Finite Group Databasebeta

← #35462852

Subgroups — #35462852

Order 21248.

834 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128001C211yesmaximal, characteristic
2C2×#19127981C211yesmaximal, characteristic
3#24912484C211yesmaximal, characteristic, complemented
4#24908959C211yesmaximal, characteristic, complemented
5#24908954C211yesmaximal, characteristic, complemented
6#24908801C211yesmaximal, characteristic, complemented
7#24908796C211yesmaximal, characteristic, complemented
8#24908229C211yesmaximal, characteristic, complemented
9#24908182C211yesmaximal, characteristic, complemented
10#24907724C211yesmaximal, characteristic
11#24907724C211yesmaximal, characteristic
12#24907625C211yesmaximal, characteristic, complemented
13#24907473C211yesmaximal, characteristic
14C4×#14581713C211yesmaximal, characteristic
15C83×#2349C211yesmaximal, characteristic
16C2×#14581792C2211yescharacteristic
17C2×#14581784C2211yescharacteristic
18#19128001—24no—
19#19127981—24no—
20C2×#14581725C2211yescharacteristic
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×C4⋊(C83⋊C8)C2211yescharacteristic
27C2×C4⋊(C83⋊C8)C2211yescharacteristic
28C2×C4⋊(C83⋊C8)C2211yescharacteristic
29C2×#14581713C2211yescharacteristic
30C2×#14581713C2211yescharacteristic
31C2×#14581713C2211yescharacteristic
32C2×#14581713C2211yescharacteristic
33C2×C4×C83⋊C8C2211yescharacteristic
34#19127634—21no—
35#19127621—21no—
36#19127605—21no—
37#19127599C2211yescharacteristic
38#19127599C2211yescharacteristic
39#19127598—21no—
40#19127567C2211yescharacteristic
41C4×C4⋊Dic83C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×C4×D4×C83C2211yescharacteristic
51C2×C4×D4×C83C2211yescharacteristic
52C2×C4×D4×C83C2211yescharacteristic
53#389×C83C2211yescharacteristic
54#389×C83C2211yescharacteristic
55#385×C83—21no—
56#385×C83—21no—
57#382×C83C2211yescharacteristic
58#380×C83—21no—
59C4×C4⋊C4×C83C2211yescharacteristic
60C2×#95×C83—21no—
61C22×Dic166—21no—
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66#14581792—22no—
67#14581792—41no—
68#14581784—22no—
69#14581784—41no—
70C2×C4⋊Dic83—21no—
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×Dic83⋊C4—21no—
76C2×Dic83⋊C4—21no—
77C2×Dic83⋊C4—21no—
78C2×Dic83⋊C4—21no—
79C2×C4×Dic83—21no—
80#14581725—22no—
81#14581725—22no—
82#14581725—22no—
83#14581723—22no—
84#14581723—41no—
85#14581723—41no—
86C4⋊(C83⋊C8)—22no—
87C4⋊(C83⋊C8)—22no—
88C4⋊(C83⋊C8)—22no—
89#14581713—22no—
90#14581713—22no—
91#14581713D414yes—
92#14581713Q814yescomplemented
93C4×C83⋊C8—22no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C22×D4×C83C2311yescharacteristic
100C23×C4×C83—21no—
101C23×C4×C83—21no—
102C23×C4×C83—21no—
103C4×D4×C83—22no—
104C4×D4×C83—22no—
105C4×D4×C83—22no—
106C4×D4×C83—41no—
107C4×D4×C83—41no—
108C4×D4×C83Q814yescomplemented
109C2×C4⋊C4×C83—21no—
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83D411yescharacteristic
112C2×C4⋊C4×C83D411yescharacteristic
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C22⋊C4×C83—21no—
117C2×C22⋊C4×C83—21no—
118C2×C22⋊C4×C83—21no—
119C2×C42×C83C2311yescharacteristic
120C2×C42×C83C2311yescharacteristic
121C2×C42×C83C2311yescharacteristic
122#95×C83—21no—
123#95×C83—21no—
124#95×C83—21no—
125#95×C83—21no—
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—
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⋊C8C2×D412yes—
146C2×C83⋊C8C2×Q812yes—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—42no—
154C4⋊Dic83C2×D412yes—
155C4⋊Dic83C2×Q812yes—
156Dic83⋊C4—42no—
157Dic83⋊C4—42no—
158Dic83⋊C4—42no—
159Dic83⋊C4—81no—
160C4×Dic83—81no—
161C24×C83—21no—
162C2×D4×C83—21no—
163C2×D4×C83—21no—
164C2×D4×C83—21no—
165C2×D4×C83C2×D412yes—
166C2×D4×C83C2×Q812yes—
167C2×D4×C83C2×Q812yes—
168C22×C4×C83—21no—
169C22×C4×C83—24no—
170C22×C4×C83—21no—
171C22×C4×C83—21no—
172C22×C4×C83—42no—
173C22×C4×C83—42no—
174C22×C4×C83—42no—
175C22×C4×C83—42no—
176C22×C4×C83—81no—
177C22×C4×C83—81no—
178C22×C4×C83—81no—
179C22×C4×C83—81no—
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2×D411yescharacteristic
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83C2411yescharacteristic
187C4⋊C4×C83—22no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—21no—
190C4⋊C4×C83—21no—
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—41no—
194C4⋊C4×C83—81no—
195C4⋊C4×C83—81no—
196C4⋊C4×C83C2×Q812yes—
197C22⋊C4×C83—24no—
198C22⋊C4×C83—42no—
199C22⋊C4×C83—42no—
200C42×C83—21no—
201C42×C83—21no—
202C42×C83—21no—
203C42×C83—21no—
204C42×C83C2×Q812yes—
205C42×C83#4112yes—
206C23×C83—22no—
207C23×C83—22no—
208C23×C83—22no—
209C23×C83—41no—
210C23×C83—41no—
211C23×C83—41no—
212C23×C83—41no—
213C23×C83C22×D411yescharacteristic
214D4×C83—22no—
215D4×C83—22no—
216D4×C83—22no—
217D4×C83C4⋊Q814yes—
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—22no—
225C2×C4×C83—21no—
226C2×C4×C83—21no—
227C2×C4×C83—21no—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
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×C83—41no—
242C2×C4×C83—84no—
243C2×C4×C83—84no—
244C2×C4×C83—84no—
245C2×C4×C83C22⋊Q812yes—
246C2×C4×C83C22⋊Q812yes—
247C2×C4×C83#13512yes—
248C2×C4×C83C22×D411yescharacteristic
249C2×C4×C83C22×Q811yescharacteristic
250C2×C4×C83C22×Q811yescharacteristic
251C2×C4×C83C2×#4111yescharacteristic
252C2×C4×C83H(5,𝔽2)11yescharacteristic
253C2×C4×C832−1+411yescharacteristic
254C2×Dic83—22no—
255C2×Dic83—22no—
256C2×Dic83—41no—
257C2×Dic83—41no—
258C2×Dic83—41no—
259C2×Dic83—41no—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—41no—
265C2×Dic83—41no—
266Dic166—84no—
267C83⋊C8—22no—
268C83⋊C8—22no—
269C83⋊C8—41no—
270C83⋊C8—41no—
271C22×C83—24no—
272C22×C83—42no—
273C22×C83—42no—
274C22×C83—42no—
275C22×C83—42no—
276C22×C83—42no—
277C22×C83—42no—
278C22×C83C2×C22⋊Q811yescharacteristic
279C22×C83D4211yescharacteristic
280C22×C83D4×Q811yescharacteristic
281C22×C83C2×#13511yescharacteristic
282C22×C83#57511yescharacteristic
283C22×C83#57611yescharacteristic
284C22×C83#57711yescharacteristic
285C4×C83—22no—
286C4×C83—22no—
287C4×C83—22no—
288C4×C83—42no—
289C4×C83—41no—
290C4×C83—41no—
291C4×C83—41no—
292C4×C83—81no—
293C4×C83—81no—
294C4×C83C2×C4⋊Q811yescharacteristic
295C4×C83#54411yescharacteristic
296C4×C83#54711yescharacteristic
297C4×C83#55411yescharacteristic
298Dic83—42no—
299Dic83—42no—
300Dic83—81no—
301Dic83—81no—
302#26049—831nomaximal, Sylow
303C2×C83—44no—
304C2×C83#244011yescharacteristic
305C2×C83#307611yescharacteristic
306C2×C83#308111yescharacteristic
307C2×C83#327511yescharacteristic
308C2×C83#328011yescharacteristic
309C2×C83#338111yescharacteristic
310C2×C83#338911yescharacteristic
311C2×#478—831no—
312C2×#474—831no—
313#2550—831no—
314#2349D8311yescharacteristic
315#2087—831no—
316#2086—831no—
317#2059—831no—
318#2058—831no—
319#1949—831no—
320#1939—831no—
321#1864—831no—
322#1864—831no—
323#1846—831no—
324#1804—831no—
325C4×#102—831no—
326C83#2604911yescharacteristic, complemented, Sylow
327C2×C4⋊Q8—831no—
328C2×#125—831no—
329C2×C4×D4C2×D8311yescharacteristic
330C2×C4×D4C2×D8311yescharacteristic
331C2×C4×D4C2×D8311yescharacteristic
332#478—1664no—
333#474—1664no—
334C2×#107—831no—
335C2×#107—831no—
336C2×#107—831no—
337C2×#106—831no—
338C2×#106—831no—
339C2×#106—831no—
340C2×C4⋊C8—831no—
341C2×C4⋊C8—831no—
342C2×C4⋊C8—831no—
343C2×#102—831no—
344C2×#102—831no—
345C2×#102—831no—
346C2×#102—831no—
347C2×C4×C8—831no—
348#398—1661no—
349#395—1661no—
350#391—1661no—
351#389—831no—
352#389—831no—
353#389—1661no—
354#389C2×D8311yescharacteristic
355#389C2×D8311yescharacteristic
356#385—21no—
357#385—21no—
358#382—831no—
359#382C2×D8311yescharacteristic
360#380—21no—
361C4×C4⋊C4—831no—
362C4×C4⋊C4C2×D8311yescharacteristic
363C2×#95—21no—
364#340—831no—
365#340—831no—
366#340—831no—
367#340—831no—
368#340—831no—
369#340—831no—
370#336—831no—
371#336—831no—
372C22×Q8—1661no—
373C22×D4C22×D8311yescharacteristic
374C23×C4—21no—
375C23×C4—21no—
376C23×C4—21no—
377C22×C8—831no—
378C22×C8—831no—
379C22×C8—831no—
380C22×C8—831no—
381C4⋊Q8—3321no—
382C4⋊Q8—1662no—
383#125—3321no—
384#125—1662no—
385C4×D4—22no—
386C4×D4—22no—
387C4×D4—22no—
388C4×D4—41no—
389C4×D4—41no—
390C4×D4Dic16614yescomplemented
391C2×C4⋊C4—21no—
392C2×C4⋊C4—21no—
393C2×C4⋊C4—831no—
394C2×C4⋊C4—831no—
395C2×C4⋊C4—831no—
396C2×C4⋊C4—831no—
397C2×C4⋊C4—1661no—
398C2×C4⋊C4—1661no—
399C2×C4⋊C4—1661no—
400C2×C4⋊C4—1661no—
401C2×C4⋊C4—1661no—
402C2×C4⋊C4C22⋊D8311yescharacteristic
403C2×C4⋊C4C22⋊D8311yescharacteristic
404C2×C4⋊C4C22×D8311yescharacteristic
405C2×C4⋊C4C22×D8311yescharacteristic
406C2×C4⋊C4C22×D8311yescharacteristic
407C2×C22⋊C4—21no—
408C2×C22⋊C4—21no—
409C2×C22⋊C4—21no—
410C2×C42—1661no—
411C2×C42C22×D8311yescharacteristic
412C2×C42C22×D8311yescharacteristic
413C2×C42C22×D8311yescharacteristic
414#107—1662no—
415#107—1662no—
416#107—1662no—
417#106—3321no—
418#106—3321no—
419#106—1662no—
420C4⋊C8—1662no—
421C4⋊C8—1662no—
422C4⋊C8—1662no—
423#102—834no—
424#102—834no—
425#102—1662no—
426#102—1662no—
427C4×C8—1662no—
428#95—21no—
429#95—21no—
430#95—21no—
431#95—21no—
432#95—1661no—
433#95—1661no—
434#95—1661no—
435#95—1661no—
436#95—1661no—
437#95C22⋊D8311yescharacteristic
438#95C22⋊D8311yescharacteristic
439C24—21no—
440C2×Q8—3322no—
441C2×Q8—3322no—
442C2×Q8—6641no—
443C2×D4—21no—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4D4×D8312yes—
447C2×D4Q8×D8312yes—
448C2×D4C2×Dic16612yes—
449C22×C4—21no—
450C22×C4—24no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—42no—
454C22×C4—42no—
455C22×C4—42no—
456C22×C4—42no—
457C22×C4—81no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4—81no—
461C22×C4—1661no—
462C22×C4—1661no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4C2×C22⋊D8311yescharacteristic
466C22×C4C2×C22⋊D8311yescharacteristic
467C22×C4C2×C22⋊D8311yescharacteristic
468C22×C4C2×C22⋊D8311yescharacteristic
469C22×C4C2×C22⋊D8311yescharacteristic
470C22×C4C2×C22⋊D8311yescharacteristic
471C22×C4C23×D8311yescharacteristic
472C2×C8—832no—
473C2×C8—832no—
474C2×C8—1661no—
475C2×C8—1661no—
476C2×C8—1661no—
477C2×C8—1661no—
478C2×C8—1661no—
479C2×C8—1661no—
480C2×C8—1661no—
481C2×C8—1661no—
482C2×C8—1661no—
483C2×C8—1661no—
484C4⋊C4—22no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—41no—
491C4⋊C4—81no—
492C4⋊C4—81no—
493C4⋊C4—3322no—
494C4⋊C4—3322no—
495C4⋊C4—3322no—
496C4⋊C4—3322no—
497C4⋊C4—832no—
498C4⋊C4—832no—
499C4⋊C4—6641no—
500C4⋊C4—1661no—
501C4⋊C4—1661no—
502C4⋊C4—1661no—
503C4⋊C4—1661no—
504C4⋊C4—1661no—
505C4⋊C4—1661no—
506C4⋊C4C2×Dic16612yes—
507C22⋊C4—24no—
508C22⋊C4—42no—
509C22⋊C4—42no—
510C42—21no—
511C42—21no—
512C42—21no—
513C42—21no—
514C42—6641no—
515C42C2×Dic16612yes—
516C42#1305059012yes—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—41no—
521C23—41no—
522C23—41no—
523C23—41no—
524C23C22×C22⋊D8311yescharacteristic
525Q8—6644no—
526D4—22no—
527D4—22no—
528D4—22no—
529D4#1458179114yes—
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—22no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—41no—
549C2×C4—41no—
550C2×C4—41no—
551C2×C4—41no—
552C2×C4—41no—
553C2×C4—41no—
554C2×C4—84no—
555C2×C4—84no—
556C2×C4—84no—
557C2×C4—3321no—
558C2×C4—3321no—
559C2×C4—3321no—
560C2×C4—3321no—
561C2×C4—3321no—
562C2×C4—3321no—
563C2×C4—3321no—
564C2×C4—3321no—
565C2×C4—3321no—
566C2×C4—3321no—
567C2×C4—1662no—
568C2×C4—1662no—
569C2×C4#1458177112yes—
570C2×C4#1458177112yes—
571C2×C4#1458181212yes—
572C2×C4C2×D4×D8311yescharacteristic
573C2×C4C2×Q8×D8311yescharacteristic
574C2×C4C22×Dic16611yescharacteristic
575C2×C4C2×#1305059011yescharacteristic
576C2×C4#1458183711yescharacteristic
577C2×C4#1458184011yescharacteristic
578C8—3321no—
579C8—3321no—
580C8—1662no—
581C8—1662no—
582C22—24no—
583C22—42no—
584C22—42no—
585C22—42no—
586C22—42no—
587C22—42no—
588C22—42no—
589C22C2×#1458177111yescharacteristic
590C22D4×C22⋊D8311yescharacteristic
591C22Q8×C22⋊D8311yescharacteristic
592C22C2×#1458181211yescharacteristic
593C22#1912849911yescharacteristic
594C22#1912850411yescharacteristic
595C22#1912851211yescharacteristic
596C4—22no—
597C4—22no—
598C4—22no—
599C4—42no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—81no—
604C4—81no—
605C4—3322no—
606C4—3322no—
607C4—6641no—
608C4—6641no—
609C4C2×#1458179111yescharacteristic
610C4#1912833211yescharacteristic
611C4#1912835211yescharacteristic
612C4#1912840611yescharacteristic
613C2—44no—
614C2#2491115911yescharacteristic
615C2#2491725111yescharacteristic
616C2#2491729411yescharacteristic
617C2#2491909011yescharacteristic
618C2#2491914411yescharacteristic
619C2#2492033811yescharacteristic
620C2#2492041711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2349C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
hypercenter#2349D83
hypocenterC83#26049
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2349

Complements

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

NH ≅ G/Ntype
3. #24912484C2semidirect
4. #24908959C2semidirect
5. #24908954C2semidirect
6. #24908801C2semidirect
7. #24908796C2semidirect
8. #24908229C2semidirect
9. #24908182C2semidirect
12. #24907625C2semidirect
92. #14581713Q8semidirect
108. C4×D4×C83Q8semidirect
326. C83#26049semidirect
390. C4×D4Dic166semidirect

Maximal subgroups

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

subgroupindex
C2×#191280012
C2×#191279812
#249124842
#249089592
#249089542
#249088012
#249087962
#249082292
#249081822
#249077242
#249076252
#249074732
C4×#145817132
C83×#23492
#2604983