Finite Group Databasebeta

← #26184264

Subgroups — #26184264

Order 12032.

1028 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257349C211yesmaximal, characteristic
2#20251816C211yesmaximal, characteristic, complemented
3#20242009C212yesmaximal, complemented
4#20242000C211yesmaximal, characteristic, complemented
5#20241996C211yesmaximal, characteristic, complemented
6#20241987C212yesmaximal, complemented
7#20241979C211yesmaximal, characteristic, complemented
8#20241977C211yesmaximal, characteristic, complemented
9#20241974C212yesmaximal, complemented
10C2×#15688862C212yesmaximal, complemented
11C47×#2970C211yesmaximal, characteristic, complemented
12C2×#13094031—21no—
13#15689858C411yescharacteristic
14#15689846C411yescharacteristic
15#15689836—21no—
16#15689797—22no—
17D4×Dic94C411yescharacteristic
18#15689782C412yes—
19#15689769C412yes—
20#15689761—21no—
21D4×D188C411yescharacteristic
22#15689745—21no—
23#15689706—22no—
24#15689673C2211yescharacteristic
25#15689657C2212yes—
26#15689657C2211yescharacteristic
27C2×#13094012C2212yes—
28#15689595—21no—
29#15689079—22no—
30#15689079—21no—
31C4×#10625329C2212yescomplemented
32C4×#10625329C2211yescharacteristic, complemented
33C2×#13093935C2212yescomplemented
34C2×#13093935C2211yescharacteristic, complemented
35#15688864C2211yescharacteristic, complemented
36#15688864C2212yescomplemented
37#15688864C2211yescharacteristic, complemented
38#15688862C2211yescharacteristic
39#15688862C2212yes—
40#15688862C2211yescharacteristic
41#15688862C2212yescomplemented
42#15688862C2212yescomplemented
43#15688862C2212yescomplemented
44#15688862C2212yescomplemented
45C4×#41×C47C2211yescharacteristic
46C47×#446—21no—
47C47×#435—22no—
48C8×D4×C47—21no—
49C47×#432C2212yescomplemented
50C47×#432C2211yescharacteristic, complemented
51C4×M16×C47C2211yescharacteristic, complemented
52C2×#97×C47C2212yescomplemented
53C2×#10625350C2×C412yes—
54C22×Dic94C2×C412yes—
55C22×D188C2×C412yes—
56C2×D4×D47—21no—
57#13094031—28no—
58C2×#10625329—22no—
59C2×#10625329—21no—
60C22×C47⋊C8—22no—
61C22×C47⋊C8—21no—
62#13094024C2×C411yescharacteristic
63#13094023—21no—
64#13094021C2×C411yescharacteristic
65#13094019—22no—
66#13094018—22no—
67#13094018—21no—
68#13094016C2311yescharacteristic
69#13094013—21no—
70#13094012—22no—
71#13094012—22no—
72#13094012C2×C412yes—
73#13094012C2×C412yes—
74#13094012C2311yescharacteristic
75#13094012C2312yes—
76#13094010—22no—
77#13094009—22no—
78#13094009—22no—
79#13094007—22no—
80#13094007—21no—
81#13094003C2×C412yes—
82#13094003C2×C411yescharacteristic
83#13094002—22no—
84#13094001—22no—
85#13094000—21no—
86#13093999—21no—
87#13093998—21no—
88#13093998—21no—
89#13093998C2×C412yes—
90#13093998C2×C411yescharacteristic
91#13093997—22no—
92#13093996—22no—
93#13093994—22no—
94#13093992—22no—
95C4×Dic94C2×C412yes—
96C4×Dic94C2×C411yescharacteristic
97C4×D188—21no—
98C4×D188—21no—
99C4×D188C2×C411yescharacteristic
100C4×D188C2×C412yes—
101D4×Dic47—21no—
102#13093986—21no—
103C4×C22⋊D47—21no—
104C4×C22⋊D47—22no—
105(C22⋊D47)⋊C4—22no—
106#13093982—21no—
107#13093981—21no—
108#13093981—22no—
109C2×(C2×D47)⋊C4—22no—
110C2×(C2×D47)⋊C4—22no—
111C22⋊C4×D47—22no—
112#13093935—22no—
113#13093935—21no—
114#13093935C2311yescharacteristic
115#13093935C2312yes—
116#13093935C2311yescharacteristic
117C4×C47⋊C8—22no—
118C4×C47⋊C8—21no—
119C2×#41×C47D411yescharacteristic
120C2×M16×C47—21no—
121C22×C8×C47—22no—
122C4×Q8×C47C2×C411yescharacteristic
123C4×Q8×C47D411yescharacteristic
124C4×D4×C47C2×C411yescharacteristic
125C4×D4×C47C2×C412yes—
126C4×D4×C47D411yescharacteristic
127C4×D4×C47D412yes—
128#117×C47D412yes—
129#117×C47D411yescharacteristic
130C2×C42×C47C2312yes—
131C2×C42×C47C2311yescharacteristic
132C4⋊C8×C47—21no—
133C4⋊C8×C47—22no—
134C22⋊C8×C47—22no—
135C22⋊C8×C47—21no—
136#97×C47—22no—
137#97×C47C2311yescharacteristic
138#97×C47C2312yes—
139#97×C47C2311yescharacteristic
140C4×C8×C47—21no—
141C23×D47—22no—
142#41×C47—24no—
143C2×Q8×C47C22⋊C411yescharacteristic
144C2×D4×C47C22⋊C412yes—
145C2×D4×C47C22⋊C411yescharacteristic
146C22×C4×C47C22×C412yes—
147C22×C4×C47C22×C411yescharacteristic
148C22×C4×C47C2×D412yes—
149C22×C4×C47C2×D411yescharacteristic
150C22×C4×C47C2×D412yes—
151C22×C4×C47C2×D411yescharacteristic
152C2×C22⋊D47—22no—
153C2×C22⋊D47—21no—
154C2×C22⋊D47—22no—
155C22×Dic47—22no—
156D4×D47—44no—
157#10625350—82no—
158C2×D188—21no—
159C2×D188—21no—
160C2×D188—22no—
161C2×D188—82no—
162C2×D188C22×C411yescharacteristic
163C2×D188C22×C412yes—
164C2×C4×D47—21no—
165C2×C4×D47—22no—
166C2×C4×D47—22no—
167C2×Dic94—22no—
168C2×Dic94—82no—
169C2×Dic94C22×C411yescharacteristic
170C2×Dic94C22×C412yes—
171M16×C47—42no—
172C2×C8×C47—21no—
173C2×C8×C47—22no—
174C2×C8×C47—21no—
175C2×C8×C47—44no—
176C4⋊C4×C47C22⋊C411yescharacteristic
177C4⋊C4×C47C22⋊C412yes—
178C4⋊C4×C47C22⋊C411yescharacteristic
179C4⋊C4×C47C22⋊C412yes—
180C22⋊C4×C47C22⋊C412yes—
181C22⋊C4×C47C22⋊C412yes—
182C22⋊C4×C47C22⋊C412yes—
183C42×C47C22×C412yes—
184C42×C47C22×C411yescharacteristic
185C42×C47C2×D412yes—
186C42×C47C2×D411yescharacteristic
187C42×C47C2×D411yescharacteristic
188C42×C47C2×D412yes—
189C42×C47C2411yescharacteristic
190C22⋊Dic47—22no—
191C22⋊Dic47—21no—
192(C2×D47)⋊C4—21no—
193(C2×D47)⋊C4—22no—
194(C2×D47)⋊C4—22no—
195(C2×D47)⋊C4—22no—
196(C2×D47)⋊C4—22no—
197(C2×D47)⋊C4—22no—
198(C2×D47)⋊C4—22no—
199(C2×D47)⋊C4—21no—
200(C2×D47)⋊C4—21no—
201(C2×D47)⋊C4—21no—
202(C2×D47)⋊C4—22no—
203(C2×D47)⋊C4—22no—
204C4⋊Dic47—21no—
205C4⋊Dic47—21no—
206C4⋊Dic47C22×C411yescharacteristic
207C4⋊Dic47C22×C411yescharacteristic
208Dic47⋊C4—21no—
209Dic47⋊C4—22no—
210Dic47⋊C4—22no—
211Dic47⋊C4—21no—
212Dic47⋊C4—21no—
213Dic47⋊C4—21no—
214C4×Dic47—22no—
215C4×Dic47—21no—
216#10625329—28no—
217#10625329—24no—
218C2×C47⋊C8—28no—
219C2×C47⋊C8—24no—
220C2×C47⋊C8—21no—
221C2×C47⋊C8—22no—
222C2×C47⋊C8—21no—
223C23×C47C2×C22⋊C411yescharacteristic
224C23×C47C2×C22⋊C412yes—
225C22×D47—21no—
226C22×D47—22no—
227C22×D47—44no—
228C22×D47—82no—
229C22×D47—82no—
230Q8×C47—22no—
231D4×C47—24no—
232D4×C47#10414yes—
233C2×C4×C47—24no—
234C2×C4×C47—22no—
235C2×C4×C47—42no—
236C2×C4×C47—41no—
237C2×C4×C47—42no—
238C2×C4×C47—41no—
239C2×C4×C47C2×C22⋊C412yes—
240C2×C4×C47C2×C22⋊C411yescharacteristic
241C2×C4×C47C2×C22⋊C412yes—
242C2×C4×C47C2×C22⋊C412yes—
243C2×C4×C47C2×C22⋊C411yescharacteristic
244C2×C4×C47C2×C22⋊C411yescharacteristic
245C2×C4×C47C23×C411yescharacteristic
246C2×C4×C47C22×D411yescharacteristic
247C2×C4×C47C22×D411yescharacteristic
248C22⋊D47—44no—
249C22⋊D47—82no—
250C22⋊D47—81no—
251C2×Dic47—21no—
252C2×Dic47—21no—
253C2×Dic47—22no—
254C2×Dic47—21no—
255C2×Dic47—82no—
256D188—41no—
257D188—42no—
258D188—42no—
259D188—81no—
260D188—82no—
261C4×D47—42no—
262C4×D47—82no—
263C4×D47—82no—
264Dic94—41no—
265Dic94—42no—
266Dic94—82no—
267C8×C47—22no—
268C8×C47—42no—
269C8×C47—41no—
270C47⋊C8—22no—
271C47⋊C8—24no—
272C47⋊C8—22no—
273#17652—471nomaximal, Sylow
274C22×C47—22no—
275C22×C47—22no—
276C22×C47—21no—
277C22×C47—21no—
278C22×C47—21no—
279C22×C47C2×#10414yes—
280C22×C47C22×C22⋊C411yescharacteristic
281C2×D47—42no—
282C2×D47—41no—
283C2×D47—41no—
284C2×D47—42no—
285C2×D47—42no—
286C2×D47—41no—
287C2×D47—164no—
288C4×C47—21no—
289C4×C47—22no—
290C4×C47—21no—
291C4×C47—21no—
292C4×C47—22no—
293C4×C47—21no—
294C4×C47—22no—
295C4×C47—21no—
296C4×C47C2×#10412yes—
297Dic47—41no—
298Dic47—42no—
299Dic47—41no—
300Dic47—41no—
301#3514—471no—
302#2970D4711yescharacteristic, complemented
303#2875—471no—
304#1532—472no—
305#1530—471no—
306#1529—471no—
307#1527—472no—
308#1525—471no—
309#1524—471no—
310#1524—472no—
311C2×#329—472no—
312C2×C47—24no—
313C2×C47—41no—
314C2×C47#288811yescharacteristic
315C2×C47C22×#10411yescharacteristic
316C2×C47#290111yescharacteristic
317D47—82no—
318D47—81no—
319C2×#131—941no—
320#559—471no—
321#556—471no—
322#555—941no—
323#551—942no—
324D4×Q8—471no—
325#548—472no—
326#546—472no—
327#546—941no—
328D42—471no—
329#542—941no—
330#538—942no—
331#532—471no—
332#529—471no—
333#529—472no—
334C2×#124—472no—
335#518—941no—
336C4×#41C2×D4711yescharacteristic
337#446—21no—
338#435—22no—
339C8×D4—21no—
340#432C2×D4712yescomplemented
341#432C2×D4711yescharacteristic, complemented
342#405—942no—
343#405—941no—
344C4×M16—472no—
345C4×M16—471no—
346C4×M16C2×D4711yescharacteristic, complemented
347C2×#97—472no—
348C2×#97—471no—
349C2×#97C2×D4712yescomplemented
350#330—471no—
351#330—472no—
352#330—471no—
353#329—471no—
354#329—472no—
355#329—471no—
356#329—472no—
357#329—472no—
358#329—472no—
359#329—472no—
360C47#1765211yescharacteristic, complemented, Sylow
361C2×#41—472no—
362C2×#41D18811yescharacteristic
363C22×Q8—472no—
364C22×D4—472no—
365C22×D4—941no—
366#131—948no—
367C2×M16—21no—
368C2×M16—942no—
369C2×M16—941no—
370C22×C8—22no—
371C22×C8—942no—
372C22×C8—941no—
373C4⋊Q8—471no—
374C4⋊Q8—941no—
375#127—471no—
376#126—942no—
377#126—941no—
378#126—942no—
379#125—471no—
380#124—472no—
381#124—472no—
382#124—471no—
383#124—472no—
384#124—941no—
385#124—942no—
386#124—942no—
387#124—942no—
388#123—942no—
389#123—941no—
390#123—942no—
391#123—942no—
392C22⋊Q8—472no—
393C22⋊Q8—471no—
394C22⋊Q8—941no—
395C22⋊Q8—942no—
396C22⋊Q8—942no—
397#121—472no—
398#121—471no—
399#121—942no—
400#121—941no—
401#121—941no—
402#121—941no—
403#121—942no—
404#120—942no—
405#120—942no—
406C4×Q8—472no—
407C4×Q8—471no—
408C4×Q8C4×D4711yescharacteristic
409C4×Q8C22⋊D4711yescharacteristic
410C4×D4—471no—
411C4×D4—472no—
412C4×D4—941no—
413C4×D4—941no—
414C4×D4—942no—
415C4×D4—941no—
416C4×D4—942no—
417C4×D4—941no—
418C4×D4—941no—
419C4×D4C4×D4711yescharacteristic
420C4×D4C4×D4712yes—
421C4×D4C22⋊D4711yescharacteristic
422C4×D4C22⋊D4712yes—
423#117—941no—
424#117—942no—
425#117—941no—
426#117D18812yes—
427#117D18811yescharacteristic
428C2×C22⋊C4—942no—
429C2×C22⋊C4—942no—
430C2×C22⋊C4—942no—
431C2×C42C22×D4712yes—
432C2×C42C22×D4711yescharacteristic
433C4⋊C8—21no—
434C4⋊C8—22no—
435C22⋊C8—22no—
436C22⋊C8—21no—
437#97—22no—
438#97—471no—
439#97—472no—
440#97—471no—
441#97—942no—
442#97—941no—
443#97C22×D4711yescharacteristic
444#97C22×D4712yes—
445#97C22×D4711yescharacteristic
446C4×C8—21no—
447C4×C8—942no—
448C4×C8—941no—
449C24—942no—
450#41—24no—
451#41—3762no—
452C2×Q8—471no—
453C2×Q8—472no—
454C2×Q8—942no—
455C2×Q8—3762no—
456C2×Q8(C2×D47)⋊C411yescharacteristic
457C2×D4—471no—
458C2×D4—472no—
459C2×D4—941no—
460C2×D4—942no—
461C2×D4—942no—
462C2×D4—941no—
463C2×D4—942no—
464C2×D4—941no—
465C2×D4—3762no—
466C2×D4—1884no—
467C2×D4(C2×D47)⋊C412yes—
468C2×D4(C2×D47)⋊C411yescharacteristic
469C22×C4—942no—
470C22×C4—942no—
471C22×C4—942no—
472C22×C4—941no—
473C22×C4C2×C4×D4712yes—
474C22×C4C2×C4×D4711yescharacteristic
475C22×C4C2×D18812yes—
476C22×C4C2×D18811yescharacteristic
477C22×C4C2×C22⋊D4712yes—
478C22×C4C2×C22⋊D4711yescharacteristic
479M16—42no—
480M16—948no—
481M16—944no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—22no—
485C2×C8—44no—
486C2×C8—948no—
487C2×C8—944no—
488C2×C8—941no—
489C2×C8—942no—
490C2×C8—941no—
491C4⋊C4—471no—
492C4⋊C4—471no—
493C4⋊C4—941no—
494C4⋊C4—942no—
495C4⋊C4—942no—
496C4⋊C4—941no—
497C4⋊C4—941no—
498C4⋊C4—941no—
499C4⋊C4—941no—
500C4⋊C4—941no—
501C4⋊C4(C2×D47)⋊C411yescharacteristic
502C4⋊C4(C2×D47)⋊C411yescharacteristic
503C4⋊C4(C2×D47)⋊C412yes—
504C4⋊C4(C2×D47)⋊C412yes—
505C22⋊C4—942no—
506C22⋊C4—941no—
507C22⋊C4—941no—
508C22⋊C4—942no—
509C22⋊C4—942no—
510C22⋊C4—942no—
511C22⋊C4—942no—
512C22⋊C4—942no—
513C22⋊C4—941no—
514C22⋊C4—942no—
515C22⋊C4—941no—
516C22⋊C4—942no—
517C22⋊C4—941no—
518C22⋊C4—942no—
519C22⋊C4(C2×D47)⋊C412yes—
520C22⋊C4(C2×D47)⋊C412yes—
521C22⋊C4(C2×D47)⋊C412yes—
522C42—942no—
523C42—941no—
524C42C2×C4×D4712yes—
525C42C2×C4×D4711yescharacteristic
526C42C2×D18812yes—
527C42C2×D18811yescharacteristic
528C42C2×C22⋊D4711yescharacteristic
529C42C2×C22⋊D4712yes—
530C42C23×D4711yescharacteristic
531C23—941no—
532C23—942no—
533C23—3762no—
534C23—3762no—
535C23—1884no—
536C23C2×(C2×D47)⋊C411yescharacteristic
537C23C2×(C2×D47)⋊C412yes—
538Q8—22no—
539Q8—3762no—
540Q8—1881no—
541Q8—1882no—
542D4—24no—
543D4—3761no—
544D4—3762no—
545D4—3761no—
546D4—3762no—
547D4—1881no—
548D4—1882no—
549D4—1884no—
550D4—1882no—
551D4#1309395014yes—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—42no—
555C2×C4—41no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—941no—
559C2×C4—941no—
560C2×C4—941no—
561C2×C4—942no—
562C2×C4—3762no—
563C2×C4—3762no—
564C2×C4—3762no—
565C2×C4—1882no—
566C2×C4C2×(C2×D47)⋊C412yes—
567C2×C4C2×(C2×D47)⋊C411yescharacteristic
568C2×C4C2×(C2×D47)⋊C412yes—
569C2×C4C2×(C2×D47)⋊C411yescharacteristic
570C2×C4C2×(C2×D47)⋊C412yes—
571C2×C4C2×(C2×D47)⋊C411yescharacteristic
572C2×C4C22×C4×D4711yescharacteristic
573C2×C4C22×C22⋊D4711yescharacteristic
574C2×C4C22×D18811yescharacteristic
575C8—22no—
576C8—42no—
577C8—41no—
578C8—942no—
579C8—944no—
580C8—942no—
581C22—22no—
582C22—22no—
583C22—21no—
584C22—21no—
585C22—21no—
586C22—1881no—
587C22—1882no—
588C22—1882no—
589C22—1881no—
590C22—1882no—
591C22—1881no—
592C22—7524no—
593C22C2×#1309395014yes—
594C22C22×(C2×D47)⋊C411yescharacteristic
595C4—21no—
596C4—22no—
597C4—21no—
598C4—22no—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—22no—
603C4—1881no—
604C4—1881no—
605C4—1882no—
606C4—1881no—
607C4C2×#1309395012yes—
608C2—24no—
609C2—41no—
610C2—3761no—
611C2—3762no—
612C2#2025186611yescharacteristic
613C2C22×#1309395011yescharacteristic
614C2#2025193611yescharacteristic
subgroupG/H
centerC22C22×(C2×D47)⋊C4
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
FittingC47×#2970C2
socleC22×C47C22×C22⋊C4
prime radical#20257349C2
cosocleC42×C47C24
hypercenter#2970D47
hypocenterC47#17652
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C4×#41 ◁ #2970

Complements

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

NH ≅ G/Ntype
2. #20251816C2semidirect
3. #20242009C2semidirect
4. #20242000C2semidirect
5. #20241996C2semidirect
6. #20241987C2semidirect
7. #20241979C2semidirect
8. #20241977C2semidirect
9. #20241974C2semidirect
10. C2×#15688862C2semidirect
11. C47×#2970C2semidirect
31. C4×#10625329C22semidirect
32. C4×#10625329C22semidirect
33. C2×#13093935C22semidirect
34. C2×#13093935C22semidirect
35. #15688864C22semidirect
36. #15688864C22semidirect
37. #15688864C22semidirect
41. #15688862C22semidirect
42. #15688862C22semidirect
43. #15688862C22semidirect
44. #15688862C22semidirect
49. C47×#432C22semidirect
50. C47×#432C22semidirect
51. C4×M16×C47C22semidirect
52. C2×#97×C47C22semidirect
302. #2970D47semidirect
340. #432C2×D47semidirect
341. #432C2×D47semidirect
346. C4×M16C2×D47semidirect
349. C2×#97C2×D47semidirect
360. C47#17652semidirect

Maximal subgroups

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

subgroupindex
#202573492
#202518162
#202420092
#202420002
#202419962
#202419872
#202419792
#202419772
#202419742
C2×#156888622
C47×#29702
#1765247