Finite Group Databasebeta

← #26187229

Subgroups — #26187229

Order 12032.

616 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20253408C211yesmaximal, characteristic
2#20252406C211yesmaximal, characteristic, complemented
3#20252392C211yesmaximal, characteristic, complemented
4#20242280C211yesmaximal, characteristic, complemented
5#20242274C211yesmaximal, characteristic, complemented
6#20242240C211yesmaximal, characteristic, complemented
7#20242233C211yesmaximal, characteristic, complemented
8#20242233C211yesmaximal, characteristic, complemented
9#20242219C211yesmaximal, characteristic, complemented
10C8×Dic188C211yesmaximal, characteristic, complemented
11C8×#10625325C211yesmaximal, characteristic, complemented
12#20242168C211yesmaximal, characteristic, complemented
13#20241849C211yesmaximal, characteristic, complemented
14#20241771C211yesmaximal, characteristic, complemented
15C8×M16×C47C211yesmaximal, characteristic, complemented
16#15689657C2211yescharacteristic
17C4×#10625350C2211yescharacteristic
18#15689465C411yescharacteristic
19#15689448C411yescharacteristic
20#15689431C411yescharacteristic
21#15689424C411yescharacteristic
22#15689346C411yescharacteristic
23#15689339C411yescharacteristic
24#15689287C411yescharacteristic
25#15689270C411yescharacteristic
26#15689253C2211yescharacteristic
27C8×Dic94C2211yescharacteristic
28C8×Dic94C2211yescharacteristic
29C8×Dic94C2211yescharacteristic
30C4×Dic188C2211yescharacteristic
31C4×#10625325C2211yescharacteristic
32C4×#10625325C2211yescharacteristic
33C4×D376C2211yescharacteristic
34#15689202C2211yescharacteristic
35#15689202C2211yescharacteristic
36#15689202C2211yescharacteristic
37#15689199—21no—
38#15689199—21no—
39#15689199—21no—
40#15689192—21no—
41C8×D188C2211yescharacteristic
42#15689185—21no—
43#15689180C2211yescharacteristic
44#15689178—21no—
45#15689176C2211yescharacteristic
46#15689080—21no—
47#15689075—21no—
48#15688876C2211yescharacteristic
49#15688876C2211yescharacteristic
50#15688874C2211yescharacteristic
51#15688874C2211yescharacteristic
52#15688855C2211yescharacteristic
53#15688855C2211yescharacteristic
54#15688855C2211yescharacteristic
55#15688855C2211yescharacteristic
56#15688852C2211yescharacteristic
57#15688852C2211yescharacteristic
58#15688852C2211yescharacteristic
59#15688852C2211yescharacteristic
60C47×#431C2211yescharacteristic
61C4×M16×C47C2211yescharacteristic
62C2×C4×C8×C47C2211yescharacteristic
63C47×#322C2211yescharacteristic
64C47×#322C2211yescharacteristic
65C82×C47C2211yescharacteristic
66C82×C47C2211yescharacteristic
67C2×#10625350C2×C411yescharacteristic
68C2×Dic188C2×C411yescharacteristic
69C2×#10625325C2×C411yescharacteristic
70C2×#10625325C2×C411yescharacteristic
71C2×D376C2×C411yescharacteristic
72C2×#10625324—21no—
73C2×#10625324—21no—
74C2×#10625324—21no—
75C2×C8×D47—21no—
76#13094018—21no—
77#13094016C2×C411yescharacteristic
78#13094012C2×C411yescharacteristic
79#13094007—21no—
80#13094003C2×C411yescharacteristic
81#13093998C2×C411yescharacteristic
82C4×Dic94C2×C411yescharacteristic
83C4×Dic94C2311yescharacteristic
84C4×Dic94C2311yescharacteristic
85C4×D188C2×C411yescharacteristic
86C4×D188C2311yescharacteristic
87C4×D188C2311yescharacteristic
88C4×C22⋊D47—21no—
89C4×C22⋊D47—21no—
90C4×C22⋊D47—21no—
91#13093983C2×C411yescharacteristic
92#13093981—21no—
93#13093981—21no—
94C42×D47—21no—
95#13093958C2×C411yescharacteristic
96#13093958C2×C411yescharacteristic
97#13093956C2×C411yescharacteristic
98#13093956C2×C411yescharacteristic
99C4⋊(C47⋊C8)C2311yescharacteristic
100C4⋊(C47⋊C8)C2311yescharacteristic
101C4⋊(C47⋊C8)C2311yescharacteristic
102C4⋊(C47⋊C8)C2311yescharacteristic
103Dic47⋊C8—21no—
104Dic47⋊C8—21no—
105Dic47⋊C8—21no—
106Dic47⋊C8—21no—
107#13093949C2×C411yescharacteristic
108#13093949C2×C411yescharacteristic
109#13093949C2×C411yescharacteristic
110#13093949C2×C411yescharacteristic
111#13093946C2×C411yescharacteristic
112#13093946C2×C411yescharacteristic
113#13093946C2×C411yescharacteristic
114#13093946C2×C411yescharacteristic
115C22⋊(C47⋊C8)—21no—
116C22⋊(C47⋊C8)—21no—
117(C2×D47)⋊C8—21no—
118(C2×D47)⋊C8—21no—
119(C2×D47)⋊C8—21no—
120(C2×D47)⋊C8—21no—
121#13093935—21no—
122#13093934—21no—
123C4×C47⋊C8—21no—
124C8×Dic47—21no—
125C8×Dic47—21no—
126C8×Dic47—21no—
127C2×M16×C47D411yescharacteristic
128C2×M16×C47D411yescharacteristic
129C22×C8×C47—21no—
130C2×C42×C47C2311yescharacteristic
131C4⋊C8×C47C2×C411yescharacteristic
132C4⋊C8×C47C2×C411yescharacteristic
133C22⋊C8×C47C2×C411yescharacteristic
134C22⋊C8×C47C2×C411yescharacteristic
135#97×C47D411yescharacteristic
136#97×C47D411yescharacteristic
137C4×C8×C47—21no—
138C4×C8×C47C2311yescharacteristic
139C4×C8×C47C2311yescharacteristic
140C4×C8×C47C2311yescharacteristic
141C4×C8×C47C2311yescharacteristic
142C4×C8×C47C2311yescharacteristic
143C4×C8×C47C2311yescharacteristic
144C22×C4×C47C22×C411yescharacteristic
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47#4111yescharacteristic
147C2×C22⋊D47—21no—
148C2×C22⋊D47—21no—
149#10625350—22no—
150#10625350—41no—
151C2×D188C22×C411yescharacteristic
152C2×D188C22×C411yescharacteristic
153C2×C4×D47—21no—
154C2×C4×D47—21no—
155C2×C4×D47—21no—
156C2×C4×D47—21no—
157C2×Dic94C22×C411yescharacteristic
158C2×Dic94C22×C411yescharacteristic
159M16×C47—22no—
160M16×C47—22no—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47—42no—
164C2×C8×C47C22×C411yescharacteristic
165C2×C8×C47C22×C411yescharacteristic
166C2×C8×C47C22×C411yescharacteristic
167C2×C8×C47C22×C411yescharacteristic
168C2×C8×C47C2×D411yescharacteristic
169C2×C8×C47C2×D411yescharacteristic
170C2×C8×C47C2×D411yescharacteristic
171C2×C8×C47C2×D411yescharacteristic
172C42×C47C22×C411yescharacteristic
173C42×C47C2×D411yescharacteristic
174C42×C47#4111yescharacteristic
175C42×C47C2411yescharacteristic
176C22⋊Dic47—21no—
177C22⋊Dic47—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⋊Dic47C22×C411yescharacteristic
185C4⋊Dic47C22×C411yescharacteristic
186C4⋊Dic47C22×C411yescharacteristic
187C4⋊Dic47C22×C411yescharacteristic
188Dic47⋊C4—21no—
189Dic47⋊C4—21no—
190Dic47⋊C4—21no—
191Dic47⋊C4—21no—
192Dic47⋊C4—21no—
193Dic47⋊C4—21no—
194C4×Dic47—21no—
195C4×Dic47—21no—
196C4×Dic47—21no—
197C4×Dic47—21no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202Dic188—22no—
203D376—41no—
204#10625325—22no—
205#10625325—41no—
206#10625324—42no—
207#10625324—42no—
208#10625324—42no—
209C8×D47—42no—
210C23×C47C4×D411yescharacteristic
211C22×D47—21no—
212C22×D47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—41no—
218C2×C4×C47C4×D411yescharacteristic
219C2×C4×C47C4×D411yescharacteristic
220C2×C4×C47C4×D411yescharacteristic
221C2×C4×C47C23×C411yescharacteristic
222C2×C4×C47C22×D411yescharacteristic
223C2×C4×C47C2×#4111yescharacteristic
224C22⋊D47—42no—
225C22⋊D47—81no—
226C2×Dic47—21no—
227C2×Dic47—21no—
228C2×Dic47—21no—
229C2×Dic47—21no—
230C2×Dic47—21no—
231C2×Dic47—21no—
232D188—21no—
233D188—21no—
234D188—21no—
235D188—21no—
236C4×D47—42no—
237C4×D47—41no—
238C4×D47—41no—
239C4×D47—41no—
240C4×D47—41no—
241C4×D47—81no—
242Dic94—21no—
243Dic94—21no—
244Dic94—21no—
245Dic94#13112yes—
246C8×C47—22no—
247C8×C47—21no—
248C8×C47—21no—
249C8×C47—21no—
250C8×C47—21no—
251C8×C47—21no—
252C8×C47—21no—
253C8×C47—41no—
254C8×C47#13512yes—
255C8×C47#13512yes—
256C47⋊C8—41no—
257C47⋊C8—41no—
258C47⋊C8—41no—
259C47⋊C8—41no—
260#17999—471nomaximal, Sylow
261C22×C47—21no—
262C22×C47—21no—
263C22×C47—21no—
264C22×C47C2×C4×D411yescharacteristic
265C2×D47—41no—
266C2×D47—41no—
267C2×D47—41no—
268C2×D47—41no—
269C2×D47—41no—
270C2×D47—41no—
271C4×C47—21no—
272C4×C47—21no—
273C4×C47—21no—
274C4×C47—21no—
275C4×C47C2×#13111yescharacteristic
276C4×C47#56811yescharacteristic
277C4×C47C2×#13511yescharacteristic
278C4×C47C2×#13511yescharacteristic
279Dic47—22no—
280Dic47—41no—
281Dic47—41no—
282Dic47—41no—
283Dic47—41no—
284Dic47—41no—
285#3102—471no—
286#2971—471no—
287#2969—471no—
288#1590—471no—
289#1589—471no—
290#1584—471no—
291#1583—471no—
292#1583—471no—
293#1581—471no—
294C8×Q16—471no—
295C8×SD16—471no—
296#1572—471no—
297#1494—471no—
298#1476—471no—
299C8×M16D4711yescharacteristic, complemented
300C2×C47—41no—
301C2×C47#293311yescharacteristic
302C2×C47C4×#13511yescharacteristic
303C2×C47#295611yescharacteristic
304D47—81no—
305D47—81no—
306#529—471no—
307C4×#41—471no—
308#485—471no—
309#482—471no—
310#479—471no—
311#478—471no—
312#464—471no—
313#463—471no—
314#452—471no—
315#449—471no—
316#446—471no—
317C8×Q8—471no—
318C8×Q8—471no—
319C8×Q8—471no—
320C4×Q16—471no—
321C4×SD16—471no—
322C4×SD16—471no—
323C4×D8—471no—
324#436—471no—
325#436—471no—
326#436—471no—
327#436—941no—
328#436—941no—
329#436—941no—
330#435—941no—
331C8×D4—471no—
332#433—941no—
333#432—471no—
334#432—941no—
335#431—471no—
336#431C2×D4711yescharacteristic
337#405—941no—
338C4×M16—941no—
339C4×M16C2×D4711yescharacteristic
340C2×C4×C8C2×D4711yescharacteristic
341#335—471no—
342#335—471no—
343#334—471no—
344#334—471no—
345Q8⋊C8—471no—
346Q8⋊C8—471no—
347Q8⋊C8—471no—
348Q8⋊C8—471no—
349D4⋊C8—471no—
350D4⋊C8—471no—
351D4⋊C8—471no—
352D4⋊C8—471no—
353#322C2×D4711yescharacteristic
354#322C2×D4711yescharacteristic
355C82C2×D4711yescharacteristic
356C82C2×D4711yescharacteristic
357C47#1799911yescharacteristic, complemented, Sylow
358C2×#41—471no—
359C2×Q16—471no—
360C2×SD16—471no—
361C2×SD16—471no—
362C2×D8—471no—
363C2×M16—941no—
364C2×M16—941no—
365C2×M16—941no—
366C2×M16D18811yescharacteristic
367C2×M16D18811yescharacteristic
368C22×C8—21no—
369C22×C8—941no—
370#126—941no—
371#125—471no—
372#124—471no—
373#123—941no—
374C22⋊Q8—471no—
375#121—471no—
376C4×Q8—471no—
377C4×Q8—471no—
378C4×Q8—471no—
379C4×D4—471no—
380C4×D4—471no—
381C4×D4—471no—
382C4×D4—941no—
383C4×D4—941no—
384C4×D4—941no—
385#117—471no—
386#117—941no—
387#117—941no—
388C2×C42—941no—
389C2×C42C22×D4711yescharacteristic
390#107—471no—
391#107—471no—
392#106—471no—
393#106—471no—
394C4⋊C8—471no—
395C4⋊C8—471no—
396C4⋊C8—471no—
397C4⋊C8—471no—
398C4⋊C8—941no—
399C4⋊C8—941no—
400C4⋊C8—941no—
401C4⋊C8—941no—
402C4⋊C8C4×D4711yescharacteristic
403C4⋊C8C4×D4711yescharacteristic
404#103—471no—
405#103—471no—
406#103—471no—
407#103—471no—
408#102—471no—
409#102—471no—
410#102—471no—
411#102—471no—
412C22⋊C8—941no—
413C22⋊C8—941no—
414C22⋊C8—941no—
415C22⋊C8—941no—
416C22⋊C8—941no—
417C22⋊C8—941no—
418C22⋊C8C4×D4711yescharacteristic
419C22⋊C8C4×D4711yescharacteristic
420#97—941no—
421#97—941no—
422#97D18811yescharacteristic
423#97D18811yescharacteristic
424C4×C8—21no—
425C4×C8—941no—
426C4×C8—941no—
427C4×C8—941no—
428C4×C8—941no—
429C4×C8C22×D4711yescharacteristic
430C4×C8C22×D4711yescharacteristic
431C4×C8C22×D4711yescharacteristic
432C4×C8C22×D4711yescharacteristic
433C4×C8C22×D4711yescharacteristic
434C4×C8C22×D4711yescharacteristic
435#41—942no—
436#41—1881no—
437C2×Q8—471no—
438C2×Q8—471no—
439C2×D4—471no—
440C2×D4—471no—
441C2×D4—941no—
442C2×D4—941no—
443C22×C4—941no—
444C22×C4—941no—
445C22×C4—941no—
446C22×C4—941no—
447C22×C4C2×C4×D4711yescharacteristic
448C22×C4C2×D18811yescharacteristic
449C22×C4#1062535011yescharacteristic
450Q16—942no—
451SD16—942no—
452SD16—1881no—
453D8—1881no—
454M16—22no—
455M16—22no—
456M16—1882no—
457M16—1882no—
458M16—1882no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—42no—
462C2×C8—941no—
463C2×C8—941no—
464C2×C8—941no—
465C2×C8—941no—
466C2×C8—1882no—
467C2×C8C2×C4×D4711yescharacteristic
468C2×C8C2×C4×D4711yescharacteristic
469C2×C8C2×C4×D4711yescharacteristic
470C2×C8C2×C4×D4711yescharacteristic
471C2×C8C2×D18811yescharacteristic
472C2×C8C2×D18811yescharacteristic
473C2×C8C2×D18811yescharacteristic
474C2×C8C2×D18811yescharacteristic
475C4⋊C4—471no—
476C4⋊C4—471no—
477C4⋊C4—471no—
478C4⋊C4—471no—
479C4⋊C4—941no—
480C4⋊C4—941no—
481C4⋊C4—941no—
482C4⋊C4—941no—
483C4⋊C4—941no—
484C4⋊C4—941no—
485C22⋊C4—941no—
486C22⋊C4—941no—
487C22⋊C4—941no—
488C22⋊C4—941no—
489C22⋊C4—941no—
490C22⋊C4—941no—
491C22⋊C4—941no—
492C22⋊C4—941no—
493C42—941no—
494C42—941no—
495C42—941no—
496C42—941no—
497C42C2×C4×D4711yescharacteristic
498C42C2×D18811yescharacteristic
499C42#1062535011yescharacteristic
500C42C23×D4711yescharacteristic
501C23—941no—
502C23—941no—
503C23C4×D18811yescharacteristic
504Q8—472no—
505Q8—941no—
506Q8—941no—
507Q8—941no—
508D4—941no—
509D4—941no—
510D4—941no—
511D4—941no—
512D4—3761no—
513D4—1882no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—41no—
519C2×C4—941no—
520C2×C4—941no—
521C2×C4—941no—
522C2×C4—941no—
523C2×C4—941no—
524C2×C4—941no—
525C2×C4—3761no—
526C2×C4—1882no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4—1881no—
530C2×C4—1881no—
531C2×C4C4×D18811yescharacteristic
532C2×C4C4×D18811yescharacteristic
533C2×C4C4×D18811yescharacteristic
534C2×C4C22×C4×D4711yescharacteristic
535C2×C4C22×D18811yescharacteristic
536C2×C4C2×#1062535011yescharacteristic
537C8—22no—
538C8—21no—
539C8—21no—
540C8—21no—
541C8—21no—
542C8—21no—
543C8—21no—
544C8—41no—
545C8—1881no—
546C8—1881no—
547C8—1881no—
548C8—1881no—
549C8#1309404612yes—
550C8#1309404612yes—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—1881no—
555C22—1881no—
556C22—1881no—
557C22—1881no—
558C22—1881no—
559C22—1881no—
560C22C2×C4×D18811yescharacteristic
561C4—21no—
562C4—21no—
563C4—21no—
564C4—21no—
565C4—942no—
566C4—1881no—
567C4—1881no—
568C4—1881no—
569C4—1881no—
570C4—1881no—
571C4#131×D4711yescharacteristic
572C4#1568988811yescharacteristic
573C4C2×#1309404611yescharacteristic
574C4C2×#1309404611yescharacteristic
575C2—41no—
576C2—3761no—
577C2—3761no—
578C2#2025211611yescharacteristic
579C2C4×#1309404611yescharacteristic
580C2#2025229911yescharacteristic
subgroupG/H
centerC2×C4C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC8×M16×C47C2
socleC22×C47C2×C4×D4
prime radical#15689270C4
cosocleC42×C47C24
hypercenterC8×M16D47
hypocenterC47#17999
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C4×M16 ◁ C8×M16

Complements

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

NH ≅ G/Ntype
2. #20252406C2semidirect
3. #20252392C2semidirect
4. #20242280C2semidirect
5. #20242274C2semidirect
6. #20242240C2semidirect
7. #20242233C2semidirect
8. #20242233C2semidirect
9. #20242219C2semidirect
10. C8×Dic188C2semidirect
11. C8×#10625325C2semidirect
12. #20242168C2semidirect
13. #20241849C2semidirect
14. #20241771C2semidirect
15. C8×M16×C47C2semidirect
299. C8×M16D47semidirect
357. C47#17999semidirect

Maximal subgroups

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

subgroupindex
#202534082
#202524062
#202523922
#202422802
#202422742
#202422402
#202422332
#202422192
C8×Dic1882
C8×#106253252
#202421682
#202418492
#202417712
C8×M16×C472
#1799947