Finite Group Databasebeta

← #27106721

Subgroups — #27106721

Order 12032.

1648 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257799C211yesmaximal, characteristic
2#20256320C212yesmaximal, complemented
3#20256305C214yesmaximal, complemented
4Q8×#10625337C212yesmaximal, complemented
5#20256230C214yesmaximal, complemented
6#20256202C212yesmaximal, complemented
7#20256084C212yesmaximal, complemented
8#20254671C214yesmaximal, complemented
9#20254602C211yesmaximal, characteristic, complemented
10#20254524C211yesmaximal, characteristic
11#20253617C214yesmaximal, complemented
12#20253510C211yesmaximal, characteristic
13#20253427C211yesmaximal, characteristic, complemented
14#20252483C211yesmaximal, characteristic, complemented
15C47×#3502C211yesmaximal, characteristic, complemented
16#15689877—22no—
17#15689858C2212yes—
18Q8×Dic94C2212yes—
19#15689847—21no—
20#15689846C2212yes—
21Q8×D188C2212yes—
22#15689788—21no—
23#15689746—22no—
24#15689744C2214yes—
25#15689696—22no—
26#15689680C2211yescharacteristic
27#15689673C2211yescharacteristic
28#15689533C2212yes—
29#15689533C2214yes—
30#15689526C2211yescharacteristic
31#15689526C2214yes—
32#15689519C2211yescharacteristic
33#15689493C2212yes—
34#15689485C2214yes—
35#15689480C2214yes—
36#15689471C2211yescharacteristic
37#15689471C2211yescharacteristic
38#15689464C2214yes—
39#15689458C2212yes—
40#15689447C2214yes—
41#15689436C2212yes—
42#15689427C2214yes—
43#15689420C2214yes—
44#15689420C2214yes—
45#15689418C2214yes—
46#15689418C2212yes—
47#15689418C2212yes—
48#15689415C2212yes—
49#15689415C2214yes—
50#15689408C2214yes—
51#15689405C2212yes—
52#15689400C2214yes—
53#15689345C2214yes—
54#15689345C2214yes—
55#15689333C2212yes—
56#15689323C2212yes—
57#15689279C2212yes—
58#15689262C2212yes—
59#15689254C2214yes—
60#15689254C2212yes—
61Q8×C47⋊C8C2212yes—
62#15689228C2214yes—
63#15689221C2214yes—
64#15689221C2214yes—
65C4×#10625337C2212yes—
66C4×#10625337C2214yes—
67C4×#10625337C2212yes—
68C4×#10625336C2214yes—
69#15689184C2214yes—
70#15689180C2211yescharacteristic
71#15689176C2211yescharacteristic
72Q82×C47—21no—
73Q82×C47C2212yes—
74C47×#557—21no—
75C47×#557—22no—
76C47×#557C2214yes—
77C47×#557C2212yes—
78C47×#544—22no—
79C47×#544C2212yes—
80C47×#533—22no—
81C47×#533C2211yescharacteristic
82C47×#533C2211yescharacteristic
83C47×#519C2212yes—
84C2×C4×Q8×C47C2211yescharacteristic
85C2×#10625350—21no—
86C2×Q8×D47—22no—
87C2×#10625337C2312yes—
88C2×#10625337C2312yes—
89C2×#10625336C2314yes—
90#13094024C2311yescharacteristic
91#13094024C2314yes—
92#13094024C2311yescharacteristic
93#13094024C2311yescharacteristic
94#13094023—22no—
95#13094023—24no—
96#13094021C2311yescharacteristic
97#13094019—24no—
98#13094019—24no—
99#13094016—21no—
100#13094013—22no—
101#13094012—21no—
102#13094012C2314yes—
103#13094010—24no—
104#13094009—24no—
105#13094006—24no—
106#13094004—24no—
107#13094004—22no—
108#13094003—21no—
109#13094003C2312yes—
110#13094002—24no—
111#13094000—22no—
112#13093998—21no—
113#13093998C2312yes—
114Q8×Dic47—22no—
115C4×Dic94—21no—
116C4×Dic94C2312yes—
117C4×Dic94C2314yes—
118C4×D188—21no—
119C4×D188C2312yes—
120C4×D188C2314yes—
121#13093986—22no—
122#13093983—21no—
123#13093982—22no—
124#13093957C2314yes—
125#13093957C2312yes—
126#13093957C2312yes—
127#13093957C2314yes—
128#13093957C2314yes—
129#13093955C2314yes—
130#13093955C2314yes—
131C4⋊(C47⋊C8)C2312yes—
132C4⋊(C47⋊C8)C2314yes—
133C4⋊(C47⋊C8)C2312yes—
134C4⋊(C47⋊C8)C2314yes—
135#13093948C2312yes—
136#13093948C2314yes—
137#13093948C2312yes—
138#13093947C2312yes—
139#13093947C2312yes—
140#13093947C2314yes—
141#13093947C2314yes—
142#13093944C2314yes—
143#13093944C2314yes—
144#13093944C2314yes—
145C22⋊(C47⋊C8)C2312yes—
146C22⋊(C47⋊C8)C2312yes—
147#13093935C2312yes—
148#13093935C2314yes—
149C4×C47⋊C8C2311yescharacteristic
150C4×C47⋊C8C2311yescharacteristic
151C4×C47⋊C8C2314yes—
152C22×Q8×C47C2311yescharacteristic
153C4⋊Q8×C47—22no—
154C4⋊Q8×C47—24no—
155C4⋊Q8×C47—22no—
156C4⋊Q8×C47—24no—
157C4⋊Q8×C47—22no—
158C4⋊Q8×C47D411yescharacteristic
159C4⋊Q8×C47D411yescharacteristic
160C4⋊Q8×C47C2314yes—
161C4⋊Q8×C47C2311yescharacteristic
162C4⋊Q8×C47C2311yescharacteristic
163#125×C47—24no—
164#125×C47—24no—
165#125×C47—24no—
166#125×C47—22no—
167#125×C47D412yes—
168#125×C47C2314yes—
169#125×C47C2312yes—
170C22⋊Q8×C47—21no—
171C22⋊Q8×C47—21no—
172C22⋊Q8×C47—24no—
173C22⋊Q8×C47D412yes—
174C22⋊Q8×C47D412yes—
175C4×Q8×C47—22no—
176C4×Q8×C47—22no—
177C4×Q8×C47—22no—
178C4×Q8×C47—24no—
179C4×Q8×C47—22no—
180C4×Q8×C47—21no—
181C4×Q8×C47—21no—
182C4×Q8×C47—21no—
183C4×Q8×C47—21no—
184C4×Q8×C47C2312yes—
185C4×Q8×C47C2314yes—
186C4×Q8×C47C2312yes—
187C4×Q8×C47C2314yes—
188C4×Q8×C47C2312yes—
189C4×Q8×C47C2312yes—
190#117×C47—24no—
191#117×C47C2312yes—
192#117×C47C2312yes—
193C2×C4⋊C4×C47—21no—
194C2×C4⋊C4×C47C2311yescharacteristic
195C2×C42×C47—21no—
196C2×C42×C47C2311yescharacteristic
197C2×Q8×C47—24no—
198C2×Q8×C47—21no—
199C2×Q8×C47—21no—
200C2×Q8×C47—21no—
201C2×Q8×C47—24no—
202C2×Q8×C47C2×D412yes—
203C2×Q8×C47C2×D412yes—
204C2×Q8×C47C2412yes—
205C22×C4×C47—21no—
206C22×C4×C47—21no—
207C22×C4×C47C2×D411yescharacteristic
208C22×C4×C47C2×D411yescharacteristic
209C22×C4×C47C2411yescharacteristic
210C2×C22⋊D47—22no—
211Q8×D47—84no—
212#10625350—82no—
213C2×D188—21no—
214C2×D188—21no—
215C2×D188C2412yes—
216C2×C4×D47—22no—
217C2×C4×D47—24no—
218C2×Dic94—21no—
219C2×Dic94—24no—
220C2×Dic94—21no—
221C2×Dic94C2412yes—
222C4⋊C4×C47—22no—
223C4⋊C4×C47—22no—
224C4⋊C4×C47—21no—
225C4⋊C4×C47—24no—
226C4⋊C4×C47—24no—
227C4⋊C4×C47—22no—
228C4⋊C4×C47—21no—
229C4⋊C4×C47—21no—
230C4⋊C4×C47—24no—
231C4⋊C4×C47—24no—
232C4⋊C4×C47—22no—
233C4⋊C4×C47—24no—
234C4⋊C4×C47C2×D412yes—
235C4⋊C4×C47C2×D412yes—
236C4⋊C4×C47C2×D414yes—
237C4⋊C4×C47C2×D412yes—
238C4⋊C4×C47C2×D412yes—
239C4⋊C4×C47C2×D414yes—
240C4⋊C4×C47C2412yes—
241C4⋊C4×C47C2412yes—
242C4⋊C4×C47C2412yes—
243C22⋊C4×C47—22no—
244C22⋊C4×C47C2×D414yes—
245C42×C47—22no—
246C42×C47—22no—
247C42×C47—24no—
248C42×C47—24no—
249C42×C47—22no—
250C42×C47C2×D411yescharacteristic
251C42×C47C2×D411yescharacteristic
252C42×C47C2411yescharacteristic
253C42×C47C2411yescharacteristic
254C42×C47C2414yes—
255C22⋊Dic47—22no—
256#10625337—24no—
257#10625337—24no—
258#10625336—28no—
259(C2×D47)⋊C4—24no—
260(C2×D47)⋊C4—24no—
261(C2×D47)⋊C4—22no—
262(C2×D47)⋊C4—22no—
263(C2×D47)⋊C4—24no—
264C4⋊Dic47—21no—
265C4⋊Dic47—21no—
266C4⋊Dic47—21no—
267C4⋊Dic47—21no—
268C4⋊Dic47—24no—
269C4⋊Dic47C2412yes—
270C4⋊Dic47C2412yes—
271Dic47⋊C4—22no—
272Dic47⋊C4—24no—
273Dic47⋊C4—22no—
274C4×Dic47—24no—
275C4×Dic47—22no—
276C2×C47⋊C8C2412yes—
277C2×C47⋊C8C2412yes—
278C2×C47⋊C8C2414yes—
279C23×C47C22×D411yescharacteristic
280C22×D47—22no—
281Q8×C47—24no—
282Q8×C47—24no—
283Q8×C47—22no—
284Q8×C47—42no—
285Q8×C47—42no—
286Q8×C47—84no—
287Q8×C47—81no—
288Q8×C472−1+414yes—
289C2×C4×C47—24no—
290C2×C4×C47—22no—
291C2×C4×C47—22no—
292C2×C4×C47—22no—
293C2×C4×C47—22no—
294C2×C4×C47—22no—
295C2×C4×C47—22no—
296C2×C4×C47—41no—
297C2×C4×C47—81no—
298C2×C4×C47C22×D414yes—
299C2×C4×C47C22×D411yescharacteristic
300C2×C4×C47C22×D411yescharacteristic
301C2×C4×C47C22×D414yes—
302C2×C4×C47C22×D411yescharacteristic
303C2×C4×C47C22×D411yescharacteristic
304C2×C4×C47C22×D411yescharacteristic
305C2×C4×C47C2511yescharacteristic
306C22⋊D47—82no—
307C2×Dic47—22no—
308C2×Dic47—22no—
309C2×Dic47—22no—
310D188—24no—
311D188—42no—
312D188—81no—
313C4×D47—44no—
314C4×D47—84no—
315Dic94—22no—
316Dic94—22no—
317Dic94—42no—
318Dic94—81no—
319Dic94—84no—
320C47⋊C8—24no—
321C47⋊C8—22no—
322C47⋊C8—22no—
323#61854—471nomaximal, Sylow
324C22×C47—21no—
325C22×C47—21no—
326C22×C47—21no—
327C22×C47C23×D411yescharacteristic
328C2×D47—42no—
329C2×D47—42no—
330C2×D47—42no—
331C4×C47—21no—
332C4×C47—24no—
333C4×C47—21no—
334C4×C47—24no—
335C4×C47—21no—
336C4×C47—21no—
337C4×C47—24no—
338C4×C47—42no—
339C4×C47—42no—
340C4×C47—42no—
341C4×C47#57812yes—
342C4×C47C2×2−1+412yes—
343Dic47—42no—
344Dic47—42no—
345Dic47—42no—
346#3550—471no—
347#3502D4711yescharacteristic, complemented
348#3387—472no—
349#3386—474no—
350Q8×Q16—472no—
351#3377—474no—
352#3375—472no—
353#3365—472no—
354#3243—474no—
355#3237—471no—
356#3228—471no—
357#3121—474no—
358#3112—471no—
359#3104—471no—
360#2985—471no—
361C2×C47—41no—
362C2×C47#348611yescharacteristic
363C2×C47C2×#57811yescharacteristic
364C2×C47#358511yescharacteristic
365D47—82no—
366#563—942no—
367#559—472no—
368Q82—21no—
369Q82—472no—
370Q82C2×D4712yes—
371#557—21no—
372#557—22no—
373#557C2×D4712yes—
374#557C2×D4714yes—
375#556—472no—
376#556—941no—
377D4×Q8—472no—
378D4×Q8—941no—
379#544—22no—
380#544C2×D4712yes—
381#542—474no—
382#542—942no—
383#536—942no—
384#533—22no—
385#533—471no—
386#533C2×D4711yescharacteristic
387#533C2×D4711yescharacteristic
388#532—471no—
389#519C2×D4712yes—
390C2×C4×Q8C2×D4711yescharacteristic
391#501—472no—
392#501—474no—
393#500—471no—
394#500—474no—
395#498—471no—
396#491—472no—
397#489—474no—
398#488—474no—
399#486—471no—
400#486—471no—
401#485—474no—
402#484—472no—
403#482—474no—
404#480—472no—
405#479—474no—
406#478—474no—
407#478—474no—
408#477—472no—
409#477—474no—
410#477—472no—
411#477—472no—
412#477—474no—
413#475—472no—
414#475—474no—
415#474—474no—
416#464—474no—
417#464—474no—
418#462—472no—
419#460—472no—
420#451—472no—
421#448—472no—
422#446—472no—
423#446—474no—
424C8×Q8—472no—
425#441—474no—
426#440—474no—
427#440—474no—
428C4×Q16—472no—
429C4×Q16—472no—
430C4×Q16—474no—
431C4×SD16—474no—
432#433—474no—
433#432—471no—
434#431—471no—
435C47#6185411yescharacteristic, complemented, Sylow
436C2×#41—941no—
437C22×Q8—942no—
438C22×Q8C22×D4711yescharacteristic
439C2×Q16—472no—
440C2×Q16—472no—
441C2×SD16—474no—
442C4⋊Q8—22no—
443C4⋊Q8—24no—
444C4⋊Q8—22no—
445C4⋊Q8—24no—
446C4⋊Q8—22no—
447C4⋊Q8—471no—
448C4⋊Q8—471no—
449C4⋊Q8—474no—
450C4⋊Q8—471no—
451C4⋊Q8—944no—
452C4⋊Q8—942no—
453C4⋊Q8C22⋊D4711yescharacteristic
454C4⋊Q8C22⋊D4711yescharacteristic
455C4⋊Q8C22×D4711yescharacteristic
456C4⋊Q8C22×D4711yescharacteristic
457C4⋊Q8C22×D4714yes—
458#127—471no—
459#126—944no—
460#126—944no—
461#125—24no—
462#125—24no—
463#125—24no—
464#125—22no—
465#125—941no—
466#125C22⋊D4712yes—
467#125C22×D4712yes—
468#125C22×D4714yes—
469#124—474no—
470#124—942no—
471#124—944no—
472#124—941no—
473#123—944no—
474#123—944no—
475C22⋊Q8—21no—
476C22⋊Q8—21no—
477C22⋊Q8—24no—
478C22⋊Q8—472no—
479C22⋊Q8—942no—
480C22⋊Q8—941no—
481C22⋊Q8—942no—
482C22⋊Q8—944no—
483C22⋊Q8—944no—
484C22⋊Q8C22⋊D4712yes—
485C22⋊Q8C22⋊D4712yes—
486#121—472no—
487#121—941no—
488C4×Q8—22no—
489C4×Q8—22no—
490C4×Q8—22no—
491C4×Q8—24no—
492C4×Q8—22no—
493C4×Q8—21no—
494C4×Q8—21no—
495C4×Q8—21no—
496C4×Q8—21no—
497C4×Q8—472no—
498C4×Q8—474no—
499C4×Q8—942no—
500C4×Q8—941no—
501C4×Q8C22×D4712yes—
502C4×Q8C22×D4712yes—
503C4×Q8C22×D4714yes—
504C4×Q8C22×D4714yes—
505C4×Q8C22×D4712yes—
506C4×Q8C22×D4712yes—
507C4×D4—472no—
508C4×D4—474no—
509C4×D4—942no—
510C4×D4—941no—
511#117—24no—
512#117—941no—
513#117—942no—
514#117C22×D4712yes—
515#117C22×D4712yes—
516C2×C4⋊C4—21no—
517C2×C4⋊C4C22×D4711yescharacteristic
518C2×C42—21no—
519C2×C42C22×D4711yescharacteristic
520#107—472no—
521#107—474no—
522#107—472no—
523#107—474no—
524#107—474no—
525#106—474no—
526#106—474no—
527C4⋊C8—472no—
528C4⋊C8—472no—
529C4⋊C8—474no—
530C4⋊C8—474no—
531#103—472no—
532#103—472no—
533#103—474no—
534#103—472no—
535#103—472no—
536#103—474no—
537#103—474no—
538#102—474no—
539#102—474no—
540#102—474no—
541C22⋊C8—472no—
542C22⋊C8—472no—
543#97—472no—
544#97—474no—
545C4×C8—471no—
546C4×C8—471no—
547C4×C8—474no—
548#41—3762no—
549C2×Q8—24no—
550C2×Q8—21no—
551C2×Q8—21no—
552C2×Q8—21no—
553C2×Q8—24no—
554C2×Q8—472no—
555C2×Q8—944no—
556C2×Q8—941no—
557C2×Q8—941no—
558C2×Q8—3764no—
559C2×Q8C2×C22⋊D4712yes—
560C2×Q8C2×C22⋊D4712yes—
561C2×Q8C23×D4712yes—
562C2×D4—472no—
563C2×D4—942no—
564C2×D4—941no—
565C2×D4—941no—
566C22×C4—21no—
567C22×C4—21no—
568C22×C4—944no—
569C22×C4—942no—
570C22×C4C2×C22⋊D4711yescharacteristic
571C22×C4C2×C22⋊D4711yescharacteristic
572C22×C4C23×D4711yescharacteristic
573Q16—944no—
574Q16—944no—
575SD16—948no—
576C2×C8—472no—
577C2×C8—472no—
578C2×C8—474no—
579C4⋊C4—22no—
580C4⋊C4—22no—
581C4⋊C4—21no—
582C4⋊C4—24no—
583C4⋊C4—24no—
584C4⋊C4—22no—
585C4⋊C4—21no—
586C4⋊C4—21no—
587C4⋊C4—24no—
588C4⋊C4—24no—
589C4⋊C4—22no—
590C4⋊C4—24no—
591C4⋊C4—472no—
592C4⋊C4—472no—
593C4⋊C4—944no—
594C4⋊C4—942no—
595C4⋊C4—942no—
596C4⋊C4—941no—
597C4⋊C4—941no—
598C4⋊C4—941no—
599C4⋊C4—941no—
600C4⋊C4—944no—
601C4⋊C4C2×C22⋊D4712yes—
602C4⋊C4C2×C22⋊D4712yes—
603C4⋊C4C2×C22⋊D4712yes—
604C4⋊C4C2×C22⋊D4712yes—
605C4⋊C4C2×C22⋊D4714yes—
606C4⋊C4C2×C22⋊D4714yes—
607C4⋊C4C23×D4712yes—
608C4⋊C4C23×D4712yes—
609C4⋊C4C23×D4712yes—
610C22⋊C4—22no—
611C22⋊C4—942no—
612C22⋊C4—944no—
613C22⋊C4—942no—
614C22⋊C4—944no—
615C22⋊C4—942no—
616C22⋊C4—944no—
617C22⋊C4C2×C22⋊D4714yes—
618C42—22no—
619C42—22no—
620C42—24no—
621C42—24no—
622C42—22no—
623C42—942no—
624C42—944no—
625C42C2×C22⋊D4711yescharacteristic
626C42C2×C22⋊D4711yescharacteristic
627C42C23×D4711yescharacteristic
628C42C23×D4714yes—
629C42C23×D4711yescharacteristic
630C23—942no—
631C23C22×C22⋊D4711yescharacteristic
632Q8—24no—
633Q8—24no—
634Q8—22no—
635Q8—42no—
636Q8—42no—
637Q8—81no—
638Q8—84no—
639Q8—942no—
640Q8—942no—
641Q8—3764no—
642Q8—3761no—
643Q8—1882no—
644Q8#1309407214yes—
645D4—944no—
646D4—3761no—
647D4—3762no—
648D4—1882no—
649C2×C4—24no—
650C2×C4—22no—
651C2×C4—22no—
652C2×C4—22no—
653C2×C4—22no—
654C2×C4—22no—
655C2×C4—22no—
656C2×C4—41no—
657C2×C4—81no—
658C2×C4—942no—
659C2×C4—942no—
660C2×C4—942no—
661C2×C4—3764no—
662C2×C4—1884no—
663C2×C4C22×C22⋊D4711yescharacteristic
664C2×C4C22×C22⋊D4711yescharacteristic
665C2×C4C22×C22⋊D4714yes—
666C2×C4C22×C22⋊D4714yes—
667C2×C4C22×C22⋊D4711yescharacteristic
668C2×C4C22×C22⋊D4711yescharacteristic
669C2×C4C22×C22⋊D4711yescharacteristic
670C2×C4C24×D4711yescharacteristic
671C8—944no—
672C8—942no—
673C8—942no—
674C22—21no—
675C22—21no—
676C22—21no—
677C22—1882no—
678C22—1882no—
679C22—1882no—
680C22C23×C22⋊D4711yescharacteristic
681C4—24no—
682C4—21no—
683C4—21no—
684C4—24no—
685C4—21no—
686C4—21no—
687C4—24no—
688C4—42no—
689C4—42no—
690C4—42no—
691C4—1882no—
692C4—1882no—
693C4—1882no—
694C4#1568993812yes—
695C4C2×#1309407212yes—
696C2—41no—
697C2—3762no—
698C2#2025703911yescharacteristic
699C2C2×#1568993811yescharacteristic
700C2#2025816711yescharacteristic
subgroupG/H
centerC22C23×C22⋊D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC47×#3502C2
socleC22×C47C23×D4
prime radical#15689673C22
hypercenter#3502D47
hypocenterC47#61854
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ #533 ◁ #3502

Complements

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

NH ≅ G/Ntype
2. #20256320C2semidirect
3. #20256305C2semidirect
4. Q8×#10625337C2semidirect
5. #20256230C2semidirect
6. #20256202C2semidirect
7. #20256084C2semidirect
8. #20254671C2semidirect
9. #20254602C2semidirect
11. #20253617C2semidirect
13. #20253427C2semidirect
14. #20252483C2semidirect
15. C47×#3502C2semidirect
347. #3502D47semidirect
435. C47#61854semidirect

Maximal subgroups

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

subgroupindex
#202577992
#202563202
#202563052
Q8×#106253372
#202562302
#202562022
#202560842
#202546712
#202546022
#202545242
#202536172
#202535102
#202534272
#202524832
C47×#35022
#6185447