Finite Group Databasebeta

← #26250220

Subgroups — #26250220

Order 12032.

1154 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689415C211yesmaximal, characteristic, complemented
2C2×C4×#10625335C211yesmaximal, characteristic, complemented
3#20248310C211yesmaximal, characteristic, complemented
4#20244567C212yesmaximal, complemented
5#20244391C212yesmaximal, complemented
6#20244350C211yesmaximal, characteristic, complemented
7#20244183C211yesmaximal, characteristic, complemented
8#20243700C212yesmaximal, complemented
9#20243587C212yesmaximal, complemented
10#20243388C211yesmaximal, characteristic, complemented
11C2×C47×#555C211yesmaximal, characteristic
12C22×#10625335C2211yescharacteristic
13C2×#13094003C2212yes—
14C2×C4×Dic94C2211yescharacteristic
15#15689415—24no—
16C4×#10625335—24no—
17C2×#13093957C2212yes—
18C2×#13093955C2211yescharacteristic
19C2×C4⋊(C47⋊C8)C2211yescharacteristic
20C2×#13093947C2212yes—
21C2×#13093947C2211yescharacteristic
22C2×#13093945C2211yescharacteristic
23C2×#13093945C2212yes—
24C2×C22⋊(C47⋊C8)C2212yes—
25C2×C22⋊(C47⋊C8)C2212yes—
26C2×C4×C47⋊C8C2211yescharacteristic
27#15689046—22no—
28#15689026—21no—
29#15689019—21no—
30#15688993C2211yescharacteristic
31#15688983C2212yes—
32#15688885C2212yes—
33#15688885C2212yes—
34#15688885C2212yes—
35C47×#555—24no—
36C47×#555—24no—
37C2×C4⋊Q8×C47C2211yescharacteristic
38C2×#124×C47—22no—
39C2×#121×C47—21no—
40C2×#121×C47C2212yes—
41C2×C4×D4×C47C2211yescharacteristic
42C2×C4×D4×C47C2211yescharacteristic
43C2×#117×C47—21no—
44C2×#117×C47C2212yes—
45C22×Dic94C2311yescharacteristic
46C2×#10625335—22no—
47C2×#10625335—22no—
48C2×#10625335—22no—
49C22×C47⋊C8C2311yescharacteristic
50C22×C47⋊C8C2312yes—
51C22×C47⋊C8C2311yescharacteristic
52#13094003—24no—
53#13094003—42no—
54C4×Dic94—22no—
55C4×Dic94—22no—
56C2×C4⋊Dic47—21no—
57C2×C4⋊Dic47—21no—
58C2×C4⋊Dic47C2311yescharacteristic
59C2×C4⋊Dic47C2312yes—
60C2×Dic47⋊C4—22no—
61C2×Dic47⋊C4—21no—
62C2×C22⋊Dic47—22no—
63C2×C22⋊Dic47—22no—
64C2×C4×Dic47—21no—
65#13093957—24no—
66#13093955—22no—
67C4⋊(C47⋊C8)—22no—
68#13093947—24no—
69#13093947—22no—
70#13093945—22no—
71#13093945—42no—
72C22⋊(C47⋊C8)—24no—
73C22⋊(C47⋊C8)—42no—
74C4×C47⋊C8—22no—
75C47⋊#95—22no—
76C22×Q8×C47—21no—
77C22×D4×C47—22no—
78C22×D4×C47D411yescharacteristic
79C22×D4×C47C2311yescharacteristic
80C23×C4×C47—21no—
81C23×C4×C47C2312yes—
82C4⋊Q8×C47—22no—
83C4⋊Q8×C47D414yes—
84#124×C47—216no—
85#121×C47—28no—
86#121×C47—24no—
87#121×C47—24no—
88C4×D4×C47—22no—
89C4×D4×C47—22no—
90C4×D4×C47—22no—
91C4×D4×C47D414yes—
92#117×C47—28no—
93#117×C47—24no—
94#117×C47—24no—
95C2×C4⋊C4×C47—21no—
96C2×C4⋊C4×C47D411yescharacteristic
97C2×C4⋊C4×C47C2311yescharacteristic
98C2×C4⋊C4×C47C2312yes—
99C2×C22⋊C4×C47—22no—
100C2×C22⋊C4×C47—22no—
101C2×C22⋊C4×C47—21no—
102C2×C22⋊C4×C47D412yes—
103C2×C42×C47—22no—
104C2×C42×C47C2311yescharacteristic
105C24×C47—21no—
106C24×C47C2×D412yes—
107C2×Q8×C47—24no—
108C2×Q8×C47—81no—
109C2×Q8×C47—81no—
110C2×D4×C47—28no—
111C2×D4×C47—22no—
112C2×D4×C47—22no—
113C2×D4×C47—22no—
114C2×D4×C47—21no—
115C2×D4×C47—21no—
116C2×D4×C47—21no—
117C2×D4×C47—21no—
118C2×D4×C47—21no—
119C2×D4×C47—82no—
120C2×D4×C47—82no—
121C2×D4×C47C2×D412yes—
122C22×C4×C47—21no—
123C22×C4×C47—22no—
124C22×C4×C47—24no—
125C22×C4×C47—22no—
126C22×C4×C47—22no—
127C22×C4×C47—22no—
128C22×C4×C47—22no—
129C22×C4×C47—22no—
130C22×C4×C47—22no—
131C22×C4×C47—42no—
132C22×C4×C47—42no—
133C22×C4×C47C2×D412yes—
134C22×C4×C47C2×D411yescharacteristic
135C22×C4×C47C2×D411yescharacteristic
136C22×C4×C47C2411yescharacteristic
137C22×Dic47—21no—
138C22×Dic47—21no—
139C22×Dic47—22no—
140C2×Dic94—21no—
141C2×Dic94—21no—
142C2×Dic94—21no—
143C2×Dic94—21no—
144C2×Dic94—21no—
145C2×Dic94#4112yes—
146C4⋊C4×C47—24no—
147C4⋊C4×C47—22no—
148C4⋊C4×C47—21no—
149C4⋊C4×C47—22no—
150C4⋊C4×C47C2×D412yes—
151C4⋊C4×C47C2×D414yes—
152C22⋊C4×C47—28no—
153C22⋊C4×C47—28no—
154C22⋊C4×C47—24no—
155C22⋊C4×C47—24no—
156C42×C47—28no—
157C42×C47C2×D412yes—
158C42×C47C2×D412yes—
159C22⋊Dic47—44no—
160C22⋊Dic47—82no—
161#10625335—28no—
162C4⋊Dic47—21no—
163C4⋊Dic47—21no—
164C4⋊Dic47—22no—
165C4⋊Dic47—22no—
166C4⋊Dic47—42no—
167C4⋊Dic47—81no—
168Dic47⋊C4—44no—
169Dic47⋊C4—42no—
170C4×Dic47—42no—
171C2×C47⋊C8—21no—
172C2×C47⋊C8—21no—
173C2×C47⋊C8—21no—
174C2×C47⋊C8—22no—
175C2×C47⋊C8—22no—
176C2×C47⋊C8—21no—
177C2×C47⋊C8—22no—
178C2×C47⋊C8—21no—
179C2×C47⋊C8#4112yes—
180C23×C47—22no—
181C23×C47—22no—
182C23×C47—22no—
183C23×C47—22no—
184C23×C47—22no—
185C23×C47—22no—
186C23×C47—22no—
187C23×C47—22no—
188C23×C47—41no—
189C23×C47—41no—
190C23×C47—41no—
191C23×C47—41no—
192C23×C47—41no—
193C23×C47—41no—
194C23×C47C22×D411yescharacteristic
195Q8×C47—84no—
196D4×C47—22no—
197D4×C47—22no—
198D4×C47—41no—
199D4×C47—41no—
200D4×C47—44no—
201D4×C47—88no—
202C2×C4×C47—21no—
203C2×C4×C47—21no—
204C2×C4×C47—22no—
205C2×C4×C47—22no—
206C2×C4×C47—24no—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
209C2×C4×C47—41no—
210C2×C4×C47—41no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47—42no—
216C2×C4×C47—42no—
217C2×C4×C47—48no—
218C2×C4×C47#12012yes—
219C2×C4×C47#12012yes—
220C2×C4×C47#13512yes—
221C2×C4×C47#13712yes—
222C2×C4×C47C22×D411yescharacteristic
223C2×C4×C47C22×D411yescharacteristic
224C2×C4×C47C2×#4111yescharacteristic
225C2×C4×C47C2×#4111yescharacteristic
226C2×C4×C47H(5,𝔽2)11yescharacteristic
227C2×C4×C472−1+411yescharacteristic
228C2×Dic47—22no—
229C2×Dic47—41no—
230C2×Dic47—41no—
231C2×Dic47—41no—
232C2×Dic47—42no—
233C2×Dic47—42no—
234C2×Dic47—42no—
235C2×Dic47—41no—
236C2×Dic47—41no—
237Dic94—22no—
238Dic94—22no—
239Dic94—41no—
240Dic94—41no—
241C47⋊C8—22no—
242C47⋊C8—41no—
243C47⋊C8—42no—
244#24121—471nomaximal, Sylow
245C22×C47—42no—
246C22×C47—42no—
247C22×C47—42no—
248C22×C47—42no—
249C22×C47—42no—
250C22×C47—42no—
251C22×C47—42no—
252C22×C47—42no—
253C22×C47—42no—
254C22×C47—42no—
255C22×C47—81no—
256C22×C47—81no—
257C22×C47—81no—
258C22×C47—81no—
259C22×C47C2×#12011yescharacteristic
260C22×C47#54611yescharacteristic
261C22×C47#54711yescharacteristic
262C22×C47C2×#13511yescharacteristic
263C22×C47C2×#13711yescharacteristic
264C22×C47#57611yescharacteristic
265C22×C47#57711yescharacteristic
266C4×C47—22no—
267C4×C47—42no—
268C4×C47—42no—
269C4×C47—44no—
270C4×C47#45214yes—
271C4×C47#52511yescharacteristic
272C4×C47#53711yescharacteristic
273C4×C47#54811yescharacteristic
274C4×C47#55011yescharacteristic
275Dic47—42no—
276Dic47—81no—
277Dic47—82no—
278C2×#555D4711yescharacteristic
279C2×#477—471no—
280C2×C4×SD16—471no—
281#2538—471no—
282#2033—472no—
283#2000—472no—
284#1990—471no—
285#1954—471no—
286#1873—472no—
287#1854—472no—
288#1802—471no—
289C2×C47—82no—
290C2×C47—82no—
291C2×C47#240711yescharacteristic
292C2×C47C2×#45211yescharacteristic
293C2×C47#301211yescharacteristic
294C2×C47#330511yescharacteristic
295C2×C47#332911yescharacteristic
296C2×C47#334011yescharacteristic
297C2×C47#335711yescharacteristic
298C22×SD16—471no—
299#555—24no—
300#555—24no—
301C2×C4⋊Q8C2×D4711yescharacteristic
302C2×#124—22no—
303C2×C22⋊Q8—472no—
304C2×#121—21no—
305C2×#121C2×D4712yes—
306C2×C4×Q8—471no—
307C2×C4×D4C2×D4711yescharacteristic
308C2×C4×D4C2×D4711yescharacteristic
309C2×#117—21no—
310C2×#117C2×D4712yes—
311#477—944no—
312C4×SD16—944no—
313C2×#107—472no—
314C2×#106—471no—
315C2×C4⋊C8—471no—
316C2×#103—472no—
317C2×#103—471no—
318C2×#102—471no—
319C2×#102—472no—
320C2×C22⋊C8—472no—
321C2×C22⋊C8—472no—
322C2×C4×C8—471no—
323#396—942no—
324#390—941no—
325#389—941no—
326#384—471no—
327#380—472no—
328#340—472no—
329#340—472no—
330#340—472no—
331C47#2412111yescharacteristic, complemented, Sylow
332C22×Q8—21no—
333C22×Q8—471no—
334C22×D4—22no—
335C22×D4C22⋊D4711yescharacteristic
336C22×D4C22×D4711yescharacteristic
337C23×C4—21no—
338C23×C4C22×D4712yes—
339C2×SD16—942no—
340C2×SD16—942no—
341C2×SD16—942no—
342C22×C8—471no—
343C22×C8—471no—
344C22×C8—472no—
345C4⋊Q8—22no—
346C4⋊Q8C22⋊D4714yes—
347#124—216no—
348C22⋊Q8—944no—
349C22⋊Q8—1882no—
350#121—28no—
351#121—24no—
352#121—24no—
353C4×Q8—942no—
354C4×Q8—942no—
355C4×D4—22no—
356C4×D4—22no—
357C4×D4—22no—
358C4×D4C22⋊D4714yes—
359#117—28no—
360#117—24no—
361#117—24no—
362C2×C4⋊C4—21no—
363C2×C4⋊C4—471no—
364C2×C4⋊C4—472no—
365C2×C4⋊C4—941no—
366C2×C4⋊C4—941no—
367C2×C4⋊C4—942no—
368C2×C4⋊C4—941no—
369C2×C4⋊C4C22⋊D4711yescharacteristic
370C2×C4⋊C4C22×D4711yescharacteristic
371C2×C4⋊C4C22×D4712yes—
372C2×C22⋊C4—22no—
373C2×C22⋊C4—22no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—942no—
376C2×C22⋊C4—942no—
377C2×C22⋊C4C22⋊D4712yes—
378C2×C42—22no—
379C2×C42—941no—
380C2×C42C22×D4711yescharacteristic
381#107—944no—
382#106—942no—
383C4⋊C8—942no—
384#103—942no—
385#103—944no—
386#102—942no—
387#102—1882no—
388C22⋊C8—944no—
389C22⋊C8—1882no—
390C4×C8—942no—
391#95—942no—
392C24—21no—
393C24C2×C22⋊D4712yes—
394C2×Q8—24no—
395C2×Q8—81no—
396C2×Q8—81no—
397C2×Q8—472no—
398C2×Q8—941no—
399C2×Q8—941no—
400C2×Q8—941no—
401C2×Q8—941no—
402C2×Q8—941no—
403C2×D4—28no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—22no—
407C2×D4—21no—
408C2×D4—21no—
409C2×D4—21no—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—82no—
413C2×D4—82no—
414C2×D4C2×C22⋊D4712yes—
415C22×C4—21no—
416C22×C4—22no—
417C22×C4—24no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—42no—
425C22×C4—42no—
426C22×C4—941no—
427C22×C4—941no—
428C22×C4—942no—
429C22×C4C2×C22⋊D4712yes—
430C22×C4C2×C22⋊D4711yescharacteristic
431C22×C4C2×C22⋊D4711yescharacteristic
432C22×C4C23×D4711yescharacteristic
433SD16—948no—
434C2×C8—472no—
435C2×C8—941no—
436C2×C8—941no—
437C2×C8—941no—
438C2×C8—942no—
439C2×C8—942no—
440C2×C8—941no—
441C2×C8—942no—
442C2×C8—941no—
443C4⋊C4—24no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—22no—
447C4⋊C4—941no—
448C4⋊C4—941no—
449C4⋊C4—942no—
450C4⋊C4—942no—
451C4⋊C4—3761no—
452C4⋊C4—1884no—
453C4⋊C4—1882no—
454C4⋊C4—1882no—
455C4⋊C4C2×C22⋊D4712yes—
456C4⋊C4C2×C22⋊D4714yes—
457C22⋊C4—28no—
458C22⋊C4—28no—
459C22⋊C4—24no—
460C22⋊C4—24no—
461C22⋊C4—3762no—
462C22⋊C4—1884no—
463C42—28no—
464C42—1882no—
465C42C2×C22⋊D4712yes—
466C42C2×C22⋊D4712yes—
467C23—22no—
468C23—22no—
469C23—22no—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—22no—
474C23—22no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23C22×C22⋊D4711yescharacteristic
482Q8—84no—
483Q8—942no—
484Q8—942no—
485Q8—1881no—
486Q8—1881no—
487D4—22no—
488D4—22no—
489D4—41no—
490D4—41no—
491D4—44no—
492D4—88no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—24no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—48no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—942no—
510C2×C4—1882no—
511C2×C4—1881no—
512C2×C4—1882no—
513C2×C4—1881no—
514C2×C4—1882no—
515C2×C4—1881no—
516C2×C4—1881no—
517C2×C4—1881no—
518C2×C4#1309399312yes—
519C2×C4#1309399312yes—
520C2×C4#1309404412yes—
521C2×C4#1309405412yes—
522C2×C4C22×C22⋊D4711yescharacteristic
523C2×C4C22×C22⋊D4711yescharacteristic
524C2×C4#41×D4711yescharacteristic
525C2×C4#41×D4711yescharacteristic
526C2×C4#1309407011yescharacteristic
527C2×C4#1309407111yescharacteristic
528C8—942no—
529C8—1882no—
530C8—1881no—
531C22—42no—
532C22—42no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—42no—
539C22—42no—
540C22—42no—
541C22—81no—
542C22—81no—
543C22—81no—
544C22—81no—
545C22C2×#1309399311yescharacteristic
546C22#1568976111yescharacteristic
547C22#1568977211yescharacteristic
548C22C2×#1309404411yescharacteristic
549C22C2×#1309405411yescharacteristic
550C22#1568992611yescharacteristic
551C22#1568993411yescharacteristic
552C4—22no—
553C4—44no—
554C4—42no—
555C4—42no—
556C4—3761no—
557C4—3762no—
558C4—1882no—
559C4#1568928514yes—
560C4#1568963011yescharacteristic
561C4#1568970111yescharacteristic
562C4#1568978011yescharacteristic
563C4#1568979211yescharacteristic
564C2—82no—
565C2—82no—
566C2#2024672711yescharacteristic
567C2C2×#1568928511yescharacteristic
568C2#2025269211yescharacteristic
569C2#2025533211yescharacteristic
570C2#2025567611yescharacteristic
571C2#2025580111yescharacteristic
572C2#2025600811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
Fitting, prime radicalC2×C47×#555C2
socleC23×C47C22×D4
hypercenterC2×#555D47
hypocenterC47#24121
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#555

Complements

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

NH ≅ G/Ntype
1. C2×#15689415C2semidirect
2. C2×C4×#10625335C2semidirect
3. #20248310C2semidirect
4. #20244567C2semidirect
5. #20244391C2semidirect
6. #20244350C2semidirect
7. #20244183C2semidirect
8. #20243700C2semidirect
9. #20243587C2semidirect
10. #20243388C2semidirect
331. C47#24121semidirect

Maximal subgroups

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

subgroupindex
C2×#156894152
C2×C4×#106253352
#202483102
#202445672
#202443912
#202443502
#202441832
#202437002
#202435872
#202433882
C2×C47×#5552
#2412147