Finite Group Databasebeta

← #35326608

Subgroups — #35326608

Order 21248.

1818 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×#19127568C211yesmaximal, characteristic, complemented
2C2×C4×C22⋊Dic83C211yesmaximal, characteristic
3#24905724C212yesmaximal, complemented
4#24905723C212yesmaximal, complemented
5#24905689C211yesmaximal, characteristic, complemented
6#24905688C211yesmaximal, characteristic, complemented
7#24905687C211yesmaximal, characteristic, complemented
8#24905686C211yesmaximal, characteristic
9#24905686C212yesmaximal
10#24905686C212yesmaximal
11C22×#95×C83C211yesmaximal, characteristic
12C22×C22⋊Dic83C2212yes—
13C22×C22⋊Dic83C2211yescharacteristic
14C22×C22⋊Dic83C2211yescharacteristic
15C22×C4×Dic83C2211yescharacteristic
16C22×C4×Dic83C2211yescharacteristic
17#19127568—24no—
18C4×C22⋊Dic83C418yescomplemented
19C2×C83⋊#95C2211yescharacteristic
20C2×C83⋊#95C2212yes—
21C2×C83⋊#95C2212yes—
22C2×C83⋊#95C2211yescharacteristic
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×C83C2212yes—
33C24×C4×C83C2211yescharacteristic
34C2×#95×C83—24no—
35C2×#95×C83C2211yescharacteristic
36C2×#95×C83C2212yes—
37C2×#95×C83C2211yescharacteristic
38C23×Dic83C2312yes—
39C23×Dic83C2312yes—
40C23×Dic83C2311yescharacteristic
41C23×Dic83C2311yescharacteristic
42C23×Dic83C2311yescharacteristic
43C23×Dic83C2311yescharacteristic
44C2×C22⋊Dic83—22no—
45C2×C22⋊Dic83—24no—
46C2×C22⋊Dic83—22no—
47C2×C22⋊Dic83—24no—
48C2×C22⋊Dic83—24no—
49C2×C22⋊Dic83—24no—
50C2×C22⋊Dic83C2×C414yes—
51C2×C22⋊Dic83C2×C414yes—
52C2×C4×Dic83—22no—
53C2×C4×Dic83—22no—
54C2×C4×Dic83—42no—
55C2×C4×Dic83C2×C414yescomplemented
56C2×C4×Dic83C2×C414yescomplemented
57C83⋊#95—24no—
58C83⋊#95—24no—
59C83⋊#95—22no—
60C83⋊#95—24no—
61C83⋊#95—24no—
62C83⋊#95—22no—
63C83⋊#95—42no—
64C83⋊#95—42no—
65C83⋊#95—42no—
66C83⋊#95—42no—
67C83⋊#95—42no—
68C83⋊#95C2×C414yescomplemented
69C83⋊#95C2×C414yescomplemented
70C25×C83C2311yescharacteristic
71C23×C4×C83—24no—
72C23×C4×C83—24no—
73C23×C4×C83—24no—
74C23×C4×C83—22no—
75C23×C4×C83—24no—
76C23×C4×C83—22no—
77C23×C4×C83—24no—
78C23×C4×C83—24no—
79C23×C4×C83—42no—
80C23×C4×C83—42no—
81C23×C4×C83—42no—
82C23×C4×C83C2×C414yes—
83C23×C4×C83C2311yescharacteristic
84C23×C4×C83C2312yes—
85C23×C4×C83C2312yes—
86C23×C4×C83C2311yescharacteristic
87#95×C83—24no—
88#95×C83—22no—
89#95×C83D414yes—
90C22×Dic83—21no—
91C22×Dic83—21no—
92C22×Dic83—22no—
93C22×Dic83—22no—
94C22×Dic83—21no—
95C22×Dic83—21no—
96C22×Dic83—21no—
97C22×Dic83—21no—
98C22×Dic83—22no—
99C22×Dic83—22no—
100C22×Dic83—21no—
101C22×Dic83—21no—
102C22×Dic83—21no—
103C22×Dic83—21no—
104C22×Dic83—22no—
105C22×Dic83—22no—
106C22×Dic83—21no—
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—22no—
121C22×Dic83—22no—
122C22×Dic83C22×C412yes—
123C22×Dic83C22×C412yes—
124C22×Dic83C22×C412yes—
125C22×Dic83C22×C412yes—
126C22⋊Dic83—28no—
127C22⋊Dic83—28no—
128C22⋊Dic83—48no—
129C4×Dic83—28no—
130C4×Dic83—44no—
131C24×C83—21no—
132C24×C83—21no—
133C24×C83—22no—
134C24×C83—22no—
135C24×C83—21no—
136C24×C83—21no—
137C24×C83C22×C412yes—
138C24×C83#4112yes—
139C24×C83#4114yes—
140C24×C83#4114yes—
141C24×C83#4112yes—
142C24×C83C2411yescharacteristic
143C22×C4×C83—22no—
144C22×C4×C83—22no—
145C22×C4×C83—216no—
146C22×C4×C83—28no—
147C22×C4×C83—21no—
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—21no—
155C22×C4×C83—22no—
156C22×C4×C83—22no—
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—28no—
164C22×C4×C83—44no—
165C22×C4×C83—48no—
166C22×C4×C83—48no—
167C22×C4×C83—48no—
168C22×C4×C83—48no—
169C22×C4×C83—48no—
170C22×C4×C83—48no—
171C22×C4×C83—48no—
172C22×C4×C83—48no—
173C22×C4×C83C22×C412yes—
174C22×C4×C83C22×C412yes—
175C22×C4×C83C2×D414yes—
176C22×C4×C83C2×D412yes—
177C22×C4×C83#4114yes—
178C22×C4×C83#4112yes—
179C23×C83—22no—
180C23×C83—22no—
181C23×C83—22no—
182C23×C83—22no—
183C23×C83—24no—
184C23×C83—24no—
185C23×C83—24no—
186C23×C83—24no—
187C23×C83—24no—
188C23×C83—22no—
189C23×C83—22no—
190C23×C83—24no—
191C23×C83—24no—
192C23×C83—24no—
193C23×C83—24no—
194C23×C83—22no—
195C23×C83—22no—
196C23×C83—24no—
197C23×C83#11714yes—
198C23×C83#11714yes—
199C23×C83#11714yes—
200C23×C83#12618yes—
201C23×C83#12618yes—
202C23×C83C23×C411yescharacteristic
203C23×C83C22×D411yescharacteristic
204C23×C83C2×#4112yes—
205C23×C83C2×#4111yescharacteristic
206C23×C83C2×#4112yes—
207C23×C83C2×#4111yescharacteristic
208C23×C83C2×#4112yes—
209C23×C83C2×#4111yescharacteristic
210C23×C83H(5,𝔽2)12yes—
211C23×C83H(5,𝔽2)11yescharacteristic
212C23×C832−1+411yescharacteristic
213C2×C4×C83—22no—
214C2×C4×C83—22no—
215C2×C4×C83—24no—
216C2×C4×C83—24no—
217C2×C4×C83—22no—
218C2×C4×C83—22no—
219C2×C4×C83—22no—
220C2×C4×C83—24no—
221C2×C4×C83—24no—
222C2×C4×C83—24no—
223C2×C4×C83—24no—
224C2×C4×C83—24no—
225C2×C4×C83—24no—
226C2×C4×C83—416no—
227C2×C4×C83—432no—
228C2×C4×C83—42no—
229C2×C4×C83—42no—
230C2×C4×C83—42no—
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×C83C4×D414yes—
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—22no—
248C2×Dic83—22no—
249C2×Dic83—22no—
250C2×Dic83—22no—
251C2×Dic83—42no—
252C2×Dic83—42no—
253C2×Dic83—42no—
254C2×Dic83—42no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—42no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—42no—
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×Dic83—42no—
274C2×Dic83—42no—
275C22×C83—24no—
276C22×C83—24no—
277C22×C83—24no—
278C22×C83—24no—
279C22×C83—24no—
280C22×C83—28no—
281C22×C83—28no—
282C22×C83—28no—
283C22×C83—28no—
284C22×C83—24no—
285C22×C83#38318yescomplemented
286C22×C83C2×#11711yescharacteristic
287C22×C83C2×#11711yescharacteristic
288C22×C83C2×#11711yescharacteristic
289C22×C83C2×C4×D411yescharacteristic
290C22×C83C4×#4111yescharacteristic
291C22×C83#51811yescharacteristic
292C22×C83#52011yescharacteristic
293C22×C83#52512yes—
294C22×C83C2×#12612yes—
295C22×C83C2×#12612yes—
296C22×C83#52912yes—
297C22×C83#53812yes—
298C22×C83#53912yes—
299C22×C83#54612yes—
300C22×C83#54611yescharacteristic
301C22×C83#54811yescharacteristic
302C22×C83#55112yes—
303C22×C83#55112yes—
304C22×C83#55112yes—
305C22×C83#55212yes—
306C22×C83#55211yescharacteristic
307C22×C83#55311yescharacteristic
308C22×C83#55312yes—
309C4×C83—24no—
310C4×C83—42no—
311C4×C83—44no—
312C4×C83—44no—
313Dic83—42no—
314Dic83—42no—
315Dic83—42no—
316Dic83—42no—
317Dic83—82no—
318Dic83—82no—
319#13901—831nomaximal, Sylow
320C2×C83—28no—
321C2×C83C2×#38311yescharacteristic
322C2×C83#229011yescharacteristic
323C2×C83#231811yescharacteristic
324C2×C83#234211yescharacteristic
325C2×C83#234311yescharacteristic
326C2×C83#235211yescharacteristic
327C2×C83#235311yescharacteristic
328C2×C83#248012yes—
329C2×C83#248112yes—
330C2×C83#249312yes—
331C2×C83#251012yes—
332C2×#383—831no—
333C2×C4×C22⋊C4—831no—
334C22×#95D8311yescharacteristic
335#1443—832no—
336#1443—832no—
337#1434—831no—
338#1434—831no—
339#1434—831no—
340C22⋊#95—832no—
341C22⋊#95—832no—
342C22⋊#95—831no—
343C83#1390111yescharacteristic, complemented, Sylow
344C24×C4C2×D8312yes—
345C24×C4C2×D8311yescharacteristic
346C22×C22⋊C4—831no—
347C22×C22⋊C4—831no—
348C22×C22⋊C4—832no—
349C22×C42—831no—
350C22×C42—831no—
351#383—1664no—
352C4×C22⋊C4—838no—
353C2×#95—24no—
354C2×#95—832no—
355C2×#95—832no—
356C2×#95—832no—
357C2×#95—832no—
358C2×#95—832no—
359C2×#95—831no—
360C2×#95—831no—
361C2×#95—831no—
362C2×#95—832no—
363C2×#95—832no—
364C2×#95—831no—
365C2×#95—832no—
366C2×#95—832no—
367C2×#95C2×D8311yescharacteristic
368C2×#95C2×D8312yes—
369C2×#95C2×D8311yescharacteristic
370C25C22×D8311yescharacteristic
371C23×C4—24no—
372C23×C4—24no—
373C23×C4—24no—
374C23×C4—22no—
375C23×C4—24no—
376C23×C4—22no—
377C23×C4—24no—
378C23×C4—24no—
379C23×C4—42no—
380C23×C4—42no—
381C23×C4—42no—
382C23×C4—831no—
383C23×C4—831no—
384C23×C4—831no—
385C23×C4—831no—
386C23×C4—832no—
387C23×C4—832no—
388C23×C4C4×D8314yes—
389C23×C4C22×D8311yescharacteristic
390C23×C4C22×D8312yes—
391C23×C4C22×D8312yes—
392C23×C4C22×D8311yescharacteristic
393C2×C22⋊C4—834no—
394C2×C22⋊C4—834no—
395C2×C22⋊C4—1664no—
396C2×C22⋊C4—1664no—
397C2×C22⋊C4—1664no—
398C2×C22⋊C4—1662no—
399C2×C22⋊C4—1664no—
400C2×C22⋊C4—1662no—
401C2×C42—3322no—
402C2×C42—834no—
403C2×C42—834no—
404C2×C42—1662no—
405C2×C42—1662no—
406#95—24no—
407#95—22no—
408#95—3322no—
409#95—3322no—
410#95—3322no—
411#95—3322no—
412#95—3322no—
413#95—834no—
414#95—834no—
415#95—1662no—
416#95—1664no—
417#95—1664no—
418#95—1662no—
419#95—1664no—
420#95—1664no—
421#95C22⋊D8314yes—
422C24—21no—
423C24—21no—
424C24—22no—
425C24—22no—
426C24—21no—
427C24—21no—
428C24C2×C4×D8312yes—
429C24#1305058912yes—
430C24#1305058914yes—
431C24#1305059014yes—
432C24#1305059112yes—
433C24C23×D8311yescharacteristic
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—216no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—21no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—28no—
454C22×C4—28no—
455C22×C4—44no—
456C22×C4—48no—
457C22×C4—48no—
458C22×C4—48no—
459C22×C4—48no—
460C22×C4—48no—
461C22×C4—48no—
462C22×C4—48no—
463C22×C4—48no—
464C22×C4—832no—
465C22×C4—832no—
466C22×C4—832no—
467C22×C4—832no—
468C22×C4—1662no—
469C22×C4—1662no—
470C22×C4—1662no—
471C22×C4—1662no—
472C22×C4—1662no—
473C22×C4—1662no—
474C22×C4—1662no—
475C22×C4—1662no—
476C22×C4—1662no—
477C22×C4—1662no—
478C22×C4—1662no—
479C22×C4—1662no—
480C22×C4—1662no—
481C22×C4—1662no—
482C22×C4—1661no—
483C22×C4—1661no—
484C22×C4—1661no—
485C22×C4—1661no—
486C22×C4—1662no—
487C22×C4—1662no—
488C22×C4—1661no—
489C22×C4—1661no—
490C22×C4—1662no—
491C22×C4—1662no—
492C22×C4—1661no—
493C22×C4—1661no—
494C22×C4—1661no—
495C22×C4—1661no—
496C22×C4—1662no—
497C22×C4—1662no—
498C22×C4—1661no—
499C22×C4—1661no—
500C22×C4C2×C4×D8312yes—
501C22×C4C2×C4×D8312yes—
502C22×C4C2×C22⋊D8314yes—
503C22×C4C2×C22⋊D8312yes—
504C22×C4#1305059014yes—
505C22×C4#1305059012yes—
506C22⋊C4—3328no—
507C22⋊C4—1668no—
508C22⋊C4—1668no—
509C42—3324no—
510C42—1668no—
511C23—22no—
512C23—24no—
513C23—22no—
514C23—24no—
515C23—24no—
516C23—24no—
517C23—22no—
518C23—24no—
519C23—22no—
520C23—24no—
521C23—22no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—24no—
526C23—24no—
527C23—24no—
528C23—24no—
529C23#1458174814yes—
530C23#1458174914yes—
531C23#1458175014yes—
532C23#1458178518yes—
533C23#1458178718yes—
534C23C22×C4×D8311yescharacteristic
535C23C22×C22⋊D8311yescharacteristic
536C23C2×#1305058912yes—
537C23C2×#1305058911yescharacteristic
538C23C2×#1305059011yescharacteristic
539C23C2×#1305059012yes—
540C23C2×#1305059012yes—
541C23C2×#1305059111yescharacteristic
542C23#1458183712yes—
543C23#1458183711yescharacteristic
544C23#1458184011yescharacteristic
545C2×C4—24no—
546C2×C4—24no—
547C2×C4—22no—
548C2×C4—22no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—22no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—416no—
559C2×C4—432no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—42no—
565C2×C4—42no—
566C2×C4—42no—
567C2×C4—42no—
568C2×C4—42no—
569C2×C4—42no—
570C2×C4—3321no—
571C2×C4—3321no—
572C2×C4—3322no—
573C2×C4—3321no—
574C2×C4—3321no—
575C2×C4—3322no—
576C2×C4—3322no—
577C2×C4—3322no—
578C2×C4—3322no—
579C2×C4—3322no—
580C2×C4—3322no—
581C2×C4—3322no—
582C2×C4—3322no—
583C2×C4—3322no—
584C2×C4—3322no—
585C2×C4—3322no—
586C2×C4—3322no—
587C2×C4—3322no—
588C2×C4—3322no—
589C2×C4—3322no—
590C2×C4—3322no—
591C2×C4—3322no—
592C2×C4—3322no—
593C2×C4—3322no—
594C2×C4—1662no—
595C2×C4—1662no—
596C2×C4—1662no—
597C2×C4—1662no—
598C2×C4—1662no—
599C2×C4—1662no—
600C2×C4—1662no—
601C2×C4—1662no—
602C2×C4—1662no—
603C2×C4—1662no—
604C2×C4—1662no—
605C2×C4—1662no—
606C2×C4C4×C22⋊D8314yes—
607C22—24no—
608C22—24no—
609C22—24no—
610C22—24no—
611C22—28no—
612C22—28no—
613C22—28no—
614C22—24no—
615C22—28no—
616C22—24no—
617C22#1912756818yescomplemented
618C22C2×#1458174811yescharacteristic
619C22C2×#1458174911yescharacteristic
620C22C2×#1458175011yescharacteristic
621C22C2×C4×C22⋊D8311yescharacteristic
622C22C4×#1305059011yescharacteristic
623C22#1912817411yescharacteristic
624C22#1912818211yescharacteristic
625C22#1912821212yes—
626C22C2×#1458178512yes—
627C22C2×#1458178712yes—
628C22#1912823512yes—
629C22#1912828912yes—
630C22#1912829812yes—
631C22#1912834511yescharacteristic
632C22#1912834512yes—
633C22#1912836211yescharacteristic
634C22#1912837712yes—
635C22#1912837712yes—
636C22#1912837712yes—
637C22#1912838611yescharacteristic
638C22#1912838612yes—
639C22#1912839911yescharacteristic
640C22#1912839912yes—
641C4—24no—
642C4—44no—
643C4—44no—
644C4—42no—
645C4—3322no—
646C4—3322no—
647C4—3322no—
648C4—3322no—
649C4—6642no—
650C4—6642no—
651C2—28no—
652C2C2×#1912756811yescharacteristic
653C2#2490990611yescharacteristic
654C2#2491009711yescharacteristic
655C2#2491030811yescharacteristic
656C2#2491031911yescharacteristic
657C2#2491038811yescharacteristic
658C2#2491039911yescharacteristic
659C2#2491170312yes—
660C2#2491171812yes—
661C2#2491184812yes—
662C2#2491202312yes—
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC22×#95×C83C2
socle, cosocleC24×C83C24
prime radicalC25×C83C23
hypercenterC22×#95D83
hypocenterC83#13901
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C22×#95

Complements

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

NH ≅ G/Ntype
1. C2×#19127568C2semidirect
3. #24905724C2semidirect
4. #24905723C2semidirect
5. #24905689C2semidirect
6. #24905688C2semidirect
7. #24905687C2semidirect
18. C4×C22⋊Dic83C4semidirect
55. C2×C4×Dic83C2×C4semidirect
56. C2×C4×Dic83C2×C4semidirect
68. C83⋊#95C2×C4semidirect
69. C83⋊#95C2×C4semidirect
285. C22×C83#383semidirect
343. C83#13901semidirect
617. C22#19127568semidirect

Maximal subgroups

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

subgroupindex
C2×#191275682
C2×C4×C22⋊Dic832
#249057242
#249057232
#249056892
#249056882
#249056872
#249056862
C22×#95×C832
#1390183