Finite Group Databasebeta

← #35422912

Subgroups — #35422912

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
1#24916347C211yesmaximal, characteristic
2#24916339C211yesmaximal, characteristic
3#24916063C211yesmaximal, characteristic, complemented
4#24916058C211yesmaximal, characteristic, complemented
5#24908249C211yesmaximal, characteristic, complemented
6#24908244C211yesmaximal, characteristic, complemented
7#24907698C211yesmaximal, characteristic, complemented
8#24907698C211yesmaximal, characteristic, complemented
9#24907691C211yesmaximal, characteristic, complemented
10#24907691C211yesmaximal, characteristic, complemented
11#24907631C211yesmaximal, characteristic, complemented
12#24907527C211yesmaximal, characteristic, complemented
13#24907522C211yesmaximal, characteristic, complemented
14#24907347C211yesmaximal, characteristic, complemented
15C2×#444×C83C211yesmaximal, characteristic
16#19128184C2211yescharacteristic
17#19128184C2211yescharacteristic
18#19128184C2211yescharacteristic
19#19128184C2211yescharacteristic
20#19127813—21no—
21#19127808—21no—
22#19127802—21no—
23#19127802—21no—
24Q16×Dic83—21no—
25SD16×Dic83—21no—
26SD16×Dic83—21no—
27D8×Dic83—21no—
28C2×#14581728C2211yescharacteristic
29#19127743—21no—
30#19127743—21no—
31#19127743—21no—
32#19127743—21no—
33#19127743C2211yescharacteristic
34#19127743C2211yescharacteristic
35#19127743C2211yescharacteristic
36#19127743C2211yescharacteristic
37#19127739—21no—
38#19127739—21no—
39#19127739C2211yescharacteristic
40C2×#14581726—21no—
41C2×#14581726C2211yescharacteristic
42C2×#14581726C2211yescharacteristic
43C2×#14581724—21no—
44C2×#14581724C2211yescharacteristic
45C2×#14581724C2211yescharacteristic
46#19127702C2211yescharacteristic
47#19127702C2211yescharacteristic
48#19127702C2211yescharacteristic
49#19127702C2211yescharacteristic
50#19127656C2211yescharacteristic
51#19127656C2211yescharacteristic
52#19127571C2211yescharacteristic
53#19127567C2211yescharacteristic
54#19127469C2211yescharacteristic
55#19127469C2211yescharacteristic
56#19127469C2211yescharacteristic
57#19127469C2211yescharacteristic
58#19127461C2211yescharacteristic
59#19127461C2211yescharacteristic
60C2×#135×C83C2211yescharacteristic
61C2×#131×C83C2211yescharacteristic
62C2×#131×C83C2211yescharacteristic
63#444×C83C418yescomplemented
64C2×#108×C83C2211yescharacteristic
65C2×#104×C83C2211yescharacteristic
66C2×#104×C83C2211yescharacteristic
67C2×#97×C83C2211yescharacteristic
68C2×#13050574C2311yescharacteristic
69C2×#13050574C2311yescharacteristic
70Q8×Dic83—21no—
71Q8×Dic83—21no—
72Q8×Dic83—21no—
73Q8×Dic83—21no—
74D4×Dic83—21no—
75D4×Dic83—21no—
76D4×Dic83—21no—
77D4×Dic83—21no—
78D4×Dic83—41no—
79D4×Dic83—41no—
80D4×Dic83—41no—
81D4×Dic83—41no—
82#14581751—21no—
83#14581751—21no—
84#14581751—21no—
85#14581751—21no—
86#14581751C2311yescharacteristic
87#14581751C2311yescharacteristic
88C2×C4⋊Dic83—21no—
89C2×C4⋊Dic83—21no—
90C2×C4⋊Dic83—21no—
91C2×C4⋊Dic83—21no—
92C2×C4⋊Dic83—41no—
93C2×C4⋊Dic83C2311yescharacteristic
94C2×C4⋊Dic83C2311yescharacteristic
95C2×C4⋊Dic83C2311yescharacteristic
96C2×C4⋊Dic83C2311yescharacteristic
97#14581728—22no—
98#14581726—21no—
99#14581726—21no—
100#14581726—21no—
101#14581726—21no—
102#14581726—21no—
103#14581726—21no—
104#14581726—21no—
105#14581726—21no—
106#14581726D411yescharacteristic
107#14581726D411yescharacteristic
108#14581724—21no—
109#14581724—21no—
110#14581724—21no—
111#14581724—21no—
112#14581724—21no—
113#14581724—21no—
114#14581724—21no—
115#14581724—21no—
116#14581724D411yescharacteristic
117#14581724D411yescharacteristic
118#14581716—21no—
119#14581716—21no—
120#14581716—21no—
121#14581716—21no—
122#14581713—21no—
123#14581713—21no—
124#14581713—21no—
125#14581713—21no—
126#14581702—21no—
127#14581702—21no—
128C8×Dic83—21no—
129C8×Dic83—21no—
130C83⋊#95—41no—
131C2×#41×C83C2311yescharacteristic
132C2×#41×C83C2311yescharacteristic
133#135×C83C2×C414yes—
134#135×C83Q814yes—
135C2×Q16×C83D411yescharacteristic
136C2×SD16×C83D411yescharacteristic
137C2×SD16×C83D411yescharacteristic
138C2×D8×C83D411yescharacteristic
139#131×C83—22no—
140#131×C83—22no—
141#131×C83C2×C414yescomplemented
142#131×C83C2×C414yescomplemented
143C2×M16×C83—21no—
144C2×M16×C83—21no—
145C2×M16×C83C2311yescharacteristic
146C2×M16×C83C2311yescharacteristic
147C22×C8×C83—21no—
148C22×C8×C83—21no—
149C22×C8×C83C2311yescharacteristic
150C22×C8×C83C2311yescharacteristic
151C2×C42×C83C2311yescharacteristic
152#108×C83C2×C414yescomplemented
153#104×C83C2×C414yescomplemented
154#104×C83C2×C414yescomplemented
155#97×C83C2×C414yescomplemented
156C22×Dic83—21no—
157C22×Dic83—21no—
158C22×Dic83—21no—
159C22×Dic83—21no—
160C22×Dic83—41no—
161C22×Dic83—41no—
162C22×Dic83—41no—
163C22×Dic83—41no—
164#13050574—21no—
165#13050574—21no—
166#13050574—21no—
167#13050574—21no—
168C2×C83⋊C8—21no—
169C2×C83⋊C8—21no—
170C4⋊Dic83—21no—
171C4⋊Dic83—21no—
172C4⋊Dic83—21no—
173C4⋊Dic83—21no—
174C4⋊Dic83—21no—
175C4⋊Dic83—21no—
176C4⋊Dic83—21no—
177C4⋊Dic83—21no—
178C4⋊Dic83—41no—
179C4⋊Dic83—41no—
180C4⋊Dic83—41no—
181C4⋊Dic83—41no—
182C4⋊Dic83—82no—
183C4⋊Dic83C2×D411yescharacteristic
184C4⋊Dic83C2×D411yescharacteristic
185C4⋊Dic83C2×D411yescharacteristic
186C4⋊Dic83C2×D411yescharacteristic
187C22⋊Dic83—21no—
188C22⋊Dic83—21no—
189C22⋊Dic83—41no—
190C22⋊Dic83—41no—
191C22⋊Dic83—41no—
192C22⋊Dic83—41no—
193C4×Dic83—21no—
194C4×Dic83—21no—
195C4×Dic83—41no—
196C4×Dic83—41no—
197C4×Dic83—41no—
198C4×Dic83—41no—
199#41×C83—21no—
200#41×C83—21no—
201#41×C83—21no—
202#41×C83—21no—
203#41×C83C22×C412yes—
204#41×C83C22×C412yes—
205#41×C83C2×Q812yes—
206#41×C83C2×Q812yes—
207C2×Q8×C83C2×D411yescharacteristic
208C2×Q8×C83C2×D411yescharacteristic
209C2×D4×C83—21no—
210C2×D4×C83—21no—
211C2×D4×C83C2×D411yescharacteristic
212C2×D4×C83C2×D411yescharacteristic
213C22×C4×C83—21no—
214C22×C4×C83—21no—
215C22×C4×C83—21no—
216C22×C4×C83C2411yescharacteristic
217Q16×C83C4⋊C414yescomplemented
218SD16×C83C4⋊C414yescomplemented
219SD16×C83C4⋊C414yescomplemented
220D8×C83C4⋊C414yescomplemented
221M16×C83—21no—
222M16×C83—21no—
223M16×C83—24no—
224M16×C83—24no—
225M16×C83C22×C412yes—
226M16×C83C22×C412yes—
227C2×C8×C83—21no—
228C2×C8×C83—21no—
229C2×C8×C83—24no—
230C2×C8×C83—24no—
231C2×C8×C83C22×C412yes—
232C2×C8×C83C22×C412yes—
233C2×C8×C83C2×D411yescharacteristic
234C2×C8×C83C2×D411yescharacteristic
235C2×C8×C83C2×D411yescharacteristic
236C2×C8×C83C2×D411yescharacteristic
237C2×C8×C83C2×Q812yes—
238C2×C8×C83#4112yes—
239C42×C83—21no—
240C42×C83C22×C412yes—
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—22no—
263C2×C4×C83—41no—
264C2×C4×C83—41no—
265C2×C4×C83C23×C411yescharacteristic
266C2×C4×C83C22×D411yescharacteristic
267C2×C4×C83C22×D411yescharacteristic
268C2×C4×C83C22×Q811yescharacteristic
269C2×C4×C83C2×#4111yescharacteristic
270C2×C4×C832−1+411yescharacteristic
271C2×Dic83—21no—
272C2×Dic83—21no—
273C2×Dic83—21no—
274C2×Dic83—21no—
275C2×Dic83—41no—
276C2×Dic83—41no—
277C2×Dic83—41no—
278C2×Dic83—41no—
279C2×Dic83—81no—
280C2×Dic83—81no—
281C2×Dic83—81no—
282C2×Dic83—81no—
283C2×Dic83—161no—
284C2×Dic83—161no—
285C2×Dic83—161no—
286C2×Dic83—161no—
287C8×C83—22no—
288C8×C83—22no—
289C8×C83C2×C4⋊C412yes—
290C8×C83C2×C4⋊C412yes—
291C8×C83C4×D412yes—
292C8×C83C4×D412yes—
293C83⋊C8—41no—
294C83⋊C8—41no—
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—42no—
309C4×C83C22×C4⋊C411yescharacteristic
310C4×C83C2×C4×D411yescharacteristic
311C4×C83#54711yescharacteristic
312C4×C83D4×Q811yescharacteristic
313Dic83—41no—
314Dic83—41no—
315Dic83—41no—
316Dic83—41no—
317Dic83—81no—
318Dic83—81no—
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×#444D8311yescharacteristic
330#2952—831no—
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×#131C2×D8311yescharacteristic
347C2×#131C2×D8311yescharacteristic
348#520—831no—
349#520—831no—
350#520—831no—
351#520—831no—
352#444Dic8318yescomplemented
353#442—1661no—
354#441—1661no—
355#440—1661no—
356#440—1661no—
357C4×Q16—1661no—
358C4×SD16—1661no—
359C4×SD16—1661no—
360C4×D8—1661no—
361C2×#108—831no—
362C2×#108C2×D8311yescharacteristic
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—831no—
379C2×#106—1661no—
380C2×#104C2×D8311yescharacteristic
381C2×#104C2×D8311yescharacteristic
382#417—831no—
383#417—831no—
384#417—831no—
385#417—831no—
386#405—831no—
387#405—831no—
388C2×#97C2×D8311yescharacteristic
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#135C2×Dic8314yes—
401C2×Q16D33211yescharacteristic
402C2×SD16D33211yescharacteristic
403C2×SD16D33211yescharacteristic
404C2×D8D33211yescharacteristic
405#131—22no—
406#131—22no—
407#131C2×Dic8314yescomplemented
408#131C2×Dic8314yescomplemented
409C2×M16—21no—
410C2×M16—21no—
411C2×M16—831no—
412C2×M16—831no—
413C2×M16C22×D8311yescharacteristic
414C2×M16C22×D8311yescharacteristic
415C22×C8—21no—
416C22×C8—21no—
417C22×C8C22×D8311yescharacteristic
418C22×C8C22×D8311yescharacteristic
419C4×Q8—1661no—
420C4×Q8—1661no—
421C4×Q8—1661no—
422C4×Q8—1661no—
423C4×D4—3321no—
424C4×D4—3321no—
425C4×D4—3321no—
426C4×D4—3321no—
427C4×D4—1661no—
428C4×D4—1661no—
429C4×D4—1661no—
430C4×D4—1661no—
431#117—831no—
432#117—831no—
433#117—1661no—
434#117—1661no—
435#117—1661no—
436#117—1661no—
437C2×C4⋊C4—3321no—
438C2×C4⋊C4—831no—
439C2×C4⋊C4—831no—
440C2×C4⋊C4—831no—
441C2×C4⋊C4—831no—
442C2×C4⋊C4—1661no—
443C2×C4⋊C4—1661no—
444C2×C4⋊C4—1661no—
445C2×C4⋊C4—1661no—
446C2×C42C22×D8311yescharacteristic
447#108—1662no—
448#108C2×Dic8314yescomplemented
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—831no—
460#106—831no—
461#106—1661no—
462#106—1661no—
463#106—1661no—
464#106—1661no—
465#106—1661no—
466#106—1661no—
467#106—1661no—
468#106—1661no—
469#104C2×Dic8314yescomplemented
470#104C2×Dic8314yescomplemented
471#103—1661no—
472#103—1661no—
473#103—1661no—
474#103—1661no—
475#102—1661no—
476#102—1661no—
477#102—1661no—
478#102—1661no—
479#97—1661no—
480#97—1661no—
481#97C2×Dic8314yescomplemented
482C4×C8—1661no—
483C4×C8—1661no—
484#95—3321no—
485#41—21no—
486#41—21no—
487#41—21no—
488#41—21no—
489#41C22×Dic8312yes—
490#41C22×Dic8312yes—
491#41C2×Dic16612yes—
492#41C2×Dic16612yes—
493C2×Q8C2×D33211yescharacteristic
494C2×Q8C2×D33211yescharacteristic
495C2×D4—21no—
496C2×D4—21no—
497C2×D4C2×D33211yescharacteristic
498C2×D4C2×D33211yescharacteristic
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—3321no—
503C22×C4—3321no—
504C22×C4—3321no—
505C22×C4—3321no—
506C22×C4—1661no—
507C22×C4—1661no—
508C22×C4—1661no—
509C22×C4—1661no—
510C22×C4C23×D8311yescharacteristic
511Q16C4⋊Dic8314yescomplemented
512SD16C4⋊Dic8314yescomplemented
513SD16C4⋊Dic8314yescomplemented
514D8C4⋊Dic8314yescomplemented
515M16—21no—
516M16—21no—
517M16—24no—
518M16—24no—
519M16—1661no—
520M16—1661no—
521M16—1661no—
522M16—1661no—
523M16C22×Dic8312yes—
524M16C22×Dic8312yes—
525C2×C8—21no—
526C2×C8—21no—
527C2×C8—24no—
528C2×C8—24no—
529C2×C8—1661no—
530C2×C8—1661no—
531C2×C8C22×Dic8312yes—
532C2×C8C22×Dic8312yes—
533C2×C8D4×D8311yescharacteristic
534C2×C8D4×D8311yescharacteristic
535C2×C8C2×D33211yescharacteristic
536C2×C8C2×D33211yescharacteristic
537C2×C8C2×Dic16612yes—
538C2×C8#1305058912yes—
539C4⋊C4—3321no—
540C4⋊C4—3321no—
541C4⋊C4—3321no—
542C4⋊C4—3321no—
543C4⋊C4—831no—
544C4⋊C4—831no—
545C4⋊C4—831no—
546C4⋊C4—831no—
547C4⋊C4—6642no—
548C4⋊C4—1661no—
549C4⋊C4—1661no—
550C4⋊C4—1661no—
551C4⋊C4—1661no—
552C4⋊C4—1661no—
553C4⋊C4—1661no—
554C4⋊C4—1661no—
555C4⋊C4—1661no—
556C22⋊C4—3321no—
557C22⋊C4—3321no—
558C22⋊C4—3321no—
559C22⋊C4—3321no—
560C22⋊C4—1661no—
561C22⋊C4—1661no—
562C42—21no—
563C42—3321no—
564C42—3321no—
565C42—3321no—
566C42—3321no—
567C42—1661no—
568C42—1661no—
569C42C22×Dic8312yes—
570C23—21no—
571C23—21no—
572C23#1458183811yescharacteristic
573Q8—21no—
574Q8—21no—
575Q8C2×C4⋊Dic8312yes—
576Q8C2×C4⋊Dic8312yes—
577D4—22no—
578D4—22no—
579D4—22no—
580D4—22no—
581D4—21no—
582D4—21no—
583D4C2×C4⋊Dic8312yes—
584D4C2×C4⋊Dic8312yes—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—13281no—
595C2×C4—13281no—
596C2×C4—13281no—
597C2×C4—13281no—
598C2×C4—3321no—
599C2×C4—3321no—
600C2×C4—3321no—
601C2×C4—3321no—
602C2×C4—6641no—
603C2×C4—6641no—
604C2×C4—6641no—
605C2×C4—6641no—
606C2×C4—1661no—
607C2×C4—1661no—
608C2×C4—1661no—
609C2×C4—1661no—
610C2×C4C23×Dic8311yescharacteristic
611C2×C4C2×D4×D8311yescharacteristic
612C2×C4C22×D33211yescharacteristic
613C2×C4C22×Dic16611yescharacteristic
614C2×C4C2×#1305058911yescharacteristic
615C2×C4#1458183911yescharacteristic
616C8—22no—
617C8—22no—
618C8—3321no—
619C8—3321no—
620C8C2×C4⋊Dic8312yes—
621C8C2×C4⋊Dic8312yes—
622C8D4×Dic8312yes—
623C8D4×Dic8312yes—
624C22—21no—
625C22—21no—
626C22—22no—
627C22—22no—
628C22—41no—
629C22—41no—
630C22—41no—
631C22—41no—
632C22#1912818411yescharacteristic
633C22D4×D33211yescharacteristic
634C22#1912840811yescharacteristic
635C4—22no—
636C4—22no—
637C4—42no—
638C4—3321no—
639C4—3321no—
640C4—3321no—
641C4—3321no—
642C4—6641no—
643C4—6641no—
644C4—6641no—
645C4—6641no—
646C4C22×C4⋊Dic8311yescharacteristic
647C4C2×D4×Dic8311yescharacteristic
648C4#1912835511yescharacteristic
649C4D4×Dic16611yescharacteristic
650C2—21no—
651C2—21no—
652C2—42no—
653C2—42no—
654C2D4×C4⋊Dic8311yescharacteristic
655C2#2491923011yescharacteristic
656C2#2491923911yescharacteristic
subgroupG/H
centerC22D4×D332
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#444×C83C2
socleC22×C83D42
prime radicalC2×#135×C83C22
cosocleC22×C4×C83C24
hypercenterC2×#444D83
hypocenterC83#22693
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#444

Complements

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

NH ≅ G/Ntype
3. #24916063C2semidirect
4. #24916058C2semidirect
5. #24908249C2semidirect
6. #24908244C2semidirect
7. #24907698C2semidirect
8. #24907698C2semidirect
9. #24907691C2semidirect
10. #24907691C2semidirect
11. #24907631C2semidirect
12. #24907527C2semidirect
13. #24907522C2semidirect
14. #24907347C2semidirect
63. #444×C83C4semidirect
141. #131×C83C2×C4semidirect
142. #131×C83C2×C4semidirect
152. #108×C83C2×C4semidirect
153. #104×C83C2×C4semidirect
154. #104×C83C2×C4semidirect
155. #97×C83C2×C4semidirect
217. Q16×C83C4⋊C4semidirect
218. SD16×C83C4⋊C4semidirect
219. SD16×C83C4⋊C4semidirect
220. D8×C83C4⋊C4semidirect
344. C83#22693semidirect
352. #444Dic83semidirect
407. #131C2×Dic83semidirect
408. #131C2×Dic83semidirect
448. #108C2×Dic83semidirect
469. #104C2×Dic83semidirect
470. #104C2×Dic83semidirect
481. #97C2×Dic83semidirect
511. Q16C4⋊Dic83semidirect
512. SD16C4⋊Dic83semidirect
513. SD16C4⋊Dic83semidirect
514. D8C4⋊Dic83semidirect

Maximal subgroups

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

subgroupindex
#249163472
#249163392
#249160632
#249160582
#249082492
#249082442
#249076982
#249076912
#249076312
#249075272
#249075222
#249073472
C2×#444×C832
#2269383