Finite Group Databasebeta

← #35564546

Subgroups — #35564546

Order 21248.

2542 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128169C211yesmaximal, characteristic
2#24910417C214yesmaximal, complemented
3#24910409C214yesmaximal, complemented
4#24910391C214yesmaximal, complemented
5#24910387C214yesmaximal, complemented
6#24910073C211yesmaximal, characteristic, complemented
7#24910072C211yesmaximal, characteristic, complemented
8#24910071C211yesmaximal, characteristic, complemented
9#24910052C212yesmaximal
10#24909869C211yesmaximal, characteristic, complemented
11#24909867C212yesmaximal, complemented
12#24909865C212yesmaximal, complemented
13#384×D83C211yesmaximal, characteristic, complemented
14#24909817C212yesmaximal, complemented
15C83×#2314C211yesmaximal, characteristic, complemented
16C22×#13050590C2211yescharacteristic
17#19128169C4116yescomplemented
18C2×#14581757C2212yes—
19C2×#14581754C2212yes—
20C2×(C22⋊D83)⋊C4C2212yes—
21C2×(C22⋊D83)⋊C4C2212yes—
22C2×#14581750C2212yes—
23C2×#14581749C2214yes—
24C2×#14581749C2211yescharacteristic
25C2×#14581749C2211yescharacteristic
26C2×#14581748C2214yes—
27C2×C4⋊C4×D83C2212yes—
28C2×C4⋊C4×D83C2214yes—
29C22×C4×Dic83C2211yescharacteristic
30C2×C42×D83C2211yescharacteristic
31C2×C42×D83C2211yescharacteristic
32#19127605C2214yes—
33#19127601C2214yes—
34#19127594C2214yes—
35#19127593C2214yes—
36#19127591C2214yes—
37#19127588C2214yes—
38#19127587C2214yes—
39#19127582C2214yes—
40#19127579C2214yes—
41#19127577C2214yes—
42#19127571C2211yescharacteristic
43#19127570C2212yes—
44#19127570C2214yes—
45#19127570C2212yes—
46#19127567C2211yescharacteristic
47#19127565C2214yes—
48#19127565C2214yes—
49#19127561C2212yes—
50#19127560C2214yes—
51#19127560C2214yes—
52#19127559C2214yes—
53#19127558C2212yes—
54#19127558C2212yes—
55C4⋊C4×Dic83C2212yes—
56C4⋊C4×Dic83C2212yes—
57C4×Dic83⋊C4C2214yes—
58C4×Dic83⋊C4C2214yes—
59C4×C4⋊Dic83C2212yes—
60#19127552C2214yes—
61#19127552C2214yes—
62#19127552C2214yes—
63C4×(C2×D83)⋊C4C2214yes—
64#19127548C2214yes—
65C2×#117×C83C2211yescharacteristic
66#388×C83C2214yes—
67#387×C83C2214yes—
68#384×C83C2211yescharacteristic
69#382×C83C2211yescharacteristic
70#380×C83C2212yes—
71C4×C4⋊C4×C83C2212yes—
72C2×#13050590—24no—
73C2×#13050590—24no—
74C2×#13050590C2×C418yes—
75C22×Dic166C2311yescharacteristic
76C22×D332C2311yescharacteristic
77C22×C22⋊D83C2311yescharacteristic
78C22×C22⋊D83C2311yescharacteristic
79C23×Dic83C2312yes—
80C22×C4×D83C2311yescharacteristic
81C22×C4×D83C2312yes—
82C22×C4×D83C2312yes—
83C22×C4×D83C2314yes—
84C22×C4×D83C2311yescharacteristic
85#14581757C2×C4116yescomplemented
86#14581754C2×C4116yes—
87(C22⋊D83)⋊C4C2×C4116yes—
88(C22⋊D83)⋊C4C2×C4116yescomplemented
89#14581750—28no—
90#14581749—48no—
91#14581749C2×C418yes—
92#14581749C2×C418yescomplemented
93#14581748—48no—
94C2×C4⋊Dic83C2312yes—
95C2×C4⋊Dic83C2312yes—
96C2×C4⋊Dic83C2314yes—
97C2×Dic83⋊C4C2314yes—
98C2×Dic83⋊C4C2314yes—
99C2×Dic83⋊C4C2314yes—
100C2×Dic83⋊C4C2314yes—
101C2×Dic83⋊C4C2314yes—
102C2×Dic83⋊C4C2312yes—
103C2×Dic83⋊C4C2312yes—
104C4⋊C4×D83—28no—
105C4⋊C4×D83—48no—
106C2×C22⋊Dic83C2314yes—
107C2×(C2×D83)⋊C4C2314yes—
108C2×(C2×D83)⋊C4C2314yes—
109C2×(C2×D83)⋊C4C2314yes—
110C2×(C2×D83)⋊C4C2312yes—
111C2×(C2×D83)⋊C4C2312yes—
112C2×C4×Dic83C2×C418yescomplemented
113C2×C4×Dic83C2314yes—
114C2×C4×Dic83C2314yes—
115C2×C4×Dic83C2314yes—
116C2×C4×Dic83C2314yes—
117C2×C4×Dic83C2312yes—
118C2×C4×Dic83C2312yes—
119C2×C4×Dic83C2312yes—
120C2×C4×Dic83C2311yescharacteristic
121C2×C4×Dic83C2311yescharacteristic
122C42×D83C2×C418yes—
123C42×D83C2×C418yescomplemented
124C83⋊#95C2314yes—
125C83⋊#95C2314yes—
126C83⋊#95C2314yes—
127C83⋊#95C2314yes—
128C83⋊#95C2314yes—
129C83⋊#95C2314yes—
130C83⋊#95C2314yes—
131C23×C4×C83C2311yescharacteristic
132#117×C83C2×C418yescomplemented
133C2×C4⋊C4×C83C2312yes—
134C2×C4⋊C4×C83C2312yes—
135C2×C4⋊C4×C83C2314yes—
136C2×C4⋊C4×C83C2312yes—
137C2×C22⋊C4×C83C2314yes—
138C2×C22⋊C4×C83C2312yes—
139C2×C42×C83C2314yes—
140C2×C42×C83C2311yescharacteristic
141C2×C42×C83C2311yescharacteristic
142#95×C83C2314yes—
143#95×C83C2314yes—
144#95×C83C2314yes—
145C23×D83C2411yescharacteristic
146C23×D83C2411yescharacteristic
147#13050590—232no—
148C2×Dic166—22no—
149C2×Dic166—22no—
150C2×Dic166C22×C414yes—
151C2×D332—22no—
152C2×D332—22no—
153C2×D332C22×C414yes—
154C2×C22⋊D83—22no—
155C2×C22⋊D83—22no—
156C2×C22⋊D83—22no—
157C2×C22⋊D83—22no—
158C2×C22⋊D83C22×C414yes—
159C2×C22⋊D83C22×C414yes—
160C22×Dic83—24no—
161C22×Dic83—24no—
162C22×Dic83C22×C418yes—
163C22×Dic83C2414yes—
164C22×Dic83C2414yes—
165C22×Dic83C2411yescharacteristic
166C22×Dic83C2412yes—
167C22×Dic83C2411yescharacteristic
168C22×Dic83C2412yes—
169C2×C4×D83—22no—
170C2×C4×D83—22no—
171C2×C4×D83—24no—
172C2×C4×D83—24no—
173C2×C4×D83—24no—
174C2×C4×D83—24no—
175C2×C4×D83—28no—
176C2×C4×D83—44no—
177C2×C4×D83—44no—
178C2×C4×D83C22×C414yes—
179C2×C4×D83C22×C418yes—
180C2×C4×D83C22×C418yes—
181C2×C4×D83C22×C414yes—
182C2×C4×D83#4114yes—
183C2×C4×D83#4114yes—
184C4⋊Dic83—24no—
185C4⋊Dic83—24no—
186C4⋊Dic83—44no—
187Dic83⋊C4—28no—
188Dic83⋊C4—28no—
189Dic83⋊C4—28no—
190Dic83⋊C4—28no—
191Dic83⋊C4—28no—
192Dic83⋊C4C22×C418yes—
193Dic83⋊C4C22×C418yes—
194C22⋊Dic83—44no—
195(C2×D83)⋊C4—28no—
196(C2×D83)⋊C4—44no—
197(C2×D83)⋊C4—44no—
198(C2×D83)⋊C4C22×C418yes—
199(C2×D83)⋊C4C22×C418yes—
200C4×Dic83—28no—
201C4×Dic83—44no—
202C4×Dic83—44no—
203C4×Dic83—44no—
204C4×Dic83C22×C414yes—
205C4×Dic83C22×C418yes—
206C4×Dic83C22×C418yes—
207C4×Dic83C22×C414yes—
208C4×Dic83C22×C418yes—
209C24×C83C2411yescharacteristic
210C22×C4×C83—22no—
211C22×C4×C83—22no—
212C22×C4×C83C22×C414yes—
213C22×C4×C83C2414yes—
214C22×C4×C83C2414yes—
215C22×C4×C83C2412yes—
216C22×C4×C83C2412yes—
217C22×C4×C83C2411yescharacteristic
218C22×C4×C83C2411yescharacteristic
219C4⋊C4×C83—24no—
220C4⋊C4×C83—24no—
221C4⋊C4×C83—44no—
222C4⋊C4×C83C22×C418yes—
223C22⋊C4×C83—44no—
224C22⋊C4×C83C22×C418yes—
225C42×C83—44no—
226C42×C83C22×C414yes—
227C42×C83C22×C414yes—
228C23×C83—21no—
229C23×C83—21no—
230C23×C83—22no—
231C23×C83—22no—
232C23×C83C23×C412yes—
233C23×C83C2511yescharacteristic
234C22×D83—22no—
235C22×D83—22no—
236C22×D83—21no—
237C22×D83—21no—
238C22×D83—22no—
239C22×D83—22no—
240C22×D83C23×C412yes—
241C22×D83C23×C412yes—
242C22×D83C2×#4112yes—
243C22×D83C2×#4112yes—
244C2×C4×C83—21no—
245C2×C4×C83—21no—
246C2×C4×C83—22no—
247C2×C4×C83—22no—
248C2×C4×C83—22no—
249C2×C4×C83—22no—
250C2×C4×C83—28no—
251C2×C4×C83—24no—
252C2×C4×C83—24no—
253C2×C4×C83—24no—
254C2×C4×C83—24no—
255C2×C4×C83—24no—
256C2×C4×C83—24no—
257C2×C4×C83C23×C414yes—
258C2×C4×C83C23×C414yes—
259C2×C4×C83C23×C412yes—
260C2×C4×C83C23×C412yes—
261C2×C4×C83C2×#4112yes—
262C2×C4×C83C2×#4112yes—
263C22⋊D83—28no—
264C22⋊D83—28no—
265C2×Dic83—21no—
266C2×Dic83—21no—
267C2×Dic83—22no—
268C2×Dic83—22no—
269C2×Dic83—216no—
270C2×Dic83—24no—
271C2×Dic83—24no—
272C2×Dic83—24no—
273C2×Dic83—24no—
274C2×Dic83—24no—
275C2×Dic83—24no—
276C2×Dic83C23×C414yes—
277C2×Dic83C23×C412yes—
278C2×Dic83C23×C412yes—
279C2×Dic83C23×C414yes—
280C2×Dic83C2×#4112yes—
281C2×Dic83C2×#4112yes—
282C2×Dic83H(5,𝔽2)14yes—
283C2×Dic832−1+414yes—
284D332—28no—
285C4×D83—28no—
286C4×D83—216no—
287C4×D83—216no—
288C4×D83—88no—
289C4×D83#117116yescomplemented
290Dic166—28no—
291C22×C83—22no—
292C22×C83—24no—
293C22×C83—41no—
294C22×C83—41no—
295C22×C83—42no—
296C22×C83C24×C411yescharacteristic
297C22×C83C22×#4111yescharacteristic
298C22×C83C22×#4111yescharacteristic
299C22×C83C2×H(5,𝔽2)12yes—
300C22×C83C2×2−1+412yes—
301C2×D83—24no—
302C2×D83—22no—
303C2×D83—22no—
304C2×D83—24no—
305C2×D83—22no—
306C2×D83—24no—
307C2×D83—41no—
308C2×D83—41no—
309C2×D83—42no—
310C2×D83C2×#11714yes—
311C4×C83—22no—
312C4×C83—24no—
313C4×C83—24no—
314C4×C83—44no—
315C4×C83—44no—
316C4×C83C2×#11714yes—
317Dic83—22no—
318Dic83—24no—
319Dic83—44no—
320Dic83—44no—
321Dic83C2×#11714yes—
322Dic83#52018yes—
323#34188—831nomaximal, Sylow
324C2×C83—42no—
325C2×C83C22×#11711yescharacteristic
326C2×C83C2×#52012yes—
327C2×C83#350811yescharacteristic
328C2×C83#350911yescharacteristic
329C2×C83#351112yes—
330D83—24no—
331D83—42no—
332C2×C4×#41—831no—
333#2355—834no—
334#2354—834no—
335#2352—834no—
336#2352—834no—
337#2314—831no—
338#2314—831no—
339#2314—831no—
340#2314D8311yescharacteristic, complemented
341#2311—832no—
342#2285—831no—
343#2285—832no—
344#2285—832no—
345C2×#384—831no—
346#2274—832no—
347C83#3418811yescharacteristic, complemented, Sylow
348C22×#41—831no—
349C4×#41—8316no—
350C2×C4×Q8—832no—
351C2×C4×D4—832no—
352C2×C4×D4—832no—
353C2×C4×D4—832no—
354C2×#117—832no—
355C2×#117—834no—
356C2×#117—834no—
357C2×#117—831no—
358C2×#117—831no—
359C2×#117C2×D8311yescharacteristic
360C22×C4⋊C4—834no—
361C22×C4⋊C4—832no—
362C22×C42—831no—
363C22×C42—831no—
364C22×C42—831no—
365#391—834no—
366#390—834no—
367#388—834no—
368#388—834no—
369#388—834no—
370#388C2×D8314yes—
371#387—834no—
372#387—834no—
373#387—834no—
374#387C2×D8314yes—
375#386—834no—
376#386—834no—
377#384—834no—
378#384—832no—
379#384—832no—
380#384—831no—
381#384C2×D8311yescharacteristic
382#382—831no—
383#382—834no—
384#382—834no—
385#382C2×D8311yescharacteristic
386#380—832no—
387#380—832no—
388#380—832no—
389#380—834no—
390#380—834no—
391#380—834no—
392#380C2×D8312yes—
393C4×C4⋊C4—832no—
394C4×C4⋊C4—832no—
395C4×C4⋊C4—832no—
396C4×C4⋊C4—834no—
397C4×C4⋊C4—834no—
398C4×C4⋊C4—834no—
399C4×C4⋊C4—834no—
400C4×C4⋊C4—834no—
401C4×C4⋊C4C2×D8312yes—
402C4×C22⋊C4—834no—
403C4×C22⋊C4—834no—
404C2×#41—838no—
405C2×#41—1664no—
406C2×#41—1664no—
407C22×Q8—831no—
408C22×D4—831no—
409C22×D4—831no—
410C22×D4—831no—
411C23×C4—834no—
412C23×C4—831no—
413C23×C4—832no—
414C23×C4—832no—
415C23×C4—832no—
416C23×C4—831no—
417C23×C4C22×D8311yescharacteristic
418C4×Q8—8316no—
419C4×D4—8316no—
420C4×D4—8316no—
421C4×D4—8316no—
422#117—3328no—
423#117—3328no—
424#117—838no—
425#117—838no—
426#117—1668no—
427#117C4×D8318yescomplemented
428C2×C4⋊C4—3328no—
429C2×C4⋊C4—834no—
430C2×C4⋊C4—832no—
431C2×C4⋊C4—832no—
432C2×C4⋊C4—834no—
433C2×C4⋊C4—834no—
434C2×C4⋊C4—832no—
435C2×C4⋊C4—834no—
436C2×C4⋊C4—834no—
437C2×C4⋊C4—832no—
438C2×C4⋊C4—834no—
439C2×C4⋊C4—1668no—
440C2×C4⋊C4C22×D8312yes—
441C2×C4⋊C4C22×D8312yes—
442C2×C4⋊C4C22×D8314yes—
443C2×C4⋊C4C22×D8312yes—
444C2×C22⋊C4—834no—
445C2×C22⋊C4—834no—
446C2×C22⋊C4—834no—
447C2×C22⋊C4—832no—
448C2×C22⋊C4—832no—
449C2×C22⋊C4—834no—
450C2×C22⋊C4C22×D8314yes—
451C2×C22⋊C4C22×D8312yes—
452C2×C42—834no—
453C2×C42—834no—
454C2×C42—834no—
455C2×C42—838no—
456C2×C42—831no—
457C2×C42—832no—
458C2×C42—832no—
459C2×C42—832no—
460C2×C42—831no—
461C2×C42—838no—
462C2×C42—838no—
463C2×C42—834no—
464C2×C42C22×D8314yes—
465C2×C42C22×D8311yescharacteristic
466C2×C42C22×D8311yescharacteristic
467#95—834no—
468#95—834no—
469#95—834no—
470#95—834no—
471#95—834no—
472#95—834no—
473#95—834no—
474#95C22×D8314yes—
475#95C22×D8314yes—
476#95C22×D8314yes—
477C24—831no—
478C24—831no—
479C24C23×D8311yescharacteristic
480#41—16632no—
481C2×Q8—834no—
482C2×Q8—1662no—
483C2×Q8—1662no—
484C2×D4—834no—
485C2×D4—834no—
486C2×D4—834no—
487C2×D4—1662no—
488C2×D4—1662no—
489C2×D4—1662no—
490C2×D4—1662no—
491C2×D4—1662no—
492C2×D4—1662no—
493C22×C4—22no—
494C22×C4—22no—
495C22×C4—3324no—
496C22×C4—3324no—
497C22×C4—831no—
498C22×C4—832no—
499C22×C4—834no—
500C22×C4—834no—
501C22×C4—834no—
502C22×C4—838no—
503C22×C4—831no—
504C22×C4—832no—
505C22×C4—834no—
506C22×C4—838no—
507C22×C4—838no—
508C22×C4—834no—
509C22×C4—834no—
510C22×C4—1664no—
511C22×C4—1664no—
512C22×C4—1664no—
513C22×C4—1664no—
514C22×C4—1668no—
515C22×C4—1664no—
516C22×C4—1664no—
517C22×C4—1662no—
518C22×C4—1662no—
519C22×C4C2×C4×D8314yes—
520C22×C4C23×D8314yes—
521C22×C4C23×D8314yes—
522C22×C4C23×D8311yescharacteristic
523C22×C4C23×D8312yes—
524C22×C4C23×D8312yes—
525C22×C4C23×D8311yescharacteristic
526C4⋊C4—24no—
527C4⋊C4—24no—
528C4⋊C4—44no—
529C4⋊C4—3324no—
530C4⋊C4—838no—
531C4⋊C4—838no—
532C4⋊C4—1668no—
533C4⋊C4—1668no—
534C4⋊C4—1668no—
535C4⋊C4—1668no—
536C4⋊C4—1668no—
537C4⋊C4—1664no—
538C4⋊C4—1664no—
539C4⋊C4C2×C4×D8318yes—
540C22⋊C4—44no—
541C22⋊C4—3324no—
542C22⋊C4—3324no—
543C22⋊C4—3324no—
544C22⋊C4—838no—
545C22⋊C4—838no—
546C22⋊C4—1668no—
547C22⋊C4C2×C4×D8318yes—
548C42—44no—
549C42—3324no—
550C42—3324no—
551C42—3324no—
552C42—834no—
553C42—834no—
554C42—838no—
555C42—838no—
556C42—838no—
557C42—1668no—
558C42C2×C4×D8314yes—
559C42C2×C4×D8314yes—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—22no—
564C23—832no—
565C23—832no—
566C23—832no—
567C23—832no—
568C23—1661no—
569C23—1661no—
570C23—1662no—
571C23—1662no—
572C23—1662no—
573C23—1662no—
574C23C22×C4×D8312yes—
575C23C24×D8311yescharacteristic
576Q8—1668no—
577D4—1668no—
578D4—1668no—
579D4—1668no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—22no—
583C2×C4—22no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—28no—
587C2×C4—24no—
588C2×C4—24no—
589C2×C4—24no—
590C2×C4—24no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—834no—
594C2×C4—832no—
595C2×C4—832no—
596C2×C4—832no—
597C2×C4—834no—
598C2×C4—834no—
599C2×C4—8316no—
600C2×C4—834no—
601C2×C4—832no—
602C2×C4—6648no—
603C2×C4—16616no—
604C2×C4—16616no—
605C2×C4—1664no—
606C2×C4—1664no—
607C2×C4—1664no—
608C2×C4—1664no—
609C2×C4—1664no—
610C2×C4—1664no—
611C2×C4—16616no—
612C2×C4—1661no—
613C2×C4—1661no—
614C2×C4—1662no—
615C2×C4—1662no—
616C2×C4—1668no—
617C2×C4C22×C4×D8312yes—
618C2×C4C22×C4×D8314yes—
619C2×C4C22×C4×D8314yes—
620C2×C4C22×C4×D8312yes—
621C2×C4#41×D8312yes—
622C2×C4#41×D8312yes—
623C22—22no—
624C22—24no—
625C22—41no—
626C22—42no—
627C22—41no—
628C22—3321no—
629C22—3322no—
630C22—3321no—
631C22—834no—
632C22—1662no—
633C22—1664no—
634C22—1662no—
635C22—1664no—
636C22—1664no—
637C22—1662no—
638C22C23×C4×D8311yescharacteristic
639C22C2×#41×D8311yescharacteristic
640C22C2×#41×D8311yescharacteristic
641C22H(5,𝔽2)×D8312yes—
642C222−1+4×D8312yes—
643C4—22no—
644C4—24no—
645C4—24no—
646C4—44no—
647C4—44no—
648C4—3324no—
649C4—3324no—
650C4—834no—
651C4—838no—
652C4—1662no—
653C4—1664no—
654C4#117×D8314yes—
655C2—42no—
656C2—3322no—
657C2—1664no—
658C2C2×#117×D8311yescharacteristic
659C2#520×D8312yes—
660C2#2492133511yescharacteristic
661C2#2492134511yescharacteristic
662C2#2492136812yes—
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingC83×#2314C2
socle, cosocleC23×C83C25
prime radicalC22×#13050590C22
hypercenter#2314D83
hypocenterC83#34188
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #24910417C2semidirect
3. #24910409C2semidirect
4. #24910391C2semidirect
5. #24910387C2semidirect
6. #24910073C2semidirect
7. #24910072C2semidirect
8. #24910071C2semidirect
10. #24909869C2semidirect
11. #24909867C2semidirect
12. #24909865C2semidirect
13. #384×D83C2semidirect
14. #24909817C2semidirect
15. C83×#2314C2semidirect
17. #19128169C4semidirect
85. #14581757C2×C4semidirect
88. (C22⋊D83)⋊C4C2×C4semidirect
92. #14581749C2×C4semidirect
112. C2×C4×Dic83C2×C4semidirect
123. C42×D83C2×C4semidirect
132. #117×C83C2×C4semidirect
289. C4×D83#117semidirect
340. #2314D83semidirect
347. C83#34188semidirect
427. #117C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191281692
#249104172
#249104092
#249103912
#249103872
#249100732
#249100722
#249100712
#249100522
#249098692
#249098672
#249098652
#384×D832
#249098172
C83×#23142
#3418883