Finite Group Databasebeta

← #35330629

Subgroups — #35330629

Order 21248.

1686 conjugacy classes of subgroups.

The subgroup lattice has 664 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127608C211yesmaximal, characteristic, complemented
2C2×C22⋊C4×Dic83C211yesmaximal, characteristic
3#24905734C211yesmaximal, characteristic, complemented
4#24905720C211yesmaximal, characteristic, complemented
5#24905708C211yesmaximal, characteristic
6#24905700C212yesmaximal
7#24905688C212yesmaximal, complemented
8#24905688C212yesmaximal, complemented
9#24905685C212yesmaximal
10#95×Dic83C211yesmaximal, characteristic, complemented
11C2×#393×C83C211yesmaximal, characteristic
12C24×Dic83C2211yescharacteristic
13C22×C4⋊Dic83C2211yescharacteristic
14C22×C22⋊Dic83C2212yes—
15C22×C22⋊Dic83C2211yescharacteristic
16C22×C4×Dic83C2211yescharacteristic
17C22×C4×Dic83C2212yes—
18C22×C4×Dic83C2211yescharacteristic
19#19127608—24no—
20C22⋊C4×Dic83—24no—
21C2×C83⋊#95C2211yescharacteristic
22C2×C83⋊#95C2211yescharacteristic
23C2×C83⋊#95C2212yes—
24C2×C83⋊#95C2212yes—
25C2×C83⋊#95C2212yes—
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2211yescharacteristic
28C2×C83⋊#95C2212yes—
29C2×C83⋊#95C2212yes—
30C2×C83⋊#95C2212yes—
31C2×C83⋊#95C2212yes—
32C23×Q8×C83C2211yescharacteristic
33C22×C22⋊C4×C83C2211yescharacteristic
34C22×C22⋊C4×C83C2212yes—
35#393×C83C418yescomplemented
36C2×#95×C83C2211yescharacteristic
37C2×#95×C83C2212yes—
38C23×Dic83—22no—
39C23×Dic83—22no—
40C23×Dic83—24no—
41C23×Dic83—22no—
42C23×Dic83—42no—
43C23×Dic83C2312yes—
44C23×Dic83C2311yescharacteristic
45C23×Dic83C2312yes—
46C23×Dic83C2311yescharacteristic
47C23×Dic83C2311yescharacteristic
48C23×Dic83C2311yescharacteristic
49C2×C4⋊Dic83—22no—
50C2×C4⋊Dic83—42no—
51C2×C22⋊Dic83—22no—
52C2×C22⋊Dic83—22no—
53C2×C22⋊Dic83—24no—
54C2×C22⋊Dic83—24no—
55C2×C22⋊Dic83—42no—
56C2×C22⋊Dic83Q814yescomplemented
57C2×C4×Dic83—22no—
58C2×C4×Dic83—24no—
59C2×C4×Dic83—24no—
60C2×C4×Dic83—24no—
61C2×C4×Dic83—42no—
62C2×C4×Dic83—42no—
63C2×C4×Dic83D414yes—
64C83⋊#95—24no—
65C83⋊#95—24no—
66C83⋊#95—22no—
67C83⋊#95—24no—
68C83⋊#95—24no—
69C83⋊#95—24no—
70C83⋊#95—22no—
71C83⋊#95—22no—
72C83⋊#95—42no—
73C83⋊#95—42no—
74C83⋊#95—42no—
75C25×C83C2311yescharacteristic
76C22×Q8×C83—22no—
77C22×Q8×C83—22no—
78C22×Q8×C83—42no—
79C22×Q8×C83—42no—
80C22×Q8×C83C2×C414yescomplemented
81C23×C4×C83C2311yescharacteristic
82C23×C4×C83C2312yes—
83C23×C4×C83C2312yes—
84C23×C4×C83C2311yescharacteristic
85C2×C22⋊C4×C83—22no—
86C2×C22⋊C4×C83—22no—
87C2×C22⋊C4×C83—42no—
88C2×C22⋊C4×C83—42no—
89C2×C22⋊C4×C83C2×C414yes—
90C2×C22⋊C4×C83C2×C418yes—
91#95×C83C2×C418yescomplemented
92#95×C83C2×C414yescomplemented
93C22×Dic83—21no—
94C22×Dic83—21no—
95C22×Dic83—22no—
96C22×Dic83—22no—
97C22×Dic83—22no—
98C22×Dic83—28no—
99C22×Dic83—21no—
100C22×Dic83—21no—
101C22×Dic83—22no—
102C22×Dic83—21no—
103C22×Dic83—22no—
104C22×Dic83—22no—
105C22×Dic83—22no—
106C22×Dic83—22no—
107C22×Dic83—21no—
108C22×Dic83—22no—
109C22×Dic83—22no—
110C22×Dic83—22no—
111C22×Dic83—22no—
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×Dic83—48no—
121C22×Dic83—48no—
122C22×Dic83—44no—
123C22×Dic83—44no—
124C22×Dic83C2×D412yes—
125C22×Dic83C2×D412yes—
126C22×Dic83C2×Q812yes—
127C22×Dic83C2×Q812yes—
128C22×Dic83#4112yes—
129C22×Dic83#4114yes—
130C22×Dic83#4112yes—
131C22×Dic83#4114yes—
132C4⋊Dic83—44no—
133C22⋊Dic83—28no—
134C22⋊Dic83—48no—
135C4×Dic83—44no—
136C4×Dic83—48no—
137C4×Dic83—44no—
138C24×C83—21no—
139C24×C83—21no—
140C24×C83—22no—
141C24×C83—21no—
142C24×C83—21no—
143C24×C83—22no—
144C24×C83—22no—
145C24×C83—22no—
146C24×C83C22×C412yes—
147C24×C83C2×Q812yes—
148C24×C83#4112yes—
149C24×C83C2411yescharacteristic
150C2×Q8×C83—28no—
151C2×Q8×C83—48no—
152C2×Q8×C83—44no—
153C2×Q8×C83—44no—
154C2×Q8×C83—48no—
155C22×C4×C83—22no—
156C22×C4×C83—22no—
157C22×C4×C83—21no—
158C22×C4×C83—22no—
159C22×C4×C83—21no—
160C22×C4×C83—22no—
161C22×C4×C83—22no—
162C22×C4×C83—22no—
163C22×C4×C83—22no—
164C22×C4×C83—22no—
165C22×C4×C83—22no—
166C22×C4×C83—22no—
167C22×C4×C83—22no—
168C22×C4×C83—22no—
169C22×C4×C83—22no—
170C22×C4×C83—22no—
171C22×C4×C83—22no—
172C22×C4×C83—22no—
173C22×C4×C83C22×C412yes—
174C22×C4×C83C22×C414yes—
175C22×C4×C83C22×C414yes—
176C22×C4×C83C22×C412yes—
177C22×C4×C83C2×D412yes—
178C22×C4×C83#4112yes—
179C22⋊C4×C83—28no—
180C22⋊C4×C83—48no—
181C23×C83—22no—
182C23×C83—24no—
183C23×C83—22no—
184C23×C83—24no—
185C23×C83—22no—
186C23×C83—22no—
187C23×C83—22no—
188C23×C83—22no—
189C23×C83—24no—
190C23×C83—24no—
191C23×C83—24no—
192C23×C83—24no—
193C23×C83—41no—
194C23×C83—42no—
195C23×C83—41no—
196C23×C83—41no—
197C23×C83—41no—
198C23×C83—42no—
199C23×C83—42no—
200C23×C83—42no—
201C23×C83—42no—
202C23×C83—42no—
203C23×C83C4×Q814yes—
204C23×C83C23×C411yescharacteristic
205C23×C83C22×D411yescharacteristic
206C23×C83C22×Q811yescharacteristic
207C23×C83C2×#4111yescharacteristic
208C23×C83C2×#4111yescharacteristic
209C23×C83C2×#4112yes—
210C23×C83C2×#4112yes—
211C23×C83C2×#4111yescharacteristic
212C23×C83C2×#4111yescharacteristic
213C23×C83H(5,𝔽2)12yes—
214C23×C83H(5,𝔽2)12yes—
215Q8×C83—416no—
216C2×C4×C83—24no—
217C2×C4×C83—22no—
218C2×C4×C83—22no—
219C2×C4×C83—24no—
220C2×C4×C83—22no—
221C2×C4×C83—24no—
222C2×C4×C83—24no—
223C2×C4×C83—24no—
224C2×C4×C83—24no—
225C2×C4×C83—24no—
226C2×C4×C83—24no—
227C2×C4×C83—24no—
228C2×C4×C83—41no—
229C2×C4×C83—42no—
230C2×C4×C83—41no—
231C2×C4×C83—42no—
232C2×C4×C83—42no—
233C2×C4×C83—42no—
234C2×C4×C83—42no—
235C2×C4×C83—42no—
236C2×C4×C83—42no—
237C2×C4×C83—42no—
238C2×C4×C83—42no—
239C2×C4×C83C4×D414yes—
240C2×Dic83—22no—
241C2×Dic83—22no—
242C2×Dic83—22no—
243C2×Dic83—22no—
244C2×Dic83—24no—
245C2×Dic83—24no—
246C2×Dic83—24no—
247C2×Dic83—24no—
248C2×Dic83—22no—
249C2×Dic83—22no—
250C2×Dic83—24no—
251C2×Dic83—24no—
252C2×Dic83—24no—
253C2×Dic83—24no—
254C2×Dic83—24no—
255C2×Dic83—416no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—42no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C2×Dic83—42no—
262C2×Dic83—42no—
263C2×Dic83—42no—
264C2×Dic83—42no—
265C2×Dic83—42no—
266C2×Dic83—42no—
267C2×Dic83—42no—
268C2×Dic83—42no—
269C2×Dic83—42no—
270C2×Dic83—42no—
271C2×Dic83—42no—
272C2×Dic83—42no—
273C2×Dic83C22⋊Q814yes—
274C2×Dic83#12414yes—
275C22×C83—28no—
276C22×C83—24no—
277C22×C83—42no—
278C22×C83—42no—
279C22×C83—42no—
280C22×C83—44no—
281C22×C83—44no—
282C22×C83—42no—
283C22×C83—42no—
284C22×C83—44no—
285C22×C83—42no—
286C22×C83C2×C4×D411yescharacteristic
287C22×C83C2×C4×Q811yescharacteristic
288C22×C83C4×#4112yes—
289C22×C83C4×#4111yescharacteristic
290C22×C83#51812yes—
291C22×C83C2×C22⋊Q811yescharacteristic
292C22×C83#52511yescharacteristic
293C22×C83C2×#12411yescharacteristic
294C22×C83#52912yes—
295C22×C83#52911yescharacteristic
296C22×C83#53211yescharacteristic
297C22×C83#53311yescharacteristic
298C22×C83#53411yescharacteristic
299C22×C83#53511yescharacteristic
300C22×C83#53812yes—
301C22×C83#55112yes—
302C22×C83#55112yes—
303C22×C83#55112yes—
304C22×C83#55112yes—
305C22×C83#55412yes—
306C22×C83#55412yes—
307C22×C83#55512yes—
308C22×C83#55512yes—
309C4×C83—24no—
310C4×C83—44no—
311C4×C83—44no—
312C4×C83—42no—
313Dic83—24no—
314Dic83—24no—
315Dic83—42no—
316Dic83—42no—
317Dic83—44no—
318Dic83—82no—
319#14243—831nomaximal, Sylow
320C2×C83—44no—
321C2×C83C4×C22⋊Q811yescharacteristic
322C2×C83C4×#12411yescharacteristic
323C2×C83#232111yescharacteristic
324C2×C83#236812yes—
325C2×C83#237212yes—
326C2×C83#256111yescharacteristic
327C2×C83#256311yescharacteristic
328C2×C83#256411yescharacteristic
329C2×C83#257212yes—
330C2×C83#257412yes—
331C2×C83#261011yescharacteristic
332C2×#393D8311yescharacteristic
333C2×#391—831no—
334C2×C4×C22⋊C4—831no—
335#1446—831no—
336#1442—831no—
337#1439—831no—
338#1437—832no—
339#1434—832no—
340#1434—832no—
341C22⋊#95—832no—
342C4×#95—831no—
343C83#1424311yescharacteristic, complemented, Sylow
344C23×Q8C2×D8311yescharacteristic
345C24×C4—831no—
346C22×C4⋊C4—831no—
347C22×C22⋊C4—832no—
348C22×C22⋊C4—831no—
349C22×C22⋊C4C2×D8311yescharacteristic
350C22×C22⋊C4C2×D8312yes—
351C22×C42—831no—
352C22×C42—832no—
353C22×C42—831no—
354#393Dic8318yescomplemented
355#391—1664no—
356C4×C22⋊C4—1664no—
357C2×#95—832no—
358C2×#95—832no—
359C2×#95—832no—
360C2×#95—832no—
361C2×#95—831no—
362C2×#95—832no—
363C2×#95—832no—
364C2×#95—832no—
365C2×#95—831no—
366C2×#95—832no—
367C2×#95—831no—
368C2×#95C2×D8311yescharacteristic
369C2×#95C2×D8312yes—
370C25C22×D8311yescharacteristic
371C22×Q8—22no—
372C22×Q8—22no—
373C22×Q8—42no—
374C22×Q8—42no—
375C22×Q8C2×Dic8314yescomplemented
376C23×C4—3322no—
377C23×C4—831no—
378C23×C4—831no—
379C23×C4—831no—
380C23×C4—832no—
381C23×C4—831no—
382C23×C4—832no—
383C23×C4—1664no—
384C23×C4—1662no—
385C23×C4—1662no—
386C23×C4—1662no—
387C23×C4C22×D8312yes—
388C23×C4C22×D8311yescharacteristic
389C23×C4C22×D8312yes—
390C23×C4C22×D8311yescharacteristic
391C2×C4⋊C4—3322no—
392C2×C4⋊C4—1662no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—22no—
395C2×C22⋊C4—42no—
396C2×C22⋊C4—42no—
397C2×C22⋊C4—3322no—
398C2×C22⋊C4—834no—
399C2×C22⋊C4—1664no—
400C2×C22⋊C4—1664no—
401C2×C22⋊C4—1662no—
402C2×C22⋊C4—1662no—
403C2×C22⋊C4C2×Dic8314yes—
404C2×C22⋊C4C2×Dic8318yes—
405C2×C42—3322no—
406C2×C42—3322no—
407C2×C42—834no—
408C2×C42—1662no—
409C2×C42—1664no—
410C2×C42—1664no—
411C2×C42—1664no—
412#95—3322no—
413#95—3322no—
414#95—3322no—
415#95—1662no—
416#95—1664no—
417#95—1664no—
418#95—1664no—
419#95—1664no—
420#95—1664no—
421#95—1662no—
422#95—1662no—
423#95C2×Dic8318yescomplemented
424#95C2×Dic8314yescomplemented
425C24—21no—
426C24—22no—
427C24—21no—
428C24—22no—
429C24—21no—
430C24—21no—
431C24—22no—
432C24—22no—
433C24C22×Dic8312yes—
434C24Q8×D8312yes—
435C24#1305059112yes—
436C24C23×D8311yescharacteristic
437C2×Q8—28no—
438C2×Q8—48no—
439C2×Q8—44no—
440C2×Q8—44no—
441C2×Q8—48no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—21no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—21no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—3324no—
461C22×C4—3328no—
462C22×C4—3324no—
463C22×C4—3328no—
464C22×C4—832no—
465C22×C4—832no—
466C22×C4—832no—
467C22×C4—834no—
468C22×C4—832no—
469C22×C4—832no—
470C22×C4—832no—
471C22×C4—834no—
472C22×C4—1662no—
473C22×C4—1662no—
474C22×C4—1662no—
475C22×C4—1662no—
476C22×C4—1662no—
477C22×C4—1662no—
478C22×C4—1662no—
479C22×C4—1668no—
480C22×C4—1661no—
481C22×C4—1661no—
482C22×C4—1662no—
483C22×C4—1662no—
484C22×C4—1662no—
485C22×C4—1661no—
486C22×C4—1662no—
487C22×C4—1662no—
488C22×C4—1662no—
489C22×C4—1661no—
490C22×C4—1662no—
491C22×C4—1662no—
492C22×C4—1662no—
493C22×C4—1662no—
494C22×C4—1662no—
495C22×C4—1661no—
496C22×C4—1661no—
497C22×C4—1662no—
498C22×C4—1662no—
499C22×C4C22×Dic8312yes—
500C22×C4C22×Dic8314yes—
501C22×C4C22×Dic8314yes—
502C22×C4C22×Dic8312yes—
503C22×C4D4×D8312yes—
504C22×C4#1305058912yes—
505C4⋊C4—3324no—
506C22⋊C4—28no—
507C22⋊C4—48no—
508C22⋊C4—3328no—
509C22⋊C4—1668no—
510C42—3324no—
511C42—3324no—
512C42—3328no—
513C23—24no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—22no—
519C23—24no—
520C23—24no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23—42no—
531C23—41no—
532C23—41no—
533C23—41no—
534C23—42no—
535C23Q8×Dic8314yes—
536C23C23×Dic8311yescharacteristic
537C23C2×D4×D8311yescharacteristic
538C23C2×Q8×D8311yescharacteristic
539C23#41×D8312yes—
540C23#41×D8312yes—
541C23#41×D8311yescharacteristic
542C23#41×D8311yescharacteristic
543C23C2×#1305058911yescharacteristic
544C23C2×#1305059111yescharacteristic
545C23#1458183712yes—
546C23#1458183712yes—
547Q8—416no—
548C2×C4—24no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—22no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—24no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—42no—
566C2×C4—41no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—3322no—
572C2×C4—3322no—
573C2×C4—3322no—
574C2×C4—3322no—
575C2×C4—3321no—
576C2×C4—3321no—
577C2×C4—3322no—
578C2×C4—3321no—
579C2×C4—3321no—
580C2×C4—3322no—
581C2×C4—33216no—
582C2×C4—3322no—
583C2×C4—3322no—
584C2×C4—3322no—
585C2×C4—3322no—
586C2×C4—3322no—
587C2×C4—3322no—
588C2×C4—3322no—
589C2×C4—834no—
590C2×C4—834no—
591C2×C4—1664no—
592C2×C4—1664no—
593C2×C4—1664no—
594C2×C4—1664no—
595C2×C4—1662no—
596C2×C4—1662no—
597C2×C4—1664no—
598C2×C4—1664no—
599C2×C4—1664no—
600C2×C4—1664no—
601C2×C4—1664no—
602C2×C4—1662no—
603C2×C4—1662no—
604C2×C4—1662no—
605C2×C4—1662no—
606C2×C4D4×Dic8314yes—
607C22—24no—
608C22—28no—
609C22—42no—
610C22—42no—
611C22—44no—
612C22—44no—
613C22—42no—
614C22—42no—
615C22—44no—
616C22—42no—
617C22—42no—
618C22C2×D4×Dic8311yescharacteristic
619C22C2×Q8×Dic8311yescharacteristic
620C22#41×Dic8311yescharacteristic
621C22#41×Dic8312yes—
622C22#1912817612yes—
623C22C22⋊Q8×D8311yescharacteristic
624C22#1912821411yescharacteristic
625C22#124×D8311yescharacteristic
626C22#1912823012yes—
627C22#1912823711yescharacteristic
628C22#1912825011yescharacteristic
629C22#1912825311yescharacteristic
630C22#1912826111yescharacteristic
631C22#1912826611yescharacteristic
632C22#1912828612yes—
633C22#1912837812yes—
634C22#1912837812yes—
635C22#1912838112yes—
636C22#1912838112yes—
637C22#1912840512yes—
638C22#1912840512yes—
639C22#1912841312yes—
640C22#1912841312yes—
641C4—24no—
642C4—44no—
643C4—42no—
644C4—44no—
645C4—3324no—
646C4—3322no—
647C4—3322no—
648C4—6642no—
649C4—1664no—
650C4—1664no—
651C2—44no—
652C2C22⋊Q8×Dic8311yescharacteristic
653C2#124×Dic8311yescharacteristic
654C2#2491012411yescharacteristic
655C2#2491054812yes—
656C2#2491057512yes—
657C2#2491261711yescharacteristic
658C2#2491263111yescharacteristic
659C2#2491264011yescharacteristic
660C2#2491270812yes—
661C2#2491273812yes—
662C2#2491315111yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC2×#393×C83C2
socle, cosocleC24×C83C24
prime radicalC25×C83C23
hypercenterC2×#393D83
hypocenterC83#14243
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#393

Complements

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

NH ≅ G/Ntype
1. C2×#19127608C2semidirect
3. #24905734C2semidirect
4. #24905720C2semidirect
7. #24905688C2semidirect
8. #24905688C2semidirect
10. #95×Dic83C2semidirect
35. #393×C83C4semidirect
56. C2×C22⋊Dic83Q8semidirect
80. C22×Q8×C83C2×C4semidirect
91. #95×C83C2×C4semidirect
92. #95×C83C2×C4semidirect
343. C83#14243semidirect
354. #393Dic83semidirect
375. C22×Q8C2×Dic83semidirect
423. #95C2×Dic83semidirect
424. #95C2×Dic83semidirect

Maximal subgroups

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

subgroupindex
C2×#191276082
C2×C22⋊C4×Dic832
#249057342
#249057202
#249057082
#249057002
#249056882
#249056852
#95×Dic832
C2×#393×C832
#1424383