Finite Group Databasebeta

← #15833957

Subgroups — #15833957

Order 3328.

670 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#13123841C211yesmaximal, characteristic, complemented
2D4×#6099C211yesmaximal, characteristic, complemented
3#13113750C211yesmaximal, characteristic, complemented
4#13113727C211yesmaximal, characteristic, complemented
5#13113715C211yesmaximal, characteristic, complemented
6#13113701C211yesmaximal, characteristic, complemented
7#13113673C211yesmaximal, characteristic, complemented
8#13113666C211yesmaximal, characteristic, complemented
9SD16⋊#1055C211yesmaximal, characteristic, complemented
10D8×#1055C211yesmaximal, characteristic, complemented
11#13113625C211yesmaximal, characteristic, complemented
12#13113301C211yesmaximal, characteristic, complemented
13#13113227C211yesmaximal, characteristic, complemented
14#13113131C211yesmaximal, characteristic, complemented
15C13×#3105C211yesmaximal, characteristic
16C13×#529C2211yescharacteristic
17C2×C4×D4×C13C2211yescharacteristic
18C22×#6099—21no—
19C13×#484C411yescharacteristic
20C13×#481C411yescharacteristic
21C13×#478C411yescharacteristic
22C13×#478C411yescharacteristic
23C13×#463C411yescharacteristic
24C13×#463C411yescharacteristic
25C13×#450C411yescharacteristic
26C13×#447C411yescharacteristic
27C4×C13×SD16C2211yescharacteristic
28C4×C13×SD16C2211yescharacteristic
29C4×C13×D8C2211yescharacteristic
30C4×C13×D8C2211yescharacteristic
31C13×#431C2211yescharacteristic
32#11718868—21no—
33#11718685C2211yescharacteristic, complemented
34Q8×#1055C2211yescharacteristic, complemented
35#11718673—21no—
36#11718673C2211yescharacteristic
37#11718673C2211yescharacteristic
38#11718673C2211yescharacteristic, complemented
39#11718672—21no—
40#11718672—21no—
41#11718672—21no—
42D4×#1055C2211yescharacteristic
43D4×#1055C2211yescharacteristic
44D4×#1055C2211yescharacteristic, complemented
45#11718661—21no—
46#11718658—21no—
47#11718586C2211yescharacteristic
48#11718586C2211yescharacteristic
49#11718585C2211yescharacteristic
50C4×#6099C2211yescharacteristic
51Q8⋊#1055C2211yescharacteristic, complemented
52Q8⋊#1055C2211yescharacteristic, complemented
53D4⋊#1055C2211yescharacteristic
54D4⋊#1055C2211yescharacteristic
55D4⋊#1055C2211yescharacteristic
56D4⋊#1055C2211yescharacteristic
57D4⋊#1055C2211yescharacteristic, complemented
58D4⋊#1055C2211yescharacteristic, complemented
59#11718050C2211yescharacteristic, complemented
60#11718050C2211yescharacteristic, complemented
61#11718049C2211yescharacteristic, complemented
62#11718049C2211yescharacteristic, complemented
63#11718047C2211yescharacteristic, complemented
64#11718046C2211yescharacteristic, complemented
65#11718046C2211yescharacteristic, complemented
66C8×#1055C2211yescharacteristic, complemented
67C22×D4×C13C2×C411yescharacteristic
68C23×C4×C13—21no—
69C2×C13×SD16C2×C411yescharacteristic
70C2×C13×SD16C2×C411yescharacteristic
71C2×C13×D8C2×C411yescharacteristic
72C2×C13×D8C2×C411yescharacteristic
73C13×#126—21no—
74C13×#125C2×C411yescharacteristic
75C13×#124C2×C411yescharacteristic
76C13×#123—21no—
77C13×C22⋊Q8C2×C411yescharacteristic
78C13×#121C2×C411yescharacteristic
79C4×Q8×C13C2311yescharacteristic
80C4×D4×C13—21no—
81C4×D4×C13—21no—
82C4×D4×C13—21no—
83C4×D4×C13C2×C411yescharacteristic
84C4×D4×C13C2×C411yescharacteristic
85C4×D4×C13C2311yescharacteristic
86C4×D4×C13C2311yescharacteristic
87C4×D4×C13C2311yescharacteristic
88C13×#117—21no—
89C2×C13×C4⋊C4C2×C411yescharacteristic
90C2×C13×C22⋊C4—21no—
91C2×C42×C13C2311yescharacteristic
92C2×#6099—21no—
93C2×#6099—21no—
94C2×#6099—21no—
95C2×#6099—21no—
96C2×#6099—44no—
97C2×#6099D411yescharacteristic
98C2×#6099D411yescharacteristic
99C22×#1055—21no—
100C22×#1055—21no—
101C22×#1055—21no—
102C13×#107C2×C411yescharacteristic
103C13×#107C2×C411yescharacteristic
104C13×#106C2×C411yescharacteristic
105C13×#106C2×C411yescharacteristic
106C13×C4⋊C8C2×C411yescharacteristic
107C13×C4⋊C8C2×C411yescharacteristic
108C13×#103C2×C411yescharacteristic
109C13×#103C2×C411yescharacteristic
110C13×#102C2×C411yescharacteristic
111C13×#102C2×C411yescharacteristic
112C13×#102C2×C411yescharacteristic
113C13×#102C2×C411yescharacteristic
114C13×#102C2×C411yescharacteristic
115C13×#102C2×C411yescharacteristic
116C13×C22⋊C8C2×C411yescharacteristic
117C13×C22⋊C8C2×C411yescharacteristic
118C4×C8×C13C2311yescharacteristic
119C4×C8×C13C2311yescharacteristic
120C22⋊#1055—21no—
121C22⋊#1055—21no—
122C22⋊#1055—21no—
123C22⋊#1055—21no—
124C22⋊#1055—21no—
125C22⋊#1055—21no—
126C22⋊#1055—21no—
127C22⋊#1055—21no—
128C4⋊#1055—21no—
129C4⋊#1055—21no—
130C4⋊#1055—21no—
131C4⋊#1055—21no—
132C4⋊#1055C2311yescharacteristic
133C4⋊#1055C2311yescharacteristic
134C4⋊#1055C2311yescharacteristic
135C4⋊#1055C2311yescharacteristic
136#88043—21no—
137#88043D411yescharacteristic
138#88043D411yescharacteristic
139C4×#1055—21no—
140C4×#1055—21no—
141C4×#1055C2311yescharacteristic
142C4×#1055C2311yescharacteristic
143C4×#1055C2311yescharacteristic
144C4×#1055C2311yescharacteristic
145#18006—131nomaximal, Sylow
146C24×C13—21no—
147C2×Q8×C13C22×C411yescharacteristic
148C2×D4×C13—22no—
149C2×D4×C13—21no—
150C2×D4×C13—21no—
151C2×D4×C13—41no—
152C2×D4×C13C22×C411yescharacteristic
153C2×D4×C13C22×C411yescharacteristic
154C2×D4×C13C22×C411yescharacteristic
155C22×C4×C13—21no—
156C22×C4×C13—21no—
157C22×C4×C13—21no—
158C22×C4×C13—21no—
159C22×C4×C13—21no—
160C22×C4×C13—21no—
161C22×C4×C13—42no—
162C22×C4×C13—42no—
163C22×C4×C13C22×C411yescharacteristic
164C22×C4×C13C2×D411yescharacteristic
165C22×C4×C13#4111yescharacteristic
166C13×SD16—22no—
167C13×SD16—41no—
168C13×D8—22no—
169C13×D8—41no—
170C2×C8×C13C22×C411yescharacteristic
171C2×C8×C13C22×C411yescharacteristic
172C2×C8×C13C22×C411yescharacteristic
173C2×C8×C13C22×C411yescharacteristic
174C13×C4⋊C4—21no—
175C13×C4⋊C4—21no—
176C13×C4⋊C4—21no—
177C13×C4⋊C4—21no—
178C13×C4⋊C4—21no—
179C13×C4⋊C4C22×C411yescharacteristic
180C13×C4⋊C4C22×C411yescharacteristic
181C13×C4⋊C4C22×C411yescharacteristic
182C13×C4⋊C4C22×C411yescharacteristic
183C13×C22⋊C4—21no—
184C13×C22⋊C4—21no—
185C13×C22⋊C4—21no—
186C13×C22⋊C4—21no—
187C13×C22⋊C4—21no—
188C13×C22⋊C4—21no—
189C13×C22⋊C4—21no—
190C13×C22⋊C4—21no—
191C13×C22⋊C4—21no—
192C42×C13—21no—
193C42×C13C22×C411yescharacteristic
194C42×C13C2×D411yescharacteristic
195C42×C13#4111yescharacteristic
196C42×C13C2411yescharacteristic
197#6099—22no—
198#6099—41no—
199#6099—42no—
200#6099—42no—
201#6099—42no—
202#6099—42no—
203C2×#1055—21no—
204C2×#1055—21no—
205C2×#1055—21no—
206C2×#1055—21no—
207C2×#1055—21no—
208C2×#1055—21no—
209C2×#1055—42no—
210C2×#1055—42no—
211C2×#1055—42no—
212C2×#1055C2×D411yescharacteristic
213C2×#1055C2×D411yescharacteristic
214C2×#1055C2×D411yescharacteristic
215C2×#1055C2×D411yescharacteristic
216#3105D1311yescharacteristic
217#2969—131no—
218D4×M16—131no—
219#1594—131no—
220#1590—131no—
221#1588—131no—
222#1586—131no—
223#1582—131no—
224#1581—131no—
225SD16⋊C8—131no—
226C8×D8—131no—
227#1574—131no—
228#1495—131no—
229#1478—131no—
230#1453—131no—
231C23×C13—21no—
232C23×C13—21no—
233C23×C13—21no—
234C23×C13—42no—
235C23×C13—42no—
236C23×C13—81no—
237C23×C13C4×D411yescharacteristic
238Q8×C13—21no—
239Q8×C13—21no—
240D4×C13—21no—
241D4×C13—21no—
242D4×C13—21no—
243D4×C13—21no—
244D4×C13—21no—
245D4×C13—42no—
246D4×C13—81no—
247D4×C13#13112yes—
248C2×C4×C13—21no—
249C2×C4×C13—21no—
250C2×C4×C13—21no—
251C2×C4×C13—21no—
252C2×C4×C13—21no—
253C2×C4×C13—21no—
254C2×C4×C13—21no—
255C2×C4×C13—21no—
256C2×C4×C13—21no—
257C2×C4×C13—41no—
258C2×C4×C13—42no—
259C2×C4×C13—42no—
260C2×C4×C13—42no—
261C2×C4×C13—41no—
262C2×C4×C13—41no—
263C2×C4×C13—41no—
264C2×C4×C13—41no—
265C2×C4×C13—41no—
266C2×C4×C13—41no—
267C2×C4×C13C4×D411yescharacteristic
268C2×C4×C13C4×D411yescharacteristic
269C2×C4×C13C4×D411yescharacteristic
270C2×C4×C13C23×C411yescharacteristic
271C2×C4×C13C22×D411yescharacteristic
272C2×C4×C13C2×#4111yescharacteristic
273C8×C13—21no—
274C8×C13—21no—
275C8×C13—21no—
276C8×C13—21no—
277#1055—21no—
278#1055—21no—
279#1055—21no—
280#1055—41no—
281#1055—41no—
282#1055—41no—
283#1055—41no—
284#1055—41no—
285#1055—41no—
286#1055AGL(1,ℤ/8)12yes—
287C22×M16—261no—
288#529C2×D1311yescharacteristic
289C2×C4×D4C2×D1311yescharacteristic
290#484Dic1311yescharacteristic
291#481Dic1311yescharacteristic
292#478Dic1311yescharacteristic
293#478Dic1311yescharacteristic
294#463Dic1311yescharacteristic
295#463Dic1311yescharacteristic
296#450Dic1311yescharacteristic
297#447Dic1311yescharacteristic
298#446—131no—
299C8×Q8—131no—
300C4×SD16C2×D1311yescharacteristic
301C4×SD16C2×D1311yescharacteristic
302C4×D8C2×D1311yescharacteristic
303C4×D8C2×D1311yescharacteristic
304#436—131no—
305#436—131no—
306#436—131no—
307#436—261no—
308#435—261no—
309#435—261no—
310#435—261no—
311C8×D4—131no—
312C8×D4—131no—
313C8×D4—131no—
314#433—261no—
315#431—131no—
316#431—131no—
317#431C2×D1311yescharacteristic
318#423—131no—
319#407—261no—
320#405—261no—
321C4×M16—131no—
322#335—131no—
323#335—131no—
324#334—131no—
325#334—131no—
326Q8⋊C8—131no—
327Q8⋊C8—131no—
328D4⋊C8—131no—
329D4⋊C8—131no—
330D4⋊C8—131no—
331D4⋊C8—131no—
332D4⋊C8—131no—
333D4⋊C8—131no—
334#322—131no—
335#322—131no—
336#322—131no—
337C82—131no—
338C22×C13—21no—
339C22×C13—22no—
340C22×C13—21no—
341C22×C13—21no—
342C22×C13—41no—
343C22×C13—41no—
344C22×C13—41no—
345C22×C13—41no—
346C22×C13—41no—
347C22×C13—41no—
348C22×C13—41no—
349C22×C13—41no—
350C22×C13—84no—
351C22×C13C2×C4×D411yescharacteristic
352C4×C13—21no—
353C4×C13—21no—
354C4×C13—21no—
355C4×C13—21no—
356C4×C13—41no—
357C4×C13—41no—
358C4×C13—41no—
359C4×C13—41no—
360C4×C13—41no—
361C4×C13C2×#13111yescharacteristic
362C4×C13#56811yescharacteristic
363C4×C13C2×AGL(1,ℤ/8)11yescharacteristic
364C4×C13#57511yescharacteristic
365C22×D4C2×Dic1311yescharacteristic
366C23×C4—21no—
367C2×SD16C2×Dic1311yescharacteristic
368C2×SD16C2×Dic1311yescharacteristic
369C2×D8C2×Dic1311yescharacteristic
370C2×D8C2×Dic1311yescharacteristic
371C2×M16—131no—
372C2×M16—131no—
373C2×M16—261no—
374C2×M16—261no—
375C2×M16—261no—
376C2×M16—261no—
377C2×M16—524no—
378C22×C8—261no—
379C22×C8—261no—
380C22×C8—261no—
381#126—21no—
382#125C2×Dic1311yescharacteristic
383#124C2×Dic1311yescharacteristic
384#123—21no—
385C22⋊Q8C2×Dic1311yescharacteristic
386#121C2×Dic1311yescharacteristic
387C4×Q8C22×D1311yescharacteristic
388C4×D4—21no—
389C4×D4—21no—
390C4×D4—21no—
391C4×D4C2×Dic1311yescharacteristic
392C4×D4C2×Dic1311yescharacteristic
393C4×D4C22×D1311yescharacteristic
394C4×D4C22×D1311yescharacteristic
395C4×D4C22×D1311yescharacteristic
396#117—21no—
397C2×C4⋊C4C2×Dic1311yescharacteristic
398C2×C22⋊C4—21no—
399C2×C42C22×D1311yescharacteristic
400#107C2×Dic1311yescharacteristic
401#107C2×Dic1311yescharacteristic
402#106C2×Dic1311yescharacteristic
403#106C2×Dic1311yescharacteristic
404C4⋊C8—131no—
405C4⋊C8—131no—
406C4⋊C8—131no—
407C4⋊C8—131no—
408C4⋊C8—261no—
409C4⋊C8—261no—
410C4⋊C8—261no—
411C4⋊C8—261no—
412C4⋊C8C2×Dic1311yescharacteristic
413C4⋊C8C2×Dic1311yescharacteristic
414#103C2×Dic1311yescharacteristic
415#103C2×Dic1311yescharacteristic
416#102C2×Dic1311yescharacteristic
417#102C2×Dic1311yescharacteristic
418#102C2×Dic1311yescharacteristic
419#102C2×Dic1311yescharacteristic
420#102C2×Dic1311yescharacteristic
421#102C2×Dic1311yescharacteristic
422C22⋊C8—261no—
423C22⋊C8—261no—
424C22⋊C8—261no—
425C22⋊C8—261no—
426C22⋊C8—261no—
427C22⋊C8—261no—
428C22⋊C8—261no—
429C22⋊C8—261no—
430C22⋊C8C2×Dic1311yescharacteristic
431C22⋊C8C2×Dic1311yescharacteristic
432#97—131no—
433#97—131no—
434#97—261no—
435C4×C8—131no—
436C4×C8—131no—
437C4×C8—131no—
438C4×C8—131no—
439C4×C8—261no—
440C4×C8—261no—
441C4×C8C22×D1311yescharacteristic
442C4×C8C22×D1311yescharacteristic
443C2×C13—41no—
444C2×C13—42no—
445C2×C13—81no—
446C2×C13—81no—
447C2×C13#293311yescharacteristic
448C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
449C2×C13#295711yescharacteristic
450C24—21no—
451C2×Q8C22×Dic1311yescharacteristic
452C2×D4—22no—
453C2×D4—21no—
454C2×D4—21no—
455C2×D4—41no—
456C2×D4C22×Dic1311yescharacteristic
457C2×D4C22×Dic1311yescharacteristic
458C2×D4C22×Dic1311yescharacteristic
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4D4×D1311yescharacteristic
468C22×C4#612911yescharacteristic
469C22×C4C22×Dic1311yescharacteristic
470SD16—22no—
471SD16—41no—
472D8—22no—
473D8—41no—
474M16—262no—
475M16—522no—
476M16—521no—
477M16—522no—
478M16—522no—
479M16—522no—
480C2×C8—131no—
481C2×C8—131no—
482C2×C8—131no—
483C2×C8—131no—
484C2×C8—261no—
485C2×C8—261no—
486C2×C8—261no—
487C2×C8—261no—
488C2×C8—261no—
489C2×C8—261no—
490C2×C8—522no—
491C2×C8—522no—
492C2×C8—522no—
493C2×C8C22×Dic1311yescharacteristic
494C2×C8C22×Dic1311yescharacteristic
495C2×C8C22×Dic1311yescharacteristic
496C2×C8C22×Dic1311yescharacteristic
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4C22×Dic1311yescharacteristic
503C4⋊C4C22×Dic1311yescharacteristic
504C4⋊C4C22×Dic1311yescharacteristic
505C4⋊C4C22×Dic1311yescharacteristic
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C42—21no—
516C42D4×D1311yescharacteristic
517C42#612911yescharacteristic
518C42C22×Dic1311yescharacteristic
519C42C23×D1311yescharacteristic
520C13#1800611yescharacteristic, complemented, Sylow
521C23—21no—
522C23—21no—
523C23—21no—
524C23—42no—
525C23—42no—
526C23—81no—
527C23D4×Dic1311yescharacteristic
528Q8—21no—
529Q8—21no—
530D4—21no—
531D4—21no—
532D4—21no—
533D4—21no—
534D4—21no—
535D4—42no—
536D4—81no—
537D4#8820212yes—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4D4×Dic1311yescharacteristic
558C2×C4D4×Dic1311yescharacteristic
559C2×C4D4×Dic1311yescharacteristic
560C2×C4C2×D4×D1311yescharacteristic
561C2×C4C2×#612911yescharacteristic
562C2×C4C23×Dic1311yescharacteristic
563C8—21no—
564C8—21no—
565C8—21no—
566C8—21no—
567C8—132no—
568C8—261no—
569C8—261no—
570C8—261no—
571C8—521no—
572C8—521no—
573C8—521no—
574C8—521no—
575C8—521no—
576C8—521no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—84no—
590C22C2×D4×Dic1311yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4D13×AGL(1,ℤ/8)11yescharacteristic
601C4#1171947811yescharacteristic
602C4C2×#8820211yescharacteristic
603C4#1171952411yescharacteristic
604C2—41no—
605C2—42no—
606C2—81no—
607C2—81no—
608C2#1312356311yescharacteristic
609C2AGL(1,ℤ/8)×Dic1311yescharacteristic
610C2#1312375311yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D13
commutatorC2×C4×C13C23×C4
FrattiniC42C23×D13
FittingC13×#3105C2
socleC22×C13C2×C4×D4
prime radicalC13×#447C4
cosocleC42×C13C24
hypercenter#3105D13
hypocenterC13#18006
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3105

Complements

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

NH ≅ G/Ntype
1. #13123841C2semidirect
2. D4×#6099C2semidirect
3. #13113750C2semidirect
4. #13113727C2semidirect
5. #13113715C2semidirect
6. #13113701C2semidirect
7. #13113673C2semidirect
8. #13113666C2semidirect
9. SD16⋊#1055C2semidirect
10. D8×#1055C2semidirect
11. #13113625C2semidirect
12. #13113301C2semidirect
13. #13113227C2semidirect
14. #13113131C2semidirect
33. #11718685C22semidirect
34. Q8×#1055C22semidirect
38. #11718673C22semidirect
44. D4×#1055C22semidirect
51. Q8⋊#1055C22semidirect
52. Q8⋊#1055C22semidirect
57. D4⋊#1055C22semidirect
58. D4⋊#1055C22semidirect
59. #11718050C22semidirect
60. #11718050C22semidirect
61. #11718049C22semidirect
62. #11718049C22semidirect
63. #11718047C22semidirect
64. #11718046C22semidirect
65. #11718046C22semidirect
66. C8×#1055C22semidirect
520. C13#18006semidirect

Maximal subgroups

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

subgroupindex
#131238412
D4×#60992
#131137502
#131137272
#131137152
#131137012
#131136732
#131136662
SD16⋊#10552
D8×#10552
#131136252
#131133012
#131132272
#131131312
C13×#31052
#1800613