Finite Group Databasebeta

← #26369038

Subgroups — #26369038

Order 12032.

1240 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#20257028C211yesmaximal, characteristic
2#20251867C211yesmaximal, characteristic, complemented
3#20245041C212yesmaximal, complemented
4#20245032C214yesmaximal, complemented
5#20245027C212yesmaximal, complemented
6C2×#15688912C214yesmaximal, complemented
7C47×#2883C211yesmaximal, characteristic, complemented
8C22×#10625352C411yescharacteristic, complemented
9#15689890—21no—
10#15689772—22no—
11#15689772—21no—
12#15689772—21no—
13#15689767—21no—
14#15689767—22no—
15#15689767—21no—
16#15689694C411yescharacteristic, complemented
17#15689694C2211yescharacteristic
18#15689682C412yescomplemented
19#15689682C2212yes—
20#15689630C411yescharacteristic, complemented
21#15689630C2211yescharacteristic
22C2×#13094005C412yescomplemented
23C2×#13094005C2212yes—
24#15689598C411yescharacteristic, complemented
25#15689107—22no—
26#15689107C2212yes—
27C2×#13093941—21no—
28C2×#13093941C2214yes—
29#15688912C2214yescomplemented
30#15688912C2218yescomplemented
31#15688909C2214yescomplemented
32C2×H(5,𝔽2)×C47C2211yescharacteristic
33C47×#518—21no—
34C47×#410—21no—
35C47×#410C2212yes—
36C2×#99×C47—22no—
37C2×#99×C47C2214yes—
38C2×#10625352—22no—
39C2×#10625352—21no—
40C2×#10625352—21no—
41C2×#10625352—44no—
42C22×Dic94C2×C411yescharacteristic, complemented
43C22×C22⋊D47C2×C412yescomplemented
44C23×Dic47C2×C412yescomplemented
45C22×C4×D47C2×C411yescharacteristic, complemented
46#13094031—44no—
47C2×#10625329—22no—
48C2×#10625329—24no—
49#13094022—21no—
50#13094022—21no—
51#13094011—21no—
52#13094011—21no—
53#13094007—22no—
54#13094007—22no—
55#13094005—22no—
56#13094005—22no—
57#13094005—22no—
58#13094005—22no—
59#13094005C2×C411yescharacteristic, complemented
60#13094005C2×C412yescomplemented
61#13094005C2×C412yescomplemented
62#13094005C2×C412yescomplemented
63#13094005C2×C412yescomplemented
64#13094005C2311yescharacteristic
65#13094005C2312yes—
66#13094003—22no—
67#13094003—21no—
68#13094003—21no—
69#13094000—21no—
70#13094000—21no—
71#13093996—22no—
72#13093996—22no—
73#13093996—22no—
74#13093996—22no—
75#13093995—21no—
76#13093995—21no—
77#13093993—22no—
78#13093993—22no—
79#13093992C2×C412yescomplemented
80#13093992C2×C411yescharacteristic, complemented
81#13093992C2311yescharacteristic
82D4×Dic47—22no—
83D4×Dic47—21no—
84D4×Dic47—21no—
85C4×C22⋊D47—22no—
86C4×C22⋊D47—21no—
87C4×C22⋊D47—21no—
88#13093983C2×C411yescharacteristic, complemented
89#13093983C2×C411yescharacteristic, complemented
90C2×Dic47⋊C4—22no—
91C2×Dic47⋊C4—22no—
92C2×C22⋊Dic47—22no—
93C2×C22⋊Dic47—22no—
94C2×C22⋊Dic47C2×C412yescomplemented
95C2×C22⋊Dic47C2×C412yescomplemented
96#13093943—22no—
97#13093941—24no—
98#13093941C2314yes—
99H(5,𝔽2)×C47—44no—
100C2×#41×C47C2×C411yescharacteristic
101C2×#41×C47D412yes—
102C2×#41×C47D411yescharacteristic
103C2×#41×C47D411yescharacteristic
104C2×#41×C47C2311yescharacteristic
105C22×D4×C47C2×C412yes—
106C22×D4×C47C2×C411yescharacteristic
107C22×D4×C47D412yes—
108C22×D4×C47D411yescharacteristic
109C22×D4×C47D411yescharacteristic
110C22×D4×C47C2312yes—
111C4×D4×C47—44no—
112#117×C47—22no—
113C2×C22⋊C4×C47—24no—
114#99×C47—22no—
115#99×C47—24no—
116#99×C47C2314yes—
117C24×C47C22⋊C412yes—
118C24×C47C22×C412yes—
119C24×C47C2×D412yes—
120C23×D47C22×C411yescharacteristic
121#41×C47—24no—
122#41×C47—42no—
123#41×C47—44no—
124#41×C47—42no—
125#41×C47—42no—
126C2×Q8×C47C22⋊C411yescharacteristic
127C2×Q8×C47C2×D411yescharacteristic
128C2×D4×C47—24no—
129C2×D4×C47—24no—
130C2×D4×C47—42no—
131C2×D4×C47—44no—
132C2×D4×C47—44no—
133C2×D4×C47—44no—
134C2×D4×C47C22⋊C412yes—
135C2×D4×C47C22⋊C411yescharacteristic
136C2×D4×C47C22⋊C412yes—
137C2×D4×C47C22⋊C412yes—
138C2×D4×C47C22⋊C411yescharacteristic
139C2×D4×C47C22⋊C411yescharacteristic
140C2×D4×C47C22×C411yescharacteristic
141C2×D4×C47C22×C412yes—
142C2×D4×C47C2×D411yescharacteristic
143C2×D4×C47C2×D412yes—
144C2×D4×C47C2×D412yes—
145C2×D4×C47C2411yescharacteristic
146C22×C4×C47—44no—
147C22×C4×C47C22⋊C412yes—
148C22×C4×C47C22⋊C412yes—
149C22×C4×C47C22×C411yescharacteristic
150C22×C4×C47C2×D411yescharacteristic
151C22×C4×C47C2×D411yescharacteristic
152C22×C4×C47C2×D412yes—
153C2×C22⋊D47—22no—
154C2×C22⋊D47—22no—
155C2×C22⋊D47—22no—
156C2×C22⋊D47—21no—
157C2×C22⋊D47—21no—
158C2×C22⋊D47—44no—
159C22×Dic47—22no—
160C22×Dic47—22no—
161C22×Dic47—22no—
162C22×Dic47—22no—
163C22×Dic47—44no—
164C22×Dic47C22×C411yescharacteristic
165C22×Dic47C22×C412yes—
166#10625352—48no—
167#10625352—44no—
168#10625352—44no—
169C2×C4×D47—21no—
170C2×C4×D47—21no—
171C2×C4×D47—42no—
172C2×Dic94—21no—
173C2×Dic94—21no—
174C2×Dic94—42no—
175C4⋊C4×C47—42no—
176C22⋊C4×C47—24no—
177C22⋊C4×C47—44no—
178C42×C47—42no—
179C22⋊Dic47—22no—
180C22⋊Dic47—21no—
181C22⋊Dic47—22no—
182C22⋊Dic47—22no—
183C22⋊Dic47—22no—
184C22⋊Dic47—22no—
185C22⋊Dic47—22no—
186C22⋊Dic47—21no—
187C22⋊Dic47—22no—
188C22⋊Dic47C22×C412yes—
189C22⋊Dic47C22×C411yescharacteristic
190C22⋊Dic47C22×C411yescharacteristic
191(C2×D47)⋊C4—21no—
192(C2×D47)⋊C4—22no—
193(C2×D47)⋊C4—21no—
194C4⋊Dic47—21no—
195C4⋊Dic47—21no—
196Dic47⋊C4—21no—
197Dic47⋊C4—22no—
198Dic47⋊C4—22no—
199Dic47⋊C4—22no—
200Dic47⋊C4—22no—
201Dic47⋊C4—21no—
202Dic47⋊C4—22no—
203C4×Dic47—21no—
204C4×Dic47—21no—
205#10625329—24no—
206#10625329—44no—
207#10625329—42no—
208C2×C47⋊C8—44no—
209C2×C47⋊C8—42no—
210C23×C47—24no—
211C23×C47—24no—
212C23×C47—42no—
213C23×C47—82no—
214C23×C47—82no—
215C23×C47—82no—
216C23×C47C2×C22⋊C412yes—
217C23×C47C2×C22⋊C411yescharacteristic
218C23×C47C2×C22⋊C412yes—
219C23×C47C2×C22⋊C412yes—
220C23×C47C23×C411yescharacteristic
221C23×C47C22×D411yescharacteristic
222C22×D47—21no—
223C22×D47—41no—
224C22×D47—81no—
225Q8×C47—22no—
226Q8×C47—41no—
227D4×C47—24no—
228D4×C47—24no—
229D4×C47—24no—
230D4×C47—22no—
231D4×C47—22no—
232D4×C47—41no—
233D4×C47—42no—
234D4×C47—42no—
235D4×C47—41no—
236D4×C47—42no—
237D4×C47—41no—
238D4×C47#9914yes—
239C2×C4×C47—24no—
240C2×C4×C47—22no—
241C2×C4×C47—22no—
242C2×C4×C47—22no—
243C2×C4×C47—42no—
244C2×C4×C47—42no—
245C2×C4×C47—44no—
246C2×C4×C47—164no—
247C2×C4×C47C2×C22⋊C411yescharacteristic
248C2×C4×C47C2×C22⋊C411yescharacteristic
249C2×C4×C47C2×C22⋊C412yes—
250C2×C4×C47C2×C22⋊C411yescharacteristic
251C2×C4×C47C22×D411yescharacteristic
252C22⋊D47—44no—
253C22⋊D47—44no—
254C22⋊D47—82no—
255C22⋊D47—81no—
256C22⋊D47—81no—
257C2×Dic47—21no—
258C2×Dic47—22no—
259C2×Dic47—21no—
260C2×Dic47—22no—
261C2×Dic47—21no—
262C2×Dic47—44no—
263C2×Dic47—44no—
264C2×Dic47—42no—
265C2×Dic47—41no—
266C2×Dic47—82no—
267C2×Dic47—82no—
268C4×D47—42no—
269C4×D47—42no—
270Dic94—42no—
271Dic94—42no—
272C47⋊C8—44no—
273#33350—471nomaximal, Sylow
274C22×C47—21no—
275C22×C47—22no—
276C22×C47—21no—
277C22×C47—21no—
278C22×C47—22no—
279C22×C47—42no—
280C22×C47—42no—
281C22×C47—42no—
282C22×C47—41no—
283C22×C47—41no—
284C22×C47—41no—
285C22×C47—84no—
286C22×C47—84no—
287C22×C47—162no—
288C22×C47C2×#9914yes—
289C22×C47C22×C22⋊C411yescharacteristic
290C2×D47—41no—
291C2×D47—41no—
292C2×D47—41no—
293C2×D47—82no—
294C4×C47—21no—
295C4×C47—21no—
296C4×C47—21no—
297C4×C47—22no—
298C4×C47—84no—
299C4×C47C2×#9912yes—
300Dic47—41no—
301Dic47—42no—
302Dic47—41no—
303Dic47—42no—
304Dic47—41no—
305#3485—471no—
306#2888—471no—
307#2883D4711yescharacteristic, complemented
308#2123—472no—
309#2121—474no—
310#2120—472no—
311C2×#352—474no—
312C2×C47—21no—
313C2×C47—24no—
314C2×C47—41no—
315C2×C47—41no—
316C2×C47—42no—
317C2×C47—42no—
318C2×C47—41no—
319C2×C47C22×#9911yescharacteristic
320D47—81no—
321C2×H(5,𝔽2)C2×D4711yescharacteristic
322C22×#41—471no—
323#568—941no—
324#547—942no—
325#547—941no—
326#547—941no—
327#546—941no—
328#546—942no—
329#546—941no—
330#536—471no—
331#536—471no—
332#534—472no—
333#534—472no—
334#525—471no—
335#525—471no—
336C2×#123—472no—
337C2×#123—472no—
338#518—21no—
339#518—471no—
340#413—472no—
341#413—942no—
342C2×#100—474no—
343C2×#100—941no—
344#410—21no—
345#410C2×D4712yes—
346C2×#99—22no—
347C2×#99C2×D4714yes—
348#352—474no—
349#352—478no—
350AΣL(1,𝔽16)—474no—
351C47#3335011yescharacteristic, complemented, Sylow
352H(5,𝔽2)—44no—
353C2×#41—941no—
354C2×#41—942no—
355C2×#41—941no—
356C2×#41—1884no—
357C2×#41C4×D4711yescharacteristic
358C2×#41D18812yes—
359C2×#41D18811yescharacteristic
360C2×#41C22⋊D4711yescharacteristic
361C2×#41C22×D4711yescharacteristic
362C22×Q8—471no—
363C22×D4—472no—
364C22×D4C4×D4712yes—
365C22×D4C4×D4711yescharacteristic
366C22×D4D18811yescharacteristic
367C22×D4C22⋊D4712yes—
368C22×D4C22⋊D4711yescharacteristic
369C22×D4C22×D4712yes—
370C23×C4—472no—
371C23×C4—471no—
372#131—1884no—
373C2×M16—944no—
374C2×M16—942no—
375C4⋊Q8—941no—
376C4⋊Q8—941no—
377#124—941no—
378#124—941no—
379#123—471no—
380#123—472no—
381#123—472no—
382#123—472no—
383#123—472no—
384#123—471no—
385#123—472no—
386#123—942no—
387#123—942no—
388#123—942no—
389#123—942no—
390#123—942no—
391#123—942no—
392C22⋊Q8—941no—
393C22⋊Q8—941no—
394C22⋊Q8—942no—
395C22⋊Q8—941no—
396C22⋊Q8—941no—
397#121—941no—
398#121—942no—
399#121—942no—
400#121—942no—
401#121—941no—
402#121—942no—
403#120—471no—
404#120—472no—
405#120—471no—
406#120—942no—
407#120—942no—
408C4×D4—44no—
409C4×D4—941no—
410C4×D4—942no—
411C4×D4—941no—
412C4×D4—942no—
413C4×D4—941no—
414C4×D4—941no—
415#117—22no—
416#117—471no—
417#117—471no—
418C2×C4⋊C4—942no—
419C2×C4⋊C4—942no—
420C2×C22⋊C4—24no—
421C2×C22⋊C4—472no—
422C2×C22⋊C4—472no—
423C2×C22⋊C4—942no—
424C2×C22⋊C4—942no—
425#101—942no—
426#100—474no—
427#100—944no—
428#99—22no—
429#99—24no—
430#99C22×D4714yes—
431C24—471no—
432C24(C2×D47)⋊C412yes—
433C24C2×C4×D4712yes—
434C24C2×C22⋊D4712yes—
435#41—24no—
436#41—42no—
437#41—44no—
438#41—42no—
439#41—42no—
440#41—1888no—
441#41—1884no—
442#41—1884no—
443C2×Q8—941no—
444C2×Q8—941no—
445C2×Q8—1882no—
446C2×Q8(C2×D47)⋊C411yescharacteristic
447C2×Q8C2×D18811yescharacteristic
448C2×D4—24no—
449C2×D4—24no—
450C2×D4—42no—
451C2×D4—44no—
452C2×D4—44no—
453C2×D4—44no—
454C2×D4—942no—
455C2×D4—941no—
456C2×D4—941no—
457C2×D4—942no—
458C2×D4—942no—
459C2×D4—1884no—
460C2×D4(C2×D47)⋊C411yescharacteristic
461C2×D4(C2×D47)⋊C412yes—
462C2×D4(C2×D47)⋊C412yes—
463C2×D4(C2×D47)⋊C412yes—
464C2×D4(C2×D47)⋊C411yescharacteristic
465C2×D4(C2×D47)⋊C411yescharacteristic
466C2×D4C2×C4×D4711yescharacteristic
467C2×D4C2×C4×D4712yes—
468C2×D4C2×D18812yes—
469C2×D4C2×C22⋊D4712yes—
470C2×D4C2×C22⋊D4711yescharacteristic
471C2×D4C23×D4711yescharacteristic
472C22×C4—44no—
473C22×C4—471no—
474C22×C4—472no—
475C22×C4—942no—
476C22×C4—942no—
477C22×C4—942no—
478C22×C4—942no—
479C22×C4—941no—
480C22×C4—941no—
481C22×C4—1882no—
482C22×C4—1884no—
483C22×C4(C2×D47)⋊C412yes—
484C22×C4(C2×D47)⋊C412yes—
485C22×C4C2×C4×D4711yescharacteristic
486C22×C4C2×D18812yes—
487C22×C4C2×D18811yescharacteristic
488C22×C4C2×C22⋊D4711yescharacteristic
489M16—944no—
490M16—1884no—
491M16—1882no—
492C2×C8—1884no—
493C2×C8—1882no—
494C4⋊C4—42no—
495C4⋊C4—942no—
496C4⋊C4—941no—
497C4⋊C4—942no—
498C4⋊C4—942no—
499C4⋊C4—942no—
500C4⋊C4—941no—
501C4⋊C4—941no—
502C4⋊C4—941no—
503C4⋊C4—942no—
504C22⋊C4—24no—
505C22⋊C4—44no—
506C22⋊C4—472no—
507C22⋊C4—471no—
508C22⋊C4—471no—
509C22⋊C4—942no—
510C22⋊C4—942no—
511C22⋊C4—942no—
512C22⋊C4—941no—
513C22⋊C4—942no—
514C22⋊C4—942no—
515C22⋊C4—942no—
516C22⋊C4—941no—
517C22⋊C4—942no—
518C22⋊C4—942no—
519C22⋊C4—941no—
520C22⋊C4—941no—
521C42—42no—
522C42—941no—
523C42—941no—
524C23—24no—
525C23—24no—
526C23—42no—
527C23—82no—
528C23—82no—
529C23—82no—
530C23—941no—
531C23—3761no—
532C23—1881no—
533C23C2×(C2×D47)⋊C412yes—
534C23C2×(C2×D47)⋊C411yescharacteristic
535C23C2×(C2×D47)⋊C412yes—
536C23C2×(C2×D47)⋊C412yes—
537C23C22×C4×D4711yescharacteristic
538C23C22×C22⋊D4711yescharacteristic
539Q8—22no—
540Q8—41no—
541Q8—1882no—
542Q8—1882no—
543D4—24no—
544D4—24no—
545D4—24no—
546D4—22no—
547D4—22no—
548D4—41no—
549D4—42no—
550D4—42no—
551D4—41no—
552D4—42no—
553D4—41no—
554D4—3762no—
555D4—3761no—
556D4—3761no—
557D4—1884no—
558D4—1884no—
559D4#1309393814yes—
560C2×C4—24no—
561C2×C4—22no—
562C2×C4—22no—
563C2×C4—22no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—44no—
567C2×C4—164no—
568C2×C4—941no—
569C2×C4—942no—
570C2×C4—941no—
571C2×C4—941no—
572C2×C4—942no—
573C2×C4—3762no—
574C2×C4—3762no—
575C2×C4—1882no—
576C2×C4—1884no—
577C2×C4—1884no—
578C2×C4—1882no—
579C2×C4—1881no—
580C2×C4—1882no—
581C2×C4C2×(C2×D47)⋊C411yescharacteristic
582C2×C4C2×(C2×D47)⋊C412yes—
583C2×C4C2×(C2×D47)⋊C411yescharacteristic
584C2×C4C2×(C2×D47)⋊C411yescharacteristic
585C2×C4C22×D18811yescharacteristic
586C8—1884no—
587C22—21no—
588C22—21no—
589C22—21no—
590C22—22no—
591C22—22no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—84no—
599C22—84no—
600C22—162no—
601C22—3762no—
602C22—1881no—
603C22—1881no—
604C22—1881no—
605C22C2×#1309393814yes—
606C22C22×(C2×D47)⋊C411yescharacteristic
607C4—21no—
608C4—21no—
609C4—21no—
610C4—22no—
611C4—84no—
612C4—1881no—
613C4—1882no—
614C4—1881no—
615C4—1881no—
616C4—1882no—
617C4C2×#1309393812yes—
618C2—21no—
619C2—24no—
620C2—41no—
621C2—41no—
622C2—42no—
623C2—42no—
624C2—41no—
625C2—3761no—
626C2C22×#1309393811yescharacteristic
subgroupG/H
centerC2C22×#13093938
commutatorC23×C47C23×C4
FrattiniC2×D4C23×D47
FittingC47×#2883C2
socleC2×C47C22×#99
prime radical#20257028C2
cosocleC2×D4×C47C24
hypercenter#2883D47
hypocenterC47#33350
LCSG ▷ C23×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2 ◁ C2×D4 ◁ C2×H(5,𝔽2) ◁ #2883

Complements

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

NH ≅ G/Ntype
2. #20251867C2semidirect
3. #20245041C2semidirect
4. #20245032C2semidirect
5. #20245027C2semidirect
6. C2×#15688912C2semidirect
7. C47×#2883C2semidirect
8. C22×#10625352C4semidirect
16. #15689694C4semidirect
18. #15689682C4semidirect
20. #15689630C4semidirect
22. C2×#13094005C4semidirect
24. #15689598C4semidirect
29. #15688912C22semidirect
30. #15688912C22semidirect
31. #15688909C22semidirect
42. C22×Dic94C2×C4semidirect
43. C22×C22⋊D47C2×C4semidirect
44. C23×Dic47C2×C4semidirect
45. C22×C4×D47C2×C4semidirect
59. #13094005C2×C4semidirect
60. #13094005C2×C4semidirect
61. #13094005C2×C4semidirect
62. #13094005C2×C4semidirect
63. #13094005C2×C4semidirect
79. #13093992C2×C4semidirect
80. #13093992C2×C4semidirect
88. #13093983C2×C4semidirect
89. #13093983C2×C4semidirect
94. C2×C22⋊Dic47C2×C4semidirect
95. C2×C22⋊Dic47C2×C4semidirect
307. #2883D47semidirect
351. C47#33350semidirect

Maximal subgroups

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

subgroupindex
#202570282
#202518672
#202450412
#202450322
#202450272
C2×#156889122
C47×#28832
#3335047