Finite Group Databasebeta

← #25070511

Subgroups — #25070511

Order 11008.

1002 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583528C211yesmaximal, characteristic, complemented
2#19145352C211yesmaximal, characteristic
3#19140339C211yesmaximal, characteristic
4#19133601C214yesmaximal, complemented
5#19133533C214yesmaximal, complemented
6#19133168C211yesmaximal, characteristic, complemented
7#19133167C211yesmaximal, characteristic, complemented
8C2×#14582624C211yesmaximal, characteristic, complemented
9C43×#3187C211yesmaximal, characteristic
10#14583528—22no—
11#14583528—22no—
12#14583528—22no—
13#14583528C2214yes—
14C2×#13063324—21no—
15C2×#13063323—21no—
16C2×#13063315—21no—
17C2×#13063315—21no—
18C2×#13063306—21no—
19C2×#13063304C2211yescharacteristic
20C2×#13063302—21no—
21C2×#13063281—21no—
22C22×Dic43⋊C4C2211yescharacteristic
23#14583229C2212yes—
24#14583215—21no—
25#14583126—22no—
26#14583050C2212yes—
27#14583037—21no—
28C2×C22⋊(C43⋊C8)C2211yescharacteristic
29#14582830C2212yes—
30#14582822—22no—
31#14582813C2212yes—
32#14582779—21no—
33#14582759C2211yescharacteristic
34#14582637C2214yes—
35#14582636C2214yes—
36#14582624C2214yes—
37C2×C22⋊Q8×C43C2211yescharacteristic
38C2×#121×C43C2211yescharacteristic
39C43×#483C2212yes—
40C43×#481—21no—
41C43×#465—22no—
42C43×#450C2212yes—
43C43×#448—21no—
44C2×C22⋊C8×C43C2211yescharacteristic
45C22×Dic86—21no—
46C22×Dic86D411yescharacteristic
47C23×Dic43D411yescharacteristic
48C23×Dic43D411yescharacteristic
49C2×#10621883—22no—
50C22×C43⋊C8—21no—
51#13063324—24no—
52#13063324—24no—
53#13063323—24no—
54#13063323—24no—
55#13063315—24no—
56#13063315—24no—
57#13063315—24no—
58#13063315—24no—
59#13063306—44no—
60#13063304—21no—
61#13063304—21no—
62#13063304D412yes—
63#13063304C2312yes—
64#13063302—24no—
65#13063302—24no—
66#13063281—24no—
67#13063281—24no—
68C2×C4⋊Dic43—21no—
69C2×C4⋊Dic43—21no—
70C2×C4⋊Dic43D411yescharacteristic
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—21no—
78C2×Dic43⋊C4—21no—
79C2×Dic43⋊C4—21no—
80C2×Dic43⋊C4—22no—
81C2×Dic43⋊C4D412yes—
82C2×Dic43⋊C4C2312yes—
83C2×C22⋊Dic43—21no—
84C2×C22⋊Dic43—21no—
85C2×C22⋊Dic43—22no—
86C2×C22⋊Dic43—22no—
87C2×C4×Dic43—21no—
88C2×C4×Dic43—21no—
89#13063256—22no—
90#13063248—22no—
91#13063246—22no—
92C22⋊(C43⋊C8)—21no—
93C22⋊(C43⋊C8)C2312yes—
94C43⋊#95—22no—
95C43⋊#95C2312yes—
96C22×Q8×C43D411yescharacteristic
97C22×D4×C43—21no—
98C22×D4×C43C2311yescharacteristic
99C23×C4×C43C2311yescharacteristic
100C2×SD16×C43—22no—
101C22×C8×C43—21no—
102C22⋊Q8×C43—21no—
103C22⋊Q8×C43—21no—
104C22⋊Q8×C43D412yes—
105C22⋊Q8×C43C2312yes—
106#121×C43—21no—
107#121×C43—21no—
108#121×C43—21no—
109#121×C43—21no—
110C2×C4⋊C4×C43—21no—
111C2×C4⋊C4×C43D411yescharacteristic
112C2×C4⋊C4×C43C2311yescharacteristic
113C2×C22⋊C4×C43—21no—
114C2×C22⋊C4×C43—21no—
115#106×C43—22no—
116#103×C43—22no—
117#102×C43—22no—
118C22⋊C8×C43—21no—
119C22⋊C8×C43C2312yes—
120C24×C43—21no—
121C24×C43C2×D411yescharacteristic
122C2×Q8×C43—21no—
123C2×Q8×C43—81no—
124C2×Q8×C43C2×D412yes—
125C2×D4×C43—21no—
126C2×D4×C43—21no—
127C2×D4×C43—41no—
128C2×D4×C43—41no—
129C2×D4×C43—42no—
130C2×D4×C43—42no—
131C2×D4×C43—81no—
132C22×C4×C43—21no—
133C22×C4×C43—21no—
134C22×C4×C43—21no—
135C22×C4×C43—81no—
136C22×C4×C43C2×D411yescharacteristic
137C22×C4×C43C2×D411yescharacteristic
138C22×C4×C43C2×D411yescharacteristic
139C22×C4×C43C2×D411yescharacteristic
140C22×C4×C43C2×D411yescharacteristic
141C22×C4×C43C2411yescharacteristic
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—81no—
151C22×Dic43SD1614yescomplemented
152C22×Dic43C2×D412yes—
153C22×Dic43C2×D412yes—
154C2×Dic86—24no—
155C2×Dic86—21no—
156C2×Dic86—81no—
157C2×Dic86—82no—
158C2×Dic86C2×D412yes—
159SD16×C43—82no—
160C2×C8×C43—22no—
161C2×C8×C43—81no—
162C4⋊C4×C43—22no—
163C4⋊C4×C43—21no—
164C4⋊C4×C43—22no—
165C4⋊C4×C43—21no—
166C4⋊C4×C43—21no—
167C4⋊C4×C43C2×D412yes—
168C22⋊C4×C43—22no—
169C22⋊C4×C43—22no—
170C22⋊C4×C43—41no—
171C22⋊C4×C43—41no—
172C22⋊Dic43—22no—
173C22⋊Dic43—22no—
174C22⋊Dic43—44no—
175C22⋊Dic43—42no—
176C22⋊Dic43—82no—
177#10621883—82no—
178C4⋊Dic43—24no—
179C4⋊Dic43—21no—
180C4⋊Dic43—81no—
181C4⋊Dic43C2×D412yes—
182Dic43⋊C4—24no—
183Dic43⋊C4—24no—
184Dic43⋊C4—24no—
185Dic43⋊C4—24no—
186Dic43⋊C4—22no—
187Dic43⋊C4—22no—
188Dic43⋊C4—22no—
189Dic43⋊C4—22no—
190Dic43⋊C4—22no—
191Dic43⋊C4—22no—
192Dic43⋊C4—44no—
193C4×Dic43—24no—
194C4×Dic43—24no—
195C2×C43⋊C8—22no—
196C2×C43⋊C8—81no—
197C23×C43—41no—
198C23×C43—41no—
199C23×C43—42no—
200C23×C43—42no—
201C23×C43—41no—
202C23×C43—41no—
203C23×C43—41no—
204C23×C43#12011yescharacteristic
205C23×C43#12011yescharacteristic
206C23×C43C2×SD1612yes—
207C23×C43#13712yes—
208C23×C43C22×D411yescharacteristic
209Q8×C43—42no—
210Q8×C43—81no—
211D4×C43—42no—
212D4×C43—42no—
213D4×C43—82no—
214D4×C43—82no—
215C2×C4×C43—22no—
216C2×C4×C43—22no—
217C2×C4×C43—41no—
218C2×C4×C43—41no—
219C2×C4×C43—41no—
220C2×C4×C43—41no—
221C2×C4×C43—41no—
222C2×C4×C43—41no—
223C2×C4×C43—42no—
224C2×C4×C43—42no—
225C2×C4×C43—41no—
226C2×C4×C43—41no—
227C2×C4×C43—41no—
228C2×C4×C43#12011yescharacteristic
229C2×C4×C43#12011yescharacteristic
230C2×C4×C43C22×D411yescharacteristic
231C2×C4×C43C22×D411yescharacteristic
232C2×Dic43—22no—
233C2×Dic43—22no—
234C2×Dic43—24no—
235C2×Dic43—24no—
236C2×Dic43—22no—
237C2×Dic43—22no—
238C2×Dic43—22no—
239C2×Dic43—42no—
240C2×Dic43—42no—
241C2×Dic43—42no—
242C2×Dic43—42no—
243C2×Dic43—42no—
244C2×Dic43—42no—
245C2×Dic43—42no—
246C2×Dic43—81no—
247C2×Dic43C2×SD1614yes—
248Dic86—42no—
249Dic86—81no—
250Dic86—84no—
251C8×C43—42no—
252C43⋊C8—42no—
253#18294—431nomaximal, Sylow
254C22×C43—21no—
255C22×C43—21no—
256C22×C43—21no—
257C22×C43—21no—
258C22×C43—21no—
259C22×C43—21no—
260C22×C43—42no—
261C22×C43—42no—
262C22×C43—82no—
263C22×C43—82no—
264C22×C43—81no—
265C22×C43—81no—
266C22×C43—81no—
267C22×C43#44812yes—
268C22×C43#44812yes—
269C22×C43C2×#12011yescharacteristic
270C22×C43C22×SD1611yescharacteristic
271C22×C43C2×#13711yescharacteristic
272C4×C43—21no—
273C4×C43—21no—
274C4×C43—21no—
275C4×C43—21no—
276C4×C43—42no—
277C4×C43—42no—
278C4×C43—81no—
279Dic43—24no—
280Dic43—42no—
281Dic43—42no—
282Dic43—42no—
283Dic43—82no—
284C2×#544—431no—
285#3187—431no—
286#3187D4311yescharacteristic
287#2618—431no—
288#1611—434no—
289#1599—434no—
290#1520—431no—
291#1520—431no—
292C2×#324—431no—
293C2×C43—21no—
294C2×C43—21no—
295C2×C43—22no—
296C2×C43—22no—
297C2×C43—161no—
298C2×C43C2×#44811yescharacteristic
299C2×C43#302011yescharacteristic
300C2×C43#302611yescharacteristic
301#544—434no—
302#544—862no—
303#544—862no—
304#544—862no—
305C2×C4⋊Q8—861no—
306C2×C4⋊Q8—861no—
307C2×#125—861no—
308C2×#125—861no—
309C2×#123—861no—
310C2×C22⋊Q8—431no—
311C2×C22⋊Q8—861no—
312C2×C22⋊Q8C2×D4311yescharacteristic
313C2×#121C2×D4311yescharacteristic
314C2×#117—861no—
315C22×C4⋊C4—431no—
316#483—432no—
317#483C2×D4312yes—
318#481—21no—
319#481—861no—
320#465—22no—
321#465—862no—
322#450—432no—
323#450C2×D4312yes—
324#448—21no—
325#448—861no—
326C2×C22⋊C8—431no—
327C2×C22⋊C8C2×D4311yescharacteristic
328#398—432no—
329#396—862no—
330#394—432no—
331#386—861no—
332#380—431no—
333#328—434no—
334#328—434no—
335#324—434no—
336C43#1829411yescharacteristic, complemented, Sylow
337C22×Q8—431no—
338C22×Q8—861no—
339C22×Q8C22⋊D4311yescharacteristic
340C22×D4—21no—
341C22×D4C22×D4311yescharacteristic
342C23×C4—431no—
343C23×C4—431no—
344C23×C4C22×D4311yescharacteristic
345C2×SD16—22no—
346C2×SD16—862no—
347C22×C8—21no—
348C22×C8—861no—
349C4⋊Q8—864no—
350C4⋊Q8—864no—
351C4⋊Q8—864no—
352C4⋊Q8—864no—
353#125—864no—
354#125—864no—
355#125—864no—
356#125—864no—
357#123—1724no—
358C22⋊Q8—21no—
359C22⋊Q8—21no—
360C22⋊Q8—432no—
361C22⋊Q8—432no—
362C22⋊Q8—864no—
363C22⋊Q8—864no—
364C22⋊Q8—861no—
365C22⋊Q8—861no—
366C22⋊Q8C22⋊D4312yes—
367C22⋊Q8C22×D4312yes—
368#121—21no—
369#121—21no—
370#121—21no—
371#121—21no—
372#117—864no—
373#117—864no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—431no—
376C2×C4⋊C4—432no—
377C2×C4⋊C4—432no—
378C2×C4⋊C4—861no—
379C2×C4⋊C4—861no—
380C2×C4⋊C4—861no—
381C2×C4⋊C4—861no—
382C2×C4⋊C4—861no—
383C2×C4⋊C4—861no—
384C2×C4⋊C4—861no—
385C2×C4⋊C4—861no—
386C2×C4⋊C4—861no—
387C2×C4⋊C4—861no—
388C2×C4⋊C4—861no—
389C2×C4⋊C4—862no—
390C2×C4⋊C4C22⋊D4311yescharacteristic
391C2×C4⋊C4C22×D4311yescharacteristic
392C2×C22⋊C4—21no—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—861no—
395C2×C22⋊C4—862no—
396C2×C22⋊C4—862no—
397C2×C22⋊C4—861no—
398C2×C42—861no—
399C2×C42—861no—
400#106—22no—
401#106—862no—
402#103—22no—
403#103—862no—
404#102—22no—
405#102—862no—
406C22⋊C8—21no—
407C22⋊C8—432no—
408C22⋊C8—861no—
409C22⋊C8C22×D4312yes—
410#95—432no—
411#95—862no—
412C24—21no—
413C24D4×D4311yescharacteristic
414C2×Q8—21no—
415C2×Q8—81no—
416C2×Q8—432no—
417C2×Q8—864no—
418C2×Q8—861no—
419C2×Q8—3442no—
420C2×Q8—3441no—
421C2×Q8C2×C22⋊D4312yes—
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—41no—
425C2×D4—41no—
426C2×D4—42no—
427C2×D4—42no—
428C2×D4—81no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—81no—
433C22×C4—434no—
434C22×C4—432no—
435C22×C4—432no—
436C22×C4—861no—
437C22×C4—861no—
438C22×C4—862no—
439C22×C4—861no—
440C22×C4—861no—
441C22×C4—861no—
442C22×C4—861no—
443C22×C4—862no—
444C22×C4—3441no—
445C22×C4D4×D4311yescharacteristic
446C22×C4D4×D4311yescharacteristic
447C22×C4D4×D4311yescharacteristic
448C22×C4C2×C22⋊D4311yescharacteristic
449C22×C4C2×C22⋊D4311yescharacteristic
450C22×C4C23×D4311yescharacteristic
451SD16—82no—
452SD16—3442no—
453C2×C8—22no—
454C2×C8—81no—
455C2×C8—862no—
456C2×C8—3441no—
457C4⋊C4—22no—
458C4⋊C4—21no—
459C4⋊C4—22no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—432no—
463C4⋊C4—864no—
464C4⋊C4—864no—
465C4⋊C4—864no—
466C4⋊C4—864no—
467C4⋊C4—864no—
468C4⋊C4—862no—
469C4⋊C4—862no—
470C4⋊C4—861no—
471C4⋊C4—862no—
472C4⋊C4—862no—
473C4⋊C4—862no—
474C4⋊C4—862no—
475C4⋊C4—3441no—
476C4⋊C4—1724no—
477C4⋊C4C2×C22⋊D4312yes—
478C22⋊C4—22no—
479C22⋊C4—22no—
480C22⋊C4—41no—
481C22⋊C4—41no—
482C22⋊C4—862no—
483C22⋊C4—862no—
484C22⋊C4—3442no—
485C22⋊C4—1724no—
486C22⋊C4—1722no—
487C42—864no—
488C42—864no—
489C23—41no—
490C23—41no—
491C23—42no—
492C23—42no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23#1306329311yescharacteristic
497C23#1306329311yescharacteristic
498C23SD16×D4312yes—
499C23#1306335412yes—
500C23C2×D4×D4311yescharacteristic
501Q8—42no—
502Q8—81no—
503Q8—3441no—
504Q8—3444no—
505Q8—1722no—
506D4—42no—
507D4—42no—
508D4—82no—
509D4—82no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—434no—
524C2×C4—862no—
525C2×C4—862no—
526C2×C4—864no—
527C2×C4—864no—
528C2×C4—862no—
529C2×C4—862no—
530C2×C4—862no—
531C2×C4—3441no—
532C2×C4—1722no—
533C2×C4—1722no—
534C2×C4—1722no—
535C2×C4—1722no—
536C2×C4—1722no—
537C2×C4—1722no—
538C2×C4—1722no—
539C2×C4#1306329311yescharacteristic
540C2×C4#1306329311yescharacteristic
541C2×C4C2×D4×D4311yescharacteristic
542C2×C4C22×C22⋊D4311yescharacteristic
543C8—42no—
544C8—1722no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—42no—
552C22—42no—
553C22—82no—
554C22—82no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22#1458303612yes—
559C22#1458303612yes—
560C22C2×#1306329311yescharacteristic
561C22C2×SD16×D4311yescharacteristic
562C22C2×#1306335411yescharacteristic
563C4—21no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—42no—
568C4—42no—
569C4—81no—
570C4—864no—
571C4—3442no—
572C4—1722no—
573C4—1722no—
574C4—1722no—
575C2—21no—
576C2—21no—
577C2—22no—
578C2—22no—
579C2—161no—
580C2C2×#1458303611yescharacteristic
581C2#1914395711yescharacteristic
582C2#1914399111yescharacteristic
subgroupG/H
centerC22C2×#13063293
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
FittingC43×#3187C2
socleC22×C43C2×#120
prime radicalC2×#121×C43C22
hypercenter#3187D43
hypocenterC43#18294
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
1. C2×#14583528C2semidirect
4. #19133601C2semidirect
5. #19133533C2semidirect
6. #19133168C2semidirect
7. #19133167C2semidirect
8. C2×#14582624C2semidirect
151. C22×Dic43SD16semidirect
336. C43#18294semidirect

Maximal subgroups

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

subgroupindex
C2×#145835282
#191453522
#191403392
#191336012
#191335332
#191331682
#191331672
C2×#145826242
C43×#31872
#1829443