Finite Group Databasebeta

← #35440530

Subgroups — #35440530

Order 21248.

1162 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127998C212yesmaximal, complemented
2C4×#14581766C211yesmaximal, characteristic
3#24908798C212yesmaximal, complemented
4#24908606C212yesmaximal, complemented
5#24908421C212yesmaximal, complemented
6#24907780C211yesmaximal, characteristic
7#24907780C211yesmaximal, characteristic
8#24907656C211yesmaximal, characteristic, complemented
9#24907651C211yesmaximal, characteristic, complemented
10#24907443C211yesmaximal, characteristic, complemented
11C22×#125×C83C211yesmaximal, characteristic, complemented
12C2×#14581766C2211yescharacteristic
13C2×C4×D332C2211yescharacteristic
14C2×C4×C22⋊D83—21no—
15#19127998—28no—
16C2×#14581725C2212yes—
17C2×#14581723C2212yes—
18C2×C4⋊(C83⋊C8)C2212yes—
19C2×#14581712C2212yes—
20C2×#14581712C2212yes—
21C2×C22⋊(C83⋊C8)C2211yescharacteristic
22C2×C22⋊(C83⋊C8)C2211yescharacteristic
23#19127603C2211yescharacteristic
24#19127599C2211yescharacteristic
25#19127592—21no—
26#19127580—21no—
27#19127561C2211yescharacteristic
28C4×C4⋊Dic83C2211yescharacteristic
29C4×(C2×D83)⋊C4—21no—
30#19127463C2212yes—
31#19127463C2212yes—
32#19127463C2212yes—
33#19127463C2212yes—
34#19127456C2212yes—
35C2×#125×C83—28no—
36C2×#125×C83—21no—
37C2×#125×C83—21no—
38C2×#125×C83C2212yes—
39C2×#125×C83C2212yes—
40C22×C4⋊C4×C83—21no—
41C22×C4⋊C4×C83—21no—
42C22×C4⋊C4×C83C2211yescharacteristic
43C22×C4⋊C4×C83C2211yescharacteristic
44C22×C42×C83C2211yescharacteristic
45C22×D332C2311yescharacteristic
46C22×C22⋊D83—21no—
47C22×C4×D83—21no—
48C22×C83⋊C8C2312yes—
49C22×C83⋊C8C2312yes—
50#14581766—22no—
51#14581766Q814yes—
52C4×D332—24no—
53C4×C22⋊D83—44no—
54C2×C4⋊Dic83C2311yescharacteristic
55C2×C4⋊Dic83C2311yescharacteristic
56C2×C4⋊Dic83C2311yescharacteristic
57C2×Dic83⋊C4—21no—
58C2×C22⋊Dic83—21no—
59C2×(C2×D83)⋊C4—21no—
60C2×(C2×D83)⋊C4—21no—
61C2×(C2×D83)⋊C4—21no—
62C2×(C2×D83)⋊C4—21no—
63C2×C4×Dic83—21no—
64#14581725—24no—
65#14581723—24no—
66C4⋊(C83⋊C8)—24no—
67#14581712—24no—
68#14581712—24no—
69C22⋊(C83⋊C8)—22no—
70C22⋊(C83⋊C8)—22no—
71C83⋊#95—21no—
72C83⋊#95—21no—
73C83⋊#95—21no—
74C23×C4×C83—21no—
75C23×C4×C83—21no—
76C23×C4×C83D411yescharacteristic
77C23×C4×C83D411yescharacteristic
78C23×C4×C83C2311yescharacteristic
79#125×C83—28no—
80#125×C83—28no—
81#125×C83—24no—
82#125×C83—24no—
83#125×C83—24no—
84#125×C83D418yescomplemented
85C2×C4⋊C4×C83—28no—
86C2×C4⋊C4×C83—28no—
87C2×C4⋊C4×C83—22no—
88C2×C4⋊C4×C83—22no—
89C2×C4⋊C4×C83—22no—
90C2×C4⋊C4×C83—22no—
91C2×C4⋊C4×C83—21no—
92C2×C4⋊C4×C83—21no—
93C2×C4⋊C4×C83—24no—
94C2×C4⋊C4×C83—24no—
95C2×C4⋊C4×C83C2312yes—
96C2×C4⋊C4×C83C2312yes—
97C2×C42×C83—24no—
98C2×C42×C83D411yescharacteristic
99C2×C42×C83D411yescharacteristic
100C2×C42×C83C2311yescharacteristic
101C2×C42×C83C2311yescharacteristic
102C23×D83—21no—
103C2×D332—22no—
104C2×D332—22no—
105C2×D332—21no—
106C2×D332C2×Q812yes—
107C2×C22⋊D83—24no—
108C2×C22⋊D83—42no—
109C2×C22⋊D83—42no—
110C22×Dic83—21no—
111C22×Dic83—21no—
112C22×Dic83—21no—
113C2×C4×D83—42no—
114C2×C4×D83—42no—
115C2×C4×D83—42no—
116C2×C83⋊C8—22no—
117C2×C83⋊C8—22no—
118C2×C83⋊C8—22no—
119C2×C83⋊C8—22no—
120C2×C83⋊C8—22no—
121C2×C83⋊C8—22no—
122C4⋊Dic83—21no—
123C4⋊Dic83—22no—
124C4⋊Dic83—22no—
125C4⋊Dic83C2×Q812yes—
126Dic83⋊C4—42no—
127C22⋊Dic83—42no—
128(C2×D83)⋊C4—24no—
129(C2×D83)⋊C4—42no—
130(C2×D83)⋊C4—42no—
131(C2×D83)⋊C4—42no—
132C4×Dic83—42no—
133C24×C83C2×D411yescharacteristic
134C22×C4×C83—22no—
135C22×C4×C83—22no—
136C22×C4×C83—24no—
137C22×C4×C83—24no—
138C22×C4×C83—22no—
139C22×C4×C83—22no—
140C22×C4×C83—22no—
141C22×C4×C83—22no—
142C22×C4×C83—22no—
143C22×C4×C83—22no—
144C22×C4×C83—41no—
145C22×C4×C83—41no—
146C22×C4×C83—42no—
147C22×C4×C83—42no—
148C22×C4×C83—42no—
149C22×C4×C83—42no—
150C22×C4×C83C2×D411yescharacteristic
151C22×C4×C83C2×D411yescharacteristic
152C22×C4×C83C2×D411yescharacteristic
153C22×C4×C83C2×D411yescharacteristic
154C22×C4×C83C2×D411yescharacteristic
155C22×C4×C83C2×Q812yes—
156C22×C4×C83#4112yes—
157C22×C4×C83C2411yescharacteristic
158C4⋊C4×C83—24no—
159C4⋊C4×C83—24no—
160C4⋊C4×C83—28no—
161C4⋊C4×C83—28no—
162C4⋊C4×C83—28no—
163C4⋊C4×C83—28no—
164C4⋊C4×C83—22no—
165C4⋊C4×C83—22no—
166C4⋊C4×C83C2×D414yes—
167C4⋊C4×C83C2×D414yes—
168C42×C83—22no—
169C42×C83—22no—
170C42×C83—22no—
171C42×C83C2×D414yes—
172C23×C83—21no—
173C23×C83—21no—
174C23×C83—21no—
175C23×C83—21no—
176C23×C83C22⋊Q812yes—
177C23×C83#13512yes—
178C23×C83AGL(1,ℤ/8)12yes—
179C23×C83C22×D411yescharacteristic
180C22×D83—22no—
181C22×D83—41no—
182C22×D83—41no—
183C22×D83—41no—
184C22×D83—41no—
185C22×D83—41no—
186C22×D83—41no—
187C2×C4×C83—21no—
188C2×C4×C83—21no—
189C2×C4×C83—21no—
190C2×C4×C83—21no—
191C2×C4×C83—21no—
192C2×C4×C83—21no—
193C2×C4×C83—24no—
194C2×C4×C83—24no—
195C2×C4×C83—21no—
196C2×C4×C83—21no—
197C2×C4×C83—24no—
198C2×C4×C83—24no—
199C2×C4×C83—21no—
200C2×C4×C83—21no—
201C2×C4×C83—21no—
202C2×C4×C83—21no—
203C2×C4×C83—44no—
204C2×C4×C83—44no—
205C2×C4×C83—48no—
206C2×C4×C83—48no—
207C2×C4×C83—42no—
208C2×C4×C83—42no—
209C2×C4×C83—42no—
210C2×C4×C83—42no—
211C2×C4×C83#12012yes—
212C2×C4×C83#12012yes—
213C2×C4×C83C22⋊Q812yes—
214C2×C4×C83C22×D412yes—
215C2×C4×C83C22×Q811yescharacteristic
216C2×C4×C83C2×#4111yescharacteristic
217C2×C4×C832−1+412yes—
218C22⋊D83—48no—
219C2×Dic83—22no—
220C2×Dic83—41no—
221C2×Dic83—41no—
222C2×Dic83—41no—
223C2×Dic83—41no—
224C2×Dic83—41no—
225C2×Dic83—41no—
226C2×Dic83—41no—
227C2×Dic83—41no—
228D332—24no—
229D332—42no—
230C4×D83—84no—
231C83⋊C8—42no—
232C83⋊C8—42no—
233C22×C83—22no—
234C22×C83—22no—
235C22×C83—22no—
236C22×C83—22no—
237C22×C83—22no—
238C22×C83—22no—
239C22×C83#47814yescomplemented
240C22×C83C2×#12011yescharacteristic
241C22×C83C2×C22⋊Q811yescharacteristic
242C22×C83#53611yescharacteristic
243C22×C83C2×#13511yescharacteristic
244C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
245C22×C83#57511yescharacteristic
246C22×C83#57511yescharacteristic
247C2×D83—42no—
248C2×D83—42no—
249C2×D83—42no—
250C2×D83—81no—
251C2×D83—81no—
252C2×D83—81no—
253C2×D83—81no—
254C4×C83—22no—
255C4×C83—22no—
256C4×C83—22no—
257C4×C83—41no—
258C4×C83—44no—
259C4×C83—41no—
260C4×C83—44no—
261C4×C83#54812yes—
262C4×C83D4×Q812yes—
263Dic83—42no—
264Dic83—81no—
265Dic83—81no—
266#24305—831nomaximal, Sylow
267C2×C83—24no—
268C2×C83#240911yescharacteristic
269C2×C83#300211yescharacteristic
270C2×C83#300611yescharacteristic
271C2×C83C2×#47811yescharacteristic
272C2×C83#308011yescharacteristic
273C2×C83#323211yescharacteristic
274C2×C83#323411yescharacteristic
275D83—82no—
276C22×#125D8311yescharacteristic, complemented
277C2×#478—832no—
278C4×#121—831no—
279#2059—832no—
280#2025—832no—
281#1989—832no—
282#1874—831no—
283#1874—831no—
284#1853—831no—
285#1852—831no—
286#1797—831no—
287C83#2430511yescharacteristic, complemented, Sylow
288C2×#125—28no—
289C2×#125—21no—
290C2×#125—21no—
291C2×#125C2×D8312yes—
292C2×#125C2×D8312yes—
293C2×#121—831no—
294C2×C4×D4—831no—
295C2×C4×D4—1661no—
296C22×C4⋊C4—21no—
297C22×C4⋊C4—21no—
298C22×C4⋊C4C2×D8311yescharacteristic
299C22×C4⋊C4C2×D8311yescharacteristic
300C22×C42C2×D8311yescharacteristic
301#478—1668no—
302C2×#107—832no—
303C2×#106—832no—
304C2×C4⋊C8—832no—
305C2×#102—832no—
306C2×#102—832no—
307C2×C22⋊C8—831no—
308C2×C22⋊C8—831no—
309#390—831no—
310#389—831no—
311#388—1661no—
312#386—1661no—
313#380—831no—
314C4×C4⋊C4—831no—
315C4×C22⋊C4—1661no—
316#340—832no—
317#340—832no—
318#340—832no—
319#340—832no—
320#336—832no—
321C22×D4—831no—
322C22×D4—1661no—
323C23×C4—21no—
324C23×C4—21no—
325C23×C4—1661no—
326C23×C4C22⋊D8311yescharacteristic
327C23×C4C22⋊D8311yescharacteristic
328C23×C4C22×D8311yescharacteristic
329C22×C8—832no—
330C22×C8—832no—
331#125—28no—
332#125—28no—
333#125—24no—
334#125—24no—
335#125—24no—
336#125C22⋊D8318yescomplemented
337#121—834no—
338#121—1662no—
339C4×D4—3324no—
340C4×D4—1664no—
341C2×C4⋊C4—28no—
342C2×C4⋊C4—28no—
343C2×C4⋊C4—22no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—22no—
346C2×C4⋊C4—22no—
347C2×C4⋊C4—21no—
348C2×C4⋊C4—21no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—24no—
351C2×C4⋊C4—831no—
352C2×C4⋊C4—831no—
353C2×C4⋊C4—831no—
354C2×C4⋊C4—1661no—
355C2×C4⋊C4C22×D8312yes—
356C2×C4⋊C4C22×D8312yes—
357C2×C22⋊C4—1661no—
358C2×C22⋊C4—1661no—
359C2×C22⋊C4—1661no—
360C2×C22⋊C4—1661no—
361C2×C22⋊C4—1661no—
362C2×C42—24no—
363C2×C42—1661no—
364C2×C42C22⋊D8311yescharacteristic
365C2×C42C22⋊D8311yescharacteristic
366C2×C42C22×D8311yescharacteristic
367C2×C42C22×D8311yescharacteristic
368#107—1664no—
369#106—1664no—
370C4⋊C8—1664no—
371#102—1664no—
372#102—1664no—
373C22⋊C8—1662no—
374C22⋊C8—1662no—
375#95—1661no—
376#95—1661no—
377#95—1661no—
378C24—1661no—
379C24C2×C22⋊D8311yescharacteristic
380C2×D4—3322no—
381C2×D4—3322no—
382C2×D4—832no—
383C2×D4—1664no—
384C2×D4—1662no—
385C2×D4—1661no—
386C2×D4—1662no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—24no—
390C22×C4—24no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—42no—
402C22×C4—42no—
403C22×C4—3322no—
404C22×C4—3322no—
405C22×C4—3322no—
406C22×C4—1661no—
407C22×C4—1661no—
408C22×C4—1661no—
409C22×C4C2×C22⋊D8311yescharacteristic
410C22×C4C2×C22⋊D8311yescharacteristic
411C22×C4C2×C22⋊D8311yescharacteristic
412C22×C4C2×C22⋊D8311yescharacteristic
413C22×C4C2×C22⋊D8311yescharacteristic
414C22×C4Q8×D8312yes—
415C22×C4#1305059112yes—
416C22×C4C23×D8311yescharacteristic
417C2×C8—1662no—
418C2×C8—1662no—
419C2×C8—1662no—
420C2×C8—1662no—
421C2×C8—1662no—
422C2×C8—1662no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—28no—
426C4⋊C4—28no—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—3322no—
432C4⋊C4—832no—
433C4⋊C4—1662no—
434C4⋊C4—1662no—
435C4⋊C4—1661no—
436C4⋊C4C2×C22⋊D8314yes—
437C4⋊C4C2×C22⋊D8314yes—
438C22⋊C4—3322no—
439C22⋊C4—3322no—
440C22⋊C4—3322no—
441C22⋊C4—3322no—
442C22⋊C4—1664no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—3322no—
447C42C2×C22⋊D8314yes—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—3321no—
453C23—3321no—
454C23—3321no—
455C23—3321no—
456C23—3321no—
457C23—3321no—
458C23—1662no—
459C23#1458176812yes—
460C23#1458181212yes—
461C23#1458181712yes—
462C23C22×C22⋊D8311yescharacteristic
463D4—3328no—
464D4—3322no—
465D4—1664no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—24no—
477C2×C4—24no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—48no—
485C2×C4—48no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—3321no—
491C2×C4—3321no—
492C2×C4—3321no—
493C2×C4—3321no—
494C2×C4—3321no—
495C2×C4—3321no—
496C2×C4—3321no—
497C2×C4—3321no—
498C2×C4—6644no—
499C2×C4—1662no—
500C2×C4#1458176112yes—
501C2×C4#1458176112yes—
502C2×C4#1458176812yes—
503C2×C4C22×C22⋊D8312yes—
504C2×C4C2×Q8×D8311yescharacteristic
505C2×C4C2×#1305059111yescharacteristic
506C2×C4#1458184012yes—
507C8—3322no—
508C8—3322no—
509C22—22no—
510C22—22no—
511C22—22no—
512C22—22no—
513C22—22no—
514C22—22no—
515C22—3322no—
516C22—3322no—
517C22—3322no—
518C22—6641no—
519C22—6641no—
520C22—6641no—
521C22—6641no—
522C22#1912799814yescomplemented
523C22C2×#1458176111yescharacteristic
524C22C2×#1458176811yescharacteristic
525C22#1912827511yescharacteristic
526C22C2×#1458181211yescharacteristic
527C22C2×#1458181711yescharacteristic
528C22#1912849911yescharacteristic
529C22#1912849911yescharacteristic
530C4—22no—
531C4—22no—
532C4—22no—
533C4—41no—
534C4—44no—
535C4—41no—
536C4—44no—
537C4—3322no—
538C4—6641no—
539C4—6641no—
540C4#1912836212yes—
541C4Q8×C22⋊D8312yes—
542C2—24no—
543C2—6642no—
544C2#2491082211yescharacteristic
545C2#2491668111yescharacteristic
546C2#2491671511yescharacteristic
547C2C2×#1912799811yescharacteristic
548C2#2491728111yescharacteristic
549C2#2491863111yescharacteristic
550C2#2491865311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC22×#125×C83C2
socleC23×C83C22×D4
prime radicalC2×#14581766C22
hypercenterC22×#125D83
hypocenterC83#24305
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ C22×#125

Complements

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

NH ≅ G/Ntype
1. C2×#19127998C2semidirect
3. #24908798C2semidirect
4. #24908606C2semidirect
5. #24908421C2semidirect
8. #24907656C2semidirect
9. #24907651C2semidirect
10. #24907443C2semidirect
11. C22×#125×C83C2semidirect
84. #125×C83D4semidirect
239. C22×C83#478semidirect
276. C22×#125D83semidirect
287. C83#24305semidirect
336. #125C22⋊D83semidirect
522. C22#19127998semidirect

Maximal subgroups

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

subgroupindex
C2×#191279982
C4×#145817662
#249087982
#249086062
#249084212
#249077802
#249076562
#249076512
#249074432
C22×#125×C832
#2430583