Finite Group Databasebeta

← #25253071

Subgroups — #25253071

Order 11008.

738 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#14583334C211yesmaximal, characteristic, complemented
2C22×#13063268C211yesmaximal, characteristic, complemented
3D4×D344C211yesmaximal, characteristic, complemented
4#19136862C211yesmaximal, characteristic, complemented
5#19136710C211yesmaximal, characteristic, complemented
6#19136679C211yesmaximal, characteristic, complemented
7#19136647C211yesmaximal, characteristic, complemented
8#19136635C211yesmaximal, characteristic, complemented
9#19136523C211yesmaximal, characteristic, complemented
10#19136518C211yesmaximal, characteristic, complemented
11#19136512C211yesmaximal, characteristic, complemented
12#19136506C211yesmaximal, characteristic, complemented
13#19136457C211yesmaximal, characteristic, complemented
14#19136421C211yesmaximal, characteristic, complemented
15C43×#3285C211yesmaximal, characteristic, complemented
16C22×D344C2211yescharacteristic
17C22×D344C2211yescharacteristic
18D4×D172—21no—
19#14583334—24no—
20C2×#13063268—24no—
21C2×#13063268C2211yescharacteristic
22C2×#13063268C2211yescharacteristic
23C2×#13063268C2211yescharacteristic, complemented
24C2×#13063268C2211yescharacteristic, complemented
25C2×#13063268C2211yescharacteristic, complemented
26C2×#13063268C2211yescharacteristic, complemented
27C2×#13063266C2211yescharacteristic
28C2×#13063266C2211yescharacteristic, complemented
29C2×#13063266C2211yescharacteristic, complemented
30C2×#13063265C2211yescharacteristic, complemented
31C22×C43⋊C16C2211yescharacteristic, complemented
32#14583276C2211yescharacteristic
33#14583135C2211yescharacteristic
34#14583135C2211yescharacteristic
35#14583095—21no—
36#14583035—21no—
37#14583035—21no—
38C4×D344C2211yescharacteristic
39#14582724C2211yescharacteristic, complemented
40C4⋊(C43⋊C16)C2211yescharacteristic, complemented
41#14582703C2211yescharacteristic, complemented
42#14582703C2211yescharacteristic, complemented
43#14582700C2211yescharacteristic, complemented
44#14582699C2211yescharacteristic
45#14582699C2211yescharacteristic, complemented
46#14582699C2211yescharacteristic, complemented
47C22⋊(C43⋊C16)C2211yescharacteristic, complemented
48C22⋊(C43⋊C16)C2211yescharacteristic, complemented
49#14582675C2211yescharacteristic, complemented
50C2×#135×C43C2211yescharacteristic, complemented
51C22×Q16×C43C2211yescharacteristic, complemented
52C43×#548—21no—
53C43×#495C2211yescharacteristic, complemented
54C43×#467C2211yescharacteristic, complemented
55C43×#466C2211yescharacteristic, complemented
56C43×#464—21no—
57C43×#452—21no—
58C43×#448—21no—
59C4×Q16×C43C2211yescharacteristic, complemented
60C8×D4×C43C2211yescharacteristic
61C22×D172—21no—
62C22×D172—21no—
63C2×D4×D43—41no—
64C2×D344—22no—
65C2×D344—22no—
66C2×D344—21no—
67C2×D344—21no—
68C2×D344—21no—
69C2×D344—41no—
70C2×D344—41no—
71C2×D344C2311yescharacteristic
72C2×D344C2311yescharacteristic
73C2×D344C2311yescharacteristic
74#13063322—21no—
75#13063300—41no—
76#13063299—21no—
77#13063299—21no—
78#13063295—41no—
79#13063295—41no—
80C4×D172—21no—
81#13063268—22no—
82#13063268—22no—
83#13063268—22no—
84#13063268—22no—
85#13063268—41no—
86#13063268D414yescomplemented
87#13063266—22no—
88#13063266—22no—
89#13063266—41no—
90#13063265—22no—
91C2×C43⋊C16—22no—
92C2×C43⋊C16C2311yescharacteristic
93C2×C43⋊C16C2311yescharacteristic
94C2×C43⋊C16C2311yescharacteristic
95C2×C43⋊C16C2311yescharacteristic
96#13063259C2311yescharacteristic
97#13063247—21no—
98#13063247—21no—
99#13063247—21no—
100C2×#41×C43—21no—
101C22×Q8×C43—21no—
102#135×C43—22no—
103#135×C43—41no—
104C2×Q16×C43—22no—
105C2×Q16×C43—21no—
106C2×Q16×C43—41no—
107C2×Q16×C43C2311yescharacteristic
108C2×Q16×C43C2311yescharacteristic
109C2×SD16×C43—21no—
110C2×SD16×C43—21no—
111C2×D8×C43C2311yescharacteristic
112C22×C8×C43C2311yescharacteristic
113C22×C8×C43C2311yescharacteristic
114#124×C43—21no—
115#124×C43—41no—
116C22⋊Q8×C43—21no—
117C22⋊Q8×C43—41no—
118#121×C43—21no—
119#121×C43—41no—
120C4×Q8×C43—21no—
121C4×D4×C43—41no—
122C4×D4×C43D411yescharacteristic
123#107×C43C2311yescharacteristic
124C4⋊C8×C43D411yescharacteristic
125#103×C43—21no—
126#103×C43—21no—
127#102×C43—21no—
128C22⋊C8×C43D411yescharacteristic
129C22⋊C8×C43D411yescharacteristic
130C4×C8×C43C2311yescharacteristic
131C23×D43—41no—
132C23×D43—41no—
133#41×C43—42no—
134#41×C43—81no—
135C2×Q8×C43—21no—
136C2×Q8×C43—21no—
137C2×Q8×C43—42no—
138C2×Q8×C43—41no—
139C2×Q8×C43—81no—
140C2×D4×C43—21no—
141C2×D4×C43—41no—
142C2×D4×C43—41no—
143C2×D4×C43D811yescharacteristic
144C22×C4×C43—41no—
145C22×C4×C43—41no—
146C22×C4×C43C2×D411yescharacteristic
147C22×C4×C43C2×D411yescharacteristic
148C2×C22⋊D43—41no—
149C2×C22⋊D43—41no—
150D4×D43—84no—
151C2×D172—21no—
152C2×D172—21no—
153C2×D172—21no—
154C2×D172—41no—
155C2×D172—41no—
156C2×D172—41no—
157C2×D172—41no—
158C2×D172—42no—
159C2×D172—42no—
160C2×D172—81no—
161C2×D172—81no—
162C2×C4×D43—41no—
163Q16×C43—21no—
164Q16×C43—21no—
165Q16×C43—21no—
166Q16×C43—42no—
167Q16×C43C2×D412yescomplemented
168SD16×C43—42no—
169SD16×C43—81no—
170D8×C43—21no—
171D8×C43—21no—
172C2×C8×C43—21no—
173C2×C8×C43—21no—
174C2×C8×C43—21no—
175C2×C8×C43—21no—
176C2×C8×C43C2×D411yescharacteristic
177C2×C8×C43C2×D411yescharacteristic
178C2×C8×C43C2×D411yescharacteristic
179C2×C8×C43C2411yescharacteristic
180C4⋊C4×C43—21no—
181C4⋊C4×C43—41no—
182C4⋊C4×C43—41no—
183C4⋊C4×C43D811yescharacteristic
184C22⋊C4×C43—41no—
185C22⋊C4×C43—41no—
186C22⋊C4×C43—41no—
187C22⋊C4×C43—41no—
188C22⋊C4×C43D811yescharacteristic
189C22⋊C4×C43D811yescharacteristic
190C42×C43—41no—
191C42×C43C2×D411yescharacteristic
192(C2×D43)⋊C4—41no—
193(C2×D43)⋊C4—41no—
194(C2×D43)⋊C4—41no—
195C4⋊Dic43—21no—
196D344—21no—
197D344—21no—
198D344—21no—
199D344—21no—
200D344—21no—
201D344—42no—
202D344—42no—
203D344—81no—
204D344C2×D412yes—
205C43⋊C16—21no—
206C43⋊C16—21no—
207C43⋊C16—21no—
208C43⋊C16C2×D412yescomplemented
209C23×C43—41no—
210C23×C43C2×D811yescharacteristic
211C23×C43C2×D811yescharacteristic
212C22×D43—41no—
213C22×D43—41no—
214C22×D43—41no—
215C22×D43—82no—
216C22×D43—82no—
217C22×D43—82no—
218C22×D43—82no—
219C22×D43—161no—
220C22×D43—161no—
221Q8×C43—22no—
222Q8×C43—41no—
223Q8×C43—41no—
224Q8×C43—41no—
225Q8×C43—82no—
226D4×C43—22no—
227D4×C43—41no—
228D4×C43—41no—
229D4×C43—82no—
230D4×C43—82no—
231C2×C4×C43—21no—
232C2×C4×C43—21no—
233C2×C4×C43—21no—
234C2×C4×C43—21no—
235C2×C4×C43—41no—
236C2×C4×C43—41no—
237C2×C4×C43—41no—
238C2×C4×C43—82no—
239C2×C4×C43—82no—
240C2×C4×C43C2×D811yescharacteristic
241C2×C4×C43C2×D811yescharacteristic
242C2×C4×C43C2×D811yescharacteristic
243C2×C4×C43C2×D811yescharacteristic
244C2×C4×C43C22×D411yescharacteristic
245C22⋊D43—82no—
246C22⋊D43—82no—
247C2×Dic43—41no—
248D172—22no—
249D172—41no—
250D172—41no—
251D172—41no—
252D172—41no—
253D172—41no—
254D172—82no—
255D172—82no—
256D172—82no—
257D172—161no—
258C4×D43—82no—
259C8×C43—21no—
260C8×C43—21no—
261C8×C43—21no—
262C8×C43C22×D411yescharacteristic
263C8×C43H(5,𝔽2)11yescharacteristic
264#33831—431nomaximal, Sylow
265C22×C43—21no—
266C22×C43—21no—
267C22×C43—21no—
268C22×C43—21no—
269C22×C43—21no—
270C22×C43—81no—
271C22×C43—81no—
272C22×C43—81no—
273C22×C43#50912yescomplemented
274C22×C43C22×D811yescharacteristic
275C2×D43—42no—
276C2×D43—81no—
277C2×D43—81no—
278C2×D43—81no—
279C2×D43—81no—
280C2×D43—81no—
281C2×D43—81no—
282C2×D43—81no—
283C2×D43—81no—
284C2×D43—164no—
285C2×D43—164no—
286C4×C43—21no—
287C4×C43—21no—
288C4×C43—21no—
289C4×C43—21no—
290C4×C43—42no—
291C4×C43—81no—
292C4×C43—81no—
293C4×C43D4211yescharacteristic
294C4×C43#57611yescharacteristic
295Dic43—81no—
296C2×#509—431no—
297C22×SD32—431no—
298#3285D4311yescharacteristic, complemented
299D4×D8—431no—
300#2250—431no—
301#2220—431no—
302#2215—431no—
303#2210—431no—
304#2207—431no—
305#2187—431no—
306#2186—431no—
307#2185—431no—
308#2184—431no—
309#2175—431no—
310#2168—431no—
311C2×C43—22no—
312C2×C43—41no—
313C2×C43—161no—
314C2×C43D4×D811yescharacteristic
315C2×C43C2×#50911yescharacteristic
316C2×C43#341611yescharacteristic
317D43—82no—
318D43—161no—
319D43—161no—
320C2×#135C2×D4311yescharacteristic, complemented
321C22×Q16C2×D4311yescharacteristic, complemented
322C22×D8—431no—
323C22×D8—431no—
324#548—21no—
325D42—861no—
326#509—864no—
327C2×SD32—431no—
328C2×SD32—431no—
329C2×SD32—431no—
330C2×SD32—431no—
331C2×SD32—431no—
332C2×SD32—431no—
333C2×SD32—864no—
334C2×D16—431no—
335C2×D16—431no—
336C2×D16—431no—
337C2×M32—431no—
338C22×C16—431no—
339#495C2×D4311yescharacteristic, complemented
340#493—431no—
341#467C2×D4311yescharacteristic, complemented
342#466—431no—
343#466—431no—
344#466C2×D4311yescharacteristic, complemented
345#464—21no—
346#459—861no—
347#452—21no—
348#448—21no—
349#447—861no—
350#447—861no—
351C4×Q16C2×D4311yescharacteristic, complemented
352C4×D8—431no—
353C8×D4C2×D4311yescharacteristic
354#366—431no—
355C4⋊C16—431no—
356#358—431no—
357#358—431no—
358#357—431no—
359#357—431no—
360#357—431no—
361#357—431no—
362C22⋊C16—431no—
363C22⋊C16—431no—
364#346—431no—
365C43#3383111yescharacteristic, complemented, Sylow
366C2×#41—21no—
367C22×Q8—21no—
368C22×D4—861no—
369C22×D4—861no—
370C22×D4—1721no—
371#135—22no—
372#135—41no—
373C2×Q16—22no—
374C2×Q16—21no—
375C2×Q16—41no—
376C2×Q16C22×D4311yescharacteristic
377C2×Q16C22×D4311yescharacteristic
378C2×SD16—21no—
379C2×SD16—21no—
380C2×D8—431no—
381C2×D8—431no—
382C2×D8—431no—
383C2×D8—862no—
384C2×D8—862no—
385C2×D8—861no—
386C2×D8—861no—
387C2×D8—861no—
388C2×D8—1721no—
389C2×D8—1721no—
390C2×D8C22×D4311yescharacteristic
391C22×C8C22×D4311yescharacteristic
392C22×C8C22×D4311yescharacteristic
393#127—861no—
394#124—21no—
395#124—41no—
396C22⋊Q8—21no—
397C22⋊Q8—41no—
398#121—21no—
399#121—41no—
400#121—861no—
401#121—861no—
402#121—1721no—
403#120—1721no—
404#120—1721no—
405C4×Q8—21no—
406C4×D4—41no—
407C4×D4—861no—
408C4×D4C22⋊D4311yescharacteristic
409SD32—434no—
410SD32—862no—
411SD32—862no—
412SD32—862no—
413SD32—862no—
414SD32—1721no—
415D16—862no—
416D16—862no—
417D16—1721no—
418M32—862no—
419C2×C16—431no—
420C2×C16—431no—
421C2×C16—431no—
422C2×C16—431no—
423C2×C16—862no—
424#107—431no—
425#107C22×D4311yescharacteristic
426C4⋊C8C22⋊D4311yescharacteristic
427#103—21no—
428#103—21no—
429#102—21no—
430#102—861no—
431#102—861no—
432#102—861no—
433C22⋊C8C22⋊D4311yescharacteristic
434C22⋊C8C22⋊D4311yescharacteristic
435C4×C8C22×D4311yescharacteristic
436C24—1721no—
437C24—1721no—
438#41—42no—
439#41—81no—
440C2×Q8—21no—
441C2×Q8—21no—
442C2×Q8—42no—
443C2×Q8—41no—
444C2×Q8—81no—
445C2×D4—21no—
446C2×D4—41no—
447C2×D4—41no—
448C2×D4—861no—
449C2×D4—861no—
450C2×D4—861no—
451C2×D4—3441no—
452C2×D4—3441no—
453C2×D4—3444no—
454C2×D4—1722no—
455C2×D4—1722no—
456C2×D4—1721no—
457C2×D4—1721no—
458C2×D4—1721no—
459C2×D4—1721no—
460C2×D4—1721no—
461C2×D4—1721no—
462C2×D4#1062188211yescharacteristic
463C22×C4—41no—
464C22×C4—41no—
465C22×C4—1721no—
466C22×C4C2×C22⋊D4311yescharacteristic
467C22×C4C2×C22⋊D4311yescharacteristic
468Q16—21no—
469Q16—21no—
470Q16—21no—
471Q16—42no—
472Q16D4×D4312yescomplemented
473SD16—42no—
474SD16—81no—
475D8—21no—
476D8—21no—
477D8—432no—
478D8—861no—
479D8—861no—
480D8—861no—
481D8—861no—
482D8—861no—
483D8—3441no—
484D8—1722no—
485D8—1722no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8C2×C22⋊D4311yescharacteristic
491C2×C8C2×C22⋊D4311yescharacteristic
492C2×C8C2×C22⋊D4311yescharacteristic
493C2×C8C23×D4311yescharacteristic
494C4⋊C4—21no—
495C4⋊C4—41no—
496C4⋊C4—41no—
497C4⋊C4—861no—
498C4⋊C4#1062188211yescharacteristic
499C22⋊C4—41no—
500C22⋊C4—41no—
501C22⋊C4—41no—
502C22⋊C4—41no—
503C22⋊C4—1721no—
504C22⋊C4—1721no—
505C22⋊C4—1721no—
506C22⋊C4#1062188211yescharacteristic
507C22⋊C4#1062188211yescharacteristic
508C42—41no—
509C42C2×C22⋊D4311yescharacteristic
510C16—432no—
511C16—861no—
512C16—861no—
513C16—861no—
514C23—41no—
515C23—3442no—
516C23—3442no—
517C23—3442no—
518C23—3442no—
519C23—1721no—
520C23—1721no—
521C23—1721no—
522C23—6881no—
523C23—6881no—
524C23C2×#1062188211yescharacteristic
525C23C2×#1062188211yescharacteristic
526Q8—22no—
527Q8—41no—
528Q8—41no—
529Q8—41no—
530Q8—82no—
531D4—22no—
532D4—41no—
533D4—41no—
534D4—82no—
535D4—82no—
536D4—862no—
537D4—3442no—
538D4—3442no—
539D4—3442no—
540D4—3442no—
541D4—3442no—
542D4—1721no—
543D4—1721no—
544D4—1721no—
545D4—1721no—
546D4—1721no—
547D4—6881no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—82no—
556C2×C4—82no—
557C2×C4—3442no—
558C2×C4—1721no—
559C2×C4C2×#1062188211yescharacteristic
560C2×C4C2×#1062188211yescharacteristic
561C2×C4C2×#1062188211yescharacteristic
562C2×C4C2×#1062188211yescharacteristic
563C2×C4C22×C22⋊D4311yescharacteristic
564C8—21no—
565C8—21no—
566C8—21no—
567C8C2×D4×D4311yescharacteristic
568C8#1306337011yescharacteristic
569C22—21no—
570C22—21no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22—3441no—
578C22—3441no—
579C22—3441no—
580C22—3441no—
581C22—3441no—
582C22—3441no—
583C22—3441no—
584C22—3441no—
585C22—1722no—
586C22—6884no—
587C22—6884no—
588C22#1458333412yescomplemented
589C22C22×#1062188211yescharacteristic
590C4—21no—
591C4—21no—
592C4—21no—
593C4—21no—
594C4—42no—
595C4—81no—
596C4—81no—
597C4—3441no—
598C4D4×C22⋊D4311yescharacteristic
599C4#1458370211yescharacteristic
600C2—22no—
601C2—41no—
602C2—161no—
603C2—3442no—
604C2—6881no—
605C2—6881no—
606C2D4×#1062188211yescharacteristic
607C2C2×#1458333411yescharacteristic
608C2#1914773911yescharacteristic
subgroupG/H
centerC22C22×#10621882
commutator, cosocleC2×C8×C43C24
FrattiniC2×C8C23×D43
FittingC43×#3285C2
socleC22×C43C22×D8
hypercenter#3285D43
hypocenterC43#33831
LCSG ▷ C2×C8×C43 ▷ C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3285

Complements

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

NH ≅ G/Ntype
1. C2×#14583334C2semidirect
2. C22×#13063268C2semidirect
3. D4×D344C2semidirect
4. #19136862C2semidirect
5. #19136710C2semidirect
6. #19136679C2semidirect
7. #19136647C2semidirect
8. #19136635C2semidirect
9. #19136523C2semidirect
10. #19136518C2semidirect
11. #19136512C2semidirect
12. #19136506C2semidirect
13. #19136457C2semidirect
14. #19136421C2semidirect
15. C43×#3285C2semidirect
23. C2×#13063268C22semidirect
24. C2×#13063268C22semidirect
25. C2×#13063268C22semidirect
26. C2×#13063268C22semidirect
28. C2×#13063266C22semidirect
29. C2×#13063266C22semidirect
30. C2×#13063265C22semidirect
31. C22×C43⋊C16C22semidirect
39. #14582724C22semidirect
40. C4⋊(C43⋊C16)C22semidirect
41. #14582703C22semidirect
42. #14582703C22semidirect
43. #14582700C22semidirect
45. #14582699C22semidirect
46. #14582699C22semidirect
47. C22⋊(C43⋊C16)C22semidirect
48. C22⋊(C43⋊C16)C22semidirect
49. #14582675C22semidirect
50. C2×#135×C43C22semidirect
51. C22×Q16×C43C22semidirect
53. C43×#495C22semidirect
54. C43×#467C22semidirect
55. C43×#466C22semidirect
59. C4×Q16×C43C22semidirect
86. #13063268D4semidirect
167. Q16×C43C2×D4semidirect
208. C43⋊C16C2×D4semidirect
273. C22×C43#509semidirect
298. #3285D43semidirect
320. C2×#135C2×D43semidirect
321. C22×Q16C2×D43semidirect
339. #495C2×D43semidirect
341. #467C2×D43semidirect
344. #466C2×D43semidirect
351. C4×Q16C2×D43semidirect
365. C43#33831semidirect
472. Q16D4×D43semidirect
588. C22#14583334semidirect

Maximal subgroups

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

subgroupindex
C2×#145833342
C22×#130632682
D4×D3442
#191368622
#191367102
#191366792
#191366472
#191366352
#191365232
#191365182
#191365122
#191365062
#191364572
#191364212
C43×#32852
#3383143