Finite Group Databasebeta

← #35422906

Subgroups — #35422906

Order 21248.

850 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127823C211yesmaximal, characteristic
2#24916339C211yesmaximal, characteristic
3#24916061C211yesmaximal, characteristic, complemented
4#24916056C211yesmaximal, characteristic, complemented
5#24908248C211yesmaximal, characteristic, complemented
6#24908241C211yesmaximal, characteristic, complemented
7#24907696C211yesmaximal, characteristic, complemented
8#24907694C211yesmaximal, characteristic, complemented
9#24907689C211yesmaximal, characteristic, complemented
10#24907687C211yesmaximal, characteristic, complemented
11#24907628C211yesmaximal, characteristic, complemented
12#24907525C211yesmaximal, characteristic, complemented
13#24907520C211yesmaximal, characteristic, complemented
14#24907348C211yesmaximal, characteristic, complemented
15C83×#2952C211yesmaximal, characteristic
16C2×#14581799C2211yescharacteristic
17C2×#14581799C2211yescharacteristic
18#19128184C2211yescharacteristic
19#19128184C2211yescharacteristic
20#19127823C418yescomplemented
21#19127802—21no—
22#19127802—21no—
23Q16×Dic83—21no—
24D8×Dic83—21no—
25C2×#14581728C2211yescharacteristic
26C2×#14581727C2211yescharacteristic
27#19127744—21no—
28#19127744—21no—
29#19127744C2211yescharacteristic
30#19127744C2211yescharacteristic
31#19127741—21no—
32#19127741—21no—
33#19127741C2211yescharacteristic
34#19127741C2211yescharacteristic
35#19127737—21no—
36#19127737—21no—
37#19127737C2211yescharacteristic
38C2×#14581726C2211yescharacteristic
39C2×#14581725—21no—
40C2×#14581725C2211yescharacteristic
41C2×#14581723—21no—
42C2×#14581723C2211yescharacteristic
43C2×#14581719C2211yescharacteristic
44C2×#14581719C2211yescharacteristic
45#19127702C2211yescharacteristic
46#19127702C2211yescharacteristic
47#19127656C2211yescharacteristic
48C2×#14581702C2211yescharacteristic
49#19127570C2211yescharacteristic
50#19127566C2211yescharacteristic
51#19127470C2211yescharacteristic
52#19127470C2211yescharacteristic
53#19127469C2211yescharacteristic
54#19127469C2211yescharacteristic
55#19127462C2211yescharacteristic
56#19127460C2211yescharacteristic
57C2×#135×C83C2211yescharacteristic
58#520×C83C2211yescharacteristic
59#520×C83C2211yescharacteristic
60#442×C83—21no—
61#441×C83—21no—
62C4×SD16×C83—21no—
63C4×SD16×C83—21no—
64C2×#106×C83C2211yescharacteristic
65#417×C83C2211yescharacteristic
66#417×C83C2211yescharacteristic
67#405×C83C2211yescharacteristic
68#14581799—22no—
69#14581799—22no—
70#14581799C2×C414yescomplemented
71#14581799C2×C414yescomplemented
72C2×#13050574—21no—
73C2×#13050574—21no—
74C2×#13050574C2311yescharacteristic
75C2×#13050574C2311yescharacteristic
76C2×#13050574C2311yescharacteristic
77C22×C83⋊C8—21no—
78C22×C83⋊C8—21no—
79C22×C83⋊C8C2311yescharacteristic
80Q8×Dic83—21no—
81Q8×Dic83—21no—
82D4×Dic83—21no—
83D4×Dic83—21no—
84D4×Dic83—41no—
85D4×Dic83—41no—
86#14581751—21no—
87#14581751—21no—
88#14581751C2311yescharacteristic
89C2×C4⋊Dic83—21no—
90C2×C4⋊Dic83—21no—
91C2×C4⋊Dic83C2311yescharacteristic
92C2×C4⋊Dic83C2311yescharacteristic
93C2×Dic83⋊C4—41no—
94C2×C4×Dic83C2311yescharacteristic
95#14581728C2×C414yescomplemented
96#14581727—22no—
97#14581726—21no—
98#14581726—21no—
99#14581725—21no—
100#14581725—21no—
101#14581725—21no—
102#14581725—21no—
103#14581725—21no—
104#14581725—21no—
105#14581725D411yescharacteristic
106#14581725D411yescharacteristic
107#14581723—21no—
108#14581723—21no—
109#14581723—21no—
110#14581723—21no—
111#14581723—21no—
112#14581723—21no—
113#14581723D411yescharacteristic
114#14581723D411yescharacteristic
115#14581719C2×C414yescomplemented
116#14581719C2×C414yescomplemented
117#14581716—21no—
118#14581716—21no—
119#14581713—21no—
120#14581713—21no—
121#14581702—21no—
122#14581702C2×C414yescomplemented
123C8×Dic83—21no—
124C83⋊#95—41no—
125C2×#41×C83C2311yescharacteristic
126C2×#41×C83C2311yescharacteristic
127#135×C83C2×C414yes—
128#135×C83Q814yes—
129C2×Q16×C83D411yescharacteristic
130C2×SD16×C83D411yescharacteristic
131C2×SD16×C83D411yescharacteristic
132C2×D8×C83D411yescharacteristic
133C2×M16×C83C2311yescharacteristic
134C22×C8×C83C2311yescharacteristic
135C4×Q8×C83—21no—
136C4×Q8×C83—21no—
137C4×D4×C83—21no—
138C4×D4×C83—21no—
139C4×D4×C83—41no—
140C4×D4×C83—41no—
141#117×C83—21no—
142#117×C83—21no—
143#117×C83C2311yescharacteristic
144C2×C4⋊C4×C83—21no—
145C2×C4⋊C4×C83—21no—
146C2×C4⋊C4×C83C2311yescharacteristic
147C2×C4⋊C4×C83C2311yescharacteristic
148#106×C83—21no—
149#106×C83—21no—
150#103×C83—21no—
151#103×C83—21no—
152#102×C83—21no—
153#102×C83—21no—
154#97×C83—21no—
155C4×C8×C83—21no—
156C22×Dic83—21no—
157C22×Dic83—21no—
158C22×Dic83—21no—
159C22×Dic83—41no—
160C22×Dic83—41no—
161#13050574—21no—
162#13050574—21no—
163#13050574—24no—
164#13050574—24no—
165#13050574—21no—
166#13050574—21no—
167#13050574C22×C412yes—
168#13050574C22×C412yes—
169C2×C83⋊C8—21no—
170C2×C83⋊C8—21no—
171C2×C83⋊C8—24no—
172C2×C83⋊C8—24no—
173C2×C83⋊C8—21no—
174C2×C83⋊C8C22×C412yes—
175C2×C83⋊C8C2×D411yescharacteristic
176C2×C83⋊C8C2×D411yescharacteristic
177C2×C83⋊C8#4112yes—
178C4⋊Dic83—21no—
179C4⋊Dic83—21no—
180C4⋊Dic83—21no—
181C4⋊Dic83—21no—
182C4⋊Dic83—41no—
183C4⋊Dic83—41no—
184C4⋊Dic83C2×D411yescharacteristic
185C4⋊Dic83C2×D411yescharacteristic
186Dic83⋊C4—82no—
187C22⋊Dic83—21no—
188C22⋊Dic83—41no—
189C22⋊Dic83—41no—
190C4×Dic83—21no—
191C4×Dic83—21no—
192C4×Dic83—41no—
193C4×Dic83—41no—
194C4×Dic83C22×C412yes—
195#41×C83—21no—
196#41×C83—21no—
197#41×C83—21no—
198#41×C83—21no—
199#41×C83C22×C412yes—
200#41×C83C22×C412yes—
201#41×C83C2×Q812yes—
202#41×C83C2×Q812yes—
203C2×Q8×C83C2×D411yescharacteristic
204C2×Q8×C83C2×D411yescharacteristic
205C2×D4×C83—21no—
206C2×D4×C83—21no—
207C2×D4×C83C2×D411yescharacteristic
208C2×D4×C83C2×D411yescharacteristic
209C22×C4×C83—21no—
210C22×C4×C83—21no—
211C22×C4×C83—21no—
212C22×C4×C83—21no—
213C22×C4×C83—41no—
214C22×C4×C83—41no—
215C22×C4×C83C2411yescharacteristic
216Q16×C83C4⋊C414yescomplemented
217SD16×C83C4⋊C414yescomplemented
218SD16×C83C4⋊C414yescomplemented
219D8×C83C4⋊C414yescomplemented
220M16×C83—21no—
221M16×C83—21no—
222C2×C8×C83—21no—
223C2×C8×C83C22×C412yes—
224C2×C8×C83C2×D411yescharacteristic
225C2×C8×C83C2×D411yescharacteristic
226C2×C8×C83C2×Q812yes—
227C4⋊C4×C83—21no—
228C4⋊C4×C83—21no—
229C4⋊C4×C83—21no—
230C4⋊C4×C83—21no—
231C4⋊C4×C83—41no—
232C4⋊C4×C83—41no—
233C4⋊C4×C83C2×D411yescharacteristic
234C4⋊C4×C83C2×D411yescharacteristic
235C22⋊C4×C83—21no—
236C22⋊C4×C83—41no—
237C22⋊C4×C83—41no—
238C42×C83—21no—
239C42×C83—41no—
240C42×C83—41no—
241C23×C83—21no—
242C23×C83—21no—
243C23×C83H(5,𝔽2)11yescharacteristic
244Q8×C83—21no—
245Q8×C83—21no—
246Q8×C83C2×C4⋊C412yes—
247Q8×C83C2×C4⋊C412yes—
248D4×C83—22no—
249D4×C83—22no—
250D4×C83—22no—
251D4×C83—22no—
252D4×C83—21no—
253D4×C83—21no—
254D4×C83C2×C4⋊C412yes—
255D4×C83C2×C4⋊C412yes—
256C2×C4×C83—21no—
257C2×C4×C83—21no—
258C2×C4×C83—22no—
259C2×C4×C83—22no—
260C2×C4×C83—22no—
261C2×C4×C83—22no—
262C2×C4×C83—21no—
263C2×C4×C83—21no—
264C2×C4×C83—41no—
265C2×C4×C83—41no—
266C2×C4×C83—81no—
267C2×C4×C83—81no—
268C2×C4×C83—161no—
269C2×C4×C83—161no—
270C2×C4×C83C23×C411yescharacteristic
271C2×C4×C83C22×D411yescharacteristic
272C2×C4×C83C22×D411yescharacteristic
273C2×C4×C83C22×Q811yescharacteristic
274C2×C4×C83C2×#4111yescharacteristic
275C2×C4×C832−1+411yescharacteristic
276C2×Dic83—22no—
277C2×Dic83—21no—
278C2×Dic83—21no—
279C2×Dic83—41no—
280C2×Dic83—41no—
281C2×Dic83—41no—
282C2×Dic83—41no—
283C2×Dic83—81no—
284C2×Dic83—81no—
285C2×Dic83—161no—
286C2×Dic83—161no—
287C8×C83—41no—
288C8×C83C2×C4⋊C412yes—
289C8×C83C2×C4⋊C412yes—
290C83⋊C8—22no—
291C83⋊C8—22no—
292C83⋊C8—41no—
293C83⋊C8C4×D412yes—
294C83⋊C8C4×D412yes—
295C22×C83—21no—
296C22×C83—21no—
297C22×C83—22no—
298C22×C83—22no—
299C22×C83—41no—
300C22×C83—41no—
301C22×C83—41no—
302C22×C83—41no—
303C22×C83#52011yescharacteristic
304C22×C83D4211yescharacteristic
305C22×C83#55411yescharacteristic
306C4×C83—22no—
307C4×C83—22no—
308C4×C83—41no—
309C4×C83—41no—
310C4×C83—81no—
311C4×C83—81no—
312C4×C83C22×C4⋊C411yescharacteristic
313C4×C83C2×C4×D411yescharacteristic
314C4×C83#54711yescharacteristic
315C4×C83D4×Q811yescharacteristic
316Dic83—42no—
317Dic83—41no—
318Dic83—41no—
319Dic83—81no—
320Dic83—81no—
321#22693—831nomaximal, Sylow
322C2×C83—21no—
323C2×C83—21no—
324C2×C83—42no—
325C2×C83—42no—
326C2×C83D4×C4⋊C411yescharacteristic
327C2×C83#328811yescharacteristic
328C2×C83#328911yescharacteristic
329C2×#444—831no—
330#2952D8311yescharacteristic
331#2951—831no—
332#2913—831no—
333#2912—831no—
334#1952—831no—
335#1951—831no—
336#1860—831no—
337#1860—831no—
338#1859—831no—
339#1859—831no—
340#1847—831no—
341#1816—831no—
342#1815—831no—
343#1773—831no—
344C83#2269311yescharacteristic, complemented, Sylow
345C2×#135C2×D8311yescharacteristic
346C2×#131—831no—
347C2×#131—831no—
348#520—831no—
349#520—831no—
350#520C2×D8311yescharacteristic
351#520C2×D8311yescharacteristic
352#444—838no—
353#442—21no—
354#441—21no—
355#440—1661no—
356#440—1661no—
357C4×Q16—1661no—
358C4×SD16—21no—
359C4×SD16—21no—
360C4×D8—1661no—
361C2×#108—831no—
362C2×#108—831no—
363#428—831no—
364#428—831no—
365#428—831no—
366#428—831no—
367#428—1661no—
368#428—1661no—
369#428—1661no—
370#428—1661no—
371#427—831no—
372#427—1661no—
373#427—1661no—
374C2×#107—831no—
375C2×#107—831no—
376C2×#107—1661no—
377C2×#106—831no—
378C2×#106—1661no—
379C2×#106C2×D8311yescharacteristic
380C2×#104—831no—
381C2×#104—831no—
382#417—831no—
383#417—831no—
384#417C2×D8311yescharacteristic
385#417C2×D8311yescharacteristic
386#405—831no—
387#405C2×D8311yescharacteristic
388C2×#97—831no—
389#384—831no—
390#382—831no—
391#343—831no—
392#343—831no—
393#343—831no—
394#343—831no—
395#339—831no—
396#339—831no—
397C2×#41C22×D8311yescharacteristic
398C2×#41C22×D8311yescharacteristic
399#135Dic16614yes—
400#135C4×D8314yes—
401C2×Q16C22⋊D8311yescharacteristic
402C2×SD16C22⋊D8311yescharacteristic
403C2×SD16C22⋊D8311yescharacteristic
404C2×D8C22⋊D8311yescharacteristic
405#131—834no—
406#131—834no—
407#131—1662no—
408#131—1662no—
409C2×M16—831no—
410C2×M16—831no—
411C2×M16—831no—
412C2×M16—1661no—
413C2×M16—1661no—
414C2×M16C22×D8311yescharacteristic
415C22×C8—831no—
416C22×C8—1661no—
417C22×C8—1661no—
418C22×C8C22×D8311yescharacteristic
419C4×Q8—21no—
420C4×Q8—21no—
421C4×Q8—1661no—
422C4×Q8—1661no—
423C4×D4—21no—
424C4×D4—21no—
425C4×D4—41no—
426C4×D4—41no—
427C4×D4—3321no—
428C4×D4—3321no—
429C4×D4—1661no—
430C4×D4—1661no—
431#117—21no—
432#117—21no—
433#117—831no—
434#117—1661no—
435#117—1661no—
436#117C22×D8311yescharacteristic
437C2×C4⋊C4—21no—
438C2×C4⋊C4—21no—
439C2×C4⋊C4—3321no—
440C2×C4⋊C4—831no—
441C2×C4⋊C4—831no—
442C2×C4⋊C4—1661no—
443C2×C4⋊C4—1661no—
444C2×C4⋊C4C22×D8311yescharacteristic
445C2×C4⋊C4C22×D8311yescharacteristic
446C2×C42—831no—
447#108—834no—
448#108—1662no—
449#107—831no—
450#107—831no—
451#107—1661no—
452#107—1661no—
453#107—1661no—
454#107—1661no—
455#107—1661no—
456#107—1661no—
457#107—1661no—
458#107—1661no—
459#106—21no—
460#106—21no—
461#106—831no—
462#106—831no—
463#106—1661no—
464#106—1661no—
465#106—1661no—
466#106—1661no—
467#106—1661no—
468#106—1661no—
469#104—834no—
470#104—834no—
471#103—21no—
472#103—21no—
473#103—1661no—
474#103—1661no—
475#102—21no—
476#102—21no—
477#102—1661no—
478#102—1661no—
479#97—21no—
480#97—834no—
481#97—1661no—
482C4×C8—21no—
483C4×C8—1661no—
484#95—3321no—
485#41—21no—
486#41—21no—
487#41—21no—
488#41—21no—
489#41C2×C4×D8312yes—
490#41C2×C4×D8312yes—
491#41C2×Dic16612yes—
492#41C2×Dic16612yes—
493C2×Q8C2×C22⋊D8311yescharacteristic
494C2×Q8C2×C22⋊D8311yescharacteristic
495C2×D4—21no—
496C2×D4—21no—
497C2×D4C2×C22⋊D8311yescharacteristic
498C2×D4C2×C22⋊D8311yescharacteristic
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—41no—
504C22×C4—41no—
505C22×C4—3321no—
506C22×C4—3321no—
507C22×C4—1661no—
508C22×C4—1661no—
509C22×C4—1661no—
510C22×C4C23×D8311yescharacteristic
511Q16Dic83⋊C414yescomplemented
512SD16Dic83⋊C414yescomplemented
513SD16Dic83⋊C414yescomplemented
514D8Dic83⋊C414yescomplemented
515M16—21no—
516M16—21no—
517M16—832no—
518M16—832no—
519M16—1661no—
520M16—1661no—
521M16—1664no—
522M16—1664no—
523M16—1661no—
524M16—1661no—
525C2×C8—21no—
526C2×C8—832no—
527C2×C8—831no—
528C2×C8—831no—
529C2×C8—832no—
530C2×C8—1661no—
531C2×C8—1664no—
532C2×C8—1664no—
533C2×C8—1661no—
534C2×C8—1661no—
535C2×C8C2×C4×D8312yes—
536C2×C8C2×C22⋊D8311yescharacteristic
537C2×C8C2×C22⋊D8311yescharacteristic
538C2×C8C2×Dic16612yes—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—41no—
544C4⋊C4—41no—
545C4⋊C4—3321no—
546C4⋊C4—3321no—
547C4⋊C4—831no—
548C4⋊C4—831no—
549C4⋊C4—6642no—
550C4⋊C4—1661no—
551C4⋊C4—1661no—
552C4⋊C4—1661no—
553C4⋊C4—1661no—
554C4⋊C4D4×D8311yescharacteristic
555C4⋊C4D4×D8311yescharacteristic
556C22⋊C4—21no—
557C22⋊C4—41no—
558C22⋊C4—41no—
559C22⋊C4—3321no—
560C22⋊C4—3321no—
561C22⋊C4—1661no—
562C42—21no—
563C42—41no—
564C42—41no—
565C42—3321no—
566C42—3321no—
567C42—832no—
568C42—1661no—
569C42—1661no—
570C23—21no—
571C23—21no—
572C23#1458183711yescharacteristic
573Q8—21no—
574Q8—21no—
575Q8C2×Dic83⋊C412yes—
576Q8C2×Dic83⋊C412yes—
577D4—22no—
578D4—22no—
579D4—22no—
580D4—22no—
581D4—21no—
582D4—21no—
583D4C2×Dic83⋊C412yes—
584D4C2×Dic83⋊C412yes—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—81no—
596C2×C4—81no—
597C2×C4—161no—
598C2×C4—161no—
599C2×C4—13281no—
600C2×C4—13281no—
601C2×C4—3321no—
602C2×C4—3321no—
603C2×C4—3321no—
604C2×C4—3321no—
605C2×C4—6641no—
606C2×C4—6641no—
607C2×C4—1661no—
608C2×C4—1661no—
609C2×C4—1662no—
610C2×C4C22×C4×D8311yescharacteristic
611C2×C4C2×D4×D8311yescharacteristic
612C2×C4C22×C22⋊D8311yescharacteristic
613C2×C4C22×Dic16611yescharacteristic
614C2×C4#41×D8311yescharacteristic
615C2×C4#1458183911yescharacteristic
616C8—41no—
617C8—3321no—
618C8—832no—
619C8—832no—
620C8—1662no—
621C8—1662no—
622C8C2×Dic83⋊C412yes—
623C8C2×Dic83⋊C412yes—
624C22—21no—
625C22—21no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22#1912818011yescharacteristic
633C22D4×C22⋊D8311yescharacteristic
634C22#1912841011yescharacteristic
635C4—22no—
636C4—22no—
637C4—41no—
638C4—41no—
639C4—81no—
640C4—81no—
641C4—3322no—
642C4—3321no—
643C4—3321no—
644C4—6641no—
645C4—6641no—
646C4C22×Dic83⋊C411yescharacteristic
647C4C4×D4×D8311yescharacteristic
648C4#1912835011yescharacteristic
649C4D4×Dic16611yescharacteristic
650C2—21no—
651C2—21no—
652C2—42no—
653C2—42no—
654C2D4×Dic83⋊C411yescharacteristic
655C2#2491922511yescharacteristic
656C2#2491923611yescharacteristic
subgroupG/H
centerC22D4×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2952C2
socleC22×C83D42
prime radicalC2×#135×C83C22
cosocleC22×C4×C83C24
hypercenter#2952D83
hypocenterC83#22693
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2952

Complements

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

NH ≅ G/Ntype
3. #24916061C2semidirect
4. #24916056C2semidirect
5. #24908248C2semidirect
6. #24908241C2semidirect
7. #24907696C2semidirect
8. #24907694C2semidirect
9. #24907689C2semidirect
10. #24907687C2semidirect
11. #24907628C2semidirect
12. #24907525C2semidirect
13. #24907520C2semidirect
14. #24907348C2semidirect
20. #19127823C4semidirect
70. #14581799C2×C4semidirect
71. #14581799C2×C4semidirect
95. #14581728C2×C4semidirect
115. #14581719C2×C4semidirect
116. #14581719C2×C4semidirect
122. #14581702C2×C4semidirect
216. Q16×C83C4⋊C4semidirect
217. SD16×C83C4⋊C4semidirect
218. SD16×C83C4⋊C4semidirect
219. D8×C83C4⋊C4semidirect
344. C83#22693semidirect
511. Q16Dic83⋊C4semidirect
512. SD16Dic83⋊C4semidirect
513. SD16Dic83⋊C4semidirect
514. D8Dic83⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#191278232
#249163392
#249160612
#249160562
#249082482
#249082412
#249076962
#249076942
#249076892
#249076872
#249076282
#249075252
#249075202
#249073482
C83×#29522
#2269383