Finite Group Databasebeta

← #26216351

Subgroups — #26216351

Order 12032.

1338 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#10625354C211yesmaximal, characteristic
2C2×#15689179C211yesmaximal, characteristic, complemented
3#20245781C211yesmaximal, characteristic, complemented
4#20243482C212yesmaximal, complemented
5#20243403C211yesmaximal, characteristic, complemented
6#20243399C211yesmaximal, characteristic, complemented
7#20243395C211yesmaximal, characteristic, complemented
8#20243327C212yesmaximal
9#20243220C212yesmaximal, complemented
10#20243212C212yesmaximal, complemented
11C47×#1805C211yesmaximal, characteristic, complemented
12C22×#10625354C2211yescharacteristic
13C4×#10625354—24no—
14C4×#10625354C418yescomplemented
15C2×Q8×Dic47C2211yescharacteristic
16C2×C4×D188—21no—
17C2×C4×D188C2211yescharacteristic
18C2×#13093986—21no—
19C2×#13093986C2211yescharacteristic
20C2×#13093982—21no—
21C2×#13093982C2211yescharacteristic
22C2×C42×D47—21no—
23C2×C42×D47C2211yescharacteristic
24#15689179—24no—
25C2×C4⋊(C47⋊C8)C2211yescharacteristic
26C2×#13093948C2212yes—
27C2×#13093946C2212yes—
28C2×#13093944C2212yes—
29C2×(C2×D47)⋊C8C2212yes—
30C2×#13093934C2212yes—
31#15688990—21no—
32#15688989C2211yescharacteristic
33#15688985—21no—
34#15688980C2212yes—
35C4⋊C4×Dic47C2212yes—
36#15688970—21no—
37#15688968—21no—
38#15688886C2212yes—
39#15688885C2212yes—
40#15688885C2212yes—
41C2×C4×Q8×C47C2211yescharacteristic
42C2×C4⋊C8×C47C2211yescharacteristic
43C2×#103×C47C2212yes—
44C47×#382C2211yescharacteristic
45C47×#340C2212yes—
46C2×#10625354—22no—
47C2×#10625354—22no—
48C2×#10625354—22no—
49C2×#10625354—22no—
50C2×#10625354—22no—
51C2×#10625354C2×C414yes—
52C22×D188—21no—
53C22×D188C2311yescharacteristic
54C22×C4×D47—21no—
55C22×C4×D47—21no—
56C22×C4×D47—21no—
57C22×C4×D47D411yescharacteristic
58C22×C4×D47D411yescharacteristic
59C22×C4×D47C2311yescharacteristic
60C22×C47⋊C8C2312yes—
61Q8×Dic47—22no—
62Q8×Dic47C2×C414yes—
63C4×D188—28no—
64C4×D188—22no—
65C4×D188C2×C414yes—
66#13093986—28no—
67#13093986—22no—
68#13093986C2×C414yes—
69#13093982—22no—
70#13093982—28no—
71#13093982C2×C414yes—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47C2312yes—
74C2×C4⋊Dic47C2311yescharacteristic
75C2×(C2×D47)⋊C4—21no—
76C2×(C2×D47)⋊C4—21no—
77C2×(C2×D47)⋊C4—21no—
78C2×(C2×D47)⋊C4—22no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47—21no—
81C2×C4×Dic47—22no—
82C2×C4×Dic47D411yescharacteristic
83C2×C4×Dic47D411yescharacteristic
84C2×C4×Dic47C2311yescharacteristic
85C42×D47—28no—
86C42×D47C2×C414yes—
87C42×D47D414yes—
88C4⋊(C47⋊C8)—22no—
89#13093948—24no—
90#13093946—24no—
91#13093944—24no—
92(C2×D47)⋊C8—24no—
93#13093934—24no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—22no—
97C47⋊#95—22no—
98C22×Q8×C47C2311yescharacteristic
99C22×C8×C47C2312yes—
100C4×Q8×C47—22no—
101C4×Q8×C47C2×C414yes—
102C2×C4⋊C4×C47—21no—
103C2×C4⋊C4×C47C2312yes—
104C2×C4⋊C4×C47C2311yescharacteristic
105C2×C42×C47—21no—
106C2×C42×C47C2311yescharacteristic
107C4⋊C8×C47—22no—
108#103×C47—24no—
109#95×C47—21no—
110#95×C47—21no—
111C23×D47—21no—
112C23×D47C2×D411yescharacteristic
113C2×Q8×C47—21no—
114C2×Q8×C47—21no—
115C2×Q8×C47—21no—
116C2×Q8×C47C22×C412yes—
117C2×Q8×C47#4112yes—
118C2×Q8×C47#4112yes—
119C22×C4×C47—21no—
120C22×C4×C47—21no—
121C22×C4×C47—22no—
122C22×C4×C47C2×D411yescharacteristic
123C22×C4×C47C2×D411yescharacteristic
124C22×C4×C47C2411yescharacteristic
125C22×Dic47—21no—
126C22×Dic47—22no—
127C22×Dic47C2×D411yescharacteristic
128C22×Dic47C2×D411yescharacteristic
129C22×Dic47C2×D411yescharacteristic
130#10625354—28no—
131#10625354—28no—
132#10625354—28no—
133#10625354—28no—
134C2×D188—24no—
135C2×D188—24no—
136C2×D188—24no—
137C2×D188—21no—
138C2×D188—21no—
139C2×D188—21no—
140C2×D188C22×C412yes—
141C2×D188#4112yes—
142C2×D188#4112yes—
143C2×C4×D47—24no—
144C2×C4×D47—24no—
145C2×C4×D47—21no—
146C2×C4×D47—21no—
147C2×C4×D47—21no—
148C2×C4×D47—21no—
149C2×C4×D47—24no—
150C2×C4×D47—24no—
151C2×C4×D47—24no—
152C2×C4×D47—42no—
153C2×C4×D47—42no—
154C2×C4×D47—44no—
155C2×C4×D47—44no—
156C2×C4×D47C22⋊C414yes—
157C2×C4×D47C22⋊C414yes—
158C2×C4×D47C22×C412yes—
159C2×C4×D47C2×D412yes—
160C2×C8×C47—22no—
161C2×C8×C47—22no—
162C2×C8×C47—22no—
163C4⋊C4×C47—24no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—22no—
166C4⋊C4×C47—22no—
167C4⋊C4×C47C22×C412yes—
168C42×C47—24no—
169C42×C47C22×C412yes—
170C42×C47C2×D412yes—
171(C2×D47)⋊C4—24no—
172(C2×D47)⋊C4—24no—
173(C2×D47)⋊C4—24no—
174(C2×D47)⋊C4—44no—
175C4⋊Dic47—21no—
176C4⋊Dic47—24no—
177C4⋊Dic47—22no—
178C4⋊Dic47—22no—
179C4⋊Dic47C22×C412yes—
180C4×Dic47—24no—
181C4×Dic47—24no—
182C4×Dic47—44no—
183C4×Dic47C22⋊C414yes—
184C4×Dic47C22⋊C414yes—
185C4×Dic47C22×C412yes—
186C4×Dic47C2×D412yes—
187C2×C47⋊C8—22no—
188C2×C47⋊C8—22no—
189C2×C47⋊C8—22no—
190C23×C47C22×D411yescharacteristic
191C22×D47—22no—
192C22×D47—21no—
193C22×D47—21no—
194C22×D47—22no—
195C22×D47—22no—
196C22×D47—22no—
197C22×D47—22no—
198C22×D47—42no—
199C22×D47—42no—
200C22×D47C2×C22⋊C412yes—
201Q8×C47—22no—
202Q8×C47—22no—
203Q8×C47—22no—
204Q8×C47#11714yes—
205C2×C4×C47—22no—
206C2×C4×C47—22no—
207C2×C4×C47—22no—
208C2×C4×C47—22no—
209C2×C4×C47—22no—
210C2×C4×C47—22no—
211C2×C4×C47—42no—
212C2×C4×C47—42no—
213C2×C4×C47—42no—
214C2×C4×C47—42no—
215C2×C4×C47C2×C22⋊C412yes—
216C2×C4×C47C2×C22⋊C412yes—
217C2×C4×C47C23×C411yescharacteristic
218C2×C4×C47C22×D411yescharacteristic
219C2×C4×C47C2×#4111yescharacteristic
220C2×C4×C47C2×#4111yescharacteristic
221C2×C4×C47C2×#4111yescharacteristic
222C2×C4×C47C2×#4111yescharacteristic
223C2×Dic47—21no—
224C2×Dic47—21no—
225C2×Dic47—22no—
226C2×Dic47—22no—
227C2×Dic47—22no—
228C2×Dic47—42no—
229C2×Dic47—42no—
230C2×Dic47—42no—
231C2×Dic47—42no—
232C2×Dic47C2×C22⋊C412yes—
233C2×Dic47C2×C22⋊C412yes—
234C2×Dic47C2×C22⋊C412yes—
235D188—216no—
236D188—22no—
237D188—22no—
238D188—22no—
239D188#11714yes—
240C4×D47—216no—
241C4×D47—22no—
242C4×D47—22no—
243C4×D47—22no—
244C4×D47—22no—
245C4×D47—44no—
246C4×D47—44no—
247C4×D47—48no—
248C4×D47—48no—
249C8×C47—42no—
250C47⋊C8—42no—
251#20742—471nomaximal, Sylow
252C22×C47C22×C22⋊C411yescharacteristic
253C22×C47#52511yescharacteristic
254C22×C47#52511yescharacteristic
255C22×C47#57512yes—
256C22×C47#57512yes—
257C2×D47—22no—
258C2×D47—22no—
259C2×D47—22no—
260C2×D47—24no—
261C2×D47—42no—
262C2×D47—42no—
263C2×D47—44no—
264C2×D47—44no—
265C2×D47—42no—
266C2×D47—42no—
267C4×C47—22no—
268C4×C47—22no—
269C4×C47—24no—
270C4×C47—42no—
271C4×C47—82no—
272C4×C47C2×#11711yescharacteristic
273C4×C47C2×#11711yescharacteristic
274C4×C47#52511yescharacteristic
275C4×C47#52511yescharacteristic
276Dic47—22no—
277Dic47—22no—
278Dic47—22no—
279Dic47—42no—
280Dic47—82no—
281C2×C4×#41—471no—
282C2×#432—471no—
283#2283—471no—
284#1825—472no—
285#1805—471no—
286#1805D4711yescharacteristic, complemented
287#1804—471no—
288#1804—471no—
289#1788—472no—
290#1763—472no—
291#1762—472no—
292C2×C47#228011yescharacteristic
293C2×C47#289611yescharacteristic
294C2×C47#289611yescharacteristic
295C2×C47#311112yes—
296C2×C47#311112yes—
297D47—42no—
298D47—44no—
299C22×#41—471no—
300C4×#41—478no—
301C4×#41—944no—
302C2×C4×Q8—471no—
303C2×C4×Q8C2×D4711yescharacteristic
304C2×C4×D4—471no—
305C2×C4×D4—471no—
306C2×C4×D4—941no—
307C2×C4×D4—941no—
308C2×#117—471no—
309C2×#117—941no—
310C22×C42—471no—
311C22×C42—941no—
312#432—944no—
313C2×C4⋊C8—471no—
314C2×C4⋊C8C2×D4711yescharacteristic
315C2×#103—472no—
316C2×#103C2×D4712yes—
317C2×#102—472no—
318C2×#102—472no—
319C2×C22⋊C8—472no—
320C2×#97—472no—
321#383—941no—
322#382—471no—
323#382C2×D4711yescharacteristic
324#381—941no—
325#380—472no—
326C4×C4⋊C4—472no—
327C4×C22⋊C4—941no—
328#376—941no—
329#340—472no—
330#340—472no—
331#340—472no—
332#340C2×D4712yes—
333C47#2074211yescharacteristic, complemented, Sylow
334C2×#41—474no—
335C2×#41—942no—
336C2×#41—942no—
337C2×#41—942no—
338C2×#41—942no—
339C2×#41—942no—
340C22×Q8C22×D4711yescharacteristic
341C22×D4—471no—
342C22×D4—941no—
343C23×C4—471no—
344C23×C4—471no—
345C23×C4—471no—
346C23×C4—941no—
347C23×C4—941no—
348C23×C4—941no—
349C22×C8—472no—
350C22×C8C22×D4712yes—
351C4×Q8—22no—
352C4×Q8—474no—
353C4×Q8—942no—
354C4×Q8C4×D4714yes—
355C4×D4—474no—
356C4×D4—474no—
357C4×D4—948no—
358C4×D4—942no—
359C4×D4—948no—
360C4×D4—942no—
361#117—474no—
362#117—948no—
363#117—942no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—471no—
366C2×C4⋊C4—472no—
367C2×C4⋊C4—941no—
368C2×C4⋊C4C22×D4712yes—
369C2×C4⋊C4C22×D4711yescharacteristic
370C2×C22⋊C4—941no—
371C2×C22⋊C4—941no—
372C2×C22⋊C4—941no—
373C2×C22⋊C4—942no—
374C2×C42—21no—
375C2×C42—471no—
376C2×C42—471no—
377C2×C42—471no—
378C2×C42—474no—
379C2×C42—474no—
380C2×C42—948no—
381C2×C42—941no—
382C2×C42—941no—
383C2×C42—942no—
384C2×C42C22×D4711yescharacteristic
385C4⋊C8—22no—
386C4⋊C8—942no—
387#103—24no—
388#103—944no—
389#102—944no—
390#102—944no—
391C22⋊C8—944no—
392#97—944no—
393#95—21no—
394#95—21no—
395#95—942no—
396#95—942no—
397#95—941no—
398#95—941no—
399C24—471no—
400C24—941no—
401#41—948no—
402#41—948no—
403#41—948no—
404#41—948no—
405C2×Q8—21no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8C2×C4×D4712yes—
409C2×Q8#1062535212yes—
410C2×Q8#1062535412yes—
411C2×D4—472no—
412C2×D4—472no—
413C2×D4—472no—
414C2×D4—944no—
415C2×D4—944no—
416C2×D4—944no—
417C2×D4—941no—
418C2×D4—941no—
419C2×D4—941no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—471no—
424C22×C4—471no—
425C22×C4—471no—
426C22×C4—472no—
427C22×C4—472no—
428C22×C4—474no—
429C22×C4—474no—
430C22×C4—944no—
431C22×C4—944no—
432C22×C4—944no—
433C22×C4—944no—
434C22×C4—941no—
435C22×C4—941no—
436C22×C4—941no—
437C22×C4—941no—
438C22×C4—941no—
439C22×C4—944no—
440C22×C4—942no—
441C22×C4—1882no—
442C22×C4—1882no—
443C22×C4—1884no—
444C22×C4—1884no—
445C22×C4D4×D4711yescharacteristic
446C22×C4D4×D4711yescharacteristic
447C22×C4C23×D4711yescharacteristic
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—22no—
451C2×C8—942no—
452C2×C8—942no—
453C2×C8—942no—
454C4⋊C4—24no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—472no—
459C4⋊C4—944no—
460C4⋊C4—941no—
461C4⋊C4—942no—
462C4⋊C4—942no—
463C4⋊C4C2×C4×D4712yes—
464C22⋊C4—944no—
465C22⋊C4—944no—
466C22⋊C4—944no—
467C22⋊C4—1884no—
468C42—24no—
469C42—474no—
470C42—474no—
471C42—472no—
472C42—472no—
473C42—944no—
474C42—944no—
475C42—1884no—
476C42C2×C4×D4712yes—
477C42D4×D4712yes—
478C23—472no—
479C23—942no—
480C23—942no—
481C23—941no—
482C23—942no—
483C23—941no—
484C23—942no—
485C23—942no—
486C23—1882no—
487C23—1882no—
488C23C2×D4×D4711yescharacteristic
489Q8—22no—
490Q8—22no—
491Q8—22no—
492Q8#1309398214yes—
493D4—474no—
494D4—9416no—
495D4—942no—
496D4—942no—
497D4—942no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—472no—
509C2×C4—472no—
510C2×C4—472no—
511C2×C4—9416no—
512C2×C4—941no—
513C2×C4—941no—
514C2×C4—942no—
515C2×C4—942no—
516C2×C4—942no—
517C2×C4—942no—
518C2×C4—942no—
519C2×C4—942no—
520C2×C4—942no—
521C2×C4—1884no—
522C2×C4—1884no—
523C2×C4—1888no—
524C2×C4—1888no—
525C2×C4—1882no—
526C2×C4—1882no—
527C2×C4—1882no—
528C2×C4—1882no—
529C2×C4C22⋊C4×D4712yes—
530C2×C4C22⋊C4×D4712yes—
531C2×C4C22×C4×D4711yescharacteristic
532C2×C4C2×D4×D4711yescharacteristic
533C2×C4#41×D4711yescharacteristic
534C2×C4#41×D4711yescharacteristic
535C2×C4C2×#1062535211yescharacteristic
536C2×C4C2×#1062535411yescharacteristic
537C8—42no—
538C8—1882no—
539C22—942no—
540C22—944no—
541C22—942no—
542C22—942no—
543C22—1882no—
544C22—1882no—
545C22—1884no—
546C22—1882no—
547C22—1884no—
548C22—1882no—
549C22C2×C22⋊C4×D4711yescharacteristic
550C22#1568963311yescharacteristic
551C22#1568963611yescharacteristic
552C22#1568992012yes—
553C22#1568992212yes—
554C4—22no—
555C4—24no—
556C4—22no—
557C4—42no—
558C4—82no—
559C4—942no—
560C4—942no—
561C4—942no—
562C4—3762no—
563C4—1882no—
564C4#117×D4711yescharacteristic
565C4C2×#1309398211yescharacteristic
566C4#1568963311yescharacteristic
567C4#1568963611yescharacteristic
568C2—1882no—
569C2—1884no—
570C2#2024577011yescharacteristic
571C2#2025190311yescharacteristic
572C2#2025190611yescharacteristic
573C2#2025349612yes—
574C2#2025350412yes—
subgroupG/H
centerC23C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1805C2
socleC23×C47C22×D4
prime radicalC22×#10625354C22
cosocleC22×C4×C47C24
hypercenter#1805D47
hypocenterC47#20742
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
2. C2×#15689179C2semidirect
3. #20245781C2semidirect
4. #20243482C2semidirect
5. #20243403C2semidirect
6. #20243399C2semidirect
7. #20243395C2semidirect
9. #20243220C2semidirect
10. #20243212C2semidirect
11. C47×#1805C2semidirect
14. C4×#10625354C4semidirect
286. #1805D47semidirect
333. C47#20742semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#106253542
C2×#156891792
#202457812
#202434822
#202434032
#202433992
#202433952
#202433272
#202432202
#202432122
C47×#18052
#2074247