Finite Group Databasebeta

← #16808607

Subgroups — #16808607

Order 3328.

1950 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13131682C211yesmaximal, characteristic, complemented
2#13131680C211yesmaximal, characteristic, complemented
3#13131669C213yesmaximal, complemented
4#13131668C213yesmaximal, complemented
5#13131652C213yesmaximal, complemented
6#13131650C213yesmaximal, complemented
7#13131642C211yesmaximal, characteristic, complemented
8#41×#1057C211yesmaximal, characteristic, complemented
9Q8×#6122C211yesmaximal, characteristic, complemented
10C2×Q8×#1057C211yesmaximal, characteristic, complemented
11#13131578C213yesmaximal, complemented
12#13131559C213yesmaximal, complemented
13#13131531C213yesmaximal, complemented
14C4×#88238C213yesmaximal, complemented
15#13128426C211yesmaximal, characteristic
16C2×#88238C2213yes—
17C2×#88238C2213yes—
18C13×#548C411yescharacteristic
19#11719539C411yescharacteristic
20#41×Dic13C2211yescharacteristic
21#11719522C411yescharacteristic
22C2×Q8×Dic13C2211yescharacteristic
23#11719386C413yes—
24#11719385C2213yes—
25#11719346C413yes—
26#11719329C2213yes—
27#11719323C413yes—
28#11719301C2213yes—
29C4×#6131C2211yescharacteristic
30#11719274C411yescharacteristic
31#11719270C413yes—
32C4×#6129C2213yes—
33#11719158C2213yes—
34#11719158C2213yes—
35Q8×#1057C2211yescharacteristic
36Q8×#1057C2213yes—
37Q8×#1057C2213yes—
38Q8×#1057C2211yescharacteristic
39Q8×#1057C2211yescharacteristic
40Q8×#1057C2211yescharacteristic
41#11719146C2216yes—
42#11719145C2213yes—
43#11719145C2213yes—
44D4×#1057C2213yes—
45D4×#1057C2213yes—
46#11719130C2213yes—
47#11719129C2213yes—
48C4×#6122C2213yes—
49C4×#6122C2213yes—
50#11719119C2213yes—
51C2×C4⋊#1057C2213yes—
52C2×C4×#1057C2213yes—
53C2×C4×#1057C2213yes—
54#11719077C2216yes—
55#11719076C2213yes—
56#11719076C2213yes—
57#11719073C2213yes—
58#11719073C2213yes—
59#11719072C2213yes—
60#11719072C2213yes—
61#11719068C2216yes—
62#11719066C2216yes—
63#11719065C2216yes—
64#11719064C2216yes—
65#11719063C2216yes—
66#11719062C2213yes—
67#11719062C2213yes—
68#11719062C2211yescharacteristic
69#11719062C2213yes—
70#11719054C2213yes—
71#11719051C2213yes—
72C4×#6118C2213yes—
73C4×D13⋊C8C2213yes—
74C2×C13×#41C2×C411yescharacteristic
75C22×Q8×C13C2×C411yescharacteristic
76C2×#6131C2×C411yescharacteristic
77C2×#6129C2313yes—
78C2×#6122C2313yes—
79C2×#6122C2313yes—
80C2×#6122C2313yes—
81C2×#6122C2311yescharacteristic
82C22×#1057C2313yes—
83C22×#1057C2313yes—
84C22×#1057C2313yes—
85C22×#1057C2311yescharacteristic
86#88238—212no—
87#88238—212no—
88C2×#6118C2313yes—
89C2×#6118C2313yes—
90C2×D13⋊C8C2313yes—
91C2×D13⋊C8C2313yes—
92C13×#124C2×C413yes—
93C13×C22⋊Q8C2×C413yes—
94C13×#121C2×C413yes—
95C4×Q8×C13C2×C411yescharacteristic
96C4×D4×C13C2×C413yes—
97#88201C2×C413yes—
98#88200C2×C413yes—
99Q8×Dic13C2×C411yescharacteristic
100Q8×Dic13C2×C413yes—
101Q8×Dic13C2×C411yescharacteristic
102Q8×Dic13C2311yescharacteristic
103Q8×Dic13C2313yes—
104#88194C2×C413yes—
105#88193C2×C416yes—
106#88192C2×C413yes—
107D4×Dic13C2×C413yes—
108D4×Dic13C2×C413yes—
109D4×Dic13C2313yes—
110#88184C2×C413yes—
111#88184C2×C413yes—
112C4×C22⋊D13C2×C413yes—
113C4×C22⋊D13C2×C413yes—
114#88180C2×C413yes—
115C2×C4⋊Dic13C2×C413yes—
116C2×C4×Dic13C2313yes—
117C2×C4×Dic13C2313yes—
118#88152C2×C416yes—
119#88147C2313yes—
120#88146C2×C413yes—
121#88146C2313yes—
122#88144C2×C413yes—
123#88142C2×C413yes—
124#88141C2×C413yes—
125#88140C2×C416yes—
126#88139C2×C416yes—
127#88137C2×C416yes—
128#88135C2×C413yes—
129#88135C2×C413yes—
130#88135C2316yes—
131#88133C2×C416yes—
132#88131C2311yescharacteristic
133#88131C2313yes—
134C4×D52C2×C413yes—
135C42×D13C2313yes—
136#88124C2×C413yes—
137C4×Dic26C2×C413yes—
138C22⋊#1057C2316yes—
139C22⋊#1057C2316yes—
140#88112C2316yes—
141#88111C2316yes—
142#88111C2316yes—
143#88110C2311yescharacteristic
144#88110C2313yes—
145#88110C2313yes—
146#88110C2313yes—
147#88110C2311yescharacteristic
148#88110C2313yes—
149C4⋊#1057C2313yes—
150C4⋊#1057C2313yes—
151C4⋊#1057C2313yes—
152C4⋊#1057C2313yes—
153C4⋊#1057C2316yes—
154C4×#1057C2311yescharacteristic
155C4×#1057C2313yes—
156C4×#1057C2313yes—
157C4×#1057C2313yes—
158C4×#1057C2313yes—
159C4×#1057C2313yes—
160C4×#1057C2313yes—
161C4×#1057C2313yes—
162C4×#1057C2311yescharacteristic
163C4×#1057C2313yes—
164#60218—131nomaximal, Sylow
165C13×#41—24no—
166C2×Q8×C13—24no—
167C2×Q8×C13C22×C413yes—
168C2×Q8×C13C22×C411yescharacteristic
169C2×D4×C13C22×C413yes—
170C2×D4×C13C22×C413yes—
171C22×C4×C13C22×C413yes—
172C22×C4×C13C22×C413yes—
173C2×C22⋊D13C22×C413yes—
174C2×C22⋊D13C22×C413yes—
175C2×C22⋊D13C22×C416yes—
176C22×Dic13C22×C413yes—
177C22×Dic13C22×C413yes—
178C22×Dic13C2413yes—
179C22×Dic13C2411yescharacteristic
180#6131—24no—
181#6129—26no—
182#6129—26no—
183C2×D52C22×C413yes—
184C2×C4×D13C22×C413yes—
185C2×C4×D13C22×C413yes—
186C2×C4×D13C2413yes—
187C2×Dic26C22×C413yes—
188#6122—26no—
189#6122—26no—
190#6122—26no—
191#6122#4114yescomplemented
192C2×#1057—26no—
193C2×#1057—26no—
194C2×#1057—26no—
195C2×#1057#4114yescomplemented
196C2×#1057C2413yes—
197C2×#1057C2413yes—
198C2×#1057C2413yes—
199C2×#1057C2413yes—
200C2×#1057C2411yescharacteristic
201C2×#1057C2411yescharacteristic
202C2×#1057C2411yescharacteristic
203C2×#1057C2411yescharacteristic
204#6118—26no—
205#6118—26no—
206D13⋊C8—26no—
207D13⋊C8—26no—
208C13×C4⋊C4C22×C413yes—
209C13×C4⋊C4C22×C413yes—
210C13×C22⋊C4C22×C411yescharacteristic
211C13×C22⋊C4C22×C413yes—
212C13×C22⋊C4C22×C416yes—
213C42×C13C22×C413yes—
214C22⋊Dic13C22×C413yes—
215C22⋊Dic13C22×C411yescharacteristic
216C22⋊Dic13C22×C416yes—
217#6103C22×C416yes—
218#6103C22×C413yes—
219#6103C22×C413yes—
220#6103C22×C416yes—
221C4⋊Dic13C22×C413yes—
222C4⋊Dic13C22×C413yes—
223C4⋊Dic13C22×C413yes—
224C4⋊Dic13C22×C413yes—
225C4⋊Dic13C22×C416yes—
226#6101C22×C411yescharacteristic
227#6101C22×C413yes—
228#6101C22×C416yes—
229#6101C22×C413yes—
230#6101C22×C411yescharacteristic
231C4×Dic13C22×C413yes—
232C4×Dic13C22×C413yes—
233C4×Dic13C22×C413yes—
234C4×Dic13C22×C413yes—
235C4×Dic13C22×C413yes—
236C4×Dic13C2413yes—
237C4×Dic13C2411yescharacteristic
238C4×Dic13C2411yescharacteristic
239C4×Dic13C2413yes—
240#3480—131no—
241#2971—131no—
242#2971—131no—
243#2969—133no—
244#2969—133no—
245#2966—133no—
246#2966—133no—
247C8×#41—131no—
248C8×#41—131no—
249Q8×M16—131no—
250C2×C8×Q8—131no—
251#2935—133no—
252#2931—133no—
253#2925—133no—
254C4×#131—133no—
255C23×C13C23×C413yes—
256C23×C13C23×C411yescharacteristic
257C22×D13C23×C413yes—
258Q8×C13—26no—
259Q8×C13#13114yescomplemented
260D4×C13—26no—
261D4×C13—26no—
262C2×C4×C13—26no—
263C2×C4×C13—26no—
264C2×C4×C13C23×C411yescharacteristic
265C2×C4×C13C23×C413yes—
266C2×C4×C13C23×C413yes—
267C2×C4×C13C23×C411yescharacteristic
268C2×C4×C13C23×C413yes—
269C22⋊D13—26no—
270C22⋊D13—26no—
271C22⋊D13—46no—
272C2×Dic13—23no—
273C2×Dic13—23no—
274C2×Dic13—43no—
275C2×Dic13—43no—
276C2×Dic13C23×C411yescharacteristic
277C2×Dic13C23×C411yescharacteristic
278C2×Dic13C23×C413yes—
279C2×Dic13C23×C413yes—
280C2×Dic13C23×C413yes—
281C2×Dic13C23×C411yescharacteristic
282C2×Dic13C2×#4112yes—
283C2×Dic13C2×#4112yes—
284C2×Dic13C2511yescharacteristic
285D52—26no—
286C4×D13—26no—
287C4×D13—23no—
288C4×D13—23no—
289C4×D13—43no—
290Dic26—26no—
291#1057—23no—
292#1057—23no—
293#1057—23no—
294#1057—23no—
295#1057C2×#4112yes—
296#1057C2×#4112yes—
297#1057C2×#4112yes—
298#1057C2×#4112yes—
299C2×#131—133no—
300C2×#131—133no—
301#559—133no—
302#557—131no—
303#556—133no—
304#553—133no—
305#553—133no—
306#550—131no—
307#548—131no—
308#548#26111yescharacteristic
309#532—133no—
310#529—133no—
311#529—133no—
312C4×#41—131no—
313C4×#41—131no—
314C4×#41—133no—
315C2×C4×Q8—131no—
316#446—133no—
317#446—133no—
318C8×Q8—131no—
319C8×Q8—133no—
320C8×Q8—133no—
321C8×Q8—131no—
322C8×Q8—131no—
323C8×Q8—131no—
324#436—136no—
325#436—133no—
326#436—133no—
327#436—133no—
328#436—133no—
329#435—136no—
330#435—136no—
331C8×D4—133no—
332C8×D4—133no—
333C8×D4—136no—
334C8×D4—133no—
335C8×D4—133no—
336#433—136no—
337#433—136no—
338#432—133no—
339#432—133no—
340#431—133no—
341#431—133no—
342#423—133no—
343C2×C4⋊C8—133no—
344#408—136no—
345#405—131no—
346#405—133no—
347#405—133no—
348#405—133no—
349#405—133no—
350#405—133no—
351C4×M16—133no—
352C4×M16—133no—
353C4×M16—133no—
354C2×C4×C8—133no—
355C2×C4×C8—133no—
356C2×C4×C8—133no—
357C22×C13—21no—
358C22×C13—21no—
359C22×C13—23no—
360C22×C13—23no—
361C22×C13—23no—
362C22×C13C4×#4112yes—
363C22×C13C24×C411yescharacteristic
364C2×D13—23no—
365C2×D13—23no—
366C2×D13—23no—
367C4×C13—23no—
368C4×C13—21no—
369C4×C13—23no—
370C4×C13—21no—
371C4×C13C2×#13116yes—
372Dic13—23no—
373Dic13—21no—
374Dic13—23no—
375Dic13—21no—
376Dic13—23no—
377Dic13C4×#4112yes—
378Dic13C22×#4111yescharacteristic
379Dic13C22×#4111yescharacteristic
380C2×#41—131no—
381C2×#41—133no—
382C2×#41C2×#26111yescharacteristic
383C22×Q8C2×#26111yescharacteristic
384#131—2612no—
385#131—2612no—
386C2×M16—131no—
387C2×M16—133no—
388C2×M16—133no—
389C2×M16—133no—
390C2×M16—133no—
391C2×M16—133no—
392C22×C8—131no—
393C22×C8—133no—
394C22×C8—133no—
395C22×C8—133no—
396C22×C8—133no—
397C22×C8—133no—
398C4⋊Q8—133no—
399#127—133no—
400#126—136no—
401#126—136no—
402#125—133no—
403#125—133no—
404#124—133no—
405#124—136no—
406#124C2×#26113yes—
407#123—136no—
408#123—136no—
409C22⋊Q8—133no—
410C22⋊Q8C2×#26113yes—
411#121—133no—
412#121—136no—
413#121—133no—
414#121—133no—
415#121C2×#26113yes—
416C4×Q8—131no—
417C4×Q8—133no—
418C4×Q8—131no—
419C4×Q8—133no—
420C4×Q8—131no—
421C4×Q8—133no—
422C4×Q8—133no—
423C4×Q8C2×#26111yescharacteristic
424C4×D4—133no—
425C4×D4—133no—
426C4×D4—133no—
427C4×D4—133no—
428C4×D4—133no—
429C4×D4—133no—
430C4×D4—133no—
431C4×D4—133no—
432C4×D4—136no—
433C4×D4—133no—
434C4×D4C2×#26113yes—
435#117—133no—
436#117—133no—
437#117—133no—
438#117—131no—
439#117—133no—
440C2×C4⋊C4—133no—
441C2×C42—133no—
442C2×C42—133no—
443C2×C42—133no—
444C4⋊C8—133no—
445C4⋊C8—133no—
446C4⋊C8—136no—
447C4⋊C8—133no—
448C4⋊C8—133no—
449C4⋊C8—136no—
450C22⋊C8—136no—
451C22⋊C8—136no—
452C22⋊C8—136no—
453C22⋊C8—136no—
454#97—131no—
455#97—131no—
456#97—133no—
457#97—133no—
458#97—133no—
459#97—133no—
460C4×C8—131no—
461C4×C8—131no—
462C4×C8—133no—
463C4×C8—133no—
464C4×C8—133no—
465C4×C8—133no—
466C4×C8—133no—
467C4×C8—133no—
468C4×C8—133no—
469C4×C8—133no—
470C2×C13—22no—
471C2×C13—43no—
472C2×C13C2×C4×#4111yescharacteristic
473C2×C13C22×#13111yescharacteristic
474C2×C13#357311yescharacteristic
475D13—43no—
476#41—24no—
477#41—266no—
478#41—266no—
479#41—264no—
480C2×Q8—24no—
481C2×Q8—133no—
482C2×Q8C22×#26113yes—
483C2×Q8C22×#26111yescharacteristic
484C2×D4—133no—
485C2×D4—133no—
486C2×D4—136no—
487C2×D4—133no—
488C2×D4C22×#26113yes—
489C2×D4C22×#26113yes—
490C22×C4—131no—
491C22×C4—133no—
492C22×C4—133no—
493C22×C4—133no—
494C22×C4—133no—
495C22×C4—133no—
496C22×C4—133no—
497C22×C4C22×#26113yes—
498C22×C4C22×#26113yes—
499M16—134no—
500M16—266no—
501M16—266no—
502M16—266no—
503M16—266no—
504M16—266no—
505C2×C8—131no—
506C2×C8—131no—
507C2×C8—131no—
508C2×C8—131no—
509C2×C8—134no—
510C2×C8—133no—
511C2×C8—133no—
512C2×C8—133no—
513C2×C8—133no—
514C2×C8—266no—
515C2×C8—266no—
516C2×C8—266no—
517C2×C8—266no—
518C2×C8—266no—
519C4⋊C4—133no—
520C4⋊C4—133no—
521C4⋊C4—136no—
522C4⋊C4—131no—
523C4⋊C4—131no—
524C4⋊C4—133no—
525C4⋊C4—133no—
526C4⋊C4—133no—
527C4⋊C4—133no—
528C4⋊C4—136no—
529C4⋊C4C22×#26113yes—
530C4⋊C4C22×#26113yes—
531C22⋊C4—131no—
532C22⋊C4—133no—
533C22⋊C4—136no—
534C22⋊C4—133no—
535C22⋊C4—136no—
536C22⋊C4—133no—
537C22⋊C4—136no—
538C22⋊C4C22×#26111yescharacteristic
539C22⋊C4C22×#26113yes—
540C22⋊C4C22×#26116yes—
541C42—131no—
542C42—131no—
543C42—133no—
544C42—133no—
545C42—133no—
546C42—133no—
547C42—133no—
548C42—133no—
549C42—133no—
550C42C22×#26113yes—
551C13#6021811yescharacteristic, complemented, Sylow
552C23—133no—
553C23C23×#26113yes—
554C23C23×#26111yescharacteristic
555Q8—26no—
556Q8—266no—
557Q8#8823814yescomplemented
558D4—26no—
559D4—26no—
560D4—266no—
561D4—266no—
562D4—266no—
563D4—526no—
564C2×C4—26no—
565C2×C4—26no—
566C2×C4—132no—
567C2×C4—132no—
568C2×C4—131no—
569C2×C4—131no—
570C2×C4—131no—
571C2×C4—131no—
572C2×C4—133no—
573C2×C4—133no—
574C2×C4—133no—
575C2×C4—263no—
576C2×C4—263no—
577C2×C4—263no—
578C2×C4—263no—
579C2×C4—266no—
580C2×C4—523no—
581C2×C4—523no—
582C2×C4—523no—
583C2×C4C23×#26111yescharacteristic
584C2×C4C23×#26113yes—
585C2×C4C23×#26111yescharacteristic
586C2×C4C23×#26113yes—
587C2×C4C23×#26113yes—
588C8—132no—
589C8—132no—
590C8—132no—
591C8—132no—
592C8—263no—
593C8—263no—
594C8—263no—
595C8—263no—
596C22—21no—
597C22—21no—
598C22—23no—
599C22—23no—
600C22—23no—
601C22—263no—
602C22—263no—
603C22—263no—
604C22#41×#26112yes—
605C22C24×#26111yescharacteristic
606C4—23no—
607C4—21no—
608C4—23no—
609C4—21no—
610C4—132no—
611C4—131no—
612C4—131no—
613C4—263no—
614C4—261no—
615C4—263no—
616C4—261no—
617C4—263no—
618C4C2×#8823816yes—
619C2—22no—
620C2—43no—
621C2—523no—
622C2C2×#41×#26111yescharacteristic
623C2C22×#8823811yescharacteristic
624C2#1313183211yescharacteristic
subgroupG/H
center, FrattiniC22C24×#261
commutator, socleC22×C13C24×C4
FittingC13×#548C4
prime radical#13128426C2
cosocleC2×Dic13C25
hypercenter#548#261
hypocenterC13#60218
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ #548

Complements

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

NH ≅ G/Ntype
1. #13131682C2semidirect
2. #13131680C2semidirect
3. #13131669C2semidirect
4. #13131668C2semidirect
5. #13131652C2semidirect
6. #13131650C2semidirect
7. #13131642C2semidirect
8. #41×#1057C2semidirect
9. Q8×#6122C2semidirect
10. C2×Q8×#1057C2semidirect
11. #13131578C2semidirect
12. #13131559C2semidirect
13. #13131531C2semidirect
14. C4×#88238C2semidirect
191. #6122#41semidirect
195. C2×#1057#41semidirect
259. Q8×C13#131semidirect
551. C13#60218semidirect
557. Q8#88238semidirect

Maximal subgroups

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

subgroupindex
#131316822
#131316802
#131316692
#131316682
#131316522
#131316502
#131316422
#41×#10572
Q8×#61222
C2×Q8×#10572
#131315782
#131315592
#131315312
C4×#882382
#131284262
#6021813