Finite Group Databasebeta

← #29676904

Subgroups — #29676904

Order 15616.

640 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21458533C211yesmaximal, characteristic, complemented
2#21458124C211yesmaximal, characteristic
3#21457466C211yesmaximal, characteristic, complemented
4#21445417C211yesmaximal, characteristic, complemented
5#21445371C211yesmaximal, characteristic, complemented
6#21445364C211yesmaximal, characteristic, complemented
7#21445329C211yesmaximal, characteristic, complemented
8#21445308C211yesmaximal, characteristic, complemented
9#21445103C211yesmaximal, characteristic, complemented
10#21445102C211yesmaximal, characteristic, complemented
11#21445089C211yesmaximal, characteristic, complemented
12#21445081C211yesmaximal, characteristic, complemented
13#21444710C211yesmaximal, characteristic, complemented
14#21444702C211yesmaximal, characteristic, complemented
15C61×#3353C211yesmaximal, characteristic, complemented
16C2×#14519806—21no—
17Q8×D244C2211yescharacteristic
18#16840416C2211yescharacteristic
19#16840158C2211yescharacteristic
20#16840158C2211yescharacteristic
21#16840142C2211yescharacteristic
22#16840121—21no—
23#16840112C2211yescharacteristic
24#16840112C2211yescharacteristic
25#16840095—21no—
26#16840057—21no—
27#16840042C2211yescharacteristic
28#16840042C2211yescharacteristic
29#16840021—21no—
30#16840010—21no—
31#16839974—21no—
32#16839972C2211yescharacteristic
33#16839967C2211yescharacteristic
34#16839964C2211yescharacteristic
35#16839964C2211yescharacteristic
36#16839925—21no—
37#16839919—21no—
38#16839907—21no—
39#16839893C2211yescharacteristic
40#16839860—21no—
41C4×#11758199C2211yescharacteristic
42#16839841C2211yescharacteristic
43#16839840C2211yescharacteristic
44#16839763—21no—
45#16839514C2211yescharacteristic
46#16839513C2211yescharacteristic
47#16839509C2211yescharacteristic
48#16839509C2211yescharacteristic
49#16839508C2211yescharacteristic
50#16839508C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52D4⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56C61×#556C2211yescharacteristic
57C61×#547C2211yescharacteristic
58C61×#500C2211yescharacteristic
59C61×#484—21no—
60C61×#482—21no—
61C61×#475C2211yescharacteristic
62C61×#474C2211yescharacteristic
63C61×#468—21no—
64C4×SD16×C61C2211yescharacteristic
65C61×#436C2211yescharacteristic
66C61×#428—21no—
67C2×#11758224—21no—
68C2×Q8×D61—21no—
69#14519806—44no—
70C2×Dic244—21no—
71C2×Dic244—21no—
72C2×#11758202—21no—
73C2×#11758200—21no—
74C2×#11758201—21no—
75C2×#11758201—21no—
76C2×#11758199D411yescharacteristic
77C2×#11758194—21no—
78C2×#11758189—21no—
79#14519777—21no—
80#14519777C2311yescharacteristic
81#14519777C2311yescharacteristic
82#14519761—41no—
83#14519757—21no—
84#14519757—21no—
85#14519757—41no—
86#14519751—21no—
87C4×D244—21no—
88C4×D244C2311yescharacteristic
89C2×C4⋊Dic61—21no—
90#14519710—21no—
91#14519710—21no—
92#14519710D411yescharacteristic
93#14519708—41no—
94#14519708—41no—
95C4⋊#92418—21no—
96C4⋊#92418C2311yescharacteristic
97C4⋊#92418C2311yescharacteristic
98C4⋊#92418C2311yescharacteristic
99#14519701—21no—
100#14519701—21no—
101#14519701—21no—
102#14519701—21no—
103#14519701—21no—
104#14519701—21no—
105#14519700—21no—
106#14519700—21no—
107#14519698—21no—
108#14519698—21no—
109#14519698D411yescharacteristic
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113#14519688—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C2×#41×C61—21no—
117C2×SD16×C61D411yescharacteristic
118C2×M16×C61—21no—
119C4⋊Q8×C61C2311yescharacteristic
120C4⋊Q8×C61C2311yescharacteristic
121#126×C61—41no—
122#124×C61—21no—
123#123×C61—41no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—21no—
126#121×C61—21no—
127C4×Q8×C61—21no—
128C4×Q8×C61C2311yescharacteristic
129C4×D4×C61C2311yescharacteristic
130C4×D4×C61C2311yescharacteristic
131#117×C61—21no—
132C2×C4⋊C4×C61—21no—
133#107×C61—21no—
134#107×C61—21no—
135#107×C61—21no—
136#106×C61—21no—
137#106×C61D411yescharacteristic
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61D411yescharacteristic
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145#41×C61—42no—
146#41×C61—81no—
147C2×Q8×C61—21no—
148C2×Q8×C61—21no—
149C2×Q8×C61C2×D411yescharacteristic
150C2×D4×C61—41no—
151C2×D4×C61C2×D411yescharacteristic
152C2×D4×C61C2×D411yescharacteristic
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—41no—
156C2×C22⋊D61—41no—
157C22×Dic61—41no—
158Q8×D61—42no—
159Q8×D61—81no—
160#11758224—42no—
161#11758224—81no—
162C2×D244C2×D411yescharacteristic
163C2×C4×D61—21no—
164C2×C4×D61—41no—
165C2×Dic122—21no—
166C2×Dic122—21no—
167C2×Dic122—41no—
168SD16×C61—22no—
169M16×C61—42no—
170C2×C8×C61—21no—
171C2×C8×C61C2×D411yescharacteristic
172C2×C8×C61C2×D411yescharacteristic
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—
180C4⋊C4×C61C2×D411yescharacteristic
181C4⋊C4×C61C2×D411yescharacteristic
182C4⋊C4×C61C2×D411yescharacteristic
183C22⋊C4×C61—21no—
184C22⋊C4×C61—21no—
185C22⋊C4×C61—41no—
186C22⋊C4×C61—41no—
187C22⋊C4×C61—41no—
188C42×C61—21no—
189C42×C61—41no—
190C42×C61C2411yescharacteristic
191#11758202—42no—
192#11758201—42no—
193#11758201—81no—
194#11758200—42no—
195#11758199—41no—
196#11758198—21no—
197#11758198—41no—
198#11758198—41no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61—41no—
204C4⋊Dic61—41no—
205C4⋊Dic61C2×D411yescharacteristic
206#11758194—81no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418C2×D411yescharacteristic
212C2×#92418C2×D411yescharacteristic
213Dic244—42no—
214Dic244—81no—
215#11758189—42no—
216C23×C61—21no—
217C23×C61—21no—
218C22×D61—21no—
219Q8×C61—21no—
220Q8×C61—21no—
221Q8×C61—41no—
222Q8×C61—41no—
223Q8×C61—41no—
224Q8×C61—41no—
225D4×C61—21no—
226D4×C61—21no—
227D4×C61—21no—
228D4×C61—21no—
229D4×C61—82no—
230C2×C4×C61—21no—
231C2×C4×C61—21no—
232C2×C4×C61—21no—
233C2×C4×C61—21no—
234C2×C4×C61—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—41no—
238C2×C4×C61—41no—
239C2×C4×C61—41no—
240C2×C4×C61—82no—
241C2×C4×C61C22×D411yescharacteristic
242C2×C4×C61C22×D411yescharacteristic
243C2×C4×C61H(5,𝔽2)11yescharacteristic
244C22⋊D61—82no—
245C2×Dic61—21no—
246C2×Dic61—41no—
247C2×Dic61—41no—
248C2×Dic61—82no—
249D244—21no—
250D244—21no—
251C4×D61—41no—
252C4×D61—41no—
253C4×D61—82no—
254Dic122—41no—
255Dic122—41no—
256Dic122—41no—
257Dic122—41no—
258Dic122—82no—
259C8×C61—21no—
260C8×C61—41no—
261C8×C61AGL(1,ℤ/8)12yes—
262#92418—22no—
263#92418—21no—
264#92418—21no—
265#92418—41no—
266#92418—41no—
267#92418—41no—
268#20291—611nomaximal, Sylow
269C22×C61—41no—
270C22×C61—41no—
271C22×C61—41no—
272C22×C61—41no—
273C22×C61—41no—
274C22×C61—41no—
275C22×C61D4211yescharacteristic
276C2×D61—41no—
277C2×D61—41no—
278C2×D61—41no—
279C4×C61—21no—
280C4×C61—41no—
281C4×C61—41no—
282C4×C61—41no—
283C4×C61—41no—
284C4×C61—81no—
285C4×C61—81no—
286C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
287C4×C61#57511yescharacteristic
288C4×C61#57711yescharacteristic
289C4×C61#57711yescharacteristic
290Dic61—41no—
291Dic61—81no—
292Dic61—81no—
293#3367—611no—
294#3353D6111yescharacteristic, complemented
295#3330—611no—
296#3276—611no—
297#1707—611no—
298#1700—611no—
299#1699—611no—
300#1693—611no—
301#1689—611no—
302#1654—611no—
303#1654—611no—
304#1652—611no—
305#1651—611no—
306#1590—611no—
307#1589—611no—
308C2×C61—81no—
309C2×C61—81no—
310C2×C61#327211yescharacteristic
311C2×C61#327511yescharacteristic
312C2×C61#329311yescharacteristic
313D61—81no—
314C2×#137—1221no—
315#556C2×D6111yescharacteristic
316D4×Q8—611no—
317#547—611no—
318#547C2×D6111yescharacteristic
319#500C2×D6111yescharacteristic
320#498—611no—
321#498—611no—
322#494—611no—
323#489—1221no—
324#487—611no—
325#487—611no—
326#484—21no—
327#484—1221no—
328#482—21no—
329#477—1221no—
330#475C2×D6111yescharacteristic
331#474—611no—
332#474—611no—
333#474C2×D6111yescharacteristic
334#470—1221no—
335#468—21no—
336#468—1221no—
337#463—1221no—
338#462—611no—
339#461—611no—
340#461—611no—
341#461—611no—
342#452—1221no—
343#451—1221no—
344#449—1221no—
345#446—611no—
346#440—1221no—
347C4×SD16C2×D6111yescharacteristic
348C4×D8—611no—
349#436—611no—
350#436—611no—
351#436C2×D6111yescharacteristic
352#428—21no—
353#417—1221no—
354#335—611no—
355#334—611no—
356#332—611no—
357#332—611no—
358#332—611no—
359#332—611no—
360Q8⋊C8—611no—
361D4⋊C8—611no—
362D4⋊C8—611no—
363D4⋊C8—611no—
364C82—611no—
365C61#2029111yescharacteristic, complemented, Sylow
366C2×#41—21no—
367C2×#41—1221no—
368C22×Q8—1221no—
369#137—2444no—
370C2×Q16—1221no—
371C2×Q16—1221no—
372C2×Q16—1221no—
373C2×SD16—1221no—
374C2×SD16—1221no—
375C2×SD16—1221no—
376C2×SD16C22⋊D6111yescharacteristic
377C2×D8—611no—
378C2×M16—21no—
379C2×M16—1221no—
380C2×M16—1221no—
381C4⋊Q8—611no—
382C4⋊Q8—611no—
383C4⋊Q8—1221no—
384C4⋊Q8C22×D6111yescharacteristic
385C4⋊Q8C22×D6111yescharacteristic
386#126—41no—
387#124—21no—
388#123—41no—
389#123—2441no—
390C22⋊Q8—21no—
391C22⋊Q8—21no—
392C22⋊Q8—1221no—
393C22⋊Q8—1221no—
394C22⋊Q8—2441no—
395#121—21no—
396#121—1221no—
397C4×Q8—21no—
398C4×Q8C22×D6111yescharacteristic
399C4×D4—611no—
400C4×D4—1221no—
401C4×D4C22×D6111yescharacteristic
402C4×D4C22×D6111yescharacteristic
403#117—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—1221no—
406#107—21no—
407#107—21no—
408#107—21no—
409#107—611no—
410#107—1221no—
411#107—1221no—
412#106—21no—
413#106—2441no—
414#106—2441no—
415#106C22⋊D6111yescharacteristic
416C4⋊C8—611no—
417C4⋊C8—611no—
418C4⋊C8—611no—
419C4⋊C8—1221no—
420C4⋊C8C22×D6111yescharacteristic
421#103—21no—
422#103—1221no—
423#103—1221no—
424#103—1221no—
425#103—1221no—
426#103—1221no—
427#103—1221no—
428#103—1221no—
429#103—1221no—
430#103C22⋊D6111yescharacteristic
431#102—21no—
432#102—611no—
433#102—611no—
434#102—1221no—
435#102—1221no—
436#102C22⋊D6111yescharacteristic
437C22⋊C8—21no—
438C22⋊C8—1221no—
439C22⋊C8—1221no—
440#97—1221no—
441C4×C8—611no—
442C4×C8—611no—
443C4×C8C22×D6111yescharacteristic
444#41—42no—
445#41—81no—
446#41—4881no—
447#41—2442no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—1221no—
451C2×Q8—1221no—
452C2×Q8—4881no—
453C2×Q8—2441no—
454C2×Q8—2442no—
455C2×Q8C2×C22⋊D6111yescharacteristic
456C2×D4—41no—
457C2×D4—611no—
458C2×D4—2441no—
459C2×D4D4×D6111yescharacteristic
460C2×D4C2×C22⋊D6111yescharacteristic
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—41no—
464C22×C4—1221no—
465C22×C4—2441no—
466C22×C4—2441no—
467Q16—4881no—
468Q16—2442no—
469Q16—2442no—
470SD16—22no—
471SD16—4881no—
472SD16—2442no—
473SD16—2442no—
474D8—2441no—
475M16—42no—
476M16—4881no—
477M16—2442no—
478C2×C8—21no—
479C2×C8—611no—
480C2×C8—611no—
481C2×C8—1221no—
482C2×C8—1221no—
483C2×C8—1221no—
484C2×C8—1221no—
485C2×C8C2×C22⋊D6111yescharacteristic
486C2×C8C2×C22⋊D6111yescharacteristic
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—41no—
493C4⋊C4—41no—
494C4⋊C4—611no—
495C4⋊C4—1221no—
496C4⋊C4—1221no—
497C4⋊C4—1221no—
498C4⋊C4—1221no—
499C4⋊C4—2441no—
500C4⋊C4—2441no—
501C4⋊C4D4×D6111yescharacteristic
502C4⋊C4C2×C22⋊D6111yescharacteristic
503C4⋊C4C2×C22⋊D6111yescharacteristic
504C22⋊C4—21no—
505C22⋊C4—21no—
506C22⋊C4—41no—
507C22⋊C4—41no—
508C22⋊C4—41no—
509C22⋊C4—1221no—
510C22⋊C4—2441no—
511C22⋊C4—2441no—
512C42—21no—
513C42—41no—
514C42C23×D6111yescharacteristic
515C23—21no—
516C23—21no—
517C23—1221no—
518Q8—21no—
519Q8—21no—
520Q8—41no—
521Q8—41no—
522Q8—41no—
523Q8—41no—
524Q8—4882no—
525Q8—2441no—
526Q8—2441no—
527Q8—2441no—
528Q8—2441no—
529D4—21no—
530D4—21no—
531D4—21no—
532D4—21no—
533D4—82no—
534D4—1221no—
535D4—1221no—
536D4—4882no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—82no—
548C2×C4—1221no—
549C2×C4—4882no—
550C2×C4—4882no—
551C2×C4—2441no—
552C2×C4—2441no—
553C2×C4—2441no—
554C2×C4—2441no—
555C2×C4C2×D4×D6111yescharacteristic
556C2×C4C22×C22⋊D6111yescharacteristic
557C2×C4#1451982211yescharacteristic
558C8—21no—
559C8—41no—
560C8—1222no—
561C8—1221no—
562C8—1221no—
563C8—2441no—
564C8—2441no—
565C8—2441no—
566C8#1451980212yes—
567C22—41no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—2441no—
574C22—2441no—
575C22—2441no—
576C22D4×C22⋊D6111yescharacteristic
577C4—21no—
578C4—41no—
579C4—41no—
580C4—41no—
581C4—41no—
582C4—81no—
583C4—81no—
584C4—4881no—
585C4—4881no—
586C4—2441no—
587C4C2×#1451980211yescharacteristic
588C4#1684055811yescharacteristic
589C4#1684057011yescharacteristic
590C4#1684057211yescharacteristic
591C2—81no—
592C2—81no—
593C2—4881no—
594C2#2145741911yescharacteristic
595C2#2145745011yescharacteristic
596C2#2145762111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3353C2
socleC22×C61D42
prime radical#21458124C2
hypercenter#3353D61
hypocenterC61#20291
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3353

Complements

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

NH ≅ G/Ntype
1. #21458533C2semidirect
3. #21457466C2semidirect
4. #21445417C2semidirect
5. #21445371C2semidirect
6. #21445364C2semidirect
7. #21445329C2semidirect
8. #21445308C2semidirect
9. #21445103C2semidirect
10. #21445102C2semidirect
11. #21445089C2semidirect
12. #21445081C2semidirect
13. #21444710C2semidirect
14. #21444702C2semidirect
15. C61×#3353C2semidirect
294. #3353D61semidirect
365. C61#20291semidirect

Maximal subgroups

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

subgroupindex
#214585332
#214581242
#214574662
#214454172
#214453712
#214453642
#214453292
#214453082
#214451032
#214451022
#214450892
#214450812
#214447102
#214447022
C61×#33532
#2029161