Finite Group Databasebeta

← #26284393

Subgroups — #26284393

Order 12032.

764 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#20247762C211yesmaximal, characteristic
2#20244710C211yesmaximal, characteristic, complemented
3#20244588C211yesmaximal, characteristic
4#20244529C211yesmaximal, characteristic, complemented
5#20244396C211yesmaximal, characteristic
6#20244336C211yesmaximal, characteristic, complemented
7#20244063C211yesmaximal, characteristic, complemented
8#20244029C211yesmaximal, characteristic, complemented
9#20243996C211yesmaximal, characteristic
10#20243977C211yesmaximal, characteristic, complemented
11#20243729C211yesmaximal, characteristic, complemented
12#20243695C211yesmaximal, characteristic
13#20243478C211yesmaximal, characteristic
14C4×#13093948C211yesmaximal, characteristic, complemented
15C47×#2360C211yesmaximal, characteristic
16C22×#10625337C2211yescharacteristic
17C2×#13094024C2211yescharacteristic
18C2×#13094003C2211yescharacteristic
19C2×#13093957C2211yescharacteristic
20C2×C4⋊(C47⋊C8)C2211yescharacteristic
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
30#15689065—21no—
31#15689035—21no—
32#15689029C2211yescharacteristic
33#15689029C2211yescharacteristic
34#15689027—21no—
35#15689016—21no—
36#15688983C2211yescharacteristic
37C4×C4⋊Dic47C2211yescharacteristic
38#15688885C2211yescharacteristic
39#15688885C2211yescharacteristic
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×C4⋊C4×C47C2211yescharacteristic
51C47×#391C2211yescharacteristic
52C47×#389—21no—
53C47×#389C2211yescharacteristic
54C47×#387—21no—
55C47×#385—21no—
56C4×C4⋊C4×C47C2211yescharacteristic
57C4×C22⋊C4×C47—21no—
58C4×C22⋊C4×C47C2211yescharacteristic
59C22×Dic94—21no—
60C22×Dic94C2311yescharacteristic
61C2×#10625337—22no—
62C2×#10625337—41no—
63C2×#10625337—41no—
64C22×C47⋊C8C2311yescharacteristic
65C22×C47⋊C8C2311yescharacteristic
66C22×C47⋊C8C2311yescharacteristic
67C22×C47⋊C8C2311yescharacteristic
68#13094024—22no—
69#13094024—41no—
70#13094003—22no—
71#13094003—41no—
72C2×C4⋊Dic47—21no—
73C2×C4⋊Dic47C2311yescharacteristic
74C2×C4⋊Dic47C2311yescharacteristic
75C2×C4⋊Dic47C2311yescharacteristic
76C2×Dic47⋊C4—21no—
77C2×C22⋊Dic47—21no—
78C2×C22⋊Dic47—21no—
79C2×C4×Dic47—21no—
80#13093957—22no—
81C4⋊(C47⋊C8)—41no—
82#13093948—22no—
83#13093948—22no—
84#13093948—41no—
85#13093947—22no—
86#13093947—22no—
87#13093947—41no—
88C22⋊(C47⋊C8)—41no—
89C22⋊(C47⋊C8)—41no—
90C4×C47⋊C8—41no—
91C47⋊#95—21no—
92C47⋊#95—21no—
93C47⋊#95—21no—
94C47⋊#95—21no—
95C47⋊#95—21no—
96C47⋊#95—21no—
97C22×Q8×C47C2311yescharacteristic
98C23×C4×C47—21no—
99C23×C4×C47C2311yescharacteristic
100C22⋊Q8×C47—22no—
101C22⋊Q8×C47—22no—
102C4×Q8×C47—22no—
103C4×Q8×C47—41no—
104C2×C4⋊C4×C47—21no—
105C2×C4⋊C4×C47—22no—
106C2×C4⋊C4×C47—21no—
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47—21no—
109C2×C4⋊C4×C47—41no—
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C22⋊C4×C47—21no—
114C2×C22⋊C4×C47D411yescharacteristic
115C2×C22⋊C4×C47D411yescharacteristic
116C2×C42×C47—21no—
117C2×C42×C47—21no—
118C2×C42×C47C2311yescharacteristic
119C2×C42×C47C2311yescharacteristic
120#95×C47—21no—
121#95×C47—21no—
122#95×C47—21no—
123#95×C47—21no—
124#95×C47D411yescharacteristic
125#95×C47D411yescharacteristic
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—21no—
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—42no—
144C22×C4×C47—42no—
145C22×C4×C47—81no—
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2411yescharacteristic
152C22×Dic47—21no—
153C22×Dic47—21no—
154C22×Dic47—21no—
155C22×Dic47—21no—
156C2×Dic94—21no—
157C2×Dic94—21no—
158C2×Dic94—21no—
159C2×Dic94—21no—
160C2×Dic94—21no—
161C2×Dic94—21no—
162C2×Dic94—42no—
163C2×Dic94—42no—
164C2×Dic94—81no—
165C4⋊C4×C47—24no—
166C4⋊C4×C47—21no—
167C4⋊C4×C47—21no—
168C4⋊C4×C47—21no—
169C4⋊C4×C47—21no—
170C4⋊C4×C47—42no—
171C4⋊C4×C47—42no—
172C4⋊C4×C47—81no—
173C4⋊C4×C47#4112yes—
174C4⋊C4×C47#4112yes—
175C22⋊C4×C47—24no—
176C22⋊C4×C47—41no—
177C22⋊C4×C47Q1614yescomplemented
178C42×C47—21no—
179C42×C47—21no—
180C42×C47—21no—
181C42×C47—21no—
182C42×C47—42no—
183C42×C47—81no—
184C22⋊Dic47—42no—
185C22⋊Dic47—81no—
186#10625337—44no—
187C4⋊Dic47—21no—
188C4⋊Dic47—21no—
189C4⋊Dic47—21no—
190C4⋊Dic47—21no—
191C4⋊Dic47—21no—
192C4⋊Dic47—42no—
193C4⋊Dic47#4112yes—
194Dic47⋊C4—42no—
195C4×Dic47—81no—
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—21no—
207C2×C47⋊C8—21no—
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—22no—
225C2×C4×C47—22no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
228C2×C4×C47—21no—
229C2×C4×C47—21no—
230C2×C4×C47—22no—
231C2×C4×C47—21no—
232C2×C4×C47—21no—
233C2×C4×C47—21no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
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—84no—
250C2×C4×C47C2×Q1612yes—
251C2×C4×C47C2×Q1612yes—
252C2×C4×C47C2×#4111yescharacteristic
253C2×C4×C47C2×#4111yescharacteristic
254C2×C4×C47C2×#4111yescharacteristic
255C2×C4×C47C2×#4111yescharacteristic
256C2×C4×C47H(5,𝔽2)11yescharacteristic
257C2×C4×C47H(5,𝔽2)11yescharacteristic
258C2×Dic47—22no—
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—
270Dic94—41no—
271Dic94—41no—
272Dic94—41no—
273Dic94—41no—
274Dic94—84no—
275C47⋊C8—41no—
276C47⋊C8—41no—
277C47⋊C8—41no—
278C47⋊C8—41no—
279#26726—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#54611yescharacteristic
289C22×C47#54611yescharacteristic
290C22×C47C22×Q1611yescharacteristic
291C22×C47#57511yescharacteristic
292C22×C47#57711yescharacteristic
293C22×C47#57711yescharacteristic
294C4×C47—22no—
295C4×C47—22no—
296C4×C47—42no—
297C4×C47—41no—
298C4×C47—42no—
299C4×C47—41no—
300C4×C47—42no—
301C4×C47—41no—
302C4×C47—81no—
303C4×C47#52911yescharacteristic
304C4×C47#53811yescharacteristic
305C4×C47#55511yescharacteristic
306C4×C47#55911yescharacteristic
307Dic47—42no—
308Dic47—81no—
309Dic47—81no—
310Dic47—81no—
311#2492—471no—
312#2360D4711yescharacteristic
313#2056—471no—
314#2036—471no—
315#2024—471no—
316#2001—471no—
317#1987—471no—
318#1933—471no—
319#1928—471no—
320#1923—471no—
321#1921—471no—
322#1878—471no—
323#1872—471no—
324#1824—471no—
325C4×#103—471no—
326C2×C47—42no—
327C2×C47#245911yescharacteristic
328C2×C47#310411yescharacteristic
329C2×C47#312511yescharacteristic
330C2×C47#329911yescharacteristic
331C2×C47#329911yescharacteristic
332C2×C47#331111yescharacteristic
333C2×C47#332311yescharacteristic
334C22×Q16—471no—
335C2×C4⋊Q8—471no—
336C2×C22⋊Q8—471no—
337C2×C22⋊Q8C2×D4711yescharacteristic
338C2×C4×Q8C2×D4711yescharacteristic
339C22×C4⋊C4C2×D4711yescharacteristic
340C2×#107—471no—
341C2×C4⋊C8—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—
351#400—941no—
352#393—941no—
353#391—471no—
354#391—471no—
355#391—941no—
356#391C2×D4711yescharacteristic
357#389—21no—
358#389C2×D4711yescharacteristic
359#388—941no—
360#387—21no—
361#385—21no—
362#380—471no—
363C4×C4⋊C4—471no—
364C4×C4⋊C4C2×D4711yescharacteristic
365C4×C22⋊C4—21no—
366C4×C22⋊C4C2×D4711yescharacteristic
367#340—471no—
368#340—471no—
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#2672611yescharacteristic, complemented, Sylow
378C22×Q8—471no—
379C22×Q8—941no—
380C22×Q8C22×D4711yescharacteristic
381C23×C4—21no—
382C23×C4C22×D4711yescharacteristic
383C2×Q16—942no—
384C2×Q16—1881no—
385C2×Q16—1881no—
386C22×C8—471no—
387C22×C8—471no—
388C22×C8—471no—
389C22×C8—471no—
390C4⋊Q8—942no—
391C4⋊Q8—1881no—
392C22⋊Q8—22no—
393C22⋊Q8—22no—
394C22⋊Q8—942no—
395C22⋊Q8—1881no—
396C4×Q8—22no—
397C4×Q8—41no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—22no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—41no—
404C2×C4⋊C4—471no—
405C2×C4⋊C4—471no—
406C2×C4⋊C4—471no—
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—941no—
414C2×C22⋊C4—941no—
415C2×C22⋊C4C22⋊D4711yescharacteristic
416C2×C22⋊C4C22⋊D4711yescharacteristic
417C2×C42—21no—
418C2×C42—21no—
419C2×C42—941no—
420C2×C42C22×D4711yescharacteristic
421C2×C42C22×D4711yescharacteristic
422#107—942no—
423C4⋊C8—1881no—
424#103—942no—
425#103—942no—
426#103—942no—
427#103—942no—
428#103—1881no—
429#103—1881no—
430C22⋊C8—1881no—
431C22⋊C8—1881no—
432C4×C8—1881no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—21no—
437#95—941no—
438#95—941no—
439#95—941no—
440#95—941no—
441#95—941no—
442#95—941no—
443#95C22⋊D4711yescharacteristic
444#95C22⋊D4711yescharacteristic
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—3761no—
458C2×Q8—1882no—
459C2×Q8—1882no—
460C2×Q8#1062535012yes—
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—21no—
470C22×C4—21no—
471C22×C4—42no—
472C22×C4—42no—
473C22×C4—81no—
474C22×C4—941no—
475C22×C4—941no—
476C22×C4—941no—
477C22×C4—941no—
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—941no—
486C2×C8—941no—
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—42no—
503C4⋊C4—42no—
504C4⋊C4—81no—
505C4⋊C4—472no—
506C4⋊C4—941no—
507C4⋊C4—941no—
508C4⋊C4—941no—
509C4⋊C4—941no—
510C4⋊C4—941no—
511C4⋊C4—1882no—
512C4⋊C4—1882no—
513C4⋊C4#1062535012yes—
514C4⋊C4#1062535212yes—
515C22⋊C4—24no—
516C22⋊C4—41no—
517C22⋊C4—3761no—
518C22⋊C4—1882no—
519C22⋊C4#1062533714yescomplemented
520C42—21no—
521C42—21no—
522C42—21no—
523C42—21no—
524C42—42no—
525C42—81no—
526C42—3761no—
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—3764no—
540Q8—1881no—
541Q8—1881no—
542Q8—1881no—
543Q8—1881no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—22no—
552C2×C4—21no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
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—84no—
574C2×C4—942no—
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×C4—1881no—
586C2×C4C2×#1062533712yes—
587C2×C4C2×#1062533712yes—
588C2×C4#41×D4711yescharacteristic
589C2×C4C2×#1062535211yescharacteristic
590C2×C4C2×#1062535011yescharacteristic
591C2×C4C2×#1062535011yescharacteristic
592C2×C4#1309406911yescharacteristic
593C2×C4#1309407011yescharacteristic
594C8—1881no—
595C8—1881no—
596C8—1881no—
597C8—1881no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22#1568963411yescharacteristic
606C22#1568976111yescharacteristic
607C22#1568976711yescharacteristic
608C22C22×#1062533711yescharacteristic
609C22#1568992111yescharacteristic
610C22#1568993111yescharacteristic
611C22#1568993411yescharacteristic
612C4—22no—
613C4—22no—
614C4—41no—
615C4—42no—
616C4—41no—
617C4—41no—
618C4—42no—
619C4—42no—
620C4—81no—
621C4—3761no—
622C4—3761no—
623C4—3761no—
624C4—1882no—
625C4#1568965311yescharacteristic
626C4#1568970611yescharacteristic
627C4#1568983811yescharacteristic
628C4#1568985611yescharacteristic
629C2—42no—
630C2#2024731911yescharacteristic
631C2#2025342711yescharacteristic
632C2#2025366511yescharacteristic
633C2#2025525811yescharacteristic
634C2#2025526411yescharacteristic
635C2#2025542411yescharacteristic
636C2#2025559811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2360C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenter#2360D47
hypocenterC47#26726
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
2. #20244710C2semidirect
4. #20244529C2semidirect
6. #20244336C2semidirect
7. #20244063C2semidirect
8. #20244029C2semidirect
10. #20243977C2semidirect
11. #20243729C2semidirect
14. C4×#13093948C2semidirect
177. C22⋊C4×C47Q16semidirect
377. C47#26726semidirect
519. C22⋊C4#10625337semidirect

Maximal subgroups

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

subgroupindex
#202477622
#202447102
#202445882
#202445292
#202443962
#202443362
#202440632
#202440292
#202439962
#202439772
#202437292
#202436952
#202434782
C4×#130939482
C47×#23602
#2672647