Finite Group Databasebeta

← #26246169

Subgroups — #26246169

Order 12032.

988 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
1C2×#15689171C211yesmaximal, characteristic, complemented
2#20251951C211yesmaximal, characteristic, complemented
3C2×#15689141C211yesmaximal, characteristic
4C2×#15689139C211yesmaximal, characteristic
5#20251832C211yesmaximal, characteristic
6#20244944C218yesmaximal, complemented
7C2×#15688879C211yesmaximal, characteristic, complemented
8C47×#2905C211yesmaximal, characteristic, complemented
9C22×#10625350C2211yescharacteristic
10C22×#10625329C2211yescharacteristic
11C22×#10625329C2211yescharacteristic
12C2×#13093983C2211yescharacteristic
13#15689171—22no—
14#15689171C2214yescomplemented
15C2×#13093959—22no—
16#15689151—22no—
17#15689149C2214yescomplemented
18#15689146—22no—
19#15689141—22no—
20#15689141C2214yes—
21#15689139—22no—
22#15689139C2214yes—
23C2×#13093952—22no—
24C2×#13093950—22no—
25#15689091—22no—
26#15689089C2214yes—
27#15689086—22no—
28#15688879C2214yescomplemented
29C22×M16×C47C2211yescharacteristic
30C22×M16×C47C2211yescharacteristic
31C2×#117×C47C2211yescharacteristic
32C47×#424—22no—
33C47×#424C2214yescomplemented
34C47×#423—22no—
35C2×#10625350—21no—
36C2×#10625350—21no—
37C2×#10625350—21no—
38C2×#10625350—21no—
39C2×#10625350—21no—
40C2×#10625350—21no—
41C2×#10625350—21no—
42C2×#10625350—21no—
43C2×#10625350—21no—
44C2×#10625350—21no—
45C2×#10625350Q812yes—
46C2×#10625350C2312yes—
47C22×Dic94Q811yescharacteristic
48C22×D188Q811yescharacteristic
49C22×C22⋊D47—21no—
50C22×C4×D47—21no—
51C2×#10625329—21no—
52C2×#10625329—21no—
53C2×#10625329—21no—
54C2×#10625329—21no—
55C2×#10625329—21no—
56C2×#10625329—21no—
57C2×#10625329—21no—
58C2×#10625329—21no—
59C2×#10625329—21no—
60C2×#10625329D412yes—
61C2×#10625329C2312yes—
62C2×#10625329C2312yes—
63C22×C47⋊C8—21no—
64C22×C47⋊C8D412yes—
65#13093983—21no—
66#13093983—21no—
67#13093983D412yes—
68#13093983C2312yes—
69C2×C4⋊Dic47—21no—
70C2×C22⋊Dic47—21no—
71C2×C4×Dic47D412yes—
72#13093959—28no—
73Dic47⋊C8—28no—
74#13093952—28no—
75#13093950—28no—
76(C2×D47)⋊C8—28no—
77C23×C4×C47C2311yescharacteristic
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×C47—21no—
85C2×M16×C47—21no—
86C2×M16×C47—21no—
87C2×M16×C47—21no—
88C2×M16×C47—21no—
89C2×M16×C47C2312yes—
90C22×C8×C47—21no—
91C22×C8×C47C2312yes—
92#117×C47—21no—
93#117×C47—21no—
94#117×C47D412yes—
95#117×C47C2312yes—
96C2×C4⋊C4×C47—21no—
97C2×C22⋊C4×C47—21no—
98C2×C42×C47D412yes—
99C4⋊C8×C47—28no—
100C24×C47#4111yescharacteristic
101C23×D47—21no—
102C22×C4×C47—22no—
103C22×C4×C47C2×D411yescharacteristic
104C22×C4×C47C2×D411yescharacteristic
105C22×C4×C47C2×D411yescharacteristic
106C22×C4×C47C2×D411yescharacteristic
107C22×C4×C47C2×D411yescharacteristic
108C22×C4×C47C2×D411yescharacteristic
109C22×C4×C47C2×Q811yescharacteristic
110C22×C4×C47C2×Q811yescharacteristic
111C22×C4×C47#4111yescharacteristic
112C22×C4×C47#4111yescharacteristic
113C22×C4×C47#4111yescharacteristic
114C22×C4×C47#4111yescharacteristic
115C22×C4×C47#4111yescharacteristic
116C22×C4×C47C2411yescharacteristic
117C2×C22⋊D47—22no—
118C2×C22⋊D47—22no—
119C2×C22⋊D47—41no—
120C2×C22⋊D47—41no—
121C2×C22⋊D47—41no—
122C2×C22⋊D47—41no—
123C22×Dic47—21no—
124C22×Dic47—22no—
125#10625350—22no—
126#10625350—22no—
127#10625350—22no—
128#10625350—22no—
129#10625350—22no—
130#10625350—22no—
131#10625350—22no—
132#10625350—22no—
133#10625350—41no—
134#10625350—41no—
135#10625350—41no—
136#10625350—41no—
137#10625350—41no—
138#10625350—41no—
139#10625350—41no—
140#10625350—41no—
141C2×D188—21no—
142C2×D188—21no—
143C2×D188—21no—
144C2×D188—21no—
145C2×D188—21no—
146C2×D188C2×Q812yes—
147C2×C4×D47—22no—
148C2×C4×D47—22no—
149C2×C4×D47—41no—
150C2×C4×D47—41no—
151C2×C4×D47—41no—
152C2×C4×D47—41no—
153C2×Dic94—21no—
154C2×Dic94—21no—
155C2×Dic94—21no—
156C2×Dic94—21no—
157C2×Dic94—21no—
158C2×Dic94C2×Q812yes—
159M16×C47—22no—
160M16×C47—22no—
161M16×C47—41no—
162M16×C47—41no—
163M16×C47—41no—
164M16×C47—41no—
165M16×C47—41no—
166M16×C47—41no—
167C2×C8×C47—22no—
168C2×C8×C47—22no—
169C2×C8×C47—22no—
170C2×C8×C47—22no—
171C2×C8×C47—41no—
172C2×C8×C47—41no—
173C4⋊C4×C47—22no—
174C4⋊C4×C47—22no—
175C22⋊C4×C47—22no—
176C22⋊C4×C47—22no—
177C42×C47—22no—
178C42×C47C2×D414yes—
179C22⋊Dic47—22no—
180C22⋊Dic47—22no—
181C4⋊Dic47—22no—
182C4⋊Dic47—22no—
183C4×Dic47—22no—
184C4×Dic47C2×D414yes—
185#10625329—22no—
186#10625329—22no—
187#10625329—22no—
188#10625329—22no—
189#10625329—22no—
190#10625329—22no—
191#10625329—41no—
192#10625329—41no—
193C2×C47⋊C8—22no—
194C2×C47⋊C8—22no—
195C2×C47⋊C8—22no—
196C2×C47⋊C8—41no—
197C2×C47⋊C8—41no—
198C2×C47⋊C8C2×D414yes—
199C23×C47—81no—
200C23×C47C22⋊Q811yescharacteristic
201C23×C47C22⋊Q811yescharacteristic
202C23×C47#12311yescharacteristic
203C23×C47#12311yescharacteristic
204C23×C47#12311yescharacteristic
205C23×C47#12311yescharacteristic
206C23×C47C2×#4111yescharacteristic
207C22×D47—22no—
208C22×D47—41no—
209C22×D47—41no—
210C22×D47—82no—
211C2×C4×C47—24no—
212C2×C4×C47—82no—
213C2×C4×C47C22⋊Q811yescharacteristic
214C2×C4×C47C22⋊Q811yescharacteristic
215C2×C4×C47C22⋊Q811yescharacteristic
216C2×C4×C47C22⋊Q811yescharacteristic
217C2×C4×C47C22⋊Q811yescharacteristic
218C2×C4×C47C22⋊Q811yescharacteristic
219C2×C4×C47C22⋊Q811yescharacteristic
220C2×C4×C47C22⋊Q811yescharacteristic
221C2×C4×C47C22⋊Q811yescharacteristic
222C2×C4×C47C22⋊Q811yescharacteristic
223C2×C4×C47#12311yescharacteristic
224C2×C4×C47#12311yescharacteristic
225C2×C4×C47#12311yescharacteristic
226C2×C4×C47#12311yescharacteristic
227C2×C4×C47#12311yescharacteristic
228C2×C4×C47#12311yescharacteristic
229C2×C4×C47#12311yescharacteristic
230C2×C4×C47#12311yescharacteristic
231C2×C4×C47#12711yescharacteristic
232C2×C4×C47#12711yescharacteristic
233C2×C4×C47#12711yescharacteristic
234C2×C4×C47#12711yescharacteristic
235C2×C4×C47C22×D411yescharacteristic
236C2×C4×C47C22×D411yescharacteristic
237C2×C4×C47C22×D411yescharacteristic
238C2×C4×C47C22×Q811yescharacteristic
239C2×C4×C47C2×#4111yescharacteristic
240C2×C4×C47C2×#4111yescharacteristic
241C22⋊D47—42no—
242C22⋊D47—42no—
243C22⋊D47—42no—
244C22⋊D47—42no—
245C2×Dic47—22no—
246C2×Dic47—24no—
247C2×Dic47—41no—
248C2×Dic47—41no—
249C2×Dic47—82no—
250D188—22no—
251D188—22no—
252D188—41no—
253D188—41no—
254C4×D47—42no—
255C4×D47—42no—
256C4×D47—42no—
257C4×D47—42no—
258Dic94—22no—
259Dic94—22no—
260Dic94—41no—
261Dic94—41no—
262C8×C47—42no—
263C8×C47—42no—
264C47⋊C8—24no—
265C47⋊C8—42no—
266#23740—471nomaximal, Sylow
267C22×C47—41no—
268C22×C47—41no—
269C22×C47—41no—
270C22×C47—41no—
271C22×C47—41no—
272C22×C47—41no—
273C22×C47—41no—
274C22×C47#39911yescharacteristic
275C22×C47#39911yescharacteristic
276C22×C47#39911yescharacteristic
277C22×C47#39911yescharacteristic
278C22×C47C2×C22⋊Q811yescharacteristic
279C22×C47C2×#12311yescharacteristic
280C22×C47C2×#12311yescharacteristic
281C2×D47—44no—
282C2×D47—42no—
283C2×D47—82no—
284C2×D47—82no—
285C4×C47—44no—
286C4×C47#39911yescharacteristic
287C4×C47#39911yescharacteristic
288C4×C47#39911yescharacteristic
289C4×C47#39911yescharacteristic
290C4×C47C2×C22⋊Q811yescharacteristic
291C4×C47C2×C22⋊Q811yescharacteristic
292C4×C47C2×#12311yescharacteristic
293C4×C47C2×#12711yescharacteristic
294Dic47—44no—
295Dic47—42no—
296C2×#430—471no—
297#2905—471no—
298#2905D4711yescharacteristic, complemented
299C2×#421—471no—
300C2×#421—471no—
301#2879—471no—
302#2098—478no—
303C2×H(3,ℤ/4)—471no—
304C2×C47—21no—
305C2×C47—21no—
306C2×C47—21no—
307C2×C47—21no—
308C2×C47—21no—
309C2×C47—21no—
310C2×C47—21no—
311C2×C47C2×#39911yescharacteristic
312D47—82no—
313C22×#41—471no—
314C22×M16—471no—
315C22×M16—471no—
316C22×M16C2×D4711yescharacteristic
317C22×M16C2×D4711yescharacteristic
318C2×#117—471no—
319C2×#117C2×D4711yescharacteristic
320#430—474no—
321#430—942no—
322C2×#108—942no—
323#424—22no—
324#424—474no—
325#424—942no—
326#424C2×D4714yescomplemented
327#423—22no—
328#423—942no—
329#421—474no—
330#421—474no—
331#421—942no—
332#421—942no—
333C2×#104—942no—
334C2×#104—942no—
335#408—474no—
336#408—942no—
337#407—942no—
338H(3,ℤ/4)—474no—
339C47#2374011yescharacteristic, complemented, Sylow
340C2×#41—472no—
341C2×#41—472no—
342C2×#41—941no—
343C2×#41—941no—
344C2×#41—941no—
345C2×#41—941no—
346C2×#41—941no—
347C2×#41—941no—
348C2×#41—941no—
349C2×#41—941no—
350C2×#41—941no—
351C2×#41—941no—
352C22×Q8—471no—
353C22×D4—471no—
354C22×D4—941no—
355C23×C4—941no—
356C23×C4C22×D4711yescharacteristic
357C2×M16—21no—
358C2×M16—21no—
359C2×M16—21no—
360C2×M16—21no—
361C2×M16—21no—
362C2×M16—21no—
363C2×M16—21no—
364C2×M16—21no—
365C2×M16—21no—
366C2×M16—21no—
367C2×M16—21no—
368C2×M16—472no—
369C2×M16—472no—
370C2×M16—472no—
371C2×M16—941no—
372C2×M16—941no—
373C2×M16—941no—
374C2×M16—941no—
375C2×M16—941no—
376C2×M16—941no—
377C2×M16—941no—
378C2×M16—941no—
379C2×M16—941no—
380C2×M16C22×D4712yes—
381C22×C8—21no—
382C22×C8—472no—
383C22×C8—941no—
384C22×C8C22×D4712yes—
385#117—21no—
386#117—21no—
387#117—472no—
388#117—472no—
389#117—941no—
390#117—941no—
391#117C22⋊D4712yes—
392#117C22×D4712yes—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—941no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—941no—
397C2×C42—472no—
398C2×C42C22⋊D4712yes—
399#108—948no—
400C4⋊C8—28no—
401C4⋊C8—948no—
402#104—948no—
403#104—948no—
404C22⋊C8—948no—
405C24—941no—
406C24#1062535011yescharacteristic
407#41—942no—
408#41—942no—
409#41—942no—
410#41—942no—
411#41—942no—
412#41—942no—
413#41—942no—
414#41—942no—
415#41—1881no—
416#41—1881no—
417#41—1881no—
418#41—1881no—
419#41—1881no—
420#41—1881no—
421#41—1881no—
422#41—1881no—
423C2×Q8—472no—
424C2×Q8—941no—
425C2×Q8—941no—
426C2×Q8—941no—
427C2×Q8—941no—
428C2×Q8—941no—
429C2×D4—472no—
430C2×D4—942no—
431C2×D4—942no—
432C2×D4—941no—
433C2×D4—941no—
434C2×D4—941no—
435C2×D4—941no—
436C2×D4—941no—
437C2×D4—1881no—
438C2×D4—1881no—
439C2×D4—1881no—
440C2×D4—1881no—
441C22×C4—22no—
442C22×C4—942no—
443C22×C4—941no—
444C22×C4—942no—
445C22×C4—942no—
446C22×C4—1881no—
447C22×C4—1881no—
448C22×C4—1881no—
449C22×C4—1881no—
450C22×C4#1062535011yescharacteristic
451C22×C4#1062535011yescharacteristic
452C22×C4#1062535011yescharacteristic
453C22×C4D4×D4711yescharacteristic
454C22×C4D4×D4711yescharacteristic
455C22×C4D4×D4711yescharacteristic
456C22×C4D4×D4711yescharacteristic
457C22×C4Q8×D4711yescharacteristic
458C22×C4Q8×D4711yescharacteristic
459C22×C4#1062535411yescharacteristic
460C22×C4#1062535411yescharacteristic
461C22×C4C2×C22⋊D4711yescharacteristic
462C22×C4C2×C22⋊D4711yescharacteristic
463C22×C4C23×D4711yescharacteristic
464M16—22no—
465M16—22no—
466M16—41no—
467M16—41no—
468M16—41no—
469M16—41no—
470M16—41no—
471M16—41no—
472M16—942no—
473M16—942no—
474M16—942no—
475M16—942no—
476M16—942no—
477M16—942no—
478M16—1881no—
479M16—1881no—
480C2×C8—22no—
481C2×C8—22no—
482C2×C8—22no—
483C2×C8—22no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—474no—
487C2×C8—942no—
488C2×C8—942no—
489C2×C8—942no—
490C2×C8—1881no—
491C2×C8—1881no—
492C4⋊C4—22no—
493C4⋊C4—22no—
494C4⋊C4—942no—
495C4⋊C4—942no—
496C22⋊C4—22no—
497C22⋊C4—22no—
498C22⋊C4—942no—
499C22⋊C4—942no—
500C42—22no—
501C42—474no—
502C42—942no—
503C42C2×C22⋊D4714yes—
504C23—81no—
505C23—942no—
506C23—3762no—
507C23—1881no—
508C23—1881no—
509C23#1309400211yescharacteristic
510C23#1309400211yescharacteristic
511C23#1309400711yescharacteristic
512C23#1309400711yescharacteristic
513C23#1309400911yescharacteristic
514C23#1309400911yescharacteristic
515C23C2×#1062535011yescharacteristic
516Q8—942no—
517Q8—942no—
518Q8—1881no—
519Q8—1881no—
520D4—942no—
521D4—942no—
522D4—1882no—
523D4—1882no—
524D4—1882no—
525D4—1882no—
526D4—1881no—
527D4—1881no—
528C2×C4—24no—
529C2×C4—82no—
530C2×C4—942no—
531C2×C4—944no—
532C2×C4—3762no—
533C2×C4—1881no—
534C2×C4—1881no—
535C2×C4—1882no—
536C2×C4—1882no—
537C2×C4—1882no—
538C2×C4—1882no—
539C2×C4#1309400011yescharacteristic
540C2×C4#1309400011yescharacteristic
541C2×C4#1309400011yescharacteristic
542C2×C4#1309400011yescharacteristic
543C2×C4#1309400211yescharacteristic
544C2×C4#1309400211yescharacteristic
545C2×C4#1309400211yescharacteristic
546C2×C4#1309400211yescharacteristic
547C2×C4#1309400211yescharacteristic
548C2×C4#1309400211yescharacteristic
549C2×C4#1309400711yescharacteristic
550C2×C4#1309400711yescharacteristic
551C2×C4#1309400911yescharacteristic
552C2×C4#1309400911yescharacteristic
553C2×C4#1309400911yescharacteristic
554C2×C4#1309400911yescharacteristic
555C2×C4#1309400911yescharacteristic
556C2×C4#1309400911yescharacteristic
557C2×C4#1309402011yescharacteristic
558C2×C4#1309402011yescharacteristic
559C2×C4#1309402011yescharacteristic
560C2×C4#1309402011yescharacteristic
561C2×C4C2×D4×D4711yescharacteristic
562C2×C4C2×D4×D4711yescharacteristic
563C2×C4C22×C22⋊D4711yescharacteristic
564C2×C4C2×Q8×D4711yescharacteristic
565C2×C4C2×#1062535011yescharacteristic
566C2×C4C2×#1062535411yescharacteristic
567C8—42no—
568C8—42no—
569C8—944no—
570C8—1882no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—3762no—
579C22—3762no—
580C22—1884no—
581C22—1882no—
582C22#1568905811yescharacteristic
583C22#1568905811yescharacteristic
584C22#1568905811yescharacteristic
585C22#1568905811yescharacteristic
586C22C2×#1309400211yescharacteristic
587C22C2×#1309400711yescharacteristic
588C22C2×#1309400911yescharacteristic
589C4—44no—
590C4—1884no—
591C4—1882no—
592C4#1568905811yescharacteristic
593C4#1568905811yescharacteristic
594C4#1568905811yescharacteristic
595C4#1568905811yescharacteristic
596C4C2×#1309400011yescharacteristic
597C4C2×#1309400211yescharacteristic
598C4C2×#1309400911yescharacteristic
599C4C2×#1309402011yescharacteristic
600C2—21no—
601C2—21no—
602C2—21no—
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—3762no—
608C2C2×#1568905811yescharacteristic
subgroupG/H
centerC4C2×#13094020
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2905C2
socleC2×C47C2×#399
prime radicalC22×#10625350C22
hypercenter#2905D47
hypocenterC47#23740
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C4 ◁ C23×C4 ◁ #2905

Complements

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

NH ≅ G/Ntype
1. C2×#15689171C2semidirect
2. #20251951C2semidirect
6. #20244944C2semidirect
7. C2×#15688879C2semidirect
8. C47×#2905C2semidirect
14. #15689171C22semidirect
17. #15689149C22semidirect
28. #15688879C22semidirect
33. C47×#424C22semidirect
298. #2905D47semidirect
326. #424C2×D47semidirect
339. C47#23740semidirect

Maximal subgroups

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

subgroupindex
C2×#156891712
#202519512
C2×#156891412
C2×#156891392
#202518322
#202449442
C2×#156888792
C47×#29052
#2374047