Finite Group Databasebeta

← #35567187

Subgroups — #35567187

Order 21248.

2458 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×#19128213C211yesmaximal, characteristic
2#24910603C214yesmaximal, complemented
3#24910593C214yesmaximal, complemented
4#24910414C212yesmaximal, complemented
5#24910410C212yesmaximal, complemented
6#24910395C212yesmaximal, complemented
7#24910389C212yesmaximal, complemented
8#24910227C214yesmaximal, complemented
9#24910215C214yesmaximal, complemented
10#24910072C211yesmaximal, characteristic, complemented
11#24910052C212yesmaximal, complemented
12#24909845C212yesmaximal, complemented
13C83×#2314C211yesmaximal, characteristic, complemented
14C22×#13050590C2211yescharacteristic
15C23×C4×D83C2211yescharacteristic
16#19128213C4116yescomplemented
17C2×#14581776C2212yes—
18C2×#14581772C2212yes—
19C2×#14581767C2212yes—
20C2×#14581762C2212yes—
21C2×C4×D332C2212yes—
22C2×C4×D332C2212yes—
23C2×#14581750C2212yes—
24C2×#14581750C2212yes—
25C2×#14581750C2212yescomplemented
26C2×C4⋊C4×D83C2212yescomplemented
27#19127605C2214yescomplemented
28#19127603C2214yes—
29#19127602C2214yes—
30#19127601C2214yes—
31#19127594C2214yes—
32#19127593C2214yes—
33#19127593C2214yescomplemented
34#19127592C2214yes—
35#19127591C2214yes—
36#19127591C2214yescomplemented
37#19127590C2214yes—
38#19127587C2214yescomplemented
39#19127586C2214yescomplemented
40#19127584C2214yescomplemented
41#19127579C2214yes—
42#19127579C2214yes—
43#19127579C2214yescomplemented
44#19127577C2214yes—
45#19127571C2211yescharacteristic, complemented
46#19127567C2211yescharacteristic, complemented
47#19127563C2214yes—
48#19127561C2212yes—
49#19127558C2212yes—
50#19127558C2212yes—
51#19127558C2212yescomplemented
52#19127558C2212yescomplemented
53#19127556C2214yes—
54C4⋊C4×Dic83C2214yescomplemented
55C4×C4⋊Dic83C2214yescomplemented
56C4×C4⋊Dic83C2212yescomplemented
57C22⋊C4×Dic83C2214yes—
58C4×(C2×D83)⋊C4C2214yescomplemented
59#19127548C2214yes—
60#19127548C2214yescomplemented
61C2×#117×C83C2211yescharacteristic
62#388×C83C2214yes—
63#387×C83C2214yescomplemented
64#384×C83C2211yescharacteristic, complemented
65#382×C83C2211yescharacteristic, complemented
66#380×C83C2212yes—
67C4×C4⋊C4×C83C2212yescomplemented
68C24×D83C2311yescharacteristic
69C2×#13050590—42no—
70C2×#13050590—42no—
71C2×#13050590C2×C418yes—
72C22×Dic166C2311yescharacteristic
73C22×D332C2312yes—
74C22×D332C2312yes—
75C22×D332C2311yescharacteristic
76C22×C22⋊D83C2312yes—
77C23×Dic83C2311yescharacteristic
78C22×C4×D83—42no—
79C22×C4×D83—42no—
80C22×C4×D83C2×C418yescomplemented
81C22×C4×D83C2312yes—
82C22×C4×D83C2312yes—
83C22×C4×D83C2314yes—
84C22×C4×D83C2312yes—
85#14581776C2×C4116yescomplemented
86#14581772C2×C4116yescomplemented
87#14581767C2×C4116yes—
88#14581762C2×C4116yescomplemented
89C4×D332C2×C4116yescomplemented
90C4×D332C2×C4116yescomplemented
91#14581750—44no—
92#14581750—44no—
93#14581750—44no—
94C2×C4⋊Dic83C2312yes—
95C2×C4⋊Dic83C2312yes—
96C2×C4⋊Dic83C2314yes—
97C2×C4⋊Dic83C2312yes—
98C2×C4⋊Dic83C2312yes—
99C2×Dic83⋊C4C2314yes—
100C4⋊C4×D83—44no—
101C2×C22⋊Dic83C2314yes—
102C2×(C2×D83)⋊C4C2314yes—
103C2×(C2×D83)⋊C4C2314yes—
104C2×(C2×D83)⋊C4C2314yes—
105C2×(C2×D83)⋊C4C2314yes—
106C2×(C2×D83)⋊C4C2314yes—
107C2×(C2×D83)⋊C4C2314yes—
108C2×(C2×D83)⋊C4C2314yes—
109C2×(C2×D83)⋊C4C2312yes—
110C2×(C2×D83)⋊C4C2312yes—
111C2×(C2×D83)⋊C4C2312yes—
112C2×(C2×D83)⋊C4C2312yes—
113C2×C4×Dic83C2314yes—
114C2×C4×Dic83C2314yes—
115C2×C4×Dic83C2314yes—
116C2×C4×Dic83C2314yes—
117C83⋊#95C2314yes—
118C83⋊#95C2314yes—
119C83⋊#95C2314yes—
120C83⋊#95C2314yes—
121C83⋊#95C2314yes—
122C83⋊#95C2314yes—
123C83⋊#95C2314yes—
124C23×C4×C83C2311yescharacteristic
125#117×C83C2×C418yescomplemented
126C2×C4⋊C4×C83C2312yes—
127C2×C4⋊C4×C83C2312yes—
128C2×C4⋊C4×C83C2314yes—
129C2×C4⋊C4×C83C2312yes—
130C2×C22⋊C4×C83C2314yes—
131C2×C22⋊C4×C83C2312yes—
132C2×C42×C83C2314yes—
133C2×C42×C83C2311yescharacteristic
134C2×C42×C83C2311yescharacteristic
135#95×C83C2314yes—
136#95×C83C2314yes—
137#95×C83C2314yes—
138C23×D83—41no—
139C23×D83—41no—
140C23×D83—42no—
141C23×D83—42no—
142C23×D83C22×C414yes—
143C23×D83C2412yes—
144C23×D83C2412yes—
145#13050590—416no—
146C2×Dic166—41no—
147C2×Dic166—41no—
148C2×Dic166C22×C414yes—
149C2×D332—42no—
150C2×D332—42no—
151C2×D332—42no—
152C2×D332—42no—
153C2×D332—41no—
154C2×D332—41no—
155C2×D332C22×C418yes—
156C2×D332C22×C418yes—
157C2×D332C22×C414yes—
158C2×C22⋊D83—42no—
159C2×C22⋊D83—42no—
160C2×C22⋊D83C22×C418yes—
161C22×Dic83—41no—
162C22×Dic83—41no—
163C22×Dic83C22×C414yes—
164C22×Dic83C2414yes—
165C22×Dic83C2414yes—
166C22×Dic83C2412yes—
167C22×Dic83C2412yes—
168C2×C4×D83—24no—
169C2×C4×D83—24no—
170C2×C4×D83—24no—
171C2×C4×D83—24no—
172C2×C4×D83—416no—
173C2×C4×D83—42no—
174C2×C4×D83—42no—
175C2×C4×D83—44no—
176C2×C4×D83—44no—
177C2×C4×D83—44no—
178C2×C4×D83C22×C418yes—
179C2×C4×D83C22×C418yes—
180C2×C4×D83C22×C418yes—
181C4⋊Dic83—24no—
182C4⋊Dic83—24no—
183C4⋊Dic83—44no—
184C4⋊Dic83C22×C418yes—
185C4⋊Dic83C22×C418yes—
186Dic83⋊C4—44no—
187C22⋊Dic83—44no—
188(C2×D83)⋊C4—44no—
189(C2×D83)⋊C4—44no—
190(C2×D83)⋊C4—44no—
191(C2×D83)⋊C4—44no—
192(C2×D83)⋊C4—44no—
193(C2×D83)⋊C4—44no—
194(C2×D83)⋊C4—44no—
195(C2×D83)⋊C4C22×C418yes—
196(C2×D83)⋊C4C22×C418yes—
197(C2×D83)⋊C4C22×C418yes—
198(C2×D83)⋊C4C22×C418yes—
199C4×Dic83—44no—
200C4×Dic83—44no—
201C4×Dic83—44no—
202C4×Dic83—44no—
203C24×C83C2411yescharacteristic
204C22×C4×C83—22no—
205C22×C4×C83—22no—
206C22×C4×C83C22×C414yes—
207C22×C4×C83C2414yes—
208C22×C4×C83C2414yes—
209C22×C4×C83C2411yescharacteristic
210C22×C4×C83C2412yes—
211C22×C4×C83C2412yes—
212C22×C4×C83C2411yescharacteristic
213C4⋊C4×C83—24no—
214C4⋊C4×C83—24no—
215C4⋊C4×C83—44no—
216C4⋊C4×C83C22×C418yes—
217C22⋊C4×C83—44no—
218C22⋊C4×C83C22×C418yes—
219C42×C83—44no—
220C42×C83C22×C414yes—
221C42×C83C22×C414yes—
222C23×C83—21no—
223C23×C83—21no—
224C23×C83—22no—
225C23×C83—22no—
226C23×C83C23×C412yes—
227C23×C83C2511yescharacteristic
228C22×D83—22no—
229C22×D83—22no—
230C22×D83—22no—
231C22×D83—22no—
232C22×D83—22no—
233C22×D83—22no—
234C22×D83—22no—
235C22×D83—22no—
236C22×D83—22no—
237C22×D83—22no—
238C22×D83—22no—
239C22×D83—22no—
240C22×D83—44no—
241C22×D83—48no—
242C22×D83—44no—
243C22×D83—44no—
244C22×D83—48no—
245C22×D83C23×C414yes—
246C22×D83C23×C414yes—
247C2×C4×C83—21no—
248C2×C4×C83—21no—
249C2×C4×C83—22no—
250C2×C4×C83—22no—
251C2×C4×C83—22no—
252C2×C4×C83—22no—
253C2×C4×C83—28no—
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×C83C23×C414yes—
261C2×C4×C83C23×C414yes—
262C2×C4×C83C23×C412yes—
263C2×C4×C83C23×C412yes—
264C2×C4×C83H(5,𝔽2)12yes—
265C2×C4×C83H(5,𝔽2)12yes—
266C22⋊D83—48no—
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×Dic83C23×C414yes—
279C2×Dic83C23×C414yes—
280D332—48no—
281D332—48no—
282D332—44no—
283C4×D83—216no—
284C4×D83—216no—
285C4×D83—48no—
286C4×D83—88no—
287Dic166—44no—
288C22×C83—22no—
289C22×C83—24no—
290C22×C83—41no—
291C22×C83—41no—
292C22×C83—42no—
293C22×C83C24×C411yescharacteristic
294C22×C83C2×H(5,𝔽2)11yescharacteristic
295C22×C83C2×H(5,𝔽2)11yescharacteristic
296C22×C83#58512yes—
297C22×C83#58512yes—
298C2×D83—24no—
299C2×D83—24no—
300C2×D83—24no—
301C2×D83—24no—
302C2×D83—24no—
303C2×D83—24no—
304C2×D83—416no—
305C2×D83—42no—
306C2×D83—42no—
307C2×D83—42no—
308C2×D83—42no—
309C2×D83—42no—
310C2×D83—42no—
311C2×D83—42no—
312C2×D83—42no—
313C4×C83—22no—
314C4×C83—24no—
315C4×C83—24no—
316C4×C83—44no—
317C4×C83—44no—
318C4×C83#51814yes—
319Dic83—24no—
320Dic83—24no—
321Dic83—44no—
322Dic83—44no—
323#34361—831nomaximal, Sylow
324C2×C83—42no—
325C2×C83C2×#51811yescharacteristic
326C2×C83#342812yes—
327C2×C83#353911yescharacteristic
328C2×C83#353911yescharacteristic
329C2×C83#354012yes—
330D83—44no—
331D83—44no—
332C2×#525—831no—
333#2375—834no—
334#2374—834no—
335#2354—832no—
336#2354—832no—
337#2352—832no—
338#2352—832no—
339#2333—834no—
340#2332—834no—
341#2314—831no—
342#2314D8311yescharacteristic, complemented
343#2311—832no—
344#2280—832no—
345C83#3436111yescharacteristic, complemented, Sylow
346C22×#41—831no—
347C24×C4—831no—
348#525—8316no—
349C2×#123—832no—
350C2×C22⋊Q8—832no—
351C2×#121—832no—
352C2×#120—832no—
353C2×C4×D4—832no—
354C2×C4×D4—832no—
355C2×#117—832no—
356C2×#117—832no—
357C2×#117—832no—
358C2×#117C2×D8311yescharacteristic
359C22×C4⋊C4—832no—
360#391—834no—
361#390—834no—
362#390—834no—
363#390—834no—
364#388—834no—
365#388—834no—
366#388—834no—
367#388—834no—
368#388—834no—
369#388—834no—
370#388—834no—
371#388C2×D8314yes—
372#387—834no—
373#387—834no—
374#387—834no—
375#387C2×D8314yescomplemented
376#386—834no—
377#386—834no—
378#386—834no—
379#386—834no—
380#384—831no—
381#384C2×D8311yescharacteristic, complemented
382#382—831no—
383#382C2×D8311yescharacteristic, complemented
384#381—834no—
385#380—832no—
386#380—832no—
387#380—832no—
388#380—832no—
389#380—832no—
390#380C2×D8312yes—
391#379—834no—
392C4×C4⋊C4—834no—
393C4×C4⋊C4—832no—
394C4×C4⋊C4—834no—
395C4×C4⋊C4C2×D8312yescomplemented
396C4×C22⋊C4—834no—
397C4×C22⋊C4—834no—
398C4×C22⋊C4—834no—
399C4×C22⋊C4—834no—
400C25—831no—
401C2×#41—3322no—
402C2×#41—3322no—
403C2×#41—838no—
404C22×Q8—831no—
405C22×D4—832no—
406C22×D4—832no—
407C22×D4—832no—
408C22×D4—831no—
409C23×C4—3322no—
410C23×C4—3322no—
411C23×C4—834no—
412C23×C4—831no—
413C23×C4—838no—
414C23×C4—832no—
415C23×C4—832no—
416C23×C4—832no—
417C23×C4C22×D8311yescharacteristic
418#123—8316no—
419C22⋊Q8—8316no—
420#121—8316no—
421#120—8316no—
422C4×D4—8316no—
423C4×D4—8316no—
424#117—3324no—
425#117—3324no—
426#117—3324no—
427#117C4×D8318yescomplemented
428C2×C4⋊C4—3324no—
429C2×C4⋊C4—834no—
430C2×C4⋊C4—834no—
431C2×C4⋊C4—832no—
432C2×C4⋊C4—832no—
433C2×C4⋊C4—832no—
434C2×C4⋊C4—832no—
435C2×C4⋊C4C22×D8312yes—
436C2×C4⋊C4C22×D8312yes—
437C2×C4⋊C4C22×D8314yes—
438C2×C4⋊C4C22×D8312yes—
439C2×C22⋊C4—834no—
440C2×C22⋊C4—834no—
441C2×C22⋊C4—834no—
442C2×C22⋊C4—834no—
443C2×C22⋊C4—834no—
444C2×C22⋊C4—834no—
445C2×C22⋊C4—834no—
446C2×C22⋊C4—834no—
447C2×C22⋊C4—832no—
448C2×C22⋊C4—832no—
449C2×C22⋊C4—832no—
450C2×C22⋊C4—832no—
451C2×C22⋊C4C22×D8314yes—
452C2×C22⋊C4C22×D8312yes—
453C2×C42—834no—
454C2×C42—834no—
455C2×C42—834no—
456C2×C42—834no—
457C2×C42C22×D8314yes—
458C2×C42C22×D8311yescharacteristic
459C2×C42C22×D8311yescharacteristic
460#95—834no—
461#95—834no—
462#95—834no—
463#95—834no—
464#95—834no—
465#95—834no—
466#95—834no—
467#95C22×D8314yes—
468#95C22×D8314yes—
469#95C22×D8314yes—
470C24—3321no—
471C24—3321no—
472C24—3322no—
473C24—3322no—
474C24—834no—
475C24—832no—
476C24—832no—
477C24C23×D8311yescharacteristic
478#41—33216no—
479C2×Q8—3321no—
480C2×Q8—3321no—
481C2×Q8—834no—
482C2×D4—3321no—
483C2×D4—3321no—
484C2×D4—3322no—
485C2×D4—3322no—
486C2×D4—3322no—
487C2×D4—3322no—
488C2×D4—3322no—
489C2×D4—3322no—
490C2×D4—838no—
491C2×D4—838no—
492C2×D4—838no—
493C2×D4—834no—
494C22×C4—22no—
495C22×C4—22no—
496C22×C4—3322no—
497C22×C4—3322no—
498C22×C4—33216no—
499C22×C4—3324no—
500C22×C4—3324no—
501C22×C4—3324no—
502C22×C4—3321no—
503C22×C4—3321no—
504C22×C4—834no—
505C22×C4—838no—
506C22×C4—832no—
507C22×C4—832no—
508C22×C4—838no—
509C22×C4—838no—
510C22×C4—834no—
511C22×C4—834no—
512C22×C4—1664no—
513C22×C4—1664no—
514C22×C4—1664no—
515C22×C4—1664no—
516C22×C4C2×C4×D8314yes—
517C22×C4C23×D8314yes—
518C22×C4C23×D8314yes—
519C22×C4C23×D8311yescharacteristic
520C22×C4C23×D8312yes—
521C22×C4C23×D8312yes—
522C22×C4C23×D8311yescharacteristic
523C4⋊C4—24no—
524C4⋊C4—24no—
525C4⋊C4—44no—
526C4⋊C4—3324no—
527C4⋊C4—3324no—
528C4⋊C4—838no—
529C4⋊C4—838no—
530C4⋊C4—1664no—
531C4⋊C4—1664no—
532C4⋊C4C2×C4×D8318yes—
533C22⋊C4—44no—
534C22⋊C4—3324no—
535C22⋊C4—3324no—
536C22⋊C4—3324no—
537C22⋊C4—3324no—
538C22⋊C4—3324no—
539C22⋊C4—3324no—
540C22⋊C4—3324no—
541C22⋊C4—3324no—
542C22⋊C4—838no—
543C22⋊C4—838no—
544C22⋊C4—838no—
545C22⋊C4—838no—
546C22⋊C4C2×C4×D8318yes—
547C42—44no—
548C42—3324no—
549C42—3324no—
550C42—3324no—
551C42—3324no—
552C42C2×C4×D8314yes—
553C42C2×C4×D8314yes—
554C23—21no—
555C23—21no—
556C23—22no—
557C23—22no—
558C23—3324no—
559C23—3328no—
560C23—3324no—
561C23—3324no—
562C23—3328no—
563C23—834no—
564C23—834no—
565C23—1662no—
566C23—1662no—
567C23—1662no—
568C23—1662no—
569C23—1662no—
570C23—1662no—
571C23—1662no—
572C23—1662no—
573C23—1662no—
574C23—1662no—
575C23—1662no—
576C23—1662no—
577C23C22×C4×D8312yes—
578C23C24×D8311yescharacteristic
579Q8—3324no—
580D4—3324no—
581D4—3328no—
582D4—3328no—
583D4—3328no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—28no—
591C2×C4—24no—
592C2×C4—24no—
593C2×C4—24no—
594C2×C4—24no—
595C2×C4—24no—
596C2×C4—24no—
597C2×C4—3328no—
598C2×C4—3324no—
599C2×C4—834no—
600C2×C4—834no—
601C2×C4—6648no—
602C2×C4—1664no—
603C2×C4—1664no—
604C2×C4—1664no—
605C2×C4—1664no—
606C2×C4—1664no—
607C2×C4—1664no—
608C2×C4—1662no—
609C2×C4—1662no—
610C2×C4—1662no—
611C2×C4—1662no—
612C2×C4—16616no—
613C2×C4—16616no—
614C2×C4C22×C4×D8314yes—
615C2×C4C22×C4×D8314yes—
616C2×C4C22×C4×D8312yes—
617C2×C4C22×C4×D8312yes—
618C2×C4#1458183812yes—
619C2×C4#1458183812yes—
620C22—22no—
621C22—24no—
622C22—41no—
623C22—41no—
624C22—42no—
625C22—33216no—
626C22—3322no—
627C22—3322no—
628C22—3322no—
629C22—3322no—
630C22—3322no—
631C22—3322no—
632C22—3322no—
633C22—3322no—
634C22—1664no—
635C22—1664no—
636C22—1664no—
637C22—1664no—
638C22—1664no—
639C22—1664no—
640C22C23×C4×D8311yescharacteristic
641C22C2×#1458183811yescharacteristic
642C22C2×#1458183811yescharacteristic
643C22#1912853612yes—
644C22#1912853812yes—
645C4—22no—
646C4—24no—
647C4—24no—
648C4—44no—
649C4—44no—
650C4—3324no—
651C4—3324no—
652C4—1664no—
653C4—1664no—
654C4#1912817514yes—
655C2—42no—
656C2—3324no—
657C2—3324no—
658C2C2×#1912817511yescharacteristic
659C2#2492066312yes—
660C2#2492172211yescharacteristic
661C2#2492172511yescharacteristic
662C2#2492173512yes—
subgroupG/H
center, FrattiniC23C24×D83
commutatorC22×C83C24×C4
FittingC83×#2314C2
socle, cosocleC23×C83C25
prime radicalC2×#19128213C2
hypercenter#2314D83
hypocenterC83#34361
LCSG ▷ C22×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2314

Complements

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

NH ≅ G/Ntype
2. #24910603C2semidirect
3. #24910593C2semidirect
4. #24910414C2semidirect
5. #24910410C2semidirect
6. #24910395C2semidirect
7. #24910389C2semidirect
8. #24910227C2semidirect
9. #24910215C2semidirect
10. #24910072C2semidirect
11. #24910052C2semidirect
12. #24909845C2semidirect
13. C83×#2314C2semidirect
16. #19128213C4semidirect
25. C2×#14581750C22semidirect
26. C2×C4⋊C4×D83C22semidirect
27. #19127605C22semidirect
33. #19127593C22semidirect
36. #19127591C22semidirect
38. #19127587C22semidirect
39. #19127586C22semidirect
40. #19127584C22semidirect
43. #19127579C22semidirect
45. #19127571C22semidirect
46. #19127567C22semidirect
51. #19127558C22semidirect
52. #19127558C22semidirect
54. C4⋊C4×Dic83C22semidirect
55. C4×C4⋊Dic83C22semidirect
56. C4×C4⋊Dic83C22semidirect
58. C4×(C2×D83)⋊C4C22semidirect
60. #19127548C22semidirect
63. #387×C83C22semidirect
64. #384×C83C22semidirect
65. #382×C83C22semidirect
67. C4×C4⋊C4×C83C22semidirect
80. C22×C4×D83C2×C4semidirect
85. #14581776C2×C4semidirect
86. #14581772C2×C4semidirect
88. #14581762C2×C4semidirect
89. C4×D332C2×C4semidirect
90. C4×D332C2×C4semidirect
125. #117×C83C2×C4semidirect
342. #2314D83semidirect
345. C83#34361semidirect
375. #387C2×D83semidirect
381. #384C2×D83semidirect
383. #382C2×D83semidirect
395. C4×C4⋊C4C2×D83semidirect
427. #117C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191282132
#249106032
#249105932
#249104142
#249104102
#249103952
#249103892
#249102272
#249102152
#249100722
#249100522
#249098452
C83×#23142
#3436183