Finite Group Databasebeta

← #29663026

Subgroups — #29663026

Order 15616.

614 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455873C211yesmaximal, characteristic
2#21455601C211yesmaximal, characteristic, complemented
3#21454837C211yesmaximal, characteristic, complemented
4#21445391C211yesmaximal, characteristic, complemented
5#21445349C211yesmaximal, characteristic, complemented
6#21445339C211yesmaximal, characteristic, complemented
7#21445281C211yesmaximal, characteristic, complemented
8#21445274C211yesmaximal, characteristic, complemented
9#21445232C211yesmaximal, characteristic, complemented
10#21445220C211yesmaximal, characteristic, complemented
11#21445164C211yesmaximal, characteristic, complemented
12#21444571C211yesmaximal, characteristic, complemented
13#21444529C211yesmaximal, characteristic, complemented
14#21444514C211yesmaximal, characteristic, complemented
15C61×#3182C211yesmaximal, characteristic, complemented
16C4×#11758222C2211yescharacteristic
17#16840083—21no—
18#16840066—21no—
19#16840059—21no—
20#16840054C2211yescharacteristic
21#16840054C2211yescharacteristic
22#16840049C2211yescharacteristic
23#16840049C2211yescharacteristic
24#16840044C2211yescharacteristic
25#16840039C2211yescharacteristic
26#16839959C2211yescharacteristic
27#16839954C2211yescharacteristic
28#16839905—21no—
29C8×Dic122C2211yescharacteristic
30C8×Dic122C2211yescharacteristic
31C4×#11758200—21no—
32C4×#11758199—21no—
33#16839841C2211yescharacteristic
34#16839841C2211yescharacteristic
35#16839838—21no—
36#16839838—21no—
37#16839819C2211yescharacteristic
38#16839817—21no—
39#16839815C2211yescharacteristic
40#16839798—21no—
41C2×C4⋊#92418C2211yescharacteristic
42#16839756—21no—
43#16839714—21no—
44#16839515C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839513C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839511C2211yescharacteristic
51#16839511C2211yescharacteristic
52#16839508C2211yescharacteristic
53#16839508C2211yescharacteristic
54#16839506C2211yescharacteristic
55#16839506C2211yescharacteristic
56C61×#533C2211yescharacteristic
57C61×#532C2211yescharacteristic
58C61×#500C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#487C2211yescharacteristic
61C61×#486C2211yescharacteristic
62C61×#468—21no—
63C61×#468—21no—
64C61×#440—21no—
65C61×#440—21no—
66C4×M16×C61C2211yescharacteristic
67C2×#11758222D411yescharacteristic
68C2×#11758200—21no—
69C2×#11758199—21no—
70C2×#11758189—21no—
71C2×#11758189—21no—
72C22×#92418—21no—
73C4×Dic122D411yescharacteristic
74C4×Dic122C2311yescharacteristic
75C4×D244D411yescharacteristic
76C4×D244C2311yescharacteristic
77C4×C22⋊D61—21no—
78C4×C22⋊D61—21no—
79#14519736D411yescharacteristic
80#14519734—21no—
81C42×D61—21no—
82#14519710—21no—
83#14519710—21no—
84#14519710—21no—
85#14519708—21no—
86#14519708—21no—
87#14519708—21no—
88C4⋊#92418—21no—
89C4⋊#92418C2311yescharacteristic
90C4⋊#92418C2311yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93C4⋊#92418C2311yescharacteristic
94C4⋊#92418C2311yescharacteristic
95#14519706—21no—
96#14519706—21no—
97#14519700—21no—
98#14519700—21no—
99#14519700—21no—
100#14519698—21no—
101#14519698—21no—
102#14519697—21no—
103#14519697—21no—
104#14519697—21no—
105C22⋊#92418—21no—
106C22⋊#92418—21no—
107#14519689—21no—
108#14519689—21no—
109#14519688—21no—
110C4×#92418—21no—
111C8×Dic61—21no—
112C8×Dic61—21no—
113C2×#41×C61—21no—
114C2×SD16×C61—21no—
115C2×SD16×C61—21no—
116C2×M16×C61D411yescharacteristic
117C2×M16×C61D411yescharacteristic
118C4⋊Q8×C61C2311yescharacteristic
119C4⋊Q8×C61C2311yescharacteristic
120C4⋊Q8×C61C2311yescharacteristic
121#127×C61C2311yescharacteristic
122#125×C61—21no—
123#124×C61—21no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—21no—
126#121×C61—21no—
127#121×C61—21no—
128C4×Q8×C61—21no—
129C4×Q8×C61—21no—
130C4×D4×C61—21no—
131C4×D4×C61—21no—
132#117×C61—21no—
133C2×C42×C61C2311yescharacteristic
134#107×C61—21no—
135#107×C61—21no—
136#103×C61—21no—
137#103×C61—21no—
138#102×C61—21no—
139#102×C61—21no—
140#97×C61D411yescharacteristic
141#97×C61D411yescharacteristic
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144#41×C61—82no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—41no—
152C22×C4×C61—41no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61#4111yescharacteristic
156C2×C22⋊D61—21no—
157#11758222—22no—
158#11758222—41no—
159C2×D244C2×D411yescharacteristic
160C2×C4×D61—21no—
161C2×C4×D61—21no—
162C2×C4×D61—21no—
163C2×Dic122C2×D411yescharacteristic
164SD16×C61—42no—
165SD16×C61—42no—
166M16×C61—22no—
167M16×C61—41no—
168C2×C8×C61C2×D411yescharacteristic
169C2×C8×C61C2×D411yescharacteristic
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2×D411yescharacteristic
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—21no—
178C4⋊C4×C61—41no—
179C4⋊C4×C61—41no—
180C22⋊C4×C61—41no—
181C22⋊C4×C61—41no—
182C22⋊C4×C61—41no—
183C42×C61—41no—
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D411yescharacteristic
186C42×C61#4111yescharacteristic
187C42×C61C2411yescharacteristic
188C22⋊Dic61—21no—
189#11758200—42no—
190#11758199—42no—
191#11758198—21no—
192#11758198—21no—
193C4⋊Dic61C2×D411yescharacteristic
194C4⋊Dic61C2×D411yescharacteristic
195#11758196—21no—
196#11758196—21no—
197C4×Dic61—21no—
198C4×Dic61—21no—
199C4×Dic61—21no—
200C2×#92418—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—42no—
207#11758189—42no—
208#11758189—42no—
209C23×C61—41no—
210C23×C61#12111yescharacteristic
211C22×D61—21no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—81no—
215Q8×C61—81no—
216D4×C61—41no—
217D4×C61—41no—
218D4×C61—41no—
219D4×C61—41no—
220D4×C61—81no—
221D4×C61—82no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—82no—
231C2×C4×C61#12111yescharacteristic
232C2×C4×C61#12111yescharacteristic
233C2×C4×C61#12111yescharacteristic
234C2×C4×C61C22×D411yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C2×#4111yescharacteristic
237C22⋊D61—42no—
238C2×Dic61—21no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241D244—21no—
242D244—21no—
243C4×D61—42no—
244C4×D61—41no—
245C4×D61—41no—
246C4×D61—81no—
247Dic122—21no—
248Dic122#13712yes—
249C8×C61—21no—
250C8×C61—21no—
251C8×C61—21no—
252C8×C61#13512yes—
253#92418—41no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#92418—41no—
258#92418—41no—
259#19183—611nomaximal, Sylow
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61—81no—
266C22×C61C2×#12111yescharacteristic
267C2×D61—41no—
268C2×D61—41no—
269C2×D61—41no—
270C4×C61—21no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—21no—
274C4×C61—81no—
275C4×C61—81no—
276C4×C61—81no—
277C4×C61C2×#13511yescharacteristic
278C4×C61C2×#13711yescharacteristic
279C4×C61#57511yescharacteristic
280C4×C61#57511yescharacteristic
281Dic61—22no—
282Dic61—41no—
283Dic61—41no—
284#3182D6111yescharacteristic, complemented
285#3106—611no—
286#3076—611no—
287#2971—611no—
288#1703—611no—
289#1697—611no—
290#1695—611no—
291#1685—611no—
292#1684—611no—
293#1678—611no—
294#1676—611no—
295#1666—611no—
296#1565—611no—
297#1554—611no—
298#1551—611no—
299C2×C61—41no—
300C2×C61—161no—
301C2×C61#303511yescharacteristic
302C2×C61#305611yescharacteristic
303C2×C61#306811yescharacteristic
304D61—81no—
305#533C2×D6111yescharacteristic
306#532C2×D6111yescharacteristic
307C4×#41—611no—
308#500C2×D6111yescharacteristic
309#492C2×D6111yescharacteristic
310#487C2×D6111yescharacteristic
311#486C2×D6111yescharacteristic
312#482—1221no—
313#479—1221no—
314#478—1221no—
315#477—611no—
316#477—611no—
317#476—611no—
318#476—611no—
319#475—611no—
320#474—611no—
321#468—21no—
322#468—21no—
323#460—611no—
324#459—611no—
325#449—1221no—
326C8×Q8—611no—
327C8×Q8—611no—
328#440—21no—
329#440—21no—
330C4×SD16—1221no—
331C4×D8—1221no—
332#436—611no—
333#436—611no—
334#436—1221no—
335#436—1221no—
336#432—611no—
337#432—1221no—
338#431—611no—
339#427—1221no—
340C2×C4⋊C8—611no—
341#416—1221no—
342C4×M16—1221no—
343C4×M16C2×D6111yescharacteristic
344#335—611no—
345#335—611no—
346#334—611no—
347#334—611no—
348#333—611no—
349#333—611no—
350#333—611no—
351#333—611no—
352#332—611no—
353#332—611no—
354#331—611no—
355#331—611no—
356C61#1918311yescharacteristic, complemented, Sylow
357C2×#41—21no—
358C2×#41—611no—
359C2×SD16—21no—
360C2×SD16—21no—
361C2×SD16—1221no—
362C2×D8—1221no—
363C2×M16—1221no—
364C2×M16—1221no—
365C2×M16C22⋊D6111yescharacteristic
366C2×M16C22⋊D6111yescharacteristic
367C22×C8—1221no—
368C4⋊Q8C22×D6111yescharacteristic
369C4⋊Q8C22×D6111yescharacteristic
370C4⋊Q8C22×D6111yescharacteristic
371#127C22×D6111yescharacteristic
372#125—21no—
373#124—21no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376#121—21no—
377#121—21no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8—611no—
381C4×Q8—611no—
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—611no—
385C4×D4—611no—
386C4×D4—1221no—
387C4×D4—1221no—
388#117—21no—
389#117—611no—
390#117—1221no—
391C2×C42—1221no—
392C2×C42C22×D6111yescharacteristic
393#107—21no—
394#107—21no—
395#107—1221no—
396#107—1221no—
397#107—1221no—
398#106—1221no—
399#106—1221no—
400#106—1221no—
401C4⋊C8—611no—
402C4⋊C8—611no—
403C4⋊C8—611no—
404C4⋊C8—611no—
405C4⋊C8—611no—
406C4⋊C8—611no—
407C4⋊C8—1221no—
408C4⋊C8—1221no—
409C4⋊C8—1221no—
410#103—21no—
411#103—21no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#102—21no—
416#102—21no—
417#102—1221no—
418#102—1221no—
419#102—1221no—
420#102—1221no—
421#102—1221no—
422C22⋊C8—1221no—
423C22⋊C8—1221no—
424C22⋊C8—1221no—
425C22⋊C8—1221no—
426#97—1221no—
427#97C22⋊D6111yescharacteristic
428#97C22⋊D6111yescharacteristic
429C4×C8—1221no—
430C4×C8—1221no—
431C4×C8—1221no—
432C4×C8C22×D6111yescharacteristic
433C4×C8C22×D6111yescharacteristic
434#41—82no—
435#41—1222no—
436#41—2441no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—611no—
441C2×D4—21no—
442C2×D4—21no—
443C2×D4—21no—
444C2×D4—41no—
445C2×D4—611no—
446C2×D4—1221no—
447C22×C4—41no—
448C22×C4—1221no—
449C22×C4—1221no—
450C22×C4—1221no—
451C22×C4D4×D6111yescharacteristic
452C22×C4#1175822411yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454SD16—42no—
455SD16—42no—
456SD16—2442no—
457D8—2442no—
458M16—22no—
459M16—41no—
460M16—2442no—
461M16—2442no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—1221no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—2442no—
469C2×C8C2×C22⋊D6111yescharacteristic
470C2×C8C2×C22⋊D6111yescharacteristic
471C2×C8C2×C22⋊D6111yescharacteristic
472C2×C8C2×C22⋊D6111yescharacteristic
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—41no—
480C4⋊C4—41no—
481C4⋊C4—611no—
482C4⋊C4—611no—
483C4⋊C4—1221no—
484C4⋊C4—1221no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—41no—
488C22⋊C4—1221no—
489C22⋊C4—1221no—
490C22⋊C4—1221no—
491C42—41no—
492C42—1221no—
493C42—1221no—
494C42—1221no—
495C42D4×D6111yescharacteristic
496C42#1175822411yescharacteristic
497C42C2×C22⋊D6111yescharacteristic
498C42C23×D6111yescharacteristic
499C23—41no—
500C23—1221no—
501C23#1451974811yescharacteristic
502Q8—41no—
503Q8—41no—
504Q8—81no—
505Q8—81no—
506Q8—612no—
507Q8—1221no—
508D4—41no—
509D4—41no—
510D4—41no—
511D4—41no—
512D4—81no—
513D4—82no—
514D4—1221no—
515D4—1221no—
516D4—2442no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—82no—
526C2×C4—1221no—
527C2×C4—1221no—
528C2×C4—1221no—
529C2×C4—4881no—
530C2×C4—2442no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4#1451974811yescharacteristic
534C2×C4#1451974811yescharacteristic
535C2×C4#1451974811yescharacteristic
536C2×C4C2×D4×D6111yescharacteristic
537C2×C4C22×C22⋊D6111yescharacteristic
538C2×C4C2×#1175822411yescharacteristic
539C8—21no—
540C8—21no—
541C8—21no—
542C8—2441no—
543C8—2441no—
544C8—2441no—
545C8—2441no—
546C8—2441no—
547C8—2441no—
548C8#1451979712yes—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—81no—
553C22—81no—
554C22—81no—
555C22—2441no—
556C22—2441no—
557C22—2441no—
558C22C2×#1451974811yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—81no—
564C4—81no—
565C4—81no—
566C4—1222no—
567C4—2441no—
568C4—2441no—
569C4C2×#1451979711yescharacteristic
570C4#137×D6111yescharacteristic
571C4#1684056011yescharacteristic
572C4#1684056111yescharacteristic
573C2—41no—
574C2—161no—
575C2—4881no—
576C2#2145526811yescharacteristic
577C2#2145544311yescharacteristic
578C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3182C2
socleC22×C61C2×#121
prime radical#21455873C2
hypercenter#3182D61
hypocenterC61#19183
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3182

Complements

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

NH ≅ G/Ntype
2. #21455601C2semidirect
3. #21454837C2semidirect
4. #21445391C2semidirect
5. #21445349C2semidirect
6. #21445339C2semidirect
7. #21445281C2semidirect
8. #21445274C2semidirect
9. #21445232C2semidirect
10. #21445220C2semidirect
11. #21445164C2semidirect
12. #21444571C2semidirect
13. #21444529C2semidirect
14. #21444514C2semidirect
15. C61×#3182C2semidirect
284. #3182D61semidirect
356. C61#19183semidirect

Maximal subgroups

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

subgroupindex
#214558732
#214556012
#214548372
#214453912
#214453492
#214453392
#214452812
#214452742
#214452322
#214452202
#214451642
#214445712
#214445292
#214445142
C61×#31822
#1918361