Finite Group Databasebeta

← #36711872

Subgroups — #36711872

Order 22784.

504 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26032621C211yesmaximal, characteristic, complemented
2#26022535C211yesmaximal, characteristic, complemented
3#26022454C211yesmaximal, characteristic, complemented
4#26022449C211yesmaximal, characteristic, complemented
5#26022440C211yesmaximal, characteristic, complemented
6#26022430C211yesmaximal, characteristic, complemented
7#26022409C211yesmaximal, characteristic, complemented
8#26022401C211yesmaximal, characteristic, complemented
9#26022363C211yesmaximal, characteristic, complemented
10#26022346C211yesmaximal, characteristic, complemented
11#26022116C211yesmaximal, characteristic, complemented
12#26022078C211yesmaximal, characteristic, complemented
13#26022038C211yesmaximal, characteristic, complemented
14C2×#19151423C211yesmaximal, characteristic, complemented
15C89×#3334C211yesmaximal, characteristic
16#19152288—21no—
17C2×#15667956C2211yescharacteristic
18C2×#15667952C2211yescharacteristic
19C22×#13083549C2211yescharacteristic
20#19152000C2211yescharacteristic
21#19151940—21no—
22#19151929—21no—
23#19151906—21no—
24#19151839C2211yescharacteristic
25#19151839C2211yescharacteristic
26C4×Dic356C2211yescharacteristic
27#19151636C2211yescharacteristic
28C2×#15667946C2211yescharacteristic
29#19151425C2211yescharacteristic
30#19151425C2211yescharacteristic
31#19151425C2211yescharacteristic
32#19151423C2211yescharacteristic
33#19151423C2211yescharacteristic
34#19151423C2211yescharacteristic
35#19151423C2211yescharacteristic
36#19151423C2211yescharacteristic
37#19151423C2211yescharacteristic
38C4⋊#13083549C2211yescharacteristic
39#19151401C2211yescharacteristic
40#19151401C2211yescharacteristic
41#19151401C2211yescharacteristic
42#19151399C2211yescharacteristic
43#19151399C2211yescharacteristic
44#19151399C2211yescharacteristic
45C22⋊#13083549C2211yescharacteristic
46C22⋊#13083549C2211yescharacteristic
47#19151374C2211yescharacteristic
48#547×C89—21no—
49#500×C89C2211yescharacteristic
50#480×C89—21no—
51#480×C89—21no—
52#474×C89—21no—
53#466×C89C2211yescharacteristic
54#466×C89C2211yescharacteristic
55C4×D8×C89C2211yescharacteristic
56C8×D4×C89C2211yescharacteristic
57C2×#107×C89C2211yescharacteristic
58C2×#107×C89C2211yescharacteristic
59C2×Dic356C2311yescharacteristic
60#15668005—41no—
61#15668004—21no—
62#15668002—41no—
63#15667993—41no—
64#15667991—21no—
65#15667991—21no—
66C4×Dic178—21no—
67#15667971—21no—
68#15667968—41no—
69C2×C4⋊Dic89—21no—
70#15667956—22no—
71#15667952—22no—
72C2×#13083549—22no—
73C2×#13083549C2311yescharacteristic
74C2×#13083549C2311yescharacteristic
75C2×#13083549C2311yescharacteristic
76C2×#13083549C2311yescharacteristic
77#15667946—21no—
78#15667946C2311yescharacteristic
79#15667946C2311yescharacteristic
80#15667946C2311yescharacteristic
81#15667944—21no—
82#15667944—21no—
83#15667937—21no—
84#15667937—21no—
85#15667937—21no—
86C2×#41×C89—41no—
87C2×D8×C89C2311yescharacteristic
88C22×C8×C89C2311yescharacteristic
89C22×C8×C89C2311yescharacteristic
90C4⋊Q8×C89—21no—
91#123×C89—41no—
92#123×C89—41no—
93C22⋊Q8×C89—41no—
94#121×C89—21no—
95#121×C89—21no—
96C4×D4×C89—21no—
97C4×D4×C89D411yescharacteristic
98C2×C4⋊C4×C89—21no—
99C2×C4⋊C4×C89—21no—
100#107×C89—21no—
101#107×C89—21no—
102#107×C89—21no—
103#107×C89C2311yescharacteristic
104#107×C89C2311yescharacteristic
105#107×C89C2311yescharacteristic
106C4⋊C8×C89D411yescharacteristic
107#102×C89—21no—
108#102×C89—21no—
109#102×C89—21no—
110C22⋊C8×C89D411yescharacteristic
111C22⋊C8×C89D411yescharacteristic
112C4×C8×C89C2311yescharacteristic
113C2×Dic178—21no—
114C22×Dic89—41no—
115Dic356—21no—
116Dic356—21no—
117C4⋊Dic89—21no—
118C4⋊Dic89—21no—
119C4⋊Dic89—21no—
120C4⋊Dic89—41no—
121#13083553—41no—
122#13083553—41no—
123#13083553—41no—
124#13083553—41no—
125#13083553—41no—
126C22⋊Dic89—41no—
127C22⋊Dic89—41no—
128C22⋊Dic89—41no—
129C4×Dic89—41no—
130C4×Dic89—41no—
131#13083549—21no—
132#13083549—21no—
133#13083549—21no—
134#13083549#4112yescomplemented
135#41×C89—84no—
136C2×Q8×C89—41no—
137C2×D4×C89—21no—
138C2×D4×C89—41no—
139C2×D4×C89—41no—
140C2×D4×C89D811yescharacteristic
141C22×C4×C89—41no—
142C22×C4×C89—41no—
143C22×C4×C89—41no—
144C22×C4×C89C2×D411yescharacteristic
145C22×C4×C89C2×D411yescharacteristic
146D8×C89—21no—
147D8×C89—21no—
148C2×C8×C89—21no—
149C2×C8×C89—21no—
150C2×C8×C89—21no—
151C2×C8×C89—21no—
152C2×C8×C89C2×D411yescharacteristic
153C2×C8×C89C2×D411yescharacteristic
154C2×C8×C89C2×D411yescharacteristic
155C2×C8×C89C2411yescharacteristic
156C4⋊C4×C89—21no—
157C4⋊C4×C89—21no—
158C4⋊C4×C89—21no—
159C4⋊C4×C89—41no—
160C4⋊C4×C89—41no—
161C4⋊C4×C89—41no—
162C4⋊C4×C89D811yescharacteristic
163C22⋊C4×C89—41no—
164C22⋊C4×C89—41no—
165C22⋊C4×C89—41no—
166C22⋊C4×C89D811yescharacteristic
167C22⋊C4×C89D811yescharacteristic
168C42×C89C2×D411yescharacteristic
169C23×C89—41no—
170C23×C89C2×D811yescharacteristic
171C23×C89C2×D811yescharacteristic
172Q8×C89—82no—
173D4×C89—22no—
174D4×C89—41no—
175D4×C89—41no—
176D4×C89—82no—
177D4×C89—82no—
178C2×C4×C89—21no—
179C2×C4×C89—21no—
180C2×C4×C89—21no—
181C2×C4×C89—21no—
182C2×C4×C89—41no—
183C2×C4×C89—41no—
184C2×C4×C89—41no—
185C2×C4×C89—82no—
186C2×C4×C89—82no—
187C2×C4×C89—82no—
188C2×C4×C89C2×D811yescharacteristic
189C2×C4×C89C2×D811yescharacteristic
190C2×C4×C89C2×D811yescharacteristic
191C2×C4×C89C2×D811yescharacteristic
192C2×C4×C89C22×D411yescharacteristic
193C2×Dic89—41no—
194C2×Dic89—41no—
195C2×Dic89—41no—
196C2×Dic89—41no—
197C2×Dic89—82no—
198Dic178—41no—
199Dic178—41no—
200C8×C89—21no—
201C8×C89—21no—
202C8×C89—21no—
203C8×C89C2×#4111yescharacteristic
204C8×C892−1+411yescharacteristic
205C22×C89—21no—
206C22×C89—21no—
207C22×C89—21no—
208C22×C89—21no—
209C22×C89—21no—
210C22×C89—81no—
211C22×C89—81no—
212C22×C89—81no—
213C22×C89#50912yes—
214C22×C89C22×D811yescharacteristic
215C4×C89—21no—
216C4×C89—21no—
217C4×C89—21no—
218C4×C89—21no—
219C4×C89—81no—
220C4×C89—81no—
221C4×C89—81no—
222C4×C89#54711yescharacteristic
223C4×C89#57811yescharacteristic
224Dic89—81no—
225Dic89—81no—
226Dic89—81no—
227Dic89—81no—
228#33910—891nomaximal, Sylow
229C2×C89—22no—
230C2×C89—41no—
231C2×C89—161no—
232C2×C89#333411yescharacteristic
233C2×C89C2×#50911yescharacteristic
234C2×C89#341711yescharacteristic
235#3336—891no—
236#3334D8911yescharacteristic
237#2255—891no—
238#2236—891no—
239#2235—891no—
240#2234—891no—
241#2232—891no—
242#2228—891no—
243#2227—891no—
244#2220—891no—
245#2217—891no—
246#2175—891no—
247#2168—891no—
248#2159—891no—
249C2×#366—891no—
250C89#3391011yescharacteristic, complemented, Sylow
251#552—1781no—
252#547—21no—
253C2×SD32—891no—
254C2×M32—891no—
255C22×C16—891no—
256#500C2×D8911yescharacteristic
257#499—891no—
258#485—1781no—
259#483—1781no—
260#480—21no—
261#480—21no—
262#479—1781no—
263#474—21no—
264#467—891no—
265#467—891no—
266#466C2×D8911yescharacteristic
267#466C2×D8911yescharacteristic
268C4×Q16—891no—
269C4×D8C2×D8911yescharacteristic
270C8×D4C2×D8911yescharacteristic
271#427—891no—
272C2×#107—891no—
273C2×#107C2×D8911yescharacteristic
274C2×#107C2×D8911yescharacteristic
275#367—891no—
276#367—891no—
277#367—891no—
278#366—891no—
279#366—891no—
280#366—891no—
281#366—891no—
282#366—891no—
283#366—891no—
284C4⋊C16—891no—
285#358—891no—
286#358—891no—
287#358—891no—
288#357—891no—
289#357—891no—
290#357—891no—
291C22⋊C16—891no—
292C22⋊C16—891no—
293#346—891no—
294C2×#41—41no—
295C2×Q16—891no—
296C2×D8C22×D8911yescharacteristic
297C22×C8C22×D8911yescharacteristic
298C22×C8C22×D8911yescharacteristic
299C4⋊Q8—21no—
300#126—3561no—
301#125—3561no—
302#125—1781no—
303#123—41no—
304#123—41no—
305#123—3561no—
306C22⋊Q8—41no—
307C22⋊Q8—1781no—
308C22⋊Q8—1781no—
309#121—21no—
310#121—21no—
311C4×Q8—1781no—
312C4×D4—21no—
313C4×D4C22⋊D8911yescharacteristic
314#117—3561no—
315#117—1781no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—1781no—
319SD32—1782no—
320M32—1782no—
321C2×C16—891no—
322C2×C16—891no—
323C2×C16—891no—
324C2×C16—891no—
325C2×C16—1782no—
326#107—21no—
327#107—21no—
328#107—21no—
329#107—891no—
330#107—891no—
331#107—891no—
332#107—1781no—
333#107C22×D8911yescharacteristic
334#107C22×D8911yescharacteristic
335#107C22×D8911yescharacteristic
336#106—1781no—
337#106—1781no—
338C4⋊C8C22⋊D8911yescharacteristic
339#103—1781no—
340#103—1781no—
341#103—1781no—
342#102—21no—
343#102—21no—
344#102—21no—
345C22⋊C8C22⋊D8911yescharacteristic
346C22⋊C8C22⋊D8911yescharacteristic
347C4×C8C22×D8911yescharacteristic
348#41—84no—
349C2×Q8—41no—
350C2×Q8—1781no—
351C2×D4—21no—
352C2×D4—41no—
353C2×D4—41no—
354C2×D4#1308355911yescharacteristic
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—3561no—
359C22×C4C2×C22⋊D8911yescharacteristic
360C22×C4C2×C22⋊D8911yescharacteristic
361Q16—1781no—
362Q16—1781no—
363D8—21no—
364D8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8C2×C22⋊D8911yescharacteristic
370C2×C8C2×C22⋊D8911yescharacteristic
371C2×C8C2×C22⋊D8911yescharacteristic
372C2×C8C23×D8911yescharacteristic
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—3561no—
380C4⋊C4—3561no—
381C4⋊C4—3561no—
382C4⋊C4—3561no—
383C4⋊C4—3561no—
384C4⋊C4—3561no—
385C4⋊C4—1781no—
386C4⋊C4—1781no—
387C4⋊C4—1781no—
388C4⋊C4#1308355911yescharacteristic
389C22⋊C4—41no—
390C22⋊C4—41no—
391C22⋊C4—41no—
392C22⋊C4—3561no—
393C22⋊C4—3561no—
394C22⋊C4—3561no—
395C22⋊C4#1308355911yescharacteristic
396C22⋊C4#1308355911yescharacteristic
397C42—3561no—
398C42—3561no—
399C42C2×C22⋊D8911yescharacteristic
400C16—892no—
401C16—1781no—
402C16—1781no—
403C16—1781no—
404C23—41no—
405C23C2×#1308355911yescharacteristic
406C23C2×#1308355911yescharacteristic
407Q8—82no—
408Q8—3561no—
409Q8—3561no—
410D4—22no—
411D4—41no—
412D4—41no—
413D4—82no—
414D4—82no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—3561no—
426C2×C4—3561no—
427C2×C4—3561no—
428C2×C4—3561no—
429C2×C4—7122no—
430C2×C4C2×#1308355911yescharacteristic
431C2×C4C2×#1308355911yescharacteristic
432C2×C4C2×#1308355911yescharacteristic
433C2×C4C2×#1308355911yescharacteristic
434C2×C4C22×C22⋊D8911yescharacteristic
435C8—21no—
436C8—21no—
437C8—21no—
438C8#41×D8911yescharacteristic
439C8#1566805911yescharacteristic
440C22—21no—
441C22—21no—
442C22—21no—
443C22—21no—
444C22—21no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22#1915203412yes—
449C22C22×#1308355911yescharacteristic
450C4—21no—
451C4—21no—
452C4—21no—
453C4—21no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4—7121no—
458C4—7121no—
459C4—7121no—
460C4—7121no—
461C4#1915224711yescharacteristic
462C4#1915241311yescharacteristic
463C2—22no—
464C2—41no—
465C2—161no—
466C2#2603259511yescharacteristic
467C2C2×#1915203411yescharacteristic
468C2#2603340311yescharacteristic
subgroupG/H
centerC22C22×#13083559
commutator, cosocleC2×C8×C89C24
FrattiniC2×C8C23×D89
Fitting, prime radicalC89×#3334C2
socleC22×C89C22×D8
hypercenter#3334D89
hypocenterC89#33910
LCSG ▷ C2×C8×C89 ▷ C4×C89 ▷ C2×C89 ▷ C89
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3334

Complements

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

NH ≅ G/Ntype
1. #26032621C2semidirect
2. #26022535C2semidirect
3. #26022454C2semidirect
4. #26022449C2semidirect
5. #26022440C2semidirect
6. #26022430C2semidirect
7. #26022409C2semidirect
8. #26022401C2semidirect
9. #26022363C2semidirect
10. #26022346C2semidirect
11. #26022116C2semidirect
12. #26022078C2semidirect
13. #26022038C2semidirect
14. C2×#19151423C2semidirect
134. #13083549#41semidirect
250. C89#33910semidirect

Maximal subgroups

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

subgroupindex
#260326212
#260225352
#260224542
#260224492
#260224402
#260224302
#260224092
#260224012
#260223632
#260223462
#260221162
#260220782
#260220382
C2×#191514232
C89×#33342
#3391089