Finite Group Databasebeta

← #35426419

Subgroups — #35426419

Order 21248.

870 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
1C2×#19127998C211yesmaximal, characteristic
2#24910563C211yesmaximal, characteristic, complemented
3#24910363C211yesmaximal, characteristic
4#24908224C211yesmaximal, characteristic, complemented
5#24908200C211yesmaximal, characteristic, complemented
6#24908190C211yesmaximal, characteristic, complemented
7#24908046C211yesmaximal, characteristic, complemented
8#24908032C211yesmaximal, characteristic, complemented
9#24907725C211yesmaximal, characteristic
10#24907725C211yesmaximal, characteristic
11#24907335C211yesmaximal, characteristic, complemented
12#24907289C211yesmaximal, characteristic
13#24907287C211yesmaximal, characteristic
14#24907277C211yesmaximal, characteristic
15C83×#1992C211yesmaximal, characteristic, complemented
16C2×C4×D332C2211yescharacteristic
17C2×#14581754C2211yescharacteristic
18C2×#14581754C2211yescharacteristic
19C2×C4⋊C4×D83—21no—
20#19127998C418yescomplemented
21C2×#14581725C2211yescharacteristic
22C2×#14581723C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×C4⋊(C83⋊C8)C2211yescharacteristic
26C2×#14581714C2211yescharacteristic
27C2×#14581714C2211yescharacteristic
28C2×#14581712C2211yescharacteristic
29C2×#14581712C2211yescharacteristic
30C2×#14581702C2211yescharacteristic
31C2×C8×Dic83C2211yescharacteristic
32#19127600—21no—
33#19127600C2211yescharacteristic
34#19127600C2211yescharacteristic
35#19127597C2211yescharacteristic
36#19127597C2211yescharacteristic
37#19127587—21no—
38#19127587—21no—
39#19127572—21no—
40#19127572—21no—
41#19127558—21no—
42C4⋊C4×Dic83—21no—
43C4⋊C4×Dic83C2211yescharacteristic
44C4⋊C4×Dic83C2211yescharacteristic
45C4×C4⋊Dic83C2211yescharacteristic
46C4×C4⋊Dic83C2211yescharacteristic
47#19127464C2211yescharacteristic
48#19127464C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127455C2211yescharacteristic
52#19127455C2211yescharacteristic
53C2×#125×C83C2211yescharacteristic
54C2×#97×C83C2211yescharacteristic
55#384×C83C2211yescharacteristic
56#340×C83C2211yescharacteristic
57#340×C83C2211yescharacteristic
58#340×C83C2211yescharacteristic
59#340×C83C2211yescharacteristic
60C22×D332C2311yescharacteristic
61C22×C4×D83—21no—
62C22×C4×D83—21no—
63C22×C4×D83—21no—
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66C4×D332—22no—
67C4×D332C2×C414yes—
68#14581754—22no—
69#14581754—22no—
70#14581754—41no—
71#14581754—41no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83—21no—
74C2×C4⋊Dic83D411yescharacteristic
75C2×C4⋊Dic83D411yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×C4⋊Dic83C2311yescharacteristic
79C2×Dic83⋊C4—21no—
80C2×Dic83⋊C4—21no—
81C4⋊C4×D83—44no—
82C2×(C2×D83)⋊C4—21no—
83C2×(C2×D83)⋊C4—21no—
84C2×(C2×D83)⋊C4—21no—
85C2×C4×Dic83—21no—
86C2×C4×Dic83—21no—
87C2×C4×Dic83—21no—
88C2×C4×Dic83C2311yescharacteristic
89C2×C4×Dic83C2311yescharacteristic
90#14581725C2×C414yescomplemented
91#14581723—41no—
92#14581723—41no—
93#14581723C2×C414yescomplemented
94C4⋊(C83⋊C8)C2×C414yescomplemented
95#14581714—22no—
96#14581714Q814yes—
97#14581712C2×C414yes—
98#14581712C2×C414yes—
99#14581702—41no—
100C8×Dic83—41no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C83⋊#95—21no—
104C83⋊#95—21no—
105C83⋊#95—21no—
106C83⋊#95—21no—
107C83⋊#95—21no—
108C83⋊#95D411yescharacteristic
109C83⋊#95D411yescharacteristic
110C22×C8×C83C2311yescharacteristic
111C22×C8×C83C2311yescharacteristic
112#125×C83—22no—
113#125×C83C2×C414yescomplemented
114C2×C4⋊C4×C83—21no—
115C2×C4⋊C4×C83—21no—
116C2×C4⋊C4×C83—21no—
117C2×C4⋊C4×C83—21no—
118C2×C4⋊C4×C83C2311yescharacteristic
119C2×C4⋊C4×C83C2311yescharacteristic
120C2×C4⋊C4×C83C2311yescharacteristic
121C2×C4⋊C4×C83C2311yescharacteristic
122C2×C42×C83C2311yescharacteristic
123#97×C83—22no—
124C23×D83—21no—
125C2×D332—21no—
126C2×D332—21no—
127C2×D332—21no—
128C2×D332C22×C412yes—
129C2×D332C2×Q812yes—
130C2×D332#4112yes—
131C22×Dic83—21no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83C2×D411yescharacteristic
135C22×Dic83C2×D411yescharacteristic
136C22×Dic83C2×D411yescharacteristic
137C22×Dic83C2×D411yescharacteristic
138C2×C4×D83—24no—
139C2×C4×D83—42no—
140C2×C4×D83—42no—
141C2×C4×D83—42no—
142C2×C4×D83—42no—
143C2×C4×D83—42no—
144C2×C4×D83—42no—
145C2×C4×D83—81no—
146C2×C4×D83—81no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8—21no—
149C2×C83⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8C22×C412yes—
152C2×C83⋊C8C22×C412yes—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—22no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—41no—
159C4⋊Dic83—42no—
160C4⋊Dic83—81no—
161C4⋊Dic83C22×C412yes—
162C4⋊Dic83C2×Q812yes—
163Dic83⋊C4—42no—
164Dic83⋊C4—42no—
165(C2×D83)⋊C4—24no—
166(C2×D83)⋊C4—42no—
167(C2×D83)⋊C4—42no—
168C4×Dic83—21no—
169C4×Dic83—21no—
170C4×Dic83—21no—
171C4×Dic83—21no—
172C4×Dic83—42no—
173C4×Dic83—81no—
174C4×Dic83—81no—
175C22×C4×C83—21no—
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83—21no—
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2411yescharacteristic
182C2×C8×C83—21no—
183C2×C8×C83—21no—
184C2×C8×C83—21no—
185C2×C8×C83—21no—
186C2×C8×C83—21no—
187C2×C8×C83C2×Q812yes—
188C4⋊C4×C83—24no—
189C4⋊C4×C83—24no—
190C4⋊C4×C83—21no—
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83—42no—
197C4⋊C4×C83—81no—
198C4⋊C4×C83C22×C412yes—
199C4⋊C4×C83C22×C412yes—
200C42×C83C22×C412yes—
201C42×C83#4112yes—
202C23×C83C22×D411yescharacteristic
203C22×D83—22no—
204C22×D83—22no—
205C22×D83—22no—
206C22×D83—41no—
207C22×D83—41no—
208C22×D83—41no—
209C22×D83—41no—
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—22no—
215C2×C4×C83—22no—
216C2×C4×C83—41no—
217C2×C4×C83—41no—
218C2×C4×C83—41no—
219C2×C4×C83—41no—
220C2×C4×C83—41no—
221C2×C4×C83—41no—
222C2×C4×C83—41no—
223C2×C4×C83—41no—
224C2×C4×C83—41no—
225C2×C4×C83—41no—
226C2×C4×C83C4×D412yes—
227C2×C4×C83C4×D412yes—
228C2×C4×C83C23×C411yescharacteristic
229C2×C4×C83C22×Q811yescharacteristic
230C2×C4×C83C2×#4111yescharacteristic
231C2×C4×C83C2×#4111yescharacteristic
232C2×C4×C83H(5,𝔽2)11yescharacteristic
233C2×C4×C832−1+411yescharacteristic
234C2×Dic83—21no—
235C2×Dic83—21no—
236C2×Dic83—22no—
237C2×Dic83—21no—
238C2×Dic83—21no—
239C2×Dic83—21no—
240C2×Dic83—22no—
241C2×Dic83—21no—
242C2×Dic83—21no—
243C2×Dic83—21no—
244C2×Dic83—21no—
245C2×Dic83—21no—
246C2×Dic83—21no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83AGL(1,ℤ/8)12yes—
255D332—22no—
256D332—22no—
257D332—22no—
258D332C4×Q814yes—
259C4×D83—48no—
260C4×D83—84no—
261C4×D83—84no—
262C8×C83—22no—
263C8×C83—41no—
264C83⋊C8—22no—
265C83⋊C8—22no—
266C22×C83C2×C4×D411yescharacteristic
267C22×C83#54611yescharacteristic
268C22×C83D4×Q811yescharacteristic
269C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
270C22×C83#57511yescharacteristic
271C22×C83#57711yescharacteristic
272C22×C83#57711yescharacteristic
273C2×D83—24no—
274C2×D83—42no—
275C2×D83—42no—
276C2×D83—42no—
277C2×D83—42no—
278C2×D83—42no—
279C2×D83—42no—
280C4×C83—22no—
281C4×C83—22no—
282C4×C83—42no—
283C4×C83—42no—
284C4×C83—81no—
285C4×C83—81no—
286C4×C83C2×C4×Q811yescharacteristic
287C4×C83#52011yescharacteristic
288C4×C83#55211yescharacteristic
289C4×C83#55411yescharacteristic
290Dic83—22no—
291Dic83—42no—
292Dic83—42no—
293Dic83—41no—
294Dic83—41no—
295Dic83—41no—
296Dic83—81no—
297#22978—831nomaximal, Sylow
298C2×C83#236011yescharacteristic
299C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
300C2×C83#295211yescharacteristic
301C2×C83#330711yescharacteristic
302C2×C83#331011yescharacteristic
303C2×C83#338411yescharacteristic
304C2×C83#338811yescharacteristic
305D83—44no—
306C2×#478—831no—
307#2370—831no—
308D4×C4⋊C4—831no—
309#1992D8311yescharacteristic, complemented
310#1948—831no—
311#1943—831no—
312#1941—831no—
313#1920—831no—
314#1918—831no—
315#1864—831no—
316#1864—831no—
317#1771—831no—
318#1762—831no—
319#1762—831no—
320C4×#102—831no—
321C83#2297811yescharacteristic, complemented, Sylow
322C2×#125C2×D8311yescharacteristic
323C2×C4×D4—831no—
324C2×C4×D4—831no—
325C2×C4×D4—831no—
326C22×C4⋊C4—1661no—
327#478—838no—
328C2×#107—831no—
329C2×#106—831no—
330C2×#106—831no—
331C2×#106—831no—
332C2×C4⋊C8—831no—
333C2×#102—831no—
334C2×#102—831no—
335C2×#102—831no—
336C2×#102—831no—
337C2×#97—831no—
338C2×#97C2×D8311yescharacteristic
339C2×C4×C8—831no—
340#389—831no—
341#389—831no—
342#389—831no—
343#389—831no—
344#389—1661no—
345#387—1661no—
346#387—1661no—
347#385—1661no—
348#385—1661no—
349#384C2×D8311yescharacteristic
350#380—1661no—
351C4×C4⋊C4—831no—
352C4×C4⋊C4—831no—
353C4×C4⋊C4—831no—
354C4×C4⋊C4—831no—
355C4×C4⋊C4—1661no—
356#340—831no—
357#340—831no—
358#340—831no—
359#340—831no—
360#340C2×D8311yescharacteristic
361#340C2×D8311yescharacteristic
362#340C2×D8311yescharacteristic
363#340C2×D8311yescharacteristic
364#336—831no—
365#336—831no—
366C22×D4—831no—
367C23×C4—1661no—
368C23×C4—1661no—
369C23×C4—1661no—
370C22×C8—831no—
371C22×C8—831no—
372C22×C8C22×D8311yescharacteristic
373C22×C8C22×D8311yescharacteristic
374#125—22no—
375#125C4×D8314yescomplemented
376C4×D4—3321no—
377C4×D4—3321no—
378C4×D4—834no—
379C4×D4—1662no—
380C4×D4—1662no—
381C4×D4—1662no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—3324no—
387C2×C4⋊C4—831no—
388C2×C4⋊C4—831no—
389C2×C4⋊C4—831no—
390C2×C4⋊C4—831no—
391C2×C4⋊C4—831no—
392C2×C4⋊C4—1661no—
393C2×C4⋊C4—1661no—
394C2×C4⋊C4—1661no—
395C2×C4⋊C4—1661no—
396C2×C4⋊C4C22×D8311yescharacteristic
397C2×C4⋊C4C22×D8311yescharacteristic
398C2×C4⋊C4C22×D8311yescharacteristic
399C2×C4⋊C4C22×D8311yescharacteristic
400C2×C22⋊C4—1661no—
401C2×C22⋊C4—1661no—
402C2×C22⋊C4—1661no—
403C2×C42—831no—
404C2×C42—831no—
405C2×C42—1661no—
406C2×C42—1661no—
407C2×C42—1661no—
408C2×C42C22×D8311yescharacteristic
409#107—834no—
410#106—3321no—
411#106—3321no—
412#106—834no—
413C4⋊C8—834no—
414#102—834no—
415#102—834no—
416#102—834no—
417#102—1662no—
418#97—22no—
419#97—3321no—
420C4×C8—3321no—
421#95—831no—
422#95—831no—
423#95—1661no—
424#95—1661no—
425#95—1661no—
426#95—1661no—
427#95—1661no—
428#95—1661no—
429#95—1661no—
430C24—1661no—
431C2×D4—832no—
432C2×D4—832no—
433C2×D4—832no—
434C2×D4—1661no—
435C2×D4—1661no—
436C2×D4—1661no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—3322no—
442C22×C4—3322no—
443C22×C4—3322no—
444C22×C4—3322no—
445C22×C4—3322no—
446C22×C4—3322no—
447C22×C4—831no—
448C22×C4—831no—
449C22×C4—831no—
450C22×C4—831no—
451C22×C4—6641no—
452C22×C4—6641no—
453C22×C4—1661no—
454C22×C4—1661no—
455C22×C4—1661no—
456C22×C4—1664no—
457C22×C4D4×D8311yescharacteristic
458C22×C4D4×D8311yescharacteristic
459C22×C4C23×D8311yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—832no—
466C2×C8—832no—
467C2×C8—1661no—
468C2×C8—1661no—
469C2×C8—1661no—
470C2×C8—1661no—
471C2×C8Q8×D8312yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—24no—
475C4⋊C4—24no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—42no—
481C4⋊C4—81no—
482C4⋊C4—3321no—
483C4⋊C4—3322no—
484C4⋊C4—3322no—
485C4⋊C4—3322no—
486C4⋊C4—832no—
487C4⋊C4—832no—
488C4⋊C4—6641no—
489C4⋊C4—1662no—
490C4⋊C4—1661no—
491C4⋊C4—1661no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4C2×C4×D8312yes—
495C4⋊C4C2×C4×D8312yes—
496C22⋊C4—3322no—
497C22⋊C4—3322no—
498C22⋊C4—1664no—
499C42—3322no—
500C42—6641no—
501C42—6641no—
502C42—1661no—
503C42—1661no—
504C42—1661no—
505C42—1661no—
506C42C2×C4×D8312yes—
507C42#1305058912yes—
508C23—3321no—
509C23—3321no—
510C23—3321no—
511C23—3321no—
512C23—1662no—
513C23—1662no—
514C23—1662no—
515C23C2×D4×D8311yescharacteristic
516D4—834no—
517D4—1662no—
518D4—1662no—
519D4—1662no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—3328no—
537C2×C4—3321no—
538C2×C4—3321no—
539C2×C4—3321no—
540C2×C4—3321no—
541C2×C4—3321no—
542C2×C4—3321no—
543C2×C4—3321no—
544C2×C4—832no—
545C2×C4—6644no—
546C2×C4—6644no—
547C2×C4—1661no—
548C2×C4—1661no—
549C2×C4—1661no—
550C2×C4—1662no—
551C2×C4—1662no—
552C2×C4—1661no—
553C2×C4—1661no—
554C2×C4—1661no—
555C2×C4—1661no—
556C2×C4—1661no—
557C2×C4—1661no—
558C2×C4—1661no—
559C2×C4—1661no—
560C2×C4(C22⋊D83)⋊C412yes—
561C2×C4(C22⋊D83)⋊C412yes—
562C2×C4C22×C4×D8311yescharacteristic
563C2×C4C2×Q8×D8311yescharacteristic
564C2×C4#41×D8311yescharacteristic
565C2×C4C2×#1305058911yescharacteristic
566C2×C4#1458183811yescharacteristic
567C2×C4#1458184011yescharacteristic
568C8—22no—
569C8—41no—
570C8—1662no—
571C8—1662no—
572C22—3322no—
573C22—3322no—
574C22—3322no—
575C22—3322no—
576C22—3322no—
577C22—3322no—
578C22—1664no—
579C22C2×(C22⋊D83)⋊C411yescharacteristic
580C22#1912834411yescharacteristic
581C22#1912836611yescharacteristic
582C22AGL(1,ℤ/8)×D8311yescharacteristic
583C22#1912850011yescharacteristic
584C22#1912851011yescharacteristic
585C22#1912851111yescharacteristic
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—81no—
591C4—81no—
592C4—3322no—
593C4—3322no—
594C4—3321no—
595C4—3321no—
596C4—3321no—
597C4—6641no—
598C4—1662no—
599C4C4×Q8×D8311yescharacteristic
600C4#1912818311yescharacteristic
601C4#1912838511yescharacteristic
602C4#1912840911yescharacteristic
603C2—3324no—
604C2#2491047311yescharacteristic
605C2#2491627611yescharacteristic
606C2#2491634211yescharacteristic
607C2#2491943811yescharacteristic
608C2#2491948311yescharacteristic
609C2#2492035911yescharacteristic
610C2#2492040711yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1992C2
socleC23×C83C22×D4
prime radicalC22×D332C23
cosocleC22×C4×C83C24
hypercenter#1992D83
hypocenterC83#22978
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1992

Complements

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

NH ≅ G/Ntype
2. #24910563C2semidirect
4. #24908224C2semidirect
5. #24908200C2semidirect
6. #24908190C2semidirect
7. #24908046C2semidirect
8. #24908032C2semidirect
11. #24907335C2semidirect
15. C83×#1992C2semidirect
20. #19127998C4semidirect
90. #14581725C2×C4semidirect
93. #14581723C2×C4semidirect
94. C4⋊(C83⋊C8)C2×C4semidirect
113. #125×C83C2×C4semidirect
309. #1992D83semidirect
321. C83#22978semidirect
375. #125C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191279982
#249105632
#249103632
#249082242
#249082002
#249081902
#249080462
#249080322
#249077252
#249073352
#249072892
#249072872
#249072772
C83×#19922
#2297883