Finite Group Databasebeta

← #35564698

Subgroups — #35564698

Order 21248.

3118 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128213C211yesmaximal, characteristic
2C2×#19128181C211yesmaximal, characteristic, complemented
3C22×#14581750C211yesmaximal, characteristic, complemented
4#24910500C214yesmaximal, complemented
5#24910489C214yesmaximal, complemented
6#24910414C214yesmaximal, complemented
7#24910389C214yesmaximal, complemented
8#24910120C214yesmaximal, complemented
9#24910056C211yesmaximal, characteristic, complemented
10#24909982C214yesmaximal, complemented
11#24909842C211yesmaximal, characteristic, complemented
12#24909819C211yesmaximal, characteristic, complemented
13C83×#2317C211yesmaximal, characteristic, complemented
14C22×#13050590C2211yescharacteristic
15C23×C4×D83C2211yescharacteristic
16#19128213C4116yescomplemented
17C2×#14581776C2212yes—
18C2×#14581772C2212yes—
19C2×#14581767C2212yes—
20C2×#14581762C2212yes—
21#19128181—28no—
22C2×C4×Dic166C2212yescomplemented
23C2×C4×D332C2214yes—
24C2×C4×D332C2212yescomplemented
25C2×(C22⋊D83)⋊C4C2214yes—
26C2×#14581751C2211yescharacteristic
27C2×#14581750—28no—
28C2×#14581750C2214yes—
29C2×#14581750C2214yes—
30C2×#14581750C2212yescomplemented
31C22×C4⋊Dic83C2211yescharacteristic
32C2×C4⋊C4×D83C2212yescomplemented
33C22×(C2×D83)⋊C4C2212yes—
34C22×C4×Dic83C2212yes—
35#19127606C2212yescomplemented
36#19127602C2218yes—
37#19127602C2212yescomplemented
38#19127591C2218yes—
39#19127591C2218yescomplemented
40#19127590C2218yes—
41#19127584C2218yescomplemented
42#19127580C2214yes—
43#19127579C2218yes—
44#19127562C2212yes—
45#19127558C2212yes—
46#19127558C2212yes—
47#19127558C2214yescomplemented
48#19127556C2212yes—
49C4⋊C4×Dic83C2218yescomplemented
50C4⋊C4×Dic83C2214yescomplemented
51C4×C4⋊Dic83C2214yescomplemented
52C22⋊C4×Dic83C2214yes—
53C4×(C2×D83)⋊C4C2214yes—
54#19127548C2218yescomplemented
55C2×#117×C83C2211yescharacteristic
56C2×#117×C83C2211yescharacteristic
57C22×C4⋊C4×C83C2211yescharacteristic
58#389×C83C2214yescomplemented
59#385×C83C2214yes—
60#384×C83C2212yescomplemented
61#382×C83C2212yescomplemented
62C24×D83C2311yescharacteristic
63C2×#13050590—24no—
64C2×#13050590—24no—
65C2×#13050590C2×C418yes—
66C22×Dic166C2311yescharacteristic
67C22×D332C2311yescharacteristic
68C22×D332C2314yes—
69C22×C22⋊D83C2312yes—
70C23×Dic83C2311yescharacteristic
71C23×Dic83C2312yes—
72C22×C4×D83—24no—
73C22×C4×D83—24no—
74C22×C4×D83C2×C418yescomplemented
75C22×C4×D83C2312yes—
76C22×C4×D83C2312yes—
77C22×C4×D83C2312yes—
78C22×C4×D83C2314yes—
79#14581776C2×C4116yescomplemented
80#14581772C2×C4116yescomplemented
81#14581767C2×C4116yescomplemented
82#14581762C2×C4116yescomplemented
83C4×Dic166—28no—
84C4×D332—28no—
85C4×D332C2×C4132yescomplemented
86(C22⋊D83)⋊C4—216no—
87#14581751—24no—
88#14581750—216no—
89#14581750—216no—
90#14581750—28no—
91C2×C4⋊Dic83D418yescomplemented
92C2×C4⋊Dic83C2312yes—
93C2×C4⋊Dic83C2314yes—
94C2×C4⋊Dic83C2312yes—
95C2×C4⋊Dic83C2312yes—
96C2×Dic83⋊C4C2314yes—
97C4⋊C4×D83—28no—
98C2×C22⋊Dic83C2312yes—
99C2×(C2×D83)⋊C4—28no—
100C2×(C2×D83)⋊C4C2318yes—
101C2×(C2×D83)⋊C4C2314yes—
102C2×(C2×D83)⋊C4C2314yes—
103C2×(C2×D83)⋊C4C2314yes—
104C2×(C2×D83)⋊C4C2318yes—
105C2×(C2×D83)⋊C4C2314yes—
106C2×(C2×D83)⋊C4C2314yes—
107C2×C4×Dic83—28no—
108C2×C4×Dic83C2314yes—
109C2×C4×Dic83C2312yes—
110C2×C4×Dic83C2318yes—
111C2×C4×Dic83C2318yes—
112C83⋊#95C2318yes—
113C83⋊#95C2314yes—
114C83⋊#95C2318yes—
115C83⋊#95C2314yes—
116C23×C4×C83C2312yes—
117C23×C4×C83C2311yescharacteristic
118#117×C83—24no—
119#117×C83C2×C418yescomplemented
120C2×C4⋊C4×C83—24no—
121C2×C4⋊C4×C83C2311yescharacteristic
122C2×C4⋊C4×C83C2311yescharacteristic
123C2×C4⋊C4×C83C2312yes—
124C2×C4⋊C4×C83C2312yes—
125C2×C4⋊C4×C83C2318yes—
126C2×C4⋊C4×C83C2312yes—
127C2×C22⋊C4×C83C2312yes—
128C2×C22⋊C4×C83C2312yes—
129C2×C42×C83C2312yes—
130C2×C42×C83C2312yes—
131#95×C83C2318yes—
132C23×D83—22no—
133C23×D83—22no—
134C23×D83—42no—
135C23×D83—42no—
136C23×D83C22×C414yes—
137C23×D83C2412yes—
138C23×D83C2412yes—
139#13050590—232no—
140C2×Dic166—22no—
141C2×Dic166—22no—
142C2×Dic166C22×C414yes—
143C2×D332—22no—
144C2×D332—22no—
145C2×D332—44no—
146C2×D332—44no—
147C2×D332C22×C414yes—
148C2×D332C22×C4116yes—
149C2×C22⋊D83—24no—
150C2×C22⋊D83—24no—
151C2×C22⋊D83C22×C418yes—
152C22×Dic83—22no—
153C22×Dic83—22no—
154C22×Dic83—44no—
155C22×Dic83C22×C414yes—
156C22×Dic83C2×D418yes—
157C22×Dic83C2414yes—
158C22×Dic83C2412yes—
159C22×Dic83C2412yes—
160C22×Dic83C2414yes—
161C2×C4×D83—232no—
162C2×C4×D83—24no—
163C2×C4×D83—24no—
164C2×C4×D83—24no—
165C2×C4×D83—24no—
166C2×C4×D83—24no—
167C2×C4×D83—24no—
168C2×C4×D83—28no—
169C2×C4×D83—48no—
170C2×C4×D83C22×C418yes—
171C2×C4×D83C22×C418yes—
172C2×C4×D83C22×C418yes—
173C4⋊Dic83—24no—
174C4⋊Dic83C22×C4116yes—
175C4⋊Dic83C2×D418yes—
176C4⋊Dic83C2×D418yes—
177Dic83⋊C4—28no—
178C22⋊Dic83—24no—
179(C2×D83)⋊C4—216no—
180(C2×D83)⋊C4—28no—
181(C2×D83)⋊C4—44no—
182(C2×D83)⋊C4—44no—
183(C2×D83)⋊C4—48no—
184(C2×D83)⋊C4C22×C4116yes—
185(C2×D83)⋊C4C22×C4116yes—
186C4×Dic83—24no—
187C4×Dic83—216no—
188C4×Dic83—28no—
189C4×Dic83—48no—
190C24×C83C2411yescharacteristic
191C22×C4×C83—24no—
192C22×C4×C83—44no—
193C22×C4×C83C22×C414yes—
194C22×C4×C83C2×D414yes—
195C22×C4×C83#4114yes—
196C22×C4×C83C2412yes—
197C22×C4×C83C2414yes—
198C22×C4×C83C2412yes—
199C22×C4×C83C2414yes—
200C22×C4×C83C2411yescharacteristic
201C22×C4×C83C2411yescharacteristic
202C4⋊C4×C83—24no—
203C4⋊C4×C83—24no—
204C4⋊C4×C83—22no—
205C4⋊C4×C83—22no—
206C4⋊C4×C83—48no—
207C4⋊C4×C83C22×C418yes—
208C22⋊C4×C83—24no—
209C22⋊C4×C83C22×C418yes—
210C42×C83—24no—
211C42×C83C22×C418yes—
212C23×C83—22no—
213C23×C83—22no—
214C23×C83C23×C412yes—
215C23×C83C22×D412yes—
216C23×C83C2×#4112yes—
217C23×C83C2511yescharacteristic
218C22×D83—28no—
219C22×D83—22no—
220C22×D83—22no—
221C22×D83—22no—
222C22×D83—22no—
223C22×D83—24no—
224C22×D83—22no—
225C22×D83—22no—
226C22×D83—24no—
227C22×D83—22no—
228C22×D83—22no—
229C22×D83—48no—
230C22×D83—44no—
231C22×D83—44no—
232C22×D83—48no—
233C22×D83C23×C414yes—
234C22×D83C23×C414yes—
235C2×C4×C83—24no—
236C2×C4×C83—24no—
237C2×C4×C83—22no—
238C2×C4×C83—22no—
239C2×C4×C83—24no—
240C2×C4×C83—24no—
241C2×C4×C83—28no—
242C2×C4×C83—24no—
243C2×C4×C83—48no—
244C2×C4×C83C4×D4116yes—
245C2×C4×C83C23×C418yes—
246C2×C4×C83C23×C412yes—
247C2×C4×C83C23×C412yes—
248C2×C4×C83C22×D412yes—
249C2×C4×C83C22×D412yes—
250C2×C4×C83C2×#4112yes—
251C2×C4×C83C2×#4112yes—
252C22⋊D83—216no—
253C2×Dic83—28no—
254C2×Dic83—22no—
255C2×Dic83—22no—
256C2×Dic83—22no—
257C2×Dic83—22no—
258C2×Dic83—28no—
259C2×Dic83—24no—
260C2×Dic83—24no—
261C2×Dic83—24no—
262C2×Dic83—48no—
263C2×Dic83C23×C414yes—
264C2×Dic83C23×C414yes—
265C2×Dic83C22×D418yes—
266D332—28no—
267D332—416no—
268C4×D83—216no—
269C4×D83—216no—
270C4×D83—216no—
271C4×D83—88no—
272Dic166—28no—
273C22×C83—24no—
274C22×C83—22no—
275C22×C83—22no—
276C22×C83—24no—
277C22×C83C2×C4×D414yes—
278C22×C83C24×C411yescharacteristic
279C22×C83C23×D411yescharacteristic
280C22×C83C22×#4111yescharacteristic
281C22×C83#58512yes—
282C22×C83#58512yes—
283C2×D83—24no—
284C2×D83—24no—
285C2×D83—24no—
286C2×D83—28no—
287C2×D83—24no—
288C2×D83—416no—
289C2×D83—42no—
290C2×D83—42no—
291C2×D83—42no—
292C2×D83—42no—
293C2×D83—44no—
294C2×D83—42no—
295C2×D83—42no—
296C4×C83—28no—
297C4×C83—42no—
298C4×C83—42no—
299C4×C83—44no—
300C4×C83C2×C4×D414yes—
301C4×C83C2×C4×D414yes—
302Dic83—24no—
303Dic83—24no—
304Dic83—28no—
305Dic83—44no—
306#34201—831nomaximal, Sylow
307C2×C83—24no—
308C2×C83C22×C4×D411yescharacteristic
309C2×C83#342812yes—
310C2×C83#348511yescharacteristic
311C2×C83#348711yescharacteristic
312C2×C83#349312yes—
313D83—44no—
314D83—44no—
315C2×#525—831no—
316C2×#520—831no—
317C22×#117—831no—
318#2363—834no—
319#2362—834no—
320#2354—834no—
321#2352—834no—
322#2321—834no—
323#2317D8311yescharacteristic, complemented
324#2311—831no—
325C4×#123—834no—
326#2280—831no—
327#2274—831no—
328C83#3420111yescharacteristic, complemented, Sylow
329C22×#41—831no—
330C24×C4—831no—
331#525—8316no—
332C2×#123—832no—
333C2×C22⋊Q8—832no—
334C2×#121—832no—
335C2×#120—832no—
336#520—1668no—
337C2×C4×Q8—832no—
338C2×C4×D4—834no—
339C2×C4×D4—834no—
340C2×C4×D4—832no—
341C2×#117—831no—
342C2×#117—834no—
343C2×#117—832no—
344C2×#117—834no—
345C2×#117—1668no—
346C2×#117C2×D8311yescharacteristic
347C2×#117C2×D8311yescharacteristic
348C22×C4⋊C4—831no—
349C22×C4⋊C4—832no—
350C22×C4⋊C4C2×D8311yescharacteristic
351C22×C22⋊C4—832no—
352C22×C42—832no—
353#391—832no—
354#390—832no—
355#390—838no—
356#389C2×D8314yescomplemented
357#388—838no—
358#388—838no—
359#388—838no—
360#387—838no—
361#386—834no—
362#386—838no—
363#385C2×D8314yes—
364#384C2×D8312yescomplemented
365#382C2×D8312yescomplemented
366#381—832no—
367#380—832no—
368#380—832no—
369#380—834no—
370#379—832no—
371C4×C4⋊C4—838no—
372C4×C4⋊C4—834no—
373C4×C4⋊C4—834no—
374C4×C22⋊C4—834no—
375C4×C22⋊C4—834no—
376C4×C22⋊C4—838no—
377C25—831no—
378C2×#41—838no—
379C2×#41—1664no—
380C2×#41—1664no—
381C22×Q8—831no—
382C22×D4—832no—
383C22×D4—834no—
384C22×D4—831no—
385C23×C4—832no—
386C23×C4—834no—
387C23×C4—831no—
388C23×C4—838no—
389C23×C4—832no—
390C23×C4—832no—
391C23×C4—832no—
392C23×C4—1664no—
393C23×C4—1664no—
394C23×C4C22×D8312yes—
395C23×C4C22×D8311yescharacteristic
396#123—8316no—
397C22⋊Q8—8316no—
398#121—8316no—
399#120—8316no—
400C4×Q8—1668no—
401C4×D4—8332no—
402C4×D4—1668no—
403C4×D4—16616no—
404#117—24no—
405#117—16616no—
406#117—16616no—
407#117—1668no—
408#117—1664no—
409#117C4×D8318yescomplemented
410C2×C4⋊C4—24no—
411C2×C4⋊C4—838no—
412C2×C4⋊C4—832no—
413C2×C4⋊C4—832no—
414C2×C4⋊C4—834no—
415C2×C4⋊C4—834no—
416C2×C4⋊C4—832no—
417C2×C4⋊C4—1668no—
418C2×C4⋊C4C22×D8311yescharacteristic
419C2×C4⋊C4C22×D8312yes—
420C2×C4⋊C4C22×D8312yes—
421C2×C4⋊C4C22×D8311yescharacteristic
422C2×C4⋊C4C22×D8318yes—
423C2×C4⋊C4C22×D8312yes—
424C2×C22⋊C4—834no—
425C2×C22⋊C4—834no—
426C2×C22⋊C4—838no—
427C2×C22⋊C4—834no—
428C2×C22⋊C4—834no—
429C2×C22⋊C4—832no—
430C2×C22⋊C4—838no—
431C2×C22⋊C4—834no—
432C2×C22⋊C4—1668no—
433C2×C22⋊C4C22×D8312yes—
434C2×C22⋊C4C22×D8312yes—
435C2×C42—838no—
436C2×C42—838no—
437C2×C42—834no—
438C2×C42—832no—
439C2×C42—1668no—
440C2×C42C22×D8312yes—
441C2×C42C22×D8312yes—
442#95—834no—
443#95—834no—
444#95—838no—
445#95—838no—
446#95C22×D8318yes—
447C24—3322no—
448C24—3322no—
449C24—834no—
450C24—832no—
451C24—832no—
452C24—1662no—
453C24—1662no—
454C24C23×D8311yescharacteristic
455#41—16632no—
456C2×Q8—834no—
457C2×Q8—1662no—
458C2×Q8—1662no—
459C2×D4—3324no—
460C2×D4—3324no—
461C2×D4—838no—
462C2×D4—8316no—
463C2×D4—834no—
464C2×D4—1664no—
465C2×D4—1664no—
466C2×D4—1662no—
467C2×D4—1662no—
468C22×C4—24no—
469C22×C4—44no—
470C22×C4—3328no—
471C22×C4—3324no—
472C22×C4—838no—
473C22×C4—834no—
474C22×C4—832no—
475C22×C4—832no—
476C22×C4—834no—
477C22×C4—838no—
478C22×C4—838no—
479C22×C4—838no—
480C22×C4—834no—
481C22×C4—1668no—
482C22×C4—1662no—
483C22×C4—1662no—
484C22×C4—1664no—
485C22×C4—1664no—
486C22×C4—1664no—
487C22×C4—1664no—
488C22×C4—1664no—
489C22×C4—1664no—
490C22×C4—16632no—
491C22×C4C2×C4×D8314yes—
492C22×C4D4×D8314yes—
493C22×C4#1305059114yes—
494C22×C4C23×D8312yes—
495C22×C4C23×D8314yes—
496C22×C4C23×D8312yes—
497C22×C4C23×D8314yes—
498C22×C4C23×D8311yescharacteristic
499C22×C4C23×D8311yescharacteristic
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—22no—
503C4⋊C4—22no—
504C4⋊C4—48no—
505C4⋊C4—8316no—
506C4⋊C4—838no—
507C4⋊C4—838no—
508C4⋊C4—1668no—
509C4⋊C4—1664no—
510C4⋊C4C2×C4×D8318yes—
511C22⋊C4—24no—
512C22⋊C4—3324no—
513C22⋊C4—3328no—
514C22⋊C4—3324no—
515C22⋊C4—8316no—
516C22⋊C4—8316no—
517C22⋊C4—16616no—
518C22⋊C4—1668no—
519C22⋊C4—1664no—
520C22⋊C4C2×C4×D8318yes—
521C42—24no—
522C42—3328no—
523C42—16616no—
524C42—1668no—
525C42—1664no—
526C42C2×C4×D8318yes—
527C23—22no—
528C23—22no—
529C23—3324no—
530C23—3324no—
531C23—3328no—
532C23—3328no—
533C23—834no—
534C23—834no—
535C23—1668no—
536C23—1662no—
537C23—1662no—
538C23—1662no—
539C23—1662no—
540C23—1664no—
541C23—1662no—
542C23—1662no—
543C23—1662no—
544C23—1664no—
545C23—1662no—
546C23C22×C4×D8312yes—
547C23C2×D4×D8312yes—
548C23C2×#1305059112yes—
549C23C24×D8311yescharacteristic
550Q8—1668no—
551D4—33216no—
552D4—16616no—
553D4—1668no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—28no—
561C2×C4—24no—
562C2×C4—48no—
563C2×C4—3328no—
564C2×C4—834no—
565C2×C4—834no—
566C2×C4—838no—
567C2×C4—6648no—
568C2×C4—1664no—
569C2×C4—1664no—
570C2×C4—1664no—
571C2×C4—1668no—
572C2×C4—1662no—
573C2×C4—1662no—
574C2×C4—1662no—
575C2×C4—1662no—
576C2×C4—1668no—
577C2×C4—16616no—
578C2×C4—16616no—
579C2×C4—16616no—
580C2×C4#14581754116yes—
581C2×C4C22×C4×D8318yes—
582C2×C4C22×C4×D8312yes—
583C2×C4C22×C4×D8312yes—
584C2×C4C2×D4×D8312yes—
585C2×C4C2×D4×D8312yes—
586C2×C4C2×#1305059112yes—
587C2×C4C2×#1305059112yes—
588C22—22no—
589C22—22no—
590C22—24no—
591C22—24no—
592C22—33216no—
593C22—3322no—
594C22—3322no—
595C22—3322no—
596C22—3322no—
597C22—3322no—
598C22—3322no—
599C22—3324no—
600C22—1664no—
601C22—1664no—
602C22—1664no—
603C22—1668no—
604C22—1664no—
605C22C2×#1458175414yes—
606C22C23×C4×D8311yescharacteristic
607C22C22×D4×D8311yescharacteristic
608C22C22×#1305059111yescharacteristic
609C22#1912853612yes—
610C22#1912853812yes—
611C4—28no—
612C4—42no—
613C4—44no—
614C4—42no—
615C4—3324no—
616C4—1668no—
617C4—1664no—
618C4—1664no—
619C4C2×#1458175414yes—
620C4C2×#1458175414yes—
621C2—24no—
622C2—3324no—
623C2—3324no—
624C2C22×#1458175411yescharacteristic
625C2#2492066312yes—
626C2#2492110611yescharacteristic
627C2#2492111911yescharacteristic
628C2#2492118412yes—
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingC83×#2317C2
socle, cosocleC23×C83C25
prime radicalC2×#19128213C2
hypercenter#2317D83
hypocenterC83#34201
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2317

Complements

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

NH ≅ G/Ntype
2. C2×#19128181C2semidirect
3. C22×#14581750C2semidirect
4. #24910500C2semidirect
5. #24910489C2semidirect
6. #24910414C2semidirect
7. #24910389C2semidirect
8. #24910120C2semidirect
9. #24910056C2semidirect
10. #24909982C2semidirect
11. #24909842C2semidirect
12. #24909819C2semidirect
13. C83×#2317C2semidirect
16. #19128213C4semidirect
22. C2×C4×Dic166C22semidirect
24. C2×C4×D332C22semidirect
30. C2×#14581750C22semidirect
32. C2×C4⋊C4×D83C22semidirect
35. #19127606C22semidirect
37. #19127602C22semidirect
39. #19127591C22semidirect
41. #19127584C22semidirect
47. #19127558C22semidirect
49. C4⋊C4×Dic83C22semidirect
50. C4⋊C4×Dic83C22semidirect
51. C4×C4⋊Dic83C22semidirect
54. #19127548C22semidirect
58. #389×C83C22semidirect
60. #384×C83C22semidirect
61. #382×C83C22semidirect
74. C22×C4×D83C2×C4semidirect
79. #14581776C2×C4semidirect
80. #14581772C2×C4semidirect
81. #14581767C2×C4semidirect
82. #14581762C2×C4semidirect
85. C4×D332C2×C4semidirect
91. C2×C4⋊Dic83D4semidirect
119. #117×C83C2×C4semidirect
323. #2317D83semidirect
328. C83#34201semidirect
356. #389C2×D83semidirect
364. #384C2×D83semidirect
365. #382C2×D83semidirect
409. #117C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191282132
C2×#191281812
C22×#145817502
#249105002
#249104892
#249104142
#249103892
#249101202
#249100562
#249099822
#249098422
#249098192
C83×#23172
#3420183