Finite Group Databasebeta

← #35405132

Subgroups — #35405132

Order 21248.

1590 conjugacy classes of subgroups.

The subgroup lattice has 618 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#117×D83C211yesmaximal, characteristic
2C2×#19127739C211yesmaximal, characteristic, complemented
3C2×#19127737C211yesmaximal, characteristic, complemented
4C22×(C2×D83)⋊C8C211yesmaximal, characteristic
5#24909842C211yesmaximal, characteristic
6#24907665C212yesmaximal, complemented
7#24907663C212yesmaximal, complemented
8#24907657C212yesmaximal, complemented
9#24907652C212yesmaximal, complemented
10#24907179C211yesmaximal, characteristic, complemented
11C83×#1735C211yesmaximal, characteristic, complemented
12C23×C4×D83C2211yescharacteristic
13C23×C83⋊C8C2211yescharacteristic
14C2×#14581751C2211yescharacteristic
15C2×#14581751C2211yescharacteristic
16C2×#14581750C2212yes—
17C2×#14581749—21no—
18C2×#14581748—21no—
19#117×D83—24no—
20#117×D83—24no—
21C2×C4⋊C4×D83C2212yes—
22C2×C22⋊C4×D83—21no—
23C2×C42×D83—21no—
24#19127739—24no—
25#19127737—24no—
26C2×#14581726C2212yescomplemented
27C2×#14581725C2212yescomplemented
28C2×#14581724C2212yescomplemented
29C2×#14581723C2212yescomplemented
30C2×(C2×D83)⋊C8C418yescomplemented
31C2×(C2×D83)⋊C8C2211yescharacteristic
32C2×(C2×D83)⋊C8C2211yescharacteristic
33C2×(C2×D83)⋊C8C2211yescharacteristic
34C2×(C2×D83)⋊C8C2211yescharacteristic
35#19127562—21no—
36#19127558C2212yes—
37#19127558C2212yes—
38#19127556—21no—
39C4×(C2×D83)⋊C4—21no—
40C4×(C2×D83)⋊C4—21no—
41#19127463C2212yescomplemented
42#19127463C2212yescomplemented
43C23×C8×C83C2211yescharacteristic
44C2×#117×C83C2211yescharacteristic
45C2×#117×C83C2211yescharacteristic
46#340×C83C2212yescomplemented
47#340×C83C2212yescomplemented
48C24×D83D411yescharacteristic
49C23×Dic83D411yescharacteristic
50C22×C4×D83—22no—
51C22×C4×D83—22no—
52C22×C4×D83—22no—
53C22×C4×D83—22no—
54C22×C4×D83—22no—
55C22×C4×D83C2×C414yes—
56C22×C4×D83D411yescharacteristic
57C22×C4×D83D411yescharacteristic
58C22×C4×D83Q814yes—
59C22×C4×D83C2311yescharacteristic
60C22×C4×D83C2311yescharacteristic
61C22×C83⋊C8—22no—
62C22×C83⋊C8—22no—
63C22×C83⋊C8C2×C414yescomplemented
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66#14581751—22no—
67#14581751—22no—
68#14581751—22no—
69#14581751D414yes—
70#14581750—24no—
71#14581750—24no—
72#14581749—28no—
73#14581748—28no—
74C2×C4⋊Dic83C2312yes—
75C2×C4⋊Dic83C2312yes—
76C2×Dic83⋊C4—22no—
77C4⋊C4×D83—24no—
78C4⋊C4×D83—24no—
79C2×C22⋊Dic83—21no—
80C2×C22⋊Dic83—21no—
81C2×(C2×D83)⋊C4—22no—
82C2×(C2×D83)⋊C4—22no—
83C2×(C2×D83)⋊C4—22no—
84C22⋊C4×D83—28no—
85C2×C4×Dic83—22no—
86C2×C4×Dic83—21no—
87C2×C4×Dic83—21no—
88C42×D83—28no—
89#14581726—24no—
90#14581725—24no—
91#14581724—24no—
92#14581723—24no—
93(C2×D83)⋊C8C2×C414yes—
94(C2×D83)⋊C8C2×C414yes—
95(C2×D83)⋊C8C2×C414yescomplemented
96(C2×D83)⋊C8C2×C414yescomplemented
97C83⋊#95—22no—
98C83⋊#95—22no—
99C23×C4×C83C2311yescharacteristic
100C22×C8×C83—22no—
101C22×C8×C83—22no—
102C22×C8×C83C2×C414yescomplemented
103C22×C8×C83C2311yescharacteristic
104C22×C8×C83C2311yescharacteristic
105#117×C83—22no—
106#117×C83—22no—
107#117×C83—22no—
108#117×C83D414yes—
109C2×C4⋊C4×C83C2312yes—
110C2×C4⋊C4×C83C2312yes—
111C2×C22⋊C4×C83—21no—
112C2×C22⋊C4×C83—21no—
113C2×C42×C83—21no—
114C2×C42×C83—21no—
115C23×D83—22no—
116C23×D83—22no—
117C23×D83—24no—
118C23×D83—42no—
119C23×D83C4⋊C414yes—
120C23×D83C2×D411yescharacteristic
121C23×D83C2×D411yescharacteristic
122C22×Dic83—22no—
123C22×Dic83—22no—
124C22×Dic83—22no—
125C22×Dic83—22no—
126C22×Dic83C4⋊C414yes—
127C22×Dic83C2×D411yescharacteristic
128C22×Dic83C2×D411yescharacteristic
129C2×C4×D83—28no—
130C2×C4×D83—28no—
131C2×C4×D83—22no—
132C2×C4×D83—22no—
133C2×C4×D83—28no—
134C2×C4×D83—28no—
135C2×C4×D83—22no—
136C2×C4×D83—22no—
137C2×C4×D83—21no—
138C2×C4×D83—21no—
139C2×C4×D83—21no—
140C2×C4×D83—21no—
141C2×C4×D83—21no—
142C2×C4×D83—21no—
143C2×C4×D83—21no—
144C2×C4×D83—21no—
145C2×C4×D83—28no—
146C2×C4×D83—44no—
147C2×C4×D83—44no—
148C2×C4×D83C4⋊C414yes—
149C2×C4×D83C4⋊C414yes—
150C2×C4×D83C22×C412yes—
151C2×C4×D83C22×C412yes—
152C2×C4×D83C2×Q812yes—
153C2×C4×D83C2×Q812yes—
154C2×C83⋊C8—28no—
155C2×C83⋊C8—21no—
156C2×C83⋊C8—21no—
157C2×C83⋊C8—21no—
158C2×C83⋊C8—21no—
159C2×C83⋊C8C22×C412yes—
160C2×C83⋊C8C22×C412yes—
161C4⋊Dic83—22no—
162C4⋊Dic83—22no—
163C4⋊Dic83—22no—
164C4⋊Dic83C2×D414yes—
165Dic83⋊C4—28no—
166C22⋊Dic83—24no—
167C22⋊Dic83—42no—
168(C2×D83)⋊C4—28no—
169(C2×D83)⋊C4—44no—
170(C2×D83)⋊C4—44no—
171C4×Dic83—28no—
172C4×Dic83—24no—
173C4×Dic83—42no—
174C24×C83C2×D411yescharacteristic
175C22×C4×C83—22no—
176C22×C4×C83—22no—
177C22×C4×C83C22×C412yes—
178C22×C4×C83C2×D412yes—
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D412yes—
181C22×C4×C83C2×Q812yes—
182C22×C4×C83#4112yes—
183C22×C4×C83#4112yes—
184C22×C4×C83C2411yescharacteristic
185C2×C8×C83—28no—
186C2×C8×C83—21no—
187C2×C8×C83—21no—
188C2×C8×C83—21no—
189C2×C8×C83—21no—
190C2×C8×C83C22×C412yes—
191C2×C8×C83C22×C412yes—
192C4⋊C4×C83—22no—
193C4⋊C4×C83—22no—
194C4⋊C4×C83—22no—
195C4⋊C4×C83C2×D414yes—
196C22⋊C4×C83—24no—
197C22⋊C4×C83—42no—
198C42×C83—24no—
199C42×C83—42no—
200C23×C83—22no—
201C23×C83—22no—
202C23×C83C2×C4⋊C412yes—
203C23×C83#12012yes—
204C23×C83#12312yes—
205C23×C83C22×D411yescharacteristic
206C22×D83—21no—
207C22×D83—21no—
208C22×D83—22no—
209C22×D83—21no—
210C22×D83—21no—
211C22×D83—22no—
212C22×D83—22no—
213C22×D83—28no—
214C22×D83—28no—
215C22×D83—44no—
216C22×D83—48no—
217C22×D83—48no—
218C22×D83C2×C4⋊C412yes—
219C22×D83C2×C4⋊C412yes—
220C22×D83#13514yes—
221C2×C4×C83—24no—
222C2×C4×C83—42no—
223C2×C4×C83—42no—
224C2×C4×C83—42no—
225C2×C4×C83—42no—
226C2×C4×C83—42no—
227C2×C4×C83C2×C4⋊C412yes—
228C2×C4×C83C4×D414yes—
229C2×C4×C83C4×D414yes—
230C2×C4×C83#12012yes—
231C2×C4×C83C22⋊Q814yes—
232C2×C4×C83C22⋊Q814yes—
233C2×C4×C83#12312yes—
234C2×C4×C83C23×C411yescharacteristic
235C2×C4×C83C22×D411yescharacteristic
236C2×C4×C83C22×D411yescharacteristic
237C2×C4×C83C22×Q811yescharacteristic
238C2×C4×C83C2×#4111yescharacteristic
239C2×C4×C83C2×#4111yescharacteristic
240C2×Dic83—21no—
241C2×Dic83—21no—
242C2×Dic83—21no—
243C2×Dic83—21no—
244C2×Dic83—28no—
245C2×Dic83—24no—
246C2×Dic83—42no—
247C2×Dic83—42no—
248C2×Dic83—42no—
249C2×Dic83—42no—
250C2×Dic83—42no—
251C2×Dic83C2×C4⋊C412yes—
252C2×Dic83C2×C4⋊C412yes—
253C4×D83—22no—
254C4×D83—22no—
255C4×D83—22no—
256C4×D83—22no—
257C4×D83—22no—
258C4×D83—22no—
259C4×D83—22no—
260C4×D83—22no—
261C4×D83—28no—
262C4×D83—28no—
263C4×D83—416no—
264C8×C83—22no—
265C8×C83—22no—
266C83⋊C8—22no—
267C83⋊C8—22no—
268C22×C83—24no—
269C22×C83—22no—
270C22×C83—22no—
271C22×C83—24no—
272C22×C83#38514yes—
273C22×C83C22×C4⋊C411yescharacteristic
274C22×C83C2×#12011yescharacteristic
275C22×C83C2×#12311yescharacteristic
276C22×C83C2×#13512yes—
277C22×C83#57512yes—
278C2×D83—24no—
279C2×D83—22no—
280C2×D83—24no—
281C2×D83—22no—
282C2×D83—22no—
283C2×D83—22no—
284C2×D83—22no—
285C2×D83—416no—
286C2×D83—42no—
287C2×D83—42no—
288C2×D83#42714yescomplemented
289C4×C83—44no—
290C4×C83—82no—
291C4×C83#38514yes—
292C4×C83C2×C4×D411yescharacteristic
293C4×C83C2×C4×D411yescharacteristic
294C4×C83C2×C22⋊Q811yescharacteristic
295C4×C83C2×C22⋊Q811yescharacteristic
296Dic83—22no—
297Dic83—22no—
298Dic83—44no—
299Dic83—82no—
300#20843—831nomaximal, Sylow
301C2×C83—24no—
302C2×C83C2×#38511yescharacteristic
303C2×C83C2×#42711yescharacteristic
304C2×C83#291111yescharacteristic
305C2×C83#300212yes—
306C2×C83#309112yes—
307D83—24no—
308D83—42no—
309C22×#117—831no—
310C2×#427—831no—
311C2×#427—831no—
312C22×C22⋊C8—831no—
313#2280—831no—
314#1854—832no—
315#1854—832no—
316#1853—832no—
317#1852—832no—
318#1735—831no—
319#1735D8311yescharacteristic, complemented
320C83#2084311yescharacteristic, complemented, Sylow
321C24×C4—831no—
322C23×C8—831no—
323C23×C8C2×D8311yescharacteristic
324C2×#117—831no—
325C2×#117—831no—
326C2×#117—832no—
327C2×#117—1661no—
328C2×#117—1661no—
329C2×#117—1664no—
330C2×#117—1664no—
331C2×#117C2×D8311yescharacteristic
332C2×#117C2×D8311yescharacteristic
333C22×C4⋊C4—832no—
334C22×C22⋊C4—1661no—
335C22×C42—1661no—
336#427—1664no—
337#427—1664no—
338C2×#107—832no—
339C2×#107—832no—
340C2×#106—832no—
341C2×#106—832no—
342C2×C22⋊C8—838no—
343C2×C22⋊C8—831no—
344C2×C22⋊C8—831no—
345C2×C22⋊C8—831no—
346C2×C22⋊C8—831no—
347#381—1661no—
348#380—832no—
349#380—832no—
350#379—1661no—
351C4×C22⋊C4—1661no—
352C4×C22⋊C4—1661no—
353#340—832no—
354#340—832no—
355#340C2×D8312yescomplemented
356#340C2×D8312yescomplemented
357C25—831no—
358C23×C4—831no—
359C23×C4—834no—
360C23×C4—831no—
361C23×C4—831no—
362C23×C4—834no—
363C23×C4—831no—
364C23×C4—831no—
365C23×C4—1662no—
366C23×C4—1662no—
367C23×C4—1662no—
368C23×C4—1662no—
369C23×C4—1662no—
370C23×C4C22×D8311yescharacteristic
371C22×C8—22no—
372C22×C8—22no—
373C22×C8—834no—
374C22×C8—831no—
375C22×C8—831no—
376C22×C8—1662no—
377C22×C8—1662no—
378C22×C8C4×D8314yescomplemented
379C22×C8C22×D8311yescharacteristic
380C22×C8C22×D8311yescharacteristic
381#117—22no—
382#117—22no—
383#117—22no—
384#117—834no—
385#117—1668no—
386#117—1662no—
387#117—1662no—
388#117—1668no—
389#117—1664no—
390#117—1662no—
391#117—1664no—
392#117D33214yes—
393C2×C4⋊C4—832no—
394C2×C4⋊C4—832no—
395C2×C4⋊C4—1664no—
396C2×C4⋊C4—1662no—
397C2×C4⋊C4—1664no—
398C2×C4⋊C4C22×D8312yes—
399C2×C4⋊C4C22×D8312yes—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—21no—
402C2×C22⋊C4—1668no—
403C2×C22⋊C4—1662no—
404C2×C22⋊C4—1662no—
405C2×C22⋊C4—1661no—
406C2×C22⋊C4—1662no—
407C2×C22⋊C4—1661no—
408C2×C42—21no—
409C2×C42—21no—
410C2×C42—1668no—
411C2×C42—1661no—
412C2×C42—1662no—
413C2×C42—1661no—
414#107—1664no—
415#107—1664no—
416#106—1664no—
417#106—1664no—
418C22⋊C8—834no—
419C22⋊C8—834no—
420C22⋊C8—834no—
421C22⋊C8—834no—
422#95—1662no—
423#95—1662no—
424C24—3322no—
425C24—834no—
426C24—831no—
427C24—831no—
428C24—1664no—
429C24—1662no—
430C24—1662no—
431C24D4×D8311yescharacteristic
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—3324no—
435C22×C4—3324no—
436C22×C4—831no—
437C22×C4—831no—
438C22×C4—834no—
439C22×C4—832no—
440C22×C4—832no—
441C22×C4—832no—
442C22×C4—832no—
443C22×C4—834no—
444C22×C4—834no—
445C22×C4—1668no—
446C22×C4—1662no—
447C22×C4—1662no—
448C22×C4—1662no—
449C22×C4—1662no—
450C22×C4—1662no—
451C22×C4—1662no—
452C22×C4—1668no—
453C22×C4—1668no—
454C22×C4—1662no—
455C22×C4—1662no—
456C22×C4—1661no—
457C22×C4—1661no—
458C22×C4—1661no—
459C22×C4—1661no—
460C22×C4—1661no—
461C22×C4—1661no—
462C22×C4—1661no—
463C22×C4—1661no—
464C22×C4—1668no—
465C22×C4—1668no—
466C22×C4C2×C4×D8312yes—
467C22×C4D4×D8312yes—
468C22×C4D4×D8311yescharacteristic
469C22×C4C2×D33212yes—
470C22×C4Q8×D8312yes—
471C22×C4#1305058912yes—
472C22×C4#1305059012yes—
473C22×C4C23×D8311yescharacteristic
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—28no—
479C2×C8—832no—
480C2×C8—832no—
481C2×C8—1661no—
482C2×C8—1661no—
483C2×C8—1661no—
484C2×C8—1661no—
485C2×C8—1668no—
486C2×C8C2×C4×D8312yes—
487C2×C8C2×C4×D8312yes—
488C4⋊C4—22no—
489C4⋊C4—22no—
490C4⋊C4—22no—
491C4⋊C4—834no—
492C4⋊C4—1668no—
493C4⋊C4—1662no—
494C4⋊C4—1662no—
495C4⋊C4—1662no—
496C4⋊C4C2×D33214yes—
497C22⋊C4—24no—
498C22⋊C4—42no—
499C22⋊C4—3324no—
500C22⋊C4—3324no—
501C22⋊C4—3322no—
502C22⋊C4—1668no—
503C22⋊C4—1664no—
504C42—24no—
505C42—42no—
506C42—3322no—
507C42—1668no—
508C42—1664no—
509C23—22no—
510C23—22no—
511C23—3328no—
512C23—3328no—
513C23—3324no—
514C23—832no—
515C23—834no—
516C23—832no—
517C23—1668no—
518C23—1668no—
519C23—1661no—
520C23—1661no—
521C23—1662no—
522C23—1662no—
523C23—1661no—
524C23—1661no—
525C23—1662no—
526C23C4⋊C4×D8312yes—
527C23#1458176212yes—
528C23#1458177412yes—
529C23C2×D4×D8311yescharacteristic
530C2×C4—24no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—33216no—
537C2×C4—3322no—
538C2×C4—3322no—
539C2×C4—3322no—
540C2×C4—3322no—
541C2×C4—3322no—
542C2×C4—832no—
543C2×C4—832no—
544C2×C4—1664no—
545C2×C4—1668no—
546C2×C4—1661no—
547C2×C4—1661no—
548C2×C4—1661no—
549C2×C4—1661no—
550C2×C4—1668no—
551C2×C4—1668no—
552C2×C4—1662no—
553C2×C4—1662no—
554C2×C4—1662no—
555C2×C4—1662no—
556C2×C4—1662no—
557C2×C4—1662no—
558C2×C4—1662no—
559C2×C4—1662no—
560C2×C4C4⋊C4×D8312yes—
561C2×C4(C22⋊D83)⋊C414yes—
562C2×C4C4×D33214yes—
563C2×C4#1458176212yes—
564C2×C4#1458177014yes—
565C2×C4#1458177214yes—
566C2×C4#1458177412yes—
567C2×C4C22×C4×D8311yescharacteristic
568C2×C4C2×D4×D8311yescharacteristic
569C2×C4C22×D33211yescharacteristic
570C2×C4C2×Q8×D8311yescharacteristic
571C2×C4C2×#1305058911yescharacteristic
572C2×C4C2×#1305059011yescharacteristic
573C8—22no—
574C8—22no—
575C8—1662no—
576C8—1662no—
577C22—22no—
578C22—22no—
579C22—24no—
580C22—24no—
581C22—33216no—
582C22—3322no—
583C22—3322no—
584C22—834no—
585C22—1662no—
586C22—1662no—
587C22—1662no—
588C22—1662no—
589C22—1662no—
590C22—1664no—
591C22—1664no—
592C22#1912757514yes—
593C22C2×C4⋊C4×D8311yescharacteristic
594C22C2×#1458176211yescharacteristic
595C22C2×#1458177411yescharacteristic
596C22#135×D8312yes—
597C22#1912849812yes—
598C4—44no—
599C4—82no—
600C4—3324no—
601C4—6642no—
602C4—1662no—
603C4—1662no—
604C4#1912757514yes—
605C4C2×(C22⋊D83)⋊C411yescharacteristic
606C4C2×C4×D33211yescharacteristic
607C4C2×#1458177011yescharacteristic
608C4C2×#1458177211yescharacteristic
609C2—24no—
610C2—3322no—
611C2—1664no—
612C2C2×#1912757511yescharacteristic
613C2#427×D8311yescharacteristic
614C2#2491605111yescharacteristic
615C2#2491668312yes—
616C2#2491737412yes—
subgroupG/H
centerC22×C4D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1735C2
socleC23×C83C22×D4
prime radicalC24×D83D4
cosocleC22×C4×C83C24
hypercenter#1735D83
hypocenterC83#20843
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ #1735

Complements

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

NH ≅ G/Ntype
2. C2×#19127739C2semidirect
3. C2×#19127737C2semidirect
6. #24907665C2semidirect
7. #24907663C2semidirect
8. #24907657C2semidirect
9. #24907652C2semidirect
10. #24907179C2semidirect
11. C83×#1735C2semidirect
26. C2×#14581726C22semidirect
27. C2×#14581725C22semidirect
28. C2×#14581724C22semidirect
29. C2×#14581723C22semidirect
30. C2×(C2×D83)⋊C8C4semidirect
41. #19127463C22semidirect
42. #19127463C22semidirect
46. #340×C83C22semidirect
47. #340×C83C22semidirect
63. C22×C83⋊C8C2×C4semidirect
95. (C2×D83)⋊C8C2×C4semidirect
96. (C2×D83)⋊C8C2×C4semidirect
102. C22×C8×C83C2×C4semidirect
288. C2×D83#427semidirect
319. #1735D83semidirect
320. C83#20843semidirect
355. #340C2×D83semidirect
356. #340C2×D83semidirect
378. C22×C8C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#117×D832
C2×#191277392
C2×#191277372
C22×(C2×D83)⋊C82
#249098422
#249076652
#249076632
#249076572
#249076522
#249071792
C83×#17352
#2084383