Finite Group Databasebeta

← #35463210

Subgroups — #35463210

Order 21248.

774 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×#19128102C211yesmaximal, characteristic, complemented
2#24911181C211yesmaximal, characteristic
3#24911170C211yesmaximal, characteristic
4#24908947C211yesmaximal, characteristic, complemented
5#24908942C211yesmaximal, characteristic, complemented
6#24908803C211yesmaximal, characteristic, complemented
7#24908799C211yesmaximal, characteristic, complemented
8#24908202C211yesmaximal, characteristic, complemented
9#24908189C211yesmaximal, characteristic, complemented
10#24907662C211yesmaximal, characteristic
11#24907656C211yesmaximal, characteristic
12#24907628C211yesmaximal, characteristic, complemented
13#24907496C211yesmaximal, characteristic
14#24907244C211yesmaximal, characteristic
15C2×#428×C83C211yesmaximal, characteristic
16C2×#14581783C2211yescharacteristic
17C2×#14581783C2211yescharacteristic
18#19128102—24no—
19C2×#14581726C2211yescharacteristic
20C2×#14581725C2211yescharacteristic
21C2×#14581724C2211yescharacteristic
22C2×#14581723C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×C22⋊(C83⋊C8)C2211yescharacteristic
26C2×#14581702C2211yescharacteristic
27C2×C8×Dic83C2211yescharacteristic
28#19127640—21no—
29#19127637—21no—
30#19127634—21no—
31#19127624—21no—
32#19127600C2211yescharacteristic
33#19127600C2211yescharacteristic
34#19127599—21no—
35#19127597C2211yescharacteristic
36#19127597C2211yescharacteristic
37#19127585—21no—
38#19127574—21no—
39#19127570C2211yescharacteristic
40#19127566C2211yescharacteristic
41#19127561—21no—
42#19127561C2211yescharacteristic
43#19127561C2211yescharacteristic
44C4×C22⋊Dic83C2211yescharacteristic
45#19127464C2211yescharacteristic
46#19127464C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127455C2211yescharacteristic
52#19127455C2211yescharacteristic
53C22×M16×C83C2211yescharacteristic
54C2×#117×C83C2211yescharacteristic
55C22×C4⋊C4×C83C2211yescharacteristic
56#428×C83—24no—
57C2×#107×C83C2211yescharacteristic
58C2×#107×C83C2211yescharacteristic
59C2×#106×C83C2211yescharacteristic
60C2×#106×C83C2211yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63#14581783—22no—
64#14581783—22no—
65#14581783—41no—
66#14581783—41no—
67C2×C4⋊Dic83—21no—
68C2×C4⋊Dic83—21no—
69C2×C4⋊Dic83—21no—
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×Dic83⋊C4—21no—
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—21no—
80C2×Dic83⋊C4—21no—
81C2×C22⋊Dic83—21no—
82C2×C22⋊Dic83—21no—
83C2×C22⋊Dic83D411yescharacteristic
84C2×C22⋊Dic83D411yescharacteristic
85C2×C4×Dic83C2311yescharacteristic
86C2×C4×Dic83C2311yescharacteristic
87#14581726—22no—
88#14581725—41no—
89#14581724—22no—
90#14581723—22no—
91#14581723—22no—
92#14581723—41no—
93C22⋊(C83⋊C8)Q814yes—
94#14581702—22no—
95C8×Dic83—22no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C83⋊#95—21no—
104C83⋊#95D411yescharacteristic
105C83⋊#95D411yescharacteristic
106C23×C4×C83—21no—
107C23×C4×C83C2311yescharacteristic
108C2×M16×C83—22no—
109C2×M16×C83—22no—
110C2×M16×C83Q814yes—
111C22×C8×C83C2311yescharacteristic
112C22×C8×C83C2311yescharacteristic
113#117×C83—22no—
114#117×C83—41no—
115C2×C4⋊C4×C83—22no—
116C2×C4⋊C4×C83—21no—
117C2×C4⋊C4×C83—41no—
118C2×C4⋊C4×C83C2311yescharacteristic
119C2×C4⋊C4×C83C2311yescharacteristic
120C2×C4⋊C4×C83C2311yescharacteristic
121C2×C4⋊C4×C83C2311yescharacteristic
122C2×C22⋊C4×C83—21no—
123C2×C42×C83—21no—
124#107×C83—22no—
125#107×C83D414yes—
126#106×C83—22no—
127#106×C83—22no—
128C22×Dic83—21no—
129C22×Dic83—21no—
130C22×Dic83—21no—
131C22×Dic83—21no—
132C22×Dic83C2×D411yescharacteristic
133C22×Dic83C2×D411yescharacteristic
134C22×Dic83C2×D411yescharacteristic
135C22×Dic83C2×D411yescharacteristic
136C2×C83⋊C8—21no—
137C2×C83⋊C8—21no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8C2×Q812yes—
141C2×C83⋊C8C2×Q812yes—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—42no—
151C4⋊Dic83—42no—
152C4⋊Dic83—42no—
153C4⋊Dic83—81no—
154Dic83⋊C4—42no—
155Dic83⋊C4—42no—
156Dic83⋊C4—42no—
157Dic83⋊C4—81no—
158Dic83⋊C4—81no—
159Dic83⋊C4—81no—
160C22⋊Dic83—41no—
161C22⋊Dic83—41no—
162C22⋊Dic83—42no—
163C22⋊Dic83—81no—
164C4×Dic83—21no—
165C4×Dic83—21no—
166C4×Dic83—21no—
167C4×Dic83—21no—
168C24×C83C2×D411yescharacteristic
169C22×C4×C83—21no—
170C22×C4×C83—21no—
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—42no—
178C22×C4×C83—42no—
179C22×C4×C83—81no—
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2×Q812yes—
182C22×C4×C83C2×Q812yes—
183C22×C4×C83C2411yescharacteristic
184M16×C83—28no—
185C2×C8×C83—21no—
186C2×C8×C83—21no—
187C2×C8×C83C2×D412yes—
188C2×C8×C83C2×Q812yes—
189C2×C8×C83C2×Q812yes—
190C2×C8×C83#4112yes—
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—42no—
198C4⋊C4×C83C2×D412yes—
199C4⋊C4×C83C2×D412yes—
200C22⋊C4×C83—42no—
201C42×C83—42no—
202C23×C83—21no—
203C23×C83—21no—
204C23×C83—21no—
205C23×C83—21no—
206C23×C83—21no—
207C23×C83—21no—
208C23×C83C4⋊Q812yes—
209C23×C83C22×D411yescharacteristic
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—22no—
213C2×C4×C83—22no—
214C2×C4×C83—22no—
215C2×C4×C83—22no—
216C2×C4×C83—22no—
217C2×C4×C83—22no—
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×C83—41no—
226C2×C4×C83—41no—
227C2×C4×C83—41no—
228C2×C4×C83—84no—
229C2×C4×C83C4⋊Q812yes—
230C2×C4×C83C22×D411yescharacteristic
231C2×C4×C83C22×Q811yescharacteristic
232C2×C4×C83C22×Q811yescharacteristic
233C2×C4×C83C2×#4111yescharacteristic
234C2×C4×C83H(5,𝔽2)11yescharacteristic
235C2×C4×C832−1+411yescharacteristic
236C2×Dic83—21no—
237C2×Dic83—21no—
238C2×Dic83—21no—
239C2×Dic83—21no—
240C2×Dic83—21no—
241C2×Dic83—21no—
242C2×Dic83—21no—
243C2×Dic83—21no—
244C2×Dic83—21no—
245C2×Dic83—21no—
246C2×Dic83—21no—
247C2×Dic83—21no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C8×C83—22no—
261C8×C83C22⋊Q814yes—
262C83⋊C8—22no—
263C83⋊C8—22no—
264C22×C83—22no—
265C22×C83—22no—
266C22×C83—22no—
267C22×C83—41no—
268C22×C83—41no—
269C22×C83—41no—
270C22×C83—41no—
271C22×C83C2×C4⋊Q811yescharacteristic
272C22×C83D4211yescharacteristic
273C22×C83#54711yescharacteristic
274C22×C83#57511yescharacteristic
275C22×C83#57511yescharacteristic
276C22×C83#57711yescharacteristic
277C22×C83#57711yescharacteristic
278C4×C83—22no—
279C4×C83—42no—
280C4×C83—42no—
281C4×C83—81no—
282C4×C83—81no—
283C4×C83C2×C22⋊Q811yescharacteristic
284C4×C83#54411yescharacteristic
285C4×C83D4×Q811yescharacteristic
286C4×C83#55411yescharacteristic
287Dic83—41no—
288Dic83—41no—
289Dic83—41no—
290Dic83—41no—
291Dic83—81no—
292Dic83—81no—
293Dic83—81no—
294Dic83—81no—
295#26075—831nomaximal, Sylow
296C2×C83—42no—
297C2×C83#244011yescharacteristic
298C2×C83#316311yescharacteristic
299C2×C83#316611yescharacteristic
300C2×C83#326711yescharacteristic
301C2×C83#327111yescharacteristic
302C2×C83#335111yescharacteristic
303C2×C83#335111yescharacteristic
304C2×#499—831no—
305C2×#428D8311yescharacteristic
306#2442—831no—
307#2441—831no—
308#2085—831no—
309#2084—831no—
310#2059—831no—
311#2059—831no—
312#1943—831no—
313#1941—831no—
314#1854—831no—
315#1853—831no—
316#1847—831no—
317#1809—831no—
318#1753—831no—
319C83#2607511yescharacteristic, complemented, Sylow
320C22×M16C2×D8311yescharacteristic
321C2×#125—831no—
322C2×#125—831no—
323C2×#117C2×D8311yescharacteristic
324C22×C4⋊C4C2×D8311yescharacteristic
325#499—1664no—
326#428—24no—
327C2×#107—831no—
328C2×#107—831no—
329C2×#107C2×D8311yescharacteristic
330C2×#107C2×D8311yescharacteristic
331C2×#106—831no—
332C2×#106—831no—
333C2×#106—831no—
334C2×#106—831no—
335C2×#106C2×D8311yescharacteristic
336C2×#106C2×D8311yescharacteristic
337C2×C22⋊C8—831no—
338C2×#97—831no—
339C2×C4×C8—831no—
340#400—1661no—
341#399—1661no—
342#398—1661no—
343#396—1661no—
344#389—831no—
345#389—831no—
346#389—831no—
347#389—831no—
348#389—1661no—
349#387—1661no—
350#385—1661no—
351#384—831no—
352#382—831no—
353#380—831no—
354#380—831no—
355#380—1661no—
356C4×C22⋊C4—831no—
357#340—831no—
358#340—831no—
359#340—831no—
360#340—831no—
361#340—831no—
362#340—831no—
363#336—831no—
364#336—831no—
365C23×C4—21no—
366C23×C4C22×D8311yescharacteristic
367C2×M16—22no—
368C2×M16—22no—
369C2×M16Dic16614yes—
370C22×C8—831no—
371C22×C8—831no—
372C22×C8C22×D8311yescharacteristic
373C22×C8C22×D8311yescharacteristic
374#125—3321no—
375#125—3321no—
376#125—1662no—
377#125—1662no—
378#117—22no—
379#117—41no—
380C2×C4⋊C4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—41no—
383C2×C4⋊C4—831no—
384C2×C4⋊C4—831no—
385C2×C4⋊C4—831no—
386C2×C4⋊C4—831no—
387C2×C4⋊C4—1661no—
388C2×C4⋊C4—1661no—
389C2×C4⋊C4—1661no—
390C2×C4⋊C4—1661no—
391C2×C4⋊C4—1661no—
392C2×C4⋊C4—1661no—
393C2×C4⋊C4—1661no—
394C2×C4⋊C4—1661no—
395C2×C4⋊C4—1661no—
396C2×C4⋊C4—1661no—
397C2×C4⋊C4C22×D8311yescharacteristic
398C2×C4⋊C4C22×D8311yescharacteristic
399C2×C4⋊C4C22×D8311yescharacteristic
400C2×C4⋊C4C22×D8311yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—831no—
403C2×C22⋊C4—831no—
404C2×C22⋊C4—1661no—
405C2×C22⋊C4—1661no—
406C2×C42—21no—
407C2×C42—831no—
408C2×C42—831no—
409#107—22no—
410#107—3321no—
411#107—1662no—
412#107C22⋊D8314yes—
413#106—22no—
414#106—22no—
415#106—3321no—
416#106—1662no—
417#106—1662no—
418#106—1662no—
419C22⋊C8—834no—
420#97—1662no—
421C4×C8—1662no—
422#95—831no—
423#95—831no—
424#95—1661no—
425#95—1661no—
426#95—1661no—
427#95—1661no—
428#95—1661no—
429#95—1661no—
430#95—1661no—
431#95—1661no—
432C24D4×D8311yescharacteristic
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—81no—
444C22×C4—831no—
445C22×C4—831no—
446C22×C4—831no—
447C22×C4—831no—
448C22×C4—1661no—
449C22×C4—1661no—
450C22×C4—1661no—
451C22×C4—1661no—
452C22×C4D4×D8311yescharacteristic
453C22×C4Q8×D8312yes—
454C22×C4C2×Dic16612yes—
455C22×C4C23×D8311yescharacteristic
456M16—28no—
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×C22⋊D8312yes—
466C2×C8C2×Dic16612yes—
467C2×C8C2×Dic16612yes—
468C2×C8#1305059012yes—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—42no—
476C4⋊C4—3322no—
477C4⋊C4—3322no—
478C4⋊C4—3322no—
479C4⋊C4—3322no—
480C4⋊C4—3322no—
481C4⋊C4—3322no—
482C4⋊C4—6641no—
483C4⋊C4—6641no—
484C4⋊C4—6641no—
485C4⋊C4—6641no—
486C4⋊C4—1661no—
487C4⋊C4—1661no—
488C4⋊C4—1661no—
489C4⋊C4—1661no—
490C4⋊C4—1661no—
491C4⋊C4—1661no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4C2×C22⋊D8312yes—
495C4⋊C4C2×C22⋊D8312yes—
496C22⋊C4—42no—
497C22⋊C4—3322no—
498C22⋊C4—3321no—
499C22⋊C4—3321no—
500C22⋊C4—6641no—
501C42—42no—
502C42—1661no—
503C42—1661no—
504C42—1661no—
505C42—1661no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23#1458179112yes—
513C23C2×D4×D8311yescharacteristic
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—84no—
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—3321no—
542C2×C4—3321no—
543C2×C4—3321no—
544C2×C4—3321no—
545C2×C4—1661no—
546C2×C4—1661no—
547C2×C4—1661no—
548C2×C4—1661no—
549C2×C4—1661no—
550C2×C4—1661no—
551C2×C4—1661no—
552C2×C4—1661no—
553C2×C4—1661no—
554C2×C4—1661no—
555C2×C4—1661no—
556C2×C4—1661no—
557C2×C4#1458179112yes—
558C2×C4C22×C22⋊D8311yescharacteristic
559C2×C4C2×Q8×D8311yescharacteristic
560C2×C4C22×Dic16611yescharacteristic
561C2×C4C2×#1305059011yescharacteristic
562C2×C4#1458183711yescharacteristic
563C2×C4#1458184011yescharacteristic
564C8—22no—
565C8—1662no—
566C8—1662no—
567C8#1458177114yes—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22C2×#1458179111yescharacteristic
576C22D4×C22⋊D8311yescharacteristic
577C22#1912835211yescharacteristic
578C22#1912849711yescharacteristic
579C22#1912850011yescharacteristic
580C22#1912851011yescharacteristic
581C22#1912851111yescharacteristic
582C4—22no—
583C4—42no—
584C4—42no—
585C4—81no—
586C4—81no—
587C4—3321no—
588C4—3321no—
589C4—3321no—
590C4—3321no—
591C4—6641no—
592C4—6641no—
593C4—6641no—
594C4—6641no—
595C4C2×#1458177111yescharacteristic
596C4#1912833211yescharacteristic
597C4Q8×C22⋊D8311yescharacteristic
598C4#1912840611yescharacteristic
599C2—42no—
600C2#2491115911yescharacteristic
601C2#2491799311yescharacteristic
602C2#2491801311yescharacteristic
603C2#2491898211yescharacteristic
604C2#2491904211yescharacteristic
605C2#2492001911yescharacteristic
606C2#2492002411yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC2×#428×C83C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenterC2×#428D83
hypocenterC83#26075
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
1. C2×#19128102C2semidirect
4. #24908947C2semidirect
5. #24908942C2semidirect
6. #24908803C2semidirect
7. #24908799C2semidirect
8. #24908202C2semidirect
9. #24908189C2semidirect
12. #24907628C2semidirect
319. C83#26075semidirect

Maximal subgroups

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

subgroupindex
C2×#191281022
#249111812
#249111702
#249089472
#249089422
#249088032
#249087992
#249082022
#249081892
#249076622
#249076562
#249076282
#249074962
#249072442
C2×#428×C832
#2607583