Finite Group Databasebeta

← #35577457

Subgroups — #35577457

Order 21248.

1978 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24915773C212yesmaximal, complemented
2#24915750C212yesmaximal
3#24913577C211yesmaximal, characteristic, complemented
4#24913577C211yesmaximal, characteristic, complemented
5#24913268C212yesmaximal
6#24912737C214yesmaximal, complemented
7#24912547C214yesmaximal, complemented
8#24912181C214yesmaximal, complemented
9#24911484C214yesmaximal, complemented
10#24910196C212yesmaximal, complemented
11#24910061C212yesmaximal
12C4×#14581771C212yesmaximal, complemented
13C2×C4×C22⋊C4×C83C211yesmaximal, characteristic
14C2×#14581784C2212yescomplemented
15C2×#14581771C2212yes—
16C2×#14581771C2212yes—
17C2×#14581771C2212yes—
18C2×#14581771C2212yes—
19C2×C4×Dic166C2212yescomplemented
20#19127637C2214yes—
21#19127633C2214yescomplemented
22#19127633C2214yescomplemented
23#19127624C2214yes—
24#19127624C2214yes—
25#19127622C2214yescomplemented
26#19127622C2214yescomplemented
27#19127613C2214yes—
28#19127607C2212yescomplemented
29#19127605C2214yescomplemented
30#19127599C2212yescomplemented
31#19127599C2212yescomplemented
32#19127599C2212yescomplemented
33#19127598C2214yescomplemented
34#19127594C2214yes—
35#19127594C2214yes—
36#19127594C2214yes—
37#19127594C2214yes—
38#19127588C2214yescomplemented
39#19127585C2214yescomplemented
40#19127585C2214yescomplemented
41#19127585C2214yescomplemented
42#19127584C2214yescomplemented
43#19127574C2214yes—
44#19127574C2214yes—
45#19127574C2214yes—
46#19127574C2214yes—
47#19127567C2212yescomplemented
48#19127561C2212yes—
49#19127561C2212yes—
50#19127561C2212yes—
51#19127561C2212yes—
52#19127561C2212yes—
53#19127561C2212yes—
54#19127557C2214yes—
55C4×Dic83⋊C4C2214yescomplemented
56C4×C4⋊Dic83C2212yescomplemented
57C4×C22⋊Dic83C2214yes—
58C24×C4×C83C2211yescharacteristic
59C22×C22⋊C4×C83C2211yescharacteristic
60C22×C22⋊C4×C83C2211yescharacteristic
61C22×C42×C83C2211yescharacteristic
62C22×C42×C83C2211yescharacteristic
63C4×C22⋊C4×C83C2214yes—
64C4×C22⋊C4×C83C2214yes—
65C2×#95×C83C2211yescharacteristic
66C2×#95×C83C2211yescharacteristic
67C22×Dic166C2312yes—
68C22×Dic166C2312yes—
69#14581784—44no—
70#14581771—28no—
71#14581771—28no—
72#14581771—44no—
73#14581771—44no—
74C4×Dic166—44no—
75C2×C4⋊Dic83C2312yes—
76C2×C4⋊Dic83C2312yes—
77C2×C4⋊Dic83C2312yes—
78C2×C4⋊Dic83C2312yes—
79C2×C4⋊Dic83C2312yes—
80C2×C4⋊Dic83C2312yes—
81C2×C4⋊Dic83C2312yes—
82C2×C4⋊Dic83C2312yes—
83C2×C4⋊Dic83C2312yes—
84C2×C4⋊Dic83C2312yes—
85C2×Dic83⋊C4C2314yes—
86C2×Dic83⋊C4C2314yes—
87C2×Dic83⋊C4C2314yes—
88C2×Dic83⋊C4C2314yes—
89C2×Dic83⋊C4C2314yes—
90C2×Dic83⋊C4C2314yes—
91C2×C22⋊Dic83C2314yes—
92C2×C22⋊Dic83C2314yes—
93C2×C22⋊Dic83C2314yes—
94C2×C22⋊Dic83C2314yes—
95C2×C22⋊Dic83C2314yes—
96C2×C22⋊Dic83C2314yes—
97C2×C4×Dic83C2314yes—
98C2×C4×Dic83C2314yes—
99C83⋊#95C2314yes—
100C83⋊#95C2314yes—
101C83⋊#95C2314yes—
102C83⋊#95C2314yes—
103C83⋊#95C2314yes—
104C83⋊#95C2314yes—
105C83⋊#95C2314yes—
106C83⋊#95C2314yes—
107C83⋊#95C2314yes—
108C83⋊#95C2314yes—
109C25×C83C2311yescharacteristic
110C23×C4×C83—24no—
111C23×C4×C83—42no—
112C23×C4×C83—42no—
113C23×C4×C83C2311yescharacteristic
114C23×C4×C83C2312yes—
115C23×C4×C83C2312yes—
116C23×C4×C83C2311yescharacteristic
117C23×C4×C83C2311yescharacteristic
118C23×C4×C83C2311yescharacteristic
119C23×C4×C83C2311yescharacteristic
120C23×C4×C83C2311yescharacteristic
121C2×C22⋊C4×C83—24no—
122C2×C22⋊C4×C83Q818yescomplemented
123C2×C22⋊C4×C83C2314yes—
124C2×C22⋊C4×C83C2314yes—
125C2×C42×C83—42no—
126C2×C42×C83—42no—
127C2×C42×C83C2312yes—
128C2×C42×C83C2312yes—
129C2×C42×C83C2312yes—
130C2×C42×C83C2312yes—
131#95×C83C2314yes—
132#95×C83C2314yes—
133C2×Dic166—24no—
134C2×Dic166—24no—
135C2×Dic166—42no—
136C2×Dic166—42no—
137C2×Dic166—42no—
138C2×Dic166—42no—
139C22×Dic83C2414yes—
140C22×Dic83C2414yes—
141C22×Dic83C2414yes—
142C22×Dic83C2414yes—
143C4⋊Dic83—24no—
144C4⋊Dic83—24no—
145C4⋊Dic83—42no—
146C4⋊Dic83—42no—
147C4⋊Dic83—42no—
148C4⋊Dic83—42no—
149C4⋊Dic83—42no—
150C4⋊Dic83—42no—
151C4⋊Dic83—42no—
152C4⋊Dic83—42no—
153Dic83⋊C4—28no—
154Dic83⋊C4—28no—
155Dic83⋊C4—28no—
156Dic83⋊C4—28no—
157Dic83⋊C4—44no—
158Dic83⋊C4—44no—
159C22⋊Dic83—28no—
160C22⋊Dic83—28no—
161C22⋊Dic83—44no—
162C22⋊Dic83—44no—
163C22⋊Dic83—44no—
164C22⋊Dic83—44no—
165C4×Dic83—44no—
166C4×Dic83—44no—
167C24×C83—22no—
168C24×C83—22no—
169C24×C83—41no—
170C24×C83—41no—
171C24×C83—41no—
172C24×C83—41no—
173C24×C83C2×Q814yes—
174C24×C83C2412yes—
175C24×C83C2411yescharacteristic
176C22×C4×C83—22no—
177C22×C4×C83—22no—
178C22×C4×C83—22no—
179C22×C4×C83—22no—
180C22×C4×C83—24no—
181C22×C4×C83—24no—
182C22×C4×C83—24no—
183C22×C4×C83—24no—
184C22×C4×C83—41no—
185C22×C4×C83—41no—
186C22×C4×C83—41no—
187C22×C4×C83—41no—
188C22×C4×C83—42no—
189C22×C4×C83—42no—
190C22×C4×C83—41no—
191C22×C4×C83—41no—
192C22×C4×C83—41no—
193C22×C4×C83—41no—
194C22×C4×C83—41no—
195C22×C4×C83—41no—
196C22×C4×C83—416no—
197C22×C4×C83C2×Q814yes—
198C22×C4×C83C2×Q814yes—
199C22×C4×C83#4114yes—
200C22×C4×C83#4114yes—
201C22×C4×C83C2412yes—
202C22×C4×C83C2412yes—
203C22×C4×C83C2412yes—
204C22×C4×C83C2412yes—
205C22×C4×C83C2412yes—
206C22×C4×C83C2411yescharacteristic
207C22×C4×C83C2411yescharacteristic
208C22⋊C4×C83—28no—
209C22⋊C4×C83C2×Q8116yes—
210C42×C83—24no—
211C42×C83—24no—
212C42×C83—42no—
213C42×C83—42no—
214C23×C83—21no—
215C23×C83—21no—
216C23×C83—21no—
217C23×C83—21no—
218C23×C83—28no—
219C23×C83—22no—
220C23×C83—22no—
221C23×C83—22no—
222C23×C83—22no—
223C23×C83—22no—
224C23×C83—22no—
225C23×C83—44no—
226C23×C83—44no—
227C23×C83—44no—
228C23×C83—44no—
229C23×C83—44no—
230C23×C83—44no—
231C23×C83C22×Q814yes—
232C23×C83C22×Q812yes—
233C23×C83C2×#4112yes—
234C23×C83C2×#4112yes—
235C23×C83C2511yescharacteristic
236C2×C4×C83—28no—
237C2×C4×C83—21no—
238C2×C4×C83—21no—
239C2×C4×C83—22no—
240C2×C4×C83—22no—
241C2×C4×C83—22no—
242C2×C4×C83—21no—
243C2×C4×C83—21no—
244C2×C4×C83—22no—
245C2×C4×C83—22no—
246C2×C4×C83—22no—
247C2×C4×C83—22no—
248C2×C4×C83—22no—
249C2×C4×C83—22no—
250C2×C4×C83—22no—
251C2×C4×C83—22no—
252C2×C4×C83—44no—
253C2×C4×C83—44no—
254C2×C4×C83—44no—
255C2×C4×C83—44no—
256C2×C4×C83—48no—
257C2×C4×C83—48no—
258C2×C4×C83—44no—
259C2×C4×C83C22×Q814yes—
260C2×C4×C83C22×Q814yes—
261C2×C4×C83C2×#4114yes—
262C2×C4×C83C2×#4114yes—
263C2×C4×C83H(5,𝔽2)12yes—
264C2×C4×C83H(5,𝔽2)12yes—
265C2×Dic83—24no—
266C2×Dic83—24no—
267C2×Dic83—24no—
268C2×Dic83—24no—
269C2×Dic83—24no—
270C2×Dic83—24no—
271C2×Dic83—24no—
272C2×Dic83—24no—
273C2×Dic83—24no—
274C2×Dic83—24no—
275C2×Dic83—24no—
276C2×Dic83—24no—
277Dic166—48no—
278Dic166—84no—
279C22×C83—22no—
280C22×C83—22no—
281C22×C83—22no—
282C22×C83—22no—
283C22×C83—22no—
284C22×C83—22no—
285C22×C83—24no—
286C22×C83—24no—
287C22×C83—24no—
288C22×C83—416no—
289C22×C83—42no—
290C22×C83—42no—
291C22×C83—42no—
292C22×C83—42no—
293C22×C83#53314yes—
294C22×C83C23×Q811yescharacteristic
295C22×C83C22×#4111yescharacteristic
296C22×C83C22×#4111yescharacteristic
297C22×C83C2×H(5,𝔽2)11yescharacteristic
298C22×C83C2×H(5,𝔽2)11yescharacteristic
299C22×C83#58511yescharacteristic
300C22×C83#58511yescharacteristic
301C4×C83—24no—
302C4×C83—24no—
303C4×C83—22no—
304C4×C83—22no—
305C4×C83—24no—
306C4×C83—24no—
307C4×C83—42no—
308Dic83—44no—
309Dic83—44no—
310Dic83—44no—
311Dic83—44no—
312#35012—831nomaximal, Sylow
313C2×C83—24no—
314C2×C83—44no—
315C2×C83C2×#53311yescharacteristic
316C2×C83#345911yescharacteristic
317C2×C83#346011yescharacteristic
318C2×C83#349011yescharacteristic
319C2×C83#349311yescharacteristic
320C2×C83#350111yescharacteristic
321C2×C83#350111yescharacteristic
322#2851—832no—
323#2848—832no—
324#2659—831no—
325#2659—831no—
326#2626—832no—
327#2574—834no—
328#2556—834no—
329#2522—834no—
330#2465—834no—
331#2329—832no—
332#2312—832no—
333C4×C22⋊Q8—832no—
334C2×C4×C22⋊C4D8311yescharacteristic
335C83#3501211yescharacteristic, complemented, Sylow
336C24×C4C2×D8311yescharacteristic
337C2×#125—832no—
338C2×C22⋊Q8—832no—
339C2×C22⋊Q8—832no—
340C2×C22⋊Q8—832no—
341C2×C22⋊Q8—832no—
342C2×C4×Q8—832no—
343C22×C22⋊C4C2×D8311yescharacteristic
344C22×C22⋊C4C2×D8311yescharacteristic
345C22×C42C2×D8311yescharacteristic
346C22×C42C2×D8311yescharacteristic
347#399—834no—
348#398—834no—
349#398—834no—
350#396—834no—
351#396—834no—
352#395—834no—
353#395—834no—
354#393—834no—
355#391—832no—
356#391—834no—
357#389—832no—
358#389—832no—
359#389—832no—
360#389—834no—
361#388—834no—
362#388—834no—
363#388—834no—
364#388—834no—
365#387—834no—
366#387—834no—
367#387—834no—
368#387—834no—
369#387—834no—
370#385—834no—
371#385—834no—
372#385—834no—
373#385—834no—
374#382—832no—
375#380—832no—
376#380—832no—
377#380—832no—
378#380—832no—
379#380—832no—
380#380—832no—
381#379—834no—
382C4×C4⋊C4—832no—
383C4×C4⋊C4—834no—
384C4×C22⋊C4—834no—
385C4×C22⋊C4C2×D8314yes—
386C4×C22⋊C4C2×D8314yes—
387C2×#95C2×D8311yescharacteristic
388C2×#95C2×D8311yescharacteristic
389C25C22×D8311yescharacteristic
390C22×Q8—832no—
391C22×Q8—832no—
392C23×C4—24no—
393C23×C4—42no—
394C23×C4—42no—
395C23×C4C22×D8311yescharacteristic
396C23×C4C22×D8312yes—
397C23×C4C22×D8312yes—
398C23×C4C22×D8311yescharacteristic
399C23×C4C22×D8311yescharacteristic
400C23×C4C22×D8311yescharacteristic
401C23×C4C22×D8311yescharacteristic
402C23×C4C22×D8311yescharacteristic
403#125—3324no—
404C22⋊Q8—3324no—
405C22⋊Q8—3324no—
406C22⋊Q8—1668no—
407C22⋊Q8—1668no—
408C4×Q8—3324no—
409C2×C4⋊C4—832no—
410C2×C4⋊C4—832no—
411C2×C4⋊C4—832no—
412C2×C4⋊C4—832no—
413C2×C4⋊C4—832no—
414C2×C4⋊C4—834no—
415C2×C4⋊C4—832no—
416C2×C4⋊C4—832no—
417C2×C4⋊C4—832no—
418C2×C4⋊C4—834no—
419C2×C4⋊C4—834no—
420C2×C4⋊C4—834no—
421C2×C4⋊C4—834no—
422C2×C4⋊C4—832no—
423C2×C4⋊C4—832no—
424C2×C4⋊C4—834no—
425C2×C22⋊C4—24no—
426C2×C22⋊C4—834no—
427C2×C22⋊C4—834no—
428C2×C22⋊C4—834no—
429C2×C22⋊C4—834no—
430C2×C22⋊C4—834no—
431C2×C22⋊C4—834no—
432C2×C22⋊C4Dic16618yescomplemented
433C2×C22⋊C4C22×D8314yes—
434C2×C22⋊C4C22×D8314yes—
435C2×C42—42no—
436C2×C42—42no—
437C2×C42—834no—
438C2×C42—834no—
439C2×C42C22×D8312yes—
440C2×C42C22×D8312yes—
441C2×C42C22×D8312yes—
442C2×C42C22×D8312yes—
443#95—834no—
444#95—834no—
445#95—834no—
446#95—834no—
447#95—834no—
448#95—834no—
449#95—834no—
450#95—834no—
451#95—834no—
452#95—834no—
453#95C22×D8314yes—
454#95C22×D8314yes—
455C24—22no—
456C24—22no—
457C24—41no—
458C24—41no—
459C24—41no—
460C24—41no—
461C24C2×Dic16614yes—
462C24C23×D8312yes—
463C24C23×D8311yescharacteristic
464C2×Q8—3322no—
465C2×Q8—3322no—
466C2×Q8—3322no—
467C2×Q8—3322no—
468C2×Q8—1664no—
469C2×Q8—1664no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—22no—
474C22×C4—24no—
475C22×C4—24no—
476C22×C4—24no—
477C22×C4—24no—
478C22×C4—41no—
479C22×C4—41no—
480C22×C4—41no—
481C22×C4—41no—
482C22×C4—42no—
483C22×C4—42no—
484C22×C4—41no—
485C22×C4—41no—
486C22×C4—41no—
487C22×C4—41no—
488C22×C4—41no—
489C22×C4—41no—
490C22×C4—416no—
491C22×C4—834no—
492C22×C4—834no—
493C22×C4—834no—
494C22×C4—834no—
495C22×C4C2×Dic16614yes—
496C22×C4C2×Dic16614yes—
497C22×C4#1305059014yes—
498C22×C4#1305059014yes—
499C22×C4C23×D8312yes—
500C22×C4C23×D8312yes—
501C22×C4C23×D8312yes—
502C22×C4C23×D8312yes—
503C22×C4C23×D8312yes—
504C22×C4C23×D8311yescharacteristic
505C22×C4C23×D8311yescharacteristic
506C4⋊C4—3322no—
507C4⋊C4—3322no—
508C4⋊C4—3322no—
509C4⋊C4—3324no—
510C4⋊C4—3322no—
511C4⋊C4—3322no—
512C4⋊C4—3324no—
513C4⋊C4—3322no—
514C4⋊C4—3322no—
515C4⋊C4—3322no—
516C4⋊C4—1668no—
517C4⋊C4—1668no—
518C4⋊C4—1664no—
519C4⋊C4—1664no—
520C4⋊C4—1668no—
521C4⋊C4—1668no—
522C22⋊C4—28no—
523C22⋊C4—3324no—
524C22⋊C4—3324no—
525C22⋊C4—3324no—
526C22⋊C4—3324no—
527C22⋊C4—1668no—
528C22⋊C4—1668no—
529C22⋊C4C2×Dic166116yes—
530C42—24no—
531C42—24no—
532C42—42no—
533C42—42no—
534C42—3324no—
535C42—3324no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—28no—
541C23—22no—
542C23—22no—
543C23—22no—
544C23—22no—
545C23—22no—
546C23—22no—
547C23—44no—
548C23—44no—
549C23—44no—
550C23—44no—
551C23—44no—
552C23—44no—
553C23C22×Dic16614yes—
554C23C22×Dic16612yes—
555C23C2×#1305059012yes—
556C23C2×#1305059012yes—
557C23C24×D8311yescharacteristic
558Q8—3328no—
559Q8—6644no—
560C2×C4—28no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—22no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—44no—
577C2×C4—44no—
578C2×C4—44no—
579C2×C4—44no—
580C2×C4—48no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—1664no—
584C2×C4—1664no—
585C2×C4—1664no—
586C2×C4—1664no—
587C2×C4—1664no—
588C2×C4—1664no—
589C2×C4—1664no—
590C2×C4—1664no—
591C2×C4—1664no—
592C2×C4—1664no—
593C2×C4—1664no—
594C2×C4—1664no—
595C2×C4C22×Dic16614yes—
596C2×C4C22×Dic16614yes—
597C2×C4C2×#1305059014yes—
598C2×C4C2×#1305059014yes—
599C2×C4#1458183712yes—
600C2×C4#1458183712yes—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—416no—
611C22—42no—
612C22—42no—
613C22—42no—
614C22—42no—
615C22#1912825814yes—
616C22C23×Dic16611yescharacteristic
617C22C22×#1305059011yescharacteristic
618C22C22×#1305059011yescharacteristic
619C22C2×#1458183711yescharacteristic
620C22C2×#1458183711yescharacteristic
621C22#1912853711yescharacteristic
622C22#1912853711yescharacteristic
623C4—24no—
624C4—22no—
625C4—22no—
626C4—24no—
627C4—24no—
628C4—24no—
629C4—42no—
630C4—3324no—
631C4—3324no—
632C4—3324no—
633C4—3324no—
634C2—24no—
635C2—44no—
636C2C2×#1912825811yescharacteristic
637C2#2492087211yescharacteristic
638C2#2492088511yescharacteristic
639C2#2492115211yescharacteristic
640C2#2492118311yescharacteristic
641C2#2492126611yescharacteristic
642C2#2492126611yescharacteristic
subgroupG/H
center, FrattiniC23C24×D83
commutator, socle, cosocleC23×C83C25
FittingC2×C4×C22⋊C4×C83C2
prime radicalC25×C83C23
hypercenterC2×C4×C22⋊C4D83
hypocenterC83#35012
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #24915773C2semidirect
3. #24913577C2semidirect
4. #24913577C2semidirect
6. #24912737C2semidirect
7. #24912547C2semidirect
8. #24912181C2semidirect
9. #24911484C2semidirect
10. #24910196C2semidirect
12. C4×#14581771C2semidirect
14. C2×#14581784C22semidirect
19. C2×C4×Dic166C22semidirect
21. #19127633C22semidirect
22. #19127633C22semidirect
25. #19127622C22semidirect
26. #19127622C22semidirect
28. #19127607C22semidirect
29. #19127605C22semidirect
30. #19127599C22semidirect
31. #19127599C22semidirect
32. #19127599C22semidirect
33. #19127598C22semidirect
38. #19127588C22semidirect
39. #19127585C22semidirect
40. #19127585C22semidirect
41. #19127585C22semidirect
42. #19127584C22semidirect
47. #19127567C22semidirect
55. C4×Dic83⋊C4C22semidirect
56. C4×C4⋊Dic83C22semidirect
122. C2×C22⋊C4×C83Q8semidirect
335. C83#35012semidirect
432. C2×C22⋊C4Dic166semidirect

Maximal subgroups

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

subgroupindex
#249157732
#249157502
#249135772
#249132682
#249127372
#249125472
#249121812
#249114842
#249101962
#249100612
C4×#145817712
C2×C4×C22⋊C4×C832
#3501283