Finite Group Databasebeta

← #29665168

Subgroups — #29665168

Order 15616.

738 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456756C211yesmaximal, characteristic, complemented
2#21455611C211yesmaximal, characteristic, complemented
3#459×D61C211yesmaximal, characteristic
4#21454840C211yesmaximal, characteristic, complemented
5#21445229C211yesmaximal, characteristic, complemented
6#21445228C211yesmaximal, characteristic, complemented
7#21445227C211yesmaximal, characteristic, complemented
8#21445226C211yesmaximal, characteristic, complemented
9#21445163C211yesmaximal, characteristic, complemented
10#21445162C211yesmaximal, characteristic, complemented
11#21445157C211yesmaximal, characteristic, complemented
12#21444588C211yesmaximal, characteristic, complemented
13#21444485C211yesmaximal, characteristic, complemented
14#21444475C211yesmaximal, characteristic, complemented
15C61×#1665C211yesmaximal, characteristic, complemented
16C2×D8×D61—21no—
17#16840317C2211yescharacteristic
18#127×D61C2211yescharacteristic
19C4×D4×D61C2211yescharacteristic
20#16840136C2211yescharacteristic, complemented
21#16840109C2211yescharacteristic, complemented
22#16840108C2211yescharacteristic, complemented
23#16840065—21no—
24#16840064—21no—
25#16840053C2211yescharacteristic, complemented
26#16840052C2211yescharacteristic, complemented
27#16840039C2211yescharacteristic, complemented
28#16839960—21no—
29#16839960—21no—
30#16839958C2211yescharacteristic
31#16839957C2211yescharacteristic
32#16839956—21no—
33#16839955—21no—
34#16839954C2211yescharacteristic
35#16839911—21no—
36#16839911—21no—
37C8×D244C2211yescharacteristic, complemented
38D4×#92418C2211yescharacteristic, complemented
39#16839827C2211yescharacteristic, complemented
40C8×C22⋊D61—21no—
41#16839825—21no—
42#16839814C2211yescharacteristic
43#16839813C2211yescharacteristic
44#16839803—21no—
45#16839784C2211yescharacteristic
46C4⋊C8×D61C2211yescharacteristic
47#16839765—21no—
48#102×D61—21no—
49C2×#14519689—21no—
50C22⋊C8×D61—21no—
51#16839513C2211yescharacteristic, complemented
52#16839512C2211yescharacteristic, complemented
53#16839512C2211yescharacteristic, complemented
54#16839510C2211yescharacteristic, complemented
55#16839510C2211yescharacteristic, complemented
56#16839508C2211yescharacteristic, complemented
57#16839506C2211yescharacteristic, complemented
58#16839505C2211yescharacteristic, complemented
59#16839505C2211yescharacteristic, complemented
60C61×#492C2211yescharacteristic, complemented
61C61×#474C2211yescharacteristic, complemented
62C61×#459C2211yescharacteristic
63C8×D4×C61C2211yescharacteristic, complemented
64C61×#334C2211yescharacteristic, complemented
65C61×#332C2211yescharacteristic, complemented
66C61×#331C2211yescharacteristic, complemented
67C2×D4×D61—21no—
68C2×D4×D61—21no—
69C2×D4×D61—21no—
70C2×D4×D61D411yescharacteristic
71C22×C4×D61—21no—
72C2×#11758201—21no—
73C2×#11758201—21no—
74C2×D488—21no—
75C2×#11758199—21no—
76C2×#11758199—21no—
77D8×D61—44no—
78C2×#11758189—21no—
79C22×#92418—21no—
80C2×C8×D61—21no—
81C2×C8×D61—21no—
82C2×C8×D61—21no—
83#14519777C2311yescharacteristic
84#14519774C2311yescharacteristic
85#14519773—21no—
86#14519773—21no—
87#14519773—21no—
88#14519768—21no—
89#14519757—21no—
90#14519753—21no—
91Q8×Dic61—21no—
92#14519742—21no—
93C4×D244C2311yescharacteristic
94D4×Dic61D411yescharacteristic
95#14519739D411yescharacteristic
96C4×C22⋊D61—21no—
97#14519737—21no—
98#14519735—21no—
99C4⋊C4×D61D411yescharacteristic
100C22⋊C4×D61—21no—
101C42×D61C2311yescharacteristic
102#14519710—21no—
103#14519709—21no—
104#14519708—21no—
105C4⋊#92418C2311yescharacteristic
106C4⋊#92418C2311yescharacteristic
107C4⋊#92418C2311yescharacteristic
108#14519706—21no—
109#14519706—21no—
110#14519706D411yescharacteristic
111#14519706D411yescharacteristic
112#14519699—21no—
113#14519699—21no—
114#14519699—21no—
115#14519699—21no—
116#14519698—21no—
117#14519698—21no—
118#14519698—21no—
119#14519698—21no—
120#14519697—21no—
121#14519697—21no—
122C22⋊#92418—21no—
123#14519689—21no—
124#14519689—21no—
125#14519689—21no—
126#14519689D411yescharacteristic
127#14519689D411yescharacteristic
128C4×#92418C2311yescharacteristic
129C8×Dic61—21no—
130C2×SD16×C61—21no—
131C2×SD16×C61—21no—
132C2×D8×C61—21no—
133C22×C8×C61—21no—
134C4⋊Q8×C61C2311yescharacteristic
135#127×C61C2311yescharacteristic
136C4×D4×C61C2311yescharacteristic
137#107×C61—21no—
138C4⋊C8×C61C2311yescharacteristic
139C4⋊C8×C61C2311yescharacteristic
140C4⋊C8×C61C2311yescharacteristic
141#102×C61—21no—
142#102×C61—21no—
143C22⋊C8×C61—21no—
144C4×C8×C61C2311yescharacteristic
145C23×D61—21no—
146C23×D61—41no—
147C2×Q8×C61—21no—
148C2×D4×C61—21no—
149C2×D4×C61—21no—
150C2×D4×C61—21no—
151C2×D4×C61C2×D411yescharacteristic
152C22×C4×C61—21no—
153C2×C22⋊D61—21no—
154C2×C22⋊D61—41no—
155C2×C22⋊D61—41no—
156C2×C22⋊D61—41no—
157C22×Dic61—21no—
158D4×D61—22no—
159D4×D61—42no—
160D4×D61—42no—
161D4×D61—81no—
162D4×D61—81no—
163D4×D61—82no—
164D4×D61SD1614yescomplemented
165C2×D244—21no—
166C2×D244—21no—
167C2×D244—21no—
168C2×D244C2×D411yescharacteristic
169C2×C4×D61—21no—
170C2×C4×D61—21no—
171C2×C4×D61—21no—
172C2×C4×D61—81no—
173C2×C4×D61—81no—
174C2×C4×D61C2×D411yescharacteristic
175C2×C4×D61C2×D411yescharacteristic
176C2×C4×D61#4111yescharacteristic
177C2×Dic122—21no—
178SD16×C61—81no—
179SD16×C61—81no—
180D8×C61—42no—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—81no—
185C2×C8×C61C2×D411yescharacteristic
186C2×C8×C61C2×D411yescharacteristic
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61C2×D411yescharacteristic
190C22⋊C4×C61—21no—
191C42×C61C2411yescharacteristic
192C22⋊Dic61—21no—
193#11758201—81no—
194#11758201—81no—
195#11758199—42no—
196#11758199—42no—
197#11758198—21no—
198#11758198—21no—
199#11758198—41no—
200C4⋊Dic61—21no—
201C4⋊Dic61—21no—
202C4⋊Dic61—41no—
203C4⋊Dic61C2×D411yescharacteristic
204#11758196—21no—
205#11758196—41no—
206C4×Dic61—41no—
207C4×Dic61C2×D411yescharacteristic
208C4×Dic61C2×D411yescharacteristic
209C4×Dic61#4111yescharacteristic
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—81no—
214C2×#92418C2×D411yescharacteristic
215C2×#92418C2×D411yescharacteristic
216D488—42no—
217#11758189—42no—
218C8×D61—42no—
219C8×D61—81no—
220C8×D61—81no—
221C23×C61—21no—
222C23×C61—41no—
223C22×D61—21no—
224C22×D61—24no—
225C22×D61—41no—
226C22×D61—81no—
227C22×D61—81no—
228C22×D61—82no—
229C22×D61—82no—
230C22×D61—82no—
231C22×D61#12111yescharacteristic
232Q8×C61—41no—
233Q8×C61—41no—
234D4×C61—21no—
235D4×C61—41no—
236D4×C61—41no—
237D4×C61—41no—
238D4×C61—41no—
239D4×C61—81no—
240D4×C61C2×SD1612yes—
241C2×C4×C61—21no—
242C2×C4×C61—41no—
243C2×C4×C61—41no—
244C2×C4×C61—41no—
245C2×C4×C61C22×D411yescharacteristic
246C2×C4×C61C22×D411yescharacteristic
247C2×C4×C61C2×#4111yescharacteristic
248C22⋊D61—24no—
249C22⋊D61—82no—
250C22⋊D61—82no—
251C22⋊D61—82no—
252C2×Dic61—21no—
253C2×Dic61—41no—
254C2×Dic61—81no—
255C2×Dic61#12111yescharacteristic
256C2×Dic61#12111yescharacteristic
257C2×Dic61#12111yescharacteristic
258D244—21no—
259D244—41no—
260D244—41no—
261D244—41no—
262D244—41no—
263D244—81no—
264D244C2×SD1612yes—
265C4×D61—21no—
266C4×D61—21no—
267C4×D61—41no—
268C4×D61—41no—
269C4×D61—41no—
270C4×D61—81no—
271C4×D61—81no—
272C4×D61C2×SD1612yes—
273C4×D61#13512yes—
274Dic122—81no—
275C8×C61—21no—
276C8×C61—21no—
277C8×C61—41no—
278C8×C61—41no—
279C8×C61—41no—
280#92418—21no—
281#92418—21no—
282#92418—41no—
283#92418—41no—
284#92418—41no—
285#19384—611nomaximal, Sylow
286C22×C61—22no—
287C22×C61—41no—
288C22×C61—41no—
289C22×C61—81no—
290C22×C61—81no—
291C22×C61—81no—
292C22×C61C2×#12111yescharacteristic
293C2×D61—21no—
294C2×D61—21no—
295C2×D61—22no—
296C2×D61—41no—
297C2×D61—41no—
298C2×D61—84no—
299C2×D61—81no—
300C2×D61—81no—
301C2×D61—81no—
302C2×D61—164no—
303C2×D61#46512yes—
304C4×C61—21no—
305C4×C61—41no—
306C4×C61—81no—
307C4×C61C22×SD1611yescharacteristic
308C4×C61C2×#13511yescharacteristic
309C4×C61#57611yescharacteristic
310C4×C61#57611yescharacteristic
311Dic61—21no—
312Dic61—21no—
313Dic61—41no—
314Dic61—81no—
315Dic61#46512yes—
316#3208—611no—
317#3077—611no—
318C2×#459—611no—
319#2972—611no—
320#1677—611no—
321#1677—611no—
322#1677—611no—
323#1677—611no—
324#1665—611no—
325#1665—611no—
326#1665—611no—
327#1665D6111yescharacteristic, complemented
328#1570—611no—
329#1545—611no—
330#1542—611no—
331C2×C61—42no—
332C2×C61—161no—
333C2×C61#304111yescharacteristic
334C2×C61C2×#46511yescharacteristic
335C2×C61#306811yescharacteristic
336D61—22no—
337D61—42no—
338D61—161no—
339C22×D8—1221no—
340#533—611no—
341C2×#127—611no—
342C2×C4×D4—611no—
343#492—611no—
344#492C2×D6111yescharacteristic, complemented
345#486—611no—
346#486—611no—
347#478—1221no—
348#478—1221no—
349#476—611no—
350#476—611no—
351#474—611no—
352#474C2×D6111yescharacteristic, complemented
353#460—1221no—
354#460—1221no—
355#459—611no—
356#459—611no—
357#459—611no—
358#459—1221no—
359#459—1221no—
360#459C2×D6111yescharacteristic
361#450—1221no—
362#450—1221no—
363C8×D4—611no—
364C8×D4—611no—
365C8×D4—611no—
366C8×D4—1221no—
367C8×D4—1221no—
368C8×D4C2×D6111yescharacteristic, complemented
369#431—611no—
370#431—611no—
371#428—1221no—
372#423—611no—
373C2×C4⋊C8—611no—
374#418—1221no—
375C2×#102—1221no—
376C2×C22⋊C8—1221no—
377C2×C22⋊C8—1221no—
378#334—611no—
379#334—611no—
380#334—611no—
381#334C2×D6111yescharacteristic, complemented
382#333—611no—
383#333—611no—
384#332—611no—
385#332C2×D6111yescharacteristic, complemented
386#331—611no—
387#331—611no—
388#331—611no—
389#331C2×D6111yescharacteristic, complemented
390C61#1938411yescharacteristic, complemented, Sylow
391C22×D4—611no—
392C22×D4—1221no—
393C22×D4—1221no—
394C22×D4—1221no—
395C23×C4—1221no—
396C2×SD16—21no—
397C2×SD16—21no—
398C2×SD16—1221no—
399C2×SD16—1221no—
400C2×D8—21no—
401C2×D8—1221no—
402C2×D8—1221no—
403C2×D8—1221no—
404C2×D8—2444no—
405C2×M16—1221no—
406C22×C8—21no—
407C22×C8—1221no—
408C22×C8—1221no—
409C22×C8—1221no—
410C22×C8—1221no—
411C4⋊Q8—611no—
412C4⋊Q8C22×D6111yescharacteristic
413#127—611no—
414#127—1221no—
415#127—1221no—
416#127—1221no—
417#127C22×D6111yescharacteristic
418#125—1221no—
419C22⋊Q8—1221no—
420C22⋊Q8—1221no—
421C4×Q8—1221no—
422C4×Q8—1221no—
423C4×D4—611no—
424C4×D4—611no—
425C4×D4—611no—
426C4×D4—1221no—
427C4×D4—1221no—
428C4×D4C22×D6111yescharacteristic
429#117—1221no—
430C2×C4⋊C4—611no—
431C2×C22⋊C4—1221no—
432C2×C42—611no—
433#107—21no—
434#107—1221no—
435#106—1221no—
436#106—1221no—
437C4⋊C8—611no—
438C4⋊C8—611no—
439C4⋊C8—611no—
440C4⋊C8—611no—
441C4⋊C8—611no—
442C4⋊C8—1221no—
443C4⋊C8—1221no—
444C4⋊C8C22×D6111yescharacteristic
445C4⋊C8C22×D6111yescharacteristic
446C4⋊C8C22×D6111yescharacteristic
447#102—21no—
448#102—21no—
449#102—1221no—
450#102—1221no—
451#102—1221no—
452#102—1221no—
453#102—1221no—
454#102—1221no—
455#102—1221no—
456#102—1221no—
457#102—1221no—
458#102—1221no—
459C22⋊C8—21no—
460C22⋊C8—611no—
461C22⋊C8—611no—
462C22⋊C8—1221no—
463C22⋊C8—1221no—
464C22⋊C8—1221no—
465C22⋊C8—1221no—
466C4×C8—611no—
467C4×C8—1221no—
468C4×C8C22×D6111yescharacteristic
469C24—1221no—
470C24—2441no—
471C2×Q8—21no—
472C2×Q8—1221no—
473C2×D4—21no—
474C2×D4—21no—
475C2×D4—21no—
476C2×D4—614no—
477C2×D4—611no—
478C2×D4—1222no—
479C2×D4—1221no—
480C2×D4—1221no—
481C2×D4—1221no—
482C2×D4—1221no—
483C2×D4—4881no—
484C2×D4—4881no—
485C2×D4—4882no—
486C2×D4—2442no—
487C2×D4—2442no—
488C2×D4—2441no—
489C2×D4—2441no—
490C2×D4—2441no—
491C2×D4D4×D6111yescharacteristic
492C22×C4—21no—
493C22×C4—611no—
494C22×C4—611no—
495C22×C4—611no—
496C22×C4—1221no—
497C22×C4—1221no—
498C22×C4—1221no—
499C22×C4—1221no—
500C22×C4—4881no—
501C22×C4—4881no—
502SD16—81no—
503SD16—81no—
504SD16—4881no—
505SD16—4881no—
506D8—42no—
507D8—2442no—
508D8—2442no—
509D8—2442no—
510M16—2442no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—81no—
515C2×C8—611no—
516C2×C8—611no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—4881no—
521C2×C8—4881no—
522C2×C8—4881no—
523C2×C8—2442no—
524C2×C8D4×D6111yescharacteristic
525C2×C8D4×D6111yescharacteristic
526C4⋊C4—21no—
527C4⋊C4—21no—
528C4⋊C4—611no—
529C4⋊C4—1221no—
530C4⋊C4—1221no—
531C4⋊C4—1221no—
532C4⋊C4—2441no—
533C4⋊C4—2441no—
534C4⋊C4D4×D6111yescharacteristic
535C22⋊C4—21no—
536C22⋊C4—1221no—
537C22⋊C4—1221no—
538C22⋊C4—1221no—
539C22⋊C4—2441no—
540C42—611no—
541C42—611no—
542C42—611no—
543C42—2441no—
544C42C23×D6111yescharacteristic
545C23—21no—
546C23—41no—
547C23—611no—
548C23—1224no—
549C23—1221no—
550C23—4881no—
551C23—4881no—
552C23—4882no—
553C23—4882no—
554C23—4882no—
555C23—2441no—
556Q8—41no—
557Q8—41no—
558Q8—4881no—
559D4—21no—
560D4—41no—
561D4—41no—
562D4—41no—
563D4—41no—
564D4—81no—
565D4—612no—
566D4—1224no—
567D4—1221no—
568D4—4881no—
569D4—4882no—
570D4—4882no—
571D4—4882no—
572D4—2441no—
573D4—2441no—
574D4—2441no—
575D4—2441no—
576D4SD16×D6112yes—
577C2×C4—21no—
578C2×C4—41no—
579C2×C4—41no—
580C2×C4—41no—
581C2×C4—611no—
582C2×C4—612no—
583C2×C4—612no—
584C2×C4—611no—
585C2×C4—611no—
586C2×C4—1221no—
587C2×C4—1221no—
588C2×C4—1221no—
589C2×C4—4881no—
590C2×C4—4881no—
591C2×C4—4881no—
592C2×C4—2441no—
593C2×C4—2441no—
594C2×C4—2441no—
595C2×C4—2441no—
596C2×C4C2×D4×D6111yescharacteristic
597C2×C4C2×D4×D6111yescharacteristic
598C2×C4#41×D6111yescharacteristic
599C8—21no—
600C8—21no—
601C8—41no—
602C8—41no—
603C8—41no—
604C8—1221no—
605C8—1221no—
606C8—2441no—
607C8—2441no—
608C8—2441no—
609C22—22no—
610C22—41no—
611C22—41no—
612C22—81no—
613C22—81no—
614C22—81no—
615C22—612no—
616C22—1221no—
617C22—1221no—
618C22—1222no—
619C22—9764no—
620C22—4884no—
621C22—4881no—
622C22—4881no—
623C22—4881no—
624C22—2441no—
625C22—2441no—
626C22#121×D6111yescharacteristic
627C4—21no—
628C4—41no—
629C4—81no—
630C4—612no—
631C4—1221no—
632C4—1221no—
633C4—4881no—
634C4—2441no—
635C4C2×SD16×D6111yescharacteristic
636C4#135×D6111yescharacteristic
637C4#1684056311yescharacteristic
638C4#1684056611yescharacteristic
639C2—42no—
640C2—161no—
641C2—1222no—
642C2—9761no—
643C2—2442no—
644C2#2145531711yescharacteristic
645C2#465×D6111yescharacteristic
646C2#2145554211yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1665C2
socleC22×C61C2×#121
prime radical#459×D61C2
hypercenter#1665D61
hypocenterC61#19384
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1665

Complements

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

NH ≅ G/Ntype
1. #21456756C2semidirect
2. #21455611C2semidirect
4. #21454840C2semidirect
5. #21445229C2semidirect
6. #21445228C2semidirect
7. #21445227C2semidirect
8. #21445226C2semidirect
9. #21445163C2semidirect
10. #21445162C2semidirect
11. #21445157C2semidirect
12. #21444588C2semidirect
13. #21444485C2semidirect
14. #21444475C2semidirect
15. C61×#1665C2semidirect
20. #16840136C22semidirect
21. #16840109C22semidirect
22. #16840108C22semidirect
25. #16840053C22semidirect
26. #16840052C22semidirect
27. #16840039C22semidirect
37. C8×D244C22semidirect
38. D4×#92418C22semidirect
39. #16839827C22semidirect
51. #16839513C22semidirect
52. #16839512C22semidirect
53. #16839512C22semidirect
54. #16839510C22semidirect
55. #16839510C22semidirect
56. #16839508C22semidirect
57. #16839506C22semidirect
58. #16839505C22semidirect
59. #16839505C22semidirect
60. C61×#492C22semidirect
61. C61×#474C22semidirect
63. C8×D4×C61C22semidirect
64. C61×#334C22semidirect
65. C61×#332C22semidirect
66. C61×#331C22semidirect
164. D4×D61SD16semidirect
327. #1665D61semidirect
344. #492C2×D61semidirect
352. #474C2×D61semidirect
368. C8×D4C2×D61semidirect
381. #334C2×D61semidirect
385. #332C2×D61semidirect
389. #331C2×D61semidirect
390. C61#19384semidirect

Maximal subgroups

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

subgroupindex
#214567562
#214556112
#459×D612
#214548402
#214452292
#214452282
#214452272
#214452262
#214451632
#214451622
#214451572
#214445882
#214444852
#214444752
C61×#16652
#1938461