Finite Group Databasebeta

← #35504557

Subgroups — #35504557

Order 21248.

728 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910145C211yesmaximal, characteristic
2#24909030C211yesmaximal, characteristic, complemented
3#24909014C211yesmaximal, characteristic, complemented
4#24908793C211yesmaximal, characteristic, complemented
5#24908601C211yesmaximal, characteristic, complemented
6#24908052C211yesmaximal, characteristic, complemented
7#24908045C211yesmaximal, characteristic, complemented
8#24907894C211yesmaximal, characteristic
9#24907884C211yesmaximal, characteristic
10#24907730C211yesmaximal, characteristic, complemented
11#24907730C211yesmaximal, characteristic, complemented
12#24907494C211yesmaximal, characteristic
13#24907486C211yesmaximal, characteristic
14#24907160C211yesmaximal, characteristic
15C83×#2628C211yesmaximal, characteristic
16C22×#13050574C2211yescharacteristic
17C23×C83⋊C8C2211yescharacteristic
18C2×#14581771C2211yescharacteristic
19C2×C4×Dic166C2211yescharacteristic
20C2×#14581725C2211yescharacteristic
21C2×#14581723C2211yescharacteristic
22C2×#14581723C2211yescharacteristic
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×C4⋊Q8×C83C2211yescharacteristic
49C2×#117×C83C2211yescharacteristic
50#400×C83—21no—
51#399×C83—21no—
52#395×C83—21no—
53#393×C83—21no—
54#389×C83C2211yescharacteristic
55#389×C83C2211yescharacteristic
56#382×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#14581771Q814yescomplemented
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#14581723—22no—
86#14581723—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—
97C22×Q8×C83—21no—
98C23×C4×C83C2311yescharacteristic
99C4⋊Q8×C83—41no—
100C4⋊Q8×C83—41no—
101#117×C83—41no—
102#117×C83—41no—
103C2×C4⋊C4×C83—21no—
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83—21no—
106C2×C4⋊C4×C83D411yescharacteristic
107C2×C4⋊C4×C83D411yescharacteristic
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C4⋊C4×C83C2311yescharacteristic
110C2×C4⋊C4×C83C2311yescharacteristic
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C22⋊C4×C83—21no—
113C2×C22⋊C4×C83—21no—
114C2×C22⋊C4×C83D411yescharacteristic
115C2×C22⋊C4×C83D411yescharacteristic
116C2×C42×C83C2311yescharacteristic
117C2×C42×C83C2311yescharacteristic
118#95×C83—21no—
119#95×C83—21no—
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
162C2×Q8×C83—42no—
163C2×Q8×C83—81no—
164C2×Q8×C83—81no—
165C22×C4×C83—21no—
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83—21no—
169C22×C4×C83—21no—
170C22×C4×C83—21no—
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83C2×D411yescharacteristic
174C22×C4×C83C2×D411yescharacteristic
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83C2×D411yescharacteristic
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×Q812yes—
179C22×C4×C83#4112yes—
180C22×C4×C83C2411yescharacteristic
181C4⋊C4×C83—21no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—41no—
190C4⋊C4×C83—41no—
191C4⋊C4×C83—42no—
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
210Q8×C83—84no—
211C2×C4×C83—21no—
212C2×C4×C83—21no—
213C2×C4×C83—22no—
214C2×C4×C83—22no—
215C2×C4×C83—22no—
216C2×C4×C83—22no—
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—21no—
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×C83—41no—
241C2×C4×C83C22⋊Q812yes—
242C2×C4×C83C22×Q811yescharacteristic
243C2×C4×C83C2×#4111yescharacteristic
244C2×C4×C83C2×#4111yescharacteristic
245C2×C4×C83C2×#4111yescharacteristic
246C2×C4×C83H(5,𝔽2)11yescharacteristic
247C2×C4×C83H(5,𝔽2)11yescharacteristic
248C2×Dic83—22no—
249C2×Dic83—22no—
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—
259C2×Dic83—41no—
260Dic166—22no—
261Dic166—22no—
262Dic166—41no—
263Dic166—41no—
264C83⋊C8—22no—
265C83⋊C8—22no—
266C83⋊C8—41no—
267C83⋊C8—41no—
268C22×C83—22no—
269C22×C83—22no—
270C22×C83—22no—
271C22×C83—41no—
272C22×C83—41no—
273C22×C83—41no—
274C22×C83—41no—
275C22×C83C2×C22⋊Q811yescharacteristic
276C22×C83#53211yescharacteristic
277C22×C83#53411yescharacteristic
278C22×C83#57711yescharacteristic
279C22×C83#57711yescharacteristic
280C22×C83#57711yescharacteristic
281C22×C83#57811yescharacteristic
282C4×C83—22no—
283C4×C83—41no—
284C4×C83—41no—
285C4×C83—41no—
286C4×C83—41no—
287C4×C83—81no—
288C4×C83—81no—
289C4×C83—81no—
290C4×C83—81no—
291C4×C83#55111yescharacteristic
292C4×C83#55111yescharacteristic
293C4×C83#55411yescharacteristic
294C4×C83#55411yescharacteristic
295Dic83—42no—
296Dic83—42no—
297Dic83—81no—
298Dic83—81no—
299#29170—831nomaximal, Sylow
300C2×C83—42no—
301C2×C83#256411yescharacteristic
302C2×C83#308211yescharacteristic
303C2×C83#308411yescharacteristic
304C2×C83#317311yescharacteristic
305C2×C83#317811yescharacteristic
306C2×C83#320011yescharacteristic
307C2×C83#320111yescharacteristic
308#2628D8311yescharacteristic
309#2323—831no—
310#2101—831no—
311#2097—831no—
312#2058—831no—
313#2024—831no—
314#1921—831no—
315#1920—831no—
316#1894—831no—
317#1892—831no—
318#1865—831no—
319#1865—831no—
320#1809—831no—
321#1807—831no—
322#1729—831no—
323C83#2917011yescharacteristic, complemented, Sylow
324C22×M16—831no—
325C23×C8—831no—
326C2×C4⋊Q8C2×D8311yescharacteristic
327C2×C22⋊Q8—831no—
328C2×C4×Q8—831no—
329C2×#117C2×D8311yescharacteristic
330C2×#107—831no—
331C2×#106—831no—
332C2×#106—831no—
333C2×#106—831no—
334C2×#103—831no—
335C2×#103—831no—
336C2×#103—831no—
337C2×#103—831no—
338#400—21no—
339#399—21no—
340#395—21no—
341#393—21no—
342#391—831no—
343#389—831no—
344#389—1661no—
345#389C2×D8311yescharacteristic
346#389C2×D8311yescharacteristic
347#388—1661no—
348#388—1661no—
349#387—1661no—
350#382C2×D8311yescharacteristic
351#380—831no—
352#380C2×D8311yescharacteristic
353#380C2×D8311yescharacteristic
354C4×C4⋊C4—831no—
355#340—831no—
356#340—831no—
357#340—831no—
358#340—831no—
359#340—831no—
360#340—831no—
361#340—831no—
362#340—831no—
363#336—831no—
364#336—831no—
365#336—831no—
366#336—831no—
367C22×Q8—21no—
368C22×Q8—831no—
369C23×C4C22×D8311yescharacteristic
370C2×M16—3321no—
371C2×M16—1662no—
372C2×M16—1662no—
373C22×C8—3321no—
374C22×C8—831no—
375C22×C8—831no—
376C22×C8—831no—
377C22×C8—831no—
378C22×C8—1662no—
379C22×C8—1662no—
380C4⋊Q8—41no—
381C4⋊Q8—41no—
382C22⋊Q8—834no—
383C22⋊Q8—1662no—
384C4×Q8—3321no—
385C4×Q8—3321no—
386#117—41no—
387#117—41no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—21no—
391C2×C4⋊C4—831no—
392C2×C4⋊C4—831no—
393C2×C4⋊C4—831no—
394C2×C4⋊C4—1661no—
395C2×C4⋊C4—1661no—
396C2×C4⋊C4—1661no—
397C2×C4⋊C4C22⋊D8311yescharacteristic
398C2×C4⋊C4C22⋊D8311yescharacteristic
399C2×C4⋊C4C22×D8311yescharacteristic
400C2×C4⋊C4C22×D8311yescharacteristic
401C2×C4⋊C4C22×D8311yescharacteristic
402C2×C4⋊C4C22×D8311yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—1661no—
406C2×C22⋊C4—1661no—
407C2×C22⋊C4C22⋊D8311yescharacteristic
408C2×C22⋊C4C22⋊D8311yescharacteristic
409C2×C42—1661no—
410C2×C42—1661no—
411C2×C42C22×D8311yescharacteristic
412C2×C42C22×D8311yescharacteristic
413#107—1662no—
414#106—3321no—
415#106—3321no—
416#106—1662no—
417#103—3321no—
418#103—3321no—
419#103—1662no—
420#103—1662no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—1661no—
428#95—1661no—
429#95—1661no—
430#95—1661no—
431#95—1661no—
432C24C2×C22⋊D8311yescharacteristic
433C2×Q8—42no—
434C2×Q8—81no—
435C2×Q8—81no—
436C2×Q8—832no—
437C2×Q8—1661no—
438C2×Q8—1661no—
439C2×Q8—1661no—
440C2×Q8—1661no—
441C2×Q8—1661no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—21no—
445C22×C4—21no—
446C22×C4—21no—
447C22×C4—21no—
448C22×C4—21no—
449C22×C4—21no—
450C22×C4—1661no—
451C22×C4—1661no—
452C22×C4—1661no—
453C22×C4—1661no—
454C22×C4C2×C22⋊D8311yescharacteristic
455C22×C4C2×C22⋊D8311yescharacteristic
456C22×C4C2×C22⋊D8311yescharacteristic
457C22×C4C2×C22⋊D8311yescharacteristic
458C22×C4C2×C22⋊D8311yescharacteristic
459C22×C4Q8×D8312yes—
460C22×C4#1305059112yes—
461C22×C4C23×D8311yescharacteristic
462M16—3324no—
463C2×C8—3324no—
464C2×C8—832no—
465C2×C8—832no—
466C2×C8—1661no—
467C2×C8—1661no—
468C2×C8—1661no—
469C2×C8—1661no—
470C2×C8—1661no—
471C2×C8—1661no—
472C2×C8—1661no—
473C2×C8—1661no—
474C2×C8—1661no—
475C2×C8—1661no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—41no—
485C4⋊C4—41no—
486C4⋊C4—42no—
487C4⋊C4—81no—
488C4⋊C4—81no—
489C4⋊C4—3322no—
490C4⋊C4—832no—
491C4⋊C4—6641no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4—1661no—
495C4⋊C4—1661no—
496C4⋊C4—1664no—
497C4⋊C4—1661no—
498C22⋊C4—41no—
499C22⋊C4—41no—
500C22⋊C4—81no—
501C22⋊C4—81no—
502C22⋊C4—6641no—
503C22⋊C4—1664no—
504C42—21no—
505C42—21no—
506C42—21no—
507C42—21no—
508C42—3322no—
509C42—6641no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23—21no—
514C23—21no—
515C23—21no—
516C23#1458176812yes—
517C23C22×C22⋊D8311yescharacteristic
518Q8—84no—
519Q8—3321no—
520Q8—3321no—
521Q8—1662no—
522Q8—1662no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
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—21no—
540C2×C4—21no—
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—41no—
552C2×C4—41no—
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—3321no—
562C2×C4—3321no—
563C2×C4—1662no—
564C2×C4—1662no—
565C2×C4#1458176812yes—
566C2×C4C2×Q8×D8311yescharacteristic
567C2×C4#41×D8311yescharacteristic
568C2×C4#41×D8311yescharacteristic
569C2×C4C2×#1305059111yescharacteristic
570C2×C4#1458183711yescharacteristic
571C2×C4#1458183711yescharacteristic
572C8—3321no—
573C8—3321no—
574C8—1662no—
575C8—1662no—
576C22—22no—
577C22—22no—
578C22—22no—
579C22—41no—
580C22—41no—
581C22—41no—
582C22—41no—
583C22C2×#1458176811yescharacteristic
584C22#1912824711yescharacteristic
585C22#1912826011yescharacteristic
586C22#1912850911yescharacteristic
587C22#1912851211yescharacteristic
588C22#1912851211yescharacteristic
589C22#1912851611yescharacteristic
590C4—22no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—41no—
595C4—81no—
596C4—81no—
597C4—81no—
598C4—81no—
599C4—3322no—
600C4—3322no—
601C4—6641no—
602C4—6641no—
603C4#1912837811yescharacteristic
604C4#1912837811yescharacteristic
605C4#1912840511yescharacteristic
606C4#1912840511yescharacteristic
607C2—42no—
608C2#2491264511yescharacteristic
609C2#2491729911yescharacteristic
610C2#2491732111yescharacteristic
611C2#2491807411yescharacteristic
612C2#2491812511yescharacteristic
613C2#2491833511yescharacteristic
614C2#2491834511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2628C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#2628D83
hypocenterC83#29170
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2628

Complements

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

NH ≅ G/Ntype
2. #24909030C2semidirect
3. #24909014C2semidirect
4. #24908793C2semidirect
5. #24908601C2semidirect
6. #24908052C2semidirect
7. #24908045C2semidirect
10. #24907730C2semidirect
11. #24907730C2semidirect
71. #14581771Q8semidirect
323. C83#29170semidirect

Maximal subgroups

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

subgroupindex
#249101452
#249090302
#249090142
#249087932
#249086012
#249080522
#249080452
#249078942
#249078842
#249077302
#249074942
#249074862
#249071602
C83×#26282
#2917083