Finite Group Databasebeta

← #16752859

Subgroups — #16752859

Order 3328.

1770 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13128701C211yesmaximal, characteristic, complemented
2#13127711C214yesmaximal, complemented
3#13127699C214yesmaximal, complemented
4#13127496C214yesmaximal, complemented
5#13127485C214yesmaximal, complemented
6#13126004C214yesmaximal, complemented
7#13125987C211yesmaximal, characteristic, complemented
8#13125964C211yesmaximal, characteristic, complemented
9#13124883C214yesmaximal, complemented
10#13124844C211yesmaximal, characteristic, complemented
11#13124836C211yesmaximal, characteristic
12#13123917C211yesmaximal, characteristic, complemented
13C2×C13×#554C211yesmaximal, characteristic, complemented
14C13×#554—22no—
15C13×#554—22no—
16C13×#554C2214yes—
17C13×#554C2214yes—
18C2×C13×C4⋊Q8C2211yescharacteristic
19C2×C13×#125C2212yes—
20C2×C13×C22⋊Q8—21no—
21C2×C13×C22⋊Q8—22no—
22C2×C4×Q8×C13C2211yescharacteristic
23C2×C4×D4×C13C2212yes—
24C2×C4×D4×C13C2211yescharacteristic
25C22×C13×C4⋊C4—21no—
26#11719521—21no—
27#11719290C2212yescomplemented
28#11719289C2212yescomplemented
29#11719286—22no—
30#11719284C2212yescomplemented
31#11719273—22no—
32#11719272C2212yescomplemented
33#11719220—22no—
34#11719209—21no—
35#11719184C2214yes—
36#11719182C2211yescharacteristic
37#11719175C2211yescharacteristic
38#11718879C2212yes—
39#11718878C2212yes—
40#11718754C2212yescomplemented
41#11718752C2214yescomplemented
42#11718749C2212yescomplemented
43#11718745C2214yescomplemented
44#11718740C2212yescomplemented
45#11718739C2212yescomplemented
46#11718729C2218yescomplemented
47#11718727C2218yescomplemented
48#11718725C2218yescomplemented
49#11718724C2218yescomplemented
50Q8×#1055C2214yescomplemented
51Q8×#1055C2214yescomplemented
52#11718680C2212yes—
53C4×#6105C2218yes—
54C4×#6105C2214yes—
55#11718677C2218yes—
56#11718676C2212yes—
57C4×#6104C2218yes—
58C4×#6104C2214yes—
59#11718670C2218yes—
60#11718669C2212yes—
61#11718668C2212yes—
62#11718632C2212yes—
63#11718631C2214yes—
64#11718629C2214yes—
65#11718628C2212yes—
66#11718586C2214yes—
67#11718586C2211yescharacteristic
68#11718586C2211yescharacteristic
69C22×Q8×C13—21no—
70C22×Q8×C13D411yescharacteristic
71C22×D4×C13C2311yescharacteristic
72C23×C4×C13—21no—
73C23×C4×C13—22no—
74C2×#6127—21no—
75C13×C4⋊Q8—21no—
76C13×C4⋊Q8—21no—
77C13×C4⋊Q8C2312yes—
78C13×C4⋊Q8C2312yes—
79C13×#125—24no—
80C13×#125C2318yes—
81C13×C22⋊Q8—24no—
82C13×C22⋊Q8—24no—
83C13×C22⋊Q8—28no—
84C13×C22⋊Q8—28no—
85C4×Q8×C13D412yes—
86C4×Q8×C13D412yes—
87C4×Q8×C13C2312yes—
88C4×Q8×C13C2312yes—
89C4×D4×C13—21no—
90C4×D4×C13—21no—
91C4×D4×C13—24no—
92C4×D4×C13C2312yes—
93C4×D4×C13C2312yes—
94C4×D4×C13C2318yes—
95C2×C13×C4⋊C4—22no—
96C2×C13×C4⋊C4—21no—
97C2×C13×C4⋊C4—22no—
98C2×C13×C4⋊C4—24no—
99C2×C13×C4⋊C4—24no—
100C2×C13×C4⋊C4D411yescharacteristic
101C2×C13×C4⋊C4D412yes—
102C2×C13×C4⋊C4C2311yescharacteristic
103C2×C13×C4⋊C4C2312yes—
104C2×C13×C22⋊C4—21no—
105C2×C13×C22⋊C4—22no—
106C2×C42×C13C2311yescharacteristic
107C2×C42×C13C2312yes—
108#88201—22no—
109#88200—22no—
110#88200—22no—
111#88198—22no—
112C2×#6105C2314yes—
113C2×#6104C2314yes—
114#88184—21no—
115#88184C2312yes—
116#88183—24no—
117C4×C22⋊D13—24no—
118#88180—21no—
119#88178—21no—
120#88178C2312yes—
121#88152—24no—
122#88150—24no—
123#88148—24no—
124#88147—22no—
125#88146—22no—
126C4⋊C4×D13—22no—
127#88143—24no—
128#88141—22no—
129#88130—24no—
130#88129—21no—
131#88129C2314yes—
132#88128C2311yescharacteristic
133C4×D52—21no—
134C4×D52C2312yes—
135C4×D52C2314yes—
136#88126—24no—
137#88124—21no—
138#88124C2314yes—
139#88123C2311yescharacteristic
140#88123C2311yescharacteristic
141#88123C2311yescharacteristic
142C4×Dic26—21no—
143C4×Dic26C2312yes—
144C4×Dic26C2314yes—
145#88072C2314yes—
146#88072C2314yes—
147C22⋊#1055C2312yes—
148C22⋊#1055C2312yes—
149#88050C2318yes—
150#88050C2314yes—
151#88049C2314yes—
152#88049C2318yes—
153#88048C2314yes—
154#88048C2318yes—
155#88047C2314yes—
156#88047C2318yes—
157C4⋊#1055C2312yes—
158C4⋊#1055C2312yes—
159C4⋊#1055C2318yes—
160C4×#1055C2312yes—
161C4×#1055C2318yes—
162C4×#1055C2312yes—
163#61819—131nomaximal, Sylow
164C24×C13—21no—
165C2×Q8×C13—24no—
166C2×Q8×C13—22no—
167C2×Q8×C13—22no—
168C2×Q8×C13—82no—
169C2×Q8×C13C2×D412yes—
170C2×Q8×C13C2×D412yes—
171C2×D4×C13—24no—
172C2×D4×C13—21no—
173C2×D4×C13C2412yes—
174C22×C4×C13—21no—
175C22×C4×C13—22no—
176C22×C4×C13—22no—
177C22×C4×C13—22no—
178C22×C4×C13—28no—
179C22×C4×C13—44no—
180C22×C4×C13—84no—
181C22×C4×C13C2×D412yes—
182C22×C4×C13C2×D411yescharacteristic
183C22×C4×C13C2×D412yes—
184C22×C4×C13C2×D411yescharacteristic
185C22×C4×C13C2411yescharacteristic
186C2×C22⋊D13—22no—
187#6127—44no—
188C2×D52—21no—
189C2×D52—21no—
190C2×D52C2412yes—
191C2×C4×D13—22no—
192C2×C4×D13—24no—
193C2×Dic26—21no—
194C2×Dic26—21no—
195C2×Dic26C2412yes—
196C13×C4⋊C4—28no—
197C13×C4⋊C4—22no—
198C13×C4⋊C4—22no—
199C13×C4⋊C4—28no—
200C13×C4⋊C4—21no—
201C13×C4⋊C4—22no—
202C13×C4⋊C4C2×D412yes—
203C13×C4⋊C4C2×D414yes—
204C13×C4⋊C4C2×D412yes—
205C13×C4⋊C4C2×D414yes—
206C13×C4⋊C4C2412yes—
207C13×C4⋊C4C2414yes—
208C13×C22⋊C4—22no—
209C13×C22⋊C4—22no—
210C13×C22⋊C4—28no—
211C42×C13C2×D411yescharacteristic
212C42×C13C2×D411yescharacteristic
213C42×C13C2×D414yes—
214C42×C13C2411yescharacteristic
215C42×C13C2414yes—
216C42×C13C2411yescharacteristic
217C22⋊Dic13—22no—
218#6105—28no—
219#6104—28no—
220#6103—22no—
221#6103—24no—
222#6103—22no—
223#6103—24no—
224#6103—24no—
225C4⋊Dic13—21no—
226C4⋊Dic13—21no—
227C4⋊Dic13—21no—
228C4⋊Dic13—21no—
229C4⋊Dic13C2412yes—
230C4⋊Dic13C2412yes—
231#6101—22no—
232#6101—22no—
233#6101—24no—
234#6101—24no—
235#6101—24no—
236C4×Dic13—22no—
237C4×Dic13—24no—
238C2×#1055C2414yes—
239C2×#1055C2414yes—
240#3507—131no—
241C2×#554D1311yescharacteristic, complemented
242#3381—134no—
243#3380—134no—
244#3361—134no—
245#3360—134no—
246#3232—134no—
247#3230—131no—
248#3227—131no—
249#3105—134no—
250#3101—131no—
251#3100—131no—
252#2982—131no—
253C23×C13—24no—
254C23×C13—22no—
255C23×C13—42no—
256C23×C13—42no—
257C23×C13C22×D411yescharacteristic
258C22×D13—22no—
259Q8×C13—24no—
260Q8×C13—24no—
261Q8×C13—82no—
262Q8×C13—82no—
263D4×C13—24no—
264D4×C13—22no—
265D4×C132−1+414yes—
266C2×C4×C13—24no—
267C2×C4×C13—22no—
268C2×C4×C13—22no—
269C2×C4×C13—24no—
270C2×C4×C13—42no—
271C2×C4×C13—41no—
272C2×C4×C13—44no—
273C2×C4×C13—42no—
274C2×C4×C13—44no—
275C2×C4×C13—44no—
276C2×C4×C13—88no—
277C2×C4×C13#13514yes—
278C2×C4×C13C22×D414yes—
279C2×C4×C13C22×D411yescharacteristic
280C2×C4×C13C22×D411yescharacteristic
281C2×C4×C13C22×D414yes—
282C2×C4×C13C22×D411yescharacteristic
283C2×C4×C13C22×D411yescharacteristic
284C2×C4×C13C22×D411yescharacteristic
285C2×C4×C13C2511yescharacteristic
286C22⋊D13—44no—
287C2×Dic13—22no—
288C2×Dic13—22no—
289C2×Dic13—22no—
290D52—24no—
291D52—42no—
292D52—81no—
293C4×D13—44no—
294C4×D13—84no—
295Dic26—24no—
296Dic26—42no—
297Dic26—81no—
298#1055—24no—
299#1055—24no—
300#559—132no—
301#557—132no—
302#556—132no—
303#556—262no—
304#554—22no—
305#554—22no—
306#554—132no—
307#554C2×D1314yes—
308#554C2×D1314yes—
309#552—262no—
310#539—262no—
311#536—261no—
312#533—131no—
313#532—131no—
314C2×C4⋊Q8C2×D1311yescharacteristic
315#529—134no—
316C2×#125C2×D1312yes—
317C2×C22⋊Q8—21no—
318C2×C22⋊Q8—22no—
319#520—261no—
320C2×C4×Q8C2×D1311yescharacteristic
321C2×C4×D4C2×D1311yescharacteristic
322C2×C4×D4C2×D1312yes—
323C22×C4⋊C4—21no—
324#500—132no—
325#499—138no—
326#498—132no—
327#488—138no—
328#487—132no—
329#486—132no—
330#484—134no—
331#483—132no—
332#481—134no—
333#480—132no—
334#479—138no—
335#478—138no—
336#478—138no—
337#477—134no—
338#476—134no—
339#476—132no—
340#474—132no—
341#463—138no—
342#461—132no—
343#459—132no—
344#450—132no—
345#447—132no—
346C8×Q8—134no—
347C8×Q8—134no—
348C4×SD16—134no—
349C4×SD16—138no—
350C4×D8—134no—
351C4×D8—138no—
352#431—131no—
353#431—131no—
354#431—134no—
355C22×C13—21no—
356C22×C13—21no—
357C22×C13—24no—
358C22×C13—48no—
359C22×C13—42no—
360C22×C13—42no—
361C22×C13C2×#13512yes—
362C22×C13C23×D411yescharacteristic
363C2×D13—42no—
364C2×D13—42no—
365C2×D13—42no—
366C4×C13—21no—
367C4×C13—24no—
368C4×C13—21no—
369C4×C13—21no—
370C4×C13—24no—
371C4×C13—44no—
372C4×C13—42no—
373C4×C13C2×#13512yes—
374C4×C13C2×#13512yes—
375C4×C13C2×2−1+412yes—
376Dic13—42no—
377Dic13—42no—
378Dic13—42no—
379C2×#41—261no—
380C22×Q8—21no—
381C22×Q8C22⋊D1311yescharacteristic
382C22×D4C22×D1311yescharacteristic
383C23×C4—21no—
384C23×C4—22no—
385C2×SD16—134no—
386C2×D8—134no—
387C4⋊Q8—21no—
388C4⋊Q8—21no—
389C4⋊Q8—131no—
390C4⋊Q8—131no—
391C4⋊Q8—131no—
392C4⋊Q8—262no—
393C4⋊Q8C22×D1312yes—
394C4⋊Q8C22×D1312yes—
395#127—131no—
396#126—264no—
397#126—264no—
398#125—24no—
399#125—134no—
400#125—264no—
401#125—261no—
402#125C22×D1318yes—
403#124—134no—
404#124—262no—
405#124—261no—
406#123—264no—
407#123—264no—
408C22⋊Q8—24no—
409C22⋊Q8—24no—
410C22⋊Q8—28no—
411C22⋊Q8—28no—
412C22⋊Q8—132no—
413C22⋊Q8—262no—
414C22⋊Q8—262no—
415C22⋊Q8—261no—
416C22⋊Q8—264no—
417#121—132no—
418#121—261no—
419C4×Q8—134no—
420C4×Q8—132no—
421C4×Q8—262no—
422C4×Q8—261no—
423C4×Q8C22⋊D1312yes—
424C4×Q8C22⋊D1312yes—
425C4×Q8C22×D1312yes—
426C4×Q8C22×D1312yes—
427C4×D4—24no—
428C4×D4—21no—
429C4×D4—21no—
430C4×D4—134no—
431C4×D4—132no—
432C4×D4—264no—
433C4×D4—262no—
434C4×D4—261no—
435C4×D4C22×D1312yes—
436C4×D4C22×D1318yes—
437C4×D4C22×D1312yes—
438#117—261no—
439#117—262no—
440#117—264no—
441C2×C4⋊C4—22no—
442C2×C4⋊C4—21no—
443C2×C4⋊C4—22no—
444C2×C4⋊C4—24no—
445C2×C4⋊C4—24no—
446C2×C4⋊C4—262no—
447C2×C4⋊C4C22⋊D1311yescharacteristic
448C2×C4⋊C4C22⋊D1312yes—
449C2×C4⋊C4C22×D1311yescharacteristic
450C2×C4⋊C4C22×D1312yes—
451C2×C22⋊C4—21no—
452C2×C22⋊C4—22no—
453C2×C42C22×D1312yes—
454C2×C42C22×D1311yescharacteristic
455#107—134no—
456#107—138no—
457#106—134no—
458#106—138no—
459C4⋊C8—132no—
460C4⋊C8—132no—
461C4⋊C8—138no—
462#103—134no—
463#103—138no—
464#102—134no—
465#102—134no—
466#102—134no—
467#102—138no—
468C22⋊C8—132no—
469C22⋊C8—132no—
470C4×C8—132no—
471C4×C8—138no—
472C4×C8—132no—
473C2×C13—22no—
474C2×C13—44no—
475C2×C13#348611yescharacteristic
476C2×C13C22×#13511yescharacteristic
477C2×C13#358511yescharacteristic
478D13—82no—
479C24—21no—
480#41—524no—
481C2×Q8—24no—
482C2×Q8—22no—
483C2×Q8—22no—
484C2×Q8—82no—
485C2×Q8—132no—
486C2×Q8—261no—
487C2×Q8—261no—
488C2×Q8C2×C22⋊D1312yes—
489C2×Q8C2×C22⋊D1312yes—
490C2×D4—24no—
491C2×D4—21no—
492C2×D4—132no—
493C2×D4—262no—
494C2×D4—261no—
495C2×D4—261no—
496C2×D4C23×D1312yes—
497C22×C4—21no—
498C22×C4—22no—
499C22×C4—22no—
500C22×C4—22no—
501C22×C4—28no—
502C22×C4—44no—
503C22×C4—84no—
504C22×C4—262no—
505C22×C4—264no—
506C22×C4C2×C22⋊D1312yes—
507C22×C4C2×C22⋊D1311yescharacteristic
508C22×C4C2×C22⋊D1312yes—
509C22×C4C2×C22⋊D1311yescharacteristic
510C22×C4C23×D1311yescharacteristic
511SD16—268no—
512D8—268no—
513C2×C8—134no—
514C2×C8—134no—
515C4⋊C4—28no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—28no—
519C4⋊C4—21no—
520C4⋊C4—22no—
521C4⋊C4—132no—
522C4⋊C4—132no—
523C4⋊C4—261no—
524C4⋊C4—264no—
525C4⋊C4—261no—
526C4⋊C4—262no—
527C4⋊C4—264no—
528C4⋊C4—261no—
529C4⋊C4—262no—
530C4⋊C4—264no—
531C4⋊C4—261no—
532C4⋊C4C2×C22⋊D1312yes—
533C4⋊C4C2×C22⋊D1312yes—
534C4⋊C4C2×C22⋊D1314yes—
535C4⋊C4C2×C22⋊D1314yes—
536C4⋊C4C23×D1312yes—
537C4⋊C4C23×D1314yes—
538C22⋊C4—22no—
539C22⋊C4—22no—
540C22⋊C4—28no—
541C22⋊C4—262no—
542C22⋊C4—262no—
543C22⋊C4—264no—
544C22⋊C4—264no—
545C22⋊C4—262no—
546C22⋊C4—264no—
547C42—264no—
548C42—262no—
549C42C2×C22⋊D1314yes—
550C42C2×C22⋊D1311yescharacteristic
551C42C2×C22⋊D1311yescharacteristic
552C42C23×D1311yescharacteristic
553C42C23×D1314yes—
554C42C23×D1311yescharacteristic
555C13#6181911yescharacteristic, complemented, Sylow
556C23—24no—
557C23—22no—
558C23—42no—
559C23—42no—
560C23—262no—
561C23C22×C22⋊D1311yescharacteristic
562Q8—24no—
563Q8—24no—
564Q8—82no—
565Q8—82no—
566Q8—264no—
567Q8—522no—
568Q8—1041no—
569D4—24no—
570D4—22no—
571D4—264no—
572D4—524no—
573D4—522no—
574D4—1041no—
575D4#8825414yes—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—44no—
583C2×C4—42no—
584C2×C4—44no—
585C2×C4—44no—
586C2×C4—88no—
587C2×C4—262no—
588C2×C4—262no—
589C2×C4—262no—
590C2×C4—524no—
591C2×C4—1044no—
592C2×C4#8820414yes—
593C2×C4C22×C22⋊D1314yes—
594C2×C4C22×C22⋊D1311yescharacteristic
595C2×C4C22×C22⋊D1314yes—
596C2×C4C22×C22⋊D1311yescharacteristic
597C2×C4C22×C22⋊D1311yescharacteristic
598C2×C4C22×C22⋊D1311yescharacteristic
599C2×C4C22×C22⋊D1311yescharacteristic
600C2×C4C24×D1311yescharacteristic
601C8—264no—
602C8—264no—
603C22—21no—
604C22—21no—
605C22—24no—
606C22—48no—
607C22—42no—
608C22—42no—
609C22—522no—
610C22—522no—
611C22—522no—
612C22C2×#8820412yes—
613C22C23×C22⋊D1311yescharacteristic
614C4—21no—
615C4—24no—
616C4—24no—
617C4—21no—
618C4—21no—
619C4—42no—
620C4—44no—
621C4—522no—
622C4—522no—
623C4—522no—
624C4C2×#8820412yes—
625C4C2×#8820412yes—
626C4C2×#8825412yes—
627C2—22no—
628C2—44no—
629C2—1042no—
630C2#1312848711yescharacteristic
631C2C22×#8820411yescharacteristic
632C2#1312961611yescharacteristic
subgroupG/H
centerC22C23×C22⋊D13
commutator, cosocleC2×C4×C13C25
FrattiniC2×C4C24×D13
FittingC2×C13×#554C2
socleC22×C13C23×D4
prime radical#13124836C2
hypercenterC2×#554D13
hypocenterC13#61819
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×#554

Complements

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

NH ≅ G/Ntype
1. #13128701C2semidirect
2. #13127711C2semidirect
3. #13127699C2semidirect
4. #13127496C2semidirect
5. #13127485C2semidirect
6. #13126004C2semidirect
7. #13125987C2semidirect
8. #13125964C2semidirect
9. #13124883C2semidirect
10. #13124844C2semidirect
12. #13123917C2semidirect
13. C2×C13×#554C2semidirect
27. #11719290C22semidirect
28. #11719289C22semidirect
30. #11719284C22semidirect
32. #11719272C22semidirect
40. #11718754C22semidirect
41. #11718752C22semidirect
42. #11718749C22semidirect
43. #11718745C22semidirect
44. #11718740C22semidirect
45. #11718739C22semidirect
46. #11718729C22semidirect
47. #11718727C22semidirect
48. #11718725C22semidirect
49. #11718724C22semidirect
50. Q8×#1055C22semidirect
51. Q8×#1055C22semidirect
241. C2×#554D13semidirect
555. C13#61819semidirect

Maximal subgroups

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

subgroupindex
#131287012
#131277112
#131276992
#131274962
#131274852
#131260042
#131259872
#131259642
#131248832
#131248442
#131248362
#131239172
C2×C13×#5542
#6181913