Finite Group Databasebeta

← #35564699

Subgroups — #35564699

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×#19128212C211yesmaximal, characteristic
2C2×#19128182C211yesmaximal, characteristic, complemented
3#24910501C214yesmaximal, complemented
4#24910490C214yesmaximal, complemented
5#24910413C214yesmaximal, complemented
6#24910393C214yesmaximal, complemented
7#24910121C214yesmaximal, complemented
8#24910087C211yesmaximal, characteristic, complemented
9#24910056C211yesmaximal, characteristic, complemented
10C4×#14581775C214yesmaximal, complemented
11#24909846C211yesmaximal, characteristic, complemented
12#24909820C211yesmaximal, characteristic, complemented
13C22×#117×C83C211yesmaximal, characteristic, complemented
14C22×#13050590C2211yescharacteristic
15#19128212C4116yescomplemented
16C2×#14581775C2212yes—
17C2×#14581771C2212yes—
18C2×#14581766C2212yes—
19C2×#14581761C2212yes—
20#19128182—28no—
21C2×#14581757C2212yescomplemented
22C2×#14581754C2212yescomplemented
23C2×C4×C22⋊D83C2214yes—
24C2×C4×C22⋊D83C2214yes—
25C2×#14581751C2211yescharacteristic
26C2×#14581751C2211yescharacteristic
27C2×#14581751C2211yescharacteristic
28C2×#14581750C2212yescomplemented
29C2×C4⋊C4×D83C2212yescomplemented
30#19127606C2212yescomplemented
31#19127602C2212yescomplemented
32#19127601C2218yes—
33#19127597C2214yescomplemented
34#19127595C2218yes—
35#19127592C2218yescomplemented
36#19127589C2218yes—
37#19127585C2218yescomplemented
38#19127580C2214yes—
39#19127577C2218yes—
40#19127574C2214yes—
41#19127570C2212yescomplemented
42#19127566C2212yescomplemented
43#19127564C2212yes—
44#19127561C2212yes—
45#19127561C2212yes—
46#19127558C2214yescomplemented
47#19127557C2212yes—
48C4⋊C4×Dic83C2214yescomplemented
49C4⋊C4×Dic83C2214yescomplemented
50C4×Dic83⋊C4C2218yescomplemented
51C4×(C2×D83)⋊C4C2218yescomplemented
52C4×C22⋊Dic83C2214yes—
53C4×C22⋊Dic83C2214yes—
54C24×C4×C83C2211yescharacteristic
55C2×#117×C83—28no—
56C2×#117×C83C2214yes—
57C2×#117×C83C2214yes—
58C22×C4⋊C4×C83C2211yescharacteristic
59C22×C4⋊C4×C83C2211yescharacteristic
60C22×C22⋊C4×C83C2212yes—
61C22×C42×C83C2212yes—
62C2×#13050590—24no—
63C2×#13050590—24no—
64C2×#13050590C2×C418yes—
65C22×Dic166C2311yescharacteristic
66C22×D332C2311yescharacteristic
67C22×C22⋊D83C2312yes—
68C22×C22⋊D83C2314yes—
69C22×C4×D83C2312yes—
70C22×C4×D83C2314yes—
71#14581775C2×C4116yescomplemented
72#14581771C2×C4116yescomplemented
73#14581766C2×C4116yescomplemented
74#14581761C2×C4116yescomplemented
75#14581757—28no—
76#14581754—28no—
77C4×C22⋊D83—216no—
78C4×C22⋊D83C2×C4132yescomplemented
79#14581751—24no—
80#14581751—24no—
81#14581751C2×C418yescomplemented
82#14581750—28no—
83C2×C4⋊Dic83C2312yes—
84C2×C4⋊Dic83C2311yescharacteristic
85C2×C4⋊Dic83C2311yescharacteristic
86C2×C4⋊Dic83C2312yes—
87C2×Dic83⋊C4C2314yes—
88C2×Dic83⋊C4C2318yes—
89C2×Dic83⋊C4C2314yes—
90C4⋊C4×D83—28no—
91C2×C22⋊Dic83C2312yes—
92C2×C22⋊Dic83C2312yes—
93C2×C22⋊Dic83C2314yes—
94C2×C22⋊Dic83C2314yes—
95C2×C22⋊Dic83C2312yes—
96C2×C22⋊Dic83C2314yes—
97C2×(C2×D83)⋊C4C2314yes—
98C2×(C2×D83)⋊C4C2318yes—
99C2×(C2×D83)⋊C4C2314yes—
100C2×C4×Dic83C2314yes—
101C2×C4×Dic83C2312yes—
102C2×C4×Dic83C2312yes—
103C2×C4×Dic83C2318yes—
104C2×C4×Dic83C2312yes—
105C83⋊#95C2314yes—
106C83⋊#95C2318yes—
107C83⋊#95C2318yes—
108C83⋊#95C2314yes—
109C83⋊#95C2318yes—
110C25×C83C2311yescharacteristic
111C23×C4×C83—24no—
112C23×C4×C83—24no—
113C23×C4×C83C2×C418yescomplemented
114C23×C4×C83C2312yes—
115C23×C4×C83C2311yescharacteristic
116C23×C4×C83C2312yes—
117C23×C4×C83C2312yes—
118C23×C4×C83C2312yes—
119C23×C4×C83C2311yescharacteristic
120#117×C83—216no—
121#117×C83—216no—
122C2×C4⋊C4×C83—24no—
123C2×C4⋊C4×C83D418yescomplemented
124C2×C4⋊C4×C83C2312yes—
125C2×C4⋊C4×C83C2312yes—
126C2×C4⋊C4×C83C2312yes—
127C2×C4⋊C4×C83C2312yes—
128C2×C22⋊C4×C83—28no—
129C2×C22⋊C4×C83C2318yes—
130C2×C42×C83—28no—
131C2×C42×C83C2318yes—
132C23×D83C2412yes—
133#13050590—232no—
134C2×Dic166—22no—
135C2×Dic166—22no—
136C2×Dic166C22×C414yes—
137C2×D332—22no—
138C2×D332—22no—
139C2×D332C22×C414yes—
140C2×C22⋊D83—24no—
141C2×C22⋊D83—24no—
142C2×C22⋊D83—44no—
143C2×C22⋊D83—44no—
144C2×C22⋊D83C22×C418yes—
145C2×C22⋊D83C22×C4116yes—
146C22×Dic83C2412yes—
147C22×Dic83C2414yes—
148C22×Dic83C2412yes—
149C22×Dic83C2412yes—
150C22×Dic83C2414yes—
151C2×C4×D83—24no—
152C2×C4×D83—24no—
153C2×C4×D83—28no—
154C2×C4×D83—48no—
155C2×C4×D83C22×C418yes—
156C4⋊Dic83—22no—
157C4⋊Dic83—22no—
158C4⋊Dic83—24no—
159C4⋊Dic83C22×C418yes—
160Dic83⋊C4—28no—
161Dic83⋊C4—48no—
162Dic83⋊C4C22×C4116yes—
163C22⋊Dic83—24no—
164C22⋊Dic83—24no—
165C22⋊Dic83—44no—
166C22⋊Dic83—44no—
167C22⋊Dic83C22×C4116yes—
168C22⋊Dic83C22×C418yes—
169(C2×D83)⋊C4—28no—
170(C2×D83)⋊C4—48no—
171(C2×D83)⋊C4C22×C4116yes—
172C4×Dic83—28no—
173C4×Dic83—24no—
174C4×Dic83—24no—
175C4×Dic83—48no—
176C4×Dic83C22×C418yes—
177C24×C83—22no—
178C24×C83—22no—
179C24×C83—42no—
180C24×C83—42no—
181C24×C83C22×C414yes—
182C24×C83C2412yes—
183C24×C83C2411yescharacteristic
184C22×C4×C83—22no—
185C22×C4×C83—22no—
186C22×C4×C83—232no—
187C22×C4×C83—24no—
188C22×C4×C83—24no—
189C22×C4×C83—24no—
190C22×C4×C83—24no—
191C22×C4×C83—24no—
192C22×C4×C83—44no—
193C22×C4×C83—44no—
194C22×C4×C83C22×C418yes—
195C22×C4×C83C22×C414yes—
196C22×C4×C83C22×C418yes—
197C22×C4×C83C22×C414yes—
198C22×C4×C83C2×D418yes—
199C22×C4×C83C2×D414yes—
200C22×C4×C83#4114yes—
201C22×C4×C83C2414yes—
202C22×C4×C83C2412yes—
203C22×C4×C83C2414yes—
204C22×C4×C83C2411yescharacteristic
205C22×C4×C83C2411yescharacteristic
206C4⋊C4×C83—24no—
207C4⋊C4×C83—24no—
208C4⋊C4×C83C2×D418yes—
209C4⋊C4×C83C2×D418yes—
210C22⋊C4×C83—216no—
211C42×C83—216no—
212C23×C83—22no—
213C23×C83—22no—
214C23×C83—28no—
215C23×C83—22no—
216C23×C83—22no—
217C23×C83—22no—
218C23×C83—22no—
219C23×C83—22no—
220C23×C83—22no—
221C23×C83—48no—
222C23×C83—44no—
223C23×C83—44no—
224C23×C83—48no—
225C23×C83C23×C414yes—
226C23×C83C23×C412yes—
227C23×C83C22×D412yes—
228C23×C83C2×#4112yes—
229C23×C83C2511yescharacteristic
230C22×D83—22no—
231C22×D83—22no—
232C22×D83—24no—
233C22×D83—24no—
234C22×D83C23×C414yes—
235C2×C4×C83—28no—
236C2×C4×C83—22no—
237C2×C4×C83—22no—
238C2×C4×C83—28no—
239C2×C4×C83—216no—
240C2×C4×C83—216no—
241C2×C4×C83—24no—
242C2×C4×C83—28no—
243C2×C4×C83—48no—
244C2×C4×C83—48no—
245C2×C4×C83C4×D4116yes—
246C2×C4×C83C23×C414yes—
247C2×C4×C83C23×C412yes—
248C2×C4×C83C23×C412yes—
249C2×C4×C83C22×D418yes—
250C2×C4×C83C22×D412yes—
251C2×C4×C83C22×D412yes—
252C2×C4×C83C2×#4112yes—
253C2×C4×C83C2×#4112yes—
254C22⋊D83—216no—
255C22⋊D83—416no—
256C2×Dic83—22no—
257C2×Dic83—22no—
258C2×Dic83—24no—
259C2×Dic83—24no—
260C2×Dic83—22no—
261C2×Dic83—22no—
262C2×Dic83—24no—
263C2×Dic83—24no—
264C2×Dic83—24no—
265C2×Dic83—24no—
266C2×Dic83—24no—
267C2×Dic83C23×C418yes—
268C2×Dic83C23×C414yes—
269D332—28no—
270C4×D83—216no—
271C4×D83—88no—
272Dic166—28no—
273C22×C83—24no—
274C22×C83—22no—
275C22×C83—22no—
276C22×C83—24no—
277C22×C83—24no—
278C22×C83—24no—
279C22×C83—24no—
280C22×C83—416no—
281C22×C83—42no—
282C22×C83—42no—
283C22×C83—42no—
284C22×C83—42no—
285C22×C83C2×C4×D414yes—
286C22×C83C24×C411yescharacteristic
287C22×C83C23×D411yescharacteristic
288C22×C83C22×#4111yescharacteristic
289C22×C83#58512yes—
290C22×C83#58512yes—
291C2×D83—24no—
292C2×D83—28no—
293C2×D83—42no—
294C2×D83—42no—
295C2×D83—44no—
296C4×C83—24no—
297C4×C83—28no—
298C4×C83—44no—
299C4×C83C2×C4×D414yes—
300C4×C83C2×C4×D414yes—
301Dic83—24no—
302Dic83—28no—
303Dic83—42no—
304Dic83—42no—
305Dic83—44no—
306#34201—831nomaximal, Sylow
307C2×C83—24no—
308C2×C83—44no—
309C2×C83C22×C4×D411yescharacteristic
310C2×C83#342812yes—
311C2×C83#348511yescharacteristic
312C2×C83#348711yescharacteristic
313C2×C83#349312yes—
314D83—44no—
315C2×#525—831no—
316C2×#520—831no—
317C22×#117D8311yescharacteristic, complemented
318#2363—834no—
319#2362—834no—
320#2354—834no—
321#2352—834no—
322#2321—834no—
323#2317—831no—
324#2311—831no—
325C4×#123—834no—
326#2280—831no—
327#2274—831no—
328C83#3420111yescharacteristic, complemented, Sylow
329C22×#41—831no—
330C24×C4C2×D8311yescharacteristic
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—28no—
342C2×#117—831no—
343C2×#117—831no—
344C2×#117—831no—
345C2×#117—832no—
346C2×#117C2×D8314yes—
347C2×#117C2×D8314yes—
348C22×C4⋊C4—832no—
349C22×C4⋊C4C2×D8311yescharacteristic
350C22×C4⋊C4C2×D8311yescharacteristic
351C22×C22⋊C4C2×D8312yes—
352C22×C42C2×D8312yes—
353#391—832no—
354#390—832no—
355#390—838no—
356#389—834no—
357#388—838no—
358#388—838no—
359#388—838no—
360#387—838no—
361#386—834no—
362#386—838no—
363#385—834no—
364#384—832no—
365#382—832no—
366#381—832no—
367#380—832no—
368#380—832no—
369#380—834no—
370#379—832no—
371C4×C4⋊C4—834no—
372C4×C4⋊C4—834no—
373C4×C4⋊C4—838no—
374C4×C22⋊C4—834no—
375C4×C22⋊C4—834no—
376C4×C22⋊C4—838no—
377C25C22×D8311yescharacteristic
378C2×#41—838no—
379C2×#41—1664no—
380C2×#41—1664no—
381C22×Q8—831no—
382C22×D4—834no—
383C22×D4—832no—
384C22×D4—831no—
385C23×C4—24no—
386C23×C4—24no—
387C23×C4—834no—
388C23×C4—832no—
389C23×C4C4×D8318yescomplemented
390C23×C4C22×D8312yes—
391C23×C4C22×D8312yes—
392C23×C4C22×D8311yescharacteristic
393C23×C4C22×D8312yes—
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—16616no—
403C4×D4—1668no—
404#117—216no—
405#117—216no—
406#117—838no—
407#117—1664no—
408#117—1664no—
409#117—1668no—
410C2×C4⋊C4—24no—
411C2×C4⋊C4—838no—
412C2×C4⋊C4—832no—
413C2×C4⋊C4—834no—
414C2×C4⋊C4—831no—
415C2×C4⋊C4—834no—
416C2×C4⋊C4—831no—
417C2×C4⋊C4—832no—
418C2×C4⋊C4—1668no—
419C2×C4⋊C4C22⋊D8318yescomplemented
420C2×C4⋊C4C22×D8312yes—
421C2×C4⋊C4C22×D8312yes—
422C2×C4⋊C4C22×D8312yes—
423C2×C4⋊C4C22×D8312yes—
424C2×C22⋊C4—28no—
425C2×C22⋊C4—834no—
426C2×C22⋊C4—834no—
427C2×C22⋊C4—838no—
428C2×C22⋊C4—832no—
429C2×C22⋊C4—834no—
430C2×C22⋊C4—834no—
431C2×C22⋊C4—832no—
432C2×C22⋊C4—832no—
433C2×C22⋊C4—834no—
434C2×C22⋊C4C22×D8318yes—
435C2×C42—28no—
436C2×C42—838no—
437C2×C42—832no—
438C2×C42—832no—
439C2×C42—834no—
440C2×C42—832no—
441C2×C42C22×D8318yes—
442#95—834no—
443#95—834no—
444#95—838no—
445#95—838no—
446#95—838no—
447C24—22no—
448C24—22no—
449C24—42no—
450C24—42no—
451C24—832no—
452C24C2×C4×D8314yes—
453C24C23×D8312yes—
454C24C23×D8311yescharacteristic
455#41—16632no—
456C2×Q8—834no—
457C2×Q8—1662no—
458C2×Q8—1662no—
459C2×D4—3324no—
460C2×D4—3324no—
461C2×D4—8316no—
462C2×D4—838no—
463C2×D4—834no—
464C2×D4—1664no—
465C2×D4—1664no—
466C2×D4—1662no—
467C2×D4—1662no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—24no—
471C22×C4—24no—
472C22×C4—24no—
473C22×C4—24no—
474C22×C4—232no—
475C22×C4—24no—
476C22×C4—44no—
477C22×C4—44no—
478C22×C4—3328no—
479C22×C4—834no—
480C22×C4—832no—
481C22×C4—838no—
482C22×C4—834no—
483C22×C4—832no—
484C22×C4—832no—
485C22×C4—1668no—
486C22×C4—1664no—
487C22×C4—1664no—
488C22×C4C2×C4×D8318yes—
489C22×C4C2×C4×D8318yes—
490C22×C4C2×C4×D8314yes—
491C22×C4C2×C4×D8314yes—
492C22×C4C2×C22⋊D8318yes—
493C22×C4C2×C22⋊D8314yes—
494C22×C4#1305059014yes—
495C22×C4C23×D8314yes—
496C22×C4C23×D8312yes—
497C22×C4C23×D8314yes—
498C22×C4C23×D8311yescharacteristic
499C22×C4C23×D8311yescharacteristic
500C4⋊C4—24no—
501C4⋊C4—24no—
502C4⋊C4—3328no—
503C4⋊C4—838no—
504C4⋊C4—8316no—
505C4⋊C4—1662no—
506C4⋊C4—1668no—
507C4⋊C4—1662no—
508C4⋊C4—1664no—
509C4⋊C4C2×C22⋊D8318yes—
510C4⋊C4C2×C22⋊D8318yes—
511C22⋊C4—216no—
512C22⋊C4—3328no—
513C22⋊C4—3324no—
514C22⋊C4—3324no—
515C22⋊C4—838no—
516C22⋊C4—8316no—
517C22⋊C4—8316no—
518C22⋊C4—1664no—
519C22⋊C4—1664no—
520C22⋊C4—1668no—
521C42—216no—
522C42—3328no—
523C42—838no—
524C42—1664no—
525C42—1668no—
526C42—1664no—
527C23—22no—
528C23—22no—
529C23—28no—
530C23—22no—
531C23—22no—
532C23—22no—
533C23—22no—
534C23—22no—
535C23—22no—
536C23—48no—
537C23—44no—
538C23—44no—
539C23—48no—
540C23—834no—
541C23—1662no—
542C23—1662no—
543C23—1664no—
544C23—1664no—
545C23C22×C4×D8314yes—
546C23C22×C4×D8312yes—
547C23C22×C22⋊D8312yes—
548C23C2×#1305059012yes—
549C23C24×D8311yescharacteristic
550Q8—1668no—
551D4—33216no—
552D4—16616no—
553D4—1668no—
554C2×C4—28no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—28no—
558C2×C4—216no—
559C2×C4—216no—
560C2×C4—24no—
561C2×C4—28no—
562C2×C4—48no—
563C2×C4—48no—
564C2×C4—838no—
565C2×C4—834no—
566C2×C4—6648no—
567C2×C4—1662no—
568C2×C4—1662no—
569C2×C4—1664no—
570C2×C4—1664no—
571C2×C4—1664no—
572C2×C4—1664no—
573C2×C4—1664no—
574C2×C4—1662no—
575C2×C4—1662no—
576C2×C4—1664no—
577C2×C4—1664no—
578C2×C4—16616no—
579C2×C4C4×C22⋊D83116yes—
580C2×C4C22×C4×D8314yes—
581C2×C4C22×C4×D8312yes—
582C2×C4C22×C4×D8312yes—
583C2×C4C22×C22⋊D8318yes—
584C2×C4C22×C22⋊D8312yes—
585C2×C4C22×C22⋊D8312yes—
586C2×C4C2×#1305059012yes—
587C2×C4C2×#1305059012yes—
588C22—22no—
589C22—22no—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—24no—
594C22—24no—
595C22—416no—
596C22—42no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—3322no—
601C22—3322no—
602C22—3324no—
603C22—1664no—
604C22—1668no—
605C22C2×C4×C22⋊D8314yes—
606C22C23×C4×D8311yescharacteristic
607C22C23×C22⋊D8311yescharacteristic
608C22C22×#1305059011yescharacteristic
609C22#1912853712yes—
610C22#1912853712yes—
611C4—28no—
612C4—24no—
613C4—44no—
614C4—3322no—
615C4—3322no—
616C4—3324no—
617C4—1664no—
618C4—1668no—
619C4C2×C4×C22⋊D8314yes—
620C4C2×C4×C22⋊D8314yes—
621C2—24no—
622C2—44no—
623C2—3324no—
624C2C22×C4×C22⋊D8311yescharacteristic
625C2#2492066412yes—
626C2#2492110711yescharacteristic
627C2#2492112111yescharacteristic
628C2#2492118312yes—
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingC22×#117×C83C2
socle, cosocleC23×C83C25
prime radicalC2×#19128212C2
hypercenterC22×#117D83
hypocenterC83#34201
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×#117

Complements

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

NH ≅ G/Ntype
2. C2×#19128182C2semidirect
3. #24910501C2semidirect
4. #24910490C2semidirect
5. #24910413C2semidirect
6. #24910393C2semidirect
7. #24910121C2semidirect
8. #24910087C2semidirect
9. #24910056C2semidirect
10. C4×#14581775C2semidirect
11. #24909846C2semidirect
12. #24909820C2semidirect
13. C22×#117×C83C2semidirect
15. #19128212C4semidirect
21. C2×#14581757C22semidirect
22. C2×#14581754C22semidirect
28. C2×#14581750C22semidirect
29. C2×C4⋊C4×D83C22semidirect
30. #19127606C22semidirect
31. #19127602C22semidirect
33. #19127597C22semidirect
35. #19127592C22semidirect
37. #19127585C22semidirect
41. #19127570C22semidirect
42. #19127566C22semidirect
46. #19127558C22semidirect
48. C4⋊C4×Dic83C22semidirect
49. C4⋊C4×Dic83C22semidirect
50. C4×Dic83⋊C4C22semidirect
51. C4×(C2×D83)⋊C4C22semidirect
71. #14581775C2×C4semidirect
72. #14581771C2×C4semidirect
73. #14581766C2×C4semidirect
74. #14581761C2×C4semidirect
78. C4×C22⋊D83C2×C4semidirect
81. #14581751C2×C4semidirect
113. C23×C4×C83C2×C4semidirect
123. C2×C4⋊C4×C83D4semidirect
317. C22×#117D83semidirect
328. C83#34201semidirect
389. C23×C4C4×D83semidirect
419. C2×C4⋊C4C22⋊D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191282122
C2×#191281822
#249105012
#249104902
#249104132
#249103932
#249101212
#249100872
#249100562
C4×#145817752
#249098462
#249098202
C22×#117×C832
#3420183