Finite Group Databasebeta

← #35375434

Subgroups — #35375434

Order 21248.

670 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917510C211yesmaximal, characteristic
2#24916461C211yesmaximal, characteristic, complemented
3D4×#13050574C211yesmaximal, characteristic, complemented
4#24906374C211yesmaximal, characteristic, complemented
5#24906356C211yesmaximal, characteristic, complemented
6#24906344C211yesmaximal, characteristic, complemented
7#24906324C211yesmaximal, characteristic, complemented
8#24906296C211yesmaximal, characteristic, complemented
9#24906295C211yesmaximal, characteristic, complemented
10#24906272C211yesmaximal, characteristic, complemented
11#24906255C211yesmaximal, characteristic, complemented
12#24906251C211yesmaximal, characteristic, complemented
13#24905929C211yesmaximal, characteristic, complemented
14#24905758C211yesmaximal, characteristic, complemented
15C83×#1478C211yesmaximal, characteristic, complemented
16C22×#13050574—21no—
17#19128235C2211yescharacteristic
18#19128036C411yescharacteristic
19#19128019C411yescharacteristic
20#19128001C411yescharacteristic
21#19128001C411yescharacteristic
22#19127916C411yescharacteristic
23#19127916C411yescharacteristic
24#19127852C411yescharacteristic
25#19127835C411yescharacteristic
26#19127828C2211yescharacteristic, complemented
27#19127825C2211yescharacteristic, complemented
28C4×#13050578C2211yescharacteristic
29C4×#13050578C2211yescharacteristic
30C4×#13050575C2211yescharacteristic
31C4×#13050575C2211yescharacteristic
32#19127779C2211yescharacteristic
33#19127779C2211yescharacteristic
34#19127778C2211yescharacteristic, complemented
35#19127776—21no—
36#19127775—21no—
37#19127773—21no—
38#19127773—21no—
39D4×C83⋊C8C2211yescharacteristic
40D4×C83⋊C8C2211yescharacteristic
41#19127766C2211yescharacteristic, complemented
42#19127760—21no—
43#19127754C2211yescharacteristic
44#19127725C2211yescharacteristic
45#19127665—21no—
46#19127655—21no—
47C4×#13050574C2211yescharacteristic
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#431×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×#13050574—21no—
72C2×#13050574—21no—
73C2×#13050574—21no—
74C2×#13050574—44no—
75C2×#13050574D411yescharacteristic
76C2×#13050574D411yescharacteristic
77C2×#13050573—21no—
78C22×C83⋊C8—21no—
79C22×C83⋊C8—21no—
80C2×C8×D83—21no—
81#14581786—21no—
82#14581784C2×C411yescharacteristic
83#14581780C2×C411yescharacteristic
84#14581775—21no—
85#14581771C2×C411yescharacteristic
86#14581766C2×C411yescharacteristic
87C4×Dic166C2311yescharacteristic
88C4×D332C2311yescharacteristic
89C4×C22⋊D83—21no—
90#14581749—21no—
91#14581725C2×C411yescharacteristic
92#14581725C2×C411yescharacteristic
93#14581723C2×C411yescharacteristic
94#14581723C2×C411yescharacteristic
95C4⋊(C83⋊C8)—21no—
96C4⋊(C83⋊C8)C2×C411yescharacteristic
97C4⋊(C83⋊C8)C2×C411yescharacteristic
98C4⋊(C83⋊C8)C2311yescharacteristic
99C4⋊(C83⋊C8)C2311yescharacteristic
100Dic83⋊C8—21no—
101Dic83⋊C8—21no—
102#14581715C2×C411yescharacteristic
103#14581715C2×C411yescharacteristic
104#14581713C2×C411yescharacteristic
105#14581713C2×C411yescharacteristic
106#14581713C2×C411yescharacteristic
107#14581713C2×C411yescharacteristic
108#14581712C2×C411yescharacteristic
109#14581712C2×C411yescharacteristic
110C22⋊(C83⋊C8)—21no—
111C22⋊(C83⋊C8)—21no—
112C22⋊(C83⋊C8)—21no—
113C22⋊(C83⋊C8)—21no—
114C22⋊(C83⋊C8)C2×C411yescharacteristic
115C22⋊(C83⋊C8)C2×C411yescharacteristic
116(C2×D83)⋊C8—21no—
117(C2×D83)⋊C8—21no—
118#14581703D411yescharacteristic
119#14581703D411yescharacteristic
120#14581702—21no—
121C4×C83⋊C8C2311yescharacteristic
122C4×C83⋊C8C2311yescharacteristic
123C4×C83⋊C8C2311yescharacteristic
124C4×C83⋊C8C2311yescharacteristic
125C8×Dic83—21no—
126C22×D4×C83C2×C411yescharacteristic
127C23×C4×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×C83C2311yescharacteristic
139C4⋊C8×C83C2311yescharacteristic
140C22⋊C8×C83—21no—
141C22⋊C8×C83—21no—
142C4×C8×C83—21no—
143C4×C8×C83C2311yescharacteristic
144C4×C8×C83C2311yescharacteristic
145C2×Dic166C22×C411yescharacteristic
146C2×D332C22×C411yescharacteristic
147C2×C22⋊D83—21no—
148C2×C4×D83—21no—
149#13050578—22no—
150#13050578—41no—
151#13050575—22no—
152#13050575—41no—
153#13050574—22no—
154#13050574—41no—
155#13050574—42no—
156#13050574—42no—
157#13050574—42no—
158#13050573—42no—
159C2×C83⋊C8—21no—
160C2×C83⋊C8—21no—
161C2×C83⋊C8—42no—
162C2×C83⋊C8—42no—
163C2×C83⋊C8C22×C411yescharacteristic
164C2×C83⋊C8C22×C411yescharacteristic
165C2×C83⋊C8C22×C411yescharacteristic
166C2×C83⋊C8C22×C411yescharacteristic
167C2×C83⋊C8C2×D411yescharacteristic
168C2×C83⋊C8C2×D411yescharacteristic
169C2×C83⋊C8C2×D411yescharacteristic
170C2×C83⋊C8C2×D411yescharacteristic
171C8×D83—42no—
172C4⋊Dic83C22×C411yescharacteristic
173C4⋊Dic83C22×C411yescharacteristic
174Dic83⋊C4—21no—
175Dic83⋊C4—21no—
176Dic83⋊C4—21no—
177Dic83⋊C4—21no—
178C22⋊Dic83—21no—
179(C2×D83)⋊C4—21no—
180(C2×D83)⋊C4—21no—
181(C2×D83)⋊C4—21no—
182(C2×D83)⋊C4—21no—
183C4×Dic83—21no—
184C24×C83—21no—
185C2×D4×C83—22no—
186C2×D4×C83—21no—
187C2×D4×C83—41no—
188C2×D4×C83C22×C411yescharacteristic
189C2×D4×C83C22×C411yescharacteristic
190C22×C4×C83—21no—
191C22×C4×C83—21no—
192C22×C4×C83—21no—
193C22×C4×C83—21no—
194C22×C4×C83—21no—
195C22×C4×C83—42no—
196C22×C4×C83—42no—
197C22×C4×C83C22×C411yescharacteristic
198C22×C4×C83C2×D411yescharacteristic
199C22×C4×C83#4111yescharacteristic
200C2×C8×C83—21no—
201C2×C8×C83—21no—
202C2×C8×C83—21no—
203C2×C8×C83—21no—
204C4⋊C4×C83—21no—
205C4⋊C4×C83C22×C411yescharacteristic
206C4⋊C4×C83C22×C411yescharacteristic
207C22⋊C4×C83—21no—
208C22⋊C4×C83—21no—
209C22⋊C4×C83—21no—
210C22⋊C4×C83—21no—
211C42×C83C22×C411yescharacteristic
212C42×C83C2×D411yescharacteristic
213C42×C83#4111yescharacteristic
214C42×C83C2411yescharacteristic
215C23×C83—21no—
216C23×C83—21no—
217C23×C83—42no—
218C23×C83—42no—
219C23×C83—81no—
220C23×C83C4×D411yescharacteristic
221C22×D83—21no—
222D4×C83—21no—
223D4×C83—21no—
224D4×C83—21no—
225D4×C83—42no—
226D4×C83#13112yes—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—41no—
234C2×C4×C83—42no—
235C2×C4×C83—42no—
236C2×C4×C83—42no—
237C2×C4×C83—41no—
238C2×C4×C83—41no—
239C2×C4×C83—41no—
240C2×C4×C83—41no—
241C2×C4×C83C4×D411yescharacteristic
242C2×C4×C83C4×D411yescharacteristic
243C2×C4×C83C4×D411yescharacteristic
244C2×C4×C83C23×C411yescharacteristic
245C2×C4×C83C22×D411yescharacteristic
246C2×C4×C83C2×#4111yescharacteristic
247C22⋊D83—81no—
248C2×Dic83—21no—
249C2×Dic83—21no—
250C2×Dic83—21no—
251D332—21no—
252D332—21no—
253C4×D83—41no—
254C4×D83—41no—
255Dic166—21no—
256Dic166—21no—
257C8×C83—41no—
258C8×C83—41no—
259C8×C83—41no—
260C8×C83—41no—
261C83⋊C8—21no—
262C83⋊C8—21no—
263C83⋊C8—21no—
264C83⋊C8—21no—
265C83⋊C8—21no—
266C83⋊C8—21no—
267C83⋊C8—21no—
268C83⋊C8—41no—
269C83⋊C8—41no—
270C83⋊C8AGL(1,ℤ/8)12yes—
271C22×C83—21no—
272C22×C83—22no—
273C22×C83—21no—
274C22×C83—21no—
275C22×C83—41no—
276C22×C83—41no—
277C22×C83—41no—
278C22×C83—41no—
279C22×C83—41no—
280C22×C83—84no—
281C22×C83C2×C4×D411yescharacteristic
282C2×D83—41no—
283C2×D83—41no—
284C2×D83—41no—
285C4×C83—21no—
286C4×C83—21no—
287C4×C83—21no—
288C4×C83—21no—
289C4×C83—41no—
290C4×C83—41no—
291C4×C83C2×#13111yescharacteristic
292C4×C83#56811yescharacteristic
293C4×C83C2×AGL(1,ℤ/8)11yescharacteristic
294C4×C83#57511yescharacteristic
295Dic83—41no—
296Dic83—41no—
297Dic83—41no—
298#18006—831nomaximal, Sylow
299C2×C83—41no—
300C2×C83—42no—
301C2×C83—81no—
302C2×C83#293311yescharacteristic
303C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
304C2×C83#295711yescharacteristic
305D83—81no—
306#3105—831no—
307#2969—831no—
308D4×M16—831no—
309#1594—831no—
310#1590—831no—
311#1588—831no—
312#1586—831no—
313#1582—831no—
314#1581—831no—
315SD16⋊C8—831no—
316C8×D8—831no—
317#1574—831no—
318#1495—831no—
319#1478D8311yescharacteristic, complemented
320#1453—831no—
321C83#1800611yescharacteristic, complemented, Sylow
322C22×M16—1661no—
323#529—831no—
324C2×C4×D4C2×D8311yescharacteristic
325#484—831no—
326#481—831no—
327#478—831no—
328#478—831no—
329#463—831no—
330#463—831no—
331#450—831no—
332#447—831no—
333#446—831no—
334C8×Q8—831no—
335C4×SD16—831no—
336C4×SD16—831no—
337C4×D8—831no—
338C4×D8—831no—
339#436—831no—
340#436—831no—
341#436—831no—
342#436—1661no—
343#435—1661no—
344#435—1661no—
345#435—1661no—
346C8×D4—831no—
347C8×D4—831no—
348C8×D4—831no—
349#433—1661no—
350#431—831no—
351#431C2×D8311yescharacteristic
352#431C2×D8311yescharacteristic
353#423—831no—
354#407—1661no—
355#405—1661no—
356C4×M16—831no—
357#335—831no—
358#335—831no—
359#334—831no—
360#334—831no—
361Q8⋊C8—831no—
362Q8⋊C8—831no—
363D4⋊C8—831no—
364D4⋊C8—831no—
365D4⋊C8C2×D8311yescharacteristic
366D4⋊C8C2×D8311yescharacteristic
367D4⋊C8C2×D8311yescharacteristic
368D4⋊C8C2×D8311yescharacteristic
369#322—831no—
370#322—831no—
371#322—831no—
372C82—831no—
373C22×D4C4×D8311yescharacteristic
374C23×C4—21no—
375C2×SD16—831no—
376C2×SD16—831no—
377C2×D8—831no—
378C2×D8—831no—
379C2×M16—3324no—
380C2×M16—831no—
381C2×M16—831no—
382C2×M16—1661no—
383C2×M16—1661no—
384C2×M16—1661no—
385C2×M16—1661no—
386C22×C8—1661no—
387C22×C8—1661no—
388C22×C8—1661no—
389#126—1661no—
390#125—831no—
391#124—831no—
392#123—1661no—
393C22⋊Q8—831no—
394#121—831no—
395C4×Q8—831no—
396C4×D4—21no—
397C4×D4—21no—
398C4×D4—831no—
399C4×D4—1661no—
400C4×D4C4×D8311yescharacteristic
401C4×D4C4×D8311yescharacteristic
402C4×D4C22×D8311yescharacteristic
403C4×D4C22×D8311yescharacteristic
404#117—1661no—
405C2×C4⋊C4C4×D8311yescharacteristic
406C2×C22⋊C4—21no—
407C2×C42C22×D8311yescharacteristic
408#107—831no—
409#107—831no—
410#106—831no—
411#106—831no—
412C4⋊C8—21no—
413C4⋊C8—831no—
414C4⋊C8—831no—
415C4⋊C8—831no—
416C4⋊C8—831no—
417C4⋊C8—1661no—
418C4⋊C8—1661no—
419C4⋊C8—1661no—
420C4⋊C8C22×D8311yescharacteristic
421C4⋊C8C22×D8311yescharacteristic
422#103—831no—
423#103—831no—
424#102—831no—
425#102—831no—
426#102—831no—
427#102—831no—
428#102—831no—
429#102—831no—
430C22⋊C8—21no—
431C22⋊C8—21no—
432C22⋊C8—831no—
433C22⋊C8—831no—
434C22⋊C8—1661no—
435C22⋊C8—1661no—
436C22⋊C8—1661no—
437C22⋊C8—1661no—
438C22⋊C8—1661no—
439C22⋊C8—1661no—
440#97—831no—
441#97—831no—
442#97—1661no—
443C4×C8—21no—
444C4×C8—831no—
445C4×C8—831no—
446C4×C8—831no—
447C4×C8—831no—
448C4×C8—1661no—
449C4×C8C22×D8311yescharacteristic
450C4×C8C22×D8311yescharacteristic
451C24—21no—
452C2×Q8—831no—
453C2×D4—22no—
454C2×D4—21no—
455C2×D4—41no—
456C2×D4—831no—
457C2×D4—1661no—
458C2×D4C2×C4×D8311yescharacteristic
459C2×D4C2×C4×D8311yescharacteristic
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—42no—
466C22×C4—42no—
467C22×C4—1661no—
468C22×C4C2×C4×D8311yescharacteristic
469C22×C4D4×D8311yescharacteristic
470C22×C4#1305058911yescharacteristic
471SD16—3321no—
472SD16—1662no—
473D8—3321no—
474D8—1662no—
475M16—3322no—
476M16—3322no—
477M16—3322no—
478M16—3322no—
479M16—3321no—
480M16—1662no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—3322no—
486C2×C8—3322no—
487C2×C8—3322no—
488C2×C8—831no—
489C2×C8—831no—
490C2×C8—831no—
491C2×C8—831no—
492C2×C8—831no—
493C2×C8—831no—
494C2×C8—831no—
495C2×C8—831no—
496C2×C8—1661no—
497C2×C8—1661no—
498C4⋊C4—21no—
499C4⋊C4—831no—
500C4⋊C4—831no—
501C4⋊C4—1661no—
502C4⋊C4—1661no—
503C4⋊C4—1661no—
504C4⋊C4—1661no—
505C4⋊C4C2×C4×D8311yescharacteristic
506C4⋊C4C2×C4×D8311yescharacteristic
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—1661no—
512C22⋊C4—1661no—
513C22⋊C4—1661no—
514C22⋊C4—1661no—
515C22⋊C4—1661no—
516C42—1661no—
517C42C2×C4×D8311yescharacteristic
518C42D4×D8311yescharacteristic
519C42#1305058911yescharacteristic
520C42C23×D8311yescharacteristic
521C23—21no—
522C23—21no—
523C23—42no—
524C23—42no—
525C23—81no—
526C23—1661no—
527C23(C22⋊D83)⋊C411yescharacteristic
528Q8—1661no—
529Q8—1661no—
530D4—21no—
531D4—21no—
532D4—21no—
533D4—42no—
534D4—6641no—
535D4—1661no—
536D4—1661no—
537D4#1458179812yes—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—3321no—
553C2×C4—3321no—
554C2×C4—1661no—
555C2×C4—1661no—
556C2×C4—1661no—
557C2×C4(C22⋊D83)⋊C411yescharacteristic
558C2×C4(C22⋊D83)⋊C411yescharacteristic
559C2×C4(C22⋊D83)⋊C411yescharacteristic
560C2×C4C22×C4×D8311yescharacteristic
561C2×C4C2×D4×D8311yescharacteristic
562C2×C4C2×#1305058911yescharacteristic
563C8—41no—
564C8—41no—
565C8—41no—
566C8—41no—
567C8—3321no—
568C8—3321no—
569C8—832no—
570C8—1661no—
571C8—1661no—
572C8—1661no—
573C8—1661no—
574C8—1661no—
575C8—1661no—
576C8—1661no—
577C22—21no—
578C22—21no—
579C22—21no—
580C22—22no—
581C22—41no—
582C22—41no—
583C22—41no—
584C22—41no—
585C22—41no—
586C22—84no—
587C22—3321no—
588C22—3321no—
589C22—3321no—
590C22C2×(C22⋊D83)⋊C411yescharacteristic
591C4—21no—
592C4—21no—
593C4—21no—
594C4—21no—
595C4—41no—
596C4—41no—
597C4—3321no—
598C4—3321no—
599C4—3321no—
600C4C2×#1458179811yescharacteristic
601C4#1912846711yescharacteristic
602C4AGL(1,ℤ/8)×D8311yescharacteristic
603C4#1912849711yescharacteristic
604C2—41no—
605C2—42no—
606C2—81no—
607C2—6641no—
608C2#2491618411yescharacteristic
609C2#2491626711yescharacteristic
610C2#2491637611yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC42C23×D83
FittingC83×#1478C2
socleC22×C83C2×C4×D4
prime radical#19127835C4
cosocleC42×C83C24
hypercenter#1478D83
hypocenterC83#18006
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1478

Complements

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

NH ≅ G/Ntype
2. #24916461C2semidirect
3. D4×#13050574C2semidirect
4. #24906374C2semidirect
5. #24906356C2semidirect
6. #24906344C2semidirect
7. #24906324C2semidirect
8. #24906296C2semidirect
9. #24906295C2semidirect
10. #24906272C2semidirect
11. #24906255C2semidirect
12. #24906251C2semidirect
13. #24905929C2semidirect
14. #24905758C2semidirect
15. C83×#1478C2semidirect
26. #19127828C22semidirect
27. #19127825C22semidirect
34. #19127778C22semidirect
41. #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
319. #1478D83semidirect
321. C83#18006semidirect

Maximal subgroups

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

subgroupindex
#249175102
#249164612
D4×#130505742
#249063742
#249063562
#249063442
#249063242
#249062962
#249062952
#249062722
#249062552
#249062512
#249059292
#249057582
C83×#14782
#1800683