Finite Group Databasebeta

← #35456462

Subgroups — #35456462

Order 21248.

790 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127988C211yesmaximal, characteristic
2C2×#19127832C211yesmaximal, characteristic, complemented
3#24912821C211yesmaximal, characteristic
4#24908965C211yesmaximal, characteristic, complemented
5#24908964C211yesmaximal, characteristic, complemented
6#24908848C211yesmaximal, characteristic
7#24908834C211yesmaximal, characteristic
8#24908437C211yesmaximal, characteristic, complemented
9#24908231C211yesmaximal, characteristic, complemented
10#24908198C211yesmaximal, characteristic, complemented
11#24908188C211yesmaximal, characteristic, complemented
12#24908159C211yesmaximal, characteristic
13#24908145C211yesmaximal, characteristic
14#24907291C211yesmaximal, characteristic
15C83×#2546C211yesmaximal, characteristic, complemented
16C2×C4×D332C2211yescharacteristic
17#19127988—24no—
18#19127832—24no—
19C2×#14581725C2211yescharacteristic
20C2×#14581723C2211yescharacteristic
21C2×#14581723C2211yescharacteristic
22C2×#14581723C2211yescharacteristic
23C2×C4⋊(C83⋊C8)C2211yescharacteristic
24C2×C4⋊(C83⋊C8)C2211yescharacteristic
25C2×C4⋊(C83⋊C8)C2211yescharacteristic
26C2×#14581712C2211yescharacteristic
27C2×#14581712C2211yescharacteristic
28C2×#14581712C2211yescharacteristic
29C2×#14581712C2211yescharacteristic
30C2×#14581703C2211yescharacteristic
31C2×#14581703C2211yescharacteristic
32C2×C4×C83⋊C8C2211yescharacteristic
33#19127636—21no—
34#19127612—21no—
35#19127603C2211yescharacteristic
36#19127603C2211yescharacteristic
37#19127599C2211yescharacteristic
38#19127599C2211yescharacteristic
39#19127572—21no—
40#19127572—21no—
41#19127567C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50C2×C4⋊Q8×C83—21no—
51C2×C4⋊Q8×C83C2211yescharacteristic
52C2×C4×Q8×C83C2211yescharacteristic
53#398×C83—21no—
54#398×C83—21no—
55#391×C83—21no—
56#389×C83C2211yescharacteristic
57#389×C83C2211yescharacteristic
58#389×C83C2211yescharacteristic
59#389×C83C2211yescharacteristic
60C4×C4⋊C4×C83C2211yescharacteristic
61C22×D332C2311yescharacteristic
62C22×C4×D83—21no—
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C4×D332—22no—
68C4×D332—22no—
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×Dic83⋊C4—21no—
73C2×Dic83⋊C4—21no—
74C2×(C2×D83)⋊C4—21no—
75C2×(C2×D83)⋊C4—21no—
76C2×(C2×D83)⋊C4—21no—
77C2×(C2×D83)⋊C4—21no—
78C2×(C2×D83)⋊C4—21no—
79#14581725—22no—
80#14581723—22no—
81#14581723—22no—
82#14581723—22no—
83C4⋊(C83⋊C8)—22no—
84C4⋊(C83⋊C8)—22no—
85C4⋊(C83⋊C8)D414yes—
86#14581712—22no—
87#14581712—22no—
88#14581712—22no—
89#14581712Q814yescomplemented
90#14581703—22no—
91#14581703—22no—
92C4×C83⋊C8—22no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C22×Q8×C83—21no—
98C22×Q8×C83D411yescharacteristic
99C4⋊Q8×C83—22no—
100C4⋊Q8×C83—22no—
101C4⋊Q8×C83—44no—
102C4×Q8×C83—22no—
103C4×Q8×C83—22no—
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83—21no—
106C2×C4⋊C4×C83—21no—
107C2×C4⋊C4×C83—21no—
108C2×C4⋊C4×C83—21no—
109C2×C4⋊C4×C83D411yescharacteristic
110C2×C4⋊C4×C83D411yescharacteristic
111C2×C4⋊C4×C83D411yescharacteristic
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C42×C83—21no—
117C2×C42×C83C2311yescharacteristic
118C2×C42×C83C2311yescharacteristic
119C2×C42×C83C2311yescharacteristic
120#95×C83—21no—
121#95×C83—21no—
122#95×C83—21no—
123#95×C83—21no—
124#95×C83—21no—
125C23×D83—21no—
126C2×D332—21no—
127C2×D332—21no—
128C2×D332—21no—
129C2×D332—21no—
130C2×D332—21no—
131C2×D332C2×Q812yes—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83—21no—
135C2×C4×D83—42no—
136C2×C4×D83—81no—
137C2×C4×D83—81no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8—21no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—21no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8C2×D412yes—
148C2×C83⋊C8C2×D412yes—
149C2×C83⋊C8C2×Q812yes—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156Dic83⋊C4—81no—
157Dic83⋊C4—81no—
158(C2×D83)⋊C4—42no—
159(C2×D83)⋊C4—42no—
160(C2×D83)⋊C4—42no—
161(C2×D83)⋊C4—81no—
162(C2×D83)⋊C4—81no—
163C2×Q8×C83—22no—
164C2×Q8×C83—22no—
165C2×Q8×C83—41no—
166C2×Q8×C83—42no—
167C2×Q8×C83—42no—
168C2×Q8×C83—42no—
169C22×C4×C83—21no—
170C22×C4×C83—21no—
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83C2×D411yescharacteristic
174C22×C4×C83C2×D411yescharacteristic
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83C2×D411yescharacteristic
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2411yescharacteristic
180C4⋊C4×C83—22no—
181C4⋊C4×C83—22no—
182C4⋊C4×C83—22no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—42no—
190C4⋊C4×C83—42no—
191C4⋊C4×C83—42no—
192C4⋊C4×C83—42no—
193C4⋊C4×C83—42no—
194C4⋊C4×C83C2×Q812yes—
195C4⋊C4×C83#4112yes—
196C42×C83—21no—
197C42×C83—21no—
198C42×C83—21no—
199C42×C83—21no—
200C42×C83—42no—
201C42×C83C2×D412yes—
202C42×C83#4112yes—
203C23×C83C22×D411yescharacteristic
204C22×D83—22no—
205C22×D83—41no—
206C22×D83—41no—
207C22×D83—41no—
208C22×D83—41no—
209C22×D83—41no—
210C22×D83—41no—
211Q8×C83—44no—
212Q8×C83—84no—
213C2×C4×C83—21no—
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—21no—
222C2×C4×C83—22no—
223C2×C4×C83—22no—
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—41no—
236C2×C4×C83—41no—
237C2×C4×C83—41no—
238C2×C4×C83—41no—
239C2×C4×C83—41no—
240C2×C4×C83#12112yes—
241C2×C4×C83#12112yes—
242C2×C4×C83AGL(1,ℤ/8)12yes—
243C2×C4×C83C22×D411yescharacteristic
244C2×C4×C83C22×Q811yescharacteristic
245C2×C4×C83C2×#4111yescharacteristic
246C2×C4×C83C2×#4111yescharacteristic
247C2×C4×C832−1+411yescharacteristic
248C2×C4×C832−1+411yescharacteristic
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258D332—22no—
259D332—22no—
260D332—41no—
261D332—41no—
262C4×D83—84no—
263C83⋊C8—22no—
264C83⋊C8—22no—
265C83⋊C8—22no—
266C83⋊C8—41no—
267C22×C83C2×#12111yescharacteristic
268C22×C83#54811yescharacteristic
269C22×C83D4×Q811yescharacteristic
270C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
271C22×C83#57511yescharacteristic
272C22×C83#57711yescharacteristic
273C22×C83#57711yescharacteristic
274C2×D83—42no—
275C2×D83—42no—
276C2×D83—42no—
277C2×D83—81no—
278C2×D83—81no—
279C2×D83—81no—
280C2×D83—81no—
281C4×C83—22no—
282C4×C83—22no—
283C4×C83—22no—
284C4×C83—42no—
285C4×C83—42no—
286C4×C83—41no—
287C4×C83—41no—
288C4×C83—41no—
289C4×C83—81no—
290C4×C83—81no—
291C4×C83#53311yescharacteristic
292C4×C83#54111yescharacteristic
293C4×C83#54711yescharacteristic
294C4×C83D4×Q811yescharacteristic
295Dic83—81no—
296Dic83—81no—
297Dic83—81no—
298#25485—831nomaximal, Sylow
299C2×C83#242911yescharacteristic
300C2×C83#305511yescharacteristic
301C2×C83#305711yescharacteristic
302C2×C83#337111yescharacteristic
303C2×C83#337311yescharacteristic
304C2×C83#338411yescharacteristic
305C2×C83#338811yescharacteristic
306D83—82no—
307C2×#476—831no—
308C2×#446—831no—
309#2580—831no—
310#2546D8311yescharacteristic, complemented
311#2088—831no—
312#2088—831no—
313#2067—831no—
314#2065—831no—
315#1993—831no—
316#1949—831no—
317#1943—831no—
318#1941—831no—
319#1936—831no—
320#1934—831no—
321#1762—831no—
322C83#2548511yescharacteristic, complemented, Sylow
323C2×C4⋊Q8—21no—
324C2×C4⋊Q8C2×D8311yescharacteristic
325C2×C4×Q8C2×D8311yescharacteristic
326C2×C4×D4—831no—
327#476—1664no—
328#446—1664no—
329C2×#107—831no—
330C2×#106—831no—
331C2×#106—831no—
332C2×#106—831no—
333C2×C4⋊C8—831no—
334C2×C4⋊C8—831no—
335C2×C4⋊C8—831no—
336C2×#102—831no—
337C2×#102—831no—
338C2×#102—831no—
339C2×#102—831no—
340C2×#97—831no—
341C2×#97—831no—
342C2×C4×C8—831no—
343#399—1661no—
344#398—21no—
345#398—21no—
346#393—1661no—
347#391—21no—
348#390—831no—
349#390—831no—
350#389—831no—
351#389—831no—
352#389C2×D8311yescharacteristic
353#389C2×D8311yescharacteristic
354#389C2×D8311yescharacteristic
355#389C2×D8311yescharacteristic
356#385—1661no—
357#385—1661no—
358#382—831no—
359C4×C4⋊C4C2×D8311yescharacteristic
360#340—831no—
361#340—831no—
362#340—831no—
363#340—831no—
364#340—831no—
365#340—831no—
366#340—831no—
367#340—831no—
368C22×Q8—21no—
369C22×Q8C22⋊D8311yescharacteristic
370C22×D4—831no—
371C23×C4—1661no—
372C22×C8—831no—
373C22×C8—831no—
374C22×C8—831no—
375C22×C8—831no—
376C4⋊Q8—22no—
377C4⋊Q8—22no—
378C4⋊Q8—44no—
379C4×Q8—22no—
380C4×Q8—22no—
381C4×D4—1662no—
382C4×D4—1662no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—831no—
389C2×C4⋊C4—831no—
390C2×C4⋊C4—831no—
391C2×C4⋊C4—1661no—
392C2×C4⋊C4—1661no—
393C2×C4⋊C4C22⋊D8311yescharacteristic
394C2×C4⋊C4C22⋊D8311yescharacteristic
395C2×C4⋊C4C22⋊D8311yescharacteristic
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×C22⋊C4—1661no—
404C2×C22⋊C4—1661no—
405C2×C42—21no—
406C2×C42C22×D8311yescharacteristic
407C2×C42C22×D8311yescharacteristic
408C2×C42C22×D8311yescharacteristic
409#107—1662no—
410#106—1662no—
411#106—1662no—
412#106—1662no—
413C4⋊C8—834no—
414C4⋊C8—1662no—
415C4⋊C8—1662no—
416#102—834no—
417#102—1662no—
418#102—1662no—
419#102—1662no—
420#97—1662no—
421#97—1662no—
422C4×C8—1662no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—1661no—
429#95—1661no—
430#95—1661no—
431#95—1661no—
432C24—1661no—
433C2×Q8—22no—
434C2×Q8—22no—
435C2×Q8—41no—
436C2×Q8—42no—
437C2×Q8—42no—
438C2×Q8—42no—
439C2×D4—832no—
440C2×D4—1661no—
441C2×D4—1661no—
442C2×D4—1661no—
443C2×D4—1661no—
444C2×D4—1661no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—3322no—
450C22×C4—6641no—
451C22×C4—6641no—
452C22×C4—1661no—
453C22×C4—1661no—
454C22×C4—1661no—
455C22×C4C2×C22⋊D8311yescharacteristic
456C22×C4C2×C22⋊D8311yescharacteristic
457C22×C4C2×C22⋊D8311yescharacteristic
458C22×C4C2×C22⋊D8311yescharacteristic
459C22×C4C2×C22⋊D8311yescharacteristic
460C22×C4C2×C22⋊D8311yescharacteristic
461C22×C4C23×D8311yescharacteristic
462C2×C8—832no—
463C2×C8—832no—
464C2×C8—832no—
465C2×C8—1661no—
466C2×C8—1661no—
467C2×C8—1661no—
468C2×C8—1661no—
469C2×C8—1661no—
470C2×C8—1661no—
471C2×C8—1661no—
472C2×C8—1661no—
473C2×C8—1661no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—22no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—42no—
484C4⋊C4—42no—
485C4⋊C4—42no—
486C4⋊C4—42no—
487C4⋊C4—42no—
488C4⋊C4—6641no—
489C4⋊C4—6641no—
490C4⋊C4—1661no—
491C4⋊C4—1661no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4—1661no—
495C4⋊C4—1661no—
496C4⋊C4Q8×D8312yes—
497C4⋊C4#1305059112yes—
498C22⋊C4—3322no—
499C22⋊C4—3322no—
500C22⋊C4—3322no—
501C22⋊C4—6641no—
502C22⋊C4—6641no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—42no—
508C42D4×D8312yes—
509C42#1305058912yes—
510C23—3321no—
511C23—3321no—
512C23—3321no—
513C23—3321no—
514C23—3321no—
515C23—3321no—
516C23—1662no—
517C23C22×C22⋊D8311yescharacteristic
518Q8—44no—
519Q8—84no—
520D4—3321no—
521D4—3321no—
522D4—1662no—
523D4—1662no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—3321no—
552C2×C4—3321no—
553C2×C4—3321no—
554C2×C4—3321no—
555C2×C4—3321no—
556C2×C4—3321no—
557C2×C4—3321no—
558C2×C4—3321no—
559C2×C4—3321no—
560C2×C4—6644no—
561C2×C4#1458176412yes—
562C2×C4#1458176412yes—
563C2×C4#1458181812yes—
564C2×C4C2×D4×D8311yescharacteristic
565C2×C4C2×Q8×D8311yescharacteristic
566C2×C4C2×#1305058911yescharacteristic
567C2×C4C2×#1305059111yescharacteristic
568C2×C4#1458184011yescharacteristic
569C2×C4#1458184011yescharacteristic
570C8—3321no—
571C8—1662no—
572C8—1662no—
573C8—1662no—
574C22—3322no—
575C22—3322no—
576C22—3322no—
577C22—6641no—
578C22—6641no—
579C22—6641no—
580C22—6641no—
581C22C2×#1458176411yescharacteristic
582C22#1912836211yescharacteristic
583C22Q8×C22⋊D8311yescharacteristic
584C22C2×#1458181811yescharacteristic
585C22#1912849911yescharacteristic
586C22#1912850911yescharacteristic
587C22#1912850911yescharacteristic
588C4—22no—
589C4—22no—
590C4—22no—
591C4—42no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—42no—
596C4—81no—
597C4—81no—
598C4—6641no—
599C4—6641no—
600C4—6641no—
601C4#1912825611yescharacteristic
602C4#1912831211yescharacteristic
603C4#1912835411yescharacteristic
604C4#1912836611yescharacteristic
605C2—6642no—
606C2#2491106511yescharacteristic
607C2#2491707811yescharacteristic
608C2#2491710411yescharacteristic
609C2#2492023111yescharacteristic
610C2#2492025511yescharacteristic
611C2#2492035811yescharacteristic
612C2#2492040211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2546C2
socleC23×C83C22×D4
prime radicalC22×D332C23
hypercenter#2546D83
hypocenterC83#25485
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2546

Complements

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

NH ≅ G/Ntype
2. C2×#19127832C2semidirect
4. #24908965C2semidirect
5. #24908964C2semidirect
8. #24908437C2semidirect
9. #24908231C2semidirect
10. #24908198C2semidirect
11. #24908188C2semidirect
15. C83×#2546C2semidirect
89. #14581712Q8semidirect
310. #2546D83semidirect
322. C83#25485semidirect

Maximal subgroups

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

subgroupindex
C2×#191279882
C2×#191278322
#249128212
#249089652
#249089642
#249088482
#249088342
#249084372
#249082312
#249081982
#249081882
#249081592
#249081452
#249072912
C83×#25462
#2548583