Finite Group Databasebeta

← #16726741

Subgroups — #16726741

Order 3328.

2038 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13128035C211yesmaximal, characteristic
2#13123504C213yesmaximal, complemented
3#13123501C213yesmaximal, complemented
4#13123493C213yesmaximal, complemented
5#13123493C213yesmaximal, complemented
6#13123486C213yesmaximal, complemented
7#13123479C213yesmaximal, complemented
8#13123473C213yesmaximal, complemented
9C2×#11718501C213yesmaximal, complemented
10#13123404C211yesmaximal, characteristic, complemented
11#13123283C211yesmaximal, characteristic
12#13123283C211yesmaximal, characteristic
13#13123262C211yesmaximal, characteristic, complemented
14#131×Dic13C211yesmaximal, characteristic, complemented
15C13×#2906C211yesmaximal, characteristic, complemented
16C2×#41×D13C2211yescharacteristic
17C2×C13×#131C2211yescharacteristic
18C2×C13×#131C2211yescharacteristic
19C4×C13×#41C2211yescharacteristic
20#41×Dic13C2211yescharacteristic
21C2×#88202C2211yescharacteristic
22C2×#88202C2211yescharacteristic
23#11719279C411yescharacteristic
24#11719279C411yescharacteristic
25#11719277C411yescharacteristic
26#11719277C411yescharacteristic
27#11719251C413yes—
28#11719251C413yes—
29#11719250C413yes—
30#11719250C413yes—
31#11719227C2213yes—
32D13×#117C2213yes—
33C4×#6127C2213yes—
34C2×#88126C2213yes—
35C13×#424C2213yescomplemented
36C13×#424C2213yescomplemented
37C13×#423C2213yescomplemented
38C2×C13×C4⋊C8C2213yescomplemented
39#11718790C2213yes—
40#11718790C2213yes—
41#11718787C2213yes—
42#11718787C2213yes—
43#11718783C2213yescomplemented
44#11718783C2213yescomplemented
45M16×Dic13C2213yescomplemented
46M16×Dic13C2213yescomplemented
47#11718771C2213yes—
48#11718771C2213yes—
49C2×#88060C2213yes—
50C2×#88060C2213yes—
51#11718762C2213yescomplemented
52#11718762C2213yescomplemented
53C2×#88055C2213yescomplemented
54C2×C8×Dic13C2213yescomplemented
55#11718660C2213yescomplemented
56#11718660C2213yescomplemented
57#11718585C2213yescomplemented
58C2×C4⋊#1055C2213yescomplemented
59#11718501C2211yescharacteristic
60#11718501C2213yes—
61#11718501C2213yes—
62#11718501C2213yes—
63#11718501C2216yes—
64#11718501C2213yescomplemented
65#11718501C2213yescomplemented
66#11718501C2213yescomplemented
67#11718501C2213yescomplemented
68#11718501C2216yescomplemented
69#11718501C2216yescomplemented
70#11718496C2216yescomplemented
71#11718496C2216yescomplemented
72#11718494C2216yescomplemented
73#11718494C2216yescomplemented
74C2×C13×#41C2311yescharacteristic
75#41×D13—24no—
76#41×D13—24no—
77C2×#6131C2×C411yescharacteristic
78C2×Q8×D13C2×C411yescharacteristic
79C2×#6129C2×C413yes—
80C2×D4×D13C2×C413yes—
81C2×#6127C2313yes—
82C22×C4×D13C2313yes—
83C13×#131—24no—
84C13×#131—24no—
85C2×C13×M16C2313yes—
86C2×C13×M16C2313yes—
87C22×C8×C13C2313yes—
88C22×C8×C13C2313yes—
89C4×Q8×C13C2×C411yescharacteristic
90C4×Q8×C13C2×C411yescharacteristic
91C4×D4×C13C2×C413yes—
92C4×D4×C13C2×C413yes—
93C13×#117C2313yes—
94C2×C42×C13C2313yes—
95#88202—24no—
96#88202—24no—
97Q8×Dic13C2×C411yescharacteristic
98Q8×Dic13C2×C411yescharacteristic
99D4×Dic13C2×C413yes—
100D4×Dic13C2×C413yes—
101C4×C22⋊D13C2×C416yes—
102C4×C22⋊D13C2×C416yes—
103C2×#6103C2×C413yes—
104C2×#6103C2×C413yes—
105#88180C2313yes—
106C2×#6101C2×C413yes—
107C2×#6101C2×C413yes—
108C2×C4×Dic13C2313yes—
109C2×#6099C2313yes—
110C2×#6099C2313yes—
111C22×#1055C2313yes—
112C22×#1055C2313yes—
113#88147C2×C413yes—
114#88147C2×C413yes—
115#88146C2×C413yes—
116#88146C2×C413yes—
117C4⋊C4×D13C2×C413yes—
118C4⋊C4×D13C2×C413yes—
119#88141C2×C413yes—
120#88141C2×C413yes—
121#88135C2×C416yes—
122#88135C2×C416yes—
123C22⋊C4×D13C2×C416yes—
124#88131C2×C416yes—
125C4×D52C2×C413yes—
126C4×D52C2×C413yes—
127#88126C2×C413yes—
128#88126C2×C413yes—
129#88126C2×C413yes—
130#88126C2×C413yes—
131#88126C2311yescharacteristic
132#88126C2313yes—
133#88126C2313yes—
134#88126C2313yes—
135C42×D13C2316yes—
136C4×Dic26C2×C413yes—
137C4×Dic26C2×C413yes—
138C13×C4⋊C8C2311yescharacteristic
139C13×C4⋊C8C2313yes—
140C13×C4⋊C8C2313yes—
141C13×C4⋊C8C2313yes—
142C13×C4⋊C8C2316yes—
143#88060C2313yes—
144#88060C2313yes—
145#88060C2316yes—
146#88060C2316yes—
147#88060C2313yes—
148#88060C2313yes—
149#88060C2311yescharacteristic
150#88060C2311yescharacteristic
151#88060C2313yes—
152#88060C2313yes—
153#88055C2313yes—
154#88055C2313yes—
155#88055C2316yes—
156#88055C2311yescharacteristic
157#88055C2313yes—
158C8×Dic13C2313yes—
159C8×Dic13C2313yes—
160C8×Dic13C2316yes—
161C8×Dic13C2311yescharacteristic
162C8×Dic13C2313yes—
163C4⋊#1055C2313yes—
164C4⋊#1055C2316yes—
165C4⋊#1055C2313yes—
166C4⋊#1055C2311yescharacteristic
167C4⋊#1055C2313yes—
168#60129—131nomaximal, Sylow
169C23×D13C22×C413yes—
170C13×#41#4114yes—
171C13×#41#4114yes—
172C2×Q8×C13C22×C411yescharacteristic
173C2×D4×C13C22×C413yes—
174C22×C4×C13C22×C413yes—
175C22×C4×C13C22×C413yes—
176C22×C4×C13C2413yes—
177C2×C22⋊D13C22×C416yes—
178C22×Dic13C22×C413yes—
179C22×Dic13C22×C413yes—
180C22×Dic13C22×C413yes—
181#6131—24no—
182Q8×D13—24no—
183#6129—212no—
184D4×D13—212no—
185#6127—26no—
186#6127—26no—
187C2×D52C22×C413yes—
188C2×C4×D13—26no—
189C2×C4×D13—26no—
190C2×C4×D13C22×C413yes—
191C2×C4×D13C22×C413yes—
192C2×C4×D13C22×C416yes—
193C2×C4×D13C22×C416yes—
194C2×C4×D13C2413yes—
195C2×C4×D13C2411yescharacteristic
196C2×Dic26C22×C413yes—
197C13×M16—26no—
198C13×M16—26no—
199C2×C8×C13—26no—
200C2×C8×C13—26no—
201C2×C8×C13C2411yescharacteristic
202C2×C8×C13C2411yescharacteristic
203C2×C8×C13C2413yes—
204C2×C8×C13C2413yes—
205C13×C4⋊C4C22×C413yes—
206C13×C4⋊C4C22×C413yes—
207C13×C22⋊C4C22×C416yes—
208C42×C13C22×C413yes—
209C42×C13C22×C413yes—
210C42×C13C2413yes—
211C42×C13C2411yescharacteristic
212C22⋊Dic13C22×C416yes—
213#6103C22×C411yescharacteristic
214#6103C22×C411yescharacteristic
215#6103C22×C413yes—
216#6103C22×C413yes—
217#6103C22×C413yes—
218#6103C22×C413yes—
219#6103C22×C413yes—
220#6103C22×C413yes—
221C4⋊Dic13C22×C413yes—
222C4⋊Dic13C22×C413yes—
223#6101C22×C413yes—
224#6101C22×C413yes—
225#6101C22×C411yescharacteristic
226#6101C22×C411yescharacteristic
227#6101C22×C413yes—
228#6101C22×C413yes—
229#6101C22×C413yes—
230#6101C22×C413yes—
231C4×Dic13C22×C416yes—
232C4×Dic13C22×C416yes—
233C4×Dic13C22×C413yes—
234C4×Dic13C22×C413yes—
235C4×Dic13C2413yes—
236C4×Dic13C2411yescharacteristic
237#6099—26no—
238#6099—26no—
239C2×#1055—26no—
240C2×#1055—26no—
241C2×#1055C2411yescharacteristic
242C2×#1055C2411yescharacteristic
243C2×#1055C2413yes—
244C2×#1055C2413yes—
245#3428—131no—
246#2925—133no—
247#2925—133no—
248#2924—133no—
249#2924—133no—
250#2923—133no—
251#2922—133no—
252#2921—133no—
253C2×#433—133no—
254#2906—131no—
255#2906D1311yescharacteristic, complemented
256#2880—131no—
257#2880—131no—
258#2875—131no—
259C4×#131—131no—
260C23×C13C23×C413yes—
261C22×D13—26no—
262C22×D13—43no—
263C22×D13—43no—
264C22×D13C23×C411yescharacteristic
265C22×D13C23×C413yes—
266Q8×C13#11714yes—
267D4×C13#117112yes—
268C2×C4×C13—43no—
269C2×C4×C13—43no—
270C2×C4×C13C23×C413yes—
271C2×C4×C13C23×C413yes—
272C2×C4×C13C23×C411yescharacteristic
273C2×C4×C13C23×C411yescharacteristic
274C2×C4×C13C23×C413yes—
275C2×C4×C13C2×#4116yes—
276C2×C4×C13C2×#4116yes—
277C2×C4×C13C2511yescharacteristic
278C22⋊D13—212no—
279C2×Dic13—26no—
280C2×Dic13—43no—
281C2×Dic13—43no—
282C2×Dic13C23×C413yes—
283C2×Dic13C23×C411yescharacteristic
284C2×Dic13C23×C411yescharacteristic
285C2×Dic13C23×C413yes—
286C2×Dic13C23×C413yes—
287C2×Dic13C23×C411yescharacteristic
288D52—26no—
289C4×D13—26no—
290C4×D13—26no—
291C4×D13—21no—
292C4×D13—21no—
293C4×D13—23no—
294C4×D13—23no—
295C4×D13—46no—
296C4×D13—46no—
297Dic26—26no—
298C8×C13—21no—
299C8×C13—21no—
300C8×C13—23no—
301C8×C13—23no—
302#1055—21no—
303#1055—21no—
304#1055—23no—
305#1055—23no—
306C22×#41—131no—
307C2×#131—131no—
308C2×#131—131no—
309C2×#131C2×D1311yescharacteristic
310C2×#131C2×D1311yescharacteristic
311#520—131no—
312#520—131no—
313#520—133no—
314#520—133no—
315#519—131no—
316#519—131no—
317#518—133no—
318#518—133no—
319C4×#41—131no—
320C4×#41—133no—
321C4×#41—133no—
322C4×#41C2×D1311yescharacteristic
323C2×#117—133no—
324C2×#117—133no—
325#433—131no—
326#433—133no—
327#433—133no—
328#433—133no—
329#433—136no—
330#433—133no—
331#433—133no—
332#433—133no—
333#433—133no—
334#433—136no—
335#433—136no—
336#432—136no—
337#432—136no—
338#431—136no—
339#431—136no—
340#424—133no—
341#424—133no—
342#424C2×D1313yescomplemented
343#424C2×D1313yescomplemented
344#423—133no—
345#423C2×D1313yescomplemented
346C2×C4⋊C8—133no—
347C2×C4⋊C8C2×D1313yescomplemented
348#408—133no—
349#408—133no—
350#408—133no—
351#408—133no—
352#407—133no—
353#407—133no—
354C2×C22⋊C8—133no—
355C2×C22⋊C8—133no—
356#405—133no—
357#405—133no—
358#405—133no—
359#405—133no—
360C4×M16—133no—
361C4×M16—133no—
362C2×#97—133no—
363C2×C4×C8—133no—
364C22×C13—23no—
365C22×C13—23no—
366C22×C13C2×#11716yes—
367C22×C13C24×C411yescharacteristic
368C2×D13—21no—
369C2×D13—23no—
370C2×D13—23no—
371C2×D13—23no—
372C2×D13—21no—
373C2×D13—21no—
374C2×D13—86no—
375C4×C13—23no—
376C4×C13—23no—
377C4×C13—21no—
378C4×C13—21no—
379C4×C13C2×#11716yes—
380C4×C13C22×#4111yescharacteristic
381C4×C13C22×#4111yescharacteristic
382Dic13—23no—
383Dic13—21no—
384Dic13—23no—
385Dic13—23no—
386Dic13—21no—
387Dic13—21no—
388C2×#41—131no—
389C2×#41—133no—
390C2×#41—133no—
391C2×#41—264no—
392C2×#41—264no—
393C2×#41C22×D1311yescharacteristic
394C22×Q8—131no—
395C22×D4—133no—
396C23×C4—133no—
397#131—24no—
398#131—24no—
399#131—264no—
400#131—264no—
401C2×M16—133no—
402C2×M16—133no—
403C2×M16C22×D1313yes—
404C2×M16C22×D1313yes—
405C22×C8—133no—
406C22×C8—133no—
407C22×C8C22×D1313yes—
408C22×C8C22×D1313yes—
409C4×Q8—131no—
410C4×Q8—131no—
411C4×Q8—133no—
412C4×Q8—133no—
413C4×Q8—133no—
414C4×Q8—133no—
415C4×Q8C4×D1311yescharacteristic
416C4×Q8C4×D1311yescharacteristic
417C4×D4—133no—
418C4×D4—133no—
419C4×D4—136no—
420C4×D4—136no—
421C4×D4—133no—
422C4×D4—133no—
423C4×D4—136no—
424C4×D4—136no—
425C4×D4—133no—
426C4×D4—133no—
427C4×D4C4×D1313yes—
428C4×D4C4×D1313yes—
429#117—133no—
430#117—133no—
431#117—133no—
432#117—136no—
433#117—133no—
434#117—133no—
435#117—133no—
436#117—133no—
437#117—133no—
438#117—131no—
439#117—133no—
440#117—133no—
441#117C22×D1313yes—
442C2×C4⋊C4—133no—
443C2×C4⋊C4—133no—
444C2×C4⋊C4—133no—
445C2×C4⋊C4—133no—
446C2×C22⋊C4—133no—
447C2×C22⋊C4—133no—
448C2×C22⋊C4—136no—
449C2×C42—133no—
450C2×C42—136no—
451C2×C42C22×D1313yes—
452C4⋊C8—131no—
453C4⋊C8—133no—
454C4⋊C8—133no—
455C4⋊C8—133no—
456C4⋊C8—136no—
457C4⋊C8C22×D1311yescharacteristic
458C4⋊C8C22×D1313yes—
459C4⋊C8C22×D1313yes—
460C4⋊C8C22×D1313yes—
461C4⋊C8C22×D1316yes—
462C22⋊C8—131no—
463C22⋊C8—131no—
464C22⋊C8—133no—
465C22⋊C8—133no—
466C22⋊C8—133no—
467C22⋊C8—133no—
468C22⋊C8—133no—
469C22⋊C8—133no—
470C22⋊C8—136no—
471C22⋊C8—136no—
472#97—131no—
473#97—133no—
474#97—133no—
475#97—133no—
476#97—136no—
477C4×C8—131no—
478C4×C8—133no—
479C4×C8—133no—
480C4×C8—133no—
481C4×C8—136no—
482C2×C13—26no—
483C2×C13C22×#11711yescharacteristic
484C2×C13#357311yescharacteristic
485C2×C13#357311yescharacteristic
486D13—43no—
487D13—41no—
488C24—133no—
489#41—264no—
490#41—2612no—
491#41—266no—
492#41—266no—
493#41#612914yes—
494#41#613114yes—
495C2×Q8—133no—
496C2×Q8—264no—
497C2×Q8C2×C4×D1311yescharacteristic
498C2×D4—136no—
499C2×D4—133no—
500C2×D4—2612no—
501C2×D4C2×C4×D1313yes—
502C22×C4—133no—
503C22×C4—133no—
504C22×C4—133no—
505C22×C4—133no—
506C22×C4—133no—
507C22×C4—133no—
508C22×C4—131no—
509C22×C4—136no—
510C22×C4—136no—
511C22×C4—266no—
512C22×C4—266no—
513C22×C4C2×C4×D1313yes—
514C22×C4C2×C4×D1313yes—
515C22×C4C23×D1313yes—
516M16—26no—
517M16—26no—
518M16—266no—
519M16—266no—
520C2×C8—26no—
521C2×C8—26no—
522C2×C8—131no—
523C2×C8—133no—
524C2×C8—133no—
525C2×C8—131no—
526C2×C8—266no—
527C2×C8—266no—
528C2×C8C23×D1311yescharacteristic
529C2×C8C23×D1313yes—
530C2×C8C23×D1313yes—
531C2×C8C23×D1311yescharacteristic
532C4⋊C4—133no—
533C4⋊C4—133no—
534C4⋊C4—131no—
535C4⋊C4—131no—
536C4⋊C4—133no—
537C4⋊C4—133no—
538C4⋊C4—133no—
539C4⋊C4—133no—
540C4⋊C4—133no—
541C4⋊C4—133no—
542C4⋊C4C2×C4×D1313yes—
543C4⋊C4C2×C4×D1313yes—
544C22⋊C4—136no—
545C22⋊C4—133no—
546C22⋊C4—133no—
547C22⋊C4—131no—
548C22⋊C4—131no—
549C22⋊C4—133no—
550C22⋊C4—133no—
551C22⋊C4—133no—
552C22⋊C4—133no—
553C22⋊C4C2×C4×D1316yes—
554C42—133no—
555C42—133no—
556C42—136no—
557C42—136no—
558C42—131no—
559C42—133no—
560C42C2×C4×D1313yes—
561C42C2×C4×D1313yes—
562C42C23×D1313yes—
563C42C23×D1311yescharacteristic
564C13#6012911yescharacteristic, complemented, Sylow
565C23—131no—
566C23—133no—
567C23—266no—
568C23—523no—
569C23—523no—
570C23C22×C4×D1313yes—
571Q8—266no—
572Q8#8814614yes—
573D4—2612no—
574D4—266no—
575D4#88146112yes—
576C2×C4—43no—
577C2×C4—43no—
578C2×C4—133no—
579C2×C4—131no—
580C2×C4—133no—
581C2×C4—131no—
582C2×C4—133no—
583C2×C4—131no—
584C2×C4—266no—
585C2×C4—266no—
586C2×C4—266no—
587C2×C4—261no—
588C2×C4—263no—
589C2×C4—261no—
590C2×C4—263no—
591C2×C4—526no—
592C2×C4—526no—
593C2×C4—523no—
594C2×C4—523no—
595C2×C4C22×C4×D1313yes—
596C2×C4C22×C4×D1313yes—
597C2×C4C22×C4×D1311yescharacteristic
598C2×C4C22×C4×D1311yescharacteristic
599C2×C4C22×C4×D1313yes—
600C2×C4C2×#612916yes—
601C2×C4C2×#613116yes—
602C2×C4C24×D1311yescharacteristic
603C8—21no—
604C8—21no—
605C8—23no—
606C8—23no—
607C8—261no—
608C8—261no—
609C8—263no—
610C8—263no—
611C22—23no—
612C22—23no—
613C22—261no—
614C22—263no—
615C22—263no—
616C22—261no—
617C22—263no—
618C22—261no—
619C22—1046no—
620C22C2×#8814616yes—
621C22C23×C4×D1311yescharacteristic
622C4—21no—
623C4—23no—
624C4—21no—
625C4—23no—
626C4—261no—
627C4—263no—
628C4—263no—
629C4—261no—
630C4—263no—
631C4—261no—
632C4C2×#8814616yes—
633C4C22×#612911yescharacteristic
634C4C22×#613111yescharacteristic
635C2—26no—
636C2—521no—
637C2—523no—
638C2C22×#8814611yescharacteristic
639C2#1312954711yescharacteristic
640C2#1312954911yescharacteristic
subgroupG/H
center, FrattiniC2×C4C24×D13
commutator, socleC22×C13C24×C4
FittingC13×#2906C2
prime radicalC2×#41×D13C22
cosocleC2×C4×C13C25
hypercenter#2906D13
hypocenterC13#60129
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C2×C4 ◁ #2906

Complements

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

NH ≅ G/Ntype
2. #13123504C2semidirect
3. #13123501C2semidirect
4. #13123493C2semidirect
5. #13123493C2semidirect
6. #13123486C2semidirect
7. #13123479C2semidirect
8. #13123473C2semidirect
9. C2×#11718501C2semidirect
10. #13123404C2semidirect
13. #13123262C2semidirect
14. #131×Dic13C2semidirect
15. C13×#2906C2semidirect
35. C13×#424C22semidirect
36. C13×#424C22semidirect
37. C13×#423C22semidirect
38. C2×C13×C4⋊C8C22semidirect
43. #11718783C22semidirect
44. #11718783C22semidirect
45. M16×Dic13C22semidirect
46. M16×Dic13C22semidirect
51. #11718762C22semidirect
52. #11718762C22semidirect
53. C2×#88055C22semidirect
54. C2×C8×Dic13C22semidirect
55. #11718660C22semidirect
56. #11718660C22semidirect
57. #11718585C22semidirect
58. C2×C4⋊#1055C22semidirect
64. #11718501C22semidirect
65. #11718501C22semidirect
66. #11718501C22semidirect
67. #11718501C22semidirect
68. #11718501C22semidirect
69. #11718501C22semidirect
70. #11718496C22semidirect
71. #11718496C22semidirect
72. #11718494C22semidirect
73. #11718494C22semidirect
255. #2906D13semidirect
342. #424C2×D13semidirect
343. #424C2×D13semidirect
345. #423C2×D13semidirect
347. C2×C4⋊C8C2×D13semidirect
564. C13#60129semidirect

Maximal subgroups

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

subgroupindex
#131280352
#131235042
#131235012
#131234932
#131234862
#131234792
#131234732
C2×#117185012
#131234042
#131232832
#131232622
#131×Dic132
C13×#29062
#6012913