Finite Group Databasebeta

← #26215751

Subgroups — #26215751

Order 12032.

1514 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#10625354C211yesmaximal, characteristic
2C2×C4⋊C8×D47C211yesmaximal, characteristic, complemented
3C2×#15689118C212yesmaximal
4#20245794C211yesmaximal, characteristic, complemented
5#20243403C211yesmaximal, characteristic, complemented
6#20243399C211yesmaximal, characteristic, complemented
7#20243395C211yesmaximal, characteristic, complemented
8#20243391C212yesmaximal, complemented
9#20243387C212yesmaximal, complemented
10#20243103C212yesmaximal, complemented
11C47×#1805C211yesmaximal, characteristic, complemented
12C22×#10625354C2211yescharacteristic
13C22×C8×D47C2212yes—
14C4×#10625354—24no—
15C4×#10625354C418yescomplemented
16C2×Q8×Dic47C2211yescharacteristic
17C2×C4×D188—21no—
18C2×C4×D188C2211yescharacteristic
19C2×#13093986—21no—
20C2×#13093986C2211yescharacteristic
21C2×#13093982—21no—
22C2×#13093982C2212yes—
23C2×#13093982C2211yescharacteristic
24C2×C42×D47—21no—
25C2×C42×D47C2211yescharacteristic
26C2×C4⋊(C47⋊C8)C2211yescharacteristic
27C2×Dic47⋊C8C2212yes—
28C4⋊C8×D47—24no—
29#15689118—28no—
30C2×#13093948C2212yes—
31C2×#13093946C2212yes—
32C2×#13093944C2212yes—
33#15688992C2212yes—
34#15688989C2211yescharacteristic
35#15688982—21no—
36#15688982—21no—
37#15688974—21no—
38#15688974—21no—
39#15688886C2212yes—
40#15688885C2212yes—
41#15688885C2212yes—
42C2×C4×Q8×C47C2211yescharacteristic
43C2×C4⋊C8×C47C2211yescharacteristic
44C2×#103×C47C2212yes—
45C47×#382C2211yescharacteristic
46C47×#340C2212yes—
47C2×#10625354—22no—
48C2×#10625354—22no—
49C2×#10625354—22no—
50C2×#10625354C2×C414yes—
51C2×#10625354D414yes—
52C2×#10625354Q814yes—
53C22×D188—21no—
54C22×D188C2311yescharacteristic
55C22×C4×D47—21no—
56C22×C4×D47—21no—
57C22×C4×D47—21no—
58C22×C4×D47D411yescharacteristic
59C22×C4×D47D411yescharacteristic
60C22×C4×D47C2311yescharacteristic
61C22×C47⋊C8C2312yes—
62C2×C8×D47—24no—
63C2×C8×D47—24no—
64C2×C8×D47—24no—
65Q8×Dic47—22no—
66Q8×Dic47C2×C414yes—
67C4×D188—28no—
68C4×D188—22no—
69C4×D188C2×C414yes—
70#13093986—28no—
71#13093986—22no—
72#13093986C2×C414yes—
73#13093982—22no—
74#13093982—28no—
75#13093982—24no—
76#13093982—24no—
77#13093982C2×C414yes—
78C2×C4⋊Dic47—21no—
79C2×C4⋊Dic47C2312yes—
80C2×C4⋊Dic47C2311yescharacteristic
81C2×Dic47⋊C4—22no—
82C2×Dic47⋊C4—22no—
83C2×(C2×D47)⋊C4—21no—
84C2×(C2×D47)⋊C4—21no—
85C2×(C2×D47)⋊C4—21no—
86C2×(C2×D47)⋊C4—22no—
87C2×C4×Dic47—21no—
88C2×C4×Dic47—21no—
89C2×C4×Dic47—22no—
90C2×C4×Dic47D411yescharacteristic
91C2×C4×Dic47D411yescharacteristic
92C2×C4×Dic47C2311yescharacteristic
93C42×D47—28no—
94C42×D47C2×C414yes—
95C42×D47D414yes—
96C4⋊(C47⋊C8)—22no—
97Dic47⋊C8—24no—
98#13093948—24no—
99#13093946—24no—
100#13093944—24no—
101C47⋊#95—21no—
102C47⋊#95—21no—
103C22×Q8×C47C2311yescharacteristic
104C22×C8×C47C2312yes—
105C4×Q8×C47—22no—
106C4×Q8×C47C2×C414yes—
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47C2312yes—
109C2×C4⋊C4×C47C2311yescharacteristic
110C2×C42×C47—21no—
111C2×C42×C47C2311yescharacteristic
112C4⋊C8×C47—22no—
113#103×C47—24no—
114#95×C47—21no—
115#95×C47—21no—
116C23×D47—21no—
117C23×D47C2×D411yescharacteristic
118C2×Q8×C47—21no—
119C2×Q8×C47—21no—
120C2×Q8×C47—21no—
121C2×Q8×C47C22×C412yes—
122C2×Q8×C47C2×D412yes—
123C2×Q8×C47C2×Q812yes—
124C22×C4×C47—21no—
125C22×C4×C47—21no—
126C22×C4×C47—22no—
127C22×C4×C47C2×D411yescharacteristic
128C22×C4×C47C2×D411yescharacteristic
129C22×C4×C47C2411yescharacteristic
130C22×Dic47—21no—
131C22×Dic47—22no—
132C22×Dic47C2×D411yescharacteristic
133C22×Dic47C2×D411yescharacteristic
134C22×Dic47C2×D411yescharacteristic
135#10625354—28no—
136#10625354—28no—
137#10625354—28no—
138#10625354C4⋊C4116yescomplemented
139C2×D188—24no—
140C2×D188—24no—
141C2×D188—24no—
142C2×D188—21no—
143C2×D188—21no—
144C2×D188—21no—
145C2×D188C22×C412yes—
146C2×D188C2×D412yes—
147C2×D188C2×Q812yes—
148C2×C4×D47—24no—
149C2×C4×D47—24no—
150C2×C4×D47—24no—
151C2×C4×D47—24no—
152C2×C4×D47—24no—
153C2×C4×D47—24no—
154C2×C4×D47—42no—
155C2×C4×D47—44no—
156C2×C4×D47—44no—
157C2×C4×D47C22⋊C414yes—
158C2×C4×D47C22⋊C414yes—
159C2×C4×D47C22×C412yes—
160C2×C4×D47C2×D412yes—
161C2×C4×D47C2×D412yes—
162C2×C4×D47C2×Q812yes—
163C2×C4×D47#4112yes—
164C2×C4×D47#4112yes—
165C2×C8×C47—22no—
166C2×C8×C47—22no—
167C2×C8×C47—22no—
168C4⋊C4×C47—24no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—22no—
171C4⋊C4×C47—22no—
172C4⋊C4×C47C22×C412yes—
173C42×C47—24no—
174C42×C47C22×C412yes—
175C42×C47C2×D412yes—
176(C2×D47)⋊C4—24no—
177(C2×D47)⋊C4—24no—
178(C2×D47)⋊C4—24no—
179(C2×D47)⋊C4—44no—
180C4⋊Dic47—21no—
181C4⋊Dic47—24no—
182C4⋊Dic47—22no—
183C4⋊Dic47—22no—
184C4⋊Dic47C22×C412yes—
185Dic47⋊C4—44no—
186Dic47⋊C4—44no—
187C4×Dic47—24no—
188C4×Dic47—24no—
189C4×Dic47—44no—
190C4×Dic47C22⋊C414yes—
191C4×Dic47C22⋊C414yes—
192C4×Dic47C22×C412yes—
193C4×Dic47C2×D412yes—
194C2×C47⋊C8—22no—
195C2×C47⋊C8—22no—
196C2×C47⋊C8—22no—
197C8×D47—48no—
198C23×C47C22×D411yescharacteristic
199C22×D47—22no—
200C22×D47—22no—
201C22×D47—22no—
202C22×D47—22no—
203C22×D47—22no—
204C22×D47—42no—
205C22×D47—42no—
206C22×D47C2×C22⋊C412yes—
207C22×D47#12112yes—
208C22×D47C22⋊Q812yes—
209Q8×C47—22no—
210Q8×C47—22no—
211Q8×C47—22no—
212Q8×C47C2×C4⋊C414yes—
213C2×C4×C47—22no—
214C2×C4×C47—22no—
215C2×C4×C47—22no—
216C2×C4×C47—22no—
217C2×C4×C47—22no—
218C2×C4×C47—22no—
219C2×C4×C47—42no—
220C2×C4×C47—42no—
221C2×C4×C47—42no—
222C2×C4×C47—42no—
223C2×C4×C47C2×C22⋊C412yes—
224C2×C4×C47C2×C22⋊C412yes—
225C2×C4×C47C23×C411yescharacteristic
226C2×C4×C47C22×D411yescharacteristic
227C2×C4×C47C22×D411yescharacteristic
228C2×C4×C47C22×Q811yescharacteristic
229C2×C4×C47C2×#4111yescharacteristic
230C2×C4×C47C2×#4111yescharacteristic
231C2×Dic47—22no—
232C2×Dic47—22no—
233C2×Dic47—42no—
234C2×Dic47—42no—
235C2×Dic47—42no—
236C2×Dic47—42no—
237C2×Dic47C2×C22⋊C412yes—
238C2×Dic47C2×C22⋊C412yes—
239C2×Dic47C2×C22⋊C412yes—
240C2×Dic47#12112yes—
241C2×Dic47C22⋊Q812yes—
242C2×Dic47#13514yes—
243D188—216no—
244D188—22no—
245D188—22no—
246D188—22no—
247D188C2×C4⋊C414yes—
248C4×D47—216no—
249C4×D47—28no—
250C4×D47—44no—
251C4×D47—48no—
252C4×D47—48no—
253C4×D47C2×C4⋊C414yes—
254C4×D47#11714yes—
255C4×D47#12114yes—
256C4×D47C22⋊Q814yes—
257C8×C47—42no—
258C47⋊C8—42no—
259#20645—471nomaximal, Sylow
260C22×C47C22×C22⋊C411yescharacteristic
261C22×C47C2×#12111yescharacteristic
262C22×C47C2×C22⋊Q811yescharacteristic
263C22×C47C2×#13512yes—
264C22×C47#57512yes—
265C2×D47—24no—
266C2×D47—22no—
267C2×D47—22no—
268C2×D47—24no—
269C2×D47—24no—
270C2×D47—44no—
271C2×D47—42no—
272C2×D47—42no—
273C2×D47—44no—
274C2×D47#38014yes—
275C4×C47—22no—
276C4×C47—22no—
277C4×C47—24no—
278C4×C47—42no—
279C4×C47—82no—
280C4×C47C22×C4⋊C411yescharacteristic
281C4×C47C2×#11711yescharacteristic
282C4×C47C2×#12111yescharacteristic
283C4×C47C2×C22⋊Q811yescharacteristic
284Dic47—22no—
285Dic47—42no—
286Dic47—82no—
287Dic47#38014yes—
288Dic47#41614yescomplemented
289C2×C4×#41—471no—
290C22×C4⋊C8—471no—
291C2×#416—472no—
292#2285—471no—
293#1805—471no—
294#1805D4711yescharacteristic, complemented
295#1804—471no—
296#1804—471no—
297#1803—472no—
298#1802—472no—
299#1735—472no—
300C2×C47C2×#38011yescharacteristic
301C2×C47C2×#41611yescharacteristic
302C2×C47#289611yescharacteristic
303C2×C47#303512yes—
304C2×C47#307612yes—
305D47—24no—
306D47—44no—
307C22×#41—471no—
308C23×C8—472no—
309C4×#41—478no—
310C4×#41—944no—
311C2×C4×Q8—471no—
312C2×C4×Q8C2×D4711yescharacteristic
313C2×C4×D4—471no—
314C2×C4×D4—471no—
315C2×C4×D4—941no—
316C2×C4×D4—941no—
317C2×#117—471no—
318C2×#117—472no—
319C2×#117—941no—
320C22×C42—471no—
321C22×C42—941no—
322C2×C4⋊C8—472no—
323C2×C4⋊C8—471no—
324C2×C4⋊C8—944no—
325C2×C4⋊C8C2×D4711yescharacteristic
326#416—948no—
327C2×#103—472no—
328C2×#103C2×D4712yes—
329C2×#102—472no—
330C2×#102—472no—
331#384—472no—
332#382—471no—
333#382C2×D4711yescharacteristic
334#380—941no—
335#380—941no—
336C4×C4⋊C4—941no—
337C4×C4⋊C4—941no—
338#340—472no—
339#340—472no—
340#340—472no—
341#340C2×D4712yes—
342C47#2064511yescharacteristic, complemented, Sylow
343C2×#41—474no—
344C2×#41—474no—
345C2×#41—474no—
346C2×#41—942no—
347C2×#41—942no—
348C2×#41—942no—
349C22×Q8C22×D4711yescharacteristic
350C22×D4—471no—
351C22×D4—941no—
352C23×C4—471no—
353C23×C4—471no—
354C23×C4—471no—
355C23×C4—941no—
356C23×C4—941no—
357C23×C4—941no—
358C22×C8—472no—
359C22×C8—944no—
360C22×C8—944no—
361C22×C8—944no—
362C22×C8C22×D4712yes—
363C4×Q8—22no—
364C4×Q8—474no—
365C4×Q8—942no—
366C4×Q8C4×D4714yes—
367C4×D4—474no—
368C4×D4—474no—
369C4×D4—948no—
370C4×D4—942no—
371C4×D4—948no—
372C4×D4—942no—
373#117—474no—
374#117—948no—
375#117—942no—
376#117—944no—
377#117—944no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—471no—
380C2×C4⋊C4—472no—
381C2×C4⋊C4—941no—
382C2×C4⋊C4—942no—
383C2×C4⋊C4—942no—
384C2×C4⋊C4C22×D4712yes—
385C2×C4⋊C4C22×D4711yescharacteristic
386C2×C22⋊C4—941no—
387C2×C22⋊C4—941no—
388C2×C22⋊C4—941no—
389C2×C22⋊C4—942no—
390C2×C42—21no—
391C2×C42—471no—
392C2×C42—471no—
393C2×C42—471no—
394C2×C42—474no—
395C2×C42—474no—
396C2×C42—948no—
397C2×C42—941no—
398C2×C42—941no—
399C2×C42—942no—
400C2×C42C22×D4711yescharacteristic
401C4⋊C8—22no—
402C4⋊C8—944no—
403C4⋊C8—942no—
404#103—24no—
405#103—944no—
406#102—944no—
407#102—944no—
408#95—21no—
409#95—21no—
410#95—941no—
411#95—941no—
412C24—471no—
413C24—941no—
414#41—4716no—
415#41—948no—
416#41—948no—
417#41—948no—
418C2×Q8—21no—
419C2×Q8—21no—
420C2×Q8—21no—
421C2×Q8C2×C4×D4712yes—
422C2×Q8D4×D4712yes—
423C2×Q8Q8×D4712yes—
424C2×D4—472no—
425C2×D4—472no—
426C2×D4—472no—
427C2×D4—944no—
428C2×D4—944no—
429C2×D4—944no—
430C2×D4—941no—
431C2×D4—941no—
432C2×D4—941no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—22no—
436C22×C4—471no—
437C22×C4—471no—
438C22×C4—472no—
439C22×C4—472no—
440C22×C4—472no—
441C22×C4—472no—
442C22×C4—472no—
443C22×C4—472no—
444C22×C4—471no—
445C22×C4—474no—
446C22×C4—474no—
447C22×C4—944no—
448C22×C4—944no—
449C22×C4—944no—
450C22×C4—944no—
451C22×C4—944no—
452C22×C4—941no—
453C22×C4—944no—
454C22×C4—942no—
455C22×C4—1882no—
456C22×C4—1884no—
457C22×C4—1884no—
458C22×C4D4×D4711yescharacteristic
459C22×C4D4×D4711yescharacteristic
460C22×C4C23×D4711yescharacteristic
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—22no—
464C2×C8—942no—
465C2×C8—942no—
466C2×C8—942no—
467C2×C8—1888no—
468C4⋊C4—24no—
469C4⋊C4—21no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—472no—
473C4⋊C4—944no—
474C4⋊C4—941no—
475C4⋊C4—942no—
476C4⋊C4—942no—
477C4⋊C4—1884no—
478C4⋊C4—1884no—
479C4⋊C4C2×C4×D4712yes—
480C22⋊C4—944no—
481C22⋊C4—944no—
482C22⋊C4—944no—
483C22⋊C4—1884no—
484C42—24no—
485C42—474no—
486C42—474no—
487C42—472no—
488C42—472no—
489C42—944no—
490C42—944no—
491C42—1884no—
492C42C2×C4×D4712yes—
493C42D4×D4712yes—
494C23—472no—
495C23—472no—
496C23—472no—
497C23—942no—
498C23—942no—
499C23—942no—
500C23—942no—
501C23—942no—
502C23—1882no—
503C23—1882no—
504C23C2×D4×D4711yescharacteristic
505Q8—22no—
506Q8—22no—
507Q8—22no—
508Q8C4⋊C4×D4714yes—
509D4—474no—
510D4—9416no—
511D4—942no—
512D4—942no—
513D4—942no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—472no—
525C2×C4—472no—
526C2×C4—472no—
527C2×C4—472no—
528C2×C4—474no—
529C2×C4—474no—
530C2×C4—474no—
531C2×C4—474no—
532C2×C4—474no—
533C2×C4—472no—
534C2×C4—948no—
535C2×C4—9416no—
536C2×C4—942no—
537C2×C4—942no—
538C2×C4—1884no—
539C2×C4—1888no—
540C2×C4—1888no—
541C2×C4—1882no—
542C2×C4—1882no—
543C2×C4—1882no—
544C2×C4—1882no—
545C2×C4C22⋊C4×D4712yes—
546C2×C4C22⋊C4×D4712yes—
547C2×C4C22×C4×D4711yescharacteristic
548C2×C4C2×D4×D4711yescharacteristic
549C2×C4C2×D4×D4711yescharacteristic
550C2×C4C2×Q8×D4711yescharacteristic
551C2×C4#41×D4711yescharacteristic
552C2×C4#41×D4711yescharacteristic
553C8—42no—
554C8—1882no—
555C22—474no—
556C22—942no—
557C22—944no—
558C22—942no—
559C22—944no—
560C22—944no—
561C22—1882no—
562C22—1884no—
563C22—1882no—
564C22—1884no—
565C22C2×C22⋊C4×D4711yescharacteristic
566C22#121×D4711yescharacteristic
567C22C22⋊Q8×D4711yescharacteristic
568C22#135×D4712yes—
569C22#1568992212yes—
570C4—22no—
571C4—22no—
572C4—24no—
573C4—42no—
574C4—82no—
575C4—474no—
576C4—474no—
577C4—942no—
578C4—3762no—
579C4—1882no—
580C4C2×C4⋊C4×D4711yescharacteristic
581C4#117×D4711yescharacteristic
582C4#121×D4711yescharacteristic
583C4C22⋊Q8×D4711yescharacteristic
584C2—944no—
585C2—1884no—
586C2#380×D4711yescharacteristic
587C2#416×D4711yescharacteristic
588C2#2025190611yescharacteristic
589C2#2025284212yes—
590C2#2025317512yes—
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#20645
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×C4⋊C8×D47C2semidirect
4. #20245794C2semidirect
5. #20243403C2semidirect
6. #20243399C2semidirect
7. #20243395C2semidirect
8. #20243391C2semidirect
9. #20243387C2semidirect
10. #20243103C2semidirect
11. C47×#1805C2semidirect
15. C4×#10625354C4semidirect
138. #10625354C4⋊C4semidirect
288. Dic47#416semidirect
294. #1805D47semidirect
342. C47#20645semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#106253542
C2×C4⋊C8×D472
C2×#156891182
#202457942
#202434032
#202433992
#202433952
#202433912
#202433872
#202431032
C47×#18052
#2064547