Finite Group Databasebeta

← #26228471

Subgroups — #26228471

Order 12032.

732 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689420C211yesmaximal, characteristic
2#20246130C211yesmaximal, characteristic, complemented
3#20246026C211yesmaximal, characteristic
4#20244151C211yesmaximal, characteristic, complemented
5#20244125C211yesmaximal, characteristic, complemented
6#20244115C211yesmaximal, characteristic, complemented
7#20244086C211yesmaximal, characteristic
8#20244072C211yesmaximal, characteristic
9#20244013C211yesmaximal, characteristic
10#20244013C211yesmaximal, characteristic
11#20243971C211yesmaximal, characteristic, complemented
12#20243957C211yesmaximal, characteristic, complemented
13#20243389C211yesmaximal, characteristic
14#20243247C211yesmaximal, characteristic, complemented
15C47×#1989C211yesmaximal, characteristic, complemented
16C2×C4×D188C2211yescharacteristic
17#15689420C418yescomplemented
18C2×#13093957C2211yescharacteristic
19C2×#13093955C2211yescharacteristic
20C2×#13093955C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×C4⋊(C47⋊C8)C2211yescharacteristic
23C2×#13093946C2211yescharacteristic
24C2×#13093946C2211yescharacteristic
25C2×#13093944C2211yescharacteristic
26C2×#13093944C2211yescharacteristic
27C2×#13093934C2211yescharacteristic
28C2×C8×Dic47C2211yescharacteristic
29#15689024C2211yescharacteristic
30#15689024C2211yescharacteristic
31#15689022—21no—
32#15689022—21no—
33#15689022C2211yescharacteristic
34#15689022C2211yescharacteristic
35#15689019C2211yescharacteristic
36#15689019C2211yescharacteristic
37#15689014—21no—
38#15689014—21no—
39#15689009—21no—
40#15689009—21no—
41#15688993C2211yescharacteristic
42#15688985—21no—
43#15688982—21no—
44C4⋊C4×Dic47C2211yescharacteristic
45C4⋊C4×Dic47C2211yescharacteristic
46#15688969C2211yescharacteristic
47#15688886C2211yescharacteristic
48#15688886C2211yescharacteristic
49#15688885C2211yescharacteristic
50#15688885C2211yescharacteristic
51#15688877C2211yescharacteristic
52#15688877C2211yescharacteristic
53C2×#125×C47C2211yescharacteristic
54C2×C4⋊C8×C47C2211yescharacteristic
55C4×C4⋊C4×C47C2211yescharacteristic
56C47×#340C2211yescharacteristic
57C47×#340C2211yescharacteristic
58C47×#340C2211yescharacteristic
59C47×#340C2211yescharacteristic
60C22×D188C2311yescharacteristic
61C22×C4×D47—21no—
62C22×C47⋊C8C2311yescharacteristic
63C22×C47⋊C8C2311yescharacteristic
64C4×D188—22no—
65C4×D188C2×C414yes—
66C2×C4⋊Dic47—21no—
67C2×C4⋊Dic47—21no—
68C2×C4⋊Dic47—21no—
69C2×C4⋊Dic47—21no—
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×(C2×D47)⋊C4—21no—
76C2×(C2×D47)⋊C4—21no—
77C2×(C2×D47)⋊C4—21no—
78C2×(C2×D47)⋊C4—21no—
79C2×(C2×D47)⋊C4—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—21no—
82C2×C4×Dic47C2311yescharacteristic
83C2×C4×Dic47C2311yescharacteristic
84#13093957C2×C414yes—
85#13093955—41no—
86#13093955—41no—
87#13093955C2×C414yes—
88C4⋊(C47⋊C8)C2×C414yes—
89#13093946—41no—
90#13093946—41no—
91#13093944C2×C414yes—
92#13093944C2×C414yes—
93#13093934—41no—
94C8×Dic47—41no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C47⋊#95—21no—
98C47⋊#95—21no—
99C47⋊#95—21no—
100C47⋊#95—21no—
101C47⋊#95D411yescharacteristic
102C47⋊#95D411yescharacteristic
103C47⋊#95D411yescharacteristic
104C47⋊#95D411yescharacteristic
105C22×C8×C47C2311yescharacteristic
106C22×C8×C47C2311yescharacteristic
107#125×C47—22no—
108#125×C47C2×C414yes—
109C2×C4⋊C4×C47—21no—
110C2×C4⋊C4×C47—21no—
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C4⋊C4×C47C2311yescharacteristic
114C2×C4⋊C4×C47C2311yescharacteristic
115C2×C42×C47—21no—
116C2×C42×C47C2311yescharacteristic
117C4⋊C8×C47—22no—
118#95×C47—21no—
119C23×D47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—21no—
122C22×C4×C47—21no—
123C22×C4×C47—21no—
124C22×C4×C47C2×D411yescharacteristic
125C22×C4×C47C2×D411yescharacteristic
126C22×C4×C47C2411yescharacteristic
127C22×Dic47—21no—
128C22×Dic47—21no—
129C22×Dic47—21no—
130C22×Dic47C2×D411yescharacteristic
131C22×Dic47C2×D411yescharacteristic
132C22×Dic47C2×D411yescharacteristic
133C22×Dic47C2×D411yescharacteristic
134C2×D188—21no—
135C2×D188—21no—
136C2×D188—21no—
137C2×D188—21no—
138C2×D188—21no—
139C2×D188C22×C412yes—
140C2×C4×D47—24no—
141C2×C4×D47—81no—
142C2×C4×D47—81no—
143C2×C8×C47—21no—
144C2×C8×C47—21no—
145C2×C8×C47—21no—
146C2×C8×C47—21no—
147C2×C8×C47—21no—
148C2×C8×C47—21no—
149C4⋊C4×C47—24no—
150C4⋊C4×C47—24no—
151C4⋊C4×C47—21no—
152C4⋊C4×C47—21no—
153C4⋊C4×C47—21no—
154C4⋊C4×C47—21no—
155C4⋊C4×C47—21no—
156C4⋊C4×C47—21no—
157C4⋊C4×C47C22×C412yes—
158C4⋊C4×C47C22×C412yes—
159C42×C47—81no—
160C42×C47C22×C412yes—
161C42×C47#4112yes—
162(C2×D47)⋊C4—24no—
163(C2×D47)⋊C4—81no—
164(C2×D47)⋊C4—81no—
165(C2×D47)⋊C4—81no—
166(C2×D47)⋊C4—81no—
167C4⋊Dic47—21no—
168C4⋊Dic47—21no—
169C4⋊Dic47—21no—
170C4⋊Dic47—21no—
171C4⋊Dic47—21no—
172C4⋊Dic47—81no—
173C4⋊Dic47—81no—
174C4⋊Dic47—81no—
175C4⋊Dic47—81no—
176C4⋊Dic47C22×C412yes—
177Dic47⋊C4—81no—
178Dic47⋊C4—81no—
179C4×Dic47—21no—
180C4×Dic47—21no—
181C4×Dic47—21no—
182C4×Dic47—21no—
183C4×Dic47—81no—
184C4×Dic47—81no—
185C2×C47⋊C8—21no—
186C2×C47⋊C8—21no—
187C2×C47⋊C8—21no—
188C2×C47⋊C8—21no—
189C2×C47⋊C8C22×C412yes—
190C2×C47⋊C8C22×C412yes—
191C23×C47C22×D411yescharacteristic
192C22×D47—22no—
193C22×D47—41no—
194C22×D47—41no—
195C22×D47—41no—
196C22×D47—41no—
197C22×D47—41no—
198C22×D47—41no—
199C2×C4×C47—21no—
200C2×C4×C47—21no—
201C2×C4×C47—21no—
202C2×C4×C47—21no—
203C2×C4×C47—22no—
204C2×C4×C47—22no—
205C2×C4×C47—41no—
206C2×C4×C47—41no—
207C2×C4×C47—41no—
208C2×C4×C47—41no—
209C2×C4×C47—41no—
210C2×C4×C47—41no—
211C2×C4×C47—41no—
212C2×C4×C47—41no—
213C2×C4×C47—41no—
214C2×C4×C47—41no—
215C2×C4×C47C4×D412yes—
216C2×C4×C47C4×D412yes—
217C2×C4×C47C23×C411yescharacteristic
218C2×C4×C47C2×#4111yescharacteristic
219C2×C4×C47H(5,𝔽2)11yescharacteristic
220C2×C4×C472−1+411yescharacteristic
221C2×C4×C472−1+411yescharacteristic
222C2×C4×C472−1+411yescharacteristic
223C2×Dic47—22no—
224C2×Dic47—21no—
225C2×Dic47—21no—
226C2×Dic47—21no—
227C2×Dic47—21no—
228C2×Dic47—21no—
229C2×Dic47—21no—
230C2×Dic47—21no—
231C2×Dic47—21no—
232C2×Dic47—21no—
233C2×Dic47—21no—
234C2×Dic47—21no—
235C2×Dic47—21no—
236C2×Dic47—41no—
237C2×Dic47—41no—
238C2×Dic47—41no—
239C2×Dic47—41no—
240C2×Dic47—41no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—41no—
244D188—22no—
245D188—22no—
246D188—41no—
247D188—41no—
248C4×D47—84no—
249C8×C47—41no—
250C8×C47—41no—
251C47⋊C8—22no—
252C47⋊C8—22no—
253#22122—471nomaximal, Sylow
254C22×C47C2×C4×D411yescharacteristic
255C22×C47#53611yescharacteristic
256C22×C47#53711yescharacteristic
257C22×C47#57611yescharacteristic
258C22×C47#57711yescharacteristic
259C22×C47#57711yescharacteristic
260C22×C47#57711yescharacteristic
261C2×D47—42no—
262C2×D47—42no—
263C2×D47—42no—
264C2×D47—81no—
265C2×D47—81no—
266C2×D47—81no—
267C2×D47—81no—
268C4×C47—22no—
269C4×C47—22no—
270C4×C47—42no—
271C4×C47—42no—
272C4×C47—81no—
273C4×C47—81no—
274C4×C47#51911yescharacteristic
275C4×C47#52011yescharacteristic
276C4×C47#54111yescharacteristic
277C4×C47#54211yescharacteristic
278Dic47—42no—
279Dic47—41no—
280Dic47—41no—
281Dic47—41no—
282Dic47—41no—
283Dic47—81no—
284Dic47—81no—
285C2×#478—471no—
286#2330—471no—
287#2319—471no—
288#1989D4711yescharacteristic, complemented
289#1948—471no—
290#1943—471no—
291#1941—471no—
292#1936—471no—
293#1934—471no—
294#1926—471no—
295#1926—471no—
296#1920—471no—
297#1918—471no—
298#1802—471no—
299#1768—471no—
300C2×C47#232311yescharacteristic
301C2×C47#294611yescharacteristic
302C2×C47#295211yescharacteristic
303C2×C47#324111yescharacteristic
304C2×C47#324411yescharacteristic
305C2×C47#325711yescharacteristic
306C2×C47#326011yescharacteristic
307D47—82no—
308C2×#125C2×D4711yescharacteristic
309C2×C4×D4—471no—
310#478—478no—
311C2×#107—471no—
312C2×#106—471no—
313C2×#106—471no—
314C2×#106—471no—
315C2×C4⋊C8—471no—
316C2×C4⋊C8C2×D4711yescharacteristic
317C2×#102—471no—
318C2×#102—471no—
319C2×#102—471no—
320C2×#102—471no—
321C2×#97—471no—
322C2×C4×C8—471no—
323#390—471no—
324#390—471no—
325#389—471no—
326#389—471no—
327#389—471no—
328#389—471no—
329#389—941no—
330#389—941no—
331#388—941no—
332#388—941no—
333#387—941no—
334#387—941no—
335#384—471no—
336#381—941no—
337#380—941no—
338C4×C4⋊C4—471no—
339C4×C4⋊C4—471no—
340C4×C4⋊C4C2×D4711yescharacteristic
341#376—471no—
342#340—471no—
343#340—471no—
344#340—471no—
345#340—471no—
346#340C2×D4711yescharacteristic
347#340C2×D4711yescharacteristic
348#340C2×D4711yescharacteristic
349#340C2×D4711yescharacteristic
350#336—471no—
351#336—471no—
352C47#2212211yescharacteristic, complemented, Sylow
353C22×D4—471no—
354C23×C4—941no—
355C22×C8—471no—
356C22×C8—471no—
357C22×C8C22×D4711yescharacteristic
358C22×C8C22×D4711yescharacteristic
359#125—22no—
360#125C4×D4714yes—
361C4×D4—474no—
362C4×D4—942no—
363C2×C4⋊C4—21no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—471no—
366C2×C4⋊C4—471no—
367C2×C4⋊C4—471no—
368C2×C4⋊C4—941no—
369C2×C4⋊C4—941no—
370C2×C4⋊C4—941no—
371C2×C4⋊C4—941no—
372C2×C4⋊C4—941no—
373C2×C4⋊C4—941no—
374C2×C4⋊C4C22×D4711yescharacteristic
375C2×C4⋊C4C22×D4711yescharacteristic
376C2×C4⋊C4C22×D4711yescharacteristic
377C2×C4⋊C4C22×D4711yescharacteristic
378C2×C22⋊C4—941no—
379C2×C22⋊C4—941no—
380C2×C22⋊C4—941no—
381C2×C22⋊C4—941no—
382C2×C22⋊C4—941no—
383C2×C42—21no—
384C2×C42—471no—
385C2×C42—471no—
386C2×C42—941no—
387C2×C42—941no—
388C2×C42C22×D4711yescharacteristic
389#107—474no—
390#106—474no—
391#106—1881no—
392#106—1881no—
393C4⋊C8—22no—
394C4⋊C8—474no—
395#102—474no—
396#102—474no—
397#102—1881no—
398#102—1881no—
399#97—1881no—
400C4×C8—1881no—
401#95—21no—
402#95—471no—
403#95—471no—
404#95—471no—
405#95—471no—
406#95—941no—
407#95—941no—
408#95—941no—
409#95—941no—
410#95—941no—
411#95—941no—
412C24—941no—
413C2×D4—472no—
414C2×D4—941no—
415C2×D4—941no—
416C2×D4—941no—
417C2×D4—941no—
418C2×D4—941no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—471no—
424C22×C4—471no—
425C22×C4—471no—
426C22×C4—471no—
427C22×C4—941no—
428C22×C4—944no—
429C22×C4—941no—
430C22×C4—941no—
431C22×C4—3761no—
432C22×C4—3761no—
433C22×C4D4×D4711yescharacteristic
434C22×C4D4×D4711yescharacteristic
435C22×C4C23×D4711yescharacteristic
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—472no—
443C2×C8—472no—
444C2×C8—941no—
445C2×C8—941no—
446C2×C8—941no—
447C2×C8—941no—
448C4⋊C4—24no—
449C4⋊C4—24no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—472no—
457C4⋊C4—941no—
458C4⋊C4—941no—
459C4⋊C4—941no—
460C4⋊C4—941no—
461C4⋊C4—941no—
462C4⋊C4—3761no—
463C4⋊C4—3761no—
464C4⋊C4—3761no—
465C4⋊C4—3761no—
466C4⋊C4—3761no—
467C4⋊C4—3761no—
468C4⋊C4C2×C4×D4712yes—
469C4⋊C4C2×C4×D4712yes—
470C22⋊C4—944no—
471C22⋊C4—3761no—
472C22⋊C4—3761no—
473C22⋊C4—3761no—
474C22⋊C4—3761no—
475C42—81no—
476C42—941no—
477C42—941no—
478C42—941no—
479C42—941no—
480C42—3761no—
481C42—3761no—
482C42C2×C4×D4712yes—
483C42#1062535212yes—
484C23—942no—
485C23—1881no—
486C23—1881no—
487C23—1881no—
488C23—1881no—
489C23—1881no—
490C23—1881no—
491C23C2×D4×D4711yescharacteristic
492D4—942no—
493D4—942no—
494D4—1881no—
495D4—1881no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—942no—
513C2×C4—941no—
514C2×C4—941no—
515C2×C4—941no—
516C2×C4—941no—
517C2×C4—941no—
518C2×C4—941no—
519C2×C4—941no—
520C2×C4—941no—
521C2×C4—941no—
522C2×C4—941no—
523C2×C4—941no—
524C2×C4—941no—
525C2×C4—3764no—
526C2×C4—1881no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4—1881no—
532C2×C4—1881no—
533C2×C4—1881no—
534C2×C4(C22⋊D47)⋊C412yes—
535C2×C4(C22⋊D47)⋊C412yes—
536C2×C4C22×C4×D4711yescharacteristic
537C2×C4C2×#1062535211yescharacteristic
538C2×C4#1309407011yescharacteristic
539C2×C4#1309407111yescharacteristic
540C2×C4#1309407211yescharacteristic
541C2×C4#1309407211yescharacteristic
542C8—41no—
543C8—41no—
544C8—942no—
545C8—942no—
546C22—3761no—
547C22—3761no—
548C22—3761no—
549C22—3761no—
550C22—1882no—
551C22—1882no—
552C22—1882no—
553C22C2×(C22⋊D47)⋊C411yescharacteristic
554C22#1568969611yescharacteristic
555C22#1568970311yescharacteristic
556C22#1568992511yescharacteristic
557C22#1568993011yescharacteristic
558C22#1568993211yescharacteristic
559C22#1568993311yescharacteristic
560C4—22no—
561C4—22no—
562C4—42no—
563C4—42no—
564C4—81no—
565C4—81no—
566C4—3761no—
567C4—3761no—
568C4—1882no—
569C4—1881no—
570C4—1881no—
571C4—1881no—
572C4—1881no—
573C4#1568959911yescharacteristic
574C4#1568960511yescharacteristic
575C4#1568973411yescharacteristic
576C4#1568974211yescharacteristic
577C2—3762no—
578C2#2024607311yescharacteristic
579C2#2025222311yescharacteristic
580C2#2025226711yescharacteristic
581C2#2025463811yescharacteristic
582C2#2025469311yescharacteristic
583C2#2025480511yescharacteristic
584C2#2025483611yescharacteristic
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1989C2
socleC23×C47C22×D4
prime radicalC22×D188C23
cosocleC22×C4×C47C24
hypercenter#1989D47
hypocenterC47#22122
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1989

Complements

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

NH ≅ G/Ntype
2. #20246130C2semidirect
4. #20244151C2semidirect
5. #20244125C2semidirect
6. #20244115C2semidirect
11. #20243971C2semidirect
12. #20243957C2semidirect
14. #20243247C2semidirect
15. C47×#1989C2semidirect
17. #15689420C4semidirect
288. #1989D47semidirect
352. C47#22122semidirect

Maximal subgroups

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

subgroupindex
C2×#156894202
#202461302
#202460262
#202441512
#202441252
#202441152
#202440862
#202440722
#202440132
#202439712
#202439572
#202433892
#202432472
C47×#19892
#2212247