Finite Group Databasebeta

← #35462884

Subgroups — #35462884

Order 21248.

834 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127998C211yesmaximal, characteristic
2C2×#19127988C211yesmaximal, characteristic
3#24910374C211yesmaximal, characteristic
4#24908965C211yesmaximal, characteristic, complemented
5#24908964C211yesmaximal, characteristic, complemented
6#24908798C211yesmaximal, characteristic, complemented
7#24908788C211yesmaximal, characteristic, complemented
8#24908231C211yesmaximal, characteristic, complemented
9#24908182C211yesmaximal, characteristic, complemented
10#24907709C211yesmaximal, characteristic
11#24907709C211yesmaximal, characteristic
12#24907625C211yesmaximal, characteristic, complemented
13#24907470C211yesmaximal, characteristic
14C4×#14581712C211yesmaximal, characteristic
15C83×#2550C211yesmaximal, characteristic, complemented
16C2×C4×D332C2211yescharacteristic
17C2×C4×D332C2211yescharacteristic
18C2×C4×D332C2211yescharacteristic
19#19127998—24no—
20#19127988—24no—
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×#14581723C2211yescharacteristic
27C2×C4⋊(C83⋊C8)C2211yescharacteristic
28C2×C4⋊(C83⋊C8)C2211yescharacteristic
29C2×C4⋊(C83⋊C8)C2211yescharacteristic
30C2×#14581712C2211yescharacteristic
31C2×#14581712C2211yescharacteristic
32C2×#14581712C2211yescharacteristic
33C2×#14581712C2211yescharacteristic
34C2×C4×C83⋊C8C2211yescharacteristic
35#19127599C2211yescharacteristic
36#19127599C2211yescharacteristic
37#19127572—21no—
38#19127572—21no—
39#19127567C2211yescharacteristic
40#19127560—21no—
41#95×D83—21no—
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×C4⋊Q8×C83C2211yescharacteristic
51C2×#125×C83C2211yescharacteristic
52#398×C83—21no—
53#395×C83—21no—
54#391×C83—21no—
55#389×C83—21no—
56#389×C83C2211yescharacteristic
57#389×C83C2211yescharacteristic
58#382×C83C2211yescharacteristic
59C4×C4⋊C4×C83C2211yescharacteristic
60C4×C4⋊C4×C83C2211yescharacteristic
61C22×D332C2311yescharacteristic
62C22×C4×D83—21no—
63C22×C4×D83—21no—
64C22×C4×D83—21no—
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69C4×D332—22no—
70C4×D332—22no—
71C4×D332—22no—
72C4×D332—41no—
73C4×D332—41no—
74C4×D332Q814yescomplemented
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—21no—
80C2×(C2×D83)⋊C4—21no—
81C2×(C2×D83)⋊C4—21no—
82C2×(C2×D83)⋊C4—21no—
83#14581725—22no—
84#14581725—41no—
85#14581725—41no—
86#14581723—22no—
87#14581723—22no—
88#14581723—22no—
89C4⋊(C83⋊C8)—22no—
90C4⋊(C83⋊C8)—22no—
91C4⋊(C83⋊C8)—22no—
92#14581712—22no—
93#14581712—22no—
94#14581712D414yes—
95#14581712Q814yescomplemented
96C4×C83⋊C8—22no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C22×Q8×C83—21no—
102C4⋊Q8×C83—22no—
103C4⋊Q8×C83—41no—
104#125×C83—22no—
105#125×C83—41no—
106C2×C4⋊C4×C83—21no—
107C2×C4⋊C4×C83—21no—
108C2×C4⋊C4×C83—21no—
109C2×C4⋊C4×C83—21no—
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83D411yescharacteristic
112C2×C4⋊C4×C83D411yescharacteristic
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C42×C83—21no—
118C2×C42×C83C2311yescharacteristic
119C2×C42×C83C2311yescharacteristic
120C2×C42×C83C2311yescharacteristic
121#95×C83—21no—
122#95×C83—21no—
123#95×C83—21no—
124#95×C83—21no—
125#95×C83—21no—
126#95×C83D411yescharacteristic
127#95×C83D411yescharacteristic
128C23×D83—21no—
129C2×D332—21no—
130C2×D332—21no—
131C2×D332—21no—
132C2×D332C2×D412yes—
133C2×D332C2×Q812yes—
134C2×D332C2×Q812yes—
135C22×Dic83—21no—
136C22×Dic83—21no—
137C22×Dic83—21no—
138C2×C4×D83—24no—
139C2×C4×D83—42no—
140C2×C4×D83—42no—
141C2×C4×D83—42no—
142C2×C4×D83—42no—
143C2×C4×D83—81no—
144C2×C4×D83—81no—
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⋊C8—21no—
152C2×C83⋊C8—21no—
153C2×C83⋊C8—21no—
154C2×C83⋊C8—21no—
155C2×C83⋊C8—21no—
156C2×C83⋊C8—21no—
157C2×C83⋊C8C2×D412yes—
158C2×C83⋊C8C2×Q812yes—
159C4⋊Dic83—21no—
160C4⋊Dic83—21no—
161C4⋊Dic83—21no—
162C4⋊Dic83—21no—
163C4⋊Dic83—21no—
164C4⋊Dic83C2×Q812yes—
165Dic83⋊C4—81no—
166Dic83⋊C4—81no—
167(C2×D83)⋊C4—24no—
168(C2×D83)⋊C4—42no—
169(C2×D83)⋊C4—42no—
170C2×Q8×C83—42no—
171C2×Q8×C83—42no—
172C2×Q8×C83—81no—
173C22×C4×C83—21no—
174C22×C4×C83—21no—
175C22×C4×C83—21no—
176C22×C4×C83—21no—
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2411yescharacteristic
184C4⋊C4×C83—22no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—21no—
190C4⋊C4×C83—21no—
191C4⋊C4×C83—41no—
192C4⋊C4×C83—42no—
193C4⋊C4×C83—42no—
194C4⋊C4×C83—42no—
195C4⋊C4×C83—42no—
196C4⋊C4×C83—81no—
197C4⋊C4×C83C2×D412yes—
198C4⋊C4×C83C2×Q812yes—
199C42×C83—21no—
200C42×C83—21no—
201C42×C83—21no—
202C42×C83—21no—
203C42×C83—81no—
204C42×C83C2×Q812yes—
205C42×C83#4112yes—
206C23×C83C22×D411yescharacteristic
207C22×D83—22no—
208C22×D83—22no—
209C22×D83—22no—
210C22×D83—41no—
211C22×D83—41no—
212C22×D83—41no—
213C22×D83—41no—
214Q8×C83—84no—
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—21no—
223C2×C4×C83—21no—
224C2×C4×C83—22no—
225C2×C4×C83—22no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
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—41no—
241C2×C4×C83—41no—
242C2×C4×C83C22⋊Q812yes—
243C2×C4×C83C22⋊Q812yes—
244C2×C4×C83#13512yes—
245C2×C4×C83C22×D411yescharacteristic
246C2×C4×C83C22×Q811yescharacteristic
247C2×C4×C83C22×Q811yescharacteristic
248C2×C4×C83C2×#4111yescharacteristic
249C2×C4×C83H(5,𝔽2)11yescharacteristic
250C2×C4×C832−1+411yescharacteristic
251C2×Dic83—22no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260D332—22no—
261D332—22no—
262D332—22no—
263D332C4⋊Q814yes—
264C4×D83—84no—
265C4×D83—84no—
266C4×D83—84no—
267C83⋊C8—22no—
268C83⋊C8—22no—
269C83⋊C8—41no—
270C83⋊C8—41no—
271C22×C83C2×C22⋊Q811yescharacteristic
272C22×C83D4211yescharacteristic
273C22×C83D4×Q811yescharacteristic
274C22×C83C2×#13511yescharacteristic
275C22×C83#57511yescharacteristic
276C22×C83#57611yescharacteristic
277C22×C83#57711yescharacteristic
278C2×D83—24no—
279C2×D83—42no—
280C2×D83—42no—
281C2×D83—42no—
282C2×D83—42no—
283C2×D83—42no—
284C2×D83—42no—
285C4×C83—22no—
286C4×C83—22no—
287C4×C83—22no—
288C4×C83—42no—
289C4×C83—41no—
290C4×C83—41no—
291C4×C83—42no—
292C4×C83—41no—
293C4×C83—81no—
294C4×C83—81no—
295C4×C83C2×C4⋊Q811yescharacteristic
296C4×C83#54411yescharacteristic
297C4×C83#54711yescharacteristic
298C4×C83#55411yescharacteristic
299Dic83—42no—
300Dic83—81no—
301Dic83—81no—
302#26051—831nomaximal, Sylow
303C2×C83#244011yescharacteristic
304C2×C83#307611yescharacteristic
305C2×C83#308111yescharacteristic
306C2×C83#327311yescharacteristic
307C2×C83#328211yescharacteristic
308C2×C83#338011yescharacteristic
309C2×C83#338811yescharacteristic
310D83—44no—
311C2×#478—831no—
312C2×#476—831no—
313#2550D8311yescharacteristic, complemented
314#2349—831no—
315#2088—831no—
316#2088—831no—
317#2059—831no—
318#2057—831no—
319#1949—831no—
320#1939—831no—
321#1862—831no—
322#1862—831no—
323#1846—831no—
324#1804—831no—
325C4×#102—831no—
326C83#2605111yescharacteristic, complemented, Sylow
327C2×C4⋊Q8C2×D8311yescharacteristic
328C2×#125C2×D8311yescharacteristic
329C2×C4×D4—831no—
330C2×C4×D4—831no—
331C2×C4×D4—831no—
332#478—1664no—
333#476—1664no—
334C2×#107—831no—
335C2×#107—831no—
336C2×#107—831no—
337C2×#106—831no—
338C2×#106—831no—
339C2×#106—831no—
340C2×C4⋊C8—831no—
341C2×C4⋊C8—831no—
342C2×C4⋊C8—831no—
343C2×#102—831no—
344C2×#102—831no—
345C2×#102—831no—
346C2×#102—831no—
347C2×C4×C8—831no—
348#398—21no—
349#395—21no—
350#391—21no—
351#389—21no—
352#389—831no—
353#389—831no—
354#389C2×D8311yescharacteristic
355#389C2×D8311yescharacteristic
356#385—1661no—
357#385—1661no—
358#382—831no—
359#382C2×D8311yescharacteristic
360#380—1661no—
361C4×C4⋊C4C2×D8311yescharacteristic
362C4×C4⋊C4C2×D8311yescharacteristic
363C2×#95—1661no—
364#340—831no—
365#340—831no—
366#340—831no—
367#340—831no—
368#340—831no—
369#340—831no—
370#336—831no—
371#336—831no—
372C22×Q8—21no—
373C22×D4—831no—
374C23×C4—1661no—
375C23×C4—1661no—
376C23×C4—1661no—
377C22×C8—831no—
378C22×C8—831no—
379C22×C8—831no—
380C22×C8—831no—
381C4⋊Q8—22no—
382C4⋊Q8—41no—
383#125—22no—
384#125—41no—
385C4×D4—3321no—
386C4×D4—3321no—
387C4×D4—834no—
388C4×D4—1662no—
389C4×D4—1662no—
390C4×D4—1662no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—21no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—831no—
397C2×C4⋊C4—831no—
398C2×C4⋊C4—831no—
399C2×C4⋊C4—1661no—
400C2×C4⋊C4—1661no—
401C2×C4⋊C4C22⋊D8311yescharacteristic
402C2×C4⋊C4C22⋊D8311yescharacteristic
403C2×C4⋊C4C22×D8311yescharacteristic
404C2×C4⋊C4C22×D8311yescharacteristic
405C2×C4⋊C4C22×D8311yescharacteristic
406C2×C4⋊C4C22×D8311yescharacteristic
407C2×C22⋊C4—1661no—
408C2×C22⋊C4—1661no—
409C2×C22⋊C4—1661no—
410C2×C42—21no—
411C2×C42C22×D8311yescharacteristic
412C2×C42C22×D8311yescharacteristic
413C2×C42C22×D8311yescharacteristic
414#107—3321no—
415#107—3321no—
416#107—1662no—
417#106—1662no—
418#106—1662no—
419#106—1662no—
420C4⋊C8—1662no—
421C4⋊C8—1662no—
422C4⋊C8—1662no—
423#102—834no—
424#102—834no—
425#102—1662no—
426#102—1662no—
427C4×C8—1662no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—21no—
433#95—1661no—
434#95—1661no—
435#95—1661no—
436#95—1661no—
437#95C22⋊D8311yescharacteristic
438#95C22⋊D8311yescharacteristic
439C24—1661no—
440C2×Q8—42no—
441C2×Q8—42no—
442C2×Q8—81no—
443C2×D4—832no—
444C2×D4—832no—
445C2×D4—832no—
446C2×D4—1661no—
447C2×D4—1661no—
448C2×D4—1661no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—3322no—
454C22×C4—3322no—
455C22×C4—3322no—
456C22×C4—3322no—
457C22×C4—6641no—
458C22×C4—6641no—
459C22×C4—6641no—
460C22×C4—6641no—
461C22×C4—1661no—
462C22×C4—1661no—
463C22×C4—1661no—
464C22×C4—1664no—
465C22×C4C2×C22⋊D8311yescharacteristic
466C22×C4C2×C22⋊D8311yescharacteristic
467C22×C4C2×C22⋊D8311yescharacteristic
468C22×C4C2×C22⋊D8311yescharacteristic
469C22×C4C2×C22⋊D8311yescharacteristic
470C22×C4C2×C22⋊D8311yescharacteristic
471C22×C4C23×D8311yescharacteristic
472C2×C8—832no—
473C2×C8—832no—
474C2×C8—1661no—
475C2×C8—1661no—
476C2×C8—1661no—
477C2×C8—1661no—
478C2×C8—1661no—
479C2×C8—1661no—
480C2×C8—1661no—
481C2×C8—1661no—
482C2×C8—1661no—
483C2×C8—1661no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—41no—
492C4⋊C4—42no—
493C4⋊C4—42no—
494C4⋊C4—42no—
495C4⋊C4—42no—
496C4⋊C4—81no—
497C4⋊C4—832no—
498C4⋊C4—6641no—
499C4⋊C4—6641no—
500C4⋊C4—1661no—
501C4⋊C4—1661no—
502C4⋊C4—1661no—
503C4⋊C4—1661no—
504C4⋊C4—1661no—
505C4⋊C4D4×D8312yes—
506C4⋊C4Q8×D8312yes—
507C22⋊C4—3322no—
508C22⋊C4—3322no—
509C22⋊C4—1664no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—81no—
515C42Q8×D8312yes—
516C42#1305059112yes—
517C23—3321no—
518C23—3321no—
519C23—3321no—
520C23—3321no—
521C23—1662no—
522C23—1662no—
523C23—1662no—
524C23C22×C22⋊D8311yescharacteristic
525Q8—84no—
526D4—834no—
527D4—1662no—
528D4—1662no—
529D4—1662no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—3321no—
558C2×C4—3321no—
559C2×C4—3321no—
560C2×C4—3321no—
561C2×C4—3321no—
562C2×C4—3321no—
563C2×C4—3321no—
564C2×C4—3321no—
565C2×C4—6644no—
566C2×C4—6644no—
567C2×C4—6644no—
568C2×C4—1662no—
569C2×C4#1458176812yes—
570C2×C4#1458176812yes—
571C2×C4#1458181212yes—
572C2×C4C2×D4×D8311yescharacteristic
573C2×C4C2×Q8×D8311yescharacteristic
574C2×C4C2×Q8×D8311yescharacteristic
575C2×C4C2×#1305059111yescharacteristic
576C2×C4#1458183711yescharacteristic
577C2×C4#1458184011yescharacteristic
578C8—3321no—
579C8—3321no—
580C8—1662no—
581C8—1662no—
582C22—3322no—
583C22—3322no—
584C22—3322no—
585C22—3322no—
586C22—3322no—
587C22—3322no—
588C22—1664no—
589C22C2×#1458176811yescharacteristic
590C22D4×C22⋊D8311yescharacteristic
591C22Q8×C22⋊D8311yescharacteristic
592C22C2×#1458181211yescharacteristic
593C22#1912849911yescharacteristic
594C22#1912850411yescharacteristic
595C22#1912850911yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—42no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—41no—
604C4—81no—
605C4—81no—
606C4—3322no—
607C4—6641no—
608C4—6641no—
609C4C4⋊Q8×D8311yescharacteristic
610C4#1912832811yescharacteristic
611C4#1912835411yescharacteristic
612C4#1912841211yescharacteristic
613C2—3324no—
614C2#2491116111yescharacteristic
615C2#2491724511yescharacteristic
616C2#2491728811yescharacteristic
617C2#2491907011yescharacteristic
618C2#2491917111yescharacteristic
619C2#2492032611yescharacteristic
620C2#2492040211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2550C2
socleC23×C83C22×D4
prime radicalC22×D332C23
hypercenter#2550D83
hypocenterC83#26051
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2550

Complements

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

NH ≅ G/Ntype
4. #24908965C2semidirect
5. #24908964C2semidirect
6. #24908798C2semidirect
7. #24908788C2semidirect
8. #24908231C2semidirect
9. #24908182C2semidirect
12. #24907625C2semidirect
15. C83×#2550C2semidirect
74. C4×D332Q8semidirect
95. #14581712Q8semidirect
313. #2550D83semidirect
326. C83#26051semidirect

Maximal subgroups

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

subgroupindex
C2×#191279982
C2×#191279882
#249103742
#249089652
#249089642
#249087982
#249087882
#249082312
#249081822
#249077092
#249076252
#249074702
C4×#145817122
C83×#25502
#2605183