Finite Group Databasebeta

← #35330701

Subgroups — #35330701

Order 21248.

1470 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127594C211yesmaximal, characteristic
2C2×C4⋊C4×Dic83C211yesmaximal, characteristic, complemented
3#24905731C211yesmaximal, characteristic, complemented
4#24905720C211yesmaximal, characteristic, complemented
5#24905708C211yesmaximal, characteristic
6#24905700C212yesmaximal
7#24905700C212yesmaximal
8#24905697C212yesmaximal, complemented
9#24905688C212yesmaximal, complemented
10#95×Dic83C211yesmaximal, characteristic, complemented
11C2×#397×C83C211yesmaximal, characteristic
12C22×C4⋊Dic83C2211yescharacteristic
13C22×C4⋊Dic83C2211yescharacteristic
14C22×C22⋊Dic83C2211yescharacteristic
15C22×C22⋊Dic83C2211yescharacteristic
16C22×C22⋊Dic83C2212yes—
17C22×C4×Dic83C2212yes—
18C22×C4×Dic83C2211yescharacteristic
19C22×C4×Dic83C2212yes—
20#19127594—24no—
21C4⋊C4×Dic83—24no—
22C2×C83⋊#95C2211yescharacteristic
23C2×C83⋊#95C2211yescharacteristic
24C2×C83⋊#95C2211yescharacteristic
25C2×C83⋊#95C2212yes—
26C2×C83⋊#95C2212yes—
27C2×C83⋊#95C2212yes—
28C2×C83⋊#95C2212yes—
29C2×C83⋊#95C2212yes—
30C2×C83⋊#95C2212yes—
31C2×C83⋊#95C2212yes—
32C22×C4⋊C4×C83C2211yescharacteristic
33C22×C22⋊C4×C83C2211yescharacteristic
34C22×C22⋊C4×C83C2212yes—
35#397×C83C418yescomplemented
36C2×#95×C83C2211yescharacteristic
37C2×#95×C83C2212yes—
38C23×Dic83C2311yescharacteristic
39C23×Dic83C2311yescharacteristic
40C23×Dic83C2312yes—
41C23×Dic83C2312yes—
42C23×Dic83C2311yescharacteristic
43C23×Dic83C2311yescharacteristic
44C2×C4⋊Dic83—24no—
45C2×C4⋊Dic83—22no—
46C2×C4⋊Dic83—24no—
47C2×C4⋊Dic83—22no—
48C2×C22⋊Dic83—22no—
49C2×C22⋊Dic83—22no—
50C2×C22⋊Dic83—24no—
51C2×C22⋊Dic83—24no—
52C2×C22⋊Dic83—24no—
53C2×C22⋊Dic83—24no—
54C2×C22⋊Dic83—42no—
55C2×C4×Dic83—24no—
56C2×C4×Dic83—24no—
57C2×C4×Dic83—24no—
58C2×C4×Dic83—24no—
59C2×C4×Dic83—42no—
60C2×C4×Dic83—42no—
61C2×C4×Dic83—42no—
62C2×C4×Dic83D414yes—
63C83⋊#95—24no—
64C83⋊#95—22no—
65C83⋊#95—24no—
66C83⋊#95—24no—
67C83⋊#95—24no—
68C83⋊#95—22no—
69C83⋊#95—22no—
70C83⋊#95—42no—
71C83⋊#95—42no—
72C83⋊#95—42no—
73C25×C83C2311yescharacteristic
74C23×C4×C83C2312yes—
75C23×C4×C83C2312yes—
76C23×C4×C83C2311yescharacteristic
77C23×C4×C83C2311yescharacteristic
78C2×C4⋊C4×C83—22no—
79C2×C4⋊C4×C83—22no—
80C2×C4⋊C4×C83C2×C414yes—
81C2×C22⋊C4×C83—22no—
82C2×C22⋊C4×C83—22no—
83C2×C22⋊C4×C83—42no—
84C2×C22⋊C4×C83—42no—
85C2×C22⋊C4×C83C2×C414yes—
86C2×C22⋊C4×C83C2×C418yes—
87#95×C83C2×C418yes—
88#95×C83C2×C414yes—
89C22×Dic83—22no—
90C22×Dic83—22no—
91C22×Dic83—21no—
92C22×Dic83—21no—
93C22×Dic83—21no—
94C22×Dic83—21no—
95C22×Dic83—22no—
96C22×Dic83—22no—
97C22×Dic83—22no—
98C22×Dic83—22no—
99C22×Dic83—22no—
100C22×Dic83—22no—
101C22×Dic83—22no—
102C22×Dic83—22no—
103C22×Dic83—22no—
104C22×Dic83—22no—
105C22×Dic83—22no—
106C22×Dic83—22no—
107C22×Dic83—21no—
108C22×Dic83—21no—
109C22×Dic83—22no—
110C22×Dic83—22no—
111C22×Dic83—22no—
112C22×Dic83—22no—
113C22×Dic83—21no—
114C22×Dic83—21no—
115C22×Dic83—21no—
116C22×Dic83—21no—
117C22×Dic83C2×D412yes—
118C22×Dic83C2×D412yes—
119C22×Dic83#4114yes—
120C22×Dic83#4114yes—
121C22×Dic83#4114yes—
122C22×Dic83#4114yes—
123C4⋊Dic83—28no—
124C4⋊Dic83—44no—
125C22⋊Dic83—48no—
126C22⋊Dic83—44no—
127C22⋊Dic83—44no—
128C4×Dic83—28no—
129C4×Dic83—48no—
130C4×Dic83—48no—
131C24×C83—21no—
132C24×C83—21no—
133C24×C83—22no—
134C24×C83—22no—
135C24×C83—22no—
136C24×C83—22no—
137C24×C83—21no—
138C24×C83—21no—
139C24×C83—21no—
140C24×C83—21no—
141C24×C83C22×C412yes—
142C24×C83C2411yescharacteristic
143C22×C4×C83—22no—
144C22×C4×C83—22no—
145C22×C4×C83—21no—
146C22×C4×C83—22no—
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—22no—
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×C83C22×C412yes—
162C22×C4×C83C22×C414yes—
163C22×C4×C83C22×C414yes—
164C22×C4×C83C22×C412yes—
165C22×C4×C83C2×D412yes—
166C22×C4×C83#4112yes—
167C4⋊C4×C83—28no—
168C22⋊C4×C83—28no—
169C22⋊C4×C83—48no—
170C23×C83—22no—
171C23×C83—24no—
172C23×C83—24no—
173C23×C83—22no—
174C23×C83—22no—
175C23×C83—41no—
176C23×C83—41no—
177C23×C83—41no—
178C23×C83—42no—
179C23×C83—42no—
180C23×C83—41no—
181C23×C83—41no—
182C23×C83—41no—
183C23×C83—41no—
184C23×C83—41no—
185C23×C83—42no—
186C23×C83—42no—
187C23×C83—42no—
188C23×C83—42no—
189C23×C83—42no—
190C23×C83—42no—
191C23×C83—42no—
192C23×C83—42no—
193C23×C83C23×C411yescharacteristic
194C23×C83C22×D411yescharacteristic
195C23×C83C2×#4112yes—
196C23×C83C2×#4112yes—
197C23×C83C2×#4112yes—
198C23×C83C2×#4112yes—
199C23×C83C2×#4111yescharacteristic
200C23×C83H(5,𝔽2)11yescharacteristic
201C23×C83H(5,𝔽2)11yescharacteristic
202C23×C83H(5,𝔽2)11yescharacteristic
203C23×C832−1+411yescharacteristic
204C2×C4×C83—24no—
205C2×C4×C83—22no—
206C2×C4×C83—22no—
207C2×C4×C83—24no—
208C2×C4×C83—22no—
209C2×C4×C83—24no—
210C2×C4×C83—24no—
211C2×C4×C83—24no—
212C2×C4×C83—24no—
213C2×C4×C83—24no—
214C2×C4×C83—24no—
215C2×C4×C83—24no—
216C2×C4×C83—41no—
217C2×C4×C83—42no—
218C2×C4×C83—41no—
219C2×C4×C83—42no—
220C2×C4×C83—42no—
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×C83C4×D414yes—
228C2×Dic83—22no—
229C2×Dic83—22no—
230C2×Dic83—22no—
231C2×Dic83—22no—
232C2×Dic83—24no—
233C2×Dic83—24no—
234C2×Dic83—24no—
235C2×Dic83—24no—
236C2×Dic83—24no—
237C2×Dic83—24no—
238C2×Dic83—24no—
239C2×Dic83—24no—
240C2×Dic83—24no—
241C2×Dic83—24no—
242C2×Dic83—42no—
243C2×Dic83—42no—
244C2×Dic83—42no—
245C2×Dic83—42no—
246C2×Dic83—42no—
247C2×Dic83—42no—
248C2×Dic83—42no—
249C2×Dic83—42no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—42no—
257C2×Dic83—42no—
258C2×Dic83—42no—
259C2×Dic83—42no—
260C2×Dic83#12314yes—
261C2×Dic83#12414yes—
262C22×C83—42no—
263C22×C83—44no—
264C22×C83—44no—
265C22×C83—42no—
266C22×C83—42no—
267C22×C83—81no—
268C22×C83—81no—
269C22×C83—81no—
270C22×C83—81no—
271C22×C83—82no—
272C22×C83—82no—
273C22×C83C2×C4×D411yescharacteristic
274C22×C83C4×#4112yes—
275C22×C83C4×#4112yes—
276C22×C83#51811yescharacteristic
277C22×C83#52011yescharacteristic
278C22×C83C2×#12311yescharacteristic
279C22×C83#52511yescharacteristic
280C22×C83C2×#12411yescharacteristic
281C22×C83#52912yes—
282C22×C83#52912yes—
283C22×C83#53211yescharacteristic
284C22×C83#53511yescharacteristic
285C22×C83#53711yescharacteristic
286C22×C83#54011yescharacteristic
287C22×C83#54211yescharacteristic
288C22×C83#55112yes—
289C22×C83#55112yes—
290C22×C83#55212yes—
291C22×C83#55312yes—
292C22×C83#55312yes—
293C22×C83#55312yes—
294C22×C83#55612yes—
295C22×C83#55912yes—
296C4×C83—24no—
297C4×C83—44no—
298C4×C83—44no—
299C4×C83—42no—
300Dic83—24no—
301Dic83—24no—
302Dic83—44no—
303Dic83—44no—
304Dic83—81no—
305Dic83—81no—
306#14249—831nomaximal, Sylow
307C2×C83—82no—
308C2×C83C4×#12311yescharacteristic
309C2×C83C4×#12411yescharacteristic
310C2×C83#232411yescharacteristic
311C2×C83#237312yes—
312C2×C83#237412yes—
313C2×C83#249811yescharacteristic
314C2×C83#250411yescharacteristic
315C2×C83#251612yes—
316C2×C83#252512yes—
317C2×C83#259211yescharacteristic
318C2×C83#260311yescharacteristic
319C2×#397D8311yescharacteristic
320C2×#388—831no—
321C2×C4×C4⋊C4—831no—
322#1445—831no—
323#1442—831no—
324#1439—831no—
325#1437—832no—
326#1437—832no—
327#1436—832no—
328#1434—832no—
329C4×#95—831no—
330C83#1424911yescharacteristic, complemented, Sylow
331C22×C4⋊C4—831no—
332C22×C4⋊C4—831no—
333C22×C4⋊C4C2×D8311yescharacteristic
334C22×C22⋊C4—832no—
335C22×C22⋊C4—831no—
336C22×C22⋊C4—831no—
337C22×C22⋊C4C2×D8311yescharacteristic
338C22×C22⋊C4C2×D8312yes—
339C22×C42—832no—
340C22×C42—832no—
341C22×C42—831no—
342#397Dic8318yescomplemented
343#388—1664no—
344C4×C4⋊C4—1664no—
345C2×#95—832no—
346C2×#95—832no—
347C2×#95—832no—
348C2×#95—832no—
349C2×#95—831no—
350C2×#95—831no—
351C2×#95—832no—
352C2×#95—832no—
353C2×#95—832no—
354C2×#95—831no—
355C2×#95C2×D8311yescharacteristic
356C2×#95C2×D8312yes—
357C25C22×D8311yescharacteristic
358C23×C4—831no—
359C23×C4—831no—
360C23×C4—832no—
361C23×C4—832no—
362C23×C4—831no—
363C23×C4—831no—
364C23×C4C22×D8311yescharacteristic
365C23×C4C22×D8312yes—
366C23×C4C22×D8312yes—
367C23×C4C22×D8311yescharacteristic
368C2×C4⋊C4—22no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—1662no—
371C2×C4⋊C4—1664no—
372C2×C4⋊C4—1664no—
373C2×C4⋊C4—1662no—
374C2×C4⋊C4C2×Dic8314yes—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C22⋊C4—42no—
378C2×C22⋊C4—42no—
379C2×C22⋊C4—3322no—
380C2×C22⋊C4—1662no—
381C2×C22⋊C4—1662no—
382C2×C22⋊C4—1664no—
383C2×C22⋊C4—1664no—
384C2×C22⋊C4—1664no—
385C2×C22⋊C4—1664no—
386C2×C22⋊C4C2×Dic8314yes—
387C2×C22⋊C4C2×Dic8318yes—
388C2×C42—3322no—
389C2×C42—3322no—
390C2×C42—3322no—
391C2×C42—834no—
392C2×C42—1664no—
393C2×C42—1664no—
394C2×C42—1664no—
395C2×C42—1664no—
396#95—3322no—
397#95—3322no—
398#95—3322no—
399#95—1664no—
400#95—1662no—
401#95—1664no—
402#95—1664no—
403#95—1664no—
404#95—1662no—
405#95—1662no—
406#95C2×Dic8318yes—
407#95C2×Dic8314yes—
408C24—21no—
409C24—22no—
410C24—22no—
411C24—22no—
412C24—22no—
413C24—21no—
414C24—21no—
415C24—21no—
416C24—21no—
417C24—21no—
418C24C22×Dic8312yes—
419C24C23×D8311yescharacteristic
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—22no—
438C22×C4—832no—
439C22×C4—832no—
440C22×C4—834no—
441C22×C4—834no—
442C22×C4—834no—
443C22×C4—834no—
444C22×C4—1661no—
445C22×C4—1661no—
446C22×C4—1662no—
447C22×C4—1662no—
448C22×C4—1662no—
449C22×C4—1662no—
450C22×C4—1661no—
451C22×C4—1661no—
452C22×C4—1661no—
453C22×C4—1661no—
454C22×C4—1662no—
455C22×C4—1662no—
456C22×C4—1662no—
457C22×C4—1662no—
458C22×C4—1662no—
459C22×C4—1662no—
460C22×C4—1662no—
461C22×C4—1662no—
462C22×C4—1662no—
463C22×C4—1662no—
464C22×C4—1662no—
465C22×C4—1662no—
466C22×C4—1662no—
467C22×C4—1662no—
468C22×C4—1661no—
469C22×C4—1661no—
470C22×C4—1661no—
471C22×C4—1661no—
472C22×C4C22×Dic8312yes—
473C22×C4C22×Dic8314yes—
474C22×C4C22×Dic8314yes—
475C22×C4C22×Dic8312yes—
476C22×C4D4×D8312yes—
477C22×C4#1305058912yes—
478C4⋊C4—28no—
479C4⋊C4—3324no—
480C4⋊C4—1668no—
481C22⋊C4—28no—
482C22⋊C4—48no—
483C22⋊C4—3324no—
484C22⋊C4—3324no—
485C22⋊C4—3328no—
486C42—3328no—
487C42—3328no—
488C42—1668no—
489C23—22no—
490C23—24no—
491C23—24no—
492C23—22no—
493C23—22no—
494C23—42no—
495C23—42no—
496C23—42no—
497C23—42no—
498C23—42no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—42no—
506C23—42no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23C23×Dic8311yescharacteristic
513C23C2×D4×D8311yescharacteristic
514C23#41×D8312yes—
515C23#41×D8312yes—
516C23#41×D8312yes—
517C23#41×D8312yes—
518C23C2×#1305058911yescharacteristic
519C23#1458183711yescharacteristic
520C23#1458183711yescharacteristic
521C23#1458183811yescharacteristic
522C23#1458183911yescharacteristic
523C2×C4—24no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—24no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—24no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—42no—
545C2×C4—42no—
546C2×C4—3322no—
547C2×C4—3322no—
548C2×C4—3322no—
549C2×C4—3322no—
550C2×C4—3322no—
551C2×C4—3322no—
552C2×C4—3322no—
553C2×C4—3322no—
554C2×C4—3322no—
555C2×C4—3322no—
556C2×C4—3322no—
557C2×C4—3322no—
558C2×C4—3321no—
559C2×C4—3321no—
560C2×C4—3321no—
561C2×C4—3321no—
562C2×C4—3321no—
563C2×C4—3321no—
564C2×C4—834no—
565C2×C4—834no—
566C2×C4—1664no—
567C2×C4—1664no—
568C2×C4—1664no—
569C2×C4—1664no—
570C2×C4—1664no—
571C2×C4—1664no—
572C2×C4—1664no—
573C2×C4—1664no—
574C2×C4—1664no—
575C2×C4—1664no—
576C2×C4—1662no—
577C2×C4—1662no—
578C2×C4—1662no—
579C2×C4—1662no—
580C2×C4D4×Dic8314yes—
581C22—44no—
582C22—42no—
583C22—42no—
584C22—42no—
585C22—44no—
586C22—81no—
587C22—81no—
588C22—81no—
589C22—82no—
590C22—81no—
591C22—82no—
592C22C2×D4×Dic8311yescharacteristic
593C22#41×Dic8312yes—
594C22#41×Dic8312yes—
595C22#1912817611yescharacteristic
596C22#1912818411yescharacteristic
597C22#123×D8311yescharacteristic
598C22#1912820911yescharacteristic
599C22#124×D8311yescharacteristic
600C22#1912822812yes—
601C22#1912823012yes—
602C22#1912825011yescharacteristic
603C22#1912826711yescharacteristic
604C22#1912828011yescharacteristic
605C22#1912830411yescharacteristic
606C22#1912831811yescharacteristic
607C22#1912837612yes—
608C22#1912837912yes—
609C22#1912838712yes—
610C22#1912839612yes—
611C22#1912839612yes—
612C22#1912840212yes—
613C22#1912841812yes—
614C22#1912843512yes—
615C4—24no—
616C4—44no—
617C4—44no—
618C4—42no—
619C4—3324no—
620C4—3324no—
621C4—6641no—
622C4—6641no—
623C4—1664no—
624C4—1664no—
625C2—82no—
626C2#123×Dic8311yescharacteristic
627C2#124×Dic8311yescharacteristic
628C2#2491015811yescharacteristic
629C2#2491058712yes—
630C2#2491059912yes—
631C2#2491191011yescharacteristic
632C2#2491195011yescharacteristic
633C2#2491209212yes—
634C2#2491221512yes—
635C2#2491294711yescharacteristic
636C2#2491309311yescharacteristic
subgroupG/H
center, FrattiniC24C23×D83
commutatorC23×C83C23×C4
FittingC2×#397×C83C2
socle, cosocleC24×C83C24
prime radicalC25×C83C23
hypercenterC2×#397D83
hypocenterC83#14249
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#397

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×Dic83C2semidirect
3. #24905731C2semidirect
4. #24905720C2semidirect
8. #24905697C2semidirect
9. #24905688C2semidirect
10. #95×Dic83C2semidirect
35. #397×C83C4semidirect
330. C83#14249semidirect
342. #397Dic83semidirect

Maximal subgroups

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

subgroupindex
C2×#191275942
C2×C4⋊C4×Dic832
#249057312
#249057202
#249057082
#249057002
#249056972
#249056882
#95×Dic832
C2×#397×C832
#1424983