Finite Group Databasebeta

← #26186183

Subgroups — #26186183

Order 12032.

634 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253412C211yesmaximal, characteristic
2#20252362C211yesmaximal, characteristic, complemented
3C8×#10625354C211yesmaximal, characteristic, complemented
4#20242231C211yesmaximal, characteristic, complemented
5#20242222C211yesmaximal, characteristic, complemented
6#20242219C211yesmaximal, characteristic, complemented
7Q16×C47⋊C8C211yesmaximal, characteristic, complemented
8#20242185C211yesmaximal, characteristic, complemented
9#20242185C211yesmaximal, characteristic, complemented
10#20242182C211yesmaximal, characteristic, complemented
11#20242167C211yesmaximal, characteristic, complemented
12#20241821C211yesmaximal, characteristic, complemented
13#20241795C211yesmaximal, characteristic, complemented
14#20241673C211yesmaximal, characteristic, complemented
15C47×#1584C211yesmaximal, characteristic, complemented
16#15689660C2211yescharacteristic
17C4×#10625354C2211yescharacteristic
18#15689463C411yescharacteristic
19#15689450C411yescharacteristic
20#15689432C411yescharacteristic
21#15689426C411yescharacteristic
22#15689347C411yescharacteristic
23#15689341C411yescharacteristic
24#15689284C411yescharacteristic
25#15689271C411yescharacteristic
26Q8×C47⋊C8C2211yescharacteristic
27Q8×C47⋊C8C2211yescharacteristic
28C4×#10625336C2211yescharacteristic
29C4×#10625336C2211yescharacteristic
30C4×D376C2211yescharacteristic
31#15689202C2211yescharacteristic
32#15689196—21no—
33C8×D188—21no—
34C8×D188C2211yescharacteristic
35#15689188—21no—
36#15689188—21no—
37#15689188C2211yescharacteristic
38#15689188C2211yescharacteristic
39#15689183—21no—
40#15689179C2211yescharacteristic
41#15689175—21no—
42#15689175C2211yescharacteristic
43#15689174C2211yescharacteristic
44#15689174C2211yescharacteristic
45#15689080—21no—
46C4×C8×D47—21no—
47C4×C8×D47C2211yescharacteristic
48#15688876C2211yescharacteristic
49#15688873C2211yescharacteristic
50#15688873C2211yescharacteristic
51Q8⋊(C47⋊C8)C2211yescharacteristic
52Q8⋊(C47⋊C8)C2211yescharacteristic
53#15688852C2211yescharacteristic
54#15688852C2211yescharacteristic
55#15688850C2211yescharacteristic
56#15688850C2211yescharacteristic
57#15688844C2211yescharacteristic
58C8×C47⋊C8C2211yescharacteristic
59C8×C47⋊C8C2211yescharacteristic
60C47×#446C2211yescharacteristic
61C8×Q8×C47C2211yescharacteristic
62C4×Q16×C47C2211yescharacteristic
63C47×#335C2211yescharacteristic
64C47×Q8⋊C8C2211yescharacteristic
65C47×Q8⋊C8C2211yescharacteristic
66C47×#322C2211yescharacteristic
67C2×#10625354C2×C411yescharacteristic
68C2×#10625336C2×C411yescharacteristic
69C2×#10625336C2×C411yescharacteristic
70C2×D376C2×C411yescharacteristic
71C2×#10625324—21no—
72C2×C8×D47—21no—
73C2×C8×D47—21no—
74C2×C8×D47—21no—
75C2×C8×D47D411yescharacteristic
76C2×C8×D47D411yescharacteristic
77#13094019—21no—
78#13094015C2×C411yescharacteristic
79#13094013C2×C411yescharacteristic
80#13094009—21no—
81#13094000C2×C411yescharacteristic
82#13093999C2×C411yescharacteristic
83Q8×Dic47C2×C411yescharacteristic
84C4×D188—21no—
85C4×D188—21no—
86C4×D188C2311yescharacteristic
87C4×D188C2311yescharacteristic
88#13093986—21no—
89#13093986C2×C411yescharacteristic
90#13093982—21no—
91#13093982C2×C411yescharacteristic
92#13093981—21no—
93C42×D47—21no—
94C42×D47C2311yescharacteristic
95#13093958C2×C411yescharacteristic
96#13093955C2×C411yescharacteristic
97#13093955C2×C411yescharacteristic
98C4⋊(C47⋊C8)—21no—
99C4⋊(C47⋊C8)—21no—
100C4⋊(C47⋊C8)C2311yescharacteristic
101C4⋊(C47⋊C8)C2311yescharacteristic
102Dic47⋊C8—21no—
103Dic47⋊C8C2×C411yescharacteristic
104Dic47⋊C8C2×C411yescharacteristic
105#13093948C2×C411yescharacteristic
106#13093948C2×C411yescharacteristic
107#13093946C2×C411yescharacteristic
108#13093946C2×C411yescharacteristic
109#13093944C2×C411yescharacteristic
110#13093944C2×C411yescharacteristic
111(C2×D47)⋊C8—21no—
112(C2×D47)⋊C8—21no—
113(C2×D47)⋊C8—21no—
114(C2×D47)⋊C8—21no—
115(C2×D47)⋊C8—21no—
116(C2×D47)⋊C8—21no—
117(C2×D47)⋊C8—21no—
118(C2×D47)⋊C8C2×C411yescharacteristic
119(C2×D47)⋊C8C2×C411yescharacteristic
120#13093934—21no—
121C4×C47⋊C8—21no—
122C4×C47⋊C8—21no—
123C4×C47⋊C8C2311yescharacteristic
124C4×C47⋊C8C2311yescharacteristic
125C4×C47⋊C8C2311yescharacteristic
126C8×Dic47—21no—
127C8×Dic47D411yescharacteristic
128C8×Dic47D411yescharacteristic
129C2×Q16×C47C2×C411yescharacteristic
130C4×Q8×C47C2311yescharacteristic
131C4×Q8×C47C2311yescharacteristic
132#107×C47C2×C411yescharacteristic
133C4⋊C8×C47—21no—
134C4⋊C8×C47—21no—
135C4⋊C8×C47C2311yescharacteristic
136C4⋊C8×C47C2311yescharacteristic
137#103×C47C2×C411yescharacteristic
138#103×C47C2×C411yescharacteristic
139#97×C47—21no—
140C4×C8×C47—21no—
141C4×C8×C47C2311yescharacteristic
142C4×C8×C47C2311yescharacteristic
143C4×C8×C47C2311yescharacteristic
144C2×Q8×C47C22×C411yescharacteristic
145C2×Q8×C47C22×C411yescharacteristic
146#10625354—22no—
147#10625354—22no—
148C2×D188—21no—
149C2×D188—21no—
150C2×D188C22×C411yescharacteristic
151C2×D188C22×C411yescharacteristic
152C2×C4×D47—21no—
153C2×C4×D47—21no—
154C2×C4×D47—21no—
155C2×C4×D47—21no—
156C2×C4×D47C22×C411yescharacteristic
157C2×C4×D47C2×D411yescharacteristic
158C2×C4×D47#4111yescharacteristic
159Q16×C47—22no—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47C22×C411yescharacteristic
164C2×C8×C47C22×C411yescharacteristic
165C2×C8×C47C2×D411yescharacteristic
166C2×C8×C47C2×D411yescharacteristic
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47C22×C411yescharacteristic
170C4⋊C4×C47C22×C411yescharacteristic
171C42×C47—21no—
172C42×C47—21no—
173C42×C47C2411yescharacteristic
174#10625336—22no—
175#10625336—22no—
176(C2×D47)⋊C4—21no—
177(C2×D47)⋊C4—21no—
178(C2×D47)⋊C4—21no—
179(C2×D47)⋊C4—21no—
180(C2×D47)⋊C4—21no—
181(C2×D47)⋊C4—21no—
182(C2×D47)⋊C4—21no—
183(C2×D47)⋊C4—21no—
184C4⋊Dic47—21no—
185C4⋊Dic47—21no—
186C4⋊Dic47C22×C411yescharacteristic
187C4⋊Dic47C22×C411yescharacteristic
188Dic47⋊C4—21no—
189Dic47⋊C4—21no—
190C4×Dic47—21no—
191C4×Dic47—21no—
192C4×Dic47C22×C411yescharacteristic
193C4×Dic47C2×D411yescharacteristic
194C4×Dic47#4111yescharacteristic
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
198C2×C47⋊C8C22×C411yescharacteristic
199C2×C47⋊C8C22×C411yescharacteristic
200C2×C47⋊C8C2×D411yescharacteristic
201C2×C47⋊C8C2×D411yescharacteristic
202D376—22no—
203#10625324—42no—
204C8×D47—22no—
205C8×D47—22no—
206C8×D47—42no—
207C8×D47—42no—
208C8×D47—42no—
209C22×D47—21no—
210C22×D47—21no—
211C22×D47C4×D411yescharacteristic
212Q8×C47—21no—
213Q8×C47—21no—
214Q8×C47—21no—
215Q8×C47#13112yes—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47C23×C411yescharacteristic
221C2×C4×C47C22×D411yescharacteristic
222C2×C4×C47C2×#4111yescharacteristic
223C2×Dic47—21no—
224C2×Dic47—21no—
225C2×Dic47C4×D411yescharacteristic
226C2×Dic47C4×D411yescharacteristic
227C2×Dic47C4×D411yescharacteristic
228D188—24no—
229D188—21no—
230D188—21no—
231D188—21no—
232D188—42no—
233D188#13112yes—
234C4×D47—24no—
235C4×D47—21no—
236C4×D47—21no—
237C4×D47—21no—
238C4×D47—21no—
239C4×D47—41no—
240C4×D47—42no—
241C4×D47—41no—
242C4×D47—41no—
243C4×D47—41no—
244C4×D47—41no—
245C8×C47—21no—
246C8×C47—21no—
247C8×C47—21no—
248C8×C47—41no—
249C8×C47—41no—
250C8×C47—41no—
251C8×C47#13512yes—
252C47⋊C8—21no—
253C47⋊C8—21no—
254C47⋊C8—21no—
255C47⋊C8—41no—
256C47⋊C8—41no—
257C47⋊C8—41no—
258C47⋊C8#13512yes—
259#17904—471nomaximal, Sylow
260C22×C47C2×C4×D411yescharacteristic
261C2×D47—21no—
262C2×D47—22no—
263C2×D47—21no—
264C2×D47—21no—
265C2×D47—41no—
266C2×D47—41no—
267C2×D47—41no—
268C2×D47—41no—
269C2×D47—41no—
270C4×C47—22no—
271C4×C47—21no—
272C4×C47—41no—
273C4×C47—41no—
274C4×C47—41no—
275C4×C47C2×#13111yescharacteristic
276C4×C47C2×#13111yescharacteristic
277C4×C47C2×#13511yescharacteristic
278C4×C47C2×#13511yescharacteristic
279Dic47—21no—
280Dic47—21no—
281Dic47—21no—
282Dic47—41no—
283Dic47—41no—
284#3102—471no—
285#2966—471no—
286C8×#41—471no—
287#1584D4711yescharacteristic, complemented
288#1583—471no—
289#1582—471no—
290#1581—471no—
291C8×Q16—471no—
292C8×SD16—471no—
293C8×SD16—471no—
294C8×D8—471no—
295#1572—471no—
296#1488—471no—
297#1482—471no—
298#1450—471no—
299C2×C47#292911yescharacteristic
300C2×C47C4×#13511yescharacteristic
301C2×C47#295511yescharacteristic
302D47—41no—
303D47—42no—
304D47—81no—
305#529—471no—
306C4×#41—471no—
307#485—471no—
308#482—471no—
309#479—471no—
310#478—471no—
311#464—471no—
312#463—471no—
313#452—471no—
314#449—471no—
315#446C2×D4711yescharacteristic
316C8×Q8—471no—
317C8×Q8—471no—
318C8×Q8C2×D4711yescharacteristic
319C4×Q16C2×D4711yescharacteristic
320C4×SD16—471no—
321C4×SD16—471no—
322C4×D8—471no—
323#436—471no—
324#435—941no—
325C8×D4—471no—
326C8×D4—471no—
327C8×D4—471no—
328C8×D4—941no—
329C8×D4—941no—
330C8×D4—941no—
331#433—941no—
332#432—471no—
333#431—471no—
334#431—471no—
335#431—471no—
336#431—941no—
337#405—941no—
338C2×C4×C8—471no—
339C2×C4×C8—941no—
340#335—471no—
341#335C2×D4711yescharacteristic
342#334—471no—
343#334—471no—
344Q8⋊C8—471no—
345Q8⋊C8—471no—
346Q8⋊C8C2×D4711yescharacteristic
347Q8⋊C8C2×D4711yescharacteristic
348D4⋊C8—471no—
349D4⋊C8—471no—
350D4⋊C8—471no—
351D4⋊C8—471no—
352#322—471no—
353#322C2×D4711yescharacteristic
354C82—471no—
355C82—471no—
356C47#1790411yescharacteristic, complemented, Sylow
357C2×#41—471no—
358C2×Q16C4×D4711yescharacteristic
359C2×SD16—471no—
360C2×SD16—471no—
361C2×D8—471no—
362C2×M16—941no—
363C22×C8—471no—
364C22×C8—471no—
365C22×C8—941no—
366C22×C8—941no—
367C22×C8—941no—
368#126—941no—
369#125—471no—
370#124—471no—
371#123—941no—
372C22⋊Q8—471no—
373#121—471no—
374C4×Q8—471no—
375C4×Q8C22×D4711yescharacteristic
376C4×Q8C22×D4711yescharacteristic
377C4×D4—471no—
378C4×D4—471no—
379C4×D4—471no—
380C4×D4—941no—
381C4×D4—941no—
382C4×D4—941no—
383#117—471no—
384#117—941no—
385#117—941no—
386C2×C42—471no—
387C2×C42—941no—
388#107—471no—
389#107C4×D4711yescharacteristic
390#106—471no—
391#106—471no—
392C4⋊C8—21no—
393C4⋊C8—21no—
394C4⋊C8—471no—
395C4⋊C8—471no—
396C4⋊C8—471no—
397C4⋊C8—471no—
398C4⋊C8—941no—
399C4⋊C8—941no—
400C4⋊C8—941no—
401C4⋊C8C22×D4711yescharacteristic
402C4⋊C8C22×D4711yescharacteristic
403#103—471no—
404#103—471no—
405#103C4×D4711yescharacteristic
406#103C4×D4711yescharacteristic
407#102—471no—
408#102—471no—
409#102—471no—
410#102—471no—
411C22⋊C8—471no—
412C22⋊C8—471no—
413C22⋊C8—941no—
414C22⋊C8—941no—
415C22⋊C8—941no—
416C22⋊C8—941no—
417C22⋊C8—941no—
418C22⋊C8—941no—
419C22⋊C8—941no—
420#97—21no—
421#97—941no—
422C4×C8—21no—
423C4×C8—471no—
424C4×C8—471no—
425C4×C8—471no—
426C4×C8—471no—
427C4×C8—471no—
428C4×C8—941no—
429C4×C8—941no—
430C4×C8—941no—
431C4×C8C22×D4711yescharacteristic
432C4×C8C22×D4711yescharacteristic
433C4×C8C22×D4711yescharacteristic
434#41—942no—
435#41—942no—
436C2×Q8C2×C4×D4711yescharacteristic
437C2×Q8C2×C4×D4711yescharacteristic
438C2×D4—471no—
439C2×D4—471no—
440C2×D4—941no—
441C2×D4—941no—
442C22×C4—471no—
443C22×C4—471no—
444C22×C4—471no—
445C22×C4—941no—
446C22×C4—941no—
447C22×C4—941no—
448C22×C4—941no—
449Q16—22no—
450SD16—942no—
451SD16—942no—
452D8—942no—
453M16—1882no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—471no—
458C2×C8—471no—
459C2×C8—471no—
460C2×C8—471no—
461C2×C8—942no—
462C2×C8—942no—
463C2×C8—941no—
464C2×C8—941no—
465C2×C8—941no—
466C2×C8—1882no—
467C2×C8—1882no—
468C2×C8—1882no—
469C2×C8C2×C4×D4711yescharacteristic
470C2×C8C2×C4×D4711yescharacteristic
471C2×C8D4×D4711yescharacteristic
472C2×C8D4×D4711yescharacteristic
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—471no—
476C4⋊C4—471no—
477C4⋊C4—941no—
478C4⋊C4—941no—
479C4⋊C4—941no—
480C4⋊C4—941no—
481C4⋊C4C2×C4×D4711yescharacteristic
482C4⋊C4C2×C4×D4711yescharacteristic
483C22⋊C4—941no—
484C22⋊C4—941no—
485C22⋊C4—941no—
486C22⋊C4—941no—
487C22⋊C4—941no—
488C22⋊C4—941no—
489C22⋊C4—941no—
490C22⋊C4—941no—
491C42—21no—
492C42—21no—
493C42—471no—
494C42—471no—
495C42—471no—
496C42—941no—
497C42—941no—
498C42C23×D4711yescharacteristic
499C23—471no—
500C23—941no—
501C23—941no—
502Q8—21no—
503Q8—21no—
504Q8—21no—
505Q8#1309403212yes—
506D4—472no—
507D4—944no—
508D4—941no—
509D4—941no—
510D4—941no—
511D4—1882no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—471no—
517C2×C4—471no—
518C2×C4—471no—
519C2×C4—944no—
520C2×C4—941no—
521C2×C4—941no—
522C2×C4—941no—
523C2×C4—941no—
524C2×C4—941no—
525C2×C4—941no—
526C2×C4—1881no—
527C2×C4—1882no—
528C2×C4—1881no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4—1881no—
532C2×C4C22×C4×D4711yescharacteristic
533C2×C4C2×D4×D4711yescharacteristic
534C2×C4#41×D4711yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—41no—
539C8—41no—
540C8—41no—
541C8—472no—
542C8—941no—
543C8—941no—
544C8—941no—
545C8—1881no—
546C8—1881no—
547C8—1881no—
548C8#1309404712yes—
549C22—941no—
550C22—941no—
551C22—942no—
552C22—941no—
553C22—1881no—
554C22—1881no—
555C22—1881no—
556C22—1881no—
557C22—1881no—
558C22C4×D4×D4711yescharacteristic
559C4—21no—
560C4—22no—
561C4—41no—
562C4—41no—
563C4—41no—
564C4—941no—
565C4—941no—
566C4—941no—
567C4—1881no—
568C4—1881no—
569C4#131×D4711yescharacteristic
570C4C2×#1309403211yescharacteristic
571C4#135×D4711yescharacteristic
572C4C2×#1309404711yescharacteristic
573C2—3761no—
574C2—1881no—
575C2—1882no—
576C2#2025207911yescharacteristic
577C2C4×#1309404711yescharacteristic
578C2#2025228811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1584C2
socleC22×C47C2×C4×D4
prime radical#15689271C4
cosocleC42×C47C24
hypercenter#1584D47
hypocenterC47#17904
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
2. #20252362C2semidirect
3. C8×#10625354C2semidirect
4. #20242231C2semidirect
5. #20242222C2semidirect
6. #20242219C2semidirect
7. Q16×C47⋊C8C2semidirect
8. #20242185C2semidirect
9. #20242185C2semidirect
10. #20242182C2semidirect
11. #20242167C2semidirect
12. #20241821C2semidirect
13. #20241795C2semidirect
14. #20241673C2semidirect
15. C47×#1584C2semidirect
287. #1584D47semidirect
356. C47#17904semidirect

Maximal subgroups

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

subgroupindex
#202534122
#202523622
C8×#106253542
#202422312
#202422222
#202422192
Q16×C47⋊C82
#202421852
#202421822
#202421672
#202418212
#202417952
#202416732
C47×#15842
#1790447