Finite Group Databasebeta

← #25151827

Subgroups — #25151827

Order 11008.

1550 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic86C211yesmaximal, characteristic, complemented
2C2×#14583294C211yesmaximal, characteristic, complemented
3C2×#14583118C211yesmaximal, characteristic, complemented
4C2×#14583054C212yesmaximal, complemented
5C2×#14583037C212yesmaximal, complemented
6#19135603C212yesmaximal, complemented
7#19135598C212yesmaximal, complemented
8#19135518C211yesmaximal, characteristic, complemented
9#19135394C211yesmaximal, characteristic, complemented
10#19134428C211yesmaximal, characteristic, complemented
11C43×#2540C211yesmaximal, characteristic
12C23×Dic86C2212yes—
13C22×#10621885C2211yescharacteristic
14C22×#10621885C2212yes—
15C22×#10621883C2211yescharacteristic
16C22×#10621883C2212yes—
17D4×Dic86—24no—
18D4×Dic86—24no—
19C2×#13063325C2211yescharacteristic
20C2×#13063324—21no—
21C2×#13063304C2212yes—
22C2×#13063302—22no—
23C2×C4×Dic86C2211yescharacteristic
24C2×D4×Dic43—21no—
25#14583294—24no—
26#14583118—24no—
27#14583054—28no—
28#14583037—28no—
29C2×C4⋊(C43⋊C8)C2211yescharacteristic
30C2×#13063249C2211yescharacteristic
31C2×#13063248C2212yes—
32C2×#13063248C2212yes—
33C2×#13063248C2212yes—
34C2×#13063246C2211yescharacteristic
35C2×C22⋊(C43⋊C8)C2212yes—
36C2×C22⋊(C43⋊C8)C2212yes—
37C2×#13063236C2211yescharacteristic
38#14582661C2212yes—
39C2×#124×C43C2211yescharacteristic
40C2×C22⋊Q8×C43C2212yes—
41C2×#121×C43C2212yes—
42C2×C4×D4×C43C2211yescharacteristic
43C43×#399—21no—
44C43×#392—21no—
45C43×#390—22no—
46C4×C4⋊C4×C43C2211yescharacteristic
47C22×Dic86—22no—
48C22×Dic86—24no—
49C22×Dic86—24no—
50C22×Dic86—24no—
51C22×Dic86—24no—
52C22×Dic86—42no—
53C22×Dic86—42no—
54C22×Dic86C2312yes—
55C22×Dic86C2311yescharacteristic
56C23×Dic43—22no—
57C2×#10621885—24no—
58C2×#10621885—22no—
59C2×#10621885—24no—
60C2×#10621885—24no—
61C2×#10621885—42no—
62C2×#10621883—22no—
63C2×#10621883—24no—
64C2×#10621883—24no—
65C2×#10621883—24no—
66C2×#10621883—42no—
67C22×C43⋊C8C2311yescharacteristic
68C22×C43⋊C8C2311yescharacteristic
69C22×C43⋊C8C2312yes—
70#13063325—22no—
71#13063325D414yescomplemented
72#13063324—28no—
73#13063304—24no—
74#13063304—24no—
75#13063302—216no—
76C4×Dic86—22no—
77C4×Dic86D414yescomplemented
78D4×Dic43—28no—
79C2×C4⋊Dic43—21no—
80C2×C4⋊Dic43—21no—
81C2×C4⋊Dic43C2311yescharacteristic
82C2×Dic43⋊C4—21no—
83C2×Dic43⋊C4—22no—
84C2×C22⋊Dic43—22no—
85C2×C4×Dic43—21no—
86C4⋊(C43⋊C8)—22no—
87#13063249—22no—
88#13063248—24no—
89#13063248—24no—
90#13063248D418yescomplemented
91#13063246—22no—
92C22⋊(C43⋊C8)—24no—
93C22⋊(C43⋊C8)—24no—
94#13063236—22no—
95C22×Q8×C43C2311yescharacteristic
96C22×D4×C43—22no—
97C22×D4×C43D411yescharacteristic
98C22×D4×C43C2311yescharacteristic
99C23×C4×C43C2312yes—
100#124×C43—22no—
101#124×C43—22no—
102C22⋊Q8×C43—24no—
103C22⋊Q8×C43—42no—
104#121×C43—24no—
105#121×C43—42no—
106C4×D4×C43—22no—
107C4×D4×C43—22no—
108C2×C4⋊C4×C43—22no—
109C2×C4⋊C4×C43D411yescharacteristic
110C2×C4⋊C4×C43C2312yes—
111C2×C22⋊C4×C43—22no—
112C2×C22⋊C4×C43—22no—
113C2×C22⋊C4×C43—21no—
114C2×C22⋊C4×C43—21no—
115C2×C22⋊C4×C43D412yes—
116C2×C42×C43—21no—
117C2×C42×C43C2311yescharacteristic
118#95×C43—21no—
119C24×C43—21no—
120C24×C43C2×D412yes—
121C2×Q8×C43—22no—
122C2×Q8×C43—21no—
123C2×Q8×C43—21no—
124C2×Q8×C43—22no—
125C2×D4×C43—22no—
126C2×D4×C43—24no—
127C2×D4×C43—22no—
128C2×D4×C43—21no—
129C2×D4×C43—21no—
130C2×D4×C43—22no—
131C2×D4×C43—44no—
132C2×D4×C43—44no—
133C2×D4×C43—82no—
134C22×C4×C43—22no—
135C22×C4×C43—22no—
136C22×C4×C43—22no—
137C22×C4×C43—22no—
138C22×C4×C43—22no—
139C22×C4×C43—22no—
140C22×C4×C43—22no—
141C22×C4×C43—21no—
142C22×C4×C43C2×D412yes—
143C22×C4×C43C2×D411yescharacteristic
144C22×C4×C43C2×D411yescharacteristic
145C22×C4×C43C2411yescharacteristic
146C22×Dic43—28no—
147C22×Dic43—21no—
148C22×Dic43—21no—
149C22×Dic43—22no—
150C22×Dic43—44no—
151C22×Dic43—44no—
152C2×Dic86—21no—
153C2×Dic86—22no—
154C2×Dic86—22no—
155C2×Dic86—22no—
156C2×Dic86—22no—
157C2×Dic86—22no—
158C2×Dic86—28no—
159C2×Dic86—22no—
160C2×Dic86—48no—
161C2×Dic86—48no—
162C2×Dic86—48no—
163C2×Dic86—44no—
164C2×Dic86—44no—
165C2×Dic86—48no—
166C2×Dic86C2×D414yes—
167C2×Dic86C2×D414yes—
168C2×Dic86C2×D412yes—
169C4⋊C4×C43—22no—
170C4⋊C4×C43—22no—
171C4⋊C4×C43—44no—
172C4⋊C4×C43C2×D414yes—
173C22⋊C4×C43—24no—
174C22⋊C4×C43—44no—
175C22⋊C4×C43—44no—
176C22⋊C4×C43—42no—
177C22⋊C4×C43—42no—
178C42×C43—81no—
179C42×C43C2×D412yes—
180C42×C43C2×D412yes—
181C22⋊Dic43—28no—
182#10621885—48no—
183#10621885—44no—
184#10621883—48no—
185#10621883—44no—
186C4⋊Dic43—24no—
187C4⋊Dic43—24no—
188C4⋊Dic43—21no—
189C4⋊Dic43C2×D412yes—
190Dic43⋊C4—28no—
191Dic43⋊C4—24no—
192C4×Dic43—24no—
193C2×C43⋊C8—22no—
194C2×C43⋊C8—22no—
195C2×C43⋊C8—22no—
196C2×C43⋊C8—22no—
197C2×C43⋊C8—21no—
198C2×C43⋊C8—21no—
199C2×C43⋊C8C2×D414yes—
200C23×C43—22no—
201C23×C43—22no—
202C23×C43—22no—
203C23×C43—22no—
204C23×C43—22no—
205C23×C43—22no—
206C23×C43—41no—
207C23×C43—42no—
208C23×C43—42no—
209C23×C43—41no—
210C23×C43—41no—
211C23×C43#13714yes—
212C23×C43C22×D411yescharacteristic
213Q8×C43—41no—
214Q8×C43—41no—
215Q8×C43—42no—
216D4×C43—28no—
217D4×C43—41no—
218D4×C43—41no—
219D4×C43—42no—
220D4×C43—88no—
221C2×C4×C43—22no—
222C2×C4×C43—21no—
223C2×C4×C43—21no—
224C2×C4×C43—22no—
225C2×C4×C43—24no—
226C2×C4×C43—22no—
227C2×C4×C43—22no—
228C2×C4×C43—42no—
229C2×C4×C43—42no—
230C2×C4×C43—42no—
231C2×C4×C43—42no—
232C2×C4×C43—42no—
233C2×C4×C43—42no—
234C2×C4×C43—41no—
235C2×C4×C43—42no—
236C2×C4×C43#12012yes—
237C2×C4×C43#12012yes—
238C2×C4×C43C22×D411yescharacteristic
239C2×C4×C43C22×D412yes—
240C2×C4×C43C22×D411yescharacteristic
241C2×C4×C43H(5,𝔽2)12yes—
242C2×Dic43—22no—
243C2×Dic43—24no—
244C2×Dic43—22no—
245C2×Dic43—24no—
246C2×Dic43—416no—
247C2×Dic43—42no—
248C2×Dic43—42no—
249C2×Dic43—41no—
250C2×Dic43—41no—
251Dic86—22no—
252Dic86—24no—
253Dic86—24no—
254Dic86—24no—
255Dic86—416no—
256Dic86—42no—
257Dic86—42no—
258Dic86#12714yes—
259C43⋊C8—24no—
260C43⋊C8—41no—
261C43⋊C8—41no—
262#25766—431nomaximal, Sylow
263C22×C43—24no—
264C22×C43—24no—
265C22×C43—24no—
266C22×C43—42no—
267C22×C43—42no—
268C22×C43—42no—
269C22×C43—42no—
270C22×C43—81no—
271C22×C43—81no—
272C22×C43—82no—
273C22×C43—81no—
274C22×C43—82no—
275C22×C43C2×#12011yescharacteristic
276C22×C43D4212yes—
277C22×C43C2×#13712yes—
278C22×C43#57711yescharacteristic
279C22×C43#57811yescharacteristic
280C4×C43—22no—
281C4×C43—22no—
282C4×C43—42no—
283C4×C43—44no—
284C4×C43—81no—
285C4×C43C2×#12711yescharacteristic
286C4×C43#53511yescharacteristic
287C4×C43D4212yes—
288Dic43—24no—
289Dic43—44no—
290Dic43—42no—
291C2×D4×Q8—431no—
292C2×#497—431no—
293C2×#464—431no—
294C2×#451—432no—
295C2×#448—432no—
296#2540D4311yescharacteristic
297#2001—432no—
298#2000—432no—
299#1981—431no—
300#1955—431no—
301#1763—431no—
302C2×C43—44no—
303C2×C43—161no—
304C2×C43#243311yescharacteristic
305C2×C43#300411yescharacteristic
306C2×C43#301311yescharacteristic
307C2×C43#326612yes—
308C2×C43#327012yes—
309C23×Q8—432no—
310C22×Q16—431no—
311C22×Q16—432no—
312C22×SD16—431no—
313C22×SD16—432no—
314D4×Q8—864no—
315D4×Q8—864no—
316C2×C4⋊Q8—431no—
317C2×C4⋊Q8—861no—
318C2×#124C2×D4311yescharacteristic
319C2×C22⋊Q8—432no—
320C2×C22⋊Q8—862no—
321C2×C22⋊Q8C2×D4312yes—
322C2×#121C2×D4312yes—
323C2×C4×Q8—431no—
324C2×C4×D4—861no—
325C2×C4×D4C2×D4311yescharacteristic
326#497—864no—
327#464—864no—
328#451—868no—
329#448—868no—
330C2×C4⋊C8—431no—
331C2×#103—432no—
332C2×#103—431no—
333C2×#103—432no—
334C2×#103—432no—
335C2×#102—431no—
336C2×C22⋊C8—432no—
337C2×C22⋊C8—432no—
338C2×#97—431no—
339#399—21no—
340#392—21no—
341#390—22no—
342C4×C4⋊C4C2×D4311yescharacteristic
343#340—432no—
344C43#2576611yescharacteristic, complemented, Sylow
345C22×Q8—431no—
346C22×Q8—432no—
347C22×Q8—864no—
348C22×Q8—864no—
349C22×Q8—864no—
350C22×Q8—864no—
351C22×Q8—862no—
352C22×Q8—1722no—
353C22×Q8—1722no—
354C22×Q8C22×D4311yescharacteristic
355C22×D4—22no—
356C22×D4C22⋊D4311yescharacteristic
357C22×D4C22×D4311yescharacteristic
358C23×C4—862no—
359C23×C4C22×D4312yes—
360C2×Q16—864no—
361C2×Q16—864no—
362C2×Q16—862no—
363C2×Q16—864no—
364C2×Q16—1722no—
365C2×SD16—864no—
366C2×SD16—864no—
367C2×SD16—862no—
368C2×SD16—864no—
369C2×SD16—1722no—
370C22×C8—432no—
371C22×C8—431no—
372C22×C8—431no—
373C4⋊Q8—434no—
374C4⋊Q8—868no—
375C4⋊Q8—862no—
376#124—22no—
377#124—22no—
378C22⋊Q8—24no—
379C22⋊Q8—42no—
380C22⋊Q8—8616no—
381C22⋊Q8—864no—
382C22⋊Q8—864no—
383#121—24no—
384#121—42no—
385C4×Q8—434no—
386C4×Q8—862no—
387C4×D4—22no—
388C4×D4—22no—
389C4×D4—868no—
390C2×C4⋊C4—22no—
391C2×C4⋊C4—431no—
392C2×C4⋊C4—861no—
393C2×C4⋊C4—861no—
394C2×C4⋊C4—862no—
395C2×C4⋊C4—861no—
396C2×C4⋊C4C22⋊D4311yescharacteristic
397C2×C4⋊C4C22×D4312yes—
398C2×C22⋊C4—22no—
399C2×C22⋊C4—22no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—862no—
403C2×C22⋊C4C22⋊D4312yes—
404C2×C42—21no—
405C2×C42—861no—
406C2×C42C22×D4311yescharacteristic
407C4⋊C8—862no—
408#103—438no—
409#103—862no—
410#103—864no—
411#103—864no—
412#102—862no—
413C22⋊C8—864no—
414C22⋊C8—864no—
415#97—862no—
416#95—21no—
417C24—21no—
418C24C2×C22⋊D4312yes—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8—21no—
422C2×Q8—22no—
423C2×Q8—434no—
424C2×Q8—432no—
425C2×Q8—434no—
426C2×Q8—868no—
427C2×Q8—861no—
428C2×Q8—862no—
429C2×Q8—862no—
430C2×Q8—862no—
431C2×Q8—862no—
432C2×Q8—862no—
433C2×Q8—862no—
434C2×Q8—1724no—
435C2×Q8—1724no—
436C2×Q8—1728no—
437C2×Q8—1728no—
438C2×Q8—1728no—
439C2×Q8—1728no—
440C2×D4—24no—
441C2×D4—22no—
442C2×D4—21no—
443C2×D4—22no—
444C2×D4—21no—
445C2×D4—22no—
446C2×D4—44no—
447C2×D4—44no—
448C2×D4—82no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—21no—
457C22×C4—861no—
458C22×C4—868no—
459C22×C4—861no—
460C22×C4—862no—
461C22×C4—1724no—
462C22×C4—1724no—
463C22×C4C2×C22⋊D4312yes—
464C22×C4C2×C22⋊D4311yescharacteristic
465C22×C4C2×C22⋊D4311yescharacteristic
466C22×C4C23×D4311yescharacteristic
467Q16—1728no—
468Q16—1724no—
469SD16—1728no—
470SD16—1724no—
471C2×C8—434no—
472C2×C8—862no—
473C2×C8—862no—
474C2×C8—862no—
475C2×C8—862no—
476C2×C8—861no—
477C2×C8—861no—
478C4⋊C4—22no—
479C4⋊C4—22no—
480C4⋊C4—44no—
481C4⋊C4—432no—
482C4⋊C4—868no—
483C4⋊C4—864no—
484C4⋊C4—864no—
485C4⋊C4—864no—
486C4⋊C4—861no—
487C4⋊C4D4×D4314yes—
488C22⋊C4—24no—
489C22⋊C4—44no—
490C22⋊C4—44no—
491C22⋊C4—42no—
492C22⋊C4—42no—
493C22⋊C4—868no—
494C42—81no—
495C42—864no—
496C42D4×D4312yes—
497C42D4×D4312yes—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—41no—
505C23—42no—
506C23—41no—
507C23—41no—
508C23—42no—
509C23#1306335514yes—
510C23C22×C22⋊D4311yescharacteristic
511Q8—41no—
512Q8—41no—
513Q8—42no—
514Q8—434no—
515Q8—862no—
516Q8—864no—
517Q8—864no—
518Q8—864no—
519Q8—17216no—
520Q8—1722no—
521Q8—1722no—
522D4—28no—
523D4—41no—
524D4—41no—
525D4—42no—
526D4—88no—
527C2×C4—22no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—24no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—41no—
541C2×C4—42no—
542C2×C4—862no—
543C2×C4—864no—
544C2×C4—862no—
545C2×C4—864no—
546C2×C4—17216no—
547C2×C4—1721no—
548C2×C4—1721no—
549C2×C4—1722no—
550C2×C4—1722no—
551C2×C4#1306329312yes—
552C2×C4#1306329312yes—
553C2×C4C2×D4×D4311yescharacteristic
554C2×C4C2×D4×D4312yes—
555C2×C4C2×D4×D4311yescharacteristic
556C2×C4#1306337012yes—
557C8—864no—
558C8—1721no—
559C8—1721no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—42no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—82no—
568C22—81no—
569C22—82no—
570C22—81no—
571C22—81no—
572C22C2×#1306329311yescharacteristic
573C22D4×C22⋊D4312yes—
574C22C2×#1306335512yes—
575C22#1458371011yescharacteristic
576C22#1458371411yescharacteristic
577C4—22no—
578C4—22no—
579C4—44no—
580C4—42no—
581C4—81no—
582C4—864no—
583C4—1724no—
584C4—1722no—
585C4#127×D4311yescharacteristic
586C4#1458346211yescharacteristic
587C4#1458353412yes—
588C2—44no—
589C2—161no—
590C2#1913822911yescharacteristic
591C2#1914382811yescharacteristic
592C2#1914390711yescharacteristic
593C2#1914610112yes—
594C2#1914616112yes—
subgroupG/H
centerC23C22×C22⋊D43
commutator, cosocleC22×C4×C43C24
FrattiniC22×C4C23×D43
Fitting, prime radicalC43×#2540C2
socleC23×C43C22×D4
hypercenter#2540D43
hypocenterC43#25766
LCSG ▷ C22×C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2540

Complements

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

NH ≅ G/Ntype
1. C2×D4×Dic86C2semidirect
2. C2×#14583294C2semidirect
3. C2×#14583118C2semidirect
4. C2×#14583054C2semidirect
5. C2×#14583037C2semidirect
6. #19135603C2semidirect
7. #19135598C2semidirect
8. #19135518C2semidirect
9. #19135394C2semidirect
10. #19134428C2semidirect
71. #13063325D4semidirect
77. C4×Dic86D4semidirect
90. #13063248D4semidirect
344. C43#25766semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic862
C2×#145832942
C2×#145831182
C2×#145830542
C2×#145830372
#191356032
#191355982
#191355182
#191353942
#191344282
C43×#25402
#2576643