Finite Group Databasebeta

← #2634×D83 #35571329

Subgroups — #2634×D83 #35571329

Order 21248.

2286 conjugacy classes of subgroups.

The subgroup lattice has 662 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#125×D83C211yesmaximal, characteristic
2C2×C4×Q8×D83C211yesmaximal, characteristic
3#24913345C212yesmaximal, complemented
4#24913344C212yesmaximal, complemented
5#24913343C214yesmaximal, complemented
6#24913342C211yesmaximal, characteristic, complemented
7#24913341C212yesmaximal, complemented
8#24913340C211yesmaximal, characteristic, complemented
9#24913339C211yesmaximal, characteristic, complemented
10#24913338C212yesmaximal, complemented
11#400×D83C212yesmaximal
12#400×D83C212yesmaximal
13#398×D83C212yesmaximal
14#395×D83C212yesmaximal
15#387×D83C214yesmaximal
16#383×D83C211yesmaximal, characteristic
17C83×#2634C211yesmaximal, characteristic, complemented
18C22×Q8×D83C2211yescharacteristic
19C2×#14581784C2211yescharacteristic
20C2×#14581783C2211yescharacteristic
21C2×#14581783C2211yescharacteristic
22C2×#14581782C2212yes—
23C2×#14581782C2212yes—
24#125×D83—28no—
25C2×Q8×Dic83C2211yescharacteristic
26C2×C4×Dic166C2211yescharacteristic
27C2×C4×Dic166C2212yes—
28C2×#14581757C2211yescharacteristic
29C2×#14581757C2212yes—
30C4×Q8×D83—28no—
31C2×C4⋊C4×D83C2212yes—
32C2×C4⋊C4×D83C2214yes—
33C2×C4⋊C4×D83C2212yes—
34C2×C4⋊C4×D83C2211yescharacteristic
35C2×C4⋊C4×D83C2212yes—
36C2×C4⋊C4×D83C2211yescharacteristic
37C2×C4⋊C4×D83C2211yescharacteristic
38C2×C42×D83C2212yes—
39C2×C42×D83C2211yescharacteristic
40#19127643C2214yes—
41#19127643C2214yes—
42#19127642C2214yes—
43#19127642C2214yes—
44#19127641C2212yes—
45#19127641C2212yes—
46#19127640C2212yes—
47#19127640C2212yes—
48#19127639C2212yes—
49#19127639C2212yes—
50#19127635C2214yes—
51#19127634C2214yes—
52#19127633C2212yes—
53#19127632C2212yes—
54#19127631C2212yes—
55#19127622C2212yes—
56#19127622C2214yes—
57#19127621C2212yes—
58#19127621C2214yes—
59#19127620C2212yes—
60#19127588C2214yes—
61#19127587C2214yes—
62#19127586C2214yes—
63#19127585C2214yes—
64#19127584C2214yes—
65#19127583C2214yes—
66#19127582C2214yes—
67#19127569C2211yescharacteristic
68#19127568C2212yes—
69#19127568C2212yes—
70#19127568C2212yes—
71#95×D83C2212yes—
72#95×D83C2212yes—
73#95×D83C2212yes—
74#95×D83C2214yes—
75#95×D83C2214yes—
76#95×D83C2214yes—
77C2×#125×C83C2211yescharacteristic
78C2×C4×Q8×C83C2211yescharacteristic
79#400×C83C2212yes—
80#400×C83C2212yes—
81#398×C83C2212yes—
82#395×C83C2212yes—
83#387×C83C2214yes—
84#383×C83C2211yescharacteristic
85C22×Dic166C2311yescharacteristic
86C22×Dic166C2312yes—
87C2×Q8×D83—24no—
88C2×Q8×D83—28no—
89C22×C4×D83C2312yes—
90C22×C4×D83C2312yes—
91C22×C4×D83C2312yes—
92C22×C4×D83C2314yes—
93C22×C4×D83C2311yescharacteristic
94C22×C4×D83C2311yescharacteristic
95C22×C4×D83C2311yescharacteristic
96C22×C4×D83C2312yes—
97#14581784—24no—
98#14581783—24no—
99#14581783—24no—
100#14581782—28no—
101#14581782—28no—
102Q8×Dic83—24no—
103C4×Dic166—28no—
104C4×Dic166—24no—
105#14581757—24no—
106#14581757—28no—
107C2×C4⋊Dic83C2314yes—
108C2×C4⋊Dic83C2312yes—
109C2×C4⋊Dic83C2312yes—
110C2×C4⋊Dic83C2311yescharacteristic
111C2×C4⋊Dic83C2312yes—
112C2×C4⋊Dic83C2311yescharacteristic
113C2×C4⋊Dic83C2311yescharacteristic
114C2×Dic83⋊C4C2314yes—
115C2×Dic83⋊C4C2312yes—
116C2×Dic83⋊C4C2314yes—
117C2×Dic83⋊C4C2312yes—
118C2×Dic83⋊C4C2312yes—
119C2×Dic83⋊C4C2312yes—
120C2×Dic83⋊C4C2312yes—
121C2×Dic83⋊C4C2312yes—
122C2×Dic83⋊C4C2312yes—
123C2×Dic83⋊C4C2312yes—
124C2×Dic83⋊C4C2312yes—
125C4⋊C4×D83—28no—
126C4⋊C4×D83—24no—
127C4⋊C4×D83—28no—
128C4⋊C4×D83—28no—
129C4⋊C4×D83—24no—
130C4⋊C4×D83—24no—
131C4⋊C4×D83—216no—
132C2×C4×Dic83C2312yes—
133C2×C4×Dic83C2312yes—
134C2×C4×Dic83C2312yes—
135C2×C4×Dic83C2311yescharacteristic
136C2×C4×Dic83C2311yescharacteristic
137C2×C4×Dic83C2311yescharacteristic
138C42×D83—28no—
139C42×D83D418yes—
140C83⋊#95C2314yes—
141C83⋊#95C2314yes—
142C83⋊#95C2314yes—
143C83⋊#95C2312yes—
144C83⋊#95C2312yes—
145C83⋊#95C2312yes—
146C83⋊#95C2314yes—
147C83⋊#95C2314yes—
148C83⋊#95C2314yes—
149C83⋊#95C2314yes—
150C83⋊#95C2314yes—
151C83⋊#95C2314yes—
152C83⋊#95C2314yes—
153C83⋊#95C2314yes—
154C83⋊#95C2314yes—
155C22×Q8×C83C2311yescharacteristic
156#125×C83—24no—
157C4×Q8×C83—24no—
158C2×C4⋊C4×C83C2312yes—
159C2×C4⋊C4×C83C2314yes—
160C2×C4⋊C4×C83C2312yes—
161C2×C4⋊C4×C83C2311yescharacteristic
162C2×C4⋊C4×C83C2312yes—
163C2×C4⋊C4×C83C2311yescharacteristic
164C2×C4⋊C4×C83C2311yescharacteristic
165C2×C42×C83C2312yes—
166C2×C42×C83C2311yescharacteristic
167#95×C83C2312yes—
168#95×C83C2312yes—
169#95×C83C2312yes—
170#95×C83C2314yes—
171#95×C83C2314yes—
172#95×C83C2314yes—
173C23×D83C2411yescharacteristic
174C2×Dic166—22no—
175C2×Dic166—24no—
176C2×Dic166—24no—
177C2×Dic166—24no—
178C2×Dic166—24no—
179Q8×D83—232no—
180C22×Dic83C2414yes—
181C22×Dic83C2412yes—
182C22×Dic83C2412yes—
183C22×Dic83C2412yes—
184C22×Dic83C2411yescharacteristic
185C22×Dic83C2411yescharacteristic
186C22×Dic83C2411yescharacteristic
187C22×Dic83C2412yes—
188C2×C4×D83—24no—
189C2×C4×D83—24no—
190C2×C4×D83—24no—
191C2×C4×D83—24no—
192C2×C4×D83—24no—
193C2×C4×D83—24no—
194C2×C4×D83—24no—
195C2×C4×D83—24no—
196C2×C4×D83—28no—
197C2×C4×D83—28no—
198C2×C4×D83—28no—
199C2×C4×D83—28no—
200C2×C4×D83—28no—
201C2×C4×D83—28no—
202C2×C4×D83C2×D414yes—
203C2×C4×D83C2×D414yes—
204C2×C4×D83C2×D414yes—
205C2×C4×D83#4118yes—
206C4⋊Dic83—22no—
207C4⋊Dic83—22no—
208C4⋊Dic83—22no—
209C4⋊Dic83—24no—
210C4⋊Dic83—24no—
211C4⋊Dic83—24no—
212C4⋊Dic83—28no—
213Dic83⋊C4—24no—
214Dic83⋊C4—24no—
215Dic83⋊C4—24no—
216Dic83⋊C4—28no—
217Dic83⋊C4—24no—
218Dic83⋊C4—24no—
219Dic83⋊C4—24no—
220Dic83⋊C4—24no—
221Dic83⋊C4—24no—
222Dic83⋊C4—24no—
223Dic83⋊C4—28no—
224C4×Dic83—24no—
225C4×Dic83—24no—
226C4×Dic83—24no—
227C4×Dic83C2×D414yes—
228C4×Dic83C2×D414yes—
229C4×Dic83C2×D414yes—
230C2×Q8×C83—22no—
231C2×Q8×C83—24no—
232C22×C4×C83C2414yes—
233C22×C4×C83C2412yes—
234C22×C4×C83C2412yes—
235C22×C4×C83C2412yes—
236C22×C4×C83C2411yescharacteristic
237C22×C4×C83C2411yescharacteristic
238C22×C4×C83C2411yescharacteristic
239C22×C4×C83C2412yes—
240C4⋊C4×C83—24no—
241C4⋊C4×C83—24no—
242C4⋊C4×C83—22no—
243C4⋊C4×C83—22no—
244C4⋊C4×C83—24no—
245C4⋊C4×C83—22no—
246C4⋊C4×C83—28no—
247C42×C83—24no—
248C42×C83C2×D414yes—
249C23×C83C2511yescharacteristic
250C22×D83C22×D412yes—
251C22×D83C2×#4114yes—
252C22×D832−1+414yes—
253C22×D832−1+414yes—
254Q8×C83—28no—
255C2×C4×C83—22no—
256C2×C4×C83—22no—
257C2×C4×C83—22no—
258C2×C4×C83—22no—
259C2×C4×C83—22no—
260C2×C4×C83—22no—
261C2×C4×C83—22no—
262C2×C4×C83—22no—
263C2×C4×C83—24no—
264C2×C4×C83—24no—
265C2×C4×C83—24no—
266C2×C4×C83—24no—
267C2×C4×C83—24no—
268C2×C4×C83—24no—
269C2×C4×C83C22×D412yes—
270C2×C4×C83C22×D412yes—
271C2×C4×C83C22×D412yes—
272C2×C4×C83C2×#4114yes—
273C2×Dic83—22no—
274C2×Dic83—22no—
275C2×Dic83—22no—
276C2×Dic83—22no—
277C2×Dic83—22no—
278C2×Dic83—22no—
279C2×Dic83—22no—
280C2×Dic83—22no—
281C2×Dic83—24no—
282C2×Dic83—24no—
283C2×Dic83—24no—
284C2×Dic83—24no—
285C2×Dic83—24no—
286C2×Dic83—24no—
287C2×Dic83C22×D412yes—
288C2×Dic83C22×D412yes—
289C2×Dic83C22×D412yes—
290C2×Dic83C2×#4114yes—
291C4×D83—28no—
292C4×D83—216no—
293C4×D83—28no—
294C4×D83—28no—
295C4×D83—48no—
296C4×D83—48no—
297C4×D83—48no—
298C4×D83—416no—
299Dic166—216no—
300Dic166—28no—
301C22×C83C23×D411yescharacteristic
302C22×C83C22×#4112yes—
303C22×C83C2×2−1+412yes—
304C22×C83C2×2−1+412yes—
305C2×D83#52514yes—
306C2×D83#53614yes—
307C2×D83#53614yes—
308C2×D83#541116yes—
309C4×C83—22no—
310C4×C83—24no—
311C4×C83—22no—
312C4×C83—22no—
313C4×C83—42no—
314C4×C83—42no—
315C4×C83—42no—
316C4×C83—44no—
317Dic83—22no—
318Dic83—24no—
319Dic83—22no—
320Dic83—22no—
321Dic83—42no—
322Dic83—42no—
323Dic83—42no—
324Dic83—44no—
325C2×#2634—831nomaximal, Sylow
326C2×C83C2×#52511yescharacteristic
327C2×C83C2×#53611yescharacteristic
328C2×C83C2×#53611yescharacteristic
329C2×C83C2×#54114yes—
330D83#263418yesdirect product factor
331C22×#125—831no—
332C22×C4×Q8—831no—
333#2634—832no—
334#2634—832no—
335#2634—832no—
336#2634—832no—
337#2634—834no—
338#2634—831no—
339#2634—831no—
340#2634—831no—
341#2634D8311yescharacteristic, direct product factor
342C2×#400—832no—
343C2×#400—832no—
344C2×#398—832no—
345C2×#395—832no—
346C2×#387—834no—
347C2×#383—831no—
348C83C2×#263411yescharacteristic, complemented, Sylow
349C23×Q8—831no—
350C2×#125—832no—
351C2×#125—832no—
352C2×#125—831no—
353C2×#125—831no—
354C2×#125—831no—
355C2×#125—1668no—
356C2×#125C2×D8311yescharacteristic
357C2×C4×Q8—831no—
358C2×C4×Q8—831no—
359C2×C4×Q8—832no—
360C2×C4×Q8—832no—
361C2×C4×Q8—831no—
362C2×C4×Q8—1668no—
363C2×C4×Q8C2×D8311yescharacteristic
364C22×C4⋊C4—834no—
365C22×C4⋊C4—832no—
366C22×C4⋊C4—831no—
367C22×C4⋊C4—832no—
368C22×C4⋊C4—831no—
369C22×C4⋊C4—831no—
370C22×C4⋊C4—832no—
371C22×C42—831no—
372C22×C42—832no—
373#400—832no—
374#400—832no—
375#400—832no—
376#400—832no—
377#400—832no—
378#400—832no—
379#400—834no—
380#400—834no—
381#400—834no—
382#400—834no—
383#400C2×D8312yes—
384#400C2×D8312yes—
385#398—832no—
386#398—832no—
387#398—834no—
388#398—832no—
389#398—834no—
390#398C2×D8312yes—
391#395—832no—
392#395—834no—
393#395—832no—
394#395—834no—
395#395—832no—
396#395C2×D8312yes—
397#387—834no—
398#387—834no—
399#387—834no—
400#387—834no—
401#387—834no—
402#387—834no—
403#387—834no—
404#387C2×D8314yes—
405#383—831no—
406#383—832no—
407#383—832no—
408#383—832no—
409#383C2×D8311yescharacteristic
410C2×#95—832no—
411C2×#95—834no—
412C2×#95—832no—
413C2×#95—834no—
414C2×#95—834no—
415C2×#95—832no—
416C22×Q8—831no—
417C22×Q8—832no—
418C22×Q8—1664no—
419C22×Q8—1668no—
420C22×Q8C22×D8311yescharacteristic
421C23×C4—831no—
422C23×C4—831no—
423C23×C4—832no—
424C23×C4—831no—
425C23×C4—832no—
426C23×C4—832no—
427C23×C4—832no—
428C23×C4—834no—
429#125—24no—
430#125—1668no—
431#125—1668no—
432#125—1664no—
433#125—1664no—
434#125—1664no—
435C4×Q8—24no—
436C4×Q8—1668no—
437C4×Q8—1664no—
438C4×Q8—1668no—
439C4×Q8—1664no—
440C4×Q8—1664no—
441C2×C4⋊C4—832no—
442C2×C4⋊C4—832no—
443C2×C4⋊C4—832no—
444C2×C4⋊C4—832no—
445C2×C4⋊C4—831no—
446C2×C4⋊C4—832no—
447C2×C4⋊C4—832no—
448C2×C4⋊C4—832no—
449C2×C4⋊C4—832no—
450C2×C4⋊C4—832no—
451C2×C4⋊C4—832no—
452C2×C4⋊C4—832no—
453C2×C4⋊C4—831no—
454C2×C4⋊C4—832no—
455C2×C4⋊C4—831no—
456C2×C4⋊C4—834no—
457C2×C4⋊C4—834no—
458C2×C4⋊C4—834no—
459C2×C4⋊C4—1664no—
460C2×C4⋊C4—1668no—
461C2×C4⋊C4—1664no—
462C2×C4⋊C4—1664no—
463C2×C4⋊C4—1668no—
464C2×C4⋊C4—1668no—
465C2×C4⋊C4—16616no—
466C2×C4⋊C4C22×D8312yes—
467C2×C4⋊C4C22×D8311yescharacteristic
468C2×C4⋊C4C22×D8312yes—
469C2×C4⋊C4C22×D8312yes—
470C2×C4⋊C4C22×D8311yescharacteristic
471C2×C4⋊C4C22×D8311yescharacteristic
472C2×C4⋊C4C22×D8314yes—
473C2×C42—831no—
474C2×C42—831no—
475C2×C42—831no—
476C2×C42—838no—
477C2×C42—832no—
478C2×C42—832no—
479C2×C42—832no—
480C2×C42—1668no—
481C2×C42C22×D8312yes—
482C2×C42C22×D8311yescharacteristic
483#95—834no—
484#95—832no—
485#95—834no—
486#95—832no—
487#95—834no—
488#95—832no—
489#95—834no—
490#95—834no—
491#95—834no—
492#95—834no—
493#95—834no—
494#95—834no—
495#95—834no—
496#95—834no—
497#95—834no—
498#95C22×D8312yes—
499#95C22×D8312yes—
500#95C22×D8312yes—
501#95C22×D8314yes—
502#95C22×D8314yes—
503#95C22×D8314yes—
504C24—831no—
505C2×Q8—22no—
506C2×Q8—24no—
507C2×Q8—1664no—
508C2×Q8—1664no—
509C2×Q8—1664no—
510C2×Q8—16632no—
511C2×Q8—1662no—
512C2×Q8—1664no—
513C22×C4—831no—
514C22×C4—831no—
515C22×C4—831no—
516C22×C4—832no—
517C22×C4—834no—
518C22×C4—834no—
519C22×C4—834no—
520C22×C4—838no—
521C22×C4—834no—
522C22×C4—832no—
523C22×C4—832no—
524C22×C4—832no—
525C22×C4—1664no—
526C22×C4—1664no—
527C22×C4—1664no—
528C22×C4—1668no—
529C22×C4—1668no—
530C22×C4—1668no—
531C22×C4—1668no—
532C22×C4—1668no—
533C22×C4—1668no—
534C22×C4—1664no—
535C22×C4—1664no—
536C22×C4—1664no—
537C22×C4—1664no—
538C22×C4—1664no—
539C22×C4C23×D8314yes—
540C22×C4C23×D8312yes—
541C22×C4C23×D8312yes—
542C22×C4C23×D8312yes—
543C22×C4C23×D8311yescharacteristic
544C22×C4C23×D8311yescharacteristic
545C22×C4C23×D8311yescharacteristic
546C22×C4C23×D8312yes—
547C4⋊C4—24no—
548C4⋊C4—22no—
549C4⋊C4—24no—
550C4⋊C4—22no—
551C4⋊C4—22no—
552C4⋊C4—24no—
553C4⋊C4—28no—
554C4⋊C4—1664no—
555C4⋊C4—1664no—
556C4⋊C4—1664no—
557C4⋊C4—1664no—
558C4⋊C4—1664no—
559C4⋊C4—1664no—
560C4⋊C4—1664no—
561C4⋊C4—1664no—
562C4⋊C4—1664no—
563C4⋊C4—1668no—
564C4⋊C4—1668no—
565C4⋊C4—1664no—
566C4⋊C4—1662no—
567C4⋊C4—1664no—
568C4⋊C4—1664no—
569C4⋊C4—1662no—
570C4⋊C4—1662no—
571C4⋊C4—1668no—
572C42—24no—
573C42—834no—
574C42—834no—
575C42—834no—
576C42—1664no—
577C42—1664no—
578C42—1664no—
579C42D4×D8314yes—
580C23—832no—
581C23—834no—
582C23—834no—
583C23—834no—
584C23C24×D8311yescharacteristic
585Q8—28no—
586Q8—1668no—
587Q8—16616no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—24no—
597C2×C4—24no—
598C2×C4—24no—
599C2×C4—24no—
600C2×C4—24no—
601C2×C4—24no—
602C2×C4—3328no—
603C2×C4—3328no—
604C2×C4—3328no—
605C2×C4—33216no—
606C2×C4—832no—
607C2×C4—832no—
608C2×C4—832no—
609C2×C4—834no—
610C2×C4—1668no—
611C2×C4—16616no—
612C2×C4—1668no—
613C2×C4—1668no—
614C2×C4—1662no—
615C2×C4—1662no—
616C2×C4—1662no—
617C2×C4—1664no—
618C2×C4—1664no—
619C2×C4—1664no—
620C2×C4—1664no—
621C2×C4—1664no—
622C2×C4—1664no—
623C2×C4—1662no—
624C2×C4—1662no—
625C2×C4—1662no—
626C2×C4—1662no—
627C2×C4—1662no—
628C2×C4C2×D4×D8312yes—
629C2×C4C2×D4×D8312yes—
630C2×C4C2×D4×D8312yes—
631C2×C4#41×D8314yes—
632C22—834no—
633C22—834no—
634C22—834no—
635C22—8316no—
636C22C22×D4×D8311yescharacteristic
637C22C2×#41×D8312yes—
638C222−1+4×D8312yes—
639C222−1+4×D8312yes—
640C4—22no—
641C4—24no—
642C4—22no—
643C4—22no—
644C4—42no—
645C4—42no—
646C4—42no—
647C4—44no—
648C4—3322no—
649C4—3322no—
650C4—3322no—
651C4—3324no—
652C4—1662no—
653C4—1664no—
654C4—1662no—
655C4—1662no—
656C2—838no—
657C2#525×D8311yescharacteristic
658C2#536×D8311yescharacteristic
659C2#536×D8311yescharacteristic
660C2#541×D8314yes—
subgroupG/H
center, FrattiniC23C24×D83
commutator, socle, cosocleC23×C83C25
FittingC83×#2634C2
prime radicalC23×D83C24
hypercenter#2634D83
hypocenterC83C2×#2634
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2634

Complements

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

NH ≅ G/Ntype
3. #24913345C2semidirect
4. #24913344C2semidirect
5. #24913343C2semidirect
6. #24913342C2semidirect
7. #24913341C2semidirect
8. #24913340C2semidirect
9. #24913339C2semidirect
10. #24913338C2semidirect
17. C83×#2634C2semidirect
330. D83#2634direct (N × H)
341. #2634D83direct (N × H)
348. C83C2×#2634semidirect

Maximal subgroups

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

subgroupindex
C2×#125×D832
C2×C4×Q8×D832
#249133452
#249133442
#249133432
#249133422
#249133412
#249133402
#249133392
#249133382
#400×D832
#398×D832
#395×D832
#387×D832
#383×D832
C83×#26342
C2×#263483