Finite Group Databasebeta

← #35414940

Subgroups — #35414940

Order 21248.

850 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127753C211yesmaximal, characteristic
2#24909869C211yesmaximal, characteristic
3#24909867C211yesmaximal, characteristic
4#24908233C211yesmaximal, characteristic, complemented
5#24908231C211yesmaximal, characteristic, complemented
6#24908229C211yesmaximal, characteristic, complemented
7#24907665C211yesmaximal, characteristic
8#24907663C211yesmaximal, characteristic
9#24907661C211yesmaximal, characteristic
10#24907659C211yesmaximal, characteristic
11#24907621C211yesmaximal, characteristic, complemented
12#24907618C211yesmaximal, characteristic, complemented
13#24907589C211yesmaximal, characteristic, complemented
14#24907589C211yesmaximal, characteristic, complemented
15C83×#1949C211yesmaximal, characteristic, complemented
16C2×#14581750C2211yescharacteristic
17C2×#14581750C2211yescharacteristic
18C2×#14581749—21no—
19C2×C42×D83C2211yescharacteristic
20#19127753C418yescomplemented
21C2×#14581726C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581724C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×#14581723C2211yescharacteristic
27C2×C4⋊(C83⋊C8)C2211yescharacteristic
28C2×(C2×D83)⋊C8C2211yescharacteristic
29C2×(C2×D83)⋊C8C2211yescharacteristic
30C2×C8×Dic83C2211yescharacteristic
31C2×C8×Dic83C2211yescharacteristic
32#19127570C2211yescharacteristic
33#19127570C2211yescharacteristic
34#19127570C2211yescharacteristic
35#19127570C2211yescharacteristic
36#19127567C2211yescharacteristic
37#19127565—21no—
38#19127560—21no—
39#19127560—21no—
40#19127560—21no—
41#19127558C2211yescharacteristic
42#19127558C2211yescharacteristic
43C4×Dic83⋊C4—21no—
44C4×C4⋊Dic83C2211yescharacteristic
45#19127552—21no—
46#19127552—21no—
47#19127464C2211yescharacteristic
48#19127464C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127463C2211yescharacteristic
52#19127463C2211yescharacteristic
53C2×#107×C83C2211yescharacteristic
54C2×#106×C83C2211yescharacteristic
55C2×C4⋊C8×C83C2211yescharacteristic
56#382×C83C2211yescharacteristic
57C4×C4⋊C4×C83C2211yescharacteristic
58#340×C83C2211yescharacteristic
59#340×C83C2211yescharacteristic
60C22×C4×D83D411yescharacteristic
61C22×C4×D83D411yescharacteristic
62C22×C4×D83C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65#14581750—41no—
66#14581750—41no—
67#14581750—41no—
68#14581750—41no—
69#14581749—44no—
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×Dic83⋊C4—21no—
75C2×Dic83⋊C4—21no—
76C2×Dic83⋊C4—21no—
77C2×Dic83⋊C4—21no—
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—21no—
80C2×Dic83⋊C4—21no—
81C2×Dic83⋊C4—21no—
82C2×(C2×D83)⋊C4—21no—
83C2×(C2×D83)⋊C4—21no—
84C2×(C2×D83)⋊C4—21no—
85C2×(C2×D83)⋊C4—21no—
86C2×C4×Dic83—21no—
87C2×C4×Dic83—21no—
88C2×C4×Dic83—21no—
89C2×C4×Dic83D411yescharacteristic
90C2×C4×Dic83D411yescharacteristic
91C2×C4×Dic83C2311yescharacteristic
92C42×D83C2×C414yes—
93C42×D83Q814yes—
94#14581726—41no—
95#14581725—41no—
96#14581725—41no—
97#14581724—41no—
98#14581723—41no—
99#14581723—41no—
100C4⋊(C83⋊C8)C2×C414yes—
101(C2×D83)⋊C8C2×C414yes—
102(C2×D83)⋊C8C2×C414yes—
103C8×Dic83C2×C414yes—
104C8×Dic83C2×C414yes—
105C83⋊#95—21no—
106C83⋊#95—21no—
107C83⋊#95—21no—
108C83⋊#95—21no—
109C83⋊#95—21no—
110C22×C8×C83C2311yescharacteristic
111C22×C8×C83C2311yescharacteristic
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C42×C83—21no—
117C2×C42×C83C2311yescharacteristic
118#107×C83—41no—
119#106×C83—41no—
120C4⋊C8×C83C2×C414yes—
121#95×C83—21no—
122#95×C83—21no—
123#95×C83—21no—
124C23×D83C2×D411yescharacteristic
125C22×Dic83—21no—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C22×Dic83—21no—
129C22×Dic83C2×D411yescharacteristic
130C22×Dic83C2×D411yescharacteristic
131C22×Dic83C2×D411yescharacteristic
132C2×C4×D83—21no—
133C2×C4×D83—21no—
134C2×C4×D83—21no—
135C2×C4×D83—21no—
136C2×C4×D83—22no—
137C2×C4×D83—22no—
138C2×C4×D83—22no—
139C2×C4×D83—22no—
140C2×C4×D83C4⋊C414yes—
141C2×C4×D83C4⋊C414yes—
142C2×C4×D83C22×C412yes—
143C2×C4×D83C2×Q812yes—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—21no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8C22×C412yes—
149C2×C83⋊C8C22×C412yes—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158Dic83⋊C4—42no—
159Dic83⋊C4—42no—
160Dic83⋊C4—42no—
161Dic83⋊C4—42no—
162Dic83⋊C4—42no—
163Dic83⋊C4—42no—
164Dic83⋊C4—42no—
165Dic83⋊C4—42no—
166(C2×D83)⋊C4—42no—
167(C2×D83)⋊C4—42no—
168(C2×D83)⋊C4—42no—
169(C2×D83)⋊C4—42no—
170C4×Dic83—42no—
171C4×Dic83—42no—
172C4×Dic83—42no—
173C4×Dic83C4⋊C414yes—
174C4×Dic83C4⋊C414yes—
175C4×Dic83C22×C412yes—
176C4×Dic83C2×Q812yes—
177C22×C4×C83—21no—
178C22×C4×C83—21no—
179C22×C4×C83—21no—
180C22×C4×C83—21no—
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2411yescharacteristic
184C2×C8×C83—21no—
185C2×C8×C83—21no—
186C2×C8×C83—21no—
187C2×C8×C83—21no—
188C2×C8×C83C22×C412yes—
189C2×C8×C83C22×C412yes—
190C4⋊C4×C83—21no—
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83—21no—
197C4⋊C4×C83—21no—
198C42×C83—42no—
199C42×C83C22×C412yes—
200C42×C83C2×Q812yes—
201C23×C83C22×D411yescharacteristic
202C22×D83—21no—
203C22×D83—21no—
204C22×D83—21no—
205C22×D83—21no—
206C22×D83—21no—
207C22×D83—21no—
208C22×D83C2×C4⋊C412yes—
209C2×C4×C83—21no—
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—21no—
213C2×C4×C83—41no—
214C2×C4×C83—41no—
215C2×C4×C83—41no—
216C2×C4×C83—41no—
217C2×C4×C83—41no—
218C2×C4×C83—41no—
219C2×C4×C83—41no—
220C2×C4×C83—41no—
221C2×C4×C83—41no—
222C2×C4×C83—41no—
223C2×C4×C83—41no—
224C2×C4×C83—41no—
225C2×C4×C83C2×C4⋊C412yes—
226C2×C4×C83C2×C4⋊C412yes—
227C2×C4×C83C23×C411yescharacteristic
228C2×C4×C83C22×Q811yescharacteristic
229C2×C4×C83H(5,𝔽2)11yescharacteristic
230C2×C4×C83H(5,𝔽2)11yescharacteristic
231C2×C4×C832−1+411yescharacteristic
232C2×C4×C832−1+411yescharacteristic
233C2×Dic83—21no—
234C2×Dic83—21no—
235C2×Dic83—41no—
236C2×Dic83—41no—
237C2×Dic83—41no—
238C2×Dic83—41no—
239C2×Dic83—41no—
240C2×Dic83—41no—
241C2×Dic83—41no—
242C2×Dic83—41no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83C2×C4⋊C412yes—
248C2×Dic83C2×C4⋊C412yes—
249C2×Dic83C2×C4⋊C412yes—
250C2×Dic83#13512yes—
251C2×Dic83#13512yes—
252C2×Dic83AGL(1,ℤ/8)12yes—
253C2×Dic83#13712yes—
254C4×D83—22no—
255C4×D83—22no—
256C4×D83—22no—
257C4×D83—22no—
258C4×D83—28no—
259C4×D83—28no—
260C8×C83—22no—
261C8×C83—22no—
262C83⋊C8—22no—
263C83⋊C8—22no—
264C22×C83C22×C4⋊C411yescharacteristic
265C22×C83#53511yescharacteristic
266C22×C83#53611yescharacteristic
267C22×C83C2×#13511yescharacteristic
268C22×C83C2×#13511yescharacteristic
269C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C83C2×#13711yescharacteristic
271C2×D83—22no—
272C2×D83—22no—
273C2×D83—22no—
274C2×D83—41no—
275C2×D83—41no—
276C2×D83—41no—
277C2×D83—41no—
278C4×C83—22no—
279C4×C83—22no—
280C4×C83—81no—
281C4×C83—81no—
282C4×C83—81no—
283C4×C83—81no—
284C4×C83#52011yescharacteristic
285C4×C83#52011yescharacteristic
286C4×C83#54411yescharacteristic
287C4×C83#54411yescharacteristic
288Dic83—22no—
289Dic83—81no—
290Dic83—81no—
291Dic83—81no—
292Dic83—81no—
293Dic83#42714yes—
294Dic83#42814yes—
295#21984—831nomaximal, Sylow
296C2×C83#231711yescharacteristic
297C2×C83C2×#42711yescharacteristic
298C2×C83C2×#42811yescharacteristic
299C2×C83#320711yescharacteristic
300C2×C83#321011yescharacteristic
301C2×C83#322811yescharacteristic
302C2×C83#323111yescharacteristic
303D83—42no—
304C2×#431—831no—
305#2285—831no—
306#2285—831no—
307#1949—831no—
308#1949—831no—
309#1949—831no—
310#1949D8311yescharacteristic, complemented
311#1854—831no—
312#1854—831no—
313#1854—831no—
314#1854—831no—
315#1845—831no—
316#1844—831no—
317#1834—831no—
318#1834—831no—
319C83#2198411yescharacteristic, complemented, Sylow
320C2×#117—831no—
321C2×#117—831no—
322C2×#117—1661no—
323C22×C42—831no—
324#431—838no—
325C2×#107—831no—
326C2×#107—831no—
327C2×#107—831no—
328C2×#107C2×D8311yescharacteristic
329C2×#106—831no—
330C2×#106—831no—
331C2×#106—831no—
332C2×#106C2×D8311yescharacteristic
333C2×C4⋊C8—831no—
334C2×C4⋊C8C2×D8311yescharacteristic
335C2×C22⋊C8—831no—
336C2×C22⋊C8—831no—
337C2×C4×C8—831no—
338C2×C4×C8—831no—
339#384—831no—
340#384—831no—
341#384—831no—
342#384—831no—
343#382—831no—
344#382—1661no—
345#382C2×D8311yescharacteristic
346#380—831no—
347#380—831no—
348#380—1661no—
349#380—1661no—
350#380—1661no—
351C4×C4⋊C4—831no—
352C4×C4⋊C4—1661no—
353C4×C4⋊C4—1661no—
354C4×C4⋊C4—1661no—
355C4×C4⋊C4C2×D8311yescharacteristic
356#340—831no—
357#340—831no—
358#340—831no—
359#340—831no—
360#340—831no—
361#340—831no—
362#340C2×D8311yescharacteristic
363#340C2×D8311yescharacteristic
364C23×C4—831no—
365C23×C4—831no—
366C23×C4—831no—
367C22×C8—831no—
368C22×C8—831no—
369C22×C8C22×D8311yescharacteristic
370C22×C8C22×D8311yescharacteristic
371#117—3324no—
372#117—3321no—
373#117—3321no—
374#117—3321no—
375#117—3321no—
376C2×C4⋊C4—831no—
377C2×C4⋊C4—831no—
378C2×C4⋊C4—831no—
379C2×C4⋊C4—831no—
380C2×C4⋊C4—1661no—
381C2×C4⋊C4—1661no—
382C2×C4⋊C4—1661no—
383C2×C4⋊C4—1661no—
384C2×C4⋊C4—1661no—
385C2×C4⋊C4—1661no—
386C2×C4⋊C4—1661no—
387C2×C4⋊C4—1661no—
388C2×C4⋊C4C22×D8311yescharacteristic
389C2×C4⋊C4C22×D8311yescharacteristic
390C2×C4⋊C4C22×D8311yescharacteristic
391C2×C4⋊C4C22×D8311yescharacteristic
392C2×C22⋊C4—1661no—
393C2×C22⋊C4—1661no—
394C2×C22⋊C4—1661no—
395C2×C22⋊C4—1661no—
396C2×C42—21no—
397C2×C42—831no—
398C2×C42—831no—
399C2×C42—831no—
400C2×C42—834no—
401C2×C42—834no—
402C2×C42—1661no—
403C2×C42—1661no—
404C2×C42—1661no—
405C2×C42C22×D8311yescharacteristic
406#107—41no—
407#107—3321no—
408#107—3321no—
409#107—3321no—
410#106—41no—
411#106—3321no—
412#106—3321no—
413#106—3321no—
414C4⋊C8—834no—
415C4⋊C8C4×D8314yes—
416C22⋊C8—834no—
417C22⋊C8—834no—
418C4×C8—834no—
419C4×C8—834no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—1661no—
424#95—1661no—
425#95—1661no—
426#95—1661no—
427#95—1661no—
428C24—831no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—831no—
434C22×C4—831no—
435C22×C4—831no—
436C22×C4—832no—
437C22×C4—832no—
438C22×C4—834no—
439C22×C4—834no—
440C22×C4—1662no—
441C22×C4—1662no—
442C22×C4—1662no—
443C22×C4—1662no—
444C22×C4—1661no—
445C22×C4—1661no—
446C22×C4—1661no—
447C22×C4—1661no—
448C22×C4—1661no—
449C22×C4—1661no—
450C22×C4—1661no—
451C22×C4—1661no—
452C22×C4D4×D8311yescharacteristic
453C22×C4D4×D8311yescharacteristic
454C22×C4C23×D8311yescharacteristic
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—832no—
460C2×C8—832no—
461C2×C8—1661no—
462C2×C8—1661no—
463C2×C8—1661no—
464C2×C8—1661no—
465C2×C8C2×C4×D8312yes—
466C2×C8C2×C4×D8312yes—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—3322no—
476C4⋊C4—3322no—
477C4⋊C4—3322no—
478C4⋊C4—3322no—
479C4⋊C4—3322no—
480C4⋊C4—3322no—
481C4⋊C4—3322no—
482C4⋊C4—3322no—
483C4⋊C4—1661no—
484C4⋊C4—1661no—
485C4⋊C4—1661no—
486C4⋊C4—1661no—
487C4⋊C4—1661no—
488C4⋊C4—1661no—
489C4⋊C4—1661no—
490C4⋊C4—1661no—
491C22⋊C4—3322no—
492C22⋊C4—3322no—
493C22⋊C4—3322no—
494C22⋊C4—3322no—
495C42—42no—
496C42—3322no—
497C42—3322no—
498C42—3322no—
499C42—832no—
500C42—832no—
501C42—834no—
502C42—834no—
503C42C2×C4×D8312yes—
504C42Q8×D8312yes—
505C23—832no—
506C23—1661no—
507C23—1661no—
508C23—1661no—
509C23—1661no—
510C23—1661no—
511C23—1661no—
512C23C2×D4×D8311yescharacteristic
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—3321no—
530C2×C4—3321no—
531C2×C4—3321no—
532C2×C4—3321no—
533C2×C4—3321no—
534C2×C4—3321no—
535C2×C4—3321no—
536C2×C4—3321no—
537C2×C4—3321no—
538C2×C4—3321no—
539C2×C4—3321no—
540C2×C4—3321no—
541C2×C4—832no—
542C2×C4—832no—
543C2×C4—832no—
544C2×C4—832no—
545C2×C4—832no—
546C2×C4—832no—
547C2×C4—832no—
548C2×C4—1668no—
549C2×C4—1668no—
550C2×C4—1661no—
551C2×C4—1661no—
552C2×C4—1662no—
553C2×C4—1662no—
554C2×C4—1662no—
555C2×C4—1662no—
556C2×C4C4⋊C4×D8312yes—
557C2×C4C4⋊C4×D8312yes—
558C2×C4C22×C4×D8311yescharacteristic
559C2×C4C2×Q8×D8311yescharacteristic
560C2×C4#1458183711yescharacteristic
561C2×C4#1458183811yescharacteristic
562C2×C4#1458183911yescharacteristic
563C2×C4#1458184011yescharacteristic
564C8—22no—
565C8—22no—
566C8—1662no—
567C8—1662no—
568C22—3321no—
569C22—3321no—
570C22—3321no—
571C22—3321no—
572C22—1662no—
573C22—1662no—
574C22—1662no—
575C22C2×C4⋊C4×D8311yescharacteristic
576C22#1912826711yescharacteristic
577C22#1912827411yescharacteristic
578C22#135×D8311yescharacteristic
579C22#135×D8311yescharacteristic
580C22AGL(1,ℤ/8)×D8311yescharacteristic
581C22#137×D8311yescharacteristic
582C4—22no—
583C4—22no—
584C4—81no—
585C4—81no—
586C4—81no—
587C4—81no—
588C4—834no—
589C4—834no—
590C4—6641no—
591C4—6641no—
592C4—6641no—
593C4—6641no—
594C4—1662no—
595C4#1912818011yescharacteristic
596C4#1912818311yescharacteristic
597C4#1912832811yescharacteristic
598C4#1912833111yescharacteristic
599C2—3322no—
600C2#2491008511yescharacteristic
601C2#427×D8311yescharacteristic
602C2#428×D8311yescharacteristic
603C2#2491839511yescharacteristic
604C2#2491841811yescharacteristic
605C2#2491859811yescharacteristic
606C2#2491862111yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1949C2
socleC23×C83C22×D4
prime radicalC23×D83C2×D4
cosocleC22×C4×C83C24
hypercenter#1949D83
hypocenterC83#21984
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1949

Complements

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

NH ≅ G/Ntype
4. #24908233C2semidirect
5. #24908231C2semidirect
6. #24908229C2semidirect
11. #24907621C2semidirect
12. #24907618C2semidirect
13. #24907589C2semidirect
14. #24907589C2semidirect
15. C83×#1949C2semidirect
20. #19127753C4semidirect
310. #1949D83semidirect
319. C83#21984semidirect

Maximal subgroups

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

subgroupindex
C2×#191277532
#249098692
#249098672
#249082332
#249082312
#249082292
#249076652
#249076632
#249076612
#249076592
#249076212
#249076182
#249075892
C83×#19492
#2198483