Finite Group Databasebeta

← #35456475

Subgroups — #35456475

Order 21248.

782 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×#19128001C211yesmaximal, characteristic
2C2×#19127832C211yesmaximal, characteristic, complemented
3#24912851C211yesmaximal, characteristic, complemented
4#24908965C211yesmaximal, characteristic, complemented
5#24908954C211yesmaximal, characteristic, complemented
6#24908837C211yesmaximal, characteristic
7#24908837C211yesmaximal, characteristic
8#24908431C211yesmaximal, characteristic, complemented
9#24908224C211yesmaximal, characteristic, complemented
10#24908203C211yesmaximal, characteristic, complemented
11#24908188C211yesmaximal, characteristic, complemented
12#24908163C211yesmaximal, characteristic
13#24908149C211yesmaximal, characteristic
14#24907288C211yesmaximal, characteristic
15C83×#2538C211yesmaximal, characteristic
16C2×#14581784C2211yescharacteristic
17#19128001—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×#14581713C2211yescharacteristic
27C2×#14581713C2211yescharacteristic
28C2×#14581713C2211yescharacteristic
29C2×#14581713C2211yescharacteristic
30C2×#14581703C2211yescharacteristic
31C2×#14581703C2211yescharacteristic
32C2×C4×C83⋊C8C2211yescharacteristic
33#19127632—21no—
34#19127620—21no—
35#19127599C2211yescharacteristic
36#19127599C2211yescharacteristic
37#19127599C2211yescharacteristic
38#19127599C2211yescharacteristic
39#19127596—21no—
40#19127596—21no—
41C4×C4⋊Dic83C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50C2×C22⋊Q8×C83—21no—
51C2×C4×Q8×C83C2211yescharacteristic
52C2×C4×D4×C83C2211yescharacteristic
53#396×C83—21no—
54#396×C83—21no—
55#390×C83C2211yescharacteristic
56#390×C83C2211yescharacteristic
57#389×C83C2211yescharacteristic
58#389×C83C2211yescharacteristic
59#384×C83C2211yescharacteristic
60#380×C83—21no—
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65#14581784—22no—
66#14581784—22no—
67C2×C4⋊Dic83C2311yescharacteristic
68C2×C4⋊Dic83C2311yescharacteristic
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2311yescharacteristic
71C2×Dic83⋊C4—21no—
72C2×Dic83⋊C4—21no—
73C2×Dic83⋊C4—21no—
74C2×Dic83⋊C4—21no—
75C2×Dic83⋊C4—21no—
76C2×Dic83⋊C4—21no—
77C2×C4×Dic83—21no—
78#14581725—22no—
79#14581723—22no—
80#14581723—22no—
81#14581723—22no—
82C4⋊(C83⋊C8)—22no—
83C4⋊(C83⋊C8)—22no—
84C4⋊(C83⋊C8)D414yes—
85#14581713—22no—
86#14581713—22no—
87#14581713—22no—
88#14581713Q814yescomplemented
89#14581703—22no—
90#14581703—22no—
91C4×C83⋊C8—22no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C22×Q8×C83D411yescharacteristic
98C22×D4×C83C2311yescharacteristic
99C23×C4×C83—21no—
100C22⋊Q8×C83—44no—
101C4×Q8×C83—22no—
102C4×Q8×C83—22no—
103C4×D4×C83—22no—
104C4×D4×C83—22no—
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×C22⋊C4×C83—21no—
116C2×C22⋊C4×C83—21no—
117C2×C22⋊C4×C83—21no—
118C2×C22⋊C4×C83—21no—
119C2×C22⋊C4×C83—21no—
120C2×C42×C83C2311yescharacteristic
121C2×C42×C83C2311yescharacteristic
122C2×C42×C83C2311yescharacteristic
123#95×C83—21no—
124#95×C83—21no—
125C22×Dic83—21no—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C22×Dic83—21no—
129C2×C83⋊C8—21no—
130C2×C83⋊C8—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⋊C8C2×D412yes—
139C2×C83⋊C8C2×D412yes—
140C2×C83⋊C8C2×Q812yes—
141C4⋊Dic83—21no—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83C2×Q812yes—
148C4⋊Dic83#4112yes—
149Dic83⋊C4—42no—
150Dic83⋊C4—42no—
151Dic83⋊C4—42no—
152Dic83⋊C4—42no—
153Dic83⋊C4—81no—
154Dic83⋊C4—81no—
155C4×Dic83—81no—
156C24×C83—21no—
157C2×Q8×C83—22no—
158C2×Q8×C83—22no—
159C2×Q8×C83—41no—
160C2×D4×C83—21no—
161C2×D4×C83—21no—
162C2×D4×C83—21no—
163C2×D4×C83—21no—
164C2×D4×C83—21no—
165C2×D4×C83C2×Q812yes—
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83—21no—
169C22×C4×C83—42no—
170C22×C4×C83—42no—
171C22×C4×C83—42no—
172C22×C4×C83C2×D411yescharacteristic
173C22×C4×C83C2×D411yescharacteristic
174C22×C4×C83C2×D411yescharacteristic
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83C2×D411yescharacteristic
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2411yescharacteristic
179C4⋊C4×C83—22no—
180C4⋊C4×C83—22no—
181C4⋊C4×C83—22no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—42no—
189C4⋊C4×C83—42no—
190C4⋊C4×C83—42no—
191C4⋊C4×C83—81no—
192C22⋊C4×C83—42no—
193C22⋊C4×C83—42no—
194C22⋊C4×C83—42no—
195C22⋊C4×C83—81no—
196C22⋊C4×C83—81no—
197C42×C83—21no—
198C42×C83—21no—
199C42×C83—21no—
200C42×C83—21no—
201C42×C83C2×D412yes—
202C42×C83#4112yes—
203C23×C83—22no—
204C23×C83—41no—
205C23×C83—41no—
206C23×C83—41no—
207C23×C83—41no—
208C23×C83—41no—
209C23×C83—41no—
210C23×C83C22×D411yescharacteristic
211Q8×C83—44no—
212D4×C83—22no—
213D4×C83—22no—
214D4×C83—41no—
215D4×C83—41no—
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—21no—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
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—84no—
241C2×C4×C83#12112yes—
242C2×C4×C83#12112yes—
243C2×C4×C83AGL(1,ℤ/8)12yes—
244C2×C4×C83C22×D411yescharacteristic
245C2×C4×C83C22×Q811yescharacteristic
246C2×C4×C83C2×#4111yescharacteristic
247C2×C4×C83C2×#4111yescharacteristic
248C2×C4×C832−1+411yescharacteristic
249C2×C4×C832−1+411yescharacteristic
250C2×Dic83—22no—
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—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C83⋊C8—22no—
263C83⋊C8—22no—
264C83⋊C8—22no—
265C83⋊C8—41no—
266C22×C83—42no—
267C22×C83—42no—
268C22×C83—42no—
269C22×C83—81no—
270C22×C83—81no—
271C22×C83—81no—
272C22×C83—81no—
273C22×C83C2×#12111yescharacteristic
274C22×C83#54811yescharacteristic
275C22×C83D4×Q811yescharacteristic
276C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C83#57511yescharacteristic
278C22×C83#57711yescharacteristic
279C22×C83#57711yescharacteristic
280C4×C83—22no—
281C4×C83—22no—
282C4×C83—22no—
283C4×C83—41no—
284C4×C83—41no—
285C4×C83—41no—
286C4×C83—81no—
287C4×C83—81no—
288C4×C83—81no—
289C4×C83#53311yescharacteristic
290C4×C83#54111yescharacteristic
291C4×C83#54711yescharacteristic
292C4×C83D4×Q811yescharacteristic
293Dic83—42no—
294Dic83—42no—
295Dic83—81no—
296Dic83—81no—
297#25486—831nomaximal, Sylow
298C2×C83—82no—
299C2×C83#242911yescharacteristic
300C2×C83#305511yescharacteristic
301C2×C83#306011yescharacteristic
302C2×C83#336811yescharacteristic
303C2×C83#337611yescharacteristic
304C2×C83#338411yescharacteristic
305C2×C83#338811yescharacteristic
306C2×#478—831no—
307C2×#446—831no—
308#2584—831no—
309#2538D8311yescharacteristic
310#2088—831no—
311#2086—831no—
312#2065—831no—
313#2065—831no—
314#1992—831no—
315#1948—831no—
316#1944—831no—
317#1941—831no—
318#1936—831no—
319#1934—831no—
320#1762—831no—
321C83#2548611yescharacteristic, complemented, Sylow
322C2×#125—831no—
323C2×C22⋊Q8—21no—
324C2×C4×Q8C2×D8311yescharacteristic
325C2×C4×D4C2×D8311yescharacteristic
326#478—1664no—
327#446—1664no—
328C2×#107—831no—
329C2×#106—831no—
330C2×#106—831no—
331C2×#106—831no—
332C2×C4⋊C8—831no—
333C2×C4⋊C8—831no—
334C2×C4⋊C8—831no—
335C2×#102—831no—
336C2×#102—831no—
337C2×#102—831no—
338C2×#102—831no—
339C2×#97—831no—
340C2×#97—831no—
341C2×C4×C8—831no—
342#398—1661no—
343#396—21no—
344#396—21no—
345#395—1661no—
346#390C2×D8311yescharacteristic
347#390C2×D8311yescharacteristic
348#389—831no—
349#389—831no—
350#389—831no—
351#389—831no—
352#389—1661no—
353#389—1661no—
354#389C2×D8311yescharacteristic
355#389C2×D8311yescharacteristic
356#384C2×D8311yescharacteristic
357#380—21no—
358C4×C4⋊C4—831no—
359#340—831no—
360#340—831no—
361#340—831no—
362#340—831no—
363#340—831no—
364#340—831no—
365#340—831no—
366#340—831no—
367C22×Q8C22⋊D8311yescharacteristic
368C22×D4C22×D8311yescharacteristic
369C23×C4—21no—
370C22×C8—831no—
371C22×C8—831no—
372C22×C8—831no—
373C22×C8—831no—
374#125—1662no—
375#125—1662no—
376C22⋊Q8—44no—
377C4×Q8—22no—
378C4×Q8—22no—
379C4×D4—22no—
380C4×D4—22no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—831no—
386C2×C4⋊C4—831no—
387C2×C4⋊C4—831no—
388C2×C4⋊C4—831no—
389C2×C4⋊C4—1661no—
390C2×C4⋊C4—1661no—
391C2×C4⋊C4—1661no—
392C2×C4⋊C4—1661no—
393C2×C4⋊C4—1661no—
394C2×C4⋊C4—1661no—
395C2×C4⋊C4C22⋊D8311yescharacteristic
396C2×C4⋊C4C22⋊D8311yescharacteristic
397C2×C4⋊C4C22⋊D8311yescharacteristic
398C2×C4⋊C4C22×D8311yescharacteristic
399C2×C4⋊C4C22×D8311yescharacteristic
400C2×C4⋊C4C22×D8311yescharacteristic
401C2×C22⋊C4—21no—
402C2×C22⋊C4—21no—
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C42—1661no—
407C2×C42C22×D8311yescharacteristic
408C2×C42C22×D8311yescharacteristic
409C2×C42C22×D8311yescharacteristic
410#107—1662no—
411#106—1662no—
412#106—1662no—
413#106—1662no—
414C4⋊C8—834no—
415C4⋊C8—1662no—
416C4⋊C8—1662no—
417#102—834no—
418#102—1662no—
419#102—1662no—
420#102—1662no—
421#97—1662no—
422#97—1662no—
423C4×C8—1662no—
424#95—21no—
425#95—21no—
426#95—1661no—
427#95—1661no—
428#95—1661no—
429#95—1661no—
430#95—1661no—
431C24—21no—
432C2×Q8—22no—
433C2×Q8—22no—
434C2×Q8—41no—
435C2×D4—21no—
436C2×D4—21no—
437C2×D4—21no—
438C2×D4—21no—
439C2×D4—21no—
440C2×D4Q8×D8312yes—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—1661no—
448C22×C4—1661no—
449C22×C4—1661no—
450C22×C4—1661no—
451C22×C4C2×C22⋊D8311yescharacteristic
452C22×C4C2×C22⋊D8311yescharacteristic
453C22×C4C2×C22⋊D8311yescharacteristic
454C22×C4C2×C22⋊D8311yescharacteristic
455C22×C4C2×C22⋊D8311yescharacteristic
456C22×C4C2×C22⋊D8311yescharacteristic
457C22×C4C23×D8311yescharacteristic
458C2×C8—832no—
459C2×C8—832no—
460C2×C8—832no—
461C2×C8—1661no—
462C2×C8—1661no—
463C2×C8—1661no—
464C2×C8—1661no—
465C2×C8—1661no—
466C2×C8—1661no—
467C2×C8—1661no—
468C2×C8—1661no—
469C2×C8—1661no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—22no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—42no—
480C4⋊C4—42no—
481C4⋊C4—42no—
482C4⋊C4—81no—
483C4⋊C4—3322no—
484C4⋊C4—3322no—
485C4⋊C4—3322no—
486C4⋊C4—3322no—
487C4⋊C4—832no—
488C4⋊C4—832no—
489C4⋊C4—6641no—
490C4⋊C4—6641no—
491C4⋊C4—1661no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4—1661no—
495C4⋊C4—1661no—
496C4⋊C4—1661no—
497C22⋊C4—42no—
498C22⋊C4—42no—
499C22⋊C4—42no—
500C22⋊C4—81no—
501C22⋊C4—81no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—6641no—
507C42D4×D8312yes—
508C42#1305058912yes—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23C22×C22⋊D8311yescharacteristic
517Q8—44no—
518D4—22no—
519D4—22no—
520D4—41no—
521D4—41no—
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—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—84no—
547C2×C4—3321no—
548C2×C4—3321no—
549C2×C4—3321no—
550C2×C4—3321no—
551C2×C4—3321no—
552C2×C4—3321no—
553C2×C4—3321no—
554C2×C4—3321no—
555C2×C4—3321no—
556C2×C4—3321no—
557C2×C4—1662no—
558C2×C4—1662no—
559C2×C4#1458176412yes—
560C2×C4#1458176412yes—
561C2×C4#1458181812yes—
562C2×C4C2×D4×D8311yescharacteristic
563C2×C4C2×Q8×D8311yescharacteristic
564C2×C4#41×D8311yescharacteristic
565C2×C4C2×#1305058911yescharacteristic
566C2×C4#1458183911yescharacteristic
567C2×C4#1458184011yescharacteristic
568C8—3321no—
569C8—1662no—
570C8—1662no—
571C8—1662no—
572C22—42no—
573C22—42no—
574C22—42no—
575C22—81no—
576C22—81no—
577C22—81no—
578C22—81no—
579C22C2×#1458176411yescharacteristic
580C22#1912835811yescharacteristic
581C22Q8×C22⋊D8311yescharacteristic
582C22C2×#1458181811yescharacteristic
583C22#1912849911yescharacteristic
584C22#1912850911yescharacteristic
585C22#1912851211yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—81no—
593C4—81no—
594C4—81no—
595C4—3322no—
596C4—3322no—
597C4—6641no—
598C4—6641no—
599C4#1912825311yescharacteristic
600C4#1912831111yescharacteristic
601C4#1912835311yescharacteristic
602C4#1912836411yescharacteristic
603C2—82no—
604C2#2491106411yescharacteristic
605C2#2491707811yescharacteristic
606C2#2491712211yescharacteristic
607C2#2492018811yescharacteristic
608C2#2492028411yescharacteristic
609C2#2492035811yescharacteristic
610C2#2492040811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2538C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
hypercenter#2538D83
hypocenterC83#25486
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ #2538

Complements

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

NH ≅ G/Ntype
2. C2×#19127832C2semidirect
3. #24912851C2semidirect
4. #24908965C2semidirect
5. #24908954C2semidirect
8. #24908431C2semidirect
9. #24908224C2semidirect
10. #24908203C2semidirect
11. #24908188C2semidirect
88. #14581713Q8semidirect
321. C83#25486semidirect

Maximal subgroups

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

subgroupindex
C2×#191280012
C2×#191278322
#249128512
#249089652
#249089542
#249088372
#249084312
#249082242
#249082032
#249081882
#249081632
#249081492
#249072882
C83×#25382
#2548683