Finite Group Databasebeta

← #26319807

Subgroups — #26319807

Order 12032.

766 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20251727C211yesmaximal, characteristic
2#20244819C211yesmaximal, characteristic
3#20244591C211yesmaximal, characteristic
4#20244588C211yesmaximal, characteristic
5#20244396C211yesmaximal, characteristic
6#20244248C211yesmaximal, characteristic, complemented
7#20244118C211yesmaximal, characteristic, complemented
8#20244067C211yesmaximal, characteristic, complemented
9#20244063C211yesmaximal, characteristic, complemented
10#20244029C211yesmaximal, characteristic, complemented
11#20243980C211yesmaximal, characteristic, complemented
12#20243729C211yesmaximal, characteristic, complemented
13#20243642C211yesmaximal, characteristic, complemented
14C4×C22⋊(C47⋊C8)C211yesmaximal, characteristic
15C4×C22⋊Q8×C47C211yesmaximal, characteristic
16C22×#10625337C2211yescharacteristic
17C2×#13094003C2211yescharacteristic
18C2×#13093957C2211yescharacteristic
19C2×#13093957C2211yescharacteristic
20C2×#13093957C2211yescharacteristic
21C2×#13093948C2211yescharacteristic
22C2×#13093948C2211yescharacteristic
23C2×#13093948C2211yescharacteristic
24C2×#13093947C2211yescharacteristic
25C2×#13093947C2211yescharacteristic
26C2×#13093947C2211yescharacteristic
27C2×C22⋊(C47⋊C8)C2211yescharacteristic
28C2×C22⋊(C47⋊C8)C2211yescharacteristic
29C2×C4×C47⋊C8C2211yescharacteristic
30C2×C4×C47⋊C8C2211yescharacteristic
31#15689029C2211yescharacteristic
32#15689029C2211yescharacteristic
33#15689021C2211yescharacteristic
34#15689021C2211yescharacteristic
35#15689007—21no—
36#15689007—21no—
37#15688996—21no—
38#15688986—21no—
39#15688983C2211yescharacteristic
40#15688885C2211yescharacteristic
41#15688885C2211yescharacteristic
42#15688885C2211yescharacteristic
43#15688885C2211yescharacteristic
44#15688885C2211yescharacteristic
45#15688885C2211yescharacteristic
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—41no—
68#13094003—41no—
69C2×C4⋊Dic47C2311yescharacteristic
70C2×C4⋊Dic47C2311yescharacteristic
71C2×C4⋊Dic47C2311yescharacteristic
72C2×Dic47⋊C4—21no—
73C2×Dic47⋊C4—21no—
74C2×Dic47⋊C4—21no—
75C2×C22⋊Dic47—21no—
76C2×C22⋊Dic47—21no—
77#13093957—22no—
78#13093957—22no—
79#13093957—41no—
80#13093948—22no—
81#13093948—22no—
82#13093948—41no—
83#13093947—22no—
84#13093947—22no—
85#13093947—41no—
86C22⋊(C47⋊C8)—22no—
87C22⋊(C47⋊C8)—22no—
88C4×C47⋊C8—41no—
89C4×C47⋊C8—41no—
90C47⋊#95—21no—
91C47⋊#95—21no—
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—21no—
134C22×C4×C47—21no—
135C22×C4×C47—22no—
136C22×C4×C47—21no—
137C22×C4×C47—21no—
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—41no—
145C22×C4×C47—41no—
146C22×C4×C47—41no—
147C22×C4×C47—41no—
148C22×C4×C47Q1614yescomplemented
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—21no—
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—42no—
185C22⋊Dic47—42no—
186#10625337—44no—
187C4⋊Dic47—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47—21no—
193Dic47⋊C4—42no—
194Dic47⋊C4—42no—
195Dic47⋊C4—42no—
196C2×C47⋊C8—21no—
197C2×C47⋊C8—21no—
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#4112yes—
207C2×C47⋊C8#4112yes—
208C23×C47—21no—
209C23×C47—21no—
210C23×C47—21no—
211C23×C47—21no—
212C23×C47—21no—
213C23×C47—21no—
214C23×C47C2×Q1612yes—
215C23×C47C22×D411yescharacteristic
216Q8×C47—22no—
217Q8×C47—22no—
218Q8×C47—41no—
219Q8×C47—41no—
220C2×C4×C47—21no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—22no—
227C2×C4×C47—22no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—21no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—44no—
235C2×C4×C47—44no—
236C2×C4×C47—41no—
237C2×C4×C47—41no—
238C2×C4×C47—41no—
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×C47C2×Q1612yes—
251C2×C4×C47C2×Q1612yes—
252C2×C4×C47#13512yes—
253C2×C4×C47C2×#4111yescharacteristic
254C2×C4×C47C2×#4111yescharacteristic
255C2×C4×C47C2×#4111yescharacteristic
256C2×C4×C47C2×#4111yescharacteristic
257C2×C4×C47H(5,𝔽2)11yescharacteristic
258C2×C4×C47H(5,𝔽2)11yescharacteristic
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—41no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267C2×Dic47—41no—
268C2×Dic47—41no—
269C2×Dic47—41no—
270C2×Dic47—41no—
271Dic94—41no—
272Dic94—41no—
273Dic94—41no—
274Dic94—41no—
275C47⋊C8—22no—
276C47⋊C8—22no—
277C47⋊C8—41no—
278C47⋊C8—41no—
279#29462—471nomaximal, Sylow
280C22×C47—22no—
281C22×C47—22no—
282C22×C47—22no—
283C22×C47—41no—
284C22×C47—41no—
285C22×C47—41no—
286C22×C47—41no—
287C22×C47#52511yescharacteristic
288C22×C47#53211yescharacteristic
289C22×C47#53411yescharacteristic
290C22×C47C22×Q1611yescharacteristic
291C22×C47C2×#13511yescharacteristic
292C22×C47#57511yescharacteristic
293C22×C47#57511yescharacteristic
294C4×C47—22no—
295C4×C47—22no—
296C4×C47—22no—
297C4×C47—42no—
298C4×C47—41no—
299C4×C47—42no—
300C4×C47—41no—
301C4×C47—81no—
302C4×C47—81no—
303C4×C47#55111yescharacteristic
304C4×C47#55311yescharacteristic
305C4×C47#55311yescharacteristic
306C4×C47#55911yescharacteristic
307Dic47—81no—
308Dic47—81no—
309Dic47—81no—
310Dic47—81no—
311#2855—471no—
312C4×C22⋊Q8D4711yescharacteristic
313#2076—471no—
314#2036—471no—
315#2036—471no—
316#2001—471no—
317#1969—471no—
318#1942—471no—
319#1933—471no—
320#1933—471no—
321#1928—471no—
322#1921—471no—
323#1878—471no—
324#1863—471no—
325C4×C22⋊C8—471no—
326C2×C47—42no—
327C2×C47#260211yescharacteristic
328C2×C47#310211yescharacteristic
329C2×C47#310411yescharacteristic
330C2×C47#317011yescharacteristic
331C2×C47#318311yescharacteristic
332C2×C47#318911yescharacteristic
333C2×C47#319111yescharacteristic
334C22×Q16—471no—
335C2×C22⋊Q8—471no—
336C2×C22⋊Q8C2×D4711yescharacteristic
337C2×C4×Q8C2×D4711yescharacteristic
338C22×C42C2×D4711yescharacteristic
339C2×#107—471no—
340C2×#107—471no—
341C2×#107—471no—
342C2×#103—471no—
343C2×#103—471no—
344C2×#103—471no—
345C2×#103—471no—
346C2×#103—471no—
347C2×#103—471no—
348C2×C22⋊C8—471no—
349C2×C22⋊C8—471no—
350C2×C4×C8—471no—
351C2×C4×C8—471no—
352#391—471no—
353#391—471no—
354#391C2×D4711yescharacteristic
355#389—471no—
356#389—471no—
357#389C2×D4711yescharacteristic
358#387—21no—
359#387—941no—
360#387—941no—
361#385—21no—
362#385—941no—
363#381—941no—
364#380—471no—
365#380C2×D4711yescharacteristic
366C4×C4⋊C4—21no—
367C4×C4⋊C4C2×D4711yescharacteristic
368C4×C22⋊C4—21no—
369#340—471no—
370#340—471no—
371#340—471no—
372#340—471no—
373#340—471no—
374#340—471no—
375#336—471no—
376#336—471no—
377C47#2946211yescharacteristic, complemented, Sylow
378C22×Q8—471no—
379C22×Q8C22×D4711yescharacteristic
380C23×C4C22⋊D4711yescharacteristic
381C23×C4C22⋊D4711yescharacteristic
382C23×C4C22×D4711yescharacteristic
383C2×Q16—942no—
384C2×Q16—942no—
385C2×Q16—1881no—
386C22×C8—471no—
387C22×C8—471no—
388C22×C8—471no—
389C22×C8—471no—
390C22⋊Q8—22no—
391C22⋊Q8—22no—
392C22⋊Q8—1881no—
393C22⋊Q8—1881no—
394C4×Q8—41no—
395C4×Q8—41no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—471no—
400C2×C4⋊C4—471no—
401C2×C4⋊C4—471no—
402C2×C4⋊C4—941no—
403C2×C4⋊C4—941no—
404C2×C4⋊C4—941no—
405C2×C4⋊C4C22×D4711yescharacteristic
406C2×C4⋊C4C22×D4711yescharacteristic
407C2×C4⋊C4C22×D4711yescharacteristic
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C22⋊C4—941no—
411C2×C22⋊C4—941no—
412C2×C42—21no—
413C2×C42—21no—
414C2×C42—41no—
415C2×C42—41no—
416C2×C42C22⋊D4711yescharacteristic
417C2×C42C22⋊D4711yescharacteristic
418C2×C42C22×D4711yescharacteristic
419C2×C42C22×D4711yescharacteristic
420#107—942no—
421#107—942no—
422#107—1881no—
423#103—942no—
424#103—942no—
425#103—942no—
426#103—942no—
427#103—1881no—
428#103—1881no—
429C22⋊C8—942no—
430C22⋊C8—942no—
431C4×C8—1881no—
432C4×C8—1881no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—21no—
438#95—941no—
439#95—941no—
440#95—941no—
441#95—941no—
442#95—941no—
443#95—941no—
444#95—941no—
445C24C2×C22⋊D4711yescharacteristic
446C2×Q8—21no—
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—941no—
452C2×Q8—941no—
453C2×Q8—941no—
454C2×Q8—941no—
455C2×Q8—941no—
456C2×Q8—941no—
457C2×Q8#1062535012yes—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—22no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—21no—
469C22×C4—41no—
470C22×C4—41no—
471C22×C4—41no—
472C22×C4—41no—
473C22×C4—941no—
474C22×C4—941no—
475C22×C4—941no—
476C22×C4—941no—
477C22×C4#1062533714yescomplemented
478C22×C4C2×C22⋊D4711yescharacteristic
479C22×C4C2×C22⋊D4711yescharacteristic
480C22×C4C2×C22⋊D4711yescharacteristic
481C22×C4C2×C22⋊D4711yescharacteristic
482C22×C4C2×C22⋊D4711yescharacteristic
483C22×C4C23×D4711yescharacteristic
484Q16—1884no—
485C2×C8—472no—
486C2×C8—472no—
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—
497C4⋊C4—24no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—941no—
506C4⋊C4—941no—
507C4⋊C4—941no—
508C4⋊C4—941no—
509C4⋊C4—941no—
510C4⋊C4—941no—
511C4⋊C4—1882no—
512C4⋊C4—1882no—
513C4⋊C4—1882no—
514C4⋊C4#1062535012yes—
515C22⋊C4—24no—
516C22⋊C4—42no—
517C22⋊C4—1882no—
518C22⋊C4—1882no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—41no—
524C42—41no—
525C42—42no—
526C42—42no—
527C23—21no—
528C23—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23C2×#1062533712yes—
534C23C22×C22⋊D4711yescharacteristic
535Q8—22no—
536Q8—22no—
537Q8—41no—
538Q8—41no—
539Q8—1881no—
540Q8—1881no—
541Q8—1881no—
542Q8—1881no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—44no—
558C2×C4—44no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
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—1881no—
574C2×C4—1881no—
575C2×C4—1881no—
576C2×C4—1881no—
577C2×C4—1881no—
578C2×C4—1881no—
579C2×C4—1881no—
580C2×C4—1881no—
581C2×C4—1881no—
582C2×C4—1881no—
583C2×C4—1881no—
584C2×C4—1881no—
585C2×C4C2×#1062533712yes—
586C2×C4C2×#1062533712yes—
587C2×C4#1309404412yes—
588C2×C4#41×D4711yescharacteristic
589C2×C4#41×D4711yescharacteristic
590C2×C4C2×#1062535011yescharacteristic
591C2×C4C2×#1062535011yescharacteristic
592C2×C4#1309406911yescharacteristic
593C2×C4#1309406911yescharacteristic
594C8—942no—
595C8—942no—
596C8—1881no—
597C8—1881no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22#1568963411yescharacteristic
606C22#1568966911yescharacteristic
607C22#1568968211yescharacteristic
608C22C22×#1062533711yescharacteristic
609C22C2×#1309404411yescharacteristic
610C22#1568992111yescharacteristic
611C22#1568992111yescharacteristic
612C4—22no—
613C4—22no—
614C4—22no—
615C4—41no—
616C4—42no—
617C4—42no—
618C4—41no—
619C4—81no—
620C4—81no—
621C4—3761no—
622C4—3761no—
623C4—3761no—
624C4—3761no—
625C4#1568980211yescharacteristic
626C4#1568982211yescharacteristic
627C4#1568982311yescharacteristic
628C4#1568985611yescharacteristic
629C2—42no—
630C2#2024900011yescharacteristic
631C2#2025340611yescharacteristic
632C2#2025342711yescharacteristic
633C2#2025397011yescharacteristic
634C2#2025411111yescharacteristic
635C2#2025416011yescharacteristic
636C2#2025417611yescharacteristic
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#29462
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. #20244248C2semidirect
7. #20244118C2semidirect
8. #20244067C2semidirect
9. #20244063C2semidirect
10. #20244029C2semidirect
11. #20243980C2semidirect
12. #20243729C2semidirect
13. #20243642C2semidirect
148. C22×C4×C47Q16semidirect
377. C47#29462semidirect
477. C22×C4#10625337semidirect

Maximal subgroups

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

subgroupindex
#202517272
#202448192
#202445912
#202445882
#202443962
#202442482
#202441182
#202440672
#202440632
#202440292
#202439802
#202437292
#202436422
C4×C22⋊(C47⋊C8)2
C4×C22⋊Q8×C472
#2946247