Finite Group Databasebeta

← #35411092

Subgroups — #35411092

Order 21248.

954 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127957C211yesmaximal, characteristic
2C2×C4×#13050580C211yesmaximal, characteristic, complemented
3C2×#19127657C211yesmaximal, characteristic
4#24910145C211yesmaximal, characteristic
5#24908333C211yesmaximal, characteristic, complemented
6#24908203C211yesmaximal, characteristic, complemented
7#24908142C211yesmaximal, characteristic, complemented
8#24908138C211yesmaximal, characteristic, complemented
9#24907887C211yesmaximal, characteristic
10#24907884C211yesmaximal, characteristic
11#24907494C211yesmaximal, characteristic
12C4×#14581723C211yesmaximal, characteristic, complemented
13#24907296C211yesmaximal, characteristic, complemented
14#24907293C211yesmaximal, characteristic, complemented
15C83×#2323C211yesmaximal, characteristic
16C22×#13050580C2211yescharacteristic
17C22×#13050574C2211yescharacteristic
18C23×C83⋊C8C2211yescharacteristic
19C2×#14581771C2211yescharacteristic
20C2×C4×Dic166C2211yescharacteristic
21#19127957C418yescomplemented
22C4×#13050580—24no—
23C2×#14581725C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×#14581716C2211yescharacteristic
27C2×#14581716C2211yescharacteristic
28C2×#14581716C2211yescharacteristic
29C2×#14581715C2211yescharacteristic
30C2×#14581715C2211yescharacteristic
31C2×#14581715C2211yescharacteristic
32#19127657—24no—
33C2×#14581703C2211yescharacteristic
34C2×#14581703C2211yescharacteristic
35C2×C4×C83⋊C8C2211yescharacteristic
36C2×C4×C83⋊C8C2211yescharacteristic
37#19127607C2211yescharacteristic
38#19127599C2211yescharacteristic
39#19127596—21no—
40#19127595—21no—
41#19127589—21no—
42#19127582—21no—
43#19127561C2211yescharacteristic
44C4×C4⋊Dic83C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51C2×C22⋊Q8×C83C2211yescharacteristic
52C2×C4×Q8×C83C2211yescharacteristic
53C2×#117×C83C2211yescharacteristic
54#391×C83C2211yescharacteristic
55#389×C83—21no—
56#389×C83C2211yescharacteristic
57#388×C83—21no—
58#388×C83—21no—
59#387×C83—21no—
60#380×C83C2211yescharacteristic
61C4×C4⋊C4×C83C2211yescharacteristic
62C22×Dic166C2311yescharacteristic
63C2×#13050580—22no—
64C2×#13050580—22no—
65C2×#13050580C2×C414yes—
66C2×#13050574—22no—
67C2×#13050574—22no—
68C2×#13050574C2×C414yes—
69C22×C83⋊C8—22no—
70C22×C83⋊C8—22no—
71C22×C83⋊C8—22no—
72C22×C83⋊C8C2311yescharacteristic
73C22×C83⋊C8C2311yescharacteristic
74C22×C83⋊C8C2311yescharacteristic
75C22×C83⋊C8C2311yescharacteristic
76#14581771—22no—
77#14581771C2×C414yes—
78C4×Dic166—22no—
79C4×Dic166—22no—
80C2×C4⋊Dic83C2311yescharacteristic
81C2×C4⋊Dic83C2311yescharacteristic
82C2×C4⋊Dic83C2311yescharacteristic
83C2×Dic83⋊C4—21no—
84C2×Dic83⋊C4—21no—
85C2×Dic83⋊C4—21no—
86C2×C22⋊Dic83—21no—
87C2×C22⋊Dic83—21no—
88C2×C4×Dic83—21no—
89C2×C4×Dic83—21no—
90#14581725—22no—
91#14581723—22no—
92#14581723C2×C414yes—
93#14581716—22no—
94#14581716—22no—
95#14581716C2×C414yes—
96#14581715—22no—
97#14581715—22no—
98#14581715C2×C414yes—
99#14581703—22no—
100#14581703—22no—
101C4×C83⋊C8—22no—
102C4×C83⋊C8D414yes—
103C83⋊#95—21no—
104C83⋊#95—21no—
105C83⋊#95—21no—
106C83⋊#95—21no—
107C83⋊#95—21no—
108C22×Q8×C83C2311yescharacteristic
109C23×C4×C83C2311yescharacteristic
110C22⋊Q8×C83—22no—
111C22⋊Q8×C83C2×C414yes—
112C4×Q8×C83—22no—
113C4×Q8×C83—22no—
114#117×C83—22no—
115#117×C83—22no—
116C2×C4⋊C4×C83—21no—
117C2×C4⋊C4×C83—21no—
118C2×C4⋊C4×C83—21no—
119C2×C4⋊C4×C83D411yescharacteristic
120C2×C4⋊C4×C83D411yescharacteristic
121C2×C4⋊C4×C83C2311yescharacteristic
122C2×C4⋊C4×C83C2311yescharacteristic
123C2×C4⋊C4×C83C2311yescharacteristic
124C2×C22⋊C4×C83—21no—
125C2×C22⋊C4×C83—21no—
126C2×C22⋊C4×C83D411yescharacteristic
127C2×C22⋊C4×C83D411yescharacteristic
128C2×C42×C83—21no—
129C2×C42×C83—21no—
130C2×C42×C83C2311yescharacteristic
131C2×C42×C83C2311yescharacteristic
132#95×C83—21no—
133#95×C83—21no—
134#95×C83—21no—
135#95×C83—21no—
136#95×C83—21no—
137C2×Dic166—21no—
138C2×Dic166—21no—
139C2×Dic166—21no—
140C2×Dic166—21no—
141C2×Dic166—21no—
142C2×Dic166C22×C412yes—
143C22×Dic83—21no—
144C22×Dic83—21no—
145C22×Dic83—21no—
146C22×Dic83—21no—
147#13050580—28no—
148#13050574—28no—
149C2×C83⋊C8—28no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8—21no—
152C2×C83⋊C8—21no—
153C2×C83⋊C8—21no—
154C2×C83⋊C8—21no—
155C2×C83⋊C8—21no—
156C2×C83⋊C8C22×C412yes—
157C2×C83⋊C8C22×C412yes—
158C2×C83⋊C8C2×D412yes—
159C2×C83⋊C8C2×D412yes—
160C2×C83⋊C8#4112yes—
161C2×C83⋊C8#4112yes—
162C4⋊Dic83—21no—
163C4⋊Dic83—21no—
164C4⋊Dic83—21no—
165C4⋊Dic83—21no—
166C4⋊Dic83—21no—
167C4⋊Dic83C22×C412yes—
168Dic83⋊C4—24no—
169Dic83⋊C4—42no—
170Dic83⋊C4—81no—
171C22⋊Dic83—24no—
172C22⋊Dic83—81no—
173C4×Dic83—42no—
174C4×Dic83—81no—
175C24×C83C2×D411yescharacteristic
176C2×Q8×C83—21no—
177C2×Q8×C83—21no—
178C2×Q8×C83—21no—
179C2×Q8×C83—21no—
180C2×Q8×C83—21no—
181C2×Q8×C83C22×C412yes—
182C22×C4×C83—21no—
183C22×C4×C83—21no—
184C22×C4×C83—21no—
185C22×C4×C83—21no—
186C22×C4×C83—21no—
187C22×C4×C83—21no—
188C22×C4×C83—21no—
189C22×C4×C83—21no—
190C22×C4×C83C22×C412yes—
191C22×C4×C83C2×D411yescharacteristic
192C22×C4×C83C2×D411yescharacteristic
193C22×C4×C83C2×D411yescharacteristic
194C22×C4×C83C2×D411yescharacteristic
195C22×C4×C83C2×D411yescharacteristic
196C22×C4×C83#4112yes—
197C22×C4×C83C2411yescharacteristic
198C4⋊C4×C83—24no—
199C4⋊C4×C83—22no—
200C4⋊C4×C83—22no—
201C4⋊C4×C83—21no—
202C4⋊C4×C83—21no—
203C4⋊C4×C83—21no—
204C4⋊C4×C83—21no—
205C4⋊C4×C83—21no—
206C4⋊C4×C83—42no—
207C4⋊C4×C83—81no—
208C4⋊C4×C83C22×C412yes—
209C22⋊C4×C83—24no—
210C22⋊C4×C83—22no—
211C22⋊C4×C83—22no—
212C22⋊C4×C83—81no—
213C42×C83—21no—
214C42×C83—21no—
215C42×C83—42no—
216C42×C83—81no—
217C42×C83C2×D412yes—
218C42×C83#4112yes—
219C23×C83—21no—
220C23×C83—21no—
221C23×C83—21no—
222C23×C83—21no—
223C23×C83—21no—
224C23×C83—21no—
225C23×C83C4×D412yes—
226C23×C83C22×D411yescharacteristic
227Q8×C83—22no—
228Q8×C83—22no—
229Q8×C83—41no—
230Q8×C83—41no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—22no—
238C2×C4×C83—22no—
239C2×C4×C83—22no—
240C2×C4×C83—22no—
241C2×C4×C83—22no—
242C2×C4×C83—22no—
243C2×C4×C83—21no—
244C2×C4×C83—21no—
245C2×C4×C83—21no—
246C2×C4×C83—21no—
247C2×C4×C83—21no—
248C2×C4×C83—21no—
249C2×C4×C83—41no—
250C2×C4×C83—41no—
251C2×C4×C83—41no—
252C2×C4×C83—41no—
253C2×C4×C83—41no—
254C2×C4×C83—41no—
255C2×C4×C83—41no—
256C2×C4×C83—41no—
257C2×C4×C83—41no—
258C2×C4×C83—41no—
259C2×C4×C83C4×D412yes—
260C2×C4×C83#12112yes—
261C2×C4×C83#12112yes—
262C2×C4×C83C23×C411yescharacteristic
263C2×C4×C83C22×D411yescharacteristic
264C2×C4×C83C2×#4111yescharacteristic
265C2×C4×C83C2×#4111yescharacteristic
266C2×C4×C83C2×#4111yescharacteristic
267C2×C4×C83C2×#4111yescharacteristic
268C2×Dic83—22no—
269C2×Dic83—22no—
270C2×Dic83—41no—
271C2×Dic83—41no—
272C2×Dic83—41no—
273C2×Dic83—41no—
274C2×Dic83—41no—
275C2×Dic83—41no—
276C2×Dic83—41no—
277C2×Dic83—41no—
278C2×Dic83—41no—
279C2×Dic83—41no—
280Dic166—22no—
281Dic166—22no—
282Dic166—41no—
283Dic166—41no—
284C83⋊C8—22no—
285C83⋊C8—22no—
286C83⋊C8—22no—
287C83⋊C8C4×D414yes—
288C22×C83—22no—
289C22×C83—22no—
290C22×C83—22no—
291C22×C83—41no—
292C22×C83—41no—
293C22×C83—41no—
294C22×C83—41no—
295C22×C83C2×C4×D411yescharacteristic
296C22×C83C2×#12111yescharacteristic
297C22×C83#53211yescharacteristic
298C22×C83#57711yescharacteristic
299C22×C83#57711yescharacteristic
300C22×C83#57811yescharacteristic
301C22×C83#57811yescharacteristic
302C4×C83—22no—
303C4×C83—22no—
304C4×C83—22no—
305C4×C83—42no—
306C4×C83—41no—
307C4×C83—41no—
308C4×C83—42no—
309C4×C83—81no—
310C4×C83—81no—
311C4×C83C2×C4×D411yescharacteristic
312C4×C83C4×#4111yescharacteristic
313C4×C83#52511yescharacteristic
314C4×C83#52911yescharacteristic
315Dic83—42no—
316Dic83—42no—
317Dic83—81no—
318Dic83—81no—
319#21537—831nomaximal, Sylow
320C2×C83—42no—
321C2×C83C4×#12111yescharacteristic
322C2×C83#294711yescharacteristic
323C2×C83#294711yescharacteristic
324C2×C83#305911yescharacteristic
325C2×C83#306011yescharacteristic
326C2×C83#317811yescharacteristic
327C2×C83#317811yescharacteristic
328C2×#470—831no—
329C2×C4×Q16—831no—
330C2×#405—831no—
331#2323—831no—
332#2323D8311yescharacteristic
333#1971—831no—
334#1944—831no—
335#1933—831no—
336#1933—831no—
337#1892—831no—
338#1892—831no—
339#1809—831no—
340C4×#106—831no—
341#1763—831no—
342#1763—831no—
343C83#2153711yescharacteristic, complemented, Sylow
344C22×Q16—831no—
345C22×M16—831no—
346C23×C8—831no—
347C2×C22⋊Q8—831no—
348C2×C22⋊Q8C2×D8311yescharacteristic
349C2×C4×Q8—831no—
350C2×C4×Q8C2×D8311yescharacteristic
351C2×#117C2×D8311yescharacteristic
352#470—838no—
353C4×Q16—1664no—
354C2×#107—831no—
355C2×#106—831no—
356C2×#106—831no—
357C2×#103—831no—
358C2×#103—831no—
359C2×#103—831no—
360C2×#103—831no—
361C2×#103—831no—
362C2×#103—831no—
363#405—1664no—
364C2×#97—831no—
365C2×#97—831no—
366C2×C4×C8—831no—
367C2×C4×C8—831no—
368#391—831no—
369#391C2×D8311yescharacteristic
370#389—21no—
371#389—831no—
372#389—1661no—
373#389C2×D8311yescharacteristic
374#388—21no—
375#388—21no—
376#388—1661no—
377#388—1661no—
378#387—21no—
379#387—1661no—
380#380—831no—
381#380C2×D8311yescharacteristic
382C4×C4⋊C4—831no—
383C4×C4⋊C4C2×D8311yescharacteristic
384#340—831no—
385#340—831no—
386#340—831no—
387#340—831no—
388#340—831no—
389#340—831no—
390C22×Q8—831no—
391C22×Q8C22×D8311yescharacteristic
392C23×C4C22×D8311yescharacteristic
393C2×Q16—834no—
394C2×Q16—1662no—
395C2×Q16—1662no—
396C2×M16—834no—
397C2×M16—1662no—
398C2×M16—1662no—
399C22×C8—831no—
400C22×C8—831no—
401C22×C8—831no—
402C22×C8—831no—
403C22×C8—1662no—
404C22×C8—1662no—
405C22×C8—1662no—
406C22⋊Q8—22no—
407C22⋊Q8—834no—
408C22⋊Q8—1662no—
409C22⋊Q8C4×D8314yes—
410C4×Q8—22no—
411C4×Q8—22no—
412C4×Q8—1662no—
413C4×Q8—1662no—
414#117—22no—
415#117—22no—
416C2×C4⋊C4—21no—
417C2×C4⋊C4—21no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—831no—
420C2×C4⋊C4—831no—
421C2×C4⋊C4—831no—
422C2×C4⋊C4—1661no—
423C2×C4⋊C4—1661no—
424C2×C4⋊C4—1661no—
425C2×C4⋊C4C22⋊D8311yescharacteristic
426C2×C4⋊C4C22⋊D8311yescharacteristic
427C2×C4⋊C4C22×D8311yescharacteristic
428C2×C4⋊C4C22×D8311yescharacteristic
429C2×C4⋊C4C22×D8311yescharacteristic
430C2×C22⋊C4—21no—
431C2×C22⋊C4—21no—
432C2×C22⋊C4—1661no—
433C2×C22⋊C4—1661no—
434C2×C22⋊C4C22⋊D8311yescharacteristic
435C2×C22⋊C4C22⋊D8311yescharacteristic
436C2×C42—21no—
437C2×C42—21no—
438C2×C42—1661no—
439C2×C42—1661no—
440C2×C42C22×D8311yescharacteristic
441C2×C42C22×D8311yescharacteristic
442#107—1662no—
443#106—834no—
444#106—1662no—
445#103—834no—
446#103—834no—
447#103—1662no—
448#103—1662no—
449#103—1662no—
450#103—1662no—
451#97—1662no—
452#97—1662no—
453C4×C8—834no—
454C4×C8—1662no—
455#95—21no—
456#95—21no—
457#95—21no—
458#95—21no—
459#95—21no—
460#95—1661no—
461#95—1661no—
462#95—1661no—
463#95—1661no—
464#95—1661no—
465C24C2×C22⋊D8311yescharacteristic
466C2×Q8—21no—
467C2×Q8—21no—
468C2×Q8—21no—
469C2×Q8—21no—
470C2×Q8—21no—
471C2×Q8—832no—
472C2×Q8—1661no—
473C2×Q8—1661no—
474C2×Q8—1661no—
475C2×Q8—1661no—
476C2×Q8—1661no—
477C2×Q8C2×C4×D8312yes—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—21no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—1661no—
487C22×C4—1661no—
488C22×C4—1661no—
489C22×C4—1661no—
490C22×C4C2×C4×D8312yes—
491C22×C4C2×C22⋊D8311yescharacteristic
492C22×C4C2×C22⋊D8311yescharacteristic
493C22×C4C2×C22⋊D8311yescharacteristic
494C22×C4C2×C22⋊D8311yescharacteristic
495C22×C4C2×C22⋊D8311yescharacteristic
496C22×C4#1305059012yes—
497C22×C4C23×D8311yescharacteristic
498Q16—1668no—
499M16—1668no—
500C2×C8—832no—
501C2×C8—832no—
502C2×C8—832no—
503C2×C8—832no—
504C2×C8—832no—
505C2×C8—832no—
506C2×C8—1668no—
507C2×C8—1661no—
508C2×C8—1661no—
509C2×C8—1661no—
510C2×C8—1661no—
511C2×C8—1661no—
512C2×C8—1661no—
513C4⋊C4—22no—
514C4⋊C4—22no—
515C4⋊C4—21no—
516C4⋊C4—24no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—42no—
522C4⋊C4—81no—
523C4⋊C4—3322no—
524C4⋊C4—832no—
525C4⋊C4—6641no—
526C4⋊C4—1661no—
527C4⋊C4—1661no—
528C4⋊C4—1661no—
529C4⋊C4—1661no—
530C4⋊C4—1661no—
531C4⋊C4—1664no—
532C4⋊C4C2×C4×D8312yes—
533C22⋊C4—24no—
534C22⋊C4—22no—
535C22⋊C4—22no—
536C22⋊C4—81no—
537C22⋊C4—6641no—
538C22⋊C4—1664no—
539C42—21no—
540C42—21no—
541C42—42no—
542C42—81no—
543C42—3322no—
544C42—6641no—
545C42D4×D8312yes—
546C42#1305058912yes—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—21no—
552C23—21no—
553C23C4×C22⋊D8312yes—
554C23C22×C22⋊D8311yescharacteristic
555Q8—22no—
556Q8—22no—
557Q8—41no—
558Q8—41no—
559Q8—3321no—
560Q8—3321no—
561Q8—1662no—
562Q8—1662no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—21no—
567C2×C4—21no—
568C2×C4—21no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—41no—
588C2×C4—41no—
589C2×C4—41no—
590C2×C4—41no—
591C2×C4—3321no—
592C2×C4—3321no—
593C2×C4—3321no—
594C2×C4—3321no—
595C2×C4—3321no—
596C2×C4—3321no—
597C2×C4—3321no—
598C2×C4—3321no—
599C2×C4—3321no—
600C2×C4—3321no—
601C2×C4—1662no—
602C2×C4—1662no—
603C2×C4C4×C22⋊D8312yes—
604C2×C4#1458176412yes—
605C2×C4#1458176412yes—
606C2×C4C22×C4×D8311yescharacteristic
607C2×C4C2×D4×D8311yescharacteristic
608C2×C4#41×D8311yescharacteristic
609C2×C4#41×D8311yescharacteristic
610C2×C4C2×#1305058911yescharacteristic
611C2×C4C2×#1305059011yescharacteristic
612C8—834no—
613C8—1662no—
614C8—1662no—
615C8—1662no—
616C22—22no—
617C22—22no—
618C22—22no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22—41no—
623C22C2×C4×C22⋊D8311yescharacteristic
624C22C2×#1458176411yescharacteristic
625C22#1912824711yescharacteristic
626C22#1912850911yescharacteristic
627C22#1912851211yescharacteristic
628C22#1912851611yescharacteristic
629C22#1912851611yescharacteristic
630C4—22no—
631C4—22no—
632C4—22no—
633C4—42no—
634C4—41no—
635C4—41no—
636C4—42no—
637C4—81no—
638C4—81no—
639C4—3322no—
640C4—3322no—
641C4—6641no—
642C4—6641no—
643C4C4×D4×D8311yescharacteristic
644C4C4×#1305058911yescharacteristic
645C4#1912821011yescharacteristic
646C4#1912823111yescharacteristic
647C2—42no—
648C2C4×#1458176411yescharacteristic
649C2#2491630211yescharacteristic
650C2#2491630611yescharacteristic
651C2#2491711511yescharacteristic
652C2#2491712211yescharacteristic
653C2#2491812511yescharacteristic
654C2#2491813311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2323C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
cosocleC22×C4×C83C24
hypercenter#2323D83
hypocenterC83#21537
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2323

Complements

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

NH ≅ G/Ntype
2. C2×C4×#13050580C2semidirect
5. #24908333C2semidirect
6. #24908203C2semidirect
7. #24908142C2semidirect
8. #24908138C2semidirect
12. C4×#14581723C2semidirect
13. #24907296C2semidirect
14. #24907293C2semidirect
21. #19127957C4semidirect
343. C83#21537semidirect

Maximal subgroups

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

subgroupindex
C2×#191279572
C2×C4×#130505802
C2×#191276572
#249101452
#249083332
#249082032
#249081422
#249081382
#249078872
#249078842
#249074942
C4×#145817232
#249072962
#249072932
C83×#23232
#2153783