Finite Group Databasebeta

← #26184454

Subgroups — #26184454

Order 12032.

940 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257349C211yesmaximal, characteristic
2#20252039C211yesmaximal, characteristic, complemented
3#20242017C212yesmaximal, complemented
4#20242011C212yesmaximal, complemented
5#20242000C211yesmaximal, characteristic, complemented
6#20241996C211yesmaximal, characteristic, complemented
7#20241993C212yesmaximal, complemented
8#20241985C212yesmaximal, complemented
9#20241979C211yesmaximal, characteristic, complemented
10#20241977C211yesmaximal, characteristic, complemented
11C47×#2971C211yesmaximal, characteristic, complemented
12#15689858C411yescharacteristic
13#15689846C411yescharacteristic
14#15689836—21no—
15#15689797—22no—
16D4×Dic94C411yescharacteristic
17#15689782C412yes—
18#15689769C412yes—
19#15689761—21no—
20D4×D188C411yescharacteristic
21#15689745—21no—
22#15689706—22no—
23#15689673C2211yescharacteristic
24#15689657C2211yescharacteristic
25#15689657C2212yes—
26C2×#13094012C2212yes—
27#15689595—21no—
28#15689184—22no—
29#15689180C2212yescomplemented
30#15689180C2212yescomplemented
31#15689150—21no—
32#15689090—21no—
33C4×#10625329C2211yescharacteristic, complemented
34C2×#13093935C2211yescharacteristic, complemented
35#15688864C2211yescharacteristic, complemented
36#15688864C2211yescharacteristic, complemented
37#15688864C2212yescomplemented
38#15688862C2211yescharacteristic
39#15688862C2211yescharacteristic
40#15688862C2212yes—
41#15688862C2212yescomplemented
42#15688862C2212yescomplemented
43#15688862C2212yescomplemented
44#15688862C2212yescomplemented
45C4×#41×C47C2211yescharacteristic
46C8×Q8×C47—21no—
47C47×#436—21no—
48C47×#436—22no—
49C47×#432C2211yescharacteristic, complemented
50C47×#432C2212yescomplemented
51C4×M16×C47C2211yescharacteristic, complemented
52C4×M16×C47C2212yescomplemented
53C2×#10625350C2×C412yes—
54C22×Dic94C2×C412yes—
55C22×D188C2×C412yes—
56C2×D4×D47—21no—
57C2×#10625329—21no—
58C22×C47⋊C8—21no—
59#13094024C2×C411yescharacteristic
60#13094023—21no—
61#13094021C2×C411yescharacteristic
62#13094019—22no—
63#13094018—22no—
64#13094018—21no—
65#13094016C2311yescharacteristic
66#13094013—21no—
67#13094012—22no—
68#13094012—22no—
69#13094012C2×C412yes—
70#13094012C2×C412yes—
71#13094012C2311yescharacteristic
72#13094012C2312yes—
73#13094010—22no—
74#13094009—22no—
75#13094009—22no—
76#13094007—22no—
77#13094007—21no—
78#13094003C2×C412yes—
79#13094003C2×C411yescharacteristic
80#13094002—22no—
81#13094001—22no—
82#13094000—21no—
83#13093999—21no—
84#13093998—21no—
85#13093998—21no—
86#13093998C2×C412yes—
87#13093998C2×C411yescharacteristic
88#13093997—22no—
89#13093996—22no—
90#13093994—22no—
91#13093992—22no—
92C4×Dic94C2×C412yes—
93C4×Dic94C2×C411yescharacteristic
94C4×D188—21no—
95C4×D188—21no—
96C4×D188C2×C411yescharacteristic
97C4×D188C2×C412yes—
98D4×Dic47—21no—
99#13093986—21no—
100C4×C22⋊D47—21no—
101C4×C22⋊D47—22no—
102(C22⋊D47)⋊C4—22no—
103#13093982—21no—
104#13093981—21no—
105#13093981—22no—
106C2×(C2×D47)⋊C4—22no—
107C2×(C2×D47)⋊C4—22no—
108C22⋊C4×D47—22no—
109C4⋊(C47⋊C8)—22no—
110C4⋊(C47⋊C8)—22no—
111C22⋊(C47⋊C8)—22no—
112C22⋊(C47⋊C8)—22no—
113#13093935—21no—
114#13093935C2311yescharacteristic
115#13093935C2311yescharacteristic
116#13093935C2312yes—
117C4×C47⋊C8—21no—
118C2×#41×C47D411yescharacteristic
119C2×M16×C47—22no—
120C2×M16×C47—21no—
121C4×Q8×C47C2×C411yescharacteristic
122C4×Q8×C47D411yescharacteristic
123C4×D4×C47C2×C411yescharacteristic
124C4×D4×C47C2×C412yes—
125C4×D4×C47D411yescharacteristic
126C4×D4×C47D412yes—
127#117×C47D412yes—
128#117×C47D411yescharacteristic
129C2×C42×C47C2311yescharacteristic
130C2×C42×C47C2312yes—
131C4⋊C8×C47—21no—
132C4⋊C8×C47—22no—
133C22⋊C8×C47—22no—
134C22⋊C8×C47—21no—
135#97×C47C2311yescharacteristic
136#97×C47C2311yescharacteristic
137#97×C47C2312yes—
138C4×C8×C47—21no—
139C4×C8×C47—22no—
140C23×D47—22no—
141#41×C47—42no—
142C2×Q8×C47C22⋊C411yescharacteristic
143C2×D4×C47C22⋊C411yescharacteristic
144C2×D4×C47C22⋊C412yes—
145C22×C4×C47C22×C412yes—
146C22×C4×C47C22×C411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D412yes—
149C22×C4×C47C2×D412yes—
150C22×C4×C47C2×D411yescharacteristic
151C2×C22⋊D47—22no—
152C2×C22⋊D47—21no—
153C2×C22⋊D47—22no—
154C22×Dic47—22no—
155D4×D47—44no—
156#10625350—82no—
157C2×D188—21no—
158C2×D188—21no—
159C2×D188—22no—
160C2×D188—82no—
161C2×D188C22×C411yescharacteristic
162C2×D188C22×C412yes—
163C2×C4×D47—21no—
164C2×C4×D47—22no—
165C2×C4×D47—22no—
166C2×Dic94—22no—
167C2×Dic94—82no—
168C2×Dic94C22×C411yescharacteristic
169C2×Dic94C22×C412yes—
170M16×C47—44no—
171M16×C47—42no—
172C2×C8×C47—21no—
173C2×C8×C47—22no—
174C2×C8×C47—21no—
175C4⋊C4×C47C22⋊C411yescharacteristic
176C4⋊C4×C47C22⋊C412yes—
177C4⋊C4×C47C22⋊C411yescharacteristic
178C4⋊C4×C47C22⋊C412yes—
179C22⋊C4×C47C22⋊C412yes—
180C22⋊C4×C47C22⋊C412yes—
181C22⋊C4×C47C22⋊C412yes—
182C42×C47C22×C412yes—
183C42×C47C22×C411yescharacteristic
184C42×C47C2×D411yescharacteristic
185C42×C47C2×D412yes—
186C42×C47C2×D411yescharacteristic
187C42×C47C2×D412yes—
188C42×C47C2411yescharacteristic
189C22⋊Dic47—22no—
190C22⋊Dic47—21no—
191(C2×D47)⋊C4—21no—
192(C2×D47)⋊C4—22no—
193(C2×D47)⋊C4—22no—
194(C2×D47)⋊C4—22no—
195(C2×D47)⋊C4—22no—
196(C2×D47)⋊C4—22no—
197(C2×D47)⋊C4—22no—
198(C2×D47)⋊C4—21no—
199(C2×D47)⋊C4—21no—
200(C2×D47)⋊C4—21no—
201(C2×D47)⋊C4—22no—
202(C2×D47)⋊C4—22no—
203C4⋊Dic47—21no—
204C4⋊Dic47—21no—
205C4⋊Dic47C22×C411yescharacteristic
206C4⋊Dic47C22×C411yescharacteristic
207Dic47⋊C4—21no—
208Dic47⋊C4—22no—
209Dic47⋊C4—22no—
210Dic47⋊C4—21no—
211Dic47⋊C4—21no—
212Dic47⋊C4—21no—
213C4×Dic47—22no—
214C4×Dic47—21no—
215#10625329—42no—
216C2×C47⋊C8—22no—
217C2×C47⋊C8—21no—
218C2×C47⋊C8—21no—
219C2×C47⋊C8—42no—
220C23×C47C2×C22⋊C412yes—
221C23×C47C2×C22⋊C411yescharacteristic
222C22×D47—21no—
223C22×D47—22no—
224C22×D47—44no—
225C22×D47—82no—
226C22×D47—82no—
227Q8×C47—22no—
228D4×C47—22no—
229D4×C47—42no—
230C2×C4×C47—22no—
231C2×C4×C47—42no—
232C2×C4×C47—41no—
233C2×C4×C47—42no—
234C2×C4×C47—41no—
235C2×C4×C47—42no—
236C2×C4×C47C2×C22⋊C412yes—
237C2×C4×C47C2×C22⋊C411yescharacteristic
238C2×C4×C47C2×C22⋊C412yes—
239C2×C4×C47C2×C22⋊C412yes—
240C2×C4×C47C2×C22⋊C411yescharacteristic
241C2×C4×C47C2×C22⋊C411yescharacteristic
242C2×C4×C47C23×C411yescharacteristic
243C2×C4×C47C22×D411yescharacteristic
244C2×C4×C47C22×D411yescharacteristic
245C22⋊D47—44no—
246C22⋊D47—82no—
247C22⋊D47—81no—
248C2×Dic47—21no—
249C2×Dic47—21no—
250C2×Dic47—22no—
251C2×Dic47—21no—
252C2×Dic47—82no—
253D188—41no—
254D188—42no—
255D188—42no—
256D188—81no—
257D188—82no—
258C4×D47—42no—
259C4×D47—82no—
260C4×D47—82no—
261Dic94—41no—
262Dic94—42no—
263Dic94—82no—
264C8×C47—22no—
265C8×C47—42no—
266C8×C47—41no—
267C47⋊C8—41no—
268C47⋊C8—41no—
269C47⋊C8—42no—
270#17681—471nomaximal, Sylow
271C22×C47—22no—
272C22×C47—21no—
273C22×C47—22no—
274C22×C47—21no—
275C22×C47—21no—
276C22×C47—22no—
277C22×C47C22×C22⋊C411yescharacteristic
278C2×D47—42no—
279C2×D47—41no—
280C2×D47—41no—
281C2×D47—42no—
282C2×D47—42no—
283C2×D47—41no—
284C2×D47—164no—
285C4×C47—21no—
286C4×C47—22no—
287C4×C47—21no—
288C4×C47—21no—
289C4×C47—22no—
290C4×C47—21no—
291C4×C47—22no—
292C4×C47—21no—
293C4×C47#42112yes—
294Dic47—41no—
295Dic47—42no—
296Dic47—41no—
297Dic47—41no—
298#3514—471no—
299#2971D4711yescharacteristic, complemented
300#2923—471no—
301#1535—472no—
302#1533—472no—
303#1530—471no—
304#1529—471no—
305#1529—472no—
306#1527—472no—
307#1525—471no—
308#1524—471no—
309C2×C47—42no—
310C2×C47—41no—
311C2×C47#288811yescharacteristic
312C2×C47C2×#42111yescharacteristic
313C2×C47#290111yescharacteristic
314D47—82no—
315D47—81no—
316#559—471no—
317#556—471no—
318#555—941no—
319#551—942no—
320D4×Q8—471no—
321#548—472no—
322#546—472no—
323#546—941no—
324D42—471no—
325#542—941no—
326#538—942no—
327#532—471no—
328#529—471no—
329#529—472no—
330C2×#124—472no—
331#518—941no—
332C4×#41C2×D4711yescharacteristic
333C8×Q8—21no—
334#436—21no—
335#436—22no—
336#433—942no—
337#432—472no—
338#432—472no—
339#432C2×D4711yescharacteristic, complemented
340#432C2×D4712yescomplemented
341#424—941no—
342#408—941no—
343C4×M16—471no—
344C4×M16C2×D4711yescharacteristic, complemented
345C4×M16C2×D4712yescomplemented
346C2×#97—471no—
347#330—471no—
348#330—471no—
349#330—472no—
350#329—471no—
351#329—471no—
352#329—472no—
353#329—472no—
354#329—472no—
355#329—472no—
356#329—472no—
357C47#1768111yescharacteristic, complemented, Sylow
358C2×#41—472no—
359C2×#41D18811yescharacteristic
360C22×Q8—472no—
361C22×D4—472no—
362C22×D4—941no—
363C2×M16—22no—
364C2×M16—21no—
365C2×M16—941no—
366C22×C8—941no—
367C4⋊Q8—471no—
368C4⋊Q8—941no—
369#127—471no—
370#126—942no—
371#126—941no—
372#126—942no—
373#125—471no—
374#124—472no—
375#124—472no—
376#124—471no—
377#124—472no—
378#124—941no—
379#124—942no—
380#124—942no—
381#124—942no—
382#123—942no—
383#123—941no—
384#123—942no—
385#123—942no—
386C22⋊Q8—472no—
387C22⋊Q8—471no—
388C22⋊Q8—941no—
389C22⋊Q8—942no—
390C22⋊Q8—942no—
391#121—472no—
392#121—471no—
393#121—942no—
394#121—941no—
395#121—941no—
396#121—941no—
397#121—942no—
398#120—942no—
399#120—942no—
400C4×Q8—472no—
401C4×Q8—471no—
402C4×Q8C4×D4711yescharacteristic
403C4×Q8C22⋊D4711yescharacteristic
404C4×D4—471no—
405C4×D4—472no—
406C4×D4—941no—
407C4×D4—941no—
408C4×D4—942no—
409C4×D4—941no—
410C4×D4—942no—
411C4×D4—941no—
412C4×D4—941no—
413C4×D4C4×D4711yescharacteristic
414C4×D4C4×D4712yes—
415C4×D4C22⋊D4711yescharacteristic
416C4×D4C22⋊D4712yes—
417#117—941no—
418#117—942no—
419#117—941no—
420#117D18812yes—
421#117D18811yescharacteristic
422C2×C22⋊C4—942no—
423C2×C22⋊C4—942no—
424C2×C22⋊C4—942no—
425C2×C42C22×D4712yes—
426C2×C42C22×D4711yescharacteristic
427C4⋊C8—21no—
428C4⋊C8—22no—
429C4⋊C8—942no—
430C4⋊C8—942no—
431C22⋊C8—22no—
432C22⋊C8—21no—
433C22⋊C8—942no—
434C22⋊C8—942no—
435#97—471no—
436#97—471no—
437#97—472no—
438#97—941no—
439#97C22×D4711yescharacteristic
440#97C22×D4711yescharacteristic
441#97C22×D4712yes—
442C4×C8—21no—
443C4×C8—22no—
444C4×C8—941no—
445C24—942no—
446#41—42no—
447#41—3762no—
448C2×Q8—471no—
449C2×Q8—472no—
450C2×Q8—942no—
451C2×Q8—3762no—
452C2×Q8(C2×D47)⋊C411yescharacteristic
453C2×D4—471no—
454C2×D4—472no—
455C2×D4—941no—
456C2×D4—942no—
457C2×D4—942no—
458C2×D4—941no—
459C2×D4—942no—
460C2×D4—941no—
461C2×D4—3762no—
462C2×D4—1884no—
463C2×D4(C2×D47)⋊C411yescharacteristic
464C2×D4(C2×D47)⋊C412yes—
465C22×C4—942no—
466C22×C4—942no—
467C22×C4—942no—
468C22×C4—941no—
469C22×C4C2×C4×D4712yes—
470C22×C4C2×C4×D4711yescharacteristic
471C22×C4C2×D18811yescharacteristic
472C22×C4C2×D18812yes—
473C22×C4C2×C22⋊D4712yes—
474C22×C4C2×C22⋊D4711yescharacteristic
475M16—44no—
476M16—42no—
477M16—1882no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—22no—
481C2×C8—942no—
482C2×C8—941no—
483C2×C8—941no—
484C2×C8—1882no—
485C4⋊C4—471no—
486C4⋊C4—471no—
487C4⋊C4—941no—
488C4⋊C4—942no—
489C4⋊C4—942no—
490C4⋊C4—941no—
491C4⋊C4—941no—
492C4⋊C4—941no—
493C4⋊C4—941no—
494C4⋊C4—941no—
495C4⋊C4(C2×D47)⋊C411yescharacteristic
496C4⋊C4(C2×D47)⋊C411yescharacteristic
497C4⋊C4(C2×D47)⋊C412yes—
498C4⋊C4(C2×D47)⋊C412yes—
499C22⋊C4—942no—
500C22⋊C4—941no—
501C22⋊C4—941no—
502C22⋊C4—942no—
503C22⋊C4—942no—
504C22⋊C4—942no—
505C22⋊C4—942no—
506C22⋊C4—942no—
507C22⋊C4—941no—
508C22⋊C4—942no—
509C22⋊C4—941no—
510C22⋊C4—942no—
511C22⋊C4—941no—
512C22⋊C4—942no—
513C22⋊C4(C2×D47)⋊C412yes—
514C22⋊C4(C2×D47)⋊C412yes—
515C22⋊C4(C2×D47)⋊C412yes—
516C42—942no—
517C42—941no—
518C42C2×C4×D4712yes—
519C42C2×C4×D4711yescharacteristic
520C42C2×D18811yescharacteristic
521C42C2×D18812yes—
522C42C2×C22⋊D4711yescharacteristic
523C42C2×C22⋊D4712yes—
524C42C23×D4711yescharacteristic
525C23—941no—
526C23—942no—
527C23—3762no—
528C23—3762no—
529C23—1884no—
530C23C2×(C2×D47)⋊C412yes—
531C23C2×(C2×D47)⋊C411yescharacteristic
532Q8—22no—
533Q8—3762no—
534Q8—1881no—
535Q8—1882no—
536D4—22no—
537D4—42no—
538D4—3761no—
539D4—3762no—
540D4—3761no—
541D4—3762no—
542D4—1881no—
543D4—1882no—
544D4—1884no—
545D4—1882no—
546C2×C4—22no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—42no—
552C2×C4—941no—
553C2×C4—941no—
554C2×C4—941no—
555C2×C4—942no—
556C2×C4—3762no—
557C2×C4—3762no—
558C2×C4—3762no—
559C2×C4—1882no—
560C2×C4C2×(C2×D47)⋊C412yes—
561C2×C4C2×(C2×D47)⋊C411yescharacteristic
562C2×C4C2×(C2×D47)⋊C412yes—
563C2×C4C2×(C2×D47)⋊C411yescharacteristic
564C2×C4C2×(C2×D47)⋊C412yes—
565C2×C4C2×(C2×D47)⋊C411yescharacteristic
566C2×C4C22×C4×D4711yescharacteristic
567C2×C4C22×C22⋊D4711yescharacteristic
568C2×C4C22×D18811yescharacteristic
569C8—22no—
570C8—41no—
571C8—42no—
572C8—1881no—
573C8—1881no—
574C8—1882no—
575C22—21no—
576C22—22no—
577C22—21no—
578C22—22no—
579C22—22no—
580C22—21no—
581C22—1881no—
582C22—1882no—
583C22—1882no—
584C22—1881no—
585C22—1882no—
586C22—1881no—
587C22—7524no—
588C22C22×(C2×D47)⋊C411yescharacteristic
589C4—21no—
590C4—22no—
591C4—21no—
592C4—22no—
593C4—21no—
594C4—21no—
595C4—21no—
596C4—22no—
597C4—1881no—
598C4—1881no—
599C4—1882no—
600C4—1881no—
601C4#1568913912yes—
602C2—42no—
603C2—41no—
604C2—3761no—
605C2—3762no—
606C2#2025186611yescharacteristic
607C2C2×#1568913911yescharacteristic
608C2#2025193611yescharacteristic
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#2971C2
socleC22×C47C22×C22⋊C4
prime radical#20257349C2
cosocleC42×C47C24
hypercenter#2971D47
hypocenterC47#17681
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C4×#41 ◁ #2971

Complements

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

NH ≅ G/Ntype
2. #20252039C2semidirect
3. #20242017C2semidirect
4. #20242011C2semidirect
5. #20242000C2semidirect
6. #20241996C2semidirect
7. #20241993C2semidirect
8. #20241985C2semidirect
9. #20241979C2semidirect
10. #20241977C2semidirect
11. C47×#2971C2semidirect
29. #15689180C22semidirect
30. #15689180C22semidirect
33. C4×#10625329C22semidirect
34. C2×#13093935C22semidirect
35. #15688864C22semidirect
36. #15688864C22semidirect
37. #15688864C22semidirect
41. #15688862C22semidirect
42. #15688862C22semidirect
43. #15688862C22semidirect
44. #15688862C22semidirect
49. C47×#432C22semidirect
50. C47×#432C22semidirect
51. C4×M16×C47C22semidirect
52. C4×M16×C47C22semidirect
299. #2971D47semidirect
339. #432C2×D47semidirect
340. #432C2×D47semidirect
344. C4×M16C2×D47semidirect
345. C4×M16C2×D47semidirect
357. C47#17681semidirect

Maximal subgroups

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

subgroupindex
#202573492
#202520392
#202420172
#202420112
#202420002
#202419962
#202419932
#202419852
#202419792
#202419772
C47×#29712
#1768147