Finite Group Databasebeta

← #25148747

Subgroups — #25148747

Order 11008.

648 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19145148C211yesmaximal, characteristic, complemented
2#19144388C211yesmaximal, characteristic
3C2×#14583200C211yesmaximal, characteristic
4#19136105C211yesmaximal, characteristic, complemented
5#19136100C211yesmaximal, characteristic, complemented
6#19135954C211yesmaximal, characteristic
7#19135954C211yesmaximal, characteristic
8#19135588C211yesmaximal, characteristic, complemented
9#19135380C211yesmaximal, characteristic, complemented
10#19135375C211yesmaximal, characteristic, complemented
11#19135371C211yesmaximal, characteristic, complemented
12#19135257C211yesmaximal, characteristic
13#19135257C211yesmaximal, characteristic
14C4×#13063241C211yesmaximal, characteristic
15C43×#2962C211yesmaximal, characteristic, complemented
16C2×#13063317C2211yescharacteristic
17C2×C4×D172C2211yescharacteristic
18#14583305—21no—
19#14583305—21no—
20#14583298—21no—
21#14583298—21no—
22#14583200—21no—
23#14583200—21no—
24#14583200—21no—
25#14583200—21no—
26#14583196—21no—
27#14583196—21no—
28#14583186—21no—
29#14583176—21no—
30C2×#13063261C2211yescharacteristic
31C2×#13063261C2211yescharacteristic
32#14582942C2211yescharacteristic
33#14582942C2211yescharacteristic
34#14582942C2211yescharacteristic
35#14582942C2211yescharacteristic
36#14582942C2211yescharacteristic
37#14582942C2211yescharacteristic
38C2×#13063259C2211yescharacteristic
39C2×#13063257C2211yescharacteristic
40#14582923C2211yescharacteristic
41#14582903C2211yescharacteristic
42#14582903C2211yescharacteristic
43C2×#13063247C2211yescharacteristic
44C2×#13063247C2211yescharacteristic
45C2×#13063241C2211yescharacteristic
46C2×#13063241C2211yescharacteristic
47C4×#10621877C2211yescharacteristic
48#14582667C2211yescharacteristic
49#14582667C2211yescharacteristic
50#14582667C2211yescharacteristic
51#14582667C2211yescharacteristic
52#14582667C2211yescharacteristic
53#14582667C2211yescharacteristic
54#14582664C2211yescharacteristic
55#14582664C2211yescharacteristic
56C43×#533C2211yescharacteristic
57C43×#446—21no—
58C43×#446—21no—
59C43×#446—21no—
60C8×Q8×C43—21no—
61C43×#424C2211yescharacteristic
62C43×#424C2211yescharacteristic
63C2×C4⋊C8×C43C2211yescharacteristic
64C43×#405C2211yescharacteristic
65C43×#405C2211yescharacteristic
66C4×M16×C43C2211yescharacteristic
67C22×D172C2311yescharacteristic
68C22×C4×D43—21no—
69C2×#10621877C2311yescharacteristic
70C2×#10621877C2311yescharacteristic
71C2×#10621877C2311yescharacteristic
72C2×#10621877C2311yescharacteristic
73#13063317—21no—
74#13063317—21no—
75#13063317—21no—
76#13063317—21no—
77C4×D172—21no—
78C4×D172—21no—
79C4×D172—21no—
80C4×D172—21no—
81C2×C4⋊Dic43C2311yescharacteristic
82C2×C4⋊Dic43C2311yescharacteristic
83C2×C4⋊Dic43C2311yescharacteristic
84C2×Dic43⋊C4—21no—
85C2×Dic43⋊C4—21no—
86C2×(C2×D43)⋊C4—21no—
87#13063261—22no—
88#13063261—22no—
89#13063259—21no—
90#13063259—21no—
91#13063258—21no—
92#13063258—21no—
93#13063258—21no—
94#13063258—21no—
95#13063258—21no—
96#13063258—21no—
97#13063257—21no—
98#13063257—21no—
99#13063256—21no—
100#13063256—21no—
101#13063256—21no—
102#13063256—21no—
103#13063256—21no—
104#13063256—21no—
105C4⋊(C43⋊C8)—21no—
106C4⋊(C43⋊C8)—21no—
107#13063247—21no—
108#13063247—21no—
109#13063247—21no—
110#13063247—21no—
111#13063245—21no—
112#13063245—21no—
113#13063245—21no—
114#13063245—21no—
115#13063241—22no—
116#13063241Q814yescomplemented
117#13063236—21no—
118C4×C43⋊C8—21no—
119C2×M16×C43C2311yescharacteristic
120C2×M16×C43C2311yescharacteristic
121C22×C8×C43C2311yescharacteristic
122C22×C8×C43C2311yescharacteristic
123C4⋊Q8×C43D411yescharacteristic
124C4⋊Q8×C43D411yescharacteristic
125#125×C43D411yescharacteristic
126#125×C43D411yescharacteristic
127C22⋊Q8×C43—21no—
128C22⋊Q8×C43—21no—
129C4×Q8×C43—21no—
130C4×Q8×C43—21no—
131#117×C43C2311yescharacteristic
132#117×C43C2311yescharacteristic
133C2×C42×C43C2311yescharacteristic
134C4⋊C8×C43—21no—
135C4⋊C8×C43—21no—
136C4⋊C8×C43—21no—
137C4⋊C8×C43—21no—
138C4⋊C8×C43D411yescharacteristic
139C4⋊C8×C43D411yescharacteristic
140C4⋊C8×C43D411yescharacteristic
141C4⋊C8×C43D411yescharacteristic
142#97×C43—21no—
143#97×C43—21no—
144#97×C43—21no—
145C4×C8×C43—21no—
146C4×C8×C43—21no—
147C4×C8×C43—21no—
148C23×D43—21no—
149C2×Q8×C43—21no—
150C2×Q8×C43—21no—
151C22×C4×C43#4111yescharacteristic
152C22×C4×C43#4111yescharacteristic
153C22×C4×C43C2411yescharacteristic
154C22×Dic43—21no—
155C22×Dic43—21no—
156C22×Dic43—21no—
157C2×D172—21no—
158C2×D172—21no—
159C2×D172—21no—
160C2×D172—21no—
161C2×D172—21no—
162C2×D172C2×Q812yes—
163C2×C4×D43—41no—
164C2×C4×D43—41no—
165C2×C4×D43—82no—
166M16×C43—21no—
167M16×C43—21no—
168M16×C43—21no—
169M16×C43C2×Q812yes—
170C2×C8×C43—21no—
171C2×C8×C43—21no—
172C2×C8×C43—21no—
173C2×C8×C43—21no—
174C2×C8×C43—21no—
175C2×C8×C43—21no—
176C2×C8×C43C2×D411yescharacteristic
177C2×C8×C43C2×D411yescharacteristic
178C2×C8×C43C2×D411yescharacteristic
179C2×C8×C43C2×D411yescharacteristic
180C4⋊C4×C43—21no—
181C4⋊C4×C43—21no—
182C4⋊C4×C43—21no—
183C4⋊C4×C43—21no—
184C4⋊C4×C43—21no—
185C4⋊C4×C43—21no—
186C4⋊C4×C43C2×D411yescharacteristic
187C4⋊C4×C43C2×D411yescharacteristic
188C4⋊C4×C43C2×D411yescharacteristic
189C4⋊C4×C43C2×D411yescharacteristic
190C22⋊C4×C43—21no—
191C22⋊C4×C43—21no—
192C42×C43—21no—
193C42×C43—21no—
194C42×C43C2×D411yescharacteristic
195C42×C43C2×D411yescharacteristic
196C42×C43C2×D411yescharacteristic
197C42×C43C2×D411yescharacteristic
198(C2×D43)⋊C4—41no—
199(C2×D43)⋊C4—41no—
200C4⋊Dic43—21no—
201C4⋊Dic43—21no—
202C4⋊Dic43—21no—
203C4⋊Dic43—21no—
204C4⋊Dic43—21no—
205C4⋊Dic43—21no—
206Dic43⋊C4—41no—
207Dic43⋊C4—41no—
208Dic43⋊C4—41no—
209Dic43⋊C4—41no—
210#10621877—21no—
211#10621877—21no—
212#10621877—21no—
213#10621877—21no—
214#10621877—21no—
215#10621877—21no—
216#10621877C2×Q812yes—
217#10621877#4112yes—
218C2×C43⋊C8—21no—
219C2×C43⋊C8—21no—
220C2×C43⋊C8—21no—
221C2×C43⋊C8—21no—
222C23×C43C2×#4111yescharacteristic
223C22×D43—22no—
224C22×D43—41no—
225C22×D43—41no—
226C22×D43—81no—
227C22×D43—81no—
228Q8×C43—42no—
229Q8×C43—81no—
230C2×C4×C43—21no—
231C2×C4×C43—21no—
232C2×C4×C43—21no—
233C2×C4×C43—21no—
234C2×C4×C43—41no—
235C2×C4×C43—41no—
236C2×C4×C43#12111yescharacteristic
237C2×C4×C43#12111yescharacteristic
238C2×C4×C43#12111yescharacteristic
239C2×C4×C43#12111yescharacteristic
240C2×C4×C43C22×D411yescharacteristic
241C2×C4×C43C22×D411yescharacteristic
242C2×C4×C43C22×Q811yescharacteristic
243C2×C4×C43C2×#4111yescharacteristic
244C2×C4×C432−1+411yescharacteristic
245C2×C4×C432−1+411yescharacteristic
246C2×Dic43—41no—
247C2×Dic43—41no—
248C2×Dic43—41no—
249C2×Dic43—81no—
250C2×Dic43—81no—
251C2×Dic43—81no—
252D172—22no—
253D172—22no—
254D172—41no—
255D172—41no—
256C4×D43—82no—
257C4×D43—82no—
258C8×C43—21no—
259C8×C43—21no—
260C8×C43—21no—
261C8×C43—21no—
262C8×C43—22no—
263C8×C43—41no—
264C43⋊C8—22no—
265C43⋊C8—22no—
266C43⋊C8—41no—
267C43⋊C8—41no—
268#25493—431nomaximal, Sylow
269C22×C43—21no—
270C22×C43—21no—
271C22×C43C2×#12111yescharacteristic
272C22×C43#53311yescharacteristic
273C22×C43#54111yescharacteristic
274C2×D43—42no—
275C2×D43—82no—
276C2×D43—81no—
277C2×D43—81no—
278C2×D43—82no—
279C4×C43—21no—
280C4×C43—21no—
281C4×C43—21no—
282C4×C43—21no—
283C4×C43—41no—
284C4×C43—41no—
285C4×C43—41no—
286C4×C43—41no—
287C4×C43#54711yescharacteristic
288C4×C43#54811yescharacteristic
289C4×C43D4×Q811yescharacteristic
290C4×C43D4×Q811yescharacteristic
291Dic43—81no—
292Dic43—81no—
293Dic43—81no—
294#3166—431no—
295#3077—431no—
296C2×#478—431no—
297#2962D4311yescharacteristic, complemented
298#2090—431no—
299#2089—431no—
300#2063—431no—
301#2063—431no—
302#1998—431no—
303#1952—431no—
304#1951—431no—
305#1950—431no—
306#1931—431no—
307#1931—431no—
308C4×#100—431no—
309C2×C43—21no—
310C2×C43—21no—
311C2×C43#242911yescharacteristic
312C2×C43#306711yescharacteristic
313C2×C43#306811yescharacteristic
314D43—82no—
315#533C2×D4311yescharacteristic
316C2×#125—431no—
317C2×C4×D4—431no—
318#501—861no—
319#501—861no—
320#499—861no—
321#499—861no—
322#478—861no—
323#478—861no—
324#478—861no—
325#478—861no—
326#478—861no—
327#478—861no—
328#476—861no—
329#474—861no—
330#446—21no—
331#446—21no—
332#446—21no—
333C8×Q8—21no—
334C2×#108—431no—
335C2×#108—431no—
336#428—431no—
337#428—431no—
338#428—431no—
339#428—431no—
340#428—431no—
341#428—431no—
342C2×#107—431no—
343C2×#106—431no—
344#424C2×D4311yescharacteristic
345#424C2×D4311yescharacteristic
346#423—431no—
347C2×C4⋊C8C2×D4311yescharacteristic
348#418—431no—
349#418—431no—
350C2×#102—431no—
351C2×#102—431no—
352C2×#100—431no—
353C2×#100—431no—
354#405C2×D4311yescharacteristic
355#405C2×D4311yescharacteristic
356C4×M16—431no—
357C4×M16C2×D4311yescharacteristic
358#343—431no—
359#343—431no—
360#343—431no—
361#343—431no—
362#343—431no—
363#343—431no—
364#341—431no—
365#341—431no—
366C43#2549311yescharacteristic, complemented, Sylow
367C22×D4—431no—
368C23×C4—861no—
369C2×M16—431no—
370C2×M16—431no—
371C2×M16—431no—
372C2×M16—431no—
373C2×M16C22×D4311yescharacteristic
374C2×M16C22×D4311yescharacteristic
375C22×C8C22×D4311yescharacteristic
376C22×C8C22×D4311yescharacteristic
377C4⋊Q8D17211yescharacteristic
378C4⋊Q8D17211yescharacteristic
379#125—861no—
380#125—861no—
381#125—861no—
382#125—861no—
383#125D17211yescharacteristic
384#125D17211yescharacteristic
385C22⋊Q8—21no—
386C22⋊Q8—21no—
387C4×Q8—21no—
388C4×Q8—21no—
389C4×D4—861no—
390C4×D4—861no—
391C4×D4—861no—
392C4×D4—861no—
393#117C22×D4311yescharacteristic
394#117C22×D4311yescharacteristic
395C2×C4⋊C4—431no—
396C2×C4⋊C4—431no—
397C2×C4⋊C4—431no—
398C2×C4⋊C4—861no—
399C2×C4⋊C4—861no—
400C2×C22⋊C4—861no—
401C2×C42C22×D4311yescharacteristic
402#108—862no—
403#108—862no—
404#107—861no—
405#107—861no—
406#107—861no—
407#107—861no—
408#107—861no—
409#107—861no—
410#107—861no—
411#107—861no—
412#106—861no—
413#106—861no—
414#106—861no—
415#106—861no—
416#106—861no—
417#106—861no—
418#106—861no—
419#106—861no—
420C4⋊C8—21no—
421C4⋊C8—21no—
422C4⋊C8—21no—
423C4⋊C8—21no—
424C4⋊C8—861no—
425C4⋊C8—861no—
426C4⋊C8C22⋊D4311yescharacteristic
427C4⋊C8C22⋊D4311yescharacteristic
428C4⋊C8C22⋊D4311yescharacteristic
429C4⋊C8C22⋊D4311yescharacteristic
430#102—861no—
431#102—861no—
432#102—861no—
433#102—861no—
434#102—861no—
435#102—861no—
436#102—861no—
437#102—861no—
438#100—434no—
439#100—862no—
440#97—21no—
441#97—21no—
442#97—21no—
443#97—861no—
444C4×C8—21no—
445C4×C8—21no—
446C4×C8—21no—
447C4×C8—861no—
448C24—861no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×D4—432no—
452C2×D4—861no—
453C2×D4—861no—
454C2×D4—861no—
455C2×D4—861no—
456C2×D4—861no—
457C22×C4—861no—
458C22×C4—861no—
459C22×C4—861no—
460C22×C4—3442no—
461C22×C4—1721no—
462C22×C4—1721no—
463C22×C4#1062189811yescharacteristic
464C22×C4#1062189811yescharacteristic
465C22×C4C23×D4311yescharacteristic
466M16—21no—
467M16—21no—
468M16—21no—
469M16—432no—
470M16—432no—
471M16—861no—
472M16—861no—
473M16—861no—
474M16—861no—
475M16—861no—
476M16—861no—
477M16Q8×D4312yes—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—861no—
485C2×C8—861no—
486C2×C8—861no—
487C2×C8—861no—
488C2×C8C2×C22⋊D4311yescharacteristic
489C2×C8C2×C22⋊D4311yescharacteristic
490C2×C8C2×C22⋊D4311yescharacteristic
491C2×C8C2×C22⋊D4311yescharacteristic
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—861no—
499C4⋊C4—861no—
500C4⋊C4—861no—
501C4⋊C4—861no—
502C4⋊C4—861no—
503C4⋊C4—861no—
504C4⋊C4—1721no—
505C4⋊C4—1721no—
506C4⋊C4—1721no—
507C4⋊C4—1721no—
508C4⋊C4C2×D17211yescharacteristic
509C4⋊C4C2×D17211yescharacteristic
510C4⋊C4C2×D17211yescharacteristic
511C4⋊C4C2×D17211yescharacteristic
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—1721no—
515C22⋊C4—1721no—
516C42—21no—
517C42—21no—
518C42C2×D17211yescharacteristic
519C42C2×D17211yescharacteristic
520C42C2×C22⋊D4311yescharacteristic
521C42C2×C22⋊D4311yescharacteristic
522C23—862no—
523C23—3441no—
524C23—3441no—
525C23—1721no—
526C23—1721no—
527C23C2×#1062189811yescharacteristic
528Q8—42no—
529Q8—81no—
530D4—862no—
531D4—862no—
532D4—1721no—
533D4—1721no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—3442no—
541C2×C4—3442no—
542C2×C4—3441no—
543C2×C4—3441no—
544C2×C4—3441no—
545C2×C4—1721no—
546C2×C4—1721no—
547C2×C4—1721no—
548C2×C4#1306329911yescharacteristic
549C2×C4#1306329911yescharacteristic
550C2×C4#1306329911yescharacteristic
551C2×C4#1306329911yescharacteristic
552C2×C4C22×C22⋊D4311yescharacteristic
553C2×C4C22×D17211yescharacteristic
554C2×C4C2×Q8×D4311yescharacteristic
555C2×C4#41×D4311yescharacteristic
556C2×C4#1306337211yescharacteristic
557C2×C4#1306337311yescharacteristic
558C8—22no—
559C8—21no—
560C8—21no—
561C8—21no—
562C8—21no—
563C8—41no—
564C8—862no—
565C8—862no—
566C8—1721no—
567C8—1721no—
568C22—21no—
569C22—21no—
570C22—3442no—
571C22—3442no—
572C22—3441no—
573C22—3441no—
574C22—1722no—
575C22C2×#1306329911yescharacteristic
576C22#1458345311yescharacteristic
577C22#1458350711yescharacteristic
578C4—21no—
579C4—21no—
580C4—21no—
581C4—21no—
582C4—41no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—3441no—
587C4—3441no—
588C4—3441no—
589C4#1458354811yescharacteristic
590C4#1458356111yescharacteristic
591C4Q8×C22⋊D4311yescharacteristic
592C4Q8×D17211yescharacteristic
593C2—21no—
594C2—21no—
595C2—3442no—
596C2#1913818811yescharacteristic
597C2#1914431511yescharacteristic
598C2#1914432311yescharacteristic
subgroupG/H
centerC22C2×#13063299
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#2962C2
socleC22×C43C2×#121
prime radicalC22×D172C23
hypercenter#2962D43
hypocenterC43#25493
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2962

Complements

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

NH ≅ G/Ntype
1. #19145148C2semidirect
4. #19136105C2semidirect
5. #19136100C2semidirect
8. #19135588C2semidirect
9. #19135380C2semidirect
10. #19135375C2semidirect
11. #19135371C2semidirect
15. C43×#2962C2semidirect
116. #13063241Q8semidirect
297. #2962D43semidirect
366. C43#25493semidirect

Maximal subgroups

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

subgroupindex
#191451482
#191443882
C2×#145832002
#191361052
#191361002
#191359542
#191355882
#191353802
#191353752
#191353712
#191352572
C4×#130632412
C43×#29622
#2549343