Finite Group Databasebeta

← #35416487

Subgroups — #35416487

Order 21248.

796 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128001C211yesmaximal, characteristic
2C4×#14581784C211yesmaximal, characteristic, complemented
3#24908420C211yesmaximal, characteristic, complemented
4#24908231C211yesmaximal, characteristic, complemented
5#24908193C211yesmaximal, characteristic, complemented
6#24908188C211yesmaximal, characteristic, complemented
7#24908149C211yesmaximal, characteristic
8#24908149C211yesmaximal, characteristic
9#24908087C211yesmaximal, characteristic
10#24908080C211yesmaximal, characteristic
11#24908052C211yesmaximal, characteristic, complemented
12#24908045C211yesmaximal, characteristic, complemented
13#24907473C211yesmaximal, characteristic
14C4×C4⋊(C83⋊C8)C211yesmaximal, characteristic, complemented
15C83×#2293C211yesmaximal, characteristic
16C2×#14581784C2211yescharacteristic
17#19128001C418yescomplemented
18C2×#14581725C2211yescharacteristic
19C2×#14581725C2211yescharacteristic
20C2×#14581723C2211yescharacteristic
21C2×#14581723C2211yescharacteristic
22C2×C4⋊(C83⋊C8)C2211yescharacteristic
23C2×C4⋊(C83⋊C8)C2211yescharacteristic
24C2×#14581713C2211yescharacteristic
25C2×#14581713C2211yescharacteristic
26C2×#14581713C2211yescharacteristic
27C2×#14581713C2211yescharacteristic
28C2×C4×C83⋊C8C2211yescharacteristic
29C2×C4×C83⋊C8C2211yescharacteristic
30#19127599C2211yescharacteristic
31#19127599C2211yescharacteristic
32#19127585—21no—
33#19127585—21no—
34#19127567C2211yescharacteristic
35C4×Dic83⋊C4—21no—
36C4×Dic83⋊C4—21no—
37C4×C4⋊Dic83C2211yescharacteristic
38C4×C4⋊Dic83C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127456C2211yescharacteristic
48#19127456C2211yescharacteristic
49C2×C4×D4×C83C2211yescharacteristic
50C2×#117×C83—21no—
51#390×C83C2211yescharacteristic
52#390×C83C2211yescharacteristic
53#389×C83C2211yescharacteristic
54#389×C83C2211yescharacteristic
55#388×C83—21no—
56#388×C83—21no—
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C22⋊C4×C83—21no—
59C43×C83C2211yescharacteristic
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64#14581784—22no—
65#14581784C2×C414yes—
66C2×C4⋊Dic83C2311yescharacteristic
67C2×C4⋊Dic83C2311yescharacteristic
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×Dic83⋊C4—21no—
71C2×Dic83⋊C4—21no—
72C2×Dic83⋊C4—21no—
73C2×Dic83⋊C4—21no—
74C2×C4×Dic83—21no—
75C2×C4×Dic83—21no—
76#14581725—41no—
77#14581725C2×C414yes—
78#14581723—41no—
79#14581723C2×C414yes—
80C4⋊(C83⋊C8)—22no—
81C4⋊(C83⋊C8)C2×C414yes—
82#14581713—41no—
83#14581713—41no—
84#14581713C2×C414yes—
85#14581713C2×C414yes—
86C4×C83⋊C8—41no—
87C4×C83⋊C8—41no—
88C83⋊#95—21no—
89C83⋊#95—21no—
90C83⋊#95—21no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C22×D4×C83C2311yescharacteristic
95C23×C4×C83—21no—
96C4×D4×C83—22no—
97C4×D4×C83C2×C414yes—
98#117×C83—44no—
99C2×C4⋊C4×C83—21no—
100C2×C4⋊C4×C83—21no—
101C2×C4⋊C4×C83C2311yescharacteristic
102C2×C4⋊C4×C83C2311yescharacteristic
103C2×C4⋊C4×C83C2311yescharacteristic
104C2×C22⋊C4×C83—21no—
105C2×C22⋊C4×C83—21no—
106C2×C22⋊C4×C83—21no—
107C2×C22⋊C4×C83—21no—
108C2×C22⋊C4×C83—21no—
109C2×C42×C83—21no—
110C2×C42×C83D411yescharacteristic
111C2×C42×C83D411yescharacteristic
112C2×C42×C83D411yescharacteristic
113C2×C42×C83D411yescharacteristic
114C2×C42×C83C2311yescharacteristic
115C2×C42×C83C2311yescharacteristic
116C2×C42×C83C2311yescharacteristic
117#95×C83—21no—
118#95×C83—21no—
119#95×C83—21no—
120C22×Dic83—21no—
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83—21no—
124C2×C83⋊C8—21no—
125C2×C83⋊C8—21no—
126C2×C83⋊C8—21no—
127C2×C83⋊C8—21no—
128C2×C83⋊C8—21no—
129C2×C83⋊C8—21no—
130C2×C83⋊C8—21no—
131C2×C83⋊C8—21no—
132C2×C83⋊C8—21no—
133C2×C83⋊C8—21no—
134C2×C83⋊C8C22×C412yes—
135C2×C83⋊C8C22×C412yes—
136C4⋊Dic83—21no—
137C4⋊Dic83—21no—
138C4⋊Dic83—21no—
139C4⋊Dic83—21no—
140C4⋊Dic83—21no—
141C4⋊Dic83—21no—
142C4⋊Dic83C22×C412yes—
143C4⋊Dic83C22×C412yes—
144Dic83⋊C4—24no—
145Dic83⋊C4—24no—
146Dic83⋊C4—42no—
147Dic83⋊C4—42no—
148C4×Dic83—42no—
149C4×Dic83—42no—
150C24×C83—21no—
151C2×D4×C83—21no—
152C2×D4×C83—21no—
153C2×D4×C83—21no—
154C2×D4×C83—21no—
155C2×D4×C83—21no—
156C2×D4×C83C22×C412yes—
157C22×C4×C83—21no—
158C22×C4×C83—24no—
159C22×C4×C83—21no—
160C22×C4×C83—21no—
161C22×C4×C83—42no—
162C22×C4×C83—42no—
163C22×C4×C83C2×D411yescharacteristic
164C22×C4×C83C2×D411yescharacteristic
165C22×C4×C83C2×D411yescharacteristic
166C22×C4×C83C2×D411yescharacteristic
167C22×C4×C83C2×D411yescharacteristic
168C22×C4×C83C2×D411yescharacteristic
169C22×C4×C83C2411yescharacteristic
170C4⋊C4×C83—21no—
171C4⋊C4×C83—21no—
172C4⋊C4×C83—21no—
173C4⋊C4×C83—21no—
174C4⋊C4×C83—21no—
175C4⋊C4×C83—42no—
176C4⋊C4×C83—42no—
177C4⋊C4×C83C22×C412yes—
178C22⋊C4×C83—24no—
179C22⋊C4×C83—42no—
180C22⋊C4×C83—42no—
181C22⋊C4×C83—42no—
182C22⋊C4×C83—42no—
183C42×C83—22no—
184C42×C83—22no—
185C42×C83—22no—
186C42×C83—21no—
187C42×C83—21no—
188C42×C83—21no—
189C42×C83—21no—
190C42×C83—41no—
191C42×C83—42no—
192C42×C83C22×C412yes—
193C42×C83#4112yes—
194C23×C83—22no—
195C23×C83—41no—
196C23×C83—41no—
197C23×C83—41no—
198C23×C83—41no—
199C23×C83—41no—
200C23×C83—41no—
201C23×C83C22×D411yescharacteristic
202D4×C83—22no—
203D4×C83—22no—
204D4×C83—41no—
205D4×C83—41no—
206C2×C4×C83—21no—
207C2×C4×C83—21no—
208C2×C4×C83—21no—
209C2×C4×C83—21no—
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—22no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—48no—
221C2×C4×C83—41no—
222C2×C4×C83—41no—
223C2×C4×C83—41no—
224C2×C4×C83—41no—
225C2×C4×C83—41no—
226C2×C4×C83—41no—
227C2×C4×C83—41no—
228C2×C4×C83—41no—
229C2×C4×C83C4×D412yes—
230C2×C4×C83C4×D412yes—
231C2×C4×C83#13512yes—
232C2×C4×C83#13512yes—
233C2×C4×C83AGL(1,ℤ/8)12yes—
234C2×C4×C83C23×C411yescharacteristic
235C2×C4×C83C2×#4111yescharacteristic
236C2×C4×C83H(5,𝔽2)11yescharacteristic
237C2×C4×C832−1+411yescharacteristic
238C2×C4×C832−1+411yescharacteristic
239C2×C4×C832−1+411yescharacteristic
240C2×Dic83—22no—
241C2×Dic83—22no—
242C2×Dic83—41no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C83⋊C8—22no—
253C83⋊C8—22no—
254C83⋊C8—41no—
255C83⋊C8—41no—
256C22×C83—42no—
257C22×C83—42no—
258C22×C83—42no—
259C22×C83—81no—
260C22×C83—81no—
261C22×C83—81no—
262C22×C83—81no—
263C22×C83C2×C4×D411yescharacteristic
264C22×C83#53611yescharacteristic
265C22×C83#53711yescharacteristic
266C22×C83C2×#13511yescharacteristic
267C22×C83C2×#13511yescharacteristic
268C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
269C22×C83#57511yescharacteristic
270C4×C83—22no—
271C4×C83—22no—
272C4×C83—22no—
273C4×C83—22no—
274C4×C83—22no—
275C4×C83—42no—
276C4×C83—41no—
277C4×C83—81no—
278C4×C83—81no—
279C4×C83#51911yescharacteristic
280C4×C83#52011yescharacteristic
281C4×C83#54111yescharacteristic
282C4×C83#54211yescharacteristic
283Dic83—42no—
284Dic83—42no—
285Dic83—81no—
286Dic83—81no—
287#22114—831nomaximal, Sylow
288C2×C83—82no—
289C2×C83#232311yescharacteristic
290C2×C83C4×#13511yescharacteristic
291C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
292C2×C83#322811yescharacteristic
293C2×C83#323211yescharacteristic
294C2×C83#324711yescharacteristic
295C2×C83#325011yescharacteristic
296C2×#478—831no—
297C4×#125—831no—
298#2293D8311yescharacteristic
299#1989—831no—
300#1949—831no—
301#1942—831no—
302#1941—831no—
303#1934—831no—
304#1934—831no—
305#1926—831no—
306#1925—831no—
307#1921—831no—
308#1920—831no—
309#1804—831no—
310C4×C4⋊C8—831no—
311C83#2211411yescharacteristic, complemented, Sylow
312C2×#125—831no—
313C2×C4×D4C2×D8311yescharacteristic
314C2×#117—21no—
315#478—838no—
316C2×#107—831no—
317C2×#107—831no—
318C2×#106—831no—
319C2×#106—831no—
320C2×C4⋊C8—831no—
321C2×C4⋊C8—831no—
322C2×#102—831no—
323C2×#102—831no—
324C2×#102—831no—
325C2×#102—831no—
326C2×C4×C8—831no—
327C2×C4×C8—831no—
328#390C2×D8311yescharacteristic
329#390C2×D8311yescharacteristic
330#389—831no—
331#389—831no—
332#389C2×D8311yescharacteristic
333#389C2×D8311yescharacteristic
334#388—21no—
335#388—21no—
336#387—1661no—
337#387—1661no—
338#382—831no—
339C4×C4⋊C4—831no—
340C4×C4⋊C4—831no—
341C4×C4⋊C4—1661no—
342C4×C4⋊C4—1661no—
343C4×C4⋊C4C2×D8311yescharacteristic
344C4×C22⋊C4—21no—
345C43C2×D8311yescharacteristic
346#340—831no—
347#340—831no—
348#340—831no—
349#340—831no—
350#340—831no—
351#340—831no—
352#340—831no—
353#340—831no—
354#336—831no—
355#336—831no—
356C22×D4C22×D8311yescharacteristic
357C23×C4—21no—
358C22×C8—831no—
359C22×C8—831no—
360C22×C8—831no—
361C22×C8—831no—
362#125—834no—
363#125—1662no—
364C4×D4—22no—
365C4×D4C4×D8314yes—
366#117—44no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—831no—
370C2×C4⋊C4—831no—
371C2×C4⋊C4—831no—
372C2×C4⋊C4—831no—
373C2×C4⋊C4—1661no—
374C2×C4⋊C4—1661no—
375C2×C4⋊C4—1661no—
376C2×C4⋊C4—1661no—
377C2×C4⋊C4C22×D8311yescharacteristic
378C2×C4⋊C4C22×D8311yescharacteristic
379C2×C4⋊C4C22×D8311yescharacteristic
380C2×C22⋊C4—21no—
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—21no—
384C2×C22⋊C4—21no—
385C2×C42—21no—
386C2×C42—1661no—
387C2×C42—1661no—
388C2×C42C22⋊D8311yescharacteristic
389C2×C42C22⋊D8311yescharacteristic
390C2×C42C22⋊D8311yescharacteristic
391C2×C42C22⋊D8311yescharacteristic
392C2×C42C22×D8311yescharacteristic
393C2×C42C22×D8311yescharacteristic
394C2×C42C22×D8311yescharacteristic
395#107—3321no—
396#107—834no—
397#106—3321no—
398#106—834no—
399C4⋊C8—834no—
400C4⋊C8—1662no—
401#102—3321no—
402#102—3321no—
403#102—834no—
404#102—834no—
405C4×C8—3321no—
406C4×C8—3321no—
407#95—21no—
408#95—21no—
409#95—21no—
410#95—1661no—
411#95—1661no—
412#95—1661no—
413#95—1661no—
414#95—1661no—
415#95—1661no—
416C24—21no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4C2×C4×D8312yes—
423C22×C4—21no—
424C22×C4—24no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—42no—
428C22×C4—42no—
429C22×C4—1661no—
430C22×C4—1661no—
431C22×C4—1661no—
432C22×C4—1661no—
433C22×C4C2×C22⋊D8311yescharacteristic
434C22×C4C2×C22⋊D8311yescharacteristic
435C22×C4C2×C22⋊D8311yescharacteristic
436C22×C4C2×C22⋊D8311yescharacteristic
437C22×C4C2×C22⋊D8311yescharacteristic
438C22×C4C2×C22⋊D8311yescharacteristic
439C22×C4C23×D8311yescharacteristic
440C2×C8—832no—
441C2×C8—832no—
442C2×C8—1661no—
443C2×C8—1661no—
444C2×C8—1661no—
445C2×C8—1661no—
446C2×C8—1661no—
447C2×C8—1661no—
448C2×C8—1661no—
449C2×C8—1661no—
450C2×C8—1661no—
451C2×C8—1661no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—3322no—
460C4⋊C4—3322no—
461C4⋊C4—832no—
462C4⋊C4—832no—
463C4⋊C4—1661no—
464C4⋊C4—1661no—
465C4⋊C4—1661no—
466C4⋊C4—1661no—
467C4⋊C4—1661no—
468C4⋊C4—1661no—
469C4⋊C4—1664no—
470C4⋊C4—1664no—
471C4⋊C4C2×C4×D8312yes—
472C22⋊C4—24no—
473C22⋊C4—42no—
474C22⋊C4—42no—
475C22⋊C4—42no—
476C22⋊C4—42no—
477C42—22no—
478C42—22no—
479C42—22no—
480C42—21no—
481C42—21no—
482C42—21no—
483C42—21no—
484C42—41no—
485C42—42no—
486C42—3322no—
487C42—3322no—
488C42C2×C4×D8312yes—
489C42#1305059012yes—
490C23—22no—
491C23—41no—
492C23—41no—
493C23—41no—
494C23—41no—
495C23—41no—
496C23—41no—
497C23C22×C22⋊D8311yescharacteristic
498D4—22no—
499D4—22no—
500D4—41no—
501D4—41no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—22no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—48no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—3321no—
526C2×C4—3321no—
527C2×C4—3321no—
528C2×C4—3321no—
529C2×C4—3321no—
530C2×C4—3321no—
531C2×C4—3321no—
532C2×C4—3321no—
533C2×C4—3321no—
534C2×C4—3321no—
535C2×C4—1662no—
536C2×C4—1662no—
537C2×C4C4×C22⋊D8312yes—
538C2×C4C4×C22⋊D8312yes—
539C2×C4#1458181212yes—
540C2×C4#1458181212yes—
541C2×C4#1458181812yes—
542C2×C4C22×C4×D8311yescharacteristic
543C2×C4C2×#1305059011yescharacteristic
544C2×C4#1458183811yescharacteristic
545C2×C4#1458183911yescharacteristic
546C2×C4#1458183911yescharacteristic
547C2×C4#1458183911yescharacteristic
548C8—3321no—
549C8—3321no—
550C8—1662no—
551C8—1662no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22—81no—
556C22—81no—
557C22—81no—
558C22—81no—
559C22C2×C4×C22⋊D8311yescharacteristic
560C22#1912827211yescharacteristic
561C22#1912827911yescharacteristic
562C22C2×#1458181211yescharacteristic
563C22C2×#1458181211yescharacteristic
564C22C2×#1458181811yescharacteristic
565C22#1912849911yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—22no—
570C4—22no—
571C4—42no—
572C4—41no—
573C4—81no—
574C4—81no—
575C4—3322no—
576C4—3322no—
577C4—6641no—
578C4—6641no—
579C4#1912817811yescharacteristic
580C4#1912818111yescharacteristic
581C4#1912831311yescharacteristic
582C4#1912832211yescharacteristic
583C2—82no—
584C2#2491014511yescharacteristic
585C2C4×#1458181211yescharacteristic
586C2C4×#1458181811yescharacteristic
587C2#2491859511yescharacteristic
588C2#2491862711yescharacteristic
589C2#2491878311yescharacteristic
590C2#2491881011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2293C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
cosocleC22×C4×C83C24
hypercenter#2293D83
hypocenterC83#22114
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C43 ◁ #2293

Complements

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

NH ≅ G/Ntype
2. C4×#14581784C2semidirect
3. #24908420C2semidirect
4. #24908231C2semidirect
5. #24908193C2semidirect
6. #24908188C2semidirect
11. #24908052C2semidirect
12. #24908045C2semidirect
14. C4×C4⋊(C83⋊C8)C2semidirect
17. #19128001C4semidirect
311. C83#22114semidirect

Maximal subgroups

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

subgroupindex
C2×#191280012
C4×#145817842
#249084202
#249082312
#249081932
#249081882
#249081492
#249080872
#249080802
#249080522
#249080452
#249074732
C4×C4⋊(C83⋊C8)2
C83×#22932
#2211483