Finite Group Databasebeta

← #34030302

Subgroups — #34030302

Order 18688.

646 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626710C211yesmaximal, characteristic, complemented
2#22626694C211yesmaximal, characteristic, complemented
3#22626343C212yesmaximal, complemented
4#22626340C212yesmaximal, complemented
5#22626338C212yesmaximal, complemented
6#22626337C212yesmaximal, complemented
7C2×#17989511C211yesmaximal, characteristic, complemented
8#22626259C211yesmaximal, characteristic, complemented
9D73⋊#360C211yesmaximal, characteristic, complemented
10#22626150C211yesmaximal, characteristic, complemented
11#22623064C211yesmaximal, characteristic
12#17989666—21no—
13#17989656—21no—
14#17989643C2211yescharacteristic
15#17989634C2212yes—
16#17989633C2212yes—
17#17989630—22no—
18#17989628C2212yescomplemented
19#17989627C2212yescomplemented
20SD16×#64408—22no—
21C2×#14556661C2211yescharacteristic
22C2×#14556659C2211yescharacteristic
23#17989584—21no—
24#17989578—21no—
25#17989550C2211yescharacteristic
26#17989511C2212yescomplemented
27#17989511C2212yescomplemented
28#17989506C2212yescomplemented
29#17989505C2212yescomplemented
30#17989504C2212yes—
31#17989503C2212yes—
32#17989478C2212yescomplemented
33#17989477C2212yescomplemented
34#135×D73C2211yescharacteristic
35#135×D73C2211yescharacteristic
36#17989029C411yescharacteristic, complemented
37#17989029C2211yescharacteristic
38#17989026C411yescharacteristic, complemented
39#17989026C2211yescharacteristic
40#17989024C411yescharacteristic, complemented
41#17989024C2211yescharacteristic
42#17989021C411yescharacteristic, complemented
43#17989021C2211yescharacteristic
44#17989019C411yescharacteristic, complemented
45#17989017C411yescharacteristic, complemented
46#17989017C411yescharacteristic, complemented
47C2×#14556530C2211yescharacteristic
48#508×C73C411yescharacteristic, complemented
49#14556679—22no—
50#14556678—22no—
51#14556678—42no—
52C2×#11924038—21no—
53C2×#11924038—41no—
54D73⋊M16—41no—
55D73⋊M16D411yescharacteristic
56Q8×#64408—22no—
57D4×#64408—22no—
58D4×#64408—42no—
59#14556670—41no—
60#14556670D411yescharacteristic
61C2×C4⋊#64408—21no—
62C2×C4⋊#64408—41no—
63#14556666C2312yes—
64#14556665—21no—
65#14556664—21no—
66#14556661—21no—
67#14556661C2312yes—
68#14556659—21no—
69#14556659C2312yes—
70#14556656—22no—
71#14556655—22no—
72#14556654—22no—
73#14556653—22no—
74#14556645C2312yes—
75C8×#64408D412yes—
76#41×D73—21no—
77#41×D73—21no—
78#14556614C2×C411yescharacteristic
79#14556614C2311yescharacteristic
80#14556613C2×C411yescharacteristic
81#14556613C2×C411yescharacteristic, complemented
82#14556612—21no—
83#14556612—21no—
84#14556611—21no—
85#14556611—21no—
86#14556610C2×C411yescharacteristic
87#14556610C2311yescharacteristic
88Q16×D73C2×C411yescharacteristic, complemented
89Q16×D73C2311yescharacteristic
90SD16×D73—21no—
91SD16×D73—21no—
92D8×D73C2×C411yescharacteristic, complemented
93D8×D73C2311yescharacteristic
94C2×C8×D73C2311yescharacteristic
95Dic584C2×C411yescharacteristic, complemented
96#14556537C2×C411yescharacteristic, complemented
97#14556537C2×C411yescharacteristic, complemented
98#14556536C2×C411yescharacteristic, complemented
99#14556536C2×C411yescharacteristic, complemented
100#14556535C2×C411yescharacteristic, complemented
101#14556535C2×C411yescharacteristic, complemented
102#14556534C2×C411yescharacteristic, complemented
103#14556534C2×C411yescharacteristic, complemented
104D1168C2×C411yescharacteristic, complemented
105#14556532C2×C411yescharacteristic, complemented
106#14556532C2×C411yescharacteristic, complemented
107#14556530C2×C411yescharacteristic, complemented
108#14556530C2×C411yescharacteristic, complemented
109#14556530C2311yescharacteristic
110#14556530C2311yescharacteristic
111C2×#11924003C2×C411yescharacteristic, complemented
112#135×C73C2×C411yescharacteristic
113#135×C73C2×C411yescharacteristic, complemented
114Q32×C73C2×C411yescharacteristic, complemented
115SD32×C73C2×C411yescharacteristic, complemented
116SD32×C73C2×C411yescharacteristic, complemented
117D16×C73C2×C411yescharacteristic, complemented
118C2×C16×C73C2×C411yescharacteristic, complemented
119C22×#64408—42no—
120C22×#64408—41no—
121#11924038—21no—
122#11924038—22no—
123#11924038—42no—
124#11924038—41no—
125#11924037—21no—
126#11924037—42no—
127#11924036—42no—
128#11924036C2×D412yes—
129C2×#10596833—42no—
130C2×#10596833—41no—
131C4⋊#64408—21no—
132C4⋊#64408—22no—
133C4⋊#64408—42no—
134C4⋊#64408—41no—
135C22⋊#64408—21no—
136C22⋊#64408—42no—
137C4×#64408—42no—
138C4×#64408C2×D412yes—
139#11924031—22no—
140#11924029—21no—
141#11924029—21no—
142#11924028—21no—
143#11924028—21no—
144#11924028—41no—
145#11924028—41no—
146#11924027—21no—
147#11924027—21no—
148#11924027—41no—
149#11924027—41no—
150Q8×D73—21no—
151Q8×D73—21no—
152D4×D73—21no—
153D4×D73—21no—
154D4×D73—41no—
155D4×D73—41no—
156C2×C4×D73—41no—
157C2×C4×D73—41no—
158C2×C4×D73C2×D411yescharacteristic
159Dic292C22×C411yescharacteristic
160Dic292C22×C411yescharacteristic
161#11924018—21no—
162#11924018—21no—
163#11924017—21no—
164#11924017—21no—
165#11924016—21no—
166#11924016—21no—
167#11924015—21no—
168#11924015—21no—
169D584C22×C411yescharacteristic
170D584C22×C411yescharacteristic
171#11924013—21no—
172#11924013—21no—
173C2×#10596831#4111yescharacteristic
174C8×D73C22×C411yescharacteristic
175C8×D73C2×D411yescharacteristic
176C8×D73#4111yescharacteristic
177C8×D73C2411yescharacteristic
178#11924003C22×C411yescharacteristic
179#11924003C22×C411yescharacteristic
180#41×C73—21no—
181#41×C73—21no—
182Q16×C73C22×C411yescharacteristic
183Q16×C73C22×C411yescharacteristic
184SD16×C73—21no—
185SD16×C73—21no—
186D8×C73C22×C411yescharacteristic
187D8×C73C22×C411yescharacteristic
188C2×C8×C73C22×C411yescharacteristic
189C16×C73C22×C411yescharacteristic
190C16×C73C22×C411yescharacteristic
191C22×D73—41no—
192C22×D73—41no—
193C22×D73AGL(1,ℤ/8)11yescharacteristic
194C2×#64408—22no—
195C2×#64408—42no—
196C2×#64408—162no—
197C2×#64408—161no—
198Q8×C73—21no—
199Q8×C73—21no—
200D4×C73—21no—
201D4×C73—21no—
202D4×C73—41no—
203D4×C73—41no—
204C2×C4×C73—41no—
205C2×C4×C73—41no—
206C2×C4×C73C4×D411yescharacteristic
207C22⋊D73—41no—
208C22⋊D73—41no—
209C22⋊D73—41no—
210C22⋊D73—41no—
211C22⋊D73—41no—
212C22⋊D73—41no—
213C2×Dic73—41no—
214C2×Dic73—41no—
215C2×Dic73AGL(1,ℤ/8)11yescharacteristic
216D292—21no—
217D292—21no—
218D292—41no—
219D292—41no—
220C4×D73—41no—
221C4×D73—41no—
222C4×D73—41no—
223C4×D73—41no—
224C4×D73—41no—
225C4×D73—41no—
226C4×D73C4×D411yescharacteristic
227C4×D73AGL(1,ℤ/8)11yescharacteristic
228C4×D73AGL(1,ℤ/8)11yescharacteristic
229C4×D73C22×D411yescharacteristic
230Dic146—21no—
231Dic146—21no—
232Dic146—41no—
233Dic146—41no—
234#10596833—22no—
235#10596833—42no—
236C8×C73C4×D411yescharacteristic
237C8×C73C23×C411yescharacteristic
238#10596831C4×D411yescharacteristic
239#10596831C2×#4111yescharacteristic
240C22×C73—41no—
241C22×C73—41no—
242C22×C73#44211yescharacteristic
243C2×D73—41no—
244C2×D73—41no—
245C2×D73—41no—
246C2×D73—81no—
247C2×D73—81no—
248C2×D73—161no—
249C2×D73#44211yescharacteristic
250C2×D73C2×AGL(1,ℤ/8)11yescharacteristic
251#64408—42no—
252#64408—82no—
253C4×C73—41no—
254C4×C73—41no—
255C4×C73#44211yescharacteristic
256C4×C73C2×C4×D411yescharacteristic
257Dic73—41no—
258Dic73—41no—
259Dic73#44211yescharacteristic
260Dic73C2×AGL(1,ℤ/8)11yescharacteristic
261#33469—731nomaximal, Sylow
262C2×C73—21no—
263C2×C73—81no—
264C2×C73—81no—
265C2×C73C2×#44211yescharacteristic
266D73—21no—
267D73—21no—
268D73—81no—
269D73—81no—
270#3414—731no—
271#2957—731no—
272#2952—731no—
273AGL(1,ℤ/16)—732no—
274AGL(1,ℤ/16)—732no—
275#2180—732no—
276#2180—732no—
277C2×#365—731no—
278#2148—731no—
279C2×#360—731no—
280#2110—731no—
281C73#3346911yescharacteristic, complemented, Sylow
282C2×#135—731no—
283C2×#135—731no—
284#568—1461no—
285#520—1461no—
286#510—731no—
287#510—731no—
288#510—731no—
289#510—731no—
290#509—731no—
291#509—731no—
292#509—731no—
293#509—731no—
294#508—731no—
295#508—731no—
296#508—731no—
297#508#6440811yescharacteristic, complemented
298C2×M32—731no—
299C2×M32—731no—
300#444—732no—
301#444—732no—
302#443—1462no—
303#442—732no—
304#441—732no—
305C4×SD16—1462no—
306C2×#108—731no—
307C2×#106—731no—
308#421—1461no—
309#417—1461no—
310#405—731no—
311#365—732no—
312#365—732no—
313#362—732no—
314#361—732no—
315#360—732no—
316#360—732no—
317#347—732no—
318#347—732no—
319C2×#41—1461no—
320C2×#41—1461no—
321#135—731no—
322#135—731no—
323#135—731no—
324#135—731no—
325#135—731no—
326#135—731no—
327#135—1461no—
328#135—1461no—
329#135—1461no—
330#135—1461no—
331#135C2×#6440811yescharacteristic
332#135C2×#6440811yescharacteristic, complemented
333C2×Q16—731no—
334C2×Q16—731no—
335C2×SD16—1461no—
336C2×SD16—1461no—
337C2×D8—731no—
338C2×D8—731no—
339#131—2922no—
340#131—1462no—
341#131—1462no—
342C2×M16—2921no—
343C2×M16—2921no—
344C2×M16—731no—
345C2×M16—1461no—
346C22×C8—731no—
347C4×Q8—1462no—
348C4×D4—2922no—
349C4×D4—1462no—
350#117—2921no—
351#117—731no—
352C2×C4⋊C4—2921no—
353C2×C4⋊C4—1461no—
354Q32—731no—
355Q32—731no—
356Q32—731no—
357Q32C2×#6440811yescharacteristic, complemented
358SD32—731no—
359SD32—731no—
360SD32—731no—
361SD32—731no—
362SD32—731no—
363SD32—731no—
364SD32C2×#6440811yescharacteristic, complemented
365SD32C2×#6440811yescharacteristic, complemented
366D16—731no—
367D16—731no—
368D16—731no—
369D16C2×#6440811yescharacteristic, complemented
370M32—731no—
371M32—731no—
372M32—731no—
373M32—731no—
374M32—732no—
375M32—1461no—
376C2×C16—731no—
377C2×C16—1461no—
378C2×C16C2×#6440811yescharacteristic, complemented
379#108—732no—
380#108—1461no—
381#106—732no—
382#106—1461no—
383#104—1462no—
384#104—1462no—
385#103—1462no—
386#102—1462no—
387#97—732no—
388C4×C8—732no—
389#41—21no—
390#41—21no—
391#41—2921no—
392#41—2921no—
393#41—2921no—
394#41—2921no—
395#41—1461no—
396#41—1461no—
397#41—1461no—
398#41—1461no—
399#41—1461no—
400#41—1461no—
401C2×Q8—1461no—
402C2×Q8—1461no—
403C2×D4—2921no—
404C2×D4—2921no—
405C2×D4—1461no—
406C2×D4—1461no—
407C22×C4—2921no—
408C22×C4—2921no—
409C22×C4—2921no—
410C22×C4—2922no—
411C22×C4—731no—
412Q16—731no—
413Q16—731no—
414Q16—1461no—
415Q16—1461no—
416Q16C22×#6440811yescharacteristic
417Q16C22×#6440811yescharacteristic
418SD16—21no—
419SD16—21no—
420SD16—1461no—
421SD16—1461no—
422SD16—1461no—
423SD16—1461no—
424SD16—1461no—
425SD16—1461no—
426D8—731no—
427D8—731no—
428D8—1461no—
429D8—1461no—
430D8C22×#6440811yescharacteristic
431D8C22×#6440811yescharacteristic
432M16—2921no—
433M16—2922no—
434M16—2922no—
435M16—1462no—
436M16—1461no—
437M16—1461no—
438C2×C8—2921no—
439C2×C8—2922no—
440C2×C8—2922no—
441C2×C8—731no—
442C2×C8—731no—
443C2×C8—731no—
444C2×C8—731no—
445C2×C8—731no—
446C2×C8—732no—
447C2×C8C22×#6440811yescharacteristic
448C4⋊C4—2921no—
449C4⋊C4—2922no—
450C4⋊C4—1462no—
451C4⋊C4—1461no—
452C22⋊C4—2922no—
453C22⋊C4—1461no—
454C42—2922no—
455C42—732no—
456C16—731no—
457C16—731no—
458C16—1462no—
459C16C22×#6440811yescharacteristic
460C16C22×#6440811yescharacteristic
461C23—2921no—
462C23—2921no—
463C23—731no—
464Q8—21no—
465Q8—21no—
466Q8—2921no—
467Q8—2921no—
468Q8—1461no—
469Q8—1461no—
470D4—21no—
471D4—21no—
472D4—41no—
473D4—41no—
474D4—2921no—
475D4—2921no—
476D4—2921no—
477D4—2921no—
478D4—2921no—
479D4—2921no—
480D4—2921no—
481D4—2921no—
482D4—1461no—
483D4—1461no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—2921no—
487C2×C4—2921no—
488C2×C4—2921no—
489C2×C4—2921no—
490C2×C4—2921no—
491C2×C4—2921no—
492C2×C4—2921no—
493C2×C4—2921no—
494C2×C4—2922no—
495C2×C4—731no—
496C2×C4—731no—
497C2×C4—731no—
498C2×C4—731no—
499C2×C4—731no—
500C2×C4—11682no—
501C2×C4—11681no—
502C2×C4—1462no—
503C2×C4D4×#6440811yescharacteristic
504C8—2922no—
505C8—731no—
506C8—731no—
507C8—1462no—
508C8D4×#6440811yescharacteristic
509C8C23×#6440811yescharacteristic
510C22—41no—
511C22—41no—
512C22—2921no—
513C22—2921no—
514C22—2921no—
515C22—5841no—
516C22—5841no—
517C22—731no—
518C22—731no—
519C22—11681no—
520C22#1798962811yescharacteristic
521C4—41no—
522C4—41no—
523C4—2922no—
524C4—2921no—
525C4—2921no—
526C4—5842no—
527C4—731no—
528C4—731no—
529C4#1798962811yescharacteristic
530C4C2×D4×#6440811yescharacteristic
531C2—21no—
532C2—81no—
533C2—81no—
534C2—5841no—
535C2—5841no—
536C2—1461no—
537C2—1461no—
538C2C2×#1798962811yescharacteristic
subgroupG/H
centerC2C2×#17989628
commutatorC8×C73C23×C4
FrattiniC8C23×#64408
Fitting#508×C73C4
socleC2×C73C2×#442
prime radical#22623064C2
cosocleC8×D73C24
hypercenter#508#64408
hypocenterC73#33469
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #508

Complements

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

NH ≅ G/Ntype
1. #22626710C2semidirect
2. #22626694C2semidirect
3. #22626343C2semidirect
4. #22626340C2semidirect
5. #22626338C2semidirect
6. #22626337C2semidirect
7. C2×#17989511C2semidirect
8. #22626259C2semidirect
9. D73⋊#360C2semidirect
10. #22626150C2semidirect
18. #17989628C22semidirect
19. #17989627C22semidirect
26. #17989511C22semidirect
27. #17989511C22semidirect
28. #17989506C22semidirect
29. #17989505C22semidirect
32. #17989478C22semidirect
33. #17989477C22semidirect
36. #17989029C4semidirect
38. #17989026C4semidirect
40. #17989024C4semidirect
42. #17989021C4semidirect
44. #17989019C4semidirect
45. #17989017C4semidirect
46. #17989017C4semidirect
48. #508×C73C4semidirect
81. #14556613C2×C4semidirect
88. Q16×D73C2×C4semidirect
92. D8×D73C2×C4semidirect
95. Dic584C2×C4semidirect
96. #14556537C2×C4semidirect
97. #14556537C2×C4semidirect
98. #14556536C2×C4semidirect
99. #14556536C2×C4semidirect
100. #14556535C2×C4semidirect
101. #14556535C2×C4semidirect
102. #14556534C2×C4semidirect
103. #14556534C2×C4semidirect
104. D1168C2×C4semidirect
105. #14556532C2×C4semidirect
106. #14556532C2×C4semidirect
107. #14556530C2×C4semidirect
108. #14556530C2×C4semidirect
111. C2×#11924003C2×C4semidirect
113. #135×C73C2×C4semidirect
114. Q32×C73C2×C4semidirect
115. SD32×C73C2×C4semidirect
116. SD32×C73C2×C4semidirect
117. D16×C73C2×C4semidirect
118. C2×C16×C73C2×C4semidirect
281. C73#33469semidirect
297. #508#64408semidirect
332. #135C2×#64408semidirect
357. Q32C2×#64408semidirect
364. SD32C2×#64408semidirect
365. SD32C2×#64408semidirect
369. D16C2×#64408semidirect
378. C2×C16C2×#64408semidirect

Maximal subgroups

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

subgroupindex
#226267102
#226266942
#226263432
#226263402
#226263382
#226263372
C2×#179895112
#226262592
D73⋊#3602
#226261502
#226230642
#3346973