Finite Group Databasebeta

← #25068366

Subgroups — #25068366

Order 11008.

654 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19145151C211yesmaximal, characteristic
2#19143543C211yesmaximal, characteristic, complemented
3#19143340C211yesmaximal, characteristic
4#19133484C211yesmaximal, characteristic, complemented
5#19133436C211yesmaximal, characteristic, complemented
6#19133435C211yesmaximal, characteristic, complemented
7#19133429C211yesmaximal, characteristic, complemented
8#19133428C211yesmaximal, characteristic, complemented
9C8×#10621873C211yesmaximal, characteristic, complemented
10SD16×C43⋊C8C211yesmaximal, characteristic, complemented
11#334×D43C211yesmaximal, characteristic, complemented
12#19132996C211yesmaximal, characteristic, complemented
13#19132992C211yesmaximal, characteristic
14C8×#10621872C211yesmaximal, characteristic, complemented
15C43×#1590C211yesmaximal, characteristic, complemented
16C2×#13063333—21no—
17#14583453C2211yescharacteristic
18#14583446C2211yescharacteristic
19#14583296C411yescharacteristic
20#14583270C411yescharacteristic
21#14583248C411yescharacteristic
22#14583244C411yescharacteristic
23#14583131C411yescharacteristic
24#14583129C411yescharacteristic
25#14583127C411yescharacteristic
26#14583125C411yescharacteristic
27#14583027—21no—
28#14583026—21no—
29#14583026C2211yescharacteristic
30Q8×C43⋊C8C2211yescharacteristic
31C8×Dic86C2211yescharacteristic
32C4×#10621873C2211yescharacteristic
33C4×#10621884C2211yescharacteristic
34C4×#10621883C2211yescharacteristic
35#14582976C2211yescharacteristic
36#14582968—21no—
37#14582967—21no—
38C8×D172C2211yescharacteristic
39D4×C43⋊C8C2211yescharacteristic
40#14582950C2211yescharacteristic
41#14582924—21no—
42C4⋊C8×D43C2211yescharacteristic
43C4⋊C8×D43C2211yescharacteristic
44#14582864—21no—
45#14582856—21no—
46C4×#10621872C2211yescharacteristic
47C4×C8×D43C2211yescharacteristic
48#14582650C2211yescharacteristic
49#14582649C2211yescharacteristic
50#14582649C2211yescharacteristic
51#14582631C2211yescharacteristic
52Q8⋊(C43⋊C8)C2211yescharacteristic
53#14582629C2211yescharacteristic
54#14582628C2211yescharacteristic
55D4⋊(C43⋊C8)C2211yescharacteristic
56#14582626C2211yescharacteristic
57#14582621C2211yescharacteristic
58#14582620C2211yescharacteristic
59C8×C43⋊C8C2211yescharacteristic
60C43×#446C2211yescharacteristic
61C4×SD16×C43C2211yescharacteristic
62C43×#436C2211yescharacteristic
63C43×#334C2211yescharacteristic
64C43×Q8⋊C8C2211yescharacteristic
65C43×D4⋊C8C2211yescharacteristic
66C82×C43C2211yescharacteristic
67C2×#10621898—21no—
68C2×#10621873C2×C411yescharacteristic
69C2×#10621883C2×C411yescharacteristic
70C2×#10621884C2×C411yescharacteristic
71#13063333—44no—
72C2×#10621872—21no—
73C2×#10621872D411yescharacteristic
74C2×#10621872D411yescharacteristic
75C22×C43⋊C8—21no—
76C2×C8×D43—21no—
77C2×C8×D43—21no—
78C2×C8×D43—21no—
79C2×C8×D43C2×C411yescharacteristic
80C2×C8×D43C2×C411yescharacteristic
81#13063324C2×C411yescharacteristic
82#13063324C2×C411yescharacteristic
83#13063323C2×C411yescharacteristic
84#13063321C2×C411yescharacteristic
85#13063315—21no—
86#13063311—21no—
87#13063305C2×C411yescharacteristic
88#13063303—21no—
89#13063301C2×C411yescharacteristic
90#13063300C2×C411yescharacteristic
91#13063298—21no—
92#13063296C2×C411yescharacteristic
93C4×Dic86—21no—
94C4×Dic86C2311yescharacteristic
95C4×D172C2311yescharacteristic
96C4×C22⋊D43—21no—
97#13063282—21no—
98C42×D43C2311yescharacteristic
99#13063257C2×C411yescharacteristic
100#13063256C2×C411yescharacteristic
101#13063256C2×C411yescharacteristic
102C4⋊(C43⋊C8)—21no—
103C4⋊(C43⋊C8)C2311yescharacteristic
104C4⋊(C43⋊C8)C2311yescharacteristic
105Dic43⋊C8—21no—
106Dic43⋊C8—21no—
107Dic43⋊C8—21no—
108Dic43⋊C8—21no—
109Dic43⋊C8—21no—
110#13063250C2×C411yescharacteristic
111#13063249C2×C411yescharacteristic
112#13063248C2×C411yescharacteristic
113#13063247C2×C411yescharacteristic
114#13063246C2×C411yescharacteristic
115#13063245C2×C411yescharacteristic
116C22⋊(C43⋊C8)—21no—
117(C2×D43)⋊C8—21no—
118(C2×D43)⋊C8—21no—
119(C2×D43)⋊C8—21no—
120#13063235—21no—
121#13063235D411yescharacteristic
122#13063235D411yescharacteristic
123C4×C43⋊C8—21no—
124C4×C43⋊C8C2311yescharacteristic
125C4×C43⋊C8C2311yescharacteristic
126C4×C43⋊C8C2311yescharacteristic
127C8×Dic43—21no—
128C8×Dic43C2×C411yescharacteristic
129C8×Dic43C2×C411yescharacteristic
130C2×SD16×C43C2×C411yescharacteristic
131C2×M16×C43—21no—
132C4×Q8×C43C2311yescharacteristic
133C4×D4×C43C2311yescharacteristic
134#106×C43C2×C411yescharacteristic
135C4⋊C8×C43—21no—
136C4⋊C8×C43C2311yescharacteristic
137C4⋊C8×C43C2311yescharacteristic
138#103×C43C2×C411yescharacteristic
139#102×C43C2×C411yescharacteristic
140C22⋊C8×C43—21no—
141#97×C43—21no—
142C4×C8×C43C2311yescharacteristic
143C4×C8×C43C2311yescharacteristic
144C4×C8×C43C2311yescharacteristic
145C2×Q8×C43C22×C411yescharacteristic
146C2×D4×C43C22×C411yescharacteristic
147C22×C4×C43—21no—
148C2×C22⋊D43—21no—
149C2×C22⋊D43—21no—
150C2×C22⋊D43—21no—
151#10621898—42no—
152#10621898—42no—
153C2×D172—21no—
154C2×D172C22×C411yescharacteristic
155C2×C4×D43—21no—
156C2×C4×D43C22×C411yescharacteristic
157C2×C4×D43C2×D411yescharacteristic
158C2×C4×D43#4111yescharacteristic
159C2×Dic86—21no—
160C2×Dic86—21no—
161C2×Dic86C22×C411yescharacteristic
162SD16×C43—22no—
163M16×C43—42no—
164C2×C8×C43—21no—
165C2×C8×C43—21no—
166C2×C8×C43—21no—
167C2×C8×C43C22×C411yescharacteristic
168C2×C8×C43C22×C411yescharacteristic
169C2×C8×C43C2×D411yescharacteristic
170C2×C8×C43C2×D411yescharacteristic
171C4⋊C4×C43—21no—
172C4⋊C4×C43C22×C411yescharacteristic
173C4⋊C4×C43C22×C411yescharacteristic
174C22⋊C4×C43—21no—
175C42×C43—21no—
176C42×C43C2411yescharacteristic
177C22⋊Dic43—21no—
178#10621884—41no—
179#10621883—41no—
180(C2×D43)⋊C4—21no—
181(C2×D43)⋊C4—21no—
182(C2×D43)⋊C4—21no—
183(C2×D43)⋊C4—21no—
184C4⋊Dic43—21no—
185C4⋊Dic43C22×C411yescharacteristic
186C4⋊Dic43C22×C411yescharacteristic
187Dic43⋊C4—21no—
188Dic43⋊C4—21no—
189Dic43⋊C4—21no—
190Dic43⋊C4—21no—
191Dic43⋊C4—21no—
192Dic43⋊C4—21no—
193C4×Dic43—21no—
194C4×Dic43C22×C411yescharacteristic
195C4×Dic43C2×D411yescharacteristic
196C4×Dic43#4111yescharacteristic
197C2×C43⋊C8—21no—
198C2×C43⋊C8—21no—
199C2×C43⋊C8—21no—
200C2×C43⋊C8—42no—
201C2×C43⋊C8C22×C411yescharacteristic
202C2×C43⋊C8C22×C411yescharacteristic
203C2×C43⋊C8C2×D411yescharacteristic
204C2×C43⋊C8C2×D411yescharacteristic
205#10621873—22no—
206#10621872—22no—
207#10621872—42no—
208#10621872SD1614yescomplemented
209C8×D43—22no—
210C8×D43—41no—
211C8×D43—42no—
212C8×D43—42no—
213C8×D43—42no—
214C23×C43—21no—
215C22×D43—21no—
216C22×D43C4×D411yescharacteristic
217Q8×C43—21no—
218Q8×C43—21no—
219D4×C43—21no—
220D4×C43—21no—
221C2×C4×C43—21no—
222C2×C4×C43—21no—
223C2×C4×C43—21no—
224C2×C4×C43—41no—
225C2×C4×C43—41no—
226C2×C4×C43C23×C411yescharacteristic
227C2×C4×C43C22×D411yescharacteristic
228C2×C4×C43C2×#4111yescharacteristic
229C22⋊D43—42no—
230C22⋊D43—42no—
231C22⋊D43—81no—
232C2×Dic43—21no—
233C2×Dic43—21no—
234C2×Dic43—21no—
235C2×Dic43C4×D411yescharacteristic
236C2×Dic43C4×D411yescharacteristic
237C2×Dic43C4×D411yescharacteristic
238D172—21no—
239D172—21no—
240D172—42no—
241C4×D43—21no—
242C4×D43—21no—
243C4×D43—41no—
244C4×D43—41no—
245C4×D43—41no—
246C4×D43C2×SD1612yes—
247C4×D43#13512yes—
248Dic86—21no—
249Dic86—21no—
250Dic86—42no—
251Dic86—81no—
252C8×C43—21no—
253C8×C43—21no—
254C8×C43—41no—
255C8×C43—41no—
256C8×C43—41no—
257C8×C43#13112yescomplemented
258C8×C43C2×SD1612yes—
259C43⋊C8—21no—
260C43⋊C8—21no—
261C43⋊C8—21no—
262C43⋊C8—41no—
263C43⋊C8—41no—
264C43⋊C8—41no—
265C43⋊C8C2×SD1612yes—
266#18067—431nomaximal, Sylow
267C22×C43—41no—
268C22×C43—41no—
269C22×C43—41no—
270C22×C43C2×C4×D411yescharacteristic
271C2×D43—21no—
272C2×D43—21no—
273C2×D43—41no—
274C2×D43—41no—
275C2×D43—41no—
276C2×D43C4×SD1612yes—
277C4×C43—21no—
278C4×C43—41no—
279C4×C43—41no—
280C4×C43—41no—
281C4×C43C2×#13111yescharacteristic
282C4×C43#56811yescharacteristic
283C4×C43C22×SD1611yescharacteristic
284C4×C43C2×#13511yescharacteristic
285Dic43—21no—
286Dic43—21no—
287Dic43—41no—
288Dic43—41no—
289Dic43—41no—
290Dic43C4×SD1612yes—
291#3167—431no—
292#2962—431no—
293#2935—431no—
294#1590—431no—
295#1590D4311yescharacteristic, complemented
296#1583—431no—
297#1583—431no—
298#1582—431no—
299#1582—431no—
300C8×SD16—431no—
301C8×SD16—431no—
302C2×#334—431no—
303#1481—431no—
304#1480—431no—
305C8×M16—431no—
306C2×C43—81no—
307C2×C43#293511yescharacteristic
308C2×C43C2×C4×SD1611yescharacteristic
309C2×C43#295711yescharacteristic
310D43—22no—
311D43—81no—
312C2×#131—861no—
313#533—431no—
314#532—431no—
315#498—431no—
316#492—431no—
317#487—431no—
318#486—431no—
319#465—431no—
320#465—431no—
321#465—431no—
322#465—431no—
323#446—431no—
324#446—861no—
325#446—861no—
326#446C2×D4311yescharacteristic
327C8×Q8—431no—
328C8×Q8—431no—
329C4×SD16—431no—
330C4×SD16—431no—
331C4×SD16—431no—
332C4×SD16C2×D4311yescharacteristic
333#436—431no—
334#436C2×D4311yescharacteristic
335#435—861no—
336#435—861no—
337C8×D4—431no—
338C8×D4—431no—
339#431—431no—
340#424—861no—
341C2×C4⋊C8—431no—
342C2×C4⋊C8—431no—
343#408—861no—
344#405—861no—
345C4×M16—431no—
346C2×C4×C8—431no—
347#334—431no—
348#334—431no—
349#334—431no—
350#334C2×D4311yescharacteristic
351Q8⋊C8—431no—
352Q8⋊C8—431no—
353Q8⋊C8—431no—
354Q8⋊C8C2×D4311yescharacteristic
355D4⋊C8—431no—
356D4⋊C8—431no—
357D4⋊C8—431no—
358D4⋊C8C2×D4311yescharacteristic
359#322—431no—
360#322—431no—
361C82—431no—
362C82C2×D4311yescharacteristic
363C43#1806711yescharacteristic, complemented, Sylow
364C2×#41—861no—
365C2×SD16—431no—
366C2×SD16—431no—
367C2×SD16—431no—
368C2×SD16C4×D4311yescharacteristic
369#131—1724no—
370C2×M16—21no—
371C2×M16—431no—
372C2×M16—431no—
373C2×M16—861no—
374C22×C8—431no—
375C22×C8—431no—
376C22×C8—861no—
377C22×C8—861no—
378C22×C8—861no—
379C22×C8—861no—
380C4⋊Q8—431no—
381C4⋊Q8—431no—
382C4⋊Q8—431no—
383#127—431no—
384#125—861no—
385#124—861no—
386C22⋊Q8—431no—
387C22⋊Q8—431no—
388C22⋊Q8—861no—
389#121—431no—
390#121—431no—
391#121—861no—
392C4×Q8—431no—
393C4×Q8—861no—
394C4×Q8C22×D4311yescharacteristic
395C4×D4—431no—
396C4×D4—861no—
397C4×D4C22×D4311yescharacteristic
398#117—861no—
399C2×C42—431no—
400#106—431no—
401#106—431no—
402#106—431no—
403#106C4×D4311yescharacteristic
404C4⋊C8—21no—
405C4⋊C8—431no—
406C4⋊C8—431no—
407C4⋊C8—861no—
408C4⋊C8—861no—
409C4⋊C8—861no—
410C4⋊C8—861no—
411C4⋊C8—861no—
412C4⋊C8—861no—
413C4⋊C8C22×D4311yescharacteristic
414C4⋊C8C22×D4311yescharacteristic
415#103—431no—
416#103—431no—
417#103—431no—
418#103C4×D4311yescharacteristic
419#102—431no—
420#102—431no—
421#102—431no—
422#102C4×D4311yescharacteristic
423C22⋊C8—21no—
424C22⋊C8—861no—
425C22⋊C8—861no—
426C22⋊C8—861no—
427C22⋊C8—861no—
428#97—21no—
429#97—431no—
430#97—431no—
431#97—861no—
432C4×C8—431no—
433C4×C8—431no—
434C4×C8—431no—
435C4×C8—431no—
436C4×C8—431no—
437C4×C8—861no—
438C4×C8—861no—
439C4×C8C22×D4311yescharacteristic
440C4×C8C22×D4311yescharacteristic
441C4×C8C22×D4311yescharacteristic
442#41—1722no—
443#41—1722no—
444C2×Q8—431no—
445C2×Q8—861no—
446C2×Q8—861no—
447C2×Q8C2×C4×D4311yescharacteristic
448C2×D4—431no—
449C2×D4—861no—
450C2×D4—861no—
451C2×D4—861no—
452C2×D4—861no—
453C2×D4C2×C4×D4311yescharacteristic
454C22×C4—21no—
455C22×C4—431no—
456C22×C4—431no—
457C22×C4—431no—
458C22×C4—861no—
459SD16—22no—
460SD16—862no—
461SD16—1721no—
462SD16—1721no—
463M16—42no—
464M16—434no—
465M16—862no—
466M16—1722no—
467C2×C8—21no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—431no—
471C2×C8—431no—
472C2×C8—431no—
473C2×C8—431no—
474C2×C8—862no—
475C2×C8—861no—
476C2×C8—861no—
477C2×C8—861no—
478C2×C8—1721no—
479C2×C8—1722no—
480C2×C8—1722no—
481C2×C8—1722no—
482C2×C8—1722no—
483C2×C8C2×C4×D4311yescharacteristic
484C2×C8C2×C4×D4311yescharacteristic
485C2×C8D4×D4311yescharacteristic
486C2×C8D4×D4311yescharacteristic
487C4⋊C4—21no—
488C4⋊C4—431no—
489C4⋊C4—431no—
490C4⋊C4—861no—
491C4⋊C4—861no—
492C4⋊C4—861no—
493C4⋊C4—861no—
494C4⋊C4—861no—
495C4⋊C4—861no—
496C4⋊C4—861no—
497C4⋊C4C2×C4×D4311yescharacteristic
498C4⋊C4C2×C4×D4311yescharacteristic
499C22⋊C4—21no—
500C22⋊C4—861no—
501C22⋊C4—861no—
502C22⋊C4—861no—
503C22⋊C4—861no—
504C22⋊C4—861no—
505C42—21no—
506C42—431no—
507C42—431no—
508C42—431no—
509C42—861no—
510C42C23×D4311yescharacteristic
511C23—21no—
512C23—431no—
513C23—861no—
514Q8—21no—
515Q8—21no—
516Q8—861no—
517Q8—861no—
518Q8—3441no—
519Q8—1722no—
520D4—21no—
521D4—21no—
522D4—861no—
523D4—861no—
524D4—3441no—
525D4—1722no—
526D4—1722no—
527D4—1722no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—431no—
534C2×C4—432no—
535C2×C4—432no—
536C2×C4—431no—
537C2×C4—431no—
538C2×C4—861no—
539C2×C4—861no—
540C2×C4—861no—
541C2×C4—861no—
542C2×C4—861no—
543C2×C4—1721no—
544C2×C4—1721no—
545C2×C4—1721no—
546C2×C4C22×C4×D4311yescharacteristic
547C2×C4C2×D4×D4311yescharacteristic
548C2×C4#41×D4311yescharacteristic
549C8—21no—
550C8—21no—
551C8—41no—
552C8—41no—
553C8—41no—
554C8—432no—
555C8—861no—
556C8—861no—
557C8—861no—
558C8—1721no—
559C8—1721no—
560C8—1721no—
561C8#1306333312yescomplemented
562C8SD16×D4312yes—
563C22—41no—
564C22—41no—
565C22—41no—
566C22—432no—
567C22—861no—
568C22—861no—
569C22—1721no—
570C22—1721no—
571C22—1721no—
572C22C4×D4×D4311yescharacteristic
573C4—21no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—432no—
578C4—861no—
579C4—861no—
580C4—1721no—
581C4—1721no—
582C4—1721no—
583C4C2×#1306333311yescharacteristic
584C4#1458366411yescharacteristic
585C4C2×SD16×D4311yescharacteristic
586C4#135×D4311yescharacteristic
587C2—81no—
588C2—862no—
589C2—3441no—
590C2#1914334011yescharacteristic
591C2C4×SD16×D4311yescharacteristic
592C2#1914350711yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D43
commutatorC2×C4×C43C23×C4
FrattiniC42C23×D43
FittingC43×#1590C2
socleC22×C43C2×C4×D4
prime radical#14583446C22
cosocleC42×C43C24
hypercenter#1590D43
hypocenterC43#18067
LCSG ▷ C2×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1590

Complements

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

NH ≅ G/Ntype
2. #19143543C2semidirect
4. #19133484C2semidirect
5. #19133436C2semidirect
6. #19133435C2semidirect
7. #19133429C2semidirect
8. #19133428C2semidirect
9. C8×#10621873C2semidirect
10. SD16×C43⋊C8C2semidirect
11. #334×D43C2semidirect
12. #19132996C2semidirect
14. C8×#10621872C2semidirect
15. C43×#1590C2semidirect
208. #10621872SD16semidirect
257. C8×C43#131semidirect
295. #1590D43semidirect
363. C43#18067semidirect
561. C8#13063333semidirect

Maximal subgroups

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

subgroupindex
#191451512
#191435432
#191433402
#191334842
#191334362
#191334352
#191334292
#191334282
C8×#106218732
SD16×C43⋊C82
#334×D432
#191329962
#191329922
C8×#106218722
C43×#15902
#1806743