Finite Group Databasebeta

← #26184076

Subgroups — #26184076

Order 12032.

898 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257253C211yesmaximal, characteristic
2#20252473C211yesmaximal, characteristic, complemented
3#20241963C212yesmaximal, complemented
4#20241938C211yesmaximal, characteristic, complemented
5#20241934C211yesmaximal, characteristic
6#20241924C212yesmaximal, complemented
7#20241911C212yesmaximal, complemented
8#20241899C211yesmaximal, characteristic, complemented
9#20241895C211yesmaximal, characteristic
10#20241872C212yesmaximal, complemented
11C47×#2356C211yesmaximal, characteristic, complemented
12#15689858C412yescomplemented
13#15689851C412yescomplemented
14#15689846C412yescomplemented
15#15689845—22no—
16#15689833C412yescomplemented
17#15689814—22no—
18#15689720—21no—
19#15689720—21no—
20#15689697—21no—
21#15689680C2211yescharacteristic
22#15689673C2211yescharacteristic
23#15689657C2212yes—
24#15689657C2212yes—
25#15689604—21no—
26#15689254—22no—
27#15689254—22no—
28#15689180C2212yes—
29#15689180C2211yescharacteristic
30#15689180C2211yescharacteristic
31#15689176C2212yes—
32#15688859C2212yes—
33#15688859C2212yes—
34#15688859C2212yes—
35#15688859C2212yes—
36#15688856C2212yes—
37#15688856C2212yes—
38#15688856C2212yes—
39#15688856C2212yes—
40C2×C4×Q8×C47C2211yescharacteristic
41C47×#391—21no—
42C47×#387—21no—
43C47×#387—21no—
44C47×#383C2211yescharacteristic
45C47×#383C2211yescharacteristic
46C47×#382C2211yescharacteristic
47C4×C4⋊C4×C47—21no—
48C4×C4⋊C4×C47C2211yescharacteristic
49C47×#376C2211yescharacteristic
50C47×#376C2211yescharacteristic
51C2×#10625350—21no—
52#13094024C2×C411yescharacteristic
53#13094024C2×C411yescharacteristic
54#13094024C2×C411yescharacteristic
55#13094022—22no—
56#13094021C2×C411yescharacteristic
57#13094019—22no—
58#13094019—22no—
59#13094018—22no—
60#13094018—22no—
61#13094016—21no—
62#13094016C2×C412yes—
63#13094016C2×C412yes—
64#13094014—22no—
65#13094014—22no—
66#13094013—22no—
67#13094012—21no—
68#13094012C2×C412yes—
69#13094012C2×C412yes—
70#13094009—22no—
71#13094009—22no—
72#13094007—22no—
73#13094007—22no—
74#13094003—21no—
75#13094003C2312yes—
76#13094002—22no—
77#13094002—22no—
78#13094000—22no—
79#13094000—22no—
80#13093998—21no—
81#13093998C2312yes—
82C4×Dic94—21no—
83C4×Dic94C2×C412yes—
84C4×Dic94C2×C412yes—
85C4×Dic94C2×C412yes—
86#13093989—22no—
87C4×D188—21no—
88C4×D188C2×C412yes—
89C4×D188C2×C412yes—
90C4×D188C2×C412yes—
91#13093986—22no—
92C4×C22⋊D47—22no—
93C4×C22⋊D47—22no—
94#13093983—21no—
95#13093982—22no—
96#13093981—22no—
97#13093981—22no—
98C4⋊C4×D47—22no—
99C4⋊(C47⋊C8)—22no—
100C4⋊(C47⋊C8)—22no—
101C4⋊(C47⋊C8)—21no—
102C4⋊(C47⋊C8)—21no—
103C22⋊(C47⋊C8)C2312yes—
104C22⋊(C47⋊C8)C2312yes—
105#13093935—22no—
106#13093935—21no—
107#13093935—21no—
108C4×C47⋊C8—22no—
109C22×Q8×C47D411yescharacteristic
110C4×Q8×C47C2×C412yes—
111C4×Q8×C47C2×C412yes—
112C4×Q8×C47D412yes—
113C4×Q8×C47D412yes—
114C2×C4⋊C4×C47—21no—
115C2×C4⋊C4×C47—21no—
116C2×C4⋊C4×C47D411yescharacteristic
117C2×C4⋊C4×C47D411yescharacteristic
118C2×C4⋊C4×C47D411yescharacteristic
119C2×C42×C47—21no—
120C2×C42×C47—21no—
121C2×C42×C47—21no—
122C2×C42×C47C2311yescharacteristic
123C2×C42×C47C2311yescharacteristic
124C2×C42×C47C2311yescharacteristic
125#95×C47—21no—
126#95×C47—21no—
127#95×C47—21no—
128#95×C47—21no—
129#95×C47—21no—
130#95×C47—21no—
131#95×C47—21no—
132#95×C47C2311yescharacteristic
133#95×C47C2311yescharacteristic
134#95×C47C2311yescharacteristic
135#95×C47C2311yescharacteristic
136C2×Q8×C47—41no—
137C2×Q8×C47—41no—
138C2×Q8×C47C22⋊C412yes—
139C2×Q8×C47C22⋊C412yes—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47—21no—
143C22×C4×C47—21no—
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2411yescharacteristic
151C2×C22⋊D47—22no—
152#10625350—42no—
153#10625350—42no—
154C2×D188—21no—
155C2×D188—21no—
156C2×D188C22×C412yes—
157C2×C4×D47—22no—
158C2×C4×D47—22no—
159C2×C4×D47—22no—
160C2×Dic94—21no—
161C2×Dic94—21no—
162C2×Dic94C22×C412yes—
163C4⋊C4×C47—42no—
164C4⋊C4×C47—42no—
165C4⋊C4×C47C22⋊C412yes—
166C4⋊C4×C47C22⋊C412yes—
167C4⋊C4×C47C22⋊C412yes—
168C4⋊C4×C47C22⋊C412yes—
169C4⋊C4×C47C22⋊C412yes—
170C4⋊C4×C47C22⋊C412yes—
171C42×C47—42no—
172C42×C47—42no—
173C42×C47—42no—
174C42×C47C22×C411yescharacteristic
175C42×C47C22×C412yes—
176C42×C47C22×C412yes—
177C42×C47C22×C411yescharacteristic
178C42×C47C2×D412yes—
179C42×C47C2×D412yes—
180C42×C47C2×D411yescharacteristic
181C42×C47C2×D411yescharacteristic
182C22⋊Dic47—22no—
183(C2×D47)⋊C4—22no—
184(C2×D47)⋊C4—22no—
185(C2×D47)⋊C4—22no—
186(C2×D47)⋊C4—22no—
187(C2×D47)⋊C4—22no—
188(C2×D47)⋊C4—22no—
189(C2×D47)⋊C4—22no—
190(C2×D47)⋊C4—22no—
191C4⋊Dic47—21no—
192C4⋊Dic47—21no—
193C4⋊Dic47—21no—
194C4⋊Dic47—21no—
195C4⋊Dic47C22×C412yes—
196C4⋊Dic47C22×C412yes—
197Dic47⋊C4—22no—
198Dic47⋊C4—22no—
199Dic47⋊C4—22no—
200Dic47⋊C4—22no—
201Dic47⋊C4—22no—
202Dic47⋊C4—22no—
203Dic47⋊C4—22no—
204Dic47⋊C4—22no—
205C4×Dic47—22no—
206C4×Dic47—22no—
207C4×Dic47—22no—
208C2×C47⋊C8—22no—
209C2×C47⋊C8—22no—
210C23×C47C22×D411yescharacteristic
211C22×D47—22no—
212Q8×C47—42no—
213Q8×C47—42no—
214C2×C4×C47—21no—
215C2×C4×C47—21no—
216C2×C4×C47—21no—
217C2×C4×C47—21no—
218C2×C4×C47—21no—
219C2×C4×C47—21no—
220C2×C4×C47—22no—
221C2×C4×C47—41no—
222C2×C4×C47—41no—
223C2×C4×C47—41no—
224C2×C4×C47—41no—
225C2×C4×C47—41no—
226C2×C4×C47—41no—
227C2×C4×C47—41no—
228C2×C4×C47—41no—
229C2×C4×C47—41no—
230C2×C4×C47—41no—
231C2×C4×C47—41no—
232C2×C4×C47—41no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47C2×C22⋊C411yescharacteristic
237C2×C4×C47C2×C22⋊C412yes—
238C2×C4×C47C2×C22⋊C412yes—
239C2×C4×C47C2×C22⋊C411yescharacteristic
240C2×C4×C47C2×C22⋊C412yes—
241C2×C4×C47C2×C22⋊C412yes—
242C2×C4×C47C2×C22⋊C411yescharacteristic
243C2×C4×C47C2×C22⋊C411yescharacteristic
244C2×C4×C47C23×C411yescharacteristic
245C2×C4×C47C22×D411yescharacteristic
246C22⋊D47—42no—
247C22⋊D47—42no—
248C2×Dic47—22no—
249C2×Dic47—22no—
250C2×Dic47—22no—
251D188—42no—
252D188—42no—
253D188—81no—
254C4×D47—44no—
255C4×D47—82no—
256C4×D47—82no—
257Dic94—42no—
258Dic94—42no—
259Dic94—81no—
260C47⋊C8—42no—
261C47⋊C8—42no—
262#17624—471nomaximal, Sylow
263C22×C47—21no—
264C22×C47—21no—
265C22×C47C22×C22⋊C411yescharacteristic
266C22×C47#57511yescharacteristic
267C22×C47#57511yescharacteristic
268C2×D47—42no—
269C2×D47—42no—
270C2×D47—42no—
271C4×C47—21no—
272C4×C47—21no—
273C4×C47—21no—
274C4×C47—21no—
275C4×C47—22no—
276C4×C47—22no—
277C4×C47—21no—
278C4×C47—22no—
279C4×C47—22no—
280C4×C47—21no—
281C4×C47—42no—
282C4×C47—81no—
283C4×C47—81no—
284C4×C47—81no—
285Dic47—42no—
286Dic47—42no—
287Dic47—42no—
288#3507—471no—
289#2983—471no—
290#2356D4711yescharacteristic, complemented
291#1521—472no—
292#1515—471no—
293#1514—471no—
294#1512—472no—
295#1509—472no—
296#1506—471no—
297#1505—471no—
298#1500—472no—
299C2×C47—21no—
300C2×C47—21no—
301C2×C47#288411yescharacteristic
302C2×C47#289611yescharacteristic
303C2×C47#290111yescharacteristic
304D47—82no—
305#559—472no—
306#557—472no—
307#556—472no—
308#556—942no—
309#554—472no—
310#552—942no—
311#539—941no—
312#539—941no—
313#536—941no—
314#533—471no—
315#532—471no—
316#529—472no—
317#529—472no—
318#520—941no—
319C2×C4×Q8C2×D4711yescharacteristic
320#446—942no—
321#446—942no—
322#432—471no—
323#432—471no—
324#432—472no—
325#431—472no—
326#391—21no—
327#387—21no—
328#387—21no—
329#383C2×D4711yescharacteristic
330#383C2×D4711yescharacteristic
331#382C2×D4711yescharacteristic
332C4×C4⋊C4—21no—
333C4×C4⋊C4C2×D4711yescharacteristic
334#376C2×D4711yescharacteristic
335#376C2×D4711yescharacteristic
336#328—472no—
337#328—472no—
338#328—472no—
339#328—472no—
340#327—472no—
341#327—472no—
342#327—472no—
343#327—472no—
344C47#1762411yescharacteristic, complemented, Sylow
345C2×#41—941no—
346C22×Q8C22⋊D4711yescharacteristic
347C4⋊Q8—471no—
348C4⋊Q8—471no—
349C4⋊Q8—471no—
350C4⋊Q8—942no—
351#127—471no—
352#126—942no—
353#126—942no—
354#126—942no—
355#126—942no—
356#125—472no—
357#125—472no—
358#125—942no—
359#125—942no—
360#125—941no—
361#124—472no—
362#124—472no—
363#124—942no—
364#124—941no—
365#123—942no—
366#123—942no—
367#123—942no—
368#123—942no—
369C22⋊Q8—472no—
370C22⋊Q8—942no—
371C22⋊Q8—942no—
372C22⋊Q8—941no—
373C22⋊Q8—942no—
374C22⋊Q8—942no—
375#121—472no—
376#121—941no—
377C4×Q8—472no—
378C4×Q8—472no—
379C4×Q8—472no—
380C4×Q8—942no—
381C4×Q8—941no—
382C4×Q8C4×D4712yes—
383C4×Q8C4×D4712yes—
384C4×Q8D18812yes—
385C4×Q8D18812yes—
386C4×D4—472no—
387C4×D4—472no—
388C4×D4—472no—
389C4×D4—942no—
390C4×D4—942no—
391C4×D4—942no—
392C4×D4—941no—
393#117—941no—
394#117—942no—
395#117—942no—
396#117—942no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—942no—
400C2×C4⋊C4C22⋊D4711yescharacteristic
401C2×C4⋊C4C22⋊D4711yescharacteristic
402C2×C4⋊C4C22⋊D4711yescharacteristic
403C2×C42—21no—
404C2×C42—21no—
405C2×C42—21no—
406C2×C42C22×D4711yescharacteristic
407C2×C42C22×D4711yescharacteristic
408C2×C42C22×D4711yescharacteristic
409C4⋊C8—942no—
410C4⋊C8—942no—
411C4⋊C8—941no—
412C4⋊C8—941no—
413C22⋊C8—472no—
414C22⋊C8—472no—
415#97—942no—
416#97—941no—
417#97—941no—
418C4×C8—942no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95C22×D4711yescharacteristic
427#95C22×D4711yescharacteristic
428#95C22×D4711yescharacteristic
429#95C22×D4711yescharacteristic
430#41—1882no—
431#41—1882no—
432C2×Q8—41no—
433C2×Q8—41no—
434C2×Q8—472no—
435C2×Q8—941no—
436C2×Q8—941no—
437C2×Q8(C2×D47)⋊C412yes—
438C2×Q8(C2×D47)⋊C412yes—
439C2×D4—472no—
440C2×D4—942no—
441C2×D4—941no—
442C2×D4—941no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—942no—
448C22×C4—942no—
449C22×C4—942no—
450C22×C4C2×C22⋊D4711yescharacteristic
451C22×C4C2×C22⋊D4711yescharacteristic
452C22×C4C2×C22⋊D4711yescharacteristic
453C22×C4C2×C22⋊D4711yescharacteristic
454C22×C4C2×C22⋊D4711yescharacteristic
455C22×C4C2×C22⋊D4711yescharacteristic
456C22×C4C23×D4711yescharacteristic
457C2×C8—942no—
458C2×C8—942no—
459C4⋊C4—42no—
460C4⋊C4—42no—
461C4⋊C4—472no—
462C4⋊C4—472no—
463C4⋊C4—942no—
464C4⋊C4—942no—
465C4⋊C4—942no—
466C4⋊C4—942no—
467C4⋊C4—942no—
468C4⋊C4—942no—
469C4⋊C4—942no—
470C4⋊C4—941no—
471C4⋊C4—941no—
472C4⋊C4—941no—
473C4⋊C4—941no—
474C4⋊C4—942no—
475C4⋊C4(C2×D47)⋊C412yes—
476C4⋊C4(C2×D47)⋊C412yes—
477C4⋊C4(C2×D47)⋊C412yes—
478C4⋊C4(C2×D47)⋊C412yes—
479C4⋊C4(C2×D47)⋊C412yes—
480C4⋊C4(C2×D47)⋊C412yes—
481C22⋊C4—942no—
482C22⋊C4—942no—
483C22⋊C4—942no—
484C22⋊C4—942no—
485C22⋊C4—942no—
486C22⋊C4—942no—
487C22⋊C4—942no—
488C22⋊C4—942no—
489C22⋊C4—942no—
490C42—42no—
491C42—42no—
492C42—42no—
493C42—942no—
494C42—942no—
495C42—942no—
496C42C2×C4×D4711yescharacteristic
497C42C2×C4×D4712yes—
498C42C2×C4×D4712yes—
499C42C2×C4×D4711yescharacteristic
500C42C2×D18812yes—
501C42C2×D18812yes—
502C42C2×D18811yescharacteristic
503C42C2×D18811yescharacteristic
504C23—942no—
505C23C22×C22⋊D4711yescharacteristic
506Q8—42no—
507Q8—42no—
508Q8—3761no—
509Q8—1882no—
510Q8—1882no—
511D4—3761no—
512D4—1882no—
513D4—1882no—
514D4—1882no—
515D4—1882no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—942no—
539C2×C4—942no—
540C2×C4—942no—
541C2×C4—3762no—
542C2×C4—3762no—
543C2×C4—1884no—
544C2×C4C2×(C2×D47)⋊C411yescharacteristic
545C2×C4C2×(C2×D47)⋊C412yes—
546C2×C4C2×(C2×D47)⋊C412yes—
547C2×C4C2×(C2×D47)⋊C412yes—
548C2×C4C2×(C2×D47)⋊C412yes—
549C2×C4C2×(C2×D47)⋊C411yescharacteristic
550C2×C4C2×(C2×D47)⋊C411yescharacteristic
551C2×C4C2×(C2×D47)⋊C411yescharacteristic
552C2×C4C22×C4×D4711yescharacteristic
553C2×C4C22×D18811yescharacteristic
554C8—1882no—
555C8—1882no—
556C22—21no—
557C22—21no—
558C22—1882no—
559C22—1882no—
560C22—1882no—
561C22C22×(C2×D47)⋊C411yescharacteristic
562C22#1568992111yescharacteristic
563C22#1568992111yescharacteristic
564C4—21no—
565C4—21no—
566C4—22no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—42no—
575C4—81no—
576C4—81no—
577C4—81no—
578C4—1882no—
579C4—1882no—
580C4—1882no—
581C2—21no—
582C2—21no—
583C2—3762no—
584C2#2025185111yescharacteristic
585C2#2025190511yescharacteristic
586C2#2025193611yescharacteristic
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2356C2
socleC22×C47C22×C22⋊C4
prime radical#15689673C22
cosocleC22×C4×C47C24
hypercenter#2356D47
hypocenterC47#17624
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2356

Complements

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

NH ≅ G/Ntype
2. #20252473C2semidirect
3. #20241963C2semidirect
4. #20241938C2semidirect
6. #20241924C2semidirect
7. #20241911C2semidirect
8. #20241899C2semidirect
10. #20241872C2semidirect
11. C47×#2356C2semidirect
12. #15689858C4semidirect
13. #15689851C4semidirect
14. #15689846C4semidirect
16. #15689833C4semidirect
290. #2356D47semidirect
344. C47#17624semidirect

Maximal subgroups

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

subgroupindex
#202572532
#202524732
#202419632
#202419382
#202419342
#202419242
#202419112
#202418992
#202418952
#202418722
C47×#23562
#1762447