Finite Group Databasebeta

← #35374552

Subgroups — #35374552

Order 21248.

586 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917500C211yesmaximal, characteristic
2#24916559C211yesmaximal, characteristic
3#24916542C211yesmaximal, characteristic
4#24906336C211yesmaximal, characteristic, complemented
5#24906313C211yesmaximal, characteristic, complemented
6#24906311C211yesmaximal, characteristic, complemented
7#24906282C211yesmaximal, characteristic, complemented
8#24906281C211yesmaximal, characteristic, complemented
9#24906267C211yesmaximal, characteristic, complemented
10#24906267C211yesmaximal, characteristic, complemented
11#24906240C211yesmaximal, characteristic
12#24905908C211yesmaximal, characteristic
13#24905882C211yesmaximal, characteristic
14#24905761C211yesmaximal, characteristic
15C83×#1587C211yesmaximal, characteristic, complemented
16#19128255C2211yescharacteristic
17C4×Q8×D83C2211yescharacteristic
18#19128038C411yescharacteristic
19#19128035C411yescharacteristic
20#19127995C411yescharacteristic
21#19127994C411yescharacteristic
22#19127910C411yescharacteristic
23#19127909C411yescharacteristic
24#19127858C411yescharacteristic
25#19127855C411yescharacteristic
26#19127831C2211yescharacteristic
27#19127829—21no—
28#19127828—21no—
29Q8×C83⋊C8C2211yescharacteristic
30Q8×C83⋊C8C2211yescharacteristic
31C8×Dic166—21no—
32C8×Dic166C2211yescharacteristic
33#19127825—21no—
34#19127825C2211yescharacteristic
35#19127825C2211yescharacteristic
36C4×Dic332C2211yescharacteristic
37C4×#13050580C2211yescharacteristic
38C4×#13050580C2211yescharacteristic
39#19127763—21no—
40#19127761—21no—
41#19127757C2211yescharacteristic
42#19127753—21no—
43#19127753C2211yescharacteristic
44#19127752—21no—
45#19127752C2211yescharacteristic
46#19127752C2211yescharacteristic
47C4×#13050573C2211yescharacteristic
48#19127453C2211yescharacteristic
49#19127453C2211yescharacteristic
50#19127452C2211yescharacteristic
51#19127433C2211yescharacteristic
52#19127433C2211yescharacteristic
53Q8⋊(C83⋊C8)C2211yescharacteristic
54Q8⋊(C83⋊C8)C2211yescharacteristic
55#19127431C2211yescharacteristic
56#19127431C2211yescharacteristic
57#19127422C2211yescharacteristic
58#19127422C2211yescharacteristic
59C8×C83⋊C8C2211yescharacteristic
60#446×C83C2211yescharacteristic
61C8×Q8×C83C2211yescharacteristic
62C4×Q16×C83C2211yescharacteristic
63#334×C83C2211yescharacteristic
64Q8⋊C8×C83C2211yescharacteristic
65Q8⋊C8×C83C2211yescharacteristic
66#322×C83C2211yescharacteristic
67C2×Q8×D83C2×C411yescharacteristic
68C2×Dic332C2×C411yescharacteristic
69C2×#13050580C2×C411yescharacteristic
70C2×#13050580C2×C411yescharacteristic
71C2×#13050573—21no—
72#14581791C2×C411yescharacteristic
73#14581790C2×C411yescharacteristic
74#14581782—21no—
75#14581772C2×C411yescharacteristic
76#14581770—21no—
77#14581768C2×C411yescharacteristic
78Q8×Dic83C2×C411yescharacteristic
79C4×Dic166—21no—
80C4×Dic166—21no—
81C4×Dic166C2311yescharacteristic
82C4×Dic166C2311yescharacteristic
83#14581757—21no—
84#14581757C2×C411yescharacteristic
85#14581749—21no—
86C4⋊C4×D83—21no—
87C4⋊C4×D83C2×C411yescharacteristic
88C42×D83—21no—
89C42×D83C2311yescharacteristic
90#14581726C2×C411yescharacteristic
91#14581725C2×C411yescharacteristic
92#14581725C2×C411yescharacteristic
93C4⋊(C83⋊C8)—21no—
94C4⋊(C83⋊C8)—21no—
95C4⋊(C83⋊C8)C2311yescharacteristic
96C4⋊(C83⋊C8)C2311yescharacteristic
97Dic83⋊C8—21no—
98Dic83⋊C8—21no—
99Dic83⋊C8—21no—
100Dic83⋊C8—21no—
101Dic83⋊C8C2×C411yescharacteristic
102Dic83⋊C8C2×C411yescharacteristic
103Dic83⋊C8D411yescharacteristic
104Dic83⋊C8D411yescharacteristic
105#14581717C2×C411yescharacteristic
106#14581717C2×C411yescharacteristic
107#14581716C2×C411yescharacteristic
108#14581716C2×C411yescharacteristic
109#14581715C2×C411yescharacteristic
110#14581715C2×C411yescharacteristic
111(C2×D83)⋊C8—21no—
112(C2×D83)⋊C8—21no—
113(C2×D83)⋊C8C2×C411yescharacteristic
114(C2×D83)⋊C8C2×C411yescharacteristic
115(C2×D83)⋊C8D411yescharacteristic
116(C2×D83)⋊C8D411yescharacteristic
117#14581702—21no—
118#14581702—21no—
119#14581702—21no—
120C4×C83⋊C8—21no—
121C4×C83⋊C8—21no—
122C4×C83⋊C8C2311yescharacteristic
123C4×C83⋊C8C2311yescharacteristic
124C4×C83⋊C8C2311yescharacteristic
125C8×Dic83—21no—
126C8×Dic83—21no—
127C8×Dic83—21no—
128C8×Dic83—21no—
129C2×Q16×C83C2×C411yescharacteristic
130C4×Q8×C83C2311yescharacteristic
131C4×Q8×C83C2311yescharacteristic
132#107×C83C2×C411yescharacteristic
133C4⋊C8×C83—21no—
134C4⋊C8×C83—21no—
135C4⋊C8×C83C2311yescharacteristic
136C4⋊C8×C83C2311yescharacteristic
137#103×C83C2×C411yescharacteristic
138#103×C83C2×C411yescharacteristic
139#97×C83—21no—
140C4×C8×C83—21no—
141C4×C8×C83C2311yescharacteristic
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C2×Dic166—21no—
145C2×Dic166—21no—
146C2×Dic166C22×C411yescharacteristic
147C2×Dic166C22×C411yescharacteristic
148Q8×D83—22no—
149Q8×D83—22no—
150C2×C4×D83—21no—
151C2×C4×D83—21no—
152C2×C4×D83C22×C411yescharacteristic
153C2×C4×D83C2×D411yescharacteristic
154C2×C4×D83#4111yescharacteristic
155Dic332—22no—
156#13050580—22no—
157#13050580—22no—
158#13050573—42no—
159C2×C83⋊C8—21no—
160C2×C83⋊C8—21no—
161C2×C83⋊C8—21no—
162C2×C83⋊C8C22×C411yescharacteristic
163C2×C83⋊C8C22×C411yescharacteristic
164C2×C83⋊C8C2×D411yescharacteristic
165C2×C83⋊C8C2×D411yescharacteristic
166C4⋊Dic83—21no—
167C4⋊Dic83—21no—
168C4⋊Dic83C22×C411yescharacteristic
169C4⋊Dic83C22×C411yescharacteristic
170Dic83⋊C4—21no—
171Dic83⋊C4—21no—
172Dic83⋊C4—21no—
173Dic83⋊C4—21no—
174Dic83⋊C4—21no—
175Dic83⋊C4—21no—
176Dic83⋊C4—21no—
177Dic83⋊C4—21no—
178(C2×D83)⋊C4—21no—
179(C2×D83)⋊C4—21no—
180C4×Dic83—21no—
181C4×Dic83—21no—
182C4×Dic83—21no—
183C4×Dic83—21no—
184C4×Dic83C22×C411yescharacteristic
185C4×Dic83C2×D411yescharacteristic
186C4×Dic83#4111yescharacteristic
187C2×Q8×C83C22×C411yescharacteristic
188C2×Q8×C83C22×C411yescharacteristic
189Q16×C83—22no—
190C2×C8×C83—21no—
191C2×C8×C83—21no—
192C2×C8×C83—21no—
193C2×C8×C83C22×C411yescharacteristic
194C2×C8×C83C22×C411yescharacteristic
195C2×C8×C83C2×D411yescharacteristic
196C2×C8×C83C2×D411yescharacteristic
197C4⋊C4×C83—21no—
198C4⋊C4×C83—21no—
199C4⋊C4×C83C22×C411yescharacteristic
200C4⋊C4×C83C22×C411yescharacteristic
201C42×C83—21no—
202C42×C83—21no—
203C42×C83C2411yescharacteristic
204C22×D83C4×D411yescharacteristic
205Q8×C83—21no—
206Q8×C83—21no—
207Q8×C83—21no—
208Q8×C83#13112yes—
209C2×C4×C83—21no—
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—21no—
213C2×C4×C83C23×C411yescharacteristic
214C2×C4×C83C22×D411yescharacteristic
215C2×C4×C83C2×#4111yescharacteristic
216C2×Dic83—21no—
217C2×Dic83—21no—
218C2×Dic83—21no—
219C2×Dic83—21no—
220C2×Dic83C4×D411yescharacteristic
221C2×Dic83C4×D411yescharacteristic
222C2×Dic83C4×D411yescharacteristic
223C4×D83—24no—
224C4×D83—21no—
225C4×D83—21no—
226C4×D83—21no—
227C4×D83—21no—
228C4×D83—41no—
229C4×D83—81no—
230Dic166—24no—
231Dic166—21no—
232Dic166—21no—
233Dic166—21no—
234Dic166—81no—
235Dic166#13112yes—
236C8×C83—22no—
237C8×C83—21no—
238C8×C83—21no—
239C8×C83—21no—
240C8×C83—21no—
241C8×C83—41no—
242C8×C83—41no—
243C83⋊C8—21no—
244C83⋊C8—21no—
245C83⋊C8—21no—
246C83⋊C8—21no—
247C83⋊C8—41no—
248C83⋊C8—41no—
249C83⋊C8—41no—
250C22×C83C2×C4×D411yescharacteristic
251C2×D83—21no—
252C2×D83—21no—
253C2×D83—21no—
254C4×C83—22no—
255C4×C83—21no—
256C4×C83—41no—
257C4×C83—41no—
258C4×C83—41no—
259C4×C83C2×#13111yescharacteristic
260C4×C83C2×#13111yescharacteristic
261C4×C83#57811yescharacteristic
262C4×C83#57811yescharacteristic
263Dic83—21no—
264Dic83—22no—
265Dic83—21no—
266Dic83—21no—
267Dic83—41no—
268Dic83—41no—
269Dic83—41no—
270#17925—831nomaximal, Sylow
271C2×C83#292911yescharacteristic
272C2×C83#295011yescharacteristic
273C2×C83#295611yescharacteristic
274D83—41no—
275#3104—831no—
276#2985—831no—
277#2982—831no—
278#1587—831no—
279#1587D8311yescharacteristic, complemented
280#1584—831no—
281#1584—831no—
282#1579—831no—
283#1579—831no—
284C8×Q16—831no—
285C8×Q16—831no—
286#1572—831no—
287#1491—831no—
288#1485—831no—
289#1453—831no—
290C83#1792511yescharacteristic, complemented, Sylow
291#533—831no—
292C2×C4×Q8—831no—
293#484—831no—
294#484—831no—
295#477—831no—
296#477—831no—
297#462—831no—
298#462—831no—
299#451—831no—
300#451—831no—
301#446—831no—
302#446—1661no—
303#446—1661no—
304#446C2×D8311yescharacteristic
305C8×Q8—831no—
306C8×Q8—831no—
307C8×Q8—831no—
308C8×Q8—831no—
309C8×Q8—831no—
310C8×Q8—1661no—
311C8×Q8—1661no—
312C8×Q8C2×D8311yescharacteristic
313C4×Q16—831no—
314C4×Q16—831no—
315C4×Q16—831no—
316C4×Q16C2×D8311yescharacteristic
317#433—1661no—
318#433—1661no—
319#432—831no—
320#431—831no—
321#431—831no—
322#431—831no—
323#431—1661no—
324#431—1661no—
325C4×M16—831no—
326#335—831no—
327#335—831no—
328#334—831no—
329#334C2×D8311yescharacteristic
330Q8⋊C8—831no—
331Q8⋊C8—831no—
332Q8⋊C8—831no—
333Q8⋊C8—831no—
334Q8⋊C8—831no—
335Q8⋊C8—831no—
336Q8⋊C8C2×D8311yescharacteristic
337Q8⋊C8C2×D8311yescharacteristic
338#322—831no—
339#322—831no—
340#322C2×D8311yescharacteristic
341C82—831no—
342C22×Q8—831no—
343C2×Q16—831no—
344C2×Q16—831no—
345C2×Q16—831no—
346C2×Q16C4×D8311yescharacteristic
347C2×M16—1661no—
348C4⋊Q8—831no—
349C4⋊Q8—831no—
350#125—1661no—
351C22⋊Q8—831no—
352C22⋊Q8—831no—
353C22⋊Q8—1661no—
354C4×Q8—831no—
355C4×Q8—831no—
356C4×Q8—831no—
357C4×Q8—831no—
358C4×Q8—1661no—
359C4×Q8—1661no—
360C4×Q8—1661no—
361C4×Q8C22×D8311yescharacteristic
362C4×Q8C22×D8311yescharacteristic
363#117—1661no—
364C2×C4⋊C4—831no—
365C2×C4⋊C4—1661no—
366C2×C42—831no—
367C2×C42—1661no—
368#107—831no—
369#107—831no—
370#107—831no—
371#107C4×D8311yescharacteristic
372C4⋊C8—21no—
373C4⋊C8—21no—
374C4⋊C8—831no—
375C4⋊C8—831no—
376C4⋊C8—831no—
377C4⋊C8—831no—
378C4⋊C8—831no—
379C4⋊C8—831no—
380C4⋊C8—1661no—
381C4⋊C8—1661no—
382C4⋊C8—1661no—
383C4⋊C8—1661no—
384C4⋊C8—1661no—
385C4⋊C8—1661no—
386C4⋊C8C22×D8311yescharacteristic
387C4⋊C8C22×D8311yescharacteristic
388#103—831no—
389#103—831no—
390#103—831no—
391#103—831no—
392#103—831no—
393#103—831no—
394#103C4×D8311yescharacteristic
395#103C4×D8311yescharacteristic
396C22⋊C8—831no—
397C22⋊C8—831no—
398C22⋊C8—831no—
399C22⋊C8—831no—
400C22⋊C8—1661no—
401C22⋊C8—1661no—
402#97—21no—
403#97—1661no—
404#97—1661no—
405#97—1661no—
406C4×C8—21no—
407C4×C8—831no—
408C4×C8—831no—
409C4×C8—831no—
410C4×C8—1661no—
411C4×C8—1661no—
412C4×C8—1661no—
413C4×C8—1661no—
414C4×C8—1661no—
415C4×C8—1661no—
416C4×C8C22×D8311yescharacteristic
417C4×C8C22×D8311yescharacteristic
418C4×C8C22×D8311yescharacteristic
419C2×Q8—831no—
420C2×Q8—831no—
421C2×Q8—1661no—
422C2×Q8—1661no—
423C2×Q8—1662no—
424C2×Q8—1662no—
425C2×Q8C2×C4×D8311yescharacteristic
426C2×Q8C2×C4×D8311yescharacteristic
427C22×C4—831no—
428C22×C4—831no—
429C22×C4—831no—
430C22×C4—1661no—
431C22×C4—1661no—
432Q16—22no—
433Q16—1662no—
434Q16—1662no—
435Q16—1662no—
436M16—3322no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—831no—
441C2×C8—831no—
442C2×C8—831no—
443C2×C8—831no—
444C2×C8—1661no—
445C2×C8—1661no—
446C2×C8—1661no—
447C2×C8C2×C4×D8311yescharacteristic
448C2×C8C2×C4×D8311yescharacteristic
449C2×C8D4×D8311yescharacteristic
450C2×C8D4×D8311yescharacteristic
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—831no—
454C4⋊C4—831no—
455C4⋊C4—1661no—
456C4⋊C4—1661no—
457C4⋊C4—1661no—
458C4⋊C4—1661no—
459C4⋊C4—1661no—
460C4⋊C4—1661no—
461C4⋊C4—1661no—
462C4⋊C4—1661no—
463C4⋊C4—1661no—
464C4⋊C4—1661no—
465C4⋊C4C2×C4×D8311yescharacteristic
466C4⋊C4C2×C4×D8311yescharacteristic
467C22⋊C4—1661no—
468C22⋊C4—1661no—
469C42—21no—
470C42—21no—
471C42—831no—
472C42—831no—
473C42—831no—
474C42—1661no—
475C42—1661no—
476C42—1661no—
477C42—1661no—
478C42C23×D8311yescharacteristic
479C23—831no—
480Q8—21no—
481Q8—21no—
482Q8—21no—
483Q8—832no—
484Q8—6641no—
485Q8—1664no—
486Q8—1661no—
487Q8—1661no—
488Q8—1661no—
489Q8#1458180012yes—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—3321no—
495C2×C4—831no—
496C2×C4—831no—
497C2×C4—831no—
498C2×C4—6641no—
499C2×C4—1664no—
500C2×C4—1661no—
501C2×C4—1661no—
502C2×C4—1661no—
503C2×C4—1661no—
504C2×C4—1661no—
505C2×C4—1661no—
506C2×C4—1661no—
507C2×C4—1661no—
508C2×C4C22×C4×D8311yescharacteristic
509C2×C4C2×D4×D8311yescharacteristic
510C2×C4#41×D8311yescharacteristic
511C8—22no—
512C8—21no—
513C8—21no—
514C8—21no—
515C8—21no—
516C8—41no—
517C8—41no—
518C8—3321no—
519C8—3321no—
520C8—3321no—
521C8—1661no—
522C8—1661no—
523C8—1661no—
524C8—1661no—
525C22—1661no—
526C22—1661no—
527C22—1661no—
528C22C4×D4×D8311yescharacteristic
529C4—21no—
530C4—22no—
531C4—41no—
532C4—41no—
533C4—41no—
534C4—3321no—
535C4—3321no—
536C4—3321no—
537C4—1661no—
538C4—1661no—
539C4—1661no—
540C4—1662no—
541C4#131×D8311yescharacteristic
542C4C2×#1458180011yescharacteristic
543C4#1912851711yescharacteristic
544C4#1912851711yescharacteristic
545C2—3321no—
546C2#2491615411yescharacteristic
547C2#2491632811yescharacteristic
548C2#2491636811yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1587C2
socleC22×C83C2×C4×D4
prime radicalC22×D83C4×D4
cosocleC42×C83C24
hypercenter#1587D83
hypocenterC83#17925
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C4×C8 ◁ #1587

Complements

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

NH ≅ G/Ntype
4. #24906336C2semidirect
5. #24906313C2semidirect
6. #24906311C2semidirect
7. #24906282C2semidirect
8. #24906281C2semidirect
9. #24906267C2semidirect
10. #24906267C2semidirect
15. C83×#1587C2semidirect
279. #1587D83semidirect
290. C83#17925semidirect

Maximal subgroups

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

subgroupindex
#249175002
#249165592
#249165422
#249063362
#249063132
#249063112
#249062822
#249062812
#249062672
#249062402
#249059082
#249058822
#249057612
C83×#15872
#1792583