Finite Group Databasebeta

← #35611217

Subgroups — #35611217

Order 21248.

2326 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128329C211yesmaximal, characteristic, complemented
2C22×#14581769C211yesmaximal, characteristic
3#24915748C212yesmaximal
4#24913934C214yesmaximal, complemented
5#24913930C214yesmaximal, complemented
6#24913779C214yesmaximal, complemented
7#24913775C214yesmaximal, complemented
8#24913282C212yesmaximal, complemented
9#24912618C212yesmaximal
10#24911752C211yesmaximal, characteristic, complemented
11#24911746C211yesmaximal, characteristic, complemented
12#24911737C211yesmaximal, characteristic
13#24911105C212yesmaximal, complemented
14#24910064C211yesmaximal, characteristic, complemented
15C83×#2482C211yesmaximal, characteristic
16C23×Dic166C2211yescharacteristic
17C24×Dic83C2211yescharacteristic
18#19128329—28no—
19C2×#14581792C2212yescomplemented
20C2×#14581791C2212yescomplemented
21C2×#14581791C2214yescomplemented
22C2×#14581784C2212yescomplemented
23C2×#14581783C2212yescomplemented
24C2×#14581769—28no—
25C2×#14581769C2212yes—
26C2×#14581769C2214yes—
27C2×#14581769C2214yes—
28C2×#14581769C2212yes—
29C2×#14581769C2214yes—
30C2×#14581751C2211yescharacteristic
31C22×C4⋊Dic83C2211yescharacteristic
32C22×Dic83⋊C4C2211yescharacteristic
33C22×Dic83⋊C4C2211yescharacteristic
34C22×C22⋊Dic83C2211yescharacteristic
35#19127642C2214yescomplemented
36#19127641C2212yescomplemented
37#19127639C2212yescomplemented
38#19127637C2212yes—
39#19127635C2214yescomplemented
40#19127633C2212yescomplemented
41#19127632C2214yescomplemented
42#19127632C2212yescomplemented
43#19127629C2214yes—
44#19127626C2212yes—
45#19127622C2214yescomplemented
46#19127621C2214yescomplemented
47#19127621C2214yescomplemented
48#19127613C2214yes—
49#19127605C2214yescomplemented
50#19127594C2214yes—
51#19127589C2214yes—
52#19127589C2214yes—
53#19127586C2214yescomplemented
54#19127586C2214yescomplemented
55#19127585C2214yescomplemented
56#19127584C2214yescomplemented
57#19127576C2212yes—
58#19127576C2214yes—
59#19127574C2212yes—
60#19127573C2212yes—
61#19127573C2214yes—
62#19127564C2211yescharacteristic
63#19127562C2212yes—
64#19127562C2212yes—
65#19127559C2214yes—
66#19127559C2212yes—
67#19127557C2212yes—
68C22⋊C4×Dic83C2212yes—
69C2×#117×C83C2211yescharacteristic
70C22×C22⋊C4×C83C2211yescharacteristic
71#399×C83C2212yes—
72#397×C83C2212yes—
73#388×C83C2214yes—
74#385×C83C2212yes—
75#381×C83C2211yescharacteristic
76#379×C83C2212yes—
77C22×Dic166—24no—
78C22×Dic166—44no—
79C22×Dic166C2314yes—
80C22×Dic166C2312yes—
81C22×Dic166C2312yes—
82C23×Dic83—24no—
83C23×Dic83—44no—
84C23×Dic83C2311yescharacteristic
85C23×Dic83C2311yescharacteristic
86C23×Dic83C2314yes—
87C23×Dic83C2311yescharacteristic
88C23×Dic83C2311yescharacteristic
89#14581792—28no—
90#14581791—28no—
91#14581791—48no—
92#14581784—28no—
93#14581783—28no—
94#14581769—28no—
95#14581769—28no—
96#14581769—216no—
97#14581769—216no—
98#14581769—48no—
99#14581751—24no—
100C2×C4⋊Dic83D418yes—
101C2×C4⋊Dic83C2314yes—
102C2×C4⋊Dic83C2312yes—
103C2×C4⋊Dic83C2312yes—
104C2×C4⋊Dic83C2311yescharacteristic
105C2×C4⋊Dic83C2311yescharacteristic
106C2×C4⋊Dic83C2312yes—
107C2×C4⋊Dic83C2312yes—
108C2×Dic83⋊C4—24no—
109C2×Dic83⋊C4—24no—
110C2×Dic83⋊C4C2312yes—
111C2×Dic83⋊C4C2314yes—
112C2×Dic83⋊C4C2312yes—
113C2×Dic83⋊C4C2312yes—
114C2×Dic83⋊C4C2312yes—
115C2×Dic83⋊C4C2312yes—
116C2×Dic83⋊C4C2312yes—
117C2×Dic83⋊C4C2314yes—
118C2×Dic83⋊C4C2314yes—
119C2×Dic83⋊C4C2314yes—
120C2×C22⋊Dic83—24no—
121C2×C22⋊Dic83C2312yes—
122C2×C22⋊Dic83C2314yes—
123C2×C22⋊Dic83C2312yes—
124C2×C22⋊Dic83C2314yes—
125C2×C22⋊Dic83C2314yes—
126C2×C4×Dic83C2312yes—
127C2×C4×Dic83C2314yes—
128C83⋊#95C2314yes—
129C83⋊#95C2314yes—
130C83⋊#95C2314yes—
131C83⋊#95C2314yes—
132C83⋊#95C2314yes—
133C83⋊#95C2314yes—
134C83⋊#95C2314yes—
135C83⋊#95C2312yes—
136C83⋊#95C2312yes—
137C83⋊#95C2314yes—
138C83⋊#95C2314yes—
139C25×C83C2311yescharacteristic
140C23×C4×C83C2311yescharacteristic
141C23×C4×C83C2311yescharacteristic
142#117×C83—24no—
143C2×C4⋊C4×C83C2311yescharacteristic
144C2×C4⋊C4×C83C2311yescharacteristic
145C2×C4⋊C4×C83C2314yes—
146C2×C22⋊C4×C83—24no—
147C2×C22⋊C4×C83C2312yes—
148C2×C22⋊C4×C83C2314yes—
149C2×C22⋊C4×C83C2312yes—
150C2×C22⋊C4×C83C2312yes—
151C2×C22⋊C4×C83C2312yes—
152C2×C22⋊C4×C83C2314yes—
153C2×C42×C83C2312yes—
154#95×C83C2312yes—
155#95×C83C2312yes—
156#95×C83C2314yes—
157C2×Dic166—28no—
158C2×Dic166—28no—
159C2×Dic166—24no—
160C2×Dic166—24no—
161C2×Dic166—44no—
162C2×Dic166—416no—
163C2×Dic166—44no—
164C2×Dic166—44no—
165C22×Dic83—22no—
166C22×Dic83—24no—
167C22×Dic83—22no—
168C22×Dic83—24no—
169C22×Dic83—28no—
170C22×Dic83—28no—
171C22×Dic83—42no—
172C22×Dic83—44no—
173C22×Dic83—42no—
174C22×Dic83—416no—
175C22×Dic83C2×D414yes—
176C22×Dic83C2×D414yes—
177C22×Dic83C2412yes—
178C22×Dic83C2412yes—
179C22×Dic83C2414yes—
180C22×Dic83C2412yes—
181C22×Dic83C2412yes—
182C22×Dic83C2412yes—
183C22×Dic83C2412yes—
184C4⋊Dic83—22no—
185C4⋊Dic83—22no—
186C4⋊Dic83—24no—
187C4⋊Dic83—24no—
188C4⋊Dic83—28no—
189C4⋊Dic83C2×D418yes—
190C4⋊Dic83C2×D418yes—
191Dic83⋊C4—24no—
192Dic83⋊C4—24no—
193Dic83⋊C4—28no—
194Dic83⋊C4—24no—
195Dic83⋊C4—24no—
196Dic83⋊C4—24no—
197Dic83⋊C4—24no—
198Dic83⋊C4—28no—
199Dic83⋊C4—44no—
200Dic83⋊C4—44no—
201C22⋊Dic83—24no—
202C22⋊Dic83—28no—
203C22⋊Dic83—24no—
204C22⋊Dic83—28no—
205C22⋊Dic83—44no—
206C4×Dic83—24no—
207C4×Dic83—44no—
208C24×C83—22no—
209C24×C83—24no—
210C24×C83—24no—
211C24×C83—42no—
212C24×C83C2412yes—
213C24×C83C2411yescharacteristic
214C22×C4×C83—24no—
215C22×C4×C83—22no—
216C22×C4×C83—42no—
217C22×C4×C83C2×D414yes—
218C22×C4×C83C2412yes—
219C22×C4×C83C2412yes—
220C22×C4×C83C2414yes—
221C22×C4×C83C2412yes—
222C22×C4×C83C2411yescharacteristic
223C22×C4×C83C2411yescharacteristic
224C4⋊C4×C83—22no—
225C4⋊C4×C83—22no—
226C4⋊C4×C83—44no—
227C22⋊C4×C83—24no—
228C22⋊C4×C83—28no—
229C22⋊C4×C83—24no—
230C22⋊C4×C83—24no—
231C22⋊C4×C83—24no—
232C22⋊C4×C83—44no—
233C42×C83—24no—
234C23×C83—22no—
235C23×C83—22no—
236C23×C83—28no—
237C23×C83—22no—
238C23×C83—24no—
239C23×C83—22no—
240C23×C83—24no—
241C23×C83—416no—
242C23×C83—44no—
243C23×C83—44no—
244C23×C83C22×D412yes—
245C23×C832−1+414yes—
246C23×C832−1+414yes—
247C23×C83C2511yescharacteristic
248C2×C4×C83—28no—
249C2×C4×C83—22no—
250C2×C4×C83—22no—
251C2×C4×C83—22no—
252C2×C4×C83—22no—
253C2×C4×C83—22no—
254C2×C4×C83—24no—
255C2×C4×C83—24no—
256C2×C4×C83—24no—
257C2×C4×C83—24no—
258C2×C4×C83—24no—
259C2×C4×C83—24no—
260C2×C4×C83—44no—
261C2×C4×C83C22×D412yes—
262C2×C4×C83C22×D412yes—
263C2×Dic83—28no—
264C2×Dic83—22no—
265C2×Dic83—24no—
266C2×Dic83—24no—
267C2×Dic83—22no—
268C2×Dic83—22no—
269C2×Dic83—22no—
270C2×Dic83—22no—
271C2×Dic83—24no—
272C2×Dic83—24no—
273C2×Dic83—24no—
274C2×Dic83—24no—
275C2×Dic83—24no—
276C2×Dic83—24no—
277C2×Dic83—44no—
278C2×Dic83—44no—
279C2×Dic83—416no—
280C2×Dic83—44no—
281C2×Dic83C22×D414yes—
282C2×Dic83C22×D414yes—
283C2×Dic83H(5,𝔽2)14yes—
284Dic166—416no—
285Dic166—84no—
286Dic166—84no—
287C22×C83—22no—
288C22×C83—22no—
289C22×C83—28no—
290C22×C83—24no—
291C22×C83—28no—
292C22×C83—416no—
293C22×C83—42no—
294C22×C83—42no—
295C22×C83—44no—
296C22×C83#53614yes—
297C22×C83C23×D411yescharacteristic
298C22×C83C2×H(5,𝔽2)12yes—
299C22×C83C2×2−1+412yes—
300C22×C83C2×2−1+412yes—
301C4×C83—22no—
302C4×C83—22no—
303C4×C83—42no—
304C4×C83—42no—
305C4×C83—42no—
306C4×C83—44no—
307Dic83—24no—
308Dic83—24no—
309Dic83—24no—
310Dic83—42no—
311Dic83—42no—
312Dic83—42no—
313Dic83—44no—
314#36708—831nomaximal, Sylow
315C2×C83—24no—
316C2×C83—44no—
317C2×C83C2×#53611yescharacteristic
318C2×C83#348411yescharacteristic
319C2×C83#348611yescharacteristic
320C2×C83#353814yes—
321C2×#544—831no—
322C22×C22⋊Q8—831no—
323#2848—832no—
324#2690—834no—
325#2690—834no—
326#2677—834no—
327#2677—834no—
328#2627—832no—
329#2561—832no—
330#2484—831no—
331#2484—831no—
332#2482—831no—
333#2482D8311yescharacteristic
334#2434—832no—
335#2313—831no—
336C83#3670811yescharacteristic, complemented, Sylow
337C23×Q8—831no—
338C24×C4—831no—
339#544—1668no—
340C2×C4⋊Q8—834no—
341C2×C4⋊Q8—832no—
342C2×C4⋊Q8—832no—
343C2×#125—832no—
344C2×#125—832no—
345C2×C22⋊Q8—834no—
346C2×C22⋊Q8—832no—
347C2×C22⋊Q8—832no—
348C2×C22⋊Q8—834no—
349C2×C22⋊Q8—834no—
350C2×C22⋊Q8—1668no—
351C2×#117—831no—
352C2×#117C2×D8311yescharacteristic
353C22×C4⋊C4—831no—
354C22×C4⋊C4—831no—
355C22×C4⋊C4—831no—
356C22×C22⋊C4—831no—
357C22×C22⋊C4C2×D8311yescharacteristic
358#400—832no—
359#400—834no—
360#400—832no—
361#399—832no—
362#399C2×D8312yes—
363#398—834no—
364#398—832no—
365#398—834no—
366#398—832no—
367#397—834no—
368#397—832no—
369#397C2×D8312yes—
370#395—834no—
371#395—834no—
372#395—834no—
373#393—834no—
374#391—834no—
375#388—834no—
376#388—834no—
377#388—834no—
378#388C2×D8314yes—
379#387—834no—
380#387—834no—
381#387—834no—
382#387—834no—
383#385—832no—
384#385—834no—
385#385—832no—
386#385—834no—
387#385—832no—
388#385C2×D8312yes—
389#381—831no—
390#381—832no—
391#381—832no—
392#381C2×D8311yescharacteristic
393#380—832no—
394#380—834no—
395#379—832no—
396#379C2×D8312yes—
397C4×C22⋊C4—832no—
398C25C22×D8311yescharacteristic
399C22×Q8—3324no—
400C22×Q8—832no—
401C22×Q8—832no—
402C22×Q8—834no—
403C22×Q8—1664no—
404C23×C4—3324no—
405C23×C4—831no—
406C23×C4—831no—
407C23×C4—834no—
408C23×C4—831no—
409C23×C4—831no—
410C23×C4—1664no—
411C23×C4C22×D8311yescharacteristic
412C23×C4C22×D8311yescharacteristic
413C4⋊Q8—3328no—
414C4⋊Q8—1668no—
415C4⋊Q8—1668no—
416#125—1668no—
417#125—1668no—
418C22⋊Q8—3328no—
419C22⋊Q8—16616no—
420C22⋊Q8—16616no—
421C22⋊Q8—1668no—
422C22⋊Q8—1668no—
423#117—24no—
424#117—1664no—
425C2×C4⋊C4—834no—
426C2×C4⋊C4—834no—
427C2×C4⋊C4—838no—
428C2×C4⋊C4—832no—
429C2×C4⋊C4—832no—
430C2×C4⋊C4—832no—
431C2×C4⋊C4—832no—
432C2×C4⋊C4—834no—
433C2×C4⋊C4—831no—
434C2×C4⋊C4—831no—
435C2×C4⋊C4—832no—
436C2×C4⋊C4—832no—
437C2×C4⋊C4—832no—
438C2×C4⋊C4—832no—
439C2×C4⋊C4—832no—
440C2×C4⋊C4—832no—
441C2×C4⋊C4—834no—
442C2×C4⋊C4—834no—
443C2×C4⋊C4—1664no—
444C2×C4⋊C4—1664no—
445C2×C4⋊C4C22×D8311yescharacteristic
446C2×C4⋊C4C22×D8311yescharacteristic
447C2×C4⋊C4C22×D8314yes—
448C2×C22⋊C4—24no—
449C2×C22⋊C4—834no—
450C2×C22⋊C4—834no—
451C2×C22⋊C4—832no—
452C2×C22⋊C4—832no—
453C2×C22⋊C4—834no—
454C2×C22⋊C4—1664no—
455C2×C22⋊C4C22×D8312yes—
456C2×C22⋊C4C22×D8312yes—
457C2×C22⋊C4C22×D8312yes—
458C2×C22⋊C4C22×D8312yes—
459C2×C22⋊C4C22×D8314yes—
460C2×C22⋊C4C22×D8314yes—
461C2×C42—834no—
462C2×C42—832no—
463C2×C42C22×D8312yes—
464#95—834no—
465#95—834no—
466#95—834no—
467#95—834no—
468#95—832no—
469#95—832no—
470#95—834no—
471#95—834no—
472#95—834no—
473#95—834no—
474#95—834no—
475#95C22×D8312yes—
476#95C22×D8312yes—
477#95C22×D8314yes—
478C24—22no—
479C24—24no—
480C24—24no—
481C24—42no—
482C24C23×D8312yes—
483C24C23×D8311yescharacteristic
484C2×Q8—3324no—
485C2×Q8—3324no—
486C2×Q8—3324no—
487C2×Q8—33216no—
488C2×Q8—1668no—
489C2×Q8—1668no—
490C2×Q8—1664no—
491C2×Q8—1664no—
492C22×C4—24no—
493C22×C4—22no—
494C22×C4—42no—
495C22×C4—33216no—
496C22×C4—3322no—
497C22×C4—3324no—
498C22×C4—3322no—
499C22×C4—834no—
500C22×C4—834no—
501C22×C4—832no—
502C22×C4—832no—
503C22×C4—832no—
504C22×C4—834no—
505C22×C4—832no—
506C22×C4—832no—
507C22×C4—832no—
508C22×C4—1668no—
509C22×C4—1668no—
510C22×C4—1662no—
511C22×C4—1664no—
512C22×C4—1664no—
513C22×C4—1662no—
514C22×C4D4×D8314yes—
515C22×C4C23×D8314yes—
516C22×C4C23×D8312yes—
517C22×C4C23×D8312yes—
518C22×C4C23×D8312yes—
519C22×C4C23×D8311yescharacteristic
520C22×C4C23×D8311yescharacteristic
521C4⋊C4—22no—
522C4⋊C4—22no—
523C4⋊C4—44no—
524C4⋊C4—3324no—
525C4⋊C4—3324no—
526C4⋊C4—838no—
527C4⋊C4—838no—
528C4⋊C4—1668no—
529C4⋊C4—1668no—
530C4⋊C4—1668no—
531C4⋊C4—1664no—
532C4⋊C4—1664no—
533C4⋊C4—1664no—
534C4⋊C4—1664no—
535C4⋊C4—1664no—
536C4⋊C4—1664no—
537C4⋊C4—1664no—
538C4⋊C4—1664no—
539C4⋊C4—1662no—
540C4⋊C4—1662no—
541C22⋊C4—24no—
542C22⋊C4—28no—
543C22⋊C4—24no—
544C22⋊C4—24no—
545C22⋊C4—24no—
546C22⋊C4—44no—
547C22⋊C4—3324no—
548C22⋊C4—1668no—
549C22⋊C4—1664no—
550C22⋊C4—1668no—
551C22⋊C4—1664no—
552C42—24no—
553C42—3324no—
554C42—1664no—
555C23—22no—
556C23—22no—
557C23—28no—
558C23—22no—
559C23—24no—
560C23—22no—
561C23—24no—
562C23—416no—
563C23—44no—
564C23—44no—
565C23C2×D4×D8312yes—
566C23#1458183914yes—
567C23#1458183914yes—
568C23C24×D8311yescharacteristic
569Q8—33216no—
570Q8—6644no—
571Q8—6644no—
572C2×C4—28no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—44no—
585C2×C4—3324no—
586C2×C4—3324no—
587C2×C4—3324no—
588C2×C4—33216no—
589C2×C4—834no—
590C2×C4—834no—
591C2×C4—834no—
592C2×C4—1662no—
593C2×C4—1662no—
594C2×C4—1662no—
595C2×C4—1664no—
596C2×C4—1664no—
597C2×C4—1664no—
598C2×C4—1664no—
599C2×C4—1664no—
600C2×C4—1664no—
601C2×C4—1668no—
602C2×C4—1662no—
603C2×C4—1664no—
604C2×C4—1664no—
605C2×C4—1662no—
606C2×C4C2×D4×D8312yes—
607C2×C4C2×D4×D8312yes—
608C22—22no—
609C22—22no—
610C22—28no—
611C22—24no—
612C22—28no—
613C22—416no—
614C22—42no—
615C22—44no—
616C22—42no—
617C22#1912827314yes—
618C22C22×D4×D8311yescharacteristic
619C22H(5,𝔽2)×D8312yes—
620C22C2×#1458183912yes—
621C22C2×#1458183912yes—
622C4—22no—
623C4—22no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—44no—
628C4—3322no—
629C4—3322no—
630C4—3324no—
631C4—3322no—
632C4—1664no—
633C4—1664no—
634C4—1664no—
635C2—24no—
636C2—44no—
637C2C2×#1912827311yescharacteristic
638C2#2492109711yescharacteristic
639C2#2492111011yescharacteristic
640C2#2492170314yes—
subgroupG/H
center, FrattiniC23C24×D83
commutator, socle, cosocleC23×C83C25
FittingC83×#2482C2
prime radicalC25×C83C23
hypercenter#2482D83
hypocenterC83#36708
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2482

Complements

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

NH ≅ G/Ntype
1. C2×#19128329C2semidirect
4. #24913934C2semidirect
5. #24913930C2semidirect
6. #24913779C2semidirect
7. #24913775C2semidirect
8. #24913282C2semidirect
10. #24911752C2semidirect
11. #24911746C2semidirect
13. #24911105C2semidirect
14. #24910064C2semidirect
19. C2×#14581792C22semidirect
20. C2×#14581791C22semidirect
21. C2×#14581791C22semidirect
22. C2×#14581784C22semidirect
23. C2×#14581783C22semidirect
35. #19127642C22semidirect
36. #19127641C22semidirect
37. #19127639C22semidirect
39. #19127635C22semidirect
40. #19127633C22semidirect
41. #19127632C22semidirect
42. #19127632C22semidirect
45. #19127622C22semidirect
46. #19127621C22semidirect
47. #19127621C22semidirect
49. #19127605C22semidirect
53. #19127586C22semidirect
54. #19127586C22semidirect
55. #19127585C22semidirect
56. #19127584C22semidirect
336. C83#36708semidirect

Maximal subgroups

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

subgroupindex
C2×#191283292
C22×#145817692
#249157482
#249139342
#249139302
#249137792
#249137752
#249132822
#249126182
#249117522
#249117462
#249117372
#249111052
#249100642
C83×#24822
#3670883