2	1	no	—
466	C₂²×C₄	—	2	1	no	—
467	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
468	C₂²×C₄	C₂×D₄×C₈₉	1	4	yes	—
469	C₂²×C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
470	C₂²×C₄	C₂⁴×C₈₉	1	1	yes	characteristic
471	Q₁₆	—	2	4	no	—
472	SD₁₆	—	2	4	no	—
473	M₁₆	—	2	1	no	—
474	M₁₆	—	2	2	no	—
475	M₁₆	—	2	1	no	—
476	M₁₆	—	2	1	no	—
477	M₁₆	—	2	2	no	—
478	M₁₆	C₂×Q₈×C₈₉	1	2	yes	complemented
479	C₂×C₈	—	2	1	no	—
480	C₂×C₈	—	2	1	no	—
481	C₂×C₈	—	2	2	no	—
482	C₄⋊C₄	—	2	4	no	—
483	C₄⋊C₄	—	2	4	no	—
484	C₄⋊C₄	—	2	4	no	—
485	C₄⋊C₄	—	2	4	no	—
486	C₄⋊C₄	—	2	4	no	—
487	C₄⋊C₄	—	2	1	no	—
488	C₄⋊C₄	—	2	2	no	—
489	C₄⋊C₄	—	2	2	no	—
490	C₄⋊C₄	—	2	1	no	—
491	C₄⋊C₄	—	2	1	no	—
492	C₄⋊C₄	—	2	1	no	—
493	C₄⋊C₄	—	2	2	no	—
494	C₄⋊C₄	—	2	2	no	—
495	C₄⋊C₄	—	2	2	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₄⋊C₄	—	2	1	no	—
498	C₄⋊C₄	—	2	1	no	—
499	C₄⋊C₄	—	4	1	no	—
500	C₄⋊C₄	—	4	1	no	—
501	C₄⋊C₄	—	8	4	no	—
502	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
503	C₄⋊C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
504	C₄⋊C₄	C₂×D₄×C₈₉	1	2	yes	—
505	C₄⋊C₄	C₂×D₄×C₈₉	1	1	yes	characteristic
506	C₂²⋊C₄	—	2	4	no	—
507	C₂²⋊C₄	—	2	2	no	—
508	C₂²⋊C₄	—	2	1	no	—
509	C₂²⋊C₄	—	2	1	no	—
510	C₂²⋊C₄	—	2	1	no	—
511	C₄²	—	2	4	no	—
512	C₄²	—	2	1	no	—
513	C₄²	—	2	1	no	—
514	C₄²	—	2	1	no	—
515	C₄²	—	2	2	no	—
516	C₄²	—	2	2	no	—
517	C₄²	—	2	2	no	—
518	C₄²	—	4	1	no	—
519	C₄²	—	4	1	no	—
520	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
521	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
522	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
523	C₄²	C₂×D₄×C₈₉	1	1	yes	characteristic
524	C₂³	—	2	1	no	—
525	C₂³	C₂²×D₄×C₈₉	1	1	yes	characteristic
526	Q₈	—	2	1	no	—
527	Q₈	—	2	2	no	—
528	Q₈	—	2	2	no	—
529	Q₈	—	4	4	no	—
530	Q₈	—	4	2	no	—
531	Q₈	—	4	2	no	—
532	Q₈	—	4	1	no	—
533	Q₈	—	4	1	no	—
534	Q₈	—	8	2	no	—
535	Q₈	—	8	4	no	—
536	Q₈	C₂²⋊Q₈×C₈₉	1	2	yes	—
537	D₄	—	2	2	no	—
538	D₄	—	2	2	no	—
539	D₄	—	2	1	no	—
540	D₄	C₂²⋊Q₈×C₈₉	1	2	yes	—
541	C₂×C₄	—	2	1	no	—
542	C₂×C₄	—	2	2	no	—
543	C₂×C₄	—	2	2	no	—
544	C₂×C₄	—	2	2	no	—
545	C₂×C₄	—	2	4	no	—
546	C₂×C₄	—	2	2	no	—
547	C₂×C₄	—	2	1	no	—
548	C₂×C₄	—	2	1	no	—
549	C₂×C₄	—	2	4	no	—
550	C₂×C₄	—	2	2	no	—
551	C₂×C₄	—	2	2	no	—
552	C₂×C₄	—	4	1	no	—
553	C₂×C₄	—	4	2	no	—
554	C₂×C₄	—	4	2	no	—
555	C₂×C₄	—	4	1	no	—
556	C₂×C₄	—	4	2	no	—
557	C₂×C₄	—	4	1	no	—
558	C₂×C₄	—	4	2	no	—
559	C₂×C₄	—	4	4	no	—
560	C₂×C₄	—	4	2	no	—
561	C₂×C₄	—	4	1	no	—
562	C₂×C₄	—	8	4	no	—
563	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
564	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
565	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
566	C₂×C₄	#120×C₈₉	1	1	yes	characteristic
567	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
568	C₂×C₄	C₂²×D₄×C₈₉	1	1	yes	characteristic
569	C₂×C₄	C₂²×Q₈×C₈₉	1	1	yes	characteristic
570	C₂×C₄	C₂×#41×C₈₉	1	1	yes	characteristic
571	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
572	C₂×C₄	2₋¹⁺⁴×C₈₉	1	1	yes	characteristic
573	C₈	—	2	2	no	—
574	C₈	—	4	1	no	—
575	C₈	—	4	2	no	—
576	C₂²	—	2	1	no	—
577	C₂²	—	2	1	no	—
578	C₂²	—	2	2	no	—
579	C₂²	—	4	1	no	—
580	C₂²	—	4	1	no	—
581	C₂²	C₂×#120×C₈₉	1	1	yes	characteristic
582	C₂²	#548×C₈₉	1	1	yes	characteristic
583	C₂²	D₄×Q₈×C₈₉	1	1	yes	characteristic
584	C₄	—	2	2	no	—
585	C₄	—	2	1	no	—
586	C₄	—	2	1	no	—
587	C₄	—	2	1	no	—
588	C₄	—	4	4	no	—
589	C₄	—	4	2	no	—
590	C₄	—	4	1	no	—
591	C₄	—	4	2	no	—
592	C₄	—	4	1	no	—
593	C₄	—	4	4	no	—
594	C₄	—	4	2	no	—
595	C₄	—	8	1	no	—
596	C₄	#456×C₈₉	1	2	yes	—
597	C₄	C₂×C₂²⋊Q₈×C₈₉	1	1	yes	characteristic
598	C₄	#536×C₈₉	1	1	yes	characteristic
599	C₄	#548×C₈₉	1	1	yes	characteristic
600	C₄	D₄×Q₈×C₈₉	1	1	yes	characteristic
601	C₂	—	2	1	no	—
602	C₂	—	2	1	no	—
603	C₂	—	4	2	no	—
604	C₂	C₈₉×#2409	1	1	yes	characteristic
605	C₂	C₂×#456×C₈₉	1	1	yes	characteristic
606	C₂	C₈₉×#3019	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₈₉	C₂×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₈₉
prime radical	C₂×#41×C₈₉	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈₉ ◁ C₂×C₄²×C₈₉ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₈₉×#3078	C₂	semidirect
4. C₈₉×#2567	C₂	semidirect
5. C₈₉×#2060	C₂	semidirect
6. C₈₉×#2028	C₂	semidirect
7. C₈₉×#1997	C₂	semidirect
8. C₈₉×#1986	C₂	semidirect
58. #137×C₈₉	Q₈	semidirect
153. C₂×Q₈×C₈₉	C₂×Q₈	semidirect
174. M₁₆×C₈₉	C₂×Q₈	semidirect
297. #24323	C₈₉	direct (N × H)
305. #3078	C₂×C₈₉	semidirect
307. #2567	C₂×C₈₉	semidirect
308. #2060	C₂×C₈₉	semidirect
309. #2028	C₂×C₈₉	semidirect
310. #1997	C₂×C₈₉	semidirect
311. #1986	C₂×C₈₉	semidirect
315. C₈₉	#24323	direct (N × H)
362. #137	Q₈×C₈₉	semidirect
457. C₂×Q₈	C₂×Q₈×C₈₉	semidirect
478. M₁₆	C₂×Q₈×C₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₈₉×#3503	2
C₈₉×#3078	2
C₄×#137×C₈₉	2
C₈₉×#2567	2
C₈₉×#2060	2
C₈₉×#2028	2
C₈₉×#1997	2
C₈₉×#1986	2
C₈₉×#1882	2
C₈₉×#1859	2
C₈₉×#1806	2
#24323	89

Subgroups — C89×#24323 #36402171

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₉×#24323 #36402171