Finite Group Databasebeta

← #35326229

Subgroups — #35326229

Order 21248.

1790 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127626C211yesmaximal, characteristic
2C2×C4⋊C4×Dic83C211yesmaximal, characteristic, complemented
3C2×C22⋊C4×Dic83C212yesmaximal
4C2×C22⋊C4×Dic83C211yesmaximal, characteristic
5#24905732C211yesmaximal, characteristic, complemented
6#24905724C212yesmaximal, complemented
7#24905722C211yesmaximal, characteristic, complemented
8#24905710C212yesmaximal
9#95×Dic83C212yesmaximal, complemented
10#95×Dic83C211yesmaximal, characteristic, complemented
11C83×#1439C211yesmaximal, characteristic
12C24×Dic83C2211yescharacteristic
13C22×C4⋊Dic83C2211yescharacteristic
14C22×C22⋊Dic83C2211yescharacteristic
15C22×C22⋊Dic83C2212yes—
16C22×C4×Dic83C2212yes—
17C22×C4×Dic83C2211yescharacteristic
18C22×C4×Dic83C2212yes—
19C22×C4×Dic83C2211yescharacteristic
20#19127626C418yescomplemented
21C4⋊C4×Dic83—24no—
22C22⋊C4×Dic83—24no—
23C22⋊C4×Dic83—28no—
24C2×C83⋊#95C2212yes—
25C2×C83⋊#95C2211yescharacteristic
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2212yes—
28C2×C83⋊#95C2212yes—
29C2×C83⋊#95C2212yes—
30C2×C83⋊#95C2211yescharacteristic
31C2×C83⋊#95C2212yes—
32C2×C83⋊#95C2211yescharacteristic
33C2×C83⋊#95C2212yes—
34C22×C4⋊C4×C83C2211yescharacteristic
35C22×C22⋊C4×C83C2212yes—
36C22×C22⋊C4×C83C2211yescharacteristic
37C2×#95×C83C2211yescharacteristic
38C2×#95×C83C2212yes—
39C23×Dic83—22no—
40C23×Dic83—22no—
41C23×Dic83—24no—
42C23×Dic83—24no—
43C23×Dic83D414yes—
44C23×Dic83C2312yes—
45C23×Dic83C2312yes—
46C23×Dic83C2311yescharacteristic
47C23×Dic83C2311yescharacteristic
48C23×Dic83C2311yescharacteristic
49C23×Dic83C2311yescharacteristic
50C2×C4⋊Dic83—22no—
51C2×C4⋊Dic83—22no—
52C2×C4⋊Dic83C2×C414yescomplemented
53C2×C22⋊Dic83—22no—
54C2×C22⋊Dic83—22no—
55C2×C22⋊Dic83—24no—
56C2×C22⋊Dic83—22no—
57C2×C22⋊Dic83—24no—
58C2×C22⋊Dic83C2×C418yes—
59C2×C4×Dic83—24no—
60C2×C4×Dic83—24no—
61C2×C4×Dic83—24no—
62C2×C4×Dic83—24no—
63C2×C4×Dic83—22no—
64C2×C4×Dic83—24no—
65C2×C4×Dic83—24no—
66C2×C4×Dic83—24no—
67C2×C4×Dic83—24no—
68C2×C4×Dic83D414yes—
69C83⋊#95—24no—
70C83⋊#95—22no—
71C83⋊#95—24no—
72C83⋊#95—24no—
73C83⋊#95—24no—
74C83⋊#95—22no—
75C83⋊#95—24no—
76C83⋊#95—24no—
77C83⋊#95C2×C418yescomplemented
78C83⋊#95C2×C414yescomplemented
79C25×C83C2311yescharacteristic
80C23×C4×C83C2311yescharacteristic
81C23×C4×C83C2312yes—
82C23×C4×C83C2312yes—
83C23×C4×C83C2311yescharacteristic
84C2×C4⋊C4×C83—22no—
85C2×C4⋊C4×C83—24no—
86C2×C22⋊C4×C83—24no—
87C2×C22⋊C4×C83—22no—
88C2×C22⋊C4×C83—24no—
89C2×C22⋊C4×C83—24no—
90C2×C22⋊C4×C83—22no—
91C2×C22⋊C4×C83C2×C414yes—
92#95×C83—24no—
93#95×C83—22no—
94C22×Dic83—21no—
95C22×Dic83—22no—
96C22×Dic83—22no—
97C22×Dic83—21no—
98C22×Dic83—22no—
99C22×Dic83—21no—
100C22×Dic83—22no—
101C22×Dic83—22no—
102C22×Dic83—22no—
103C22×Dic83—22no—
104C22×Dic83—22no—
105C22×Dic83—22no—
106C22×Dic83—22no—
107C22×Dic83—22no—
108C22×Dic83—22no—
109C22×Dic83—22no—
110C22×Dic83—22no—
111C22×Dic83—22no—
112C22×Dic83—21no—
113C22×Dic83—22no—
114C22×Dic83—22no—
115C22×Dic83—21no—
116C22×Dic83—21no—
117C22×Dic83—22no—
118C22×Dic83—28no—
119C22×Dic83—216no—
120C22×Dic83—44no—
121C22×Dic83—44no—
122C22×Dic83—48no—
123C22×Dic83C22×C414yes—
124C22×Dic83C22×C414yes—
125C22×Dic83C2×D412yes—
126C22×Dic83C2×D412yes—
127C22×Dic83C2×D412yes—
128C22×Dic83C2×D412yes—
129C22×Dic83#4112yes—
130C22×Dic83#4112yes—
131C22×Dic83#4114yes—
132C22×Dic83#4114yes—
133C4⋊Dic83—28no—
134C22⋊Dic83—28no—
135C22⋊Dic83—216no—
136C4×Dic83—28no—
137C4×Dic83—48no—
138C4×Dic83—48no—
139C4×Dic83—44no—
140C24×C83—21no—
141C24×C83—21no—
142C24×C83—22no—
143C24×C83—22no—
144C24×C83—21no—
145C24×C83—22no—
146C24×C83—22no—
147C24×C83—21no—
148C24×C83C22×C412yes—
149C24×C83C2×D412yes—
150C24×C83#4112yes—
151C24×C83C2411yescharacteristic
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—21no—
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×C83C22×C412yes—
171C22×C4×C83C22×C412yes—
172C22×C4×C83C2×D412yes—
173C22×C4×C83#4114yes—
174C22×C4×C83#4114yes—
175C22×C4×C83#4112yes—
176C4⋊C4×C83—28no—
177C22⋊C4×C83—216no—
178C22⋊C4×C83—28no—
179C23×C83—22no—
180C23×C83—24no—
181C23×C83—24no—
182C23×C83—22no—
183C23×C83—22no—
184C23×C83—24no—
185C23×C83—24no—
186C23×C83—22no—
187C23×C83—22no—
188C23×C83—24no—
189C23×C83—24no—
190C23×C83—22no—
191C23×C83—41no—
192C23×C83—41no—
193C23×C83—41no—
194C23×C83—42no—
195C23×C83—42no—
196C23×C83—42no—
197C23×C83—42no—
198C23×C83—42no—
199C23×C83—42no—
200C23×C83—41no—
201C23×C83C4×D414yes—
202C23×C83C23×C411yescharacteristic
203C23×C83C22×D411yescharacteristic
204C23×C83C22×D411yescharacteristic
205C23×C83C2×#4111yescharacteristic
206C23×C83C2×#4112yes—
207C23×C83C2×#4112yes—
208C23×C83C2×#4111yescharacteristic
209C23×C83C2×#4111yescharacteristic
210C23×C83C2×#4112yes—
211C23×C83C2×#4112yes—
212C23×C83C2×#4111yescharacteristic
213C2×C4×C83—24no—
214C2×C4×C83—22no—
215C2×C4×C83—22no—
216C2×C4×C83—24no—
217C2×C4×C83—22no—
218C2×C4×C83—24no—
219C2×C4×C83—24no—
220C2×C4×C83—24no—
221C2×C4×C83—24no—
222C2×C4×C83—24no—
223C2×C4×C83—24no—
224C2×C4×C83—24no—
225C2×C4×C83—42no—
226C2×C4×C83—42no—
227C2×C4×C83—42no—
228C2×C4×C83—41no—
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—41no—
236C2×C4×C83C4×D414yes—
237C2×Dic83—24no—
238C2×Dic83—24no—
239C2×Dic83—24no—
240C2×Dic83—24no—
241C2×Dic83—24no—
242C2×Dic83—24no—
243C2×Dic83—22no—
244C2×Dic83—22no—
245C2×Dic83—24no—
246C2×Dic83—22no—
247C2×Dic83—22no—
248C2×Dic83—24no—
249C2×Dic83—22no—
250C2×Dic83—42no—
251C2×Dic83—42no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—42no—
255C2×Dic83—42no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—42no—
259C2×Dic83—42no—
260C2×Dic83—42no—
261C2×Dic83—42no—
262C2×Dic83—42no—
263C2×Dic83—42no—
264C2×Dic83—42no—
265C2×Dic83—42no—
266C2×Dic83—416no—
267C2×Dic83#12114yes—
268C2×Dic83#12314yes—
269C2×Dic83#12318yes—
270C2×Dic83#12414yes—
271C2×Dic83#12618yes—
272C22×C83—24no—
273C22×C83—28no—
274C22×C83—44no—
275C22×C83—42no—
276C22×C83—42no—
277C22×C83—44no—
278C22×C83—42no—
279C22×C83—42no—
280C22×C83—44no—
281C22×C83—42no—
282C22×C83—42no—
283C22×C83C2×C4×D411yescharacteristic
284C22×C83C2×C4×D411yescharacteristic
285C22×C83C4×#4112yes—
286C22×C83C4×#4112yes—
287C22×C83C4×#4111yescharacteristic
288C22×C83C2×#12111yescharacteristic
289C22×C83C2×#12311yescharacteristic
290C22×C83C2×#12312yes—
291C22×C83#52512yes—
292C22×C83#52511yescharacteristic
293C22×C83#52511yescharacteristic
294C22×C83C2×#12411yescharacteristic
295C22×C83C2×#12612yes—
296C22×C83#52912yes—
297C22×C83#52912yes—
298C22×C83#52912yes—
299C22×C83#52912yes—
300C22×C83#52912yes—
301C22×C83#52912yes—
302C22×C83#52912yes—
303C22×C83#52911yescharacteristic
304C22×C83#53211yescharacteristic
305C22×C83#53211yescharacteristic
306C4×C83—24no—
307C4×C83—44no—
308C4×C83—44no—
309C4×C83—42no—
310Dic83—42no—
311Dic83—42no—
312Dic83—42no—
313Dic83—44no—
314Dic83—44no—
315Dic83#39718yescomplemented
316C4×#397—831nomaximal, Sylow
317C2×C83—44no—
318C2×C83C4×#12111yescharacteristic
319C2×C83C4×#12312yes—
320C2×C83C4×#12311yescharacteristic
321C2×C83C4×#12411yescharacteristic
322C2×C83C4×#12612yes—
323C2×C83C2×#39711yescharacteristic
324C2×C83#239211yescharacteristic
325C2×C83#239512yes—
326C2×C83#239511yescharacteristic
327C2×C83#239811yescharacteristic
328C2×C83#240112yes—
329C2×#397—831no—
330C2×C4×C4⋊C4—831no—
331C2×C4×C22⋊C4—832no—
332C2×C4×C22⋊C4—831no—
333#1445—831no—
334#1443—832no—
335#1442—831no—
336#1440—832no—
337#1439D8311yescharacteristic
338C4×#95—831no—
339C4×#95—832no—
340C83C4×#39711yescharacteristic, complemented, Sylow
341C24×C4—831no—
342C22×C4⋊C4—831no—
343C22×C4⋊C4C2×D8311yescharacteristic
344C22×C22⋊C4—832no—
345C22×C22⋊C4—831no—
346C22×C22⋊C4C2×D8312yes—
347C22×C22⋊C4C2×D8311yescharacteristic
348C22×C42—831no—
349C22×C42—831no—
350C22×C42—832no—
351C22×C42—832no—
352#397—838no—
353C4×C4⋊C4—1664no—
354C4×C22⋊C4—1664no—
355C4×C22⋊C4—1668no—
356C2×#95—831no—
357C2×#95—832no—
358C2×#95—832no—
359C2×#95—832no—
360C2×#95—832no—
361C2×#95—832no—
362C2×#95—832no—
363C2×#95—831no—
364C2×#95—831no—
365C2×#95—832no—
366C2×#95C2×D8312yes—
367C2×#95C2×D8311yescharacteristic
368C25C22×D8311yescharacteristic
369C23×C4—831no—
370C23×C4—834no—
371C23×C4—832no—
372C23×C4—832no—
373C23×C4—831no—
374C23×C4—831no—
375C23×C4—831no—
376C23×C4—1664no—
377C23×C4—1664no—
378C23×C4—1662no—
379C23×C4—1662no—
380C23×C4C22×D8311yescharacteristic
381C23×C4C22×D8312yes—
382C23×C4C22×D8312yes—
383C23×C4C22×D8311yescharacteristic
384C2×C4⋊C4—22no—
385C2×C4⋊C4—24no—
386C2×C4⋊C4—834no—
387C2×C4⋊C4—1662no—
388C2×C4⋊C4—1662no—
389C2×C22⋊C4—24no—
390C2×C22⋊C4—22no—
391C2×C22⋊C4—24no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—22no—
394C2×C22⋊C4—838no—
395C2×C22⋊C4—1662no—
396C2×C22⋊C4—1662no—
397C2×C22⋊C4—1664no—
398C2×C22⋊C4—1662no—
399C2×C22⋊C4—1664no—
400C2×C22⋊C4C4×D8314yes—
401C2×C42—834no—
402C2×C42—1662no—
403C2×C42—1664no—
404C2×C42—1664no—
405C2×C42—1664no—
406C2×C42—1664no—
407C2×C42—1664no—
408C2×C42—1664no—
409C2×C42—1664no—
410C2×C42—1664no—
411#95—24no—
412#95—22no—
413#95—834no—
414#95—838no—
415#95—1664no—
416#95—1664no—
417#95—1662no—
418#95—1664no—
419#95—1664no—
420#95—1662no—
421#95—1664no—
422#95—1664no—
423C24—21no—
424C24—21no—
425C24—22no—
426C24—22no—
427C24—21no—
428C24—21no—
429C24—22no—
430C24—22no—
431C24C2×C4×D8312yes—
432C24D4×D8312yes—
433C24#1305059112yes—
434C24C23×D8311yescharacteristic
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—21no—
438C22×C4—22no—
439C22×C4—22no—
440C22×C4—22no—
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—22no—
444C22×C4—22no—
445C22×C4—22no—
446C22×C4—22no—
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—21no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—3324no—
454C22×C4—3324no—
455C22×C4—3328no—
456C22×C4—832no—
457C22×C4—832no—
458C22×C4—832no—
459C22×C4—834no—
460C22×C4—834no—
461C22×C4—834no—
462C22×C4—834no—
463C22×C4—832no—
464C22×C4—832no—
465C22×C4—832no—
466C22×C4—1668no—
467C22×C4—16616no—
468C22×C4—1661no—
469C22×C4—1662no—
470C22×C4—1662no—
471C22×C4—1661no—
472C22×C4—1662no—
473C22×C4—1662no—
474C22×C4—1662no—
475C22×C4—1662no—
476C22×C4—1662no—
477C22×C4—1662no—
478C22×C4—1662no—
479C22×C4—1661no—
480C22×C4—1662no—
481C22×C4—1662no—
482C22×C4—1661no—
483C22×C4—1662no—
484C22×C4—1662no—
485C22×C4—1662no—
486C22×C4—1662no—
487C22×C4—1662no—
488C22×C4—1662no—
489C22×C4—1662no—
490C22×C4—1661no—
491C22×C4—1661no—
492C22×C4C2×C4×D8312yes—
493C22×C4C2×C4×D8312yes—
494C22×C4D4×D8312yes—
495C22×C4#1305058914yes—
496C22×C4#1305058914yes—
497C22×C4#1305058912yes—
498C4⋊C4—28no—
499C4⋊C4—1668no—
500C22⋊C4—216no—
501C22⋊C4—28no—
502C22⋊C4—1668no—
503C22⋊C4—16616no—
504C42—3328no—
505C42—3328no—
506C42—3324no—
507C42—1668no—
508C23—22no—
509C23—24no—
510C23—24no—
511C23—22no—
512C23—22no—
513C23—24no—
514C23—24no—
515C23—22no—
516C23—22no—
517C23—24no—
518C23—24no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—42no—
528C23—42no—
529C23—41no—
530C23#1458175414yes—
531C23C22×C4×D8311yescharacteristic
532C23C2×D4×D8311yescharacteristic
533C23C2×D4×D8311yescharacteristic
534C23#41×D8312yes—
535C23#41×D8312yes—
536C23#41×D8311yescharacteristic
537C23#41×D8311yescharacteristic
538C23C2×#1305058911yescharacteristic
539C23C2×#1305058912yes—
540C23C2×#1305058912yes—
541C23C2×#1305059111yescharacteristic
542C2×C4—24no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—24no—
546C2×C4—22no—
547C2×C4—24no—
548C2×C4—24no—
549C2×C4—24no—
550C2×C4—24no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—42no—
557C2×C4—41no—
558C2×C4—42no—
559C2×C4—42no—
560C2×C4—42no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—42no—
564C2×C4—41no—
565C2×C4—3322no—
566C2×C4—3322no—
567C2×C4—3321no—
568C2×C4—3321no—
569C2×C4—3322no—
570C2×C4—3322no—
571C2×C4—3321no—
572C2×C4—3321no—
573C2×C4—3322no—
574C2×C4—3322no—
575C2×C4—3322no—
576C2×C4—3322no—
577C2×C4—3322no—
578C2×C4—3322no—
579C2×C4—3322no—
580C2×C4—3322no—
581C2×C4—33216no—
582C2×C4—834no—
583C2×C4—838no—
584C2×C4—834no—
585C2×C4—834no—
586C2×C4—838no—
587C2×C4—1664no—
588C2×C4—1664no—
589C2×C4—1664no—
590C2×C4—1664no—
591C2×C4—1664no—
592C2×C4—1664no—
593C2×C4—1662no—
594C2×C4—1662no—
595C2×C4—1664no—
596C2×C4—1662no—
597C2×C4—1662no—
598C2×C4—1664no—
599C2×C4—1662no—
600C2×C4(C22⋊D83)⋊C414yes—
601C22—24no—
602C22—28no—
603C22—42no—
604C22—42no—
605C22—44no—
606C22—42no—
607C22—42no—
608C22—42no—
609C22—44no—
610C22—42no—
611C22—44no—
612C22C2×(C22⋊D83)⋊C411yescharacteristic
613C22C2×#1458175411yescharacteristic
614C22C4×#1305058912yes—
615C22C4×#1305058912yes—
616C22#1912816911yescharacteristic
617C22#121×D8311yescharacteristic
618C22#123×D8312yes—
619C22#123×D8311yescharacteristic
620C22#1912820712yes—
621C22#1912820711yescharacteristic
622C22#1912821411yescharacteristic
623C22#124×D8311yescharacteristic
624C22#126×D8312yes—
625C22#1912822812yes—
626C22#1912823012yes—
627C22#1912823212yes—
628C22#1912823212yes—
629C22#1912823212yes—
630C22#1912823712yes—
631C22#1912823711yescharacteristic
632C22#1912823712yes—
633C22#1912824611yescharacteristic
634C22#1912824911yescharacteristic
635C4—24no—
636C4—44no—
637C4—44no—
638C4—42no—
639C4—3322no—
640C4—3322no—
641C4—3324no—
642C4—3322no—
643C4—3324no—
644C4—838no—
645C2—44no—
646C2#2490996911yescharacteristic
647C2#2490998212yes—
648C2#2490998811yescharacteristic
649C2#2491001011yescharacteristic
650C2#2491001712yes—
651C2#397×D8311yescharacteristic
652C2#2491069811yescharacteristic
653C2#2491071912yes—
654C2#2491072811yescharacteristic
655C2#2491074511yescharacteristic
656C2#2491076512yes—
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC83×#1439C2
socle, cosocleC24×C83C24
prime radicalC25×C83C23
hypercenter#1439D83
hypocenterC83C4×#397
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×Dic83C2semidirect
5. #24905732C2semidirect
6. #24905724C2semidirect
7. #24905722C2semidirect
9. #95×Dic83C2semidirect
10. #95×Dic83C2semidirect
20. #19127626C4semidirect
52. C2×C4⋊Dic83C2×C4semidirect
77. C83⋊#95C2×C4semidirect
78. C83⋊#95C2×C4semidirect
315. Dic83#397semidirect
340. C83C4×#397semidirect

Maximal subgroups

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

subgroupindex
C2×#191276262
C2×C4⋊C4×Dic832
C2×C22⋊C4×Dic832
#249057322
#249057242
#249057222
#249057102
#95×Dic832
C83×#14392
C4×#39783