Finite Group Databasebeta

← #26319141

Subgroups — #26319141

Order 12032.

784 conjugacy classes of subgroups.

The subgroup lattice has 642 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13094003C211yesmaximal, characteristic
2#20244814C211yesmaximal, characteristic
3#20244584C211yesmaximal, characteristic
4#20244581C211yesmaximal, characteristic
5#20244396C211yesmaximal, characteristic
6#20244022C211yesmaximal, characteristic, complemented
7#20244019C211yesmaximal, characteristic, complemented
8#20243973C211yesmaximal, characteristic, complemented
9#20243970C211yesmaximal, characteristic, complemented
10#20243732C211yesmaximal, characteristic, complemented
11#20243729C211yesmaximal, characteristic, complemented
12#20243642C211yesmaximal, characteristic, complemented
13#20243641C211yesmaximal, characteristic, complemented
14#20243456C211yesmaximal, characteristic
15C4×C22⋊Q8×C47C211yesmaximal, characteristic
16C22×#10625337C2211yescharacteristic
17C2×#13094003C2211yescharacteristic
18C2×C4×Dic94C2211yescharacteristic
19C2×#13093957C2211yescharacteristic
20C2×#13093955C2211yescharacteristic
21C2×#13093955C2211yescharacteristic
22C2×#13093948C2211yescharacteristic
23C2×#13093948C2211yescharacteristic
24C2×#13093948C2211yescharacteristic
25C2×#13093947C2211yescharacteristic
26C2×#13093947C2211yescharacteristic
27C2×#13093947C2211yescharacteristic
28C2×C22⋊(C47⋊C8)C2211yescharacteristic
29C2×C22⋊(C47⋊C8)C2211yescharacteristic
30#15689029C2211yescharacteristic
31#15689021C2211yescharacteristic
32#15689007—21no—
33#15688996—21no—
34#15688983C2211yescharacteristic
35C4×Dic47⋊C4—21no—
36C4×C4⋊Dic47C2211yescharacteristic
37C4×C22⋊Dic47—21no—
38#15688885C2211yescharacteristic
39#15688885C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688878C2211yescharacteristic
45#15688878C2211yescharacteristic
46#15688878C2211yescharacteristic
47#15688878C2211yescharacteristic
48C2×C22⋊Q8×C47C2211yescharacteristic
49C2×C4×Q8×C47C2211yescharacteristic
50C22×C42×C47C2211yescharacteristic
51C47×#391C2211yescharacteristic
52C47×#389C2211yescharacteristic
53C47×#387—21no—
54C47×#385—21no—
55C47×#380C2211yescharacteristic
56C4×C4⋊C4×C47—21no—
57C4×C4⋊C4×C47C2211yescharacteristic
58C4×C22⋊C4×C47—21no—
59C22×Dic94C2311yescharacteristic
60C2×#10625337—22no—
61C2×#10625337—22no—
62C2×#10625337—41no—
63C22×C47⋊C8C2311yescharacteristic
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67#13094003—22no—
68#13094003—22no—
69C4×Dic94—41no—
70C4×Dic94—41no—
71C2×C4⋊Dic47C2311yescharacteristic
72C2×C4⋊Dic47C2311yescharacteristic
73C2×C4⋊Dic47C2311yescharacteristic
74C2×Dic47⋊C4—21no—
75C2×Dic47⋊C4—21no—
76C2×Dic47⋊C4—21no—
77C2×C22⋊Dic47—21no—
78C2×C22⋊Dic47—21no—
79C2×C4×Dic47—21no—
80C2×C4×Dic47—21no—
81#13093957—41no—
82#13093955—22no—
83#13093955—22no—
84#13093948—22no—
85#13093948—22no—
86#13093948—41no—
87#13093947—22no—
88#13093947—22no—
89#13093947—41no—
90C22⋊(C47⋊C8)—41no—
91C22⋊(C47⋊C8)—41no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C22×Q8×C47C2311yescharacteristic
98C23×C4×C47D411yescharacteristic
99C23×C4×C47D411yescharacteristic
100C23×C4×C47C2311yescharacteristic
101C22⋊Q8×C47—22no—
102C22⋊Q8×C47—22no—
103C4×Q8×C47—41no—
104C4×Q8×C47—41no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47—21no—
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47C2311yescharacteristic
109C2×C4⋊C4×C47C2311yescharacteristic
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C22⋊C4×C47—21no—
112C2×C22⋊C4×C47—21no—
113C2×C42×C47—21no—
114C2×C42×C47—21no—
115C2×C42×C47—41no—
116C2×C42×C47—41no—
117C2×C42×C47D411yescharacteristic
118C2×C42×C47D411yescharacteristic
119C2×C42×C47C2311yescharacteristic
120C2×C42×C47C2311yescharacteristic
121#95×C47—21no—
122#95×C47—21no—
123#95×C47—21no—
124#95×C47—21no—
125#95×C47—21no—
126C24×C47C2×D411yescharacteristic
127C2×Q8×C47—21no—
128C2×Q8×C47—21no—
129C2×Q8×C47—21no—
130C2×Q8×C47—21no—
131C2×Q8×C47—21no—
132C2×Q8×C47#4112yes—
133C22×C4×C47—22no—
134C22×C4×C47—21no—
135C22×C4×C47—21no—
136C22×C4×C47—22no—
137C22×C4×C47—22no—
138C22×C4×C47—21no—
139C22×C4×C47—21no—
140C22×C4×C47—21no—
141C22×C4×C47—21no—
142C22×C4×C47—21no—
143C22×C4×C47—21no—
144C22×C4×C47—21no—
145C22×C4×C47—21no—
146C22×C4×C47—41no—
147C22×C4×C47—41no—
148C22×C4×C47—41no—
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2411yescharacteristic
155C22×Dic47—21no—
156C22×Dic47—21no—
157C22×Dic47—21no—
158C22×Dic47—21no—
159C2×Dic94—21no—
160C2×Dic94—21no—
161C2×Dic94—21no—
162C2×Dic94—21no—
163C2×Dic94—21no—
164C2×Dic94#4112yes—
165C4⋊C4×C47—24no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—21no—
171C4⋊C4×C47—42no—
172C4⋊C4×C47—42no—
173C4⋊C4×C47#4112yes—
174C22⋊C4×C47—24no—
175C22⋊C4×C47—42no—
176C42×C47—21no—
177C42×C47—21no—
178C42×C47—21no—
179C42×C47—21no—
180C42×C47—41no—
181C42×C47—41no—
182C42×C47—42no—
183C42×C47—42no—
184C22⋊Dic47—24no—
185C22⋊Dic47—42no—
186#10625337—44no—
187C4⋊Dic47—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47#4112yes—
193Dic47⋊C4—24no—
194Dic47⋊C4—42no—
195Dic47⋊C4—42no—
196C4×Dic47—42no—
197C4×Dic47—42no—
198C2×C47⋊C8—21no—
199C2×C47⋊C8—21no—
200C2×C47⋊C8—21no—
201C2×C47⋊C8—21no—
202C2×C47⋊C8—21no—
203C2×C47⋊C8—21no—
204C2×C47⋊C8—21no—
205C2×C47⋊C8—21no—
206C2×C47⋊C8—21no—
207C2×C47⋊C8—21no—
208C2×C47⋊C8—21no—
209C2×C47⋊C8—21no—
210C23×C47—21no—
211C23×C47—21no—
212C23×C47—21no—
213C23×C47—21no—
214C23×C47—21no—
215C23×C47—21no—
216C23×C47#13512yes—
217C23×C47C22×D411yescharacteristic
218Q8×C47—22no—
219Q8×C47—22no—
220Q8×C47—41no—
221Q8×C47—41no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—22no—
228C2×C4×C47—22no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—21no—
237C2×C4×C47—44no—
238C2×C4×C47—44no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—41no—
250C2×C4×C47—41no—
251C2×C4×C47—41no—
252C2×C4×C47—41no—
253C2×C4×C47#13712yes—
254C2×C4×C47#13712yes—
255C2×C4×C47C2×#4111yescharacteristic
256C2×C4×C47C2×#4111yescharacteristic
257C2×C4×C47C2×#4111yescharacteristic
258C2×C4×C47C2×#4111yescharacteristic
259C2×C4×C47H(5,𝔽2)11yescharacteristic
260C2×C4×C47H(5,𝔽2)11yescharacteristic
261C2×Dic47—22no—
262C2×Dic47—22no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273Dic94—22no—
274Dic94—22no—
275Dic94—41no—
276Dic94—41no—
277C47⋊C8—41no—
278C47⋊C8—41no—
279C47⋊C8—41no—
280C47⋊C8—41no—
281#29416—471nomaximal, Sylow
282C22×C47—22no—
283C22×C47—22no—
284C22×C47—22no—
285C22×C47—41no—
286C22×C47—41no—
287C22×C47—41no—
288C22×C47—41no—
289C22×C47#52511yescharacteristic
290C22×C47#52511yescharacteristic
291C22×C47#53411yescharacteristic
292C22×C47C2×#13511yescharacteristic
293C22×C47C2×#13711yescharacteristic
294C22×C47C2×#13711yescharacteristic
295C22×C47#57511yescharacteristic
296C4×C47—22no—
297C4×C47—22no—
298C4×C47—41no—
299C4×C47—42no—
300C4×C47—42no—
301C4×C47—41no—
302C4×C47—41no—
303C4×C47—81no—
304C4×C47—81no—
305C4×C47#55111yescharacteristic
306C4×C47#55311yescharacteristic
307C4×C47#55311yescharacteristic
308C4×C47#55911yescharacteristic
309Dic47—42no—
310Dic47—42no—
311Dic47—81no—
312Dic47—81no—
313C4×C22⋊Q8—471no—
314C4×C22⋊Q8D4711yescharacteristic
315#2075—471no—
316#2035—471no—
317#2035—471no—
318#2001—471no—
319#1927—471no—
320#1927—471no—
321#1920—471no—
322#1920—471no—
323#1878—471no—
324#1878—471no—
325#1863—471no—
326#1863—471no—
327#1817—471no—
328C2×C47—42no—
329C2×C47#260011yescharacteristic
330C2×C47#310811yescharacteristic
331C2×C47#310811yescharacteristic
332C2×C47#311611yescharacteristic
333C2×C47#311611yescharacteristic
334C2×C47#319111yescharacteristic
335C2×C47#319511yescharacteristic
336C22×Q16—471no—
337C2×C22⋊Q8—471no—
338C2×C22⋊Q8C2×D4711yescharacteristic
339C2×C4×Q8—471no—
340C2×C4×Q8C2×D4711yescharacteristic
341C22×C42C2×D4711yescharacteristic
342C2×#107—471no—
343C2×#106—471no—
344C2×#106—471no—
345C2×#103—471no—
346C2×#103—471no—
347C2×#103—471no—
348C2×#103—471no—
349C2×#103—471no—
350C2×#103—471no—
351C2×C22⋊C8—471no—
352C2×C22⋊C8—471no—
353#391—471no—
354#391C2×D4711yescharacteristic
355#389—471no—
356#389C2×D4711yescharacteristic
357#387—21no—
358#387—941no—
359#385—21no—
360#385—941no—
361#380—471no—
362#380C2×D4711yescharacteristic
363C4×C4⋊C4—21no—
364C4×C4⋊C4—471no—
365C4×C4⋊C4—941no—
366C4×C4⋊C4C2×D4711yescharacteristic
367C4×C22⋊C4—21no—
368C4×C22⋊C4—941no—
369#340—471no—
370#340—471no—
371#340—471no—
372#340—471no—
373#340—471no—
374#340—471no—
375#336—471no—
376#336—471no—
377#336—471no—
378#336—471no—
379C47#2941611yescharacteristic, complemented, Sylow
380C22×Q8—471no—
381C22×Q8C22×D4711yescharacteristic
382C23×C4C22⋊D4711yescharacteristic
383C23×C4C22⋊D4711yescharacteristic
384C23×C4C22×D4711yescharacteristic
385C2×Q16—942no—
386C2×Q16—942no—
387C2×Q16—1881no—
388C22×C8—471no—
389C22×C8—471no—
390C22×C8—471no—
391C22×C8—471no—
392C22⋊Q8—22no—
393C22⋊Q8—22no—
394C22⋊Q8—942no—
395C22⋊Q8—942no—
396C4×Q8—41no—
397C4×Q8—41no—
398C4×Q8—1881no—
399C4×Q8—1881no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—471no—
404C2×C4⋊C4—471no—
405C2×C4⋊C4—471no—
406C2×C4⋊C4—941no—
407C2×C4⋊C4—941no—
408C2×C4⋊C4—941no—
409C2×C4⋊C4C22×D4711yescharacteristic
410C2×C4⋊C4C22×D4711yescharacteristic
411C2×C4⋊C4C22×D4711yescharacteristic
412C2×C22⋊C4—21no—
413C2×C22⋊C4—21no—
414C2×C22⋊C4—941no—
415C2×C22⋊C4—941no—
416C2×C42—21no—
417C2×C42—21no—
418C2×C42—41no—
419C2×C42—41no—
420C2×C42—941no—
421C2×C42—941no—
422C2×C42C22⋊D4711yescharacteristic
423C2×C42C22⋊D4711yescharacteristic
424C2×C42C22×D4711yescharacteristic
425C2×C42C22×D4711yescharacteristic
426#107—1881no—
427#106—942no—
428#106—942no—
429#103—942no—
430#103—942no—
431#103—942no—
432#103—942no—
433#103—1881no—
434#103—1881no—
435C22⋊C8—1881no—
436C22⋊C8—1881no—
437#95—21no—
438#95—21no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—941no—
443#95—941no—
444#95—941no—
445#95—941no—
446#95—941no—
447C24C2×C22⋊D4711yescharacteristic
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8—21no—
453C2×Q8—472no—
454C2×Q8—941no—
455C2×Q8—941no—
456C2×Q8—941no—
457C2×Q8—941no—
458C2×Q8—941no—
459C2×Q8#1062535012yes—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—21no—
471C22×C4—21no—
472C22×C4—21no—
473C22×C4—41no—
474C22×C4—41no—
475C22×C4—41no—
476C22×C4—941no—
477C22×C4—941no—
478C22×C4—941no—
479C22×C4—941no—
480C22×C4C2×C22⋊D4711yescharacteristic
481C22×C4C2×C22⋊D4711yescharacteristic
482C22×C4C2×C22⋊D4711yescharacteristic
483C22×C4C2×C22⋊D4711yescharacteristic
484C22×C4C2×C22⋊D4711yescharacteristic
485C22×C4C23×D4711yescharacteristic
486Q16—1884no—
487C2×C8—941no—
488C2×C8—941no—
489C2×C8—941no—
490C2×C8—941no—
491C2×C8—941no—
492C2×C8—941no—
493C2×C8—941no—
494C2×C8—941no—
495C2×C8—941no—
496C2×C8—941no—
497C2×C8—941no—
498C2×C8—941no—
499C4⋊C4—24no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—21no—
504C4⋊C4—21no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—472no—
508C4⋊C4—944no—
509C4⋊C4—941no—
510C4⋊C4—941no—
511C4⋊C4—941no—
512C4⋊C4—941no—
513C4⋊C4—941no—
514C4⋊C4—1882no—
515C4⋊C4—1882no—
516C4⋊C4#1062535012yes—
517C22⋊C4—24no—
518C22⋊C4—42no—
519C22⋊C4—944no—
520C22⋊C4—1882no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—21no—
525C42—41no—
526C42—41no—
527C42—42no—
528C42—42no—
529C42—1882no—
530C42—1882no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23#1309404412yes—
538C23C22×C22⋊D4711yescharacteristic
539Q8—22no—
540Q8—22no—
541Q8—41no—
542Q8—41no—
543Q8—942no—
544Q8—942no—
545Q8—1881no—
546Q8—1881no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—44no—
563C2×C4—44no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—41no—
575C2×C4—41no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—942no—
579C2×C4—942no—
580C2×C4—1881no—
581C2×C4—1881no—
582C2×C4—1881no—
583C2×C4—1881no—
584C2×C4—1881no—
585C2×C4—1881no—
586C2×C4—1881no—
587C2×C4—1881no—
588C2×C4—1881no—
589C2×C4—1881no—
590C2×C4#1309405412yes—
591C2×C4#1309405512yes—
592C2×C4#41×D4711yescharacteristic
593C2×C4#41×D4711yescharacteristic
594C2×C4C2×#1062535011yescharacteristic
595C2×C4C2×#1062535011yescharacteristic
596C2×C4#1309406911yescharacteristic
597C2×C4#1309406911yescharacteristic
598C8—1881no—
599C8—1881no—
600C8—1881no—
601C8—1881no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—41no—
609C22#1568963011yescharacteristic
610C22#1568963411yescharacteristic
611C22#1568968211yescharacteristic
612C22C2×#1309404411yescharacteristic
613C22C2×#1309405411yescharacteristic
614C22C2×#1309405511yescharacteristic
615C22#1568992111yescharacteristic
616C4—22no—
617C4—22no—
618C4—42no—
619C4—41no—
620C4—42no—
621C4—41no—
622C4—41no—
623C4—81no—
624C4—81no—
625C4—3761no—
626C4—3761no—
627C4—1882no—
628C4—1882no—
629C4#1568980211yescharacteristic
630C4#1568982211yescharacteristic
631C4#1568982311yescharacteristic
632C4#1568985611yescharacteristic
633C2—42no—
634C2#2024897011yescharacteristic
635C2#2025346411yescharacteristic
636C2#2025346811yescharacteristic
637C2#2025354411yescharacteristic
638C2#2025355011yescharacteristic
639C2#2025417611yescharacteristic
640C2#2025420911yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC4×C22⋊Q8×C47C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenterC4×C22⋊Q8D47
hypocenterC47#29416
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
6. #20244022C2semidirect
7. #20244019C2semidirect
8. #20243973C2semidirect
9. #20243970C2semidirect
10. #20243732C2semidirect
11. #20243729C2semidirect
12. #20243642C2semidirect
13. #20243641C2semidirect
379. C47#29416semidirect

Maximal subgroups

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

subgroupindex
C4×#130940032
#202448142
#202445842
#202445812
#202443962
#202440222
#202440192
#202439732
#202439702
#202437322
#202437292
#202436422
#202436412
#202434562
C4×C22⋊Q8×C472
#2941647