Finite Group Databasebeta

← #26186233

Subgroups — #26186233

Order 12032.

626 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#20253408C211yesmaximal, characteristic
2#20252357C211yesmaximal, characteristic, complemented
3#20252344C211yesmaximal, characteristic, complemented
4#20242265C211yesmaximal, characteristic, complemented
5#20242258C211yesmaximal, characteristic, complemented
6#20242251C211yesmaximal, characteristic, complemented
7#20242244C211yesmaximal, characteristic, complemented
8#20242211C211yesmaximal, characteristic, complemented
9#20242206C211yesmaximal, characteristic, complemented
10#20242199C211yesmaximal, characteristic, complemented
11#20242199C211yesmaximal, characteristic, complemented
12#20241820C211yesmaximal, characteristic, complemented
13#20241794C211yesmaximal, characteristic, complemented
14#20241681C211yesmaximal, 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#15689191—21no—
36#15689188C2211yescharacteristic
37#15689188C2211yescharacteristic
38#15689188C2211yescharacteristic
39(C22⋊D47)⋊C8—21no—
40(C22⋊D47)⋊C8—21no—
41(C22⋊D47)⋊C8—21no—
42#15689185—21no—
43#15689176C2211yescharacteristic
44#15689174—21no—
45#15689080—21no—
46C2×C4×C47⋊C8C2211yescharacteristic
47C4×C8×D47—21no—
48#15688853C2211yescharacteristic
49#15688853C2211yescharacteristic
50#15688853C2211yescharacteristic
51#15688853C2211yescharacteristic
52#15688850C2211yescharacteristic
53#15688850C2211yescharacteristic
54#15688850C2211yescharacteristic
55#15688850C2211yescharacteristic
56#15688845C2211yescharacteristic
57#15688845C2211yescharacteristic
58#15688844C2211yescharacteristic
59#15688844C2211yescharacteristic
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×#10625324—21no—
73C22×C47⋊C8D411yescharacteristic
74C22×C47⋊C8D411yescharacteristic
75C2×C8×D47—21no—
76C2×C8×D47—21no—
77C2×C8×D47—21no—
78#13094018—21no—
79#13094016C2×C411yescharacteristic
80#13094012C2×C411yescharacteristic
81#13094007—21no—
82#13094003C2×C411yescharacteristic
83#13093998C2×C411yescharacteristic
84C4×Dic94C2×C411yescharacteristic
85C4×Dic94C2311yescharacteristic
86C4×Dic94C2311yescharacteristic
87C4×D188C2×C411yescharacteristic
88C4×D188C2311yescharacteristic
89C4×D188C2311yescharacteristic
90C4×C22⋊D47—21no—
91C4×C22⋊D47—21no—
92C4×C22⋊D47—21no—
93#13093983C2×C411yescharacteristic
94#13093981—21no—
95#13093981—21no—
96C42×D47—21no—
97#13093958C2×C411yescharacteristic
98#13093958C2×C411yescharacteristic
99#13093956C2×C411yescharacteristic
100#13093956C2×C411yescharacteristic
101C4⋊(C47⋊C8)—21no—
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
114C22⋊(C47⋊C8)—21no—
115(C2×D47)⋊C8—21no—
116(C2×D47)⋊C8—21no—
117(C2×D47)⋊C8—21no—
118(C2×D47)⋊C8—21no—
119#13093934—21no—
120C4×C47⋊C8D411yescharacteristic
121C4×C47⋊C8D411yescharacteristic
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—22no—
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—
193C2×C47⋊C8—21no—
194C2×C47⋊C8—21no—
195C2×C47⋊C8—41no—
196C2×C47⋊C8—41no—
197C2×C47⋊C8C2×D411yescharacteristic
198C2×C47⋊C8C2×D411yescharacteristic
199C2×C47⋊C8C2×D411yescharacteristic
200C2×C47⋊C8C2×D411yescharacteristic
201Dic188—22no—
202D376—22no—
203#10625325—22no—
204#10625325—22no—
205#10625324—42no—
206C8×D47—42no—
207C8×D47—42no—
208C8×D47—42no—
209C23×C47C4×D411yescharacteristic
210C22×D47—21no—
211C22×D47—21no—
212C2×C4×C47—21no—
213C2×C4×C47—21no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—41no—
217C2×C4×C47C4×D411yescharacteristic
218C2×C4×C47C4×D411yescharacteristic
219C2×C4×C47C4×D411yescharacteristic
220C2×C4×C47C23×C411yescharacteristic
221C2×C4×C47C22×D411yescharacteristic
222C2×C4×C47C2×#4111yescharacteristic
223C22⋊D47—24no—
224C22⋊D47—42no—
225C2×Dic47—21no—
226C2×Dic47—21no—
227C2×Dic47—21no—
228C2×Dic47—21no—
229C2×Dic47—21no—
230C2×Dic47—21no—
231D188—21no—
232D188—21no—
233D188—21no—
234D188#13112yes—
235C4×D47—24no—
236C4×D47—42no—
237C4×D47—41no—
238C4×D47—41no—
239C4×D47—41no—
240C4×D47—41no—
241Dic94—21no—
242Dic94—21no—
243Dic94—21no—
244Dic94#13112yes—
245C8×C47—21no—
246C8×C47—21no—
247C8×C47—21no—
248C8×C47—21no—
249C8×C47—41no—
250C8×C47—41no—
251C8×C47—41no—
252C8×C47—41no—
253C47⋊C8—21no—
254C47⋊C8—21no—
255C47⋊C8—21no—
256C47⋊C8—21no—
257C47⋊C8—41no—
258C47⋊C8—41no—
259#17909—471nomaximal, Sylow
260C22×C47—21no—
261C22×C47—21no—
262C22×C47—21no—
263C22×C47C2×C4×D411yescharacteristic
264C2×D47—22no—
265C2×D47—41no—
266C2×D47—41no—
267C2×D47—41no—
268C2×D47—41no—
269C2×D47—41no—
270C4×C47—21no—
271C4×C47—21no—
272C4×C47—21no—
273C4×C47—21no—
274C4×C47C2×#13111yescharacteristic
275C4×C47C2×#13111yescharacteristic
276C4×C47#57511yescharacteristic
277C4×C47#57511yescharacteristic
278Dic47—22no—
279Dic47—41no—
280Dic47—41no—
281Dic47—41no—
282Dic47—41no—
283Dic47—41no—
284#3102—471no—
285#2966—471no—
286C8×#41—471no—
287#1588—471no—
288#1587—471no—
289#1586—471no—
290#1585—471no—
291#1580—471no—
292#1579—471no—
293SD16⋊C8—471no—
294SD16⋊C8—471no—
295#1572D4711yescharacteristic, complemented
296#1488—471no—
297#1482—471no—
298#1452—471no—
299C2×C47—41no—
300C2×C47#292911yescharacteristic
301C2×C47#294311yescharacteristic
302C2×C47#295511yescharacteristic
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#446—471no—
316C8×Q8—471no—
317C8×Q8—471no—
318C8×Q8—471no—
319C4×Q16—471no—
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#432C2×D4711yescharacteristic
333#431—471no—
334#431—941no—
335#431C2×D4711yescharacteristic
336#431C2×D4711yescharacteristic
337#405—941no—
338C2×C4×C8—471no—
339C2×C4×C8—941no—
340#335C2×D4711yescharacteristic
341#335C2×D4711yescharacteristic
342#334C2×D4711yescharacteristic
343#334C2×D4711yescharacteristic
344Q8⋊C8—471no—
345Q8⋊C8—471no—
346Q8⋊C8—471no—
347Q8⋊C8—471no—
348D4⋊C8—471no—
349D4⋊C8—471no—
350D4⋊C8—471no—
351D4⋊C8—471no—
352#322—471no—
353#322—471no—
354#322—471no—
355#322—471no—
356C47#1790911yescharacteristic, complemented, Sylow
357C2×#41—471no—
358C2×Q16—471no—
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×Q8—471no—
376C4×Q8—471no—
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—941no—
387C2×C42C22×D4711yescharacteristic
388#107—471no—
389#107—471no—
390#106—471no—
391#106—471no—
392C4⋊C8—941no—
393C4⋊C8—941no—
394C4⋊C8—941no—
395C4⋊C8—941no—
396C4⋊C8—941no—
397C4⋊C8C4×D4711yescharacteristic
398C4⋊C8C4×D4711yescharacteristic
399C4⋊C8C22×D4711yescharacteristic
400C4⋊C8C22×D4711yescharacteristic
401C4⋊C8C22×D4711yescharacteristic
402C4⋊C8C22×D4711yescharacteristic
403#103—471no—
404#103—471no—
405#103—471no—
406#103—471no—
407#102—471no—
408#102—471no—
409#102—471no—
410#102—471no—
411C22⋊C8—21no—
412C22⋊C8—21no—
413C22⋊C8—941no—
414C22⋊C8—941no—
415C22⋊C8—941no—
416C22⋊C8—941no—
417C22⋊C8—941no—
418C22⋊C8C4×D4711yescharacteristic
419C22⋊C8C4×D4711yescharacteristic
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—471no—
429C4×C8—941no—
430C4×C8—941no—
431C4×C8—941no—
432C4×C8C22×D4711yescharacteristic
433C4×C8C22×D4711yescharacteristic
434#41—942no—
435#41—942no—
436C2×Q8—471no—
437C2×Q8—471no—
438C2×D4—471no—
439C2×D4—471no—
440C2×D4—941no—
441C2×D4—941no—
442C22×C4—941no—
443C22×C4—941no—
444C22×C4—941no—
445C22×C4—941no—
446C22×C4C2×C4×D4711yescharacteristic
447C22×C4D4×D4711yescharacteristic
448C22×C4#1062535411yescharacteristic
449Q16—942no—
450SD16—942no—
451SD16—942no—
452D8—942no—
453M16—1882no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—471no—
459C2×C8—471no—
460C2×C8—471no—
461C2×C8—471no—
462C2×C8—941no—
463C2×C8—941no—
464C2×C8—1882no—
465C2×C8—1882no—
466C2×C8—1882no—
467C2×C8—1881no—
468C2×C8—1881no—
469C2×C8C2×C4×D4711yescharacteristic
470C2×C8C2×C4×D4711yescharacteristic
471C2×C8C2×C4×D4711yescharacteristic
472C2×C8C2×C4×D4711yescharacteristic
473C4⋊C4—471no—
474C4⋊C4—471no—
475C4⋊C4—471no—
476C4⋊C4—471no—
477C4⋊C4—941no—
478C4⋊C4—941no—
479C4⋊C4—941no—
480C4⋊C4—941no—
481C4⋊C4—941no—
482C4⋊C4—941no—
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—941no—
492C42—941no—
493C42—941no—
494C42—941no—
495C42C2×C4×D4711yescharacteristic
496C42D4×D4711yescharacteristic
497C42#1062535411yescharacteristic
498C42C23×D4711yescharacteristic
499C23—941no—
500C23—941no—
501C23#1309398611yescharacteristic
502Q8—472no—
503Q8—941no—
504Q8—941no—
505Q8—941no—
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—41no—
517C2×C4—944no—
518C2×C4—941no—
519C2×C4—941no—
520C2×C4—941no—
521C2×C4—941no—
522C2×C4—941no—
523C2×C4—941no—
524C2×C4—1882no—
525C2×C4—1881no—
526C2×C4—1881no—
527C2×C4—1881no—
528C2×C4—1881no—
529C2×C4#1309398611yescharacteristic
530C2×C4#1309398611yescharacteristic
531C2×C4#1309398611yescharacteristic
532C2×C4C22×C4×D4711yescharacteristic
533C2×C4C2×D4×D4711yescharacteristic
534C2×C4C2×#1062535411yescharacteristic
535C8—21no—
536C8—21no—
537C8—21no—
538C8—21no—
539C8—41no—
540C8—41no—
541C8—41no—
542C8—41no—
543C8—941no—
544C8—941no—
545C8—941no—
546C8—941no—
547C8—1881no—
548C8—1881no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—942no—
553C22—1881no—
554C22—1881no—
555C22—1881no—
556C22—1881no—
557C22—1881no—
558C22C2×#1309398611yescharacteristic
559C4—21no—
560C4—21no—
561C4—21no—
562C4—21no—
563C4—942no—
564C4—1881no—
565C4—1881no—
566C4—1881no—
567C4—1881no—
568C4—1881no—
569C4#131×D4711yescharacteristic
570C4#131×D4711yescharacteristic
571C4#1568992011yescharacteristic
572C4#1568992011yescharacteristic
573C2—41no—
574C2—3761no—
575C2—1882no—
576C2#2025208111yescharacteristic
577C2#2025218811yescharacteristic
578C2#2025229011yescharacteristic
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#17909
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. #20252357C2semidirect
3. #20252344C2semidirect
4. #20242265C2semidirect
5. #20242258C2semidirect
6. #20242251C2semidirect
7. #20242244C2semidirect
8. #20242211C2semidirect
9. #20242206C2semidirect
10. #20242199C2semidirect
11. #20242199C2semidirect
12. #20241820C2semidirect
13. #20241794C2semidirect
14. #20241681C2semidirect
15. C47×#1572C2semidirect
295. #1572D47semidirect
356. C47#17909semidirect

Maximal subgroups

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

subgroupindex
#202534082
#202523572
#202523442
#202422652
#202422582
#202422512
#202422442
#202422112
#202422062
#202421992
#202418202
#202417942
#202416812
C47×#15722
#1790947