Finite Group Databasebeta

← #35326577

Subgroups — #35326577

Order 21248.

1602 conjugacy classes of subgroups.

The subgroup lattice has 654 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊Dic83C211yesmaximal, characteristic
2C2×#19127546C211yesmaximal, characteristic, complemented
3#24905722C212yesmaximal, complemented
4#24905710C212yesmaximal
5#24905700C212yesmaximal
6#24905695C212yesmaximal, complemented
7#24905689C211yesmaximal, characteristic, complemented
8#24905688C211yesmaximal, characteristic, complemented
9#24905687C211yesmaximal, characteristic, complemented
10#24905686C211yesmaximal, characteristic
11C2×#380×C83C211yesmaximal, characteristic
12C22×C22⋊Dic83C2212yes—
13C22×C22⋊Dic83C2211yescharacteristic
14C22×C22⋊Dic83C2211yescharacteristic
15C22×C4×Dic83C2212yes—
16C22×C4×Dic83C2211yescharacteristic
17C22×C4×Dic83C2211yescharacteristic
18C4×C22⋊Dic83C418yescomplemented
19#19127546—24no—
20C2×C83⋊#95C2211yescharacteristic
21C2×C83⋊#95C2211yescharacteristic
22C2×C83⋊#95C2212yes—
23C2×C83⋊#95C2212yes—
24C2×C83⋊#95C2212yes—
25C2×C83⋊#95C2212yes—
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2212yes—
28C2×C83⋊#95C2212yes—
29C2×C83⋊#95C2212yes—
30C2×C83⋊#95C2211yescharacteristic
31C2×C83⋊#95C2211yescharacteristic
32C24×C4×C83C2211yescharacteristic
33C22×C4⋊C4×C83C2212yes—
34C22×C22⋊C4×C83C2212yes—
35#380×C83—24no—
36C2×#95×C83C2211yescharacteristic
37C2×#95×C83C2211yescharacteristic
38C23×Dic83C2312yes—
39C23×Dic83C2311yescharacteristic
40C23×Dic83C2311yescharacteristic
41C23×Dic83C2311yescharacteristic
42C23×Dic83C2312yes—
43C23×Dic83C2311yescharacteristic
44C2×C22⋊Dic83—24no—
45C2×C22⋊Dic83—24no—
46C2×C22⋊Dic83—22no—
47C2×C22⋊Dic83—22no—
48C2×C22⋊Dic83—24no—
49C2×C22⋊Dic83—42no—
50C2×C22⋊Dic83C2×C414yes—
51C2×C22⋊Dic83C2×C414yes—
52C2×C4×Dic83—24no—
53C2×C4×Dic83—24no—
54C2×C4×Dic83—24no—
55C2×C4×Dic83—22no—
56C2×C4×Dic83—22no—
57C2×C4×Dic83—42no—
58C2×C4×Dic83C2×C414yes—
59C2×C4×Dic83C2×C414yes—
60C83⋊#95—24no—
61C83⋊#95—24no—
62C83⋊#95—24no—
63C83⋊#95—22no—
64C83⋊#95—24no—
65C83⋊#95—22no—
66C83⋊#95—42no—
67C83⋊#95—42no—
68C83⋊#95—42no—
69C83⋊#95—42no—
70C83⋊#95C2×C414yes—
71C83⋊#95C2×C414yes—
72C25×C83C2311yescharacteristic
73C23×C4×C83—22no—
74C23×C4×C83—24no—
75C23×C4×C83—22no—
76C23×C4×C83—42no—
77C23×C4×C83C2×C414yes—
78C23×C4×C83C2312yes—
79C23×C4×C83C2312yes—
80C23×C4×C83C2311yescharacteristic
81C23×C4×C83C2311yescharacteristic
82C2×C4⋊C4×C83—24no—
83C2×C4⋊C4×C83—24no—
84C2×C4⋊C4×C83—42no—
85C2×C22⋊C4×C83—24no—
86C2×C22⋊C4×C83—24no—
87C2×C22⋊C4×C83—42no—
88#95×C83—22no—
89#95×C83D414yes—
90C22×Dic83—21no—
91C22×Dic83—22no—
92C22×Dic83—21no—
93C22×Dic83—21no—
94C22×Dic83—22no—
95C22×Dic83—22no—
96C22×Dic83—22no—
97C22×Dic83—22no—
98C22×Dic83—21no—
99C22×Dic83—22no—
100C22×Dic83—21no—
101C22×Dic83—21no—
102C22×Dic83—21no—
103C22×Dic83—21no—
104C22×Dic83—22no—
105C22×Dic83—21no—
106C22×Dic83—21no—
107C22×Dic83—22no—
108C22×Dic83—22no—
109C22×Dic83—22no—
110C22×Dic83—21no—
111C22×Dic83—21no—
112C22×Dic83—22no—
113C22×Dic83—22no—
114C22×Dic83—22no—
115C22×Dic83—22no—
116C22×Dic83—22no—
117C22×Dic83—22no—
118C22×Dic83—22no—
119C22×Dic83—22no—
120C22×Dic83C22×C412yes—
121C22×Dic83C22×C412yes—
122C22×Dic83C22×C412yes—
123C22×Dic83C22×C412yes—
124C22×Dic83#4114yes—
125C22×Dic83#4114yes—
126C22⋊Dic83—28no—
127C22⋊Dic83—28no—
128C22⋊Dic83—48no—
129C4×Dic83—216no—
130C4×Dic83—28no—
131C4×Dic83—44no—
132C24×C83—21no—
133C24×C83—22no—
134C24×C83—21no—
135C24×C83—22no—
136C24×C83—22no—
137C24×C83—21no—
138C24×C83—22no—
139C24×C83—21no—
140C24×C83C22×C412yes—
141C24×C83#4112yes—
142C24×C83#4112yes—
143C24×C83C2411yescharacteristic
144C22×C4×C83—21no—
145C22×C4×C83—22no—
146C22×C4×C83—21no—
147C22×C4×C83—22no—
148C22×C4×C83—22no—
149C22×C4×C83—22no—
150C22×C4×C83—22no—
151C22×C4×C83—22no—
152C22×C4×C83—22no—
153C22×C4×C83—22no—
154C22×C4×C83—22no—
155C22×C4×C83—28no—
156C22×C4×C83—28no—
157C22×C4×C83—22no—
158C22×C4×C83—22no—
159C22×C4×C83—22no—
160C22×C4×C83—22no—
161C22×C4×C83—22no—
162C22×C4×C83—22no—
163C22×C4×C83—22no—
164C22×C4×C83—44no—
165C22×C4×C83—48no—
166C22×C4×C83—48no—
167C22×C4×C83C22×C412yes—
168C22×C4×C83C22×C412yes—
169C22×C4×C83C2×D414yes—
170C22×C4×C83C2×D412yes—
171C22×C4×C83#4114yes—
172C22×C4×C83#4112yes—
173C4⋊C4×C83—48no—
174C22⋊C4×C83—48no—
175C23×C83—22no—
176C23×C83—22no—
177C23×C83—22no—
178C23×C83—22no—
179C23×C83—22no—
180C23×C83—22no—
181C23×C83—24no—
182C23×C83—24no—
183C23×C83—22no—
184C23×C83—22no—
185C23×C83—24no—
186C23×C83—22no—
187C23×C83—24no—
188C23×C83—22no—
189C23×C83—42no—
190C23×C83—42no—
191C23×C83—42no—
192C23×C83—42no—
193C23×C83—42no—
194C23×C83—42no—
195C23×C83—42no—
196C23×C83—42no—
197C23×C83#11714yes—
198C23×C83C23×C411yescharacteristic
199C23×C83C22×D411yescharacteristic
200C23×C83C2×#4112yes—
201C23×C83C2×#4112yes—
202C23×C83C2×#4112yes—
203C23×C83C2×#4111yescharacteristic
204C23×C83C2×#4111yescharacteristic
205C23×C83C2×#4111yescharacteristic
206C23×C83H(5,𝔽2)12yes—
207C23×C83H(5,𝔽2)11yescharacteristic
208C23×C832−1+411yescharacteristic
209C2×C4×C83—22no—
210C2×C4×C83—22no—
211C2×C4×C83—22no—
212C2×C4×C83—22no—
213C2×C4×C83—24no—
214C2×C4×C83—24no—
215C2×C4×C83—24no—
216C2×C4×C83—24no—
217C2×C4×C83—24no—
218C2×C4×C83—24no—
219C2×C4×C83—24no—
220C2×C4×C83—416no—
221C2×C4×C83—42no—
222C2×C4×C83—42no—
223C2×C4×C83—42no—
224C2×C4×C83—42no—
225C2×C4×C83—42no—
226C2×C4×C83—42no—
227C2×C4×C83—42no—
228C2×C4×C83—42no—
229C2×C4×C83—42no—
230C2×C4×C83—42no—
231C2×C4×C83—42no—
232C2×C4×C83#11714yes—
233C2×C4×C83C4×D414yes—
234C2×Dic83—22no—
235C2×Dic83—22no—
236C2×Dic83—24no—
237C2×Dic83—24no—
238C2×Dic83—22no—
239C2×Dic83—22no—
240C2×Dic83—22no—
241C2×Dic83—22no—
242C2×Dic83—22no—
243C2×Dic83—22no—
244C2×Dic83—22no—
245C2×Dic83—22no—
246C2×Dic83—22no—
247C2×Dic83—24no—
248C2×Dic83—24no—
249C2×Dic83—24no—
250C2×Dic83—42no—
251C2×Dic83—42no—
252C2×Dic83—42no—
253C2×Dic83—42no—
254C2×Dic83—41no—
255C2×Dic83—42no—
256C2×Dic83—41no—
257C2×Dic83—42no—
258C2×Dic83—42no—
259C2×Dic83—41no—
260C2×Dic83—42no—
261C2×Dic83—41no—
262C2×Dic83—42no—
263C2×Dic83—42no—
264C2×Dic83—42no—
265C2×Dic83—42no—
266C2×Dic83—42no—
267C2×Dic83—42no—
268C2×Dic83—42no—
269C2×Dic83#11714yes—
270C22×C83—24no—
271C22×C83—24no—
272C22×C83—24no—
273C22×C83—42no—
274C22×C83—42no—
275C22×C83—42no—
276C22×C83—44no—
277C22×C83—44no—
278C22×C83—44no—
279C22×C83—44no—
280C22×C83—42no—
281C22×C83C2×#11711yescharacteristic
282C22×C83C2×#11711yescharacteristic
283C22×C83C2×#11711yescharacteristic
284C22×C83C2×C4×D411yescharacteristic
285C22×C83C4×#4111yescharacteristic
286C22×C83#51811yescharacteristic
287C22×C83#52011yescharacteristic
288C22×C83#52512yes—
289C22×C83#52912yes—
290C22×C83#52912yes—
291C22×C83#52912yes—
292C22×C83#54012yes—
293C22×C83#54212yes—
294C22×C83#54612yes—
295C22×C83#54611yescharacteristic
296C22×C83#54811yescharacteristic
297C22×C83#55112yes—
298C22×C83#55112yes—
299C22×C83#55212yes—
300C22×C83#55211yescharacteristic
301C22×C83#55312yes—
302C22×C83#55311yescharacteristic
303C22×C83#55512yes—
304C4×C83—24no—
305C4×C83—24no—
306C4×C83—44no—
307C4×C83—82no—
308Dic83—24no—
309Dic83—42no—
310Dic83—42no—
311Dic83—42no—
312Dic83—44no—
313Dic83—82no—
314#13898—831nomaximal, Sylow
315C2×C83—44no—
316C2×C83#228311yescharacteristic
317C2×C83#229011yescharacteristic
318C2×C83#231911yescharacteristic
319C2×C83#234211yescharacteristic
320C2×C83#234611yescharacteristic
321C2×C83#235211yescharacteristic
322C2×C83#235311yescharacteristic
323C2×C83#247612yes—
324C2×C83#247712yes—
325C2×C83#253012yes—
326C2×C83#253112yes—
327C2×#380D8311yescharacteristic
328C2×C4×C22⋊C4—831no—
329C2×#376—831no—
330#1442—832no—
331#1440—832no—
332#1437—832no—
333#1436—832no—
334#1434—831no—
335#1434—831no—
336#1434—831no—
337C22⋊#95—831no—
338C83#1389811yescharacteristic, complemented, Sylow
339C24×C4C2×D8311yescharacteristic
340C22×C4⋊C4C2×D8312yes—
341C22×C22⋊C4—832no—
342C22×C22⋊C4—831no—
343C22×C22⋊C4—831no—
344C22×C22⋊C4C2×D8312yes—
345C22×C42—831no—
346C22×C42—831no—
347C22×C42—832no—
348#380—24no—
349C4×C22⋊C4—838no—
350#376—1664no—
351C2×#95—832no—
352C2×#95—832no—
353C2×#95—832no—
354C2×#95—832no—
355C2×#95—831no—
356C2×#95—831no—
357C2×#95—831no—
358C2×#95—831no—
359C2×#95—832no—
360C2×#95—832no—
361C2×#95—832no—
362C2×#95—832no—
363C2×#95C2×D8311yescharacteristic
364C2×#95C2×D8311yescharacteristic
365C25C22×D8311yescharacteristic
366C23×C4—22no—
367C23×C4—24no—
368C23×C4—22no—
369C23×C4—42no—
370C23×C4—831no—
371C23×C4—831no—
372C23×C4—831no—
373C23×C4—831no—
374C23×C4—832no—
375C23×C4—832no—
376C23×C4C4×D8314yes—
377C23×C4C22×D8312yes—
378C23×C4C22×D8312yes—
379C23×C4C22×D8311yescharacteristic
380C23×C4C22×D8311yescharacteristic
381C2×C4⋊C4—24no—
382C2×C4⋊C4—24no—
383C2×C4⋊C4—42no—
384C2×C22⋊C4—24no—
385C2×C22⋊C4—24no—
386C2×C22⋊C4—42no—
387C2×C22⋊C4—3322no—
388C2×C22⋊C4—834no—
389C2×C22⋊C4—834no—
390C2×C22⋊C4—1664no—
391C2×C22⋊C4—1664no—
392C2×C22⋊C4—1664no—
393C2×C22⋊C4—1662no—
394C2×C22⋊C4—1662no—
395C2×C42—3322no—
396C2×C42—834no—
397C2×C42—834no—
398C2×C42—1664no—
399C2×C42—1664no—
400C2×C42—1664no—
401C2×C42—1662no—
402C2×C42—1662no—
403#95—22no—
404#95—3322no—
405#95—3322no—
406#95—3322no—
407#95—3322no—
408#95—834no—
409#95—834no—
410#95—1664no—
411#95—1664no—
412#95—1664no—
413#95—1662no—
414#95—1664no—
415#95—1662no—
416#95C22⋊D8314yes—
417C24—21no—
418C24—22no—
419C24—22no—
420C24—22no—
421C24—21no—
422C24—21no—
423C24—22no—
424C24—21no—
425C24C2×C4×D8312yes—
426C24#1305058912yes—
427C24#1305059112yes—
428C24C23×D8311yescharacteristic
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—21no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—28no—
441C22×C4—28no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—44no—
450C22×C4—48no—
451C22×C4—48no—
452C22×C4—832no—
453C22×C4—834no—
454C22×C4—832no—
455C22×C4—832no—
456C22×C4—832no—
457C22×C4—834no—
458C22×C4—1662no—
459C22×C4—1662no—
460C22×C4—1662no—
461C22×C4—1662no—
462C22×C4—1662no—
463C22×C4—1662no—
464C22×C4—1662no—
465C22×C4—1661no—
466C22×C4—1662no—
467C22×C4—1662no—
468C22×C4—1662no—
469C22×C4—1662no—
470C22×C4—1662no—
471C22×C4—1661no—
472C22×C4—1661no—
473C22×C4—1661no—
474C22×C4—1661no—
475C22×C4—1662no—
476C22×C4—1661no—
477C22×C4—1661no—
478C22×C4—1661no—
479C22×C4—1662no—
480C22×C4—1662no—
481C22×C4—1662no—
482C22×C4—1661no—
483C22×C4—1662no—
484C22×C4—1661no—
485C22×C4—1661no—
486C22×C4—1662no—
487C22×C4—1661no—
488C22×C4C2×C4×D8312yes—
489C22×C4C2×C4×D8312yes—
490C22×C4C2×C22⋊D8314yes—
491C22×C4C2×C22⋊D8312yes—
492C22×C4#1305058914yes—
493C22×C4#1305059012yes—
494C4⋊C4—48no—
495C22⋊C4—48no—
496C22⋊C4—3328no—
497C22⋊C4—1668no—
498C22⋊C4—1668no—
499C42—3324no—
500C42—16616no—
501C42—1668no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—22no—
506C23—24no—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—22no—
512C23—22no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—42no—
517C23—42no—
518C23—42no—
519C23—42no—
520C23—42no—
521C23—42no—
522C23—42no—
523C23—42no—
524C23#1458175014yes—
525C23C22×C4×D8311yescharacteristic
526C23C22×C22⋊D8311yescharacteristic
527C23#41×D8312yes—
528C23#41×D8312yes—
529C23C2×#1305058912yes—
530C23C2×#1305058911yescharacteristic
531C23C2×#1305059011yescharacteristic
532C23C2×#1305059111yescharacteristic
533C23#1458183712yes—
534C23#1458183711yescharacteristic
535C23#1458184011yescharacteristic
536C2×C4—24no—
537C2×C4—24no—
538C2×C4—24no—
539C2×C4—24no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—24no—
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—416no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—42no—
558C2×C4—42no—
559C2×C4—3322no—
560C2×C4—3322no—
561C2×C4—3322no—
562C2×C4—3321no—
563C2×C4—3322no—
564C2×C4—3321no—
565C2×C4—3322no—
566C2×C4—3321no—
567C2×C4—3322no—
568C2×C4—3321no—
569C2×C4—3322no—
570C2×C4—3322no—
571C2×C4—3322no—
572C2×C4—3322no—
573C2×C4—3322no—
574C2×C4—3322no—
575C2×C4—3322no—
576C2×C4—3322no—
577C2×C4—3322no—
578C2×C4—834no—
579C2×C4—1664no—
580C2×C4—1664no—
581C2×C4—1662no—
582C2×C4—1662no—
583C2×C4—1662no—
584C2×C4—1662no—
585C2×C4—1662no—
586C2×C4—1662no—
587C2×C4—1662no—
588C2×C4—1662no—
589C2×C4—1662no—
590C2×C4—1664no—
591C2×C4—1662no—
592C2×C4—1662no—
593C2×C4—1664no—
594C2×C4—1664no—
595C2×C4#1458174814yes—
596C2×C4C4×C22⋊D8314yes—
597C22—24no—
598C22—24no—
599C22—24no—
600C22—42no—
601C22—42no—
602C22—44no—
603C22—44no—
604C22—42no—
605C22—44no—
606C22—44no—
607C22—42no—
608C22#117×D8311yescharacteristic
609C22C2×#1458174811yescharacteristic
610C22C2×#1458175011yescharacteristic
611C22C2×C4×C22⋊D8311yescharacteristic
612C22#1912816911yescharacteristic
613C22#1912817411yescharacteristic
614C22#1912818211yescharacteristic
615C22#1912820812yes—
616C22#1912822912yes—
617C22#1912823212yes—
618C22#1912823712yes—
619C22#1912830112yes—
620C22#1912832112yes—
621C22#1912834511yescharacteristic
622C22#1912834512yes—
623C22#1912836211yescharacteristic
624C22#1912837712yes—
625C22#1912837812yes—
626C22#1912838512yes—
627C22#1912838611yescharacteristic
628C22#1912839412yes—
629C22#1912839911yescharacteristic
630C22#1912841512yes—
631C4—24no—
632C4—24no—
633C4—44no—
634C4—82no—
635C4—3322no—
636C4—3324no—
637C4—3322no—
638C4—3322no—
639C4—6642no—
640C4—1664no—
641C2—44no—
642C2#2490985811yescharacteristic
643C2#2490990011yescharacteristic
644C2#2491010511yescharacteristic
645C2#2491030811yescharacteristic
646C2#2491035011yescharacteristic
647C2#2491038811yescharacteristic
648C2#2491040511yescharacteristic
649C2#2491164512yes—
650C2#2491166012yes—
651C2#2491229112yes—
652C2#2491230612yes—
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC2×#380×C83C2
socle, cosocleC24×C83C24
prime radicalC25×C83C23
hypercenterC2×#380D83
hypocenterC83#13898
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C2×#19127546C2semidirect
3. #24905722C2semidirect
6. #24905695C2semidirect
7. #24905689C2semidirect
8. #24905688C2semidirect
9. #24905687C2semidirect
18. C4×C22⋊Dic83C4semidirect
338. C83#13898semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊Dic832
C2×#191275462
#249057222
#249057102
#249057002
#249056952
#249056892
#249056882
#249056872
#249056862
C2×#380×C832
#1389883