Finite Group Databasebeta

← #29663028

Subgroups — #29663028

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#21456538C211yesmaximal, characteristic, complemented
2#21455879C211yesmaximal, characteristic
3#21454839C211yesmaximal, characteristic, complemented
4#21445392C211yesmaximal, characteristic, complemented
5#21445354C211yesmaximal, characteristic, complemented
6#21445340C211yesmaximal, characteristic, complemented
7#21445286C211yesmaximal, characteristic, complemented
8#21445275C211yesmaximal, characteristic, complemented
9#21445237C211yesmaximal, characteristic, complemented
10#21445221C211yesmaximal, characteristic, complemented
11#21445169C211yesmaximal, characteristic, complemented
12#21444569C211yesmaximal, characteristic, complemented
13#21444529C211yesmaximal, characteristic, complemented
14#21444512C211yesmaximal, characteristic, complemented
15C61×#3076C211yesmaximal, characteristic, complemented
16#16840312C2211yescharacteristic
17#16840165C2211yescharacteristic
18#16840136C2211yescharacteristic
19#16840112C2211yescharacteristic
20#16840110C2211yescharacteristic
21#16840086—21no—
22#16840057C2211yescharacteristic
23#16840052C2211yescharacteristic
24#16840010—21no—
25#16840010—21no—
26#16839962C2211yescharacteristic
27#16839957C2211yescharacteristic
28#16839907—21no—
29Q8×#92418C2211yescharacteristic
30Q8×#92418C2211yescharacteristic
31#16839860—21no—
32#16839860—21no—
33C4×#11758201—21no—
34C4×#11758199—21no—
35#16839840—21no—
36#16839840—21no—
37#16839840C2211yescharacteristic
38#16839840C2211yescharacteristic
39#16839819—21no—
40#16839819C2211yescharacteristic
41#16839815C2211yescharacteristic
42C4×#11758194—21no—
43C4×#11758194C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839510C2211yescharacteristic
51#16839510C2211yescharacteristic
52#16839508C2211yescharacteristic
53#16839508C2211yescharacteristic
54#16839505C2211yescharacteristic
55#16839505C2211yescharacteristic
56C61×#533C2211yescharacteristic
57C4×#41×C61C2211yescharacteristic
58C61×#479—21no—
59C61×#478—21no—
60C61×#477C2211yescharacteristic
61C61×#476C2211yescharacteristic
62C61×#475C2211yescharacteristic
63C61×#474C2211yescharacteristic
64C61×#427—21no—
65C2×C4⋊C8×C61C2211yescharacteristic
66C61×#416—21no—
67C2×#11758222—21no—
68C2×#11758201—21no—
69C2×#11758201—21no—
70C2×#11758201—21no—
71C2×#11758199—21no—
72C2×#11758194—21no—
73C2×#11758194—21no—
74C2×#11758194D411yescharacteristic
75C2×#11758194D411yescharacteristic
76#14519777C2311yescharacteristic
77#14519774C2311yescharacteristic
78#14519765—21no—
79#14519751—21no—
80#14519751—21no—
81C4×D244—21no—
82C4×D244—21no—
83#14519710—21no—
84#14519710—21no—
85#14519710—21no—
86#14519708—21no—
87C4⋊#92418—21no—
88C4⋊#92418—21no—
89C4⋊#92418C2311yescharacteristic
90C4⋊#92418C2311yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93#14519701—21no—
94#14519701—21no—
95#14519701—21no—
96#14519698—21no—
97#14519697—21no—
98#14519697—21no—
99#14519697—21no—
100#14519697—21no—
101C22⋊#92418—21no—
102C22⋊#92418—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105#14519688—21no—
106#14519688D411yescharacteristic
107#14519688D411yescharacteristic
108C4×#92418—21no—
109C4×#92418—21no—
110C4×#92418—21no—
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C2×#41×C61D411yescharacteristic
114C22×C8×C61—21no—
115C4⋊Q8×C61C2311yescharacteristic
116C4⋊Q8×C61C2311yescharacteristic
117#125×C61—21no—
118C22⋊Q8×C61—21no—
119C22⋊Q8×C61—21no—
120C4×Q8×C61—21no—
121C4×Q8×C61—21no—
122C4×Q8×C61D411yescharacteristic
123C4×Q8×C61C2311yescharacteristic
124C4×D4×C61—21no—
125C4×D4×C61—21no—
126C4×D4×C61D411yescharacteristic
127C4×D4×C61C2311yescharacteristic
128#117×C61—21no—
129#117×C61—21no—
130#117×C61D411yescharacteristic
131C2×C42×C61—21no—
132C2×C42×C61C2311yescharacteristic
133#107×C61—21no—
134#107×C61—21no—
135#106×C61—21no—
136#106×C61—21no—
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140#103×C61—21no—
141#103×C61—21no—
142#102×C61—21no—
143#102×C61—21no—
144#41×C61—22no—
145#41×C61—41no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61C2×D411yescharacteristic
149C2×D4×C61—21no—
150C2×D4×C61C2×D411yescharacteristic
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61#4111yescharacteristic
157C2×C22⋊D61—41no—
158#11758222—82no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—21no—
162C2×C4×D61—41no—
163C2×Dic122—21no—
164C2×C8×C61—21no—
165C2×C8×C61—21no—
166C2×C8×C61—42no—
167C4⋊C4×C61—21no—
168C4⋊C4×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—41no—
174C4⋊C4×C61—41no—
175C4⋊C4×C61C2×D411yescharacteristic
176C4⋊C4×C61C2×D411yescharacteristic
177C22⋊C4×C61—21no—
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—41no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—41no—
185C42×C61C2×D411yescharacteristic
186C42×C61C2×D411yescharacteristic
187C42×C61#4111yescharacteristic
188C42×C61C2411yescharacteristic
189#11758201—42no—
190#11758201—42no—
191#11758201—42no—
192#11758199—42no—
193#11758198—41no—
194#11758198—41no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197#11758194—22no—
198#11758194—41no—
199#11758194—42no—
200#11758194—42no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209C23×C61—21no—
210C23×C61#12111yescharacteristic
211C22×D61—41no—
212Q8×C61—21no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—81no—
216Q8×C61#13712yes—
217D4×C61—21no—
218D4×C61—21no—
219D4×C61—42no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—41no—
228C2×C4×C61—42no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—41no—
232C2×C4×C61—41no—
233C2×C4×C61—81no—
234C2×C4×C61#12111yescharacteristic
235C2×C4×C61#12111yescharacteristic
236C2×C4×C61#12111yescharacteristic
237C2×C4×C61C22×D411yescharacteristic
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C2×#4111yescharacteristic
240C22⋊D61—82no—
241C2×Dic61—41no—
242D244—41no—
243D244—41no—
244D244—41no—
245D244—41no—
246D244—81no—
247C4×D61—82no—
248Dic122—81no—
249C8×C61—41no—
250C8×C61—41no—
251#92418—21no—
252#92418—21no—
253#92418—21no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#92418—41no—
258#92418#13512yes—
259#19183—611nomaximal, Sylow
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—41no—
264C22×C61—41no—
265C22×C61—41no—
266C22×C61C2×#12111yescharacteristic
267C2×D61—81no—
268C2×D61—81no—
269C2×D61—81no—
270C4×C61—21no—
271C4×C61—21no—
272C4×C61—22no—
273C4×C61—21no—
274C4×C61—21no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—81no—
278C4×C61—81no—
279C4×C61C2×#13511yescharacteristic
280C4×C61C2×#13711yescharacteristic
281C4×C61#57511yescharacteristic
282C4×C61#57511yescharacteristic
283Dic61—81no—
284#3182—611no—
285#3106—611no—
286#3076D6111yescharacteristic, complemented
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—81no—
301C2×C61#303511yescharacteristic
302C2×C61#305611yescharacteristic
303C2×C61#306811yescharacteristic
304D61—161no—
305#533C2×D6111yescharacteristic
306#532—611no—
307C4×#41C2×D6111yescharacteristic
308#500—611no—
309#492—611no—
310#487—611no—
311#486—611no—
312#482—1221no—
313#479—21no—
314#478—21no—
315#477—611no—
316#477C2×D6111yescharacteristic
317#476—611no—
318#476C2×D6111yescharacteristic
319#475C2×D6111yescharacteristic
320#474C2×D6111yescharacteristic
321#468—1221no—
322#468—1221no—
323#460—611no—
324#459—611no—
325#449—1221no—
326C8×Q8—611no—
327C8×Q8—611no—
328#440—1221no—
329#440—1221no—
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—21no—
340C2×C4⋊C8C2×D6111yescharacteristic
341#416—21no—
342C4×M16—611no—
343C4×M16—1221no—
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—1221no—
358C2×#41C22⋊D6111yescharacteristic
359C2×SD16—1221no—
360C2×SD16—1221no—
361C2×SD16—1221no—
362C2×D8—1221no—
363C2×M16—611no—
364C2×M16—611no—
365C2×M16—1221no—
366C2×M16—1221no—
367C22×C8—21no—
368C4⋊Q8—611no—
369C4⋊Q8C22×D6111yescharacteristic
370C4⋊Q8C22×D6111yescharacteristic
371#127—611no—
372#125—21no—
373#124—1221no—
374C22⋊Q8—21no—
375C22⋊Q8—21no—
376#121—1221no—
377#121—1221no—
378C4×Q8—21no—
379C4×Q8—21no—
380C4×Q8C22⋊D6111yescharacteristic
381C4×Q8C22×D6111yescharacteristic
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—1221no—
385C4×D4—1221no—
386C4×D4C22⋊D6111yescharacteristic
387C4×D4C22×D6111yescharacteristic
388#117—21no—
389#117—21no—
390#117C22⋊D6111yescharacteristic
391C2×C42—21no—
392C2×C42C22×D6111yescharacteristic
393#107—21no—
394#107—21no—
395#107—1221no—
396#107—1221no—
397#107—1221no—
398#106—21no—
399#106—21no—
400#106—1221no—
401C4⋊C8—21no—
402C4⋊C8—611no—
403C4⋊C8—611no—
404C4⋊C8—611no—
405C4⋊C8—611no—
406C4⋊C8—1221no—
407C4⋊C8—1221no—
408C4⋊C8C22×D6111yescharacteristic
409C4⋊C8C22×D6111yescharacteristic
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—611no—
427#97—611no—
428#97—1221no—
429C4×C8—611no—
430C4×C8—611no—
431C4×C8—1221no—
432C4×C8—1221no—
433C4×C8—1221no—
434#41—22no—
435#41—41no—
436#41—4882no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—1221no—
440C2×Q8C2×C22⋊D6111yescharacteristic
441C2×D4—21no—
442C2×D4—1221no—
443C2×D4—1221no—
444C2×D4—1221no—
445C2×D4—2441no—
446C2×D4C2×C22⋊D6111yescharacteristic
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—2441no—
451C22×C4D4×D6111yescharacteristic
452C22×C4#1175822411yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454SD16—2442no—
455SD16—2442no—
456SD16—2442no—
457D8—2442no—
458M16—1222no—
459M16—2442no—
460M16—2441no—
461M16—2442no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—42no—
465C2×C8—611no—
466C2×C8—611no—
467C2×C8—611no—
468C2×C8—611no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
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—1221no—
482C4⋊C4—1221no—
483C4⋊C4C2×C22⋊D6111yescharacteristic
484C4⋊C4C2×C22⋊D6111yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—41no—
489C22⋊C4—2441no—
490C22⋊C4—2441no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—41no—
495C42D4×D6111yescharacteristic
496C42#1175822411yescharacteristic
497C42C2×C22⋊D6111yescharacteristic
498C42C23×D6111yescharacteristic
499C23—21no—
500C23—2441no—
501C23#1451974911yescharacteristic
502Q8—21no—
503Q8—41no—
504Q8—41no—
505Q8—81no—
506Q8—4881no—
507Q8#1451980812yes—
508D4—21no—
509D4—21no—
510D4—42no—
511D4—4881no—
512D4—4882no—
513D4—2441no—
514D4—2441no—
515D4—2441no—
516D4—2441no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—81no—
531C2×C4—4882no—
532C2×C4—2441no—
533C2×C4#1451974911yescharacteristic
534C2×C4#1451974911yescharacteristic
535C2×C4#1451974911yescharacteristic
536C2×C4C2×D4×D6111yescharacteristic
537C2×C4C22×C22⋊D6111yescharacteristic
538C2×C4C2×#1175822411yescharacteristic
539C8—41no—
540C8—41no—
541C8—612no—
542C8—1221no—
543C8—1221no—
544C8—1221no—
545C8—2441no—
546C8—2441no—
547C8—2441no—
548C8—2441no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—4881no—
556C22—4881no—
557C22—4881no—
558C22C2×#1451974911yescharacteristic
559C4—21no—
560C4—22no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—41no—
565C4—41no—
566C4—81no—
567C4—81no—
568C4—4881no—
569C4#135×D6111yescharacteristic
570C4C2×#1451980811yescharacteristic
571C4#1684055911yescharacteristic
572C4#1684056011yescharacteristic
573C2—41no—
574C2—81no—
575C2—9761no—
576C2#2145527011yescharacteristic
577C2#2145544511yescharacteristic
578C2#2145554911yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3076C2
socleC22×C61C2×#121
prime radical#21455879C2
hypercenter#3076D61
hypocenterC61#19183
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
1. #21456538C2semidirect
3. #21454839C2semidirect
4. #21445392C2semidirect
5. #21445354C2semidirect
6. #21445340C2semidirect
7. #21445286C2semidirect
8. #21445275C2semidirect
9. #21445237C2semidirect
10. #21445221C2semidirect
11. #21445169C2semidirect
12. #21444569C2semidirect
13. #21444529C2semidirect
14. #21444512C2semidirect
15. C61×#3076C2semidirect
286. #3076D61semidirect
356. C61#19183semidirect

Maximal subgroups

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

subgroupindex
#214565382
#214558792
#214548392
#214453922
#214453542
#214453402
#214452862
#214452752
#214452372
#214452212
#214451692
#214445692
#214445292
#214445122
C61×#30762
#1918361