Finite Group Databasebeta

← #35466989

Subgroups — #35466989

Order 21248.

838 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128008C211yesmaximal, characteristic, complemented
2C22×#14581725C211yesmaximal, characteristic
3#24911187C211yesmaximal, characteristic
4#24909011C211yesmaximal, characteristic, complemented
5#24908954C211yesmaximal, characteristic, complemented
6#24908858C211yesmaximal, characteristic, complemented
7#24908801C211yesmaximal, characteristic, complemented
8#24908231C211yesmaximal, characteristic, complemented
9#24908034C211yesmaximal, characteristic, complemented
10#24907887C211yesmaximal, characteristic
11#24907771C211yesmaximal, characteristic
12#24907717C211yesmaximal, characteristic, complemented
13#24907656C211yesmaximal, characteristic
14#24907643C211yesmaximal, characteristic
15C83×#2360C211yesmaximal, characteristic
16C23×C83⋊C8C2211yescharacteristic
17C2×#14581784C2211yescharacteristic
18C22×C4⋊Dic83C2211yescharacteristic
19#19128008—24no—
20C2×#14581725—24no—
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581725C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×#14581723C2211yescharacteristic
27C2×C4⋊(C83⋊C8)C2211yescharacteristic
28C2×C4⋊(C83⋊C8)C2211yescharacteristic
29C2×#14581716C2211yescharacteristic
30C2×#14581716C2211yescharacteristic
31C2×#14581716C2211yescharacteristic
32C2×#14581716C2211yescharacteristic
33C2×C22⋊(C83⋊C8)C2211yescharacteristic
34#19127643—21no—
35#19127635—21no—
36#19127599C2211yescharacteristic
37#19127599C2211yescharacteristic
38#19127573—21no—
39#19127567C2211yescharacteristic
40#19127561C2211yescharacteristic
41#19127559—21no—
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×C22⋊Q8×C83C2211yescharacteristic
51C2×C4×Q8×C83C2211yescharacteristic
52C22×C4⋊C4×C83C2211yescharacteristic
53#391×C83C2211yescharacteristic
54#389×C83—21no—
55#389×C83C2211yescharacteristic
56#387×C83—21no—
57#385×C83—21no—
58C4×C4⋊C4×C83C2211yescharacteristic
59C4×C22⋊C4×C83—21no—
60C4×C22⋊C4×C83C2211yescharacteristic
61C23×Dic83—21no—
62C22×C83⋊C8—22no—
63C22×C83⋊C8—22no—
64C22×C83⋊C8—22no—
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69#14581784—22no—
70#14581784—41no—
71C2×C4⋊Dic83—22no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83—41no—
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—21no—
80C2×Dic83⋊C4—21no—
81C2×Dic83⋊C4—21no—
82C2×C22⋊Dic83—21no—
83#14581725—22no—
84#14581725—22no—
85#14581725—22no—
86#14581725D414yes—
87#14581723—22no—
88#14581723—41no—
89C4⋊(C83⋊C8)—22no—
90C4⋊(C83⋊C8)—22no—
91#14581716—22no—
92#14581716—22no—
93#14581716—22no—
94#14581716—41no—
95C22⋊(C83⋊C8)—22no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C22×Q8×C83C2311yescharacteristic
102C23×C4×C83—21no—
103C23×C4×C83C2311yescharacteristic
104C22⋊Q8×C83—22no—
105C22⋊Q8×C83Q814yes—
106C4×Q8×C83—22no—
107C4×Q8×C83—41no—
108C2×C4⋊C4×C83—21no—
109C2×C4⋊C4×C83—22no—
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83—21no—
112C2×C4⋊C4×C83—21no—
113C2×C4⋊C4×C83—41no—
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C22⋊C4×C83—21no—
118C2×C22⋊C4×C83D411yescharacteristic
119C2×C22⋊C4×C83D411yescharacteristic
120C2×C42×C83—21no—
121C2×C42×C83—21no—
122C2×C42×C83C2311yescharacteristic
123C2×C42×C83C2311yescharacteristic
124#95×C83—21no—
125#95×C83—21no—
126#95×C83—21no—
127#95×C83—21no—
128#95×C83D411yescharacteristic
129#95×C83D411yescharacteristic
130C22×Dic83—21no—
131C22×Dic83—21no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83—42no—
135C22×Dic83—42no—
136C22×Dic83—81no—
137C2×C83⋊C8—28no—
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—21no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8C2×D412yes—
149C2×C83⋊C8#4112yes—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—42no—
158C4⋊Dic83C2×D412yes—
159Dic83⋊C4—42no—
160Dic83⋊C4—42no—
161Dic83⋊C4—81no—
162Dic83⋊C4—81no—
163C22⋊Dic83—42no—
164C24×C83C2×D411yescharacteristic
165C2×Q8×C83—21no—
166C2×Q8×C83—21no—
167C2×Q8×C83—21no—
168C2×Q8×C83—21no—
169C2×Q8×C83—21no—
170C2×Q8×C83C2×Q812yes—
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83—21no—
174C22×C4×C83—21no—
175C22×C4×C83—21no—
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83—21no—
179C22×C4×C83—42no—
180C22×C4×C83—42no—
181C22×C4×C83—81no—
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2×D411yescharacteristic
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83C2×Q812yes—
188C22×C4×C83#4112yes—
189C22×C4×C83C2411yescharacteristic
190C4⋊C4×C83—24no—
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—42no—
196C4⋊C4×C83—42no—
197C4⋊C4×C83—81no—
198C4⋊C4×C83C2×D412yes—
199C4⋊C4×C83C2×Q812yes—
200C22⋊C4×C83—24no—
201C22⋊C4×C83—22no—
202C22⋊C4×C83—22no—
203C42×C83—21no—
204C42×C83—21no—
205C42×C83—21no—
206C42×C83—21no—
207C42×C83—42no—
208C42×C83—81no—
209C23×C83—21no—
210C23×C83—21no—
211C23×C83—21no—
212C23×C83—21no—
213C23×C83C22⋊Q812yes—
214C23×C83#13512yes—
215C23×C83#13712yes—
216C23×C83C22×D411yescharacteristic
217Q8×C83—22no—
218Q8×C83—22no—
219Q8×C83—41no—
220Q8×C83—41no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
223C2×C4×C83—22no—
224C2×C4×C83—22no—
225C2×C4×C83—22no—
226C2×C4×C83—22no—
227C2×C4×C83—22no—
228C2×C4×C83—22no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—22no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—21no—
238C2×C4×C83—21no—
239C2×C4×C83—21no—
240C2×C4×C83—21no—
241C2×C4×C83—21no—
242C2×C4×C83—41no—
243C2×C4×C83—41no—
244C2×C4×C83—41no—
245C2×C4×C83—41no—
246C2×C4×C83—41no—
247C2×C4×C83—41no—
248C2×C4×C83—41no—
249C2×C4×C83—41no—
250C2×C4×C83—41no—
251C2×C4×C83—84no—
252C2×C4×C83C22⋊Q812yes—
253C2×C4×C83C22×D411yescharacteristic
254C2×C4×C83C22×Q811yescharacteristic
255C2×C4×C83C2×#4111yescharacteristic
256C2×C4×C83C2×#4111yescharacteristic
257C2×C4×C83H(5,𝔽2)11yescharacteristic
258C2×C4×C832−1+411yescharacteristic
259C2×Dic83—22no—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—41no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C2×Dic83—41no—
268C2×Dic83—41no—
269C2×Dic83—41no—
270C2×Dic83—41no—
271C2×Dic83—84no—
272C83⋊C8—22no—
273C83⋊C8—22no—
274C83⋊C8—41no—
275C83⋊C8—41no—
276C22×C83—22no—
277C22×C83—22no—
278C22×C83—22no—
279C22×C83—22no—
280C22×C83—22no—
281C22×C83—22no—
282C22×C83#47914yescomplemented
283C22×C83C2×C22⋊Q811yescharacteristic
284C22×C83D4211yescharacteristic
285C22×C83#54711yescharacteristic
286C22×C83C2×#13511yescharacteristic
287C22×C83C2×#13711yescharacteristic
288C22×C83#57611yescharacteristic
289C22×C83#57811yescharacteristic
290C4×C83—22no—
291C4×C83—41no—
292C4×C83—42no—
293C4×C83—41no—
294C4×C83—42no—
295C4×C83—41no—
296C4×C83—41no—
297C4×C83—42no—
298C4×C83—81no—
299C4×C83#52511yescharacteristic
300C4×C83#53911yescharacteristic
301C4×C83D4×Q811yescharacteristic
302C4×C83#55411yescharacteristic
303Dic83—42no—
304Dic83—81no—
305Dic83—81no—
306Dic83—81no—
307#26344—831nomaximal, Sylow
308C2×C83—24no—
309C2×C83#244911yescharacteristic
310C2×C83C2×#47911yescharacteristic
311C2×C83#308311yescharacteristic
312C2×C83#327011yescharacteristic
313C2×C83#328011yescharacteristic
314C2×C83#333611yescharacteristic
315C2×C83#334611yescharacteristic
316C2×#479—831no—
317C22×#107—831no—
318#2442—831no—
319#2360D8311yescharacteristic
320#2097—831no—
321#2086—831no—
322#2068—831no—
323#2059—831no—
324#1949—831no—
325#1918—831no—
326#1892—831no—
327#1872—831no—
328#1863—831no—
329#1853—831no—
330#1850—831no—
331C83#2634411yescharacteristic, complemented, Sylow
332C23×C8—831no—
333C2×#125—831no—
334C2×C22⋊Q8C2×D8311yescharacteristic
335C2×C4×Q8C2×D8311yescharacteristic
336C22×C4⋊C4—831no—
337C22×C4⋊C4C2×D8311yescharacteristic
338#479—1664no—
339C2×#107—831no—
340C2×#107—831no—
341C2×#107—831no—
342C2×#107—831no—
343C2×#107—1664no—
344C2×#106—831no—
345C2×#106—831no—
346C2×C4⋊C8—831no—
347C2×C4⋊C8—831no—
348C2×#103—831no—
349C2×#103—831no—
350C2×#103—831no—
351C2×#103—831no—
352C2×C22⋊C8—831no—
353#400—1661no—
354#398—1661no—
355#391C2×D8311yescharacteristic
356#389—21no—
357#389—831no—
358#389—831no—
359#389C2×D8311yescharacteristic
360#387—21no—
361#385—21no—
362#385—1661no—
363#382—831no—
364#380—831no—
365#380—1661no—
366C4×C4⋊C4C2×D8311yescharacteristic
367C4×C22⋊C4—21no—
368C4×C22⋊C4C2×D8311yescharacteristic
369#340—831no—
370#340—831no—
371#340—831no—
372#340—831no—
373#340—831no—
374#340—831no—
375#336—831no—
376#336—831no—
377C22×Q8C22×D8311yescharacteristic
378C23×C4—21no—
379C23×C4—1661no—
380C23×C4C22×D8311yescharacteristic
381C22×C8—831no—
382C22×C8—831no—
383C22×C8—831no—
384C22×C8—831no—
385C22×C8—1662no—
386C22×C8—1662no—
387C22×C8—1662no—
388#125—3321no—
389#125—1662no—
390C22⋊Q8—22no—
391C22⋊Q8Dic16614yes—
392C4×Q8—22no—
393C4×Q8—41no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—41no—
400C2×C4⋊C4—3321no—
401C2×C4⋊C4—831no—
402C2×C4⋊C4—831no—
403C2×C4⋊C4—831no—
404C2×C4⋊C4—831no—
405C2×C4⋊C4—1661no—
406C2×C4⋊C4—1661no—
407C2×C4⋊C4—1661no—
408C2×C4⋊C4—1662no—
409C2×C4⋊C4—1661no—
410C2×C4⋊C4—1661no—
411C2×C4⋊C4C22×D8311yescharacteristic
412C2×C4⋊C4C22×D8311yescharacteristic
413C2×C4⋊C4C22×D8311yescharacteristic
414C2×C22⋊C4—21no—
415C2×C22⋊C4—1661no—
416C2×C22⋊C4C22⋊D8311yescharacteristic
417C2×C22⋊C4C22⋊D8311yescharacteristic
418C2×C42—21no—
419C2×C42—21no—
420C2×C42C22×D8311yescharacteristic
421C2×C42C22×D8311yescharacteristic
422#107—834no—
423#107—1662no—
424#107—1662no—
425#107—1662no—
426#106—3321no—
427#106—1662no—
428C4⋊C8—1662no—
429C4⋊C8—1662no—
430#103—3321no—
431#103—1662no—
432#103—1662no—
433#103—1662no—
434C22⋊C8—1662no—
435#95—21no—
436#95—21no—
437#95—21no—
438#95—21no—
439#95—1661no—
440#95—1661no—
441#95—1661no—
442#95—1661no—
443#95—1661no—
444#95C22⋊D8311yescharacteristic
445#95C22⋊D8311yescharacteristic
446C24C2×C22⋊D8311yescharacteristic
447C2×Q8—21no—
448C2×Q8—21no—
449C2×Q8—21no—
450C2×Q8—21no—
451C2×Q8—21no—
452C2×Q8C2×Dic16612yes—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—81no—
464C22×C4—3322no—
465C22×C4—3322no—
466C22×C4—6641no—
467C22×C4—1661no—
468C22×C4—1661no—
469C22×C4—1661no—
470C22×C4—1661no—
471C22×C4C2×C22⋊D8311yescharacteristic
472C22×C4C2×C22⋊D8311yescharacteristic
473C22×C4C2×C22⋊D8311yescharacteristic
474C22×C4C2×C22⋊D8311yescharacteristic
475C22×C4C2×C22⋊D8311yescharacteristic
476C22×C4C2×Dic16612yes—
477C22×C4#1305059012yes—
478C22×C4C23×D8311yescharacteristic
479C2×C8—832no—
480C2×C8—832no—
481C2×C8—1661no—
482C2×C8—1661no—
483C2×C8—1661no—
484C2×C8—1661no—
485C2×C8—1661no—
486C2×C8—1661no—
487C2×C8—1668no—
488C2×C8—1661no—
489C2×C8—1661no—
490C2×C8—1661no—
491C2×C8—1661no—
492C4⋊C4—24no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—42no—
498C4⋊C4—42no—
499C4⋊C4—81no—
500C4⋊C4—3322no—
501C4⋊C4—3322no—
502C4⋊C4—3322no—
503C4⋊C4—832no—
504C4⋊C4—6641no—
505C4⋊C4—6641no—
506C4⋊C4—1661no—
507C4⋊C4—1661no—
508C4⋊C4—1661no—
509C4⋊C4—1661no—
510C4⋊C4—1661no—
511C4⋊C4—1661no—
512C4⋊C4—1661no—
513C4⋊C4D4×D8312yes—
514C4⋊C4C2×Dic16612yes—
515C22⋊C4—24no—
516C22⋊C4—22no—
517C22⋊C4—22no—
518C22⋊C4—3322no—
519C42—21no—
520C42—21no—
521C42—21no—
522C42—21no—
523C42—42no—
524C42—81no—
525C23—21no—
526C23—21no—
527C23—21no—
528C23—21no—
529C23#1458177112yes—
530C23#1458181212yes—
531C23#1458182312yes—
532C23C22×C22⋊D8311yescharacteristic
533Q8—22no—
534Q8—22no—
535Q8—41no—
536Q8—41no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—84no—
568C2×C4—3321no—
569C2×C4—3321no—
570C2×C4—3321no—
571C2×C4—3321no—
572C2×C4—3321no—
573C2×C4—3321no—
574C2×C4—3321no—
575C2×C4—3321no—
576C2×C4—3321no—
577C2×C4—3321no—
578C2×C4—3321no—
579C2×C4—6644no—
580C2×C4—1662no—
581C2×C4#1458177112yes—
582C2×C4C2×D4×D8311yescharacteristic
583C2×C4C22×Dic16611yescharacteristic
584C2×C4#41×D8311yescharacteristic
585C2×C4C2×#1305059011yescharacteristic
586C2×C4#1458183711yescharacteristic
587C2×C4#1458183911yescharacteristic
588C8—3321no—
589C8—3321no—
590C8—1662no—
591C8—1662no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22#1912800814yescomplemented
599C22C2×#1458177111yescharacteristic
600C22D4×C22⋊D8311yescharacteristic
601C22#1912835011yescharacteristic
602C22C2×#1458181211yescharacteristic
603C22C2×#1458182311yescharacteristic
604C22#1912850411yescharacteristic
605C22#1912851611yescharacteristic
606C4—22no—
607C4—41no—
608C4—42no—
609C4—41no—
610C4—42no—
611C4—41no—
612C4—42no—
613C4—41no—
614C4—81no—
615C4—3322no—
616C4—6641no—
617C4—6641no—
618C4—6641no—
619C4#1912821011yescharacteristic
620C4#1912829311yescharacteristic
621C4D4×Dic16611yescharacteristic
622C4#1912841011yescharacteristic
623C2—24no—
624C2#2491126611yescharacteristic
625C2C2×#1912800811yescharacteristic
626C2#2491731411yescharacteristic
627C2#2491903411yescharacteristic
628C2#2491914411yescharacteristic
629C2#2491982511yescharacteristic
630C2#2491994211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2360C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#2360D83
hypocenterC83#26344
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2360

Complements

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

NH ≅ G/Ntype
1. C2×#19128008C2semidirect
4. #24909011C2semidirect
5. #24908954C2semidirect
6. #24908858C2semidirect
7. #24908801C2semidirect
8. #24908231C2semidirect
9. #24908034C2semidirect
12. #24907717C2semidirect
282. C22×C83#479semidirect
331. C83#26344semidirect
598. C22#19128008semidirect

Maximal subgroups

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

subgroupindex
C2×#191280082
C22×#145817252
#249111872
#249090112
#249089542
#249088582
#249088012
#249082312
#249080342
#249078872
#249077712
#249077172
#249076562
#249076432
C83×#23602
#2634483