Finite Group Databasebeta

← #35467334

Subgroups — #35467334

Order 21248.

804 conjugacy classes of subgroups.

The subgroup lattice has 626 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128008C211yesmaximal, characteristic, complemented
2C2×#19127744C211yesmaximal, characteristic
3#24911187C211yesmaximal, characteristic
4#24909011C211yesmaximal, characteristic, complemented
5#24908959C211yesmaximal, characteristic, complemented
6#24908858C211yesmaximal, characteristic, complemented
7#24908801C211yesmaximal, characteristic, complemented
8#24908231C211yesmaximal, characteristic, complemented
9#24908041C211yesmaximal, characteristic, complemented
10#24907897C211yesmaximal, characteristic
11#24907771C211yesmaximal, characteristic
12#24907731C211yesmaximal, characteristic, complemented
13#24907662C211yesmaximal, characteristic
14#24907648C211yesmaximal, characteristic
15C83×#2362C211yesmaximal, characteristic
16C22×#13050574C2211yescharacteristic
17C2×#14581784C2211yescharacteristic
18C22×C4⋊Dic83C2211yescharacteristic
19#19128008—24no—
20#19127744—24no—
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×#14581723C2211yescharacteristic
27C2×C4⋊(C83⋊C8)C2211yescharacteristic
28C2×C4⋊(C83⋊C8)C2211yescharacteristic
29C2×#14581716C2211yescharacteristic
30C2×#14581716C2211yescharacteristic
31C2×#14581716C2211yescharacteristic
32C2×#14581716C2211yescharacteristic
33C2×C22⋊(C83⋊C8)C2211yescharacteristic
34#19127643—21no—
35#19127635—21no—
36#19127599C2211yescharacteristic
37#19127599C2211yescharacteristic
38#19127573—21no—
39#19127567C2211yescharacteristic
40#19127561C2211yescharacteristic
41#19127559—21no—
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×C22⋊Q8×C83C2211yescharacteristic
51C2×C4×Q8×C83C2211yescharacteristic
52C2×#117×C83C2211yescharacteristic
53#391×C83C2211yescharacteristic
54#389×C83C2211yescharacteristic
55#388×C83—21no—
56#388×C83—21no—
57#387×C83—21no—
58C4×C4⋊C4×C83—21no—
59C4×C4⋊C4×C83C2211yescharacteristic
60C4×C22⋊C4×C83C2211yescharacteristic
61C23×Dic83—21no—
62C2×#13050574—22no—
63C2×#13050574—22no—
64C2×#13050574—22no—
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69#14581784—22no—
70#14581784—41no—
71C2×C4⋊Dic83—22no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83—41no—
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—21no—
80C2×Dic83⋊C4—21no—
81C2×Dic83⋊C4—21no—
82C2×C22⋊Dic83—21no—
83#14581725—22no—
84#14581725—22no—
85#14581725—22no—
86#14581723—22no—
87#14581723—41no—
88#14581723D414yes—
89C4⋊(C83⋊C8)—22no—
90C4⋊(C83⋊C8)—22no—
91#14581716—22no—
92#14581716—22no—
93#14581716—22no—
94#14581716—41no—
95C22⋊(C83⋊C8)—22no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C22×Q8×C83C2311yescharacteristic
102C23×C4×C83C2311yescharacteristic
103C22⋊Q8×C83—22no—
104C22⋊Q8×C83Q814yes—
105C4×Q8×C83—22no—
106C4×Q8×C83—41no—
107#117×C83—22no—
108#117×C83—41no—
109C2×C4⋊C4×C83—21no—
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83—21no—
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C22⋊C4×C83—21no—
116C2×C22⋊C4×C83—21no—
117C2×C22⋊C4×C83D411yescharacteristic
118C2×C22⋊C4×C83D411yescharacteristic
119C2×C42×C83—21no—
120C2×C42×C83—21no—
121C2×C42×C83—21no—
122C2×C42×C83C2311yescharacteristic
123C2×C42×C83C2311yescharacteristic
124#95×C83—21no—
125#95×C83—21no—
126#95×C83—21no—
127#95×C83—21no—
128#95×C83D411yescharacteristic
129#95×C83D411yescharacteristic
130C22×Dic83—21no—
131C22×Dic83—21no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83—42no—
135C22×Dic83—42no—
136C22×Dic83—81no—
137#13050574—28no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8—21no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—21no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8C2×D412yes—
149C2×C83⋊C8#4112yes—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—42no—
158C4⋊Dic83C2×D412yes—
159Dic83⋊C4—42no—
160Dic83⋊C4—42no—
161Dic83⋊C4—81no—
162Dic83⋊C4—81no—
163C22⋊Dic83—42no—
164C24×C83C2×D411yescharacteristic
165C2×Q8×C83—21no—
166C2×Q8×C83—21no—
167C2×Q8×C83—21no—
168C2×Q8×C83—21no—
169C2×Q8×C83—21no—
170C2×Q8×C83C2×Q812yes—
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83—21no—
174C22×C4×C83—21no—
175C22×C4×C83—21no—
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83—21no—
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2×D411yescharacteristic
184C22×C4×C83C2×Q812yes—
185C22×C4×C83#4112yes—
186C22×C4×C83C2411yescharacteristic
187C4⋊C4×C83—24no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—21no—
190C4⋊C4×C83—21no—
191C4⋊C4×C83—21no—
192C4⋊C4×C83—42no—
193C4⋊C4×C83—42no—
194C4⋊C4×C83C2×D412yes—
195C4⋊C4×C83C2×Q812yes—
196C22⋊C4×C83—24no—
197C22⋊C4×C83—41no—
198C22⋊C4×C83—41no—
199C22⋊C4×C83—42no—
200C42×C83—21no—
201C42×C83—21no—
202C42×C83—21no—
203C42×C83—21no—
204C42×C83—42no—
205C42×C83—42no—
206C42×C83—42no—
207C23×C83—21no—
208C23×C83—21no—
209C23×C83—21no—
210C23×C83—21no—
211C23×C83—21no—
212C23×C83—21no—
213C23×C83C22⋊Q812yes—
214C23×C83C22×D411yescharacteristic
215Q8×C83—22no—
216Q8×C83—22no—
217Q8×C83—41no—
218Q8×C83—41no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—22no—
222C2×C4×C83—22no—
223C2×C4×C83—22no—
224C2×C4×C83—22no—
225C2×C4×C83—22no—
226C2×C4×C83—22no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—22no—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
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—41no—
241C2×C4×C83—41no—
242C2×C4×C83—41no—
243C2×C4×C83—41no—
244C2×C4×C83—41no—
245C2×C4×C83—41no—
246C2×C4×C83—41no—
247C2×C4×C83—41no—
248C2×C4×C83—41no—
249C2×C4×C83C22⋊Q812yes—
250C2×C4×C83C22×D411yescharacteristic
251C2×C4×C83C22×Q811yescharacteristic
252C2×C4×C83C2×#4111yescharacteristic
253C2×C4×C83C2×#4111yescharacteristic
254C2×C4×C83H(5,𝔽2)11yescharacteristic
255C2×C4×C832−1+411yescharacteristic
256C2×Dic83—22no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—41no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C2×Dic83—41no—
268C2×Dic83—84no—
269C83⋊C8—22no—
270C83⋊C8—22no—
271C83⋊C8—41no—
272C83⋊C8—41no—
273C22×C83—22no—
274C22×C83—22no—
275C22×C83—22no—
276C22×C83—41no—
277C22×C83—41no—
278C22×C83—41no—
279C22×C83—41no—
280C22×C83C2×C22⋊Q811yescharacteristic
281C22×C83D4211yescharacteristic
282C22×C83#54711yescharacteristic
283C22×C83#57511yescharacteristic
284C22×C83#57511yescharacteristic
285C22×C83#57711yescharacteristic
286C22×C83#57811yescharacteristic
287C4×C83—22no—
288C4×C83—42no—
289C4×C83—41no—
290C4×C83—41no—
291C4×C83—42no—
292C4×C83—41no—
293C4×C83—42no—
294C4×C83—41no—
295C4×C83—81no—
296C4×C83#52511yescharacteristic
297C4×C83#53911yescharacteristic
298C4×C83D4×Q811yescharacteristic
299C4×C83#55411yescharacteristic
300Dic83—42no—
301Dic83—81no—
302Dic83—81no—
303Dic83—81no—
304#26367—831nomaximal, Sylow
305C2×C83—42no—
306C2×C83#244911yescharacteristic
307C2×C83#308011yescharacteristic
308C2×C83#308211yescharacteristic
309C2×C83#326911yescharacteristic
310C2×C83#327511yescharacteristic
311C2×C83#334911yescharacteristic
312C2×C83#335111yescharacteristic
313C2×#479—831no—
314C2×#428—831no—
315#2442—831no—
316#2362D8311yescharacteristic
317#2097—831no—
318#2087—831no—
319#2068—831no—
320#2059—831no—
321#1949—831no—
322#1919—831no—
323#1894—831no—
324#1872—831no—
325#1865—831no—
326#1854—831no—
327#1851—831no—
328C83#2636711yescharacteristic, complemented, Sylow
329C22×M16—831no—
330C2×#125—831no—
331C2×C22⋊Q8C2×D8311yescharacteristic
332C2×C4×Q8C2×D8311yescharacteristic
333C2×#117C2×D8311yescharacteristic
334C22×C4⋊C4—831no—
335#479—1664no—
336#428—1664no—
337C2×#107—831no—
338C2×#107—831no—
339C2×#107—831no—
340C2×#106—831no—
341C2×#106—831no—
342C2×#106—831no—
343C2×C4⋊C8—831no—
344C2×C4⋊C8—831no—
345C2×#103—831no—
346C2×#103—831no—
347C2×#103—831no—
348C2×#103—831no—
349C2×C22⋊C8—831no—
350#400—1661no—
351#398—1661no—
352#391C2×D8311yescharacteristic
353#389—831no—
354#389—831no—
355#389C2×D8311yescharacteristic
356#388—21no—
357#388—21no—
358#387—21no—
359#385—1661no—
360#382—831no—
361#380—831no—
362#380—1661no—
363C4×C4⋊C4—21no—
364C4×C4⋊C4C2×D8311yescharacteristic
365C4×C22⋊C4C2×D8311yescharacteristic
366#340—831no—
367#340—831no—
368#340—831no—
369#340—831no—
370#340—831no—
371#340—831no—
372#336—831no—
373#336—831no—
374C22×Q8C22×D8311yescharacteristic
375C23×C4—1661no—
376C23×C4C22×D8311yescharacteristic
377C2×M16—1662no—
378C2×M16—1662no—
379C2×M16—1662no—
380C22×C8—831no—
381C22×C8—831no—
382C22×C8—831no—
383C22×C8—831no—
384#125—3321no—
385#125—1662no—
386C22⋊Q8—22no—
387C22⋊Q8Dic16614yes—
388C4×Q8—22no—
389C4×Q8—41no—
390#117—22no—
391#117—41no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—3321no—
396C2×C4⋊C4—831no—
397C2×C4⋊C4—831no—
398C2×C4⋊C4—831no—
399C2×C4⋊C4—831no—
400C2×C4⋊C4—1661no—
401C2×C4⋊C4—1661no—
402C2×C4⋊C4—1661no—
403C2×C4⋊C4—1662no—
404C2×C4⋊C4—1661no—
405C2×C4⋊C4—1661no—
406C2×C4⋊C4C22×D8311yescharacteristic
407C2×C4⋊C4C22×D8311yescharacteristic
408C2×C4⋊C4C22×D8311yescharacteristic
409C2×C22⋊C4—21no—
410C2×C22⋊C4—21no—
411C2×C22⋊C4—1661no—
412C2×C22⋊C4C22⋊D8311yescharacteristic
413C2×C22⋊C4C22⋊D8311yescharacteristic
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—21no—
417C2×C42C22×D8311yescharacteristic
418C2×C42C22×D8311yescharacteristic
419#107—1662no—
420#107—1662no—
421#107—1662no—
422#106—3321no—
423#106—834no—
424#106—1662no—
425C4⋊C8—1662no—
426C4⋊C8—1662no—
427#103—3321no—
428#103—1662no—
429#103—1662no—
430#103—1662no—
431C22⋊C8—1662no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—1661no—
437#95—1661no—
438#95—1661no—
439#95—1661no—
440#95—1661no—
441#95C22⋊D8311yescharacteristic
442#95C22⋊D8311yescharacteristic
443C24C2×C22⋊D8311yescharacteristic
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8C2×Dic16612yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—3322no—
459C22×C4—3322no—
460C22×C4—6641no—
461C22×C4—1661no—
462C22×C4—1661no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4C2×C22⋊D8311yescharacteristic
466C22×C4C2×C22⋊D8311yescharacteristic
467C22×C4C2×C22⋊D8311yescharacteristic
468C22×C4C2×C22⋊D8311yescharacteristic
469C22×C4C2×C22⋊D8311yescharacteristic
470C22×C4C2×Dic16612yes—
471C22×C4#1305059012yes—
472C22×C4C23×D8311yescharacteristic
473M16—1668no—
474C2×C8—832no—
475C2×C8—832no—
476C2×C8—1661no—
477C2×C8—1661no—
478C2×C8—1661no—
479C2×C8—1661no—
480C2×C8—1661no—
481C2×C8—1661no—
482C2×C8—1661no—
483C2×C8—1661no—
484C2×C8—1661no—
485C2×C8—1661no—
486C4⋊C4—24no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—42no—
492C4⋊C4—42no—
493C4⋊C4—3322no—
494C4⋊C4—3322no—
495C4⋊C4—3322no—
496C4⋊C4—832no—
497C4⋊C4—6641no—
498C4⋊C4—6641no—
499C4⋊C4—1661no—
500C4⋊C4—1661no—
501C4⋊C4—1661no—
502C4⋊C4—1661no—
503C4⋊C4—1661no—
504C4⋊C4—1661no—
505C4⋊C4—1661no—
506C4⋊C4D4×D8312yes—
507C4⋊C4C2×Dic16612yes—
508C22⋊C4—24no—
509C22⋊C4—41no—
510C22⋊C4—41no—
511C22⋊C4—42no—
512C22⋊C4—3322no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—42no—
518C42—42no—
519C42—42no—
520C23—21no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23#1458177112yes—
527C23C22×C22⋊D8311yescharacteristic
528Q8—22no—
529Q8—22no—
530Q8—41no—
531Q8—41no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—22no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—3321no—
563C2×C4—3321no—
564C2×C4—3321no—
565C2×C4—3321no—
566C2×C4—3321no—
567C2×C4—3321no—
568C2×C4—3321no—
569C2×C4—3321no—
570C2×C4—3321no—
571C2×C4—3321no—
572C2×C4—3321no—
573C2×C4—6644no—
574C2×C4—1662no—
575C2×C4#1458177112yes—
576C2×C4C2×D4×D8311yescharacteristic
577C2×C4C22×Dic16611yescharacteristic
578C2×C4#41×D8311yescharacteristic
579C2×C4C2×#1305059011yescharacteristic
580C2×C4#1458183711yescharacteristic
581C2×C4#1458183911yescharacteristic
582C8—3321no—
583C8—3321no—
584C8—1662no—
585C8—1662no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—41no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22C2×#1458177111yescharacteristic
594C22D4×C22⋊D8311yescharacteristic
595C22#1912835011yescharacteristic
596C22#1912849911yescharacteristic
597C22#1912849911yescharacteristic
598C22#1912850911yescharacteristic
599C22#1912851611yescharacteristic
600C4—22no—
601C4—42no—
602C4—41no—
603C4—42no—
604C4—41no—
605C4—41no—
606C4—41no—
607C4—42no—
608C4—81no—
609C4—3322no—
610C4—6641no—
611C4—6641no—
612C4—6641no—
613C4#1912821011yescharacteristic
614C4#1912829311yescharacteristic
615C4D4×Dic16611yescharacteristic
616C4#1912841011yescharacteristic
617C2—42no—
618C2#2491126611yescharacteristic
619C2#2491728511yescharacteristic
620C2#2491730511yescharacteristic
621C2#2491901411yescharacteristic
622C2#2491909511yescharacteristic
623C2#2491998711yescharacteristic
624C2#2492002311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2362C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#2362D83
hypocenterC83#26367
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2362

Complements

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

NH ≅ G/Ntype
1. C2×#19128008C2semidirect
4. #24909011C2semidirect
5. #24908959C2semidirect
6. #24908858C2semidirect
7. #24908801C2semidirect
8. #24908231C2semidirect
9. #24908041C2semidirect
12. #24907731C2semidirect
328. C83#26367semidirect

Maximal subgroups

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

subgroupindex
C2×#191280082
C2×#191277442
#249111872
#249090112
#249089592
#249088582
#249088012
#249082312
#249080412
#249078972
#249077712
#249077312
#249076622
#249076482
C83×#23622
#2636783