Finite Group Databasebeta

← #35463276

Subgroups — #35463276

Order 21248.

782 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128110C211yesmaximal, characteristic, complemented
2C2×#19127743C211yesmaximal, characteristic
3#24911187C211yesmaximal, characteristic
4#24911154C211yesmaximal, characteristic
5#24908946C211yesmaximal, characteristic, complemented
6#24908946C211yesmaximal, characteristic, complemented
7#24908802C211yesmaximal, characteristic, complemented
8#24908792C211yesmaximal, characteristic, complemented
9#24908201C211yesmaximal, characteristic, complemented
10#24908196C211yesmaximal, characteristic, complemented
11#24907664C211yesmaximal, characteristic
12#24907653C211yesmaximal, characteristic
13#24907627C211yesmaximal, characteristic, complemented
14#24907496C211yesmaximal, characteristic
15C83×#1753C211yesmaximal, characteristic
16C2×#14581792C2211yescharacteristic
17C2×#14581784C2211yescharacteristic
18C2×#14581751C2211yescharacteristic
19C22×C4⋊Dic83C2211yescharacteristic
20#19128110—24no—
21#19127743—24no—
22C2×#14581726C2211yescharacteristic
23C2×#14581726C2211yescharacteristic
24C2×#14581726C2211yescharacteristic
25C2×#14581726C2211yescharacteristic
26C2×#14581726C2211yescharacteristic
27C2×#14581726C2211yescharacteristic
28C2×#14581724C2211yescharacteristic
29C2×#14581724C2211yescharacteristic
30C2×#14581724C2211yescharacteristic
31C2×#14581724C2211yescharacteristic
32#19127643—21no—
33#19127635—21no—
34#19127624—21no—
35#19127613—21no—
36#19127605—21no—
37#19127599C2211yescharacteristic
38#19127599C2211yescharacteristic
39#19127599C2211yescharacteristic
40#19127599C2211yescharacteristic
41#19127588—21no—
42#19127573—21no—
43#19127567C2211yescharacteristic
44#19127567C2211yescharacteristic
45#19127561C2211yescharacteristic
46#19127561C2211yescharacteristic
47#19127559—21no—
48#19127464C2211yescharacteristic
49#19127464C2211yescharacteristic
50#19127464C2211yescharacteristic
51#19127464C2211yescharacteristic
52#19127464C2211yescharacteristic
53#19127464C2211yescharacteristic
54C22×M16×C83C2211yescharacteristic
55C2×C22⋊C8×C83C2211yescharacteristic
56C2×#97×C83C2211yescharacteristic
57C2×C4×C8×C83C2211yescharacteristic
58C4×C22⋊C4×C83C2211yescharacteristic
59#336×C83C2211yescharacteristic
60#336×C83C2211yescharacteristic
61C22×Dic166—21no—
62C23×Dic83—21no—
63#14581792—22no—
64#14581792—41no—
65#14581784—22no—
66#14581784—41no—
67#14581751—22no—
68#14581751—41no—
69C2×C4⋊Dic83—22no—
70C2×C4⋊Dic83—21no—
71C2×C4⋊Dic83—21no—
72C2×C4⋊Dic83—41no—
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×C4⋊Dic83C2311yescharacteristic
79C2×C4⋊Dic83C2311yescharacteristic
80C2×C4⋊Dic83C2311yescharacteristic
81C2×Dic83⋊C4—21no—
82C2×Dic83⋊C4—21no—
83C2×Dic83⋊C4—21no—
84C2×Dic83⋊C4—21no—
85C2×Dic83⋊C4—21no—
86C2×Dic83⋊C4—21no—
87C2×C22⋊Dic83—21no—
88C2×C22⋊Dic83—21no—
89C2×C22⋊Dic83—21no—
90C2×C4×Dic83—21no—
91#14581726—22no—
92#14581726—22no—
93#14581726—22no—
94#14581726—41no—
95#14581726—41no—
96#14581726D414yes—
97#14581724—22no—
98#14581724—22no—
99#14581724—22no—
100#14581724—22no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C83⋊#95—21no—
104C83⋊#95—21no—
105C83⋊#95—21no—
106C83⋊#95—21no—
107C83⋊#95—21no—
108C83⋊#95—21no—
109C83⋊#95—21no—
110C83⋊#95—21no—
111C23×C4×C83C2311yescharacteristic
112C2×M16×C83—22no—
113C2×M16×C83—22no—
114C2×M16×C83Q814yescomplemented
115C22×C8×C83C2311yescharacteristic
116C22×C8×C83C2311yescharacteristic
117C22×C8×C83C2311yescharacteristic
118C22×C8×C83C2311yescharacteristic
119C2×C22⋊C4×C83D411yescharacteristic
120C2×C22⋊C4×C83D411yescharacteristic
121C2×C42×C83C2311yescharacteristic
122C2×C42×C83C2311yescharacteristic
123C22⋊C8×C83Q814yescomplemented
124#97×C83—22no—
125C4×C8×C83—22no—
126#95×C83D411yescharacteristic
127#95×C83D411yescharacteristic
128C2×Dic166—42no—
129C2×Dic166—42no—
130C2×Dic166—81no—
131C22×Dic83—21no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83—21no—
135C22×Dic83—21no—
136C22×Dic83—21no—
137C22×Dic83—21no—
138C22×Dic83—21no—
139C22×Dic83—42no—
140C22×Dic83—42no—
141C22×Dic83—81no—
142C4⋊Dic83—21no—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—42no—
157C4⋊Dic83—42no—
158C4⋊Dic83C2×D412yes—
159C4⋊Dic83C2×D412yes—
160Dic83⋊C4—42no—
161Dic83⋊C4—42no—
162Dic83⋊C4—42no—
163Dic83⋊C4—81no—
164Dic83⋊C4—81no—
165Dic83⋊C4—81no—
166C22⋊Dic83—42no—
167C22⋊Dic83—42no—
168C22⋊Dic83—81no—
169C4×Dic83—42no—
170C24×C83C2×D411yescharacteristic
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83—21no—
174C22×C4×C83—21no—
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83C2×D411yescharacteristic
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×Q812yes—
181C22×C4×C83C2×Q812yes—
182C22×C4×C83C2411yescharacteristic
183M16×C83—28no—
184C2×C8×C83—21no—
185C2×C8×C83—21no—
186C2×C8×C83—21no—
187C2×C8×C83—21no—
188C2×C8×C83—21no—
189C2×C8×C83—21no—
190C2×C8×C83C2×D412yes—
191C2×C8×C83C2×Q812yes—
192C2×C8×C83C2×Q812yes—
193C2×C8×C83C2×Q812yes—
194C2×C8×C83C2×Q812yes—
195C2×C8×C83#4112yes—
196C22⋊C4×C83—41no—
197C22⋊C4×C83—41no—
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×C83C4⋊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×C83C4⋊Q812yes—
229C2×C4×C83C22×D411yescharacteristic
230C2×C4×C83C22×Q811yescharacteristic
231C2×C4×C83C22×Q811yescharacteristic
232C2×C4×C83C2×#4111yescharacteristic
233C2×C4×C83H(5,𝔽2)11yescharacteristic
234C2×C4×C832−1+411yescharacteristic
235C2×Dic83—22no—
236C2×Dic83—22no—
237C2×Dic83—41no—
238C2×Dic83—41no—
239C2×Dic83—41no—
240C2×Dic83—41no—
241C2×Dic83—41no—
242C2×Dic83—41no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—41no—
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—
259C2×Dic83—84no—
260Dic166—84no—
261C8×C83—22no—
262C8×C83—22no—
263C8×C83—22no—
264C8×C83C22⋊Q814yes—
265C22×C83—22no—
266C22×C83—22no—
267C22×C83—22no—
268C22×C83—41no—
269C22×C83—41no—
270C22×C83—41no—
271C22×C83—41no—
272C22×C83C2×C4⋊Q811yescharacteristic
273C22×C83D4211yescharacteristic
274C22×C83#54711yescharacteristic
275C22×C83#57511yescharacteristic
276C22×C83#57511yescharacteristic
277C22×C83#57611yescharacteristic
278C22×C83#57811yescharacteristic
279C4×C83—22no—
280C4×C83—41no—
281C4×C83—41no—
282C4×C83—41no—
283C4×C83—41no—
284C4×C83C2×C22⋊Q811yescharacteristic
285C4×C83#54411yescharacteristic
286C4×C83D4×Q811yescharacteristic
287C4×C83#55411yescharacteristic
288Dic83—42no—
289Dic83—42no—
290Dic83—81no—
291Dic83—81no—
292Dic83—81no—
293Dic83—81no—
294Dic83—81no—
295Dic83—81no—
296#26079—831nomaximal, Sylow
297C2×C83—42no—
298C2×C83#244011yescharacteristic
299C2×C83#316411yescharacteristic
300C2×C83#316611yescharacteristic
301C2×C83#327311yescharacteristic
302C2×C83#327711yescharacteristic
303C2×C83#334711yescharacteristic
304C2×C83#334911yescharacteristic
305C2×#501—831no—
306C2×#428—831no—
307#2442—831no—
308#2439—831no—
309#2085—831no—
310#2085—831no—
311#2059—831no—
312#2057—831no—
313#1943—831no—
314#1942—831no—
315#1854—831no—
316#1852—831no—
317#1846—831no—
318#1809—831no—
319#1753D8311yescharacteristic
320C83#2607911yescharacteristic, complemented, Sylow
321C22×M16C2×D8311yescharacteristic
322C2×C4⋊Q8—831no—
323C2×#125—831no—
324C2×#117—831no—
325C22×C4⋊C4—831no—
326#501—1664no—
327#428—1664no—
328C2×#107—831no—
329C2×#107—831no—
330C2×#107—831no—
331C2×#107—831no—
332C2×#107—831no—
333C2×#107—831no—
334C2×#106—831no—
335C2×#106—831no—
336C2×#106—831no—
337C2×#106—831no—
338C2×C22⋊C8C2×D8311yescharacteristic
339C2×#97C2×D8311yescharacteristic
340C2×C4×C8C2×D8311yescharacteristic
341#400—1661no—
342#398—1661no—
343#396—1661no—
344#393—1661no—
345#391—1661no—
346#389—831no—
347#389—831no—
348#389—831no—
349#389—831no—
350#387—1661no—
351#385—1661no—
352#382—831no—
353#382—831no—
354#380—831no—
355#380—831no—
356#380—1661no—
357C4×C22⋊C4C2×D8311yescharacteristic
358#340—831no—
359#340—831no—
360#340—831no—
361#340—831no—
362#340—831no—
363#340—831no—
364#336C2×D8311yescharacteristic
365#336C2×D8311yescharacteristic
366C22×Q8—1661no—
367C23×C4—1661no—
368C23×C4C22×D8311yescharacteristic
369C2×M16—22no—
370C2×M16—22no—
371C2×M16Dic16614yescomplemented
372C22×C8C22×D8311yescharacteristic
373C22×C8C22×D8311yescharacteristic
374C22×C8C22×D8311yescharacteristic
375C22×C8C22×D8311yescharacteristic
376C4⋊Q8—3321no—
377C4⋊Q8—1662no—
378#125—3321no—
379#125—1662no—
380#117—3321no—
381#117—1662no—
382C2×C4⋊C4—3321no—
383C2×C4⋊C4—831no—
384C2×C4⋊C4—831no—
385C2×C4⋊C4—831no—
386C2×C4⋊C4—831no—
387C2×C4⋊C4—831no—
388C2×C4⋊C4—831no—
389C2×C4⋊C4—831no—
390C2×C4⋊C4—831no—
391C2×C4⋊C4—1661no—
392C2×C4⋊C4—1661no—
393C2×C4⋊C4—1661no—
394C2×C4⋊C4—1661no—
395C2×C4⋊C4—1661no—
396C2×C4⋊C4—1662no—
397C2×C4⋊C4—1661no—
398C2×C4⋊C4—1661no—
399C2×C4⋊C4—1661no—
400C2×C22⋊C4—1661no—
401C2×C22⋊C4—1661no—
402C2×C22⋊C4—1661no—
403C2×C22⋊C4D33211yescharacteristic
404C2×C22⋊C4D33211yescharacteristic
405C2×C42—1661no—
406C2×C42C22×D8311yescharacteristic
407C2×C42C22×D8311yescharacteristic
408#107—3321no—
409#107—3321no—
410#107—834no—
411#107—1662no—
412#107—1662no—
413#107—1662no—
414#106—1662no—
415#106—1662no—
416#106—1662no—
417#106—1662no—
418C22⋊C8Dic16614yescomplemented
419#97—22no—
420C4×C8—22no—
421#95—1661no—
422#95—1661no—
423#95—1661no—
424#95—1661no—
425#95—1661no—
426#95—1661no—
427#95—1661no—
428#95—1661no—
429#95—1661no—
430#95—1661no—
431#95D33211yescharacteristic
432#95D33211yescharacteristic
433C24C2×D33211yescharacteristic
434C2×Q8—3322no—
435C2×Q8—3322no—
436C2×Q8—6641no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—3322no—
442C22×C4—3322no—
443C22×C4—6641no—
444C22×C4—1661no—
445C22×C4—1661no—
446C22×C4—1661no—
447C22×C4—1661no—
448C22×C4—1661no—
449C22×C4—1661no—
450C22×C4—1661no—
451C22×C4—1661no—
452C22×C4C2×D33211yescharacteristic
453C22×C4C2×D33211yescharacteristic
454C22×C4C2×D33211yescharacteristic
455C22×C4C2×D33211yescharacteristic
456C22×C4C2×D33211yescharacteristic
457C22×C4C2×Dic16612yes—
458C22×C4C2×Dic16612yes—
459C22×C4C23×D8311yescharacteristic
460M16—28no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8D4×D8312yes—
468C2×C8C2×Dic16612yes—
469C2×C8C2×Dic16612yes—
470C2×C8C2×Dic16612yes—
471C2×C8C2×Dic16612yes—
472C2×C8#1305058912yes—
473C4⋊C4—3322no—
474C4⋊C4—3322no—
475C4⋊C4—3322no—
476C4⋊C4—3322no—
477C4⋊C4—3322no—
478C4⋊C4—832no—
479C4⋊C4—832no—
480C4⋊C4—6641no—
481C4⋊C4—6641no—
482C4⋊C4—6641no—
483C4⋊C4—1661no—
484C4⋊C4—1661no—
485C4⋊C4—1661no—
486C4⋊C4—1661no—
487C4⋊C4—1661no—
488C4⋊C4—1661no—
489C4⋊C4—1661no—
490C4⋊C4—1661no—
491C4⋊C4—1661no—
492C4⋊C4—1661no—
493C4⋊C4—1661no—
494C4⋊C4—1661no—
495C4⋊C4—1661no—
496C4⋊C4—1661no—
497C22⋊C4—41no—
498C22⋊C4—41no—
499C22⋊C4—3322no—
500C22⋊C4—3322no—
501C22⋊C4—6641no—
502C42—21no—
503C42—21no—
504C42—21no—
505C42—21no—
506C42—3322no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23—21no—
513C23#1458179212yes—
514C23C22×D33211yescharacteristic
515Q8—6644no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—22no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—3321no—
535C2×C4—3321no—
536C2×C4—3321no—
537C2×C4—3321no—
538C2×C4—3321no—
539C2×C4—3321no—
540C2×C4—3321no—
541C2×C4—3321no—
542C2×C4—3321no—
543C2×C4—3321no—
544C2×C4—3321no—
545C2×C4—3321no—
546C2×C4—3321no—
547C2×C4—3321no—
548C2×C4—3321no—
549C2×C4—3321no—
550C2×C4—3321no—
551C2×C4—3321no—
552C2×C4—3321no—
553C2×C4—3321no—
554C2×C4—3321no—
555C2×C4—3321no—
556C2×C4—6644no—
557C2×C4—1662no—
558C2×C4—1662no—
559C2×C4#1458179212yes—
560C2×C4C2×D4×D8311yescharacteristic
561C2×C4C22×Dic16611yescharacteristic
562C2×C4C22×Dic16611yescharacteristic
563C2×C4C2×#1305058911yescharacteristic
564C2×C4#1458183811yescharacteristic
565C2×C4#1458183911yescharacteristic
566C8—22no—
567C8—22no—
568C8—22no—
569C8#1458176914yes—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22C2×#1458179211yescharacteristic
578C22D4×D33211yescharacteristic
579C22#1912835511yescharacteristic
580C22#1912850111yescharacteristic
581C22#1912850111yescharacteristic
582C22#1912850611yescharacteristic
583C22#1912851811yescharacteristic
584C4—22no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—3322no—
590C4—3322no—
591C4—6641no—
592C4—6641no—
593C4—6641no—
594C4—6641no—
595C4—6641no—
596C4—6641no—
597C4C2×#1458176911yescharacteristic
598C4#1912832911yescharacteristic
599C4D4×Dic16611yescharacteristic
600C4#1912840811yescharacteristic
601C2—42no—
602C2#2491116511yescharacteristic
603C2#2491800211yescharacteristic
604C2#2491801511yescharacteristic
605C2#2491907511yescharacteristic
606C2#2491912111yescharacteristic
607C2#2491996511yescharacteristic
608C2#2491999511yescharacteristic
subgroupG/H
centerC23C22×D332
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#1753C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#1753D83
hypocenterC83#26079
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #1753

Complements

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

NH ≅ G/Ntype
1. C2×#19128110C2semidirect
5. #24908946C2semidirect
6. #24908946C2semidirect
7. #24908802C2semidirect
8. #24908792C2semidirect
9. #24908201C2semidirect
10. #24908196C2semidirect
13. #24907627C2semidirect
114. C2×M16×C83Q8semidirect
123. C22⋊C8×C83Q8semidirect
320. C83#26079semidirect
371. C2×M16Dic166semidirect
418. C22⋊C8Dic166semidirect

Maximal subgroups

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

subgroupindex
C2×#191281102
C2×#191277432
#249111872
#249111542
#249089462
#249088022
#249087922
#249082012
#249081962
#249076642
#249076532
#249076272
#249074962
C83×#17532
#2607983