Finite Group Databasebeta

← #25126328

Subgroups — #25126328

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
1C4×#13063355C211yesmaximal, characteristic, complemented
2C4×#13063350C211yesmaximal, characteristic
3C2×#14583001C211yesmaximal, characteristic, complemented
4#19136100C211yesmaximal, characteristic, complemented
5#19136035C211yesmaximal, characteristic, complemented
6#19135961C211yesmaximal, characteristic, complemented
7#19135954C211yesmaximal, characteristic, complemented
8#19135893C211yesmaximal, characteristic, complemented
9#19135886C211yesmaximal, characteristic
10#19135858C211yesmaximal, characteristic, complemented
11#19135710C211yesmaximal, characteristic
12#19134501C211yesmaximal, characteristic, complemented
13C4×#13063243C211yesmaximal, characteristic, complemented
14C4×#13063241C211yesmaximal, characteristic, complemented
15C43×#2923C211yesmaximal, characteristic, complemented
16C2×#13063355C2211yescharacteristic
17C2×#13063350C2211yescharacteristic
18C22×#10621873C2211yescharacteristic
19C2×C4×Dic86C2211yescharacteristic
20C2×C4×D172C2211yescharacteristic
21#14583009—21no—
22#14583004—21no—
23#14583001—22no—
24#14583001—22no—
25#14582999—21no—
26#14582997—21no—
27C4×#10621885—21no—
28C4×#10621884—21no—
29C4×#10621883—21no—
30C4×#10621882—21no—
31C2×#13063261C2211yescharacteristic
32C2×#13063261C2211yescharacteristic
33#14582942C2211yescharacteristic
34#14582942C2211yescharacteristic
35C2×#13063259C2211yescharacteristic
36#14582907C2211yescharacteristic
37#14582903C2211yescharacteristic
38#14582900C2211yescharacteristic
39#14582900C2211yescharacteristic
40C2×#13063250C2211yescharacteristic
41C2×#13063247C2211yescharacteristic
42C2×#13063243C2211yescharacteristic
43C2×#13063243C2211yescharacteristic
44C2×#13063241C2211yescharacteristic
45C2×#13063241C2211yescharacteristic
46C4×#10621877C2211yescharacteristic
47C4×#10621877C2211yescharacteristic
48#14582667C2211yescharacteristic
49#14582667C2211yescharacteristic
50#14582667C2211yescharacteristic
51#14582667C2211yescharacteristic
52#14582664C2211yescharacteristic
53#14582664C2211yescharacteristic
54C4×#41×C43C2211yescharacteristic
55C43×#433—22no—
56C43×#432—22no—
57C43×#424C2211yescharacteristic
58C43×#424C2211yescharacteristic
59C43×#408C2211yescharacteristic
60C43×#408C2211yescharacteristic
61C4×M16×C43C2211yescharacteristic
62C2×#97×C43C2211yescharacteristic
63C22×Dic86C2311yescharacteristic
64C22×D172C2311yescharacteristic
65C22×C4×D43—21no—
66#13063355—22no—
67#13063355—22no—
68#13063350—22no—
69#13063350—22no—
70C2×#10621885—21no—
71C2×#10621873—22no—
72C2×#10621873—22no—
73C2×#10621873—22no—
74C2×#10621883—21no—
75C2×#10621884—21no—
76C2×#10621882—21no—
77C2×#10621877C2311yescharacteristic
78C2×#10621877C2311yescharacteristic
79C2×#10621877C2311yescharacteristic
80C2×#10621877C2311yescharacteristic
81C4×Dic86—21no—
82C4×Dic86—21no—
83C4×Dic86—21no—
84C4×Dic86—21no—
85C4×D172—21no—
86C4×D172—21no—
87C4×D172—21no—
88C4×D172—21no—
89C2×C4⋊Dic43C2311yescharacteristic
90C2×C4⋊Dic43C2311yescharacteristic
91C2×Dic43⋊C4—21no—
92C2×(C2×D43)⋊C4—21no—
93C2×C4×Dic43—21no—
94#13063261—22no—
95#13063261—22no—
96#13063259—22no—
97#13063258—21no—
98#13063258—21no—
99#13063256—21no—
100#13063256—21no—
101#13063250—22no—
102#13063249—21no—
103#13063249—21no—
104#13063248—21no—
105#13063248—21no—
106#13063247—22no—
107#13063246—21no—
108#13063246—21no—
109#13063245—21no—
110#13063245—21no—
111#13063243—22no—
112#13063243—22no—
113#13063241—22no—
114#13063241—22no—
115#13063236—21no—
116#13063236—21no—
117C4×C43⋊C8—21no—
118C4×C43⋊C8—21no—
119C2×#41×C43C2311yescharacteristic
120C2×M16×C43C2311yescharacteristic
121C2×M16×C43C2311yescharacteristic
122C22×C8×C43C2311yescharacteristic
123C22×C8×C43C2311yescharacteristic
124C4×Q8×C43D411yescharacteristic
125C4×Q8×C43D411yescharacteristic
126C4×D4×C43—21no—
127C4×D4×C43—21no—
128C4×D4×C43D411yescharacteristic
129C4×D4×C43D411yescharacteristic
130#117×C43—21no—
131#117×C43C2311yescharacteristic
132C2×C42×C43—21no—
133C2×C42×C43C2311yescharacteristic
134C4⋊C8×C43—22no—
135C4⋊C8×C43—22no—
136C22⋊C8×C43—22no—
137C22⋊C8×C43—22no—
138#97×C43—21no—
139#97×C43D412yes—
140#97×C43D412yes—
141C4×C8×C43—21no—
142C23×D43—21no—
143#41×C43—21no—
144#41×C43—21no—
145#41×C43—21no—
146#41×C43—21no—
147C2×Q8×C43C2×D411yescharacteristic
148C2×D4×C43—21no—
149C2×D4×C43C2×D411yescharacteristic
150C22×C4×C43—21no—
151C22×C4×C43—21no—
152C22×C4×C43—21no—
153C22×C4×C43#4111yescharacteristic
154C22×C4×C43#4111yescharacteristic
155C22×C4×C43C2411yescharacteristic
156C22×Dic43—21no—
157C22×Dic43—21no—
158C22×Dic43—21no—
159C2×D172—21no—
160C2×D172—21no—
161C2×D172—21no—
162C2×D172—21no—
163C2×D172—21no—
164C2×D172#4112yes—
165C2×C4×D43—41no—
166C2×C4×D43—41no—
167C2×C4×D43—82no—
168C2×Dic86—21no—
169C2×Dic86—21no—
170C2×Dic86—21no—
171C2×Dic86—21no—
172C2×Dic86—21no—
173C2×Dic86#4112yes—
174M16×C43—21no—
175M16×C43—21no—
176M16×C43—21no—
177M16×C43—21no—
178C2×C8×C43—21no—
179C2×C8×C43—21no—
180C2×C8×C43—21no—
181C2×C8×C43—21no—
182C2×C8×C43—21no—
183C2×C8×C43—21no—
184C2×C8×C43C2×D412yes—
185C2×C8×C43C2×D412yes—
186C4⋊C4×C43—21no—
187C4⋊C4×C43—21no—
188C4⋊C4×C43C2×D411yescharacteristic
189C4⋊C4×C43C2×D411yescharacteristic
190C22⋊C4×C43—21no—
191C22⋊C4×C43—21no—
192C22⋊C4×C43—21no—
193C42×C43—21no—
194C42×C43—21no—
195C42×C43—21no—
196C42×C43C2×D411yescharacteristic
197C42×C43C2×D411yescharacteristic
198C42×C43C2×D411yescharacteristic
199C42×C43C2×D411yescharacteristic
200#10621885—24no—
201#10621884—24no—
202#10621883—24no—
203#10621882—24no—
204(C2×D43)⋊C4—41no—
205(C2×D43)⋊C4—41no—
206C4⋊Dic43—21no—
207C4⋊Dic43—21no—
208C4⋊Dic43—21no—
209C4⋊Dic43—21no—
210Dic43⋊C4—41no—
211Dic43⋊C4—41no—
212C4×Dic43—41no—
213C4×Dic43—41no—
214#10621877—21no—
215#10621877—21no—
216#10621877—21no—
217#10621877—21no—
218#10621877—21no—
219#10621877—21no—
220#10621877#4112yes—
221#10621877#4112yes—
222C2×C43⋊C8—21no—
223C2×C43⋊C8—21no—
224C2×C43⋊C8—21no—
225C2×C43⋊C8—21no—
226#10621873—28no—
227C23×C43—21no—
228C23×C43C2×#4111yescharacteristic
229C22×D43—22no—
230C22×D43—41no—
231C22×D43—41no—
232C22×D43—81no—
233C22×D43—81no—
234Q8×C43—21no—
235Q8×C43—21no—
236D4×C43—21no—
237D4×C43—21no—
238D4×C43—41no—
239D4×C43—41no—
240C2×C4×C43—21no—
241C2×C4×C43—21no—
242C2×C4×C43—21no—
243C2×C4×C43—41no—
244C2×C4×C43—41no—
245C2×C4×C43—41no—
246C2×C4×C43—41no—
247C2×C4×C43—81no—
248C2×C4×C43—81no—
249C2×C4×C43#12111yescharacteristic
250C2×C4×C43#12111yescharacteristic
251C2×C4×C43#12111yescharacteristic
252C2×C4×C43#12111yescharacteristic
253C2×C4×C43C22×D411yescharacteristic
254C2×C4×C43C22×D411yescharacteristic
255C2×C4×C43C2×#4111yescharacteristic
256C2×C4×C43C2×#4111yescharacteristic
257C2×C4×C43C2×#4111yescharacteristic
258C2×C4×C43C2×#4111yescharacteristic
259C2×Dic43—22no—
260C2×Dic43—41no—
261C2×Dic43—41no—
262C2×Dic43—41no—
263C2×Dic43—41no—
264C2×Dic43—81no—
265C2×Dic43—81no—
266D172—22no—
267D172—22no—
268D172—41no—
269D172—41no—
270C4×D43—82no—
271C4×D43—82no—
272Dic86—22no—
273Dic86—22no—
274Dic86—41no—
275Dic86—41no—
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—21no—
299C4×C43—21no—
300C4×C43—21no—
301C4×C43—21no—
302C4×C43—41no—
303C4×C43—41no—
304C4×C43—41no—
305C4×C43#52511yescharacteristic
306C4×C43#52511yescharacteristic
307C4×C43#52511yescharacteristic
308C4×C43#53211yescharacteristic
309Dic43—42no—
310Dic43—81no—
311Dic43—81no—
312C4×#137—431no—
313C4×AGL(1,ℤ/8)—431no—
314C2×#440—431no—
315#2923D4311yescharacteristic, complemented
316#2089—431no—
317#2078—431no—
318#2064—431no—
319#2063—431no—
320#2052—431no—
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×#137—431no—
335C2×AGL(1,ℤ/8)—431no—
336C22×SD16—431no—
337C4×#41C2×D4311yescharacteristic
338C2×C4×Q8—431no—
339C2×C4×D4—431no—
340#442—861no—
341#441—861no—
342#440—861no—
343#440—861no—
344#440—862no—
345#440—862no—
346C4×Q16—861no—
347C4×SD16—861no—
348C4×SD16—861no—
349C4×D8—861no—
350#433—22no—
351#432—22no—
352C2×#108—431no—
353C2×#108—431no—
354#428—431no—
355#428—431no—
356C2×#107—431no—
357#424C2×D4311yescharacteristic
358#424C2×D4311yescharacteristic
359#419—431no—
360#418—431no—
361#417—431no—
362#417—431no—
363C2×#103—431no—
364C2×#102—431no—
365C2×#101—431no—
366C2×#101—431no—
367C2×#100—431no—
368C2×#100—431no—
369#408C2×D4311yescharacteristic
370#408C2×D4311yescharacteristic
371C4×M16—431no—
372C4×M16—431no—
373C4×M16C2×D4311yescharacteristic
374C2×#97C2×D4311yescharacteristic
375#343—431no—
376#343—431no—
377#343—431no—
378#343—431no—
379#341—431no—
380#341—431no—
381C43#2372111yescharacteristic, complemented, Sylow
382C2×#41C22×D4311yescharacteristic
383C22×Q8—431no—
384C22×D4—431no—
385C23×C4—861no—
386#137—862no—
387#137—862no—
388AGL(1,ℤ/8)—862no—
389AGL(1,ℤ/8)—862no—
390C2×Q16—861no—
391C2×SD16—862no—
392C2×SD16—862no—
393C2×SD16—861no—
394C2×SD16—861no—
395C2×SD16—862no—
396C2×D8—861no—
397C2×M16—431no—
398C2×M16—431no—
399C2×M16—431no—
400C2×M16—431no—
401C2×M16C22×D4311yescharacteristic
402C2×M16C22×D4311yescharacteristic
403C22×C8C22×D4311yescharacteristic
404C22×C8C22×D4311yescharacteristic
405C4×Q8—861no—
406C4×Q8—861no—
407C4×Q8—861no—
408C4×Q8—861no—
409C4×Q8D17211yescharacteristic
410C4×Q8D17211yescharacteristic
411C4×D4—21no—
412C4×D4—21no—
413C4×D4—861no—
414C4×D4—861no—
415C4×D4—861no—
416C4×D4—861no—
417C4×D4D17211yescharacteristic
418C4×D4D17211yescharacteristic
419#117—21no—
420#117C22×D4311yescharacteristic
421C2×C4⋊C4—431no—
422C2×C4⋊C4—431no—
423C2×C4⋊C4—861no—
424C2×C22⋊C4—861no—
425C2×C42—21no—
426C2×C42—861no—
427C2×C42C22×D4311yescharacteristic
428#108—862no—
429#108—862no—
430#107—862no—
431#107—861no—
432#107—861no—
433#106—861no—
434#106—861no—
435C4⋊C8—22no—
436C4⋊C8—22no—
437#103—861no—
438#103—861no—
439#103—862no—
440#103—861no—
441#103—861no—
442#102—861no—
443#102—861no—
444#102—862no—
445#102—861no—
446#102—861no—
447#101—862no—
448#101—862no—
449#100—862no—
450#100—862no—
451C22⋊C8—22no—
452C22⋊C8—22no—
453#97—21no—
454#97—861no—
455#97—861no—
456#97C22⋊D4312yes—
457#97C22⋊D4312yes—
458C4×C8—21no—
459C4×C8—861no—
460C4×C8—861no—
461C24—861no—
462#41—21no—
463#41—21no—
464#41—21no—
465#41—21no—
466C2×Q8—432no—
467C2×Q8—861no—
468C2×Q8—861no—
469C2×Q8—861no—
470C2×Q8—861no—
471C2×Q8—861no—
472C2×Q8C2×D17211yescharacteristic
473C2×D4—21no—
474C2×D4—432no—
475C2×D4—861no—
476C2×D4—861no—
477C2×D4—861no—
478C2×D4—861no—
479C2×D4—861no—
480C2×D4C2×D17211yescharacteristic
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—861no—
485C22×C4—861no—
486C22×C4—861no—
487C22×C4—3442no—
488C22×C4—1721no—
489C22×C4—1721no—
490C22×C4#1062189811yescharacteristic
491C22×C4#1062189811yescharacteristic
492C22×C4C23×D4311yescharacteristic
493Q16—864no—
494SD16—868no—
495SD16—864no—
496SD16—864no—
497D8—864no—
498M16—21no—
499M16—21no—
500M16—21no—
501M16—21no—
502M16—432no—
503M16—432no—
504M16—861no—
505M16—861no—
506M16—861no—
507M16—861no—
508M16—861no—
509M16—861no—
510C2×C8—21no—
511C2×C8—21no—
512C2×C8—21no—
513C2×C8—21no—
514C2×C8—21no—
515C2×C8—21no—
516C2×C8—861no—
517C2×C8—861no—
518C2×C8—861no—
519C2×C8—861no—
520C2×C8C2×C22⋊D4312yes—
521C2×C8C2×C22⋊D4312yes—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—861no—
525C4⋊C4—861no—
526C4⋊C4—861no—
527C4⋊C4—861no—
528C4⋊C4—1721no—
529C4⋊C4—1721no—
530C4⋊C4C2×D17211yescharacteristic
531C4⋊C4C2×D17211yescharacteristic
532C22⋊C4—21no—
533C22⋊C4—21no—
534C22⋊C4—21no—
535C22⋊C4—1721no—
536C22⋊C4—1721no—
537C42—21no—
538C42—21no—
539C42—21no—
540C42—1721no—
541C42—1721no—
542C42C2×D17211yescharacteristic
543C42C2×D17211yescharacteristic
544C42C2×C22⋊D4311yescharacteristic
545C42C2×C22⋊D4311yescharacteristic
546C23—21no—
547C23—862no—
548C23—3441no—
549C23—3441no—
550C23—1721no—
551C23—1721no—
552C23C2×#1062189811yescharacteristic
553Q8—21no—
554Q8—21no—
555Q8—862no—
556Q8—862no—
557Q8—1721no—
558Q8—1721no—
559D4—21no—
560D4—21no—
561D4—41no—
562D4—41no—
563D4—862no—
564D4—862no—
565D4—1721no—
566D4—1721no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—81no—
575C2×C4—81no—
576C2×C4—862no—
577C2×C4—3442no—
578C2×C4—3442no—
579C2×C4—3441no—
580C2×C4—3441no—
581C2×C4—1721no—
582C2×C4—1721no—
583C2×C4—1721no—
584C2×C4—1721no—
585C2×C4#1306329911yescharacteristic
586C2×C4#1306329911yescharacteristic
587C2×C4#1306329911yescharacteristic
588C2×C4#1306329911yescharacteristic
589C2×C4C22×C22⋊D4311yescharacteristic
590C2×C4C22×D17211yescharacteristic
591C2×C4#41×D4311yescharacteristic
592C2×C4#41×D4311yescharacteristic
593C2×C4#41×D4311yescharacteristic
594C2×C4#41×D4311yescharacteristic
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—
613C22C2×#1306329911yescharacteristic
614C22#1458342711yescharacteristic
615C22#1458342711yescharacteristic
616C4—21no—
617C4—21no—
618C4—21no—
619C4—21no—
620C4—41no—
621C4—41no—
622C4—41no—
623C4—3441no—
624C4—3441no—
625C4—1722no—
626C4#1458340611yescharacteristic
627C4#1458341111yescharacteristic
628C4#1458341111yescharacteristic
629C4#1458344511yescharacteristic
630C2—21no—
631C2—21no—
632C2—81no—
633C2—3442no—
634C2#1913783111yescharacteristic
635C2#1914432311yescharacteristic
636C2#1914433211yescharacteristic
subgroupG/H
centerC22C2×#13063299
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2923C2
socleC22×C43C2×#121
prime radicalC2×#13063350C22
hypercenter#2923D43
hypocenterC43#23721
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2923

Complements

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

NH ≅ G/Ntype
1. C4×#13063355C2semidirect
3. C2×#14583001C2semidirect
4. #19136100C2semidirect
5. #19136035C2semidirect
6. #19135961C2semidirect
7. #19135954C2semidirect
8. #19135893C2semidirect
10. #19135858C2semidirect
12. #19134501C2semidirect
13. C4×#13063243C2semidirect
14. C4×#13063241C2semidirect
15. C43×#2923C2semidirect
315. #2923D43semidirect
381. C43#23721semidirect

Maximal subgroups

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

subgroupindex
C4×#130633552
C4×#130633502
C2×#145830012
#191361002
#191360352
#191359612
#191359542
#191358932
#191358862
#191358582
#191357102
#191345012
C4×#130632432
C4×#130632412
C43×#29232
#2372143