Finite Group Databasebeta

← #35374458

Subgroups — #35374458

Order 21248.

648 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917510C211yesmaximal, characteristic
2#24916608C211yesmaximal, characteristic, complemented
3#24916487C211yesmaximal, characteristic, complemented
4#24906324C211yesmaximal, characteristic, complemented
5#24906317C211yesmaximal, characteristic, complemented
6#24906296C211yesmaximal, characteristic, complemented
7#24906289C211yesmaximal, characteristic, complemented
8#24906284C211yesmaximal, characteristic, complemented
9#24906272C211yesmaximal, characteristic, complemented
10#24906266C211yesmaximal, characteristic, complemented
11#24906255C211yesmaximal, characteristic, complemented
12#24906254C211yesmaximal, characteristic, complemented
13#24905872C211yesmaximal, characteristic, complemented
14#24905763C211yesmaximal, characteristic, complemented
15C83×#1486C211yesmaximal, characteristic, complemented
16#19128235C2211yescharacteristic
17#19128036C411yescharacteristic
18#19128019C411yescharacteristic
19#19128001C411yescharacteristic
20#19128001C411yescharacteristic
21#19127916C411yescharacteristic
22#19127916C411yescharacteristic
23#19127852C411yescharacteristic
24#19127835C411yescharacteristic
25#19127828C2211yescharacteristic, complemented
26#19127825C2211yescharacteristic, complemented
27C4×#13050578C2211yescharacteristic
28C4×#13050578C2211yescharacteristic
29C4×#13050575C2211yescharacteristic
30C4×#13050575C2211yescharacteristic
31#19127778C2211yescharacteristic, complemented
32#19127777—21no—
33#19127770—21no—
34D4×C83⋊C8—21no—
35D4×C83⋊C8—21no—
36D4×C83⋊C8C2211yescharacteristic
37D4×C83⋊C8C2211yescharacteristic
38D4×C83⋊C8C2211yescharacteristic
39D4×C83⋊C8C2211yescharacteristic
40#19127766C2211yescharacteristic, complemented
41#19127761—21no—
42#19127754C2211yescharacteristic
43#19127754C2211yescharacteristic
44#19127754C2211yescharacteristic
45#19127726—21no—
46C2×C22⋊(C83⋊C8)—21no—
47C2×C22⋊(C83⋊C8)—21no—
48#19127453C2211yescharacteristic, complemented
49#19127453C2211yescharacteristic, complemented
50#19127451C2211yescharacteristic, complemented
51#19127451C2211yescharacteristic, complemented
52#19127433C2211yescharacteristic, complemented
53#19127433C2211yescharacteristic, complemented
54#19127430C2211yescharacteristic, complemented
55#19127430C2211yescharacteristic, complemented
56#19127423C2211yescharacteristic, complemented
57#19127423C2211yescharacteristic, complemented
58#19127422C2211yescharacteristic, complemented
59C8×C83⋊C8C2211yescharacteristic, complemented
60C2×C4×D4×C83C2211yescharacteristic
61#431×C83C2211yescharacteristic
62#423×C83C2211yescharacteristic
63D4⋊C8×C83C2211yescharacteristic
64D4⋊C8×C83C2211yescharacteristic
65D4⋊C8×C83C2211yescharacteristic
66D4⋊C8×C83C2211yescharacteristic
67C2×#13050578C2×C411yescharacteristic
68C2×#13050578C2×C411yescharacteristic
69C2×#13050575C2×C411yescharacteristic
70C2×#13050575C2×C411yescharacteristic
71C2×#13050573—21no—
72C22×C83⋊C8—21no—
73C22×C83⋊C8—21no—
74C22×C83⋊C8—21no—
75C22×C83⋊C8—21no—
76C2×C8×D83—21no—
77#14581786—21no—
78#14581784C2×C411yescharacteristic
79#14581780C2×C411yescharacteristic
80#14581775—21no—
81#14581771C2×C411yescharacteristic
82#14581766C2×C411yescharacteristic
83C4×Dic166C2311yescharacteristic
84C4×D332C2311yescharacteristic
85C4×C22⋊D83—21no—
86#14581749—21no—
87#14581725C2×C411yescharacteristic
88#14581725C2×C411yescharacteristic
89#14581723C2×C411yescharacteristic
90#14581723C2×C411yescharacteristic
91C4⋊(C83⋊C8)C2×C411yescharacteristic
92C4⋊(C83⋊C8)C2×C411yescharacteristic
93C4⋊(C83⋊C8)D411yescharacteristic
94C4⋊(C83⋊C8)D411yescharacteristic
95C4⋊(C83⋊C8)C2311yescharacteristic
96C4⋊(C83⋊C8)C2311yescharacteristic
97Dic83⋊C8—21no—
98Dic83⋊C8—21no—
99#14581715C2×C411yescharacteristic
100#14581715C2×C411yescharacteristic
101#14581713C2×C411yescharacteristic
102#14581713C2×C411yescharacteristic
103#14581713C2×C411yescharacteristic
104#14581713C2×C411yescharacteristic
105#14581712C2×C411yescharacteristic
106#14581712C2×C411yescharacteristic
107C22⋊(C83⋊C8)—21no—
108C22⋊(C83⋊C8)—21no—
109C22⋊(C83⋊C8)—21no—
110C22⋊(C83⋊C8)—21no—
111C22⋊(C83⋊C8)—21no—
112C22⋊(C83⋊C8)C2×C411yescharacteristic
113C22⋊(C83⋊C8)C2×C411yescharacteristic
114C22⋊(C83⋊C8)D411yescharacteristic
115C22⋊(C83⋊C8)D411yescharacteristic
116(C2×D83)⋊C8—21no—
117(C2×D83)⋊C8—21no—
118#14581702—21no—
119C4×C83⋊C8—21no—
120C4×C83⋊C8C2311yescharacteristic
121C4×C83⋊C8C2311yescharacteristic
122C4×C83⋊C8C2311yescharacteristic
123C4×C83⋊C8C2311yescharacteristic
124C8×Dic83—21no—
125C22×D4×C83C2×C411yescharacteristic
126C23×C4×C83—21no—
127C2×M16×C83—21no—
128C4×D4×C83—21no—
129C4×D4×C83—21no—
130C4×D4×C83C2×C411yescharacteristic
131C4×D4×C83C2×C411yescharacteristic
132C4×D4×C83C2311yescharacteristic
133C4×D4×C83C2311yescharacteristic
134C2×C4⋊C4×C83C2×C411yescharacteristic
135C2×C22⋊C4×C83—21no—
136C2×C42×C83C2311yescharacteristic
137C4⋊C8×C83—21no—
138C4⋊C8×C83—21no—
139C4⋊C8×C83C2311yescharacteristic
140C4⋊C8×C83C2311yescharacteristic
141C22⋊C8×C83—21no—
142C4×C8×C83C2311yescharacteristic
143C4×C8×C83C2311yescharacteristic
144C2×Dic166C22×C411yescharacteristic
145C2×D332C22×C411yescharacteristic
146C2×C22⋊D83—21no—
147C2×C4×D83—21no—
148#13050578—22no—
149#13050578—22no—
150#13050575—22no—
151#13050575—22no—
152#13050573—81no—
153C2×C83⋊C8—21no—
154C2×C83⋊C8—21no—
155C2×C83⋊C8—81no—
156C2×C83⋊C8—81no—
157C2×C83⋊C8—81no—
158C2×C83⋊C8—81no—
159C2×C83⋊C8C22×C411yescharacteristic
160C2×C83⋊C8C22×C411yescharacteristic
161C2×C83⋊C8C22×C411yescharacteristic
162C2×C83⋊C8C22×C411yescharacteristic
163C2×C83⋊C8C2×D411yescharacteristic
164C2×C83⋊C8C2×D411yescharacteristic
165C2×C83⋊C8C2×D411yescharacteristic
166C2×C83⋊C8C2×D411yescharacteristic
167C8×D83—81no—
168C4⋊Dic83C22×C411yescharacteristic
169C4⋊Dic83C22×C411yescharacteristic
170Dic83⋊C4—21no—
171Dic83⋊C4—21no—
172Dic83⋊C4—21no—
173Dic83⋊C4—21no—
174C22⋊Dic83—21no—
175(C2×D83)⋊C4—21no—
176(C2×D83)⋊C4—21no—
177(C2×D83)⋊C4—21no—
178(C2×D83)⋊C4—21no—
179C4×Dic83—21no—
180C24×C83—21no—
181C2×D4×C83—22no—
182C2×D4×C83—22no—
183C2×D4×C83—21no—
184C2×D4×C83C22×C411yescharacteristic
185C2×D4×C83C22×C411yescharacteristic
186C22×C4×C83—21no—
187C22×C4×C83—21no—
188C22×C4×C83—21no—
189C22×C4×C83—21no—
190C22×C4×C83—21no—
191C22×C4×C83—81no—
192C22×C4×C83—81no—
193C22×C4×C83C22×C411yescharacteristic
194C22×C4×C83C2×D411yescharacteristic
195C22×C4×C83#4111yescharacteristic
196M16×C83—81no—
197C2×C8×C83—21no—
198C2×C8×C83—21no—
199C2×C8×C83—21no—
200C2×C8×C83—21no—
201C4⋊C4×C83—21no—
202C4⋊C4×C83C22×C411yescharacteristic
203C4⋊C4×C83C22×C411yescharacteristic
204C22⋊C4×C83—21no—
205C22⋊C4×C83—21no—
206C22⋊C4×C83—21no—
207C22⋊C4×C83—21no—
208C42×C83C22×C411yescharacteristic
209C42×C83C2×D411yescharacteristic
210C42×C83#4111yescharacteristic
211C42×C83C2411yescharacteristic
212C23×C83—24no—
213C23×C83—21no—
214C23×C83—21no—
215C23×C83—81no—
216C23×C83—81no—
217C23×C83C4×D411yescharacteristic
218C22×D83—21no—
219D4×C83—24no—
220D4×C83—21no—
221D4×C83—21no—
222D4×C83#13112yes—
223D4×C83#13112yes—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—41no—
231C2×C4×C83—41no—
232C2×C4×C83—41no—
233C2×C4×C83—41no—
234C2×C4×C83—41no—
235C2×C4×C83—81no—
236C2×C4×C83—81no—
237C2×C4×C83—81no—
238C2×C4×C83C4×D411yescharacteristic
239C2×C4×C83C4×D411yescharacteristic
240C2×C4×C83C4×D411yescharacteristic
241C2×C4×C83C23×C411yescharacteristic
242C2×C4×C83C22×D411yescharacteristic
243C2×C4×C83C2×#4111yescharacteristic
244C22⋊D83—81no—
245C2×Dic83—21no—
246C2×Dic83—21no—
247C2×Dic83—21no—
248D332—21no—
249D332—21no—
250C4×D83—41no—
251C4×D83—41no—
252Dic166—21no—
253Dic166—21no—
254C8×C83—41no—
255C8×C83—41no—
256C8×C83—41no—
257C8×C83—41no—
258C83⋊C8—21no—
259C83⋊C8—21no—
260C83⋊C8—21no—
261C83⋊C8—21no—
262C83⋊C8—21no—
263C83⋊C8—21no—
264C83⋊C8—21no—
265C83⋊C8—21no—
266C83⋊C8—41no—
267C83⋊C8—41no—
268C22×C83—21no—
269C22×C83—22no—
270C22×C83—22no—
271C22×C83—21no—
272C22×C83—21no—
273C22×C83—41no—
274C22×C83—41no—
275C22×C83—41no—
276C22×C83—41no—
277C22×C83—84no—
278C22×C83C2×C4×D411yescharacteristic
279C2×D83—41no—
280C2×D83—41no—
281C2×D83—41no—
282C4×C83—21no—
283C4×C83—21no—
284C4×C83—21no—
285C4×C83—21no—
286C4×C83—41no—
287C4×C83—41no—
288C4×C83C2×#13111yescharacteristic
289C4×C83C2×#13111yescharacteristic
290C4×C83#57611yescharacteristic
291C4×C83#57711yescharacteristic
292Dic83—41no—
293Dic83—41no—
294Dic83—41no—
295#17918—831nomaximal, Sylow
296C2×C83—41no—
297C2×C83—42no—
298C2×C83—42no—
299C2×C83#292911yescharacteristic
300C2×C83#294611yescharacteristic
301C2×C83#295711yescharacteristic
302D83—81no—
303#3105—831no—
304#2993—831no—
305#2972—831no—
306#1586—831no—
307#1585—831no—
308#1582—831no—
309#1581—831no—
310#1580—831no—
311SD16⋊C8—831no—
312C8×SD16—831no—
313C8×D8—831no—
314#1574—831no—
315#1486D8311yescharacteristic, complemented
316#1482—831no—
317#1454—831no—
318C83#1791811yescharacteristic, complemented, Sylow
319#529—831no—
320C2×C4×D4C2×D8311yescharacteristic
321#484—831no—
322#481—831no—
323#478—831no—
324#478—831no—
325#463—831no—
326#463—831no—
327#450—831no—
328#447—831no—
329#446—831no—
330C8×Q8—831no—
331C4×SD16—831no—
332C4×SD16—831no—
333C4×D8—831no—
334C4×D8—831no—
335#436—831no—
336#436—1661no—
337#435—1661no—
338C8×D4—831no—
339C8×D4—831no—
340C8×D4—831no—
341C8×D4—831no—
342C8×D4—831no—
343C8×D4—1661no—
344C8×D4—1661no—
345#433—1661no—
346#431—831no—
347#431—831no—
348#431—831no—
349#431C2×D8311yescharacteristic
350#424—1661no—
351#423C2×D8311yescharacteristic
352C2×C22⋊C8—1661no—
353C2×C22⋊C8—1661no—
354#335—831no—
355#335—831no—
356#334—831no—
357#334—831no—
358Q8⋊C8—831no—
359Q8⋊C8—831no—
360D4⋊C8—831no—
361D4⋊C8—831no—
362D4⋊C8C2×D8311yescharacteristic
363D4⋊C8C2×D8311yescharacteristic
364D4⋊C8C2×D8311yescharacteristic
365D4⋊C8C2×D8311yescharacteristic
366#322—831no—
367#322—831no—
368#322—831no—
369C82—831no—
370C22×D4C4×D8311yescharacteristic
371C23×C4—21no—
372C2×SD16—831no—
373C2×SD16—831no—
374C2×D8—831no—
375C2×D8—831no—
376C2×M16—21no—
377C2×M16—1661no—
378C22×C8—1661no—
379C22×C8—1661no—
380C22×C8—1661no—
381C22×C8—1661no—
382C22×C8—1661no—
383#126—1661no—
384#125—831no—
385#124—831no—
386#123—1661no—
387C22⋊Q8—831no—
388#121—831no—
389C4×Q8—831no—
390C4×D4—21no—
391C4×D4—21no—
392C4×D4—831no—
393C4×D4—1661no—
394C4×D4C4×D8311yescharacteristic
395C4×D4C4×D8311yescharacteristic
396C4×D4C22×D8311yescharacteristic
397C4×D4C22×D8311yescharacteristic
398#117—1661no—
399C2×C4⋊C4C4×D8311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C42C22×D8311yescharacteristic
402#107—831no—
403#107—831no—
404#106—831no—
405#106—831no—
406C4⋊C8—21no—
407C4⋊C8—21no—
408C4⋊C8—831no—
409C4⋊C8—831no—
410C4⋊C8—831no—
411C4⋊C8—831no—
412C4⋊C8—831no—
413C4⋊C8—831no—
414C4⋊C8—1661no—
415C4⋊C8—1661no—
416C4⋊C8C22×D8311yescharacteristic
417C4⋊C8C22×D8311yescharacteristic
418#103—831no—
419#103—831no—
420#102—831no—
421#102—831no—
422#102—831no—
423#102—831no—
424#102—831no—
425#102—831no—
426C22⋊C8—21no—
427C22⋊C8—831no—
428C22⋊C8—831no—
429C22⋊C8—831no—
430C22⋊C8—831no—
431C22⋊C8—1661no—
432C22⋊C8—1661no—
433C22⋊C8—1661no—
434C22⋊C8—1661no—
435C22⋊C8—1661no—
436C22⋊C8—1661no—
437C22⋊C8—1661no—
438#97—1661no—
439C4×C8—831no—
440C4×C8—831no—
441C4×C8—831no—
442C4×C8—831no—
443C4×C8—1661no—
444C4×C8—1661no—
445C4×C8C22×D8311yescharacteristic
446C4×C8C22×D8311yescharacteristic
447C24—21no—
448C2×Q8—831no—
449C2×D4—22no—
450C2×D4—22no—
451C2×D4—21no—
452C2×D4—831no—
453C2×D4—1661no—
454C2×D4C2×C4×D8311yescharacteristic
455C2×D4C2×C4×D8311yescharacteristic
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—81no—
462C22×C4—81no—
463C22×C4—1661no—
464C22×C4C2×C4×D8311yescharacteristic
465C22×C4D4×D8311yescharacteristic
466C22×C4#1305058911yescharacteristic
467SD16—1662no—
468SD16—1662no—
469D8—1662no—
470D8—1662no—
471M16—81no—
472M16—6641no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—831no—
478C2×C8—831no—
479C2×C8—831no—
480C2×C8—831no—
481C2×C8—831no—
482C2×C8—831no—
483C2×C8—831no—
484C2×C8—831no—
485C2×C8—6641no—
486C2×C8—6641no—
487C2×C8—6641no—
488C2×C8—6641no—
489C2×C8—6641no—
490C2×C8—1661no—
491C2×C8—1661no—
492C4⋊C4—21no—
493C4⋊C4—831no—
494C4⋊C4—831no—
495C4⋊C4—1661no—
496C4⋊C4—1661no—
497C4⋊C4—1661no—
498C4⋊C4—1661no—
499C4⋊C4C2×C4×D8311yescharacteristic
500C4⋊C4C2×C4×D8311yescharacteristic
501C22⋊C4—21no—
502C22⋊C4—21no—
503C22⋊C4—21no—
504C22⋊C4—21no—
505C22⋊C4—1661no—
506C22⋊C4—1661no—
507C22⋊C4—1661no—
508C22⋊C4—1661no—
509C22⋊C4—1661no—
510C42—1661no—
511C42C2×C4×D8311yescharacteristic
512C42D4×D8311yescharacteristic
513C42#1305058911yescharacteristic
514C42C23×D8311yescharacteristic
515C23—24no—
516C23—21no—
517C23—21no—
518C23—81no—
519C23—81no—
520C23—1661no—
521C23(C22⋊D83)⋊C411yescharacteristic
522Q8—1661no—
523Q8—1661no—
524D4—24no—
525D4—21no—
526D4—21no—
527D4—6641no—
528D4—1661no—
529D4—1661no—
530D4#1458180012yes—
531D4#1458180012yes—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—3321no—
547C2×C4—3321no—
548C2×C4—1661no—
549C2×C4—1661no—
550C2×C4—1661no—
551C2×C4(C22⋊D83)⋊C411yescharacteristic
552C2×C4(C22⋊D83)⋊C411yescharacteristic
553C2×C4(C22⋊D83)⋊C411yescharacteristic
554C2×C4C22×C4×D8311yescharacteristic
555C2×C4C2×D4×D8311yescharacteristic
556C2×C4C2×#1305058911yescharacteristic
557C8—41no—
558C8—41no—
559C8—41no—
560C8—41no—
561C8—3321no—
562C8—3321no—
563C8—1661no—
564C8—1661no—
565C8—1661no—
566C8—1661no—
567C8—1661no—
568C8—1661no—
569C8—1661no—
570C8—1661no—
571C22—21no—
572C22—21no—
573C22—21no—
574C22—22no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—84no—
581C22—3321no—
582C22—3321no—
583C22—3321no—
584C22C2×(C22⋊D83)⋊C411yescharacteristic
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—41no—
590C4—41no—
591C4—3321no—
592C4—3321no—
593C4—3321no—
594C4C2×#1458180011yescharacteristic
595C4C2×#1458180011yescharacteristic
596C4#1912850211yescharacteristic
597C4#1912850711yescharacteristic
598C2—41no—
599C2—42no—
600C2—42no—
601C2—6641no—
602C2#2491615311yescharacteristic
603C2#2491628911yescharacteristic
604C2#2491637611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1486C2
socleC22×C83C2×C4×D4
prime radical#19127835C4
cosocleC42×C83C24
hypercenter#1486D83
hypocenterC83#17918
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
2. #24916608C2semidirect
3. #24916487C2semidirect
4. #24906324C2semidirect
5. #24906317C2semidirect
6. #24906296C2semidirect
7. #24906289C2semidirect
8. #24906284C2semidirect
9. #24906272C2semidirect
10. #24906266C2semidirect
11. #24906255C2semidirect
12. #24906254C2semidirect
13. #24905872C2semidirect
14. #24905763C2semidirect
15. C83×#1486C2semidirect
25. #19127828C22semidirect
26. #19127825C22semidirect
31. #19127778C22semidirect
40. #19127766C22semidirect
48. #19127453C22semidirect
49. #19127453C22semidirect
50. #19127451C22semidirect
51. #19127451C22semidirect
52. #19127433C22semidirect
53. #19127433C22semidirect
54. #19127430C22semidirect
55. #19127430C22semidirect
56. #19127423C22semidirect
57. #19127423C22semidirect
58. #19127422C22semidirect
59. C8×C83⋊C8C22semidirect
315. #1486D83semidirect
318. C83#17918semidirect

Maximal subgroups

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

subgroupindex
#249175102
#249166082
#249164872
#249063242
#249063172
#249062962
#249062892
#249062842
#249062722
#249062662
#249062552
#249062542
#249058722
#249057632
C83×#14862
#1791883