Finite Group Databasebeta

← #29663032

Subgroups — #29663032

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#21456537C211yesmaximal, characteristic, complemented
2#21455879C211yesmaximal, characteristic
3#21455607C211yesmaximal, characteristic, complemented
4#21445394C211yesmaximal, characteristic, complemented
5#21445352C211yesmaximal, characteristic, complemented
6#21445342C211yesmaximal, characteristic, complemented
7#21445284C211yesmaximal, characteristic, complemented
8#21445277C211yesmaximal, characteristic, complemented
9#21445235C211yesmaximal, characteristic, complemented
10#21445223C211yesmaximal, characteristic, complemented
11#21445167C211yesmaximal, characteristic, complemented
12#21444571C211yesmaximal, characteristic, complemented
13#21444530C211yesmaximal, characteristic, complemented
14#21444515C211yesmaximal, characteristic, complemented
15C61×#2971C211yesmaximal, characteristic, complemented
16#16840319C2211yescharacteristic
17#16840312C2211yescharacteristic
18#16840167C2211yescharacteristic
19#16840135C2211yescharacteristic
20#16840113C2211yescharacteristic
21#16840109C2211yescharacteristic
22#16840086—21no—
23#16840069—21no—
24#16840062—21no—
25#16840057C2211yescharacteristic
26#16840057C2211yescharacteristic
27#16840052C2211yescharacteristic
28#16840052C2211yescharacteristic
29#16840047C2211yescharacteristic
30#16840042C2211yescharacteristic
31#16840008—21no—
32#16840008—21no—
33#16839962C2211yescharacteristic
34#16839957C2211yescharacteristic
35#16839907—21no—
36#16839864—21no—
37#16839864—21no—
38C4×#11758201—21no—
39C4×#11758199—21no—
40#16839815C2211yescharacteristic
41#16839798—21no—
42C2×C4⋊#92418C2211yescharacteristic
43#16839758—21no—
44#16839515C2211yescharacteristic
45#16839515C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839513C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839510C2211yescharacteristic
50#16839510C2211yescharacteristic
51#16839510C2211yescharacteristic
52#16839509C2211yescharacteristic
53#16839509C2211yescharacteristic
54#16839505C2211yescharacteristic
55#16839505C2211yescharacteristic
56C4×#41×C61C2211yescharacteristic
57C8×Q8×C61C2211yescharacteristic
58C8×Q8×C61C2211yescharacteristic
59C61×#436—21no—
60C61×#436—21no—
61C61×#436C2211yescharacteristic
62C61×#436C2211yescharacteristic
63C61×#432—21no—
64C61×#432C2211yescharacteristic
65C4×M16×C61—21no—
66C4×M16×C61C2211yescharacteristic
67C2×#11758222—21no—
68C2×#11758190—21no—
69C2×#11758190—21no—
70C2×#11758201—21no—
71C2×#11758199—21no—
72C22×#92418—21no—
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519777C2311yescharacteristic
76#14519774C2311yescharacteristic
77#14519769—21no—
78#14519765—21no—
79#14519756—21no—
80#14519756—21no—
81#14519751—21no—
82#14519751—21no—
83C4×Dic122—21no—
84C4×Dic122—21no—
85C4×D244—21no—
86C4×D244—21no—
87#14519736—21no—
88#14519711—21no—
89#14519711—21no—
90#14519710—21no—
91#14519710—21no—
92#14519710—21no—
93#14519708—21no—
94#14519708—21no—
95#14519708—21no—
96C4⋊#92418—21no—
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99C4⋊#92418C2311yescharacteristic
100C4⋊#92418C2311yescharacteristic
101#14519702—21no—
102#14519702—21no—
103#14519701—21no—
104#14519701—21no—
105#14519701—21no—
106#14519699—21no—
107#14519699—21no—
108#14519698—21no—
109#14519698—21no—
110#14519698—21no—
111#14519697—21no—
112#14519697—21no—
113C22⋊#92418—21no—
114C4×#92418—21no—
115C2×#41×C61D411yescharacteristic
116C2×M16×C61—21no—
117C2×M16×C61—21no—
118C2×M16×C61D411yescharacteristic
119C2×M16×C61D411yescharacteristic
120C4×Q8×C61D411yescharacteristic
121C4×Q8×C61C2311yescharacteristic
122C4×D4×C61—21no—
123C4×D4×C61—21no—
124C4×D4×C61D411yescharacteristic
125C4×D4×C61C2311yescharacteristic
126#117×C61—21no—
127#117×C61D411yescharacteristic
128C2×C42×C61—21no—
129C2×C42×C61C2311yescharacteristic
130C4⋊C8×C61—21no—
131C4⋊C8×C61—21no—
132C4⋊C8×C61C2311yescharacteristic
133C4⋊C8×C61C2311yescharacteristic
134C22⋊C8×C61—21no—
135C22⋊C8×C61—21no—
136C22⋊C8×C61—21no—
137#97×C61—21no—
138#97×C61D411yescharacteristic
139#97×C61D411yescharacteristic
140C4×C8×C61—21no—
141C4×C8×C61—21no—
142C4×C8×C61C2311yescharacteristic
143C4×C8×C61C2311yescharacteristic
144#41×C61—22no—
145#41×C61—41no—
146C2×Q8×C61C2×D411yescharacteristic
147C2×D4×C61—21no—
148C2×D4×C61C2×D411yescharacteristic
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61C2×D411yescharacteristic
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61#4111yescharacteristic
155C2×C22⋊D61—41no—
156#11758222—82no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×C4×D61—41no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163C2×Dic122—21no—
164M16×C61—22no—
165M16×C61—41no—
166M16×C61—42no—
167M16×C61—42no—
168C2×C8×C61—21no—
169C2×C8×C61—21no—
170C2×C8×C61C2×D411yescharacteristic
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2×D411yescharacteristic
173C2×C8×C61C2×D411yescharacteristic
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61C2×D411yescharacteristic
177C4⋊C4×C61C2×D411yescharacteristic
178C22⋊C4×C61—21no—
179C22⋊C4×C61—21no—
180C22⋊C4×C61—21no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61C2×D411yescharacteristic
185C42×C61C2×D411yescharacteristic
186C42×C61#4111yescharacteristic
187C42×C61C2411yescharacteristic
188C22⋊Dic61—41no—
189#11758201—42no—
190#11758199—42no—
191#11758198—41no—
192#11758198—41no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199#11758196—41no—
200#11758196—41no—
201C4×Dic61—41no—
202C2×#92418—21no—
203C2×#92418—21no—
204C2×#92418—21no—
205C2×#92418—21no—
206C2×#92418—42no—
207#11758190—42no—
208#11758190—42no—
209C23×C61—21no—
210C23×C61#12111yescharacteristic
211C22×D61—41no—
212Q8×C61—21no—
213Q8×C61#13712yes—
214D4×C61—21no—
215D4×C61—21no—
216D4×C61—42no—
217C2×C4×C61—21no—
218C2×C4×C61—21no—
219C2×C4×C61—21no—
220C2×C4×C61—21no—
221C2×C4×C61—21no—
222C2×C4×C61—21no—
223C2×C4×C61—21no—
224C2×C4×C61—41no—
225C2×C4×C61—42no—
226C2×C4×C61—41no—
227C2×C4×C61—41no—
228C2×C4×C61—81no—
229C2×C4×C61#12111yescharacteristic
230C2×C4×C61#12111yescharacteristic
231C2×C4×C61#12111yescharacteristic
232C2×C4×C61C22×D411yescharacteristic
233C2×C4×C61C22×D411yescharacteristic
234C2×C4×C61C2×#4111yescharacteristic
235C22⋊D61—82no—
236C2×Dic61—41no—
237C2×Dic61—41no—
238C2×Dic61—41no—
239D244—41no—
240D244—41no—
241D244—41no—
242D244—41no—
243D244—81no—
244C4×D61—82no—
245Dic122—41no—
246Dic122—41no—
247Dic122—81no—
248Dic122—81no—
249C8×C61—21no—
250C8×C61—21no—
251C8×C61—21no—
252C8×C61—41no—
253C8×C61—41no—
254C8×C61#13512yes—
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—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×C61C2×#13511yescharacteristic
278C4×C61C2×#13711yescharacteristic
279C4×C61#57511yescharacteristic
280C4×C61#57511yescharacteristic
281Dic61—81no—
282Dic61—81no—
283Dic61—81no—
284#3182—611no—
285#3106—611no—
286#3076—611no—
287#2971D6111yescharacteristic, complemented
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#533—611no—
306#532—611no—
307C4×#41C2×D6111yescharacteristic
308#500—611no—
309#492—611no—
310#487—611no—
311#486—611no—
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—1221no—
322#468—1221no—
323#460—611no—
324#459—611no—
325#449—1221no—
326C8×Q8C2×D6111yescharacteristic
327C8×Q8C2×D6111yescharacteristic
328#440—1221no—
329#440—1221no—
330C4×SD16—1221no—
331C4×D8—1221no—
332#436—21no—
333#436—21no—
334#436C2×D6111yescharacteristic
335#436C2×D6111yescharacteristic
336#432—21no—
337#432C2×D6111yescharacteristic
338#431—611no—
339#427—1221no—
340C2×C4⋊C8—611no—
341#416—1221no—
342C4×M16—21no—
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—1221no—
358C2×#41D24411yescharacteristic
359C2×SD16—1221no—
360C2×SD16—1221no—
361C2×SD16—1221no—
362C2×D8—1221no—
363C2×M16—21no—
364C2×M16—21no—
365C2×M16C22⋊D6111yescharacteristic
366C2×M16C22⋊D6111yescharacteristic
367C22×C8—1221no—
368C4⋊Q8—611no—
369C4⋊Q8—611no—
370C4⋊Q8—611no—
371#127—611no—
372#125—1221no—
373#124—1221no—
374C22⋊Q8—1221no—
375C22⋊Q8—1221no—
376#121—1221no—
377#121—1221no—
378C4×Q8—1221no—
379C4×Q8—1221no—
380C4×Q8D24411yescharacteristic
381C4×Q8C22×D6111yescharacteristic
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—1221no—
385C4×D4—1221no—
386C4×D4D24411yescharacteristic
387C4×D4C22×D6111yescharacteristic
388#117—21no—
389#117—1221no—
390#117D24411yescharacteristic
391C2×C42—21no—
392C2×C42C22×D6111yescharacteristic
393#107—1221no—
394#107—1221no—
395#107—1221no—
396#107—1221no—
397#107—1221no—
398#106—1221no—
399#106—1221no—
400#106—1221no—
401C4⋊C8—21no—
402C4⋊C8—21no—
403C4⋊C8—611no—
404C4⋊C8—611no—
405C4⋊C8—611no—
406C4⋊C8—611no—
407C4⋊C8—1221no—
408C4⋊C8C22×D6111yescharacteristic
409C4⋊C8C22×D6111yescharacteristic
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#103—1221no—
414#103—1221no—
415#102—1221no—
416#102—1221no—
417#102—1221no—
418#102—1221no—
419#102—1221no—
420#102—1221no—
421#102—1221no—
422C22⋊C8—21no—
423C22⋊C8—21no—
424C22⋊C8—21no—
425C22⋊C8—1221no—
426#97—21no—
427#97C22⋊D6111yescharacteristic
428#97C22⋊D6111yescharacteristic
429C4×C8—21no—
430C4×C8—21no—
431C4×C8—1221no—
432C4×C8C22×D6111yescharacteristic
433C4×C8C22×D6111yescharacteristic
434#41—22no—
435#41—41no—
436#41—4882no—
437C2×Q8—1221no—
438C2×Q8—1221no—
439C2×Q8—1221no—
440C2×Q8C2×D24411yescharacteristic
441C2×D4—21no—
442C2×D4—1221no—
443C2×D4—1221no—
444C2×D4—1221no—
445C2×D4—2441no—
446C2×D4C2×D24411yescharacteristic
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—2441no—
451C22×C4C2×D24411yescharacteristic
452C22×C4#1175822211yescharacteristic
453C22×C4C2×C22⋊D6111yescharacteristic
454SD16—2442no—
455SD16—2442no—
456SD16—2442no—
457D8—2442no—
458M16—22no—
459M16—41no—
460M16—42no—
461M16—42no—
462C2×C8—21no—
463C2×C8—21no—
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—1221no—
476C4⋊C4—1221no—
477C4⋊C4—1221no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—2441no—
482C4⋊C4—2441no—
483C4⋊C4C2×D24411yescharacteristic
484C4⋊C4C2×D24411yescharacteristic
485C22⋊C4—21no—
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—2441no—
489C22⋊C4—2441no—
490C22⋊C4—2441no—
491C42—21no—
492C42—21no—
493C42—21no—
494C42—2441no—
495C42C2×D24411yescharacteristic
496C42#1175822211yescharacteristic
497C42C2×C22⋊D6111yescharacteristic
498C42C23×D6111yescharacteristic
499C23—21no—
500C23—2441no—
501C23#1451975111yescharacteristic
502Q8—21no—
503Q8—4881no—
504Q8—4881no—
505Q8—2441no—
506Q8—2441no—
507Q8#1451981012yes—
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—81no—
529C2×C4—4882no—
530C2×C4—2441no—
531C2×C4—2441no—
532C2×C4—2441no—
533C2×C4#1451975111yescharacteristic
534C2×C4#1451975111yescharacteristic
535C2×C4#1451975111yescharacteristic
536C2×C4C22×C22⋊D6111yescharacteristic
537C2×C4C22×D24411yescharacteristic
538C2×C4C2×#1175822211yescharacteristic
539C8—21no—
540C8—21no—
541C8—21no—
542C8—41no—
543C8—41no—
544C8—2441no—
545C8—2441no—
546C8—2441no—
547C8—2441no—
548C8#1451979712yes—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22—4881no—
556C22—4881no—
557C22—4881no—
558C22C2×#1451975111yescharacteristic
559C4—21no—
560C4—22no—
561C4—21no—
562C4—21no—
563C4—21no—
564C4—41no—
565C4—41no—
566C4—4881no—
567C4—4881no—
568C4—4881no—
569C4C2×#1451979711yescharacteristic
570C4C2×#1451981011yescharacteristic
571C4#1684056011yescharacteristic
572C4#1684056211yescharacteristic
573C2—41no—
574C2—81no—
575C2—9761no—
576C2#2145527411yescharacteristic
577C2#2145544911yescharacteristic
578C2#2145554711yescharacteristic
subgroupG/H
centerC22C2×#14519751
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#2971C2
socleC22×C61C2×#121
prime radical#21455879C2
hypercenter#2971D61
hypocenterC61#19183
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2971

Complements

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

NH ≅ G/Ntype
1. #21456537C2semidirect
3. #21455607C2semidirect
4. #21445394C2semidirect
5. #21445352C2semidirect
6. #21445342C2semidirect
7. #21445284C2semidirect
8. #21445277C2semidirect
9. #21445235C2semidirect
10. #21445223C2semidirect
11. #21445167C2semidirect
12. #21444571C2semidirect
13. #21444530C2semidirect
14. #21444515C2semidirect
15. C61×#2971C2semidirect
287. #2971D61semidirect
356. C61#19183semidirect

Maximal subgroups

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

subgroupindex
#214565372
#214558792
#214556072
#214453942
#214453522
#214453422
#214452842
#214452772
#214452352
#214452232
#214451672
#214445712
#214445302
#214445152
C61×#29712
#1918361