Finite Group Databasebeta

← #35456139

Subgroups — #35456139

Order 21248.

764 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128111C211yesmaximal, characteristic, complemented
2C2×#19127981C211yesmaximal, characteristic
3#24910552C211yesmaximal, characteristic, complemented
4#24908945C211yesmaximal, characteristic, complemented
5#24908935C211yesmaximal, characteristic, complemented
6#24908851C211yesmaximal, characteristic
7#24908837C211yesmaximal, characteristic
8#24908421C211yesmaximal, characteristic, complemented
9#24908229C211yesmaximal, characteristic, complemented
10#24908219C211yesmaximal, characteristic, complemented
11#24908187C211yesmaximal, characteristic, complemented
12#24908087C211yesmaximal, characteristic
13#24908080C211yesmaximal, characteristic
14#24907288C211yesmaximal, characteristic
15C83×#2566C211yesmaximal, characteristic
16C2×#14581792C2211yescharacteristic
17#19128111—24no—
18#19127981—24no—
19C2×#14581725C2211yescharacteristic
20C2×#14581725C2211yescharacteristic
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×C4⋊(C83⋊C8)C2211yescharacteristic
26C2×C4⋊(C83⋊C8)C2211yescharacteristic
27C2×C4⋊(C83⋊C8)C2211yescharacteristic
28C2×#14581713C2211yescharacteristic
29C2×#14581713C2211yescharacteristic
30C2×#14581713C2211yescharacteristic
31C2×#14581713C2211yescharacteristic
32C2×#14581703C2211yescharacteristic
33#19127605—21no—
34#19127605—21no—
35#19127599C2211yescharacteristic
36#19127599C2211yescharacteristic
37#19127597—21no—
38C4⋊C4×Dic83—21no—
39C4×C4⋊Dic83C2211yescharacteristic
40C4×C4⋊Dic83C2211yescharacteristic
41C4×C4⋊Dic83C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×#125×C83C2211yescharacteristic
51C2×C4×D4×C83C2211yescharacteristic
52#396×C83—21no—
53#396×C83—21no—
54#390×C83C2211yescharacteristic
55#390×C83C2211yescharacteristic
56#385×C83—21no—
57#385×C83—21no—
58#384×C83C2211yescharacteristic
59#382×C83C2211yescharacteristic
60C4×C4⋊C4×C83C2211yescharacteristic
61C22×Dic166—21no—
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66#14581792—22no—
67#14581792D414yes—
68C2×C4⋊Dic83—21no—
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×Dic83⋊C4—21no—
74C2×Dic83⋊C4—21no—
75C2×C4×Dic83—21no—
76C2×C4×Dic83—21no—
77C2×C4×Dic83—21no—
78#14581725—22no—
79#14581725—22no—
80#14581725—22no—
81#14581725—22no—
82#14581723—22no—
83#14581723—22no—
84C4⋊(C83⋊C8)—22no—
85C4⋊(C83⋊C8)—22no—
86C4⋊(C83⋊C8)—22no—
87#14581713—22no—
88#14581713—22no—
89#14581713—22no—
90#14581713Q814yescomplemented
91#14581703—22no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C22×D4×C83C2311yescharacteristic
98C23×C4×C83—21no—
99#125×C83—22no—
100#125×C83—22no—
101C4×D4×C83—22no—
102C4×D4×C83—22no—
103C2×C4⋊C4×C83—21no—
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83—21no—
106C2×C4⋊C4×C83—21no—
107C2×C4⋊C4×C83D411yescharacteristic
108C2×C4⋊C4×C83D411yescharacteristic
109C2×C4⋊C4×C83C2311yescharacteristic
110C2×C4⋊C4×C83C2311yescharacteristic
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C22⋊C4×C83—21no—
113C2×C22⋊C4×C83—21no—
114C2×C22⋊C4×C83—21no—
115C2×C22⋊C4×C83—21no—
116C2×C22⋊C4×C83—21no—
117C2×C42×C83C2311yescharacteristic
118C2×C42×C83C2311yescharacteristic
119C2×C42×C83C2311yescharacteristic
120#95×C83—21no—
121#95×C83—21no—
122#95×C83D411yescharacteristic
123#95×C83D411yescharacteristic
124C2×Dic166—24no—
125C2×Dic166—81no—
126C2×Dic166—81no—
127C22×Dic83—21no—
128C22×Dic83—21no—
129C22×Dic83—21no—
130C22×Dic83—21no—
131C2×C83⋊C8—21no—
132C2×C83⋊C8—21no—
133C2×C83⋊C8—21no—
134C2×C83⋊C8—21no—
135C2×C83⋊C8—21no—
136C2×C83⋊C8—21no—
137C2×C83⋊C8—21no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8C2×Q812yes—
142C2×C83⋊C8#4112yes—
143C4⋊Dic83—24no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83C2×D412yes—
150C4⋊Dic83C2×D412yes—
151C4⋊Dic83C2×Q812yes—
152Dic83⋊C4—81no—
153Dic83⋊C4—81no—
154C4×Dic83—42no—
155C4×Dic83—81no—
156C4×Dic83—81no—
157C24×C83—21no—
158C2×D4×C83—21no—
159C2×D4×C83—21no—
160C2×D4×C83—21no—
161C2×D4×C83—21no—
162C2×D4×C83—21no—
163C2×D4×C83C2×Q812yes—
164C22×C4×C83—21no—
165C22×C4×C83—21no—
166C22×C4×C83—21no—
167C22×C4×C83—42no—
168C22×C4×C83—81no—
169C22×C4×C83—81no—
170C22×C4×C83C2×D411yescharacteristic
171C22×C4×C83C2×D411yescharacteristic
172C22×C4×C83C2×D411yescharacteristic
173C22×C4×C83C2×D411yescharacteristic
174C22×C4×C83C2×D411yescharacteristic
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83C2411yescharacteristic
177C4⋊C4×C83—22no—
178C4⋊C4×C83—21no—
179C4⋊C4×C83—21no—
180C4⋊C4×C83—21no—
181C4⋊C4×C83—21no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—41no—
185C4⋊C4×C83—42no—
186C4⋊C4×C83—42no—
187C4⋊C4×C83—81no—
188C4⋊C4×C83—81no—
189C22⋊C4×C83—42no—
190C22⋊C4×C83—42no—
191C22⋊C4×C83—42no—
192C22⋊C4×C83—81no—
193C22⋊C4×C83—81no—
194C42×C83—21no—
195C42×C83—21no—
196C42×C83—21no—
197C42×C83—21no—
198C42×C83C2×D412yes—
199C42×C83#4112yes—
200C23×C83—22no—
201C23×C83—41no—
202C23×C83—41no—
203C23×C83—41no—
204C23×C83—41no—
205C23×C83—41no—
206C23×C83—41no—
207C23×C83C22×D411yescharacteristic
208D4×C83—22no—
209D4×C83—22no—
210D4×C83—41no—
211D4×C83—41no—
212C2×C4×C83—21no—
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—21no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—41no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
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—84no—
237C2×C4×C83#12112yes—
238C2×C4×C83#12112yes—
239C2×C4×C83#13712yes—
240C2×C4×C83C22×D411yescharacteristic
241C2×C4×C83C22×Q811yescharacteristic
242C2×C4×C83C2×#4111yescharacteristic
243C2×C4×C83C2×#4111yescharacteristic
244C2×C4×C832−1+411yescharacteristic
245C2×C4×C832−1+411yescharacteristic
246C2×Dic83—22no—
247C2×Dic83—22no—
248C2×Dic83—22no—
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—
258Dic166—84no—
259C83⋊C8—22no—
260C83⋊C8—22no—
261C83⋊C8—41no—
262C83⋊C8—41no—
263C22×C83—42no—
264C22×C83—42no—
265C22×C83—42no—
266C22×C83—81no—
267C22×C83—81no—
268C22×C83—81no—
269C22×C83—81no—
270C22×C83C2×#12111yescharacteristic
271C22×C83#54711yescharacteristic
272C22×C83D4×Q811yescharacteristic
273C22×C83C2×#13711yescharacteristic
274C22×C83#57511yescharacteristic
275C22×C83#57611yescharacteristic
276C22×C83#57611yescharacteristic
277C4×C83—22no—
278C4×C83—22no—
279C4×C83—22no—
280C4×C83—41no—
281C4×C83—41no—
282C4×C83—41no—
283C4×C83—81no—
284C4×C83—81no—
285C4×C83—81no—
286C4×C83#53311yescharacteristic
287C4×C83#54111yescharacteristic
288C4×C83#54811yescharacteristic
289C4×C83D4×Q811yescharacteristic
290Dic83—42no—
291Dic83—42no—
292Dic83—42no—
293Dic83—81no—
294#25462—831nomaximal, Sylow
295C2×C83—82no—
296C2×C83#242911yescharacteristic
297C2×C83#303711yescharacteristic
298C2×C83#304111yescharacteristic
299C2×C83#334611yescharacteristic
300C2×C83#334711yescharacteristic
301C2×C83#338511yescharacteristic
302C2×C83#338911yescharacteristic
303C2×#501—831no—
304C2×#474—831no—
305#2566D8311yescharacteristic
306#2369—831no—
307#2085—831no—
308#2083—831no—
309#2067—831no—
310#2065—831no—
311#1989—831no—
312#1949—831no—
313#1947—831no—
314#1940—831no—
315#1926—831no—
316#1925—831no—
317#1762—831no—
318C83#2546211yescharacteristic, complemented, Sylow
319C2×C4⋊Q8—831no—
320C2×#125C2×D8311yescharacteristic
321C2×C4×D4C2×D8311yescharacteristic
322#501—1664no—
323#474—1664no—
324C2×#107—831no—
325C2×#107—831no—
326C2×#107—831no—
327C2×#107—831no—
328C2×#106—831no—
329C2×#106—831no—
330C2×C4⋊C8—831no—
331C2×C4⋊C8—831no—
332C2×C4⋊C8—831no—
333C2×#102—831no—
334C2×#102—831no—
335C2×#102—831no—
336C2×#102—831no—
337C2×#97—831no—
338#396—21no—
339#396—21no—
340#391—1661no—
341#391—1661no—
342#390C2×D8311yescharacteristic
343#390C2×D8311yescharacteristic
344#389—831no—
345#389—831no—
346#389—1661no—
347#385—21no—
348#385—21no—
349#384C2×D8311yescharacteristic
350#382C2×D8311yescharacteristic
351C4×C4⋊C4—831no—
352C4×C4⋊C4—831no—
353C4×C4⋊C4—831no—
354C4×C4⋊C4—1661no—
355C4×C4⋊C4C2×D8311yescharacteristic
356#340—831no—
357#340—831no—
358#340—831no—
359#340—831no—
360#340—831no—
361#340—831no—
362#336—831no—
363#336—831no—
364C22×Q8—1661no—
365C22×D4C22×D8311yescharacteristic
366C23×C4—21no—
367C22×C8—831no—
368C22×C8—831no—
369C22×C8—831no—
370C22×C8—831no—
371C4⋊Q8—834no—
372C4⋊Q8—1662no—
373#125—22no—
374#125—22no—
375C4×D4—22no—
376C4×D4—22no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—831no—
382C2×C4⋊C4—831no—
383C2×C4⋊C4—831no—
384C2×C4⋊C4—831no—
385C2×C4⋊C4—1661no—
386C2×C4⋊C4—1661no—
387C2×C4⋊C4—1661no—
388C2×C4⋊C4C22⋊D8311yescharacteristic
389C2×C4⋊C4C22⋊D8311yescharacteristic
390C2×C4⋊C4C22×D8311yescharacteristic
391C2×C4⋊C4C22×D8311yescharacteristic
392C2×C4⋊C4C22×D8311yescharacteristic
393C2×C22⋊C4—21no—
394C2×C22⋊C4—21no—
395C2×C22⋊C4—21no—
396C2×C22⋊C4—21no—
397C2×C22⋊C4—21no—
398C2×C42—1661no—
399C2×C42—1661no—
400C2×C42—1661no—
401C2×C42C22×D8311yescharacteristic
402C2×C42C22×D8311yescharacteristic
403C2×C42C22×D8311yescharacteristic
404#107—1662no—
405#107—1662no—
406#107—1662no—
407#107—1662no—
408#106—1662no—
409#106—1662no—
410C4⋊C8—1662no—
411C4⋊C8—1662no—
412C4⋊C8—1662no—
413#102—834no—
414#102—1662no—
415#102—1662no—
416#102—1662no—
417#97—1662no—
418#95—21no—
419#95—21no—
420#95—1661no—
421#95—1661no—
422#95—1661no—
423#95—1661no—
424#95—1661no—
425#95C22⋊D8311yescharacteristic
426#95C22⋊D8311yescharacteristic
427C24—21no—
428C2×Q8—6641no—
429C2×Q8—6641no—
430C2×Q8—1664no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4Q8×D8312yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—42no—
441C22×C4—81no—
442C22×C4—81no—
443C22×C4—1661no—
444C22×C4—1661no—
445C22×C4—1661no—
446C22×C4—1661no—
447C22×C4C2×C22⋊D8311yescharacteristic
448C22×C4C2×C22⋊D8311yescharacteristic
449C22×C4C2×C22⋊D8311yescharacteristic
450C22×C4C2×C22⋊D8311yescharacteristic
451C22×C4C2×C22⋊D8311yescharacteristic
452C22×C4C2×C22⋊D8311yescharacteristic
453C22×C4C23×D8311yescharacteristic
454C2×C8—832no—
455C2×C8—832no—
456C2×C8—1661no—
457C2×C8—1661no—
458C2×C8—1661no—
459C2×C8—1661no—
460C2×C8—1661no—
461C2×C8—1661no—
462C2×C8—1661no—
463C2×C8—1661no—
464C2×C8—1661no—
465C2×C8—1661no—
466C4⋊C4—22no—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—41no—
474C4⋊C4—42no—
475C4⋊C4—42no—
476C4⋊C4—81no—
477C4⋊C4—81no—
478C4⋊C4—832no—
479C4⋊C4—832no—
480C4⋊C4—832no—
481C4⋊C4—6641no—
482C4⋊C4—6641no—
483C4⋊C4—1661no—
484C4⋊C4—1661no—
485C4⋊C4—1664no—
486C4⋊C4—1661no—
487C4⋊C4—1661no—
488C4⋊C4—1661no—
489C22⋊C4—42no—
490C22⋊C4—42no—
491C22⋊C4—42no—
492C22⋊C4—81no—
493C22⋊C4—81no—
494C42—21no—
495C42—21no—
496C42—21no—
497C42—21no—
498C42—3322no—
499C42—6641no—
500C42—6641no—
501C42D4×D8312yes—
502C42#1305058912yes—
503C23—22no—
504C23—41no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23C22×C22⋊D8311yescharacteristic
511Q8—6644no—
512D4—22no—
513D4—22no—
514D4—41no—
515D4—41no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—84no—
541C2×C4—3321no—
542C2×C4—3321no—
543C2×C4—3321no—
544C2×C4—3321no—
545C2×C4—3321no—
546C2×C4—3321no—
547C2×C4—3321no—
548C2×C4—3321no—
549C2×C4—3321no—
550C2×C4—1662no—
551C2×C4—1662no—
552C2×C4—1662no—
553C2×C4#1458176312yes—
554C2×C4#1458176312yes—
555C2×C4#1458182212yes—
556C2×C4C2×D4×D8311yescharacteristic
557C2×C4C2×Q8×D8311yescharacteristic
558C2×C4#41×D8311yescharacteristic
559C2×C4C2×#1305058911yescharacteristic
560C2×C4#1458183911yescharacteristic
561C2×C4#1458184011yescharacteristic
562C8—3321no—
563C8—3321no—
564C8—1662no—
565C8—1662no—
566C22—42no—
567C22—42no—
568C22—42no—
569C22—81no—
570C22—81no—
571C22—81no—
572C22—81no—
573C22C2×#1458176311yescharacteristic
574C22#1912835011yescharacteristic
575C22Q8×C22⋊D8311yescharacteristic
576C22C2×#1458182211yescharacteristic
577C22#1912849911yescharacteristic
578C22#1912850411yescharacteristic
579C22#1912850411yescharacteristic
580C4—22no—
581C4—22no—
582C4—22no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—81no—
587C4—81no—
588C4—81no—
589C4—3322no—
590C4—3322no—
591C4—3322no—
592C4—6641no—
593C4#1912825311yescharacteristic
594C4#1912831111yescharacteristic
595C4#1912836111yescharacteristic
596C4#1912836411yescharacteristic
597C2—82no—
598C2#2491105811yescharacteristic
599C2#2491693111yescharacteristic
600C2#2491696211yescharacteristic
601C2#2491994811yescharacteristic
602C2#2491995711yescharacteristic
603C2#2492037311yescharacteristic
604C2#2492041711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2566C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
hypercenter#2566D83
hypocenterC83#25462
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2566

Complements

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

NH ≅ G/Ntype
1. C2×#19128111C2semidirect
3. #24910552C2semidirect
4. #24908945C2semidirect
5. #24908935C2semidirect
8. #24908421C2semidirect
9. #24908229C2semidirect
10. #24908219C2semidirect
11. #24908187C2semidirect
90. #14581713Q8semidirect
318. C83#25462semidirect

Maximal subgroups

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

subgroupindex
C2×#191281112
C2×#191279812
#249105522
#249089452
#249089352
#249088512
#249088372
#249084212
#249082292
#249082192
#249081872
#249080872
#249080802
#249072882
C83×#25662
#2546283