Finite Group Databasebeta

← #35407098

Subgroups — #35407098

Order 21248.

1122 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127700C211yesmaximal, characteristic
2C2×#19127695C211yesmaximal, characteristic
3C2×#19127657C211yesmaximal, characteristic, complemented
4#24910051C211yesmaximal, characteristic
5#24908159C212yesmaximal, complemented
6#24908152C212yesmaximal, complemented
7#24908145C212yesmaximal, complemented
8#24908138C212yesmaximal, complemented
9#24907184C211yesmaximal, characteristic, complemented
10C2×#19127463C211yesmaximal, characteristic, complemented
11C83×#2316C211yesmaximal, characteristic, complemented
12C22×#13050590C2211yescharacteristic
13C22×#13050574C2211yescharacteristic
14C23×C83⋊C8C2211yescharacteristic
15C22×C4⋊Dic83C2211yescharacteristic
16#19127700C418yescomplemented
17#19127695—24no—
18C2×#14581715C2211yescharacteristic
19C2×#14581715C2211yescharacteristic
20C2×#14581715C2212yes—
21C2×#14581712C2211yescharacteristic
22C2×#14581712C2211yescharacteristic
23C2×#14581712C2212yes—
24#19127657—24no—
25C2×#14581703C2212yes—
26C2×C4×C83⋊C8C2212yes—
27#19127607C2212yes—
28#19127603C2212yes—
29#19127578—22no—
30#19127558—21no—
31#19127558—21no—
32#19127463C2212yes—
33#19127463C2212yes—
34#19127463C2212yes—
35#19127463C2212yes—
36C2×#117×C83C2211yescharacteristic
37C2×#117×C83C2211yescharacteristic
38C22×C4⋊C4×C83C2211yescharacteristic
39#389×C83—21no—
40#389×C83—21no—
41#389×C83C2212yes—
42#389×C83C2212yes—
43#380×C83—22no—
44C2×#13050590—22no—
45C2×#13050590—22no—
46C2×#13050590—22no—
47C2×#13050590—22no—
48C2×#13050590—22no—
49C2×#13050590C2×C414yes—
50C22×Dic166C2311yescharacteristic
51C22×D332C2311yescharacteristic
52C22×C22⋊D83—21no—
53C23×Dic83—21no—
54C22×C4×D83—21no—
55C2×#13050574—22no—
56C2×#13050574—22no—
57C2×#13050574C2×C414yes—
58C22×C83⋊C8—22no—
59C22×C83⋊C8—22no—
60C22×C83⋊C8D414yes—
61C22×C83⋊C8C2312yes—
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64C2×C4⋊Dic83—21no—
65C2×C4⋊Dic83—22no—
66C2×C4⋊Dic83—22no—
67C2×C4⋊Dic83C2312yes—
68C2×(C2×D83)⋊C4—22no—
69C2×(C2×D83)⋊C4—22no—
70#14581715—22no—
71#14581715—22no—
72#14581715C2×C418yescomplemented
73#14581712—22no—
74#14581712—22no—
75#14581712C2×C418yescomplemented
76#14581703—24no—
77C4×C83⋊C8—24no—
78C83⋊#95—22no—
79C83⋊#95—22no—
80C23×C4×C83—21no—
81C23×C4×C83C2311yescharacteristic
82#117×C83—22no—
83#117×C83—22no—
84#117×C83—22no—
85#117×C83—22no—
86C2×C4⋊C4×C83—21no—
87C2×C4⋊C4×C83—22no—
88C2×C4⋊C4×C83—22no—
89C2×C4⋊C4×C83—22no—
90C2×C4⋊C4×C83C2×C414yes—
91C2×C4⋊C4×C83D411yescharacteristic
92C2×C4⋊C4×C83D411yescharacteristic
93C2×C4⋊C4×C83C2311yescharacteristic
94C2×C4⋊C4×C83C2311yescharacteristic
95C2×C4⋊C4×C83C2312yes—
96C2×C22⋊C4×C83—21no—
97C2×C22⋊C4×C83D412yes—
98C2×C42×C83—21no—
99C2×C42×C83C2312yes—
100#95×C83—22no—
101#95×C83—22no—
102C23×D83—21no—
103#13050590—28no—
104#13050590—28no—
105#13050590—44no—
106#13050590—44no—
107C2×Dic166—21no—
108C2×Dic166—21no—
109C2×Dic166—21no—
110C2×Dic166—21no—
111C2×Dic166—21no—
112C2×Dic166C22×C412yes—
113C2×D332—21no—
114C2×D332—21no—
115C2×D332—21no—
116C2×D332—21no—
117C2×D332—21no—
118C2×D332C22×C412yes—
119C2×C22⋊D83—24no—
120C2×C22⋊D83—42no—
121C2×C22⋊D83—42no—
122C22×Dic83—21no—
123C22×Dic83—22no—
124C22×Dic83—42no—
125C22×Dic83—81no—
126C22×Dic83—81no—
127C2×C4×D83—24no—
128C2×C4×D83—42no—
129C2×C4×D83—42no—
130#13050574—28no—
131C2×C83⋊C8—28no—
132C2×C83⋊C8—22no—
133C2×C83⋊C8—21no—
134C2×C83⋊C8—21no—
135C2×C83⋊C8—21no—
136C2×C83⋊C8—21no—
137C2×C83⋊C8—22no—
138C2×C83⋊C8C22×C414yes—
139C2×C83⋊C8C2×D412yes—
140C2×C83⋊C8C2×D412yes—
141C4⋊Dic83—22no—
142C4⋊Dic83—22no—
143C4⋊Dic83—42no—
144(C2×D83)⋊C4—82no—
145(C2×D83)⋊C4—82no—
146C24×C83C2×D411yescharacteristic
147C22×C4×C83—22no—
148C22×C4×C83—24no—
149C22×C4×C83—21no—
150C22×C4×C83—21no—
151C22×C4×C83—81no—
152C22×C4×C83—81no—
153C22×C4×C83C22×C412yes—
154C22×C4×C83C2×D412yes—
155C22×C4×C83C2×D412yes—
156C22×C4×C83C2×D412yes—
157C22×C4×C83C2×D411yescharacteristic
158C22×C4×C83#4112yes—
159C22×C4×C83#4112yes—
160C22×C4×C83#4112yes—
161C22×C4×C83#4112yes—
162C22×C4×C83C2411yescharacteristic
163C4⋊C4×C83—24no—
164C4⋊C4×C83—22no—
165C4⋊C4×C83—22no—
166C4⋊C4×C83—22no—
167C4⋊C4×C83—21no—
168C4⋊C4×C83—21no—
169C4⋊C4×C83—21no—
170C4⋊C4×C83—21no—
171C4⋊C4×C83—82no—
172C4⋊C4×C83—82no—
173C4⋊C4×C83C22×C414yes—
174C22⋊C4×C83—24no—
175C22⋊C4×C83—42no—
176C42×C83—22no—
177C42×C83—22no—
178C42×C83—42no—
179C23×C83—21no—
180C23×C83—21no—
181C23×C83—21no—
182C23×C83—21no—
183C23×C83C4×D412yes—
184C23×C83#12112yes—
185C23×C83#12412yes—
186C23×C83C22×D411yescharacteristic
187C22×D83—22no—
188C22×D83—41no—
189C22×D83—41no—
190C22×D83—41no—
191C22×D83—41no—
192C22×D83—41no—
193C22×D83—41no—
194C2×C4×C83—24no—
195C2×C4×C83—22no—
196C2×C4×C83—22no—
197C2×C4×C83—22no—
198C2×C4×C83—22no—
199C2×C4×C83—22no—
200C2×C4×C83—22no—
201C2×C4×C83—42no—
202C2×C4×C83—42no—
203C2×C4×C83—41no—
204C2×C4×C83—41no—
205C2×C4×C83—41no—
206C2×C4×C83—41no—
207C2×C4×C83—41no—
208C2×C4×C83—41no—
209C2×C4×C83—84no—
210C2×C4×C83#11714yes—
211C2×C4×C83C4×D412yes—
212C2×C4×C83C4×D414yes—
213C2×C4×C83#12112yes—
214C2×C4×C83#12314yes—
215C2×C4×C83#12412yes—
216C2×C4×C83#12614yes—
217C2×C4×C83C23×C411yescharacteristic
218C2×C4×C83C22×D411yescharacteristic
219C2×C4×C83C2×#4111yescharacteristic
220C2×C4×C83C2×#4111yescharacteristic
221C2×C4×C83C2×#4111yescharacteristic
222C2×C4×C83C2×#4111yescharacteristic
223C22⋊D83—48no—
224C2×Dic83—22no—
225C2×Dic83—41no—
226C2×Dic83—41no—
227C2×Dic83—42no—
228C2×Dic83—42no—
229C2×Dic83—42no—
230C2×Dic83—84no—
231D332—22no—
232D332—22no—
233D332—41no—
234D332—41no—
235C4×D83—48no—
236Dic166—22no—
237Dic166—22no—
238Dic166—41no—
239Dic166—41no—
240C83⋊C8—24no—
241C83⋊C8—22no—
242C83⋊C8—22no—
243C22×C83—22no—
244C22×C83—22no—
245C22×C83—22no—
246C22×C83—22no—
247C22×C83—22no—
248C22×C83—22no—
249C22×C83#38814yes—
250C22×C83C2×C4×D411yescharacteristic
251C22×C83C2×#12111yescharacteristic
252C22×C83C2×#12411yescharacteristic
253C22×C83#57611yescharacteristic
254C22×C83#57711yescharacteristic
255C22×C83#57711yescharacteristic
256C22×C83#57811yescharacteristic
257C2×D83—42no—
258C2×D83—42no—
259C2×D83—42no—
260C2×D83—81no—
261C2×D83—81no—
262C2×D83—81no—
263C2×D83—81no—
264C4×C83—44no—
265C4×C83—42no—
266C4×C83—42no—
267C4×C83—81no—
268C4×C83—81no—
269C4×C83#38814yes—
270C4×C83C2×#11711yescharacteristic
271C4×C83C2×C4×D411yescharacteristic
272C4×C83C2×#12311yescharacteristic
273C4×C83C2×#12611yescharacteristic
274Dic83—42no—
275Dic83—82no—
276#21092—831nomaximal, Sylow
277C2×C83—24no—
278C2×C83C2×#38811yescharacteristic
279C2×C83#294611yescharacteristic
280C2×C83#294711yescharacteristic
281C2×C83#306011yescharacteristic
282C2×C83#306111yescharacteristic
283C2×C83#313611yescharacteristic
284C2×C83#313711yescharacteristic
285D83—82no—
286C2×#417—831no—
287C2×#416—831no—
288C2×#405—831no—
289#2316D8311yescharacteristic, complemented
290#2311—831no—
291#1936—832no—
292#1935—832no—
293#1934—832no—
294#1933—832no—
295#1736—831no—
296C2×#340—831no—
297C83#2109211yescharacteristic, complemented, Sylow
298C22×#41—831no—
299C22×M16—831no—
300C23×C8—831no—
301C2×#117C2×D8311yescharacteristic
302C2×#117C2×D8311yescharacteristic
303C22×C4⋊C4—831no—
304C22×C4⋊C4C2×D8311yescharacteristic
305#417—838no—
306#416—1664no—
307C2×#103—832no—
308C2×#103—831no—
309C2×#103—831no—
310C2×#102—832no—
311C2×#102—831no—
312C2×#102—831no—
313#405—1664no—
314C2×#97—832no—
315C2×C4×C8—832no—
316#391—832no—
317#390—832no—
318#389—21no—
319#389—21no—
320#389C2×D8312yes—
321#389C2×D8312yes—
322#386—1662no—
323#380—22no—
324#380—1661no—
325#380—1661no—
326#340—832no—
327#340—832no—
328#340—832no—
329#340—832no—
330C2×#41—834no—
331C2×#41—1662no—
332C2×#41—1662no—
333C2×#41—1662no—
334C2×#41—1662no—
335C2×#41—1662no—
336C22×Q8—831no—
337C22×D4—831no—
338C22×D4—1661no—
339C23×C4—21no—
340C23×C4—1661no—
341C23×C4—1661no—
342C23×C4C22×D8311yescharacteristic
343C2×M16—834no—
344C2×M16—1662no—
345C2×M16—1662no—
346C22×C8—834no—
347C22×C8—831no—
348C22×C8—831no—
349C22×C8—832no—
350C22×C8—1662no—
351C22×C8—1662no—
352#117—22no—
353#117—22no—
354#117—22no—
355#117—22no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—832no—
361C2×C4⋊C4—1661no—
362C2×C4⋊C4—1662no—
363C2×C4⋊C4—1662no—
364C2×C4⋊C4C4×D8314yes—
365C2×C4⋊C4C22⋊D8311yescharacteristic
366C2×C4⋊C4C22⋊D8311yescharacteristic
367C2×C4⋊C4C22×D8311yescharacteristic
368C2×C4⋊C4C22×D8311yescharacteristic
369C2×C4⋊C4C22×D8312yes—
370C2×C22⋊C4—21no—
371C2×C22⋊C4—1662no—
372C2×C22⋊C4—1662no—
373C2×C22⋊C4C22⋊D8312yes—
374C2×C42—21no—
375C2×C42C22×D8312yes—
376#103—838no—
377#103—1662no—
378#103—1662no—
379#102—838no—
380#102—1662no—
381#102—1662no—
382#97—1664no—
383C4×C8—1664no—
384#95—22no—
385#95—22no—
386#95—1662no—
387#95—1662no—
388C24—1661no—
389C24C2×C22⋊D8311yescharacteristic
390#41—3324no—
391#41—3324no—
392#41—1668no—
393#41—1668no—
394C2×Q8—832no—
395C2×Q8—1661no—
396C2×Q8—1661no—
397C2×Q8—1661no—
398C2×Q8—1661no—
399C2×Q8—1661no—
400C2×D4—3322no—
401C2×D4—3322no—
402C2×D4—832no—
403C2×D4—1664no—
404C2×D4—1661no—
405C2×D4—1661no—
406C2×D4—1661no—
407C2×D4—1661no—
408C2×D4—1661no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—3322no—
416C22×C4—3322no—
417C22×C4—3322no—
418C22×C4—6641no—
419C22×C4—6641no—
420C22×C4—1662no—
421C22×C4—1661no—
422C22×C4—1664no—
423C22×C4C2×C4×D8312yes—
424C22×C4D4×D8312yes—
425C22×C4C2×C22⋊D8312yes—
426C22×C4C2×C22⋊D8312yes—
427C22×C4C2×C22⋊D8311yescharacteristic
428C22×C4#1305058912yes—
429C22×C4#1305059012yes—
430C22×C4#1305059112yes—
431C22×C4#1305059112yes—
432C22×C4C23×D8311yescharacteristic
433M16—1668no—
434C2×C8—834no—
435C2×C8—832no—
436C2×C8—832no—
437C2×C8—1668no—
438C2×C8—1662no—
439C2×C8—1661no—
440C2×C8—1661no—
441C2×C8—1661no—
442C2×C8—1661no—
443C2×C8—1662no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—82no—
453C4⋊C4—82no—
454C4⋊C4—3322no—
455C4⋊C4—1662no—
456C4⋊C4—1662no—
457C4⋊C4C2×C4×D8314yes—
458C22⋊C4—24no—
459C22⋊C4—42no—
460C22⋊C4—6642no—
461C22⋊C4—6642no—
462C42—22no—
463C42—22no—
464C42—42no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—3321no—
470C23—3321no—
471C23—3321no—
472C23—3321no—
473C23—3321no—
474C23—3321no—
475C23—1662no—
476C23C4×C22⋊D8312yes—
477C23#1458176412yes—
478C23#1458178112yes—
479C23C22×C22⋊D8311yescharacteristic
480Q8—3321no—
481Q8—3321no—
482Q8—1662no—
483Q8—1662no—
484D4—3328no—
485D4—3321no—
486D4—3321no—
487D4—1662no—
488D4—1662no—
489C2×C4—24no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—84no—
505C2×C4—3321no—
506C2×C4—3321no—
507C2×C4—3328no—
508C2×C4—3322no—
509C2×C4—3322no—
510C2×C4—3322no—
511C2×C4—6644no—
512C2×C4—1662no—
513C2×C4#1458175014yes—
514C2×C4C4×C22⋊D8312yes—
515C2×C4#1458175414yes—
516C2×C4#1458176412yes—
517C2×C4#1458177714yes—
518C2×C4#1458178112yes—
519C2×C4#1458178714yes—
520C2×C4C22×C4×D8311yescharacteristic
521C2×C4C2×D4×D8311yescharacteristic
522C2×C4C2×#1305058911yescharacteristic
523C2×C4C2×#1305059011yescharacteristic
524C2×C4C2×#1305059111yescharacteristic
525C2×C4C2×#1305059111yescharacteristic
526C8—1664no—
527C8—1662no—
528C8—1662no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—3322no—
536C22—3322no—
537C22—3322no—
538C22—6641no—
539C22—6641no—
540C22—6641no—
541C22—6641no—
542C22#1912759114yes—
543C22C2×C4×C22⋊D8311yescharacteristic
544C22C2×#1458176411yescharacteristic
545C22C2×#1458178111yescharacteristic
546C22#1912850411yescharacteristic
547C22#1912850911yescharacteristic
548C22#1912851211yescharacteristic
549C22#1912851611yescharacteristic
550C4—44no—
551C4—42no—
552C4—42no—
553C4—81no—
554C4—81no—
555C4—3322no—
556C4—6642no—
557C4#1912759114yes—
558C4C2×#1458175011yescharacteristic
559C4C2×#1458175411yescharacteristic
560C4C2×#1458177711yescharacteristic
561C4C2×#1458178711yescharacteristic
562C2—24no—
563C2—6642no—
564C2C2×#1912759111yescharacteristic
565C2#2491629511yescharacteristic
566C2#2491630611yescharacteristic
567C2#2491712211yescharacteristic
568C2#2491713011yescharacteristic
569C2#2491782211yescharacteristic
570C2#2491783311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2316C2
socleC23×C83C22×D4
prime radicalC22×#13050590C22
cosocleC22×C4×C83C24
hypercenter#2316D83
hypocenterC83#21092
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2316

Complements

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

NH ≅ G/Ntype
3. C2×#19127657C2semidirect
5. #24908159C2semidirect
6. #24908152C2semidirect
7. #24908145C2semidirect
8. #24908138C2semidirect
9. #24907184C2semidirect
10. C2×#19127463C2semidirect
11. C83×#2316C2semidirect
16. #19127700C4semidirect
72. #14581715C2×C4semidirect
75. #14581712C2×C4semidirect
289. #2316D83semidirect
297. C83#21092semidirect

Maximal subgroups

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

subgroupindex
C2×#191277002
C2×#191276952
C2×#191276572
#249100512
#249081592
#249081522
#249081452
#249081382
#249071842
C2×#191274632
C83×#23162
#2109283