Finite Group Databasebeta

← #25160072

Subgroups — #25160072

Order 11008.

706 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19144456C211yesmaximal, characteristic
2#19144437C211yesmaximal, characteristic, complemented
3#19143438C211yesmaximal, characteristic
4#19143194C211yesmaximal, characteristic
5#19136143C211yesmaximal, characteristic, complemented
6#19136103C211yesmaximal, characteristic, complemented
7#19135958C211yesmaximal, characteristic, complemented
8#19135944C211yesmaximal, characteristic, complemented
9#19135372C211yesmaximal, characteristic, complemented
10#19135192C211yesmaximal, characteristic, complemented
11#19135107C211yesmaximal, characteristic
12#19134967C211yesmaximal, characteristic
13#19134871C211yesmaximal, characteristic
14#19134829C211yesmaximal, characteristic
15C43×#1857C211yesmaximal, characteristic, complemented
16C2×#13063356C2211yescharacteristic
17C2×#13063333C2211yescharacteristic
18#14583527C2211yescharacteristic
19#14583380C2211yescharacteristic
20#14583379C2211yescharacteristic
21#14583206—21no—
22#14583206—21no—
23#14583205—21no—
24#14583202—21no—
25#14583194—21no—
26#14583188—21no—
27#14583184—21no—
28#14583183—21no—
29#14583004—21no—
30#14582997—21no—
31C4×#10621885—21no—
32C4×#10621884—21no—
33C2×#13063260C2211yescharacteristic
34#14582942C2211yescharacteristic
35#14582942C2211yescharacteristic
36#14582941C2211yescharacteristic
37#14582940—21no—
38#14582938—21no—
39#14582935C2211yescharacteristic
40#14582934—21no—
41C2×#13063258C2211yescharacteristic
42#107×D43—21no—
43C2×#13063256C2211yescharacteristic
44#14582927C2211yescharacteristic
45#14582926C2211yescharacteristic
46C2×#13063253C2211yescharacteristic
47#14582908C2211yescharacteristic
48#14582901C2211yescharacteristic
49#14582893C2211yescharacteristic
50C2×#13063249C2211yescharacteristic
51C2×#13063249C2211yescharacteristic
52#14582855C2211yescharacteristic
53#14582765C2211yescharacteristic
54#14582668C2211yescharacteristic
55#14582667C2211yescharacteristic
56#14582667C2211yescharacteristic
57#14582663C2211yescharacteristic
58#14582659C2211yescharacteristic
59#14582654C2211yescharacteristic
60C43×#519C2211yescharacteristic
61C43×#428C2211yescharacteristic
62C43×#428C2211yescharacteristic
63C2×#101×C43C2211yescharacteristic
64C43×#391C2211yescharacteristic
65C43×#339C2211yescharacteristic
66C43×#339C2211yescharacteristic
67C2×#10621898C2311yescharacteristic
68#13063356—22no—
69#13063356Q814yescomplemented
70C2×#10621885—21no—
71C2×#10621884—21no—
72#13063333—22no—
73#13063333—22no—
74C2×#10621877C2311yescharacteristic
75C2×#10621877C2311yescharacteristic
76C2×#10621872—21no—
77C22×C43⋊C8C2311yescharacteristic
78C22×C43⋊C8C2311yescharacteristic
79C2×C8×D43—21no—
80#13063325—21no—
81#13063324—21no—
82#13063317—21no—
83#13063316—21no—
84#13063302—41no—
85C4×Dic86—21no—
86#13063290—21no—
87C4×D172—21no—
88#13063287—21no—
89C4×C22⋊D43—41no—
90(C22⋊D43)⋊C4—41no—
91#13063284C2311yescharacteristic
92#13063283—21no—
93#13063283—21no—
94C2×C4⋊Dic43C2311yescharacteristic
95C2×C4⋊Dic43C2311yescharacteristic
96C4⋊C4×D43—21no—
97C4⋊C4×D43—21no—
98#13063260—22no—
99#13063259—21no—
100#13063258—21no—
101#13063258—21no—
102#13063258—21no—
103#13063258—21no—
104#13063258D411yescharacteristic
105#13063258D411yescharacteristic
106#13063257—21no—
107#13063256—21no—
108#13063256—21no—
109#13063256—21no—
110#13063256—21no—
111#13063256D411yescharacteristic
112#13063256D411yescharacteristic
113C4⋊(C43⋊C8)—21no—
114C4⋊(C43⋊C8)—21no—
115Dic43⋊C8—21no—
116Dic43⋊C8—21no—
117#13063253—22no—
118#13063250—21no—
119#13063249—21no—
120#13063249—21no—
121#13063249—21no—
122#13063249—21no—
123#13063249—21no—
124#13063249—21no—
125#13063248D411yescharacteristic
126#13063248D411yescharacteristic
127#13063247—21no—
128#13063245D411yescharacteristic
129#13063245D411yescharacteristic
130#13063236—21no—
131C4×C43⋊C8—21no—
132C43⋊#95—41no—
133C22×Q8×C43C2311yescharacteristic
134C2×M16×C43C2311yescharacteristic
135C2×M16×C43C2311yescharacteristic
136C4×Q8×C43—21no—
137C4×Q8×C43—21no—
138C4×Q8×C43—21no—
139C4×Q8×C43—21no—
140#117×C43—41no—
141#117×C43C2311yescharacteristic
142#117×C43C2311yescharacteristic
143C2×C4⋊C4×C43C2311yescharacteristic
144C2×C42×C43C2311yescharacteristic
145#107×C43—21no—
146#107×C43—21no—
147#106×C43—21no—
148#106×C43—21no—
149#101×C43—22no—
150#95×C43—41no—
151C2×Q8×C43—21no—
152C2×Q8×C43—21no—
153C2×Q8×C43—21no—
154C2×Q8×C43—21no—
155C2×Q8×C43—21no—
156C2×Q8×C43C2×Q812yes—
157C22×C4×C43—21no—
158C22×C4×C43—21no—
159C22×C4×C43—21no—
160C22×C4×C43C2411yescharacteristic
161C2×C22⋊D43—21no—
162C22×Dic43—21no—
163C22×Dic43—21no—
164#10621898—21no—
165#10621898—21no—
166#10621898C2×Q812yes—
167#10621898#4112yes—
168C2×D172C2×D411yescharacteristic
169C2×C4×D43—21no—
170C2×C4×D43—41no—
171C2×C4×D43—41no—
172C2×Dic86—41no—
173C2×Dic86C2×D411yescharacteristic
174M16×C43—21no—
175M16×C43—21no—
176M16×C43—21no—
177M16×C43#4112yes—
178C2×C8×C43—21no—
179C2×C8×C43—21no—
180C4⋊C4×C43—21no—
181C4⋊C4×C43—21no—
182C4⋊C4×C43—41no—
183C4⋊C4×C43—41no—
184C4⋊C4×C43C2×D411yescharacteristic
185C4⋊C4×C43C2×D411yescharacteristic
186C4⋊C4×C43C2×D411yescharacteristic
187C4⋊C4×C43C2×D411yescharacteristic
188C22⋊C4×C43—21no—
189C22⋊C4×C43—21no—
190C42×C43—21no—
191C42×C43—21no—
192C42×C43—21no—
193C42×C43—21no—
194C42×C43—41no—
195C42×C43—41no—
196C22⋊Dic43—21no—
197#10621885—24no—
198#10621884—24no—
199(C2×D43)⋊C4—41no—
200(C2×D43)⋊C4—41no—
201C4⋊Dic43—21no—
202C4⋊Dic43—21no—
203C4⋊Dic43—21no—
204C4⋊Dic43—21no—
205C4⋊Dic43—41no—
206C4⋊Dic43C2×D411yescharacteristic
207C4⋊Dic43C2×D411yescharacteristic
208Dic43⋊C4—41no—
209Dic43⋊C4—41no—
210Dic43⋊C4—41no—
211Dic43⋊C4—41no—
212C4×Dic43—21no—
213C4×Dic43—41no—
214C4×Dic43—41no—
215#10621877—21no—
216#10621877—21no—
217#10621877—21no—
218#10621877C2×Q812yes—
219C2×C43⋊C8—21no—
220C2×C43⋊C8—21no—
221C2×C43⋊C8—21no—
222C2×C43⋊C8—21no—
223C2×C43⋊C8—21no—
224C2×C43⋊C8—21no—
225C2×C43⋊C8C2×D411yescharacteristic
226C2×C43⋊C8C2×D411yescharacteristic
227C2×C43⋊C8C2×D411yescharacteristic
228C2×C43⋊C8C2×D411yescharacteristic
229#10621872—24no—
230C8×D43—24no—
231C23×C43H(5,𝔽2)11yescharacteristic
232C22×D43—21no—
233Q8×C43—22no—
234Q8×C43—22no—
235Q8×C43—41no—
236Q8×C43—41no—
237C2×C4×C43—21no—
238C2×C4×C43—21no—
239C2×C4×C43—21no—
240C2×C4×C43—21no—
241C2×C4×C43—22no—
242C2×C4×C43—41no—
243C2×C4×C43—41no—
244C2×C4×C43—41no—
245C2×C4×C43—41no—
246C2×C4×C43—41no—
247C2×C4×C43—41no—
248C2×C4×C43—81no—
249C2×C4×C43C22×D411yescharacteristic
250C2×C4×C43C22×D411yescharacteristic
251C2×C4×C43C22×Q811yescharacteristic
252C2×C4×C43C2×#4111yescharacteristic
253C2×C4×C43C2×#4111yescharacteristic
254C2×C4×C432−1+411yescharacteristic
255C22⋊D43—22no—
256C22⋊D43—22no—
257C2×Dic43—21no—
258C2×Dic43—21no—
259C2×Dic43—21no—
260C2×Dic43—41no—
261C2×Dic43—41no—
262C2×Dic43—81no—
263C2×Dic43—81no—
264D172—21no—
265D172C22⋊Q812yes—
266C4×D43—22no—
267C4×D43—22no—
268C4×D43—161no—
269C4×D43—161no—
270Dic86—21no—
271Dic86—161no—
272Dic86C22⋊Q812yes—
273C8×C43—22no—
274C8×C43—41no—
275C43⋊C8—22no—
276C43⋊C8—21no—
277C43⋊C8—21no—
278C43⋊C8—21no—
279C43⋊C8—21no—
280C43⋊C8—41no—
281#26403—431nomaximal, Sylow
282C22×C43—21no—
283C22×C43—21no—
284C22×C43#53911yescharacteristic
285C22×C43D4211yescharacteristic
286C22×C43#55411yescharacteristic
287C2×D43—22no—
288C2×D43—41no—
289C2×D43—41no—
290C4×C43—42no—
291C4×C43—41no—
292C4×C43—41no—
293C4×C43—41no—
294C4×C43—41no—
295C4×C43—81no—
296C4×C43—81no—
297C4×C43C2×C22⋊Q811yescharacteristic
298C4×C43#52511yescharacteristic
299C4×C43#54711yescharacteristic
300C4×C43D4×Q811yescharacteristic
301Dic43—22no—
302Dic43—41no—
303Dic43—41no—
304Dic43—81no—
305Dic43—81no—
306#3084—431no—
307#3082—431no—
308#2947—431no—
309#2913—431no—
310#2097—431no—
311#2089—431no—
312#2064—431no—
313#2061—431no—
314#1951—431no—
315#1922—431no—
316#1908—431no—
317#1882—431no—
318#1866—431no—
319#1860—431no—
320#1857D4311yescharacteristic, complemented
321C2×C43—21no—
322C2×C43—21no—
323C2×C43#244911yescharacteristic
324C2×C43#328911yescharacteristic
325C2×C43#329011yescharacteristic
326D43—42no—
327C2×#137—431no—
328C2×#131—431no—
329#544—431no—
330#520—431no—
331#520—431no—
332#519C2×D4311yescharacteristic
333#479—861no—
334#479—861no—
335#479—861no—
336#478—861no—
337#477—861no—
338#476—861no—
339#475—861no—
340#475—861no—
341#441—861no—
342#440—861no—
343C4×Q16—861no—
344C4×SD16—861no—
345C2×#108—431no—
346#428—431no—
347#428—431no—
348#428—431no—
349#428—861no—
350#428—861no—
351#428C2×D4311yescharacteristic
352#428C2×D4311yescharacteristic
353#427—431no—
354#427—861no—
355C2×#107—431no—
356C2×#107—861no—
357C2×#106—431no—
358#424—431no—
359#424—431no—
360C2×#104—431no—
361#419—431no—
362#417—431no—
363#416—431no—
364C2×#103—431no—
365C2×#103—431no—
366C2×#101C2×D4311yescharacteristic
367#405—431no—
368#391C2×D4311yescharacteristic
369#382—431no—
370#343—431no—
371#343—431no—
372#343—431no—
373#341—431no—
374#339—431no—
375#339C2×D4311yescharacteristic
376#339C2×D4311yescharacteristic
377#336—431no—
378C43#2640311yescharacteristic, complemented, Sylow
379C2×#41—431no—
380C22×Q8C22×D4311yescharacteristic
381#137—434no—
382#137—862no—
383C2×Q16—861no—
384C2×SD16—861no—
385#131—862no—
386#131—862no—
387C2×M16—431no—
388C2×M16—431no—
389C2×M16—861no—
390C2×M16C22×D4311yescharacteristic
391C2×M16C22×D4311yescharacteristic
392C22×C8—431no—
393C22×C8—431no—
394C22×C8—861no—
395C4⋊Q8—861no—
396C4⋊Q8—861no—
397#125—861no—
398#125—861no—
399C22⋊Q8—1721no—
400C4×Q8—21no—
401C4×Q8—21no—
402C4×Q8—21no—
403C4×Q8—21no—
404C4×Q8—861no—
405C4×Q8—861no—
406C4×D4—861no—
407C4×D4—861no—
408C4×D4—1721no—
409C4×D4—1721no—
410#117—41no—
411#117—431no—
412#117—861no—
413#117—861no—
414#117C22×D4311yescharacteristic
415#117C22×D4311yescharacteristic
416C2×C4⋊C4—431no—
417C2×C4⋊C4—431no—
418C2×C4⋊C4—861no—
419C2×C4⋊C4—861no—
420C2×C4⋊C4C22×D4311yescharacteristic
421C2×C42C22×D4311yescharacteristic
422#108—862no—
423#107—21no—
424#107—21no—
425#107—431no—
426#107—431no—
427#107—861no—
428#107—861no—
429#107—861no—
430#107—861no—
431#107—861no—
432#106—21no—
433#106—21no—
434#106—431no—
435#106—431no—
436#106—861no—
437#106—861no—
438#106—861no—
439#106—861no—
440#106—861no—
441C4⋊C8—861no—
442C4⋊C8—861no—
443C4⋊C8—861no—
444C4⋊C8—861no—
445#104—862no—
446#103—431no—
447#103—431no—
448#103—861no—
449#103—861no—
450#103—861no—
451#103—861no—
452#103—861no—
453#103—861no—
454#103—861no—
455#102—431no—
456#102—431no—
457#102—861no—
458#101—22no—
459#97—861no—
460C4×C8—861no—
461#95—41no—
462#95—1721no—
463#41—432no—
464#41—432no—
465#41—861no—
466#41—861no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—21no—
472C2×Q8—431no—
473C2×Q8—1721no—
474C2×Q8Q8×D4312yes—
475C2×D4—431no—
476C2×D4—861no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—861no—
481C22×C4—861no—
482C22×C4—861no—
483C22×C4—1721no—
484C22×C4—1721no—
485C22×C4C23×D4311yescharacteristic
486Q16—864no—
487SD16—864no—
488M16—21no—
489M16—21no—
490M16—21no—
491M16—432no—
492M16—864no—
493M16—861no—
494M16—861no—
495M16—861no—
496M16#1062189812yes—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—431no—
500C2×C8—431no—
501C2×C8—431no—
502C2×C8—431no—
503C2×C8—864no—
504C2×C8—861no—
505C2×C8—861no—
506C2×C8—861no—
507C2×C8—861no—
508C2×C8—861no—
509C2×C8—861no—
510C4⋊C4—21no—
511C4⋊C4—21no—
512C4⋊C4—41no—
513C4⋊C4—41no—
514C4⋊C4—431no—
515C4⋊C4—431no—
516C4⋊C4—861no—
517C4⋊C4—861no—
518C4⋊C4—861no—
519C4⋊C4—861no—
520C4⋊C4—1721no—
521C4⋊C4—1721no—
522C4⋊C4—1721no—
523C4⋊C4—1721no—
524C4⋊C4—1721no—
525C4⋊C4D4×D4311yescharacteristic
526C4⋊C4D4×D4311yescharacteristic
527C4⋊C4D4×D4311yescharacteristic
528C4⋊C4D4×D4311yescharacteristic
529C22⋊C4—21no—
530C22⋊C4—21no—
531C22⋊C4—861no—
532C22⋊C4—1721no—
533C22⋊C4—1721no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—21no—
538C42—41no—
539C42—41no—
540C42—861no—
541C42—1721no—
542C42—1721no—
543C23—861no—
544C23#1306337011yescharacteristic
545Q8—22no—
546Q8—22no—
547Q8—41no—
548Q8—41no—
549Q8—432no—
550Q8—861no—
551Q8—6881no—
552D4—432no—
553D4—862no—
554D4—862no—
555D4—861no—
556C2×C4—22no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—81no—
568C2×C4—861no—
569C2×C4—862no—
570C2×C4—862no—
571C2×C4—861no—
572C2×C4—861no—
573C2×C4—3441no—
574C2×C4—3441no—
575C2×C4—1721no—
576C2×C4—1721no—
577C2×C4—6881no—
578C2×C4—6881no—
579C2×C4C2×D4×D4311yescharacteristic
580C2×C4C2×D4×D4311yescharacteristic
581C2×C4C2×Q8×D4311yescharacteristic
582C2×C4#41×D4311yescharacteristic
583C2×C4C2×#1062189811yescharacteristic
584C2×C4#1306337211yescharacteristic
585C8—22no—
586C8—41no—
587C8—861no—
588C8—861no—
589C8—862no—
590C8—861no—
591C8—861no—
592C8—1721no—
593C22—21no—
594C22—21no—
595C22—862no—
596C22—1721no—
597C22—1721no—
598C22#1458349211yescharacteristic
599C22#1458353111yescharacteristic
600C22#1458360411yescharacteristic
601C4—42no—
602C4—41no—
603C4—41no—
604C4—41no—
605C4—41no—
606C4—81no—
607C4—81no—
608C4—862no—
609C4—3441no—
610C4—3441no—
611C4—1721no—
612C4—1721no—
613C4C22⋊Q8×D4311yescharacteristic
614C4#1458340811yescharacteristic
615C4#1458355411yescharacteristic
616C4#1458356211yescharacteristic
617C2—21no—
618C2—21no—
619C2—1722no—
620C2#1913840311yescharacteristic
621C2#1914636511yescharacteristic
622C2#1914637411yescharacteristic
subgroupG/H
centerC22#14583531
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1857C2
socleC22×C43D42
prime radicalC2×#10621898C23
hypercenter#1857D43
hypocenterC43#26403
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1857

Complements

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

NH ≅ G/Ntype
2. #19144437C2semidirect
5. #19136143C2semidirect
6. #19136103C2semidirect
7. #19135958C2semidirect
8. #19135944C2semidirect
9. #19135372C2semidirect
10. #19135192C2semidirect
15. C43×#1857C2semidirect
69. #13063356Q8semidirect
320. #1857D43semidirect
378. C43#26403semidirect

Maximal subgroups

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

subgroupindex
#191444562
#191444372
#191434382
#191431942
#191361432
#191361032
#191359582
#191359442
#191353722
#191351922
#191351072
#191349672
#191348712
#191348292
C43×#18572
#2640343