Finite Group Databasebeta

← #25126337

Subgroups — #25126337

Order 11008.

776 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#137×Dic43C211yesmaximal, characteristic, complemented
2#19143408C211yesmaximal, characteristic
3C2×#14583002C211yesmaximal, characteristic, complemented
4#19143242C211yesmaximal, characteristic, complemented
5#19136104C211yesmaximal, characteristic, complemented
6#19136038C211yesmaximal, characteristic, complemented
7#19135960C211yesmaximal, characteristic, complemented
8#19135950C211yesmaximal, characteristic, complemented
9#19135886C211yesmaximal, characteristic
10#19135854C211yesmaximal, characteristic, complemented
11#19135706C211yesmaximal, characteristic
12#19134503C211yesmaximal, characteristic, complemented
13#101×Dic43C211yesmaximal, characteristic, complemented
14#19134386C211yesmaximal, characteristic, complemented
15C43×#2052C211yesmaximal, characteristic, complemented
16C2×#13063350C2211yescharacteristic
17C22×#10621884C2211yescharacteristic
18#41×Dic43C2211yescharacteristic
19C2×Q8×Dic43C2211yescharacteristic
20C2×#13063287C2211yescharacteristic
21#14583008—21no—
22#14583006—21no—
23#14583002—21no—
24#14583002—22no—
25#14583002—22no—
26#14583000—21no—
27Q16×Dic43—21no—
28SD16×Dic43—21no—
29#14582984—21no—
30#14582979—21no—
31#14582957—22no—
32#14582953—22no—
33C2×#13063261C2211yescharacteristic
34#14582941C2211yescharacteristic
35#14582939C2211yescharacteristic
36C2×#13063258C2211yescharacteristic
37#14582927C2211yescharacteristic
38#14582925C2211yescharacteristic
39#14582908C2211yescharacteristic
40#14582905C2211yescharacteristic
41#14582900C2211yescharacteristic
42C2×#13063247C2211yescharacteristic
43C2×#13063244C2211yescharacteristic
44C2×#13063241C2211yescharacteristic
45C2×#13063241C2211yescharacteristic
46#14582866C2211yescharacteristic
47M16×Dic43C2211yescharacteristic
48M16×Dic43C2211yescharacteristic
49M16×Dic43C2211yescharacteristic
50C2×#13063235C2211yescharacteristic
51#14582668C2211yescharacteristic
52#14582667C2211yescharacteristic
53#14582667C2211yescharacteristic
54#14582663C2211yescharacteristic
55#14582663C2211yescharacteristic
56C2×#137×C43C2211yescharacteristic
57C2×#108×C43C2211yescharacteristic
58C43×#417C2211yescharacteristic
59C2×#103×C43C2211yescharacteristic
60C2×#101×C43C2211yescharacteristic
61C43×#408C2211yescharacteristic
62C43×#343C2211yescharacteristic
63C22×D172C2311yescharacteristic
64C22×C4×D43—21no—
65#13063350—22no—
66#13063350—22no—
67C2×#10621884—21no—
68C2×#10621884—22no—
69C2×#10621884—22no—
70C2×#10621884—22no—
71C2×#10621882—21no—
72C2×#10621877C2311yescharacteristic
73C2×#10621877C2311yescharacteristic
74C2×#10621877C2311yescharacteristic
75C22×C43⋊C8C2311yescharacteristic
76Q8×Dic43—21no—
77Q8×Dic43—21no—
78Q8×Dic43—21no—
79Q8×Dic43—21no—
80Q8×Dic43D411yescharacteristic
81Q8×Dic43D411yescharacteristic
82D4×Dic43—21no—
83D4×Dic43—21no—
84D4×Dic43D411yescharacteristic
85D4×Dic43D411yescharacteristic
86#13063287—21no—
87#13063287—21no—
88#13063287—21no—
89#13063287—21no—
90#13063284—21no—
91#13063284C2311yescharacteristic
92C2×C4⋊Dic43—21no—
93C2×C4⋊Dic43C2311yescharacteristic
94C2×(C2×D43)⋊C4—21no—
95C2×C4×Dic43—21no—
96C2×C4×Dic43—21no—
97C2×C4×Dic43C2311yescharacteristic
98#13063261—22no—
99#13063259—21no—
100#13063258—21no—
101#13063258—22no—
102#13063257—21no—
103#13063256—21no—
104Dic43⋊C8—22no—
105Dic43⋊C8—22no—
106#13063249—21no—
107#13063249—21no—
108#13063249—21no—
109#13063247—22no—
110#13063247—21no—
111#13063247—21no—
112#13063246—21no—
113#13063244—22no—
114#13063241—22no—
115#13063241—22no—
116C22⋊(C43⋊C8)—22no—
117#13063235—21no—
118#13063235—21no—
119#13063235—21no—
120#13063235D412yes—
121#13063235D412yes—
122C8×Dic43—21no—
123C8×Dic43—21no—
124C8×Dic43—21no—
125C2×#41×C43C2311yescharacteristic
126C22×Q8×C43C2311yescharacteristic
127#137×C43—22no—
128#137×C43—22no—
129C2×Q16×C43—21no—
130C2×SD16×C43—21no—
131C2×M16×C43C2311yescharacteristic
132C2×M16×C43C2311yescharacteristic
133C2×M16×C43C2311yescharacteristic
134C22×C8×C43C2311yescharacteristic
135C2×C4⋊C4×C43C2311yescharacteristic
136#108×C43—22no—
137#103×C43—21no—
138#103×C43—22no—
139#102×C43—21no—
140#101×C43—22no—
141C22⋊C8×C43—22no—
142C23×D43—21no—
143#41×C43—21no—
144#41×C43—21no—
145#41×C43—21no—
146#41×C43—21no—
147C2×Q8×C43—21no—
148C2×Q8×C43—21no—
149C2×Q8×C43—21no—
150C2×Q8×C43—21no—
151C2×Q8×C43—21no—
152C2×Q8×C43C2×D411yescharacteristic
153C2×Q8×C43#4112yes—
154C2×D4×C43—21no—
155C2×D4×C43C2×D411yescharacteristic
156C22×C4×C43—21no—
157C22×C4×C43—21no—
158C22×C4×C43—21no—
159C22×C4×C43C2411yescharacteristic
160C22×Dic43—21no—
161C22×Dic43—21no—
162C22×Dic43—21no—
163C22×Dic43#4111yescharacteristic
164C22×Dic43#4111yescharacteristic
165C2×D172—21no—
166C2×D172—21no—
167C2×D172—21no—
168C2×D172—21no—
169C2×D172—21no—
170C2×D172#4112yes—
171C2×C4×D43—41no—
172C2×C4×D43—41no—
173C2×C4×D43—82no—
174Q16×C43—24no—
175SD16×C43—24no—
176M16×C43—21no—
177M16×C43—21no—
178M16×C43—21no—
179M16×C43—21no—
180M16×C43—21no—
181M16×C43#4112yes—
182C2×C8×C43—21no—
183C2×C8×C43—21no—
184C2×C8×C43—21no—
185C2×C8×C43—21no—
186C2×C8×C43—21no—
187C2×C8×C43C2×D412yes—
188C4⋊C4×C43—21no—
189C4⋊C4×C43—21no—
190C22⋊Dic43—21no—
191C22⋊Dic43—21no—
192C22⋊Dic43—21no—
193#10621884—28no—
194#10621884—24no—
195#10621882—24no—
196(C2×D43)⋊C4—41no—
197(C2×D43)⋊C4—41no—
198C4⋊Dic43—21no—
199C4⋊Dic43—21no—
200C4⋊Dic43—21no—
201C4⋊Dic43—21no—
202C4⋊Dic43—41no—
203C4⋊Dic43—41no—
204C4⋊Dic43C2×D411yescharacteristic
205C4⋊Dic43C2×D411yescharacteristic
206C4×Dic43—21no—
207C4×Dic43—21no—
208C4×Dic43—21no—
209C4×Dic43—41no—
210C4×Dic43—41no—
211C4×Dic43C2×D411yescharacteristic
212C4×Dic43C2×D411yescharacteristic
213C4×Dic43C2×D411yescharacteristic
214C4×Dic43C2×D411yescharacteristic
215#10621877—21no—
216#10621877—21no—
217#10621877—21no—
218#10621877—21no—
219#10621877—21no—
220#10621877#4112yes—
221C2×C43⋊C8—21no—
222C2×C43⋊C8—21no—
223C2×C43⋊C8—21no—
224C2×C43⋊C8—21no—
225C2×C43⋊C8—21no—
226C2×C43⋊C8C2×D412yes—
227C23×C43—21no—
228C23×C43C2×#4111yescharacteristic
229C22×D43—22no—
230C22×D43—41no—
231C22×D43—41no—
232C22×D43—81no—
233C22×D43—81no—
234Q8×C43—22no—
235Q8×C43—21no—
236Q8×C43—22no—
237Q8×C43—21no—
238Q8×C43—41no—
239Q8×C43—41no—
240D4×C43—21no—
241D4×C43—21no—
242D4×C43—41no—
243D4×C43—41no—
244C2×C4×C43—21no—
245C2×C4×C43—22no—
246C2×C4×C43—41no—
247C2×C4×C43—41no—
248C2×C4×C43—41no—
249C2×C4×C43—41no—
250C2×C4×C43—41no—
251C2×C4×C43—81no—
252C2×C4×C43C22×D411yescharacteristic
253C2×C4×C43C22×D411yescharacteristic
254C2×C4×C43C2×#4111yescharacteristic
255C2×C4×C43C2×#4111yescharacteristic
256C2×C4×C43C2×#4111yescharacteristic
257C2×C4×C43C2×#4111yescharacteristic
258C2×Dic43—21no—
259C2×Dic43—21no—
260C2×Dic43—41no—
261C2×Dic43—41no—
262C2×Dic43—41no—
263C2×Dic43—81no—
264C2×Dic43—81no—
265C2×Dic43—81no—
266C2×Dic43#12111yescharacteristic
267C2×Dic43#12111yescharacteristic
268C2×Dic43#12111yescharacteristic
269C2×Dic43#12111yescharacteristic
270D172—22no—
271D172—22no—
272D172—41no—
273D172—41no—
274C4×D43—82no—
275C4×D43—82no—
276C8×C43—22no—
277C8×C43—22no—
278C8×C43—41no—
279C8×C43—41no—
280C43⋊C8—22no—
281C43⋊C8—22no—
282C43⋊C8—41no—
283C43⋊C8—41no—
284#23721—431nomaximal, Sylow
285C22×C43—21no—
286C22×C43—21no—
287C22×C43—41no—
288C22×C43—41no—
289C22×C43—41no—
290C22×C43C2×#12111yescharacteristic
291C22×C43#52911yescharacteristic
292C22×C43#52911yescharacteristic
293C2×D43—42no—
294C2×D43—82no—
295C2×D43—81no—
296C2×D43—81no—
297C2×D43—82no—
298C4×C43—41no—
299C4×C43—42no—
300C4×C43—81no—
301C4×C43#52511yescharacteristic
302C4×C43#52511yescharacteristic
303C4×C43#52511yescharacteristic
304C4×C43#53211yescharacteristic
305Dic43—21no—
306Dic43—21no—
307Dic43—21no—
308Dic43—21no—
309Dic43—41no—
310Dic43—41no—
311Dic43—81no—
312C4×#137—431no—
313C4×AGL(1,ℤ/8)—431no—
314C2×#440—431no—
315#2923—431no—
316#2089—431no—
317#2078—431no—
318#2064—431no—
319#2063—431no—
320#2052D4311yescharacteristic, complemented
321#2051—431no—
322#2046—431no—
323#2019—431no—
324#1778—431no—
325C4×#101—431no—
326C4×#100—431no—
327C2×C43—21no—
328C2×C43—21no—
329C2×C43—81no—
330C2×C43#239211yescharacteristic
331C2×C43#306811yescharacteristic
332C2×C43#306911yescharacteristic
333D43—82no—
334C2×#137C2×D4311yescharacteristic
335C2×AGL(1,ℤ/8)—431no—
336C22×SD16—431no—
337C4×#41—431no—
338C2×C4×Q8—431no—
339C2×C4×D4—431no—
340#442—861no—
341#441—861no—
342#440—862no—
343#440—862no—
344#440—861no—
345#440—861no—
346C4×Q16—861no—
347C4×SD16—861no—
348C4×SD16—861no—
349C4×D8—861no—
350#433—862no—
351#432—862no—
352C2×#108—431no—
353C2×#108C2×D4311yescharacteristic
354#428—431no—
355#428—431no—
356C2×#107—431no—
357#424—431no—
358#424—431no—
359#419—431no—
360#418—431no—
361#417—431no—
362#417C2×D4311yescharacteristic
363C2×#103C2×D4311yescharacteristic
364C2×#102—431no—
365C2×#101—431no—
366C2×#101C2×D4311yescharacteristic
367C2×#100—431no—
368C2×#100—431no—
369#408—431no—
370#408C2×D4311yescharacteristic
371C4×M16—431no—
372C4×M16—431no—
373C4×M16—431no—
374C2×#97—431no—
375#343—431no—
376#343—431no—
377#343—431no—
378#343C2×D4311yescharacteristic
379#341—431no—
380#341—431no—
381C43#2372111yescharacteristic, complemented, Sylow
382C2×#41C22×D4311yescharacteristic
383C22×Q8C22×D4311yescharacteristic
384C22×D4—431no—
385C23×C4—861no—
386#137—22no—
387#137—22no—
388AGL(1,ℤ/8)—862no—
389AGL(1,ℤ/8)—862no—
390C2×Q16—21no—
391C2×SD16—21no—
392C2×SD16—862no—
393C2×SD16—862no—
394C2×SD16—861no—
395C2×SD16—862no—
396C2×D8—861no—
397C2×M16—431no—
398C2×M16—431no—
399C2×M16—431no—
400C2×M16C22×D4311yescharacteristic
401C2×M16C22×D4311yescharacteristic
402C2×M16C22×D4311yescharacteristic
403C22×C8—431no—
404C22×C8C22×D4311yescharacteristic
405C4×Q8—431no—
406C4×Q8—431no—
407C4×Q8—861no—
408C4×Q8—861no—
409C4×Q8—861no—
410C4×Q8—861no—
411C4×D4—431no—
412C4×D4—431no—
413C4×D4—861no—
414C4×D4—861no—
415C4×D4—861no—
416C4×D4—861no—
417C4×D4—861no—
418C4×D4—861no—
419#117—431no—
420#117—861no—
421C2×C4⋊C4—431no—
422C2×C4⋊C4—861no—
423C2×C4⋊C4C22×D4311yescharacteristic
424C2×C22⋊C4—861no—
425C2×C42—431no—
426C2×C42—861no—
427C2×C42—861no—
428#108—22no—
429#108—862no—
430#107—861no—
431#107—862no—
432#107—861no—
433#106—861no—
434#106—861no—
435C4⋊C8—862no—
436C4⋊C8—862no—
437#103—21no—
438#103—22no—
439#103—861no—
440#103—861no—
441#103—861no—
442#102—21no—
443#102—861no—
444#102—861no—
445#102—861no—
446#102—862no—
447#101—22no—
448#101—862no—
449#100—862no—
450#100—862no—
451C22⋊C8—22no—
452C22⋊C8—862no—
453#97—432no—
454#97—432no—
455#97—861no—
456#97—861no—
457#97—861no—
458C4×C8—861no—
459C4×C8—861no—
460C4×C8—861no—
461C24—861no—
462#41—21no—
463#41—21no—
464#41—21no—
465#41—21no—
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8D4×D4311yescharacteristic
472C2×Q8#1062190012yes—
473C2×D4—21no—
474C2×D4—432no—
475C2×D4—861no—
476C2×D4—861no—
477C2×D4—861no—
478C2×D4—861no—
479C2×D4—861no—
480C2×D4D4×D4311yescharacteristic
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—431no—
485C22×C4—431no—
486C22×C4—861no—
487C22×C4—861no—
488C22×C4—861no—
489C22×C4—3442no—
490C22×C4—1721no—
491C22×C4—1721no—
492C22×C4C23×D4311yescharacteristic
493Q16—24no—
494SD16—24no—
495SD16—868no—
496SD16—864no—
497D8—864no—
498M16—21no—
499M16—21no—
500M16—21no—
501M16—21no—
502M16—21no—
503M16—432no—
504M16—861no—
505M16—861no—
506M16—861no—
507M16—861no—
508M16—861no—
509M16#1062190012yes—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—432no—
516C2×C8—861no—
517C2×C8—861no—
518C2×C8—861no—
519C2×C8—861no—
520C2×C8—861no—
521C2×C8D4×D4312yes—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—431no—
525C4⋊C4—431no—
526C4⋊C4—861no—
527C4⋊C4—861no—
528C4⋊C4—861no—
529C4⋊C4—861no—
530C4⋊C4—1721no—
531C4⋊C4—1721no—
532C22⋊C4—861no—
533C22⋊C4—861no—
534C22⋊C4—861no—
535C22⋊C4—1721no—
536C22⋊C4—1721no—
537C42—431no—
538C42—431no—
539C42—431no—
540C42—431no—
541C42—861no—
542C42—861no—
543C42—861no—
544C42—1721no—
545C42—1721no—
546C23—21no—
547C23—862no—
548C23—3441no—
549C23—3441no—
550C23—1721no—
551C23—1721no—
552C23#41×D4311yescharacteristic
553Q8—22no—
554Q8—21no—
555Q8—21no—
556Q8—22no—
557Q8—41no—
558Q8—41no—
559D4—21no—
560D4—21no—
561D4—41no—
562D4—41no—
563D4—862no—
564D4—862no—
565D4—1721no—
566D4—1721no—
567C2×C4—21no—
568C2×C4—22no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—81no—
575C2×C4—431no—
576C2×C4—431no—
577C2×C4—431no—
578C2×C4—431no—
579C2×C4—861no—
580C2×C4—861no—
581C2×C4—3442no—
582C2×C4—3442no—
583C2×C4—3441no—
584C2×C4—3441no—
585C2×C4—3441no—
586C2×C4—1721no—
587C2×C4—1721no—
588C2×C4—1721no—
589C2×C4C2×D4×D4311yescharacteristic
590C2×C4C2×D4×D4311yescharacteristic
591C2×C4#41×D4311yescharacteristic
592C2×C4#41×D4311yescharacteristic
593C2×C4C2×#1062190011yescharacteristic
594C2×C4C2×#1062190011yescharacteristic
595C8—22no—
596C8—22no—
597C8—41no—
598C8—41no—
599C8—862no—
600C8—862no—
601C8—1721no—
602C8—1721no—
603C22—21no—
604C22—21no—
605C22—41no—
606C22—41no—
607C22—41no—
608C22—3442no—
609C22—3442no—
610C22—3441no—
611C22—3441no—
612C22—1722no—
613C22#121×D4311yescharacteristic
614C22#1458342611yescharacteristic
615C22#1458342811yescharacteristic
616C4—41no—
617C4—42no—
618C4—81no—
619C4—861no—
620C4—861no—
621C4—861no—
622C4—861no—
623C4—3441no—
624C4—1721no—
625C4—1721no—
626C4#1458340511yescharacteristic
627C4#1458340711yescharacteristic
628C4#1458340911yescharacteristic
629C4#1458344411yescharacteristic
630C2—21no—
631C2—21no—
632C2—81no—
633C2—3442no—
634C2#1913783811yescharacteristic
635C2#1914431811yescharacteristic
636C2#1914433011yescharacteristic
subgroupG/H
centerC22#121×D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2052C2
socleC22×C43C2×#121
prime radicalC2×#13063350C22
hypercenter#2052D43
hypocenterC43#23721
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2052

Complements

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

NH ≅ G/Ntype
1. #137×Dic43C2semidirect
3. C2×#14583002C2semidirect
4. #19143242C2semidirect
5. #19136104C2semidirect
6. #19136038C2semidirect
7. #19135960C2semidirect
8. #19135950C2semidirect
10. #19135854C2semidirect
12. #19134503C2semidirect
13. #101×Dic43C2semidirect
14. #19134386C2semidirect
15. C43×#2052C2semidirect
320. #2052D43semidirect
381. C43#23721semidirect

Maximal subgroups

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

subgroupindex
#137×Dic432
#191434082
C2×#145830022
#191432422
#191361042
#191360382
#191359602
#191359502
#191358862
#191358542
#191357062
#191345032
#101×Dic432
#191343862
C43×#20522
#2372143