Finite Group Databasebeta

← #25964268

Subgroups — #25964268

Order 11008.

2054 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13063371C211yesmaximal, characteristic
2#19144453C213yesmaximal, complemented
3#19144442C213yesmaximal, complemented
4#19144420C213yesmaximal, complemented
5#19144420C213yesmaximal, complemented
6#19144413C213yesmaximal, complemented
7#19144388C213yesmaximal, complemented
8#19144377C213yesmaximal, complemented
9C2×#14583196C213yesmaximal, complemented
10#19143193C211yesmaximal, characteristic, complemented
11#19143188C211yesmaximal, characteristic, complemented
12#19143163C211yesmaximal, characteristic, complemented
13#19143118C211yesmaximal, characteristic
14#19143118C211yesmaximal, characteristic
15C43×#3504C211yesmaximal, characteristic, complemented
16C2×#13063371C2211yescharacteristic
17C2×#13063332C2211yescharacteristic
18C2×#13063332C2211yescharacteristic
19#14583382C2211yescharacteristic
20#14583381—21no—
21#14583379C2213yes—
22#14583373C2213yes—
23#14583371—23no—
24C4×#10621902—21no—
25C4×#10621898C2213yes—
26C2×C4×D172C2213yes—
27C4×D4×D43—23no—
28#14583203C2216yescomplemented
29#14583203C2211yescharacteristic, complemented
30#14583203C2213yescomplemented
31#14583196C2216yescomplemented
32#14583196C2213yescomplemented
33#14583196C2216yescomplemented
34#14583196C2216yescomplemented
35#14583196C2213yescomplemented
36#14583196C2216yescomplemented
37#14583191C2213yescomplemented
38#14583186C2216yescomplemented
39#14583186C2213yescomplemented
40#14583181C2213yescomplemented
41#14583176C2213yescomplemented
42#14583176C2216yescomplemented
43#14582942C2213yescomplemented
44#14582942C2213yescomplemented
45#14582939C2213yescomplemented
46#14582939C2213yescomplemented
47#14582935C2213yescomplemented
48#14582935C2213yescomplemented
49C2×#13063258C2213yescomplemented
50C2×#13063256C2213yescomplemented
51#14582926C2213yescomplemented
52#14582926C2213yescomplemented
53#14582923C2213yescomplemented
54C2×C4⋊(C43⋊C8)C2213yescomplemented
55#14582903C2213yes—
56#14582903C2213yes—
57#14582898C2213yes—
58#14582898C2213yes—
59#14582893C2213yes—
60#14582893C2213yes—
61C2×#13063245C2213yes—
62C2×#13063245C2213yes—
63C43×#557—21no—
64C43×#557—21no—
65C43×#554—23no—
66C43×#554—23no—
67C43×#544C2213yescomplemented
68C43×#544C2213yescomplemented
69C43×#533C2213yescomplemented
70C2×#125×C43C2213yescomplemented
71C43×#520C2211yescharacteristic
72C43×#520C2211yescharacteristic
73C4×#41×C43C2211yescharacteristic
74#13063371—24no—
75#13063371Q818yescomplemented
76C2×#10621902—21no—
77C2×#10621898C2313yes—
78C22×D172C2313yes—
79C2×D4×D43—23no—
80C22×C4×D43—23no—
81#13063332—24no—
82#13063332—24no—
83C2×#10621877C2313yes—
84C2×#10621877C2313yes—
85C22×C43⋊C8C2313yes—
86C22×C43⋊C8C2313yes—
87Q8×Dic43—21no—
88C4×Dic86C2311yescharacteristic
89C4×Dic86C2313yes—
90C4×D172—23no—
91C4×D172—23no—
92C4×D172C2313yes—
93C4×D172C2316yes—
94C4×D172C2313yes—
95D4×Dic43—23no—
96#13063287—23no—
97#13063287—23no—
98C4×C22⋊D43—23no—
99C4×C22⋊D43—23no—
100(C22⋊D43)⋊C4—26no—
101#13063284C2313yes—
102#13063283—23no—
103#13063282—23no—
104C2×C4⋊Dic43C2313yes—
105C4⋊C4×D43—23no—
106C2×(C2×D43)⋊C4—23no—
107C22⋊C4×D43—26no—
108C42×D43—23no—
109#13063258C2311yescharacteristic
110#13063258C2313yes—
111#13063258C2316yes—
112#13063258C2313yes—
113#13063258C2313yes—
114#13063256C2313yes—
115#13063256C2313yes—
116#13063256C2316yes—
117#13063256C2311yescharacteristic
118#13063256C2313yes—
119C4⋊(C43⋊C8)C2311yescharacteristic
120C4⋊(C43⋊C8)C2313yes—
121C4⋊(C43⋊C8)C2313yes—
122C4⋊(C43⋊C8)C2313yes—
123C4⋊(C43⋊C8)C2316yes—
124#13063248C2311yescharacteristic
125#13063248C2311yescharacteristic
126#13063248C2313yes—
127#13063248C2313yes—
128#13063245C2313yes—
129#13063245C2313yes—
130#13063245C2316yes—
131#13063245C2316yes—
132#13063245C2313yes—
133#13063245C2313yes—
134C2×#41×C43C2311yescharacteristic
135C4⋊Q8×C43—23no—
136C4⋊Q8×C43—23no—
137C4⋊Q8×C43C2313yes—
138C4⋊Q8×C43C2313yes—
139#125×C43—23no—
140#125×C43—23no—
141#125×C43—23no—
142#125×C43—23no—
143#125×C43C2313yes—
144#125×C43C2316yes—
145#125×C43C2311yescharacteristic
146C22⋊Q8×C43—23no—
147C22⋊Q8×C43—23no—
148C22⋊Q8×C43—26no—
149C22⋊Q8×C43—26no—
150C4×Q8×C43—21no—
151C4×Q8×C43—21no—
152C4×Q8×C43—23no—
153C4×Q8×C43—23no—
154C4×Q8×C43D411yescharacteristic
155C4×Q8×C43D411yescharacteristic
156C4×D4×C43—23no—
157C4×D4×C43—23no—
158C4×D4×C43D413yes—
159C4×D4×C43D413yes—
160#117×C43C2313yes—
161#117×C43C2313yes—
162#117×C43C2313yes—
163C2×C4⋊C4×C43—23no—
164C2×C4⋊C4×C43—23no—
165C2×C4⋊C4×C43C2313yes—
166C2×C4⋊C4×C43C2313yes—
167C2×C42×C43C2313yes—
168C23×D43—23no—
169#41×C43C2×Q814yes—
170#41×C43#4114yes—
171C2×Q8×C43—23no—
172C2×Q8×C43—23no—
173C2×Q8×C43C2×D411yescharacteristic
174C2×D4×C43C2×D413yes—
175C22×C4×C43—23no—
176C22×C4×C43—23no—
177C22×C4×C43C2×D413yes—
178C22×C4×C43C2×D413yes—
179C22×C4×C43C2413yes—
180C2×C22⋊D43—23no—
181C22×Dic43—23no—
182#10621902—28no—
183D4×D43—224no—
184#10621898—26no—
185#10621898C2×Q8112yes—
186C2×D172—23no—
187C2×D172—26no—
188C2×D172C2×Q8112yes—
189C2×D172C2413yes—
190C2×C4×D43—23no—
191C2×C4×D43—23no—
192C2×C4×D43—23no—
193C2×C4×D43—26no—
194C2×C4×D43—86no—
195C2×Dic86C2411yescharacteristic
196C4⋊C4×C43—23no—
197C4⋊C4×C43—23no—
198C4⋊C4×C43—26no—
199C4⋊C4×C43—26no—
200C4⋊C4×C43—23no—
201C4⋊C4×C43—23no—
202C4⋊C4×C43—21no—
203C4⋊C4×C43—21no—
204C4⋊C4×C43—23no—
205C4⋊C4×C43—23no—
206C4⋊C4×C43C2×D413yes—
207C4⋊C4×C43C2×D413yes—
208C4⋊C4×C43C2413yes—
209C4⋊C4×C43C2413yes—
210C4⋊C4×C43C2411yescharacteristic
211C4⋊C4×C43C2411yescharacteristic
212C22⋊C4×C43—23no—
213C22⋊C4×C43—23no—
214C22⋊C4×C43C2×D416yes—
215C42×C43—23no—
216C42×C43—23no—
217C42×C43C2×D413yes—
218C42×C43C2×D413yes—
219C42×C43C2411yescharacteristic
220C42×C43C2413yes—
221C22⋊Dic43—23no—
222(C2×D43)⋊C4—23no—
223(C2×D43)⋊C4—26no—
224(C2×D43)⋊C4—23no—
225C4⋊Dic43—23no—
226C4⋊Dic43C2413yes—
227C4⋊Dic43C2411yescharacteristic
228Dic43⋊C4—23no—
229Dic43⋊C4—21no—
230C4×Dic43—21no—
231C4×Dic43—23no—
232C4×Dic43—23no—
233#10621877—26no—
234#10621877—26no—
235C2×C43⋊C8—26no—
236C2×C43⋊C8—26no—
237C2×C43⋊C8C2411yescharacteristic
238C2×C43⋊C8C2413yes—
239C2×C43⋊C8C2413yes—
240C2×C43⋊C8C2411yescharacteristic
241C23×C43C22×D413yes—
242C22×D43—23no—
243C22×D43—212no—
244C22×D43—83no—
245C22×D43—83no—
246Q8×C43—83no—
247Q8×C43—83no—
248Q8×C43C22⋊Q814yes—
249D4×C43C22⋊Q8112yes—
250C2×C4×C43—21no—
251C2×C4×C43—21no—
252C2×C4×C43—23no—
253C2×C4×C43—23no—
254C2×C4×C43—43no—
255C2×C4×C43—43no—
256C2×C4×C43—83no—
257C2×C4×C43—83no—
258C2×C4×C43C22×D413yes—
259C2×C4×C43C22×D411yescharacteristic
260C2×C4×C43C22×D411yescharacteristic
261C2×C4×C43C22×D413yes—
262C2×C4×C43C22×D413yes—
263C2×C4×C43C22×Q816yes—
264C2×C4×C43C2×#4116yes—
265C2×C4×C43C2511yescharacteristic
266C22⋊D43—212no—
267C2×Dic43—21no—
268C2×Dic43—21no—
269C2×Dic43—23no—
270C2×Dic43—83no—
271D172—212no—
272D172—23no—
273D172C22×Q816yes—
274C4×D43—212no—
275C4×D43—46no—
276C4×D43—83no—
277C4×D43—86no—
278Dic86—21no—
279Dic86C22×Q812yes—
280C43⋊C8—21no—
281C43⋊C8—21no—
282C43⋊C8—23no—
283C43⋊C8—23no—
284#60496—431nomaximal, Sylow
285C22×C43—23no—
286C22×C43—23no—
287C22×C43C2×C22⋊Q816yes—
288C22×C43C23×D411yescharacteristic
289C2×D43—26no—
290C2×D43—43no—
291C2×D43—43no—
292C2×D43—812no—
293C4×C43—23no—
294C4×C43—21no—
295C4×C43—21no—
296C4×C43—23no—
297C4×C43—43no—
298C4×C43—41no—
299C4×C43—43no—
300C4×C43—41no—
301C4×C43C2×C22⋊Q816yes—
302C4×C43C23×Q811yescharacteristic
303C4×C43C22×#4111yescharacteristic
304Dic43—22no—
305Dic43—41no—
306Dic43—43no—
307#3504D4311yescharacteristic, complemented
308C4×H(5,𝔽2)—431no—
309#3084—433no—
310#3083—433no—
311#3081—433no—
312#3081—433no—
313#3080—433no—
314#3077—433no—
315#3076—433no—
316C2×#478—433no—
317#2913—431no—
318#2912—431no—
319#2906—431no—
320#2897—431no—
321#2897—431no—
322C2×C43—26no—
323C2×C43C22×C22⋊Q811yescharacteristic
324C2×C43#358411yescharacteristic
325C2×C43#358511yescharacteristic
326D43—46no—
327C2×H(5,𝔽2)—431no—
328C2×#131—431no—
329C2×#131—431no—
330#557—21no—
331#557—21no—
332#554—23no—
333#554—23no—
334#544C2×D4313yescomplemented
335#544C2×D4313yescomplemented
336#533C2×D4313yescomplemented
337C2×#125C2×D4313yescomplemented
338#520—431no—
339#520—433no—
340#520—861no—
341#520C2×D4311yescharacteristic
342#520C2×D4311yescharacteristic
343#518—433no—
344#518—863no—
345C4×#41—433no—
346C4×#41—861no—
347C4×#41C2×D4311yescharacteristic
348C2×C4×D4—433no—
349C2×C4×D4—863no—
350#479—431no—
351#479—433no—
352#479—436no—
353#478—433no—
354#478—436no—
355#478—436no—
356#478—436no—
357#478—433no—
358#478—436no—
359#477—433no—
360#476—433no—
361#476—436no—
362#475—433no—
363#474—433no—
364#474—436no—
365#428—433no—
366#428—433no—
367#428—433no—
368#428—433no—
369#427—433no—
370#427—433no—
371C2×#107—433no—
372C2×#106—433no—
373#424—433no—
374#424—433no—
375#423—433no—
376C2×C4⋊C8—433no—
377#418—433no—
378#418—433no—
379#417—433no—
380#417—433no—
381#416—433no—
382#416—433no—
383C2×#102—433no—
384C2×#102—433no—
385C43#6049611yescharacteristic, complemented, Sylow
386H(5,𝔽2)—438no—
387H(5,𝔽2)—864no—
388C2×#41—433no—
389C2×#41—861no—
390C2×#41C22×D4311yescharacteristic
391C22×D4—433no—
392C22×D4—863no—
393C23×C4—863no—
394#131—864no—
395#131—864no—
396C2×M16—433no—
397C2×M16—433no—
398C22×C8—433no—
399C22×C8—433no—
400C4⋊Q8—23no—
401C4⋊Q8—23no—
402C4⋊Q8C22×D4313yes—
403C4⋊Q8C22×D4313yes—
404#125—23no—
405#125—23no—
406#125—23no—
407#125—23no—
408#125C22×D4313yes—
409#125C22×D4316yes—
410#125C22×D4311yescharacteristic
411C22⋊Q8—23no—
412C22⋊Q8—23no—
413C22⋊Q8—26no—
414C22⋊Q8—26no—
415C4×Q8—21no—
416C4×Q8—21no—
417C4×Q8—23no—
418C4×Q8—23no—
419C4×Q8—431no—
420C4×Q8—433no—
421C4×Q8—861no—
422C4×Q8C22⋊D4311yescharacteristic
423C4×Q8C22⋊D4311yescharacteristic
424C4×D4—23no—
425C4×D4—23no—
426C4×D4—433no—
427C4×D4—436no—
428C4×D4—433no—
429C4×D4—863no—
430C4×D4—863no—
431C4×D4—863no—
432C4×D4—863no—
433C4×D4—863no—
434C4×D4—866no—
435C4×D4—863no—
436C4×D4—863no—
437C4×D4C22⋊D4313yes—
438C4×D4C22⋊D4313yes—
439#117—433no—
440#117—863no—
441#117—863no—
442#117C22×D4313yes—
443#117C22×D4313yes—
444#117C22×D4313yes—
445C2×C4⋊C4—23no—
446C2×C4⋊C4—23no—
447C2×C4⋊C4—433no—
448C2×C4⋊C4—863no—
449C2×C4⋊C4C22×D4313yes—
450C2×C4⋊C4C22×D4313yes—
451C2×C22⋊C4—863no—
452C2×C22⋊C4—866no—
453C2×C42—863no—
454C2×C42C22×D4313yes—
455#107—433no—
456#107—436no—
457#107—431no—
458#107—433no—
459#107—433no—
460#106—433no—
461#106—436no—
462#106—431no—
463#106—433no—
464#106—433no—
465C4⋊C8—431no—
466C4⋊C8—433no—
467C4⋊C8—433no—
468C4⋊C8—433no—
469C4⋊C8—436no—
470#103—431no—
471#103—431no—
472#103—433no—
473#103—433no—
474#102—433no—
475#102—433no—
476#102—436no—
477#102—436no—
478#102—433no—
479#102—433no—
480C24—863no—
481#41—4312no—
482#41—868no—
483#41—866no—
484#41Q8×D4314yes—
485#41#1062190214yes—
486C2×Q8—23no—
487C2×Q8—23no—
488C2×Q8—431no—
489C2×Q8C2×C22⋊D4311yescharacteristic
490C2×D4—4312no—
491C2×D4—433no—
492C2×D4—863no—
493C2×D4—8624no—
494C2×D4—863no—
495C2×D4—866no—
496C2×D4C2×C22⋊D4313yes—
497C22×C4—23no—
498C22×C4—23no—
499C22×C4—863no—
500C22×C4—863no—
501C22×C4—863no—
502C22×C4—866no—
503C22×C4—863no—
504C22×C4—3446no—
505C22×C4C2×C22⋊D4313yes—
506C22×C4C2×C22⋊D4313yes—
507C22×C4C23×D4313yes—
508M16—866no—
509M16—866no—
510C2×C8—431no—
511C2×C8—433no—
512C2×C8—433no—
513C2×C8—431no—
514C2×C8—866no—
515C2×C8—866no—
516C4⋊C4—23no—
517C4⋊C4—23no—
518C4⋊C4—26no—
519C4⋊C4—26no—
520C4⋊C4—23no—
521C4⋊C4—23no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—23no—
525C4⋊C4—23no—
526C4⋊C4—433no—
527C4⋊C4—431no—
528C4⋊C4—863no—
529C4⋊C4—863no—
530C4⋊C4—861no—
531C4⋊C4C2×C22⋊D4313yes—
532C4⋊C4C2×C22⋊D4313yes—
533C4⋊C4C23×D4313yes—
534C4⋊C4C23×D4311yescharacteristic
535C4⋊C4C23×D4313yes—
536C4⋊C4C23×D4311yescharacteristic
537C22⋊C4—23no—
538C22⋊C4—23no—
539C22⋊C4—863no—
540C22⋊C4—863no—
541C22⋊C4—866no—
542C22⋊C4—863no—
543C22⋊C4C2×C22⋊D4316yes—
544C42—23no—
545C42—23no—
546C42—861no—
547C42—863no—
548C42—863no—
549C42C2×C22⋊D4313yes—
550C42C2×C22⋊D4313yes—
551C42C23×D4311yescharacteristic
552C42C23×D4313yes—
553C23—8612no—
554C23—863no—
555C23—3443no—
556C23—3443no—
557C23C22×C22⋊D4313yes—
558Q8—83no—
559Q8—83no—
560Q8—432no—
561Q8—861no—
562Q8#1306330114yes—
563D4—436no—
564D4—8612no—
565D4—8612no—
566D4—863no—
567D4#13063301112yes—
568C2×C4—21no—
569C2×C4—21no—
570C2×C4—23no—
571C2×C4—23no—
572C2×C4—43no—
573C2×C4—43no—
574C2×C4—83no—
575C2×C4—83no—
576C2×C4—861no—
577C2×C4—8612no—
578C2×C4—861no—
579C2×C4—863no—
580C2×C4—3446no—
581C2×C4—3443no—
582C2×C4—3443no—
583C2×C4—1726no—
584C2×C4C22×C22⋊D4311yescharacteristic
585C2×C4C22×C22⋊D4313yes—
586C2×C4C22×C22⋊D4311yescharacteristic
587C2×C4C22×C22⋊D4313yes—
588C2×C4C22×C22⋊D4313yes—
589C2×C4C2×Q8×D4316yes—
590C2×C4C2×#1062190216yes—
591C2×C4C24×D4311yescharacteristic
592C8—861no—
593C8—861no—
594C8—863no—
595C8—863no—
596C22—23no—
597C22—23no—
598C22—866no—
599C22—34412no—
600C22—1723no—
601C22—1723no—
602C22C2×#1306330116yes—
603C22C23×C22⋊D4311yescharacteristic
604C4—21no—
605C4—23no—
606C4—21no—
607C4—23no—
608C4—43no—
609C4—41no—
610C4—43no—
611C4—41no—
612C4—862no—
613C4—1721no—
614C4—1723no—
615C4C2×#1306330116yes—
616C4C22×Q8×D4311yescharacteristic
617C4C22×#1062190211yescharacteristic
618C2—26no—
619C2—1726no—
620C2C22×#1306330111yescharacteristic
621C2#1914936611yescharacteristic
622C2#1914937411yescharacteristic
subgroupG/H
centerC22C23×C22⋊D43
commutator, cosocleC2×C4×C43C25
FrattiniC2×C4C24×D43
FittingC43×#3504C2
socleC22×C43C23×D4
prime radicalC2×#13063371C22
hypercenter#3504D43
hypocenterC43#60496
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C4×#41 ◁ #3504

Complements

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

NH ≅ G/Ntype
2. #19144453C2semidirect
3. #19144442C2semidirect
4. #19144420C2semidirect
5. #19144420C2semidirect
6. #19144413C2semidirect
7. #19144388C2semidirect
8. #19144377C2semidirect
9. C2×#14583196C2semidirect
10. #19143193C2semidirect
11. #19143188C2semidirect
12. #19143163C2semidirect
15. C43×#3504C2semidirect
28. #14583203C22semidirect
29. #14583203C22semidirect
30. #14583203C22semidirect
31. #14583196C22semidirect
32. #14583196C22semidirect
33. #14583196C22semidirect
34. #14583196C22semidirect
35. #14583196C22semidirect
36. #14583196C22semidirect
37. #14583191C22semidirect
38. #14583186C22semidirect
39. #14583186C22semidirect
40. #14583181C22semidirect
41. #14583176C22semidirect
42. #14583176C22semidirect
43. #14582942C22semidirect
44. #14582942C22semidirect
45. #14582939C22semidirect
46. #14582939C22semidirect
47. #14582935C22semidirect
48. #14582935C22semidirect
49. C2×#13063258C22semidirect
50. C2×#13063256C22semidirect
51. #14582926C22semidirect
52. #14582926C22semidirect
53. #14582923C22semidirect
54. C2×C4⋊(C43⋊C8)C22semidirect
67. C43×#544C22semidirect
68. C43×#544C22semidirect
69. C43×#533C22semidirect
70. C2×#125×C43C22semidirect
75. #13063371Q8semidirect
307. #3504D43semidirect
334. #544C2×D43semidirect
335. #544C2×D43semidirect
336. #533C2×D43semidirect
337. C2×#125C2×D43semidirect
385. C43#60496semidirect

Maximal subgroups

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

subgroupindex
C4×#130633712
#191444532
#191444422
#191444202
#191444132
#191443882
#191443772
C2×#145831962
#191431932
#191431882
#191431632
#191431182
C43×#35042
#6049643