Finite Group Databasebeta

← #35409283

Subgroups — #35409283

Order 21248.

894 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127755C211yesmaximal, characteristic
2#24909865C211yesmaximal, characteristic
3#117×Dic83C211yesmaximal, characteristic
4#24908225C211yesmaximal, characteristic, complemented
5#24908223C211yesmaximal, characteristic, complemented
6#24908220C211yesmaximal, characteristic, complemented
7#24908218C211yesmaximal, characteristic, complemented
8#24907664C211yesmaximal, characteristic
9#24907662C211yesmaximal, characteristic
10#24907660C211yesmaximal, characteristic
11#24907363C211yesmaximal, characteristic, complemented
12#24907362C211yesmaximal, characteristic, complemented
13#24907359C211yesmaximal, characteristic, complemented
14#24907359C211yesmaximal, characteristic, complemented
15C83×#1854C211yesmaximal, characteristic
16C2×#14581751C2211yescharacteristic
17C2×#14581748—21no—
18C22×C4×Dic83C2211yescharacteristic
19#19127755C418yescomplemented
20C2×#14581726C2211yescharacteristic
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581724C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×Dic83⋊C8C2211yescharacteristic
27C2×Dic83⋊C8C2211yescharacteristic
28C2×C22⋊(C83⋊C8)C2211yescharacteristic
29C2×#14581702C2211yescharacteristic
30C2×#14581702C2211yescharacteristic
31#19127570C2211yescharacteristic
32#19127570C2211yescharacteristic
33#19127565—21no—
34#19127561C2211yescharacteristic
35#19127559—21no—
36C4⋊C4×Dic83C2211yescharacteristic
37C4⋊C4×Dic83C2211yescharacteristic
38C4⋊C4×Dic83C2211yescharacteristic
39C4⋊C4×Dic83C2211yescharacteristic
40C4⋊C4×Dic83C2211yescharacteristic
41C4⋊C4×Dic83C2211yescharacteristic
42#19127552—21no—
43C22⋊C4×Dic83—21no—
44C22⋊C4×Dic83—21no—
45#19127547—21no—
46C42×Dic83—21no—
47#19127464C2211yescharacteristic
48#19127464C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127463C2211yescharacteristic
52#19127463C2211yescharacteristic
53C2×#117×C83C2211yescharacteristic
54C2×#107×C83C2211yescharacteristic
55C2×#106×C83C2211yescharacteristic
56C2×C22⋊C8×C83C2211yescharacteristic
57#380×C83C2211yescharacteristic
58#340×C83C2211yescharacteristic
59#340×C83C2211yescharacteristic
60C23×Dic83D411yescharacteristic
61C23×Dic83D411yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64#14581751—22no—
65#14581751—22no—
66#14581748—44no—
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×C22⋊Dic83—21no—
76C2×C22⋊Dic83—21no—
77C2×C4×Dic83—21no—
78C2×C4×Dic83—21no—
79C2×C4×Dic83—21no—
80C2×C4×Dic83—21no—
81C2×C4×Dic83—21no—
82C2×C4×Dic83—21no—
83C2×C4×Dic83—21no—
84C2×C4×Dic83C2×C414yes—
85C2×C4×Dic83D411yescharacteristic
86C2×C4×Dic83D411yescharacteristic
87C2×C4×Dic83Q814yes—
88C2×C4×Dic83C2311yescharacteristic
89C2×C4×Dic83C2311yescharacteristic
90#14581726—22no—
91#14581725—22no—
92#14581725—22no—
93#14581724—22no—
94#14581723—22no—
95#14581723—22no—
96Dic83⋊C8C2×C414yes—
97Dic83⋊C8C2×C414yes—
98C22⋊(C83⋊C8)C2×C414yes—
99#14581702C2×C414yes—
100#14581702C2×C414yes—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C83⋊#95—21no—
104C83⋊#95—21no—
105C83⋊#95—21no—
106C83⋊#95—21no—
107C83⋊#95—21no—
108C23×C4×C83C2311yescharacteristic
109C22×C8×C83C2311yescharacteristic
110C22×C8×C83C2311yescharacteristic
111#117×C83—22no—
112#117×C83—22no—
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C22⋊C4×C83—21no—
118C2×C22⋊C4×C83—21no—
119C2×C42×C83—21no—
120#107×C83—22no—
121#106×C83—22no—
122C22⋊C8×C83C2×C414yes—
123#95×C83—21no—
124C22×Dic83—22no—
125C22×Dic83—22no—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C22×Dic83—21no—
129C22×Dic83—21no—
130C22×Dic83—41no—
131C22×Dic83—41no—
132C22×Dic83C4⋊C414yes—
133C22×Dic83C4⋊C414yes—
134C22×Dic83C2×D411yescharacteristic
135C22×Dic83C2×D411yescharacteristic
136C22×Dic83C2×D411yescharacteristic
137C22×Dic83C2×D411yescharacteristic
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8C22×C412yes—
143C2×C83⋊C8C22×C412yes—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83#4112yes—
151C4⋊Dic83#4112yes—
152Dic83⋊C4—42no—
153Dic83⋊C4—42no—
154Dic83⋊C4—42no—
155Dic83⋊C4—42no—
156C22⋊Dic83—24no—
157C22⋊Dic83—42no—
158C4×Dic83—24no—
159C4×Dic83—42no—
160C4×Dic83—42no—
161C4×Dic83—42no—
162C4×Dic83—42no—
163C4×Dic83—42no—
164C4×Dic83—42no—
165C4×Dic83C4⋊C414yes—
166C4×Dic83C4⋊C414yes—
167C4×Dic83C22×C412yes—
168C4×Dic83C22×C412yes—
169C4×Dic83C2×Q812yes—
170C4×Dic83C2×Q812yes—
171C24×C83C2×D411yescharacteristic
172C22×C4×C83—21no—
173C22×C4×C83—21no—
174C22×C4×C83—21no—
175C22×C4×C83—21no—
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83—21no—
179C22×C4×C83—21no—
180C22×C4×C83C22×C412yes—
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×Q812yes—
183C22×C4×C83C2411yescharacteristic
184C2×C8×C83—21no—
185C2×C8×C83—21no—
186C2×C8×C83—21no—
187C2×C8×C83—21no—
188C2×C8×C83C22×C412yes—
189C2×C8×C83C22×C412yes—
190C4⋊C4×C83—21no—
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83#4112yes—
197C4⋊C4×C83#4112yes—
198C22⋊C4×C83—24no—
199C22⋊C4×C83—42no—
200C42×C83—24no—
201C23×C83—21no—
202C23×C83—21no—
203C23×C83—21no—
204C23×C83—21no—
205C23×C83—21no—
206C23×C83—21no—
207C23×C83C2×C4⋊C412yes—
208C23×C83C22×D411yescharacteristic
209C2×C4×C83—21no—
210C2×C4×C83—21no—
211C2×C4×C83—22no—
212C2×C4×C83—22no—
213C2×C4×C83—22no—
214C2×C4×C83—22no—
215C2×C4×C83—22no—
216C2×C4×C83—22no—
217C2×C4×C83—41no—
218C2×C4×C83—41no—
219C2×C4×C83—41no—
220C2×C4×C83—41no—
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×C83C2×C4⋊C412yes—
228C2×C4×C83C23×C411yescharacteristic
229C2×C4×C83C22×Q811yescharacteristic
230C2×C4×C83C2×#4111yescharacteristic
231C2×C4×C83C2×#4111yescharacteristic
232C2×C4×C83C2×#4111yescharacteristic
233C2×C4×C83C2×#4111yescharacteristic
234C2×Dic83—21no—
235C2×Dic83—21no—
236C2×Dic83—21no—
237C2×Dic83—21no—
238C2×Dic83—21no—
239C2×Dic83—21no—
240C2×Dic83—28no—
241C2×Dic83—22no—
242C2×Dic83—22no—
243C2×Dic83—44no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83C2×C4⋊C412yes—
255C2×Dic83C2×C4⋊C412yes—
256C2×Dic83C2×C4⋊C412yes—
257C2×Dic83C2×C4⋊C412yes—
258C2×Dic83AGL(1,ℤ/8)12yes—
259C2×Dic83#13712yes—
260C8×C83—22no—
261C8×C83—22no—
262C83⋊C8—22no—
263C83⋊C8—22no—
264C22×C83—22no—
265C22×C83—22no—
266C22×C83—22no—
267C22×C83—41no—
268C22×C83—41no—
269C22×C83—41no—
270C22×C83—41no—
271C22×C83C22×C4⋊C411yescharacteristic
272C22×C83#52511yescharacteristic
273C22×C83#52511yescharacteristic
274C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C83C2×#13711yescharacteristic
276C22×C83#57511yescharacteristic
277C22×C83#57511yescharacteristic
278C4×C83—22no—
279C4×C83—42no—
280C4×C83—42no—
281C4×C83—81no—
282C4×C83—81no—
283C4×C83C4×#4111yescharacteristic
284C4×C83C4×#4111yescharacteristic
285C4×C83#53311yescharacteristic
286C4×C83#53311yescharacteristic
287Dic83—22no—
288Dic83—22no—
289Dic83—22no—
290Dic83—42no—
291Dic83—42no—
292Dic83—81no—
293Dic83—81no—
294Dic83#42814yes—
295#21364—831nomaximal, Sylow
296C2×C83—42no—
297C2×C83#229611yescharacteristic
298C2×C83C2×#42811yescharacteristic
299C2×C83#291111yescharacteristic
300C2×C83#311511yescharacteristic
301C2×C83#311511yescharacteristic
302C2×C83#311611yescharacteristic
303C2×C83#311611yescharacteristic
304C2×#432—831no—
305#2285—831no—
306C4×#117—831no—
307#1948—831no—
308#1948—831no—
309#1947—831no—
310#1947—831no—
311#1854—831no—
312#1854—831no—
313#1854—831no—
314#1854D8311yescharacteristic
315#1776—831no—
316#1776—831no—
317#1776—831no—
318#1776—831no—
319C83#2136411yescharacteristic, complemented, Sylow
320C2×#117—831no—
321C2×#117—1661no—
322C2×#117C2×D8311yescharacteristic
323C22×C42—831no—
324#432—838no—
325C2×#107—831no—
326C2×#107—831no—
327C2×#107—831no—
328C2×#107C2×D8311yescharacteristic
329C2×#106—831no—
330C2×#106—831no—
331C2×#106—831no—
332C2×#106C2×D8311yescharacteristic
333C2×C4⋊C8—831no—
334C2×C4⋊C8—831no—
335C2×C22⋊C8—831no—
336C2×C22⋊C8C2×D8311yescharacteristic
337C2×#97—831no—
338C2×#97—831no—
339#384—831no—
340#384—831no—
341#382—1661no—
342#380—831no—
343#380—1661no—
344#380C2×D8311yescharacteristic
345C4×C4⋊C4—831no—
346C4×C4⋊C4—831no—
347C4×C4⋊C4—831no—
348C4×C4⋊C4—831no—
349C4×C4⋊C4—831no—
350C4×C4⋊C4—831no—
351C4×C4⋊C4—1661no—
352C4×C22⋊C4—1661no—
353C4×C22⋊C4—1661no—
354#376—1661no—
355C43—1661no—
356#340—831no—
357#340—831no—
358#340—831no—
359#340—831no—
360#340—831no—
361#340—831no—
362#340C2×D8311yescharacteristic
363#340C2×D8311yescharacteristic
364C23×C4—831no—
365C23×C4—831no—
366C23×C4C22×D8311yescharacteristic
367C22×C8—831no—
368C22×C8—831no—
369C22×C8C22×D8311yescharacteristic
370C22×C8C22×D8311yescharacteristic
371#117—22no—
372#117—22no—
373#117—3324no—
374#117—1662no—
375#117—1662no—
376C2×C4⋊C4—831no—
377C2×C4⋊C4—831no—
378C2×C4⋊C4—831no—
379C2×C4⋊C4—831no—
380C2×C4⋊C4—1661no—
381C2×C4⋊C4—1661no—
382C2×C4⋊C4—1661no—
383C2×C4⋊C4—1661no—
384C2×C4⋊C4C22×D8311yescharacteristic
385C2×C4⋊C4C22×D8311yescharacteristic
386C2×C4⋊C4C22×D8311yescharacteristic
387C2×C4⋊C4C22×D8311yescharacteristic
388C2×C22⋊C4—21no—
389C2×C22⋊C4—21no—
390C2×C22⋊C4—1661no—
391C2×C22⋊C4—1661no—
392C2×C42—21no—
393C2×C42—834no—
394C2×C42—831no—
395C2×C42—831no—
396C2×C42—834no—
397C2×C42—831no—
398C2×C42—831no—
399C2×C42—1661no—
400C2×C42—1661no—
401C2×C42—1661no—
402C2×C42—1661no—
403C2×C42—1661no—
404C2×C42—1661no—
405C2×C42—1661no—
406#107—22no—
407#107—1662no—
408#107—1662no—
409#107—1662no—
410#106—22no—
411#106—1662no—
412#106—1662no—
413#106—1662no—
414C4⋊C8—834no—
415C4⋊C8—834no—
416C22⋊C8—834no—
417C22⋊C8C4×D8314yes—
418#97—834no—
419#97—834no—
420#95—21no—
421#95—1661no—
422#95—1661no—
423#95—1661no—
424#95—1661no—
425#95—1661no—
426#95—1661no—
427#95—1661no—
428C24D4×D8311yescharacteristic
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—3321no—
438C22×C4—3321no—
439C22×C4—834no—
440C22×C4—831no—
441C22×C4—831no—
442C22×C4—831no—
443C22×C4—831no—
444C22×C4—834no—
445C22×C4—1661no—
446C22×C4—1661no—
447C22×C4—1661no—
448C22×C4—1661no—
449C22×C4—1662no—
450C22×C4—1662no—
451C22×C4C2×C4×D8312yes—
452C22×C4D4×D8311yescharacteristic
453C22×C4Q8×D8312yes—
454C22×C4C23×D8311yescharacteristic
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—21no—
459C2×C8—832no—
460C2×C8—832no—
461C2×C8—1661no—
462C2×C8—1661no—
463C2×C8—1661no—
464C2×C8—1661no—
465C2×C8C2×C4×D8312yes—
466C2×C8C2×C4×D8312yes—
467C4⋊C4—21no—
468C4⋊C4—21no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—3322no—
474C4⋊C4—3322no—
475C4⋊C4—3322no—
476C4⋊C4—3322no—
477C4⋊C4—832no—
478C4⋊C4—832no—
479C4⋊C4—1661no—
480C4⋊C4—1661no—
481C4⋊C4—1661no—
482C4⋊C4—1661no—
483C4⋊C4—1661no—
484C4⋊C4—1661no—
485C4⋊C4#1305058912yes—
486C4⋊C4#1305058912yes—
487C22⋊C4—24no—
488C22⋊C4—42no—
489C22⋊C4—3322no—
490C22⋊C4—1664no—
491C42—24no—
492C42—3322no—
493C42—3322no—
494C42—3322no—
495C42—3322no—
496C42—3322no—
497C42—3322no—
498C42—832no—
499C42—832no—
500C42—832no—
501C42—832no—
502C42—834no—
503C42—834no—
504C42—1664no—
505C23—21no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23C4⋊C4×D8312yes—
512C23C2×D4×D8311yescharacteristic
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—3321no—
532C2×C4—3321no—
533C2×C4—3321no—
534C2×C4—3321no—
535C2×C4—3321no—
536C2×C4—3321no—
537C2×C4—3321no—
538C2×C4—3321no—
539C2×C4—3321no—
540C2×C4—3321no—
541C2×C4—3324no—
542C2×C4—832no—
543C2×C4—832no—
544C2×C4—832no—
545C2×C4—832no—
546C2×C4—832no—
547C2×C4—832no—
548C2×C4—1662no—
549C2×C4—1662no—
550C2×C4—1668no—
551C2×C4—1661no—
552C2×C4—1661no—
553C2×C4—1661no—
554C2×C4—1661no—
555C2×C4—1661no—
556C2×C4—1661no—
557C2×C4C4⋊C4×D8312yes—
558C2×C4C22×C4×D8311yescharacteristic
559C2×C4C2×Q8×D8311yescharacteristic
560C2×C4#41×D8311yescharacteristic
561C2×C4#41×D8311yescharacteristic
562C2×C4C2×#1305058911yescharacteristic
563C2×C4C2×#1305058911yescharacteristic
564C8—22no—
565C8—22no—
566C8—1662no—
567C8—1662no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22C2×C4⋊C4×D8311yescharacteristic
576C22#1912820711yescharacteristic
577C22#1912820911yescharacteristic
578C22AGL(1,ℤ/8)×D8311yescharacteristic
579C22#137×D8311yescharacteristic
580C22#1912849811yescharacteristic
581C22#1912849811yescharacteristic
582C4—22no—
583C4—42no—
584C4—42no—
585C4—81no—
586C4—81no—
587C4—3322no—
588C4—3322no—
589C4—834no—
590C4—6641no—
591C4—6641no—
592C4—1662no—
593C4—1662no—
594C4—1662no—
595C4C4×#1305058911yescharacteristic
596C4C4×#1305058911yescharacteristic
597C4#1912825311yescharacteristic
598C4#1912825311yescharacteristic
599C2—42no—
600C2#2490994211yescharacteristic
601C2#428×D8311yescharacteristic
602C2#2491605111yescharacteristic
603C2#2491760711yescharacteristic
604C2#2491760911yescharacteristic
605C2#2491761811yescharacteristic
606C2#2491762011yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1854C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
cosocleC22×C4×C83C24
hypercenter#1854D83
hypocenterC83#21364
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1854

Complements

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

NH ≅ G/Ntype
4. #24908225C2semidirect
5. #24908223C2semidirect
6. #24908220C2semidirect
7. #24908218C2semidirect
11. #24907363C2semidirect
12. #24907362C2semidirect
13. #24907359C2semidirect
14. #24907359C2semidirect
19. #19127755C4semidirect
319. C83#21364semidirect

Maximal subgroups

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

subgroupindex
C2×#191277552
#249098652
#117×Dic832
#249082252
#249082232
#249082202
#249082182
#249076642
#249076622
#249076602
#249073632
#249073622
#249073592
C83×#18542
#2136483