Finite Group Databasebeta

← #35433214

Subgroups — #35433214

Order 21248.

1010 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127700C211yesmaximal, characteristic
2C2×#19127695C211yesmaximal, characteristic
3#24910051C211yesmaximal, characteristic
4#24908882C212yesmaximal, complemented
5#24908877C212yesmaximal, complemented
6#24908872C212yesmaximal, complemented
7#24908867C212yesmaximal, complemented
8#24907184C211yesmaximal, characteristic, complemented
9C2×#19127463C211yesmaximal, characteristic, complemented
10#24907160C211yesmaximal, characteristic, complemented
11C83×#2317C211yesmaximal, characteristic, complemented
12C22×#13050590C2211yescharacteristic
13C22×#13050574C2211yescharacteristic
14C23×C83⋊C8C2211yescharacteristic
15C22×C4⋊Dic83C2211yescharacteristic
16#19127700—24no—
17#19127695—24no—
18C2×#14581715C2212yes—
19C2×#14581715C2211yescharacteristic
20C2×#14581715C2211yescharacteristic
21C2×#14581712C2212yes—
22C2×#14581712C2211yescharacteristic
23C2×#14581712C2211yescharacteristic
24#19127607C2212yes—
25#19127603C2212yes—
26#19127578—22no—
27#19127558—21no—
28#19127558—21no—
29#19127463C2212yes—
30#19127463C2212yes—
31#19127463C2212yes—
32#19127463C2212yes—
33#19127456C2212yes—
34#19127456C2212yes—
35C2×#117×C83C2211yescharacteristic
36C2×#117×C83C2211yescharacteristic
37C22×C4⋊C4×C83C2211yescharacteristic
38#389×C83—21no—
39#389×C83—21no—
40#385×C83—22no—
41#384×C83C2212yes—
42#382×C83C2212yes—
43C2×#13050590—22no—
44C2×#13050590—22no—
45C2×#13050590—22no—
46C2×#13050590—22no—
47C2×#13050590—22no—
48C2×#13050590D414yes—
49C22×Dic166C2311yescharacteristic
50C22×D332C2311yescharacteristic
51C22×C22⋊D83—21no—
52C23×Dic83—21no—
53C22×C4×D83—21no—
54C2×#13050574—22no—
55C2×#13050574—22no—
56C2×#13050574—22no—
57C22×C83⋊C8—22no—
58C22×C83⋊C8—22no—
59C22×C83⋊C8—22no—
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2312yes—
63C2×C4⋊Dic83—21no—
64C2×C4⋊Dic83—22no—
65C2×C4⋊Dic83—22no—
66C2×C4⋊Dic83C2312yes—
67C2×(C2×D83)⋊C4—22no—
68C2×(C2×D83)⋊C4—22no—
69#14581715—22no—
70#14581715—22no—
71#14581715—24no—
72#14581712—22no—
73#14581712—22no—
74#14581712—24no—
75C83⋊#95—22no—
76C83⋊#95—22no—
77C23×C4×C83—21no—
78C23×C4×C83C2311yescharacteristic
79#117×C83—22no—
80#117×C83—22no—
81#117×C83—22no—
82#117×C83—22no—
83C2×C4⋊C4×C83—22no—
84C2×C4⋊C4×C83—22no—
85C2×C4⋊C4×C83—21no—
86C2×C4⋊C4×C83—22no—
87C2×C4⋊C4×C83—22no—
88C2×C4⋊C4×C83D411yescharacteristic
89C2×C4⋊C4×C83D411yescharacteristic
90C2×C4⋊C4×C83C2312yes—
91C2×C4⋊C4×C83C2311yescharacteristic
92C2×C4⋊C4×C83C2311yescharacteristic
93C2×C22⋊C4×C83—21no—
94C2×C22⋊C4×C83D412yes—
95C2×C42×C83—21no—
96C2×C42×C83C2312yes—
97#95×C83—22no—
98#95×C83—22no—
99C23×D83—21no—
100#13050590—28no—
101#13050590—28no—
102#13050590—44no—
103#13050590—44no—
104C2×Dic166—21no—
105C2×Dic166—21no—
106C2×Dic166—21no—
107C2×Dic166—21no—
108C2×Dic166—21no—
109C2×Dic166C2×D412yes—
110C2×D332—21no—
111C2×D332—21no—
112C2×D332—21no—
113C2×D332—21no—
114C2×D332—21no—
115C2×D332C2×D412yes—
116C2×C22⋊D83—24no—
117C2×C22⋊D83—42no—
118C2×C22⋊D83—42no—
119C22×Dic83—21no—
120C22×Dic83—22no—
121C22×Dic83—42no—
122C22×Dic83—81no—
123C22×Dic83—81no—
124C2×C4×D83—24no—
125C2×C4×D83—42no—
126C2×C4×D83—42no—
127#13050574—44no—
128C2×C83⋊C8—21no—
129C2×C83⋊C8—21no—
130C2×C83⋊C8—22no—
131C2×C83⋊C8—21no—
132C2×C83⋊C8—21no—
133C2×C83⋊C8—21no—
134C2×C83⋊C8—21no—
135C2×C83⋊C8—22no—
136C2×C83⋊C8—22no—
137C2×C83⋊C8—44no—
138C4⋊Dic83—22no—
139C4⋊Dic83—22no—
140C4⋊Dic83—42no—
141(C2×D83)⋊C4—82no—
142(C2×D83)⋊C4—82no—
143C24×C83C2×D411yescharacteristic
144C22×C4×C83—21no—
145C22×C4×C83—22no—
146C22×C4×C83—21no—
147C22×C4×C83—42no—
148C22×C4×C83—81no—
149C22×C4×C83—81no—
150C22×C4×C83C2×D412yes—
151C22×C4×C83C2×D412yes—
152C22×C4×C83C2×D412yes—
153C22×C4×C83C2×D411yescharacteristic
154C22×C4×C83#4112yes—
155C22×C4×C83#4112yes—
156C22×C4×C83#4112yes—
157C22×C4×C83#4112yes—
158C22×C4×C83#4112yes—
159C22×C4×C83C2411yescharacteristic
160C4⋊C4×C83—22no—
161C4⋊C4×C83—22no—
162C4⋊C4×C83—22no—
163C4⋊C4×C83—21no—
164C4⋊C4×C83—21no—
165C4⋊C4×C83—22no—
166C4⋊C4×C83—21no—
167C4⋊C4×C83—21no—
168C4⋊C4×C83—42no—
169C4⋊C4×C83—82no—
170C4⋊C4×C83—82no—
171C22⋊C4×C83—24no—
172C22⋊C4×C83—42no—
173C42×C83—22no—
174C42×C83—22no—
175C42×C83—42no—
176C23×C83—21no—
177C23×C83—21no—
178C23×C83—21no—
179C23×C83—21no—
180C23×C83#12112yes—
181C23×C83#12312yes—
182C23×C83#12412yes—
183C23×C83C22×D411yescharacteristic
184C22×D83—22no—
185C22×D83—41no—
186C22×D83—41no—
187C22×D83—41no—
188C22×D83—41no—
189C22×D83—41no—
190C22×D83—41no—
191C2×C4×C83—22no—
192C2×C4×C83—22no—
193C2×C4×C83—22no—
194C2×C4×C83—22no—
195C2×C4×C83—22no—
196C2×C4×C83—22no—
197C2×C4×C83—41no—
198C2×C4×C83—41no—
199C2×C4×C83—42no—
200C2×C4×C83—41no—
201C2×C4×C83—42no—
202C2×C4×C83—41no—
203C2×C4×C83—42no—
204C2×C4×C83—41no—
205C2×C4×C83—41no—
206C2×C4×C83—84no—
207C2×C4×C83#12112yes—
208C2×C4×C83#12312yes—
209C2×C4×C83#12314yes—
210C2×C4×C83#12314yes—
211C2×C4×C83#12412yes—
212C2×C4×C83#12614yes—
213C2×C4×C83#12614yes—
214C2×C4×C83C22×D411yescharacteristic
215C2×C4×C83C2×#4111yescharacteristic
216C2×C4×C83C2×#4111yescharacteristic
217C2×C4×C83C2×#4111yescharacteristic
218C2×C4×C83C2×#4111yescharacteristic
219C2×C4×C83C2×#4111yescharacteristic
220C22⋊D83—48no—
221C2×Dic83—22no—
222C2×Dic83—41no—
223C2×Dic83—41no—
224C2×Dic83—42no—
225C2×Dic83—42no—
226C2×Dic83—42no—
227C2×Dic83—84no—
228D332—22no—
229D332—22no—
230D332—41no—
231D332—41no—
232C4×D83—48no—
233Dic166—22no—
234Dic166—22no—
235Dic166—41no—
236Dic166—41no—
237C83⋊C8—41no—
238C83⋊C8—41no—
239C83⋊C8—42no—
240C22×C83—22no—
241C22×C83—22no—
242C22×C83—22no—
243C22×C83—22no—
244C22×C83—22no—
245C22×C83—22no—
246C22×C83#39714yes—
247C22×C83C2×#12111yescharacteristic
248C22×C83C2×#12311yescharacteristic
249C22×C83C2×#12411yescharacteristic
250C22×C83#57611yescharacteristic
251C22×C83#57711yescharacteristic
252C22×C83#57711yescharacteristic
253C22×C83#57811yescharacteristic
254C2×D83—42no—
255C2×D83—42no—
256C2×D83—42no—
257C2×D83—81no—
258C2×D83—81no—
259C2×D83—81no—
260C2×D83—81no—
261C4×C83—42no—
262C4×C83—42no—
263C4×C83—81no—
264C4×C83—82no—
265C4×C83—81no—
266C4×C83#39714yes—
267C4×C83C2×#12311yescharacteristic
268C4×C83C2×#12311yescharacteristic
269C4×C83C2×#12611yescharacteristic
270C4×C83C2×#12611yescharacteristic
271Dic83—42no—
272Dic83—82no—
273#23634—831nomaximal, Sylow
274C2×C83—24no—
275C2×C83C2×#39711yescharacteristic
276C2×C83#304511yescharacteristic
277C2×C83#304611yescharacteristic
278C2×C83#309811yescharacteristic
279C2×C83#309911yescharacteristic
280C2×C83#313611yescharacteristic
281C2×C83#313711yescharacteristic
282D83—82no—
283C2×#417—831no—
284C2×#416—831no—
285#2317D8311yescharacteristic, complemented
286#2311—831no—
287#2074—832no—
288#2073—832no—
289#2072—832no—
290#2071—832no—
291#1736—831no—
292C2×#340—831no—
293#1729—831no—
294C83#2363411yescharacteristic, complemented, Sylow
295C22×#41—831no—
296C22×M16—831no—
297C23×C8—831no—
298C2×#117C2×D8311yescharacteristic
299C2×#117C2×D8311yescharacteristic
300C22×C4⋊C4—831no—
301C22×C4⋊C4C2×D8311yescharacteristic
302#417—1664no—
303#416—1664no—
304C2×#103—831no—
305C2×#103—832no—
306C2×#103—831no—
307C2×#102—831no—
308C2×#102—832no—
309C2×#102—831no—
310#391—832no—
311#390—832no—
312#389—21no—
313#389—21no—
314#386—1662no—
315#385—22no—
316#384C2×D8312yes—
317#382C2×D8312yes—
318#380—1661no—
319#380—1661no—
320#340—832no—
321#340—832no—
322#340—832no—
323#340—832no—
324#336—832no—
325#336—832no—
326C2×#41—834no—
327C2×#41—1662no—
328C2×#41—1662no—
329C2×#41—1662no—
330C2×#41—1662no—
331C2×#41—1662no—
332C22×Q8—831no—
333C22×D4—831no—
334C22×D4—1661no—
335C23×C4—21no—
336C23×C4—1661no—
337C23×C4—1661no—
338C23×C4C22×D8311yescharacteristic
339C2×M16—1662no—
340C2×M16—1662no—
341C2×M16—1662no—
342C22×C8—831no—
343C22×C8—831no—
344C22×C8—832no—
345C22×C8—1662no—
346C22×C8—1662no—
347C22×C8—1662no—
348#117—22no—
349#117—22no—
350#117—22no—
351#117—22no—
352C2×C4⋊C4—22no—
353C2×C4⋊C4—22no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—832no—
358C2×C4⋊C4—1661no—
359C2×C4⋊C4—1662no—
360C2×C4⋊C4—1662no—
361C2×C4⋊C4C22⋊D8311yescharacteristic
362C2×C4⋊C4C22⋊D8311yescharacteristic
363C2×C4⋊C4C22×D8312yes—
364C2×C4⋊C4C22×D8311yescharacteristic
365C2×C4⋊C4C22×D8311yescharacteristic
366C2×C22⋊C4—21no—
367C2×C22⋊C4—1662no—
368C2×C22⋊C4—1662no—
369C2×C22⋊C4C22⋊D8312yes—
370C2×C42—21no—
371C2×C42C22×D8312yes—
372#103—1662no—
373#103—1662no—
374#103—1664no—
375#102—1662no—
376#102—1662no—
377#102—1664no—
378#95—22no—
379#95—22no—
380#95—1662no—
381#95—1662no—
382C24—1661no—
383C24C2×C22⋊D8311yescharacteristic
384#41—3324no—
385#41—3324no—
386#41—1668no—
387#41—1668no—
388C2×Q8—832no—
389C2×Q8—1661no—
390C2×Q8—1661no—
391C2×Q8—1661no—
392C2×Q8—1661no—
393C2×Q8—1661no—
394C2×D4—3322no—
395C2×D4—3322no—
396C2×D4—832no—
397C2×D4—1664no—
398C2×D4—1661no—
399C2×D4—1661no—
400C2×D4—1661no—
401C2×D4—1661no—
402C2×D4—1661no—
403C22×C4—21no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—42no—
407C22×C4—81no—
408C22×C4—81no—
409C22×C4—3322no—
410C22×C4—3322no—
411C22×C4—3322no—
412C22×C4—6641no—
413C22×C4—6641no—
414C22×C4—1662no—
415C22×C4—1661no—
416C22×C4—1664no—
417C22×C4D4×D8312yes—
418C22×C4C2×C22⋊D8312yes—
419C22×C4C2×C22⋊D8312yes—
420C22×C4C2×C22⋊D8311yescharacteristic
421C22×C4#1305058912yes—
422C22×C4#1305059012yes—
423C22×C4#1305059012yes—
424C22×C4#1305059112yes—
425C22×C4#1305059112yes—
426C22×C4C23×D8311yescharacteristic
427M16—3324no—
428C2×C8—3324no—
429C2×C8—1661no—
430C2×C8—1661no—
431C2×C8—1662no—
432C2×C8—1661no—
433C2×C8—1661no—
434C2×C8—1662no—
435C2×C8—1661no—
436C2×C8—1661no—
437C2×C8—1662no—
438C4⋊C4—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—21no—
442C4⋊C4—21no—
443C4⋊C4—22no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—42no—
447C4⋊C4—82no—
448C4⋊C4—82no—
449C4⋊C4—3322no—
450C4⋊C4—1662no—
451C4⋊C4—1662no—
452C22⋊C4—24no—
453C22⋊C4—42no—
454C22⋊C4—6642no—
455C22⋊C4—6642no—
456C42—22no—
457C42—22no—
458C42—42no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—3321no—
464C23—3321no—
465C23—3321no—
466C23—3321no—
467C23—3321no—
468C23—3321no—
469C23—1662no—
470C23#1458176312yes—
471C23#1458177512yes—
472C23#1458178112yes—
473C23C22×C22⋊D8311yescharacteristic
474Q8—3321no—
475Q8—3321no—
476Q8—1662no—
477Q8—1662no—
478D4—3328no—
479D4—3321no—
480D4—3321no—
481D4—1662no—
482D4—1662no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—41no—
495C2×C4—41no—
496C2×C4—42no—
497C2×C4—41no—
498C2×C4—84no—
499C2×C4—3321no—
500C2×C4—3321no—
501C2×C4—3328no—
502C2×C4—3322no—
503C2×C4—3322no—
504C2×C4—3322no—
505C2×C4—6644no—
506C2×C4—1662no—
507C2×C4#1458176312yes—
508C2×C4#1458177512yes—
509C2×C4#1458177714yes—
510C2×C4#1458177714yes—
511C2×C4#1458178112yes—
512C2×C4#1458178714yes—
513C2×C4#1458178714yes—
514C2×C4C2×D4×D8311yescharacteristic
515C2×C4C2×#1305058911yescharacteristic
516C2×C4C2×#1305059011yescharacteristic
517C2×C4C2×#1305059011yescharacteristic
518C2×C4C2×#1305059111yescharacteristic
519C2×C4C2×#1305059111yescharacteristic
520C8—3321no—
521C8—3321no—
522C8—3322no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—3322no—
530C22—3322no—
531C22—3322no—
532C22—6641no—
533C22—6641no—
534C22—6641no—
535C22—6641no—
536C22#1912762714yes—
537C22C2×#1458176311yescharacteristic
538C22C2×#1458177511yescharacteristic
539C22C2×#1458178111yescharacteristic
540C22#1912850411yescharacteristic
541C22#1912850911yescharacteristic
542C22#1912851211yescharacteristic
543C22#1912851611yescharacteristic
544C4—42no—
545C4—42no—
546C4—82no—
547C4—81no—
548C4—81no—
549C4—3322no—
550C4—6642no—
551C4#1912762714yes—
552C4C2×#1458177711yescharacteristic
553C4C2×#1458177711yescharacteristic
554C4C2×#1458178711yescharacteristic
555C4C2×#1458178711yescharacteristic
556C2—24no—
557C2—6642no—
558C2C2×#1912762711yescharacteristic
559C2#2491699811yescharacteristic
560C2#2491700911yescharacteristic
561C2#2491744111yescharacteristic
562C2#2491745211yescharacteristic
563C2#2491782211yescharacteristic
564C2#2491783311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2317C2
socleC23×C83C22×D4
prime radicalC22×#13050590C22
hypercenter#2317D83
hypocenterC83#23634
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2317

Complements

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

NH ≅ G/Ntype
4. #24908882C2semidirect
5. #24908877C2semidirect
6. #24908872C2semidirect
7. #24908867C2semidirect
8. #24907184C2semidirect
9. C2×#19127463C2semidirect
10. #24907160C2semidirect
11. C83×#2317C2semidirect
285. #2317D83semidirect
294. C83#23634semidirect

Maximal subgroups

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

subgroupindex
C2×#191277002
C2×#191276952
#249100512
#249088822
#249088772
#249088722
#249088672
#249071842
C2×#191274632
#249071602
C83×#23172
#2363483