Finite Group Databasebeta

← #26319507

Subgroups — #26319507

Order 12032.

660 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
1#20249348C211yesmaximal, characteristic
2#20244889C211yesmaximal, characteristic, complemented
3#20244769C211yesmaximal, characteristic, complemented
4#20244759C211yesmaximal, characteristic
5#20244703C211yesmaximal, characteristic
6#20244245C211yesmaximal, characteristic
7#20244113C211yesmaximal, characteristic, complemented
8#20244084C211yesmaximal, characteristic
9#20244081C211yesmaximal, characteristic, complemented
10#20244015C211yesmaximal, characteristic
11#20243973C211yesmaximal, characteristic, complemented
12#20243743C211yesmaximal, characteristic
13#20243656C211yesmaximal, characteristic, complemented
14#20243245C211yesmaximal, characteristic, complemented
15C47×#2320C211yesmaximal, characteristic, complemented
16C22×#10625336C2211yescharacteristic
17C2×#13093955C2211yescharacteristic
18C2×#13093955C2211yescharacteristic
19C2×#13093955C2211yescharacteristic
20C2×C4⋊(C47⋊C8)C2211yescharacteristic
21C2×C4⋊(C47⋊C8)C2211yescharacteristic
22C2×#13093948C2211yescharacteristic
23C2×#13093948C2211yescharacteristic
24C2×#13093948C2211yescharacteristic
25C2×#13093944C2211yescharacteristic
26C2×#13093944C2211yescharacteristic
27C2×#13093944C2211yescharacteristic
28C2×#13093935C2211yescharacteristic
29C2×C4×C47⋊C8C2211yescharacteristic
30#15689052—21no—
31#15689052—21no—
32#15689052—21no—
33#15689047—21no—
34#15689025C2211yescharacteristic
35#15689025C2211yescharacteristic
36#15689025C2211yescharacteristic
37#15689021C2211yescharacteristic
38#15689021C2211yescharacteristic
39#15689021C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48C2×C4×Q8×C47C2211yescharacteristic
49C47×#391C2211yescharacteristic
50C47×#391C2211yescharacteristic
51C47×#389C2211yescharacteristic
52C47×#389C2211yescharacteristic
53C47×#387—21no—
54C47×#387—21no—
55C47×#383—21no—
56C47×#382—21no—
57C4×C4⋊C4×C47C2211yescharacteristic
58C47×#376C2211yescharacteristic
59C22×D188C2311yescharacteristic
60C2×#10625336—22no—
61C2×#10625336—22no—
62C2×#10625336—41no—
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67C2×C4⋊Dic47C2311yescharacteristic
68C2×C4⋊Dic47C2311yescharacteristic
69C2×C4⋊Dic47C2311yescharacteristic
70C2×Dic47⋊C4—21no—
71C2×Dic47⋊C4—21no—
72C2×Dic47⋊C4—21no—
73C2×(C2×D47)⋊C4—21no—
74C2×(C2×D47)⋊C4—21no—
75C2×(C2×D47)⋊C4—21no—
76C2×(C2×D47)⋊C4—21no—
77C2×(C2×D47)⋊C4—21no—
78C2×(C2×D47)⋊C4—21no—
79#13093955—22no—
80#13093955—22no—
81#13093955—41no—
82C4⋊(C47⋊C8)—22no—
83C4⋊(C47⋊C8)—22no—
84#13093948—22no—
85#13093948—22no—
86#13093948—41no—
87#13093944—22no—
88#13093944—22no—
89#13093944—41no—
90#13093935—41no—
91C4×C47⋊C8—41no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C22×Q8×C47C2311yescharacteristic
96C4×Q8×C47—22no—
97C4×Q8×C47—22no—
98C2×C4⋊C4×C47—21no—
99C2×C4⋊C4×C47—21no—
100C2×C4⋊C4×C47—21no—
101C2×C4⋊C4×C47C2311yescharacteristic
102C2×C4⋊C4×C47C2311yescharacteristic
103C2×C4⋊C4×C47C2311yescharacteristic
104C2×C42×C47—21no—
105C2×C42×C47—21no—
106C2×C42×C47C2311yescharacteristic
107C2×C42×C47C2311yescharacteristic
108C2×C42×C47C2311yescharacteristic
109#95×C47—21no—
110#95×C47—21no—
111#95×C47—21no—
112#95×C47—21no—
113#95×C47—21no—
114#95×C47—21no—
115#95×C47—21no—
116#95×C47D411yescharacteristic
117#95×C47D411yescharacteristic
118#95×C47D411yescharacteristic
119#95×C47D411yescharacteristic
120C23×D47—21no—
121C2×Q8×C47—21no—
122C2×Q8×C47—21no—
123C2×Q8×C47—21no—
124C2×Q8×C47—21no—
125C2×Q8×C47—21no—
126C2×Q8×C47#4112yes—
127C22×C4×C47—21no—
128C22×C4×C47—21no—
129C22×C4×C47—21no—
130C22×C4×C47—21no—
131C22×C4×C47C2×D411yescharacteristic
132C22×C4×C47C2×D411yescharacteristic
133C22×C4×C47C2×D411yescharacteristic
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2×D411yescharacteristic
137C22×C4×C47C2411yescharacteristic
138C22×Dic47—21no—
139C22×Dic47—21no—
140C22×Dic47—21no—
141C2×D188—21no—
142C2×D188—21no—
143C2×D188—21no—
144C2×D188—21no—
145C2×D188—21no—
146C2×D188—21no—
147C4⋊C4×C47—24no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—21no—
150C4⋊C4×C47—21no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—81no—
154C4⋊C4×C47—81no—
155C4⋊C4×C47#4112yes—
156C42×C47—24no—
157C42×C47—21no—
158C42×C47—21no—
159C42×C47—21no—
160C42×C47—21no—
161C42×C47—21no—
162C42×C47—21no—
163C42×C47—81no—
164#10625336—44no—
165(C2×D47)⋊C4—81no—
166(C2×D47)⋊C4—81no—
167(C2×D47)⋊C4—81no—
168(C2×D47)⋊C4—81no—
169(C2×D47)⋊C4—81no—
170(C2×D47)⋊C4—81no—
171C4⋊Dic47—21no—
172C4⋊Dic47—21no—
173C4⋊Dic47—21no—
174C4⋊Dic47—21no—
175C4⋊Dic47—21no—
176C4⋊Dic47—21no—
177Dic47⋊C4—81no—
178Dic47⋊C4—81no—
179Dic47⋊C4—81no—
180C2×C47⋊C8—21no—
181C2×C47⋊C8—21no—
182C2×C47⋊C8—21no—
183C2×C47⋊C8—21no—
184C2×C47⋊C8—21no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8—21no—
190C2×C47⋊C8#4112yes—
191C2×C47⋊C8#4112yes—
192C23×C47C22×D411yescharacteristic
193C22×D47—41no—
194C22×D47—41no—
195C22×D47—41no—
196C22×D47—41no—
197C22×D47—41no—
198C22×D47—41no—
199C22×D47—41no—
200Q8×C47—22no—
201Q8×C47—22no—
202Q8×C47—41no—
203Q8×C47—41no—
204C2×C4×C47—21no—
205C2×C4×C47—21no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—21no—
209C2×C4×C47—21no—
210C2×C4×C47—21no—
211C2×C4×C47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47C2×#4111yescharacteristic
235C2×C4×C47C2×#4111yescharacteristic
236C2×C4×C47C2×#4111yescharacteristic
237C2×C4×C47C2×#4111yescharacteristic
238C2×C4×C47H(5,𝔽2)11yescharacteristic
239C2×C4×C472−1+411yescharacteristic
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244C2×Dic47—41no—
245C2×Dic47—41no—
246C2×Dic47—41no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249D188—41no—
250D188—41no—
251D188—41no—
252D188—41no—
253C47⋊C8—22no—
254C47⋊C8—22no—
255C47⋊C8—41no—
256C47⋊C8—41no—
257#29442—471nomaximal, Sylow
258C22×C47#52511yescharacteristic
259C22×C47#53211yescharacteristic
260C22×C47#53711yescharacteristic
261C22×C47#57711yescharacteristic
262C22×C47#57711yescharacteristic
263C22×C47#57711yescharacteristic
264C22×C47#57711yescharacteristic
265C2×D47—81no—
266C2×D47—81no—
267C2×D47—81no—
268C2×D47—81no—
269C2×D47—81no—
270C2×D47—81no—
271C2×D47—81no—
272C4×C47—42no—
273C4×C47—41no—
274C4×C47—42no—
275C4×C47—41no—
276C4×C47—41no—
277C4×C47—41no—
278C4×C47—41no—
279C4×C47—41no—
280C4×C47—81no—
281C4×C47—81no—
282C4×C47#55111yescharacteristic
283C4×C47#55211yescharacteristic
284C4×C47#55311yescharacteristic
285C4×C47#55611yescharacteristic
286Dic47—81no—
287Dic47—81no—
288Dic47—81no—
289#2644—471no—
290#2320D4711yescharacteristic, complemented
291#2088—471no—
292#2066—471no—
293#2065—471no—
294#2055—471no—
295#1968—471no—
296#1941—471no—
297#1936—471no—
298#1935—471no—
299#1926—471no—
300#1920—471no—
301#1880—471no—
302#1865—471no—
303#1768—471no—
304C2×C47#260111yescharacteristic
305C2×C47#312511yescharacteristic
306C2×C47#312611yescharacteristic
307C2×C47#317311yescharacteristic
308C2×C47#317411yescharacteristic
309C2×C47#326011yescharacteristic
310C2×C47#326311yescharacteristic
311D47—161no—
312C22×SD16—471no—
313C2×C4×Q8C2×D4711yescharacteristic
314C2×#106—471no—
315C2×#106—471no—
316C2×#106—471no—
317C2×C4⋊C8—471no—
318C2×C4⋊C8—471no—
319C2×#103—471no—
320C2×#103—471no—
321C2×#103—471no—
322C2×#102—471no—
323C2×#102—471no—
324C2×#102—471no—
325C2×#97—471no—
326C2×C4×C8—471no—
327#397—941no—
328#397—941no—
329#397—941no—
330#396—941no—
331#391C2×D4711yescharacteristic
332#391C2×D4711yescharacteristic
333#390—471no—
334#390—471no—
335#390—471no—
336#389—471no—
337#389—471no—
338#389—471no—
339#389C2×D4711yescharacteristic
340#389C2×D4711yescharacteristic
341#387—21no—
342#387—21no—
343#383—21no—
344#382—21no—
345C4×C4⋊C4C2×D4711yescharacteristic
346#376C2×D4711yescharacteristic
347#340—471no—
348#340—471no—
349#340—471no—
350#340—471no—
351#340—471no—
352#340—471no—
353#336—471no—
354#336—471no—
355C47#2944211yescharacteristic, complemented, Sylow
356C22×Q8C22×D4711yescharacteristic
357C22×D4—471no—
358C2×SD16—942no—
359C2×SD16—942no—
360C2×SD16—1881no—
361C22×C8—471no—
362C22×C8—471no—
363C22×C8—471no—
364C22×C8—471no—
365C4×Q8—22no—
366C4×Q8—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—471no—
371C2×C4⋊C4—471no—
372C2×C4⋊C4—471no—
373C2×C4⋊C4—941no—
374C2×C4⋊C4—941no—
375C2×C4⋊C4—941no—
376C2×C4⋊C4C22×D4711yescharacteristic
377C2×C4⋊C4C22×D4711yescharacteristic
378C2×C4⋊C4C22×D4711yescharacteristic
379C2×C22⋊C4—941no—
380C2×C22⋊C4—941no—
381C2×C22⋊C4—941no—
382C2×C22⋊C4—941no—
383C2×C22⋊C4—941no—
384C2×C22⋊C4—941no—
385C2×C42—21no—
386C2×C42—21no—
387C2×C42C22×D4711yescharacteristic
388C2×C42C22×D4711yescharacteristic
389C2×C42C22×D4711yescharacteristic
390#106—942no—
391#106—942no—
392#106—1881no—
393C4⋊C8—942no—
394C4⋊C8—942no—
395#103—942no—
396#103—942no—
397#103—1881no—
398#102—942no—
399#102—942no—
400#102—1881no—
401#97—1881no—
402C4×C8—1881no—
403#95—21no—
404#95—21no—
405#95—21no—
406#95—21no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—941no—
411#95—941no—
412#95—941no—
413#95C22⋊D4711yescharacteristic
414#95C22⋊D4711yescharacteristic
415#95C22⋊D4711yescharacteristic
416#95C22⋊D4711yescharacteristic
417C24—941no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8—21no—
422C2×Q8—21no—
423C2×Q8#1062535012yes—
424C2×D4—941no—
425C2×D4—941no—
426C2×D4—941no—
427C2×D4—941no—
428C2×D4—941no—
429C2×D4—941no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—941no—
435C22×C4—941no—
436C22×C4—941no—
437C22×C4C2×C22⋊D4711yescharacteristic
438C22×C4C2×C22⋊D4711yescharacteristic
439C22×C4C2×C22⋊D4711yescharacteristic
440C22×C4C2×C22⋊D4711yescharacteristic
441C22×C4C2×C22⋊D4711yescharacteristic
442C22×C4C2×C22⋊D4711yescharacteristic
443C22×C4C23×D4711yescharacteristic
444SD16—1884no—
445C2×C8—472no—
446C2×C8—472no—
447C2×C8—941no—
448C2×C8—941no—
449C2×C8—941no—
450C2×C8—941no—
451C2×C8—941no—
452C2×C8—941no—
453C2×C8—941no—
454C2×C8—941no—
455C2×C8—941no—
456C2×C8—941no—
457C4⋊C4—24no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—81no—
464C4⋊C4—81no—
465C4⋊C4—941no—
466C4⋊C4—941no—
467C4⋊C4—941no—
468C4⋊C4—941no—
469C4⋊C4—941no—
470C4⋊C4—941no—
471C4⋊C4—3761no—
472C4⋊C4—3761no—
473C4⋊C4—3761no—
474C4⋊C4#1062535012yes—
475C22⋊C4—3761no—
476C22⋊C4—3761no—
477C22⋊C4—3761no—
478C22⋊C4—3761no—
479C22⋊C4—3761no—
480C22⋊C4—3761no—
481C42—24no—
482C42—21no—
483C42—21no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—81no—
489C23—1881no—
490C23—1881no—
491C23—1881no—
492C23—1881no—
493C23—1881no—
494C23—1881no—
495C23—1881no—
496C23C22×C22⋊D4711yescharacteristic
497Q8—22no—
498Q8—22no—
499Q8—41no—
500Q8—41no—
501D4—1881no—
502D4—1881no—
503D4—1881no—
504D4—1881no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—1881no—
536C2×C4—1881no—
537C2×C4—1881no—
538C2×C4—1881no—
539C2×C4—1881no—
540C2×C4—1881no—
541C2×C4—1881no—
542C2×C4—1881no—
543C2×C4—1881no—
544C2×C4#41×D4711yescharacteristic
545C2×C4#41×D4711yescharacteristic
546C2×C4C2×#1062535011yescharacteristic
547C2×C4C2×#1062535011yescharacteristic
548C2×C4#1309407011yescharacteristic
549C2×C4#1309407111yescharacteristic
550C8—942no—
551C8—942no—
552C8—1881no—
553C8—1881no—
554C22—3761no—
555C22—3761no—
556C22—3761no—
557C22—3761no—
558C22—3761no—
559C22—3761no—
560C22—3761no—
561C22#1568963411yescharacteristic
562C22#1568966911yescharacteristic
563C22#1568970111yescharacteristic
564C22#1568993111yescharacteristic
565C22#1568993111yescharacteristic
566C22#1568993111yescharacteristic
567C22#1568993411yescharacteristic
568C4—41no—
569C4—41no—
570C4—41no—
571C4—41no—
572C4—42no—
573C4—41no—
574C4—42no—
575C4—41no—
576C4—81no—
577C4—81no—
578C4—3761no—
579C4—3761no—
580C4—3761no—
581C4#1568979711yescharacteristic
582C4#1568980611yescharacteristic
583C4#1568981711yescharacteristic
584C4#1568984111yescharacteristic
585C2—7521no—
586C2#2024898511yescharacteristic
587C2#2025366511yescharacteristic
588C2#2025367911yescharacteristic
589C2#2025399911yescharacteristic
590C2#2025401711yescharacteristic
591C2#2025482911yescharacteristic
592C2#2025485311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2320C2
socleC23×C47C22×D4
prime radicalC22×D188C23
hypercenter#2320D47
hypocenterC47#29442
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ #376 ◁ #2320

Complements

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

NH ≅ G/Ntype
2. #20244889C2semidirect
3. #20244769C2semidirect
7. #20244113C2semidirect
9. #20244081C2semidirect
11. #20243973C2semidirect
13. #20243656C2semidirect
14. #20243245C2semidirect
15. C47×#2320C2semidirect
290. #2320D47semidirect
355. C47#29442semidirect

Maximal subgroups

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

subgroupindex
#202493482
#202448892
#202447692
#202447592
#202447032
#202442452
#202441132
#202440842
#202440812
#202440152
#202439732
#202437432
#202436562
#202432452
C47×#23202
#2944247