Finite Group Databasebeta

← #26184484

Subgroups — #26184484

Order 12032.

932 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20252402C211yesmaximal, characteristic, complemented
2#20252039C211yesmaximal, characteristic, complemented
3#20242023C212yesmaximal, complemented
4#20242019C212yesmaximal, complemented
5#20242008C212yesmaximal, complemented
6#20241999C211yesmaximal, characteristic, complemented
7#20241999C212yesmaximal, complemented
8#20241995C211yesmaximal, characteristic, complemented
9#20241980C211yesmaximal, characteristic, complemented
10#20241976C211yesmaximal, characteristic, complemented
11C47×#3517C211yesmaximal, characteristic
12#15689254—21no—
13#15689195—22no—
14D4×C47⋊C8—21no—
15#15689184—22no—
16#15689180C2211yescharacteristic
17#15689180C2212yes—
18#15689180C2212yes—
19#15689180C2212yes—
20#15689150—21no—
21#15689090—21no—
22C4×#10625329C2211yescharacteristic
23C4×#10625329C2211yescharacteristic
24C2×#13093935C2211yescharacteristic
25C2×#13093935C2212yes—
26#15688865C2211yescharacteristic
27#15688865C2211yescharacteristic
28#15688865C2212yes—
29#15688863C2211yescharacteristic
30#15688863C2211yescharacteristic
31#15688863C2212yes—
32#15688863C2212yes—
33#15688863C2212yes—
34#15688863C2212yes—
35#15688863C2212yes—
36C47×#557C411yescharacteristic
37C47×#553—21no—
38C47×#553—22no—
39C47×#550—21no—
40C47×#550C411yescharacteristic
41C47×#550C412yes—
42C47×#550C411yescharacteristic
43C47×#547C411yescharacteristic
44C47×#547C412yes—
45C47×#540—21no—
46C47×#540—22no—
47C47×#532C2211yescharacteristic
48C47×#532C2212yes—
49C47×#529C2211yescharacteristic
50C47×#529C2212yes—
51C47×#520—21no—
52C4×#41×C47C2211yescharacteristic
53C2×#10625329—21no—
54C2×#10625329—21no—
55C22×C47⋊C8—21no—
56C22×C47⋊C8—22no—
57C4⋊(C47⋊C8)—21no—
58C4⋊(C47⋊C8)—22no—
59C4⋊(C47⋊C8)—22no—
60C4⋊(C47⋊C8)—22no—
61C22⋊(C47⋊C8)—22no—
62C22⋊(C47⋊C8)—22no—
63C22⋊(C47⋊C8)—22no—
64C22⋊(C47⋊C8)—21no—
65#13093935—22no—
66#13093935—21no—
67#13093935C2311yescharacteristic
68#13093935C2311yescharacteristic
69#13093935C2311yescharacteristic
70#13093935C2312yes—
71#13093935C2311yescharacteristic
72#13093935C2312yes—
73C4×C47⋊C8—21no—
74C4×C47⋊C8—21no—
75C2×#41×C47—21no—
76C2×#41×C47C2×C412yes—
77C2×#41×C47C2×C412yes—
78C2×#41×C47C2×C412yes—
79C2×#41×C47D411yescharacteristic
80C4⋊Q8×C47C2×C411yescharacteristic
81C4⋊Q8×C47C2×C412yes—
82#127×C47—21no—
83#127×C47—22no—
84#127×C47C2×C411yescharacteristic
85#127×C47C2×C412yes—
86#126×C47—21no—
87#126×C47—22no—
88#125×C47—22no—
89#125×C47—21no—
90#125×C47C2311yescharacteristic
91#124×C47C2311yescharacteristic
92#124×C47C2312yes—
93#123×C47—22no—
94#123×C47—21no—
95#123×C47—22no—
96#123×C47—22no—
97#123×C47—22no—
98C22⋊Q8×C47C2×C411yescharacteristic
99C22⋊Q8×C47C2×C412yes—
100#121×C47—21no—
101#121×C47—22no—
102#121×C47—22no—
103#121×C47—21no—
104#121×C47—21no—
105#121×C47—22no—
106#121×C47—22no—
107#121×C47—21no—
108#121×C47—22no—
109#121×C47—22no—
110#121×C47—22no—
111#121×C47C2×C411yescharacteristic
112#121×C47C2×C412yes—
113C4×Q8×C47—21no—
114C4×Q8×C47C2×C411yescharacteristic
115C4×Q8×C47C2×C412yes—
116C4×Q8×C47C2×C411yescharacteristic
117C4×Q8×C47D411yescharacteristic
118C4×D4×C47—21no—
119C4×D4×C47—22no—
120C4×D4×C47—22no—
121C4×D4×C47—21no—
122C4×D4×C47—21no—
123C4×D4×C47—22no—
124C4×D4×C47—21no—
125C4×D4×C47—22no—
126C4×D4×C47C2×C411yescharacteristic
127C4×D4×C47C2×C412yes—
128C4×D4×C47C2×C411yescharacteristic
129C4×D4×C47C2×C412yes—
130C4×D4×C47D411yescharacteristic
131C4×D4×C47D412yes—
132#117×C47—21no—
133#117×C47—22no—
134#117×C47D411yescharacteristic
135#117×C47D412yes—
136C2×C4⋊C4×C47—22no—
137C2×C4⋊C4×C47—21no—
138C2×C42×C47C2311yescharacteristic
139C2×C42×C47C2312yes—
140#41×C47—42no—
141#41×C47—44no—
142#41×C47—82no—
143#41×C47—82no—
144#41×C47—82no—
145C2×Q8×C47C22⋊C411yescharacteristic
146C2×Q8×C47C22×C411yescharacteristic
147C2×Q8×C47C22×C412yes—
148C2×D4×C47—21no—
149C2×D4×C47—22no—
150C2×D4×C47—22no—
151C2×D4×C47—22no—
152C2×D4×C47—22no—
153C2×D4×C47—22no—
154C2×D4×C47—21no—
155C2×D4×C47—21no—
156C2×D4×C47—22no—
157C2×D4×C47C22⋊C411yescharacteristic
158C2×D4×C47C22⋊C412yes—
159C2×D4×C47C22×C411yescharacteristic
160C2×D4×C47C22×C412yes—
161C22×C4×C47—22no—
162C22×C4×C47—21no—
163C22×C4×C47—22no—
164C22×C4×C47—22no—
165C22×C4×C47—22no—
166C22×C4×C47C22×C411yescharacteristic
167C22×C4×C47C22×C412yes—
168C22×C4×C47C2×D411yescharacteristic
169C22×C4×C47C2×D412yes—
170C22×C4×C47C2×D412yes—
171C22×C4×C47C2×D411yescharacteristic
172C4⋊C4×C47—21no—
173C4⋊C4×C47—22no—
174C4⋊C4×C47—22no—
175C4⋊C4×C47—21no—
176C4⋊C4×C47—21no—
177C4⋊C4×C47—22no—
178C4⋊C4×C47—21no—
179C4⋊C4×C47—22no—
180C4⋊C4×C47—21no—
181C4⋊C4×C47—21no—
182C4⋊C4×C47C22⋊C412yes—
183C4⋊C4×C47C22⋊C411yescharacteristic
184C4⋊C4×C47C22⋊C412yes—
185C4⋊C4×C47C22⋊C411yescharacteristic
186C4⋊C4×C47C22×C411yescharacteristic
187C4⋊C4×C47C22×C411yescharacteristic
188C22⋊C4×C47—21no—
189C22⋊C4×C47—22no—
190C22⋊C4×C47—21no—
191C22⋊C4×C47—22no—
192C22⋊C4×C47—21no—
193C22⋊C4×C47—22no—
194C22⋊C4×C47—21no—
195C22⋊C4×C47—22no—
196C22⋊C4×C47—21no—
197C22⋊C4×C47—22no—
198C22⋊C4×C47C22⋊C412yes—
199C22⋊C4×C47C22⋊C412yes—
200C22⋊C4×C47C22⋊C412yes—
201C42×C47—22no—
202C42×C47—21no—
203C42×C47C22×C412yes—
204C42×C47C22×C411yescharacteristic
205C42×C47C2×D411yescharacteristic
206C42×C47C2×D412yes—
207C42×C47C2×D411yescharacteristic
208C42×C47C2×D412yes—
209C42×C47C2411yescharacteristic
210#10625329—42no—
211#10625329—42no—
212C2×C47⋊C8—21no—
213C2×C47⋊C8—21no—
214C2×C47⋊C8—22no—
215C2×C47⋊C8—22no—
216C2×C47⋊C8—21no—
217C2×C47⋊C8—21no—
218C2×C47⋊C8—44no—
219C2×C47⋊C8—42no—
220C23×C47—22no—
221C23×C47—21no—
222C23×C47C2×C22⋊C411yescharacteristic
223C23×C47C2×C22⋊C412yes—
224Q8×C47—22no—
225Q8×C47—41no—
226Q8×C47—42no—
227D4×C47—22no—
228D4×C47—42no—
229D4×C47—41no—
230D4×C47—42no—
231D4×C47—44no—
232D4×C47—42no—
233D4×C47—82no—
234D4×C47—81no—
235D4×C47—82no—
236D4×C47—81no—
237D4×C47—82no—
238D4×C47—82no—
239D4×C47—82no—
240C2×C4×C47—22no—
241C2×C4×C47—21no—
242C2×C4×C47—22no—
243C2×C4×C47—21no—
244C2×C4×C47—21no—
245C2×C4×C47—42no—
246C2×C4×C47—41no—
247C2×C4×C47—41no—
248C2×C4×C47—42no—
249C2×C4×C47—42no—
250C2×C4×C47—44no—
251C2×C4×C47—42no—
252C2×C4×C47—82no—
253C2×C4×C47—82no—
254C2×C4×C47—82no—
255C2×C4×C47C2×C22⋊C412yes—
256C2×C4×C47C2×C22⋊C411yescharacteristic
257C2×C4×C47C2×C22⋊C412yes—
258C2×C4×C47C2×C22⋊C411yescharacteristic
259C2×C4×C47C2×C22⋊C412yes—
260C2×C4×C47C2×C22⋊C411yescharacteristic
261C2×C4×C47C23×C411yescharacteristic
262C2×C4×C47C22×D411yescharacteristic
263C2×C4×C47C22×D411yescharacteristic
264C47⋊C8—22no—
265C47⋊C8—41no—
266C47⋊C8—41no—
267C47⋊C8—42no—
268C47⋊C8—42no—
269C47⋊C8—41no—
270#17684—471nomaximal, Sylow
271C22×C47—21no—
272C22×C47—22no—
273C22×C47—21no—
274C22×C47—22no—
275C22×C47—22no—
276C22×C47—21no—
277C22×C47—42no—
278C22×C47—41no—
279C22×C47—42no—
280C22×C47—41no—
281C22×C47—42no—
282C22×C47—41no—
283C22×C47C22×C22⋊C411yescharacteristic
284C4×C47—21no—
285C4×C47—21no—
286C4×C47—22no—
287C4×C47—21no—
288C4×C47—22no—
289C4×C47—21no—
290C4×C47—22no—
291C4×C47—21no—
292C4×C47—42no—
293C4×C47—41no—
294C4×C47—41no—
295C4×C47—41no—
296C4×C47#42112yes—
297#3517D4711yescharacteristic
298#2970—471no—
299#2923—471no—
300#1536—472no—
301#1535—472no—
302#1532—472no—
303#1530—472no—
304#1530—471no—
305#1529—471no—
306#1525—471no—
307#1524—471no—
308C2×C47—41no—
309C2×C47—42no—
310C2×C47—82no—
311C2×C47—81no—
312C2×C47#288811yescharacteristic
313C2×C47C2×#42111yescharacteristic
314C2×C47#290111yescharacteristic
315#557Dic4711yescharacteristic
316#553—21no—
317#553—22no—
318#550—21no—
319#550Dic4711yescharacteristic
320#550Dic4712yes—
321#550Dic4711yescharacteristic
322#547Dic4711yescharacteristic
323#547Dic4712yes—
324#540—21no—
325#540—22no—
326#532C2×D4712yes—
327#532C2×D4711yescharacteristic
328#529C2×D4711yescharacteristic
329#529C2×D4712yes—
330#520—21no—
331C4×#41C2×D4711yescharacteristic
332#446—941no—
333#435—942no—
334C8×D4—941no—
335#433—942no—
336#432—472no—
337#432—472no—
338#432—472no—
339#432—471no—
340#424—941no—
341#408—941no—
342C4×M16—471no—
343C4×M16—471no—
344C2×#97—471no—
345C2×#97—472no—
346#330—471no—
347#330—471no—
348#330—472no—
349#329—471no—
350#329—472no—
351#329—472no—
352#329—472no—
353#329—471no—
354#329—472no—
355#329—472no—
356C47#1768411yescharacteristic, complemented, Sylow
357C2×#41—21no—
358C2×#41C2×Dic4712yes—
359C2×#41C2×Dic4712yes—
360C2×#41C2×Dic4712yes—
361C2×#41C22⋊D4711yescharacteristic
362C2×M16—941no—
363C2×M16—941no—
364C22×C8—941no—
365C22×C8—942no—
366C4⋊Q8C2×Dic4712yes—
367C4⋊Q8C2×Dic4711yescharacteristic
368#127—21no—
369#127—22no—
370#127C2×Dic4712yes—
371#127C2×Dic4711yescharacteristic
372#126—21no—
373#126—22no—
374#125—22no—
375#125—21no—
376#125C22×D4711yescharacteristic
377#124C22×D4711yescharacteristic
378#124C22×D4712yes—
379#123—22no—
380#123—21no—
381#123—22no—
382#123—22no—
383#123—22no—
384C22⋊Q8C2×Dic4711yescharacteristic
385C22⋊Q8C2×Dic4712yes—
386#121—21no—
387#121—22no—
388#121—22no—
389#121—21no—
390#121—21no—
391#121—22no—
392#121—22no—
393#121—21no—
394#121—22no—
395#121—22no—
396#121—22no—
397#121C2×Dic4711yescharacteristic
398#121C2×Dic4712yes—
399C4×Q8—21no—
400C4×Q8C2×Dic4711yescharacteristic
401C4×Q8C2×Dic4712yes—
402C4×Q8C2×Dic4711yescharacteristic
403C4×Q8C22⋊D4711yescharacteristic
404C4×D4—21no—
405C4×D4—22no—
406C4×D4—22no—
407C4×D4—21no—
408C4×D4—21no—
409C4×D4—22no—
410C4×D4—21no—
411C4×D4—22no—
412C4×D4C2×Dic4711yescharacteristic
413C4×D4C2×Dic4712yes—
414C4×D4C2×Dic4711yescharacteristic
415C4×D4C2×Dic4712yes—
416C4×D4C22⋊D4711yescharacteristic
417C4×D4C22⋊D4712yes—
418#117—21no—
419#117—22no—
420#117C22⋊D4711yescharacteristic
421#117C22⋊D4712yes—
422C2×C4⋊C4—22no—
423C2×C4⋊C4—21no—
424C2×C42C22×D4711yescharacteristic
425C2×C42C22×D4712yes—
426C4⋊C8—942no—
427C4⋊C8—942no—
428C4⋊C8—941no—
429C4⋊C8—942no—
430C22⋊C8—942no—
431C22⋊C8—942no—
432C22⋊C8—942no—
433C22⋊C8—941no—
434#97—471no—
435#97—471no—
436#97—472no—
437#97—471no—
438#97—471no—
439#97—472no—
440#97—941no—
441#97—942no—
442C4×C8—941no—
443C4×C8—941no—
444#41—42no—
445#41—44no—
446#41—82no—
447#41—82no—
448#41—82no—
449C2×Q8C22⋊Dic4711yescharacteristic
450C2×Q8C22×Dic4711yescharacteristic
451C2×Q8C22×Dic4712yes—
452C2×D4—21no—
453C2×D4—22no—
454C2×D4—22no—
455C2×D4—22no—
456C2×D4—22no—
457C2×D4—22no—
458C2×D4—21no—
459C2×D4—21no—
460C2×D4—22no—
461C2×D4C22⋊Dic4711yescharacteristic
462C2×D4C22⋊Dic4712yes—
463C2×D4C22×Dic4711yescharacteristic
464C2×D4C22×Dic4712yes—
465C22×C4—22no—
466C22×C4—21no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4C22×Dic4711yescharacteristic
471C22×C4C22×Dic4712yes—
472C22×C4C2×C22⋊D4711yescharacteristic
473C22×C4C2×C22⋊D4712yes—
474C22×C4C2×C22⋊D4712yes—
475C22×C4C2×C22⋊D4711yescharacteristic
476M16—1882no—
477M16—1882no—
478C2×C8—941no—
479C2×C8—942no—
480C2×C8—941no—
481C2×C8—941no—
482C2×C8—941no—
483C2×C8—942no—
484C2×C8—1884no—
485C2×C8—1882no—
486C4⋊C4—21no—
487C4⋊C4—22no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—22no—
492C4⋊C4—21no—
493C4⋊C4—22no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4C22⋊Dic4712yes—
497C4⋊C4C22⋊Dic4711yescharacteristic
498C4⋊C4C22⋊Dic4712yes—
499C4⋊C4C22⋊Dic4711yescharacteristic
500C4⋊C4C22×Dic4711yescharacteristic
501C4⋊C4C22×Dic4711yescharacteristic
502C22⋊C4—21no—
503C22⋊C4—22no—
504C22⋊C4—21no—
505C22⋊C4—22no—
506C22⋊C4—21no—
507C22⋊C4—22no—
508C22⋊C4—21no—
509C22⋊C4—22no—
510C22⋊C4—21no—
511C22⋊C4—22no—
512C22⋊C4C22⋊Dic4712yes—
513C22⋊C4C22⋊Dic4712yes—
514C22⋊C4C22⋊Dic4712yes—
515C42—22no—
516C42—21no—
517C42C22×Dic4712yes—
518C42C22×Dic4711yescharacteristic
519C42C2×C22⋊D4711yescharacteristic
520C42C2×C22⋊D4711yescharacteristic
521C42C2×C22⋊D4712yes—
522C42C2×C22⋊D4712yes—
523C42C23×D4711yescharacteristic
524C23—22no—
525C23—21no—
526C23C2×C22⋊Dic4711yescharacteristic
527C23C2×C22⋊Dic4712yes—
528Q8—22no—
529Q8—41no—
530Q8—42no—
531D4—22no—
532D4—42no—
533D4—41no—
534D4—42no—
535D4—44no—
536D4—42no—
537D4—82no—
538D4—81no—
539D4—82no—
540D4—81no—
541D4—82no—
542D4—82no—
543D4—82no—
544C2×C4—22no—
545C2×C4—21no—
546C2×C4—22no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—42no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—44no—
555C2×C4—42no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—82no—
559C2×C4C2×C22⋊Dic4712yes—
560C2×C4C2×C22⋊Dic4711yescharacteristic
561C2×C4C2×C22⋊Dic4712yes—
562C2×C4C2×C22⋊Dic4711yescharacteristic
563C2×C4C2×C22⋊Dic4712yes—
564C2×C4C2×C22⋊Dic4711yescharacteristic
565C2×C4C23×Dic4711yescharacteristic
566C2×C4C22×C22⋊D4711yescharacteristic
567C2×C4C22×C22⋊D4711yescharacteristic
568C8—942no—
569C8—1881no—
570C8—1881no—
571C8—1882no—
572C8—1882no—
573C8—1881no—
574C22—21no—
575C22—22no—
576C22—22no—
577C22—21no—
578C22—21no—
579C22—22no—
580C22—42no—
581C22—41no—
582C22—42no—
583C22—41no—
584C22—42no—
585C22—41no—
586C22C22×C22⋊Dic4711yescharacteristic
587C4—21no—
588C4—22no—
589C4—21no—
590C4—22no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—22no—
595C4—41no—
596C4—41no—
597C4—42no—
598C4—41no—
599C4#1568914012yes—
600C2—41no—
601C2—42no—
602C2—82no—
603C2—81no—
604C2#2025186711yescharacteristic
605C2C2×#1568914011yescharacteristic
606C2#2025193511yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic47
commutatorC2×C4×C47C23×C4
FrattiniC42C23×D47
Fitting, prime radicalC47×#3517C2
socleC22×C47C22×C22⋊C4
cosocleC42×C47C24
hypercenter#3517D47
hypocenterC47#17684
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3517

Complements

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

NH ≅ G/Ntype
1. #20252402C2semidirect
2. #20252039C2semidirect
3. #20242023C2semidirect
4. #20242019C2semidirect
5. #20242008C2semidirect
6. #20241999C2semidirect
7. #20241999C2semidirect
8. #20241995C2semidirect
9. #20241980C2semidirect
10. #20241976C2semidirect
356. C47#17684semidirect

Maximal subgroups

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

subgroupindex
#202524022
#202520392
#202420232
#202420192
#202420082
#202419992
#202419952
#202419802
#202419762
C47×#35172
#1768447