Finite Group Databasebeta

← #25070466

Subgroups — #25070466

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#19145367C211yesmaximal, characteristic, complemented
3#19140410C211yesmaximal, characteristic, complemented
4#19133623C214yesmaximal, complemented
5#19133555C214yesmaximal, complemented
6#19133168C211yesmaximal, characteristic, complemented
7#19133163C211yesmaximal, characteristic
8C2×#14582624C211yesmaximal, characteristic, complemented
9C43×#3187C211yesmaximal, 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×#13063302—21no—
20C2×D4×Dic43—21no—
21C22×Dic43⋊C4C2211yescharacteristic
22#14583234C2212yescomplemented
23#14583232—21no—
24#14583138—22no—
25#14583055C2212yescomplemented
26#14583054—21no—
27C2×C22⋊(C43⋊C8)C2211yescharacteristic
28#14582837C2212yescomplemented
29#14582832—22no—
30#14582818C2212yescomplemented
31#14582771—21no—
32#14582760—21no—
33#14582759C2211yescharacteristic
34#14582637C2214yescomplemented
35#14582633C2214yes—
36#14582624C2214yescomplemented
37C2×C22⋊Q8×C43C2211yescharacteristic
38C2×#121×C43C2211yescharacteristic
39C43×#483—21no—
40C43×#481C2212yes—
41C43×#465—22no—
42C43×#450—21no—
43C43×#448C2212yes—
44C2×C22⋊C8×C43C2211yescharacteristic
45C22×Dic86D411yescharacteristic
46C23×Dic43—21no—
47C23×Dic43D411yescharacteristic
48C23×Dic43D411yescharacteristic
49C2×#10621885—22no—
50C22×C43⋊C8—21no—
51#13063318—24no—
52#13063318—24no—
53#13063315—24no—
54#13063315—24no—
55#13063306—24no—
56#13063306—24no—
57#13063306—24no—
58#13063306—24no—
59#13063304—21no—
60#13063304—21no—
61#13063304D412yes—
62#13063304C2312yes—
63#13063302—24no—
64#13063302—24no—
65D4×Dic43—24no—
66D4×Dic43—24no—
67C2×C4⋊Dic43—21no—
68C2×C4⋊Dic43—21no—
69C2×C4⋊Dic43D411yescharacteristic
70C2×Dic43⋊C4—21no—
71C2×Dic43⋊C4—21no—
72C2×Dic43⋊C4—21no—
73C2×Dic43⋊C4—21no—
74C2×Dic43⋊C4—21no—
75C2×Dic43⋊C4—21no—
76C2×Dic43⋊C4—21no—
77C2×Dic43⋊C4—22no—
78C2×Dic43⋊C4D412yes—
79C2×Dic43⋊C4C2312yes—
80C2×C22⋊Dic43—21no—
81C2×C22⋊Dic43—21no—
82C2×C22⋊Dic43—21no—
83C2×C22⋊Dic43—21no—
84C2×C22⋊Dic43—21no—
85C2×C4×Dic43—21no—
86#13063258—22no—
87#13063249—22no—
88#13063248—22no—
89C22⋊(C43⋊C8)—21no—
90C22⋊(C43⋊C8)C2312yes—
91C43⋊#95—22no—
92C43⋊#95—22no—
93C43⋊#95—22no—
94C43⋊#95C2312yes—
95C22×Q8×C43C2311yescharacteristic
96C22×D4×C43—21no—
97C22×D4×C43D411yescharacteristic
98C23×C4×C43C2311yescharacteristic
99C2×SD16×C43—22no—
100C22×C8×C43—21no—
101C22⋊Q8×C43—21no—
102C22⋊Q8×C43—21no—
103C22⋊Q8×C43—21no—
104C22⋊Q8×C43—21no—
105#121×C43—21no—
106#121×C43—21no—
107#121×C43D412yes—
108#121×C43C2312yes—
109C2×C4⋊C4×C43—21no—
110C2×C4⋊C4×C43D411yescharacteristic
111C2×C4⋊C4×C43C2311yescharacteristic
112C2×C22⋊C4×C43—21no—
113C2×C22⋊C4×C43—21no—
114#106×C43—22no—
115#103×C43—22no—
116#102×C43—22no—
117C22⋊C8×C43—21no—
118C22⋊C8×C43C2312yes—
119C24×C43—21no—
120C24×C43C2×D411yescharacteristic
121C2×Q8×C43—21no—
122C2×Q8×C43—21no—
123C2×Q8×C43—81no—
124C2×D4×C43—21no—
125C2×D4×C43—22no—
126C2×D4×C43—22no—
127C2×D4×C43—42no—
128C2×D4×C43—42no—
129C2×D4×C43—81no—
130C2×D4×C43C2×D412yes—
131C22×C4×C43—21no—
132C22×C4×C43—21no—
133C22×C4×C43—21no—
134C22×C4×C43—81no—
135C22×C4×C43C2×D411yescharacteristic
136C22×C4×C43C2×D411yescharacteristic
137C22×C4×C43C2×D411yescharacteristic
138C22×C4×C43C2×D411yescharacteristic
139C22×C4×C43C2×D411yescharacteristic
140C22×C4×C43C2411yescharacteristic
141C22×Dic43—24no—
142C22×Dic43—21no—
143C22×Dic43—21no—
144C22×Dic43—21no—
145C22×Dic43—21no—
146C22×Dic43—21no—
147C22×Dic43—21no—
148C22×Dic43—22no—
149C22×Dic43—22no—
150C22×Dic43—22no—
151C22×Dic43—81no—
152C22×Dic43—82no—
153C22×Dic43C2×D412yes—
154C22×Dic43C2×D412yes—
155C2×Dic86—21no—
156C2×Dic86—81no—
157C2×Dic86C2×D412yes—
158SD16×C43—82no—
159C2×C8×C43—22no—
160C2×C8×C43—81no—
161C4⋊C4×C43—21no—
162C4⋊C4×C43—21no—
163C4⋊C4×C43—21no—
164C4⋊C4×C43—41no—
165C4⋊C4×C43—41no—
166C4⋊C4×C43C2×D412yes—
167C22⋊C4×C43—22no—
168C22⋊C4×C43—22no—
169C22⋊C4×C43—41no—
170C22⋊C4×C43—41no—
171C22⋊Dic43—24no—
172C22⋊Dic43—24no—
173C22⋊Dic43—24no—
174C22⋊Dic43—22no—
175C22⋊Dic43—22no—
176C22⋊Dic43—42no—
177#10621885—82no—
178C4⋊Dic43—24no—
179C4⋊Dic43—21no—
180C4⋊Dic43—81no—
181C4⋊Dic43C2×D412yes—
182Dic43⋊C4—24no—
183Dic43⋊C4—24no—
184Dic43⋊C4—22no—
185Dic43⋊C4—22no—
186Dic43⋊C4—22no—
187Dic43⋊C4—22no—
188Dic43⋊C4—22no—
189Dic43⋊C4—22no—
190Dic43⋊C4—82no—
191C4×Dic43—24no—
192C2×C43⋊C8—22no—
193C2×C43⋊C8—81no—
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
205Q8×C43—42no—
206Q8×C43—42no—
207D4×C43—42no—
208D4×C43—42no—
209D4×C43—42no—
210D4×C43—44no—
211D4×C43—81no—
212C2×C4×C43—22no—
213C2×C4×C43—41no—
214C2×C4×C43—41no—
215C2×C4×C43—41no—
216C2×C4×C43—41no—
217C2×C4×C43—41no—
218C2×C4×C43—41no—
219C2×C4×C43—42no—
220C2×C4×C43—42no—
221C2×C4×C43—42no—
222C2×C4×C43—41no—
223C2×C4×C43—41no—
224C2×C4×C43#12011yescharacteristic
225C2×C4×C43#12011yescharacteristic
226C2×C4×C43C22×D411yescharacteristic
227C2×C4×C43C22×D411yescharacteristic
228C2×Dic43—22no—
229C2×Dic43—22no—
230C2×Dic43—22no—
231C2×Dic43—22no—
232C2×Dic43—22no—
233C2×Dic43—22no—
234C2×Dic43—42no—
235C2×Dic43—42no—
236C2×Dic43—42no—
237C2×Dic43—42no—
238C2×Dic43—42no—
239C2×Dic43—42no—
240C2×Dic43—42no—
241C2×Dic43—42no—
242C2×Dic43—42no—
243C2×Dic43—81no—
244C2×Dic43—84no—
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—81no—
273C4×C43—81no—
274Dic43—42no—
275Dic43—42no—
276Dic43—42no—
277Dic43—42no—
278Dic43—82no—
279C2×#539—431no—
280#3189—431no—
281#3187D4311yescharacteristic
282#2627—431no—
283#1615—434no—
284#1603—434no—
285#1520—431no—
286#1519—431no—
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—861no—
306C2×C22⋊Q8C2×D4311yescharacteristic
307C2×#121C2×D4311yescharacteristic
308C2×C4×D4—861no—
309C22×C4⋊C4—431no—
310#484—432no—
311#484—861no—
312#483—21no—
313#481C2×D4312yes—
314#467—862no—
315#465—22no—
316#451—432no—
317#451—861no—
318#450—21no—
319#448C2×D4312yes—
320C2×C22⋊C8—431no—
321C2×C22⋊C8C2×D4311yescharacteristic
322#400—432no—
323#398—862no—
324#395—432no—
325#385—861no—
326#381—861no—
327#380—431no—
328#328—434no—
329#327—434no—
330#324—434no—
331C43#1828911yescharacteristic, complemented, Sylow
332C22×Q8—431no—
333C22×Q8C22×D4311yescharacteristic
334C22×D4—21no—
335C22×D4C22⋊D4311yescharacteristic
336C23×C4—431no—
337C23×C4—431no—
338C23×C4—861no—
339C23×C4C22×D4311yescharacteristic
340C2×Q16—862no—
341C2×SD16—22no—
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—21no—
353C22⋊Q8—21no—
354C22⋊Q8—21no—
355C22⋊Q8—21no—
356C22⋊Q8—432no—
357C22⋊Q8—432no—
358C22⋊Q8—864no—
359C22⋊Q8—864no—
360C22⋊Q8—861no—
361C22⋊Q8—861no—
362#121—21no—
363#121—21no—
364#121C22⋊D4312yes—
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—861no—
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—862no—
381C2×C4⋊C4—861no—
382C2×C4⋊C4C22⋊D4311yescharacteristic
383C2×C4⋊C4C22×D4311yescharacteristic
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—22no—
394#103—22no—
395#103—862no—
396#103—862no—
397#102—22no—
398C22⋊C8—21no—
399C22⋊C8—432no—
400C22⋊C8—861no—
401C22⋊C8C22×D4312yes—
402#95—432no—
403#95—862no—
404#95—862no—
405#95—862no—
406C24—21no—
407C24D4×D4311yescharacteristic
408C2×Q8—21no—
409C2×Q8—21no—
410C2×Q8—81no—
411C2×Q8—432no—
412C2×Q8—861no—
413C2×Q8—3441no—
414C2×D4—22no—
415C2×D4—22no—
416C2×D4—21no—
417C2×D4—42no—
418C2×D4—42no—
419C2×D4—81no—
420C2×D4C2×C22⋊D4312yes—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—81no—
425C22×C4—432no—
426C22×C4—432no—
427C22×C4—864no—
428C22×C4—861no—
429C22×C4—861no—
430C22×C4—861no—
431C22×C4—861no—
432C22×C4—861no—
433C22×C4—861no—
434C22×C4—862no—
435C22×C4—862no—
436C22×C4—862no—
437C22×C4—3442no—
438C22×C4—3441no—
439C22×C4D4×D4311yescharacteristic
440C22×C4D4×D4311yescharacteristic
441C22×C4D4×D4311yescharacteristic
442C22×C4C2×C22⋊D4311yescharacteristic
443C22×C4C2×C22⋊D4311yescharacteristic
444C22×C4C23×D4311yescharacteristic
445Q16—3442no—
446SD16—82no—
447C2×C8—22no—
448C2×C8—81no—
449C2×C8—862no—
450C2×C8—3441no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—41no—
455C4⋊C4—41no—
456C4⋊C4—432no—
457C4⋊C4—864no—
458C4⋊C4—864no—
459C4⋊C4—864no—
460C4⋊C4—862no—
461C4⋊C4—862no—
462C4⋊C4—861no—
463C4⋊C4—862no—
464C4⋊C4—862no—
465C4⋊C4—862no—
466C4⋊C4—862no—
467C4⋊C4—3442no—
468C4⋊C4—3441no—
469C4⋊C4C2×C22⋊D4312yes—
470C22⋊C4—22no—
471C22⋊C4—22no—
472C22⋊C4—41no—
473C22⋊C4—41no—
474C22⋊C4—864no—
475C22⋊C4—864no—
476C22⋊C4—864no—
477C22⋊C4—862no—
478C22⋊C4—862no—
479C22⋊C4—1722no—
480C42—864no—
481C23—22no—
482C23—41no—
483C23—41no—
484C23—42no—
485C23—42no—
486C23—42no—
487C23#1306329311yescharacteristic
488C23#1306329311yescharacteristic
489C23#1306334612yes—
490C23#1306335412yes—
491C23C2×D4×D4311yescharacteristic
492Q8—42no—
493Q8—42no—
494Q8—3441no—
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—41no—
512C2×C4—41no—
513C2×C4—862no—
514C2×C4—862no—
515C2×C4—862no—
516C2×C4—862no—
517C2×C4—862no—
518C2×C4—862no—
519C2×C4—3441no—
520C2×C4—3444no—
521C2×C4—1722no—
522C2×C4—1722no—
523C2×C4—1722no—
524C2×C4—1722no—
525C2×C4—1722no—
526C2×C4—1722no—
527C2×C4—1722no—
528C2×C4—1722no—
529C2×C4—1722no—
530C2×C4#1306329311yescharacteristic
531C2×C4#1306329311yescharacteristic
532C2×C4C2×D4×D4311yescharacteristic
533C2×C4C22×C22⋊D4311yescharacteristic
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#1458305812yes—
549C22#1458305812yes—
550C22C2×#1306329311yescharacteristic
551C22C2×#1306334611yescharacteristic
552C22C2×#1306335411yescharacteristic
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—42no—
558C4—81no—
559C4—81no—
560C4—3442no—
561C4—1722no—
562C4—1722no—
563C4—1722no—
564C4—1722no—
565C2—21no—
566C2—22no—
567C2—22no—
568C2—21no—
569C2—82no—
570C2C2×#1458305811yescharacteristic
571C2#1914395711yescharacteristic
572C2#1914399111yescharacteristic
subgroupG/H
centerC22C2×#13063293
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#3187C2
socleC22×C43C2×#120
prime radicalC2×#121×C43C22
hypercenter#3187D43
hypocenterC43#18289
LCSG ▷ C22×C4×C43 ▷ C22×C43 ▷ C43
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3187

Complements

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

NH ≅ G/Ntype
2. #19145367C2semidirect
3. #19140410C2semidirect
4. #19133623C2semidirect
5. #19133555C2semidirect
6. #19133168C2semidirect
8. C2×#14582624C2semidirect
22. #14583234C22semidirect
25. #14583055C22semidirect
28. #14582837C22semidirect
30. #14582818C22semidirect
34. #14582637C22semidirect
36. #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
#191453672
#191404102
#191336232
#191335552
#191331682
#191331632
C2×#145826242
C43×#31872
#1828943