Finite Group Databasebeta

← #25070471

Subgroups — #25070471

Order 11008.

972 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583494C211yesmaximal, characteristic
2#19145370C211yesmaximal, characteristic, complemented
3#19145352C211yesmaximal, characteristic
4#19140411C211yesmaximal, characteristic, complemented
5#19133627C214yesmaximal, complemented
6#19133559C214yesmaximal, complemented
7#19133166C211yesmaximal, characteristic, complemented
8C2×#14582624C211yesmaximal, characteristic, complemented
9C43×#1519C211yesmaximal, characteristic
10#14583494—22no—
11#14583494—22no—
12#14583494—22no—
13#14583494C2214yes—
14C2×#13063318—21no—
15C2×#13063315—21no—
16C2×#13063306—21no—
17C2×#13063306—21no—
18C2×#13063304C2211yescharacteristic
19C2×#13063304C2211yescharacteristic
20C2×#13063302—21no—
21C2×D4×Dic43—21no—
22C22×Dic43⋊C4C2211yescharacteristic
23#14583235—21no—
24#14583235C2212yescomplemented
25#14583229—21no—
26#14583215C2212yes—
27#14583140—22no—
28#14583126—22no—
29#14583057—21no—
30#14583057C2212yescomplemented
31#14583050—21no—
32#14583037C2212yes—
33C2×C22⋊(C43⋊C8)C2211yescharacteristic
34#14582839C2212yescomplemented
35#14582829—22no—
36#14582820C2212yescomplemented
37#14582774—21no—
38#14582760—21no—
39#14582635C2214yescomplemented
40#14582624C2214yescomplemented
41C2×#121×C43C2211yescharacteristic
42C2×C22⋊C8×C43C2211yescharacteristic
43C43×#380C2211yescharacteristic
44C43×#327C2214yes—
45C22×Dic86D411yescharacteristic
46C22×Dic86C2311yescharacteristic
47C23×Dic43—21no—
48C23×Dic43D411yescharacteristic
49C23×Dic43D411yescharacteristic
50C2×Dic172—22no—
51C2×#10621883—22no—
52C22×C43⋊C8—21no—
53#13063318—24no—
54#13063318—24no—
55#13063315—24no—
56#13063315—24no—
57#13063306—24no—
58#13063306—24no—
59#13063306—24no—
60#13063306—24no—
61#13063304—21no—
62#13063304—21no—
63#13063304—21no—
64#13063304—21no—
65#13063304—21no—
66#13063304—21no—
67#13063304D412yes—
68#13063304C2312yes—
69#13063302—24no—
70#13063302—24no—
71D4×Dic43—24no—
72D4×Dic43—24no—
73C2×C4⋊Dic43—21no—
74C2×C4⋊Dic43—22no—
75C2×C4⋊Dic43D411yescharacteristic
76C2×C4⋊Dic43C2311yescharacteristic
77C2×Dic43⋊C4—21no—
78C2×Dic43⋊C4—21no—
79C2×Dic43⋊C4—21no—
80C2×Dic43⋊C4—21no—
81C2×Dic43⋊C4—21no—
82C2×Dic43⋊C4—21no—
83C2×Dic43⋊C4—21no—
84C2×Dic43⋊C4—21no—
85C2×Dic43⋊C4D412yes—
86C2×Dic43⋊C4C2312yes—
87C2×C22⋊Dic43—21no—
88C2×C22⋊Dic43—21no—
89C2×C22⋊Dic43—21no—
90C2×C22⋊Dic43—21no—
91C2×C22⋊Dic43—21no—
92C2×C4×Dic43—21no—
93#13063259—22no—
94#13063256—22no—
95#13063250—22no—
96#13063250—22no—
97#13063248—22no—
98#13063246—22no—
99C22⋊(C43⋊C8)—21no—
100C22⋊(C43⋊C8)C2312yes—
101C43⋊#95—22no—
102C43⋊#95—22no—
103C43⋊#95—22no—
104C22×D4×C43—21no—
105C22×D4×C43D411yescharacteristic
106C23×C4×C43C2311yescharacteristic
107C22×C8×C43—21no—
108#121×C43—21no—
109#121×C43—21no—
110#121×C43D412yes—
111#121×C43C2312yes—
112C2×C4⋊C4×C43—21no—
113C2×C4⋊C4×C43D411yescharacteristic
114C2×C22⋊C4×C43—21no—
115C2×C22⋊C4×C43—21no—
116C22⋊C8×C43—21no—
117C22⋊C8×C43C2312yes—
118#95×C43C2312yes—
119C24×C43—21no—
120C24×C43C2×D411yescharacteristic
121C2×D4×C43—21no—
122C2×D4×C43—22no—
123C2×D4×C43—22no—
124C2×D4×C43—42no—
125C2×D4×C43—42no—
126C2×D4×C43—81no—
127C2×D4×C43C2×D412yes—
128C22×C4×C43—21no—
129C22×C4×C43—22no—
130C22×C4×C43—81no—
131C22×C4×C43C2×D411yescharacteristic
132C22×C4×C43C2×D411yescharacteristic
133C22×C4×C43C2×D411yescharacteristic
134C22×C4×C43C2×D411yescharacteristic
135C22×C4×C43C2×D411yescharacteristic
136C22×C4×C43C2411yescharacteristic
137C22×Dic43—24no—
138C22×Dic43—21no—
139C22×Dic43—21no—
140C22×Dic43—21no—
141C22×Dic43—21no—
142C22×Dic43—21no—
143C22×Dic43—21no—
144C22×Dic43—22no—
145C22×Dic43—22no—
146C22×Dic43—21no—
147C22×Dic43—21no—
148C22×Dic43—81no—
149C22×Dic43—82no—
150C22×Dic43C2×D412yes—
151C22×Dic43C2×D412yes—
152C2×Dic86—21no—
153C2×Dic86—21no—
154C2×Dic86—21no—
155C2×Dic86—81no—
156C2×Dic86—81no—
157C2×Dic86C2×D412yes—
158C2×C8×C43—22no—
159C2×C8×C43—81no—
160C4⋊C4×C43—21no—
161C4⋊C4×C43—81no—
162C4⋊C4×C43C2×D412yes—
163C22⋊C4×C43—22no—
164C22⋊C4×C43—22no—
165C22⋊C4×C43—42no—
166C22⋊Dic43—24no—
167C22⋊Dic43—24no—
168C22⋊Dic43—24no—
169C22⋊Dic43—22no—
170C22⋊Dic43—22no—
171C22⋊Dic43—41no—
172C22⋊Dic43—41no—
173#10621883—82no—
174C4⋊Dic43—24no—
175C4⋊Dic43—21no—
176C4⋊Dic43—21no—
177C4⋊Dic43—21no—
178C4⋊Dic43—82no—
179C4⋊Dic43C2×D412yes—
180Dic43⋊C4—24no—
181Dic43⋊C4—24no—
182Dic43⋊C4—22no—
183Dic43⋊C4—22no—
184Dic43⋊C4—22no—
185Dic43⋊C4—22no—
186Dic43⋊C4—22no—
187Dic43⋊C4—22no—
188Dic43⋊C4—41no—
189Dic43⋊C4—41no—
190C4×Dic43—24no—
191C2×C43⋊C8—22no—
192C2×C43⋊C8—81no—
193Dic172—82no—
194C23×C43—22no—
195C23×C43—41no—
196C23×C43—41no—
197C23×C43—42no—
198C23×C43—42no—
199C23×C43—42no—
200C23×C43#12011yescharacteristic
201C23×C43#12011yescharacteristic
202C23×C43#13512yes—
203C23×C43#13712yes—
204C23×C43C22×D411yescharacteristic
205D4×C43—42no—
206D4×C43—42no—
207D4×C43—42no—
208D4×C43—44no—
209D4×C43—81no—
210C2×C4×C43—22no—
211C2×C4×C43—41no—
212C2×C4×C43—41no—
213C2×C4×C43—41no—
214C2×C4×C43—41no—
215C2×C4×C43—41no—
216C2×C4×C43—41no—
217C2×C4×C43—42no—
218C2×C4×C43—42no—
219C2×C4×C43—42no—
220C2×C4×C43—42no—
221C2×C4×C43#12011yescharacteristic
222C2×C4×C43#12011yescharacteristic
223C2×C4×C43C22×D411yescharacteristic
224C2×C4×C43C22×D411yescharacteristic
225C2×Dic43—22no—
226C2×Dic43—22no—
227C2×Dic43—22no—
228C2×Dic43—22no—
229C2×Dic43—22no—
230C2×Dic43—22no—
231C2×Dic43—42no—
232C2×Dic43—42no—
233C2×Dic43—42no—
234C2×Dic43—42no—
235C2×Dic43—42no—
236C2×Dic43—42no—
237C2×Dic43—42no—
238C2×Dic43—42no—
239C2×Dic43—41no—
240C2×Dic43—41no—
241C2×Dic43—81no—
242C2×Dic43—84no—
243Dic86—42no—
244Dic86—42no—
245Dic86—42no—
246Dic86—81no—
247C8×C43—42no—
248C43⋊C8—42no—
249#18289—431nomaximal, Sylow
250C22×C43—21no—
251C22×C43—21no—
252C22×C43—21no—
253C22×C43—21no—
254C22×C43—21no—
255C22×C43—21no—
256C22×C43—42no—
257C22×C43—42no—
258C22×C43—42no—
259C22×C43—42no—
260C22×C43—42no—
261C22×C43—84no—
262C22×C43#45212yes—
263C22×C43#45212yes—
264C22×C43C2×#12011yescharacteristic
265C22×C43C2×#13511yescharacteristic
266C22×C43C2×#13711yescharacteristic
267C4×C43—21no—
268C4×C43—21no—
269C4×C43—21no—
270C4×C43—21no—
271C4×C43—42no—
272C4×C43—82no—
273Dic43—42no—
274Dic43—42no—
275Dic43—42no—
276Dic43—42no—
277Dic43—81no—
278Dic43—81no—
279C2×#539—431no—
280#3189—431no—
281#3187—431no—
282#2627—431no—
283#1615—434no—
284#1603—434no—
285#1520—431no—
286#1519D4311yescharacteristic
287C2×#324—431no—
288C2×C43—21no—
289C2×C43—22no—
290C2×C43—21no—
291C2×C43—22no—
292C2×C43—82no—
293C2×C43C2×#45211yescharacteristic
294C2×C43#302011yescharacteristic
295C2×C43#302611yescharacteristic
296#539—434no—
297#539—862no—
298#539—862no—
299#539—862no—
300C2×#126—861no—
301C2×#125—861no—
302C2×#123—861no—
303C2×#123—861no—
304C2×C22⋊Q8—431no—
305C2×C22⋊Q8—431no—
306C2×C22⋊Q8—861no—
307C2×#121C2×D4311yescharacteristic
308C2×C4×D4—861no—
309C22×C4⋊C4—431no—
310#484—432no—
311#484—861no—
312#483—861no—
313#481—432no—
314#467—862no—
315#465—862no—
316#451—432no—
317#451—861no—
318#450—861no—
319#448—432no—
320C2×C22⋊C8—431no—
321C2×C22⋊C8C2×D4311yescharacteristic
322#400—432no—
323#398—862no—
324#395—432no—
325#385—861no—
326#381—861no—
327#380C2×D4311yescharacteristic
328#328—434no—
329#327C2×D4314yes—
330#324—434no—
331C43#1828911yescharacteristic, complemented, Sylow
332C22×Q8—431no—
333C22×Q8—431no—
334C22×D4—21no—
335C22×D4D17211yescharacteristic
336C23×C4—431no—
337C23×C4—431no—
338C23×C4—861no—
339C23×C4C22×D4311yescharacteristic
340C2×Q16—862no—
341C2×SD16—862no—
342C22×C8—21no—
343C22×C8—861no—
344#126—864no—
345#126—864no—
346#125—864no—
347#125—864no—
348#123—864no—
349#123—864no—
350#123—864no—
351#123—864no—
352C22⋊Q8—432no—
353C22⋊Q8—432no—
354C22⋊Q8—864no—
355C22⋊Q8—864no—
356C22⋊Q8—861no—
357C22⋊Q8—861no—
358C22⋊Q8—861no—
359C22⋊Q8—861no—
360C22⋊Q8—861no—
361C22⋊Q8—861no—
362#121—21no—
363#121—21no—
364#121D17212yes—
365#121C22×D4312yes—
366C4×D4—864no—
367C4×D4—864no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—431no—
370C2×C4⋊C4—432no—
371C2×C4⋊C4—432no—
372C2×C4⋊C4—431no—
373C2×C4⋊C4—861no—
374C2×C4⋊C4—861no—
375C2×C4⋊C4—861no—
376C2×C4⋊C4—861no—
377C2×C4⋊C4—861no—
378C2×C4⋊C4—861no—
379C2×C4⋊C4—861no—
380C2×C4⋊C4—861no—
381C2×C4⋊C4—862no—
382C2×C4⋊C4—861no—
383C2×C4⋊C4D17211yescharacteristic
384C2×C22⋊C4—21no—
385C2×C22⋊C4—21no—
386C2×C22⋊C4—861no—
387C2×C22⋊C4—861no—
388C2×C22⋊C4—861no—
389C2×C22⋊C4—861no—
390C2×C22⋊C4—861no—
391C2×C42—861no—
392#107—862no—
393#106—862no—
394#103—862no—
395#103—862no—
396#103—862no—
397#102—862no—
398C22⋊C8—21no—
399C22⋊C8—432no—
400C22⋊C8—861no—
401C22⋊C8C22×D4312yes—
402#95—862no—
403#95—862no—
404#95—862no—
405#95C22×D4312yes—
406C24—21no—
407C24D4×D4311yescharacteristic
408C2×Q8—432no—
409C2×Q8—861no—
410C2×Q8—861no—
411C2×Q8—861no—
412C2×Q8—3441no—
413C2×Q8—3441no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—81no—
420C2×D4C2×D17212yes—
421C22×C4—21no—
422C22×C4—22no—
423C22×C4—81no—
424C22×C4—432no—
425C22×C4—432no—
426C22×C4—864no—
427C22×C4—861no—
428C22×C4—861no—
429C22×C4—861no—
430C22×C4—861no—
431C22×C4—861no—
432C22×C4—861no—
433C22×C4—862no—
434C22×C4—862no—
435C22×C4—861no—
436C22×C4—861no—
437C22×C4—3442no—
438C22×C4—3441no—
439C22×C4C2×D17211yescharacteristic
440C22×C4C2×D17211yescharacteristic
441C22×C4D4×D4311yescharacteristic
442C22×C4D4×D4311yescharacteristic
443C22×C4D4×D4311yescharacteristic
444C22×C4C23×D4311yescharacteristic
445Q16—3442no—
446SD16—3442no—
447C2×C8—22no—
448C2×C8—81no—
449C2×C8—862no—
450C2×C8—3441no—
451C4⋊C4—21no—
452C4⋊C4—81no—
453C4⋊C4—432no—
454C4⋊C4—864no—
455C4⋊C4—864no—
456C4⋊C4—864no—
457C4⋊C4—862no—
458C4⋊C4—862no—
459C4⋊C4—861no—
460C4⋊C4—862no—
461C4⋊C4—862no—
462C4⋊C4—862no—
463C4⋊C4—862no—
464C4⋊C4—861no—
465C4⋊C4—861no—
466C4⋊C4—3442no—
467C4⋊C4—1721no—
468C4⋊C4—1721no—
469C4⋊C4C2×D17212yes—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—42no—
473C22⋊C4—864no—
474C22⋊C4—864no—
475C22⋊C4—864no—
476C22⋊C4—862no—
477C22⋊C4—862no—
478C22⋊C4—1721no—
479C22⋊C4—1721no—
480C42—864no—
481C23—22no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23#1306329511yescharacteristic
488C23#1306329511yescharacteristic
489C23#1306334412yes—
490C23#1306335412yes—
491C23C2×D4×D4311yescharacteristic
492Q8—3441no—
493Q8—1722no—
494Q8—1722no—
495Q8—1722no—
496D4—42no—
497D4—44no—
498D4—42no—
499D4—42no—
500D4—81no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—862no—
513C2×C4—862no—
514C2×C4—862no—
515C2×C4—862no—
516C2×C4—862no—
517C2×C4—862no—
518C2×C4—3441no—
519C2×C4—3444no—
520C2×C4—1722no—
521C2×C4—1722no—
522C2×C4—1721no—
523C2×C4—1721no—
524C2×C4—1722no—
525C2×C4—1722no—
526C2×C4—1722no—
527C2×C4—1722no—
528C2×C4—1722no—
529C2×C4—1722no—
530C2×C4#1306329511yescharacteristic
531C2×C4#1306329511yescharacteristic
532C2×C4C2×D4×D4311yescharacteristic
533C2×C4C22×D17211yescharacteristic
534C8—42no—
535C8—1722no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—42no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—42no—
547C22—84no—
548C22#1458306212yes—
549C22#1458306212yes—
550C22C2×#1306329511yescharacteristic
551C22C2×#1306334411yescharacteristic
552C22C2×#1306335411yescharacteristic
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—42no—
558C4—82no—
559C4—3441no—
560C4—3441no—
561C4—1722no—
562C4—1722no—
563C4—1722no—
564C4—1722no—
565C2—21no—
566C2—22no—
567C2—22no—
568C2—21no—
569C2—82no—
570C2C2×#1458306211yescharacteristic
571C2#1914396111yescharacteristic
572C2#1914399411yescharacteristic
subgroupG/H
centerC22C2×#13063295
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#1519C2
socleC22×C43C2×#120
prime radicalC2×#121×C43C22
hypercenter#1519D43
hypocenterC43#18289
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1519

Complements

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

NH ≅ G/Ntype
2. #19145370C2semidirect
4. #19140411C2semidirect
5. #19133627C2semidirect
6. #19133559C2semidirect
7. #19133166C2semidirect
8. C2×#14582624C2semidirect
24. #14583235C22semidirect
30. #14583057C22semidirect
34. #14582839C22semidirect
36. #14582820C22semidirect
39. #14582635C22semidirect
40. #14582624C22semidirect
331. C43#18289semidirect

Maximal subgroups

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

subgroupindex
C2×#145834942
#191453702
#191453522
#191404112
#191336272
#191335592
#191331662
C2×#145826242
C43×#15192
#1828943