Finite Group Databasebeta

← #26184269

Subgroups — #26184269

Order 12032.

964 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257366C211yesmaximal, characteristic
2#20252402C211yesmaximal, characteristic, complemented
3#20242006C212yesmaximal, complemented
4#20242002C211yesmaximal, characteristic, complemented
5#20241993C211yesmaximal, characteristic, complemented
6#20241989C212yesmaximal, complemented
7#20241982C211yesmaximal, characteristic, complemented
8#20241977C212yesmaximal, complemented
9#20241974C211yesmaximal, characteristic, complemented
10C2×#15688864C212yesmaximal, complemented
11C47×#2875C211yesmaximal, characteristic, complemented
12#15689851C411yescharacteristic
13#15689833C411yescharacteristic
14#15689830C411yescharacteristic
15#15689819C412yes—
16#15689813C412yes—
17#15689812—21no—
18#15689806—22no—
19#15689777C411yescharacteristic
20#15689772—21no—
21#15689731—21no—
22#15689715—22no—
23#15689680C2211yescharacteristic
24#15689657C2211yescharacteristic
25#15689657C2212yes—
26C2×#13094016C2212yes—
27#15689602—21no—
28#15689254—21no—
29#15689195—22no—
30D4×C47⋊C8—21no—
31#15689180C2212yes—
32#15689180C2211yescharacteristic
33C4×#10625329C2211yescharacteristic
34C2×#13093935C2212yes—
35#15688864C2211yescharacteristic
36#15688864C2212yes—
37#15688864C2212yes—
38#15688864C2212yes—
39#15688864C2211yescharacteristic
40#15688864C2212yes—
41#15688864C2212yes—
42#15688862C2211yescharacteristic
43#15688862C2211yescharacteristic
44#15688862C2212yes—
45C2×#131×C47—21no—
46C4×#41×C47C2211yescharacteristic
47C47×#405—22no—
48C47×#405—21no—
49C4×M16×C47C2212yes—
50C4×M16×C47C2211yescharacteristic
51C2×#97×C47C2212yes—
52C2×#97×C47C2211yescharacteristic
53C2×#10625352—21no—
54C2×#10625329—21no—
55C22×C47⋊C8—22no—
56#13094024C2×C411yescharacteristic
57#13094024C2×C411yescharacteristic
58#13094022—21no—
59#13094019—22no—
60#13094018—22no—
61#13094018—21no—
62#13094017—22no—
63#13094016—22no—
64#13094016—22no—
65#13094016C2×C412yes—
66#13094016C2×C412yes—
67#13094016C2311yescharacteristic
68#13094016C2312yes—
69#13094015—21no—
70#13094014—22no—
71#13094014—22no—
72#13094012C2311yescharacteristic
73#13094008—22no—
74#13094007—22no—
75#13094007—21no—
76#13094006—22no—
77#13094005—22no—
78#13094004—21no—
79#13094003—21no—
80#13094003—21no—
81#13094003C2×C411yescharacteristic
82#13094003C2×C412yes—
83#13094002—22no—
84#13094001—22no—
85#13094000—21no—
86#13093998C2×C412yes—
87#13093998C2×C411yescharacteristic
88#13093996—22no—
89C4×Dic94—21no—
90C4×Dic94—21no—
91C4×Dic94C2×C412yes—
92C4×Dic94C2×C411yescharacteristic
93#13093989—21no—
94C4×D188C2×C411yescharacteristic
95C4×D188C2×C412yes—
96D4×Dic47—21no—
97C4×C22⋊D47—21no—
98C4×C22⋊D47—22no—
99(C22⋊D47)⋊C4—22no—
100#13093983C2×C412yes—
101#13093981—21no—
102#13093981—22no—
103#13093980—22no—
104C2×C4⋊Dic47C2×C412yes—
105C2×C4⋊Dic47C2×C412yes—
106C2×Dic47⋊C4—22no—
107C2×Dic47⋊C4—22no—
108C4⋊C4×D47—21no—
109C4⋊(C47⋊C8)—21no—
110C4⋊(C47⋊C8)—22no—
111C22⋊(C47⋊C8)—22no—
112C22⋊(C47⋊C8)—21no—
113#13093935—22no—
114#13093935C2311yescharacteristic
115#13093935C2312yes—
116#13093935C2311yescharacteristic
117C4×C47⋊C8—21no—
118C2×#41×C47D411yescharacteristic
119#131×C47—28no—
120C2×M16×C47—22no—
121C2×M16×C47—21no—
122C22×C8×C47—22no—
123C22×C8×C47—21no—
124C4×Q8×C47C2×C411yescharacteristic
125C4×Q8×C47D411yescharacteristic
126C4×D4×C47C2×C411yescharacteristic
127C4×D4×C47C2×C412yes—
128C4×D4×C47D411yescharacteristic
129C4×D4×C47D412yes—
130#117×C47D412yes—
131#117×C47D411yescharacteristic
132C2×C42×C47C2312yes—
133C2×C42×C47C2311yescharacteristic
134#97×C47—22no—
135#97×C47—21no—
136#97×C47C2311yescharacteristic
137#97×C47C2311yescharacteristic
138#97×C47C2312yes—
139C4×C8×C47—22no—
140C4×C8×C47—21no—
141#41×C47—24no—
142C2×Q8×C47C22⋊C411yescharacteristic
143C2×D4×C47C22⋊C412yes—
144C2×D4×C47C22⋊C411yescharacteristic
145C22×C4×C47C22×C412yes—
146C22×C4×C47C22×C411yescharacteristic
147C22×C4×C47C2×D412yes—
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D412yes—
150C22×C4×C47C2×D411yescharacteristic
151C2×C22⋊D47—22no—
152C2×C22⋊D47—21no—
153C22×Dic47—22no—
154C22×Dic47—22no—
155#10625352—44no—
156C2×D188C22×C411yescharacteristic
157C2×C4×D47—21no—
158C2×C4×D47—22no—
159C2×Dic94—21no—
160C2×Dic94—21no—
161C2×Dic94C22×C411yescharacteristic
162M16×C47—28no—
163M16×C47—24no—
164C2×C8×C47—28no—
165C2×C8×C47—24no—
166C2×C8×C47—21no—
167C2×C8×C47—22no—
168C2×C8×C47—21no—
169C4⋊C4×C47C22⋊C411yescharacteristic
170C4⋊C4×C47C22⋊C412yes—
171C4⋊C4×C47C22⋊C411yescharacteristic
172C4⋊C4×C47C22⋊C412yes—
173C22⋊C4×C47C22⋊C412yes—
174C22⋊C4×C47C22⋊C412yes—
175C22⋊C4×C47C22⋊C412yes—
176C42×C47C22×C412yes—
177C42×C47C22×C411yescharacteristic
178C42×C47C2×D411yescharacteristic
179C42×C47C2×D412yes—
180C42×C47C2×D412yes—
181C42×C47C2×D411yescharacteristic
182C42×C47C2411yescharacteristic
183C22⋊Dic47—22no—
184C22⋊Dic47—22no—
185C22⋊Dic47—21no—
186(C2×D47)⋊C4—21no—
187(C2×D47)⋊C4—22no—
188(C2×D47)⋊C4—22no—
189(C2×D47)⋊C4—21no—
190(C2×D47)⋊C4—21no—
191(C2×D47)⋊C4—21no—
192C4⋊Dic47—21no—
193C4⋊Dic47—22no—
194C4⋊Dic47—22no—
195C4⋊Dic47—21no—
196C4⋊Dic47C22×C411yescharacteristic
197C4⋊Dic47C22×C411yescharacteristic
198C4⋊Dic47C22×C412yes—
199C4⋊Dic47C22×C412yes—
200Dic47⋊C4—22no—
201Dic47⋊C4—22no—
202Dic47⋊C4—22no—
203Dic47⋊C4—22no—
204Dic47⋊C4—21no—
205Dic47⋊C4—22no—
206Dic47⋊C4—21no—
207Dic47⋊C4—22no—
208Dic47⋊C4—21no—
209Dic47⋊C4—21no—
210Dic47⋊C4—22no—
211Dic47⋊C4—22no—
212C4×Dic47—22no—
213C4×Dic47—22no—
214C4×Dic47—21no—
215#10625329—42no—
216C2×C47⋊C8—21no—
217C2×C47⋊C8—21no—
218C2×C47⋊C8—22no—
219C2×C47⋊C8—44no—
220C23×C47C2×C22⋊C411yescharacteristic
221C23×C47C2×C22⋊C412yes—
222C22×D47—21no—
223Q8×C47—22no—
224D4×C47—24no—
225D4×C47#10414yes—
226C2×C4×C47—24no—
227C2×C4×C47—22no—
228C2×C4×C47—42no—
229C2×C4×C47—41no—
230C2×C4×C47—42no—
231C2×C4×C47—41no—
232C2×C4×C47C2×C22⋊C411yescharacteristic
233C2×C4×C47C2×C22⋊C412yes—
234C2×C4×C47C2×C22⋊C412yes—
235C2×C4×C47C2×C22⋊C412yes—
236C2×C4×C47C2×C22⋊C411yescharacteristic
237C2×C4×C47C2×C22⋊C411yescharacteristic
238C2×C4×C47C23×C411yescharacteristic
239C2×C4×C47C22×D411yescharacteristic
240C2×C4×C47C22×D411yescharacteristic
241C22⋊D47—44no—
242C22⋊D47—81no—
243C2×Dic47—21no—
244C2×Dic47—22no—
245C2×Dic47—21no—
246C2×Dic47—22no—
247C2×Dic47—21no—
248C2×Dic47—44no—
249C2×Dic47—82no—
250D188—41no—
251C4×D47—42no—
252C4×D47—82no—
253Dic94—42no—
254Dic94—41no—
255Dic94—81no—
256C8×C47—22no—
257C8×C47—24no—
258C8×C47—22no—
259C47⋊C8—22no—
260C47⋊C8—42no—
261C47⋊C8—41no—
262#17653—471nomaximal, Sylow
263C22×C47—22no—
264C22×C47—22no—
265C22×C47—21no—
266C22×C47—21no—
267C22×C47—21no—
268C22×C47C2×#10414yes—
269C22×C47C22×C22⋊C411yescharacteristic
270C2×D47—41no—
271C2×D47—41no—
272C2×D47—41no—
273C4×C47—21no—
274C4×C47—21no—
275C4×C47—22no—
276C4×C47—21no—
277C4×C47—22no—
278C4×C47—21no—
279C4×C47—21no—
280C4×C47—22no—
281C4×C47C2×#10412yes—
282Dic47—41no—
283Dic47—42no—
284Dic47—41no—
285Dic47—42no—
286Dic47—41no—
287#3515—471no—
288#2970—471no—
289#2875D4711yescharacteristic, complemented
290#1532—472no—
291#1531—471no—
292#1529—471no—
293#1528—472no—
294#1526—471no—
295#1524—472no—
296#1524—471no—
297C2×#330—472no—
298C2×C47—24no—
299C2×C47—41no—
300C2×C47#288911yescharacteristic
301C2×C47C22×#10411yescharacteristic
302C2×C47#290111yescharacteristic
303D47—81no—
304C2×#131—21no—
305#557—471no—
306#554—471no—
307#554—471no—
308#553—472no—
309#552—472no—
310#552—942no—
311#552—941no—
312#547—471no—
313#547—941no—
314#541—941no—
315#539—942no—
316#533—471no—
317#529—471no—
318#529—472no—
319C2×#125—472no—
320#520—941no—
321C4×#41C2×D4711yescharacteristic
322#446—941no—
323#435—942no—
324C8×D4—941no—
325#432—472no—
326#432—471no—
327#405—21no—
328#405—22no—
329C4×M16—471no—
330C4×M16C2×D4712yes—
331C4×M16C2×D4711yescharacteristic
332C2×#97—472no—
333C2×#97C2×D4712yes—
334C2×#97C2×D4711yescharacteristic
335#330—471no—
336#330—472no—
337#330—472no—
338#330—472no—
339#330—471no—
340#330—472no—
341#330—472no—
342#329—471no—
343#329—472no—
344#329—471no—
345C47#1765311yescharacteristic, complemented, Sylow
346C2×#41—941no—
347C2×#41C22⋊D4711yescharacteristic
348#131—28no—
349C2×M16—22no—
350C2×M16—21no—
351C2×M16—941no—
352C22×C8—22no—
353C22×C8—21no—
354C22×C8—942no—
355C4⋊Q8—471no—
356C4⋊Q8—471no—
357C4⋊Q8—941no—
358#126—942no—
359#126—942no—
360#126—942no—
361#126—941no—
362#125—472no—
363#125—472no—
364#125—471no—
365#125—472no—
366#125—942no—
367#125—942no—
368#125—941no—
369#125—942no—
370#125—942no—
371#124—471no—
372#123—942no—
373#123—941no—
374#123—942no—
375#123—942no—
376#123—942no—
377C22⋊Q8—471no—
378C22⋊Q8—472no—
379C22⋊Q8—941no—
380C22⋊Q8—941no—
381C22⋊Q8—941no—
382C22⋊Q8—942no—
383C22⋊Q8—941no—
384C22⋊Q8—942no—
385#121—472no—
386#121—471no—
387#121—942no—
388C4×Q8—472no—
389C4×Q8—471no—
390C4×Q8—941no—
391C4×Q8—941no—
392C4×Q8—941no—
393C4×Q8C4×D4711yescharacteristic
394C4×Q8D18811yescharacteristic
395C4×D4—471no—
396C4×D4—472no—
397C4×D4—941no—
398C4×D4—942no—
399C4×D4—941no—
400C4×D4—942no—
401C4×D4C4×D4711yescharacteristic
402C4×D4C4×D4712yes—
403C4×D4D18811yescharacteristic
404C4×D4D18812yes—
405#117—472no—
406#117—942no—
407#117—941no—
408#117—942no—
409#117C22⋊D4711yescharacteristic
410#117C22⋊D4712yes—
411C2×C4⋊C4—472no—
412C2×C4⋊C4—472no—
413C2×C4⋊C4—942no—
414C2×C4⋊C4—942no—
415C2×C4⋊C4—941no—
416C2×C42C22×D4712yes—
417C2×C42C22×D4711yescharacteristic
418C4⋊C8—941no—
419C4⋊C8—942no—
420C22⋊C8—942no—
421C22⋊C8—941no—
422#97—22no—
423#97—21no—
424#97—471no—
425#97—472no—
426#97—471no—
427#97—942no—
428#97C22×D4711yescharacteristic
429#97C22×D4711yescharacteristic
430#97C22×D4712yes—
431C4×C8—22no—
432C4×C8—21no—
433C4×C8—941no—
434#41—24no—
435#41—1884no—
436C2×Q8—471no—
437C2×Q8—941no—
438C2×Q8—941no—
439C2×Q8(C2×D47)⋊C411yescharacteristic
440C2×D4—471no—
441C2×D4—941no—
442C2×D4—942no—
443C2×D4(C2×D47)⋊C412yes—
444C2×D4(C2×D47)⋊C411yescharacteristic
445C22×C4—942no—
446C22×C4—942no—
447C22×C4—942no—
448C22×C4—941no—
449C22×C4C2×C4×D4712yes—
450C22×C4C2×C4×D4711yescharacteristic
451C22×C4C2×D18812yes—
452C22×C4C2×D18811yescharacteristic
453C22×C4C2×C22⋊D4712yes—
454C22×C4C2×C22⋊D4711yescharacteristic
455M16—28no—
456M16—24no—
457M16—1882no—
458C2×C8—28no—
459C2×C8—24no—
460C2×C8—21no—
461C2×C8—22no—
462C2×C8—21no—
463C2×C8—941no—
464C2×C8—941no—
465C2×C8—942no—
466C2×C8—1884no—
467C4⋊C4—471no—
468C4⋊C4—471no—
469C4⋊C4—472no—
470C4⋊C4—472no—
471C4⋊C4—942no—
472C4⋊C4—942no—
473C4⋊C4—942no—
474C4⋊C4—942no—
475C4⋊C4—941no—
476C4⋊C4—942no—
477C4⋊C4—941no—
478C4⋊C4—942no—
479C4⋊C4—942no—
480C4⋊C4—942no—
481C4⋊C4—941no—
482C4⋊C4—941no—
483C4⋊C4—941no—
484C4⋊C4—942no—
485C4⋊C4—941no—
486C4⋊C4—942no—
487C4⋊C4(C2×D47)⋊C411yescharacteristic
488C4⋊C4(C2×D47)⋊C411yescharacteristic
489C4⋊C4(C2×D47)⋊C412yes—
490C4⋊C4(C2×D47)⋊C412yes—
491C22⋊C4—942no—
492C22⋊C4—942no—
493C22⋊C4—941no—
494C22⋊C4—941no—
495C22⋊C4—942no—
496C22⋊C4—942no—
497C22⋊C4—941no—
498C22⋊C4—941no—
499C22⋊C4—941no—
500C22⋊C4(C2×D47)⋊C412yes—
501C22⋊C4(C2×D47)⋊C412yes—
502C22⋊C4(C2×D47)⋊C412yes—
503C42—942no—
504C42—942no—
505C42—941no—
506C42C2×C4×D4712yes—
507C42C2×C4×D4711yescharacteristic
508C42C2×D18811yescharacteristic
509C42C2×D18812yes—
510C42C2×C22⋊D4712yes—
511C42C2×C22⋊D4711yescharacteristic
512C42C23×D4711yescharacteristic
513C23—941no—
514C23C2×(C2×D47)⋊C411yescharacteristic
515C23C2×(C2×D47)⋊C412yes—
516Q8—22no—
517Q8—3761no—
518Q8—1881no—
519Q8—1882no—
520D4—24no—
521D4—3761no—
522D4—1881no—
523D4—1884no—
524D4#1309395214yes—
525C2×C4—24no—
526C2×C4—22no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—941no—
532C2×C4—942no—
533C2×C4—942no—
534C2×C4—941no—
535C2×C4—941no—
536C2×C4—3762no—
537C2×C4—3762no—
538C2×C4—1884no—
539C2×C4—1882no—
540C2×C4C2×(C2×D47)⋊C411yescharacteristic
541C2×C4C2×(C2×D47)⋊C412yes—
542C2×C4C2×(C2×D47)⋊C412yes—
543C2×C4C2×(C2×D47)⋊C411yescharacteristic
544C2×C4C2×(C2×D47)⋊C412yes—
545C2×C4C2×(C2×D47)⋊C411yescharacteristic
546C2×C4C22×C4×D4711yescharacteristic
547C2×C4C22×C22⋊D4711yescharacteristic
548C2×C4C22×D18811yescharacteristic
549C8—22no—
550C8—24no—
551C8—22no—
552C8—942no—
553C8—1882no—
554C8—1881no—
555C22—22no—
556C22—22no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—1881no—
561C22—1881no—
562C22—1881no—
563C22C2×#1309395214yes—
564C22C22×(C2×D47)⋊C411yescharacteristic
565C4—21no—
566C4—22no—
567C4—21no—
568C4—22no—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—22no—
573C4—1881no—
574C4—1882no—
575C4—1881no—
576C4—1882no—
577C4—1881no—
578C4C2×#1309395212yes—
579C2—24no—
580C2—41no—
581C2—3761no—
582C2#2025187011yescharacteristic
583C2C22×#1309395211yescharacteristic
584C2#2025193311yescharacteristic
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#2875C2
socleC22×C47C22×C22⋊C4
prime radical#20257366C2
cosocleC42×C47C24
hypercenter#2875D47
hypocenterC47#17653
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C4×#41 ◁ #2875

Complements

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

NH ≅ G/Ntype
2. #20252402C2semidirect
3. #20242006C2semidirect
4. #20242002C2semidirect
5. #20241993C2semidirect
6. #20241989C2semidirect
7. #20241982C2semidirect
8. #20241977C2semidirect
9. #20241974C2semidirect
10. C2×#15688864C2semidirect
11. C47×#2875C2semidirect
289. #2875D47semidirect
345. C47#17653semidirect

Maximal subgroups

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

subgroupindex
#202573662
#202524022
#202420062
#202420022
#202419932
#202419892
#202419822
#202419772
#202419742
C2×#156888642
C47×#28752
#1765347