Finite Group Databasebeta

← #26184121

Subgroups — #26184121

Order 12032.

1010 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257240C211yesmaximal, characteristic
2#20252417C211yesmaximal, characteristic, complemented
3#20241942C212yesmaximal, complemented
4#20241938C211yesmaximal, characteristic, complemented
5#20241934C211yesmaximal, characteristic, complemented
6#20241929C212yesmaximal, complemented
7#20241899C211yesmaximal, characteristic, complemented
8#20241895C211yesmaximal, characteristic, complemented
9#20241890C212yesmaximal, complemented
10#20241877C212yesmaximal, complemented
11C47×#2353C211yesmaximal, characteristic, complemented
12#15689856—22no—
13#15689843C412yescomplemented
14#15689838C412yescomplemented
15#15689823—22no—
16#15689794C412yescomplemented
17#15689778C412yescomplemented
18#15689711—21no—
19#15689711—21no—
20#15689690—21no—
21#15689680C2211yescharacteristic
22#15689673C2211yescharacteristic
23#15689634C2212yes—
24#15689634C2212yes—
25#15689604—21no—
26#15689195—22no—
27#15689195—22no—
28#15689180C2211yescharacteristic
29#15689180C2211yescharacteristic
30#15689087C2212yes—
31C2×C22⋊(C47⋊C8)C2212yes—
32#15688859C2212yes—
33#15688859C2212yes—
34#15688859C2212yes—
35#15688859C2212yes—
36#15688856C2212yes—
37#15688856C2212yes—
38#15688856C2212yes—
39#15688856C2212yes—
40C2×C4×D4×C47C2211yescharacteristic
41C47×#388—21no—
42C47×#387—21no—
43C47×#386—21no—
44C47×#383C2211yescharacteristic
45C47×#381C2211yescharacteristic
46C47×#380C2211yescharacteristic
47C4×C22⋊C4×C47—21no—
48C4×C22⋊C4×C47C2211yescharacteristic
49C47×#376C2211yescharacteristic
50C2×#95×C47C2211yescharacteristic
51C2×#10625350—21no—
52C2×#10625350C2×C412yes—
53C2×#10625350C2×C412yescomplemented
54C2×#10625329—22no—
55C22×C47⋊C8—22no—
56#13094024C2×C411yescharacteristic, complemented
57#13094024C2×C411yescharacteristic, complemented
58#13094024C2×C411yescharacteristic, complemented
59#13094021C2×C411yescharacteristic, complemented
60#13094020—22no—
61#13094017—22no—
62#13094017—22no—
63#13094016—21no—
64#13094012—21no—
65#13094011—22no—
66#13094010—22no—
67#13094010—22no—
68#13094007—22no—
69#13094007—22no—
70#13094006—22no—
71#13094006—22no—
72#13094003—21no—
73#13094003C2×C412yes—
74#13094003C2×C412yescomplemented
75#13094003C2312yes—
76#13093998—21no—
77#13093998C2×C412yes—
78#13093998C2×C412yescomplemented
79#13093998C2312yes—
80#13093997—22no—
81#13093997—22no—
82#13093995—22no—
83#13093995—22no—
84#13093993—22no—
85#13093993—22no—
86C4×Dic94—21no—
87C4×Dic94C2×C412yescomplemented
88#13093989—22no—
89C4×D188—21no—
90C4×D188C2×C412yescomplemented
91#13093986—22no—
92C4×C22⋊D47—22no—
93C4×C22⋊D47—22no—
94C4×C22⋊D47—22no—
95C4×C22⋊D47—22no—
96#13093983—21no—
97#13093983C2×C412yes—
98#13093983C2×C412yescomplemented
99#13093982—22no—
100C4⋊C4×D47—22no—
101C4⋊(C47⋊C8)—21no—
102C4⋊(C47⋊C8)—21no—
103C22⋊(C47⋊C8)—22no—
104C22⋊(C47⋊C8)—22no—
105C22⋊(C47⋊C8)C2312yes—
106C22⋊(C47⋊C8)C2312yes—
107#13093935—21no—
108#13093935—21no—
109C22×D4×C47D411yescharacteristic
110C23×C4×C47—21no—
111C23×C4×C47C2311yescharacteristic
112C23×C4×C47C2311yescharacteristic
113C4×D4×C47C2×C412yes—
114C4×D4×C47C2×C412yes—
115C4×D4×C47D412yes—
116C4×D4×C47D412yes—
117C2×C4⋊C4×C47—21no—
118C2×C4⋊C4×C47D411yescharacteristic
119C2×C22⋊C4×C47—21no—
120C2×C22⋊C4×C47—21no—
121C2×C22⋊C4×C47—21no—
122C2×C22⋊C4×C47D411yescharacteristic
123C2×C22⋊C4×C47D411yescharacteristic
124C2×C42×C47—21no—
125C2×C42×C47—21no—
126C2×C42×C47C2311yescharacteristic
127#95×C47—21no—
128#95×C47—21no—
129#95×C47—21no—
130#95×C47—21no—
131#95×C47—21no—
132#95×C47C2311yescharacteristic
133#95×C47C2311yescharacteristic
134#95×C47C2311yescharacteristic
135#95×C47C2311yescharacteristic
136C24×C47C2×D411yescharacteristic
137C24×C47C2×D411yescharacteristic
138C2×D4×C47—41no—
139C2×D4×C47—41no—
140C2×D4×C47C22⋊C412yes—
141C2×D4×C47C22⋊C412yes—
142C22×C4×C47—21no—
143C22×C4×C47—21no—
144C22×C4×C47—21no—
145C22×C4×C47—21no—
146C22×C4×C47—42no—
147C22×C4×C47—42no—
148C22×C4×C47—81no—
149C22×C4×C47—81no—
150C22×C4×C47—81no—
151C22×C4×C47C22×C412yes—
152C22×C4×C47C22×C412yes—
153C22×C4×C47C2×D411yescharacteristic
154C22×C4×C47C2×D411yescharacteristic
155C22×C4×C47C2×D411yescharacteristic
156C22×C4×C47C2×D411yescharacteristic
157C22×C4×C47C2×D412yes—
158C22×C4×C47C2×D412yes—
159C22×C4×C47C2411yescharacteristic
160C2×C22⋊D47—22no—
161C2×C22⋊D47—22no—
162C2×C22⋊D47—22no—
163C2×C22⋊D47—22no—
164C2×C22⋊D47—22no—
165#10625350—42no—
166#10625350—42no—
167#10625350—82no—
168#10625350—82no—
169C2×D188—21no—
170C2×D188—21no—
171C2×D188C22×C412yes—
172C2×C4×D47—22no—
173C2×C4×D47—22no—
174C2×C4×D47—22no—
175C2×Dic94—21no—
176C2×Dic94—21no—
177C2×Dic94C22×C412yes—
178C4⋊C4×C47—81no—
179C4⋊C4×C47C22⋊C412yes—
180C4⋊C4×C47C22⋊C412yes—
181C22⋊C4×C47—42no—
182C22⋊C4×C47—42no—
183C22⋊C4×C47—81no—
184C22⋊C4×C47C22⋊C412yes—
185C22⋊C4×C47C22⋊C412yes—
186C22⋊C4×C47C22⋊C412yes—
187C22⋊C4×C47C22⋊C412yes—
188C42×C47—42no—
189C42×C47—81no—
190C42×C47C22×C411yescharacteristic
191C42×C47C22×C411yescharacteristic
192C42×C47C2×D411yescharacteristic
193C42×C47C2×D411yescharacteristic
194C22⋊Dic47—22no—
195C22⋊Dic47—22no—
196C22⋊Dic47—22no—
197C22⋊Dic47—22no—
198C22⋊Dic47—22no—
199(C2×D47)⋊C4—22no—
200(C2×D47)⋊C4—22no—
201(C2×D47)⋊C4—22no—
202(C2×D47)⋊C4—22no—
203C4⋊Dic47—21no—
204C4⋊Dic47—21no—
205C4⋊Dic47—21no—
206C4⋊Dic47—21no—
207C4⋊Dic47C22×C412yes—
208C4⋊Dic47C22×C412yes—
209Dic47⋊C4—22no—
210Dic47⋊C4—22no—
211Dic47⋊C4—22no—
212Dic47⋊C4—22no—
213C4×Dic47—22no—
214C4×Dic47—22no—
215C4×Dic47—22no—
216#10625329—82no—
217C2×C47⋊C8—22no—
218C2×C47⋊C8—22no—
219C2×C47⋊C8—82no—
220C23×C47—21no—
221C23×C47—21no—
222C23×C47—21no—
223C23×C47—21no—
224C23×C47—41no—
225C23×C47—41no—
226C23×C47—41no—
227C23×C47—41no—
228C23×C47C2×C22⋊C412yes—
229C23×C47C2×C22⋊C412yes—
230C23×C47C22×D411yescharacteristic
231C22×D47—22no—
232D4×C47—42no—
233D4×C47—42no—
234C2×C4×C47—21no—
235C2×C4×C47—21no—
236C2×C4×C47—22no—
237C2×C4×C47—42no—
238C2×C4×C47—42no—
239C2×C4×C47—41no—
240C2×C4×C47—41no—
241C2×C4×C47—41no—
242C2×C4×C47—42no—
243C2×C4×C47—42no—
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—41no—
254C2×C4×C47—41no—
255C2×C4×C47—41no—
256C2×C4×C47—84no—
257C2×C4×C47C2×C22⋊C411yescharacteristic
258C2×C4×C47C2×C22⋊C412yes—
259C2×C4×C47C2×C22⋊C412yes—
260C2×C4×C47C2×C22⋊C411yescharacteristic
261C2×C4×C47C2×C22⋊C411yescharacteristic
262C2×C4×C47C2×C22⋊C411yescharacteristic
263C2×C4×C47C23×C411yescharacteristic
264C2×C4×C47C22×D411yescharacteristic
265C22⋊D47—42no—
266C22⋊D47—42no—
267C22⋊D47—82no—
268C22⋊D47—82no—
269C22⋊D47—82no—
270C22⋊D47—82no—
271C2×Dic47—22no—
272C2×Dic47—22no—
273C2×Dic47—22no—
274D188—42no—
275D188—42no—
276D188—81no—
277C4×D47—44no—
278C4×D47—82no—
279C4×D47—82no—
280Dic94—42no—
281Dic94—42no—
282Dic94—81no—
283C47⋊C8—42no—
284C47⋊C8—42no—
285#17629—471nomaximal, Sylow
286C22×C47—21no—
287C22×C47—21no—
288C22×C47—22no—
289C22×C47—22no—
290C22×C47—22no—
291C22×C47—22no—
292C22×C47—22no—
293C22×C47—22no—
294C22×C47—42no—
295C22×C47—42no—
296C22×C47—81no—
297C22×C47—81no—
298C22×C47C22×C22⋊C411yescharacteristic
299C22×C47#57511yescharacteristic
300C22×C47#57511yescharacteristic
301C2×D47—42no—
302C2×D47—42no—
303C2×D47—42no—
304C4×C47—21no—
305C4×C47—21no—
306C4×C47—22no—
307C4×C47—22no—
308C4×C47—21no—
309C4×C47—21no—
310C4×C47—21no—
311C4×C47—21no—
312C4×C47—42no—
313C4×C47—81no—
314C4×C47—81no—
315C4×C47—81no—
316Dic47—42no—
317Dic47—42no—
318Dic47—42no—
319#3506—471no—
320#2973—471no—
321#2353D4711yescharacteristic, complemented
322#1516—472no—
323#1515—471no—
324#1514—471no—
325#1513—472no—
326#1506—471no—
327#1505—471no—
328#1504—472no—
329#1501—472no—
330C2×C47—21no—
331C2×C47—21no—
332C2×C47—42no—
333C2×C47—42no—
334C2×C47#288311yescharacteristic
335C2×C47#289611yescharacteristic
336C2×C47#290111yescharacteristic
337D47—82no—
338#559—942no—
339#556—472no—
340#555—472no—
341#553—942no—
342#550—472no—
343#547—472no—
344#538—941no—
345#538—941no—
346#535—941no—
347#533—471no—
348#532—471no—
349#525—472no—
350#525—472no—
351#520—941no—
352C2×C4×D4C2×D4711yescharacteristic
353#435—942no—
354#435—942no—
355#432—471no—
356#432—471no—
357#407—472no—
358C2×C22⋊C8—472no—
359#388—21no—
360#387—21no—
361#386—21no—
362#383C2×D4711yescharacteristic
363#381C2×D4711yescharacteristic
364#380C2×D4711yescharacteristic
365C4×C22⋊C4—21no—
366C4×C22⋊C4C2×D4711yescharacteristic
367#376C2×D4711yescharacteristic
368C2×#95C2×D4711yescharacteristic
369#328—472no—
370#328—472no—
371#328—472no—
372#328—472no—
373#327—472no—
374#327—472no—
375#327—472no—
376#327—472no—
377C47#1762911yescharacteristic, complemented, Sylow
378C2×#41—472no—
379C2×#41—472no—
380C2×#41—941no—
381C22×D4C22⋊D4711yescharacteristic
382C23×C4—21no—
383C23×C4C22×D4711yescharacteristic
384C23×C4C22×D4711yescharacteristic
385C2×M16—942no—
386C22×C8—942no—
387C4⋊Q8—471no—
388C4⋊Q8—471no—
389C4⋊Q8—471no—
390#127—471no—
391#127—942no—
392#126—942no—
393#126—942no—
394#125—941no—
395#124—942no—
396#124—942no—
397#124—942no—
398#124—941no—
399#123—942no—
400#123—942no—
401#123—942no—
402#123—942no—
403C22⋊Q8—472no—
404C22⋊Q8—472no—
405C22⋊Q8—472no—
406C22⋊Q8—941no—
407#121—472no—
408#121—472no—
409#121—472no—
410#121—942no—
411#121—942no—
412#121—941no—
413#121—942no—
414#121—942no—
415#120—942no—
416#120—942no—
417C4×Q8—472no—
418C4×Q8—942no—
419C4×Q8—941no—
420C4×D4—472no—
421C4×D4—942no—
422C4×D4—942no—
423C4×D4—942no—
424C4×D4—942no—
425C4×D4—942no—
426C4×D4—941no—
427C4×D4C4×D4712yes—
428C4×D4C4×D4712yes—
429C4×D4D18812yes—
430C4×D4D18812yes—
431#117—472no—
432#117—472no—
433#117—941no—
434#117—942no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—942no—
437C2×C4⋊C4C22⋊D4711yescharacteristic
438C2×C22⋊C4—21no—
439C2×C22⋊C4—21no—
440C2×C22⋊C4—21no—
441C2×C22⋊C4C22⋊D4711yescharacteristic
442C2×C22⋊C4C22⋊D4711yescharacteristic
443C2×C42—21no—
444C2×C42—21no—
445C2×C42C22×D4711yescharacteristic
446C4⋊C8—941no—
447C4⋊C8—941no—
448C22⋊C8—472no—
449C22⋊C8—472no—
450C22⋊C8—942no—
451C22⋊C8—942no—
452#97—941no—
453#97—941no—
454#95—21no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95C22×D4711yescharacteristic
460#95C22×D4711yescharacteristic
461#95C22×D4711yescharacteristic
462#95C22×D4711yescharacteristic
463C24C2×C22⋊D4711yescharacteristic
464C24C2×C22⋊D4711yescharacteristic
465#41—3762no—
466#41—3762no—
467#41—1882no—
468#41—1882no—
469C2×Q8—472no—
470C2×Q8—941no—
471C2×Q8—941no—
472C2×D4—41no—
473C2×D4—41no—
474C2×D4—472no—
475C2×D4—942no—
476C2×D4—942no—
477C2×D4—942no—
478C2×D4—942no—
479C2×D4—942no—
480C2×D4—941no—
481C2×D4—941no—
482C2×D4(C2×D47)⋊C412yes—
483C2×D4(C2×D47)⋊C412yes—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—42no—
489C22×C4—42no—
490C22×C4—81no—
491C22×C4—81no—
492C22×C4—81no—
493C22×C4—942no—
494C22×C4—942no—
495C22×C4—942no—
496C22×C4C2×C4×D4712yes—
497C22×C4C2×C4×D4712yes—
498C22×C4C2×D18812yes—
499C22×C4C2×D18812yes—
500C22×C4C2×C22⋊D4711yescharacteristic
501C22×C4C2×C22⋊D4711yescharacteristic
502C22×C4C2×C22⋊D4711yescharacteristic
503C22×C4C2×C22⋊D4711yescharacteristic
504C22×C4C23×D4711yescharacteristic
505M16—3762no—
506C2×C8—942no—
507C2×C8—942no—
508C2×C8—3762no—
509C4⋊C4—81no—
510C4⋊C4—472no—
511C4⋊C4—472no—
512C4⋊C4—942no—
513C4⋊C4—942no—
514C4⋊C4—942no—
515C4⋊C4—941no—
516C4⋊C4—941no—
517C4⋊C4—941no—
518C4⋊C4—941no—
519C4⋊C4—942no—
520C4⋊C4(C2×D47)⋊C412yes—
521C4⋊C4(C2×D47)⋊C412yes—
522C22⋊C4—42no—
523C22⋊C4—42no—
524C22⋊C4—81no—
525C22⋊C4—942no—
526C22⋊C4—942no—
527C22⋊C4—942no—
528C22⋊C4—942no—
529C22⋊C4—942no—
530C22⋊C4—942no—
531C22⋊C4—942no—
532C22⋊C4—942no—
533C22⋊C4—942no—
534C22⋊C4(C2×D47)⋊C412yes—
535C22⋊C4(C2×D47)⋊C412yes—
536C22⋊C4(C2×D47)⋊C412yes—
537C22⋊C4(C2×D47)⋊C412yes—
538C42—42no—
539C42—81no—
540C42—942no—
541C42—942no—
542C42—942no—
543C42C2×C4×D4711yescharacteristic
544C42C2×C4×D4711yescharacteristic
545C42C2×D18811yescharacteristic
546C42C2×D18811yescharacteristic
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—942no—
556C23C2×(C2×D47)⋊C412yes—
557C23C2×(C2×D47)⋊C412yes—
558C23C22×C22⋊D4711yescharacteristic
559Q8—3761no—
560Q8—1882no—
561Q8—1882no—
562D4—42no—
563D4—42no—
564D4—3761no—
565D4—3762no—
566D4—3762no—
567D4—3762no—
568D4—3762no—
569D4—1882no—
570D4—1882no—
571D4—1882no—
572D4—1882no—
573C2×C4—21no—
574C2×C4—21no—
575C2×C4—22no—
576C2×C4—41no—
577C2×C4—41no—
578C2×C4—41no—
579C2×C4—42no—
580C2×C4—42no—
581C2×C4—41no—
582C2×C4—42no—
583C2×C4—42no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—41no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—84no—
596C2×C4—942no—
597C2×C4—942no—
598C2×C4—942no—
599C2×C4—3762no—
600C2×C4—3762no—
601C2×C4—1884no—
602C2×C4C2×(C2×D47)⋊C411yescharacteristic
603C2×C4C2×(C2×D47)⋊C412yes—
604C2×C4C2×(C2×D47)⋊C412yes—
605C2×C4C2×(C2×D47)⋊C411yescharacteristic
606C2×C4C2×(C2×D47)⋊C411yescharacteristic
607C2×C4C2×(C2×D47)⋊C411yescharacteristic
608C2×C4C22×C4×D4711yescharacteristic
609C2×C4C22×D18811yescharacteristic
610C8—1882no—
611C8—1882no—
612C22—21no—
613C22—21no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—22no—
620C22—42no—
621C22—42no—
622C22—81no—
623C22—81no—
624C22—1882no—
625C22—1882no—
626C22—1882no—
627C22C22×(C2×D47)⋊C411yescharacteristic
628C22#1568992111yescharacteristic
629C22#1568992111yescharacteristic
630C4—21no—
631C4—21no—
632C4—21no—
633C4—21no—
634C4—21no—
635C4—21no—
636C4—22no—
637C4—22no—
638C4—42no—
639C4—81no—
640C4—81no—
641C4—81no—
642C4—1882no—
643C4—1882no—
644C4—1882no—
645C2—21no—
646C2—21no—
647C2—42no—
648C2—42no—
649C2—3762no—
650C2#2025184811yescharacteristic
651C2#2025190511yescharacteristic
652C2#2025193611yescharacteristic
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2353C2
socleC22×C47C22×C22⋊C4
prime radical#20257240C2
cosocleC22×C4×C47C24
hypercenter#2353D47
hypocenterC47#17629
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #2353

Complements

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

NH ≅ G/Ntype
2. #20252417C2semidirect
3. #20241942C2semidirect
4. #20241938C2semidirect
5. #20241934C2semidirect
6. #20241929C2semidirect
7. #20241899C2semidirect
8. #20241895C2semidirect
9. #20241890C2semidirect
10. #20241877C2semidirect
11. C47×#2353C2semidirect
13. #15689843C4semidirect
14. #15689838C4semidirect
16. #15689794C4semidirect
17. #15689778C4semidirect
53. C2×#10625350C2×C4semidirect
56. #13094024C2×C4semidirect
57. #13094024C2×C4semidirect
58. #13094024C2×C4semidirect
59. #13094021C2×C4semidirect
74. #13094003C2×C4semidirect
78. #13093998C2×C4semidirect
87. C4×Dic94C2×C4semidirect
90. C4×D188C2×C4semidirect
98. #13093983C2×C4semidirect
321. #2353D47semidirect
377. C47#17629semidirect

Maximal subgroups

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

subgroupindex
#202572402
#202524172
#202419422
#202419382
#202419342
#202419292
#202418992
#202418952
#202418902
#202418772
C47×#23532
#1762947