Finite Group Databasebeta

← #26123325

Subgroups — #26123325

Order 12032.

870 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689139C211yesmaximal, characteristic
2#20251905C211yesmaximal, characteristic
3C2×#15689091C211yesmaximal, characteristic
4C2×#15688894C211yesmaximal, characteristic, complemented
5C2×#15688887C211yesmaximal, characteristic, complemented
6#20243104C211yesmaximal, characteristic, complemented
7C47×#2905C211yesmaximal, characteristic, complemented
8C22×#10625350C2211yescharacteristic
9C22×#10625329C2211yescharacteristic
10C23×C47⋊C8C2211yescharacteristic
11C2×#13093983C2211yescharacteristic
12#15689139—22no—
13#15689139—22no—
14C2×#13093950—22no—
15#15689131C412yescomplemented
16#15689127C412yescomplemented
17#15689122—22no—
18#15689117—22no—
19#15689091—22no—
20#15689091—22no—
21C2×(C2×D47)⋊C8—22no—
22#15688894—22no—
23#15688887—22no—
24#15688885—22no—
25C22×M16×C47C2211yescharacteristic
26C22×M16×C47C2211yescharacteristic
27C2×#117×C47C2211yescharacteristic
28C47×#424—22no—
29C47×#424—22no—
30C47×#423—22no—
31C2×#10625350—21no—
32C2×#10625350—21no—
33C2×#10625350—21no—
34C2×#10625350—21no—
35C2×#10625350—21no—
36C2×#10625350—21no—
37C2×#10625350—21no—
38C2×#10625350—21no—
39C2×#10625350—21no—
40C2×#10625350—21no—
41C2×#10625350D412yes—
42C2×#10625350Q812yes—
43C22×Dic94C2×C411yescharacteristic
44C22×D188C2×C411yescharacteristic
45C22×C22⋊D47—21no—
46C22×C4×D47—21no—
47C2×#10625329—21no—
48C2×#10625329—21no—
49C2×#10625329—21no—
50C2×#10625329—21no—
51C2×#10625329—21no—
52C2×#10625329C2×C412yes—
53C22×C47⋊C8—21no—
54C22×C47⋊C8—21no—
55C22×C47⋊C8—21no—
56C22×C47⋊C8—21no—
57C22×C47⋊C8—21no—
58C22×C47⋊C8—21no—
59C22×C47⋊C8—21no—
60C22×C47⋊C8D412yes—
61#13093983—21no—
62#13093983—21no—
63#13093983—21no—
64#13093983D412yes—
65C2×C4⋊Dic47—21no—
66C2×C22⋊Dic47—21no—
67C2×C4×Dic47—21no—
68#13093950—28no—
69#13093947—24no—
70#13093944—24no—
71(C2×D47)⋊C8—28no—
72C23×C4×C47C2311yescharacteristic
73C2×M16×C47—21no—
74C2×M16×C47—21no—
75C2×M16×C47—21no—
76C2×M16×C47—21no—
77C2×M16×C47—21no—
78C2×M16×C47—21no—
79C2×M16×C47—21no—
80C2×M16×C47—21no—
81C2×M16×C47—21no—
82C2×M16×C47—21no—
83C2×M16×C47—21no—
84C2×M16×C47D412yes—
85C22×C8×C47—21no—
86C22×C8×C47D412yes—
87#117×C47—21no—
88#117×C47—21no—
89#117×C47C2×C412yes—
90#117×C47D412yes—
91C2×C4⋊C4×C47D412yes—
92C2×C22⋊C4×C47—21no—
93C2×C42×C47—21no—
94C4⋊C8×C47—28no—
95C24×C47#4111yescharacteristic
96C23×D47—21no—
97C22×C4×C47—22no—
98C22×C4×C47C22×C411yescharacteristic
99C22×C4×C47C2×D411yescharacteristic
100C22×C4×C47C2×D411yescharacteristic
101C22×C4×C47C2×D411yescharacteristic
102C22×C4×C47C2×D411yescharacteristic
103C22×C4×C47C2×D411yescharacteristic
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47C2×Q811yescharacteristic
107C22×C4×C47#4111yescharacteristic
108C22×C4×C47#4111yescharacteristic
109C22×C4×C47#4111yescharacteristic
110C22×C4×C47#4111yescharacteristic
111C22×C4×C47#4111yescharacteristic
112C2×C22⋊D47—22no—
113C2×C22⋊D47—22no—
114C2×C22⋊D47—41no—
115C2×C22⋊D47—41no—
116C2×C22⋊D47—41no—
117C2×C22⋊D47—41no—
118C22×Dic47—21no—
119C22×Dic47—41no—
120#10625350—22no—
121#10625350—22no—
122#10625350—22no—
123#10625350—22no—
124#10625350—22no—
125#10625350—22no—
126#10625350—22no—
127#10625350—22no—
128#10625350—41no—
129#10625350—41no—
130#10625350—41no—
131#10625350—41no—
132#10625350—41no—
133#10625350—41no—
134#10625350—41no—
135#10625350—41no—
136C2×D188—21no—
137C2×D188—21no—
138C2×D188—21no—
139C2×D188—21no—
140C2×D188—21no—
141C2×D188C4⋊C412yes—
142C2×C4×D47—22no—
143C2×C4×D47—22no—
144C2×C4×D47—41no—
145C2×C4×D47—41no—
146C2×C4×D47—41no—
147C2×C4×D47—41no—
148C2×Dic94—21no—
149C2×Dic94—21no—
150C2×Dic94—21no—
151C2×Dic94—21no—
152C2×Dic94—21no—
153C2×Dic94C4⋊C412yes—
154M16×C47—22no—
155M16×C47—22no—
156M16×C47—41no—
157M16×C47—41no—
158M16×C47—41no—
159M16×C47—41no—
160M16×C47—41no—
161M16×C47—41no—
162C2×C8×C47—22no—
163C2×C8×C47—22no—
164C2×C8×C47—22no—
165C2×C8×C47—22no—
166C2×C8×C47—41no—
167C2×C8×C47—41no—
168C4⋊C4×C47—22no—
169C4⋊C4×C47C22⋊C414yes—
170C22⋊C4×C47—22no—
171C22⋊C4×C47—22no—
172C42×C47—22no—
173C42×C47—22no—
174C22⋊Dic47—22no—
175C22⋊Dic47—41no—
176C4⋊Dic47—22no—
177C4⋊Dic47—41no—
178C4×Dic47—22no—
179C4×Dic47—41no—
180#10625329—22no—
181#10625329—22no—
182#10625329—41no—
183#10625329—41no—
184C2×C47⋊C8—22no—
185C2×C47⋊C8—22no—
186C2×C47⋊C8—22no—
187C2×C47⋊C8—22no—
188C2×C47⋊C8—41no—
189C2×C47⋊C8—41no—
190C2×C47⋊C8—41no—
191C2×C47⋊C8—41no—
192C2×C47⋊C8—41no—
193C2×C47⋊C8—41no—
194C23×C47—42no—
195C23×C47C4×D411yescharacteristic
196C23×C47#12111yescharacteristic
197C23×C47C22⋊Q811yescharacteristic
198C23×C47#12311yescharacteristic
199C23×C47#12311yescharacteristic
200C23×C47#12311yescharacteristic
201C23×C47#12611yescharacteristic
202C22×D47—22no—
203C22×D47—44no—
204C22×D47—41no—
205C22×D47—41no—
206C2×C4×C47—24no—
207C2×C4×C47—82no—
208C2×C4×C47C2×C22⋊C411yescharacteristic
209C2×C4×C47C2×C4⋊C411yescharacteristic
210C2×C4×C47#11711yescharacteristic
211C2×C4×C47C4×D411yescharacteristic
212C2×C4×C47C4×D411yescharacteristic
213C2×C4×C47C4×D411yescharacteristic
214C2×C4×C47#12011yescharacteristic
215C2×C4×C47#12011yescharacteristic
216C2×C4×C47#12011yescharacteristic
217C2×C4×C47#12111yescharacteristic
218C2×C4×C47#12111yescharacteristic
219C2×C4×C47#12111yescharacteristic
220C2×C4×C47#12111yescharacteristic
221C2×C4×C47#12111yescharacteristic
222C2×C4×C47#12111yescharacteristic
223C2×C4×C47#12111yescharacteristic
224C2×C4×C47C22⋊Q811yescharacteristic
225C2×C4×C47C22⋊Q811yescharacteristic
226C2×C4×C47C22⋊Q811yescharacteristic
227C2×C4×C47C22⋊Q811yescharacteristic
228C2×C4×C47C22⋊Q811yescharacteristic
229C2×C4×C47#12311yescharacteristic
230C2×C4×C47#12311yescharacteristic
231C2×C4×C47#12411yescharacteristic
232C2×C4×C47#12411yescharacteristic
233C2×C4×C47#12411yescharacteristic
234C2×C4×C47#12611yescharacteristic
235C2×C4×C47#12711yescharacteristic
236C22⋊D47—42no—
237C22⋊D47—42no—
238C22⋊D47—42no—
239C22⋊D47—42no—
240C2×Dic47—22no—
241C2×Dic47—41no—
242C2×Dic47—41no—
243C2×Dic47—42no—
244C2×Dic47—82no—
245D188—22no—
246D188—22no—
247D188—41no—
248D188—41no—
249C4×D47—42no—
250C4×D47—42no—
251C4×D47—42no—
252C4×D47—42no—
253Dic94—22no—
254Dic94—22no—
255Dic94—41no—
256Dic94—41no—
257C8×C47—42no—
258C8×C47—42no—
259C47⋊C8—42no—
260C47⋊C8—42no—
261#11065—471nomaximal, Sylow
262C22×C47—22no—
263C22×C47—22no—
264C22×C47—22no—
265C22×C47—41no—
266C22×C47—41no—
267C22×C47—41no—
268C22×C47—41no—
269C22×C47#38511yescharacteristic
270C22×C47#38611yescharacteristic
271C22×C47#38811yescharacteristic
272C22×C47#39411yescharacteristic
273C22×C47#39611yescharacteristic
274C22×C47#39711yescharacteristic
275C22×C47#39911yescharacteristic
276C2×D47—44no—
277C2×D47—42no—
278C2×D47—82no—
279C2×D47—82no—
280C4×C47—44no—
281C4×C47#38011yescharacteristic
282C4×C47#38511yescharacteristic
283C4×C47#38811yescharacteristic
284C4×C47#39011yescharacteristic
285C4×C47#39211yescharacteristic
286C4×C47#39311yescharacteristic
287C4×C47#39411yescharacteristic
288C4×C47#39611yescharacteristic
289Dic47—42no—
290Dic47—82no—
291#2905D4711yescharacteristic, complemented
292C2×#421—471no—
293#2896—471no—
294C2×#408—471no—
295C2×#344—471no—
296C2×#341—471no—
297#1735—471no—
298C2×C47—21no—
299C2×C47—21no—
300C2×C47—21no—
301C2×C47—21no—
302C2×C47—21no—
303C2×C47—21no—
304C2×C47#143911yescharacteristic
305C2×C47#188112yes—
306D47—82no—
307C22×#41—471no—
308C22×M16—471no—
309C22×M16C2×D4711yescharacteristic
310C22×M16C2×D4711yescharacteristic
311C23×C8—471no—
312C2×#117—471no—
313C2×#117C2×D4711yescharacteristic
314#424—22no—
315#424—22no—
316#423—22no—
317#421—942no—
318#421—942no—
319C2×#104—942no—
320#419—472no—
321#418—472no—
322#417—942no—
323#416—942no—
324#408—942no—
325#408—942no—
326C2×C22⋊C8—942no—
327#344—942no—
328#341—942no—
329#340—942no—
330C47#1106511yescharacteristic, complemented, Sylow
331C2×#41—472no—
332C2×#41—472no—
333C2×#41—941no—
334C2×#41—941no—
335C2×#41—941no—
336C2×#41—941no—
337C2×#41—941no—
338C2×#41—941no—
339C2×#41—941no—
340C2×#41—941no—
341C2×#41—941no—
342C2×#41—941no—
343C22×Q8—471no—
344C22×D4—471no—
345C22×D4—941no—
346C23×C4—941no—
347C23×C4C22×D4711yescharacteristic
348C2×M16—21no—
349C2×M16—21no—
350C2×M16—21no—
351C2×M16—21no—
352C2×M16—21no—
353C2×M16—21no—
354C2×M16—21no—
355C2×M16—21no—
356C2×M16—21no—
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—472no—
360C2×M16—941no—
361C2×M16—941no—
362C2×M16—941no—
363C2×M16—941no—
364C2×M16—941no—
365C2×M16C22⋊D4712yes—
366C22×C8—21no—
367C22×C8—472no—
368C22×C8—941no—
369C22×C8—941no—
370C22×C8—941no—
371C22×C8—941no—
372C22×C8—941no—
373C22×C8—941no—
374C22×C8—941no—
375C22×C8C22⋊D4712yes—
376#117—21no—
377#117—21no—
378#117—472no—
379#117—941no—
380#117—941no—
381#117—941no—
382#117C4×D4712yes—
383#117C22⋊D4712yes—
384C2×C4⋊C4—941no—
385C2×C4⋊C4D18812yes—
386C2×C22⋊C4—21no—
387C2×C22⋊C4—941no—
388C2×C42—21no—
389C2×C42—941no—
390C4⋊C8—28no—
391#104—948no—
392#103—944no—
393#102—944no—
394C22⋊C8—948no—
395C24—941no—
396C24#1062535011yescharacteristic
397#41—942no—
398#41—942no—
399#41—942no—
400#41—942no—
401#41—942no—
402#41—942no—
403#41—942no—
404#41—942no—
405#41—1881no—
406#41—1881no—
407#41—1881no—
408#41—1881no—
409#41—1881no—
410#41—1881no—
411#41—1881no—
412#41—1881no—
413C2×Q8—472no—
414C2×Q8—941no—
415C2×Q8—941no—
416C2×Q8—941no—
417C2×Q8—941no—
418C2×Q8—941no—
419C2×D4—472no—
420C2×D4—942no—
421C2×D4—942no—
422C2×D4—941no—
423C2×D4—941no—
424C2×D4—941no—
425C2×D4—941no—
426C2×D4—941no—
427C2×D4—1881no—
428C2×D4—1881no—
429C2×D4—1881no—
430C2×D4—1881no—
431C22×C4—22no—
432C22×C4—941no—
433C22×C4—942no—
434C22×C4—942no—
435C22×C4—1881no—
436C22×C4—1881no—
437C22×C4—1881no—
438C22×C4—1881no—
439C22×C4—1881no—
440C22×C4C2×C4×D4711yescharacteristic
441C22×C4C2×D18811yescharacteristic
442C22×C4#1062535011yescharacteristic
443C22×C4D4×D4711yescharacteristic
444C22×C4D4×D4711yescharacteristic
445C22×C4D4×D4711yescharacteristic
446C22×C4#1062535211yescharacteristic
447C22×C4Q8×D4711yescharacteristic
448C22×C4#1062535411yescharacteristic
449C22×C4#1062535411yescharacteristic
450C22×C4#1062535411yescharacteristic
451C22×C4C2×C22⋊D4711yescharacteristic
452C22×C4C2×C22⋊D4711yescharacteristic
453C22×C4C2×C22⋊D4711yescharacteristic
454M16—22no—
455M16—22no—
456M16—41no—
457M16—41no—
458M16—41no—
459M16—41no—
460M16—41no—
461M16—41no—
462M16—942no—
463M16—942no—
464M16—1881no—
465M16—1881no—
466C2×C8—22no—
467C2×C8—22no—
468C2×C8—22no—
469C2×C8—22no—
470C2×C8—41no—
471C2×C8—41no—
472C2×C8—942no—
473C2×C8—942no—
474C2×C8—942no—
475C2×C8—942no—
476C2×C8—1881no—
477C2×C8—1881no—
478C2×C8—1881no—
479C2×C8—1881no—
480C2×C8—1881no—
481C2×C8—1881no—
482C4⋊C4—22no—
483C4⋊C4—942no—
484C4⋊C4—1881no—
485C4⋊C4(C2×D47)⋊C414yes—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—942no—
489C22⋊C4—1881no—
490C42—22no—
491C42—22no—
492C42—942no—
493C42—1881no—
494C23—42no—
495C23—942no—
496C23—1884no—
497C23—1881no—
498C23—1881no—
499C23C4×C22⋊D4711yescharacteristic
500C23#1309399811yescharacteristic
501C23#1309400211yescharacteristic
502C23#1309400711yescharacteristic
503C23#1309400911yescharacteristic
504C23#1309400911yescharacteristic
505C23#1309401911yescharacteristic
506Q8—942no—
507Q8—942no—
508Q8—1881no—
509Q8—1881no—
510D4—942no—
511D4—942no—
512D4—1882no—
513D4—1882no—
514D4—1882no—
515D4—1882no—
516D4—1881no—
517D4—1881no—
518C2×C4—24no—
519C2×C4—82no—
520C2×C4—942no—
521C2×C4—3762no—
522C2×C4—1882no—
523C2×C4—1881no—
524C2×C4—1881no—
525C2×C4—1882no—
526C2×C4—1882no—
527C2×C4—1882no—
528C2×C4—1882no—
529C2×C4C2×(C2×D47)⋊C411yescharacteristic
530C2×C4C4⋊C4×D4711yescharacteristic
531C2×C4#1309398211yescharacteristic
532C2×C4C4×C22⋊D4711yescharacteristic
533C2×C4#1309398611yescharacteristic
534C2×C4#1309398611yescharacteristic
535C2×C4#1309399211yescharacteristic
536C2×C4#1309399211yescharacteristic
537C2×C4#1309399311yescharacteristic
538C2×C4#1309399511yescharacteristic
539C2×C4#1309399611yescharacteristic
540C2×C4#1309399611yescharacteristic
541C2×C4#1309399811yescharacteristic
542C2×C4#1309399911yescharacteristic
543C2×C4#1309399911yescharacteristic
544C2×C4#1309399911yescharacteristic
545C2×C4#1309400011yescharacteristic
546C2×C4#1309400011yescharacteristic
547C2×C4#1309400011yescharacteristic
548C2×C4#1309400211yescharacteristic
549C2×C4#1309400411yescharacteristic
550C2×C4#1309400911yescharacteristic
551C2×C4#1309400911yescharacteristic
552C2×C4#1309401311yescharacteristic
553C2×C4#1309401311yescharacteristic
554C2×C4#1309401311yescharacteristic
555C2×C4#1309401911yescharacteristic
556C2×C4#1309402011yescharacteristic
557C8—42no—
558C8—42no—
559C8—1882no—
560C8—1882no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—41no—
565C22—41no—
566C22—41no—
567C22—41no—
568C22—3762no—
569C22—3762no—
570C22—1884no—
571C22—1882no—
572C22#1568899411yescharacteristic
573C22#1568900211yescharacteristic
574C22#1568901311yescharacteristic
575C22#1568904011yescharacteristic
576C22#1568904511yescharacteristic
577C22#1568904911yescharacteristic
578C22#1568905811yescharacteristic
579C4—44no—
580C4—3762no—
581C4—1882no—
582C4#1568898011yescharacteristic
583C4#1568899411yescharacteristic
584C4#1568901311yescharacteristic
585C4#1568902411yescharacteristic
586C4#1568903111yescharacteristic
587C4#1568903411yescharacteristic
588C4#1568904011yescharacteristic
589C4#1568904511yescharacteristic
590C2—21no—
591C2—21no—
592C2—21no—
593C2—21no—
594C2—21no—
595C2—21no—
596C2—3762no—
597C2#2024163011yescharacteristic
598C2#2024375012yes—
subgroupG/H
centerC2×C4#13093993
commutatorC22×C4×C47C22×C4
FrattiniC23×C4C22×D47
FittingC47×#2905C2
socleC22×C47#386
prime radicalC22×#10625350C22
cosocleC23×C4×C47C23
hypercenter#2905D47
hypocenterC47#11065
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #2905

Complements

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

NH ≅ G/Ntype
4. C2×#15688894C2semidirect
5. C2×#15688887C2semidirect
6. #20243104C2semidirect
7. C47×#2905C2semidirect
15. #15689131C4semidirect
16. #15689127C4semidirect
291. #2905D47semidirect
330. C47#11065semidirect

Maximal subgroups

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

subgroupindex
C2×#156891392
#202519052
C2×#156890912
C2×#156888942
C2×#156888872
#202431042
C47×#29052
#1106547