Finite Group Databasebeta

← #16807984

Subgroups — #16807984

Order 3328.

2216 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13131382C211yesmaximal, characteristic, complemented
2#13131379C212yesmaximal, complemented
3#13131368C214yesmaximal, complemented
4#13131363C214yesmaximal, complemented
5#13131359C214yesmaximal, complemented
6#13131335C214yesmaximal, complemented
7#13131325C214yesmaximal, complemented
8#13131323C214yesmaximal, complemented
9#125×#261C211yesmaximal, characteristic, complemented
10#124×#261C212yesmaximal, complemented
11D13×#562C211yesmaximal, characteristic
12C2×Q8×#261C2212yes—
13C2×D4×#261C2212yes—
14C2×D4×#261C2214yes—
15C2×#88237C2214yes—
16C13×#562C411yescharacteristic, complemented
17#11719405C411yescharacteristic, complemented
18D13×#125C2211yescharacteristic
19#11719389C412yescomplemented
20D13×#124C2212yes—
21#11719372C414yescomplemented
22D13×#123C2214yes—
23#11719352C414yescomplemented
24D13×C22⋊Q8C2214yes—
25#11719326C414yescomplemented
26D13×#121C2214yes—
27(C2×#261)⋊Q8C2212yes—
28(C2×#261)⋊Q8C2214yes—
29#11719151C2212yes—
30#11719151C2214yes—
31#11719150C2214yes—
32#11719150C2214yes—
33#11719150C2214yes—
34#11719136C2214yes—
35C4×C22⋊#261C2214yes—
36C2×(C2×#261)⋊C4C2214yes—
37#11719083C2212yes—
38#11719083C2214yes—
39#11719083C2214yes—
40#11719082C2214yes—
41#11719082C2214yes—
42#11719082C2214yes—
43#11719082C2214yes—
44#11719082C2214yes—
45C4⋊C4×#261C2212yes—
46C4⋊C4×#261C2214yes—
47C4⋊C4×#261C2214yes—
48#11719071C2214yes—
49#11719071C2214yes—
50#11719071C2214yes—
51#11719071C2214yes—
52#11719071C2214yes—
53#11719071C2214yes—
54#11719070C2214yes—
55#11719070C2214yes—
56#11719070C2214yes—
57C22⋊C4×#261C2214yes—
58C22⋊C4×#261C2214yes—
59C22⋊C4×#261C2214yes—
60#11719060C2212yes—
61#11719060C2211yescharacteristic
62C42×#261C2211yescharacteristic
63C23×#261C2314yes—
64C23×#261C2314yes—
65C2×Q8×D13C2312yes—
66C2×D4×D13C2314yes—
67C2×D4×D13C2312yes—
68C22×C4×D13C2314yes—
69C2×C22⋊#261C2314yes—
70C2×C22⋊#261C2314yes—
71C2×C22⋊#261C2314yes—
72C2×C22⋊#261C2314yes—
73C2×C22⋊#261C2314yes—
74C2×C22⋊#261C2314yes—
75Q8×#261—44no—
76D4×#261—44no—
77D4×#261—48no—
78#88237—48no—
79C2×C4⋊#261C2314yes—
80C2×C4⋊#261C2312yes—
81C2×C4⋊#261C2312yes—
82C2×C4⋊#261C2312yes—
83C2×C4⋊#261C2314yes—
84C2×C4⋊#261C2314yes—
85C2×C4⋊#261C2314yes—
86C2×C4×#261C2311yescharacteristic
87C2×C4×#261C2312yes—
88C2×C4×#261C2314yes—
89C2×C4×#261C2311yescharacteristic
90C2×C4×#261C2312yes—
91C2×C4×#261C2314yes—
92C2×C4×#261C2314yes—
93C2×C4×#261C2314yes—
94C13×#125C2×C411yescharacteristic, complemented
95C13×#124C2×C412yescomplemented
96C13×#123C2×C414yescomplemented
97C13×C22⋊Q8C2×C414yescomplemented
98C13×#121C2×C414yescomplemented
99#88201C2×C412yescomplemented
100#88200C2×C414yescomplemented
101#88193C2×C414yescomplemented
102#88193C2×C414yescomplemented
103#88192C2×C414yescomplemented
104#88190C2×C412yescomplemented
105#88189C2×C414yescomplemented
106#88184C2×C414yescomplemented
107#88183C2×C414yescomplemented
108#88178C2×C414yescomplemented
109#88151C2×C414yescomplemented
110#88150C2×C414yescomplemented
111#88150C2×C414yescomplemented
112#88149C2×C414yescomplemented
113#88148C2×C414yescomplemented
114#88148C2×C414yescomplemented
115C4⋊C4×D13C2314yes—
116C4⋊C4×D13C2314yes—
117C4⋊C4×D13C2312yes—
118#88144C2×C412yescomplemented
119#88143C2×C414yescomplemented
120#88140C2×C414yescomplemented
121#88139C2×C414yescomplemented
122#88139C2×C414yescomplemented
123#88138C2×C414yescomplemented
124#88138C2×C414yescomplemented
125#88137C2×C414yescomplemented
126#88137C2×C414yescomplemented
127C22⋊C4×D13C2314yes—
128C22⋊C4×D13C2314yes—
129C22⋊C4×D13C2314yes—
130#88132C2×C414yescomplemented
131#88132C2×C414yescomplemented
132#88129C2×C412yescomplemented
133C42×D13C2311yescharacteristic
134#88124C2×C411yescharacteristic, complemented
135(C2×#261)⋊C4C2314yes—
136(C2×#261)⋊C4C2314yes—
137(C2×#261)⋊C4C2314yes—
138(C2×#261)⋊C4C2314yes—
139(C2×#261)⋊C4C2314yes—
140(C2×#261)⋊C4C2314yes—
141(C2×#261)⋊C4C2314yes—
142(C2×#261)⋊C4C2314yes—
143(C2×#261)⋊C4C2314yes—
144(C2×#261)⋊C4C2314yes—
145(C2×#261)⋊C4C2314yes—
146#34522—131nomaximal, Sylow
147C23×D13C2414yes—
148C22×#261—28no—
149C22×#261—44no—
150C22×#261—44no—
151C22×#261—44no—
152C22×#261—44no—
153C22×#261—44no—
154C22×#261C2414yes—
155C22×#261C2414yes—
156C22×#261C2414yes—
157C22×#261C2411yescharacteristic
158C22×#261C2412yes—
159C22×#261C2411yescharacteristic
160C2×Q8×C13C22×C412yes—
161C2×D4×C13C22×C414yes—
162C2×D4×C13C22×C412yes—
163C22×C4×C13C22×C414yes—
164C2×C22⋊D13C22×C414yes—
165C2×C22⋊D13C22×C414yes—
166C2×C22⋊D13C22×C414yes—
167C22×Dic13C22×C414yes—
168Q8×D13—42no—
169Q8×D13—42no—
170D4×D13—42no—
171D4×D13—44no—
172D4×D13—44no—
173D4×D13—42no—
174C2×D52C22×C412yes—
175C2×D52C22×C414yes—
176C2×C4×D13—44no—
177C2×C4×D13—44no—
178C2×C4×D13C22×C414yes—
179C2×C4×D13C22×C414yes—
180C2×C4×D13C2411yescharacteristic
181C2×C4×D13C2412yes—
182C2×C4×D13C2414yes—
183C2×C4×D13C2414yes—
184C2×Dic26C22×C412yes—
185C2×Dic26C22×C414yes—
186C22⋊#261—28no—
187C22⋊#261—28no—
188C22⋊#261—44no—
189C22⋊#261—44no—
190C22⋊#261—44no—
191C22⋊#261—44no—
192C4⋊#261—28no—
193C4⋊#261—42no—
194C4⋊#261—44no—
195C4⋊#261—44no—
196C4⋊#261—42no—
197C4⋊#261—44no—
198C4⋊#261—42no—
199C4×#261—28no—
200C4×#261—24no—
201C4×#261—22no—
202C4×#261—41no—
203C4×#261—44no—
204C4×#261—42no—
205C4×#261—44no—
206C4×#261—44no—
207C13×C4⋊C4C22×C414yes—
208C13×C4⋊C4C22×C412yes—
209C13×C4⋊C4C22×C414yes—
210C13×C22⋊C4C22×C414yes—
211C13×C22⋊C4C22×C414yes—
212C13×C22⋊C4C22×C414yes—
213C42×C13C22×C411yescharacteristic
214C22⋊Dic13C22×C414yes—
215C22⋊Dic13C22×C414yes—
216C22⋊Dic13C22×C414yes—
217#6103C22×C414yes—
218#6103C22×C414yes—
219#6103C22×C414yes—
220#6103C22×C414yes—
221#6103C22×C414yes—
222#6103C22×C414yes—
223C4⋊Dic13C22×C412yes—
224C4⋊Dic13C22×C414yes—
225C4⋊Dic13C22×C414yes—
226#6101C22×C414yes—
227#6101C22×C414yes—
228#6101C22×C414yes—
229#6101C22×C414yes—
230#6101C22×C414yes—
231C4×Dic13C22×C411yescharacteristic
232C4×Dic13C22×C412yes—
233C2×#562—131no—
234#2382—131no—
235#2379—132no—
236#2368—134no—
237#2363—134no—
238#2359—134no—
239#2335—134no—
240#2325—134no—
241#2323—134no—
242C4×#125—131no—
243C4×#124—132no—
244C23×C13C23×C414yes—
245C22×D13—24no—
246C22×D13—24no—
247C22×D13—24no—
248C22×D13—24no—
249C22×D13—24no—
250C22×D13—24no—
251C22×D13C23×C414yes—
252C22×D13C2511yescharacteristic
253C2×#261—22no—
254C2×#261—22no—
255C2×#261—22no—
256C2×#261—24no—
257C2×#261—22no—
258C2×#261—24no—
259C2×#261—24no—
260C2×#261—24no—
261C2×#261—21no—
262C2×#261—24no—
263C2×#261—24no—
264C2×#261—24no—
265C2×#261—24no—
266C2×#261—24no—
267C2×#261—88no—
268C2×#261—88no—
269C2×#261H(5,𝔽2)14yes—
270C2×#2612−1+412yes—
271Q8×C13—42no—
272D4×C13—42no—
273D4×C13—44no—
274C2×C4×C13—44no—
275C2×C4×C13C23×C412yes—
276C2×C4×C13C23×C411yescharacteristic
277C2×C4×C13C23×C414yes—
278C2×C4×C13C23×C414yes—
279C22⋊D13—44no—
280C22⋊D13—44no—
281C22⋊D13—44no—
282C2×Dic13—44no—
283C2×Dic13C23×C411yescharacteristic
284C2×Dic13C23×C412yes—
285C2×Dic13C23×C414yes—
286C2×Dic13C23×C414yes—
287D52—42no—
288D52—44no—
289C4×D13—24no—
290C4×D13—22no—
291C4×D13—24no—
292C4×D13—21no—
293C4×D13—21no—
294C4×D13—24no—
295C4×D13—22no—
296C4×D13—24no—
297C4×D13—44no—
298C4×D13—44no—
299Dic26—42no—
300Dic26—44no—
301#562—131no—
302#562—132no—
303#562—134no—
304#562—134no—
305#562—134no—
306#562#26111yescharacteristic, complemented
307C2×#125—131no—
308C2×#124—132no—
309C2×#123—134no—
310C2×C22⋊Q8—134no—
311C2×#121—134no—
312C2×C4×Q8—132no—
313C2×C4×D4—132no—
314C2×C4×D4—134no—
315C2×#117—134no—
316#391—134no—
317#391—132no—
318#390—134no—
319#390—132no—
320#389—132no—
321#389—134no—
322#389—134no—
323#388—134no—
324#388—134no—
325#388—134no—
326#388—134no—
327#388—134no—
328#388—134no—
329#387—134no—
330#387—134no—
331#387—134no—
332#387—134no—
333#387—134no—
334#386—134no—
335#386—134no—
336#386—134no—
337#385—134no—
338#385—134no—
339#385—134no—
340#382—131no—
341#382—132no—
342#380—134no—
343C4×C4⋊C4—132no—
344C4×C4⋊C4—134no—
345C4×C4⋊C4—134no—
346C4×C22⋊C4—134no—
347C4×C22⋊C4—134no—
348C4×C22⋊C4—134no—
349C4×C22⋊C4—134no—
350C2×#95—134no—
351C43—131no—
352C22×C13—24no—
353C22×C13—24no—
354C22×C13—24no—
355C22×C13C24×C411yescharacteristic
356C2×D13—24no—
357C2×D13—24no—
358C2×D13—24no—
359C2×D13—44no—
360C2×D13—44no—
361C2×D13—44no—
362C2×D13—44no—
363C2×D13C2×H(5,𝔽2)12yes—
364C2×D13C2×2−1+411yescharacteristic
365C2×D13#58511yescharacteristic
366C2×D13#58512yes—
367#261—24no—
368#261—22no—
369#261—44no—
370#261—44no—
371#261—44no—
372#261—41no—
373C4×C13—24no—
374C4×C13—22no—
375C4×C13—24no—
376C4×C13—21no—
377Dic13—24no—
378Dic13—22no—
379Dic13—24no—
380Dic13—21no—
381C22×Q8—132no—
382C22×D4—134no—
383C22×D4—132no—
384C23×C4—134no—
385C23×C4—134no—
386C23×C4—134no—
387#125—134no—
388#125—132no—
389#125—131no—
390#125C2×#26111yescharacteristic, complemented
391#124—132no—
392#124—132no—
393#124—134no—
394#124—134no—
395#124—132no—
396#124C2×#26112yescomplemented
397#123—134no—
398#123—134no—
399#123—134no—
400#123—134no—
401#123—134no—
402#123—134no—
403#123—134no—
404#123C2×#26114yescomplemented
405C22⋊Q8—134no—
406C22⋊Q8—134no—
407C22⋊Q8—134no—
408C22⋊Q8—134no—
409C22⋊Q8—134no—
410C22⋊Q8—134no—
411C22⋊Q8—134no—
412C22⋊Q8C2×#26114yescomplemented
413#121—134no—
414#121—134no—
415#121—134no—
416#121—134no—
417#121—134no—
418#121—134no—
419#121—134no—
420#121C2×#26114yescomplemented
421C4×Q8—524no—
422C4×D4—524no—
423C4×D4—528no—
424#117—528no—
425C2×C4⋊C4—134no—
426C2×C4⋊C4—134no—
427C2×C4⋊C4—132no—
428C2×C4⋊C4—134no—
429C2×C4⋊C4—134no—
430C2×C4⋊C4—132no—
431C2×C4⋊C4—134no—
432C2×C4⋊C4—132no—
433C2×C4⋊C4—134no—
434C2×C4⋊C4—132no—
435C2×C22⋊C4—134no—
436C2×C22⋊C4—134no—
437C2×C22⋊C4—134no—
438C2×C22⋊C4—134no—
439C2×C22⋊C4—134no—
440C2×C22⋊C4—134no—
441C2×C22⋊C4—134no—
442C2×C22⋊C4—134no—
443C2×C22⋊C4—134no—
444C2×C42—131no—
445C2×C42—134no—
446C2×C42—132no—
447C2×C42—131no—
448C2×C42—134no—
449C2×C42—134no—
450C2×C42—131no—
451C2×C42—132no—
452C2×C42—134no—
453#95—134no—
454#95—134no—
455#95—134no—
456#95—134no—
457#95—134no—
458#95—134no—
459#95—134no—
460#95—134no—
461#95—134no—
462#95—134no—
463#95—134no—
464C2×C13—44no—
465C2×C13C4×H(5,𝔽2)12yes—
466C2×C13C4×2−1+411yescharacteristic
467D13—44no—
468D13#354512yes—
469D13#354811yescharacteristic
470D13#355311yescharacteristic
471C24—134no—
472C2×Q8—134no—
473C2×Q8—132no—
474C2×Q8—522no—
475C2×Q8—522no—
476C2×Q8C22×#26112yes—
477C2×D4—134no—
478C2×D4—134no—
479C2×D4—134no—
480C2×D4—134no—
481C2×D4—132no—
482C2×D4—522no—
483C2×D4—524no—
484C2×D4—524no—
485C2×D4—522no—
486C2×D4C22×#26114yes—
487C2×D4C22×#26112yes—
488C22×C4—131no—
489C22×C4—132no—
490C22×C4—134no—
491C22×C4—131no—
492C22×C4—132no—
493C22×C4—134no—
494C22×C4—134no—
495C22×C4—134no—
496C22×C4—134no—
497C22×C4—134no—
498C22×C4—134no—
499C22×C4—131no—
500C22×C4—134no—
501C22×C4—268no—
502C22×C4—524no—
503C22×C4—524no—
504C22×C4—524no—
505C22×C4—524no—
506C22×C4—524no—
507C22×C4—524no—
508C22×C4—524no—
509C22×C4C22×#26114yes—
510C4⋊C4—134no—
511C4⋊C4—134no—
512C4⋊C4—134no—
513C4⋊C4—134no—
514C4⋊C4—134no—
515C4⋊C4—134no—
516C4⋊C4—134no—
517C4⋊C4—132no—
518C4⋊C4—268no—
519C4⋊C4—524no—
520C4⋊C4—522no—
521C4⋊C4—524no—
522C4⋊C4—522no—
523C4⋊C4—522no—
524C4⋊C4—524no—
525C4⋊C4C22×#26114yes—
526C4⋊C4C22×#26114yes—
527C4⋊C4C22×#26112yes—
528C22⋊C4—134no—
529C22⋊C4—134no—
530C22⋊C4—134no—
531C22⋊C4—134no—
532C22⋊C4—134no—
533C22⋊C4—134no—
534C22⋊C4—134no—
535C22⋊C4—134no—
536C22⋊C4—134no—
537C22⋊C4—268no—
538C22⋊C4—268no—
539C22⋊C4—524no—
540C22⋊C4—524no—
541C22⋊C4—524no—
542C22⋊C4—524no—
543C22⋊C4C22×#26114yes—
544C22⋊C4C22×#26114yes—
545C22⋊C4C22×#26114yes—
546C42—132no—
547C42—131no—
548C42—268no—
549C42—264no—
550C42—262no—
551C42—524no—
552C42—524no—
553C42—521no—
554C42—524no—
555C42—522no—
556C42C22×#26111yescharacteristic
557C13#3452211yescharacteristic, complemented, Sylow
558C23—131no—
559C23—134no—
560C23—264no—
561C23—264no—
562C23—264no—
563C23—264no—
564C23—264no—
565C23—264no—
566C23C23×#26114yes—
567Q8—42no—
568Q8—524no—
569Q8—522no—
570D4—44no—
571D4—42no—
572D4—524no—
573D4—522no—
574D4—524no—
575D4—524no—
576D4—524no—
577C2×C4—44no—
578C2×C4—134no—
579C2×C4—132no—
580C2×C4—131no—
581C2×C4—132no—
582C2×C4—134no—
583C2×C4—134no—
584C2×C4—264no—
585C2×C4—262no—
586C2×C4—261no—
587C2×C4—264no—
588C2×C4—261no—
589C2×C4—264no—
590C2×C4—262no—
591C2×C4—264no—
592C2×C4—262no—
593C2×C4—262no—
594C2×C4—262no—
595C2×C4—262no—
596C2×C4—264no—
597C2×C4—261no—
598C2×C4—264no—
599C2×C4—264no—
600C2×C4—264no—
601C2×C4—264no—
602C2×C4—264no—
603C2×C4—264no—
604C2×C4—264no—
605C2×C4—264no—
606C2×C4—524no—
607C2×C4—524no—
608C2×C4—524no—
609C2×C4—1048no—
610C2×C4—1048no—
611C2×C4C23×#26111yescharacteristic
612C2×C4C23×#26112yes—
613C2×C4C23×#26114yes—
614C2×C4C23×#26114yes—
615C22—24no—
616C22—24no—
617C22—24no—
618C22—132no—
619C22—132no—
620C22—131no—
621C22—131no—
622C22—264no—
623C22—264no—
624C22—264no—
625C22—524no—
626C22—524no—
627C22—524no—
628C22—524no—
629C22C24×#26111yescharacteristic
630C4—24no—
631C4—24no—
632C4—22no—
633C4—21no—
634C4—262no—
635C4—264no—
636C4—264no—
637C4—264no—
638C4—262no—
639C4—261no—
640C4—521no—
641C4—524no—
642C4—524no—
643C4—524no—
644C2—44no—
645C2—132no—
646C2—131no—
647C2—131no—
648C2—524no—
649C2H(5,𝔽2)×#26112yes—
650C22−1+4×#26111yescharacteristic
subgroupG/H
center, FrattiniC22C24×#261
commutator, socleC22×C13C24×C4
FittingC13×#562C4
prime radicalD13×#562C2
cosocleC22×D13C25
hypercenter#562#261
hypocenterC13#34522
LCSG ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #13131382C2semidirect
2. #13131379C2semidirect
3. #13131368C2semidirect
4. #13131363C2semidirect
5. #13131359C2semidirect
6. #13131335C2semidirect
7. #13131325C2semidirect
8. #13131323C2semidirect
9. #125×#261C2semidirect
10. #124×#261C2semidirect
16. C13×#562C4semidirect
17. #11719405C4semidirect
19. #11719389C4semidirect
21. #11719372C4semidirect
23. #11719352C4semidirect
25. #11719326C4semidirect
94. C13×#125C2×C4semidirect
95. C13×#124C2×C4semidirect
96. C13×#123C2×C4semidirect
97. C13×C22⋊Q8C2×C4semidirect
98. C13×#121C2×C4semidirect
99. #88201C2×C4semidirect
100. #88200C2×C4semidirect
101. #88193C2×C4semidirect
102. #88193C2×C4semidirect
103. #88192C2×C4semidirect
104. #88190C2×C4semidirect
105. #88189C2×C4semidirect
106. #88184C2×C4semidirect
107. #88183C2×C4semidirect
108. #88178C2×C4semidirect
109. #88151C2×C4semidirect
110. #88150C2×C4semidirect
111. #88150C2×C4semidirect
112. #88149C2×C4semidirect
113. #88148C2×C4semidirect
114. #88148C2×C4semidirect
118. #88144C2×C4semidirect
119. #88143C2×C4semidirect
120. #88140C2×C4semidirect
121. #88139C2×C4semidirect
122. #88139C2×C4semidirect
123. #88138C2×C4semidirect
124. #88138C2×C4semidirect
125. #88137C2×C4semidirect
126. #88137C2×C4semidirect
130. #88132C2×C4semidirect
131. #88132C2×C4semidirect
132. #88129C2×C4semidirect
134. #88124C2×C4semidirect
306. #562#261semidirect
390. #125C2×#261semidirect
396. #124C2×#261semidirect
404. #123C2×#261semidirect
412. C22⋊Q8C2×#261semidirect
420. #121C2×#261semidirect
557. C13#34522semidirect

Maximal subgroups

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

subgroupindex
#131313822
#131313792
#131313682
#131313632
#131313592
#131313352
#131313252
#131313232
#125×#2612
#124×#2612
D13×#5622
#3452213