Finite Group Databasebeta

← #35389805

Subgroups — #35389805

Order 21248.

704 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24918401C211yesmaximal, characteristic, complemented
2#24916943C211yesmaximal, characteristic
3#24916574C211yesmaximal, characteristic, complemented
4#24906881C212yesmaximal, complemented
5#24906874C212yesmaximal, complemented
6#24906813C212yesmaximal, complemented
7#24906806C212yesmaximal, complemented
8#24906233C211yesmaximal, characteristic, complemented
9#24906153C211yesmaximal, characteristic, complemented
10C2×#19127444C211yesmaximal, characteristic, complemented
11C83×#3076C211yesmaximal, characteristic, complemented
12C2×#14581817—21no—
13C2×#14581789C2211yescharacteristic
14#19128075C2212yes—
15#19128049C2212yes—
16#19127901—21no—
17#19127901—21no—
18#19127901—21no—
19#19127901C2212yes—
20#19127896—21no—
21#19127896C2212yes—
22#19127851—22no—
23Q8×C83⋊C8C2212yescomplemented
24D4×C83⋊C8—22no—
25D4×C83⋊C8C2212yescomplemented
26#19127754—21no—
27#19127754C2211yescharacteristic
28#19127725C2211yescharacteristic
29C2×C4⋊(C83⋊C8)—21no—
30C2×C4⋊(C83⋊C8)C2211yescharacteristic
31#19127705—21no—
32#19127451C2212yescomplemented
33#19127451C2212yescomplemented
34#19127450C2212yescomplemented
35#19127447C2212yescomplemented
36#19127444C2212yes—
37#19127444C2212yes—
38#533×C83C2211yescharacteristic
39C4×#41×C83C2211yescharacteristic
40#479×C83—21no—
41#478×C83—21no—
42#477×C83C2211yescharacteristic, complemented
43#476×C83C2211yescharacteristic, complemented
44#475×C83C2211yescharacteristic, complemented
45#474×C83C2211yescharacteristic, complemented
46#427×C83—21no—
47C2×C4⋊C8×C83C2211yescharacteristic
48#416×C83—21no—
49C22×D332—21no—
50C22×D332—21no—
51C22×D332—21no—
52#14581817—44no—
53C2×#13050577—22no—
54C2×#13050577—22no—
55C2×#13050577—22no—
56C2×#13050575—22no—
57C2×#13050574—21no—
58C22×C83⋊C8—22no—
59C22×C83⋊C8—21no—
60#14581789—21no—
61#14581789—21no—
62#14581789—21no—
63#14581789C2312yes—
64C4⋊(C83⋊C8)—22no—
65C4⋊(C83⋊C8)—21no—
66C4⋊(C83⋊C8)—21no—
67C4⋊(C83⋊C8)D411yescharacteristic
68C4⋊(C83⋊C8)D411yescharacteristic
69C4⋊(C83⋊C8)C2312yes—
70C4⋊(C83⋊C8)C2312yes—
71#14581712—22no—
72#14581712—22no—
73#14581712—22no—
74#14581712—22no—
75C22⋊(C83⋊C8)—22no—
76C22⋊(C83⋊C8)D412yes—
77C4×C83⋊C8—22no—
78C4×C83⋊C8C2312yes—
79C2×#41×C83D411yescharacteristic
80C22×C8×C83—21no—
81C4⋊Q8×C83C2311yescharacteristic
82C4⋊Q8×C83C2311yescharacteristic
83#125×C83—21no—
84C22⋊Q8×C83—21no—
85C22⋊Q8×C83—21no—
86C4×Q8×C83—21no—
87C4×Q8×C83—21no—
88C4×Q8×C83D411yescharacteristic
89C4×Q8×C83C2311yescharacteristic
90C4×D4×C83—21no—
91C4×D4×C83—21no—
92C4×D4×C83D411yescharacteristic
93C4×D4×C83C2311yescharacteristic
94#117×C83—21no—
95#117×C83—21no—
96#117×C83D411yescharacteristic
97C2×C42×C83—21no—
98C2×C42×C83C2311yescharacteristic
99#107×C83—21no—
100#107×C83—21no—
101#106×C83—21no—
102#106×C83—21no—
103C4⋊C8×C83—21no—
104C4⋊C8×C83C2311yescharacteristic
105C4⋊C8×C83C2311yescharacteristic
106#103×C83—21no—
107#103×C83—21no—
108#102×C83—21no—
109#102×C83—21no—
110C23×D83—41no—
111C2×D332—22no—
112C2×D332—22no—
113C2×D332—22no—
114C2×D332—42no—
115C2×D332—42no—
116C2×D332—41no—
117C2×D332—41no—
118C2×D332—41no—
119C2×D332—81no—
120C2×D332—81no—
121C2×D332—82no—
122#13050577—44no—
123#13050577—82no—
124#13050577—82no—
125#13050575—44no—
126#13050574—42no—
127C2×C83⋊C8—22no—
128C2×C83⋊C8—22no—
129C2×C83⋊C8—82no—
130C2×C83⋊C8—81no—
131C2×C83⋊C8C2×D412yes—
132C2×C83⋊C8C2×D412yes—
133#41×C83—22no—
134#41×C83SD1614yescomplemented
135C2×Q8×C83—21no—
136C2×Q8×C83—21no—
137C2×Q8×C83C2×D411yescharacteristic
138C2×D4×C83—21no—
139C2×D4×C83C2×D411yescharacteristic
140C22×C4×C83—21no—
141C22×C4×C83—21no—
142C22×C4×C83—21no—
143C22×C4×C83C2×D411yescharacteristic
144C22×C4×C83C2×D411yescharacteristic
145C22×C4×C83#4111yescharacteristic
146C2×C8×C83—21no—
147C2×C8×C83—21no—
148C2×C8×C83—42no—
149C4⋊C4×C83—21no—
150C4⋊C4×C83—21no—
151C4⋊C4×C83—21no—
152C4⋊C4×C83—21no—
153C4⋊C4×C83—21no—
154C4⋊C4×C83—21no—
155C4⋊C4×C83—41no—
156C4⋊C4×C83—41no—
157C4⋊C4×C83C2×D411yescharacteristic
158C4⋊C4×C83C2×D411yescharacteristic
159C22⋊C4×C83—21no—
160C22⋊C4×C83—21no—
161C22⋊C4×C83—21no—
162C22⋊C4×C83—41no—
163C42×C83—21no—
164C42×C83—21no—
165C42×C83—21no—
166C42×C83—41no—
167C42×C83C2×D411yescharacteristic
168C42×C83C2×D411yescharacteristic
169C42×C83#4111yescharacteristic
170C42×C83C2411yescharacteristic
171C23×C83—21no—
172C23×C83#12111yescharacteristic
173C22×D83—42no—
174C22×D83—82no—
175C22×D83—82no—
176C22×D83—82no—
177Q8×C83—21no—
178Q8×C83—41no—
179Q8×C83—41no—
180Q8×C83—81no—
181Q8×C83C2×SD1612yes—
182D4×C83—24no—
183D4×C83—21no—
184D4×C83C2×SD1612yes—
185C2×C4×C83—21no—
186C2×C4×C83—21no—
187C2×C4×C83—21no—
188C2×C4×C83—24no—
189C2×C4×C83—21no—
190C2×C4×C83—21no—
191C2×C4×C83—41no—
192C2×C4×C83—41no—
193C2×C4×C83—41no—
194C2×C4×C83—41no—
195C2×C4×C83—41no—
196C2×C4×C83—81no—
197C2×C4×C83#12111yescharacteristic
198C2×C4×C83#12111yescharacteristic
199C2×C4×C83#12111yescharacteristic
200C2×C4×C83C2×SD1612yes—
201C2×C4×C83#13512yes—
202C2×C4×C83C22×D411yescharacteristic
203C2×C4×C83C22×D411yescharacteristic
204C2×C4×C83C2×#4111yescharacteristic
205D332—42no—
206D332—42no—
207D332—42no—
208D332—42no—
209D332—82no—
210D332—82no—
211D332—82no—
212D332—82no—
213C8×C83—41no—
214C8×C83—41no—
215C83⋊C8—22no—
216C83⋊C8—22no—
217C83⋊C8—42no—
218C83⋊C8—42no—
219C22×C83—21no—
220C22×C83—21no—
221C22×C83—22no—
222C22×C83—41no—
223C22×C83—41no—
224C22×C83#46512yes—
225C22×C83C2×#12111yescharacteristic
226C2×D83—82no—
227C2×D83—82no—
228C2×D83—82no—
229C2×D83—164no—
230C4×C83—21no—
231C4×C83—22no—
232C4×C83—21no—
233C4×C83—21no—
234C4×C83—41no—
235C4×C83—41no—
236C4×C83—81no—
237C4×C83—81no—
238C4×C83#46512yes—
239C4×C83C22×SD1611yescharacteristic
240C4×C83C2×#13511yescharacteristic
241C4×C83#57611yescharacteristic
242C4×C83#57711yescharacteristic
243#19409—831nomaximal, Sylow
244C2×C83—22no—
245C2×C83—42no—
246C2×C83#304511yescharacteristic
247C2×C83C2×#46511yescharacteristic
248C2×C83#306811yescharacteristic
249D83—162no—
250#3208—831no—
251#3076D8311yescharacteristic, complemented
252#3038—831no—
253#2988—831no—
254#1678—832no—
255#1677—832no—
256#1666—832no—
257#1665—832no—
258#1570—831no—
259#1550—831no—
260C2×#331—831no—
261C83#1940911yescharacteristic, complemented, Sylow
262C2×AGL(1,ℤ/8)—1661no—
263#533C2×D8311yescharacteristic
264C2×#127—831no—
265C4×#41C2×D8311yescharacteristic
266#492—832no—
267#486—832no—
268#479—21no—
269#478—21no—
270#477C2×D8311yescharacteristic, complemented
271#476C2×D8311yescharacteristic, complemented
272#475C2×D8311yescharacteristic, complemented
273#474C2×D8311yescharacteristic, complemented
274#460—832no—
275#460—1661no—
276#460—1661no—
277#460—1661no—
278#459—832no—
279#459—1661no—
280#450—1662no—
281C8×Q8—832no—
282C8×D4—832no—
283C8×D4—1662no—
284#431—831no—
285#431—1661no—
286#427—21no—
287#423—831no—
288C2×C4⋊C8—831no—
289C2×C4⋊C8—1661no—
290C2×C4⋊C8C2×D8311yescharacteristic
291#418—1661no—
292#416—21no—
293#334—832no—
294#334—832no—
295#333—832no—
296#332—832no—
297#331—832no—
298#331—832no—
299C2×#41C22⋊D8311yescharacteristic
300C22×D4—1661no—
301C22×D4—1661no—
302C22×D4—1661no—
303AGL(1,ℤ/8)—3324no—
304C2×SD16—1662no—
305C2×SD16—1662no—
306C2×SD16—1662no—
307C2×D8—1662no—
308C2×M16—1661no—
309C22×C8—21no—
310C22×C8—1662no—
311C22×C8—1661no—
312C4⋊Q8C22×D8311yescharacteristic
313C4⋊Q8C22×D8311yescharacteristic
314#127—832no—
315#127—1661no—
316#127—1661no—
317#127—1661no—
318#125—21no—
319C22⋊Q8—21no—
320C22⋊Q8—21no—
321C4×Q8—21no—
322C4×Q8—21no—
323C4×Q8C22⋊D8311yescharacteristic
324C4×Q8C22×D8311yescharacteristic
325C4×D4—21no—
326C4×D4—21no—
327C4×D4C22⋊D8311yescharacteristic
328C4×D4C22×D8311yescharacteristic
329#117—21no—
330#117—21no—
331#117C22⋊D8311yescharacteristic
332C2×C42—21no—
333C2×C42C22×D8311yescharacteristic
334#107—21no—
335#107—21no—
336#106—21no—
337#106—21no—
338C4⋊C8—21no—
339C4⋊C8—831no—
340C4⋊C8—831no—
341C4⋊C8—832no—
342C4⋊C8—832no—
343C4⋊C8—1662no—
344C4⋊C8—1661no—
345C4⋊C8—1661no—
346C4⋊C8C22×D8311yescharacteristic
347C4⋊C8C22×D8311yescharacteristic
348#103—21no—
349#103—21no—
350#102—21no—
351#102—21no—
352#102—1662no—
353#102—1662no—
354#102—1662no—
355#102—1662no—
356C22⋊C8—832no—
357C22⋊C8—1662no—
358C4×C8—832no—
359C4×C8—1662no—
360C24—3321no—
361#41—22no—
362#41#1305057714yescomplemented
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8C2×C22⋊D8311yescharacteristic
366C2×D4—21no—
367C2×D4—3321no—
368C2×D4—3321no—
369C2×D4—3321no—
370C2×D4—3322no—
371C2×D4—3322no—
372C2×D4—6641no—
373C2×D4—6641no—
374C2×D4—6642no—
375C2×D4—1662no—
376C2×D4—1662no—
377C2×D4—1662no—
378C2×D4C2×C22⋊D8311yescharacteristic
379C22×C4—21no—
380C22×C4—21no—
381C22×C4—21no—
382C22×C4D4×D8311yescharacteristic
383C22×C4C2×C22⋊D8311yescharacteristic
384C22×C4#1305058911yescharacteristic
385SD16—3324no—
386SD16—6642no—
387SD16—6642no—
388D8—3324no—
389M16—3322no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—42no—
393C2×C8—832no—
394C2×C8—832no—
395C2×C8—6641no—
396C2×C8—6642no—
397C2×C8—1662no—
398C2×C8—1662no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4C2×C22⋊D8311yescharacteristic
408C4⋊C4C2×C22⋊D8311yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—21no—
411C22⋊C4—21no—
412C22⋊C4—41no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—41no—
417C42D4×D8311yescharacteristic
418C42C2×C22⋊D8311yescharacteristic
419C42#1305058911yescharacteristic
420C42C23×D8311yescharacteristic
421C23—21no—
422C23—3322no—
423C23—6642no—
424C23—6642no—
425C23—6642no—
426C23#1458176411yescharacteristic
427Q8—21no—
428Q8—41no—
429Q8—41no—
430Q8—81no—
431Q8C2×#1305057712yes—
432D4—24no—
433D4—21no—
434D4—3322no—
435D4—3322no—
436D4—3322no—
437D4—3322no—
438D4—6642no—
439D4—6642no—
440D4—6642no—
441D4—6642no—
442D4C2×#1305057712yes—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—81no—
455C2×C4#1458176411yescharacteristic
456C2×C4#1458176411yescharacteristic
457C2×C4#1458176411yescharacteristic
458C2×C4C2×#1305057712yes—
459C2×C4#1458181212yes—
460C2×C4C2×D4×D8311yescharacteristic
461C2×C4C22×C22⋊D8311yescharacteristic
462C2×C4C2×#1305058911yescharacteristic
463C8—41no—
464C8—41no—
465C8—3322no—
466C8—3322no—
467C8—1662no—
468C8—1662no—
469C22—21no—
470C22—21no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—13284no—
475C22—6642no—
476C22—6642no—
477C22—6642no—
478C22#1912793212yes—
479C22C2×#1458176411yescharacteristic
480C4—21no—
481C4—22no—
482C4—21no—
483C4—21no—
484C4—41no—
485C4—41no—
486C4—81no—
487C4—81no—
488C4#1912793212yes—
489C4C22×#1305057711yescharacteristic
490C4C2×#1458181211yescharacteristic
491C4#1912850311yescharacteristic
492C4#1912850811yescharacteristic
493C2—22no—
494C2—42no—
495C2—13282no—
496C2#2491700011yescharacteristic
497C2C2×#1912793211yescharacteristic
498C2#2491719311yescharacteristic
subgroupG/H
centerC22C2×#14581764
commutator, cosocleC42×C83C24
FrattiniC42C23×D83
FittingC83×#3076C2
socleC22×C83C2×#121
prime radical#24916943C2
hypercenter#3076D83
hypocenterC83#19409
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3076

Complements

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

NH ≅ G/Ntype
1. #24918401C2semidirect
3. #24916574C2semidirect
4. #24906881C2semidirect
5. #24906874C2semidirect
6. #24906813C2semidirect
7. #24906806C2semidirect
8. #24906233C2semidirect
9. #24906153C2semidirect
10. C2×#19127444C2semidirect
11. C83×#3076C2semidirect
23. Q8×C83⋊C8C22semidirect
25. D4×C83⋊C8C22semidirect
32. #19127451C22semidirect
33. #19127451C22semidirect
34. #19127450C22semidirect
35. #19127447C22semidirect
42. #477×C83C22semidirect
43. #476×C83C22semidirect
44. #475×C83C22semidirect
45. #474×C83C22semidirect
134. #41×C83SD16semidirect
251. #3076D83semidirect
261. C83#19409semidirect
270. #477C2×D83semidirect
271. #476C2×D83semidirect
272. #475C2×D83semidirect
273. #474C2×D83semidirect
362. #41#13050577semidirect

Maximal subgroups

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

subgroupindex
#249184012
#249169432
#249165742
#249068812
#249068742
#249068132
#249068062
#249062332
#249061532
C2×#191274442
C83×#30762
#1940983