Finite Group Databasebeta

← #29664001

Subgroups — #29664001

Order 15616.

630 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456141C211yesmaximal, characteristic, complemented
2#21455981C211yesmaximal, characteristic, complemented
3#41×#92418C211yesmaximal, characteristic, complemented
4#21445275C211yesmaximal, characteristic, complemented
5#21445268C211yesmaximal, characteristic, complemented
6#21445261C211yesmaximal, characteristic, complemented
7#21445254C211yesmaximal, characteristic, complemented
8#21445247C211yesmaximal, characteristic, complemented
9#21445240C211yesmaximal, characteristic, complemented
10#21445233C211yesmaximal, characteristic, complemented
11#21445226C211yesmaximal, characteristic, complemented
12#21444580C211yesmaximal, characteristic, complemented
13#21444530C211yesmaximal, characteristic, complemented
14#21444524C211yesmaximal, characteristic, complemented
15C61×#3115C211yesmaximal, characteristic
16#16840319C2211yescharacteristic
17#16840116—21no—
18#16840116—21no—
19#16840112C2211yescharacteristic
20#16840112C2211yescharacteristic
21#16840108C2211yescharacteristic
22#16840108C2211yescharacteristic
23#16840103—21no—
24#16840057C2211yescharacteristic
25#16840052C2211yescharacteristic
26#16839972C2211yescharacteristic
27#16839967C2211yescharacteristic
28#16839924—21no—
29Q8×#92418C2211yescharacteristic
30Q8×#92418C2211yescharacteristic
31#16839867—21no—
32#16839862—21no—
33D4×#92418—21no—
34D4×#92418—21no—
35D4×#92418C2211yescharacteristic
36D4×#92418C2211yescharacteristic
37#16839819C2211yescharacteristic
38#16839815—21no—
39#16839815C2211yescharacteristic
40#16839758—21no—
41#16839758—21no—
42C2×C4×#92418—21no—
43C2×C4×#92418C2211yescharacteristic
44#16839514C2211yescharacteristic
45#16839514C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839512C2211yescharacteristic
48#16839510C2211yescharacteristic
49#16839510C2211yescharacteristic
50#16839509C2211yescharacteristic
51#16839509C2211yescharacteristic
52#16839508C2211yescharacteristic
53#16839508C2211yescharacteristic
54#16839506C2211yescharacteristic
55#16839506C2211yescharacteristic
56C61×#532C2211yescharacteristic
57C4×#41×C61C2211yescharacteristic
58C61×#482—21no—
59C61×#475C2211yescharacteristic
60C61×#474C2211yescharacteristic
61C61×#460C2211yescharacteristic
62C61×#459C2211yescharacteristic
63C61×#449—21no—
64C61×#442—21no—
65C61×#440—21no—
66C61×#432C2211yescharacteristic
67C2×#11758202—21no—
68C2×#11758200—21no—
69C22×#92418—21no—
70C22×#92418—21no—
71C22×#92418D411yescharacteristic
72C22×#92418D411yescharacteristic
73#14519777C2311yescharacteristic
74#14519777C2311yescharacteristic
75#14519769—21no—
76#14519756—21no—
77#14519756—21no—
78C4×Dic122—21no—
79C4×Dic122—21no—
80#14519736—21no—
81#14519710—21no—
82#14519708—21no—
83C4⋊#92418—21no—
84C4⋊#92418—21no—
85C4⋊#92418C2311yescharacteristic
86C4⋊#92418C2311yescharacteristic
87C4⋊#92418C2311yescharacteristic
88C4⋊#92418C2311yescharacteristic
89#14519701—21no—
90#14519701—21no—
91#14519701—21no—
92#14519701—21no—
93#14519701—21no—
94#14519700—21no—
95#14519700—21no—
96#14519698—21no—
97#14519698—21no—
98#14519698—21no—
99#14519698—21no—
100#14519698—21no—
101C22⋊#92418—21no—
102C22⋊#92418—21no—
103C22⋊#92418—21no—
104C22⋊#92418—21no—
105#14519688—21no—
106C4×#92418—21no—
107C4×#92418—21no—
108C4×#92418—21no—
109C4×#92418D411yescharacteristic
110C4×#92418D411yescharacteristic
111C4×#92418C2311yescharacteristic
112C4×#92418C2311yescharacteristic
113C2×#41×C61—21no—
114C2×#41×C61D411yescharacteristic
115C2×SD16×C61—21no—
116C2×D8×C61—21no—
117C4⋊Q8×C61C2311yescharacteristic
118#127×C61C2311yescharacteristic
119#124×C61—21no—
120#121×C61—21no—
121#121×C61—21no—
122C4×Q8×C61D411yescharacteristic
123C4×Q8×C61C2311yescharacteristic
124C4×D4×C61—21no—
125C4×D4×C61—21no—
126C4×D4×C61—21no—
127C4×D4×C61—21no—
128C4×D4×C61D411yescharacteristic
129C4×D4×C61C2311yescharacteristic
130#117×C61—21no—
131#117×C61D411yescharacteristic
132C2×C42×C61—21no—
133C2×C42×C61C2311yescharacteristic
134#107×C61—21no—
135#106×C61—21no—
136C4⋊C8×C61C2311yescharacteristic
137C4⋊C8×C61C2311yescharacteristic
138#103×C61—21no—
139#102×C61—21no—
140#102×C61—21no—
141#102×C61—21no—
142C22⋊C8×C61—21no—
143#97×C61—21no—
144#41×C61—22no—
145#41×C61—22no—
146#41×C61—42no—
147#41×C61—42no—
148C2×Q8×C61—21no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×D4×C61—21no—
151C2×D4×C61—21no—
152C2×D4×C61—21no—
153C2×D4×C61—21no—
154C2×D4×C61—41no—
155C2×D4×C61C2×D411yescharacteristic
156C22×C4×C61—21no—
157C22×C4×C61—21no—
158C22×C4×C61—21no—
159C22×C4×C61—41no—
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61#4111yescharacteristic
163C2×Dic122—21no—
164C2×Dic122—21no—
165SD16×C61—42no—
166D8×C61—42no—
167C2×C8×C61—21no—
168C2×C8×C61—21no—
169C4⋊C4×C61—21no—
170C4⋊C4×C61—21no—
171C4⋊C4×C61—21no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61C2×D411yescharacteristic
174C4⋊C4×C61C2×D411yescharacteristic
175C22⋊C4×C61—21no—
176C22⋊C4×C61—21no—
177C22⋊C4×C61—21no—
178C22⋊C4×C61—41no—
179C22⋊C4×C61—41no—
180C42×C61—21no—
181C42×C61—21no—
182C42×C61—21no—
183C42×C61C2×D411yescharacteristic
184C42×C61C2×D411yescharacteristic
185C42×C61#4111yescharacteristic
186C42×C61C2411yescharacteristic
187C22⋊Dic61—41no—
188#11758202—42no—
189#11758200—42no—
190C4⋊Dic61—21no—
191C4⋊Dic61—21no—
192C4⋊Dic61—21no—
193C4⋊Dic61—21no—
194#11758196—41no—
195#11758196—41no—
196C4×Dic61—41no—
197C2×#92418—21no—
198C2×#92418—21no—
199C2×#92418—21no—
200C2×#92418—21no—
201C2×#92418—41no—
202C2×#92418—41no—
203C2×#92418—42no—
204C2×#92418—42no—
205C2×#92418C2×D411yescharacteristic
206C2×#92418C2×D411yescharacteristic
207C2×#92418C2×D411yescharacteristic
208C2×#92418C2×D411yescharacteristic
209C23×C61—21no—
210C23×C61—41no—
211C23×C61#12111yescharacteristic
212Q8×C61—21no—
213Q8×C61—41no—
214Q8×C61—41no—
215Q8×C61#13512yes—
216D4×C61—24no—
217D4×C61—21no—
218D4×C61—41no—
219D4×C61—41no—
220D4×C61—41no—
221D4×C61—41no—
222D4×C61—81no—
223D4×C61—82no—
224D4×C61#13512yes—
225C2×C4×C61—24no—
226C2×C4×C61—21no—
227C2×C4×C61—21no—
228C2×C4×C61—21no—
229C2×C4×C61—21no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—41no—
234C2×C4×C61—42no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—82no—
239C2×C4×C61#12111yescharacteristic
240C2×C4×C61#12111yescharacteristic
241C2×C4×C61#12111yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61C22×D411yescharacteristic
244C2×C4×C61C2×#4111yescharacteristic
245C2×Dic61—41no—
246C2×Dic61—41no—
247Dic122—41no—
248Dic122—41no—
249Dic122—81no—
250C8×C61—41no—
251C8×C61—41no—
252#92418—21no—
253#92418—21no—
254#92418—21no—
255#92418—21no—
256#92418—41no—
257#92418—41no—
258#92418—41no—
259#92418—41no—
260#19278—611nomaximal, Sylow
261C22×C61—21no—
262C22×C61—21no—
263C22×C61—22no—
264C22×C61—21no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61—81no—
268C22×C61—81no—
269C22×C61—81no—
270C22×C61C2×#12111yescharacteristic
271C4×C61—21no—
272C4×C61—21no—
273C4×C61—22no—
274C4×C61—21no—
275C4×C61—21no—
276C4×C61—41no—
277C4×C61—41no—
278C4×C61—81no—
279C4×C61C2×#13511yescharacteristic
280C4×C61C2×#13511yescharacteristic
281C4×C61#57511yescharacteristic
282C4×C61#57511yescharacteristic
283Dic61—81no—
284Dic61—81no—
285#3131—611no—
286#3116—611no—
287#3115D6111yescharacteristic
288C8×#41—611no—
289#1684—611no—
290#1683—611no—
291#1682—611no—
292#1681—611no—
293#1680—611no—
294#1679—611no—
295#1678—611no—
296#1677—611no—
297#1568—611no—
298#1554—611no—
299#1553—611no—
300C2×C61—41no—
301C2×C61—42no—
302C2×C61—161no—
303C2×C61#304011yescharacteristic
304C2×C61#305011yescharacteristic
305C2×C61#306711yescharacteristic
306#533—611no—
307#532C2×D6111yescharacteristic
308C4×#41C2×D6111yescharacteristic
309#488—1221no—
310#488—1221no—
311#487—611no—
312#487—611no—
313#486—611no—
314#486—611no—
315#485—1221no—
316#482—21no—
317#477—611no—
318#476—611no—
319#475C2×D6111yescharacteristic
320#474C2×D6111yescharacteristic
321#462—611no—
322#461—611no—
323#460C2×D6111yescharacteristic
324#459C2×D6111yescharacteristic
325#452—1221no—
326#449—21no—
327C8×Q8—611no—
328C8×Q8—611no—
329#442—21no—
330#441—1221no—
331#440—21no—
332#440—1221no—
333C8×D4—611no—
334C8×D4—611no—
335C8×D4—1221no—
336C8×D4—1221no—
337#432—611no—
338#432C2×D6111yescharacteristic
339#431—611no—
340#431—1221no—
341#416—1221no—
342#416—1221no—
343C2×C4×C8—611no—
344C2×C4×C8—1221no—
345#335—611no—
346#335—611no—
347#334—611no—
348#334—611no—
349#333—611no—
350#333—611no—
351#332—611no—
352#332—611no—
353#332—611no—
354#332—611no—
355#331—611no—
356#331—611no—
357C61#1927811yescharacteristic, complemented, Sylow
358C2×#41—21no—
359C2×#41C22⋊D6111yescharacteristic
360C2×Q16—1221no—
361C2×SD16—21no—
362C2×SD16—1221no—
363C2×D8—21no—
364C22×C8—611no—
365C22×C8—611no—
366C22×C8—1221no—
367C22×C8—1221no—
368C4⋊Q8—611no—
369C4⋊Q8—611no—
370C4⋊Q8C22×D6111yescharacteristic
371#127C22×D6111yescharacteristic
372#125—1221no—
373#124—21no—
374C22⋊Q8—1221no—
375C22⋊Q8—1221no—
376#121—21no—
377#121—21no—
378C4×Q8—1221no—
379C4×Q8—1221no—
380C4×Q8C22⋊D6111yescharacteristic
381C4×Q8C22×D6111yescharacteristic
382C4×D4—21no—
383C4×D4—21no—
384C4×D4—21no—
385C4×D4—21no—
386C4×D4C22⋊D6111yescharacteristic
387C4×D4C22×D6111yescharacteristic
388#117—21no—
389#117—1221no—
390#117C22⋊D6111yescharacteristic
391C2×C42—21no—
392C2×C42C22×D6111yescharacteristic
393#107—21no—
394#107—1221no—
395#106—21no—
396#106—1221no—
397C4⋊C8—611no—
398C4⋊C8—611no—
399C4⋊C8—611no—
400C4⋊C8—611no—
401C4⋊C8—1221no—
402C4⋊C8—1221no—
403C4⋊C8C22×D6111yescharacteristic
404C4⋊C8C22×D6111yescharacteristic
405#103—21no—
406#103—1221no—
407#103—1221no—
408#103—1221no—
409#103—1221no—
410#103—1221no—
411#103—1221no—
412#103—1221no—
413#102—21no—
414#102—21no—
415#102—21no—
416#102—1221no—
417#102—1221no—
418#102—1221no—
419#102—1221no—
420#102—1221no—
421C22⋊C8—21no—
422C22⋊C8—1221no—
423C22⋊C8—1221no—
424C22⋊C8—1221no—
425C22⋊C8—1221no—
426#97—21no—
427#97—1221no—
428C4×C8—611no—
429C4×C8—611no—
430C4×C8—611no—
431C4×C8—611no—
432C4×C8—1221no—
433C4×C8—1221no—
434C4×C8—1221no—
435#41—22no—
436#41—22no—
437#41—42no—
438#41—42no—
439C2×Q8—21no—
440C2×Q8—1221no—
441C2×Q8—1221no—
442C2×Q8C2×C22⋊D6111yescharacteristic
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—41no—
448C2×D4C2×C22⋊D6111yescharacteristic
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—41no—
453C22×C4D4×D6111yescharacteristic
454C22×C4#1175822411yescharacteristic
455C22×C4C2×C22⋊D6111yescharacteristic
456Q16—2442no—
457SD16—42no—
458SD16—2442no—
459D8—42no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—611no—
463C2×C8—611no—
464C2×C8—611no—
465C2×C8—611no—
466C2×C8—1221no—
467C2×C8—1221no—
468C2×C8—1221no—
469C2×C8—1221no—
470C2×C8—2441no—
471C2×C8—2441no—
472C2×C8—2442no—
473C2×C8—2442no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—1221no—
479C4⋊C4—1221no—
480C4⋊C4—1221no—
481C4⋊C4—1221no—
482C4⋊C4—2441no—
483C4⋊C4—2441no—
484C4⋊C4C2×C22⋊D6111yescharacteristic
485C4⋊C4C2×C22⋊D6111yescharacteristic
486C22⋊C4—21no—
487C22⋊C4—21no—
488C22⋊C4—21no—
489C22⋊C4—41no—
490C22⋊C4—41no—
491C22⋊C4—2441no—
492C42—21no—
493C42—21no—
494C42—21no—
495C42—2441no—
496C42D4×D6111yescharacteristic
497C42#1175822411yescharacteristic
498C42C2×C22⋊D6111yescharacteristic
499C42C23×D6111yescharacteristic
500C23—21no—
501C23—41no—
502C23#1451974911yescharacteristic
503Q8—21no—
504Q8—41no—
505Q8—41no—
506Q8—4881no—
507Q8—2441no—
508Q8—2441no—
509Q8#1451979712yes—
510D4—24no—
511D4—21no—
512D4—41no—
513D4—41no—
514D4—41no—
515D4—41no—
516D4—81no—
517D4—82no—
518D4#1451979712yes—
519C2×C4—24no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—41no—
528C2×C4—42no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—82no—
533C2×C4—2441no—
534C2×C4—2441no—
535C2×C4#1451974911yescharacteristic
536C2×C4#1451974911yescharacteristic
537C2×C4#1451974911yescharacteristic
538C2×C4C2×D4×D6111yescharacteristic
539C2×C4C22×C22⋊D6111yescharacteristic
540C2×C4C2×#1175822411yescharacteristic
541C8—41no—
542C8—41no—
543C8—1221no—
544C8—1221no—
545C8—1221no—
546C8—1221no—
547C8—2441no—
548C8—2441no—
549C8—2441no—
550C8—2441no—
551C22—21no—
552C22—22no—
553C22—21no—
554C22—21no—
555C22—41no—
556C22—41no—
557C22—81no—
558C22—81no—
559C22—81no—
560C22C2×#1451974911yescharacteristic
561C4—21no—
562C4—22no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—41no—
567C4—41no—
568C4—81no—
569C4—4881no—
570C4—4881no—
571C4C2×#1451979711yescharacteristic
572C4C2×#1451979711yescharacteristic
573C4#1684055811yescharacteristic
574C4#1684055811yescharacteristic
575C2—41no—
576C2—42no—
577C2—161no—
578C2#2145531111yescharacteristic
579C2#2145539711yescharacteristic
580C2#2145553611yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
Fitting, prime radicalC61×#3115C2
socleC22×C61C2×#121
hypercenter#3115D61
hypocenterC61#19278
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3115

Complements

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

NH ≅ G/Ntype
1. #21456141C2semidirect
2. #21455981C2semidirect
3. #41×#92418C2semidirect
4. #21445275C2semidirect
5. #21445268C2semidirect
6. #21445261C2semidirect
7. #21445254C2semidirect
8. #21445247C2semidirect
9. #21445240C2semidirect
10. #21445233C2semidirect
11. #21445226C2semidirect
12. #21444580C2semidirect
13. #21444530C2semidirect
14. #21444524C2semidirect
357. C61#19278semidirect

Maximal subgroups

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

subgroupindex
#214561412
#214559812
#41×#924182
#214452752
#214452682
#214452612
#214452542
#214452472
#214452402
#214452332
#214452262
#214445802
#214445302
#214445242
C61×#31152
#1927861