Finite Group Databasebeta

← #35332897

Subgroups — #35332897

Order 21248.

1810 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
1C22×#14581771C211yesmaximal, characteristic
2C2×#19127584C212yesmaximal, complemented
3#24905733C212yesmaximal, complemented
4#24905729C212yesmaximal, complemented
5#24905706C212yesmaximal
6#24905700C212yesmaximal
7Dic83⋊#95C212yesmaximal, complemented
8#24905686C211yesmaximal, characteristic
9C83×C22⋊#95C211yesmaximal, characteristic
10C23×Dic166C2211yescharacteristic
11C2×#14581771—28no—
12C2×#14581771C418yescomplemented
13C22×C4⋊Dic83C2211yescharacteristic
14C22×Dic83⋊C4C2212yes—
15C22×Dic83⋊C4C2212yes—
16C22×C22⋊Dic83C2212yes—
17C22×C22⋊Dic83C2212yes—
18C22×C4×Dic83C2212yes—
19#19127584—28no—
20C2×C83⋊#95C2212yes—
21C2×C83⋊#95C2212yes—
22C2×C83⋊#95C2212yes—
23C2×C83⋊#95C2211yescharacteristic
24C2×C83⋊#95C2212yes—
25C2×C83⋊#95C2212yes—
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2212yes—
28C2×C83⋊#95C2211yescharacteristic
29C24×C4×C83C2211yescharacteristic
30C22×C22⋊C4×C83C2211yescharacteristic
31C22×C22⋊C4×C83C2211yescharacteristic
32C2×#95×C83C2212yes—
33C2×#95×C83C2212yes—
34C22×Dic166—24no—
35C22×Dic166—41no—
36C22×Dic166—41no—
37C22×Dic166—42no—
38C22×Dic166C2×C414yes—
39C23×Dic83C2312yes—
40C23×Dic83C2312yes—
41C23×Dic83C2312yes—
42C23×Dic83C2312yes—
43#14581771—232no—
44C2×C4⋊Dic83C2×C414yes—
45C2×C4⋊Dic83D418yes—
46C2×Dic83⋊C4—24no—
47C2×Dic83⋊C4—24no—
48C2×Dic83⋊C4—24no—
49C2×Dic83⋊C4—24no—
50C2×Dic83⋊C4—42no—
51C2×Dic83⋊C4C2×C418yes—
52C2×C22⋊Dic83—24no—
53C2×C22⋊Dic83—24no—
54C2×C22⋊Dic83—24no—
55C2×C22⋊Dic83—24no—
56C2×C22⋊Dic83—42no—
57C2×C22⋊Dic83C2×C418yes—
58C2×C4×Dic83—24no—
59C2×C4×Dic83—24no—
60C2×C4×Dic83—42no—
61C83⋊#95—24no—
62C83⋊#95—24no—
63C83⋊#95—24no—
64C83⋊#95—42no—
65C83⋊#95—42no—
66C83⋊#95—41no—
67C83⋊#95—41no—
68C83⋊#95—42no—
69C83⋊#95—42no—
70C25×C83C2311yescharacteristic
71C23×C4×C83—24no—
72C23×C4×C83—42no—
73C23×C4×C83—41no—
74C23×C4×C83—41no—
75C23×C4×C83C2×C414yes—
76C23×C4×C83C2312yes—
77C23×C4×C83C2311yescharacteristic
78C23×C4×C83C2312yes—
79C23×C4×C83C2311yescharacteristic
80C2×C22⋊C4×C83—22no—
81C2×C22⋊C4×C83—41no—
82C2×C22⋊C4×C83—41no—
83C2×C22⋊C4×C83—41no—
84C2×C22⋊C4×C83—41no—
85C2×C22⋊C4×C83Q814yescomplemented
86#95×C83—24no—
87#95×C83—42no—
88C2×Dic166—28no—
89C2×Dic166—44no—
90C2×Dic166—44no—
91C2×Dic166—48no—
92C2×Dic166—48no—
93C22×Dic83—22no—
94C22×Dic83—22no—
95C22×Dic83—22no—
96C22×Dic83—22no—
97C22×Dic83—22no—
98C22×Dic83—22no—
99C22×Dic83—22no—
100C22×Dic83—22no—
101C22×Dic83—22no—
102C22×Dic83—22no—
103C22×Dic83—22no—
104C22×Dic83—22no—
105C22×Dic83—22no—
106C22×Dic83—22no—
107C22×Dic83—22no—
108C22×Dic83—22no—
109C22×Dic83—22no—
110C22×Dic83—22no—
111C22×Dic83—22no—
112C22×Dic83—22no—
113C22×Dic83—22no—
114C22×Dic83—22no—
115C22×Dic83—22no—
116C22×Dic83C22×C414yes—
117C22×Dic83C22×C414yes—
118C22×Dic83C2×D414yes—
119C22×Dic83C2×D414yes—
120C22×Dic83#4114yes—
121C4⋊Dic83C22⋊C4116yescomplemented
122Dic83⋊C4—216no—
123Dic83⋊C4—48no—
124C22⋊Dic83—216no—
125C22⋊Dic83—48no—
126C4×Dic83—48no—
127C24×C83—22no—
128C24×C83—22no—
129C24×C83—21no—
130C24×C83—21no—
131C24×C83—21no—
132C24×C83—22no—
133C24×C83—22no—
134C24×C83—21no—
135C24×C83C22×C412yes—
136C24×C83C2×Q812yes—
137C24×C83#4112yes—
138C24×C83C2411yescharacteristic
139C22×C4×C83—21no—
140C22×C4×C83—21no—
141C22×C4×C83—22no—
142C22×C4×C83—22no—
143C22×C4×C83—22no—
144C22×C4×C83—22no—
145C22×C4×C83—22no—
146C22×C4×C83—22no—
147C22×C4×C83—22no—
148C22×C4×C83—22no—
149C22×C4×C83—21no—
150C22×C4×C83—21no—
151C22×C4×C83—22no—
152C22×C4×C83—28no—
153C22×C4×C83—22no—
154C22×C4×C83—22no—
155C22×C4×C83—22no—
156C22×C4×C83—22no—
157C22×C4×C83—22no—
158C22×C4×C83—22no—
159C22×C4×C83—22no—
160C22×C4×C83—44no—
161C22×C4×C83—44no—
162C22×C4×C83—48no—
163C22×C4×C83—48no—
164C22×C4×C83C22×C412yes—
165C22×C4×C83C22×C412yes—
166C22×C4×C83C2×D414yes—
167C22×C4×C83C2×Q814yes—
168C22⋊C4×C83—44no—
169C22⋊C4×C83—44no—
170C23×C83—22no—
171C23×C83—22no—
172C23×C83—22no—
173C23×C83—22no—
174C23×C83—22no—
175C23×C83—22no—
176C23×C83—22no—
177C23×C83—22no—
178C23×C83—22no—
179C23×C83—22no—
180C23×C83—24no—
181C23×C83—24no—
182C23×C83—24no—
183C23×C83—24no—
184C23×C83—42no—
185C23×C83—42no—
186C23×C83—42no—
187C23×C83—42no—
188C23×C83—42no—
189C23×C83—42no—
190C23×C83—42no—
191C23×C83—42no—
192C23×C83C4×Q814yes—
193C23×C83C23×C411yescharacteristic
194C23×C83C22×D412yes—
195C23×C83C22×Q811yescharacteristic
196C23×C83C2×#4112yes—
197C23×C83C2×#4111yescharacteristic
198C23×C83H(5,𝔽2)11yescharacteristic
199C23×C83H(5,𝔽2)11yescharacteristic
200C23×C83H(5,𝔽2)12yes—
201C23×C83H(5,𝔽2)11yescharacteristic
202C23×C832−1+412yes—
203C23×C832−1+411yescharacteristic
204C2×C4×C83—22no—
205C2×C4×C83—22no—
206C2×C4×C83—24no—
207C2×C4×C83—24no—
208C2×C4×C83—22no—
209C2×C4×C83—22no—
210C2×C4×C83—22no—
211C2×C4×C83—24no—
212C2×C4×C83—24no—
213C2×C4×C83—24no—
214C2×C4×C83—416no—
215C2×C4×C83—42no—
216C2×C4×C83—42no—
217C2×C4×C83—42no—
218C2×C4×C83—42no—
219C2×C4×C83—42no—
220C2×C4×C83—42no—
221C2×C4×C83—42no—
222C2×C4×C83—42no—
223C2×C4×C83—42no—
224C2×C4×C83—42no—
225C2×C4×C83—42no—
226C2×C4×C83—42no—
227C2×C4×C83—42no—
228C2×C4×C83C2×C22⋊C414yes—
229C2×Dic83—24no—
230C2×Dic83—24no—
231C2×Dic83—24no—
232C2×Dic83—24no—
233C2×Dic83—24no—
234C2×Dic83—24no—
235C2×Dic83—24no—
236C2×Dic83—24no—
237C2×Dic83—24no—
238C2×Dic83—24no—
239C2×Dic83—24no—
240C2×Dic83—24no—
241C2×Dic83—42no—
242C2×Dic83—42no—
243C2×Dic83—42no—
244C2×Dic83—42no—
245C2×Dic83—42no—
246C2×Dic83—42no—
247C2×Dic83—42no—
248C2×Dic83—42no—
249C2×Dic83—42no—
250C2×Dic83—42no—
251C2×Dic83—42no—
252C2×Dic83—42no—
253C2×Dic83—42no—
254C2×Dic83—42no—
255C2×Dic83—42no—
256C2×Dic83C2×C22⋊C418yes—
257Dic166—416no—
258C22×C83—24no—
259C22×C83—24no—
260C22×C83—24no—
261C22×C83—42no—
262C22×C83—42no—
263C22×C83—42no—
264C22×C83—42no—
265C22×C83—44no—
266C22×C83—44no—
267C22×C83—44no—
268C22×C83—44no—
269C22×C83C22×C22⋊C411yescharacteristic
270C22×C83C2×C4×Q811yescharacteristic
271C22×C83C4×#4111yescharacteristic
272C22×C83#51811yescharacteristic
273C22×C83#51811yescharacteristic
274C22×C83#51911yescharacteristic
275C22×C83#52011yescharacteristic
276C22×C83#53512yes—
277C22×C83#53512yes—
278C22×C83#53811yescharacteristic
279C22×C83#53912yes—
280C22×C83#53911yescharacteristic
281C22×C83#54212yes—
282C22×C83#54311yescharacteristic
283C22×C83#54411yescharacteristic
284C22×C83#54612yes—
285C22×C83#54712yes—
286C22×C83#54812yes—
287C22×C83D4×Q812yes—
288C22×C83#55112yes—
289C22×C83#55412yes—
290C22×C83#56212yes—
291C22×C83#56312yes—
292C4×C83—24no—
293C4×C83—42no—
294C4×C83—44no—
295C4×C83—82no—
296Dic83—28no—
297Dic83—44no—
298Dic83—44no—
299Dic83—82no—
300#14448—831nomaximal, Sylow
301C2×C83—44no—
302C2×C83#230911yescharacteristic
303C2×C83#232911yescharacteristic
304C2×C83#233211yescharacteristic
305C2×C83#233911yescharacteristic
306C2×C83Q8×C22⋊C411yescharacteristic
307C2×C83#236811yescharacteristic
308C2×C83#237811yescharacteristic
309C2×C83#269212yes—
310C2×C83#269612yes—
311C2×C83#272012yes—
312C2×C83#272312yes—
313C22×C22⋊Q8—831no—
314C2×#387—832no—
315#1446—832no—
316#1445—832no—
317#1439—832no—
318#1437—832no—
319C4⋊#95—832no—
320C22⋊#95—831no—
321C22⋊#95D8311yescharacteristic
322C83#1444811yescharacteristic, complemented, Sylow
323C23×Q8—831no—
324C24×C4C2×D8311yescharacteristic
325C2×C22⋊Q8—838no—
326C2×C22⋊Q8—1668no—
327C22×C4⋊C4—832no—
328C22×C4⋊C4—831no—
329C22×C4⋊C4—832no—
330C22×C22⋊C4—832no—
331C22×C22⋊C4—832no—
332C22×C22⋊C4C2×D8311yescharacteristic
333C22×C22⋊C4C2×D8311yescharacteristic
334C22×C42—832no—
335#387—1668no—
336C2×#95—831no—
337C2×#95—831no—
338C2×#95—832no—
339C2×#95—832no—
340C2×#95—832no—
341C2×#95—832no—
342C2×#95—832no—
343C2×#95—832no—
344C2×#95—832no—
345C2×#95C2×D8312yes—
346C2×#95C2×D8312yes—
347C25C22×D8311yescharacteristic
348C22×Q8—3321no—
349C22×Q8—3321no—
350C22×Q8—3322no—
351C22×Q8—834no—
352C22×Q8—1664no—
353C23×C4—24no—
354C23×C4—42no—
355C23×C4—41no—
356C23×C4—41no—
357C23×C4—832no—
358C23×C4—832no—
359C23×C4—832no—
360C23×C4—832no—
361C23×C4C4×D8314yes—
362C23×C4C22×D8312yes—
363C23×C4C22×D8311yescharacteristic
364C23×C4C22×D8312yes—
365C23×C4C22×D8311yescharacteristic
366C22⋊Q8—16632no—
367C2×C4⋊C4—3322no—
368C2×C4⋊C4—834no—
369C2×C4⋊C4—838no—
370C2×C4⋊C4—838no—
371C2×C4⋊C4—1664no—
372C2×C4⋊C4—1664no—
373C2×C4⋊C4—1664no—
374C2×C4⋊C4—1664no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—41no—
377C2×C22⋊C4—41no—
378C2×C22⋊C4—41no—
379C2×C22⋊C4—41no—
380C2×C22⋊C4—3322no—
381C2×C22⋊C4—838no—
382C2×C22⋊C4—1664no—
383C2×C22⋊C4—1664no—
384C2×C22⋊C4—1664no—
385C2×C22⋊C4—1664no—
386C2×C22⋊C4Dic16614yescomplemented
387C2×C42—3322no—
388C2×C42—1664no—
389C2×C42—1664no—
390#95—24no—
391#95—42no—
392#95—3322no—
393#95—3322no—
394#95—3321no—
395#95—3321no—
396#95—3322no—
397#95—3322no—
398#95—1664no—
399#95—1664no—
400#95—1664no—
401C24—22no—
402C24—22no—
403C24—21no—
404C24—21no—
405C24—21no—
406C24—22no—
407C24—22no—
408C24—21no—
409C24C2×C4×D8312yes—
410C24C2×Dic16612yes—
411C24#1305059012yes—
412C24C23×D8311yescharacteristic
413C2×Q8—3328no—
414C2×Q8—3328no—
415C2×Q8—3324no—
416C2×Q8—3324no—
417C2×Q8—1668no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—28no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—44no—
440C22×C4—44no—
441C22×C4—48no—
442C22×C4—48no—
443C22×C4—834no—
444C22×C4—834no—
445C22×C4—834no—
446C22×C4—834no—
447C22×C4—834no—
448C22×C4—1662no—
449C22×C4—1662no—
450C22×C4—1662no—
451C22×C4—1662no—
452C22×C4—1662no—
453C22×C4—1662no—
454C22×C4—1662no—
455C22×C4—1662no—
456C22×C4—1662no—
457C22×C4—1662no—
458C22×C4—1662no—
459C22×C4—1662no—
460C22×C4—1662no—
461C22×C4—1662no—
462C22×C4—1662no—
463C22×C4—1662no—
464C22×C4—1662no—
465C22×C4—1662no—
466C22×C4—1662no—
467C22×C4—1662no—
468C22×C4—1662no—
469C22×C4—1662no—
470C22×C4—1662no—
471C22×C4C2×C4×D8312yes—
472C22×C4C2×C4×D8312yes—
473C22×C4D4×D8314yes—
474C22×C4C2×Dic16614yes—
475C4⋊C4—3328no—
476C4⋊C4—8316no—
477C4⋊C4—16616no—
478C22⋊C4—44no—
479C22⋊C4—44no—
480C22⋊C4—3328no—
481C22⋊C4—16616no—
482C42—3328no—
483C23—22no—
484C23—22no—
485C23—24no—
486C23—24no—
487C23—24no—
488C23—22no—
489C23—24no—
490C23—22no—
491C23—22no—
492C23—22no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—42no—
504C23—42no—
505C23C4×Dic16614yes—
506C23C22×C4×D8311yescharacteristic
507C23C2×D4×D8312yes—
508C23C22×Dic16611yescharacteristic
509C23#41×D8312yes—
510C23C2×#1305059011yescharacteristic
511C23#1458183711yescharacteristic
512C23#1458183711yescharacteristic
513C23#1458183711yescharacteristic
514C23#1458183712yes—
515C23#1458183912yes—
516C23#1458184011yescharacteristic
517Q8—33216no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—24no—
521C2×C4—24no—
522C2×C4—24no—
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—416no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—3322no—
543C2×C4—3322no—
544C2×C4—3322no—
545C2×C4—3322no—
546C2×C4—3322no—
547C2×C4—3322no—
548C2×C4—3322no—
549C2×C4—3322no—
550C2×C4—3322no—
551C2×C4—3322no—
552C2×C4—3322no—
553C2×C4—3322no—
554C2×C4—3322no—
555C2×C4—3322no—
556C2×C4—3322no—
557C2×C4—838no—
558C2×C4—1664no—
559C2×C4—1664no—
560C2×C4—1664no—
561C2×C4—1664no—
562C2×C4—1664no—
563C2×C4—1664no—
564C2×C4—1664no—
565C2×C4—1664no—
566C2×C4—1664no—
567C2×C4—1664no—
568C2×C4—1664no—
569C2×C4—1664no—
570C2×C4C22⋊C4×D8314yes—
571C22—24no—
572C22—24no—
573C22—24no—
574C22—42no—
575C22—44no—
576C22—44no—
577C22—42no—
578C22—44no—
579C22—44no—
580C22—42no—
581C22—42no—
582C22C2×C22⋊C4×D8311yescharacteristic
583C22C2×C4×Dic16611yescharacteristic
584C22#1912816911yescharacteristic
585C22#1912817411yescharacteristic
586C22#1912817411yescharacteristic
587C22#1912817811yescharacteristic
588C22#1912818211yescharacteristic
589C22#1912826612yes—
590C22#1912826612yes—
591C22#1912828911yescharacteristic
592C22#1912829412yes—
593C22#1912829811yescharacteristic
594C22#1912831912yes—
595C22#1912832711yescharacteristic
596C22#1912833211yescharacteristic
597C22#1912834812yes—
598C22#1912835312yes—
599C22#1912836012yes—
600C22D4×Dic16612yes—
601C22#1912838012yes—
602C22#1912841012yes—
603C22#1912844612yes—
604C22#1912845212yes—
605C4—24no—
606C4—44no—
607C4—42no—
608C4—82no—
609C4—3324no—
610C4—3324no—
611C4—6642no—
612C4—1668no—
613C2—44no—
614C2#2491004411yescharacteristic
615C2#2491019611yescharacteristic
616C2#2491021611yescharacteristic
617C2#2491029111yescharacteristic
618C2C22⋊C4×Dic16611yescharacteristic
619C2#2491054611yescharacteristic
620C2#2491062611yescharacteristic
621C2#2491396012yes—
622C2#2491401412yes—
623C2#2491434412yes—
624C2#2491438312yes—
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC83×C22⋊#95C2
socle, cosocleC24×C83C24
prime radicalC25×C83C23
hypercenterC22⋊#95D83
hypocenterC83#14448
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C22⋊#95

Complements

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

NH ≅ G/Ntype
2. C2×#19127584C2semidirect
3. #24905733C2semidirect
4. #24905729C2semidirect
7. Dic83⋊#95C2semidirect
12. C2×#14581771C4semidirect
85. C2×C22⋊C4×C83Q8semidirect
121. C4⋊Dic83C22⋊C4semidirect
322. C83#14448semidirect
386. C2×C22⋊C4Dic166semidirect

Maximal subgroups

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

subgroupindex
C22×#145817712
C2×#191275842
#249057332
#249057292
#249057062
#249057002
Dic83⋊#952
#249056862
C83×C22⋊#952
#1444883