Finite Group Databasebeta

← #35375331

Subgroups — #35375331

Order 21248.

670 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917492C211yesmaximal, characteristic
2#24916415C211yesmaximal, characteristic, complemented
3#24916208C211yesmaximal, characteristic
4#24906350C211yesmaximal, characteristic, complemented
5#24906305C211yesmaximal, characteristic, complemented
6#24906304C211yesmaximal, characteristic, complemented
7#24906300C211yesmaximal, characteristic, complemented
8#24906297C211yesmaximal, characteristic, complemented
9#24906266C211yesmaximal, characteristic, complemented
10C8×#13050579C211yesmaximal, characteristic, complemented
11#24906232C211yesmaximal, characteristic, complemented
12#24905912C211yesmaximal, characteristic
13Q8⋊C8×D83C211yesmaximal, characteristic, complemented
14C8×#13050573C211yesmaximal, characteristic, complemented
15C83×#1590C211yesmaximal, characteristic, complemented
16C2×#14581800—21no—
17#19128248C2211yescharacteristic
18C4×Q8×D83C2211yescharacteristic
19#19128021C411yescharacteristic
20#19128018C411yescharacteristic
21#19127985C411yescharacteristic
22#19127984C411yescharacteristic
23#19127900C411yescharacteristic
24#19127899C411yescharacteristic
25#19127841C411yescharacteristic
26#19127838C411yescharacteristic
27#19127832C2211yescharacteristic
28#19127830—21no—
29#19127829—21no—
30C8×Dic166C2211yescharacteristic
31#19127825C2211yescharacteristic
32C4×#13050579C2211yescharacteristic
33C4×#13050577C2211yescharacteristic
34C4×#13050578C2211yescharacteristic
35#19127778C2211yescharacteristic
36#19127770—21no—
37#19127769—21no—
38C8×D332C2211yescharacteristic
39D4×C83⋊C8C2211yescharacteristic
40#19127752C2211yescharacteristic
41#19127723—21no—
42#19127723C2211yescharacteristic
43C4⋊C8×D83C2211yescharacteristic
44#19127666—21no—
45#19127653—21no—
46C4×#13050573C2211yescharacteristic
47C4×C8×D83C2211yescharacteristic
48#19127452C2211yescharacteristic
49#19127451C2211yescharacteristic
50#19127451C2211yescharacteristic
51#19127433C2211yescharacteristic
52Q8⋊(C83⋊C8)C2211yescharacteristic
53#19127431C2211yescharacteristic
54#19127430C2211yescharacteristic
55D4⋊(C83⋊C8)C2211yescharacteristic
56#19127428C2211yescharacteristic
57#19127423C2211yescharacteristic
58#19127422C2211yescharacteristic
59C8×C83⋊C8C2211yescharacteristic
60#446×C83C2211yescharacteristic
61C4×SD16×C83C2211yescharacteristic
62#436×C83C2211yescharacteristic
63#334×C83C2211yescharacteristic
64Q8⋊C8×C83C2211yescharacteristic
65D4⋊C8×C83C2211yescharacteristic
66C82×C83C2211yescharacteristic
67C2×#13050590—21no—
68C2×Q8×D83C2×C411yescharacteristic
69C2×#13050579C2×C411yescharacteristic
70C2×#13050578C2×C411yescharacteristic
71C2×#13050577C2×C411yescharacteristic
72#14581800—44no—
73C2×#13050573—21no—
74C2×#13050573—21no—
75C2×#13050573D411yescharacteristic
76C2×#13050573D411yescharacteristic
77C22×C83⋊C8—21no—
78C2×C8×D83—21no—
79C2×C8×D83—21no—
80C2×C8×D83—21no—
81#14581791C2×C411yescharacteristic
82#14581788C2×C411yescharacteristic
83#14581778—21no—
84#14581767C2×C411yescharacteristic
85#14581765—21no—
86#14581763C2×C411yescharacteristic
87Q8×Dic83C2×C411yescharacteristic
88C4×Dic166—21no—
89C4×Dic166C2311yescharacteristic
90#14581757—21no—
91#14581757C2×C411yescharacteristic
92C4×D332C2311yescharacteristic
93C4×C22⋊D83—21no—
94C4⋊C4×D83—21no—
95C4⋊C4×D83C2×C411yescharacteristic
96C42×D83—21no—
97C42×D83C2311yescharacteristic
98#14581724C2×C411yescharacteristic
99#14581723C2×C411yescharacteristic
100#14581723C2×C411yescharacteristic
101C4⋊(C83⋊C8)—21no—
102C4⋊(C83⋊C8)C2311yescharacteristic
103C4⋊(C83⋊C8)C2311yescharacteristic
104Dic83⋊C8—21no—
105Dic83⋊C8—21no—
106Dic83⋊C8—21no—
107Dic83⋊C8—21no—
108Dic83⋊C8C2×C411yescharacteristic
109Dic83⋊C8C2×C411yescharacteristic
110#14581717C2×C411yescharacteristic
111#14581716C2×C411yescharacteristic
112#14581715C2×C411yescharacteristic
113#14581714C2×C411yescharacteristic
114#14581713C2×C411yescharacteristic
115#14581712C2×C411yescharacteristic
116C22⋊(C83⋊C8)—21no—
117(C2×D83)⋊C8—21no—
118(C2×D83)⋊C8—21no—
119(C2×D83)⋊C8C2×C411yescharacteristic
120(C2×D83)⋊C8C2×C411yescharacteristic
121#14581703—21no—
122#14581702D411yescharacteristic
123#14581702D411yescharacteristic
124C4×C83⋊C8C2311yescharacteristic
125C4×C83⋊C8C2311yescharacteristic
126C4×C83⋊C8C2311yescharacteristic
127C8×Dic83—21no—
128C8×Dic83—21no—
129C8×Dic83—21no—
130C2×SD16×C83C2×C411yescharacteristic
131C2×M16×C83—21no—
132C4×Q8×C83C2311yescharacteristic
133C4×D4×C83C2311yescharacteristic
134#106×C83C2×C411yescharacteristic
135C4⋊C8×C83—21no—
136C4⋊C8×C83C2311yescharacteristic
137C4⋊C8×C83C2311yescharacteristic
138#103×C83C2×C411yescharacteristic
139#102×C83C2×C411yescharacteristic
140C22⋊C8×C83—21no—
141#97×C83—21no—
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C4×C8×C83C2311yescharacteristic
145#13050590—42no—
146#13050590—42no—
147C2×Dic166—21no—
148C2×Dic166—21no—
149C2×Dic166C22×C411yescharacteristic
150Q8×D83—22no—
151Q8×D83—41no—
152C2×D332—21no—
153C2×D332C22×C411yescharacteristic
154C2×C22⋊D83—21no—
155C2×C22⋊D83—21no—
156C2×C22⋊D83—21no—
157C2×C4×D83—21no—
158C2×C4×D83—21no—
159C2×C4×D83—21no—
160C2×C4×D83C22×C411yescharacteristic
161C2×C4×D83C2×D411yescharacteristic
162C2×C4×D83#4111yescharacteristic
163#13050579—22no—
164#13050578—22no—
165#13050577—41no—
166#13050573—22no—
167#13050573—42no—
168#13050573—42no—
169#13050573SD1614yescomplemented
170C2×C83⋊C8—21no—
171C2×C83⋊C8—21no—
172C2×C83⋊C8—21no—
173C2×C83⋊C8—42no—
174C2×C83⋊C8C22×C411yescharacteristic
175C2×C83⋊C8C22×C411yescharacteristic
176C2×C83⋊C8C2×D411yescharacteristic
177C2×C83⋊C8C2×D411yescharacteristic
178C8×D83—42no—
179C8×D83—42no—
180C8×D83—42no—
181C4⋊Dic83—21no—
182C4⋊Dic83C22×C411yescharacteristic
183C4⋊Dic83C22×C411yescharacteristic
184Dic83⋊C4—21no—
185Dic83⋊C4—21no—
186Dic83⋊C4—21no—
187Dic83⋊C4—21no—
188C22⋊Dic83—21no—
189(C2×D83)⋊C4—21no—
190(C2×D83)⋊C4—21no—
191C4×Dic83—21no—
192C4×Dic83—21no—
193C4×Dic83—21no—
194C4×Dic83C22×C411yescharacteristic
195C4×Dic83C2×D411yescharacteristic
196C4×Dic83#4111yescharacteristic
197C2×Q8×C83C22×C411yescharacteristic
198C2×D4×C83C22×C411yescharacteristic
199C22×C4×C83—21no—
200SD16×C83—41no—
201M16×C83—42no—
202C2×C8×C83—21no—
203C2×C8×C83—21no—
204C2×C8×C83—21no—
205C2×C8×C83C22×C411yescharacteristic
206C2×C8×C83C22×C411yescharacteristic
207C2×C8×C83C2×D411yescharacteristic
208C2×C8×C83C2×D411yescharacteristic
209C4⋊C4×C83—21no—
210C4⋊C4×C83C22×C411yescharacteristic
211C4⋊C4×C83C22×C411yescharacteristic
212C22⋊C4×C83—21no—
213C42×C83—21no—
214C42×C83C2411yescharacteristic
215C23×C83—21no—
216C22×D83—21no—
217C22×D83C4×D411yescharacteristic
218Q8×C83—21no—
219Q8×C83—21no—
220D4×C83—21no—
221D4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—41no—
226C2×C4×C83—41no—
227C2×C4×C83C23×C411yescharacteristic
228C2×C4×C83C22×D411yescharacteristic
229C2×C4×C83C2×#4111yescharacteristic
230C22⋊D83—42no—
231C22⋊D83—42no—
232C22⋊D83—81no—
233C2×Dic83—21no—
234C2×Dic83—21no—
235C2×Dic83—21no—
236C2×Dic83C4×D411yescharacteristic
237C2×Dic83C4×D411yescharacteristic
238C2×Dic83C4×D411yescharacteristic
239D332—21no—
240D332—21no—
241D332—42no—
242C4×D83—21no—
243C4×D83—21no—
244C4×D83—41no—
245C4×D83—42no—
246C4×D83—41no—
247C4×D83—41no—
248C4×D83—81no—
249C4×D83C2×SD1612yes—
250C4×D83#13512yes—
251Dic166—21no—
252Dic166—42no—
253Dic166—42no—
254Dic166#13112yescomplemented
255C8×C83—22no—
256C8×C83—21no—
257C8×C83—21no—
258C8×C83—21no—
259C8×C83—41no—
260C8×C83—41no—
261C8×C83C2×SD1612yes—
262C83⋊C8—21no—
263C83⋊C8—21no—
264C83⋊C8—21no—
265C83⋊C8—41no—
266C83⋊C8—41no—
267C83⋊C8—41no—
268C83⋊C8C2×SD1612yes—
269C22×C83—41no—
270C22×C83—41no—
271C22×C83—41no—
272C22×C83C2×C4×D411yescharacteristic
273C2×D83—21no—
274C2×D83—21no—
275C2×D83—41no—
276C2×D83—41no—
277C2×D83—41no—
278C2×D83C4×SD1612yes—
279C4×C83—21no—
280C4×C83—41no—
281C4×C83—41no—
282C4×C83—41no—
283C4×C83C2×#13111yescharacteristic
284C4×C83#56811yescharacteristic
285C4×C83C22×SD1611yescharacteristic
286C4×C83C2×#13511yescharacteristic
287Dic83—22no—
288Dic83—21no—
289Dic83—21no—
290Dic83—41no—
291Dic83—41no—
292Dic83C4×SD1612yes—
293#17997—831nomaximal, Sylow
294C2×C83—81no—
295C2×C83#293311yescharacteristic
296C2×C83C2×C4×SD1611yescharacteristic
297C2×C83#295611yescharacteristic
298D83—22no—
299D83—81no—
300#3103—831no—
301Q8×M16—831no—
302#2935—831no—
303#1590—831no—
304#1590D8311yescharacteristic, complemented
305#1583—831no—
306#1583—831no—
307#1582—831no—
308#1582—831no—
309C8×SD16—831no—
310C8×SD16—831no—
311#1570—831no—
312#1492—831no—
313C2×Q8⋊C8—831no—
314C8×M16—831no—
315C83#1799711yescharacteristic, complemented, Sylow
316C2×#131—1661no—
317#532—831no—
318C2×C4×Q8—831no—
319#481—831no—
320#481—831no—
321#475—831no—
322#475—831no—
323#460—831no—
324#460—831no—
325#448—831no—
326#448—831no—
327#446—831no—
328#446—1661no—
329#446—1661no—
330#446C2×D8311yescharacteristic
331C8×Q8—831no—
332C8×Q8—831no—
333C4×SD16—831no—
334C4×SD16—831no—
335C4×SD16—831no—
336C4×SD16C2×D8311yescharacteristic
337#436—831no—
338#436C2×D8311yescharacteristic
339#435—1661no—
340#435—1661no—
341C8×D4—831no—
342C8×D4—831no—
343#431—831no—
344#423—831no—
345#423—1661no—
346C2×C4⋊C8—831no—
347#408—1661no—
348C4×M16—831no—
349C4×M16—1661no—
350C2×C4×C8—831no—
351#334—831no—
352#334—831no—
353#334—831no—
354#334C2×D8311yescharacteristic
355Q8⋊C8—831no—
356Q8⋊C8—831no—
357Q8⋊C8—831no—
358Q8⋊C8C2×D8311yescharacteristic
359D4⋊C8—831no—
360D4⋊C8—831no—
361D4⋊C8—831no—
362D4⋊C8C2×D8311yescharacteristic
363#322—831no—
364#322—831no—
365C82—831no—
366C82C2×D8311yescharacteristic
367C2×#41—1661no—
368C22×Q8—831no—
369C2×SD16—831no—
370C2×SD16—831no—
371C2×SD16—831no—
372C2×SD16C4×D8311yescharacteristic
373#131—3324no—
374C2×M16—21no—
375C2×M16—831no—
376C2×M16—831no—
377C2×M16—1661no—
378C2×M16—1661no—
379C22×C8—1661no—
380C22×C8—1661no—
381C22×C8—1661no—
382C22×C8—1661no—
383C4⋊Q8—831no—
384#127—831no—
385#124—1661no—
386#121—831no—
387#121—831no—
388#121—1661no—
389C4×Q8—831no—
390C4×Q8—831no—
391C4×Q8—831no—
392C4×Q8—1661no—
393C4×Q8—1661no—
394C4×Q8C22×D8311yescharacteristic
395C4×D4—831no—
396C4×D4—1661no—
397C4×D4C22×D8311yescharacteristic
398C2×C4⋊C4—831no—
399C2×C4⋊C4—1661no—
400C2×C42—831no—
401C2×C42—1661no—
402#106—831no—
403#106—831no—
404#106—831no—
405#106C4×D8311yescharacteristic
406C4⋊C8—21no—
407C4⋊C8—831no—
408C4⋊C8—831no—
409C4⋊C8—831no—
410C4⋊C8—831no—
411C4⋊C8—1661no—
412C4⋊C8—1661no—
413C4⋊C8—1661no—
414C4⋊C8—1661no—
415C4⋊C8—1661no—
416C4⋊C8C22×D8311yescharacteristic
417C4⋊C8C22×D8311yescharacteristic
418#103—831no—
419#103—831no—
420#103—831no—
421#103C4×D8311yescharacteristic
422#102—831no—
423#102—831no—
424#102—831no—
425#102C4×D8311yescharacteristic
426C22⋊C8—21no—
427C22⋊C8—831no—
428C22⋊C8—831no—
429C22⋊C8—1661no—
430C22⋊C8—1661no—
431C22⋊C8—1661no—
432#97—21no—
433#97—831no—
434#97—831no—
435#97—1661no—
436C4×C8—831no—
437C4×C8—831no—
438C4×C8—831no—
439C4×C8—1661no—
440C4×C8—1661no—
441C4×C8—1661no—
442C4×C8C22×D8311yescharacteristic
443C4×C8C22×D8311yescharacteristic
444C4×C8C22×D8311yescharacteristic
445#41—3322no—
446#41—3322no—
447C2×Q8—3321no—
448C2×Q8—831no—
449C2×Q8—1661no—
450C2×Q8—1661no—
451C2×Q8—1662no—
452C2×Q8C2×C4×D8311yescharacteristic
453C2×D4—831no—
454C2×D4—1661no—
455C2×D4—1661no—
456C2×D4—1661no—
457C2×D4—1661no—
458C2×D4C2×C4×D8311yescharacteristic
459C22×C4—21no—
460C22×C4—831no—
461C22×C4—831no—
462C22×C4—831no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4—1661no—
466SD16—41no—
467SD16—3321no—
468SD16—1662no—
469SD16—1662no—
470M16—42no—
471M16—3322no—
472M16—3322no—
473M16—834no—
474M16—1662no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—3322no—
479C2×C8—3322no—
480C2×C8—3322no—
481C2×C8—3322no—
482C2×C8—831no—
483C2×C8—831no—
484C2×C8—831no—
485C2×C8—831no—
486C2×C8—1661no—
487C2×C8—1661no—
488C2×C8—1661no—
489C2×C8C2×C4×D8311yescharacteristic
490C2×C8C2×C4×D8311yescharacteristic
491C2×C8D4×D8311yescharacteristic
492C2×C8D4×D8311yescharacteristic
493C4⋊C4—21no—
494C4⋊C4—831no—
495C4⋊C4—831no—
496C4⋊C4—1661no—
497C4⋊C4—1661no—
498C4⋊C4—1661no—
499C4⋊C4—1661no—
500C4⋊C4—1661no—
501C4⋊C4C2×C4×D8311yescharacteristic
502C4⋊C4C2×C4×D8311yescharacteristic
503C22⋊C4—21no—
504C22⋊C4—1661no—
505C22⋊C4—1661no—
506C22⋊C4—1661no—
507C42—21no—
508C42—831no—
509C42—831no—
510C42—831no—
511C42—1661no—
512C42—1661no—
513C42—1661no—
514C42C23×D8311yescharacteristic
515C23—21no—
516C23—831no—
517C23—1661no—
518Q8—21no—
519Q8—21no—
520Q8—3322no—
521Q8—3322no—
522Q8—832no—
523Q8—1661no—
524D4—21no—
525D4—21no—
526D4—3322no—
527D4—3322no—
528D4—3322no—
529D4—6641no—
530D4—1661no—
531D4—1661no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—3321no—
538C2×C4—3322no—
539C2×C4—3321no—
540C2×C4—3321no—
541C2×C4—831no—
542C2×C4—832no—
543C2×C4—832no—
544C2×C4—831no—
545C2×C4—831no—
546C2×C4—6641no—
547C2×C4—1661no—
548C2×C4—1661no—
549C2×C4—1661no—
550C2×C4—1661no—
551C2×C4—1661no—
552C2×C4C22×C4×D8311yescharacteristic
553C2×C4C2×D4×D8311yescharacteristic
554C2×C4#41×D8311yescharacteristic
555C8—22no—
556C8—21no—
557C8—21no—
558C8—21no—
559C8—41no—
560C8—41no—
561C8—3321no—
562C8—3321no—
563C8—3321no—
564C8—832no—
565C8—1661no—
566C8—1661no—
567C8—1661no—
568C8SD16×D8312yes—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—3321no—
573C22—3321no—
574C22—3321no—
575C22—832no—
576C22—1661no—
577C22—1661no—
578C22C4×D4×D8311yescharacteristic
579C4—21no—
580C4—41no—
581C4—41no—
582C4—41no—
583C4—3321no—
584C4—3321no—
585C4—832no—
586C4—1661no—
587C4—1662no—
588C4—1661no—
589C4#131×D8311yescharacteristic
590C4#1912846711yescharacteristic
591C4C2×SD16×D8311yescharacteristic
592C4#135×D8311yescharacteristic
593C2—81no—
594C2—6641no—
595C2—1662no—
596C2#2491619211yescharacteristic
597C2C4×SD16×D8311yescharacteristic
598C2#2491636811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1590C2
socleC22×C83C2×C4×D4
prime radical#19128248C22
cosocleC42×C83C24
hypercenter#1590D83
hypocenterC83#17997
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1590

Complements

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

NH ≅ G/Ntype
2. #24916415C2semidirect
4. #24906350C2semidirect
5. #24906305C2semidirect
6. #24906304C2semidirect
7. #24906300C2semidirect
8. #24906297C2semidirect
9. #24906266C2semidirect
10. C8×#13050579C2semidirect
11. #24906232C2semidirect
13. Q8⋊C8×D83C2semidirect
14. C8×#13050573C2semidirect
15. C83×#1590C2semidirect
169. #13050573SD16semidirect
254. Dic166#131semidirect
304. #1590D83semidirect
315. C83#17997semidirect

Maximal subgroups

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

subgroupindex
#249174922
#249164152
#249162082
#249063502
#249063052
#249063042
#249063002
#249062972
#249062662
C8×#130505792
#249062322
#249059122
Q8⋊C8×D832
C8×#130505732
C83×#15902
#1799783