Finite Group Databasebeta

← #29677166

Subgroups — #29677166

Order 15616.

622 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#21458846C211yesmaximal, characteristic, complemented
2#21458405C211yesmaximal, characteristic
3#21457465C211yesmaximal, characteristic
4#21445436C211yesmaximal, characteristic, complemented
5#21445337C211yesmaximal, characteristic, complemented
6#21445329C211yesmaximal, characteristic, complemented
7#21445311C211yesmaximal, characteristic, complemented
8#21445306C211yesmaximal, characteristic, complemented
9#21445074C211yesmaximal, characteristic, complemented
10#21445068C211yesmaximal, characteristic, complemented
11#21445061C211yesmaximal, characteristic, complemented
12#21445053C211yesmaximal, characteristic
13#21444710C211yesmaximal, characteristic, complemented
14#21444709C211yesmaximal, characteristic, complemented
15C61×#3386C211yesmaximal, characteristic, complemented
16C2×#14519808—21no—
17#16840485C2211yescharacteristic
18#16840417C2211yescharacteristic
19#16840167C2211yescharacteristic
20#16840165C2211yescharacteristic
21#16840165C2211yescharacteristic
22#16840154—21no—
23#16840141C2211yescharacteristic
24#16840141C2211yescharacteristic
25#16840096—21no—
26#16840084—21no—
27#16840047C2211yescharacteristic
28#16840042C2211yescharacteristic
29#16840018—21no—
30#16840005—21no—
31#16839984—21no—
32#16839972C2211yescharacteristic
33#16839972C2211yescharacteristic
34#16839969—21no—
35#16839964C2211yescharacteristic
36#16839964C2211yescharacteristic
37#16839922—21no—
38#16839918—21no—
39#16839893C2211yescharacteristic
40#16839867—21no—
41C4×#11758201C2211yescharacteristic
42#16839841C2211yescharacteristic
43#16839840C2211yescharacteristic
44#16839801—21no—
45#16839513C2211yescharacteristic
46#16839512C2211yescharacteristic
47#16839508C2211yescharacteristic
48#16839508C2211yescharacteristic
49#16839507C2211yescharacteristic
50#16839507C2211yescharacteristic
51Q8⋊#92418C2211yescharacteristic
52Q8⋊#92418C2211yescharacteristic
53D4⋊#92418C2211yescharacteristic
54#16839489C2211yescharacteristic
55C8×#92418C2211yescharacteristic
56C61×#557C2211yescharacteristic
57D4×Q8×C61C2211yescharacteristic
58C61×#501—21no—
59C61×#500C2211yescharacteristic
60C61×#478—21no—
61C61×#477—21no—
62C61×#475C2211yescharacteristic
63C61×#474C2211yescharacteristic
64C61×#446C2211yescharacteristic
65C61×#440—21no—
66C4×SD16×C61C2211yescharacteristic
67C2×#11758222—21no—
68#14519808—44no—
69C2×#11758202—21no—
70C2×#11758202—21no—
71C2×#11758202—21no—
72C2×#11758202—21no—
73C2×#11758202—21no—
74C2×#11758202—21no—
75C2×#11758201—21no—
76C2×#11758201—21no—
77C2×#11758201D411yescharacteristic
78C2×#11758194—21no—
79C2×#11758189—21no—
80#14519777C2311yescharacteristic
81#14519777C2311yescharacteristic
82#14519772—41no—
83#14519765—21no—
84#14519759—41no—
85#14519756—21no—
86#14519753—21no—
87#14519748—21no—
88C4×Dic122—21no—
89C4×D244C2311yescharacteristic
90#14519735—21no—
91C4⋊C4×D61—21no—
92#14519711—21no—
93#14519711—21no—
94#14519710—21no—
95#14519710—41no—
96#14519710—41no—
97#14519708—21no—
98#14519708D411yescharacteristic
99C4⋊#92418—21no—
100C4⋊#92418C2311yescharacteristic
101C4⋊#92418C2311yescharacteristic
102C4⋊#92418C2311yescharacteristic
103#14519701—21no—
104#14519701—21no—
105#14519701—21no—
106#14519701D411yescharacteristic
107#14519700—21no—
108#14519700—21no—
109#14519698—21no—
110#14519697D411yescharacteristic
111C22⋊#92418—21no—
112#14519689—21no—
113#14519688—21no—
114C4×#92418C2311yescharacteristic
115C4×#92418C2311yescharacteristic
116C22×Q8×C61—21no—
117C2×SD16×C61D411yescharacteristic
118C4⋊Q8×C61—21no—
119C4⋊Q8×C61—21no—
120C4⋊Q8×C61C2311yescharacteristic
121C4⋊Q8×C61C2311yescharacteristic
122#125×C61—21no—
123#125×C61—41no—
124C22⋊Q8×C61—21no—
125C22⋊Q8×C61—41no—
126C4×Q8×C61—21no—
127C4×Q8×C61—21no—
128C4×Q8×C61C2311yescharacteristic
129C4×Q8×C61C2311yescharacteristic
130C4×D4×C61—21no—
131C4×D4×C61C2311yescharacteristic
132#107×C61—21no—
133#107×C61—21no—
134#107×C61—21no—
135#106×C61—21no—
136#106×C61D411yescharacteristic
137C4⋊C8×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139#103×C61—21no—
140#103×C61D411yescharacteristic
141#102×C61—21no—
142#102×C61D411yescharacteristic
143#97×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—41no—
148C2×Q8×C61—42no—
149C2×Q8×C61—81no—
150C2×Q8×C61C2×D411yescharacteristic
151C2×Q8×C61C2×D411yescharacteristic
152C2×D4×C61C2×D411yescharacteristic
153C22×C4×C61—21no—
154C22×C4×C61—41no—
155C2×C22⋊D61—41no—
156#11758222—42no—
157#11758222—81no—
158C2×D244C2×D411yescharacteristic
159C2×C4×D61—21no—
160C2×C4×D61—41no—
161C2×Dic122—21no—
162C2×Dic122—21no—
163SD16×C61—22no—
164C2×C8×C61—21no—
165C2×C8×C61C2×D411yescharacteristic
166C2×C8×C61C2×D411yescharacteristic
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×C61—41no—
176C4⋊C4×C61—41no—
177C4⋊C4×C61C2×D411yescharacteristic
178C4⋊C4×C61C2×D411yescharacteristic
179C4⋊C4×C61C2×D411yescharacteristic
180C22⋊C4×C61—21no—
181C22⋊C4×C61—41no—
182C42×C61—21no—
183C42×C61—21no—
184C42×C61—41no—
185C42×C61C2411yescharacteristic
186C22⋊Dic61—41no—
187#11758202—42no—
188#11758202—42no—
189#11758202—42no—
190#11758202—81no—
191#11758202—81no—
192#11758202—81no—
193#11758201—22no—
194#11758201—42no—
195#11758201—81no—
196#11758198—21no—
197#11758198—41no—
198#11758198—41no—
199C4⋊Dic61—21no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—21no—
203C4⋊Dic61C2×D411yescharacteristic
204#11758196—41no—
205#11758196—41no—
206C4×Dic61—41no—
207#11758194—42no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418C2×D411yescharacteristic
213C2×#92418C2×D411yescharacteristic
214#11758189—42no—
215C23×C61—21no—
216C22×D61—21no—
217Q8×C61—21no—
218Q8×C61—21no—
219Q8×C61—21no—
220Q8×C61—21no—
221Q8×C61—41no—
222Q8×C61—41no—
223Q8×C61—41no—
224Q8×C61—41no—
225Q8×C61—82no—
226D4×C61—21no—
227D4×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—41no—
235C2×C4×C61—41no—
236C2×C4×C61—41no—
237C2×C4×C61—82no—
238C2×C4×C61C22×D411yescharacteristic
239C2×C4×C61C22×D411yescharacteristic
240C2×C4×C61H(5,𝔽2)11yescharacteristic
241C22⋊D61—82no—
242C2×Dic61—21no—
243C2×Dic61—41no—
244C2×Dic61—41no—
245D244—21no—
246D244—21no—
247C4×D61—41no—
248C4×D61—41no—
249C4×D61—82no—
250Dic122—41no—
251Dic122—41no—
252Dic122—41no—
253Dic122—41no—
254C8×C61—21no—
255C8×C61—41no—
256C8×C61#13712yescomplemented
257#92418—22no—
258#92418—21no—
259#92418—41no—
260#92418—41no—
261#92418—41no—
262#92418AGL(1,ℤ/8)12yes—
263#20311—611nomaximal, Sylow
264C22×C61—41no—
265C22×C61—41no—
266C22×C61—41no—
267C22×C61D4211yescharacteristic
268C2×D61—41no—
269C2×D61—41no—
270C2×D61—41no—
271C4×C61—21no—
272C4×C61—41no—
273C4×C61—41no—
274C4×C61—41no—
275C4×C61—41no—
276C4×C61—41no—
277C4×C61—81no—
278C4×C61—81no—
279C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
280C4×C61C2×#13711yescharacteristic
281C4×C61#57611yescharacteristic
282C4×C61#57811yescharacteristic
283Dic61—41no—
284Dic61—81no—
285Dic61—81no—
286#3398—611no—
287#3386D6111yescharacteristic, complemented
288#3353—611no—
289#3276—611no—
290#1713—611no—
291#1694—611no—
292#1693—611no—
293#1690—611no—
294#1689—611no—
295#1650—611no—
296#1649—611no—
297#1648—611no—
298#1647—611no—
299#1590—611no—
300#1590—611no—
301C2×C61—81no—
302C2×C61#327411yescharacteristic
303C2×C61#327611yescharacteristic
304C2×C61#329311yescharacteristic
305D61—81no—
306C2×#137—1221no—
307#557C2×D6111yescharacteristic
308#556—611no—
309D4×Q8C2×D6111yescharacteristic
310#547—611no—
311#501—21no—
312#500—611no—
313#500—611no—
314#500—611no—
315#500C2×D6111yescharacteristic
316#497—1221no—
317#494—611no—
318#494—611no—
319#484—1221no—
320#482—1221no—
321#478—21no—
322#477—21no—
323#475—611no—
324#475C2×D6111yescharacteristic
325#474—611no—
326#474C2×D6111yescharacteristic
327#470—1221no—
328#468—1221no—
329#464—1221no—
330#462—611no—
331#462—611no—
332#462—1221no—
333#461—611no—
334#461—611no—
335#452—1221no—
336#451—1221no—
337#446—611no—
338#446C2×D6111yescharacteristic
339#441—1221no—
340#440—21no—
341C4×SD16—611no—
342C4×SD16C2×D6111yescharacteristic
343#436—611no—
344#436—611no—
345#428—1221no—
346#334—611no—
347#334—611no—
348#332—611no—
349#332—611no—
350#332—611no—
351#332—611no—
352Q8⋊C8—611no—
353Q8⋊C8—611no—
354D4⋊C8—611no—
355D4⋊C8—611no—
356C82—611no—
357C61#2031111yescharacteristic, complemented, Sylow
358C2×#41—1221no—
359C22×Q8—21no—
360#137—2444no—
361C2×Q16—1221no—
362C2×Q16—1221no—
363C2×Q16—1221no—
364C2×Q16—1221no—
365C2×Q16—1221no—
366C2×Q16—1221no—
367C2×SD16—611no—
368C2×SD16—1221no—
369C2×SD16—1221no—
370C2×SD16C22⋊D6111yescharacteristic
371C2×M16—1221no—
372C2×M16—1221no—
373C4⋊Q8—21no—
374C4⋊Q8—21no—
375C4⋊Q8—611no—
376C4⋊Q8—611no—
377C4⋊Q8C22×D6111yescharacteristic
378C4⋊Q8C22×D6111yescharacteristic
379#126—2441no—
380#125—21no—
381#125—41no—
382#124—1221no—
383#123—2441no—
384C22⋊Q8—21no—
385C22⋊Q8—41no—
386C22⋊Q8—1221no—
387C22⋊Q8—1221no—
388#121—1221no—
389C4×Q8—21no—
390C4×Q8—21no—
391C4×Q8—1221no—
392C4×Q8C22×D6111yescharacteristic
393C4×Q8C22×D6111yescharacteristic
394C4×D4—21no—
395C4×D4—611no—
396C4×D4C22×D6111yescharacteristic
397#117—1221no—
398C2×C4⋊C4—1221no—
399#107—21no—
400#107—21no—
401#107—21no—
402#107—1221no—
403#107—1221no—
404#107—1221no—
405#107—2441no—
406#107—2441no—
407#106—21no—
408#106—611no—
409#106—1221no—
410#106C22⋊D6111yescharacteristic
411C4⋊C8—21no—
412C4⋊C8—611no—
413C4⋊C8—611no—
414C4⋊C8—611no—
415C4⋊C8—1221no—
416C4⋊C8C22×D6111yescharacteristic
417#103—21no—
418#103—611no—
419#103—1221no—
420#103—1221no—
421#103—1221no—
422#103—1221no—
423#103—1221no—
424#103C22⋊D6111yescharacteristic
425#102—21no—
426#102—611no—
427#102—1221no—
428#102C22⋊D6111yescharacteristic
429C22⋊C8—1221no—
430C22⋊C8—1221no—
431#97—21no—
432#97—1221no—
433C4×C8—611no—
434C4×C8—611no—
435C4×C8C22×D6111yescharacteristic
436#41—4881no—
437#41—2442no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—41no—
441C2×Q8—42no—
442C2×Q8—81no—
443C2×Q8—1221no—
444C2×Q8—1221no—
445C2×Q8D4×D6111yescharacteristic
446C2×Q8C2×C22⋊D6111yescharacteristic
447C2×D4—611no—
448C2×D4—2441no—
449C2×D4C2×C22⋊D6111yescharacteristic
450C22×C4—21no—
451C22×C4—41no—
452C22×C4—1221no—
453C22×C4—2441no—
454Q16—4881no—
455Q16—4881no—
456Q16—4881no—
457Q16—2442no—
458Q16—2442no—
459Q16—2442no—
460SD16—22no—
461SD16—1222no—
462SD16—4881no—
463SD16—2442no—
464M16—2442no—
465M16—2442no—
466C2×C8—21no—
467C2×C8—611no—
468C2×C8—611no—
469C2×C8—1221no—
470C2×C8—1221no—
471C2×C8—1221no—
472C2×C8—1221no—
473C2×C8C2×C22⋊D6111yescharacteristic
474C2×C8C2×C22⋊D6111yescharacteristic
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—41no—
482C4⋊C4—41no—
483C4⋊C4—41no—
484C4⋊C4—41no—
485C4⋊C4—611no—
486C4⋊C4—1221no—
487C4⋊C4—1221no—
488C4⋊C4—1221no—
489C4⋊C4—1221no—
490C4⋊C4—2441no—
491C4⋊C4—2441no—
492C4⋊C4D4×D6111yescharacteristic
493C4⋊C4C2×C22⋊D6111yescharacteristic
494C4⋊C4C2×C22⋊D6111yescharacteristic
495C22⋊C4—21no—
496C22⋊C4—41no—
497C22⋊C4—1221no—
498C22⋊C4—2441no—
499C22⋊C4—2441no—
500C22⋊C4—2441no—
501C42—21no—
502C42—21no—
503C42—41no—
504C42—2441no—
505C42C23×D6111yescharacteristic
506C23—21no—
507C23—1221no—
508Q8—21no—
509Q8—21no—
510Q8—21no—
511Q8—21no—
512Q8—41no—
513Q8—41no—
514Q8—41no—
515Q8—41no—
516Q8—82no—
517Q8—2441no—
518Q8—2441no—
519Q8—2441no—
520Q8—2441no—
521D4—21no—
522D4—21no—
523D4—1221no—
524D4—1221no—
525D4—4882no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—82no—
536C2×C4—1221no—
537C2×C4—4882no—
538C2×C4—2441no—
539C2×C4—2441no—
540C2×C4—2441no—
541C2×C4—2441no—
542C2×C4C2×D4×D6111yescharacteristic
543C2×C4C22×C22⋊D6111yescharacteristic
544C2×C4#1451982211yescharacteristic
545C8—21no—
546C8—41no—
547C8—612no—
548C8—1222no—
549C8—1221no—
550C8—2441no—
551C8—2441no—
552C8—2441no—
553C8#1451980812yescomplemented
554C22—41no—
555C22—41no—
556C22—41no—
557C22—2441no—
558C22—2441no—
559C22—2441no—
560C22D4×C22⋊D6111yescharacteristic
561C4—21no—
562C4—41no—
563C4—41no—
564C4—41no—
565C4—41no—
566C4—41no—
567C4—81no—
568C4—81no—
569C4—4881no—
570C4—4881no—
571C4—2441no—
572C4AGL(1,ℤ/8)×D6111yescharacteristic
573C4C2×#1451980811yescharacteristic
574C4#1684056611yescharacteristic
575C4#1684057711yescharacteristic
576C2—81no—
577C2—4881no—
578C2#2145743311yescharacteristic
579C2#2145746511yescharacteristic
580C2#2145762111yescharacteristic
subgroupG/H
centerC22D4×C22⋊D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3386C2
socleC22×C61D42
prime radical#16840417C22
hypercenter#3386D61
hypocenterC61#20311
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #3386

Complements

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

NH ≅ G/Ntype
1. #21458846C2semidirect
4. #21445436C2semidirect
5. #21445337C2semidirect
6. #21445329C2semidirect
7. #21445311C2semidirect
8. #21445306C2semidirect
9. #21445074C2semidirect
10. #21445068C2semidirect
11. #21445061C2semidirect
13. #21444710C2semidirect
14. #21444709C2semidirect
15. C61×#3386C2semidirect
256. C8×C61#137semidirect
287. #3386D61semidirect
357. C61#20311semidirect
553. C8#14519808semidirect

Maximal subgroups

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

subgroupindex
#214588462
#214584052
#214574652
#214454362
#214453372
#214453292
#214453112
#214453062
#214450742
#214450682
#214450612
#214450532
#214447102
#214447092
C61×#33862
#2031161