Finite Group Databasebeta

← #26184071

Subgroups — #26184071

Order 12032.

1014 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257227C211yesmaximal, characteristic
2#20246328C211yesmaximal, characteristic, complemented
3#20241966C212yesmaximal, complemented
4#20241932C212yesmaximal, complemented
5#20241922C211yesmaximal, characteristic, complemented
6#20241918C211yesmaximal, characteristic, complemented
7#20241909C211yesmaximal, characteristic, complemented
8#20241905C211yesmaximal, characteristic, complemented
9#20241893C212yesmaximal, complemented
10#20241875C212yesmaximal, complemented
11C47×#2975C211yesmaximal, characteristic, complemented
12#15689822—22no—
13#15689819C412yescomplemented
14#15689813C412yescomplemented
15#15689802—22no—
16#15689782C412yescomplemented
17#15689769C412yescomplemented
18#15689720—21no—
19#15689711—21no—
20#15689682—21no—
21#15689657C2212yes—
22#15689634C2212yes—
23C2×#13094003C2211yescharacteristic
24C2×#13093998C2211yescharacteristic
25#15689596—21no—
26#15689016—21no—
27#15689009—21no—
28#15689003—21no—
29#15688991C2211yescharacteristic, complemented
30#15688986C2211yescharacteristic
31#15688983C2211yescharacteristic, complemented
32C22⋊C4×Dic47—21no—
33C4×C22⋊Dic47C2211yescharacteristic
34#15688969C2211yescharacteristic, complemented
35C2×C47⋊#95C2211yescharacteristic, complemented
36#15688861C2212yescomplemented
37#15688861C2212yescomplemented
38#15688861C2212yescomplemented
39#15688861C2212yescomplemented
40#15688858C2212yes—
41#15688858C2212yes—
42#15688858C2212yes—
43#15688858C2212yes—
44C2×C4×D4×C47C2211yescharacteristic
45C47×#436—22no—
46C47×#436—22no—
47C47×#431C2212yescomplemented
48C47×#407C2211yescharacteristic
49C47×#407C2211yescharacteristic
50C47×#407C2212yescomplemented
51C2×#10625350C2×C412yes—
52C22×Dic94C2×C411yescharacteristic, complemented
53C22×D188C2×C411yescharacteristic, complemented
54C22×C22⋊D47—21no—
55C23×Dic47—21no—
56#13094019—22no—
57#13094018—22no—
58#13094017—22no—
59#13094016C2×C412yescomplemented
60#13094014—22no—
61#13094012C2×C412yescomplemented
62#13094010—22no—
63#13094009—22no—
64#13094007—22no—
65#13094007—22no—
66#13094006—22no—
67#13094005—22no—
68#13094003—21no—
69#13094003—21no—
70#13094003C2×C412yes—
71#13094003C2×C412yescomplemented
72#13094003C2312yes—
73#13094002—22no—
74#13093998—21no—
75#13093998—21no—
76#13093998C2×C412yes—
77#13093998C2×C412yescomplemented
78#13093998C2312yes—
79#13093997—22no—
80#13093996—22no—
81#13093996—22no—
82#13093993—22no—
83#13093992—22no—
84C4×Dic94C2×C412yescomplemented
85C4×D188C2×C412yescomplemented
86C4×C22⋊D47—22no—
87C4×C22⋊D47—22no—
88C4×C22⋊D47—22no—
89(C22⋊D47)⋊C4—22no—
90(C22⋊D47)⋊C4—22no—
91#13093983C2×C412yes—
92#13093981—22no—
93#13093980—22no—
94C2×C4⋊Dic47—21no—
95C2×C4⋊Dic47C2×C411yescharacteristic, complemented
96C2×C4⋊Dic47C2×C411yescharacteristic, complemented
97C2×Dic47⋊C4—21no—
98C2×C22⋊Dic47—21no—
99C2×C22⋊Dic47—21no—
100C2×C22⋊Dic47—21no—
101C2×C22⋊Dic47—21no—
102C2×(C2×D47)⋊C4—21no—
103C22⋊C4×D47—22no—
104C2×C4×Dic47—21no—
105C2×C4×Dic47—21no—
106C47⋊#95—21no—
107C47⋊#95—21no—
108C47⋊#95—21no—
109C47⋊#95—21no—
110C47⋊#95—21no—
111C47⋊#95C2311yescharacteristic
112C47⋊#95C2311yescharacteristic
113C47⋊#95C2311yescharacteristic
114C47⋊#95C2311yescharacteristic
115C22×D4×C47D411yescharacteristic
116C23×C4×C47C2311yescharacteristic
117C23×C4×C47C2311yescharacteristic
118C2×M16×C47—21no—
119C2×M16×C47—21no—
120C2×M16×C47—22no—
121C4×D4×C47C2×C412yes—
122C4×D4×C47C2×C412yes—
123C4×D4×C47D412yes—
124C4×D4×C47D412yes—
125C2×C4⋊C4×C47D411yescharacteristic
126C2×C22⋊C4×C47D411yescharacteristic
127C2×C22⋊C4×C47D411yescharacteristic
128C2×C42×C47C2311yescharacteristic
129C4⋊C8×C47—22no—
130C22⋊C8×C47—21no—
131C22⋊C8×C47—21no—
132C22⋊C8×C47—22no—
133C22⋊C8×C47C2312yes—
134C22⋊C8×C47C2312yes—
135C4×C8×C47—22no—
136C24×C47C2×D411yescharacteristic
137C24×C47C2×D411yescharacteristic
138C23×D47—21no—
139C2×D4×C47—41no—
140C2×D4×C47—41no—
141C2×D4×C47C22⋊C412yes—
142C2×D4×C47C22⋊C412yes—
143C22×C4×C47—81no—
144C22×C4×C47—81no—
145C22×C4×C47C22×C411yescharacteristic
146C22×C4×C47C22×C411yescharacteristic
147C22×C4×C47C22×C412yes—
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D411yescharacteristic
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D412yes—
155C22×C4×C47C2411yescharacteristic
156C2×C22⋊D47—22no—
157C2×C22⋊D47—22no—
158C2×C22⋊D47—22no—
159C2×C22⋊D47—22no—
160C2×C22⋊D47—42no—
161C2×C22⋊D47—42no—
162C22×Dic47—21no—
163C22×Dic47—21no—
164C22×Dic47—21no—
165C22×Dic47—21no—
166C22×Dic47—21no—
167C22×Dic47—21no—
168C22×Dic47—21no—
169C22×Dic47—42no—
170C22×Dic47—42no—
171C22×Dic47—81no—
172#10625350—82no—
173C2×D188—21no—
174C2×D188—81no—
175C2×D188C22×C412yes—
176C2×C4×D47—22no—
177C2×C4×D47—22no—
178C2×Dic94—21no—
179C2×Dic94—81no—
180C2×Dic94C22×C412yes—
181M16×C47—81no—
182M16×C47—81no—
183M16×C47—82no—
184C2×C8×C47—22no—
185C2×C8×C47—22no—
186C4⋊C4×C47C22⋊C412yes—
187C4⋊C4×C47C22⋊C412yes—
188C22⋊C4×C47C22⋊C412yes—
189C22⋊C4×C47C22⋊C412yes—
190C22⋊C4×C47C22⋊C412yes—
191C22⋊C4×C47C22⋊C412yes—
192C42×C47C22×C412yes—
193C42×C47C2×D412yes—
194C22⋊Dic47—22no—
195C22⋊Dic47—22no—
196C22⋊Dic47—22no—
197C22⋊Dic47—22no—
198C22⋊Dic47—42no—
199C22⋊Dic47—42no—
200C22⋊Dic47—81no—
201(C2×D47)⋊C4—22no—
202(C2×D47)⋊C4—22no—
203(C2×D47)⋊C4—22no—
204(C2×D47)⋊C4—22no—
205(C2×D47)⋊C4—22no—
206(C2×D47)⋊C4—22no—
207C4⋊Dic47—21no—
208C4⋊Dic47—21no—
209C4⋊Dic47—81no—
210C4⋊Dic47C22×C412yes—
211C4⋊Dic47C22×C412yes—
212Dic47⋊C4—22no—
213Dic47⋊C4—22no—
214Dic47⋊C4—22no—
215Dic47⋊C4—22no—
216Dic47⋊C4—22no—
217Dic47⋊C4—22no—
218C4×Dic47—22no—
219C4×Dic47—22no—
220C4×Dic47—42no—
221C4×Dic47—81no—
222C23×C47—21no—
223C23×C47—21no—
224C23×C47—21no—
225C23×C47—21no—
226C23×C47—41no—
227C23×C47—41no—
228C23×C47—41no—
229C23×C47—41no—
230C23×C47C2×C22⋊C411yescharacteristic
231C23×C47C2×C22⋊C412yes—
232C23×C47C2×C22⋊C411yescharacteristic
233C23×C47C22×D411yescharacteristic
234C22×D47—22no—
235C22×D47—42no—
236C22×D47—81no—
237C22×D47—81no—
238D4×C47—42no—
239D4×C47—42no—
240C2×C4×C47—21no—
241C2×C4×C47—21no—
242C2×C4×C47—41no—
243C2×C4×C47—41no—
244C2×C4×C47—41no—
245C2×C4×C47—41no—
246C2×C4×C47—42no—
247C2×C4×C47—41no—
248C2×C4×C47—41no—
249C2×C4×C47—41no—
250C2×C4×C47—42no—
251C2×C4×C47—41no—
252C2×C4×C47C2×C22⋊C412yes—
253C2×C4×C47C2×C22⋊C412yes—
254C2×C4×C47C2×C22⋊C411yescharacteristic
255C2×C4×C47C2×C22⋊C412yes—
256C2×C4×C47C2×C22⋊C411yescharacteristic
257C2×C4×C47C23×C411yescharacteristic
258C2×C4×C47C22×D411yescharacteristic
259C22⋊D47—44no—
260C22⋊D47—42no—
261C22⋊D47—42no—
262C22⋊D47—82no—
263C22⋊D47—82no—
264C2×Dic47—22no—
265C2×Dic47—22no—
266C2×Dic47—22no—
267C2×Dic47—22no—
268C2×Dic47—41no—
269C2×Dic47—42no—
270C2×Dic47—41no—
271C2×Dic47—41no—
272C2×Dic47—41no—
273C2×Dic47—41no—
274C2×Dic47—41no—
275C2×Dic47—41no—
276C2×Dic47—41no—
277C2×Dic47—41no—
278C2×Dic47—41no—
279C2×Dic47—41no—
280C2×Dic47—81no—
281C2×Dic47—81no—
282C2×Dic47—84no—
283D188—42no—
284D188—81no—
285C4×D47—82no—
286C4×D47—82no—
287Dic94—42no—
288Dic94—81no—
289C8×C47—42no—
290C8×C47—42no—
291#17623—471nomaximal, Sylow
292C22×C47—21no—
293C22×C47—21no—
294C22×C47—21no—
295C22×C47—22no—
296C22×C47—21no—
297C22×C47—21no—
298C22×C47—21no—
299C22×C47—22no—
300C22×C47—21no—
301C22×C47—22no—
302C22×C47—21no—
303C22×C47—42no—
304C22×C47—42no—
305C22×C47—81no—
306C22×C47—81no—
307C22×C47C22×C22⋊C411yescharacteristic
308C22×C47#57511yescharacteristic
309C22×C47#57511yescharacteristic
310C2×D47—42no—
311C2×D47—42no—
312C2×D47—42no—
313C2×D47—162no—
314C4×C47—21no—
315C4×C47—21no—
316C4×C47—21no—
317C4×C47—22no—
318C4×C47—21no—
319C4×C47—22no—
320C4×C47—22no—
321Dic47—42no—
322Dic47—42no—
323Dic47—42no—
324Dic47—42no—
325Dic47—81no—
326Dic47—81no—
327Dic47—81no—
328#3505—471no—
329#2975D4711yescharacteristic, complemented
330#2353—471no—
331#1521—472no—
332#1513—472no—
333#1511—471no—
334#1510—471no—
335#1508—471no—
336#1507—471no—
337#1504—472no—
338#1500—472no—
339C2×C47—21no—
340C2×C47—21no—
341C2×C47—41no—
342C2×C47—42no—
343C2×C47—41no—
344C2×C47#288311yescharacteristic
345C2×C47#289611yescharacteristic
346C2×C47#290111yescharacteristic
347D47—82no—
348#553—472no—
349#553—942no—
350#552—472no—
351#551—942no—
352#548—472no—
353#546—472no—
354#539—941no—
355#538—941no—
356#534—941no—
357#529—472no—
358#525—472no—
359C2×C22⋊Q8—471no—
360C2×#121—471no—
361#518—941no—
362C2×C4×D4C2×D4711yescharacteristic
363#436—22no—
364#436—22no—
365#431C2×D4712yescomplemented
366#407C2×D4711yescharacteristic
367#407C2×D4711yescharacteristic
368#407C2×D4712yescomplemented
369#388—941no—
370#387—941no—
371#386—941no—
372#383—471no—
373#381—471no—
374#380—471no—
375C4×C22⋊C4—471no—
376C4×C22⋊C4—941no—
377#376—471no—
378C2×#95—471no—
379#328—472no—
380#328—472no—
381#328—472no—
382#328—472no—
383#327—472no—
384#327—472no—
385#327—472no—
386#327—472no—
387C47#1762311yescharacteristic, complemented, Sylow
388C2×#41—472no—
389C22×Q8—471no—
390C22×D4—471no—
391C22×D4—941no—
392C22×D4D18811yescharacteristic
393C23×C4—941no—
394C23×C4C22×D4711yescharacteristic
395C23×C4C22×D4711yescharacteristic
396C2×M16—21no—
397C2×M16—21no—
398C2×M16—22no—
399#126—942no—
400#126—942no—
401#126—942no—
402#125—472no—
403#125—942no—
404#124—472no—
405#124—942no—
406#123—942no—
407#123—942no—
408#123—942no—
409#123—942no—
410#123—942no—
411C22⋊Q8—472no—
412C22⋊Q8—472no—
413C22⋊Q8—472no—
414C22⋊Q8—941no—
415C22⋊Q8—941no—
416C22⋊Q8—942no—
417#121—472no—
418#121—472no—
419#121—472no—
420#121—942no—
421#121—942no—
422#121—941no—
423#121—941no—
424#121—942no—
425#120—942no—
426#120—942no—
427C4×Q8—472no—
428C4×D4—472no—
429C4×D4—942no—
430C4×D4—942no—
431C4×D4—942no—
432C4×D4—942no—
433C4×D4—942no—
434C4×D4C4×D4712yes—
435C4×D4C4×D4712yes—
436C4×D4C22⋊D4712yes—
437C4×D4C22⋊D4712yes—
438#117—472no—
439#117—942no—
440#117—942no—
441C2×C4⋊C4—471no—
442C2×C4⋊C4—471no—
443C2×C4⋊C4—941no—
444C2×C4⋊C4—941no—
445C2×C4⋊C4D18811yescharacteristic
446C2×C22⋊C4—941no—
447C2×C22⋊C4—941no—
448C2×C22⋊C4—942no—
449C2×C22⋊C4—941no—
450C2×C22⋊C4—941no—
451C2×C22⋊C4—941no—
452C2×C22⋊C4D18811yescharacteristic
453C2×C22⋊C4D18811yescharacteristic
454C2×C42—941no—
455C2×C42—941no—
456C2×C42C22×D4711yescharacteristic
457C4⋊C8—22no—
458C22⋊C8—21no—
459C22⋊C8—21no—
460C22⋊C8—22no—
461C22⋊C8C22×D4712yes—
462C22⋊C8C22×D4712yes—
463C4×C8—22no—
464#95—471no—
465#95—471no—
466#95—471no—
467#95—471no—
468#95—941no—
469#95—941no—
470#95—941no—
471#95—941no—
472#95—941no—
473C24—941no—
474C24C2×D18811yescharacteristic
475C24C2×D18811yescharacteristic
476#41—3762no—
477C2×Q8—472no—
478C2×Q8—941no—
479C2×Q8—3761no—
480C2×D4—41no—
481C2×D4—41no—
482C2×D4—472no—
483C2×D4—942no—
484C2×D4—942no—
485C2×D4—942no—
486C2×D4—942no—
487C2×D4—941no—
488C2×D4—3761no—
489C2×D4—1882no—
490C2×D4—1882no—
491C2×D4(C2×D47)⋊C412yes—
492C2×D4(C2×D47)⋊C412yes—
493C22×C4—81no—
494C22×C4—81no—
495C22×C4—941no—
496C22×C4—941no—
497C22×C4—941no—
498C22×C4—941no—
499C22×C4—942no—
500C22×C4—942no—
501C22×C4—941no—
502C22×C4—941no—
503C22×C4—941no—
504C22×C4—3761no—
505C22×C4—1882no—
506C22×C4—1882no—
507C22×C4C2×C4×D4711yescharacteristic
508C22×C4C2×C4×D4711yescharacteristic
509C22×C4C2×C4×D4712yes—
510C22×C4C2×D18811yescharacteristic
511C22×C4C2×D18811yescharacteristic
512C22×C4C2×D18811yescharacteristic
513C22×C4C2×D18811yescharacteristic
514C22×C4C2×C22⋊D4711yescharacteristic
515C22×C4C2×C22⋊D4711yescharacteristic
516C22×C4C2×C22⋊D4712yes—
517C22×C4C23×D4711yescharacteristic
518M16—81no—
519M16—81no—
520M16—82no—
521C2×C8—22no—
522C2×C8—22no—
523C4⋊C4—472no—
524C4⋊C4—472no—
525C4⋊C4—942no—
526C4⋊C4—942no—
527C4⋊C4—942no—
528C4⋊C4—942no—
529C4⋊C4—942no—
530C4⋊C4—941no—
531C4⋊C4—941no—
532C4⋊C4—942no—
533C4⋊C4—3761no—
534C4⋊C4(C2×D47)⋊C412yes—
535C4⋊C4(C2×D47)⋊C412yes—
536C22⋊C4—942no—
537C22⋊C4—942no—
538C22⋊C4—942no—
539C22⋊C4—942no—
540C22⋊C4—942no—
541C22⋊C4—942no—
542C22⋊C4—942no—
543C22⋊C4—942no—
544C22⋊C4—942no—
545C22⋊C4—942no—
546C22⋊C4—3761no—
547C22⋊C4—1882no—
548C22⋊C4—1882no—
549C22⋊C4(C2×D47)⋊C412yes—
550C22⋊C4(C2×D47)⋊C412yes—
551C22⋊C4(C2×D47)⋊C412yes—
552C22⋊C4(C2×D47)⋊C412yes—
553C42—942no—
554C42—942no—
555C42—3761no—
556C42—1882no—
557C42C2×C4×D4712yes—
558C42C2×C22⋊D4712yes—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—942no—
568C23—3761no—
569C23—3761no—
570C23—1882no—
571C23C2×(C2×D47)⋊C411yescharacteristic
572C23C2×(C2×D47)⋊C412yes—
573C23C2×(C2×D47)⋊C411yescharacteristic
574C23C22×D18811yescharacteristic
575Q8—3761no—
576Q8—1882no—
577D4—42no—
578D4—42no—
579D4—3761no—
580D4—3762no—
581D4—3762no—
582D4—1882no—
583D4—1884no—
584D4—1882no—
585D4—1882no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—42no—
598C2×C4—942no—
599C2×C4—942no—
600C2×C4—942no—
601C2×C4—942no—
602C2×C4—3762no—
603C2×C4—3762no—
604C2×C4—3764no—
605C2×C4—3761no—
606C2×C4—3761no—
607C2×C4—1881no—
608C2×C4—1881no—
609C2×C4—1881no—
610C2×C4—1881no—
611C2×C4—1881no—
612C2×C4—1881no—
613C2×C4—1881no—
614C2×C4—1881no—
615C2×C4—1881no—
616C2×C4—1881no—
617C2×C4—1882no—
618C2×C4—1881no—
619C2×C4C2×(C2×D47)⋊C412yes—
620C2×C4C2×(C2×D47)⋊C412yes—
621C2×C4C2×(C2×D47)⋊C412yes—
622C2×C4C2×(C2×D47)⋊C411yescharacteristic
623C2×C4C2×(C2×D47)⋊C411yescharacteristic
624C2×C4C22×C4×D4711yescharacteristic
625C2×C4C22×C22⋊D4711yescharacteristic
626C8—42no—
627C8—42no—
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—22no—
633C22—21no—
634C22—22no—
635C22—21no—
636C22—21no—
637C22—22no—
638C22—21no—
639C22—42no—
640C22—42no—
641C22—81no—
642C22—81no—
643C22—1882no—
644C22—1882no—
645C22—1882no—
646C22—7522no—
647C22C22×(C2×D47)⋊C411yescharacteristic
648C22#1568992311yescharacteristic
649C22#1568992311yescharacteristic
650C4—22no—
651C4—22no—
652C4—21no—
653C4—21no—
654C4—22no—
655C4—21no—
656C4—21no—
657C4—3761no—
658C4—3761no—
659C4—3761no—
660C4—1882no—
661C4—1882no—
662C4—1882no—
663C4—1882no—
664C2—21no—
665C2—21no—
666C2—42no—
667C2—41no—
668C2—41no—
669C2—3762no—
670C2#2025184811yescharacteristic
671C2#2025190811yescharacteristic
672C2#2025193311yescharacteristic
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2975C2
socleC22×C47C22×C22⋊C4
prime radical#20257227C2
cosocleC22×C4×C47C24
hypercenter#2975D47
hypocenterC47#17623
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2975

Complements

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

NH ≅ G/Ntype
2. #20246328C2semidirect
3. #20241966C2semidirect
4. #20241932C2semidirect
5. #20241922C2semidirect
6. #20241918C2semidirect
7. #20241909C2semidirect
8. #20241905C2semidirect
9. #20241893C2semidirect
10. #20241875C2semidirect
11. C47×#2975C2semidirect
13. #15689819C4semidirect
14. #15689813C4semidirect
16. #15689782C4semidirect
17. #15689769C4semidirect
29. #15688991C22semidirect
31. #15688983C22semidirect
34. #15688969C22semidirect
35. C2×C47⋊#95C22semidirect
36. #15688861C22semidirect
37. #15688861C22semidirect
38. #15688861C22semidirect
39. #15688861C22semidirect
47. C47×#431C22semidirect
50. C47×#407C22semidirect
52. C22×Dic94C2×C4semidirect
53. C22×D188C2×C4semidirect
59. #13094016C2×C4semidirect
61. #13094012C2×C4semidirect
71. #13094003C2×C4semidirect
77. #13093998C2×C4semidirect
84. C4×Dic94C2×C4semidirect
85. C4×D188C2×C4semidirect
95. C2×C4⋊Dic47C2×C4semidirect
96. C2×C4⋊Dic47C2×C4semidirect
329. #2975D47semidirect
365. #431C2×D47semidirect
368. #407C2×D47semidirect
387. C47#17623semidirect

Maximal subgroups

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

subgroupindex
#202572272
#202463282
#202419662
#202419322
#202419222
#202419182
#202419092
#202419052
#202418932
#202418752
C47×#29752
#1762347