Finite Group Databasebeta

← #35422123

Subgroups — #35422123

Order 21248.

766 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127736C211yesmaximal, characteristic
2#24910355C211yesmaximal, characteristic
3#24910352C211yesmaximal, characteristic
4#24908200C211yesmaximal, characteristic, complemented
5#24908198C211yesmaximal, characteristic, complemented
6#24908050C211yesmaximal, characteristic, complemented
7#24908049C211yesmaximal, characteristic, complemented
8#24908047C211yesmaximal, characteristic, complemented
9#24907557C211yesmaximal, characteristic
10#24907556C211yesmaximal, characteristic
11#24907487C211yesmaximal, characteristic
12#24907361C211yesmaximal, characteristic, complemented
13#24907359C211yesmaximal, characteristic, complemented
14#24907238C211yesmaximal, characteristic
15C83×#1920C211yesmaximal, characteristic, complemented
16C22×C8×D83C2211yescharacteristic
17C2×#14581750C2211yescharacteristic
18C2×#14581750C2211yescharacteristic
19#19127736C418yescomplemented
20C2×#14581725C2211yescharacteristic
21C2×#14581725C2211yescharacteristic
22C2×#14581724C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×(C2×D83)⋊C8C2211yescharacteristic
26C2×#14581703C2211yescharacteristic
27C2×#14581702C2211yescharacteristic
28#19127600C2211yescharacteristic
29#19127599C2211yescharacteristic
30#19127597C2211yescharacteristic
31#19127593—21no—
32#19127592—21no—
33#19127591—21no—
34#19127590—21no—
35#19127568—21no—
36#19127568—21no—
37#19127558C2211yescharacteristic
38#19127558C2211yescharacteristic
39C4⋊C4×Dic83—21no—
40C4⋊C4×Dic83C2211yescharacteristic
41C4⋊C4×Dic83C2211yescharacteristic
42C4×Dic83⋊C4—21no—
43C4×C4⋊Dic83C2211yescharacteristic
44C4×(C2×D83)⋊C4C2211yescharacteristic
45#19127464C2211yescharacteristic
46#19127464C2211yescharacteristic
47#19127464C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127463C2211yescharacteristic
52#19127455C2211yescharacteristic
53C2×#106×C83C2211yescharacteristic
54#389×C83C2211yescharacteristic
55C4×C4⋊C4×C83C2211yescharacteristic
56#340×C83C2211yescharacteristic
57#340×C83C2211yescharacteristic
58#340×C83C2211yescharacteristic
59#336×C83C2211yescharacteristic
60C22×C4×D83C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63C2×C8×D83—22no—
64C2×C8×D83—22no—
65C2×C8×D83C2×C414yes—
66#14581750—22no—
67#14581750—22no—
68#14581750C2×C414yes—
69#14581750C2×C414yes—
70C2×C4⋊Dic83—21no—
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×Dic83⋊C4—21no—
76C2×Dic83⋊C4—21no—
77C2×(C2×D83)⋊C4—21no—
78C2×(C2×D83)⋊C4—21no—
79C2×(C2×D83)⋊C4—21no—
80C2×(C2×D83)⋊C4—21no—
81C2×(C2×D83)⋊C4D411yescharacteristic
82C2×(C2×D83)⋊C4D411yescharacteristic
83C2×C4×Dic83—21no—
84C2×C4×Dic83—21no—
85C2×C4×Dic83—21no—
86C2×C4×Dic83—21no—
87C2×C4×Dic83—21no—
88C2×C4×Dic83C2311yescharacteristic
89#14581725C2×C414yes—
90#14581725C2×C414yes—
91#14581724C2×C414yes—
92#14581723—22no—
93#14581723—22no—
94(C2×D83)⋊C8—22no—
95#14581703—22no—
96#14581702—22no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C83⋊#95—21no—
104C83⋊#95—21no—
105C83⋊#95D411yescharacteristic
106C83⋊#95D411yescharacteristic
107C22×C8×C83C2311yescharacteristic
108C22×C8×C83C2311yescharacteristic
109C2×C4⋊C4×C83—21no—
110C2×C4⋊C4×C83C2311yescharacteristic
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C42×C83—21no—
115C2×C42×C83C2311yescharacteristic
116#106×C83C2×C414yes—
117#95×C83—21no—
118#95×C83—21no—
119C23×D83C2×D411yescharacteristic
120C22×Dic83—21no—
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83—21no—
124C22×Dic83C2×D411yescharacteristic
125C22×Dic83C2×D411yescharacteristic
126C22×Dic83C2×D411yescharacteristic
127C2×C4×D83—21no—
128C2×C4×D83—21no—
129C2×C4×D83—21no—
130C2×C4×D83—21no—
131C2×C4×D83C22×C412yes—
132C2×C4×D83#4112yes—
133C2×C83⋊C8—21no—
134C2×C83⋊C8—21no—
135C2×C83⋊C8—21no—
136C2×C83⋊C8C22×C412yes—
137C2×C83⋊C8#4112yes—
138C2×C83⋊C8#4112yes—
139C8×D83—28no—
140C4⋊Dic83—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—42no—
147C4⋊Dic83C22×C412yes—
148C4⋊Dic83C22×C412yes—
149Dic83⋊C4—42no—
150Dic83⋊C4—42no—
151(C2×D83)⋊C4—24no—
152(C2×D83)⋊C4—24no—
153(C2×D83)⋊C4—41no—
154(C2×D83)⋊C4—41no—
155(C2×D83)⋊C4—81no—
156(C2×D83)⋊C4—81no—
157C4×Dic83—21no—
158C4×Dic83—21no—
159C4×Dic83—24no—
160C4×Dic83—24no—
161C4×Dic83—81no—
162C4×Dic83—81no—
163C4×Dic83—81no—
164C22×C4×C83—21no—
165C22×C4×C83—21no—
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83C2×D411yescharacteristic
169C22×C4×C83C2×D411yescharacteristic
170C22×C4×C83C2411yescharacteristic
171C2×C8×C83—21no—
172C2×C8×C83—21no—
173C2×C8×C83—21no—
174C2×C8×C83—21no—
175C2×C8×C83—21no—
176C2×C8×C83C22×C412yes—
177C4⋊C4×C83—21no—
178C4⋊C4×C83—21no—
179C4⋊C4×C83—21no—
180C4⋊C4×C83—21no—
181C4⋊C4×C83—21no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—42no—
184C4⋊C4×C83C22×C412yes—
185C4⋊C4×C83C22×C412yes—
186C42×C83—21no—
187C42×C83—21no—
188C42×C83—81no—
189C23×C83C22×D411yescharacteristic
190C22×D83—21no—
191C22×D83—21no—
192C22×D83—21no—
193C22×D83—21no—
194C22×D83—21no—
195C22×D83—21no—
196C22×D83C4×D412yes—
197C2×C4×C83—22no—
198C2×C4×C83—22no—
199C2×C4×C83—21no—
200C2×C4×C83—21no—
201C2×C4×C83—21no—
202C2×C4×C83—21no—
203C2×C4×C83—21no—
204C2×C4×C83—21no—
205C2×C4×C83—41no—
206C2×C4×C83—41no—
207C2×C4×C83—41no—
208C2×C4×C83—41no—
209C2×C4×C83—41no—
210C2×C4×C83—41no—
211C2×C4×C83—41no—
212C2×C4×C83—41no—
213C2×C4×C83—41no—
214C2×C4×C83—41no—
215C2×C4×C83C23×C411yescharacteristic
216C2×C4×C83C2×#4111yescharacteristic
217C2×C4×C83C2×#4111yescharacteristic
218C2×C4×C83C2×#4111yescharacteristic
219C2×C4×C832−1+411yescharacteristic
220C2×C4×C832−1+411yescharacteristic
221C2×Dic83—21no—
222C2×Dic83—21no—
223C2×Dic83—22no—
224C2×Dic83—22no—
225C2×Dic83—21no—
226C2×Dic83—21no—
227C2×Dic83—21no—
228C2×Dic83—21no—
229C2×Dic83—21no—
230C2×Dic83—21no—
231C2×Dic83—41no—
232C2×Dic83—41no—
233C2×Dic83—41no—
234C2×Dic83—41no—
235C2×Dic83—41no—
236C2×Dic83—41no—
237C2×Dic83—41no—
238C2×Dic83—41no—
239C2×Dic83—41no—
240C2×Dic83—41no—
241C2×Dic83C4×D412yes—
242C4×D83—22no—
243C4×D83—22no—
244C4×D83—22no—
245C4×D83—22no—
246C8×C83—22no—
247C8×C83—41no—
248C83⋊C8—41no—
249C83⋊C8#11714yes—
250C22×C83C2×C4×D411yescharacteristic
251C22×C83#54711yescharacteristic
252C22×C83#54711yescharacteristic
253C22×C83#57511yescharacteristic
254C22×C83#57511yescharacteristic
255C22×C83#57711yescharacteristic
256C22×C83#57711yescharacteristic
257C2×D83—22no—
258C2×D83—22no—
259C2×D83—22no—
260C2×D83—41no—
261C2×D83—41no—
262C2×D83—41no—
263C2×D83—41no—
264C4×C83—41no—
265C4×C83—41no—
266C4×C83—42no—
267C4×C83—42no—
268C4×C83—81no—
269C4×C83—81no—
270C4×C83C2×#11711yescharacteristic
271C4×C83#51911yescharacteristic
272C4×C83#55211yescharacteristic
273C4×C83#55211yescharacteristic
274Dic83—22no—
275Dic83—41no—
276Dic83—41no—
277Dic83—42no—
278Dic83—42no—
279Dic83—81no—
280Dic83—81no—
281#22620—831nomaximal, Sylow
282C2×C83#234411yescharacteristic
283C2×C83#294311yescharacteristic
284C2×C83#295211yescharacteristic
285C2×C83#335011yescharacteristic
286C2×C83#335011yescharacteristic
287C2×C83#335011yescharacteristic
288C2×C83#335011yescharacteristic
289D83—42no—
290C2×#427—831no—
291#2347—831no—
292#2347—831no—
293#1943—831no—
294#1943—831no—
295#1920—831no—
296#1920—831no—
297#1920—831no—
298#1920D8311yescharacteristic, complemented
299#1825—831no—
300#1825—831no—
301#1807—831no—
302#1776—831no—
303#1776—831no—
304#1752—831no—
305C83#2262011yescharacteristic, complemented, Sylow
306C23×C8—831no—
307C2×#117—831no—
308C2×#117—831no—
309#427—838no—
310C2×#107—831no—
311C2×#107—831no—
312C2×#106—831no—
313C2×#106—831no—
314C2×#106—831no—
315C2×#106C2×D8311yescharacteristic
316C2×C22⋊C8—831no—
317C2×#97—831no—
318C2×#97—831no—
319#389—831no—
320#389—831no—
321#389—831no—
322#389C2×D8311yescharacteristic
323#388—1661no—
324#388—1661no—
325#388—1661no—
326#388—1661no—
327#383—1661no—
328#383—1661no—
329#380—831no—
330#380—831no—
331C4×C4⋊C4—831no—
332C4×C4⋊C4—831no—
333C4×C4⋊C4—831no—
334C4×C4⋊C4—1661no—
335C4×C4⋊C4—1661no—
336C4×C4⋊C4C2×D8311yescharacteristic
337C4×C22⋊C4—831no—
338#340—831no—
339#340—831no—
340#340—831no—
341#340—831no—
342#340—831no—
343#340—831no—
344#340—831no—
345#340C2×D8311yescharacteristic
346#340C2×D8311yescharacteristic
347#340C2×D8311yescharacteristic
348#336—831no—
349#336C2×D8311yescharacteristic
350C23×C4—831no—
351C22×C8—831no—
352C22×C8—834no—
353C22×C8—831no—
354C22×C8—1662no—
355C22×C8—1662no—
356C22×C8C22×D8311yescharacteristic
357C22×C8C22×D8311yescharacteristic
358#117—834no—
359#117—834no—
360#117—1662no—
361#117—1662no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—831no—
364C2×C4⋊C4—831no—
365C2×C4⋊C4—831no—
366C2×C4⋊C4—831no—
367C2×C4⋊C4—1661no—
368C2×C4⋊C4—1661no—
369C2×C4⋊C4—1661no—
370C2×C4⋊C4C22×D8311yescharacteristic
371C2×C4⋊C4C22×D8311yescharacteristic
372C2×C4⋊C4C22×D8311yescharacteristic
373C2×C4⋊C4C22×D8311yescharacteristic
374C2×C22⋊C4—831no—
375C2×C22⋊C4—831no—
376C2×C22⋊C4—1661no—
377C2×C22⋊C4—1661no—
378C2×C22⋊C4—1661no—
379C2×C22⋊C4—1661no—
380C2×C42—21no—
381C2×C42—831no—
382C2×C42—1661no—
383C2×C42—1661no—
384C2×C42—1661no—
385C2×C42—1661no—
386C2×C42—1661no—
387C2×C42C22×D8311yescharacteristic
388#107—834no—
389#107—834no—
390#106—834no—
391#106—1662no—
392#106—1662no—
393#106C4×D8314yes—
394C22⋊C8—1662no—
395#97—1662no—
396#97—1662no—
397#95—21no—
398#95—21no—
399#95—831no—
400#95—831no—
401#95—1661no—
402#95—1661no—
403#95—1661no—
404#95—1661no—
405#95—1661no—
406#95—1661no—
407#95—1661no—
408#95—1661no—
409C24—831no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—831no—
415C22×C4—832no—
416C22×C4—832no—
417C22×C4—831no—
418C22×C4—831no—
419C22×C4—1661no—
420C22×C4—1661no—
421C22×C4—1661no—
422C22×C4—1661no—
423C22×C4—1661no—
424C22×C4—1661no—
425C22×C4—1661no—
426C22×C4—1661no—
427C22×C4D4×D8311yescharacteristic
428C22×C4D4×D8311yescharacteristic
429C22×C4C23×D8311yescharacteristic
430C2×C8—21no—
431C2×C8—21no—
432C2×C8—21no—
433C2×C8—21no—
434C2×C8—21no—
435C2×C8—832no—
436C2×C8—832no—
437C2×C8—832no—
438C2×C8—1668no—
439C2×C8—1661no—
440C2×C8—1661no—
441C2×C8—1661no—
442C2×C8C2×C4×D8312yes—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—42no—
450C4⋊C4—3322no—
451C4⋊C4—3322no—
452C4⋊C4—3322no—
453C4⋊C4—832no—
454C4⋊C4—832no—
455C4⋊C4—1661no—
456C4⋊C4—1661no—
457C4⋊C4—1661no—
458C4⋊C4—1661no—
459C4⋊C4—1661no—
460C4⋊C4—1661no—
461C4⋊C4C2×C4×D8312yes—
462C4⋊C4C2×C4×D8312yes—
463C22⋊C4—3321no—
464C22⋊C4—3321no—
465C22⋊C4—6641no—
466C22⋊C4—6641no—
467C22⋊C4—1664no—
468C22⋊C4—1664no—
469C42—21no—
470C42—21no—
471C42—81no—
472C42—6641no—
473C42—6641no—
474C42—6641no—
475C42—1664no—
476C42—1664no—
477C42—1661no—
478C42—1661no—
479C23—832no—
480C23—1661no—
481C23—1661no—
482C23—1661no—
483C23—1661no—
484C23—1661no—
485C23—1661no—
486C23C2×D4×D8311yescharacteristic
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—3321no—
506C2×C4—3321no—
507C2×C4—3321no—
508C2×C4—3321no—
509C2×C4—3321no—
510C2×C4—3321no—
511C2×C4—3321no—
512C2×C4—3321no—
513C2×C4—3321no—
514C2×C4—3321no—
515C2×C4—832no—
516C2×C4—1661no—
517C2×C4—1661no—
518C2×C4—1662no—
519C2×C4—1662no—
520C2×C4—1661no—
521C2×C4—1661no—
522C2×C4—1661no—
523C2×C4—1661no—
524C2×C4—1661no—
525C2×C4—1661no—
526C2×C4—1662no—
527C2×C4—1662no—
528C2×C4—1662no—
529C2×C4—1662no—
530C2×C4C22×C4×D8311yescharacteristic
531C2×C4#41×D8311yescharacteristic
532C2×C4#41×D8311yescharacteristic
533C2×C4#41×D8311yescharacteristic
534C2×C4#1458183911yescharacteristic
535C2×C4#1458184011yescharacteristic
536C8—22no—
537C8—41no—
538C8—3321no—
539C8—834no—
540C22—3321no—
541C22—3321no—
542C22—3321no—
543C22—3321no—
544C22—1662no—
545C22—1662no—
546C22—1662no—
547C22C4×D4×D8311yescharacteristic
548C22#1912834911yescharacteristic
549C22#1912835311yescharacteristic
550C22#1912849711yescharacteristic
551C22#1912850011yescharacteristic
552C22#1912850711yescharacteristic
553C22#1912850811yescharacteristic
554C4—42no—
555C4—42no—
556C4—41no—
557C4—41no—
558C4—81no—
559C4—81no—
560C4—3322no—
561C4—3322no—
562C4—3321no—
563C4—3321no—
564C4—6641no—
565C4—6641no—
566C4—1662no—
567C4#117×D8311yescharacteristic
568C4#1912817711yescharacteristic
569C4#1912838311yescharacteristic
570C4#1912838711yescharacteristic
571C2—3322no—
572C2#2491032911yescharacteristic
573C2#2491625811yescharacteristic
574C2#2491634111yescharacteristic
575C2#2492000111yescharacteristic
576C2#2492000511yescharacteristic
577C2#2492001111yescharacteristic
578C2#2492001511yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1920C2
socleC23×C83C22×D4
prime radicalC23×D83C2×D4
cosocleC22×C4×C83C24
hypercenter#1920D83
hypocenterC83#22620
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1920

Complements

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

NH ≅ G/Ntype
4. #24908200C2semidirect
5. #24908198C2semidirect
6. #24908050C2semidirect
7. #24908049C2semidirect
8. #24908047C2semidirect
12. #24907361C2semidirect
13. #24907359C2semidirect
15. C83×#1920C2semidirect
19. #19127736C4semidirect
298. #1920D83semidirect
305. C83#22620semidirect

Maximal subgroups

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

subgroupindex
C2×#191277362
#249103552
#249103522
#249082002
#249081982
#249080502
#249080492
#249080472
#249075572
#249075562
#249074872
#249073612
#249073592
#249072382
C83×#19202
#2262083