Subgroups — #36525902

Order 22784.

708 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×#13083561	C₂	1	1	yes	maximal, characteristic
2	M₁₆×D₃₅₆	C₂	1	1	yes	maximal, characteristic
3	#26019152	C₂	1	2	yes	maximal, complemented
4	#26019130	C₂	1	2	yes	maximal, complemented
5	#26019107	C₂	1	2	yes	maximal, complemented
6	#26019062	C₂	1	2	yes	maximal, complemented
7	#26019010	C₂	1	1	yes	maximal, characteristic, complemented
8	#26018681	C₂	1	1	yes	maximal, characteristic, complemented
9	#26018676	C₂	1	1	yes	maximal, characteristic
10	#26018539	C₂	1	1	yes	maximal, characteristic, complemented
11	Q₈×M₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×#13083561	C₄	1	1	yes	characteristic
13	C₂×M₁₆×D₈₉	—	2	1	no	—
14	C₂×C₄×D₃₅₆	C₂²	1	1	yes	characteristic
15	C₄×#13083561	C₄	1	1	yes	characteristic
16	C₄×#13083561	C₄	1	1	yes	characteristic
17	C₄×#13083561	C₄	1	1	yes	characteristic
18	C₄×#13083561	C₄	1	1	yes	characteristic
19	C₄×#13083561	C₂²	1	1	yes	characteristic
20	C₄×#13083561	C₂²	1	1	yes	characteristic
21	C₄×#13083561	C₂²	1	1	yes	characteristic
22	C₄×#13083561	C₂²	1	1	yes	characteristic
23	#19151677	C₂²	1	2	yes	—
24	#19151671	—	2	2	no	—
25	C₈×D₃₅₆	C₂²	1	2	yes	—
26	#19151651	C₂²	1	1	yes	characteristic
27	C₂×#15667943	C₄	1	1	yes	characteristic
28	#19151622	C₂²	1	1	yes	characteristic
29	C₂×#15667936	C₄	1	1	yes	characteristic
30	C₂×#15667932	C₄	1	1	yes	characteristic
31	#19151561	—	2	1	no	—
32	C₂×C₄×#10623494	C₂²	1	1	yes	characteristic
33	#19151350	C₂²	1	2	yes	complemented
34	#19151350	C₂²	1	2	yes	complemented
35	Q₈⋊#10623494	C₂²	1	2	yes	complemented
36	Q₈⋊#10623494	C₂²	1	2	yes	complemented
37	#19151325	C₂²	1	2	yes	—
38	#19151325	C₂²	1	2	yes	—
39	#19151320	C₂²	1	2	yes	complemented
40	#19151320	C₂²	1	2	yes	complemented
41	C₂×C₄×Q₈×C₈₉	C₂²	1	1	yes	characteristic
42	#446×C₈₉	—	2	2	no	—
43	#446×C₈₉	C₂²	1	2	yes	complemented
44	C₈×Q₈×C₈₉	C₂²	1	2	yes	complemented
45	#423×C₈₉	—	2	1	no	—
46	#423×C₈₉	C₂²	1	1	yes	characteristic
47	C₄×M₁₆×C₈₉	—	2	1	no	—
48	C₄×M₁₆×C₈₉	C₂²	1	1	yes	characteristic
49	C₂²×D₃₅₆	C₂×C₄	1	1	yes	characteristic
50	C₂²×C₄×D₈₉	—	2	1	no	—
51	C₂×#13083561	—	2	4	no	—
52	C₂×#13083561	C₂×C₄	1	1	yes	characteristic
53	C₂×#13083561	C₂×C₄	1	1	yes	characteristic
54	C₂×#13083561	C₂×C₄	1	1	yes	characteristic
55	C₂×#13083561	C₂×C₄	1	1	yes	characteristic
56	C₂×#13083558	—	2	1	no	—
57	C₂×#13083557	—	2	2	no	—
58	M₁₆×D₈₉	—	4	4	no	—
59	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
60	C₂²×#10623494	C₂×C₄	1	1	yes	characteristic
61	C₂×C₈×D₈₉	—	2	2	no	—
62	C₄×D₃₅₆	—	2	1	no	—
63	C₄×D₃₅₆	—	2	1	no	—
64	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
65	C₄×D₃₅₆	C₂×C₄	1	1	yes	characteristic
66	C₄×D₃₅₆	C₂³	1	1	yes	characteristic
67	C₄×D₃₅₆	C₂³	1	1	yes	characteristic
68	C₂×C₄⋊Dic₈₉	C₂×C₄	1	1	yes	characteristic
69	C₂×#13083551	—	2	1	no	—
70	#15667943	C₂×C₄	1	1	yes	characteristic
71	#15667943	C₂×C₄	1	1	yes	characteristic
72	#15667943	C₂×C₄	1	1	yes	characteristic
73	#15667943	C₂×C₄	1	1	yes	characteristic
74	C₄⋊#10623494	—	2	1	no	—
75	C₄⋊#10623494	—	2	1	no	—
76	C₄⋊#10623494	C₂³	1	2	yes	—
77	#15667936	C₂×C₄	1	1	yes	characteristic
78	#15667936	C₂×C₄	1	1	yes	characteristic
79	#15667936	C₂×C₄	1	1	yes	characteristic
80	#15667936	C₂×C₄	1	1	yes	characteristic
81	#15667932	C₂×C₄	1	1	yes	characteristic
82	#15667932	C₂×C₄	1	1	yes	characteristic
83	#15667932	C₂×C₄	1	1	yes	characteristic
84	#15667932	C₂×C₄	1	1	yes	characteristic
85	C₂²⋊#10623494	—	2	1	no	—
86	#15667924	—	2	2	no	—
87	#15667924	—	2	2	no	—
88	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
89	C₄×#10623494	C₂×C₄	1	1	yes	characteristic
90	C₄×#10623494	C₂³	1	2	yes	—
91	C₄×#10623494	C₂³	1	1	yes	characteristic
92	C₄×#10623494	C₂³	1	1	yes	characteristic
93	C₂²×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
94	C₂×M₁₆×C₈₉	—	2	1	no	—
95	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
96	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
97	C₄×Q₈×C₈₉	—	2	1	no	—
98	C₄×Q₈×C₈₉	—	2	1	no	—
99	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
100	C₄×Q₈×C₈₉	C₂×C₄	1	1	yes	characteristic
101	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
102	C₄×Q₈×C₈₉	C₂³	1	1	yes	characteristic
103	C₂×C₄⋊C₄×C₈₉	—	2	1	no	—
104	C₂×C₄⋊C₄×C₈₉	C₂×C₄	1	1	yes	characteristic
105	C₂×C₄²×C₈₉	—	2	1	no	—
106	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
107	C₄⋊C₈×C₈₉	—	2	2	no	—
108	C₄⋊C₈×C₈₉	—	2	2	no	—
109	C₄⋊C₈×C₈₉	—	2	1	no	—
110	C₄⋊C₈×C₈₉	C₂³	1	2	yes	—
111	#97×C₈₉	—	2	2	no	—
112	#97×C₈₉	D₄	1	2	yes	—
113	C₄×C₈×C₈₉	—	2	2	no	—
114	C₄×C₈×C₈₉	C₂³	1	2	yes	—
115	C₂³×D₈₉	—	2	1	no	—
116	C₂×D₃₅₆	—	2	2	no	—
117	C₂×D₃₅₆	—	2	1	no	—
118	C₂×D₃₅₆	—	4	1	no	—
119	C₂×D₃₅₆	C₂²×C₄	1	1	yes	characteristic
120	C₂×D₃₅₆	C₂²×C₄	1	1	yes	characteristic
121	C₂²×Dic₈₉	—	2	1	no	—
122	C₂×C₄×D₈₉	—	2	1	no	—
123	C₂×C₄×D₈₉	—	2	1	no	—
124	C₂×C₄×D₈₉	—	2	1	no	—
125	C₂×C₄×D₈₉	—	2	1	no	—
126	C₂×C₄×D₈₉	—	4	2	no	—
127	C₂×C₄×D₈₉	—	4	2	no	—
128	#13083561	—	2	2	no	—
129	#13083561	—	2	2	no	—
130	#13083561	—	2	2	no	—
131	#13083561	M₁₆	1	4	yes	complemented
132	#13083558	—	4	2	no	—
133	#13083557	—	4	4	no	—
134	C₂×#10623494	—	2	2	no	—
135	C₂×#10623494	—	2	2	no	—
136	C₂×#10623494	—	2	2	no	—
137	C₂×#10623494	—	4	1	no	—
138	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
139	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
140	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
141	C₂×#10623494	C₂²×C₄	1	1	yes	characteristic
142	C₈×D₈₉	—	4	4	no	—
143	C₄⋊Dic₈₉	—	2	1	no	—
144	C₄⋊Dic₈₉	C₂²×C₄	1	1	yes	characteristic
145	C₄⋊Dic₈₉	C₂²×C₄	1	1	yes	characteristic
146	#13083551	—	2	1	no	—
147	#13083551	—	2	1	no	—
148	#13083551	—	2	1	no	—
149	#13083551	—	2	1	no	—
150	C₂×Q₈×C₈₉	—	2	1	no	—
151	C₂×Q₈×C₈₉	—	2	2	no	—
152	C₂×Q₈×C₈₉	—	4	1	no	—
153	C₂×Q₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
154	C₂×Q₈×C₈₉	C₂²×C₄	1	1	yes	characteristic
155	C₂²×C₄×C₈₉	—	2	1	no	—
156	C₂²×C₄×C₈₉	—	2	1	no	—
157	C₂²×C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
158	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
159	C₂²×C₄×C₈₉	#41	1	1	yes	characteristic
160	M₁₆×C₈₉	—	2	2	no	—
161	M₁₆×C₈₉	—	2	2	no	—
162	M₁₆×C₈₉	—	4	2	no	—
163	C₂×C₈×C₈₉	—	2	2	no	—
164	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
165	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
166	C₄⋊C₄×C₈₉	—	2	1	no	—
167	C₄⋊C₄×C₈₉	—	2	1	no	—
168	C₄⋊C₄×C₈₉	—	2	1	no	—
169	C₄⋊C₄×C₈₉	—	2	1	no	—
170	C₄⋊C₄×C₈₉	—	2	1	no	—
171	C₄⋊C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
172	C₄⋊C₄×C₈₉	C₂²×C₄	1	1	yes	characteristic
173	C₄²×C₈₉	—	2	1	no	—
174	C₄²×C₈₉	—	2	1	no	—
175	C₄²×C₈₉	—	2	1	no	—
176	C₄²×C₈₉	—	2	1	no	—
177	C₄²×C₈₉	C₂²×C₄	1	1	yes	characteristic
178	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
179	C₄²×C₈₉	#41	1	1	yes	characteristic
180	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
181	C₂³×C₈₉	C₄×D₄	1	1	yes	characteristic
182	C₂²×D₈₉	—	2	1	no	—
183	C₂²×D₈₉	—	2	1	no	—
184	C₂²×D₈₉	—	4	2	no	—
185	C₂²×D₈₉	—	4	2	no	—
186	C₂²×D₈₉	—	8	1	no	—
187	Q₈×C₈₉	—	2	1	no	—
188	Q₈×C₈₉	—	2	1	no	—
189	Q₈×C₈₉	—	2	1	no	—
190	Q₈×C₈₉	—	4	2	no	—
191	Q₈×C₈₉	C₂×M₁₆	1	2	yes	complemented
192	C₂×C₄×C₈₉	—	2	1	no	—
193	C₂×C₄×C₈₉	—	2	1	no	—
194	C₂×C₄×C₈₉	—	2	1	no	—
195	C₂×C₄×C₈₉	—	2	1	no	—
196	C₂×C₄×C₈₉	—	2	1	no	—
197	C₂×C₄×C₈₉	—	2	1	no	—
198	C₂×C₄×C₈₉	—	4	1	no	—
199	C₂×C₄×C₈₉	—	4	2	no	—
200	C₂×C₄×C₈₉	—	8	1	no	—
201	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
202	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
203	C₂×C₄×C₈₉	C₄×D₄	1	1	yes	characteristic
204	C₂×C₄×C₈₉	AGL(1,ℤ/8)	1	2	yes	—
205	C₂×C₄×C₈₉	#137	1	2	yes	—
206	C₂×C₄×C₈₉	C₂³×C₄	1	1	yes	characteristic
207	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₄×C₈₉	C₂×#41	1	1	yes	characteristic
209	C₂×Dic₈₉	—	2	1	no	—
210	C₂×Dic₈₉	—	2	1	no	—
211	C₂×Dic₈₉	—	4	2	no	—
212	D₃₅₆	—	2	1	no	—
213	D₃₅₆	—	2	1	no	—
214	D₃₅₆	—	2	1	no	—
215	D₃₅₆	—	4	2	no	—
216	D₃₅₆	C₂×M₁₆	1	2	yes	—
217	C₄×D₈₉	—	4	1	no	—
218	C₄×D₈₉	—	4	1	no	—
219	C₄×D₈₉	—	4	1	no	—
220	C₄×D₈₉	—	4	1	no	—
221	C₄×D₈₉	—	4	2	no	—
222	C₄×D₈₉	—	4	2	no	—
223	C₈×C₈₉	—	2	2	no	—
224	C₈×C₈₉	—	2	2	no	—
225	C₈×C₈₉	—	4	2	no	—
226	#10623494	—	2	1	no	—
227	#10623494	—	2	1	no	—
228	#10623494	—	2	1	no	—
229	#10623494	—	4	2	no	—
230	#10623494	—	4	2	no	—
231	#10623494	C₂×M₁₆	1	2	yes	complemented
232	C₂²×C₈₉	—	2	1	no	—
233	C₂²×C₈₉	—	2	1	no	—
234	C₂²×C₈₉	#440	1	2	yes	—
235	C₂²×C₈₉	C₂×C₄×D₄	1	1	yes	characteristic
236	C₂×D₈₉	—	2	2	no	—
237	C₂×D₈₉	—	4	1	no	—
238	C₂×D₈₉	—	4	1	no	—
239	C₂×D₈₉	—	4	1	no	—
240	C₂×D₈₉	—	4	1	no	—
241	C₂×D₈₉	—	4	1	no	—
242	C₂×D₈₉	—	8	4	no	—
243	C₄×C₈₉	—	2	1	no	—
244	C₄×C₈₉	—	2	2	no	—
245	C₄×C₈₉	—	2	1	no	—
246	C₄×C₈₉	—	2	1	no	—
247	C₄×C₈₉	—	4	1	no	—
248	C₄×C₈₉	—	4	1	no	—
249	C₄×C₈₉	—	4	1	no	—
250	C₄×C₈₉	#440	1	2	yes	—
251	C₄×C₈₉	C₂²×M₁₆	1	1	yes	characteristic
252	C₄×C₈₉	#568	1	1	yes	characteristic
253	C₄×C₈₉	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
254	C₄×C₈₉	C₂×#137	1	1	yes	characteristic
255	Dic₈₉	—	4	1	no	—
256	Dic₈₉	—	4	1	no	—
257	#18040	—	89	1	no	maximal, Sylow
258	C₂×C₈₉	—	2	2	no	—
259	C₂×C₈₉	D₄×M₁₆	1	1	yes	characteristic
260	C₂×C₈₉	C₂×#440	1	1	yes	characteristic
261	C₂×C₈₉	#2957	1	1	yes	characteristic
262	D₈₉	—	4	2	no	—
263	D₈₉	—	8	1	no	—
264	Q₈×M₁₆	D₈₉	1	1	yes	characteristic, complemented
265	C₂×C₄×SD₁₆	—	89	1	no	—
266	D₄×M₁₆	—	89	1	no	—
267	#1592	—	89	2	no	—
268	#1588	—	89	2	no	—
269	#1585	—	89	2	no	—
270	SD₁₆⋊C₈	—	89	2	no	—
271	#1567	—	89	1	no	—
272	#1487	—	89	1	no	—
273	#1486	—	89	1	no	—
274	#1451	—	89	1	no	—
275	C₈₉	#18040	1	1	yes	characteristic, complemented, Sylow
276	C₂²×SD₁₆	—	89	1	no	—
277	C₂²×M₁₆	—	178	1	no	—
278	C₂×C₄×Q₈	C₂×D₈₉	1	1	yes	characteristic
279	C₂×C₄×D₄	—	89	1	no	—
280	#446	—	2	2	no	—
281	#446	C₂×D₈₉	1	2	yes	complemented
282	C₈×Q₈	C₂×D₈₉	1	2	yes	complemented
283	C₄×SD₁₆	—	89	1	no	—
284	C₄×SD₁₆	—	89	1	no	—
285	C₄×SD₁₆	—	89	1	no	—
286	C₄×SD₁₆	—	89	1	no	—
287	C₄×SD₁₆	—	89	1	no	—
288	C₄×SD₁₆	—	89	1	no	—
289	C₄×SD₁₆	—	89	1	no	—
290	C₄×SD₁₆	—	89	1	no	—
291	#436	—	89	2	no	—
292	#435	—	178	2	no	—
293	C₈×D₄	—	89	2	no	—
294	#431	—	89	1	no	—
295	C₂×#106	—	89	1	no	—
296	#423	—	2	1	no	—
297	#423	—	89	1	no	—
298	#423	C₂×D₈₉	1	1	yes	characteristic
299	C₂×#103	—	89	1	no	—
300	C₂×#102	—	89	1	no	—
301	#407	—	178	1	no	—
302	C₄×M₁₆	—	2	1	no	—
303	C₄×M₁₆	C₂×D₈₉	1	1	yes	characteristic
304	C₂×C₄×C₈	—	89	1	no	—
305	#335	—	89	2	no	—
306	#335	—	89	2	no	—
307	Q₈⋊C₈	—	89	2	no	—
308	Q₈⋊C₈	—	89	2	no	—
309	D₄⋊C₈	—	89	2	no	—
310	D₄⋊C₈	—	89	2	no	—
311	#322	—	89	2	no	—
312	#322	—	89	2	no	—
313	C₂²×Q₈	C₄×D₈₉	1	1	yes	characteristic
314	C₂²×D₄	—	89	1	no	—
315	C₂³×C₄	—	178	1	no	—
316	C₂×SD₁₆	—	89	1	no	—
317	C₂×SD₁₆	—	89	1	no	—
318	C₂×SD₁₆	—	89	1	no	—
319	C₂×SD₁₆	—	89	1	no	—
320	C₂×SD₁₆	—	178	4	no	—
321	C₂×M₁₆	—	2	1	no	—
322	C₂×M₁₆	—	356	4	no	—
323	C₂×M₁₆	—	178	1	no	—
324	C₂×M₁₆	—	178	2	no	—
325	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
326	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
327	C₂²×C₈	—	89	1	no	—
328	C₂²×C₈	—	89	1	no	—
329	C₂²×C₈	—	178	2	no	—
330	C₄×Q₈	—	2	1	no	—
331	C₄×Q₈	—	2	1	no	—
332	C₄×Q₈	C₄×D₈₉	1	1	yes	characteristic
333	C₄×Q₈	C₄×D₈₉	1	1	yes	characteristic
334	C₄×Q₈	C₂²×D₈₉	1	1	yes	characteristic
335	C₄×Q₈	C₂²×D₈₉	1	1	yes	characteristic
336	C₄×D₄	—	89	1	no	—
337	C₄×D₄	—	89	1	no	—
338	C₄×D₄	—	89	1	no	—
339	C₄×D₄	—	89	1	no	—
340	C₄×D₄	—	178	1	no	—
341	C₄×D₄	—	178	1	no	—
342	C₂×C₄⋊C₄	—	2	1	no	—
343	C₂×C₄⋊C₄	—	89	1	no	—
344	C₂×C₄⋊C₄	C₄×D₈₉	1	1	yes	characteristic
345	C₂×C₂²⋊C₄	—	178	1	no	—
346	C₂×C₄²	—	2	1	no	—
347	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
348	#106	—	89	1	no	—
349	#106	—	89	1	no	—
350	#106	—	89	1	no	—
351	#106	—	89	1	no	—
352	C₄⋊C₈	—	2	2	no	—
353	C₄⋊C₈	—	2	2	no	—
354	C₄⋊C₈	—	2	1	no	—
355	C₄⋊C₈	—	89	2	no	—
356	C₄⋊C₈	—	178	1	no	—
357	C₄⋊C₈	—	178	1	no	—
358	C₄⋊C₈	C₂²×D₈₉	1	2	yes	—
359	#103	—	89	1	no	—
360	#103	—	89	1	no	—
361	#103	—	89	1	no	—
362	#103	—	89	1	no	—
363	#102	—	89	1	no	—
364	#102	—	89	1	no	—
365	#102	—	89	1	no	—
366	#102	—	89	1	no	—
367	C₂²⋊C₈	—	178	1	no	—
368	C₂²⋊C₈	—	178	2	no	—
369	C₂²⋊C₈	—	178	2	no	—
370	#97	—	2	2	no	—
371	#97	C₂²⋊D₈₉	1	2	yes	—
372	C₄×C₈	—	2	2	no	—
373	C₄×C₈	—	89	1	no	—
374	C₄×C₈	—	89	1	no	—
375	C₄×C₈	—	89	1	no	—
376	C₄×C₈	—	89	1	no	—
377	C₄×C₈	—	89	2	no	—
378	C₄×C₈	C₂²×D₈₉	1	2	yes	—
379	C₂⁴	—	178	1	no	—
380	C₂×Q₈	—	2	1	no	—
381	C₂×Q₈	—	2	2	no	—
382	C₂×Q₈	—	4	1	no	—
383	C₂×Q₈	C₂×C₄×D₈₉	1	1	yes	characteristic
384	C₂×Q₈	C₂×C₄×D₈₉	1	1	yes	characteristic
385	C₂×D₄	—	89	1	no	—
386	C₂×D₄	—	89	1	no	—
387	C₂×D₄	—	356	1	no	—
388	C₂×D₄	—	178	2	no	—
389	C₂×D₄	—	178	1	no	—
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	356	2	no	—
393	C₂²×C₄	—	356	2	no	—
394	C₂²×C₄	—	178	1	no	—
395	C₂²×C₄	—	178	1	no	—
396	C₂²×C₄	—	178	1	no	—
397	C₂²×C₄	—	178	1	no	—
398	C₂²×C₄	—	178	1	no	—
399	C₂²×C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
400	C₂²×C₄	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
401	C₂²×C₄	#13083574	1	1	yes	characteristic
402	SD₁₆	—	89	4	no	—
403	SD₁₆	—	178	2	no	—
404	SD₁₆	—	178	2	no	—
405	SD₁₆	—	178	2	no	—
406	M₁₆	—	2	2	no	—
407	M₁₆	—	2	2	no	—
408	M₁₆	—	4	2	no	—
409	M₁₆	—	356	4	no	—
410	M₁₆	—	356	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	89	1	no	—
413	C₂×C₈	—	89	1	no	—
414	C₂×C₈	—	89	1	no	—
415	C₂×C₈	—	89	1	no	—
416	C₂×C₈	—	356	4	no	—
417	C₂×C₈	—	356	1	no	—
418	C₂×C₈	—	178	2	no	—
419	C₂×C₈	—	178	2	no	—
420	C₂×C₈	—	178	2	no	—
421	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
422	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
423	C₄⋊C₄	—	2	1	no	—
424	C₄⋊C₄	—	2	1	no	—
425	C₄⋊C₄	—	2	1	no	—
426	C₄⋊C₄	—	2	1	no	—
427	C₄⋊C₄	—	2	1	no	—
428	C₄⋊C₄	—	89	1	no	—
429	C₄⋊C₄	—	89	1	no	—
430	C₄⋊C₄	—	178	1	no	—
431	C₄⋊C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
432	C₄⋊C₄	C₂×C₄×D₈₉	1	1	yes	characteristic
433	C₂²⋊C₄	—	178	1	no	—
434	C₂²⋊C₄	—	178	1	no	—
435	C₂²⋊C₄	—	178	1	no	—
436	C₂²⋊C₄	—	178	1	no	—
437	C₄²	—	2	1	no	—
438	C₄²	—	2	1	no	—
439	C₄²	—	2	1	no	—
440	C₄²	—	2	1	no	—
441	C₄²	C₂×C₄×D₈₉	1	1	yes	characteristic
442	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
443	C₄²	#13083574	1	1	yes	characteristic
444	C₄²	C₂³×D₈₉	1	1	yes	characteristic
445	C₂³	—	356	2	no	—
446	C₂³	—	356	2	no	—
447	C₂³	—	178	1	no	—
448	C₂³	—	178	1	no	—
449	C₂³	—	712	1	no	—
450	C₂³	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
451	Q₈	—	2	1	no	—
452	Q₈	—	2	1	no	—
453	Q₈	—	2	1	no	—
454	Q₈	—	4	2	no	—
455	Q₈	M₁₆×D₈₉	1	2	yes	complemented
456	D₄	—	89	2	no	—
457	D₄	—	356	2	no	—
458	D₄	—	178	1	no	—
459	D₄	—	178	1	no	—
460	D₄	—	178	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	1	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	4	1	no	—
468	C₂×C₄	—	4	2	no	—
469	C₂×C₄	—	8	1	no	—
470	C₂×C₄	—	356	2	no	—
471	C₂×C₄	—	356	1	no	—
472	C₂×C₄	—	356	1	no	—
473	C₂×C₄	—	356	1	no	—
474	C₂×C₄	—	356	1	no	—
475	C₂×C₄	—	356	2	no	—
476	C₂×C₄	—	356	2	no	—
477	C₂×C₄	—	178	1	no	—
478	C₂×C₄	—	178	1	no	—
479	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
480	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
481	C₂×C₄	C₄×C₂²⋊D₈₉	1	1	yes	characteristic
482	C₂×C₄	#15668037	1	2	yes	—
483	C₂×C₄	#15668043	1	2	yes	—
484	C₂×C₄	C₂²×C₄×D₈₉	1	1	yes	characteristic
485	C₂×C₄	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
486	C₂×C₄	C₂×#13083574	1	1	yes	characteristic
487	C₈	—	2	2	no	—
488	C₈	—	2	2	no	—
489	C₈	—	4	2	no	—
490	C₈	—	89	2	no	—
491	C₈	—	356	2	no	—
492	C₈	—	356	2	no	—
493	C₈	—	178	1	no	—
494	C₈	—	178	1	no	—
495	C₈	—	178	1	no	—
496	C₂²	—	2	1	no	—
497	C₂²	—	2	1	no	—
498	C₂²	—	356	1	no	—
499	C₂²	—	356	1	no	—
500	C₂²	—	356	1	no	—
501	C₂²	—	356	1	no	—
502	C₂²	—	356	1	no	—
503	C₂²	—	178	2	no	—
504	C₂²	—	712	4	no	—
505	C₂²	#19151696	1	2	yes	—
506	C₂²	C₂×C₄×C₂²⋊D₈₉	1	1	yes	characteristic
507	C₄	—	2	1	no	—
508	C₄	—	2	1	no	—
509	C₄	—	2	1	no	—
510	C₄	—	2	2	no	—
511	C₄	—	4	1	no	—
512	C₄	—	4	1	no	—
513	C₄	—	4	1	no	—
514	C₄	—	356	1	no	—
515	C₄	—	356	1	no	—
516	C₄	#19151696	1	2	yes	—
517	C₄	C₂×M₁₆×D₈₉	1	1	yes	characteristic
518	C₄	#19152364	1	1	yes	characteristic
519	C₄	C₂×#15668037	1	1	yes	characteristic
520	C₄	C₂×#15668043	1	1	yes	characteristic
521	C₂	—	2	2	no	—
522	C₂	—	356	2	no	—
523	C₂	—	712	1	no	—
524	C₂	M₁₆×C₂²⋊D₈₉	1	1	yes	characteristic
525	C₂	C₂×#19151696	1	1	yes	characteristic
526	C₂	#26029167	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊D₈₉
commutator	C₂×C₄×C₈₉	C₂³×C₄
Frattini	C₄²	C₂³×D₈₉
Fitting	Q₈×M₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×C₄×D₄
prime radical	C₂²×D₃₅₆	C₂×C₄
cosocle	C₄²×C₈₉	C₂⁴
hypercenter	Q₈×M₁₆	D₈₉
hypocenter	C₈₉	#18040
LCS	G ▷ C₂×C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆ ◁ Q₈×M₁₆

Complements

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

N	H ≅ G/N	type
3. #26019152	C₂	semidirect
4. #26019130	C₂	semidirect
5. #26019107	C₂	semidirect
6. #26019062	C₂	semidirect
7. #26019010	C₂	semidirect
8. #26018681	C₂	semidirect
10. #26018539	C₂	semidirect
11. Q₈×M₁₆×C₈₉	C₂	semidirect
33. #19151350	C₂²	semidirect
34. #19151350	C₂²	semidirect
35. Q₈⋊#10623494	C₂²	semidirect
36. Q₈⋊#10623494	C₂²	semidirect
39. #19151320	C₂²	semidirect
40. #19151320	C₂²	semidirect
43. #446×C₈₉	C₂²	semidirect
44. C₈×Q₈×C₈₉	C₂²	semidirect
131. #13083561	M₁₆	semidirect
191. Q₈×C₈₉	C₂×M₁₆	semidirect
231. #10623494	C₂×M₁₆	semidirect
264. Q₈×M₁₆	D₈₉	semidirect
275. C₈₉	#18040	semidirect
281. #446	C₂×D₈₉	semidirect
282. C₈×Q₈	C₂×D₈₉	semidirect
455. Q₈	M₁₆×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×#13083561	2
M₁₆×D₃₅₆	2
#26019152	2
#26019130	2
#26019107	2
#26019062	2
#26019010	2
#26018681	2
#26018676	2
#26018539	2
Q₈×M₁₆×C₈₉	2
#18040	89