Finite Group Databasebeta

← #26243630

Subgroups — #26243630

Order 12032.

1250 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689751C211yesmaximal, characteristic, complemented
2C2×#15689125C211yesmaximal, characteristic
3C2×#15689120C211yesmaximal, characteristic
4#20245976C211yesmaximal, characteristic
5#20244540C212yesmaximal, complemented
6#20244535C212yesmaximal, complemented
7#20244530C212yesmaximal, complemented
8#20244525C212yesmaximal, complemented
9#20243108C211yesmaximal, characteristic, complemented
10C2×#15688886C211yesmaximal, characteristic, complemented
11C47×#1729C211yesmaximal, characteristic, complemented
12C22×#10625350C2211yescharacteristic
13#15689751—24no—
14#15689751—24no—
15C2×#13094024C2212yes—
16C2×#13094023—21no—
17C2×#13094016C2212yes—
18C2×#13094015—21no—
19C2×#13094001—22no—
20C2×#13093983C2211yescharacteristic
21C22×C4⋊Dic47C2211yescharacteristic
22C22×C4⋊Dic47C2211yescharacteristic
23#15689125—24no—
24#15689120—24no—
25C2×#13093949C2211yescharacteristic
26C2×#13093949C2212yes—
27C2×#13093949C2211yescharacteristic
28C2×#13093946C2211yescharacteristic
29C2×#13093946C2212yes—
30C2×#13093946C2211yescharacteristic
31#15689029C2212yes—
32#15689025C2212yes—
33#15689000—22no—
34#15688980—21no—
35#15688980—21no—
36#15688886C2212yes—
37#15688886C2212yes—
38#15688886C2212yes—
39#15688886C2212yes—
40C22×M16×C47C2211yescharacteristic
41C23×C8×C47C2211yescharacteristic
42C2×#117×C47C2211yescharacteristic
43C47×#336C2212yes—
44C47×#336C2212yes—
45C2×#10625350—22no—
46C2×#10625350—22no—
47C2×#10625350—22no—
48C2×#10625350—22no—
49C2×#10625350—22no—
50C2×#10625350Q814yescomplemented
51C22×Dic94—22no—
52C22×Dic94C2311yescharacteristic
53C22×D188C2311yescharacteristic
54C22×C22⋊D47—21no—
55C23×Dic47—21no—
56C23×Dic47—21no—
57C22×C4×D47—21no—
58#13094024—24no—
59#13094024—24no—
60#13094023—28no—
61#13094016—24no—
62#13094016—24no—
63#13094015—28no—
64#13094001—216no—
65#13093983—22no—
66#13093983D414yes—
67C2×C4⋊Dic47—22no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—22no—
70C2×C4⋊Dic47—22no—
71C2×C4⋊Dic47—22no—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47D414yes—
74C2×C4⋊Dic47C2312yes—
75C2×C4⋊Dic47C2312yes—
76C2×C4⋊Dic47C2311yescharacteristic
77C2×C4⋊Dic47C2311yescharacteristic
78C2×Dic47⋊C4—22no—
79C2×Dic47⋊C4—22no—
80C2×C22⋊Dic47—21no—
81C2×(C2×D47)⋊C4—22no—
82C2×(C2×D47)⋊C4—22no—
83C2×C4×Dic47—21no—
84#13093949—22no—
85#13093949—22no—
86#13093949—24no—
87#13093946—22no—
88#13093946—22no—
89#13093946—24no—
90C47⋊#95—22no—
91C47⋊#95—22no—
92C23×C4×C47C2311yescharacteristic
93C2×M16×C47—22no—
94C2×M16×C47—22no—
95C2×M16×C47—22no—
96C22×C8×C47—22no—
97C22×C8×C47—22no—
98C22×C8×C47—22no—
99C22×C8×C47C2311yescharacteristic
100C22×C8×C47C2311yescharacteristic
101C22×C8×C47C2312yes—
102#117×C47—22no—
103#117×C47—22no—
104C2×C4⋊C4×C47D411yescharacteristic
105C2×C4⋊C4×C47D411yescharacteristic
106C2×C22⋊C4×C47D412yes—
107C2×C42×C47C2312yes—
108C24×C47C2×D411yescharacteristic
109C23×D47—21no—
110C22×C4×C47C2×D412yes—
111C22×C4×C47C2×D412yes—
112C22×C4×C47C2×D411yescharacteristic
113C22×C4×C47C2×D412yes—
114C22×C4×C47C2×D412yes—
115C22×C4×C47C2×Q812yes—
116C22×C4×C47#4112yes—
117C22×C4×C47#4112yes—
118C22×C4×C47#4112yes—
119C22×C4×C47C2411yescharacteristic
120C2×C22⋊D47—24no—
121C2×C22⋊D47—42no—
122C2×C22⋊D47—42no—
123C22×Dic47—21no—
124C22×Dic47—21no—
125C22×Dic47—21no—
126C22×Dic47—22no—
127C22×Dic47—24no—
128C22×Dic47—22no—
129C22×Dic47—42no—
130C22×Dic47—81no—
131C22×Dic47—81no—
132C22×Dic47—81no—
133C22×Dic47—81no—
134#10625350—28no—
135#10625350—28no—
136#10625350—44no—
137#10625350—44no—
138C2×D188—21no—
139C2×D188—21no—
140C2×D188—21no—
141C2×D188—21no—
142C2×D188—21no—
143C2×D188C2×Q812yes—
144C2×C4×D47—24no—
145C2×C4×D47—42no—
146C2×C4×D47—42no—
147C2×Dic94—21no—
148C2×Dic94—21no—
149C2×Dic94—21no—
150C2×Dic94—21no—
151C2×Dic94—21no—
152C2×Dic94—28no—
153C2×Dic94—82no—
154C2×Dic94—82no—
155C2×Dic94C2×Q812yes—
156M16×C47—44no—
157C2×C8×C47—21no—
158C2×C8×C47—21no—
159C2×C8×C47—22no—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47—21no—
164C2×C8×C47—22no—
165C2×C8×C47—22no—
166C2×C8×C47—44no—
167C4⋊C4×C47—22no—
168C4⋊C4×C47—22no—
169C22⋊C4×C47—24no—
170C42×C47—22no—
171C42×C47—22no—
172C22⋊Dic47—24no—
173(C2×D47)⋊C4—82no—
174(C2×D47)⋊C4—82no—
175C4⋊Dic47—28no—
176C4⋊Dic47—24no—
177C4⋊Dic47—22no—
178C4⋊Dic47—21no—
179C4⋊Dic47—21no—
180C4⋊Dic47—22no—
181C4⋊Dic47—22no—
182C4⋊Dic47—42no—
183C4⋊Dic47C2×D414yes—
184C4⋊Dic47C2×D412yes—
185C4⋊Dic47C2×D412yes—
186Dic47⋊C4—28no—
187Dic47⋊C4—28no—
188C4×Dic47—24no—
189C23×C47—21no—
190C23×C47—21no—
191C23×C47—21no—
192C23×C47—21no—
193C23×C47#12012yes—
194C23×C47C22⋊Q812yes—
195C23×C47#12412yes—
196C23×C47C22×D411yescharacteristic
197C22×D47—22no—
198C22×D47—41no—
199C22×D47—41no—
200C22×D47—41no—
201C22×D47—41no—
202C22×D47—41no—
203C22×D47—41no—
204C2×C4×C47—22no—
205C2×C4×C47—22no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47#12012yes—
211C2×C4×C47C22⋊Q814yes—
212C2×C4×C47C22⋊Q814yes—
213C2×C4×C47C22⋊Q812yes—
214C2×C4×C47#12414yes—
215C2×C4×C47#12414yes—
216C2×C4×C47#12412yes—
217C2×C4×C47C22×D411yescharacteristic
218C2×C4×C47C22×D411yescharacteristic
219C2×C4×C47C22×Q811yescharacteristic
220C2×C4×C47C2×#4111yescharacteristic
221C2×C4×C47C2×#4111yescharacteristic
222C2×C4×C47C2×#4111yescharacteristic
223C22⋊D47—48no—
224C2×Dic47—22no—
225C2×Dic47—22no—
226C2×Dic47—24no—
227C2×Dic47—22no—
228C2×Dic47—41no—
229C2×Dic47—41no—
230C2×Dic47—41no—
231C2×Dic47—42no—
232C2×Dic47—41no—
233C2×Dic47—42no—
234C2×Dic47—41no—
235C2×Dic47—41no—
236C2×Dic47—42no—
237C2×Dic47—42no—
238C2×Dic47—42no—
239C2×Dic47—84no—
240C2×Dic47—84no—
241D188—22no—
242D188—22no—
243D188—41no—
244D188—41no—
245C4×D47—48no—
246Dic94—22no—
247Dic94—22no—
248Dic94—41no—
249Dic94—41no—
250Dic94—88no—
251C8×C47—41no—
252C8×C47—41no—
253C8×C47—42no—
254#23458—471nomaximal, Sylow
255C22×C47—22no—
256C22×C47—22no—
257C22×C47—22no—
258C22×C47—22no—
259C22×C47—22no—
260C22×C47—22no—
261C22×C47#39314yes—
262C22×C47C2×#12011yescharacteristic
263C22×C47C2×C22⋊Q811yescharacteristic
264C22×C47C2×#12411yescharacteristic
265C22×C47#57611yescharacteristic
266C22×C47#57711yescharacteristic
267C22×C47#57711yescharacteristic
268C22×C47#57811yescharacteristic
269C2×D47—42no—
270C2×D47—42no—
271C2×D47—42no—
272C2×D47—81no—
273C2×D47—81no—
274C2×D47—81no—
275C2×D47—81no—
276C4×C47—42no—
277C4×C47—42no—
278C4×C47#39314yes—
279C4×C47C2×C22⋊Q811yescharacteristic
280C4×C47C2×C22⋊Q811yescharacteristic
281C4×C47C2×#12411yescharacteristic
282C4×C47C2×#12411yescharacteristic
283Dic47—42no—
284Dic47—42no—
285Dic47—44no—
286Dic47—42no—
287Dic47—82no—
288C2×#544—471no—
289C2×#417—471no—
290C2×#416—471no—
291#2311—471no—
292#2026—472no—
293#2025—472no—
294#2024—472no—
295#2023—472no—
296#1736—471no—
297C2×#340—471no—
298#1729D4711yescharacteristic, complemented
299C2×C47—24no—
300C2×C47C2×#39311yescharacteristic
301C2×C47#301211yescharacteristic
302C2×C47#301311yescharacteristic
303C2×C47#308311yescharacteristic
304C2×C47#308411yescharacteristic
305C2×C47#313611yescharacteristic
306C2×C47#313711yescharacteristic
307D47—82no—
308C22×#41—471no—
309C22×M16C2×D4711yescharacteristic
310C23×C8C2×D4711yescharacteristic
311#544—944no—
312#544—944no—
313C2×C4⋊Q8—472no—
314C2×C4⋊Q8—941no—
315C2×#125—472no—
316C2×#125—941no—
317C2×C22⋊Q8—942no—
318C2×#117—471no—
319C2×#117C2×D4711yescharacteristic
320C22×C4⋊C4—471no—
321C22×C4⋊C4—471no—
322#417—944no—
323#416—944no—
324C2×#103—471no—
325C2×#103—471no—
326C2×#103—472no—
327C2×#102—471no—
328C2×#102—471no—
329C2×#102—472no—
330#391—472no—
331#390—472no—
332#386—942no—
333#380—941no—
334#380—941no—
335#340—472no—
336#340—472no—
337#340—472no—
338#340—472no—
339#336C2×D4712yes—
340#336C2×D4712yes—
341C47#2345811yescharacteristic, complemented, Sylow
342C2×#41—474no—
343C2×#41—942no—
344C2×#41—942no—
345C2×#41—942no—
346C2×#41—942no—
347C2×#41—942no—
348C22×Q8—471no—
349C22×Q8—942no—
350C22×D4—471no—
351C22×D4—941no—
352C23×C4—941no—
353C23×C4—941no—
354C23×C4—941no—
355C23×C4C22×D4711yescharacteristic
356C2×M16—22no—
357C2×M16—22no—
358C2×M16—22no—
359C22×C8—22no—
360C22×C8—22no—
361C22×C8—22no—
362C22×C8C22×D4711yescharacteristic
363C22×C8C22×D4711yescharacteristic
364C22×C8C22×D4712yes—
365C4⋊Q8—944no—
366C4⋊Q8—944no—
367C4⋊Q8—948no—
368#125—944no—
369#125—944no—
370#125—948no—
371C22⋊Q8—9416no—
372#117—22no—
373#117—22no—
374#117—474no—
375#117—942no—
376C2×C4⋊C4—474no—
377C2×C4⋊C4—472no—
378C2×C4⋊C4—471no—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—472no—
381C2×C4⋊C4—941no—
382C2×C4⋊C4—942no—
383C2×C4⋊C4—942no—
384C2×C4⋊C4—942no—
385C2×C4⋊C4—942no—
386C2×C4⋊C4—942no—
387C2×C4⋊C4—942no—
388C2×C4⋊C4—941no—
389C2×C4⋊C4D18811yescharacteristic
390C2×C4⋊C4D18811yescharacteristic
391C2×C22⋊C4—941no—
392C2×C22⋊C4—942no—
393C2×C22⋊C4—942no—
394C2×C22⋊C4D18812yes—
395C2×C42—941no—
396C2×C42C22×D4712yes—
397#103—942no—
398#103—942no—
399#103—944no—
400#102—942no—
401#102—942no—
402#102—944no—
403#95—942no—
404#95—942no—
405C24—941no—
406C24C2×D18811yescharacteristic
407#41—948no—
408#41—948no—
409#41—1884no—
410#41—1884no—
411C2×Q8—472no—
412C2×Q8—948no—
413C2×Q8—941no—
414C2×Q8—941no—
415C2×Q8—941no—
416C2×Q8—941no—
417C2×Q8—941no—
418C2×Q8—3762no—
419C2×Q8—3762no—
420C2×D4—472no—
421C2×D4—944no—
422C2×D4—941no—
423C2×D4—941no—
424C2×D4—941no—
425C2×D4—941no—
426C2×D4—941no—
427C2×D4—1882no—
428C2×D4—1882no—
429C22×C4—944no—
430C22×C4—941no—
431C22×C4—941no—
432C22×C4—941no—
433C22×C4—942no—
434C22×C4—944no—
435C22×C4—942no—
436C22×C4—3761no—
437C22×C4—3761no—
438C22×C4—3761no—
439C22×C4—3761no—
440C22×C4—1882no—
441C22×C4—1882no—
442C22×C4—1882no—
443C22×C4C2×D18812yes—
444C22×C4C2×D18812yes—
445C22×C4C2×D18811yescharacteristic
446C22×C4#1062535012yes—
447C22×C4#1062535012yes—
448C22×C4D4×D4712yes—
449C22×C4D4×D4712yes—
450C22×C4#1062535212yes—
451C22×C4Q8×D4712yes—
452C22×C4C23×D4711yescharacteristic
453M16—44no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—22no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—22no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—22no—
463C2×C8—44no—
464C4⋊C4—22no—
465C4⋊C4—22no—
466C4⋊C4—474no—
467C4⋊C4—472no—
468C4⋊C4—472no—
469C4⋊C4—948no—
470C4⋊C4—948no—
471C4⋊C4—948no—
472C4⋊C4—944no—
473C4⋊C4—942no—
474C4⋊C4—941no—
475C4⋊C4—941no—
476C4⋊C4—942no—
477C4⋊C4—942no—
478C4⋊C4—1882no—
479C22⋊C4—24no—
480C22⋊C4—944no—
481C22⋊C4—3762no—
482C22⋊C4—3762no—
483C42—22no—
484C42—22no—
485C42—944no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—942no—
491C23—1881no—
492C23—1881no—
493C23—1881no—
494C23—1881no—
495C23—1881no—
496C23—1881no—
497C23#1309399412yes—
498C23#1309400412yes—
499C23#1309401212yes—
500C23C22×D18811yescharacteristic
501Q8—942no—
502Q8—942no—
503Q8—3768no—
504Q8—1881no—
505Q8—1881no—
506D4—942no—
507D4—942no—
508D4—1888no—
509D4—1881no—
510D4—1881no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—942no—
518C2×C4—942no—
519C2×C4—944no—
520C2×C4—942no—
521C2×C4—3764no—
522C2×C4—3764no—
523C2×C4—1882no—
524C2×C4—1882no—
525C2×C4—1882no—
526C2×C4—1881no—
527C2×C4—1881no—
528C2×C4—1882no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4—1882no—
532C2×C4—1881no—
533C2×C4—1881no—
534C2×C4—1888no—
535C2×C4#1309399412yes—
536C2×C4#1309400214yes—
537C2×C4#1309400214yes—
538C2×C4#1309400412yes—
539C2×C4#1309401014yes—
540C2×C4#1309401014yes—
541C2×C4#1309401212yes—
542C2×C4C2×D4×D4711yescharacteristic
543C2×C4C2×D4×D4711yescharacteristic
544C2×C4C2×Q8×D4711yescharacteristic
545C2×C4C2×#1062535211yescharacteristic
546C2×C4C2×#1062535011yescharacteristic
547C2×C4C2×#1062535011yescharacteristic
548C8—41no—
549C8—41no—
550C8—42no—
551C22—22no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—3761no—
558C22—3761no—
559C22—3761no—
560C22—3761no—
561C22—1882no—
562C22—1882no—
563C22—1882no—
564C22#1568903614yes—
565C22C2×#1309399411yescharacteristic
566C22C2×#1309400411yescharacteristic
567C22C2×#1309401211yescharacteristic
568C22#1568992811yescharacteristic
569C22#1568993511yescharacteristic
570C22#1568993511yescharacteristic
571C22#1568994011yescharacteristic
572C4—42no—
573C4—42no—
574C4—3762no—
575C4—1884no—
576C4—1882no—
577C4—1882no—
578C4—1882no—
579C4#1568903614yes—
580C4C2×#1309400211yescharacteristic
581C4C2×#1309400211yescharacteristic
582C4C2×#1309401011yescharacteristic
583C4C2×#1309401011yescharacteristic
584C2—24no—
585C2—3762no—
586C2C2×#1568903611yescharacteristic
587C2#2025269511yescharacteristic
588C2#2025270611yescharacteristic
589C2#2025324111yescharacteristic
590C2#2025325211yescharacteristic
591C2#2025374411yescharacteristic
592C2#2025375511yescharacteristic
subgroupG/H
centerC23C22×D188
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#1729C2
socleC23×C47C22×D4
prime radicalC22×#10625350C22
hypercenter#1729D47
hypocenterC47#23458
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1729

Complements

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

NH ≅ G/Ntype
1. C2×#15689751C2semidirect
5. #20244540C2semidirect
6. #20244535C2semidirect
7. #20244530C2semidirect
8. #20244525C2semidirect
9. #20243108C2semidirect
10. C2×#15688886C2semidirect
11. C47×#1729C2semidirect
50. C2×#10625350Q8semidirect
298. #1729D47semidirect
341. C47#23458semidirect

Maximal subgroups

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

subgroupindex
C2×#156897512
C2×#156891252
C2×#156891202
#202459762
#202445402
#202445352
#202445302
#202445252
#202431082
C2×#156888862
C47×#17292
#2345847