Finite Group Databasebeta

← #26187724

Subgroups — #26187724

Order 12032.

654 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252407C211yesmaximal, characteristic, complemented
2#20252401C211yesmaximal, characteristic, complemented
3C4×#13094047C211yesmaximal, characteristic
4#20242278C211yesmaximal, characteristic, complemented
5#20242275C211yesmaximal, characteristic, complemented
6#20242239C211yesmaximal, characteristic, complemented
7#20242219C211yesmaximal, characteristic, complemented
8#20242216C211yesmaximal, characteristic, complemented
9C8×#10625336C211yesmaximal, characteristic, complemented
10#20242185C211yesmaximal, characteristic, complemented
11#20242150C211yesmaximal, characteristic, complemented
12#20241821C211yesmaximal, characteristic, complemented
13#20241821C211yesmaximal, characteristic, complemented
14#20241667C211yesmaximal, characteristic, complemented
15C47×#1584C211yesmaximal, characteristic, complemented
16C2×#13094047C411yescharacteristic
17C4×#10625354C2211yescharacteristic
18C4×#10625354C2211yescharacteristic
19#15689254C2211yescharacteristic
20Q8×C47⋊C8C2211yescharacteristic
21Q16×Dic47C411yescharacteristic
22C4×#10625336C2211yescharacteristic
23C4×#10625336C2211yescharacteristic
24#15689208C411yescharacteristic
25#15689208C411yescharacteristic
26C4×D376C2211yescharacteristic
27#15689205C411yescharacteristic
28#15689202—21no—
29#15689202C2211yescharacteristic
30#15689200—21no—
31#15689200C2211yescharacteristic
32#15689196—21no—
33#15689193—21no—
34C8×D188C2211yescharacteristic
35#15689188C2211yescharacteristic
36#15689179—21no—
37#15689179—21no—
38#15689179C2211yescharacteristic
39#15689179C2211yescharacteristic
40#15689174C2211yescharacteristic
41#15689160C411yescharacteristic
42#15689118C411yescharacteristic
43#15689118C411yescharacteristic
44C4×#10625324—21no—
45C4×#10625324C2211yescharacteristic
46#97×D47—21no—
47C4×C8×D47C2211yescharacteristic
48#15688876C2211yescharacteristic
49#15688873C2211yescharacteristic
50#15688873C2211yescharacteristic
51Q8⋊(C47⋊C8)C2211yescharacteristic
52Q8⋊(C47⋊C8)C2211yescharacteristic
53#15688852C2211yescharacteristic
54#15688852C2211yescharacteristic
55#15688850C2211yescharacteristic
56#15688850C2211yescharacteristic
57#15688845C2211yescharacteristic
58C8×C47⋊C8C2211yescharacteristic
59C8×C47⋊C8C2211yescharacteristic
60C47×#446C2211yescharacteristic
61C8×Q8×C47C2211yescharacteristic
62C4×Q16×C47C2211yescharacteristic
63C47×#335C2211yescharacteristic
64C47×Q8⋊C8C2211yescharacteristic
65C47×Q8⋊C8C2211yescharacteristic
66C47×#322C2211yescharacteristic
67C2×#10625354C2×C411yescharacteristic
68C2×#10625354C2×C411yescharacteristic
69#13094047—24no—
70C2×#10625336C2×C411yescharacteristic
71C2×#10625336C2×C411yescharacteristic
72C2×D376C2×C411yescharacteristic
73C2×#10625324—21no—
74C2×#10625324—21no—
75C2×#10625324D411yescharacteristic
76C2×#10625324D411yescharacteristic
77C2×C8×D47—21no—
78C2×C8×D47—21no—
79C2×C8×D47C2×C411yescharacteristic
80C2×C8×D47C2×C411yescharacteristic
81Q8×Dic47C2×C411yescharacteristic
82Q8×Dic47C2×C411yescharacteristic
83C4×D188—21no—
84C4×D188—21no—
85C4×D188C2311yescharacteristic
86C4×D188C2311yescharacteristic
87#13093986—21no—
88#13093986—21no—
89#13093986C2×C411yescharacteristic
90#13093986C2×C411yescharacteristic
91#13093982—21no—
92#13093982—21no—
93#13093982C2×C411yescharacteristic
94#13093982C2×C411yescharacteristic
95C42×D47—21no—
96C42×D47—21no—
97C42×D47C2311yescharacteristic
98#13093958C2×C411yescharacteristic
99#13093955C2×C411yescharacteristic
100#13093955C2×C411yescharacteristic
101C4⋊(C47⋊C8)—21no—
102C4⋊(C47⋊C8)—21no—
103C4⋊(C47⋊C8)C2311yescharacteristic
104C4⋊(C47⋊C8)C2311yescharacteristic
105Dic47⋊C8—21no—
106#13093948C2×C411yescharacteristic
107#13093948C2×C411yescharacteristic
108#13093946C2×C411yescharacteristic
109#13093946C2×C411yescharacteristic
110#13093944C2×C411yescharacteristic
111#13093944C2×C411yescharacteristic
112(C2×D47)⋊C8—21no—
113(C2×D47)⋊C8—21no—
114(C2×D47)⋊C8—21no—
115(C2×D47)⋊C8—21no—
116(C2×D47)⋊C8—21no—
117(C2×D47)⋊C8—21no—
118(C2×D47)⋊C8—21no—
119#13093935—21no—
120#13093934—21no—
121#13093934—21no—
122#13093934D411yescharacteristic
123#13093934D411yescharacteristic
124C4×C47⋊C8—21no—
125C4×C47⋊C8C2311yescharacteristic
126C4×C47⋊C8C2311yescharacteristic
127C4×C47⋊C8C2311yescharacteristic
128C8×Dic47C2×C411yescharacteristic
129C8×Dic47C2×C411yescharacteristic
130C2×Q16×C47C2×C411yescharacteristic
131C4×Q8×C47C2311yescharacteristic
132C4×Q8×C47C2311yescharacteristic
133#107×C47C2×C411yescharacteristic
134C4⋊C8×C47—21no—
135C4⋊C8×C47—21no—
136C4⋊C8×C47C2311yescharacteristic
137C4⋊C8×C47C2311yescharacteristic
138#103×C47C2×C411yescharacteristic
139#103×C47C2×C411yescharacteristic
140#97×C47—21no—
141C4×C8×C47—21no—
142C4×C8×C47C2311yescharacteristic
143C4×C8×C47C2311yescharacteristic
144C4×C8×C47C2311yescharacteristic
145C2×Q8×C47C22×C411yescharacteristic
146C2×Q8×C47C22×C411yescharacteristic
147#10625354—22no—
148#10625354—22no—
149#10625354—41no—
150#10625354—41no—
151C2×D188—21no—
152C2×D188—21no—
153C2×D188C22×C411yescharacteristic
154C2×D188C22×C411yescharacteristic
155C2×C4×D47—21no—
156C2×C4×D47—21no—
157C2×C4×D47—21no—
158C2×C4×D47—21no—
159C2×C4×D47—21no—
160C2×C4×D47—21no—
161C2×C4×D47C22×C411yescharacteristic
162C2×C4×D47C2×D411yescharacteristic
163C2×C4×D47#4111yescharacteristic
164Q16×C47—22no—
165C2×C8×C47—21no—
166C2×C8×C47—21no—
167C2×C8×C47—21no—
168C2×C8×C47C22×C411yescharacteristic
169C2×C8×C47C22×C411yescharacteristic
170C2×C8×C47C2×D411yescharacteristic
171C2×C8×C47C2×D411yescharacteristic
172C4⋊C4×C47—21no—
173C4⋊C4×C47—21no—
174C4⋊C4×C47C22×C411yescharacteristic
175C4⋊C4×C47C22×C411yescharacteristic
176C42×C47—21no—
177C42×C47—21no—
178C42×C47C2411yescharacteristic
179#10625336—22no—
180#10625336M1614yescomplemented
181(C2×D47)⋊C4—21no—
182(C2×D47)⋊C4—21no—
183(C2×D47)⋊C4—21no—
184(C2×D47)⋊C4—21no—
185(C2×D47)⋊C4—21no—
186(C2×D47)⋊C4—21no—
187C4⋊Dic47—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47C22×C411yescharacteristic
190C4⋊Dic47C22×C411yescharacteristic
191C4×Dic47—21no—
192C4×Dic47—21no—
193C4×Dic47—21no—
194C4×Dic47—21no—
195C4×Dic47C22×C411yescharacteristic
196C4×Dic47C2×D411yescharacteristic
197C4×Dic47#4111yescharacteristic
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8C22×C411yescharacteristic
202C2×C47⋊C8C22×C411yescharacteristic
203C2×C47⋊C8C2×D411yescharacteristic
204C2×C47⋊C8C2×D411yescharacteristic
205D376—22no—
206#10625324—22no—
207#10625324—22no—
208#10625324—42no—
209#10625324—42no—
210C8×D47—22no—
211C8×D47—41no—
212C8×D47—42no—
213C8×D47—42no—
214C22×D47—21no—
215C22×D47—21no—
216C22×D47C4×D411yescharacteristic
217Q8×C47—21no—
218Q8×C47—21no—
219Q8×C47—21no—
220Q8×C47C2×M1612yes—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47C23×C411yescharacteristic
226C2×C4×C47C22×D411yescharacteristic
227C2×C4×C47C2×#4111yescharacteristic
228C2×Dic47—21no—
229C2×Dic47—21no—
230C2×Dic47C4×D411yescharacteristic
231C2×Dic47C4×D411yescharacteristic
232C2×Dic47C4×D411yescharacteristic
233D188—21no—
234D188—21no—
235D188—21no—
236D188—42no—
237D188—42no—
238D188C2×M1612yes—
239C4×D47—21no—
240C4×D47—21no—
241C4×D47—21no—
242C4×D47—21no—
243C4×D47—41no—
244C4×D47—42no—
245C4×D47—42no—
246C4×D47—41no—
247C4×D47—41no—
248C4×D47—41no—
249C4×D47—41no—
250C4×D47—81no—
251C4×D47—81no—
252C8×C47—21no—
253C8×C47—21no—
254C8×C47—21no—
255C8×C47—41no—
256C8×C47—41no—
257C8×C47—41no—
258C8×C47#13512yescomplemented
259C47⋊C8—21no—
260C47⋊C8—21no—
261C47⋊C8—41no—
262C47⋊C8—41no—
263C47⋊C8—41no—
264C47⋊C8C2×M1612yes—
265C47⋊C8#13512yescomplemented
266#18037—471nomaximal, Sylow
267C22×C47C2×C4×D411yescharacteristic
268C2×D47—21no—
269C2×D47—22no—
270C2×D47—21no—
271C2×D47—21no—
272C2×D47—41no—
273C2×D47—41no—
274C2×D47—41no—
275C2×D47—41no—
276C2×D47—41no—
277C4×C47—22no—
278C4×C47—21no—
279C4×C47—41no—
280C4×C47—41no—
281C4×C47—41no—
282C4×C47C22×M1611yescharacteristic
283C4×C47#56811yescharacteristic
284C4×C47C2×#13511yescharacteristic
285C4×C47C2×#13511yescharacteristic
286Dic47—21no—
287Dic47—21no—
288Dic47—21no—
289Dic47—41no—
290Dic47—41no—
291#2971—471no—
292#2970—471no—
293C4×#135—471no—
294#1590—471no—
295#1590—471no—
296#1584—471no—
297#1584D4711yescharacteristic, complemented
298#1581—471no—
299#1581—471no—
300C8×SD16—471no—
301C8×SD16—471no—
302#1568—471no—
303#1488—471no—
304#1488—471no—
305C8×M16—471no—
306C2×C47D4×M1611yescharacteristic
307C2×C47C4×#13511yescharacteristic
308C2×C47#295711yescharacteristic
309D47—41no—
310D47—42no—
311D47—81no—
312C2×#135—471no—
313C4×#41—471no—
314C4×#41—471no—
315#446—471no—
316#446C2×D4711yescharacteristic
317C8×Q8—471no—
318C8×Q8C2×D4711yescharacteristic
319C4×Q16—471no—
320C4×Q16C2×D4711yescharacteristic
321C4×SD16—471no—
322C4×SD16—471no—
323C4×SD16—471no—
324C4×SD16—471no—
325C4×D8—471no—
326C4×D8—471no—
327#436—471no—
328#436—471no—
329#436—941no—
330#436—941no—
331#435—941no—
332#435—941no—
333C8×D4—471no—
334C8×D4—471no—
335#432—471no—
336#432—471no—
337#432—941no—
338#432—941no—
339#431—471no—
340#427—471no—
341#416—471no—
342#416—471no—
343C4×M16—471no—
344C4×M16—941no—
345C2×#97—941no—
346C2×C4×C8—471no—
347#335—471no—
348#335C2×D4711yescharacteristic
349#334—471no—
350#334—471no—
351Q8⋊C8—471no—
352Q8⋊C8—471no—
353Q8⋊C8C2×D4711yescharacteristic
354Q8⋊C8C2×D4711yescharacteristic
355D4⋊C8—471no—
356D4⋊C8—471no—
357D4⋊C8—471no—
358D4⋊C8—471no—
359#322—471no—
360#322C2×D4711yescharacteristic
361C82—471no—
362C82—471no—
363C47#1803711yescharacteristic, complemented, Sylow
364C2×#41—471no—
365C2×#41—471no—
366#135—944no—
367C2×Q16C4×D4711yescharacteristic
368C2×SD16—471no—
369C2×SD16—471no—
370C2×D8—471no—
371C2×M16—471no—
372C2×M16—471no—
373C2×M16—941no—
374C2×M16—941no—
375C22×C8—471no—
376C22×C8—471no—
377C22×C8—941no—
378C22×C8—941no—
379C4×Q8—471no—
380C4×Q8—471no—
381C4×Q8C22×D4711yescharacteristic
382C4×Q8C22×D4711yescharacteristic
383C4×D4—471no—
384C4×D4—471no—
385C4×D4—471no—
386C4×D4—471no—
387C4×D4—941no—
388C4×D4—941no—
389C4×D4—941no—
390C4×D4—941no—
391#117—471no—
392#117—471no—
393#117—941no—
394#117—941no—
395C2×C42—471no—
396C2×C42—941no—
397C2×C42—941no—
398#107—471no—
399#107C4×D4711yescharacteristic
400#106—471no—
401#106—471no—
402C4⋊C8—21no—
403C4⋊C8—21no—
404C4⋊C8—471no—
405C4⋊C8—471no—
406C4⋊C8—941no—
407C4⋊C8—941no—
408C4⋊C8—941no—
409C4⋊C8C22×D4711yescharacteristic
410C4⋊C8C22×D4711yescharacteristic
411#103—471no—
412#103—471no—
413#103C4×D4711yescharacteristic
414#103C4×D4711yescharacteristic
415#102—471no—
416#102—471no—
417#102—471no—
418#102—471no—
419C22⋊C8—941no—
420C22⋊C8—941no—
421C22⋊C8—941no—
422C22⋊C8—941no—
423C22⋊C8—941no—
424C22⋊C8—941no—
425C22⋊C8—941no—
426#97—21no—
427#97—471no—
428#97—471no—
429#97—941no—
430#97—941no—
431#97—941no—
432C4×C8—21no—
433C4×C8—471no—
434C4×C8—471no—
435C4×C8—471no—
436C4×C8—471no—
437C4×C8—471no—
438C4×C8—941no—
439C4×C8C22×D4711yescharacteristic
440C4×C8C22×D4711yescharacteristic
441C4×C8C22×D4711yescharacteristic
442#41—942no—
443#41—942no—
444#41—1881no—
445#41—1881no—
446C2×Q8C2×C4×D4711yescharacteristic
447C2×Q8C2×C4×D4711yescharacteristic
448C2×D4—471no—
449C2×D4—471no—
450C2×D4—941no—
451C2×D4—941no—
452C22×C4—471no—
453C22×C4—471no—
454C22×C4—471no—
455C22×C4—941no—
456C22×C4—941no—
457C22×C4—941no—
458C22×C4—941no—
459C22×C4—941no—
460C22×C4—941no—
461Q16—22no—
462SD16—474no—
463SD16—942no—
464D8—942no—
465M16—942no—
466M16—942no—
467M16—1882no—
468M16—1882no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—471no—
473C2×C8—471no—
474C2×C8—471no—
475C2×C8—471no—
476C2×C8—942no—
477C2×C8—941no—
478C2×C8—941no—
479C2×C8—941no—
480C2×C8—1881no—
481C2×C8—1882no—
482C2×C8—1882no—
483C2×C8C2×C4×D4711yescharacteristic
484C2×C8C2×C4×D4711yescharacteristic
485C2×C8D4×D4711yescharacteristic
486C2×C8D4×D4711yescharacteristic
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—471no—
490C4⋊C4—471no—
491C4⋊C4—941no—
492C4⋊C4—941no—
493C4⋊C4C2×C4×D4711yescharacteristic
494C4⋊C4C2×C4×D4711yescharacteristic
495C22⋊C4—941no—
496C22⋊C4—941no—
497C22⋊C4—941no—
498C22⋊C4—941no—
499C22⋊C4—941no—
500C22⋊C4—941no—
501C42—21no—
502C42—21no—
503C42—471no—
504C42—471no—
505C42—471no—
506C42—941no—
507C42—941no—
508C42—941no—
509C42—941no—
510C42C23×D4711yescharacteristic
511C23—471no—
512C23—941no—
513C23—941no—
514Q8—21no—
515Q8—21no—
516Q8—21no—
517Q8M16×D4712yes—
518D4—472no—
519D4—941no—
520D4—941no—
521D4—941no—
522D4—1882no—
523D4—1882no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—471no—
529C2×C4—471no—
530C2×C4—471no—
531C2×C4—941no—
532C2×C4—941no—
533C2×C4—941no—
534C2×C4—941no—
535C2×C4—941no—
536C2×C4—941no—
537C2×C4—3761no—
538C2×C4—3761no—
539C2×C4—1881no—
540C2×C4—1882no—
541C2×C4—1882no—
542C2×C4—1881no—
543C2×C4—1881no—
544C2×C4—1881no—
545C2×C4—1881no—
546C2×C4C22×C4×D4711yescharacteristic
547C2×C4C2×D4×D4711yescharacteristic
548C2×C4#41×D4711yescharacteristic
549C8—21no—
550C8—21no—
551C8—21no—
552C8—41no—
553C8—41no—
554C8—41no—
555C8—472no—
556C8—472no—
557C8—941no—
558C8—941no—
559C8—1881no—
560C8—1881no—
561C8—1881no—
562C8#1309404712yescomplemented
563C22—941no—
564C22—941no—
565C22—941no—
566C22—942no—
567C22—1881no—
568C22—1881no—
569C22—1881no—
570C22—1881no—
571C22—1881no—
572C22C4×D4×D4711yescharacteristic
573C4—21no—
574C4—22no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—941no—
579C4—941no—
580C4—941no—
581C4—1881no—
582C4—1881no—
583C4C2×M16×D4711yescharacteristic
584C4#1568988811yescharacteristic
585C4#135×D4711yescharacteristic
586C4C2×#1309404711yescharacteristic
587C2—3761no—
588C2—1881no—
589C2—1882no—
590C2#2025212111yescharacteristic
591C2C4×#1309404711yescharacteristic
592C2#2025230211yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#1584C2
socleC22×C47C2×C4×D4
prime radicalC2×#13094047C4
cosocleC42×C47C24
hypercenter#1584D47
hypocenterC47#18037
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1584

Complements

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

NH ≅ G/Ntype
1. #20252407C2semidirect
2. #20252401C2semidirect
4. #20242278C2semidirect
5. #20242275C2semidirect
6. #20242239C2semidirect
7. #20242219C2semidirect
8. #20242216C2semidirect
9. C8×#10625336C2semidirect
10. #20242185C2semidirect
11. #20242150C2semidirect
12. #20241821C2semidirect
13. #20241821C2semidirect
14. #20241667C2semidirect
15. C47×#1584C2semidirect
180. #10625336M16semidirect
258. C8×C47#135semidirect
265. C47⋊C8#135semidirect
297. #1584D47semidirect
363. C47#18037semidirect
562. C8#13094047semidirect

Maximal subgroups

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

subgroupindex
#202524072
#202524012
C4×#130940472
#202422782
#202422752
#202422392
#202422192
#202422162
C8×#106253362
#202421852
#202421502
#202418212
#202416672
C47×#15842
#1803747