Finite Group Databasebeta

← #35504602

Subgroups — #35504602

Order 21248.

720 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910145C211yesmaximal, characteristic
2#24909020C211yesmaximal, characteristic, complemented
3#24909002C211yesmaximal, characteristic, complemented
4#24908798C211yesmaximal, characteristic, complemented
5#24908611C211yesmaximal, characteristic, complemented
6#24908052C211yesmaximal, characteristic, complemented
7#24908031C211yesmaximal, characteristic, complemented
8#24907894C211yesmaximal, characteristic
9#24907884C211yesmaximal, characteristic
10#24907730C211yesmaximal, characteristic, complemented
11#24907716C211yesmaximal, characteristic, complemented
12#24907494C211yesmaximal, characteristic
13#24907490C211yesmaximal, characteristic
14#24907160C211yesmaximal, characteristic
15C83×#2629C211yesmaximal, characteristic
16C22×#13050574C2211yescharacteristic
17C23×C83⋊C8C2211yescharacteristic
18C2×#14581771C2211yescharacteristic
19C2×C4×Dic166C2211yescharacteristic
20C2×#14581725C2211yescharacteristic
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×#14581715C2211yescharacteristic
25C2×#14581715C2211yescharacteristic
26C2×#14581715C2211yescharacteristic
27C2×#14581715C2211yescharacteristic
28#19127607C2211yescharacteristic
29#19127599C2211yescharacteristic
30#19127596—21no—
31#19127595—21no—
32#19127589—21no—
33#19127582—21no—
34#19127561C2211yescharacteristic
35C4×C4⋊Dic83C2211yescharacteristic
36#19127463C2211yescharacteristic
37#19127463C2211yescharacteristic
38#19127463C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127456C2211yescharacteristic
45#19127456C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48C2×#125×C83C2211yescharacteristic
49C2×#117×C83C2211yescharacteristic
50#398×C83—21no—
51#398×C83—21no—
52#396×C83—21no—
53#396×C83—21no—
54#389×C83C2211yescharacteristic
55#389×C83C2211yescharacteristic
56#384×C83C2211yescharacteristic
57#380×C83C2211yescharacteristic
58#380×C83C2211yescharacteristic
59C22×Dic166C2311yescharacteristic
60C2×#13050574—22no—
61C2×#13050574—22no—
62C2×#13050574—41no—
63C22×C83⋊C8—22no—
64C22×C83⋊C8—22no—
65C22×C83⋊C8—41no—
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69C22×C83⋊C8C2311yescharacteristic
70#14581771—22no—
71#14581771Q814yes—
72C4×Dic166—41no—
73C4×Dic166—41no—
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×Dic83⋊C4—21no—
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—21no—
80C2×C22⋊Dic83—21no—
81C2×C22⋊Dic83—21no—
82C2×C4×Dic83—21no—
83C2×C4×Dic83—21no—
84#14581725—22no—
85#14581725—22no—
86#14581725—41no—
87#14581723—41no—
88#14581715—22no—
89#14581715—22no—
90#14581715—41no—
91#14581715—41no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C23×C4×C83C2311yescharacteristic
98#125×C83—41no—
99#125×C83—41no—
100#117×C83—41no—
101#117×C83—41no—
102C2×C4⋊C4×C83—21no—
103C2×C4⋊C4×C83—21no—
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83—21no—
106C2×C4⋊C4×C83—21no—
107C2×C4⋊C4×C83—21no—
108C2×C4⋊C4×C83D411yescharacteristic
109C2×C4⋊C4×C83D411yescharacteristic
110C2×C4⋊C4×C83C2311yescharacteristic
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C22⋊C4×C83—21no—
115C2×C22⋊C4×C83—21no—
116C2×C22⋊C4×C83D411yescharacteristic
117C2×C22⋊C4×C83D411yescharacteristic
118C2×C42×C83C2311yescharacteristic
119C2×C42×C83C2311yescharacteristic
120#95×C83—21no—
121#95×C83—21no—
122#95×C83—21no—
123#95×C83—21no—
124C2×Dic166—21no—
125C2×Dic166—21no—
126C2×Dic166—21no—
127C2×Dic166—21no—
128C2×Dic166—21no—
129C2×Dic166C2×Q812yes—
130C22×Dic83—21no—
131C22×Dic83—21no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134#13050574—44no—
135C2×C83⋊C8—21no—
136C2×C83⋊C8—21no—
137C2×C83⋊C8—21no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8—21no—
143C2×C83⋊C8—21no—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—44no—
146C2×C83⋊C8#4112yes—
147C2×C83⋊C8#4112yes—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83C2×Q812yes—
154Dic83⋊C4—24no—
155Dic83⋊C4—42no—
156Dic83⋊C4—81no—
157C22⋊Dic83—24no—
158C22⋊Dic83—81no—
159C4×Dic83—42no—
160C4×Dic83—81no—
161C24×C83C2×D411yescharacteristic
162C22×C4×C83—21no—
163C22×C4×C83—21no—
164C22×C4×C83—21no—
165C22×C4×C83—21no—
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83—21no—
169C22×C4×C83—21no—
170C22×C4×C83C2×D411yescharacteristic
171C22×C4×C83C2×D411yescharacteristic
172C22×C4×C83C2×D411yescharacteristic
173C22×C4×C83C2×D411yescharacteristic
174C22×C4×C83C2×D411yescharacteristic
175C22×C4×C83C2×Q812yes—
176C22×C4×C83#4112yes—
177C22×C4×C83C2411yescharacteristic
178C4⋊C4×C83—21no—
179C4⋊C4×C83—21no—
180C4⋊C4×C83—21no—
181C4⋊C4×C83—21no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—41no—
187C4⋊C4×C83—41no—
188C4⋊C4×C83—42no—
189C4⋊C4×C83—42no—
190C4⋊C4×C83—81no—
191C4⋊C4×C83—81no—
192C4⋊C4×C83—81no—
193C4⋊C4×C83—81no—
194C22⋊C4×C83—41no—
195C22⋊C4×C83—41no—
196C22⋊C4×C83—81no—
197C22⋊C4×C83—81no—
198C42×C83—21no—
199C42×C83—21no—
200C42×C83—21no—
201C42×C83—21no—
202C23×C83—21no—
203C23×C83—21no—
204C23×C83—21no—
205C23×C83—21no—
206C23×C83—21no—
207C23×C83—21no—
208C23×C83C22⋊Q812yes—
209C23×C83C22×D411yescharacteristic
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—22no—
213C2×C4×C83—22no—
214C2×C4×C83—22no—
215C2×C4×C83—22no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—21no—
224C2×C4×C83—21no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83—41no—
232C2×C4×C83—41no—
233C2×C4×C83—41no—
234C2×C4×C83—41no—
235C2×C4×C83—41no—
236C2×C4×C83—41no—
237C2×C4×C83—41no—
238C2×C4×C83—41no—
239C2×C4×C83—41no—
240C2×C4×C83C22⋊Q812yes—
241C2×C4×C83C22×Q811yescharacteristic
242C2×C4×C83C2×#4111yescharacteristic
243C2×C4×C83C2×#4111yescharacteristic
244C2×C4×C83C2×#4111yescharacteristic
245C2×C4×C83H(5,𝔽2)11yescharacteristic
246C2×C4×C83H(5,𝔽2)11yescharacteristic
247C2×Dic83—22no—
248C2×Dic83—22no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259Dic166—22no—
260Dic166—22no—
261Dic166—41no—
262Dic166—41no—
263C83⋊C8—22no—
264C83⋊C8—22no—
265C83⋊C8—41no—
266C83⋊C8—41no—
267C22×C83—22no—
268C22×C83—22no—
269C22×C83—22no—
270C22×C83—41no—
271C22×C83—41no—
272C22×C83—41no—
273C22×C83—41no—
274C22×C83C2×C22⋊Q811yescharacteristic
275C22×C83#53211yescharacteristic
276C22×C83#53411yescharacteristic
277C22×C83#57611yescharacteristic
278C22×C83#57711yescharacteristic
279C22×C83#57811yescharacteristic
280C22×C83#57811yescharacteristic
281C4×C83—22no—
282C4×C83—41no—
283C4×C83—41no—
284C4×C83—41no—
285C4×C83—41no—
286C4×C83—81no—
287C4×C83—81no—
288C4×C83—81no—
289C4×C83—81no—
290C4×C83#55111yescharacteristic
291C4×C83#55111yescharacteristic
292C4×C83#55411yescharacteristic
293C4×C83#55411yescharacteristic
294Dic83—42no—
295Dic83—42no—
296Dic83—81no—
297Dic83—81no—
298#29173—831nomaximal, Sylow
299C2×C83—42no—
300C2×C83#256411yescharacteristic
301C2×C83#308211yescharacteristic
302C2×C83#308311yescharacteristic
303C2×C83#317711yescharacteristic
304C2×C83#317911yescharacteristic
305C2×C83#320111yescharacteristic
306C2×C83#320311yescharacteristic
307#2629D8311yescharacteristic
308#2323—831no—
309#2099—831no—
310#2095—831no—
311#2059—831no—
312#2026—831no—
313#1921—831no—
314#1918—831no—
315#1894—831no—
316#1892—831no—
317#1865—831no—
318#1863—831no—
319#1809—831no—
320#1808—831no—
321#1729—831no—
322C83#2917311yescharacteristic, complemented, Sylow
323C22×M16—831no—
324C23×C8—831no—
325C2×#125C2×D8311yescharacteristic
326C2×C22⋊Q8—831no—
327C2×C4×Q8—831no—
328C2×#117C2×D8311yescharacteristic
329C2×#107—831no—
330C2×#107—831no—
331C2×#107—831no—
332C2×#106—831no—
333C2×#103—831no—
334C2×#103—831no—
335C2×#103—831no—
336C2×#103—831no—
337#398—21no—
338#398—21no—
339#396—21no—
340#396—21no—
341#391—831no—
342#389—831no—
343#389—1661no—
344#389C2×D8311yescharacteristic
345#389C2×D8311yescharacteristic
346#388—1661no—
347#388—1661no—
348#387—1661no—
349#384C2×D8311yescharacteristic
350#380—831no—
351#380C2×D8311yescharacteristic
352#380C2×D8311yescharacteristic
353C4×C4⋊C4—831no—
354#340—831no—
355#340—831no—
356#340—831no—
357#340—831no—
358#340—831no—
359#340—831no—
360#340—831no—
361#340—831no—
362#336—831no—
363#336—831no—
364#336—831no—
365#336—831no—
366C22×Q8—831no—
367C23×C4C22×D8311yescharacteristic
368C2×M16—3321no—
369C2×M16—1662no—
370C2×M16—1662no—
371C22×C8—3321no—
372C22×C8—831no—
373C22×C8—831no—
374C22×C8—831no—
375C22×C8—831no—
376C22×C8—1662no—
377C22×C8—1662no—
378#125—41no—
379#125—41no—
380C22⋊Q8—834no—
381C22⋊Q8—1662no—
382C4×Q8—3321no—
383C4×Q8—3321no—
384#117—41no—
385#117—41no—
386C2×C4⋊C4—21no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—831no—
393C2×C4⋊C4—831no—
394C2×C4⋊C4—831no—
395C2×C4⋊C4—1661no—
396C2×C4⋊C4—1661no—
397C2×C4⋊C4—1661no—
398C2×C4⋊C4C22⋊D8311yescharacteristic
399C2×C4⋊C4C22⋊D8311yescharacteristic
400C2×C4⋊C4C22×D8311yescharacteristic
401C2×C4⋊C4C22×D8311yescharacteristic
402C2×C4⋊C4C22×D8311yescharacteristic
403C2×C4⋊C4C22×D8311yescharacteristic
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—1661no—
407C2×C22⋊C4—1661no—
408C2×C22⋊C4C22⋊D8311yescharacteristic
409C2×C22⋊C4C22⋊D8311yescharacteristic
410C2×C42—1661no—
411C2×C42—1661no—
412C2×C42C22×D8311yescharacteristic
413C2×C42C22×D8311yescharacteristic
414#107—3321no—
415#107—1662no—
416#107—1662no—
417#106—3321no—
418#103—3321no—
419#103—3321no—
420#103—1662no—
421#103—1662no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—1661no—
427#95—1661no—
428#95—1661no—
429#95—1661no—
430#95—1661no—
431C24C2×C22⋊D8311yescharacteristic
432C2×Q8—832no—
433C2×Q8—1661no—
434C2×Q8—1661no—
435C2×Q8—1661no—
436C2×Q8—1661no—
437C2×Q8—1661no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—1661no—
447C22×C4—1661no—
448C22×C4—1661no—
449C22×C4—1661no—
450C22×C4C2×C22⋊D8311yescharacteristic
451C22×C4C2×C22⋊D8311yescharacteristic
452C22×C4C2×C22⋊D8311yescharacteristic
453C22×C4C2×C22⋊D8311yescharacteristic
454C22×C4C2×C22⋊D8311yescharacteristic
455C22×C4Q8×D8312yes—
456C22×C4#1305059112yes—
457C22×C4C23×D8311yescharacteristic
458M16—3324no—
459C2×C8—3324no—
460C2×C8—832no—
461C2×C8—832no—
462C2×C8—1661no—
463C2×C8—1661no—
464C2×C8—1661no—
465C2×C8—1661no—
466C2×C8—1661no—
467C2×C8—1661no—
468C2×C8—1661no—
469C2×C8—1661no—
470C2×C8—1661no—
471C2×C8—1661no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—41no—
481C4⋊C4—41no—
482C4⋊C4—42no—
483C4⋊C4—42no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4—81no—
487C4⋊C4—81no—
488C4⋊C4—3322no—
489C4⋊C4—832no—
490C4⋊C4—6641no—
491C4⋊C4—1661no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4—1661no—
495C4⋊C4—1664no—
496C4⋊C4—1661no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—81no—
500C22⋊C4—81no—
501C22⋊C4—6641no—
502C22⋊C4—1664no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—3322no—
508C42—6641no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23#1458176812yes—
516C23C22×C22⋊D8311yescharacteristic
517Q8—3321no—
518Q8—3321no—
519Q8—1662no—
520Q8—1662no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—3321no—
552C2×C4—3321no—
553C2×C4—3321no—
554C2×C4—3321no—
555C2×C4—3321no—
556C2×C4—3321no—
557C2×C4—3321no—
558C2×C4—3321no—
559C2×C4—3321no—
560C2×C4—3321no—
561C2×C4—1662no—
562C2×C4—1662no—
563C2×C4#1458176812yes—
564C2×C4C2×Q8×D8311yescharacteristic
565C2×C4#41×D8311yescharacteristic
566C2×C4#41×D8311yescharacteristic
567C2×C4C2×#1305059111yescharacteristic
568C2×C4#1458183711yescharacteristic
569C2×C4#1458183711yescharacteristic
570C8—3321no—
571C8—3321no—
572C8—1662no—
573C8—1662no—
574C22—22no—
575C22—22no—
576C22—22no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22—41no—
581C22C2×#1458176811yescharacteristic
582C22#1912824711yescharacteristic
583C22#1912826011yescharacteristic
584C22#1912850411yescharacteristic
585C22#1912851211yescharacteristic
586C22#1912851611yescharacteristic
587C22#1912851611yescharacteristic
588C4—22no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—81no—
594C4—81no—
595C4—81no—
596C4—81no—
597C4—3322no—
598C4—3322no—
599C4—6641no—
600C4—6641no—
601C4#1912837811yescharacteristic
602C4#1912837811yescharacteristic
603C4#1912840511yescharacteristic
604C4#1912840511yescharacteristic
605C2—42no—
606C2#2491264511yescharacteristic
607C2#2491729911yescharacteristic
608C2#2491731011yescharacteristic
609C2#2491811611yescharacteristic
610C2#2491814811yescharacteristic
611C2#2491833911yescharacteristic
612C2#2491836011yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2629C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#2629D83
hypocenterC83#29173
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2629

Complements

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

NH ≅ G/Ntype
2. #24909020C2semidirect
3. #24909002C2semidirect
4. #24908798C2semidirect
5. #24908611C2semidirect
6. #24908052C2semidirect
7. #24908031C2semidirect
10. #24907730C2semidirect
11. #24907716C2semidirect
322. C83#29173semidirect

Maximal subgroups

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

subgroupindex
#249101452
#249090202
#249090022
#249087982
#249086112
#249080522
#249080312
#249078942
#249078842
#249077302
#249077162
#249074942
#249074902
#249071602
C83×#26292
#2917383