Finite Group Databasebeta

← #36524345

Subgroups — #36524345

Order 22784.

698 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26030256C211yesmaximal, characteristic
2C2×C8×D356C211yesmaximal, characteristic
3#26019107C214yesmaximal, complemented
4#26019062C214yesmaximal, complemented
5#26019028C211yesmaximal, characteristic, complemented
6#26018676C211yesmaximal, characteristic
7#26018655C211yesmaximal, characteristic, complemented
8#26018544C211yesmaximal, characteristic, complemented
9C89×#2960C211yesmaximal, characteristic, complemented
10C22×C8×D89—21no—
11C2×C4×D356C2211yescharacteristic
12#19151926C412yes—
13#19151885C412yes—
14#19151800C412yes—
15#19151748C412yes—
16C4×#13083561C2214yes—
17C8×D356—21no—
18C8×D356—21no—
19C8×D356C2212yes—
20C8×D356C2212yes—
21#19151651C2211yescharacteristic
22#19151651C2211yescharacteristic
23C2×C4⋊#10623494C2211yescharacteristic
24C2×#15667924—21no—
25#19151350C2214yescomplemented
26Q8⋊#10623494C2214yescomplemented
27#19151325C2214yes—
28#19151320C2212yescomplemented
29#19151320C2212yescomplemented
30#533×C89C2211yescharacteristic
31#446×C89—21no—
32#446×C89C2212yescomplemented
33C8×Q8×C89—21no—
34C8×Q8×C89C2212yescomplemented
35#424×C89—21no—
36#423×C89C2211yescharacteristic
37#405×C89—21no—
38C2×C4×C8×C89C2211yescharacteristic
39C22×D356C2×C411yescharacteristic
40C22×C4×D89—21no—
41C2×#13083561C2×C414yes—
42C22×#10623494—21no—
43C2×C8×D89—22no—
44C2×C8×D89—22no—
45C2×C8×D89—44no—
46C4×D356—21no—
47C4×D356—21no—
48C4×D356C2×C412yes—
49C4×D356C2312yes—
50C2×C4⋊Dic89C2×C411yescharacteristic
51C2×#13083551—21no—
52#15667943C2×C414yes—
53C4⋊#10623494—21no—
54C4⋊#10623494—21no—
55C4⋊#10623494C2×C412yes—
56C4⋊#10623494C2312yes—
57#15667936C2×C414yes—
58#15667932C2×C414yes—
59C22⋊#10623494—21no—
60C22⋊#10623494C2×C412yes—
61#15667924—22no—
62#15667924—22no—
63C4×#10623494C2312yes—
64C4×#10623494C2312yes—
65C2×M16×C89—21no—
66C22×C8×C89D411yescharacteristic
67C22×C8×C89D411yescharacteristic
68C4⋊Q8×C89C2×C411yescharacteristic
69C4⋊Q8×C89C2×C411yescharacteristic
70#125×C89—21no—
71C22⋊Q8×C89—21no—
72C22⋊Q8×C89C2×C412yes—
73C4×Q8×C89—21no—
74C4×Q8×C89C2312yes—
75#117×C89—21no—
76C2×C42×C89C2311yescharacteristic
77C4⋊C8×C89—22no—
78C4⋊C8×C89—22no—
79C4⋊C8×C89—21no—
80C4⋊C8×C89C2312yes—
81#97×C89—22no—
82C4×C8×C89—22no—
83C4×C8×C89D411yescharacteristic
84C4×C8×C89D411yescharacteristic
85C4×C8×C89C2311yescharacteristic
86C4×C8×C89C2311yescharacteristic
87C23×D89—21no—
88C2×D356—21no—
89C2×D356—24no—
90C2×D356C22×C412yes—
91C22×Dic89—21no—
92C2×C4×D89—22no—
93C2×C4×D89—22no—
94C2×C4×D89—42no—
95C2×C4×D89—42no—
96#13083561—28no—
97C2×#10623494—22no—
98C2×#10623494—22no—
99C2×#10623494—42no—
100C2×#10623494C22×C414yes—
101C8×D89—44no—
102C8×D89—44no—
103C4⋊Dic89—21no—
104C4⋊Dic89C22×C412yes—
105#13083551—22no—
106#13083551—22no—
107C2×Q8×C89—21no—
108C2×Q8×C89C22×C412yes—
109C22×C4×C89C22×C411yescharacteristic
110C22×C4×C89C2×D411yescharacteristic
111C22×C4×C89#4111yescharacteristic
112M16×C89—42no—
113C2×C8×C89—22no—
114C2×C8×C89—22no—
115C2×C8×C89—22no—
116C2×C8×C89C2×D411yescharacteristic
117C2×C8×C89C2×D411yescharacteristic
118C2×C8×C89C2×D411yescharacteristic
119C2×C8×C89C2×D411yescharacteristic
120C4⋊C4×C89—21no—
121C4⋊C4×C89—21no—
122C4⋊C4×C89—21no—
123C4⋊C4×C89—22no—
124C4⋊C4×C89—22no—
125C4⋊C4×C89C22×C412yes—
126C22⋊C4×C89—22no—
127C42×C89—22no—
128C42×C89C22×C411yescharacteristic
129C42×C89C2×D411yescharacteristic
130C42×C89#4111yescharacteristic
131C42×C89C2411yescharacteristic
132C23×C89C4×D411yescharacteristic
133C22×D89—24no—
134C22×D89—22no—
135C22×D89—42no—
136C22×D89—42no—
137Q8×C89—24no—
138Q8×C89—42no—
139C2×C4×C89—22no—
140C2×C4×C89—22no—
141C2×C4×C89—22no—
142C2×C4×C89—41no—
143C2×C4×C89C4×D411yescharacteristic
144C2×C4×C89C4×D411yescharacteristic
145C2×C4×C89C4×D411yescharacteristic
146C2×C4×C89AGL(1,ℤ/8)12yes—
147C2×C4×C89#13712yes—
148C2×C4×C89C23×C411yescharacteristic
149C2×C4×C89C22×D411yescharacteristic
150C2×C4×C89C2×#4111yescharacteristic
151C2×Dic89—22no—
152C2×Dic89—42no—
153D356—24no—
154D356—22no—
155D356#13114yes—
156C4×D89—42no—
157C4×D89—42no—
158C4×D89—44no—
159C8×C89—21no—
160C8×C89—21no—
161C8×C89—21no—
162C8×C89—21no—
163C8×C89—42no—
164#10623494—24no—
165#10623494—42no—
166#10623494—42no—
167C22×C89—21no—
168C22×C89—21no—
169C22×C89#44012yes—
170C22×C89C2×C4×D411yescharacteristic
171C2×D89—24no—
172C2×D89—48no—
173C2×D89—42no—
174C2×D89—42no—
175C4×C89—21no—
176C4×C89—21no—
177C4×C89—21no—
178C4×C89—42no—
179C4×C89—42no—
180C4×C89#44012yes—
181C4×C89C2×#13112yes—
182C4×C89C2×AGL(1,ℤ/8)11yescharacteristic
183C4×C89C2×#13711yescharacteristic
184Dic89—42no—
185#17905—891nomaximal, Sylow
186C2×C89—22no—
187C2×C89#292911yescharacteristic
188C2×C89C2×#44011yescharacteristic
189C2×C89#295511yescharacteristic
190D89—44no—
191#3101—891no—
192#2960D8911yescharacteristic, complemented
193C2×C8×D4—891no—
194#1585—894no—
195SD16⋊C8—894no—
196#1571—891no—
197#1486—891no—
198#1481—891no—
199#1452—891no—
200C89#1790511yescharacteristic, complemented, Sylow
201C23×C8—1781no—
202#533C2×D8911yescharacteristic
203C2×C4×D4—891no—
204#483—892no—
205#476—892no—
206#461—892no—
207#450—892no—
208#446—21no—
209#446C2×D8912yescomplemented
210C8×Q8—21no—
211C8×Q8C2×D8912yescomplemented
212C4×SD16—894no—
213C8×D4—892no—
214C8×D4—892no—
215C8×D4—1781no—
216C8×D4—1781no—
217#431—891no—
218#431—891no—
219#424—21no—
220#423C2×D8911yescharacteristic
221C2×C4⋊C8—891no—
222C2×C22⋊C8—1781no—
223#405—21no—
224C2×C4×C8C2×D8911yescharacteristic
225#335—894no—
226Q8⋊C8—894no—
227D4⋊C8—894no—
228#322—892no—
229#322—892no—
230C22×D4—891no—
231C23×C4—1781no—
232C2×SD16—894no—
233C2×M16—21no—
234C22×C8—3564no—
235C22×C8—1781no—
236C22×C8—1782no—
237C22×C8—1782no—
238C22×C8C22⋊D8911yescharacteristic
239C22×C8C22⋊D8911yescharacteristic
240C4⋊Q8C4×D8911yescharacteristic
241C4⋊Q8C4×D8911yescharacteristic
242#125—21no—
243C22⋊Q8—21no—
244C22⋊Q8C4×D8912yes—
245C4×Q8—21no—
246C4×Q8C22×D8912yes—
247C4×D4—892no—
248C4×D4—892no—
249C4×D4—1781no—
250C4×D4—1781no—
251#117—21no—
252C2×C4⋊C4—891no—
253C2×C22⋊C4—1781no—
254C2×C42C22×D8911yescharacteristic
255#106—894no—
256C4⋊C8—22no—
257C4⋊C8—22no—
258C4⋊C8—21no—
259C4⋊C8—892no—
260C4⋊C8—892no—
261C4⋊C8—1781no—
262C4⋊C8—1781no—
263C4⋊C8C22×D8912yes—
264#103—894no—
265#102—894no—
266C22⋊C8—892no—
267C22⋊C8—1781no—
268C22⋊C8—1782no—
269C22⋊C8—1782no—
270#97—22no—
271C4×C8—22no—
272C4×C8—892no—
273C4×C8—892no—
274C4×C8C22⋊D8911yescharacteristic
275C4×C8C22⋊D8911yescharacteristic
276C4×C8C22×D8911yescharacteristic
277C4×C8C22×D8911yescharacteristic
278C24—1781no—
279C2×Q8—21no—
280C2×Q8C2×C4×D8912yes—
281C2×D4—892no—
282C2×D4—1784no—
283C2×D4—1781no—
284C22×C4—3562no—
285C22×C4—3562no—
286C22×C4—1781no—
287C22×C4—1782no—
288C22×C4—1782no—
289C22×C4C2×C4×D8911yescharacteristic
290C22×C4C2×C22⋊D8911yescharacteristic
291C22×C4#1308357411yescharacteristic
292SD16—1788no—
293M16—42no—
294C2×C8—22no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—894no—
298C2×C8—3564no—
299C2×C8—3564no—
300C2×C8—3562no—
301C2×C8—1782no—
302C2×C8—1782no—
303C2×C8C2×C22⋊D8911yescharacteristic
304C2×C8C2×C22⋊D8911yescharacteristic
305C2×C8C2×C22⋊D8911yescharacteristic
306C2×C8C2×C22⋊D8911yescharacteristic
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—892no—
313C4⋊C4—1781no—
314C4⋊C4C2×C4×D8912yes—
315C22⋊C4—22no—
316C22⋊C4—1782no—
317C22⋊C4—1782no—
318C42—22no—
319C42C2×C4×D8911yescharacteristic
320C42C2×C22⋊D8911yescharacteristic
321C42#1308357411yescharacteristic
322C42C23×D8911yescharacteristic
323C23—3562no—
324C23—3562no—
325C23—1784no—
326C23—1782no—
327C23C4×C22⋊D8911yescharacteristic
328Q8—24no—
329Q8—42no—
330D4—894no—
331D4—1784no—
332D4—1782no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—3562no—
338C2×C4—3562no—
339C2×C4—3562no—
340C2×C4—3564no—
341C2×C4—1782no—
342C2×C4C4×C22⋊D8911yescharacteristic
343C2×C4C4×C22⋊D8911yescharacteristic
344C2×C4C4×C22⋊D8911yescharacteristic
345C2×C4#1566803712yes—
346C2×C4#1566804312yes—
347C2×C4C22×C4×D8911yescharacteristic
348C2×C4C22×C22⋊D8911yescharacteristic
349C2×C4C2×#1308357411yescharacteristic
350C8—21no—
351C8—21no—
352C8—21no—
353C8—21no—
354C8—42no—
355C8—3562no—
356C8—3562no—
357C8—1784no—
358C22—21no—
359C22—21no—
360C22—3568no—
361C22—3562no—
362C22—3562no—
363C22—1784no—
364C22#1915169612yes—
365C22C2×C4×C22⋊D8911yescharacteristic
366C4—21no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—42no—
371C4—3562no—
372C4#1915169612yes—
373C4#131×D8912yes—
374C4C2×#1566803711yescharacteristic
375C4C2×#1566804311yescharacteristic
376C2—22no—
377C2—3564no—
378C2#2602894411yescharacteristic
379C2C2×#1915169611yescharacteristic
380C2#2602915311yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D89
commutatorC2×C4×C89C23×C4
FrattiniC42C23×D89
FittingC89×#2960C2
socleC22×C89C2×C4×D4
prime radicalC22×D356C2×C4
cosocleC42×C89C24
hypercenter#2960D89
hypocenterC89#17905
LCSG ▷ C2×C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #2960

Complements

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

NH ≅ G/Ntype
3. #26019107C2semidirect
4. #26019062C2semidirect
5. #26019028C2semidirect
7. #26018655C2semidirect
8. #26018544C2semidirect
9. C89×#2960C2semidirect
25. #19151350C22semidirect
26. Q8⋊#10623494C22semidirect
28. #19151320C22semidirect
29. #19151320C22semidirect
32. #446×C89C22semidirect
34. C8×Q8×C89C22semidirect
192. #2960D89semidirect
200. C89#17905semidirect
209. #446C2×D89semidirect
211. C8×Q8C2×D89semidirect

Maximal subgroups

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

subgroupindex
#260302562
C2×C8×D3562
#260191072
#260190622
#260190282
#260186762
#260186552
#260185442
C89×#29602
#1790589