Finite Group Databasebeta

← #35374509

Subgroups — #35374509

Order 21248.

598 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917534C211yesmaximal, characteristic
2#24916557C211yesmaximal, characteristic, complemented
3#24906334C211yesmaximal, characteristic, complemented
4#24906320C211yesmaximal, characteristic, complemented
5#24906313C211yesmaximal, characteristic, complemented
6#24906299C211yesmaximal, characteristic, complemented
7#24906280C211yesmaximal, characteristic, complemented
8#24906275C211yesmaximal, characteristic, complemented
9C8×#13050580C211yesmaximal, characteristic, complemented
10C8×#13050578C211yesmaximal, characteristic, complemented
11#24906246C211yesmaximal, characteristic, complemented
12#24905897C211yesmaximal, characteristic
13#24905865C211yesmaximal, characteristic, complemented
14#24905763C211yesmaximal, characteristic, complemented
15C83×#2988C211yesmaximal, characteristic
16#19128258C2211yescharacteristic
17#19128042C411yescharacteristic
18#19128042C411yescharacteristic
19#19127986C411yescharacteristic
20#19127981C411yescharacteristic
21#19127911C411yescharacteristic
22#19127906C411yescharacteristic
23#19127863C411yescharacteristic
24#19127863C411yescharacteristic
25#19127831C2211yescharacteristic
26#19127831C2211yescharacteristic
27#19127830—21no—
28#19127830—21no—
29C8×Dic166C2211yescharacteristic
30C8×Dic166C2211yescharacteristic
31C4×#13050580C2211yescharacteristic
32C4×#13050580C2211yescharacteristic
33C4×#13050578C2211yescharacteristic
34C4×#13050578C2211yescharacteristic
35#19127759—21no—
36#19127759—21no—
37#19127758C2211yescharacteristic
38#19127754C2211yescharacteristic
39#19127754C2211yescharacteristic
40#19127453C2211yescharacteristic
41#19127453C2211yescharacteristic
42#19127451C2211yescharacteristic
43#19127451C2211yescharacteristic
44Q8⋊(C83⋊C8)C2211yescharacteristic
45Q8⋊(C83⋊C8)C2211yescharacteristic
46#19127431C2211yescharacteristic
47#19127431C2211yescharacteristic
48#19127431C2211yescharacteristic
49#19127431C2211yescharacteristic
50D4⋊(C83⋊C8)C2211yescharacteristic
51D4⋊(C83⋊C8)C2211yescharacteristic
52#19127423C2211yescharacteristic
53#19127423C2211yescharacteristic
54#19127422C2211yescharacteristic
55C8×C83⋊C8C2211yescharacteristic
56C4×#41×C83C2211yescharacteristic
57C8×Q8×C83C2211yescharacteristic
58C8×Q8×C83C2211yescharacteristic
59C8×D4×C83—21no—
60C8×D4×C83—21no—
61C8×D4×C83C2211yescharacteristic
62C8×D4×C83C2211yescharacteristic
63#431×C83—21no—
64#431×C83C2211yescharacteristic
65C2×C4⋊C8×C83—21no—
66C2×C4⋊C8×C83C2211yescharacteristic
67C2×#13050580C2×C411yescharacteristic
68C2×#13050580C2×C411yescharacteristic
69C2×#13050578C2×C411yescharacteristic
70C2×#13050578C2×C411yescharacteristic
71#14581792C2×C411yescharacteristic
72#14581792C2×C411yescharacteristic
73#14581784—21no—
74#14581771—21no—
75#14581771C2×C411yescharacteristic
76#14581771C2×C411yescharacteristic
77C4×Dic166—21no—
78C4×Dic166C2311yescharacteristic
79C4×Dic166C2311yescharacteristic
80#14581751—21no—
81#14581725C2×C411yescharacteristic
82#14581725C2×C411yescharacteristic
83#14581723C2×C411yescharacteristic
84#14581723C2×C411yescharacteristic
85C4⋊(C83⋊C8)—21no—
86C4⋊(C83⋊C8)C2×C411yescharacteristic
87C4⋊(C83⋊C8)C2×C411yescharacteristic
88C4⋊(C83⋊C8)C2311yescharacteristic
89C4⋊(C83⋊C8)C2311yescharacteristic
90Dic83⋊C8—21no—
91Dic83⋊C8—21no—
92Dic83⋊C8—21no—
93Dic83⋊C8—21no—
94#14581716C2×C411yescharacteristic
95#14581716C2×C411yescharacteristic
96#14581715C2×C411yescharacteristic
97#14581715C2×C411yescharacteristic
98#14581715C2×C411yescharacteristic
99#14581715C2×C411yescharacteristic
100#14581713C2×C411yescharacteristic
101#14581713C2×C411yescharacteristic
102C22⋊(C83⋊C8)—21no—
103C22⋊(C83⋊C8)—21no—
104C22⋊(C83⋊C8)C2×C411yescharacteristic
105C22⋊(C83⋊C8)C2×C411yescharacteristic
106#14581703—21no—
107#14581702—21no—
108#14581702—21no—
109C4×C83⋊C8C2311yescharacteristic
110C4×C83⋊C8C2311yescharacteristic
111C4×C83⋊C8C2311yescharacteristic
112C4×C83⋊C8C2311yescharacteristic
113C8×Dic83—21no—
114C8×Dic83—21no—
115C2×#41×C83C2×C411yescharacteristic
116C22×C8×C83—21no—
117C22×C8×C83—21no—
118C22×C8×C83—21no—
119C4×Q8×C83C2×C411yescharacteristic
120C4×Q8×C83C2311yescharacteristic
121C4×D4×C83—21no—
122C4×D4×C83—21no—
123C4×D4×C83C2×C411yescharacteristic
124C4×D4×C83C2311yescharacteristic
125#117×C83—21no—
126#117×C83C2×C411yescharacteristic
127C2×C42×C83—21no—
128C2×C42×C83C2311yescharacteristic
129C4⋊C8×C83—21no—
130C4⋊C8×C83—21no—
131C4⋊C8×C83—21no—
132C4⋊C8×C83D411yescharacteristic
133C4⋊C8×C83D411yescharacteristic
134C4⋊C8×C83C2311yescharacteristic
135C4⋊C8×C83C2311yescharacteristic
136C22⋊C8×C83—21no—
137C22⋊C8×C83—21no—
138C22⋊C8×C83D411yescharacteristic
139C22⋊C8×C83D411yescharacteristic
140C4×C8×C83—21no—
141C4×C8×C83—21no—
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C2×Dic166—21no—
145C2×Dic166C22×C411yescharacteristic
146C2×Dic166C22×C411yescharacteristic
147#13050580—22no—
148#13050580—22no—
149#13050578—22no—
150#13050578—22no—
151C2×C83⋊C8—21no—
152C2×C83⋊C8—21no—
153C2×C83⋊C8—21no—
154C2×C83⋊C8—21no—
155C2×C83⋊C8C22×C411yescharacteristic
156C2×C83⋊C8C22×C411yescharacteristic
157C2×C83⋊C8C22×C411yescharacteristic
158C2×C83⋊C8C22×C411yescharacteristic
159C4⋊Dic83—21no—
160C4⋊Dic83—21no—
161C4⋊Dic83—21no—
162C4⋊Dic83C22×C411yescharacteristic
163C4⋊Dic83C22×C411yescharacteristic
164Dic83⋊C4—21no—
165Dic83⋊C4—21no—
166Dic83⋊C4—21no—
167Dic83⋊C4—21no—
168C22⋊Dic83—21no—
169C22⋊Dic83—21no—
170C4×Dic83—21no—
171C4×Dic83—21no—
172#41×C83—22no—
173#41×C83—22no—
174C2×Q8×C83C22×C411yescharacteristic
175C2×D4×C83—21no—
176C2×D4×C83C22×C411yescharacteristic
177C22×C4×C83—21no—
178C22×C4×C83—21no—
179C22×C4×C83—21no—
180C22×C4×C83C22×C411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83#4111yescharacteristic
183C2×C8×C83—21no—
184C2×C8×C83—21no—
185C2×C8×C83—81no—
186C2×C8×C83—81no—
187C2×C8×C83—81no—
188C2×C8×C83C2×D411yescharacteristic
189C2×C8×C83C2×D411yescharacteristic
190C2×C8×C83C2×D411yescharacteristic
191C2×C8×C83C2×D411yescharacteristic
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83C22×C411yescharacteristic
195C4⋊C4×C83C22×C411yescharacteristic
196C22⋊C4×C83—21no—
197C22⋊C4×C83—21no—
198C22⋊C4×C83—21no—
199C42×C83—21no—
200C42×C83—21no—
201C42×C83—21no—
202C42×C83C22×C411yescharacteristic
203C42×C83C2×D411yescharacteristic
204C42×C83#4111yescharacteristic
205C42×C83C2411yescharacteristic
206C23×C83—21no—
207C23×C83C4×D411yescharacteristic
208Q8×C83—21no—
209Q8×C83#13112yes—
210D4×C83—24no—
211D4×C83—21no—
212D4×C83#13112yes—
213C2×C4×C83—24no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—41no—
222C2×C4×C83—41no—
223C2×C4×C83—41no—
224C2×C4×C83—81no—
225C2×C4×C83C4×D411yescharacteristic
226C2×C4×C83C4×D411yescharacteristic
227C2×C4×C83C4×D411yescharacteristic
228C2×C4×C83C23×C411yescharacteristic
229C2×C4×C83C22×D411yescharacteristic
230C2×C4×C83C2×#4111yescharacteristic
231C2×Dic83—21no—
232C2×Dic83—21no—
233C2×Dic83—21no—
234C2×Dic83—21no—
235Dic166—21no—
236Dic166—21no—
237Dic166—21no—
238Dic166—21no—
239Dic166—81no—
240C8×C83—21no—
241C8×C83—21no—
242C8×C83—21no—
243C8×C83—21no—
244C8×C83—41no—
245C8×C83—41no—
246C83⋊C8—21no—
247C83⋊C8—21no—
248C83⋊C8—21no—
249C83⋊C8—21no—
250C83⋊C8—41no—
251C83⋊C8—41no—
252C83⋊C8—41no—
253C83⋊C8—41no—
254C22×C83—21no—
255C22×C83—22no—
256C22×C83—21no—
257C22×C83—21no—
258C22×C83—41no—
259C22×C83—41no—
260C22×C83C2×C4×D411yescharacteristic
261C4×C83—21no—
262C4×C83—22no—
263C4×C83—21no—
264C4×C83—21no—
265C4×C83—21no—
266C4×C83—41no—
267C4×C83—41no—
268C4×C83C2×#13111yescharacteristic
269C4×C83C2×#13111yescharacteristic
270C4×C83#57711yescharacteristic
271C4×C83#57811yescharacteristic
272Dic83—41no—
273Dic83—41no—
274Dic83—41no—
275Dic83—41no—
276#17921—831nomaximal, Sylow
277C2×C83—41no—
278C2×C83—42no—
279C2×C83#292911yescharacteristic
280C2×C83#294711yescharacteristic
281C2×C83#295711yescharacteristic
282#3107—831no—
283#2988D8311yescharacteristic
284#2985—831no—
285#1587—831no—
286#1585—831no—
287#1584—831no—
288#1582—831no—
289#1579—831no—
290SD16⋊C8—831no—
291C8×Q16—831no—
292C8×SD16—831no—
293#1573—831no—
294#1488—831no—
295#1481—831no—
296#1454—831no—
297C83#1792111yescharacteristic, complemented, Sylow
298#533—831no—
299C4×#41C2×D8311yescharacteristic
300#485—831no—
301#485—831no—
302#475—831no—
303#474—831no—
304#462—831no—
305#461—831no—
306#452—831no—
307#452—831no—
308#446—831no—
309#446—831no—
310#446—1661no—
311#446—1661no—
312C8×Q8—831no—
313C8×Q8—831no—
314C8×Q8C2×D8311yescharacteristic
315C8×Q8C2×D8311yescharacteristic
316C4×Q16—831no—
317C4×Q16—831no—
318C4×SD16—831no—
319C4×SD16—831no—
320C8×D4—21no—
321C8×D4—21no—
322C8×D4C2×D8311yescharacteristic
323C8×D4C2×D8311yescharacteristic
324#433—1661no—
325#433—1661no—
326#432—831no—
327#431—21no—
328#431—831no—
329#431—831no—
330#431C2×D8311yescharacteristic
331C2×C4⋊C8—21no—
332C2×C4⋊C8C2×D8311yescharacteristic
333#335—831no—
334#335—831no—
335#334—831no—
336#334—831no—
337Q8⋊C8—831no—
338Q8⋊C8—831no—
339Q8⋊C8—831no—
340Q8⋊C8—831no—
341Q8⋊C8—831no—
342Q8⋊C8—831no—
343D4⋊C8—831no—
344D4⋊C8—831no—
345#322—831no—
346#322—831no—
347#322—831no—
348C82—831no—
349C2×#41C4×D8311yescharacteristic
350C2×Q16—831no—
351C2×Q16—831no—
352C2×SD16—831no—
353C2×SD16—831no—
354C22×C8—21no—
355C22×C8—21no—
356C22×C8—21no—
357C4⋊Q8—831no—
358C4⋊Q8—831no—
359#125—1661no—
360C22⋊Q8—831no—
361C22⋊Q8—831no—
362C22⋊Q8—1661no—
363C4×Q8—831no—
364C4×Q8—831no—
365C4×Q8—1661no—
366C4×Q8C4×D8311yescharacteristic
367C4×Q8C22×D8311yescharacteristic
368C4×D4—21no—
369C4×D4—21no—
370C4×D4C4×D8311yescharacteristic
371C4×D4C22×D8311yescharacteristic
372#117—21no—
373#117—1661no—
374#117C4×D8311yescharacteristic
375C2×C42—21no—
376C2×C42C22×D8311yescharacteristic
377#107—831no—
378#107—831no—
379#106—831no—
380#106—831no—
381C4⋊C8—21no—
382C4⋊C8—21no—
383C4⋊C8—21no—
384C4⋊C8—831no—
385C4⋊C8—831no—
386C4⋊C8—831no—
387C4⋊C8—831no—
388C4⋊C8—1661no—
389C4⋊C8—1661no—
390C4⋊C8—1661no—
391C4⋊C8—1661no—
392C4⋊C8—1661no—
393C4⋊C8C22⋊D8311yescharacteristic
394C4⋊C8C22⋊D8311yescharacteristic
395C4⋊C8C22×D8311yescharacteristic
396C4⋊C8C22×D8311yescharacteristic
397#103—831no—
398#103—831no—
399#103—831no—
400#103—831no—
401#103—831no—
402#103—831no—
403#102—831no—
404#102—831no—
405C22⋊C8—21no—
406C22⋊C8—21no—
407C22⋊C8—831no—
408C22⋊C8—831no—
409C22⋊C8—1661no—
410C22⋊C8—1661no—
411C22⋊C8C22⋊D8311yescharacteristic
412C22⋊C8C22⋊D8311yescharacteristic
413#97—1661no—
414#97—1661no—
415#97—1661no—
416C4×C8—21no—
417C4×C8—21no—
418C4×C8—831no—
419C4×C8—831no—
420C4×C8—831no—
421C4×C8—831no—
422C4×C8—1661no—
423C4×C8—1661no—
424C4×C8C22×D8311yescharacteristic
425C4×C8C22×D8311yescharacteristic
426#41—22no—
427#41—22no—
428C2×Q8—831no—
429C2×Q8—831no—
430C2×Q8—1661no—
431C2×Q8C2×C4×D8311yescharacteristic
432C2×D4—21no—
433C2×D4C2×C4×D8311yescharacteristic
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4C2×C4×D8311yescharacteristic
438C22×C4C2×C22⋊D8311yescharacteristic
439C22×C4#1305059011yescharacteristic
440Q16—1662no—
441Q16—1662no—
442SD16—1662no—
443SD16—1662no—
444C2×C8—21no—
445C2×C8—21no—
446C2×C8—81no—
447C2×C8—81no—
448C2×C8—81no—
449C2×C8—831no—
450C2×C8—831no—
451C2×C8—831no—
452C2×C8—831no—
453C2×C8—1661no—
454C2×C8—1661no—
455C2×C8—1661no—
456C2×C8—1661no—
457C2×C8C2×C22⋊D8311yescharacteristic
458C2×C8C2×C22⋊D8311yescharacteristic
459C2×C8C2×C22⋊D8311yescharacteristic
460C2×C8C2×C22⋊D8311yescharacteristic
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—831no—
464C4⋊C4—831no—
465C4⋊C4—1661no—
466C4⋊C4—1661no—
467C4⋊C4—1661no—
468C4⋊C4—1661no—
469C4⋊C4—1661no—
470C4⋊C4—1661no—
471C4⋊C4—1661no—
472C4⋊C4C2×C4×D8311yescharacteristic
473C4⋊C4C2×C4×D8311yescharacteristic
474C22⋊C4—21no—
475C22⋊C4—21no—
476C22⋊C4—21no—
477C22⋊C4—1661no—
478C22⋊C4—1661no—
479C42—21no—
480C42—21no—
481C42—21no—
482C42—1661no—
483C42—1661no—
484C42C2×C4×D8311yescharacteristic
485C42C2×C22⋊D8311yescharacteristic
486C42#1305059011yescharacteristic
487C42C23×D8311yescharacteristic
488C23—21no—
489C23C4×C22⋊D8311yescharacteristic
490Q8—21no—
491Q8—6641no—
492Q8—1661no—
493Q8—1661no—
494Q8—1661no—
495Q8—1661no—
496Q8#1458179812yes—
497D4—24no—
498D4—21no—
499D4#1458179812yes—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—81no—
512C2×C4—1661no—
513C2×C4—1661no—
514C2×C4—1661no—
515C2×C4—1661no—
516C2×C4C4×C22⋊D8311yescharacteristic
517C2×C4C4×C22⋊D8311yescharacteristic
518C2×C4C4×C22⋊D8311yescharacteristic
519C2×C4C22×C4×D8311yescharacteristic
520C2×C4C22×C22⋊D8311yescharacteristic
521C2×C4C2×#1305059011yescharacteristic
522C8—21no—
523C8—21no—
524C8—21no—
525C8—21no—
526C8—41no—
527C8—41no—
528C8—3321no—
529C8—3321no—
530C8—3321no—
531C8—3321no—
532C8—1661no—
533C8—1661no—
534C8—1661no—
535C8—1661no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—22no—
540C22—41no—
541C22—41no—
542C22C2×C4×C22⋊D8311yescharacteristic
543C4—21no—
544C4—21no—
545C4—21no—
546C4—21no—
547C4—22no—
548C4—41no—
549C4—41no—
550C4—3321no—
551C4—3321no—
552C4—3321no—
553C4—3321no—
554C4C2×#1458179811yescharacteristic
555C4C2×#1458179811yescharacteristic
556C4#1912851211yescharacteristic
557C4#1912851611yescharacteristic
558C2—41no—
559C2—42no—
560C2#2491615711yescharacteristic
561C2#2491630211yescharacteristic
562C2#2491637811yescharacteristic
subgroupG/H
centerC2×C4C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#2988C2
socleC22×C83C2×C4×D4
prime radicalC2×#41×C83C2×C4
cosocleC42×C83C24
hypercenter#2988D83
hypocenterC83#17921
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ #431 ◁ #2988

Complements

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

NH ≅ G/Ntype
2. #24916557C2semidirect
3. #24906334C2semidirect
4. #24906320C2semidirect
5. #24906313C2semidirect
6. #24906299C2semidirect
7. #24906280C2semidirect
8. #24906275C2semidirect
9. C8×#13050580C2semidirect
10. C8×#13050578C2semidirect
11. #24906246C2semidirect
13. #24905865C2semidirect
14. #24905763C2semidirect
297. C83#17921semidirect

Maximal subgroups

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

subgroupindex
#249175342
#249165572
#249063342
#249063202
#249063132
#249062992
#249062802
#249062752
C8×#130505802
C8×#130505782
#249062462
#249058972
#249058652
#249057632
C83×#29882
#1792183