Finite Group Databasebeta

← #27101027

Subgroups — #27101027

Order 12032.

2142 conjugacy classes of subgroups.

The subgroup lattice has 678 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689778C211yesmaximal, characteristic, complemented
2#20255310C214yesmaximal, complemented
3#20255299C214yesmaximal, complemented
4#20255118C214yesmaximal, complemented
5#20255106C214yesmaximal, complemented
6#20254333C211yesmaximal, characteristic, complemented
7#20254310C211yesmaximal, characteristic, complemented
8#20254165C214yesmaximal, complemented
9#20253392C211yesmaximal, characteristic, complemented
10#20253384C211yesmaximal, characteristic, complemented
11#20252604C214yesmaximal, complemented
12#20252412C211yesmaximal, characteristic, complemented
13C47×#3506C211yesmaximal, characteristic, complemented
14C22×#10625350C2212yes—
15C2×#13094044C2218yescomplemented
16#15689778—22no—
17#15689778—22no—
18#15689778C2214yes—
19#15689778C2214yes—
20C2×#13094024C2211yescharacteristic
21C2×#13094006—22no—
22C2×#13094003C2212yes—
23C2×#13093995—21no—
24C2×C4×D188C2211yescharacteristic
25C2×C4⋊C4×D47—21no—
26#15689502C2212yescomplemented
27#15689495C2212yescomplemented
28#15689475C2212yescomplemented
29#15689475C2212yescomplemented
30#15689464C2218yescomplemented
31#15689451C2212yes—
32#15689434C2212yes—
33#15689410C2212yescomplemented
34#15689400C2212yescomplemented
35#15689362C2218yescomplemented
36#15689350C2218yescomplemented
37#15689333C2214yescomplemented
38#15689328C2214yescomplemented
39#15689325C2212yescomplemented
40#15689315C2212yescomplemented
41#15689285C2218yescomplemented
42#15689283C2218yescomplemented
43#15689278C2214yes—
44#15689273C2212yes—
45#15689266C2218yescomplemented
46#15689261C2214yes—
47#15689256C2212yes—
48C4×#10625336C2214yescomplemented
49C4×#10625334C2214yescomplemented
50D4×C47⋊C8C2214yescomplemented
51D4×C47⋊C8C2214yescomplemented
52#15689176C2211yescharacteristic
53#15689176C2211yescharacteristic
54#15689117C2218yescomplemented
55C2×C22⋊(C47⋊C8)C2214yes—
56C47×#559—22no—
57C47×#556C2212yescomplemented
58C47×#555C2212yescomplemented
59C47×#553—22no—
60C47×#550C2212yescomplemented
61C47×#547C2212yescomplemented
62C47×#538—22no—
63C47×#535—21no—
64C47×#533C2211yescharacteristic
65C47×#532C2211yescharacteristic
66C47×#525C2214yes—
67C47×#520—21no—
68C2×C4×D4×C47C2211yescharacteristic
69C2×#10625350—24no—
70C2×#10625350—48no—
71C2×#10625350C2318yes—
72C22×Dic94C2312yes—
73C22×D188C2311yescharacteristic
74C22×C22⋊D47—22no—
75C22×C4×D47—21no—
76C22×C4×D47—22no—
77#13094044—416no—
78C2×#10625337C2318yes—
79C2×#10625335C2318yes—
80C2×#10625336C2314yes—
81C2×#10625334C2314yes—
82C22×C47⋊C8C2318yes—
83#13094024—21no—
84#13094024—21no—
85#13094024C2312yes—
86#13094024C2312yes—
87#13094006—28no—
88#13094006—28no—
89#13094003—24no—
90#13094003C2318yes—
91#13093995—24no—
92#13093995—24no—
93C4×D188—21no—
94C4×D188—21no—
95C4×D188C2312yes—
96C4×D188C2312yes—
97C2×C4⋊Dic47—21no—
98C2×C4⋊Dic47—21no—
99C2×C4⋊Dic47C2311yescharacteristic
100C2×Dic47⋊C4—22no—
101C4⋊C4×D47—24no—
102C4⋊C4×D47—24no—
103C2×C22⋊Dic47—22no—
104C2×(C2×D47)⋊C4—21no—
105#13093957C2314yes—
106#13093955C2314yes—
107C4⋊(C47⋊C8)C2312yes—
108C4⋊(C47⋊C8)C2312yes—
109#13093948C2314yes—
110#13093947C2318yes—
111#13093947C2318yes—
112#13093945C2314yes—
113#13093944C2314yes—
114#13093944C2314yes—
115C22⋊(C47⋊C8)C2312yes—
116C22⋊(C47⋊C8)C2312yes—
117C22⋊(C47⋊C8)C2318yes—
118C4×C47⋊C8C2312yes—
119C4×C47⋊C8C2312yes—
120C2×#41×C47—21no—
121C2×#41×C47C2314yes—
122C22×D4×C47D411yescharacteristic
123C23×C4×C47C2312yes—
124C4⋊Q8×C47C2311yescharacteristic
125C4⋊Q8×C47C2311yescharacteristic
126C4⋊Q8×C47C2311yescharacteristic
127#127×C47—22no—
128#127×C47C2311yescharacteristic
129#126×C47—24no—
130#125×C47—21no—
131#124×C47—22no—
132#124×C47—24no—
133#124×C47—21no—
134#123×C47—24no—
135#123×C47—24no—
136C22⋊Q8×C47—21no—
137C22⋊Q8×C47C2312yes—
138C22⋊Q8×C47C2314yes—
139#121×C47—22no—
140#121×C47—22no—
141#121×C47—21no—
142#121×C47—24no—
143#121×C47C2312yes—
144#121×C47C2314yes—
145#120×C47—24no—
146C4×Q8×C47—22no—
147C4×Q8×C47—21no—
148C4×Q8×C47C2312yes—
149C4×D4×C47—24no—
150C4×D4×C47—24no—
151C4×D4×C47—22no—
152C4×D4×C47—21no—
153C4×D4×C47D412yes—
154C4×D4×C47D412yes—
155C4×D4×C47C2312yes—
156C4×D4×C47C2312yes—
157C4×D4×C47C2312yes—
158#117×C47—21no—
159#117×C47—22no—
160#117×C47C2314yes—
161C2×C4⋊C4×C47—22no—
162C2×C4⋊C4×C47D411yescharacteristic
163C2×C22⋊C4×C47D412yes—
164C2×C42×C47C2311yescharacteristic
165C24×C47C2×D412yes—
166C23×D47—21no—
167#41×C47—48no—
168#41×C47—44no—
169C2×Q8×C47—21no—
170C2×Q8×C47—21no—
171C2×Q8×C47C2412yes—
172C2×D4×C47—22no—
173C2×D4×C47—24no—
174C2×D4×C47—24no—
175C2×D4×C47—24no—
176C2×D4×C47—21no—
177C2×D4×C47—21no—
178C2×D4×C47C2×D412yes—
179C2×D4×C47C2×D412yes—
180C2×D4×C47C2412yes—
181C22×C4×C47—22no—
182C22×C4×C47—24no—
183C22×C4×C47—44no—
184C22×C4×C47C2×D411yescharacteristic
185C22×C4×C47C2×D412yes—
186C22×C4×C47C2×D414yes—
187C22×C4×C47C2×D411yescharacteristic
188C22×C4×C47C2411yescharacteristic
189C22×C4×C47C2414yes—
190C2×C22⋊D47—28no—
191C2×C22⋊D47—48no—
192C22×Dic47—21no—
193C22×Dic47—22no—
194#10625350—216no—
195#10625350—48no—
196#10625350—416no—
197C2×D188—21no—
198C2×D188—24no—
199C2×D188C2412yes—
200C2×C4×D47—22no—
201C2×C4×D47—22no—
202C2×C4×D47—28no—
203C2×C4×D47—48no—
204C2×C4×D47—44no—
205C2×Dic94—22no—
206C2×Dic94—44no—
207C2×Dic94C2414yes—
208C4⋊C4×C47—21no—
209C4⋊C4×C47—24no—
210C4⋊C4×C47—21no—
211C4⋊C4×C47—22no—
212C4⋊C4×C47—21no—
213C4⋊C4×C47—21no—
214C4⋊C4×C47—22no—
215C4⋊C4×C47C2×D412yes—
216C4⋊C4×C47C2×D412yes—
217C4⋊C4×C47C2412yes—
218C4⋊C4×C47C2412yes—
219C22⋊C4×C47—22no—
220C22⋊C4×C47—24no—
221C22⋊C4×C47—24no—
222C22⋊C4×C47—24no—
223C22⋊C4×C47—22no—
224C22⋊C4×C47—22no—
225C22⋊C4×C47C2×D414yes—
226C22⋊C4×C47C2×D414yes—
227C42×C47—24no—
228C42×C47—22no—
229C42×C47C2×D411yescharacteristic
230C42×C47C2×D411yescharacteristic
231C42×C47C2411yescharacteristic
232C42×C47C2411yescharacteristic
233C22⋊Dic47—28no—
234#10625337—48no—
235#10625336—28no—
236#10625335—48no—
237#10625334—28no—
238(C2×D47)⋊C4—22no—
239(C2×D47)⋊C4—22no—
240C4⋊Dic47—22no—
241C4⋊Dic47—22no—
242C4⋊Dic47—22no—
243C4⋊Dic47—22no—
244C4⋊Dic47—21no—
245C4⋊Dic47C2412yes—
246Dic47⋊C4—28no—
247C2×C47⋊C8—48no—
248C2×C47⋊C8C2414yes—
249C2×C47⋊C8C2414yes—
250C23×C47—24no—
251C23×C47—22no—
252C23×C47—42no—
253C23×C47—42no—
254C23×C47C22×D414yes—
255C23×C47C22×D411yescharacteristic
256C22×D47—24no—
257C22×D47—22no—
258C22×D47—42no—
259C22×D47—42no—
260Q8×C47—24no—
261Q8×C47—42no—
262Q8×C47—81no—
263D4×C47—24no—
264D4×C47—24no—
265D4×C47—24no—
266D4×C47—44no—
267D4×C47—42no—
268D4×C47—81no—
269D4×C47—84no—
270D4×C47—84no—
271C2×C4×C47—28no—
272C2×C4×C47—22no—
273C2×C4×C47—22no—
274C2×C4×C47—22no—
275C2×C4×C47—22no—
276C2×C4×C47—42no—
277C2×C4×C47—44no—
278C2×C4×C47—41no—
279C2×C4×C47—44no—
280C2×C4×C47—84no—
281C2×C4×C47#13514yescomplemented
282C2×C4×C47C22×D414yes—
283C2×C4×C47C22×D411yescharacteristic
284C2×C4×C47C22×D411yescharacteristic
285C2×C4×C47C22×D411yescharacteristic
286C2×C4×C47C22×D411yescharacteristic
287C2×C4×C47C22×D411yescharacteristic
288C2×C4×C47C2511yescharacteristic
289C22⋊D47—416no—
290C2×Dic47—22no—
291C2×Dic47—24no—
292C2×Dic47—44no—
293C2×Dic47—42no—
294D188—24no—
295D188—22no—
296D188H(5,𝔽2)14yes—
297C4×D47—416no—
298C4×D47—44no—
299C4×D47—44no—
300Dic94—28no—
301Dic94—44no—
302C47⋊C8—24no—
303C47⋊C8—24no—
304#61565—471nomaximal, Sylow
305C22×C47—21no—
306C22×C47—21no—
307C22×C47—24no—
308C22×C47—24no—
309C22×C47—24no—
310C22×C47—42no—
311C22×C47—42no—
312C22×C47—42no—
313C22×C47—84no—
314C22×C47C2×#13512yes—
315C22×C47C23×D411yescharacteristic
316C2×D47—24no—
317C2×D47—48no—
318C2×D47—42no—
319C2×D47—42no—
320C4×C47—21no—
321C4×C47—21no—
322C4×C47—21no—
323C4×C47—24no—
324C4×C47—42no—
325C4×C47—42no—
326C4×C47—42no—
327C4×C47C2×#13512yes—
328C4×C47C2×#13512yes—
329C4×C47C2×H(5,𝔽2)12yes—
330Dic47—44no—
331Dic47—42no—
332#3506D4711yescharacteristic, complemented
333C2×#547—471no—
334#3303—474no—
335#3302—474no—
336#3284—474no—
337#3283—474no—
338#3209—471no—
339#3206—471no—
340#3190—474no—
341#3101—471no—
342#3100—471no—
343#3002—474no—
344#2972—471no—
345C2×C47—22no—
346C2×C47—44no—
347C2×C47—82no—
348C2×C47#348411yescharacteristic
349C2×C47C22×#13511yescharacteristic
350C2×C47#358611yescharacteristic
351D47—44no—
352C22×#41—472no—
353C2×#135—478no—
354#559—22no—
355#556C2×D4712yescomplemented
356#555C2×D4712yescomplemented
357#553—22no—
358#550C2×D4712yescomplemented
359#547—474no—
360#547—474no—
361#547—942no—
362#547—942no—
363#547C2×D4712yescomplemented
364#538—22no—
365#535—21no—
366#533C2×D4711yescharacteristic
367#532C2×D4711yescharacteristic
368C2×C4⋊Q8—471no—
369#525C2×D4714yes—
370C2×#123—942no—
371C2×C22⋊Q8—472no—
372C2×#121—941no—
373#520—21no—
374C2×C4×D4—471no—
375C2×C4×D4C2×D4711yescharacteristic
376C22×C4⋊C4—941no—
377#494—472no—
378#492—472no—
379#487—472no—
380#487—472no—
381#485—478no—
382#483—472no—
383#480—472no—
384#476—472no—
385#474—472no—
386#467—478no—
387#465—478no—
388#462—474no—
389#461—472no—
390#461—474no—
391#459—472no—
392#452—478no—
393#452—478no—
394#451—474no—
395#450—472no—
396#449—478no—
397#448—474no—
398#447—472no—
399C4×SD16—474no—
400C4×D8—474no—
401C8×D4—474no—
402C8×D4—474no—
403#431—471no—
404#431—471no—
405#416—478no—
406C2×C22⋊C8—474no—
407C47#6156511yescharacteristic, complemented, Sylow
408C2×#41—21no—
409C2×#41—478no—
410C2×#41—944no—
411C2×#41—1888no—
412C2×#41C22×D4714yes—
413C22×Q8—472no—
414C22×D4—471no—
415C22×D4—942no—
416C22×D4C22⋊D4711yescharacteristic
417C23×C4—941no—
418C23×C4—942no—
419C23×C4C22×D4712yes—
420#135—18816no—
421C2×Q16—478no—
422C2×SD16—474no—
423C2×SD16—478no—
424C2×D8—474no—
425C22×C8—478no—
426C4⋊Q8—472no—
427C4⋊Q8—472no—
428C4⋊Q8—941no—
429C4⋊Q8—941no—
430C4⋊Q8C22×D4711yescharacteristic
431C4⋊Q8C22×D4711yescharacteristic
432C4⋊Q8C22×D4711yescharacteristic
433#127—22no—
434#127C22×D4711yescharacteristic
435#126—24no—
436#125—21no—
437#124—22no—
438#124—24no—
439#124—21no—
440#123—24no—
441#123—24no—
442#123—948no—
443#123—948no—
444C22⋊Q8—21no—
445C22⋊Q8—478no—
446C22⋊Q8—944no—
447C22⋊Q8C22×D4712yes—
448C22⋊Q8C22×D4714yes—
449#121—22no—
450#121—22no—
451#121—21no—
452#121—24no—
453#121—944no—
454#121—944no—
455#121C22×D4712yes—
456#121C22×D4714yes—
457#120—24no—
458C4×Q8—22no—
459C4×Q8—21no—
460C4×Q8C22×D4712yes—
461C4×D4—24no—
462C4×D4—24no—
463C4×D4—22no—
464C4×D4—21no—
465C4×D4—472no—
466C4×D4—472no—
467C4×D4—941no—
468C4×D4—941no—
469C4×D4C22⋊D4712yes—
470C4×D4C22⋊D4712yes—
471C4×D4C22×D4712yes—
472C4×D4C22×D4712yes—
473C4×D4C22×D4712yes—
474#117—21no—
475#117—22no—
476#117C22×D4714yes—
477C2×C4⋊C4—22no—
478C2×C4⋊C4—471no—
479C2×C4⋊C4—941no—
480C2×C4⋊C4—942no—
481C2×C4⋊C4—941no—
482C2×C4⋊C4—944no—
483C2×C4⋊C4—944no—
484C2×C4⋊C4C22⋊D4711yescharacteristic
485C2×C22⋊C4—942no—
486C2×C22⋊C4—941no—
487C2×C22⋊C4C22⋊D4712yes—
488C2×C42C22×D4711yescharacteristic
489#107—474no—
490#106—474no—
491C4⋊C8—472no—
492C4⋊C8—472no—
493#103—474no—
494#103—478no—
495#103—478no—
496#102—474no—
497#102—474no—
498#102—474no—
499C22⋊C8—472no—
500C22⋊C8—472no—
501C22⋊C8—478no—
502C4×C8—472no—
503C4×C8—472no—
504C24—941no—
505C24C2×C22⋊D4712yes—
506#41—48no—
507#41—44no—
508#41—9416no—
509#41—1888no—
510#41—18816no—
511C2×Q8—21no—
512C2×Q8—21no—
513C2×Q8—474no—
514C2×Q8—942no—
515C2×Q8—1884no—
516C2×Q8C23×D4712yes—
517C2×D4—22no—
518C2×D4—24no—
519C2×D4—24no—
520C2×D4—24no—
521C2×D4—21no—
522C2×D4—21no—
523C2×D4—472no—
524C2×D4—944no—
525C2×D4—948no—
526C2×D4—941no—
527C2×D4—1888no—
528C2×D4C2×C22⋊D4712yes—
529C2×D4C2×C22⋊D4712yes—
530C2×D4C23×D4712yes—
531C22×C4—22no—
532C22×C4—24no—
533C22×C4—44no—
534C22×C4—941no—
535C22×C4—942no—
536C22×C4—942no—
537C22×C4—942no—
538C22×C4—948no—
539C22×C4—1888no—
540C22×C4—1884no—
541C22×C4C2×C22⋊D4711yescharacteristic
542C22×C4C2×C22⋊D4712yes—
543C22×C4C2×C22⋊D4714yes—
544C22×C4C2×C22⋊D4711yescharacteristic
545C22×C4C23×D4711yescharacteristic
546C22×C4C23×D4714yes—
547Q16—1888no—
548SD16—948no—
549SD16—1888no—
550D8—948no—
551C2×C8—474no—
552C2×C8—474no—
553C2×C8—1888no—
554C4⋊C4—21no—
555C4⋊C4—24no—
556C4⋊C4—21no—
557C4⋊C4—22no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—22no—
561C4⋊C4—472no—
562C4⋊C4—948no—
563C4⋊C4—941no—
564C4⋊C4—942no—
565C4⋊C4—942no—
566C4⋊C4—942no—
567C4⋊C4—942no—
568C4⋊C4C2×C22⋊D4712yes—
569C4⋊C4C2×C22⋊D4712yes—
570C4⋊C4C23×D4712yes—
571C4⋊C4C23×D4712yes—
572C22⋊C4—22no—
573C22⋊C4—24no—
574C22⋊C4—24no—
575C22⋊C4—24no—
576C22⋊C4—22no—
577C22⋊C4—22no—
578C22⋊C4—948no—
579C22⋊C4—942no—
580C22⋊C4—942no—
581C22⋊C4C2×C22⋊D4714yes—
582C22⋊C4C2×C22⋊D4714yes—
583C42—24no—
584C42—22no—
585C42C2×C22⋊D4711yescharacteristic
586C42C2×C22⋊D4711yescharacteristic
587C42C23×D4711yescharacteristic
588C42C23×D4711yescharacteristic
589C23—24no—
590C23—22no—
591C23—42no—
592C23—42no—
593C23—944no—
594C23—942no—
595C23—1882no—
596C23—1882no—
597C23C22×C22⋊D4714yes—
598C23C22×C22⋊D4711yescharacteristic
599Q8—24no—
600Q8—42no—
601Q8—81no—
602Q8—948no—
603Q8—1884no—
604D4—24no—
605D4—24no—
606D4—24no—
607D4—44no—
608D4—42no—
609D4—81no—
610D4—84no—
611D4—84no—
612D4—474no—
613D4—944no—
614D4—942no—
615D4—18816no—
616C2×C4—28no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—22no—
620C2×C4—22no—
621C2×C4—42no—
622C2×C4—44no—
623C2×C4—41no—
624C2×C4—44no—
625C2×C4—84no—
626C2×C4—942no—
627C2×C4—944no—
628C2×C4—18816no—
629C2×C4—1884no—
630C2×C4—1882no—
631C2×C4—1884no—
632C2×C4—1884no—
633C2×C4#1309404414yescomplemented
634C2×C4C22×C22⋊D4714yes—
635C2×C4C22×C22⋊D4711yescharacteristic
636C2×C4C22×C22⋊D4711yescharacteristic
637C2×C4C22×C22⋊D4711yescharacteristic
638C2×C4C22×C22⋊D4711yescharacteristic
639C2×C4C22×C22⋊D4711yescharacteristic
640C2×C4C24×D4711yescharacteristic
641C8—944no—
642C8—944no—
643C22—21no—
644C22—21no—
645C22—24no—
646C22—24no—
647C22—24no—
648C22—42no—
649C22—42no—
650C22—42no—
651C22—84no—
652C22—944no—
653C22—1888no—
654C22—1882no—
655C22—1882no—
656C22C2×#1309404412yes—
657C22C23×C22⋊D4711yescharacteristic
658C4—21no—
659C4—21no—
660C4—21no—
661C4—24no—
662C4—42no—
663C4—42no—
664C4—42no—
665C4—1882no—
666C4—1884no—
667C4C2×#1309404412yes—
668C4C2×#1309404412yes—
669C4H(5,𝔽2)×D4712yes—
670C2—22no—
671C2—44no—
672C2—82no—
673C2—1884no—
674C2#2025701911yescharacteristic
675C2C22×#1309404411yescharacteristic
676C2#2025817411yescharacteristic
subgroupG/H
centerC22C23×C22⋊D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC47×#3506C2
socleC22×C47C23×D4
hypercenter#3506D47
hypocenterC47#61565
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. C2×#15689778C2semidirect
2. #20255310C2semidirect
3. #20255299C2semidirect
4. #20255118C2semidirect
5. #20255106C2semidirect
6. #20254333C2semidirect
7. #20254310C2semidirect
8. #20254165C2semidirect
9. #20253392C2semidirect
10. #20253384C2semidirect
11. #20252604C2semidirect
12. #20252412C2semidirect
13. C47×#3506C2semidirect
15. C2×#13094044C22semidirect
26. #15689502C22semidirect
27. #15689495C22semidirect
28. #15689475C22semidirect
29. #15689475C22semidirect
30. #15689464C22semidirect
33. #15689410C22semidirect
34. #15689400C22semidirect
35. #15689362C22semidirect
36. #15689350C22semidirect
37. #15689333C22semidirect
38. #15689328C22semidirect
39. #15689325C22semidirect
40. #15689315C22semidirect
41. #15689285C22semidirect
42. #15689283C22semidirect
45. #15689266C22semidirect
48. C4×#10625336C22semidirect
49. C4×#10625334C22semidirect
50. D4×C47⋊C8C22semidirect
51. D4×C47⋊C8C22semidirect
54. #15689117C22semidirect
57. C47×#556C22semidirect
58. C47×#555C22semidirect
60. C47×#550C22semidirect
61. C47×#547C22semidirect
281. C2×C4×C47#135semidirect
332. #3506D47semidirect
355. #556C2×D47semidirect
356. #555C2×D47semidirect
358. #550C2×D47semidirect
363. #547C2×D47semidirect
407. C47#61565semidirect
633. C2×C4#13094044semidirect

Maximal subgroups

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

subgroupindex
C2×#156897782
#202553102
#202552992
#202551182
#202551062
#202543332
#202543102
#202541652
#202533922
#202533842
#202526042
#202524122
C47×#35062
#6156547