Finite Group Databasebeta

← #35463022

Subgroups — #35463022

Order 21248.

894 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128098C211yesmaximal, characteristic, complemented
2C22×#14581723C211yesmaximal, characteristic
3#24911166C211yesmaximal, characteristic
4#24911159C211yesmaximal, characteristic
5#24908943C211yesmaximal, characteristic, complemented
6#24908942C211yesmaximal, characteristic, complemented
7#24908795C211yesmaximal, characteristic, complemented
8#24908794C211yesmaximal, characteristic, complemented
9#24908190C211yesmaximal, characteristic, complemented
10#24908190C211yesmaximal, characteristic, complemented
11#24907658C211yesmaximal, characteristic
12#24907618C211yesmaximal, characteristic, complemented
13#24907486C211yesmaximal, characteristic
14#24907237C211yesmaximal, characteristic
15C83×#1853C211yesmaximal, characteristic
16C23×C83⋊C8C2211yescharacteristic
17C2×#14581791C2211yescharacteristic
18C2×#14581791C2211yescharacteristic
19C2×#14581771—21no—
20C2×#14581769—21no—
21C22×C4⋊Dic83C2211yescharacteristic
22#19128098—24no—
23C2×#14581724C2211yescharacteristic
24C2×#14581724C2211yescharacteristic
25C2×#14581723—24no—
26C2×#14581723C2211yescharacteristic
27C2×#14581723C2211yescharacteristic
28C2×#14581723C2211yescharacteristic
29C2×#14581723C2211yescharacteristic
30C2×#14581723C2211yescharacteristic
31C2×#14581723C2211yescharacteristic
32C2×C8×Dic83C2211yescharacteristic
33C2×C8×Dic83C2211yescharacteristic
34#19127635—21no—
35#19127634—21no—
36#19127622—21no—
37#19127621—21no—
38#19127600C2211yescharacteristic
39#19127600C2211yescharacteristic
40#19127597C2211yescharacteristic
41#19127597C2211yescharacteristic
42#19127574—21no—
43#19127573—21no—
44#19127570C2211yescharacteristic
45#19127570C2211yescharacteristic
46#19127561C2211yescharacteristic
47C4×C22⋊Dic83C2211yescharacteristic
48#19127464C2211yescharacteristic
49#19127464C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127463C2211yescharacteristic
52#19127455C2211yescharacteristic
53#19127455C2211yescharacteristic
54C22×C4⋊C4×C83C2211yescharacteristic
55C2×#106×C83C2211yescharacteristic
56C2×#106×C83C2211yescharacteristic
57C2×C22⋊C8×C83C2211yescharacteristic
58#380×C83C2211yescharacteristic
59#340×C83C2211yescharacteristic
60#340×C83C2211yescharacteristic
61C22×Dic166—21no—
62C22×Dic166—21no—
63C23×Dic83—21no—
64C22×C83⋊C8—22no—
65C22×C83⋊C8—22no—
66C22×C83⋊C8Q814yescomplemented
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69#14581791—22no—
70#14581791—22no—
71#14581791—41no—
72#14581791—41no—
73#14581771—44no—
74#14581769—44no—
75C2×C4⋊Dic83—22no—
76C2×C4⋊Dic83—21no—
77C2×C4⋊Dic83—21no—
78C2×C4⋊Dic83—21no—
79C2×C4⋊Dic83—41no—
80C2×C4⋊Dic83C2311yescharacteristic
81C2×C4⋊Dic83C2311yescharacteristic
82C2×C4⋊Dic83C2311yescharacteristic
83C2×C4⋊Dic83C2311yescharacteristic
84C2×Dic83⋊C4—21no—
85C2×Dic83⋊C4—21no—
86C2×Dic83⋊C4—21no—
87C2×Dic83⋊C4—21no—
88C2×Dic83⋊C4—21no—
89C2×Dic83⋊C4—21no—
90C2×Dic83⋊C4—21no—
91C2×Dic83⋊C4—21no—
92C2×C22⋊Dic83—21no—
93C2×C22⋊Dic83D411yescharacteristic
94C2×C22⋊Dic83D411yescharacteristic
95C2×C4×Dic83C2311yescharacteristic
96C2×C4×Dic83C2311yescharacteristic
97#14581724—22no—
98#14581724—41no—
99#14581723—22no—
100#14581723—22no—
101#14581723—22no—
102#14581723—22no—
103#14581723—22no—
104#14581723D414yes—
105C8×Dic83—22no—
106C8×Dic83—22no—
107C83⋊#95—21no—
108C83⋊#95—21no—
109C83⋊#95—21no—
110C83⋊#95—21no—
111C83⋊#95—21no—
112C83⋊#95D411yescharacteristic
113C83⋊#95D411yescharacteristic
114C23×C4×C83—21no—
115C23×C4×C83C2311yescharacteristic
116C22×C8×C83C2311yescharacteristic
117C22×C8×C83C2311yescharacteristic
118C2×C4⋊C4×C83—22no—
119C2×C4⋊C4×C83—21no—
120C2×C4⋊C4×C83—41no—
121C2×C4⋊C4×C83C2311yescharacteristic
122C2×C4⋊C4×C83C2311yescharacteristic
123C2×C4⋊C4×C83C2311yescharacteristic
124C2×C4⋊C4×C83C2311yescharacteristic
125C2×C22⋊C4×C83—21no—
126#106×C83—22no—
127#106×C83—41no—
128C22⋊C8×C83Q814yescomplemented
129#95×C83—21no—
130C2×Dic166—42no—
131C2×Dic166—42no—
132C2×Dic166—42no—
133C2×Dic166—42no—
134C2×Dic166—81no—
135C2×Dic166—81no—
136C22×Dic83—21no—
137C22×Dic83—21no—
138C22×Dic83—21no—
139C22×Dic83—21no—
140C22×Dic83—42no—
141C22×Dic83—42no—
142C22×Dic83—81no—
143C22×Dic83C2×D411yescharacteristic
144C22×Dic83C2×D411yescharacteristic
145C22×Dic83C2×D411yescharacteristic
146C22×Dic83C2×D411yescharacteristic
147C2×C83⋊C8—21no—
148C2×C83⋊C8—21no—
149C2×C83⋊C8—28no—
150C2×C83⋊C8C2×D412yes—
151C2×C83⋊C8C2×Q812yes—
152C2×C83⋊C8C2×Q812yes—
153C2×C83⋊C8#4112yes—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—21no—
159C4⋊Dic83—21no—
160C4⋊Dic83—21no—
161C4⋊Dic83—42no—
162C4⋊Dic83—42no—
163C4⋊Dic83—42no—
164C4⋊Dic83C2×D412yes—
165Dic83⋊C4—42no—
166Dic83⋊C4—42no—
167Dic83⋊C4—42no—
168Dic83⋊C4—42no—
169Dic83⋊C4—42no—
170Dic83⋊C4—42no—
171Dic83⋊C4—81no—
172Dic83⋊C4—81no—
173C22⋊Dic83—42no—
174C22⋊Dic83SD1614yes—
175C22⋊Dic83SD1614yes—
176C4×Dic83—21no—
177C4×Dic83—21no—
178C4×Dic83—21no—
179C4×Dic83—21no—
180C24×C83C2×D411yescharacteristic
181C22×C4×C83—21no—
182C22×C4×C83—21no—
183C22×C4×C83—21no—
184C22×C4×C83—21no—
185C22×C4×C83—21no—
186C22×C4×C83—21no—
187C22×C4×C83—21no—
188C22×C4×C83—21no—
189C22×C4×C83—42no—
190C22×C4×C83—42no—
191C22×C4×C83—81no—
192C22×C4×C83C2×D411yescharacteristic
193C22×C4×C83C2×Q812yes—
194C22×C4×C83C2×Q812yes—
195C22×C4×C83C2411yescharacteristic
196C2×C8×C83—21no—
197C2×C8×C83—21no—
198C2×C8×C83—21no—
199C2×C8×C83—21no—
200C2×C8×C83C2×Q812yes—
201C2×C8×C83C2×Q812yes—
202C4⋊C4×C83—21no—
203C4⋊C4×C83—21no—
204C4⋊C4×C83—21no—
205C4⋊C4×C83—21no—
206C4⋊C4×C83—21no—
207C4⋊C4×C83—21no—
208C4⋊C4×C83—21no—
209C4⋊C4×C83—42no—
210C4⋊C4×C83C2×D412yes—
211C22⋊C4×C83—42no—
212C23×C83—21no—
213C23×C83—21no—
214C23×C83—21no—
215C23×C83—21no—
216C23×C83C4⋊Q812yes—
217C23×C83C2×SD1612yes—
218C23×C83C2×SD1612yes—
219C23×C83C22×D411yescharacteristic
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—22no—
223C2×C4×C83—22no—
224C2×C4×C83—22no—
225C2×C4×C83—22no—
226C2×C4×C83—22no—
227C2×C4×C83—41no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83—41no—
232C2×C4×C83—41no—
233C2×C4×C83—41no—
234C2×C4×C83—41no—
235C2×C4×C83—41no—
236C2×C4×C83—41no—
237C2×C4×C83—41no—
238C2×C4×C83—84no—
239C2×C4×C83C4⋊Q812yes—
240C2×C4×C83C22×D411yescharacteristic
241C2×C4×C83C22×Q811yescharacteristic
242C2×C4×C83C22×Q811yescharacteristic
243C2×C4×C83C2×#4111yescharacteristic
244C2×C4×C83H(5,𝔽2)11yescharacteristic
245C2×C4×C832−1+411yescharacteristic
246C2×Dic83—21no—
247C2×Dic83—21no—
248C2×Dic83—21no—
249C2×Dic83—21no—
250C2×Dic83—21no—
251C2×Dic83—21no—
252C2×Dic83—21no—
253C2×Dic83—21no—
254C2×Dic83—22no—
255C2×Dic83—41no—
256C2×Dic83—41no—
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—84no—
267C2×Dic83C2×SD1612yes—
268C2×Dic83C2×SD1612yes—
269C2×Dic83C2×SD1612yes—
270C2×Dic83C2×SD1612yes—
271Dic166—84no—
272Dic166—84no—
273C8×C83—22no—
274C8×C83—22no—
275C83⋊C8—22no—
276C83⋊C8C22⋊Q814yescomplemented
277C22×C83—22no—
278C22×C83—22no—
279C22×C83—22no—
280C22×C83—22no—
281C22×C83—22no—
282C22×C83—22no—
283C22×C83#49814yescomplemented
284C22×C83C2×C4⋊Q811yescharacteristic
285C22×C83D4211yescharacteristic
286C22×C83#54711yescharacteristic
287C22×C83C22×SD1611yescharacteristic
288C22×C83C22×SD1611yescharacteristic
289C22×C83#57711yescharacteristic
290C22×C83#57711yescharacteristic
291C4×C83—22no—
292C4×C83—42no—
293C4×C83—81no—
294C4×C83—81no—
295C4×C83—81no—
296C4×C83C2×C22⋊Q811yescharacteristic
297C4×C83#54411yescharacteristic
298C4×C83D4×Q811yescharacteristic
299C4×C83#55411yescharacteristic
300Dic83—22no—
301Dic83—22no—
302Dic83—22no—
303Dic83—22no—
304Dic83—42no—
305Dic83—81no—
306Dic83—81no—
307Dic83—81no—
308#26060—831nomaximal, Sylow
309C2×C83—24no—
310C2×C83#244011yescharacteristic
311C2×C83C2×#49811yescharacteristic
312C2×C83#316311yescharacteristic
313C2×C83D4×SD1611yescharacteristic
314C2×C83D4×SD1611yescharacteristic
315C2×C83#333511yescharacteristic
316C2×C83#333511yescharacteristic
317C2×#498—831no—
318C22×#106—831no—
319#2440—831no—
320#2440—831no—
321#2084—831no—
322#2084—831no—
323#2058—831no—
324#2058—831no—
325#1941—831no—
326#1941—831no—
327#1853—831no—
328#1853D8311yescharacteristic
329#1844—831no—
330#1807—831no—
331C8×C22⋊C4—831no—
332C83#2606011yescharacteristic, complemented, Sylow
333C23×C8—831no—
334C2×C4⋊Q8—831no—
335C2×C4⋊Q8—831no—
336C2×C22⋊Q8—1661no—
337C2×C22⋊Q8—1661no—
338C22×C4⋊C4—831no—
339C22×C4⋊C4C2×D8311yescharacteristic
340#498—1664no—
341C2×#106—831no—
342C2×#106—831no—
343C2×#106—831no—
344C2×#106—831no—
345C2×#106—831no—
346C2×#106—831no—
347C2×#106—831no—
348C2×#106—831no—
349C2×#106—1664no—
350C2×#106C2×D8311yescharacteristic
351C2×#106C2×D8311yescharacteristic
352C2×C22⋊C8C2×D8311yescharacteristic
353C2×C4×C8—831no—
354C2×C4×C8—831no—
355#398—1661no—
356#398—1661no—
357#395—1661no—
358#395—1661no—
359#389—831no—
360#389—831no—
361#389—831no—
362#389—831no—
363#385—1661no—
364#385—1661no—
365#384—831no—
366#384—831no—
367#380—831no—
368#380C2×D8311yescharacteristic
369C4×C22⋊C4—831no—
370#340—831no—
371#340—831no—
372#340—831no—
373#340—831no—
374#340C2×D8311yescharacteristic
375#340C2×D8311yescharacteristic
376#336—831no—
377#336—831no—
378C22×Q8—1661no—
379C22×Q8—1661no—
380C23×C4—21no—
381C23×C4—1661no—
382C23×C4C22×D8311yescharacteristic
383C22×C8—831no—
384C22×C8—834no—
385C22×C8—831no—
386C22×C8—1662no—
387C22×C8—1662no—
388C22×C8C22×D8311yescharacteristic
389C22×C8C22×D8311yescharacteristic
390C4⋊Q8—3321no—
391C4⋊Q8—3321no—
392C4⋊Q8—1662no—
393C4⋊Q8—1662no—
394C22⋊Q8—3324no—
395C22⋊Q8—3324no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—41no—
399C2×C4⋊C4—3321no—
400C2×C4⋊C4—831no—
401C2×C4⋊C4—831no—
402C2×C4⋊C4—831no—
403C2×C4⋊C4—831no—
404C2×C4⋊C4—1661no—
405C2×C4⋊C4—1661no—
406C2×C4⋊C4—1661no—
407C2×C4⋊C4—1661no—
408C2×C4⋊C4—1661no—
409C2×C4⋊C4—1661no—
410C2×C4⋊C4—1661no—
411C2×C4⋊C4—1661no—
412C2×C4⋊C4—1661no—
413C2×C4⋊C4—1662no—
414C2×C4⋊C4—1661no—
415C2×C4⋊C4—1661no—
416C2×C4⋊C4C22×D8311yescharacteristic
417C2×C4⋊C4C22×D8311yescharacteristic
418C2×C4⋊C4C22×D8311yescharacteristic
419C2×C4⋊C4C22×D8311yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—831no—
422C2×C22⋊C4—831no—
423C2×C22⋊C4—1661no—
424C2×C42—831no—
425C2×C42—831no—
426#106—22no—
427#106—41no—
428#106—3321no—
429#106—834no—
430#106—1662no—
431#106—1662no—
432#106—1662no—
433#106—1662no—
434#106—1662no—
435#106—1662no—
436C22⋊C8Dic16614yescomplemented
437C4×C8—1662no—
438C4×C8—1662no—
439#95—21no—
440#95—831no—
441#95—831no—
442#95—1661no—
443#95—1661no—
444#95—1661no—
445#95—1661no—
446#95—1661no—
447C24D4×D8311yescharacteristic
448C2×Q8—3322no—
449C2×Q8—3322no—
450C2×Q8—3322no—
451C2×Q8—3322no—
452C2×Q8—6641no—
453C2×Q8—6641no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—81no—
465C22×C4—3322no—
466C22×C4—3322no—
467C22×C4—831no—
468C22×C4—831no—
469C22×C4—831no—
470C22×C4—831no—
471C22×C4—6641no—
472C22×C4—1661no—
473C22×C4—1661no—
474C22×C4—1661no—
475C22×C4—1661no—
476C22×C4D4×D8311yescharacteristic
477C22×C4Q8×D8312yes—
478C22×C4C2×Dic16612yes—
479C22×C4C23×D8311yescharacteristic
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—832no—
485C2×C8—832no—
486C2×C8—832no—
487C2×C8—832no—
488C2×C8—1668no—
489C2×C8—1661no—
490C2×C8—1661no—
491C2×C8C2×Dic16612yes—
492C2×C8C2×Dic16612yes—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—42no—
501C4⋊C4—3322no—
502C4⋊C4—3322no—
503C4⋊C4—3322no—
504C4⋊C4—3322no—
505C4⋊C4—3322no—
506C4⋊C4—3322no—
507C4⋊C4—3322no—
508C4⋊C4—3322no—
509C4⋊C4—3322no—
510C4⋊C4—832no—
511C4⋊C4—6641no—
512C4⋊C4—6641no—
513C4⋊C4—1661no—
514C4⋊C4—1661no—
515C4⋊C4—1661no—
516C4⋊C4—1661no—
517C4⋊C4—1661no—
518C4⋊C4—1661no—
519C4⋊C4—1661no—
520C4⋊C4D4×D8312yes—
521C22⋊C4—42no—
522C22⋊C4—3322no—
523C22⋊C4—834no—
524C22⋊C4—834no—
525C42—1661no—
526C42—1661no—
527C42—1661no—
528C42—1661no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23#1458179112yes—
534C23SD16×D8312yes—
535C23SD16×D8312yes—
536C23C2×D4×D8311yescharacteristic
537Q8—6644no—
538Q8—6644no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—84no—
558C2×C4—3321no—
559C2×C4—3321no—
560C2×C4—3321no—
561C2×C4—3321no—
562C2×C4—3321no—
563C2×C4—3321no—
564C2×C4—3321no—
565C2×C4—3321no—
566C2×C4—3321no—
567C2×C4—3321no—
568C2×C4—3321no—
569C2×C4—832no—
570C2×C4—832no—
571C2×C4—832no—
572C2×C4—832no—
573C2×C4—6644no—
574C2×C4—1662no—
575C2×C4—1661no—
576C2×C4—1661no—
577C2×C4—1661no—
578C2×C4—1661no—
579C2×C4—1661no—
580C2×C4—1661no—
581C2×C4—1661no—
582C2×C4—1661no—
583C2×C4#1458179112yes—
584C2×C4C2×D4×D8311yescharacteristic
585C2×C4C2×Q8×D8311yescharacteristic
586C2×C4C22×Dic16611yescharacteristic
587C2×C4#41×D8311yescharacteristic
588C2×C4#1458183711yescharacteristic
589C2×C4#1458183911yescharacteristic
590C8—22no—
591C8—22no—
592C8—834no—
593C8—1662no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22#1912809814yescomplemented
601C22C2×#1458179111yescharacteristic
602C22#1912833311yescharacteristic
603C22#1912834911yescharacteristic
604C22C2×SD16×D8311yescharacteristic
605C22C2×SD16×D8311yescharacteristic
606C22#1912851011yescharacteristic
607C22#1912851111yescharacteristic
608C4—22no—
609C4—42no—
610C4—81no—
611C4—81no—
612C4—81no—
613C4—3322no—
614C4—6641no—
615C4—6641no—
616C4—6641no—
617C4—1662no—
618C4—1662no—
619C4—1662no—
620C4—1662no—
621C4C22⋊Q8×D8311yescharacteristic
622C4#1912833011yescharacteristic
623C4D4×Dic16611yescharacteristic
624C4#1912841011yescharacteristic
625C2—24no—
626C2#2491116411yescharacteristic
627C2C2×#1912809811yescharacteristic
628C2#2491799311yescharacteristic
629C2#2491915211yescharacteristic
630C2#2491915711yescharacteristic
631C2#2491981311yescharacteristic
632C2#2491981811yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#1853C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#1853D83
hypocenterC83#26060
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
1. C2×#19128098C2semidirect
5. #24908943C2semidirect
6. #24908942C2semidirect
7. #24908795C2semidirect
8. #24908794C2semidirect
9. #24908190C2semidirect
10. #24908190C2semidirect
12. #24907618C2semidirect
66. C22×C83⋊C8Q8semidirect
128. C22⋊C8×C83Q8semidirect
276. C83⋊C8C22⋊Q8semidirect
283. C22×C83#498semidirect
332. C83#26060semidirect
436. C22⋊C8Dic166semidirect
600. C22#19128098semidirect

Maximal subgroups

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

subgroupindex
C2×#191280982
C22×#145817232
#249111662
#249111592
#249089432
#249089422
#249087952
#249087942
#249081902
#249076582
#249076182
#249074862
#249072372
C83×#18532
#2606083