Finite Group Databasebeta

← #36285113

Subgroups — #36285113

Order 21248.

1706 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24921284C211yesmaximal, characteristic, complemented
2#24920408C214yesmaximal, complemented
3#24920402C214yesmaximal, complemented
4#24920350C214yesmaximal, complemented
5#24920326C214yesmaximal, complemented
6#24917997C212yesmaximal, complemented
7C2×#19128100C211yesmaximal, characteristic, complemented
8#24917325C212yesmaximal, complemented
9#24917321C212yesmaximal, complemented
10#24917285C211yesmaximal, characteristic, complemented
11#24917281C211yesmaximal, characteristic, complemented
12#24916398C211yesmaximal, characteristic, complemented
13#24916336C212yesmaximal
14C4×#14581812C211yesmaximal, characteristic
15C83×#3504C211yesmaximal, characteristic, complemented
16C2×#14581812C2211yescharacteristic
17#19128429C2214yes—
18#19128411C2214yes—
19#19128405—24no—
20#19128329C2212yes—
21#19128328—22no—
22#19128257—21no—
23C2×#14581784C2211yescharacteristic
24#125×D83—21no—
25#19128181C2212yes—
26C4×#13050590C2211yescharacteristic
27#19128111C2214yes—
28#19128107C2214yes—
29#19128104C2214yes—
30#19128100C2214yes—
31#19128100C2214yes—
32#19128100C2214yes—
33#19128008C2214yes—
34#19128008C2214yes—
35#19128005C2214yes—
36#19128005C2214yes—
37#19128001C2214yes—
38#19128001C2214yes—
39#19127998C2214yes—
40#19127998C2214yes—
41#19127991C2214yes—
42#19127988C2214yes—
43#19127986C2214yes—
44#19127983C2214yes—
45#19127832C2214yes—
46Q8×C83⋊C8C2214yes—
47#19127801C2214yes—
48#19127799C2214yes—
49C4×#13050580C2212yes—
50C4×#13050580C2214yes—
51C4×#13050577C2212yes—
52C4×#13050578C2212yes—
53C4×#13050575C2212yes—
54C4×#13050575C2214yes—
55#19127744C2212yes—
56#19127741C2212yes—
57#19127737C2211yescharacteristic
58C2×#14581725C2212yes—
59C2×#14581723C2212yes—
60#19127728C2212yes—
61#19127725C2211yescharacteristic
62#19127702C2212yes—
63#19127700C2212yes—
64#19127697C2211yescharacteristic
65#19127695C2211yescharacteristic
66#19127657C2212yes—
67C2×C4×C83⋊C8C2211yescharacteristic
68#557×C83C2214yes—
69#554×C83—24no—
70#554×C83C2214yes—
71#544×C83—22no—
72#544×C83C2212yes—
73#533×C83—21no—
74#533×C83C2211yescharacteristic
75C2×#125×C83—21no—
76C2×#125×C83C2211yescharacteristic
77#520×C83C2212yes—
78C4×#41×C83C2211yescharacteristic
79C2×#13050590C2311yescharacteristic
80#14581812Q818yescomplemented
81C2×#13050580C2311yescharacteristic
82C2×#13050578C2311yescharacteristic
83C2×#13050577C2311yescharacteristic
84C2×#13050575C2311yescharacteristic
85C2×#13050574C2312yes—
86C22×C83⋊C8C2311yescharacteristic
87C22×C83⋊C8C2311yescharacteristic
88#14581792C2314yes—
89#14581791—22no—
90#14581790—24no—
91#14581784—21no—
92#14581784—21no—
93#14581784C2314yes—
94#14581784C2314yes—
95#14581783—22no—
96#14581782—24no—
97#14581782—24no—
98#14581770—24no—
99#14581770—24no—
100#14581769—24no—
101#14581768—24no—
102C4×Dic166C2312yes—
103C4×Dic166C2314yes—
104#14581757—24no—
105C4×D332C2312yes—
106C4×D332C2314yes—
107C4×C22⋊D83—22no—
108(C22⋊D83)⋊C4—24no—
109#14581751C2311yescharacteristic
110#14581750—22no—
111#14581749—21no—
112C2×C4⋊Dic83C2312yes—
113C2×Dic83⋊C4—22no—
114C4⋊C4×D83—22no—
115C4⋊C4×D83—24no—
116C42×D83—21no—
117#14581725C2314yes—
118#14581725C2312yes—
119#14581725C2314yes—
120#14581725C2314yes—
121#14581725C2314yes—
122#14581723C2314yes—
123#14581723C2314yes—
124#14581723C2314yes—
125#14581723C2314yes—
126#14581723C2312yes—
127C4⋊(C83⋊C8)C2314yes—
128C4⋊(C83⋊C8)C2314yes—
129C4⋊(C83⋊C8)C2314yes—
130#14581716C2312yes—
131#14581716C2314yes—
132#14581715C2312yes—
133#14581715C2314yes—
134#14581713C2312yes—
135#14581713C2314yes—
136#14581712C2312yes—
137#14581712C2314yes—
138#14581703C2314yes—
139C4×C83⋊C8C2312yes—
140C4×C83⋊C8C2314yes—
141C4×C83⋊C8C2312yes—
142C2×#41×C83C2311yescharacteristic
143C4⋊Q8×C83—24no—
144C4⋊Q8×C83—22no—
145C4⋊Q8×C83D412yes—
146C4⋊Q8×C83C2314yes—
147#125×C83—22no—
148#125×C83—24no—
149#125×C83—24no—
150#125×C83—21no—
151#125×C83—21no—
152#125×C83D411yescharacteristic
153#125×C83D411yescharacteristic
154#125×C83C2314yes—
155#125×C83C2314yes—
156C22⋊Q8×C83—24no—
157C22⋊Q8×C83—24no—
158C22⋊Q8×C83—24no—
159C22⋊Q8×C83—24no—
160C22⋊Q8×C83D414yes—
161C4×Q8×C83—24no—
162C4×Q8×C83C2312yes—
163C4×Q8×C83C2314yes—
164C4×Q8×C83C2314yes—
165C4×D4×C83—22no—
166C4×D4×C83—24no—
167C4×D4×C83C2312yes—
168C4×D4×C83C2314yes—
169#117×C83—22no—
170#117×C83—21no—
171#117×C83C2311yescharacteristic
172#117×C83C2312yes—
173C2×C4⋊C4×C83—22no—
174C2×C4⋊C4×C83—22no—
175C2×C4⋊C4×C83—24no—
176C2×C4⋊C4×C83C2312yes—
177C2×C42×C83—21no—
178C2×C42×C83C2311yescharacteristic
179#13050590—22no—
180#13050590C2×Q814yes—
181C2×Dic166—24no—
182C2×Dic166C2411yescharacteristic
183C2×D332C2411yescharacteristic
184C2×C22⋊D83—21no—
185C22×Dic83—22no—
186C2×C4×D83—21no—
187C2×C4×D83—24no—
188C2×C4×D83—22no—
189#13050580C2×Q814yes—
190#13050578C2×Q814yes—
191#13050577C2×Q814yes—
192#13050575C2×Q814yes—
193#13050574—24no—
194C2×C83⋊C8—22no—
195C2×C83⋊C8C2×Q814yes—
196C2×C83⋊C8C2412yes—
197C2×C83⋊C8C2414yes—
198C2×C83⋊C8C2411yescharacteristic
199C2×C83⋊C8C2411yescharacteristic
200C4⋊Dic83—24no—
201C4⋊Dic83—22no—
202C4⋊Dic83C2414yes—
203C4⋊Dic83C2412yes—
204Dic83⋊C4—24no—
205Dic83⋊C4—24no—
206Dic83⋊C4—24no—
207Dic83⋊C4—24no—
208Dic83⋊C4—24no—
209Dic83⋊C4—22no—
210C22⋊Dic83—21no—
211(C2×D83)⋊C4—22no—
212(C2×D83)⋊C4—24no—
213C4×Dic83—22no—
214C4×Dic83—24no—
215C4×Dic83—21no—
216#41×C83—22no—
217#41×C83C2×Q814yes—
218C2×Q8×C83—24no—
219C2×Q8×C83C2×D414yes—
220C2×Q8×C83C2411yescharacteristic
221C2×D4×C83—21no—
222C2×D4×C83C2411yescharacteristic
223C22×C4×C83—21no—
224C22×C4×C83—22no—
225C22×C4×C83—22no—
226C22×C4×C83—24no—
227C22×C4×C83C2×D412yes—
228C22×C4×C83C2411yescharacteristic
229C4⋊C4×C83—24no—
230C4⋊C4×C83—22no—
231C4⋊C4×C83—24no—
232C4⋊C4×C83—24no—
233C4⋊C4×C83—24no—
234C4⋊C4×C83—24no—
235C4⋊C4×C83—22no—
236C4⋊C4×C83—24no—
237C4⋊C4×C83C2×D414yes—
238C4⋊C4×C83C2×D414yes—
239C4⋊C4×C83C2×D414yes—
240C4⋊C4×C83C2×D412yes—
241C4⋊C4×C83C2×D412yes—
242C4⋊C4×C83C2414yes—
243C4⋊C4×C83C2412yes—
244C22⋊C4×C83—21no—
245C22⋊C4×C83—22no—
246C22⋊C4×C83—24no—
247C22⋊C4×C83C2×D414yes—
248C42×C83—21no—
249C42×C83—22no—
250C42×C83—24no—
251C42×C83C2×D411yescharacteristic
252C42×C83C2×D411yescharacteristic
253C42×C83C2412yes—
254C42×C83C2414yes—
255C23×C83—21no—
256C23×C83C22×D411yescharacteristic
257C22×D83—21no—
258Q8×C83—21no—
259Q8×C83—44no—
260Q8×C83—84no—
261Q8×C83C22×Q812yes—
262D4×C83—24no—
263D4×C83—21no—
264D4×C83C22×Q812yes—
265C2×C4×C83—21no—
266C2×C4×C83—24no—
267C2×C4×C83—22no—
268C2×C4×C83—24no—
269C2×C4×C83—42no—
270C2×C4×C83—82no—
271C2×C4×C83—84no—
272C2×C4×C83—82no—
273C2×C4×C83C22×D412yes—
274C2×C4×C83C22×D414yes—
275C2×C4×C83C22×D412yes—
276C2×C4×C83C22×D414yes—
277C2×C4×C83C22×D411yescharacteristic
278C2×C4×C83C22×Q812yes—
279C2×C4×C832−1+412yes—
280C2×C4×C83C2511yescharacteristic
281C22⋊D83—24no—
282C2×Dic83—21no—
283C2×Dic83—22no—
284C2×Dic83—24no—
285C2×Dic83—82no—
286D332—21no—
287D332C22×Q812yes—
288C4×D83—24no—
289C4×D83—82no—
290C4×D83—84no—
291Dic166—21no—
292Dic166—84no—
293Dic166C22×Q812yes—
294C83⋊C8—22no—
295C83⋊C8—24no—
296C83⋊C8C22×Q812yes—
297C83⋊C8C22×Q812yes—
298C22×C83—22no—
299C22×C83—22no—
300C22×C83—42no—
301C22×C83D4×Q812yes—
302C22×C83C23×D411yescharacteristic
303C2×D83—22no—
304C2×D83—42no—
305C4×C83—24no—
306C4×C83—22no—
307C4×C83—24no—
308C4×C83—22no—
309C4×C83—22no—
310C4×C83—44no—
311C4×C83—42no—
312C4×C83D4×Q812yes—
313C4×C83C23×Q811yescharacteristic
314C4×C83C2×2−1+411yescharacteristic
315Dic83—22no—
316Dic83—42no—
317Dic83—44no—
318#61376—831nomaximal, Sylow
319C2×C83—22no—
320C2×C83—42no—
321C2×C83C2×D4×Q811yescharacteristic
322C2×C83#358412yes—
323D83—42no—
324#3504—831no—
325#3504D8311yescharacteristic, complemented
326#3388—834no—
327#3388—834no—
328#3383—834no—
329#3380—834no—
330#3164—832no—
331C2×#499—831no—
332#3084—832no—
333#3084—832no—
334#3080—831no—
335#3080—831no—
336#2960—831no—
337#2951—832no—
338C4×#135—831no—
339C83#6137611yescharacteristic, complemented, Sylow
340C2×#135—831no—
341#557—834no—
342#557C2×D8314yes—
343#554—24no—
344#554—834no—
345#554—1664no—
346#554C2×D8314yes—
347#544—22no—
348#544—832no—
349#544—1662no—
350#544C2×D8312yes—
351#533—21no—
352#533—1661no—
353#533C2×D8311yescharacteristic
354C2×#125—21no—
355C2×#125—831no—
356C2×#125—1661no—
357C2×#125C2×D8311yescharacteristic
358#520—832no—
359#520C2×D8312yes—
360C4×#41—831no—
361C4×#41C2×D8311yescharacteristic
362#501—834no—
363#501—834no—
364#500—834no—
365#499—834no—
366#499—834no—
367#499—834no—
368#479—834no—
369#479—834no—
370#479—834no—
371#479—834no—
372#478—834no—
373#478—834no—
374#478—834no—
375#478—834no—
376#476—834no—
377#476—834no—
378#475—834no—
379#475—834no—
380#446—834no—
381C8×Q8—834no—
382#440—834no—
383#440—834no—
384C4×Q16—832no—
385C4×Q16—834no—
386C4×SD16—832no—
387C4×SD16—832no—
388C4×D8—832no—
389C4×D8—834no—
390#428—832no—
391#428—832no—
392#427—831no—
393C2×#107—832no—
394C2×#106—832no—
395#424—832no—
396#423—831no—
397#417—832no—
398#417—832no—
399#416—831no—
400#416—831no—
401#405—832no—
402C2×C4×C8—831no—
403C2×#41—831no—
404C2×#41C22×D8311yescharacteristic
405#135—838no—
406C2×Q16—831no—
407C2×SD16—831no—
408C2×SD16—831no—
409C2×D8—831no—
410C2×M16—832no—
411C22×C8—831no—
412C22×C8—831no—
413C4⋊Q8—24no—
414C4⋊Q8—22no—
415C4⋊Q8—834no—
416C4⋊Q8—1664no—
417C4⋊Q8—1662no—
418C4⋊Q8C22⋊D8312yes—
419C4⋊Q8C22×D8314yes—
420#125—22no—
421#125—24no—
422#125—24no—
423#125—21no—
424#125—21no—
425#125—834no—
426#125—834no—
427#125—1664no—
428#125—1662no—
429#125—1664no—
430#125—1661no—
431#125—1661no—
432#125C22⋊D8311yescharacteristic
433#125C22⋊D8311yescharacteristic
434#125C22×D8314yes—
435#125C22×D8314yes—
436C22⋊Q8—24no—
437C22⋊Q8—24no—
438C22⋊Q8—24no—
439C22⋊Q8—24no—
440C22⋊Q8—1664no—
441C22⋊Q8—1664no—
442C22⋊Q8—1664no—
443C22⋊Q8—1664no—
444C22⋊Q8C22⋊D8314yes—
445C4×Q8—24no—
446C4×Q8—832no—
447C4×Q8—834no—
448C4×Q8—1664no—
449C4×Q8C22×D8312yes—
450C4×Q8C22×D8314yes—
451C4×Q8C22×D8314yes—
452C4×D4—22no—
453C4×D4—24no—
454C4×D4—832no—
455C4×D4—834no—
456C4×D4—1662no—
457C4×D4—1664no—
458C4×D4C22×D8312yes—
459C4×D4C22×D8314yes—
460#117—22no—
461#117—21no—
462#117—831no—
463#117—1662no—
464#117—1661no—
465#117C22×D8311yescharacteristic
466#117C22×D8312yes—
467C2×C4⋊C4—22no—
468C2×C4⋊C4—22no—
469C2×C4⋊C4—24no—
470C2×C4⋊C4—832no—
471C2×C4⋊C4—1662no—
472C2×C4⋊C4—1662no—
473C2×C4⋊C4—1664no—
474C2×C4⋊C4C22×D8312yes—
475C2×C42—21no—
476C2×C42—1661no—
477C2×C42C22×D8311yescharacteristic
478#107—834no—
479#107—834no—
480#107—832no—
481#107—834no—
482#107—834no—
483#106—834no—
484#106—834no—
485#106—832no—
486#106—834no—
487#106—834no—
488C4⋊C8—834no—
489C4⋊C8—834no—
490C4⋊C8—834no—
491#103—832no—
492#103—834no—
493#103—832no—
494#103—834no—
495#102—832no—
496#102—834no—
497#102—832no—
498#102—834no—
499#97—834no—
500C4×C8—832no—
501C4×C8—832no—
502C4×C8—834no—
503#41—22no—
504#41—834no—
505#41—1662no—
506#41Q8×D8314yes—
507C2×Q8—24no—
508C2×Q8—831no—
509C2×Q8—1664no—
510C2×Q8C2×C22⋊D8314yes—
511C2×Q8C23×D8311yescharacteristic
512C2×D4—21no—
513C2×D4—831no—
514C2×D4—1661no—
515C2×D4C23×D8311yescharacteristic
516C22×C4—21no—
517C22×C4—22no—
518C22×C4—22no—
519C22×C4—24no—
520C22×C4—1662no—
521C22×C4—1662no—
522C22×C4—1664no—
523C22×C4—1661no—
524C22×C4C2×C22⋊D8312yes—
525C22×C4C23×D8311yescharacteristic
526Q16—834no—
527SD16—834no—
528SD16—834no—
529D8—834no—
530M16—1664no—
531C2×C8—834no—
532C2×C8—831no—
533C2×C8—831no—
534C2×C8—832no—
535C2×C8—834no—
536C2×C8—1662no—
537C4⋊C4—24no—
538C4⋊C4—22no—
539C4⋊C4—24no—
540C4⋊C4—24no—
541C4⋊C4—24no—
542C4⋊C4—24no—
543C4⋊C4—22no—
544C4⋊C4—24no—
545C4⋊C4—834no—
546C4⋊C4—832no—
547C4⋊C4—1662no—
548C4⋊C4—1664no—
549C4⋊C4—1664no—
550C4⋊C4—1664no—
551C4⋊C4—1664no—
552C4⋊C4—1664no—
553C4⋊C4—1662no—
554C4⋊C4—1664no—
555C4⋊C4C2×C22⋊D8312yes—
556C4⋊C4C2×C22⋊D8314yes—
557C4⋊C4C2×C22⋊D8312yes—
558C4⋊C4C2×C22⋊D8314yes—
559C4⋊C4C2×C22⋊D8314yes—
560C4⋊C4C23×D8314yes—
561C4⋊C4C23×D8312yes—
562C22⋊C4—21no—
563C22⋊C4—22no—
564C22⋊C4—24no—
565C22⋊C4—1661no—
566C22⋊C4—1662no—
567C22⋊C4—1664no—
568C22⋊C4C2×C22⋊D8314yes—
569C42—21no—
570C42—22no—
571C42—24no—
572C42—1661no—
573C42—1662no—
574C42—1664no—
575C42C2×C22⋊D8311yescharacteristic
576C42C2×C22⋊D8311yescharacteristic
577C42C23×D8312yes—
578C42C23×D8314yes—
579C23—21no—
580C23—1661no—
581C23C22×C22⋊D8311yescharacteristic
582Q8—21no—
583Q8—44no—
584Q8—84no—
585Q8—832no—
586Q8—6644no—
587Q8—1661no—
588Q8C2×Q8×D8312yes—
589D4—24no—
590D4—21no—
591D4—832no—
592D4—1664no—
593D4—1661no—
594D4C2×Q8×D8312yes—
595C2×C4—21no—
596C2×C4—24no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—42no—
600C2×C4—82no—
601C2×C4—84no—
602C2×C4—82no—
603C2×C4—6642no—
604C2×C4—6644no—
605C2×C4—6642no—
606C2×C4—1662no—
607C2×C4—1664no—
608C2×C4—1661no—
609C2×C4—1664no—
610C2×C4C22×C22⋊D8312yes—
611C2×C4C22×C22⋊D8314yes—
612C2×C4C22×C22⋊D8312yes—
613C2×C4C22×C22⋊D8314yes—
614C2×C4C22×C22⋊D8311yescharacteristic
615C2×C4C2×Q8×D8312yes—
616C2×C4#1458184012yes—
617C2×C4C24×D8311yescharacteristic
618C8—832no—
619C8—832no—
620C8—1662no—
621C8—1664no—
622C22—22no—
623C22—22no—
624C22—42no—
625C22—3322no—
626C22—1662no—
627C22Q8×C22⋊D8312yes—
628C22C23×C22⋊D8311yescharacteristic
629C4—22no—
630C4—22no—
631C4—22no—
632C4—24no—
633C4—24no—
634C4—44no—
635C4—42no—
636C4—3324no—
637C4—3322no—
638C4—1662no—
639C4Q8×C22⋊D8312yes—
640C4C22×Q8×D8311yescharacteristic
641C4C2×#1458184011yescharacteristic
642C2—22no—
643C2—42no—
644C2—3322no—
645C2C2×Q8×C22⋊D8311yescharacteristic
646C2#2492223412yes—
subgroupG/H
centerC22C23×C22⋊D83
commutator, cosocleC2×C4×C83C25
FrattiniC2×C4C24×D83
FittingC83×#3504C2
socleC22×C83C23×D4
prime radicalC2×#14581812C22
hypercenter#3504D83
hypocenterC83#61376
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ #533 ◁ #3504

Complements

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

NH ≅ G/Ntype
1. #24921284C2semidirect
2. #24920408C2semidirect
3. #24920402C2semidirect
4. #24920350C2semidirect
5. #24920326C2semidirect
6. #24917997C2semidirect
7. C2×#19128100C2semidirect
8. #24917325C2semidirect
9. #24917321C2semidirect
10. #24917285C2semidirect
11. #24917281C2semidirect
12. #24916398C2semidirect
15. C83×#3504C2semidirect
80. #14581812Q8semidirect
325. #3504D83semidirect
339. C83#61376semidirect

Maximal subgroups

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

subgroupindex
#249212842
#249204082
#249204022
#249203502
#249203262
#249179972
C2×#191281002
#249173252
#249173212
#249172852
#249172812
#249163982
#249163362
C4×#145818122
C83×#35042
#6137683