Finite Group Databasebeta

← #29663343

Subgroups — #29663343

Order 15616.

596 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456998C211yesmaximal, characteristic, complemented
2#21456795C211yesmaximal, characteristic
3#21454794C211yesmaximal, characteristic, complemented
4#21445370C211yesmaximal, characteristic, complemented
5#21445363C211yesmaximal, characteristic, complemented
6#21445356C211yesmaximal, characteristic, complemented
7#21445349C211yesmaximal, characteristic, complemented
8#21445198C211yesmaximal, characteristic, complemented
9#21445191C211yesmaximal, characteristic, complemented
10#21445164C211yesmaximal, characteristic, complemented
11#21445157C211yesmaximal, characteristic, complemented
12#21444567C211yesmaximal, characteristic, complemented
13#21444529C211yesmaximal, characteristic, complemented
14#21444524C211yesmaximal, characteristic, complemented
15C61×#3167C211yesmaximal, characteristic, complemented
16#16840296C2211yescharacteristic
17#16840161—21no—
18#16840144—21no—
19#16840119—21no—
20#16840116—21no—
21#16840103—21no—
22#16840086—21no—
23#16840054C2211yescharacteristic
24#16840049C2211yescharacteristic
25#16840044C2211yescharacteristic
26#16840039C2211yescharacteristic
27#16839969C2211yescharacteristic
28#16839964C2211yescharacteristic
29#16839959C2211yescharacteristic
30#16839954C2211yescharacteristic
31#16839924—21no—
32#16839907—21no—
33#16839892C2211yescharacteristic
34C8×Dic122C2211yescharacteristic
35#16839841C2211yescharacteristic
36#16839833—21no—
37C8×D244C2211yescharacteristic
38C8×C22⋊D61—21no—
39#16839821—21no—
40#16839819C2211yescharacteristic
41#16839815C2211yescharacteristic
42#16839815C2211yescharacteristic
43#16839716—21no—
44#16839515C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839513C2211yescharacteristic
48#16839511C2211yescharacteristic
49#16839511C2211yescharacteristic
50#16839508C2211yescharacteristic
51#16839508C2211yescharacteristic
52#16839508C2211yescharacteristic
53#16839508C2211yescharacteristic
54#16839506C2211yescharacteristic
55#16839506C2211yescharacteristic
56C61×#533C2211yescharacteristic
57C61×#532C2211yescharacteristic
58C61×#498C2211yescharacteristic
59C61×#492C2211yescharacteristic
60C61×#487C2211yescharacteristic
61C61×#486C2211yescharacteristic
62C61×#465—21no—
63C61×#465—21no—
64C4×SD16×C61—21no—
65C4×SD16×C61—21no—
66C2×C4×C8×C61C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758200—21no—
69C2×#11758201—21no—
70C2×#11758199—21no—
71C2×#11758189—21no—
72C2×C8×D61—21no—
73#14519771—21no—
74#14519769D411yescharacteristic
75#14519765D411yescharacteristic
76#14519760—21no—
77#14519756D411yescharacteristic
78#14519751D411yescharacteristic
79C4×Dic122C2311yescharacteristic
80C4×D244C2311yescharacteristic
81C4×C22⋊D61—21no—
82#14519734—21no—
83#14519710—21no—
84#14519710—21no—
85#14519708—21no—
86#14519708—21no—
87C4⋊#92418C2311yescharacteristic
88C4⋊#92418C2311yescharacteristic
89C4⋊#92418C2311yescharacteristic
90C4⋊#92418C2311yescharacteristic
91C4⋊#92418C2311yescharacteristic
92C4⋊#92418C2311yescharacteristic
93#14519706—21no—
94#14519706—21no—
95#14519701—21no—
96#14519701—21no—
97#14519700—21no—
98#14519700—21no—
99#14519698—21no—
100#14519698—21no—
101#14519697—21no—
102#14519697—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105C22⋊#92418—21no—
106#14519689—21no—
107#14519689—21no—
108#14519688—21no—
109#14519687—21no—
110C4×#92418—21no—
111C4×#92418—21no—
112C8×Dic61—21no—
113C2×#41×C61—21no—
114C2×SD16×C61—21no—
115C2×SD16×C61—21no—
116C22×C8×C61D411yescharacteristic
117C22×C8×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#106×C61—21no—
135#106×C61—21no—
136#103×C61—21no—
137#103×C61—21no—
138#102×C61—21no—
139#102×C61—21no—
140C4×C8×C61D411yescharacteristic
141C4×C8×C61D411yescharacteristic
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144#41×C61—81no—
145#41×C61—81no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—41no—
153C22×C4×C61—41no—
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2×D411yescharacteristic
156C22×C4×C61#4111yescharacteristic
157C2×C22⋊D61—21no—
158C2×D244C2×D411yescharacteristic
159C2×C4×D61—21no—
160C2×Dic122C2×D411yescharacteristic
161SD16×C61—42no—
162SD16×C61—42no—
163C2×C8×C61—22no—
164C2×C8×C61—41no—
165C2×C8×C61C2×D411yescharacteristic
166C2×C8×C61C2×D411yescharacteristic
167C2×C8×C61C2×D411yescharacteristic
168C2×C8×C61C2×D411yescharacteristic
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—41no—
176C4⋊C4×C61—41no—
177C22⋊C4×C61—41no—
178C22⋊C4×C61—41no—
179C22⋊C4×C61—41no—
180C42×C61—41no—
181C42×C61C2×D411yescharacteristic
182C42×C61C2×D411yescharacteristic
183C42×C61#4111yescharacteristic
184C42×C61C2411yescharacteristic
185C22⋊Dic61—21no—
186#11758202—81no—
187#11758201—81no—
188#11758200—81no—
189#11758199—81no—
190#11758198—21no—
191#11758198—21no—
192#11758198—21no—
193#11758198—21no—
194C4⋊Dic61C2×D411yescharacteristic
195C4⋊Dic61C2×D411yescharacteristic
196#11758196—21no—
197#11758196—21no—
198#11758196—21no—
199#11758196—21no—
200C4×Dic61—21no—
201C2×#92418—21no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—21no—
207#11758189—42no—
208C8×D61—42no—
209C23×C61—41no—
210C23×C61#12111yescharacteristic
211C22×D61—21no—
212Q8×C61—41no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61—41no—
216Q8×C61—81no—
217D4×C61—41no—
218D4×C61—41no—
219D4×C61—41no—
220D4×C61—41no—
221D4×C61—81no—
222D4×C61—82no—
223C2×C4×C61—21no—
224C2×C4×C61—21no—
225C2×C4×C61—21no—
226C2×C4×C61—21no—
227C2×C4×C61—41no—
228C2×C4×C61—41no—
229C2×C4×C61—41no—
230C2×C4×C61—41no—
231C2×C4×C61—82no—
232C2×C4×C61#12111yescharacteristic
233C2×C4×C61#12111yescharacteristic
234C2×C4×C61#12111yescharacteristic
235C2×C4×C61C22×D411yescharacteristic
236C2×C4×C61C22×D411yescharacteristic
237C2×C4×C61C2×#4111yescharacteristic
238C22⋊D61—42no—
239C2×Dic61—21no—
240C2×Dic61—21no—
241C2×Dic61—21no—
242D244—21no—
243D244—21no—
244C4×D61—41no—
245C4×D61—41no—
246Dic122—21no—
247Dic122—21no—
248C8×C61—21no—
249C8×C61—21no—
250C8×C61—21no—
251C8×C61#13512yes—
252#92418—41no—
253#92418—41no—
254#92418—41no—
255#92418—41no—
256#92418—41no—
257#92418—41no—
258#19218—611nomaximal, Sylow
259C22×C61—21no—
260C22×C61—21no—
261C22×C61—21no—
262C22×C61—81no—
263C22×C61—81no—
264C22×C61—81no—
265C22×C61C2×#12111yescharacteristic
266C2×D61—41no—
267C2×D61—41no—
268C2×D61—41no—
269C4×C61—21no—
270C4×C61—21no—
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—81no—
274C4×C61—81no—
275C4×C61—81no—
276C4×C61C2×#13511yescharacteristic
277C4×C61#57511yescharacteristic
278C4×C61#57711yescharacteristic
279C4×C61#57711yescharacteristic
280Dic61—41no—
281Dic61—41no—
282Dic61—41no—
283#3233—611no—
284#3212—611no—
285#3167D6111yescharacteristic, complemented
286#2966—611no—
287#1700—611no—
288#1699—611no—
289#1698—611no—
290#1697—611no—
291#1672—611no—
292#1671—611no—
293#1666—611no—
294#1665—611no—
295#1564—611no—
296#1554—611no—
297#1553—611no—
298C2×C61—41no—
299C2×C61—161no—
300C2×C61#303511yescharacteristic
301C2×C61#305711yescharacteristic
302C2×C61#306711yescharacteristic
303D61—81no—
304#533C2×D6111yescharacteristic
305#532C2×D6111yescharacteristic
306#529—611no—
307#499—1221no—
308#498C2×D6111yescharacteristic
309#495—1221no—
310#492C2×D6111yescharacteristic
311#488—1221no—
312#488—1221no—
313#487C2×D6111yescharacteristic
314#486C2×D6111yescharacteristic
315#485—1221no—
316#482—1221no—
317#477—611no—
318#476—611no—
319#475—611no—
320#474—611no—
321#465—21no—
322#465—21no—
323#462—611no—
324#461—611no—
325#460—611no—
326#459—611no—
327#452—1221no—
328#449—1221no—
329#446—611no—
330C8×Q8—611no—
331C4×SD16—21no—
332C4×SD16—21no—
333#436—611no—
334#435—1221no—
335C8×D4—611no—
336C8×D4—1221no—
337#433—1221no—
338#432—611no—
339#431—611no—
340#431—611no—
341#405—1221no—
342C2×C4×C8C2×D6111yescharacteristic
343#335—611no—
344#335—611no—
345#334—611no—
346#334—611no—
347#333—611no—
348#333—611no—
349#332—611no—
350#332—611no—
351#332—611no—
352#332—611no—
353#331—611no—
354#331—611no—
355C61#1921811yescharacteristic, complemented, Sylow
356C2×#41—21no—
357C2×Q16—1221no—
358C2×SD16—21no—
359C2×SD16—21no—
360C2×SD16—1221no—
361C2×SD16—1221no—
362C2×D8—1221no—
363C2×M16—1221no—
364C22×C8—1221no—
365C22×C8C22⋊D6111yescharacteristic
366C22×C8C22⋊D6111yescharacteristic
367C4⋊Q8C22×D6111yescharacteristic
368C4⋊Q8C22×D6111yescharacteristic
369C4⋊Q8C22×D6111yescharacteristic
370#127C22×D6111yescharacteristic
371#126—1221no—
372#125—21no—
373#125—611no—
374#124—21no—
375#124—611no—
376#123—1221no—
377C22⋊Q8—21no—
378C22⋊Q8—21no—
379C22⋊Q8—611no—
380#121—21no—
381#121—21no—
382#121—611no—
383C4×Q8—21no—
384C4×Q8—21no—
385C4×Q8—611no—
386C4×D4—21no—
387C4×D4—21no—
388C4×D4—611no—
389C4×D4—1221no—
390#117—21no—
391#117—1221no—
392C2×C42C22×D6111yescharacteristic
393#107—1221no—
394#107—1221no—
395#106—21no—
396#106—21no—
397#106—1221no—
398#106—1221no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—611no—
402C4⋊C8—611no—
403C4⋊C8—611no—
404C4⋊C8—611no—
405C4⋊C8—1221no—
406C4⋊C8—1221no—
407#103—21no—
408#103—21no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#102—21no—
414#102—21no—
415#102—1221no—
416#102—1221no—
417#102—1221no—
418#102—1221no—
419C22⋊C8—1221no—
420C22⋊C8—1221no—
421C22⋊C8—1221no—
422C22⋊C8—1221no—
423C22⋊C8—1221no—
424#97—1221no—
425#97—1221no—
426C4×C8—1221no—
427C4×C8—1221no—
428C4×C8—1221no—
429C4×C8C22⋊D6111yescharacteristic
430C4×C8C22⋊D6111yescharacteristic
431C4×C8C22×D6111yescharacteristic
432C4×C8C22×D6111yescharacteristic
433#41—81no—
434#41—81no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—611no—
439C2×D4—21no—
440C2×D4—21no—
441C2×D4—21no—
442C2×D4—41no—
443C2×D4—611no—
444C2×D4—1221no—
445C22×C4—41no—
446C22×C4—1221no—
447C22×C4D4×D6111yescharacteristic
448C22×C4#1175822411yescharacteristic
449C22×C4C2×C22⋊D6111yescharacteristic
450Q16—4881no—
451SD16—42no—
452SD16—42no—
453SD16—4881no—
454SD16—4881no—
455D8—4881no—
456M16—2442no—
457C2×C8—22no—
458C2×C8—41no—
459C2×C8—1221no—
460C2×C8—1221no—
461C2×C8—1221no—
462C2×C8—1221no—
463C2×C8—1221no—
464C2×C8—1221no—
465C2×C8—2442no—
466C2×C8C2×C22⋊D6111yescharacteristic
467C2×C8C2×C22⋊D6111yescharacteristic
468C2×C8C2×C22⋊D6111yescharacteristic
469C2×C8C2×C22⋊D6111yescharacteristic
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—611no—
479C4⋊C4—611no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—1221no—
483C4⋊C4—1221no—
484C22⋊C4—41no—
485C22⋊C4—41no—
486C22⋊C4—41no—
487C22⋊C4—1221no—
488C22⋊C4—1221no—
489C22⋊C4—1221no—
490C22⋊C4—1221no—
491C22⋊C4—1221no—
492C42—41no—
493C42—1221no—
494C42D4×D6111yescharacteristic
495C42#1175822411yescharacteristic
496C42C2×C22⋊D6111yescharacteristic
497C42C23×D6111yescharacteristic
498C23—41no—
499C23—1221no—
500C23#1451974811yescharacteristic
501Q8—41no—
502Q8—41no—
503Q8—41no—
504Q8—41no—
505Q8—81no—
506Q8—1221no—
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—2441no—
530C2×C4—2441no—
531C2×C4#1451974811yescharacteristic
532C2×C4#1451974811yescharacteristic
533C2×C4#1451974811yescharacteristic
534C2×C4C2×D4×D6111yescharacteristic
535C2×C4C22×C22⋊D6111yescharacteristic
536C2×C4C2×#1175822411yescharacteristic
537C8—21no—
538C8—21no—
539C8—21no—
540C8—2441no—
541C8—2441no—
542C8—2441no—
543C8—2441no—
544C8—2441no—
545C8—2441no—
546C8#1451979712yes—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—81no—
551C22—81no—
552C22—81no—
553C22—2441no—
554C22—2441no—
555C22—2441no—
556C22C2×#1451974811yescharacteristic
557C4—21no—
558C4—21no—
559C4—21no—
560C4—21no—
561C4—81no—
562C4—81no—
563C4—81no—
564C4—2441no—
565C4—2441no—
566C4—2441no—
567C4C2×#1451979711yescharacteristic
568C4#1684056011yescharacteristic
569C4#1684056811yescharacteristic
570C4#1684056811yescharacteristic
571C2—41no—
572C2—161no—
573C2—4881no—
574C2#2145526811yescharacteristic
575C2#2145545411yescharacteristic
576C2#2145553511yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3167C2
socleC22×C61C2×#121
prime radical#21456795C2
hypercenter#3167D61
hypocenterC61#19218
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3167

Complements

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

NH ≅ G/Ntype
1. #21456998C2semidirect
3. #21454794C2semidirect
4. #21445370C2semidirect
5. #21445363C2semidirect
6. #21445356C2semidirect
7. #21445349C2semidirect
8. #21445198C2semidirect
9. #21445191C2semidirect
10. #21445164C2semidirect
11. #21445157C2semidirect
12. #21444567C2semidirect
13. #21444529C2semidirect
14. #21444524C2semidirect
15. C61×#3167C2semidirect
285. #3167D61semidirect
355. C61#19218semidirect

Maximal subgroups

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

subgroupindex
#214569982
#214567952
#214547942
#214453702
#214453632
#214453562
#214453492
#214451982
#214451912
#214451642
#214451572
#214445672
#214445292
#214445242
C61×#31672
#1921861