Finite Group Databasebeta

← #26183861

Subgroups — #26183861

Order 12032.

938 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257473C211yesmaximal, characteristic
2#20252477C211yesmaximal, characteristic, complemented
3#20241963C212yesmaximal, complemented
4#20241947C212yesmaximal, complemented
5#20241934C212yesmaximal, complemented
6#20241924C212yesmaximal, complemented
7#20241911C211yesmaximal, characteristic, complemented
8#20241895C211yesmaximal, characteristic, complemented
9#20241882C211yesmaximal, characteristic, complemented
10#20241872C211yesmaximal, characteristic, complemented
11C47×#2320C211yesmaximal, characteristic, complemented
12#15689858C411yescharacteristic, complemented
13#15689858C412yescomplemented
14Q8×Dic94C411yescharacteristic, complemented
15#15689833C411yescharacteristic, complemented
16#15689833C412yescomplemented
17#15689795C411yescharacteristic, complemented
18#15689784—21no—
19#15689774—22no—
20#15689774—21no—
21#15689745—21no—
22#15689745—22no—
23#15689673C2212yes—
24#15689673C2211yescharacteristic
25#15689657C2212yes—
26#15689657C2211yescharacteristic
27#15689605—21no—
28#15689254—22no—
29#15689254—21no—
30Q8×C47⋊C8—21no—
31#15689180C2212yes—
32#15689180C2211yescharacteristic
33#15689176C2212yes—
34#15689176C2211yescharacteristic
35#15688859C2211yescharacteristic
36#15688859C2212yes—
37#15688859C2211yescharacteristic
38#15688856C2212yes—
39#15688856C2212yes—
40#15688856C2211yescharacteristic
41#15688856C2212yes—
42#15688856C2212yes—
43#15688856C2211yescharacteristic
44#15688856C2212yes—
45C2×C4×Q8×C47C2211yescharacteristic
46C47×#391—21no—
47C47×#389—21no—
48C47×#387—22no—
49C47×#383C2212yes—
50C47×#382C2212yes—
51C4×C4⋊C4×C47C2211yescharacteristic
52C47×#376C2211yescharacteristic
53C2×#10625354—21no—
54C2×#10625350—21no—
55C2×#10625350—22no—
56#13094024C2×C412yes—
57#13094024C2×C411yescharacteristic
58#13094023—22no—
59#13094023—21no—
60#13094021C2×C412yes—
61#13094021C2×C411yescharacteristic
62#13094018—22no—
63#13094018—21no—
64#13094016C2×C412yes—
65#13094016C2×C411yescharacteristic
66#13094013—21no—
67#13094013—22no—
68#13094012—22no—
69#13094012—21no—
70#13094012C2×C412yes—
71#13094012C2×C411yescharacteristic
72#13094009—22no—
73#13094009—22no—
74#13094009—22no—
75#13094007—21no—
76#13094007—22no—
77#13094003C2311yescharacteristic
78#13094002—22no—
79#13094002—22no—
80#13094002—22no—
81#13094000—21no—
82#13094000—22no—
83#13093999—22no—
84#13093999—21no—
85#13093998—21no—
86#13093998—22no—
87#13093998C2312yes—
88#13093998C2311yescharacteristic
89Q8×Dic47—21no—
90C4×Dic94C2×C412yes—
91C4×Dic94C2×C411yescharacteristic
92C4×D188—21no—
93C4×D188—22no—
94C4×D188C2×C412yes—
95C4×D188C2×C411yescharacteristic
96C4×D188C2×C412yes—
97C4×D188C2×C412yes—
98C4×D188C2×C412yes—
99#13093986—22no—
100#13093986—21no—
101C4×C22⋊D47—21no—
102C4×C22⋊D47—22no—
103#13093981—21no—
104#13093981—22no—
105C4⋊C4×D47—22no—
106C4⋊C4×D47—21no—
107C4⋊(C47⋊C8)—21no—
108C4⋊(C47⋊C8)—22no—
109C4⋊(C47⋊C8)—22no—
110C4⋊(C47⋊C8)—21no—
111C22⋊(C47⋊C8)C2311yescharacteristic
112C22⋊(C47⋊C8)C2311yescharacteristic
113C22⋊(C47⋊C8)C2312yes—
114#13093935—22no—
115#13093935—21no—
116C4×C47⋊C8—21no—
117C4×C47⋊C8—22no—
118C22×Q8×C47D411yescharacteristic
119C4×Q8×C47C2×C411yescharacteristic
120C4×Q8×C47C2×C411yescharacteristic
121C4×Q8×C47C2×C412yes—
122C4×Q8×C47D411yescharacteristic
123C4×Q8×C47D412yes—
124C4×Q8×C47D411yescharacteristic
125C2×C4⋊C4×C47—21no—
126C2×C4⋊C4×C47—22no—
127C2×C4⋊C4×C47D412yes—
128C2×C4⋊C4×C47D411yescharacteristic
129C2×C42×C47—21no—
130C2×C42×C47—21no—
131C2×C42×C47C2311yescharacteristic
132C2×C42×C47C2312yes—
133#95×C47—21no—
134#95×C47—22no—
135#95×C47—22no—
136#95×C47—22no—
137#95×C47C2311yescharacteristic
138#95×C47C2311yescharacteristic
139#95×C47C2312yes—
140C2×Q8×C47—42no—
141C2×Q8×C47C22⋊C411yescharacteristic
142C2×Q8×C47C22⋊C411yescharacteristic
143C2×Q8×C47C22⋊C412yes—
144C22×C4×C47—21no—
145C22×C4×C47—22no—
146C22×C4×C47—21no—
147C22×C4×C47C2×D411yescharacteristic
148C22×C4×C47C2×D411yescharacteristic
149C22×C4×C47C2×D412yes—
150C22×C4×C47C2×D412yes—
151C22×C4×C47C2411yescharacteristic
152C2×C22⋊D47—21no—
153C2×C22⋊D47—22no—
154#10625354—44no—
155#10625350—81no—
156#10625350—82no—
157#10625350—81no—
158#10625350—82no—
159C2×D188—22no—
160C2×D188—21no—
161C2×D188—22no—
162C2×D188—21no—
163C2×D188C22×C412yes—
164C2×D188C22×C411yescharacteristic
165C2×C4×D47—22no—
166C2×C4×D47—21no—
167C2×C4×D47—22no—
168C2×C4×D47—22no—
169C2×C4×D47—22no—
170C2×Dic94—22no—
171C2×Dic94—21no—
172C2×Dic94C22×C411yescharacteristic
173C4⋊C4×C47—42no—
174C4⋊C4×C47—82no—
175C4⋊C4×C47C22⋊C412yes—
176C4⋊C4×C47C22⋊C411yescharacteristic
177C4⋊C4×C47C22⋊C412yes—
178C4⋊C4×C47C22⋊C411yescharacteristic
179C4⋊C4×C47C22⋊C412yes—
180C4⋊C4×C47C22⋊C412yes—
181C4⋊C4×C47C22⋊C412yes—
182C42×C47—42no—
183C42×C47—81no—
184C42×C47C22×C411yescharacteristic
185C42×C47C22×C412yes—
186C42×C47C22×C412yes—
187C42×C47C22×C411yescharacteristic
188C42×C47C2×D411yescharacteristic
189C42×C47C2×D412yes—
190C42×C47C2×D412yes—
191C42×C47C2×D411yescharacteristic
192C22⋊Dic47—21no—
193(C2×D47)⋊C4—21no—
194(C2×D47)⋊C4—22no—
195(C2×D47)⋊C4—21no—
196(C2×D47)⋊C4—22no—
197(C2×D47)⋊C4—22no—
198(C2×D47)⋊C4—22no—
199(C2×D47)⋊C4—22no—
200(C2×D47)⋊C4—22no—
201(C2×D47)⋊C4—21no—
202(C2×D47)⋊C4—21no—
203(C2×D47)⋊C4—22no—
204C4⋊Dic47—22no—
205C4⋊Dic47—21no—
206C4⋊Dic47C22×C411yescharacteristic
207C4⋊Dic47C22×C411yescharacteristic
208C4⋊Dic47C22×C412yes—
209Dic47⋊C4—21no—
210Dic47⋊C4—22no—
211Dic47⋊C4—21no—
212Dic47⋊C4—22no—
213Dic47⋊C4—22no—
214Dic47⋊C4—21no—
215Dic47⋊C4—21no—
216C4×Dic47—22no—
217C4×Dic47—21no—
218C2×C47⋊C8—21no—
219C2×C47⋊C8—21no—
220C2×C47⋊C8—22no—
221C23×C47C22×D411yescharacteristic
222C22×D47—22no—
223C22×D47—21no—
224Q8×C47—22no—
225Q8×C47—22no—
226Q8×C47—42no—
227C2×C4×C47—21no—
228C2×C4×C47—22no—
229C2×C4×C47—21no—
230C2×C4×C47—22no—
231C2×C4×C47—22no—
232C2×C4×C47—42no—
233C2×C4×C47—41no—
234C2×C4×C47—41no—
235C2×C4×C47—41no—
236C2×C4×C47—42no—
237C2×C4×C47—41no—
238C2×C4×C47—42no—
239C2×C4×C47—41no—
240C2×C4×C47—42no—
241C2×C4×C47—41no—
242C2×C4×C47—41no—
243C2×C4×C47C2×C22⋊C412yes—
244C2×C4×C47C2×C22⋊C411yescharacteristic
245C2×C4×C47C2×C22⋊C412yes—
246C2×C4×C47C2×C22⋊C412yes—
247C2×C4×C47C2×C22⋊C411yescharacteristic
248C2×C4×C47C2×C22⋊C411yescharacteristic
249C2×C4×C47C2×C22⋊C412yes—
250C2×C4×C47C2×C22⋊C411yescharacteristic
251C2×C4×C47C23×C411yescharacteristic
252C2×C4×C47C22×D411yescharacteristic
253C22⋊D47—42no—
254C22⋊D47—41no—
255C22⋊D47—42no—
256C22⋊D47—41no—
257C2×Dic47—21no—
258C2×Dic47—22no—
259C2×Dic47—21no—
260C2×Dic47—21no—
261D188—42no—
262D188—41no—
263D188—44no—
264D188—42no—
265D188—81no—
266D188—82no—
267C4×D47—44no—
268C4×D47—42no—
269C4×D47—82no—
270C4×D47—82no—
271C4×D47—82no—
272Dic94—41no—
273Dic94—81no—
274Dic94—82no—
275C47⋊C8—41no—
276C47⋊C8—41no—
277C47⋊C8—42no—
278#17602—471nomaximal, Sylow
279C22×C47—21no—
280C22×C47—21no—
281C22×C47C22×C22⋊C411yescharacteristic
282C22×C47#57611yescharacteristic
283C22×C47#57711yescharacteristic
284C2×D47—42no—
285C2×D47—41no—
286C2×D47—42no—
287C2×D47—41no—
288C2×D47—42no—
289C2×D47—41no—
290C4×C47—21no—
291C4×C47—21no—
292C4×C47—21no—
293C4×C47—22no—
294C4×C47—21no—
295C4×C47—22no—
296C4×C47—22no—
297C4×C47—21no—
298C4×C47—22no—
299C4×C47—81no—
300C4×C47—82no—
301C4×C47—81no—
302C4×C47#42112yes—
303Dic47—41no—
304Dic47—42no—
305Dic47—41no—
306Dic47—41no—
307#3522—471no—
308#2984—471no—
309#2320D4711yescharacteristic, complemented
310#1521—472no—
311#1517—472no—
312#1514—472no—
313#1512—472no—
314#1509—471no—
315#1505—471no—
316#1502—471no—
317#1500—471no—
318C2×C47—21no—
319C2×C47—21no—
320C2×C47#288411yescharacteristic
321C2×C47#289711yescharacteristic
322C2×C47C2×#42111yescharacteristic
323D47—82no—
324D47—81no—
325#559—472no—
326#559—471no—
327Q82—471no—
328#554—471no—
329#554—472no—
330#550—471no—
331#548—941no—
332#547—942no—
333#547—941no—
334#542—941no—
335#542—942no—
336#532—471no—
337#532—472no—
338#529—471no—
339#529—472no—
340#520—941no—
341C2×C4×Q8C2×D4711yescharacteristic
342#446—941no—
343#446—942no—
344C8×Q8—941no—
345#432—471no—
346#432—472no—
347#431—471no—
348#431—472no—
349#391—21no—
350#389—21no—
351#387—22no—
352#383C2×D4712yes—
353#382C2×D4712yes—
354C4×C4⋊C4C2×D4711yescharacteristic
355#376C2×D4711yescharacteristic
356#328—471no—
357#328—471no—
358#328—472no—
359#327—471no—
360#327—471no—
361#327—472no—
362#327—472no—
363#327—472no—
364#327—472no—
365#327—472no—
366C47#1760211yescharacteristic, complemented, Sylow
367C2×#41—941no—
368C2×#41—941no—
369C2×#41—942no—
370C22×Q8C22⋊D4711yescharacteristic
371C4⋊Q8—471no—
372C4⋊Q8—472no—
373C4⋊Q8—942no—
374C4⋊Q8—941no—
375#127—471no—
376#127—472no—
377#126—942no—
378#126—941no—
379#125—471no—
380#125—472no—
381#124—471no—
382#124—472no—
383#124—941no—
384#124—942no—
385#124—942no—
386#124—941no—
387#123—941no—
388#123—942no—
389#123—942no—
390#123—942no—
391#123—942no—
392C22⋊Q8—471no—
393C22⋊Q8—941no—
394C22⋊Q8—942no—
395C22⋊Q8—942no—
396C22⋊Q8—942no—
397C22⋊Q8—942no—
398#121—471no—
399#121—472no—
400#121—941no—
401#121—942no—
402#121—942no—
403#121—941no—
404C4×Q8—471no—
405C4×Q8—472no—
406C4×Q8—941no—
407C4×Q8C4×D4711yescharacteristic
408C4×Q8C4×D4711yescharacteristic
409C4×Q8C4×D4712yes—
410C4×Q8D18811yescharacteristic
411C4×Q8D18811yescharacteristic
412C4×Q8D18812yes—
413C4×D4—471no—
414C4×D4—472no—
415C4×D4—472no—
416C4×D4—472no—
417C4×D4—472no—
418C4×D4—941no—
419C4×D4—942no—
420C4×D4—942no—
421C4×D4—941no—
422C4×D4—941no—
423C4×D4—942no—
424#117—941no—
425#117—942no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—22no—
428C2×C4⋊C4—942no—
429C2×C4⋊C4—941no—
430C2×C4⋊C4C22⋊D4712yes—
431C2×C4⋊C4C22⋊D4711yescharacteristic
432C2×C42—21no—
433C2×C42—21no—
434C2×C42C22×D4711yescharacteristic
435C2×C42C22×D4712yes—
436C4⋊C8—942no—
437C4⋊C8—941no—
438C4⋊C8—941no—
439C4⋊C8—942no—
440C22⋊C8—471no—
441C22⋊C8—471no—
442C22⋊C8—472no—
443#97—942no—
444#97—941no—
445C4×C8—941no—
446C4×C8—942no—
447#95—21no—
448#95—22no—
449#95—22no—
450#95—22no—
451#95C22×D4711yescharacteristic
452#95C22×D4711yescharacteristic
453#95C22×D4712yes—
454#41—3761no—
455#41—3762no—
456#41—3761no—
457#41—3762no—
458#41—1884no—
459C2×Q8—42no—
460C2×Q8—471no—
461C2×Q8—942no—
462C2×Q8—941no—
463C2×Q8(C2×D47)⋊C411yescharacteristic
464C2×Q8(C2×D47)⋊C411yescharacteristic
465C2×Q8(C2×D47)⋊C412yes—
466C2×D4—471no—
467C2×D4—472no—
468C2×D4—941no—
469C2×D4—942no—
470C2×D4—942no—
471C2×D4—941no—
472C2×D4—942no—
473C2×D4—941no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—22no—
477C22×C4—942no—
478C22×C4—942no—
479C22×C4—942no—
480C22×C4—942no—
481C22×C4—941no—
482C22×C4C2×C22⋊D4711yescharacteristic
483C22×C4C2×C22⋊D4712yes—
484C22×C4C2×C22⋊D4711yescharacteristic
485C22×C4C2×C22⋊D4712yes—
486C22×C4C23×D4711yescharacteristic
487C2×C8—941no—
488C2×C8—941no—
489C2×C8—942no—
490C4⋊C4—42no—
491C4⋊C4—82no—
492C4⋊C4—471no—
493C4⋊C4—472no—
494C4⋊C4—471no—
495C4⋊C4—941no—
496C4⋊C4—942no—
497C4⋊C4—941no—
498C4⋊C4—942no—
499C4⋊C4—942no—
500C4⋊C4—941no—
501C4⋊C4—942no—
502C4⋊C4—941no—
503C4⋊C4—941no—
504C4⋊C4(C2×D47)⋊C412yes—
505C4⋊C4(C2×D47)⋊C411yescharacteristic
506C4⋊C4(C2×D47)⋊C412yes—
507C4⋊C4(C2×D47)⋊C411yescharacteristic
508C4⋊C4(C2×D47)⋊C412yes—
509C4⋊C4(C2×D47)⋊C412yes—
510C4⋊C4(C2×D47)⋊C412yes—
511C22⋊C4—941no—
512C22⋊C4—941no—
513C22⋊C4—942no—
514C22⋊C4—942no—
515C22⋊C4—941no—
516C22⋊C4—942no—
517C22⋊C4—942no—
518C22⋊C4—942no—
519C22⋊C4—941no—
520C22⋊C4—942no—
521C22⋊C4—941no—
522C22⋊C4—942no—
523C42—42no—
524C42—81no—
525C42—942no—
526C42—941no—
527C42C2×C4×D4711yescharacteristic
528C42C2×C4×D4712yes—
529C42C2×C4×D4712yes—
530C42C2×C4×D4711yescharacteristic
531C42C2×D18811yescharacteristic
532C42C2×D18812yes—
533C42C2×D18812yes—
534C42C2×D18811yescharacteristic
535C23—942no—
536C23—941no—
537C23C22×C22⋊D4711yescharacteristic
538Q8—22no—
539Q8—22no—
540Q8—42no—
541Q8—3761no—
542Q8—3762no—
543Q8—1881no—
544D4—3761no—
545D4—3762no—
546D4—1881no—
547D4—1882no—
548D4—1884no—
549D4—1882no—
550D4—1881no—
551D4—1881no—
552D4—1882no—
553D4—1882no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—42no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—941no—
571C2×C4—942no—
572C2×C4—941no—
573C2×C4—941no—
574C2×C4—3762no—
575C2×C4—3762no—
576C2×C4—3762no—
577C2×C4—1884no—
578C2×C4—1882no—
579C2×C4C2×(C2×D47)⋊C412yes—
580C2×C4C2×(C2×D47)⋊C411yescharacteristic
581C2×C4C2×(C2×D47)⋊C412yes—
582C2×C4C2×(C2×D47)⋊C411yescharacteristic
583C2×C4C2×(C2×D47)⋊C412yes—
584C2×C4C2×(C2×D47)⋊C412yes—
585C2×C4C2×(C2×D47)⋊C411yescharacteristic
586C2×C4C2×(C2×D47)⋊C411yescharacteristic
587C2×C4C22×C4×D4711yescharacteristic
588C2×C4C22×D18811yescharacteristic
589C8—1881no—
590C8—1881no—
591C8—1882no—
592C22—21no—
593C22—21no—
594C22—1882no—
595C22—1881no—
596C22—1882no—
597C22—1881no—
598C22—1882no—
599C22—1881no—
600C22C22×(C2×D47)⋊C411yescharacteristic
601C22#1568992611yescharacteristic
602C22#1568993111yescharacteristic
603C4—21no—
604C4—22no—
605C4—21no—
606C4—22no—
607C4—21no—
608C4—22no—
609C4—22no—
610C4—21no—
611C4—21no—
612C4—81no—
613C4—82no—
614C4—81no—
615C4—1882no—
616C4—1881no—
617C4—1881no—
618C4—1881no—
619C4#1568914112yes—
620C2—21no—
621C2—21no—
622C2—3762no—
623C2—3761no—
624C2#2025185111yescharacteristic
625C2#2025191111yescharacteristic
626C2C2×#1568914111yescharacteristic
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#2320C2
socleC22×C47C22×C22⋊C4
prime radical#20257473C2
cosocleC22×C4×C47C24
hypercenter#2320D47
hypocenterC47#17602
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2320

Complements

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

NH ≅ G/Ntype
2. #20252477C2semidirect
3. #20241963C2semidirect
4. #20241947C2semidirect
5. #20241934C2semidirect
6. #20241924C2semidirect
7. #20241911C2semidirect
8. #20241895C2semidirect
9. #20241882C2semidirect
10. #20241872C2semidirect
11. C47×#2320C2semidirect
12. #15689858C4semidirect
13. #15689858C4semidirect
14. Q8×Dic94C4semidirect
15. #15689833C4semidirect
16. #15689833C4semidirect
17. #15689795C4semidirect
309. #2320D47semidirect
366. C47#17602semidirect

Maximal subgroups

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

subgroupindex
#202574732
#202524772
#202419632
#202419472
#202419342
#202419242
#202419112
#202418952
#202418822
#202418722
C47×#23202
#1760247