Finite Group Databasebeta

← #27101032

Subgroups — #27101032

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
1#20257240C211yesmaximal, characteristic, complemented
2#20255316C214yesmaximal, complemented
3#20255305C214yesmaximal, complemented
4#20255114C214yesmaximal, complemented
5#20255106C214yesmaximal, complemented
6#20254336C211yesmaximal, characteristic, complemented
7#20254313C211yesmaximal, characteristic, complemented
8#20254169C214yesmaximal, complemented
9#20253396C211yesmaximal, characteristic, complemented
10#20253388C211yesmaximal, characteristic, complemented
11#20252606C214yesmaximal, complemented
12#20252412C211yesmaximal, characteristic, complemented
13C2×C47×#547C211yesmaximal, characteristic, complemented
14C2×#13094044C2218yescomplemented
15#15689856—22no—
16#15689843C2212yescomplemented
17#15689838C2212yescomplemented
18#15689823—22no—
19#15689794C2212yescomplemented
20#15689778C2212yescomplemented
21#15689711—22no—
22#15689690—21no—
23#15689680C2211yescharacteristic
24#15689673C2211yescharacteristic
25#15689634C2214yes—
26#15689604—21no—
27#15689502C2212yescomplemented
28#15689497C2212yescomplemented
29#15689473C2212yescomplemented
30#15689473C2212yescomplemented
31#15689461C2218yescomplemented
32#15689453C2212yes—
33#15689436C2212yes—
34#15689413C2212yescomplemented
35#15689403C2212yescomplemented
36#15689362C2218yescomplemented
37#15689354C2218yescomplemented
38#15689330C2214yescomplemented
39#15689328C2212yescomplemented
40#15689325C2214yescomplemented
41#15689318C2212yescomplemented
42#15689285C2218yescomplemented
43#15689283C2218yescomplemented
44#15689279C2214yes—
45#15689274C2212yes—
46#15689268C2218yescomplemented
47#15689262C2214yes—
48#15689257C2212yes—
49C4×#10625335C2214yescomplemented
50C4×#10625334C2214yescomplemented
51D4×C47⋊C8C2214yescomplemented
52D4×C47⋊C8C2214yescomplemented
53#15689176C2211yescharacteristic
54#15689176C2211yescharacteristic
55#15689119C2218yescomplemented
56C2×C22⋊(C47⋊C8)C2214yes—
57C22×#41×C47C2212yes—
58C47×#547—22no—
59C47×#547—22no—
60C47×#547C2214yes—
61C47×#547C2214yes—
62C2×C4⋊Q8×C47C2211yescharacteristic
63C2×#123×C47—22no—
64C2×C22⋊Q8×C47C2212yes—
65C2×#121×C47—21no—
66C2×C4×D4×C47C2211yescharacteristic
67C2×C4×D4×C47C2211yescharacteristic
68C22×C4⋊C4×C47—21no—
69C2×#10625350—21no—
70C2×#10625350C2314yes—
71#13094044—416no—
72C2×#10625337C2318yes—
73C2×#10625335C2314yes—
74C2×#10625336C2318yes—
75C2×#10625334C2314yes—
76C22×C47⋊C8C2318yes—
77#13094024C2311yescharacteristic
78#13094024C2311yescharacteristic
79#13094024C2311yescharacteristic
80#13094021C2311yescharacteristic
81#13094020—22no—
82#13094017—24no—
83#13094016—21no—
84#13094012—21no—
85#13094011—22no—
86#13094010—24no—
87#13094007—24no—
88#13094006—24no—
89#13094003—21no—
90#13094003C2314yes—
91#13094003C2312yes—
92#13093998—21no—
93#13093998C2312yes—
94#13093998C2314yes—
95#13093997—24no—
96#13093995—22no—
97#13093995—22no—
98#13093993—24no—
99C4×Dic94—21no—
100C4×Dic94C2312yes—
101#13093989—22no—
102C4×D188—21no—
103C4×D188C2312yes—
104#13093986—22no—
105C4×C22⋊D47—24no—
106C4×C22⋊D47—24no—
107#13093983—21no—
108#13093983C2314yes—
109#13093982—22no—
110C4⋊C4×D47—22no—
111#13093957C2314yes—
112#13093955C2314yes—
113C4⋊(C47⋊C8)C2312yes—
114C4⋊(C47⋊C8)C2312yes—
115#13093948C2318yes—
116#13093948C2318yes—
117#13093947C2314yes—
118#13093945C2314yes—
119#13093945C2314yes—
120#13093944C2314yes—
121C22⋊(C47⋊C8)C2312yes—
122C22⋊(C47⋊C8)C2312yes—
123C22⋊(C47⋊C8)C2318yes—
124C4×C47⋊C8C2312yes—
125C4×C47⋊C8C2312yes—
126C2×#41×C47—24no—
127C2×#41×C47—48no—
128C2×#41×C47C2318yes—
129C22×Q8×C47C2312yes—
130C22×D4×C47—22no—
131C22×D4×C47D411yescharacteristic
132C22×D4×C47C2311yescharacteristic
133C23×C4×C47—21no—
134C23×C4×C47—22no—
135C23×C4×C47C2312yes—
136C4⋊Q8×C47—21no—
137C4⋊Q8×C47—21no—
138C4⋊Q8×C47C2312yes—
139C4⋊Q8×C47C2312yes—
140#123×C47—28no—
141#123×C47—28no—
142C22⋊Q8×C47—24no—
143C22⋊Q8×C47C2318yes—
144#121×C47—24no—
145#121×C47—24no—
146C4×D4×C47—21no—
147C4×D4×C47—21no—
148C4×D4×C47D412yes—
149C4×D4×C47D412yes—
150C4×D4×C47C2312yes—
151C4×D4×C47C2312yes—
152C4×D4×C47C2312yes—
153C4×D4×C47C2312yes—
154C2×C4⋊C4×C47—21no—
155C2×C4⋊C4×C47—22no—
156C2×C4⋊C4×C47—21no—
157C2×C4⋊C4×C47—24no—
158C2×C4⋊C4×C47—24no—
159C2×C4⋊C4×C47D411yescharacteristic
160C2×C4⋊C4×C47C2311yescharacteristic
161C2×C22⋊C4×C47—22no—
162C2×C22⋊C4×C47—21no—
163C2×C22⋊C4×C47D412yes—
164C2×C42×C47C2311yescharacteristic
165C24×C47—21no—
166C24×C47C2×D412yes—
167#41×C47—216no—
168#41×C47—48no—
169#41×C47—416no—
170C2×Q8×C47—22no—
171C2×Q8×C47—44no—
172C2×Q8×C47C2414yes—
173C2×D4×C47—24no—
174C2×D4×C47—28no—
175C2×D4×C47—24no—
176C2×D4×C47—21no—
177C2×D4×C47—48no—
178C2×D4×C47C2×D412yes—
179C2×D4×C47C2×D412yes—
180C2×D4×C47C2412yes—
181C22×C4×C47—21no—
182C22×C4×C47—22no—
183C22×C4×C47—22no—
184C22×C4×C47—22no—
185C22×C4×C47—28no—
186C22×C4×C47—44no—
187C22×C4×C47—48no—
188C22×C4×C47—44no—
189C22×C4×C47C2×D411yescharacteristic
190C22×C4×C47C2×D412yes—
191C22×C4×C47C2×D414yes—
192C22×C4×C47C2×D411yescharacteristic
193C22×C4×C47C2411yescharacteristic
194C22×C4×C47C2414yes—
195C2×C22⋊D47—22no—
196C2×C22⋊D47—24no—
197C2×C22⋊D47—24no—
198#10625350—48no—
199#10625350—44no—
200C2×D188—21no—
201C2×D188—21no—
202C2×D188C2412yes—
203C2×C4×D47—22no—
204C2×C4×D47—24no—
205C2×Dic94—21no—
206C2×Dic94—21no—
207C2×Dic94C2412yes—
208C4⋊C4×C47—22no—
209C4⋊C4×C47—22no—
210C4⋊C4×C47—28no—
211C4⋊C4×C47—22no—
212C4⋊C4×C47—22no—
213C4⋊C4×C47—21no—
214C4⋊C4×C47C2×D412yes—
215C4⋊C4×C47C2×D412yes—
216C4⋊C4×C47C2412yes—
217C22⋊C4×C47—28no—
218C22⋊C4×C47—22no—
219C22⋊C4×C47—22no—
220C22⋊C4×C47C2×D414yes—
221C22⋊C4×C47C2×D414yes—
222C42×C47C2×D411yescharacteristic
223C42×C47C2×D411yescharacteristic
224C42×C47C2411yescharacteristic
225C42×C47C2411yescharacteristic
226C22⋊Dic47—22no—
227C22⋊Dic47—24no—
228C22⋊Dic47—24no—
229#10625337—48no—
230#10625336—48no—
231#10625335—28no—
232#10625334—28no—
233(C2×D47)⋊C4—24no—
234(C2×D47)⋊C4—22no—
235(C2×D47)⋊C4—22no—
236C4⋊Dic47—21no—
237C4⋊Dic47—21no—
238C4⋊Dic47—21no—
239C4⋊Dic47—21no—
240C4⋊Dic47C2412yes—
241C4⋊Dic47C2412yes—
242Dic47⋊C4—24no—
243Dic47⋊C4—22no—
244Dic47⋊C4—22no—
245C4×Dic47—24no—
246C4×Dic47—22no—
247C2×C47⋊C8—48no—
248C2×C47⋊C8C2414yes—
249C2×C47⋊C8C2414yes—
250C23×C47—24no—
251C23×C47—22no—
252C23×C47—24no—
253C23×C47—42no—
254C23×C47—42no—
255C23×C47—42no—
256C23×C47—42no—
257C23×C47C22×D414yes—
258C23×C47C22×D411yescharacteristic
259C22×D47—22no—
260Q8×C47—28no—
261Q8×C47—44no—
262D4×C47—24no—
263D4×C47—24no—
264D4×C47—24no—
265D4×C47—22no—
266D4×C47—416no—
267D4×C47H(5,𝔽2)14yes—
268C2×C4×C47—28no—
269C2×C4×C47—22no—
270C2×C4×C47—22no—
271C2×C4×C47—24no—
272C2×C4×C47—42no—
273C2×C4×C47—44no—
274C2×C4×C47—41no—
275C2×C4×C47—416no—
276C2×C4×C47—44no—
277C2×C4×C47—42no—
278C2×C4×C47—44no—
279C2×C4×C47—44no—
280C2×C4×C47#13514yescomplemented
281C2×C4×C47C22×D414yes—
282C2×C4×C47C22×D411yescharacteristic
283C2×C4×C47C22×D411yescharacteristic
284C2×C4×C47C22×D411yescharacteristic
285C2×C4×C47C22×D411yescharacteristic
286C2×C4×C47C22×D411yescharacteristic
287C2×C4×C47C2511yescharacteristic
288C22⋊D47—44no—
289C22⋊D47—84no—
290C22⋊D47—84no—
291C2×Dic47—22no—
292C2×Dic47—22no—
293C2×Dic47—22no—
294D188—24no—
295D188—42no—
296D188—81no—
297C4×D47—44no—
298C4×D47—84no—
299Dic94—24no—
300Dic94—42no—
301Dic94—81no—
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—24no—
311C22×C47—48no—
312C22×C47—42no—
313C22×C47—42no—
314C22×C47—84no—
315C22×C47C2×#13512yes—
316C22×C47C23×D411yescharacteristic
317C2×D47—42no—
318C2×D47—42no—
319C2×D47—42no—
320C4×C47—21no—
321C4×C47—21no—
322C4×C47—24no—
323C4×C47—21no—
324C4×C47—44no—
325C4×C47—42no—
326C4×C47C2×#13512yes—
327C4×C47C2×#13512yes—
328C4×C47C2×H(5,𝔽2)12yes—
329Dic47—42no—
330Dic47—42no—
331Dic47—42no—
332#3506—471no—
333C2×#547D4711yescharacteristic, complemented
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—44no—
348C2×C47#348411yescharacteristic
349C2×C47C22×#13511yescharacteristic
350C2×C47#358611yescharacteristic
351D47—82no—
352C22×#41C2×D4712yes—
353C2×#135—478no—
354#559—942no—
355#556—472no—
356#555—472no—
357#553—942no—
358#550—472no—
359#547—22no—
360#547—22no—
361#547—472no—
362#547C2×D4714yes—
363#547C2×D4714yes—
364#538—942no—
365#535—941no—
366#533—471no—
367#532—471no—
368C2×C4⋊Q8C2×D4711yescharacteristic
369#525—474no—
370C2×#123—22no—
371C2×C22⋊Q8C2×D4712yes—
372C2×#121—21no—
373#520—941no—
374C2×C4×D4C2×D4711yescharacteristic
375C2×C4×D4C2×D4711yescharacteristic
376C22×C4⋊C4—21no—
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—474no—
390#461—472no—
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—24no—
409C2×#41—48no—
410C2×#41—474no—
411C2×#41—941no—
412C2×#41C22×D4718yes—
413C22×Q8C22×D4712yes—
414C22×D4—22no—
415C22×D4C22⋊D4711yescharacteristic
416C22×D4C22×D4711yescharacteristic
417C23×C4—21no—
418C23×C4—22no—
419C23×C4C22×D4712yes—
420#135—18816no—
421C2×Q16—478no—
422C2×SD16—474no—
423C2×SD16—478no—
424C2×D8—474no—
425C22×C8—478no—
426C4⋊Q8—21no—
427C4⋊Q8—21no—
428C4⋊Q8—471no—
429C4⋊Q8—471no—
430C4⋊Q8—471no—
431C4⋊Q8C22×D4712yes—
432C4⋊Q8C22×D4712yes—
433#127—471no—
434#127—942no—
435#126—944no—
436#125—941no—
437#124—942no—
438#124—944no—
439#124—941no—
440#123—28no—
441#123—28no—
442#123—944no—
443#123—944no—
444C22⋊Q8—24no—
445C22⋊Q8—472no—
446C22⋊Q8—474no—
447C22⋊Q8—941no—
448C22⋊Q8C22×D4718yes—
449#121—24no—
450#121—24no—
451#121—472no—
452#121—474no—
453#121—942no—
454#121—942no—
455#121—941no—
456#121—944no—
457#120—944no—
458C4×Q8—472no—
459C4×Q8—942no—
460C4×Q8—941no—
461C4×D4—21no—
462C4×D4—21no—
463C4×D4—472no—
464C4×D4—944no—
465C4×D4—944no—
466C4×D4—942no—
467C4×D4—941no—
468C4×D4C22⋊D4712yes—
469C4×D4C22⋊D4712yes—
470C4×D4C22×D4712yes—
471C4×D4C22×D4712yes—
472C4×D4C22×D4712yes—
473C4×D4C22×D4712yes—
474#117—474no—
475#117—941no—
476#117—942no—
477C2×C4⋊C4—21no—
478C2×C4⋊C4—22no—
479C2×C4⋊C4—21no—
480C2×C4⋊C4—24no—
481C2×C4⋊C4—24no—
482C2×C4⋊C4—942no—
483C2×C4⋊C4C22⋊D4711yescharacteristic
484C2×C4⋊C4C22×D4711yescharacteristic
485C2×C22⋊C4—22no—
486C2×C22⋊C4—21no—
487C2×C22⋊C4C22⋊D4712yes—
488C2×C42C22×D4711yescharacteristic
489#107—474no—
490#106—474no—
491C4⋊C8—472no—
492C4⋊C8—472no—
493#103—478no—
494#103—478no—
495#103—474no—
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—21no—
505C24C2×C22⋊D4712yes—
506#41—216no—
507#41—48no—
508#41—416no—
509#41—1888no—
510#41—1884no—
511C2×Q8—22no—
512C2×Q8—44no—
513C2×Q8—472no—
514C2×Q8—941no—
515C2×Q8—941no—
516C2×Q8C23×D4714yes—
517C2×D4—24no—
518C2×D4—28no—
519C2×D4—24no—
520C2×D4—21no—
521C2×D4—48no—
522C2×D4—472no—
523C2×D4—944no—
524C2×D4—944no—
525C2×D4—942no—
526C2×D4—941no—
527C2×D4—941no—
528C2×D4C2×C22⋊D4712yes—
529C2×D4C2×C22⋊D4712yes—
530C2×D4C23×D4712yes—
531C22×C4—21no—
532C22×C4—22no—
533C22×C4—22no—
534C22×C4—22no—
535C22×C4—28no—
536C22×C4—44no—
537C22×C4—48no—
538C22×C4—44no—
539C22×C4—944no—
540C22×C4—942no—
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—22no—
555C4⋊C4—22no—
556C4⋊C4—28no—
557C4⋊C4—22no—
558C4⋊C4—22no—
559C4⋊C4—21no—
560C4⋊C4—472no—
561C4⋊C4—472no—
562C4⋊C4—944no—
563C4⋊C4—942no—
564C4⋊C4—942no—
565C4⋊C4—941no—
566C4⋊C4—941no—
567C4⋊C4—941no—
568C4⋊C4—941no—
569C4⋊C4C2×C22⋊D4712yes—
570C4⋊C4C2×C22⋊D4712yes—
571C4⋊C4C23×D4712yes—
572C22⋊C4—28no—
573C22⋊C4—22no—
574C22⋊C4—22no—
575C22⋊C4—944no—
576C22⋊C4—942no—
577C22⋊C4—944no—
578C22⋊C4—944no—
579C22⋊C4—942no—
580C22⋊C4—942no—
581C22⋊C4C2×C22⋊D4714yes—
582C22⋊C4C2×C22⋊D4714yes—
583C42—944no—
584C42—942no—
585C42C2×C22⋊D4711yescharacteristic
586C42C2×C22⋊D4711yescharacteristic
587C42C23×D4711yescharacteristic
588C42C23×D4711yescharacteristic
589C23—24no—
590C23—22no—
591C23—24no—
592C23—42no—
593C23—42no—
594C23—42no—
595C23—42no—
596C23—942no—
597C23C22×C22⋊D4714yes—
598C23C22×C22⋊D4711yescharacteristic
599Q8—28no—
600Q8—44no—
601Q8—944no—
602Q8—3761no—
603Q8—1882no—
604D4—24no—
605D4—24no—
606D4—24no—
607D4—22no—
608D4—416no—
609D4—944no—
610D4—3761no—
611D4—3764no—
612D4—3764no—
613D4—1884no—
614D4—1882no—
615D4#1309406914yes—
616C2×C4—28no—
617C2×C4—22no—
618C2×C4—22no—
619C2×C4—24no—
620C2×C4—42no—
621C2×C4—44no—
622C2×C4—41no—
623C2×C4—416no—
624C2×C4—44no—
625C2×C4—42no—
626C2×C4—44no—
627C2×C4—44no—
628C2×C4—942no—
629C2×C4—942no—
630C2×C4—942no—
631C2×C4—3764no—
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—24no—
649C22—48no—
650C22—42no—
651C22—42no—
652C22—84no—
653C22—1882no—
654C22—1882no—
655C22—1882no—
656C22C2×#1309404412yes—
657C22C23×C22⋊D4711yescharacteristic
658C4—21no—
659C4—21no—
660C4—24no—
661C4—21no—
662C4—42no—
663C4—44no—
664C4—1882no—
665C4—1882no—
666C4—1882no—
667C4C2×#1309404412yes—
668C4C2×#1309404412yes—
669C4C2×#1309406912yes—
670C2—22no—
671C2—44no—
672C2—44no—
673C2—3762no—
674C2#2025702311yescharacteristic
675C2C22×#1309404411yescharacteristic
676C2#2025817511yescharacteristic
subgroupG/H
centerC22C23×C22⋊D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC2×C47×#547C2
socleC22×C47C23×D4
hypercenterC2×#547D47
hypocenterC47#61565
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #20257240C2semidirect
2. #20255316C2semidirect
3. #20255305C2semidirect
4. #20255114C2semidirect
5. #20255106C2semidirect
6. #20254336C2semidirect
7. #20254313C2semidirect
8. #20254169C2semidirect
9. #20253396C2semidirect
10. #20253388C2semidirect
11. #20252606C2semidirect
12. #20252412C2semidirect
13. C2×C47×#547C2semidirect
14. C2×#13094044C22semidirect
16. #15689843C22semidirect
17. #15689838C22semidirect
19. #15689794C22semidirect
20. #15689778C22semidirect
27. #15689502C22semidirect
28. #15689497C22semidirect
29. #15689473C22semidirect
30. #15689473C22semidirect
31. #15689461C22semidirect
34. #15689413C22semidirect
35. #15689403C22semidirect
36. #15689362C22semidirect
37. #15689354C22semidirect
38. #15689330C22semidirect
39. #15689328C22semidirect
40. #15689325C22semidirect
41. #15689318C22semidirect
42. #15689285C22semidirect
43. #15689283C22semidirect
46. #15689268C22semidirect
49. C4×#10625335C22semidirect
50. C4×#10625334C22semidirect
51. D4×C47⋊C8C22semidirect
52. D4×C47⋊C8C22semidirect
55. #15689119C22semidirect
280. C2×C4×C47#135semidirect
333. C2×#547D47semidirect
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
#202572402
#202553162
#202553052
#202551142
#202551062
#202543362
#202543132
#202541692
#202533962
#202533882
#202526062
#202524122
C2×C47×#5472
#6156547