Finite Group Databasebeta

← #16727617

Subgroups — #16727617

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#13128434C211yesmaximal, characteristic
2#13123855C211yesmaximal, characteristic, complemented
3#13123843C213yesmaximal, complemented
4#13123840C213yesmaximal, complemented
5#13123814C213yesmaximal, complemented
6#13123805C213yesmaximal, complemented
7#41×#1055C211yesmaximal, characteristic, complemented
8C8×#6131C211yesmaximal, characteristic, complemented
9Q8×#6094C211yesmaximal, characteristic, complemented
10C8×Q8×D13C211yesmaximal, characteristic, complemented
11#13123581C213yesmaximal, complemented
12#13123545C213yesmaximal, complemented
13#13123503C213yesmaximal, complemented
14C4×#88161C213yesmaximal, complemented
15C13×#2971C211yesmaximal, characteristic, complemented
16C4×C13×#41C2211yescharacteristic
17#11719539C411yescharacteristic
18#11719536C411yescharacteristic
19C2×#88161C2213yes—
20C2×#88161C2213yes—
21#11719281C411yescharacteristic
22C4×#6131C2211yescharacteristic
23C4×Q8×D13C2211yescharacteristic
24#11719274C411yescharacteristic
25#11719264C413yes—
26#11719263C413yes—
27#11719262C2213yes—
28#11719253C2213yes—
29#11719238C413yes—
30#11719235C413yes—
31#11719184C2213yes—
32C4×#6127C2213yes—
33C8×Q8×C13C2211yescharacteristic
34C8×Q8×C13C2211yescharacteristic
35C13×#436C2213yes—
36C13×#436C2213yes—
37C13×#432C2213yes—
38C4×C13×M16C2213yes—
39#11718789C2216yes—
40#11718788C2216yes—
41#11718783C2213yes—
42#11718783C2213yes—
43Q8×#1055C2211yescharacteristic
44Q8×#1055C2211yescharacteristic
45D4×#1055C2213yes—
46D4×#1055C2213yes—
47#11718586C2213yes—
48C2×C4×#1055C2213yes—
49#11718499C2213yes—
50#11718498C2213yes—
51#11718496C2213yes—
52#11718495C2213yes—
53#11718494C2213yes—
54D13×C4⋊C8C2213yes—
55#11718492C2213yes—
56#11718491C2213yes—
57#11718391C2216yes—
58#11718388C2216yes—
59#11718386C2216yes—
60#11718381C2216yes—
61#11718370C2216yes—
62#11718369C2213yes—
63#11718369C2213yes—
64#11718369C2211yescharacteristic
65#11718369C2213yes—
66#11718349C2213yes—
67C4×#6094C2213yes—
68C4×#6094C2213yes—
69C8×D52C2213yes—
70C4×C8×D13C2213yes—
71C4×C8×D13C2213yes—
72#11718340C2213yes—
73C8×Dic26C2213yes—
74C2×C13×#41C2×C411yescharacteristic
75C2×#6131C2×C411yescharacteristic
76C2×Q8×D13C2×C411yescharacteristic
77C2×#6127C2313yes—
78C2×C13×M16C2313yes—
79C2×C13×M16C2313yes—
80C4×Q8×C13C2×C411yescharacteristic
81C4×Q8×C13C2311yescharacteristic
82C4×D4×C13C2×C413yes—
83C4×D4×C13C2313yes—
84C13×#117C2×C413yes—
85C2×C42×C13C2313yes—
86Q8×Dic13C2×C411yescharacteristic
87Q8×Dic13C2×C411yescharacteristic
88#88194C2×C413yes—
89#88192C2×C413yes—
90#88192C2×C413yes—
91#88190C2×C413yes—
92#88184C2×C413yes—
93#88183C2×C416yes—
94C4×C22⋊D13C2×C413yes—
95C4×C22⋊D13C2×C413yes—
96C4×C22⋊D13C2×C413yes—
97C4×C22⋊D13C2×C413yes—
98C4×C22⋊D13C2316yes—
99#88180C2313yes—
100#88178C2×C413yes—
101C22×#1055C2313yes—
102C22×#1055C2313yes—
103#88161—212no—
104#88161—212no—
105C2×#6094C2313yes—
106C2×#6094C2313yes—
107C2×#6094C2313yes—
108C2×#6094C2311yescharacteristic
109C2×C8×D13C2313yes—
110C2×C8×D13C2313yes—
111C2×C8×D13C2313yes—
112C2×C8×D13C2311yescharacteristic
113#88151C2×C413yes—
114#88149C2×C413yes—
115#88147C2×C413yes—
116#88146C2×C413yes—
117C4⋊C4×D13C2×C413yes—
118#88144C2×C413yes—
119#88142C2×C413yes—
120#88141C2×C413yes—
121#88139C2×C416yes—
122#88138C2×C416yes—
123#88137C2×C416yes—
124#88133C2×C416yes—
125#88130C2×C416yes—
126#88129C2×C413yes—
127C4×D52C2×C413yes—
128C4×D52C2×C413yes—
129C4×D52C2313yes—
130#88126C2311yescharacteristic
131#88126C2313yes—
132C42×D13C2313yes—
133C42×D13C2313yes—
134#88124C2×C413yes—
135C4×Dic26C2×C413yes—
136C4×Dic26C2×C413yes—
137C4×Dic26C2313yes—
138C13×C4⋊C8C2313yes—
139C13×C4⋊C8C2313yes—
140C13×C22⋊C8C2316yes—
141C13×#97C2311yescharacteristic
142C13×#97C2313yes—
143C4×C8×C13C2313yes—
144C4×C8×C13C2313yes—
145C22⋊#1055C2316yes—
146#88060C2316yes—
147#88060C2316yes—
148#88055C2313yes—
149#88055C2313yes—
150#88055C2311yescharacteristic
151#88055C2313yes—
152#88054C2316yes—
153#88054C2316yes—
154C8×Dic13C2313yes—
155C8×Dic13C2313yes—
156C8×Dic13C2311yescharacteristic
157C8×Dic13C2313yes—
158C4⋊#1055C2313yes—
159C4⋊#1055C2313yes—
160C4×#1055C2313yes—
161C4×#1055C2313yes—
162C4×#1055C2311yescharacteristic
163C4×#1055C2313yes—
164#60218—131nomaximal, Sylow
165C13×#41—24no—
166C2×Q8×C13C22×C411yescharacteristic
167C2×D4×C13C22×C413yes—
168C22×C4×C13C22×C413yes—
169C22×C4×C13C22×C413yes—
170C22×C4×C13C2413yes—
171C2×C22⋊D13C22×C413yes—
172C2×C22⋊D13C22×C413yes—
173C2×C22⋊D13C22×C416yes—
174#6131—24no—
175Q8×D13—24no—
176#6127—26no—
177#6127—26no—
178C2×D52C22×C413yes—
179C2×D52C22×C413yes—
180C2×C4×D13C22×C413yes—
181C2×C4×D13C22×C413yes—
182C2×C4×D13C22×C413yes—
183C2×C4×D13C22×C413yes—
184C2×C4×D13C2413yes—
185C2×C4×D13C2411yescharacteristic
186C2×Dic26C22×C413yes—
187C2×Dic26C22×C413yes—
188C13×M16—26no—
189C13×M16—26no—
190C2×C8×C13C2413yes—
191C2×C8×C13C2413yes—
192C2×C8×C13C2411yescharacteristic
193C2×C8×C13C2411yescharacteristic
194C13×C4⋊C4C22×C413yes—
195C13×C4⋊C4C22×C413yes—
196C13×C22⋊C4C22×C416yes—
197C42×C13C22×C413yes—
198C42×C13C22×C413yes—
199C42×C13C2413yes—
200C42×C13C2411yescharacteristic
201C22⋊Dic13C22×C413yes—
202C22⋊Dic13C22×C416yes—
203C22⋊Dic13C22×C413yes—
204#6103C22×C411yescharacteristic
205#6103C22×C411yescharacteristic
206#6103C22×C413yes—
207#6103C22×C413yes—
208#6103C22×C416yes—
209#6103C22×C416yes—
210C4⋊Dic13C22×C413yes—
211C4⋊Dic13C22×C413yes—
212C4⋊Dic13C22×C413yes—
213C4⋊Dic13C22×C413yes—
214#6101C22×C411yescharacteristic
215#6101C22×C411yescharacteristic
216#6101C22×C413yes—
217#6101C22×C413yes—
218#6101C22×C416yes—
219#6101C22×C416yes—
220C4×Dic13C22×C413yes—
221C4×Dic13C22×C413yes—
222C4×Dic13C22×C413yes—
223C4×Dic13C22×C413yes—
224C4×Dic13C2413yes—
225C4×Dic13C2411yescharacteristic
226C2×#1055—26no—
227C2×#1055—26no—
228C2×#1055C2413yes—
229C2×#1055C2413yes—
230C2×#1055C2411yescharacteristic
231C2×#1055C2411yescharacteristic
232#6094—26no—
233#6094—26no—
234#6094—26no—
235#6094#4114yescomplemented
236C8×D13—26no—
237C8×D13—26no—
238C8×D13—26no—
239C8×D13#4114yescomplemented
240#3480—131no—
241#2971—131no—
242#2971D1311yescharacteristic, complemented
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—
256C22×D13C23×C413yes—
257C22×D13C23×C411yescharacteristic
258Q8×C13#13114yescomplemented
259D4×C13—26no—
260C2×C4×C13—26no—
261C2×C4×C13—23no—
262C2×C4×C13—23no—
263C2×C4×C13—43no—
264C2×C4×C13C23×C413yes—
265C2×C4×C13C23×C411yescharacteristic
266C2×C4×C13C23×C411yescharacteristic
267C2×C4×C13C23×C413yes—
268C2×C4×C13C23×C413yes—
269C2×C4×C13C2511yescharacteristic
270C22⋊D13—26no—
271C22⋊D13—26no—
272C22⋊D13—46no—
273C2×Dic13C23×C411yescharacteristic
274C2×Dic13C23×C411yescharacteristic
275C2×Dic13C23×C413yes—
276C2×Dic13C23×C413yes—
277C2×Dic13C23×C413yes—
278C2×Dic13C23×C411yescharacteristic
279D52—26no—
280D52—26no—
281C4×D13—26no—
282C4×D13—26no—
283C4×D13—23no—
284C4×D13—23no—
285C4×D13—43no—
286C4×D13—43no—
287C4×D13C2×#4112yes—
288C4×D13C2×#4112yes—
289Dic26—26no—
290Dic26—26no—
291C8×C13—23no—
292C8×C13—23no—
293C8×C13C2×#4112yes—
294C8×C13C2×#4112yes—
295#1055—23no—
296#1055—23no—
297#1055C2×#4112yes—
298#1055C2×#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—131no—
309#532—133no—
310#529—133no—
311#529—133no—
312C4×#41—131no—
313C4×#41—133no—
314C4×#41C2×D1311yescharacteristic
315C2×C4×Q8—131no—
316#446—133no—
317#446—133no—
318C8×Q8—131no—
319C8×Q8—131no—
320C8×Q8—133no—
321C8×Q8—133no—
322C8×Q8C2×D1311yescharacteristic
323C8×Q8C2×D1311yescharacteristic
324#436—136no—
325#436—133no—
326#436—133no—
327#436C2×D1313yes—
328#436C2×D1313yes—
329#435—136no—
330#435—136no—
331C8×D4—133no—
332C8×D4—133no—
333C8×D4—133no—
334C8×D4—136no—
335C8×D4—133no—
336#433—136no—
337#433—136no—
338#432—133no—
339#432C2×D1313yes—
340#431—133no—
341#431—133no—
342#423—133no—
343C2×C4⋊C8—133no—
344#408—136no—
345#405—133no—
346#405—133no—
347#405—131no—
348#405—133no—
349#405—133no—
350#405—133no—
351C4×M16—133no—
352C4×M16—133no—
353C4×M16C2×D1313yes—
354C2×C4×C8—133no—
355C2×C4×C8—133no—
356C2×C4×C8—133no—
357C22×C13—23no—
358C22×C13—23no—
359C22×C13—23no—
360C22×C13C24×C411yescharacteristic
361C2×D13—21no—
362C2×D13—21no—
363C2×D13—23no—
364C2×D13—23no—
365C2×D13—23no—
366C2×D13C4×#4112yes—
367C4×C13—23no—
368C4×C13—21no—
369C4×C13—23no—
370C4×C13—21no—
371C4×C13C2×#13116yes—
372C4×C13C22×#4111yescharacteristic
373C4×C13C22×#4111yescharacteristic
374Dic13—23no—
375Dic13—23no—
376Dic13—21no—
377Dic13—21no—
378Dic13—23no—
379Dic13C4×#4112yes—
380C2×#41—131no—
381C2×#41—133no—
382C2×#41C4×D1311yescharacteristic
383C22×Q8—131no—
384#131—2612no—
385#131—2612no—
386C2×M16—131no—
387C2×M16—133no—
388C2×M16—133no—
389C2×M16—133no—
390C2×M16C22×D1313yes—
391C2×M16C22×D1313yes—
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#124—133no—
407#123—136no—
408#123—136no—
409C22⋊Q8—133no—
410C22⋊Q8—133no—
411#121—133no—
412#121—133no—
413#121—133no—
414#121—133no—
415#121—136no—
416C4×Q8—131no—
417C4×Q8—131no—
418C4×Q8—133no—
419C4×Q8—133no—
420C4×Q8—133no—
421C4×Q8—133no—
422C4×Q8C4×D1311yescharacteristic
423C4×Q8C22×D1311yescharacteristic
424C4×D4—133no—
425C4×D4—133no—
426C4×D4—133no—
427C4×D4—133no—
428C4×D4—136no—
429C4×D4—133no—
430C4×D4—133no—
431C4×D4—133no—
432C4×D4—133no—
433C4×D4C4×D1313yes—
434C4×D4C22×D1313yes—
435#117—133no—
436#117—133no—
437#117—131no—
438#117—133no—
439#117C4×D1313yes—
440C2×C4⋊C4—133no—
441C2×C42—133no—
442C2×C42—133no—
443C2×C42C22×D1313yes—
444C4⋊C8—136no—
445C4⋊C8—136no—
446C4⋊C8—133no—
447C4⋊C8—133no—
448C4⋊C8C22×D1313yes—
449C4⋊C8C22×D1313yes—
450C22⋊C8—136no—
451C22⋊C8—136no—
452C22⋊C8—136no—
453C22⋊C8C22×D1316yes—
454#97—131no—
455#97—133no—
456#97—133no—
457#97—133no—
458#97C22×D1311yescharacteristic
459#97C22×D1313yes—
460C4×C8—131no—
461C4×C8—133no—
462C4×C8—133no—
463C4×C8—133no—
464C4×C8—131no—
465C4×C8—133no—
466C4×C8—133no—
467C4×C8—133no—
468C4×C8C22×D1313yes—
469C4×C8C22×D1313yes—
470C2×C13—43no—
471C2×C13C2×C4×#4111yescharacteristic
472C2×C13C22×#13111yescharacteristic
473C2×C13#357311yescharacteristic
474D13—22no—
475D13—43no—
476#41—24no—
477#41—266no—
478#41—266no—
479#41—264no—
480C2×Q8—133no—
481C2×Q8—133no—
482C2×Q8—264no—
483C2×Q8C2×C4×D1311yescharacteristic
484C2×D4—133no—
485C2×D4—133no—
486C2×D4—136no—
487C2×D4—133no—
488C2×D4—133no—
489C2×D4C2×C4×D1313yes—
490C22×C4—131no—
491C22×C4—133no—
492C22×C4—133no—
493C22×C4—133no—
494C22×C4—133no—
495C22×C4—133no—
496C22×C4C2×C4×D1313yes—
497C22×C4C2×C4×D1313yes—
498C22×C4C23×D1313yes—
499M16—26no—
500M16—26no—
501M16—134no—
502M16—266no—
503M16—266no—
504M16—266no—
505C2×C8—131no—
506C2×C8—131no—
507C2×C8—134no—
508C2×C8—133no—
509C2×C8—133no—
510C2×C8—266no—
511C2×C8—266no—
512C2×C8—266no—
513C2×C8—266no—
514C2×C8—266no—
515C2×C8C23×D1313yes—
516C2×C8C23×D1313yes—
517C2×C8C23×D1311yescharacteristic
518C2×C8C23×D1311yescharacteristic
519C4⋊C4—133no—
520C4⋊C4—136no—
521C4⋊C4—131no—
522C4⋊C4—131no—
523C4⋊C4—133no—
524C4⋊C4—133no—
525C4⋊C4—133no—
526C4⋊C4—133no—
527C4⋊C4—133no—
528C4⋊C4—136no—
529C4⋊C4C2×C4×D1313yes—
530C4⋊C4C2×C4×D1313yes—
531C22⋊C4—133no—
532C22⋊C4—133no—
533C22⋊C4—136no—
534C22⋊C4—131no—
535C22⋊C4—131no—
536C22⋊C4—133no—
537C22⋊C4—136no—
538C22⋊C4—133no—
539C22⋊C4—136no—
540C22⋊C4C2×C4×D1316yes—
541C42—133no—
542C42—133no—
543C42—133no—
544C42—133no—
545C42—131no—
546C42—133no—
547C42C2×C4×D1313yes—
548C42C2×C4×D1313yes—
549C42C23×D1311yescharacteristic
550C42C23×D1313yes—
551C13#6021811yescharacteristic, complemented, Sylow
552C23—131no—
553C23—133no—
554C23C22×C4×D1313yes—
555Q8—266no—
556Q8—266no—
557Q8#8816114yescomplemented
558D4—26no—
559D4—266no—
560D4—266no—
561D4—266no—
562D4—266no—
563D4—526no—
564C2×C4—26no—
565C2×C4—23no—
566C2×C4—23no—
567C2×C4—43no—
568C2×C4—131no—
569C2×C4—132no—
570C2×C4—132no—
571C2×C4—131no—
572C2×C4—131no—
573C2×C4—133no—
574C2×C4—133no—
575C2×C4—133no—
576C2×C4—266no—
577C2×C4—266no—
578C2×C4—263no—
579C2×C4—263no—
580C2×C4—523no—
581C2×C4—523no—
582C2×C4C22×C4×D1313yes—
583C2×C4C22×C4×D1311yescharacteristic
584C2×C4C22×C4×D1311yescharacteristic
585C2×C4C22×C4×D1313yes—
586C2×C4C22×C4×D1313yes—
587C2×C4C24×D1311yescharacteristic
588C8—23no—
589C8—23no—
590C8—132no—
591C8—132no—
592C8—263no—
593C8—263no—
594C8#41×D1312yes—
595C8#41×D1312yes—
596C22—23no—
597C22—23no—
598C22—23no—
599C22—132no—
600C22—261no—
601C22—261no—
602C22—263no—
603C22—263no—
604C22—263no—
605C22C23×C4×D1311yescharacteristic
606C4—21no—
607C4—23no—
608C4—21no—
609C4—23no—
610C4—132no—
611C4—261no—
612C4—263no—
613C4—263no—
614C4—261no—
615C4—263no—
616C4C2×#8816116yes—
617C4C2×#41×D1311yescharacteristic
618C4C2×#41×D1311yescharacteristic
619C2—43no—
620C2—262no—
621C2—523no—
622C2C4×#41×D1311yescharacteristic
623C2C22×#8816111yescharacteristic
624C2#1312954911yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D13
commutator, socleC22×C13C24×C4
FittingC13×#2971C2
prime radical#13128434C2
cosocleC2×C4×C13C25
hypercenter#2971D13
hypocenterC13#60218
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C2×C4 ◁ #2971

Complements

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

NH ≅ G/Ntype
2. #13123855C2semidirect
3. #13123843C2semidirect
4. #13123840C2semidirect
5. #13123814C2semidirect
6. #13123805C2semidirect
7. #41×#1055C2semidirect
8. C8×#6131C2semidirect
9. Q8×#6094C2semidirect
10. C8×Q8×D13C2semidirect
11. #13123581C2semidirect
12. #13123545C2semidirect
13. #13123503C2semidirect
14. C4×#88161C2semidirect
15. C13×#2971C2semidirect
235. #6094#41semidirect
239. C8×D13#41semidirect
242. #2971D13semidirect
258. Q8×C13#131semidirect
551. C13#60218semidirect
557. Q8#88161semidirect

Maximal subgroups

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

subgroupindex
#131284342
#131238552
#131238432
#131238402
#131238142
#131238052
#41×#10552
C8×#61312
Q8×#60942
C8×Q8×D132
#131235812
#131235452
#131235032
C4×#881612
C13×#29712
#6021813