Finite Group Databasebeta

← #26187223

Subgroups — #26187223

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#20252405C211yesmaximal, characteristic, complemented
3#20252386C211yesmaximal, characteristic, complemented
4#20242277C211yesmaximal, characteristic, complemented
5#20242272C211yesmaximal, characteristic, complemented
6#20242237C211yesmaximal, characteristic, complemented
7#20242230C211yesmaximal, characteristic, complemented
8#20242230C211yesmaximal, characteristic, complemented
9#20242216C211yesmaximal, characteristic, complemented
10#20242194C211yesmaximal, characteristic, complemented
11#20242187C211yesmaximal, characteristic, complemented
12#20241846C211yesmaximal, characteristic, complemented
13#20241768C211yesmaximal, characteristic, complemented
14#20241669C211yesmaximal, characteristic, complemented
15C47×#1572C211yesmaximal, characteristic, complemented
16#15689657C2211yescharacteristic
17C4×#10625350C2211yescharacteristic
18#15689465C411yescharacteristic
19#15689448C411yescharacteristic
20#15689431C411yescharacteristic
21#15689424C411yescharacteristic
22#15689346C411yescharacteristic
23#15689339C411yescharacteristic
24#15689287C411yescharacteristic
25#15689270C411yescharacteristic
26#15689250C2211yescharacteristic
27#15689247C2211yescharacteristic
28#15689247C2211yescharacteristic
29#15689247C2211yescharacteristic
30C4×Dic188C2211yescharacteristic
31C4×#10625325C2211yescharacteristic
32C4×#10625325C2211yescharacteristic
33C4×D376C2211yescharacteristic
34#15689200C2211yescharacteristic
35#15689200C2211yescharacteristic
36#15689200C2211yescharacteristic
37#15689198—21no—
38#15689198—21no—
39#15689198—21no—
40#15689197—21no—
41#15689188C2211yescharacteristic
42#15689182—21no—
43#15689178—21no—
44#15689077—21no—
45C4×#10625329C2211yescharacteristic
46#15689075—21no—
47C2×C4×C47⋊C8C2211yescharacteristic
48#15688853C2211yescharacteristic
49#15688853C2211yescharacteristic
50#15688853C2211yescharacteristic
51#15688853C2211yescharacteristic
52#15688850C2211yescharacteristic
53#15688850C2211yescharacteristic
54#15688850C2211yescharacteristic
55#15688850C2211yescharacteristic
56#15688845C2211yescharacteristic
57#15688845C2211yescharacteristic
58C8×C47⋊C8C2211yescharacteristic
59C8×C47⋊C8C2211yescharacteristic
60C47×#432C2211yescharacteristic
61C47×#431C2211yescharacteristic
62C47×#431C2211yescharacteristic
63C47×#335C2211yescharacteristic
64C47×#335C2211yescharacteristic
65C47×#334C2211yescharacteristic
66C47×#334C2211yescharacteristic
67C2×#10625350C2×C411yescharacteristic
68C2×Dic188C2×C411yescharacteristic
69C2×#10625325C2×C411yescharacteristic
70C2×#10625325C2×C411yescharacteristic
71C2×D376C2×C411yescharacteristic
72C2×#10625329D411yescharacteristic
73C2×#10625329D411yescharacteristic
74C2×#10625324—21no—
75C2×#10625324—21no—
76C2×#10625324—21no—
77C22×C47⋊C8—21no—
78C2×C8×D47—21no—
79#13094018—21no—
80#13094016C2×C411yescharacteristic
81#13094012C2×C411yescharacteristic
82#13094007—21no—
83#13094003C2×C411yescharacteristic
84#13093998C2×C411yescharacteristic
85C4×Dic94C2×C411yescharacteristic
86C4×Dic94C2311yescharacteristic
87C4×Dic94C2311yescharacteristic
88C4×D188C2×C411yescharacteristic
89C4×D188C2311yescharacteristic
90C4×D188C2311yescharacteristic
91C4×C22⋊D47—21no—
92C4×C22⋊D47—21no—
93C4×C22⋊D47—21no—
94#13093983C2×C411yescharacteristic
95#13093981—21no—
96#13093981—21no—
97C42×D47—21no—
98#13093958C2×C411yescharacteristic
99#13093958C2×C411yescharacteristic
100#13093956C2×C411yescharacteristic
101#13093956C2×C411yescharacteristic
102Dic47⋊C8—21no—
103Dic47⋊C8—21no—
104Dic47⋊C8—21no—
105Dic47⋊C8—21no—
106#13093949C2×C411yescharacteristic
107#13093949C2×C411yescharacteristic
108#13093949C2×C411yescharacteristic
109#13093949C2×C411yescharacteristic
110#13093946C2×C411yescharacteristic
111#13093946C2×C411yescharacteristic
112#13093946C2×C411yescharacteristic
113#13093946C2×C411yescharacteristic
114(C2×D47)⋊C8—21no—
115(C2×D47)⋊C8—21no—
116(C2×D47)⋊C8—21no—
117(C2×D47)⋊C8—21no—
118#13093935D411yescharacteristic
119#13093935D411yescharacteristic
120#13093934—21no—
121C4×C47⋊C8—21no—
122C4×C47⋊C8C2311yescharacteristic
123C4×C47⋊C8C2311yescharacteristic
124C4×C47⋊C8C2311yescharacteristic
125C4×C47⋊C8C2311yescharacteristic
126C8×Dic47—21no—
127C8×Dic47—21no—
128C8×Dic47—21no—
129C2×C42×C47C2311yescharacteristic
130C4⋊C8×C47C2×C411yescharacteristic
131C4⋊C8×C47C2×C411yescharacteristic
132C4⋊C8×C47C2311yescharacteristic
133C4⋊C8×C47C2311yescharacteristic
134C4⋊C8×C47C2311yescharacteristic
135C4⋊C8×C47C2311yescharacteristic
136C22⋊C8×C47—21no—
137C22⋊C8×C47—21no—
138C22⋊C8×C47C2×C411yescharacteristic
139C22⋊C8×C47C2×C411yescharacteristic
140#97×C47—21no—
141C4×C8×C47—21no—
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
159C2×C8×C47—21no—
160C2×C8×C47—21no—
161C2×C8×C47—21no—
162C2×C8×C47—21no—
163C2×C8×C47C22×C411yescharacteristic
164C2×C8×C47C22×C411yescharacteristic
165C2×C8×C47C22×C411yescharacteristic
166C2×C8×C47C22×C411yescharacteristic
167C42×C47C22×C411yescharacteristic
168C42×C47C2×D411yescharacteristic
169C42×C47#4111yescharacteristic
170C42×C47C2411yescharacteristic
171C22⋊Dic47—21no—
172C22⋊Dic47—21no—
173(C2×D47)⋊C4—21no—
174(C2×D47)⋊C4—21no—
175(C2×D47)⋊C4—21no—
176(C2×D47)⋊C4—21no—
177(C2×D47)⋊C4—21no—
178(C2×D47)⋊C4—21no—
179C4⋊Dic47C22×C411yescharacteristic
180C4⋊Dic47C22×C411yescharacteristic
181C4⋊Dic47C22×C411yescharacteristic
182C4⋊Dic47C22×C411yescharacteristic
183Dic47⋊C4—21no—
184Dic47⋊C4—21no—
185Dic47⋊C4—21no—
186Dic47⋊C4—21no—
187Dic47⋊C4—21no—
188Dic47⋊C4—21no—
189C4×Dic47—21no—
190C4×Dic47—21no—
191C4×Dic47—21no—
192C4×Dic47—21no—
193#10625329—22no—
194#10625329—22no—
195C2×C47⋊C8—21no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—42no—
198C2×C47⋊C8C2×D411yescharacteristic
199C2×C47⋊C8C2×D411yescharacteristic
200C2×C47⋊C8C2×D411yescharacteristic
201C2×C47⋊C8C2×D411yescharacteristic
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—21no—
247C8×C47—21no—
248C8×C47—21no—
249C8×C47—21no—
250C8×C47—41no—
251C8×C47—41no—
252C8×C47—41no—
253C8×C47—41no—
254C47⋊C8—22no—
255C47⋊C8—21no—
256C47⋊C8—21no—
257C47⋊C8—41no—
258C47⋊C8#13512yes—
259C47⋊C8#13512yes—
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#1572D4711yescharacteristic, complemented
297#1494—471no—
298#1476—471no—
299C8×M16—471no—
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—941no—
334#432C2×D4711yescharacteristic
335#431C2×D4711yescharacteristic
336#431C2×D4711yescharacteristic
337#405—941no—
338C4×M16—471no—
339C4×M16—941no—
340C2×C4×C8—471no—
341#335C2×D4711yescharacteristic
342#335C2×D4711yescharacteristic
343#334C2×D4711yescharacteristic
344#334C2×D4711yescharacteristic
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#322—471no—
354#322—471no—
355C82—471no—
356C82—471no—
357C47#1799911yescharacteristic, complemented, Sylow
358C2×#41—471no—
359C2×Q16—471no—
360C2×SD16—471no—
361C2×SD16—471no—
362C2×D8—471no—
363C2×M16—471no—
364C2×M16—471no—
365C2×M16—941no—
366C2×M16—941no—
367C2×M16—941no—
368C22×C8—941no—
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—941no—
395C4⋊C8—941no—
396C4⋊C8—941no—
397C4⋊C8—941no—
398C4⋊C8C4×D4711yescharacteristic
399C4⋊C8C4×D4711yescharacteristic
400C4⋊C8C22×D4711yescharacteristic
401C4⋊C8C22×D4711yescharacteristic
402C4⋊C8C22×D4711yescharacteristic
403C4⋊C8C22×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—21no—
413C22⋊C8—21no—
414C22⋊C8—941no—
415C22⋊C8—941no—
416C22⋊C8—941no—
417C22⋊C8—941no—
418C22⋊C8C4×D4711yescharacteristic
419C22⋊C8C4×D4711yescharacteristic
420#97—21no—
421#97—471no—
422#97—471no—
423#97—941no—
424C4×C8—21no—
425C4×C8—471no—
426C4×C8—471no—
427C4×C8—471no—
428C4×C8—471no—
429C4×C8—941no—
430C4×C8—941no—
431C4×C8—941no—
432C4×C8—941no—
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×C4D4×D4711yescharacteristic
449C22×C4#1062535411yescharacteristic
450Q16—942no—
451SD16—942no—
452SD16—1881no—
453D8—1881no—
454M16—942no—
455M16—942no—
456M16—1882no—
457M16—1882no—
458M16—1882no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—471no—
464C2×C8—471no—
465C2×C8—471no—
466C2×C8—471no—
467C2×C8—941no—
468C2×C8—941no—
469C2×C8—1882no—
470C2×C8—1882no—
471C2×C8C2×C4×D4711yescharacteristic
472C2×C8C2×C4×D4711yescharacteristic
473C2×C8C2×C4×D4711yescharacteristic
474C2×C8C2×C4×D4711yescharacteristic
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
498C42D4×D4711yescharacteristic
499C42#1062535411yescharacteristic
500C42C23×D4711yescharacteristic
501C23—941no—
502C23—941no—
503C23#1309398611yescharacteristic
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×C4#1309398611yescharacteristic
532C2×C4#1309398611yescharacteristic
533C2×C4#1309398611yescharacteristic
534C2×C4C22×C4×D4711yescharacteristic
535C2×C4C2×D4×D4711yescharacteristic
536C2×C4C2×#1062535411yescharacteristic
537C8—21no—
538C8—21no—
539C8—21no—
540C8—21no—
541C8—41no—
542C8—41no—
543C8—41no—
544C8—41no—
545C8—472no—
546C8—472no—
547C8—942no—
548C8—941no—
549C8—941no—
550C8—1881no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—1881no—
555C22—1881no—
556C22—1881no—
557C22—1881no—
558C22—1881no—
559C22—1881no—
560C22C2×#1309398611yescharacteristic
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
573C4#135×D4711yescharacteristic
574C4#135×D4711yescharacteristic
575C2—41no—
576C2—3761no—
577C2—3761no—
578C2#2025211211yescharacteristic
579C2#2025215911yescharacteristic
580C2#2025229711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1572C2
socleC22×C47C2×C4×D4
prime radical#15689270C4
cosocleC42×C47C24
hypercenter#1572D47
hypocenterC47#17999
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1572

Complements

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

NH ≅ G/Ntype
2. #20252405C2semidirect
3. #20252386C2semidirect
4. #20242277C2semidirect
5. #20242272C2semidirect
6. #20242237C2semidirect
7. #20242230C2semidirect
8. #20242230C2semidirect
9. #20242216C2semidirect
10. #20242194C2semidirect
11. #20242187C2semidirect
12. #20241846C2semidirect
13. #20241768C2semidirect
14. #20241669C2semidirect
15. C47×#1572C2semidirect
296. #1572D47semidirect
357. C47#17999semidirect

Maximal subgroups

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

subgroupindex
#202534082
#202524052
#202523862
#202422772
#202422722
#202422372
#202422302
#202422162
#202421942
#202421872
#202418462
#202417682
#202416692
C47×#15722
#1799947