Finite Group Databasebeta

← #26319282

Subgroups — #26319282

Order 12032.

724 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20246070C211yesmaximal, characteristic
2#20244814C211yesmaximal, characteristic
3#20244591C211yesmaximal, characteristic
4#20244588C211yesmaximal, characteristic
5#20244396C211yesmaximal, characteristic
6#20244029C211yesmaximal, characteristic, complemented
7#20244026C211yesmaximal, characteristic, complemented
8#20243959C211yesmaximal, characteristic, complemented
9#20243956C211yesmaximal, characteristic, complemented
10#20243732C211yesmaximal, characteristic, complemented
11#20243729C211yesmaximal, characteristic, complemented
12#20243642C211yesmaximal, characteristic, complemented
13#20243641C211yesmaximal, characteristic, complemented
14#20243459C211yesmaximal, characteristic
15C47×#2323C211yesmaximal, characteristic
16C22×#10625337C2211yescharacteristic
17C2×#13094003C2211yescharacteristic
18C2×C4×Dic94C2211yescharacteristic
19C2×#13093957C2211yescharacteristic
20C2×#13093957C2211yescharacteristic
21C2×#13093957C2211yescharacteristic
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#15689018—21no—
33#15689017—21no—
34#15689011—21no—
35#15689004—21no—
36#15688983C2211yescharacteristic
37C4×C4⋊Dic47C2211yescharacteristic
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
50C2×#117×C47C2211yescharacteristic
51C47×#391C2211yescharacteristic
52C47×#389—21no—
53C47×#389C2211yescharacteristic
54C47×#388—21no—
55C47×#388—21no—
56C47×#387—21no—
57C47×#380C2211yescharacteristic
58C4×C4⋊C4×C47C2211yescharacteristic
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—22no—
82#13093957—22no—
83#13093957—41no—
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×C47C2311yescharacteristic
99C22⋊Q8×C47—22no—
100C22⋊Q8×C47—22no—
101C4×Q8×C47—41no—
102C4×Q8×C47—41no—
103#117×C47—41no—
104#117×C47—41no—
105C2×C4⋊C4×C47—21no—
106C2×C4⋊C4×C47—21no—
107C2×C4⋊C4×C47—21no—
108C2×C4⋊C4×C47D411yescharacteristic
109C2×C4⋊C4×C47D411yescharacteristic
110C2×C4⋊C4×C47C2311yescharacteristic
111C2×C4⋊C4×C47C2311yescharacteristic
112C2×C4⋊C4×C47C2311yescharacteristic
113C2×C22⋊C4×C47—21no—
114C2×C22⋊C4×C47—21no—
115C2×C22⋊C4×C47D411yescharacteristic
116C2×C22⋊C4×C47D411yescharacteristic
117C2×C42×C47—21no—
118C2×C42×C47—21no—
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—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×C47C2×D411yescharacteristic
144C22×C4×C47C2×D411yescharacteristic
145C22×C4×C47C2×D411yescharacteristic
146C22×C4×C47C2×D411yescharacteristic
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2411yescharacteristic
149C22×Dic47—21no—
150C22×Dic47—21no—
151C22×Dic47—21no—
152C22×Dic47—21no—
153C2×Dic94—21no—
154C2×Dic94—21no—
155C2×Dic94—21no—
156C2×Dic94—21no—
157C2×Dic94—21no—
158C2×Dic94#4112yes—
159C4⋊C4×C47—24no—
160C4⋊C4×C47—21no—
161C4⋊C4×C47—21no—
162C4⋊C4×C47—21no—
163C4⋊C4×C47—21no—
164C4⋊C4×C47—21no—
165C4⋊C4×C47—41no—
166C4⋊C4×C47—41no—
167C4⋊C4×C47—42no—
168C4⋊C4×C47—81no—
169C4⋊C4×C47#4112yes—
170C22⋊C4×C47—24no—
171C22⋊C4×C47—41no—
172C22⋊C4×C47—41no—
173C22⋊C4×C47—81no—
174C42×C47—21no—
175C42×C47—21no—
176C42×C47—21no—
177C42×C47—21no—
178C42×C47—42no—
179C42×C47—81no—
180C22⋊Dic47—24no—
181C22⋊Dic47—81no—
182#10625337—44no—
183C4⋊Dic47—21no—
184C4⋊Dic47—21no—
185C4⋊Dic47—21no—
186C4⋊Dic47—21no—
187C4⋊Dic47—21no—
188C4⋊Dic47#4112yes—
189Dic47⋊C4—24no—
190Dic47⋊C4—42no—
191Dic47⋊C4—81no—
192C4×Dic47—42no—
193C4×Dic47—81no—
194C2×C47⋊C8—21no—
195C2×C47⋊C8—21no—
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—
206C23×C47—21no—
207C23×C47—21no—
208C23×C47—21no—
209C23×C47—21no—
210C23×C47—21no—
211C23×C47—21no—
212C23×C47—21no—
213C23×C47C22×D411yescharacteristic
214Q8×C47—22no—
215Q8×C47—22no—
216Q8×C47—41no—
217Q8×C47—41no—
218C2×C4×C47—21no—
219C2×C4×C47—22no—
220C2×C4×C47—22no—
221C2×C4×C47—21no—
222C2×C4×C47—21no—
223C2×C4×C47—21no—
224C2×C4×C47—21no—
225C2×C4×C47—21no—
226C2×C4×C47—21no—
227C2×C4×C47—21no—
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—21no—
235C2×C4×C47—41no—
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×C47C2×#4111yescharacteristic
250C2×C4×C47C2×#4111yescharacteristic
251C2×C4×C47C2×#4111yescharacteristic
252C2×C4×C47C2×#4111yescharacteristic
253C2×C4×C47H(5,𝔽2)11yescharacteristic
254C2×C4×C47H(5,𝔽2)11yescharacteristic
255C2×Dic47—22no—
256C2×Dic47—22no—
257C2×Dic47—41no—
258C2×Dic47—41no—
259C2×Dic47—41no—
260C2×Dic47—41no—
261C2×Dic47—41no—
262C2×Dic47—41no—
263C2×Dic47—41no—
264C2×Dic47—41no—
265C2×Dic47—41no—
266C2×Dic47—41no—
267Dic94—22no—
268Dic94—22no—
269Dic94—41no—
270Dic94—41no—
271C47⋊C8—41no—
272C47⋊C8—41no—
273C47⋊C8—41no—
274C47⋊C8—41no—
275#29427—471nomaximal, Sylow
276C22×C47—41no—
277C22×C47—41no—
278C22×C47—41no—
279C22×C47—41no—
280C22×C47—41no—
281C22×C47—41no—
282C22×C47—41no—
283C22×C47#52511yescharacteristic
284C22×C47#52511yescharacteristic
285C22×C47#53411yescharacteristic
286C22×C47#57611yescharacteristic
287C22×C47#57811yescharacteristic
288C22×C47#57811yescharacteristic
289C22×C47#57811yescharacteristic
290C4×C47—41no—
291C4×C47—42no—
292C4×C47—42no—
293C4×C47—41no—
294C4×C47—41no—
295C4×C47—41no—
296C4×C47—41no—
297C4×C47—81no—
298C4×C47—81no—
299C4×C47#55111yescharacteristic
300C4×C47#55311yescharacteristic
301C4×C47#55311yescharacteristic
302C4×C47#55911yescharacteristic
303Dic47—42no—
304Dic47—42no—
305Dic47—81no—
306Dic47—81no—
307#2323—471no—
308#2323D4711yescharacteristic
309#2075—471no—
310#2036—471no—
311#2036—471no—
312#2001—471no—
313#1928—471no—
314#1928—471no—
315#1918—471no—
316#1918—471no—
317#1878—471no—
318#1878—471no—
319#1863—471no—
320#1863—471no—
321#1818—471no—
322C2×C47—81no—
323C2×C47#260011yescharacteristic
324C2×C47#312411yescharacteristic
325C2×C47#312411yescharacteristic
326C2×C47#312711yescharacteristic
327C2×C47#312711yescharacteristic
328C2×C47#320211yescharacteristic
329C2×C47#320311yescharacteristic
330C22×Q16—471no—
331C2×C22⋊Q8—471no—
332C2×C22⋊Q8C2×D4711yescharacteristic
333C2×C4×Q8—471no—
334C2×C4×Q8C2×D4711yescharacteristic
335C2×#117C2×D4711yescharacteristic
336C2×#107—471no—
337C2×#107—471no—
338C2×#107—471no—
339C2×#103—471no—
340C2×#103—471no—
341C2×#103—471no—
342C2×#103—471no—
343C2×#103—471no—
344C2×#103—471no—
345C2×C22⋊C8—471no—
346C2×C22⋊C8—471no—
347#391—471no—
348#391C2×D4711yescharacteristic
349#389—21no—
350#389—471no—
351#389—941no—
352#389C2×D4711yescharacteristic
353#388—21no—
354#388—21no—
355#388—941no—
356#388—941no—
357#387—21no—
358#387—941no—
359#380—471no—
360#380C2×D4711yescharacteristic
361C4×C4⋊C4—471no—
362C4×C4⋊C4C2×D4711yescharacteristic
363#340—471no—
364#340—471no—
365#340—471no—
366#340—471no—
367#340—471no—
368#340—471no—
369#336—471no—
370#336—471no—
371#336—471no—
372#336—471no—
373C47#2942711yescharacteristic, complemented, Sylow
374C22×Q8—471no—
375C22×Q8C22×D4711yescharacteristic
376C23×C4C22×D4711yescharacteristic
377C2×Q16—942no—
378C2×Q16—942no—
379C2×Q16—1881no—
380C22×C8—471no—
381C22×C8—471no—
382C22×C8—471no—
383C22×C8—471no—
384C22⋊Q8—22no—
385C22⋊Q8—22no—
386C22⋊Q8—942no—
387C22⋊Q8—942no—
388C4×Q8—41no—
389C4×Q8—41no—
390C4×Q8—1881no—
391C4×Q8—1881no—
392#117—41no—
393#117—41no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—471no—
398C2×C4⋊C4—471no—
399C2×C4⋊C4—471no—
400C2×C4⋊C4—941no—
401C2×C4⋊C4—941no—
402C2×C4⋊C4—941no—
403C2×C4⋊C4C22⋊D4711yescharacteristic
404C2×C4⋊C4C22⋊D4711yescharacteristic
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×C22⋊C4C22⋊D4711yescharacteristic
413C2×C22⋊C4C22⋊D4711yescharacteristic
414C2×C42—21no—
415C2×C42—21no—
416C2×C42—941no—
417C2×C42—941no—
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—1881no—
430C22⋊C8—1881no—
431#95—21no—
432#95—21no—
433#95—21no—
434#95—21no—
435#95—21no—
436#95—941no—
437#95—941no—
438#95—941no—
439#95—941no—
440#95—941no—
441C24C2×C22⋊D4711yescharacteristic
442C2×Q8—21no—
443C2×Q8—21no—
444C2×Q8—21no—
445C2×Q8—21no—
446C2×Q8—21no—
447C2×Q8—472no—
448C2×Q8—941no—
449C2×Q8—941no—
450C2×Q8—941no—
451C2×Q8—941no—
452C2×Q8—941no—
453C2×Q8#1062535012yes—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—941no—
465C22×C4—941no—
466C22×C4—941no—
467C22×C4—941no—
468C22×C4C2×C22⋊D4711yescharacteristic
469C22×C4C2×C22⋊D4711yescharacteristic
470C22×C4C2×C22⋊D4711yescharacteristic
471C22×C4C2×C22⋊D4711yescharacteristic
472C22×C4C2×C22⋊D4711yescharacteristic
473C22×C4C23×D4711yescharacteristic
474Q16—1884no—
475C2×C8—941no—
476C2×C8—941no—
477C2×C8—941no—
478C2×C8—941no—
479C2×C8—941no—
480C2×C8—941no—
481C2×C8—941no—
482C2×C8—941no—
483C2×C8—941no—
484C2×C8—941no—
485C2×C8—941no—
486C2×C8—941no—
487C4⋊C4—24no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—41no—
494C4⋊C4—41no—
495C4⋊C4—42no—
496C4⋊C4—81no—
497C4⋊C4—472no—
498C4⋊C4—944no—
499C4⋊C4—941no—
500C4⋊C4—941no—
501C4⋊C4—941no—
502C4⋊C4—941no—
503C4⋊C4—941no—
504C4⋊C4—3761no—
505C4⋊C4—1882no—
506C4⋊C4#1062535012yes—
507C22⋊C4—24no—
508C22⋊C4—41no—
509C22⋊C4—41no—
510C22⋊C4—81no—
511C22⋊C4—944no—
512C22⋊C4—3761no—
513C42—21no—
514C42—21no—
515C42—21no—
516C42—21no—
517C42—42no—
518C42—81no—
519C42—3761no—
520C42—1882no—
521C23—21no—
522C23—21no—
523C23—21no—
524C23—21no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23C22×C22⋊D4711yescharacteristic
529Q8—22no—
530Q8—22no—
531Q8—41no—
532Q8—41no—
533Q8—942no—
534Q8—942no—
535Q8—1881no—
536Q8—1881no—
537C2×C4—21no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
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—942no—
569C2×C4—942no—
570C2×C4—1881no—
571C2×C4—1881no—
572C2×C4—1881no—
573C2×C4—1881no—
574C2×C4—1881no—
575C2×C4—1881no—
576C2×C4—1881no—
577C2×C4—1881no—
578C2×C4—1881no—
579C2×C4—1881no—
580C2×C4#41×D4711yescharacteristic
581C2×C4#41×D4711yescharacteristic
582C2×C4C2×#1062535011yescharacteristic
583C2×C4C2×#1062535011yescharacteristic
584C2×C4#1309406911yescharacteristic
585C2×C4#1309406911yescharacteristic
586C8—1881no—
587C8—1881no—
588C8—1881no—
589C8—1881no—
590C22—41no—
591C22—41no—
592C22—41no—
593C22—41no—
594C22—41no—
595C22—41no—
596C22—41no—
597C22#1568963011yescharacteristic
598C22#1568963411yescharacteristic
599C22#1568968211yescharacteristic
600C22#1568992611yescharacteristic
601C22#1568993811yescharacteristic
602C22#1568993811yescharacteristic
603C22#1568993811yescharacteristic
604C4—41no—
605C4—41no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—42no—
610C4—42no—
611C4—81no—
612C4—81no—
613C4—3761no—
614C4—3761no—
615C4—1882no—
616C4—1882no—
617C4#1568980211yescharacteristic
618C4#1568982211yescharacteristic
619C4#1568982311yescharacteristic
620C4#1568985611yescharacteristic
621C2—81no—
622C2#2024897011yescharacteristic
623C2#2025364411yescharacteristic
624C2#2025365011yescharacteristic
625C2#2025368311yescharacteristic
626C2#2025368811yescharacteristic
627C2#2025427911yescharacteristic
628C2#2025428511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D47
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#2323C2
socleC23×C47C22×D4
prime radicalC24×C47C2×D4
hypercenter#2323D47
hypocenterC47#29427
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
6. #20244029C2semidirect
7. #20244026C2semidirect
8. #20243959C2semidirect
9. #20243956C2semidirect
10. #20243732C2semidirect
11. #20243729C2semidirect
12. #20243642C2semidirect
13. #20243641C2semidirect
373. C47#29427semidirect

Maximal subgroups

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

subgroupindex
#202460702
#202448142
#202445912
#202445882
#202443962
#202440292
#202440262
#202439592
#202439562
#202437322
#202437292
#202436422
#202436412
#202434592
C47×#23232
#2942747