Finite Group Databasebeta

← #15833951

Subgroups — #15833951

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#13124879C211yesmaximal, characteristic
2M16×D52C211yesmaximal, characteristic, complemented
3#13113748C211yesmaximal, characteristic, complemented
4#13113726C211yesmaximal, characteristic, complemented
5#13113714C211yesmaximal, characteristic, complemented
6#13113698C211yesmaximal, characteristic, complemented
7#13113670C211yesmaximal, characteristic, complemented
8#13113665C211yesmaximal, characteristic, complemented
9#13113646C211yesmaximal, characteristic, complemented
10C8×#6104C211yesmaximal, characteristic, complemented
11#13113624C211yesmaximal, characteristic, complemented
12#13113301C211yesmaximal, characteristic, complemented
13#13113226C211yesmaximal, characteristic, complemented
14#13113133C211yesmaximal, characteristic, complemented
15C13×#2969C211yesmaximal, characteristic, complemented
16C13×#529C2211yescharacteristic
17C2×M16×D13—21no—
18C2×C4×D52C2211yescharacteristic
19C13×#446C2211yescharacteristic, complemented
20C8×Q8×C13C2211yescharacteristic, complemented
21C13×#436—21no—
22C13×#436C2211yescharacteristic, complemented
23C13×#435—21no—
24C8×D4×C13C2211yescharacteristic, complemented
25C13×#433—21no—
26C13×#431C2211yescharacteristic
27C13×#405—21no—
28C4×C13×M16C2211yescharacteristic
29#11718787—21no—
30#11718727C411yescharacteristic
31#11718727C411yescharacteristic
32#11718718C411yescharacteristic
33#11718718C411yescharacteristic
34#11718707C411yescharacteristic
35#11718705C411yescharacteristic
36#11718697C411yescharacteristic
37#11718695C411yescharacteristic
38C4×#6106C2211yescharacteristic
39C4×#6106C2211yescharacteristic
40C4×#6104C2211yescharacteristic
41C4×#6104C2211yescharacteristic
42#11718586C2211yescharacteristic
43#11718586C2211yescharacteristic
44#11718585C2211yescharacteristic
45#11718367—21no—
46#11718367—21no—
47#11718349C2211yescharacteristic
48#11718349C2211yescharacteristic
49C8×D52C2211yescharacteristic
50C8×D52C2211yescharacteristic
51Q8⋊#1055C2211yescharacteristic, complemented
52Q8⋊#1055C2211yescharacteristic, complemented
53D4⋊#1055C2211yescharacteristic, complemented
54D4⋊#1055C2211yescharacteristic, complemented
55#11718075C2211yescharacteristic
56#11718075C2211yescharacteristic
57#11718075C2211yescharacteristic
58#11718075C2211yescharacteristic
59#11718072C2211yescharacteristic, complemented
60#11718072C2211yescharacteristic, complemented
61#11718071C2211yescharacteristic, complemented
62#11718071C2211yescharacteristic, complemented
63#11718047C2211yescharacteristic, complemented
64#11718046C2211yescharacteristic, complemented
65#11718046C2211yescharacteristic, complemented
66C8×#1055C2211yescharacteristic, complemented
67C22×D52C2×C411yescharacteristic
68C22×C4×D13—21no—
69C2×C13×M16—21no—
70C2×C13×M16D411yescharacteristic
71C2×C13×M16D411yescharacteristic
72C22×C8×C13—21no—
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×C13C2311yescharacteristic
82C13×#117—21no—
83C2×C42×C13C2311yescharacteristic
84C2×#6106C2×C411yescharacteristic
85C2×#6106C2×C411yescharacteristic
86C2×#6104C2×C411yescharacteristic
87C2×#6104C2×C411yescharacteristic
88C2×#6103—21no—
89C2×C4⋊Dic13C2×C411yescharacteristic
90C2×#6099—21no—
91M16×D13—44no—
92C2×#6094—21no—
93C2×#6094—21no—
94C2×C8×D13—21no—
95C2×C8×D13—21no—
96C4×D52—21no—
97C4×D52—21no—
98C4×D52C2×C411yescharacteristic
99C4×D52C2×C411yescharacteristic
100C4×D52C2311yescharacteristic
101C4×D52C2311yescharacteristic
102C13×C4⋊C8—21no—
103C13×C4⋊C8—21no—
104C13×C4⋊C8C2311yescharacteristic
105C13×C4⋊C8C2311yescharacteristic
106C13×C22⋊C8—21no—
107C13×C22⋊C8—21no—
108C13×C22⋊C8—21no—
109C13×#97—21no—
110C13×#97D411yescharacteristic
111C13×#97D411yescharacteristic
112C4×C8×C13—21no—
113C4×C8×C13—21no—
114C4×C8×C13C2311yescharacteristic
115C4×C8×C13C2311yescharacteristic
116#88075C2×C411yescharacteristic
117#88075C2×C411yescharacteristic
118#88072C2×C411yescharacteristic
119#88072C2×C411yescharacteristic
120C22⋊#1055—21no—
121C22⋊#1055C2×C411yescharacteristic
122C22⋊#1055C2×C411yescharacteristic
123#88060—21no—
124#88060—21no—
125#88060—21no—
126#88060—21no—
127#88049C2×C411yescharacteristic
128#88049C2×C411yescharacteristic
129#88049C2×C411yescharacteristic
130#88049C2×C411yescharacteristic
131#88048C2×C411yescharacteristic
132#88048C2×C411yescharacteristic
133#88047C2×C411yescharacteristic
134#88047C2×C411yescharacteristic
135C4⋊#1055—21no—
136C4⋊#1055—21no—
137C4⋊#1055C2×C411yescharacteristic
138C4⋊#1055C2×C411yescharacteristic
139C4⋊#1055C2311yescharacteristic
140C4⋊#1055C2311yescharacteristic
141C4×#1055C2311yescharacteristic
142C4×#1055C2311yescharacteristic
143C4×#1055C2311yescharacteristic
144C4×#1055C2311yescharacteristic
145#18006—131nomaximal, Sylow
146C23×D13—21no—
147C2×Q8×C13C22×C411yescharacteristic
148C2×D4×C13—21no—
149C2×D4×C13C22×C411yescharacteristic
150C22×C4×C13—21no—
151C22×C4×C13C22×C411yescharacteristic
152C22×C4×C13C2×D411yescharacteristic
153C22×C4×C13#4111yescharacteristic
154C22×Dic13—21no—
155C2×D52—22no—
156C2×D52—21no—
157C2×D52—41no—
158C2×D52C22×C411yescharacteristic
159C2×D52C22×C411yescharacteristic
160C2×C4×D13—21no—
161C2×C4×D13—21no—
162C2×C4×D13—21no—
163C2×C4×D13—21no—
164C2×C4×D13—42no—
165C2×C4×D13—42no—
166C13×M16—22no—
167C13×M16—41no—
168C13×M16—42no—
169C2×C8×C13—21no—
170C2×C8×C13—21no—
171C2×C8×C13—42no—
172C2×C8×C13C2×D411yescharacteristic
173C2×C8×C13C2×D411yescharacteristic
174C2×C8×C13C2×D411yescharacteristic
175C2×C8×C13C2×D411yescharacteristic
176C13×C4⋊C4—21no—
177C13×C4⋊C4—21no—
178C13×C4⋊C4—21no—
179C13×C4⋊C4—21no—
180C13×C4⋊C4C22×C411yescharacteristic
181C13×C4⋊C4C22×C411yescharacteristic
182C13×C22⋊C4—21no—
183C13×C22⋊C4—21no—
184C13×C22⋊C4—21no—
185C13×C22⋊C4—21no—
186C13×C22⋊C4—21no—
187C42×C13—21no—
188C42×C13C22×C411yescharacteristic
189C42×C13C2×D411yescharacteristic
190C42×C13#4111yescharacteristic
191C42×C13C2411yescharacteristic
192#6106—22no—
193#6106—41no—
194#6104—22no—
195#6104—41no—
196#6103—21no—
197#6103—21no—
198#6103—21no—
199#6103—21no—
200C4⋊Dic13—21no—
201C4⋊Dic13C22×C411yescharacteristic
202C4⋊Dic13C22×C411yescharacteristic
203#6099—42no—
204C2×#1055—21no—
205C2×#1055—21no—
206C2×#1055—21no—
207C2×#1055—21no—
208C2×#1055C22×C411yescharacteristic
209C2×#1055C22×C411yescharacteristic
210C2×#1055C22×C411yescharacteristic
211C2×#1055C22×C411yescharacteristic
212#6094—42no—
213#6094—42no—
214C8×D13—42no—
215C8×D13—42no—
216#3105—131no—
217#2969D1311yescharacteristic, complemented
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×C13C4×D411yescharacteristic
233C22×D13—21no—
234C22×D13—21no—
235C22×D13—42no—
236C22×D13—42no—
237C22×D13—81no—
238Q8×C13—21no—
239Q8×C13—21no—
240D4×C13—21no—
241D4×C13—21no—
242D4×C13—81no—
243C2×C4×C13—21no—
244C2×C4×C13—21no—
245C2×C4×C13—21no—
246C2×C4×C13—21no—
247C2×C4×C13—21no—
248C2×C4×C13—21no—
249C2×C4×C13—21no—
250C2×C4×C13—41no—
251C2×C4×C13—41no—
252C2×C4×C13—41no—
253C2×C4×C13C4×D411yescharacteristic
254C2×C4×C13C4×D411yescharacteristic
255C2×C4×C13C4×D411yescharacteristic
256C2×C4×C13C23×C411yescharacteristic
257C2×C4×C13C22×D411yescharacteristic
258C2×C4×C13C2×#4111yescharacteristic
259C2×Dic13—21no—
260C2×Dic13—21no—
261C2×Dic13—42no—
262D52—21no—
263D52—21no—
264D52—21no—
265D52—42no—
266D52#13112yes—
267C4×D13—42no—
268C4×D13—42no—
269C4×D13—41no—
270C4×D13—41no—
271C4×D13—41no—
272C4×D13—41no—
273C8×C13—21no—
274C8×C13—21no—
275C8×C13—21no—
276C8×C13—41no—
277C8×C13—41no—
278C8×C13AGL(1,ℤ/8)12yes—
279#1055—21no—
280#1055—21no—
281#1055—21no—
282#1055—21no—
283#1055—41no—
284#1055—41no—
285#1055—41no—
286#1055—41no—
287C22×M16—261no—
288#529C2×D1311yescharacteristic
289C2×C4×D4—131no—
290#484—131no—
291#481—131no—
292#478—131no—
293#478—131no—
294#463—131no—
295#463—131no—
296#450—131no—
297#447—131no—
298#446C2×D1311yescharacteristic, complemented
299C8×Q8C2×D1311yescharacteristic, complemented
300C4×SD16—131no—
301C4×SD16—131no—
302C4×D8—131no—
303C4×D8—131no—
304#436—21no—
305#436—131no—
306#436—131no—
307#436C2×D1311yescharacteristic, complemented
308#435—21no—
309#435—261no—
310#435—261no—
311C8×D4—131no—
312C8×D4—131no—
313C8×D4C2×D1311yescharacteristic, complemented
314#433—21no—
315#431—131no—
316#431—131no—
317#431C2×D1311yescharacteristic
318#423—131no—
319#407—261no—
320#405—21no—
321C4×M16C2×D1311yescharacteristic
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—21no—
340C22×C13—21no—
341C22×C13—41no—
342C22×C13—41no—
343C22×C13—41no—
344C22×C13C2×C4×D411yescharacteristic
345C2×D13—22no—
346C2×D13—41no—
347C2×D13—41no—
348C2×D13—41no—
349C2×D13—41no—
350C2×D13—41no—
351C2×D13—84no—
352C4×C13—21no—
353C4×C13—21no—
354C4×C13—21no—
355C4×C13—21no—
356C4×C13—41no—
357C4×C13—41no—
358C4×C13—41no—
359C4×C13C2×#13111yescharacteristic
360C4×C13#56811yescharacteristic
361C4×C13C2×AGL(1,ℤ/8)11yescharacteristic
362C4×C13#57511yescharacteristic
363Dic13—41no—
364Dic13—41no—
365C22×D4—131no—
366C23×C4—261no—
367C2×SD16—131no—
368C2×SD16—131no—
369C2×D8—131no—
370C2×D8—131no—
371C2×M16—21no—
372C2×M16—261no—
373C2×M16—261no—
374C2×M16—261no—
375C2×M16—524no—
376C2×M16C22⋊D1311yescharacteristic
377C2×M16C22⋊D1311yescharacteristic
378C22×C8—21no—
379C22×C8—261no—
380C22×C8—261no—
381#126—21no—
382#125C4×D1311yescharacteristic
383#124C4×D1311yescharacteristic
384#123—21no—
385C22⋊Q8C4×D1311yescharacteristic
386#121C4×D1311yescharacteristic
387C4×Q8C22×D1311yescharacteristic
388C4×D4—21no—
389C4×D4—131no—
390C4×D4—131no—
391C4×D4—131no—
392C4×D4—131no—
393C4×D4—261no—
394C4×D4—261no—
395C4×D4C22×D1311yescharacteristic
396#117—21no—
397C2×C4⋊C4—131no—
398C2×C22⋊C4—261no—
399C2×C42C22×D1311yescharacteristic
400#107—131no—
401#107—131no—
402#106—131no—
403#106—131no—
404C4⋊C8—21no—
405C4⋊C8—21no—
406C4⋊C8—131no—
407C4⋊C8—131no—
408C4⋊C8—131no—
409C4⋊C8—131no—
410C4⋊C8—261no—
411C4⋊C8—261no—
412C4⋊C8C22×D1311yescharacteristic
413C4⋊C8C22×D1311yescharacteristic
414#103—131no—
415#103—131no—
416#102—131no—
417#102—131no—
418#102—131no—
419#102—131no—
420#102—131no—
421#102—131no—
422C22⋊C8—21no—
423C22⋊C8—21no—
424C22⋊C8—21no—
425C22⋊C8—131no—
426C22⋊C8—131no—
427C22⋊C8—261no—
428C22⋊C8—261no—
429C22⋊C8—261no—
430C22⋊C8—261no—
431C22⋊C8—261no—
432#97—21no—
433#97C22⋊D1311yescharacteristic
434#97C22⋊D1311yescharacteristic
435C4×C8—21no—
436C4×C8—21no—
437C4×C8—131no—
438C4×C8—131no—
439C4×C8—131no—
440C4×C8—131no—
441C4×C8C22×D1311yescharacteristic
442C4×C8C22×D1311yescharacteristic
443C2×C13—41no—
444C2×C13—81no—
445C2×C13#293311yescharacteristic
446C2×C13C4×AGL(1,ℤ/8)11yescharacteristic
447C2×C13#295711yescharacteristic
448D13—42no—
449D13—81no—
450C24—261no—
451C2×Q8C2×C4×D1311yescharacteristic
452C2×D4—21no—
453C2×D4—131no—
454C2×D4—131no—
455C2×D4—262no—
456C2×D4—261no—
457C2×D4—521no—
458C2×D4C2×C4×D1311yescharacteristic
459C22×C4—21no—
460C22×C4—261no—
461C22×C4—261no—
462C22×C4—261no—
463C22×C4—261no—
464C22×C4—261no—
465C22×C4—522no—
466C22×C4—522no—
467C22×C4C2×C4×D1311yescharacteristic
468C22×C4#612711yescharacteristic
469C22×C4C2×C22⋊D1311yescharacteristic
470SD16—262no—
471SD16—521no—
472D8—262no—
473D8—521no—
474M16—22no—
475M16—41no—
476M16—42no—
477M16—522no—
478M16—522no—
479M16—522no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—42no—
483C2×C8—131no—
484C2×C8—131no—
485C2×C8—131no—
486C2×C8—131no—
487C2×C8—261no—
488C2×C8—261no—
489C2×C8—261no—
490C2×C8—261no—
491C2×C8—522no—
492C2×C8—522no—
493C2×C8C2×C22⋊D1311yescharacteristic
494C2×C8C2×C22⋊D1311yescharacteristic
495C2×C8C2×C22⋊D1311yescharacteristic
496C2×C8C2×C22⋊D1311yescharacteristic
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—131no—
502C4⋊C4—131no—
503C4⋊C4—261no—
504C4⋊C4C2×C4×D1311yescharacteristic
505C4⋊C4C2×C4×D1311yescharacteristic
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—261no—
512C22⋊C4—261no—
513C22⋊C4—261no—
514C22⋊C4—261no—
515C42—21no—
516C42C2×C4×D1311yescharacteristic
517C42#612711yescharacteristic
518C42C2×C22⋊D1311yescharacteristic
519C42C23×D1311yescharacteristic
520C13#1800611yescharacteristic, complemented, Sylow
521C23—21no—
522C23—261no—
523C23—261no—
524C23—522no—
525C23—522no—
526C23—1041no—
527C23C4×C22⋊D1311yescharacteristic
528Q8—21no—
529Q8—21no—
530D4—21no—
531D4—21no—
532D4—81no—
533D4—132no—
534D4—261no—
535D4—261no—
536D4—261no—
537D4—522no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—261no—
549C2×C4—261no—
550C2×C4—522no—
551C2×C4—522no—
552C2×C4—522no—
553C2×C4—521no—
554C2×C4—521no—
555C2×C4—521no—
556C2×C4—521no—
557C2×C4C4×C22⋊D1311yescharacteristic
558C2×C4C4×C22⋊D1311yescharacteristic
559C2×C4C4×C22⋊D1311yescharacteristic
560C2×C4C22×C4×D1311yescharacteristic
561C2×C4C2×#612711yescharacteristic
562C2×C4C22×C22⋊D1311yescharacteristic
563C8—21no—
564C8—21no—
565C8—21no—
566C8—41no—
567C8—41no—
568C8—261no—
569C8—261no—
570C8—261no—
571C8—261no—
572C8—521no—
573C8—521no—
574C8—521no—
575C8—521no—
576C8#8820312yes—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—262no—
584C22—521no—
585C22—521no—
586C22—521no—
587C22—521no—
588C22—521no—
589C22—1044no—
590C22C2×C4×C22⋊D1311yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4—521no—
599C4—521no—
600C4D13×#13111yescharacteristic
601C4#1171945511yescharacteristic
602C4C2×#8820311yescharacteristic
603C4#1171952711yescharacteristic
604C2—41no—
605C2—81no—
606C2—522no—
607C2—1041no—
608C2#1312355811yescharacteristic
609C2C4×#8820311yescharacteristic
610C2#1312375111yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D13
commutatorC2×C4×C13C23×C4
FrattiniC42C23×D13
FittingC13×#2969C2
socleC22×C13C2×C4×D4
prime radical#11718697C4
cosocleC42×C13C24
hypercenter#2969D13
hypocenterC13#18006
LCSG ▷ C2×C4×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ #2969

Complements

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

NH ≅ G/Ntype
2. M16×D52C2semidirect
3. #13113748C2semidirect
4. #13113726C2semidirect
5. #13113714C2semidirect
6. #13113698C2semidirect
7. #13113670C2semidirect
8. #13113665C2semidirect
9. #13113646C2semidirect
10. C8×#6104C2semidirect
11. #13113624C2semidirect
12. #13113301C2semidirect
13. #13113226C2semidirect
14. #13113133C2semidirect
15. C13×#2969C2semidirect
19. C13×#446C22semidirect
20. C8×Q8×C13C22semidirect
22. C13×#436C22semidirect
24. C8×D4×C13C22semidirect
51. Q8⋊#1055C22semidirect
52. Q8⋊#1055C22semidirect
53. D4⋊#1055C22semidirect
54. D4⋊#1055C22semidirect
59. #11718072C22semidirect
60. #11718072C22semidirect
61. #11718071C22semidirect
62. #11718071C22semidirect
63. #11718047C22semidirect
64. #11718046C22semidirect
65. #11718046C22semidirect
66. C8×#1055C22semidirect
217. #2969D13semidirect
298. #446C2×D13semidirect
299. C8×Q8C2×D13semidirect
307. #436C2×D13semidirect
313. C8×D4C2×D13semidirect
520. C13#18006semidirect

Maximal subgroups

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

subgroupindex
#131248792
M16×D522
#131137482
#131137262
#131137142
#131136982
#131136702
#131136652
#131136462
C8×#61042
#131136242
#131133012
#131132262
#131131332
C13×#29692
#1800613