Finite Group Databasebeta

← #35504393

Subgroups — #35504393

Order 21248.

784 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
1#24915773C211yesmaximal, characteristic
2#24909031C211yesmaximal, characteristic, complemented
3#24909011C211yesmaximal, characteristic, complemented
4#24908801C211yesmaximal, characteristic, complemented
5#24908614C211yesmaximal, characteristic, complemented
6#24908048C211yesmaximal, characteristic, complemented
7#24908034C211yesmaximal, characteristic, complemented
8#24907887C211yesmaximal, characteristic
9#24907887C211yesmaximal, characteristic
10#24907731C211yesmaximal, characteristic, complemented
11#24907717C211yesmaximal, characteristic, complemented
12#24907490C211yesmaximal, characteristic
13#24907486C211yesmaximal, characteristic
14C2×#19127456C211yesmaximal, characteristic
15C4×C22⋊Q8×C83C211yesmaximal, characteristic
16C23×C83⋊C8C2211yescharacteristic
17C23×C83⋊C8C2211yescharacteristic
18C2×#14581784C2211yescharacteristic
19C2×#14581725C2211yescharacteristic
20C2×#14581725C2211yescharacteristic
21C2×#14581723C2211yescharacteristic
22C2×#14581723C2211yescharacteristic
23C2×#14581716C2211yescharacteristic
24C2×#14581716C2211yescharacteristic
25C2×#14581716C2211yescharacteristic
26C2×#14581716C2211yescharacteristic
27#19127599C2211yescharacteristic
28#19127599C2211yescharacteristic
29#19127585—21no—
30#19127585—21no—
31#19127574—21no—
32#19127574—21no—
33#19127567C2211yescharacteristic
34#19127561C2211yescharacteristic
35#19127561C2211yescharacteristic
36#19127463C2211yescharacteristic
37#19127463C2211yescharacteristic
38#19127463C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127456C2211yescharacteristic
45#19127456C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48C2×C22⋊Q8×C83C2211yescharacteristic
49C2×C4×Q8×C83C2211yescharacteristic
50C22×C42×C83C2211yescharacteristic
51#391×C83C2211yescharacteristic
52#389×C83C2211yescharacteristic
53#387×C83—21no—
54#385×C83—21no—
55#380×C83C2211yescharacteristic
56C4×C4⋊C4×C83—21no—
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C22⋊C4×C83—21no—
59C22×C83⋊C8—22no—
60C22×C83⋊C8—22no—
61C22×C83⋊C8—22no—
62C22×C83⋊C8—22no—
63C22×C83⋊C8—41no—
64C22×C83⋊C8—41no—
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69#14581784—41no—
70#14581784—41no—
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×Dic83⋊C4—21no—
76C2×Dic83⋊C4—21no—
77C2×Dic83⋊C4—21no—
78C2×Dic83⋊C4—21no—
79C2×C22⋊Dic83—21no—
80C2×C22⋊Dic83—21no—
81#14581725—22no—
82#14581725—41no—
83#14581723—22no—
84#14581723—41no—
85#14581716—22no—
86#14581716—22no—
87#14581716—41no—
88#14581716—41no—
89C83⋊#95—21no—
90C83⋊#95—21no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C22×Q8×C83C2311yescharacteristic
96C23×C4×C83D411yescharacteristic
97C23×C4×C83D411yescharacteristic
98C23×C4×C83C2311yescharacteristic
99C22⋊Q8×C83—22no—
100C22⋊Q8×C83Q814yes—
101C4×Q8×C83—41no—
102C4×Q8×C83—41no—
103C2×C4⋊C4×C83—21no—
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83—21no—
106C2×C4⋊C4×C83C2311yescharacteristic
107C2×C4⋊C4×C83C2311yescharacteristic
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C22⋊C4×C83—21no—
110C2×C22⋊C4×C83—21no—
111C2×C42×C83—21no—
112C2×C42×C83—21no—
113C2×C42×C83—41no—
114C2×C42×C83—41no—
115C2×C42×C83D411yescharacteristic
116C2×C42×C83D411yescharacteristic
117C2×C42×C83C2311yescharacteristic
118C2×C42×C83C2311yescharacteristic
119#95×C83—21no—
120#95×C83—21no—
121#95×C83—21no—
122#95×C83—21no—
123#95×C83—21no—
124C22×Dic83—21no—
125C22×Dic83—21no—
126C22×Dic83—21no—
127C22×Dic83—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⋊C8—21no—
135C2×C83⋊C8—21no—
136C2×C83⋊C8—21no—
137C2×C83⋊C8—21no—
138C2×C83⋊C8—44no—
139C2×C83⋊C8—44no—
140C2×C83⋊C8#4112yes—
141C2×C83⋊C8#4112yes—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150Dic83⋊C4—42no—
151Dic83⋊C4—42no—
152Dic83⋊C4—42no—
153Dic83⋊C4—42no—
154C22⋊Dic83—42no—
155C22⋊Dic83—42no—
156C24×C83C2×D411yescharacteristic
157C2×Q8×C83—21no—
158C2×Q8×C83—21no—
159C2×Q8×C83—21no—
160C2×Q8×C83—21no—
161C2×Q8×C83—21no—
162C2×Q8×C83C2×Q812yes—
163C22×C4×C83—21no—
164C22×C4×C83—21no—
165C22×C4×C83—21no—
166C22×C4×C83—21no—
167C22×C4×C83—22no—
168C22×C4×C83—22no—
169C22×C4×C83—21no—
170C22×C4×C83—21no—
171C22×C4×C83—22no—
172C22×C4×C83—21no—
173C22×C4×C83—21no—
174C22×C4×C83—41no—
175C22×C4×C83—41no—
176C22×C4×C83—41no—
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×Q812yes—
183C22×C4×C83#4112yes—
184C22×C4×C83C2411yescharacteristic
185C4⋊C4×C83—24no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—21no—
190C4⋊C4×C83—21no—
191C4⋊C4×C83—42no—
192C4⋊C4×C83—42no—
193C4⋊C4×C83C2×Q812yes—
194C22⋊C4×C83—24no—
195C22⋊C4×C83—42no—
196C42×C83—21no—
197C42×C83—21no—
198C42×C83—21no—
199C42×C83—21no—
200C42×C83—41no—
201C42×C83—41no—
202C42×C83—42no—
203C42×C83—42no—
204C23×C83—21no—
205C23×C83—21no—
206C23×C83—21no—
207C23×C83—21no—
208C23×C83C22⋊Q812yes—
209C23×C83#13512yes—
210C23×C83#13712yes—
211C23×C83C22×D411yescharacteristic
212Q8×C83—22no—
213Q8×C83—22no—
214Q8×C83—41no—
215Q8×C83—41no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—22no—
221C2×C4×C83—22no—
222C2×C4×C83—22no—
223C2×C4×C83—22no—
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—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—44no—
236C2×C4×C83—44no—
237C2×C4×C83—41no—
238C2×C4×C83—41no—
239C2×C4×C83—41no—
240C2×C4×C83—41no—
241C2×C4×C83—41no—
242C2×C4×C83—41no—
243C2×C4×C83—41no—
244C2×C4×C83—41no—
245C2×C4×C83—41no—
246C2×C4×C83—41no—
247C2×C4×C83C22⋊Q812yes—
248C2×C4×C83#13512yes—
249C2×C4×C83C22×Q811yescharacteristic
250C2×C4×C83C2×#4111yescharacteristic
251C2×C4×C83C2×#4111yescharacteristic
252C2×C4×C83C2×#4111yescharacteristic
253C2×C4×C83H(5,𝔽2)11yescharacteristic
254C2×C4×C83H(5,𝔽2)11yescharacteristic
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—41no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C83⋊C8—22no—
268C83⋊C8—22no—
269C83⋊C8—41no—
270C83⋊C8—41no—
271C22×C83—22no—
272C22×C83—22no—
273C22×C83—22no—
274C22×C83—22no—
275C22×C83—22no—
276C22×C83—22no—
277C22×C83#47914yes—
278C22×C83C2×C22⋊Q811yescharacteristic
279C22×C83#53211yescharacteristic
280C22×C83#53411yescharacteristic
281C22×C83C2×#13511yescharacteristic
282C22×C83C2×#13511yescharacteristic
283C22×C83C2×#13711yescharacteristic
284C22×C83#57511yescharacteristic
285C4×C83—22no—
286C4×C83—22no—
287C4×C83—42no—
288C4×C83—42no—
289C4×C83—41no—
290C4×C83—41no—
291C4×C83—41no—
292C4×C83—81no—
293C4×C83—81no—
294C4×C83#55111yescharacteristic
295C4×C83#55111yescharacteristic
296C4×C83#55411yescharacteristic
297C4×C83#55411yescharacteristic
298Dic83—81no—
299Dic83—81no—
300Dic83—81no—
301Dic83—81no—
302#29158—831nomaximal, Sylow
303C2×C83—24no—
304C2×C83#256411yescharacteristic
305C2×C83C2×#47911yescharacteristic
306C2×C83#307611yescharacteristic
307C2×C83#316811yescharacteristic
308C2×C83#318111yescharacteristic
309C2×C83#319011yescharacteristic
310C2×C83#319111yescharacteristic
311#2851—831no—
312C4×C22⋊Q8D8311yescharacteristic
313#2101—831no—
314#2097—831no—
315#2059—831no—
316#2026—831no—
317#1920—831no—
318#1918—831no—
319#1892—831no—
320#1892—831no—
321#1865—831no—
322#1863—831no—
323#1808—831no—
324#1807—831no—
325C2×#336—831no—
326C83#2915811yescharacteristic, complemented, Sylow
327C23×C8—831no—
328C23×C8—831no—
329C2×#125—831no—
330C2×C22⋊Q8C2×D8311yescharacteristic
331C2×C4×Q8C2×D8311yescharacteristic
332C22×C42C2×D8311yescharacteristic
333C2×#107—831no—
334C2×#107—831no—
335C2×#106—831no—
336C2×#106—831no—
337C2×#103—831no—
338C2×#103—831no—
339C2×#103—831no—
340C2×#103—831no—
341#391C2×D8311yescharacteristic
342#389—831no—
343#389—831no—
344#389C2×D8311yescharacteristic
345#387—21no—
346#387—1661no—
347#387—1661no—
348#385—21no—
349#385—1661no—
350#385—1661no—
351#382—831no—
352#380—831no—
353#380—831no—
354#380C2×D8311yescharacteristic
355C4×C4⋊C4—21no—
356C4×C4⋊C4C2×D8311yescharacteristic
357C4×C22⋊C4—21no—
358#340—831no—
359#340—831no—
360#340—831no—
361#340—831no—
362#340—831no—
363#340—831no—
364#340—831no—
365#340—831no—
366#336—831no—
367#336—831no—
368#336—831no—
369#336—831no—
370C22×Q8C22×D8311yescharacteristic
371C23×C4C22⋊D8311yescharacteristic
372C23×C4C22⋊D8311yescharacteristic
373C23×C4C22×D8311yescharacteristic
374C22×C8—3321no—
375C22×C8—3321no—
376C22×C8—831no—
377C22×C8—831no—
378C22×C8—831no—
379C22×C8—831no—
380C22×C8—1662no—
381C22×C8—1662no—
382C22×C8—1662no—
383C22×C8—1662no—
384#125—3321no—
385#125—3321no—
386C22⋊Q8—22no—
387C22⋊Q8Dic16614yes—
388C4×Q8—41no—
389C4×Q8—41no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—831no—
394C2×C4⋊C4—831no—
395C2×C4⋊C4—831no—
396C2×C4⋊C4—831no—
397C2×C4⋊C4—1661no—
398C2×C4⋊C4—1661no—
399C2×C4⋊C4—1661no—
400C2×C4⋊C4—1661no—
401C2×C4⋊C4C22×D8311yescharacteristic
402C2×C4⋊C4C22×D8311yescharacteristic
403C2×C4⋊C4C22×D8311yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1661no—
407C2×C22⋊C4—1661no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—41no—
411C2×C42—41no—
412C2×C42C22⋊D8311yescharacteristic
413C2×C42C22⋊D8311yescharacteristic
414C2×C42C22×D8311yescharacteristic
415C2×C42C22×D8311yescharacteristic
416#107—3321no—
417#107—1662no—
418#106—3321no—
419#106—1662no—
420#103—3321no—
421#103—3321no—
422#103—1662no—
423#103—1662no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—21no—
428#95—21no—
429#95—1661no—
430#95—1661no—
431#95—1661no—
432#95—1661no—
433#95—1661no—
434#95—1661no—
435C24C2×C22⋊D8311yescharacteristic
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—21no—
441C2×Q8C2×Dic16612yes—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—1661no—
457C22×C4—1661no—
458C22×C4—1661no—
459C22×C4—1661no—
460C22×C4C2×C22⋊D8311yescharacteristic
461C22×C4C2×C22⋊D8311yescharacteristic
462C22×C4C2×C22⋊D8311yescharacteristic
463C22×C4C2×C22⋊D8311yescharacteristic
464C22×C4C2×C22⋊D8311yescharacteristic
465C22×C4C2×Dic16612yes—
466C22×C4#1305059012yes—
467C22×C4C23×D8311yescharacteristic
468C2×C8—3324no—
469C2×C8—3324no—
470C2×C8—832no—
471C2×C8—832no—
472C2×C8—1661no—
473C2×C8—1661no—
474C2×C8—1661no—
475C2×C8—1661no—
476C2×C8—1661no—
477C2×C8—1661no—
478C2×C8—1661no—
479C2×C8—1661no—
480C2×C8—1661no—
481C2×C8—1661no—
482C4⋊C4—24no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—42no—
490C4⋊C4—3322no—
491C4⋊C4—3322no—
492C4⋊C4—3322no—
493C4⋊C4—3322no—
494C4⋊C4—1661no—
495C4⋊C4—1661no—
496C4⋊C4—1661no—
497C4⋊C4—1661no—
498C4⋊C4—1661no—
499C4⋊C4—1661no—
500C4⋊C4—1661no—
501C4⋊C4—1661no—
502C4⋊C4C2×Dic16612yes—
503C22⋊C4—24no—
504C22⋊C4—42no—
505C22⋊C4—3322no—
506C22⋊C4—3322no—
507C42—21no—
508C42—21no—
509C42—21no—
510C42—21no—
511C42—41no—
512C42—41no—
513C42—42no—
514C42—42no—
515C23—21no—
516C23—21no—
517C23—21no—
518C23—21no—
519C23#1458177112yes—
520C23#1458181212yes—
521C23#1458182312yes—
522C23C22×C22⋊D8311yescharacteristic
523Q8—22no—
524Q8—22no—
525Q8—41no—
526Q8—41no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—22no—
532C2×C4—22no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—22no—
536C2×C4—22no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—44no—
547C2×C4—44no—
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—41no—
558C2×C4—3321no—
559C2×C4—3321no—
560C2×C4—3321no—
561C2×C4—3321no—
562C2×C4—3321no—
563C2×C4—3321no—
564C2×C4—3321no—
565C2×C4—3321no—
566C2×C4—3321no—
567C2×C4—3321no—
568C2×C4—3321no—
569C2×C4—3321no—
570C2×C4#1458177112yes—
571C2×C4#1458181212yes—
572C2×C4C22×Dic16611yescharacteristic
573C2×C4#41×D8311yescharacteristic
574C2×C4#41×D8311yescharacteristic
575C2×C4C2×#1305059011yescharacteristic
576C2×C4#1458183711yescharacteristic
577C2×C4#1458183711yescharacteristic
578C8—3321no—
579C8—3321no—
580C8—1662no—
581C8—1662no—
582C22—22no—
583C22—22no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22#1912800814yes—
589C22C2×#1458177111yescharacteristic
590C22#1912824711yescharacteristic
591C22#1912826011yescharacteristic
592C22C2×#1458181211yescharacteristic
593C22C2×#1458181211yescharacteristic
594C22C2×#1458182311yescharacteristic
595C22#1912849911yescharacteristic
596C4—22no—
597C4—22no—
598C4—42no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—42no—
603C4—81no—
604C4—81no—
605C4—6641no—
606C4—6641no—
607C4—6641no—
608C4—6641no—
609C4#1912838011yescharacteristic
610C4#1912838011yescharacteristic
611C4#1912841011yescharacteristic
612C4#1912841011yescharacteristic
613C2—24no—
614C2#2491263911yescharacteristic
615C2C2×#1912800811yescharacteristic
616C2#2491725111yescharacteristic
617C2#2491802711yescharacteristic
618C2#2491816611yescharacteristic
619C2#2491824411yescharacteristic
620C2#2491825111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC4×C22⋊Q8×C83C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenterC4×C22⋊Q8D83
hypocenterC83#29158
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C42 ◁ C4×C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. #24909031C2semidirect
3. #24909011C2semidirect
4. #24908801C2semidirect
5. #24908614C2semidirect
6. #24908048C2semidirect
7. #24908034C2semidirect
10. #24907731C2semidirect
11. #24907717C2semidirect
326. C83#29158semidirect

Maximal subgroups

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

subgroupindex
#249157732
#249090312
#249090112
#249088012
#249086142
#249080482
#249080342
#249078872
#249077312
#249077172
#249074902
#249074862
C2×#191274562
C4×C22⋊Q8×C832
#2915883