Finite Group Databasebeta

← #25253622

Subgroups — #25253622

Order 11008.

610 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×D344C211yesmaximal, characteristic
2#19136838C211yesmaximal, characteristic, complemented
3#19136786C211yesmaximal, characteristic, complemented
4#19136775C211yesmaximal, characteristic, complemented
5#19136749C211yesmaximal, characteristic, complemented
6#19136710C211yesmaximal, characteristic, complemented
7#19136693C211yesmaximal, characteristic, complemented
8#19136635C211yesmaximal, characteristic, complemented
9#19136523C211yesmaximal, characteristic, complemented
10#19136523C211yesmaximal, characteristic, complemented
11#19136457C211yesmaximal, characteristic, complemented
12#19136421C211yesmaximal, characteristic, complemented
13#19136310C211yesmaximal, characteristic, complemented
14C2×#14582699C211yesmaximal, characteristic, complemented
15C43×#3336C211yesmaximal, characteristic, complemented
16C22×D344C2211yescharacteristic
17C22×D344C2211yescharacteristic
18D4×D172—21no—
19C2×#13063268C2211yescharacteristic
20C2×#13063268C2211yescharacteristic, complemented
21C2×#13063268C2211yescharacteristic, complemented
22C2×#13063265C2211yescharacteristic, complemented
23C22×C43⋊C16C2211yescharacteristic, complemented
24#14583276C2211yescharacteristic
25#14583135C2211yescharacteristic
26#14583135C2211yescharacteristic
27#14583095—21no—
28#14583035—21no—
29#14583035—21no—
30C4×D344C2211yescharacteristic
31#14582726C2211yescharacteristic, complemented
32#14582724C2211yescharacteristic, complemented
33#14582724C2211yescharacteristic, complemented
34C4⋊(C43⋊C16)C2211yescharacteristic, complemented
35#14582703C2211yescharacteristic, complemented
36#14582699C2211yescharacteristic
37#14582699C2211yescharacteristic
38#14582699C2211yescharacteristic
39#14582699C2211yescharacteristic, complemented
40#14582699C2211yescharacteristic, complemented
41#14582699C2211yescharacteristic, complemented
42#14582699C2211yescharacteristic, complemented
43#14582699C2211yescharacteristic, complemented
44#14582699C2211yescharacteristic, complemented
45C22⋊(C43⋊C16)C2211yescharacteristic, complemented
46C22⋊(C43⋊C16)C2211yescharacteristic, complemented
47#14582675C2211yescharacteristic, complemented
48C43×#552—21no—
49C43×#499C2211yescharacteristic, complemented
50C43×#485—21no—
51C43×#483—21no—
52C43×#479—21no—
53C43×#467C2211yescharacteristic, complemented
54C43×#467C2211yescharacteristic, complemented
55C4×Q16×C43C2211yescharacteristic, complemented
56C8×D4×C43C2211yescharacteristic
57C43×#427C2211yescharacteristic, complemented
58C2×#107×C43C2211yescharacteristic, complemented
59C22×D172—21no—
60C22×D172—21no—
61C2×D4×D43—41no—
62C2×D344—22no—
63C2×D344—22no—
64C2×D344—21no—
65C2×D344—21no—
66C2×D344—21no—
67C2×D344—41no—
68C2×D344—41no—
69C2×D344C2311yescharacteristic
70C2×D344C2311yescharacteristic
71C2×D344C2311yescharacteristic
72#13063322—21no—
73#13063300—41no—
74#13063299—21no—
75#13063299—21no—
76#13063295—41no—
77#13063295—41no—
78C4×D172—21no—
79#13063268—22no—
80#13063268—41no—
81#13063268—41no—
82#13063265—41no—
83C2×C43⋊C16—41no—
84C2×C43⋊C16C2311yescharacteristic
85C2×C43⋊C16C2311yescharacteristic
86C2×C43⋊C16C2311yescharacteristic
87C2×C43⋊C16C2311yescharacteristic
88#13063259C2311yescharacteristic
89#13063247—21no—
90#13063247—21no—
91#13063247—21no—
92C2×Q16×C43C2311yescharacteristic
93C22×C8×C43C2311yescharacteristic
94C22×C8×C43C2311yescharacteristic
95#126×C43—41no—
96#125×C43—21no—
97#125×C43—41no—
98#123×C43—41no—
99C22⋊Q8×C43—21no—
100C22⋊Q8×C43—21no—
101C4×Q8×C43—21no—
102C4×D4×C43D411yescharacteristic
103#117×C43—21no—
104#117×C43—41no—
105C2×C4⋊C4×C43—21no—
106#107×C43—21no—
107#107×C43C2311yescharacteristic
108#107×C43C2311yescharacteristic
109#107×C43C2311yescharacteristic
110#106×C43—21no—
111#106×C43—21no—
112C4⋊C8×C43D411yescharacteristic
113#103×C43—21no—
114#103×C43—21no—
115#103×C43—21no—
116C22⋊C8×C43D411yescharacteristic
117C22⋊C8×C43D411yescharacteristic
118C4×C8×C43C2311yescharacteristic
119C23×D43—41no—
120C23×D43—41no—
121C2×Q8×C43—21no—
122C2×D4×C43D811yescharacteristic
123C22×C4×C43—41no—
124C22×C4×C43C2×D411yescharacteristic
125C22×C4×C43C2×D411yescharacteristic
126C2×C22⋊D43—41no—
127C2×C22⋊D43—41no—
128D4×D43—84no—
129C2×D172—21no—
130C2×D172—21no—
131C2×D172—21no—
132C2×D172—41no—
133C2×D172—41no—
134C2×D172—41no—
135C2×D172—41no—
136C2×D172—42no—
137C2×D172—42no—
138C2×D172—81no—
139C2×D172—81no—
140C2×C4×D43—41no—
141Q16×C43—21no—
142Q16×C43—21no—
143C2×C8×C43—21no—
144C2×C8×C43—21no—
145C2×C8×C43—21no—
146C2×C8×C43—21no—
147C2×C8×C43C2×D411yescharacteristic
148C2×C8×C43C2×D411yescharacteristic
149C2×C8×C43C2×D411yescharacteristic
150C2×C8×C43C2411yescharacteristic
151C4⋊C4×C43—21no—
152C4⋊C4×C43—21no—
153C4⋊C4×C43—21no—
154C4⋊C4×C43—41no—
155C4⋊C4×C43—41no—
156C4⋊C4×C43—41no—
157C4⋊C4×C43—41no—
158C4⋊C4×C43—41no—
159C4⋊C4×C43—41no—
160C4⋊C4×C43D811yescharacteristic
161C22⋊C4×C43—41no—
162C22⋊C4×C43—41no—
163C22⋊C4×C43—41no—
164C22⋊C4×C43D811yescharacteristic
165C22⋊C4×C43D811yescharacteristic
166C42×C43—41no—
167C42×C43—41no—
168C42×C43C2×D411yescharacteristic
169(C2×D43)⋊C4—41no—
170(C2×D43)⋊C4—41no—
171(C2×D43)⋊C4—41no—
172C4⋊Dic43—21no—
173D344—21no—
174D344—21no—
175D344—21no—
176D344—21no—
177D344—21no—
178D344—42no—
179D344—42no—
180D344—81no—
181D344#4112yes—
182C43⋊C16—21no—
183C43⋊C16—21no—
184C43⋊C16—21no—
185C43⋊C16—21no—
186C23×C43C2×D811yescharacteristic
187C23×C43C2×D811yescharacteristic
188C22×D43—41no—
189C22×D43—41no—
190C22×D43—41no—
191C22×D43—82no—
192C22×D43—82no—
193C22×D43—82no—
194C22×D43—82no—
195C22×D43—161no—
196C22×D43—161no—
197Q8×C43—41no—
198Q8×C43—41no—
199D4×C43—22no—
200C2×C4×C43—21no—
201C2×C4×C43—21no—
202C2×C4×C43—21no—
203C2×C4×C43—21no—
204C2×C4×C43—41no—
205C2×C4×C43—41no—
206C2×C4×C43—41no—
207C2×C4×C43—41no—
208C2×C4×C43—82no—
209C2×C4×C43C2×D811yescharacteristic
210C2×C4×C43C2×D811yescharacteristic
211C2×C4×C43C2×D811yescharacteristic
212C2×C4×C43C2×D811yescharacteristic
213C2×C4×C43C22×D411yescharacteristic
214C22⋊D43—82no—
215C22⋊D43—82no—
216C2×Dic43—41no—
217D172—22no—
218D172—41no—
219D172—41no—
220D172—41no—
221D172—41no—
222D172—41no—
223D172—82no—
224D172—82no—
225D172—82no—
226D172—161no—
227C4×D43—82no—
228C8×C43—21no—
229C8×C43—21no—
230C8×C43—21no—
231C8×C43C2×#4111yescharacteristic
232C8×C43H(5,𝔽2)11yescharacteristic
233#33877—431nomaximal, Sylow
234C22×C43—21no—
235C22×C43—21no—
236C22×C43—21no—
237C22×C43—21no—
238C22×C43—21no—
239C22×C43#50912yes—
240C22×C43C22×D811yescharacteristic
241C2×D43—42no—
242C2×D43—81no—
243C2×D43—81no—
244C2×D43—81no—
245C2×D43—81no—
246C2×D43—81no—
247C2×D43—81no—
248C2×D43—81no—
249C2×D43—81no—
250C2×D43—164no—
251C2×D43—164no—
252C4×C43—21no—
253C4×C43—21no—
254C4×C43—21no—
255C4×C43—21no—
256C4×C43—81no—
257C4×C43—81no—
258C4×C43—81no—
259C4×C43—81no—
260C4×C43#54611yescharacteristic
261C4×C43#57711yescharacteristic
262Dic43—81no—
263#3336D4311yescharacteristic, complemented
264D4×D8—431no—
265#2244—431no—
266#2234—431no—
267#2232—431no—
268#2228—431no—
269#2220—431no—
270#2217—431no—
271#2207—431no—
272#2187—431no—
273#2187—431no—
274#2175—431no—
275#2168—431no—
276#2140—431no—
277C2×#357—431no—
278C2×C43—22no—
279C2×C43—41no—
280C2×C43#329411yescharacteristic
281C2×C43C2×#50911yescharacteristic
282C2×C43#341611yescharacteristic
283D43—82no—
284D43—161no—
285D43—161no—
286C22×D8—431no—
287C22×D8—431no—
288#552—21no—
289D42—861no—
290C2×SD32—431no—
291C2×SD32—431no—
292C2×SD32—431no—
293C2×M32—431no—
294C22×C16—431no—
295#499C2×D4311yescharacteristic, complemented
296#493—431no—
297#485—21no—
298#483—21no—
299#479—21no—
300#467C2×D4311yescharacteristic, complemented
301#467C2×D4311yescharacteristic, complemented
302#466—431no—
303#466—431no—
304#459—861no—
305#447—861no—
306#447—861no—
307C4×Q16C2×D4311yescharacteristic, complemented
308C4×D8—431no—
309C8×D4C2×D4311yescharacteristic
310#427C2×D4311yescharacteristic, complemented
311C2×#107C2×D4311yescharacteristic, complemented
312#367—431no—
313#366—431no—
314#366—431no—
315C4⋊C16—431no—
316#358—431no—
317#357—431no—
318#357—431no—
319#357—431no—
320#357—431no—
321#357—431no—
322#357—431no—
323#357—431no—
324#357—431no—
325#357—431no—
326C22⋊C16—431no—
327C22⋊C16—431no—
328#346—431no—
329C43#3387711yescharacteristic, complemented, Sylow
330C22×D4—861no—
331C22×D4—861no—
332C22×D4—1721no—
333C2×Q16C22×D4311yescharacteristic
334C2×D8—431no—
335C2×D8—431no—
336C2×D8—431no—
337C2×D8—862no—
338C2×D8—862no—
339C2×D8—861no—
340C2×D8—861no—
341C2×D8—861no—
342C2×D8—1721no—
343C2×D8—1721no—
344C22×C8C22×D4311yescharacteristic
345C22×C8C22×D4311yescharacteristic
346#127—861no—
347#126—41no—
348#125—21no—
349#125—41no—
350#123—41no—
351C22⋊Q8—21no—
352C22⋊Q8—21no—
353#121—861no—
354#121—861no—
355#121—1721no—
356#120—1721no—
357#120—1721no—
358C4×Q8—21no—
359C4×D4—861no—
360C4×D4C22⋊D4311yescharacteristic
361#117—21no—
362#117—41no—
363C2×C4⋊C4—21no—
364SD32—862no—
365SD32—1721no—
366SD32—1721no—
367M32—1721no—
368C2×C16—431no—
369C2×C16—431no—
370C2×C16—431no—
371C2×C16—431no—
372C2×C16—1721no—
373#107—21no—
374#107—431no—
375#107C22×D4311yescharacteristic
376#107C22×D4311yescharacteristic
377#107C22×D4311yescharacteristic
378#106—21no—
379#106—21no—
380C4⋊C8C22⋊D4311yescharacteristic
381#103—21no—
382#103—21no—
383#103—21no—
384#102—861no—
385#102—861no—
386#102—861no—
387C22⋊C8C22⋊D4311yescharacteristic
388C22⋊C8C22⋊D4311yescharacteristic
389C4×C8C22×D4311yescharacteristic
390C24—1721no—
391C24—1721no—
392C2×Q8—21no—
393C2×D4—861no—
394C2×D4—861no—
395C2×D4—861no—
396C2×D4—3441no—
397C2×D4—3441no—
398C2×D4—3444no—
399C2×D4—1722no—
400C2×D4—1722no—
401C2×D4—1721no—
402C2×D4—1721no—
403C2×D4—1721no—
404C2×D4—1721no—
405C2×D4—1721no—
406C2×D4—1721no—
407C2×D4#1062188211yescharacteristic
408C22×C4—41no—
409C22×C4—1721no—
410C22×C4C2×C22⋊D4311yescharacteristic
411C22×C4C2×C22⋊D4311yescharacteristic
412Q16—21no—
413Q16—21no—
414D8—432no—
415D8—861no—
416D8—861no—
417D8—861no—
418D8—861no—
419D8—861no—
420D8—3441no—
421D8—1722no—
422D8—1722no—
423C2×C8—21no—
424C2×C8—21no—
425C2×C8—21no—
426C2×C8—21no—
427C2×C8C2×C22⋊D4311yescharacteristic
428C2×C8C2×C22⋊D4311yescharacteristic
429C2×C8C2×C22⋊D4311yescharacteristic
430C2×C8C23×D4311yescharacteristic
431C4⋊C4—21no—
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—861no—
441C4⋊C4#1062188211yescharacteristic
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—41no—
445C22⋊C4—1721no—
446C22⋊C4—1721no—
447C22⋊C4—1721no—
448C22⋊C4#1062188211yescharacteristic
449C22⋊C4#1062188211yescharacteristic
450C42—41no—
451C42—41no—
452C42C2×C22⋊D4311yescharacteristic
453C16—861no—
454C16—861no—
455C16—861no—
456C16—861no—
457C23—3442no—
458C23—3442no—
459C23—3442no—
460C23—3442no—
461C23—1721no—
462C23—1721no—
463C23—1721no—
464C23—6881no—
465C23—6881no—
466C23C2×#1062188211yescharacteristic
467C23C2×#1062188211yescharacteristic
468Q8—41no—
469Q8—41no—
470D4—22no—
471D4—862no—
472D4—3442no—
473D4—3442no—
474D4—3442no—
475D4—3442no—
476D4—3442no—
477D4—1721no—
478D4—1721no—
479D4—1721no—
480D4—1721no—
481D4—1721no—
482D4—6881no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—82no—
492C2×C4—3442no—
493C2×C4—1721no—
494C2×C4C2×#1062188211yescharacteristic
495C2×C4C2×#1062188211yescharacteristic
496C2×C4C2×#1062188211yescharacteristic
497C2×C4C2×#1062188211yescharacteristic
498C2×C4C22×C22⋊D4311yescharacteristic
499C8—21no—
500C8—21no—
501C8—21no—
502C8#41×D4311yescharacteristic
503C8#1306337111yescharacteristic
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—21no—
509C22—3441no—
510C22—3441no—
511C22—3441no—
512C22—3441no—
513C22—3441no—
514C22—3441no—
515C22—3441no—
516C22—3441no—
517C22—1722no—
518C22—6884no—
519C22—6884no—
520C22#1458333412yes—
521C22C22×#1062188211yescharacteristic
522C4—21no—
523C4—21no—
524C4—21no—
525C4—21no—
526C4—81no—
527C4—81no—
528C4—81no—
529C4—81no—
530C4—3441no—
531C4#1458353711yescharacteristic
532C4#1458370711yescharacteristic
533C2—22no—
534C2—41no—
535C2—3442no—
536C2—6881no—
537C2—6881no—
538C2#1914640211yescharacteristic
539C2C2×#1458333411yescharacteristic
540C2#1914773911yescharacteristic
subgroupG/H
centerC22C22×#10621882
commutator, cosocleC2×C8×C43C24
FrattiniC2×C8C23×D43
FittingC43×#3336C2
socleC22×C43C22×D8
prime radicalD4×D344C2
hypercenter#3336D43
hypocenterC43#33877
LCSG ▷ C2×C8×C43 ▷ C4×C43 ▷ C2×C43 ▷ C43
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3336

Complements

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

NH ≅ G/Ntype
2. #19136838C2semidirect
3. #19136786C2semidirect
4. #19136775C2semidirect
5. #19136749C2semidirect
6. #19136710C2semidirect
7. #19136693C2semidirect
8. #19136635C2semidirect
9. #19136523C2semidirect
10. #19136523C2semidirect
11. #19136457C2semidirect
12. #19136421C2semidirect
13. #19136310C2semidirect
14. C2×#14582699C2semidirect
15. C43×#3336C2semidirect
20. C2×#13063268C22semidirect
21. C2×#13063268C22semidirect
22. C2×#13063265C22semidirect
23. C22×C43⋊C16C22semidirect
31. #14582726C22semidirect
32. #14582724C22semidirect
33. #14582724C22semidirect
34. C4⋊(C43⋊C16)C22semidirect
35. #14582703C22semidirect
39. #14582699C22semidirect
40. #14582699C22semidirect
41. #14582699C22semidirect
42. #14582699C22semidirect
43. #14582699C22semidirect
44. #14582699C22semidirect
45. C22⋊(C43⋊C16)C22semidirect
46. C22⋊(C43⋊C16)C22semidirect
47. #14582675C22semidirect
49. C43×#499C22semidirect
53. C43×#467C22semidirect
54. C43×#467C22semidirect
55. C4×Q16×C43C22semidirect
57. C43×#427C22semidirect
58. C2×#107×C43C22semidirect
263. #3336D43semidirect
295. #499C2×D43semidirect
300. #467C2×D43semidirect
301. #467C2×D43semidirect
307. C4×Q16C2×D43semidirect
310. #427C2×D43semidirect
311. C2×#107C2×D43semidirect
329. C43#33877semidirect

Maximal subgroups

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

subgroupindex
D4×D3442
#191368382
#191367862
#191367752
#191367492
#191367102
#191366932
#191366352
#191365232
#191364572
#191364212
#191363102
C2×#145826992
C43×#33362
#3387743