Finite Group Databasebeta

← #35375457

Subgroups — #35375457

Order 21248.

662 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24916477C211yesmaximal, characteristic, complemented
2#24916207C211yesmaximal, characteristic, complemented
3#24906366C211yesmaximal, characteristic, complemented
4#24906354C211yesmaximal, characteristic, complemented
5#24906328C211yesmaximal, characteristic, complemented
6#24906321C211yesmaximal, characteristic, complemented
7#24906300C211yesmaximal, characteristic, complemented
8#24906293C211yesmaximal, characteristic, complemented
9#24906287C211yesmaximal, characteristic, complemented
10D8×C83⋊C8C211yesmaximal, characteristic, complemented
11#24906248C211yesmaximal, characteristic, complemented
12#24905932C211yesmaximal, characteristic, complemented
13#24905850C211yesmaximal, characteristic, complemented
14#24905758C211yesmaximal, characteristic, complemented
15C83×#3106C211yesmaximal, characteristic
16C2×#14581799—21no—
17#19127832C2211yescharacteristic
18#19127832C2211yescharacteristic
19#19127779C2211yescharacteristic
20#19127779C2211yescharacteristic
21#19127773—21no—
22#19127773—21no—
23#19127773—21no—
24#19127773—21no—
25D4×C83⋊C8C2211yescharacteristic
26D4×C83⋊C8C2211yescharacteristic
27D4×C83⋊C8C2211yescharacteristic
28D4×C83⋊C8C2211yescharacteristic
29#19127758—21no—
30#19127758C2211yescharacteristic
31#19127754C2211yescharacteristic
32C2×C4⋊(C83⋊C8)C2211yescharacteristic
33#19127668—21no—
34C4×#13050574C2211yescharacteristic
35C2×#14581703—21no—
36#19127454C2211yescharacteristic
37#19127454C2211yescharacteristic
38#19127452C2211yescharacteristic
39#19127452C2211yescharacteristic
40Q8⋊(C83⋊C8)C2211yescharacteristic
41Q8⋊(C83⋊C8)C2211yescharacteristic
42D4⋊(C83⋊C8)C2211yescharacteristic
43D4⋊(C83⋊C8)C2211yescharacteristic
44D4⋊(C83⋊C8)C2211yescharacteristic
45D4⋊(C83⋊C8)C2211yescharacteristic
46D4⋊(C83⋊C8)C2211yescharacteristic
47D4⋊(C83⋊C8)C2211yescharacteristic
48#19127423C2211yescharacteristic
49#19127423C2211yescharacteristic
50#19127422C2211yescharacteristic
51C8×C83⋊C8C2211yescharacteristic
52#532×C83C2211yescharacteristic
53C4×#41×C83C2211yescharacteristic
54#482×C83C411yescharacteristic
55#482×C83C411yescharacteristic
56#477×C83C411yescharacteristic
57#476×C83C411yescharacteristic
58#460×C83C411yescharacteristic
59#459×C83C411yescharacteristic
60#449×C83C411yescharacteristic
61#449×C83C411yescharacteristic
62C4×SD16×C83C2211yescharacteristic
63C4×SD16×C83C2211yescharacteristic
64C4×D8×C83C2211yescharacteristic
65C4×D8×C83C2211yescharacteristic
66#431×C83C2211yescharacteristic
67#14581799—44no—
68C2×#13050574—21no—
69C2×#13050574—21no—
70C2×#13050574D411yescharacteristic
71C2×#13050574D411yescharacteristic
72C22×C83⋊C8—21no—
73C22×C83⋊C8—21no—
74C22×C83⋊C8—21no—
75C22×C83⋊C8—21no—
76C22×C83⋊C8—21no—
77C4⋊(C83⋊C8)—21no—
78C4⋊(C83⋊C8)—21no—
79C4⋊(C83⋊C8)—21no—
80C4⋊(C83⋊C8)—21no—
81C4⋊(C83⋊C8)C2311yescharacteristic
82C4⋊(C83⋊C8)C2311yescharacteristic
83C4⋊(C83⋊C8)C2311yescharacteristic
84C4⋊(C83⋊C8)C2311yescharacteristic
85C22⋊(C83⋊C8)—21no—
86C22⋊(C83⋊C8)—21no—
87C22⋊(C83⋊C8)—21no—
88C22⋊(C83⋊C8)—21no—
89C22⋊(C83⋊C8)—21no—
90C22⋊(C83⋊C8)—21no—
91C22⋊(C83⋊C8)—21no—
92C22⋊(C83⋊C8)—21no—
93#14581703—21no—
94#14581703—21no—
95#14581703—21no—
96#14581703D411yescharacteristic
97#14581703D411yescharacteristic
98C4×C83⋊C8C2311yescharacteristic
99C4×C83⋊C8C2311yescharacteristic
100C4×C83⋊C8C2311yescharacteristic
101C4×C83⋊C8C2311yescharacteristic
102C2×#41×C83—21no—
103C2×#41×C83C2×C411yescharacteristic
104C2×SD16×C83C2×C411yescharacteristic
105C2×SD16×C83C2×C411yescharacteristic
106C2×D8×C83C2×C411yescharacteristic
107C2×D8×C83C2×C411yescharacteristic
108C4⋊Q8×C83C2×C411yescharacteristic
109#127×C83C2×C411yescharacteristic
110#124×C83—21no—
111#121×C83—21no—
112#121×C83C2×C411yescharacteristic
113#121×C83C2×C411yescharacteristic
114C4×Q8×C83C2×C411yescharacteristic
115C4×Q8×C83C2311yescharacteristic
116C4×D4×C83—21no—
117C4×D4×C83—21no—
118C4×D4×C83—21no—
119C4×D4×C83C2×C411yescharacteristic
120C4×D4×C83C2311yescharacteristic
121C4×D4×C83C2311yescharacteristic
122C4×D4×C83C2311yescharacteristic
123#117×C83—21no—
124#117×C83C2×C411yescharacteristic
125C2×C42×C83—21no—
126C2×C42×C83C2311yescharacteristic
127#107×C83C2×C411yescharacteristic
128#107×C83C2×C411yescharacteristic
129#106×C83C2×C411yescharacteristic
130#106×C83C2×C411yescharacteristic
131C4⋊C8×C83C2×C411yescharacteristic
132C4⋊C8×C83C2×C411yescharacteristic
133#103×C83C2×C411yescharacteristic
134#103×C83C2×C411yescharacteristic
135#102×C83C2×C411yescharacteristic
136#102×C83C2×C411yescharacteristic
137#102×C83C2×C411yescharacteristic
138#102×C83C2×C411yescharacteristic
139#102×C83C2×C411yescharacteristic
140#102×C83C2×C411yescharacteristic
141C22⋊C8×C83C2×C411yescharacteristic
142C22⋊C8×C83C2×C411yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C4×C8×C83C2311yescharacteristic
145#13050574—22no—
146#13050574—41no—
147#13050574—42no—
148#13050574—42no—
149C2×C83⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8—21no—
152C2×C83⋊C8—21no—
153C2×C83⋊C8—21no—
154C2×C83⋊C8—21no—
155C2×C83⋊C8—42no—
156C2×C83⋊C8—42no—
157C2×C83⋊C8—42no—
158C2×C83⋊C8—42no—
159C2×C83⋊C8—42no—
160C2×C83⋊C8C2×D411yescharacteristic
161C2×C83⋊C8C2×D411yescharacteristic
162C2×C83⋊C8C2×D411yescharacteristic
163C2×C83⋊C8C2×D411yescharacteristic
164#41×C83—22no—
165#41×C83—41no—
166#41×C83—42no—
167#41×C83—42no—
168C2×Q8×C83—21no—
169C2×Q8×C83C22×C411yescharacteristic
170C2×D4×C83—21no—
171C2×D4×C83—21no—
172C2×D4×C83—21no—
173C2×D4×C83—21no—
174C2×D4×C83—21no—
175C2×D4×C83C22×C411yescharacteristic
176C2×D4×C83C22×C411yescharacteristic
177C2×D4×C83C22×C411yescharacteristic
178C22×C4×C83—21no—
179C22×C4×C83—21no—
180C22×C4×C83—21no—
181C22×C4×C83—21no—
182C22×C4×C83—21no—
183C22×C4×C83C22×C411yescharacteristic
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83#4111yescharacteristic
186SD16×C83—41no—
187SD16×C83—41no—
188D8×C83—22no—
189D8×C83—22no—
190C2×C8×C83C22×C411yescharacteristic
191C2×C8×C83C22×C411yescharacteristic
192C2×C8×C83C22×C411yescharacteristic
193C2×C8×C83C22×C411yescharacteristic
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83—21no—
197C4⋊C4×C83C22×C411yescharacteristic
198C4⋊C4×C83C22×C411yescharacteristic
199C4⋊C4×C83C22×C411yescharacteristic
200C4⋊C4×C83C22×C411yescharacteristic
201C22⋊C4×C83—21no—
202C22⋊C4×C83—21no—
203C22⋊C4×C83—21no—
204C22⋊C4×C83—21no—
205C22⋊C4×C83—21no—
206C22⋊C4×C83—21no—
207C22⋊C4×C83—21no—
208C42×C83—21no—
209C42×C83—21no—
210C42×C83—21no—
211C42×C83C22×C411yescharacteristic
212C42×C83C2×D411yescharacteristic
213C42×C83#4111yescharacteristic
214C42×C83C2411yescharacteristic
215C23×C83—21no—
216C23×C83—21no—
217C23×C83—21no—
218C23×C83C4×D411yescharacteristic
219Q8×C83—21no—
220Q8×C83—21no—
221Q8×C83—42no—
222D4×C83—21no—
223D4×C83—21no—
224D4×C83—21no—
225D4×C83—21no—
226D4×C83—21no—
227D4×C83—42no—
228D4×C83—42no—
229D4×C83—42no—
230D4×C83—42no—
231D4×C83—81no—
232D4×C83#13112yescomplemented
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—21no—
238C2×C4×C83—21no—
239C2×C4×C83—21no—
240C2×C4×C83—21no—
241C2×C4×C83—21no—
242C2×C4×C83—41no—
243C2×C4×C83—42no—
244C2×C4×C83—41no—
245C2×C4×C83—41no—
246C2×C4×C83—41no—
247C2×C4×C83—41no—
248C2×C4×C83—41no—
249C2×C4×C83—41no—
250C2×C4×C83—81no—
251C2×C4×C83C4×D411yescharacteristic
252C2×C4×C83C4×D411yescharacteristic
253C2×C4×C83C4×D411yescharacteristic
254C2×C4×C83C23×C411yescharacteristic
255C2×C4×C83C22×D411yescharacteristic
256C2×C4×C83C2×#4111yescharacteristic
257C8×C83—21no—
258C8×C83—21no—
259C8×C83—21no—
260C8×C83—21no—
261C83⋊C8—21no—
262C83⋊C8—21no—
263C83⋊C8—21no—
264C83⋊C8—41no—
265C83⋊C8—41no—
266C83⋊C8—41no—
267C83⋊C8—41no—
268C83⋊C8—41no—
269C83⋊C8—41no—
270C83⋊C8AGL(1,ℤ/8)12yes—
271C22×C83—21no—
272C22×C83—22no—
273C22×C83—21no—
274C22×C83—21no—
275C22×C83—41no—
276C22×C83—41no—
277C22×C83—41no—
278C22×C83—41no—
279C22×C83—41no—
280C22×C83—41no—
281C22×C83—41no—
282C22×C83—41no—
283C22×C83C2×C4×D411yescharacteristic
284C4×C83—21no—
285C4×C83—21no—
286C4×C83—21no—
287C4×C83—21no—
288C4×C83—41no—
289C4×C83—41no—
290C4×C83—41no—
291C4×C83—41no—
292C4×C83—41no—
293C4×C83C2×#13111yescharacteristic
294C4×C83#56811yescharacteristic
295C4×C83C2×AGL(1,ℤ/8)11yescharacteristic
296C4×C83#57511yescharacteristic
297#18007—831nomaximal, Sylow
298C2×C83—41no—
299C2×C83—42no—
300C2×C83—81no—
301C2×C83—81no—
302C2×C83#293311yescharacteristic
303C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
304C2×C83#295711yescharacteristic
305#3106D8311yescharacteristic
306#2970—831no—
307#2935—831no—
308#1592—831no—
309#1590—831no—
310#1586—831no—
311#1585—831no—
312#1582—831no—
313#1581—831no—
314#1580—831no—
315C8×D8—831no—
316#1573—831no—
317#1496—831no—
318#1477—831no—
319#1453—831no—
320C83#1800711yescharacteristic, complemented, Sylow
321C2×#131—1661no—
322#532C2×D8311yescharacteristic
323C4×#41C2×D8311yescharacteristic
324#482Dic8311yescharacteristic
325#482Dic8311yescharacteristic
326#477Dic8311yescharacteristic
327#476Dic8311yescharacteristic
328#460Dic8311yescharacteristic
329#459Dic8311yescharacteristic
330#449Dic8311yescharacteristic
331#449Dic8311yescharacteristic
332#446—831no—
333#446—831no—
334C4×SD16C2×D8311yescharacteristic
335C4×SD16C2×D8311yescharacteristic
336C4×D8C2×D8311yescharacteristic
337C4×D8C2×D8311yescharacteristic
338#436—831no—
339#436—831no—
340#435—1661no—
341#435—1661no—
342#435—1661no—
343#435—1661no—
344C8×D4—831no—
345C8×D4—831no—
346C8×D4—831no—
347C8×D4—831no—
348#432—831no—
349#432—1661no—
350#431—831no—
351#431C2×D8311yescharacteristic
352C2×C4⋊C8—831no—
353#408—1661no—
354C4×M16—831no—
355C2×#97—1661no—
356#335—831no—
357#335—831no—
358#334—831no—
359#334—831no—
360Q8⋊C8—831no—
361Q8⋊C8—831no—
362D4⋊C8—831no—
363D4⋊C8—831no—
364D4⋊C8—831no—
365D4⋊C8—831no—
366D4⋊C8—831no—
367D4⋊C8—831no—
368#322—831no—
369#322—831no—
370#322—831no—
371C82—831no—
372C2×#41—21no—
373C2×#41C2×Dic8311yescharacteristic
374C2×SD16C2×Dic8311yescharacteristic
375C2×SD16C2×Dic8311yescharacteristic
376C2×D8C2×Dic8311yescharacteristic
377C2×D8C2×Dic8311yescharacteristic
378#131—3324no—
379C2×M16—831no—
380C2×M16—831no—
381C2×M16—1661no—
382C2×M16—1661no—
383C22×C8—1661no—
384C22×C8—1661no—
385C22×C8—1661no—
386C22×C8—1661no—
387C22×C8—1661no—
388C4⋊Q8C2×Dic8311yescharacteristic
389#127C2×Dic8311yescharacteristic
390#124—21no—
391#121—21no—
392#121C2×Dic8311yescharacteristic
393#121C2×Dic8311yescharacteristic
394C4×Q8C2×Dic8311yescharacteristic
395C4×Q8C22×D8311yescharacteristic
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—21no—
399C4×D4C2×Dic8311yescharacteristic
400C4×D4C22×D8311yescharacteristic
401C4×D4C22×D8311yescharacteristic
402C4×D4C22×D8311yescharacteristic
403#117—21no—
404#117C2×Dic8311yescharacteristic
405C2×C42—21no—
406C2×C42C22×D8311yescharacteristic
407#107C2×Dic8311yescharacteristic
408#107C2×Dic8311yescharacteristic
409#106C2×Dic8311yescharacteristic
410#106C2×Dic8311yescharacteristic
411C4⋊C8—831no—
412C4⋊C8—831no—
413C4⋊C8—831no—
414C4⋊C8—831no—
415C4⋊C8—1661no—
416C4⋊C8—1661no—
417C4⋊C8—1661no—
418C4⋊C8—1661no—
419C4⋊C8C2×Dic8311yescharacteristic
420C4⋊C8C2×Dic8311yescharacteristic
421#103C2×Dic8311yescharacteristic
422#103C2×Dic8311yescharacteristic
423#102C2×Dic8311yescharacteristic
424#102C2×Dic8311yescharacteristic
425#102C2×Dic8311yescharacteristic
426#102C2×Dic8311yescharacteristic
427#102C2×Dic8311yescharacteristic
428#102C2×Dic8311yescharacteristic
429C22⋊C8—1661no—
430C22⋊C8—1661no—
431C22⋊C8—1661no—
432C22⋊C8—1661no—
433C22⋊C8—1661no—
434C22⋊C8—1661no—
435C22⋊C8—1661no—
436C22⋊C8—1661no—
437C22⋊C8C2×Dic8311yescharacteristic
438C22⋊C8C2×Dic8311yescharacteristic
439#97—831no—
440#97—831no—
441#97—1661no—
442#97—1661no—
443#97—1661no—
444C4×C8—831no—
445C4×C8—831no—
446C4×C8—831no—
447C4×C8—831no—
448C4×C8C22×D8311yescharacteristic
449C4×C8C22×D8311yescharacteristic
450#41—22no—
451#41—41no—
452#41—42no—
453#41—42no—
454C2×Q8—21no—
455C2×Q8C22×Dic8311yescharacteristic
456C2×D4—21no—
457C2×D4—21no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—21no—
461C2×D4C22×Dic8311yescharacteristic
462C2×D4C22×Dic8311yescharacteristic
463C2×D4C22×Dic8311yescharacteristic
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4C22×Dic8311yescharacteristic
470C22×C4D4×D8311yescharacteristic
471C22×C4#1305058911yescharacteristic
472SD16—41no—
473SD16—41no—
474D8—22no—
475D8—22no—
476M16—3322no—
477M16—3322no—
478M16—3321no—
479M16—1662no—
480C2×C8—3322no—
481C2×C8—3322no—
482C2×C8—3322no—
483C2×C8—3322no—
484C2×C8—3322no—
485C2×C8—831no—
486C2×C8—831no—
487C2×C8—831no—
488C2×C8—831no—
489C2×C8—1661no—
490C2×C8—1661no—
491C2×C8—1661no—
492C2×C8—1661no—
493C2×C8—1661no—
494C2×C8—1661no—
495C2×C8C22×Dic8311yescharacteristic
496C2×C8C22×Dic8311yescharacteristic
497C2×C8C22×Dic8311yescharacteristic
498C2×C8C22×Dic8311yescharacteristic
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4C22×Dic8311yescharacteristic
503C4⋊C4C22×Dic8311yescharacteristic
504C4⋊C4C22×Dic8311yescharacteristic
505C4⋊C4C22×Dic8311yescharacteristic
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42C22×Dic8311yescharacteristic
517C42D4×D8311yescharacteristic
518C42#1305058911yescharacteristic
519C42C23×D8311yescharacteristic
520C23—21no—
521C23—21no—
522C23—21no—
523C23D4×Dic8311yescharacteristic
524Q8—21no—
525Q8—21no—
526Q8—42no—
527D4—21no—
528D4—21no—
529D4—21no—
530D4—21no—
531D4—21no—
532D4—42no—
533D4—42no—
534D4—42no—
535D4—42no—
536D4—81no—
537D4#1458179912yescomplemented
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—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—81no—
556C2×C4D4×Dic8311yescharacteristic
557C2×C4D4×Dic8311yescharacteristic
558C2×C4D4×Dic8311yescharacteristic
559C2×C4C23×Dic8311yescharacteristic
560C2×C4C2×D4×D8311yescharacteristic
561C2×C4C2×#1305058911yescharacteristic
562C8—21no—
563C8—21no—
564C8—21no—
565C8—21no—
566C8—3321no—
567C8—3321no—
568C8—3321no—
569C8—3321no—
570C8—3321no—
571C8—3321no—
572C8—832no—
573C8—1661no—
574C8—1661no—
575C8—1661no—
576C22—21no—
577C22—21no—
578C22—21no—
579C22—22no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22C2×D4×Dic8311yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—21no—
593C4—41no—
594C4—41no—
595C4—41no—
596C4—41no—
597C4—41no—
598C4C2×#1458179911yescharacteristic
599C4#1912846811yescharacteristic
600C4AGL(1,ℤ/8)×D8311yescharacteristic
601C4#1912849711yescharacteristic
602C2—41no—
603C2—42no—
604C2—81no—
605C2—81no—
606C2#2491619011yescharacteristic
607C2AGL(1,ℤ/8)×Dic8311yescharacteristic
608C2#2491638011yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
Fitting, prime radicalC83×#3106C2
socleC22×C83C2×C4×D4
cosocleC42×C83C24
hypercenter#3106D83
hypocenterC83#18007
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3106

Complements

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

NH ≅ G/Ntype
1. #24916477C2semidirect
2. #24916207C2semidirect
3. #24906366C2semidirect
4. #24906354C2semidirect
5. #24906328C2semidirect
6. #24906321C2semidirect
7. #24906300C2semidirect
8. #24906293C2semidirect
9. #24906287C2semidirect
10. D8×C83⋊C8C2semidirect
11. #24906248C2semidirect
12. #24905932C2semidirect
13. #24905850C2semidirect
14. #24905758C2semidirect
232. D4×C83#131semidirect
320. C83#18007semidirect
537. D4#14581799semidirect

Maximal subgroups

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

subgroupindex
#249164772
#249162072
#249063662
#249063542
#249063282
#249063212
#249063002
#249062932
#249062872
D8×C83⋊C82
#249062482
#249059322
#249058502
#249057582
C83×#31062
#1800783