Finite Group Databasebeta

← #35504324

Subgroups — #35504324

Order 21248.

734 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24913582C211yesmaximal, characteristic, complemented
2#24908614C211yesmaximal, characteristic, complemented
3#24908609C211yesmaximal, characteristic
4#24908604C211yesmaximal, characteristic, complemented
5#24908599C211yesmaximal, characteristic
6#24908280C211yesmaximal, characteristic
7#24908275C211yesmaximal, characteristic
8#24907598C211yesmaximal, characteristic, complemented
9#24907590C211yesmaximal, characteristic, complemented
10#24907478C211yesmaximal, characteristic, complemented
11#24907473C211yesmaximal, characteristic
12#24907468C211yesmaximal, characteristic, complemented
13#24907463C211yesmaximal, characteristic
14#24907320C211yesmaximal, characteristic, complemented
15C83×#2326C211yesmaximal, characteristic
16C2×#14581792C2211yescharacteristic
17C2×#14581784C2211yescharacteristic
18C2×C4⋊(C83⋊C8)C2211yescharacteristic
19C2×C4⋊(C83⋊C8)C2211yescharacteristic
20C2×#14581716C2211yescharacteristic
21C2×#14581716C2211yescharacteristic
22C2×#14581716C2211yescharacteristic
23C2×#14581716C2211yescharacteristic
24C2×#14581713C2211yescharacteristic
25C2×#14581713C2211yescharacteristic
26C2×#14581713C2211yescharacteristic
27C2×#14581713C2211yescharacteristic
28C2×#14581703C2211yescharacteristic
29C2×C4×C83⋊C8C2211yescharacteristic
30#19127633—21no—
31#19127633—21no—
32#19127622—21no—
33#19127622—21no—
34#19127571C2211yescharacteristic
35#19127571C2211yescharacteristic
36#19127567C2211yescharacteristic
37#19127567C2211yescharacteristic
38#19127463C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48C2×#124×C83C2211yescharacteristic
49C2×C4×Q8×C83C2211yescharacteristic
50C2×C4×D4×C83C2211yescharacteristic
51#391×C83C2211yescharacteristic
52#390×C83C2211yescharacteristic
53#388×C83—21no—
54#388×C83—21no—
55#385×C83—21no—
56#385×C83—21no—
57#384×C83C2211yescharacteristic
58#376×C83C2211yescharacteristic
59C22×Dic166—21no—
60C22×C83⋊C8C2311yescharacteristic
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64#14581792—41no—
65#14581792—41no—
66#14581784—41no—
67#14581784—41no—
68C2×C4⋊Dic83—21no—
69C2×C4⋊Dic83—21no—
70C2×C4⋊Dic83—21no—
71C2×C4⋊Dic83—21no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×Dic83⋊C4—21no—
78C2×Dic83⋊C4—21no—
79C4⋊(C83⋊C8)—22no—
80C4⋊(C83⋊C8)—22no—
81#14581716—41no—
82#14581716—41no—
83#14581716—41no—
84#14581716—41no—
85#14581713—41no—
86#14581713—41no—
87#14581713—41no—
88#14581713—41no—
89#14581703—22no—
90C4×C83⋊C8—22no—
91C83⋊#95—21no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C22×Q8×C83C2311yescharacteristic
96C22×D4×C83C2311yescharacteristic
97C23×C4×C83—21no—
98#124×C83—22no—
99#124×C83Q814yescomplemented
100C4×Q8×C83—22no—
101C4×Q8×C83—22no—
102C4×D4×C83—22no—
103C4×D4×C83—22no—
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83—21no—
106C2×C4⋊C4×C83—21no—
107C2×C4⋊C4×C83C2311yescharacteristic
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C22⋊C4×C83—21no—
110C2×C22⋊C4×C83—21no—
111C2×C22⋊C4×C83—21no—
112C2×C22⋊C4×C83—21no—
113C2×C22⋊C4×C83—21no—
114C2×C42×C83—21no—
115C2×C42×C83C2311yescharacteristic
116C2×C42×C83C2311yescharacteristic
117C2×C42×C83C2311yescharacteristic
118#95×C83—21no—
119#95×C83—21no—
120#95×C83D411yescharacteristic
121#95×C83D411yescharacteristic
122#95×C83D411yescharacteristic
123#95×C83D411yescharacteristic
124C2×Dic166—42no—
125C2×Dic166—81no—
126C2×Dic166—81no—
127C22×Dic83—21no—
128C22×Dic83—21no—
129C22×Dic83—21no—
130C22×Dic83—21no—
131C2×C83⋊C8—21no—
132C2×C83⋊C8—21no—
133C2×C83⋊C8—21no—
134C2×C83⋊C8—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—
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—42no—
152C4⋊Dic83—81no—
153C4⋊Dic83—81no—
154C4⋊Dic83—81no—
155C4⋊Dic83—81no—
156Dic83⋊C4—42no—
157Dic83⋊C4—42no—
158C24×C83—21no—
159C2×Q8×C83—21no—
160C2×Q8×C83—21no—
161C2×Q8×C83—21no—
162C2×Q8×C83—21no—
163C2×Q8×C83—21no—
164C2×Q8×C83C2×Q812yes—
165C2×D4×C83—21no—
166C2×D4×C83—21no—
167C2×D4×C83—21no—
168C2×D4×C83—21no—
169C2×D4×C83—21no—
170C2×D4×C83C2×Q812yes—
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83—21no—
174C22×C4×C83—42no—
175C22×C4×C83—81no—
176C22×C4×C83—81no—
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2×D411yescharacteristic
180C22×C4×C83C2×D411yescharacteristic
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2411yescharacteristic
184C4⋊C4×C83—21no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—42no—
189C4⋊C4×C83—81no—
190C4⋊C4×C83—81no—
191C22⋊C4×C83—24no—
192C22⋊C4×C83—24no—
193C22⋊C4×C83—42no—
194C22⋊C4×C83—81no—
195C22⋊C4×C83—81no—
196C42×C83—21no—
197C42×C83—21no—
198C42×C83—42no—
199C42×C83C2×Q812yes—
200C42×C83#4112yes—
201C42×C83#4112yes—
202C42×C83#4112yes—
203C23×C83—22no—
204C23×C83—41no—
205C23×C83—41no—
206C23×C83—41no—
207C23×C83—41no—
208C23×C83—41no—
209C23×C83—41no—
210C23×C83C22×D411yescharacteristic
211Q8×C83—22no—
212Q8×C83—22no—
213Q8×C83—41no—
214Q8×C83—41no—
215D4×C83—22no—
216D4×C83—22no—
217D4×C83—41no—
218D4×C83—41no—
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—22no—
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×C83C22⋊Q812yes—
244C2×C4×C83C22⋊Q812yes—
245C2×C4×C83#12412yes—
246C2×C4×C83#12412yes—
247C2×C4×C83C22×Q811yescharacteristic
248C2×C4×C83C2×#4111yescharacteristic
249C2×C4×C83C2×#4111yescharacteristic
250C2×C4×C83C2×#4111yescharacteristic
251C2×C4×C83H(5,𝔽2)11yescharacteristic
252C2×C4×C83H(5,𝔽2)11yescharacteristic
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
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—
265Dic166—84no—
266C83⋊C8—41no—
267C83⋊C8—41no—
268C83⋊C8—41no—
269C83⋊C8—41no—
270C22×C83—42no—
271C22×C83—42no—
272C22×C83—42no—
273C22×C83—81no—
274C22×C83—81no—
275C22×C83—81no—
276C22×C83—81no—
277C22×C83C2×C22⋊Q811yescharacteristic
278C22×C83C2×#12411yescharacteristic
279C22×C83#53511yescharacteristic
280C22×C83#57611yescharacteristic
281C22×C83#57711yescharacteristic
282C22×C83#57711yescharacteristic
283C22×C83#57811yescharacteristic
284C4×C83—22no—
285C4×C83—22no—
286C4×C83—22no—
287C4×C83—22no—
288C4×C83—42no—
289C4×C83—41no—
290C4×C83—41no—
291C4×C83—81no—
292C4×C83—81no—
293C4×C83#55411yescharacteristic
294C4×C83#55411yescharacteristic
295C4×C83#55511yescharacteristic
296C4×C83#55511yescharacteristic
297Dic83—81no—
298Dic83—81no—
299Dic83—81no—
300Dic83—81no—
301#29151—831nomaximal, Sylow
302C2×C83—82no—
303C2×C83#256311yescharacteristic
304C2×C83#308311yescharacteristic
305C2×C83#308411yescharacteristic
306C2×C83#313611yescharacteristic
307C2×C83#313711yescharacteristic
308C2×C83#322411yescharacteristic
309C2×C83#322511yescharacteristic
310#2660—831no—
311#2326D8311yescharacteristic
312#2026—831no—
313#2025—831no—
314#2024—831no—
315#2023—831no—
316#1959—831no—
317#1958—831no—
318#1836—831no—
319#1834—831no—
320#1805—831no—
321#1804—831no—
322#1803—831no—
323#1802—831no—
324#1768—831no—
325C83#2915111yescharacteristic, complemented, Sylow
326C2×C4⋊Q8—831no—
327C2×#125—831no—
328C2×#124C2×D8311yescharacteristic
329C2×C4×Q8C2×D8311yescharacteristic
330C2×C4×D4C2×D8311yescharacteristic
331C2×C4⋊C8—831no—
332C2×C4⋊C8—831no—
333C2×#103—831no—
334C2×#103—831no—
335C2×#103—831no—
336C2×#103—831no—
337C2×#102—831no—
338C2×#102—831no—
339C2×#102—831no—
340C2×#102—831no—
341C2×#97—831no—
342C2×C4×C8—831no—
343#398—1661no—
344#398—1661no—
345#395—1661no—
346#395—1661no—
347#391C2×D8311yescharacteristic
348#390C2×D8311yescharacteristic
349#388—21no—
350#388—21no—
351#385—21no—
352#385—21no—
353#384—831no—
354#384—831no—
355#384C2×D8311yescharacteristic
356#382—831no—
357#382—831no—
358#376C2×D8311yescharacteristic
359#340—831no—
360#340—831no—
361#340—831no—
362#340—831no—
363#340—831no—
364#340—831no—
365#340—831no—
366#340—831no—
367#336—831no—
368#336—831no—
369C22×Q8—1661no—
370C22×Q8C22×D8311yescharacteristic
371C22×D4C22×D8311yescharacteristic
372C23×C4—21no—
373C22×C8—831no—
374C22×C8—831no—
375C22×C8—831no—
376C22×C8—831no—
377C4⋊Q8—3321no—
378C4⋊Q8—3321no—
379#125—3321no—
380#125—3321no—
381#124—22no—
382#124Dic16614yescomplemented
383C4×Q8—22no—
384C4×Q8—22no—
385C4×D4—22no—
386C4×D4—22no—
387C2×C4⋊C4—21no—
388C2×C4⋊C4—21no—
389C2×C4⋊C4—21no—
390C2×C4⋊C4—831no—
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⋊C4—1661no—
398C2×C4⋊C4—1661no—
399C2×C4⋊C4—1661no—
400C2×C4⋊C4—1661no—
401C2×C4⋊C4C22×D8311yescharacteristic
402C2×C4⋊C4C22×D8311yescharacteristic
403C2×C22⋊C4—21no—
404C2×C22⋊C4—21no—
405C2×C22⋊C4—21no—
406C2×C22⋊C4—21no—
407C2×C22⋊C4—21no—
408C2×C42—21no—
409C2×C42C22×D8311yescharacteristic
410C2×C42C22×D8311yescharacteristic
411C2×C42C22×D8311yescharacteristic
412C4⋊C8—1662no—
413C4⋊C8—1662no—
414#103—3321no—
415#103—3321no—
416#103—3321no—
417#103—3321no—
418#102—3321no—
419#102—3321no—
420#102—3321no—
421#102—3321no—
422#97—1662no—
423C4×C8—1662no—
424#95—21no—
425#95—21no—
426#95—1661no—
427#95—1661no—
428#95—1661no—
429#95—1661no—
430#95C22⋊D8311yescharacteristic
431#95C22⋊D8311yescharacteristic
432#95C22⋊D8311yescharacteristic
433#95C22⋊D8311yescharacteristic
434C24—21no—
435C2×Q8—21no—
436C2×Q8—21no—
437C2×Q8—21no—
438C2×Q8—21no—
439C2×Q8—21no—
440C2×Q8—3322no—
441C2×Q8—6641no—
442C2×Q8—6641no—
443C2×Q8C2×Dic16612yes—
444C2×D4—21no—
445C2×D4—21no—
446C2×D4—21no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4C2×Dic16612yes—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—42no—
454C22×C4—81no—
455C22×C4—81no—
456C22×C4—1661no—
457C22×C4—1661no—
458C22×C4—1661no—
459C22×C4—1661no—
460C22×C4C2×C22⋊D8311yescharacteristic
461C22×C4C2×C22⋊D8311yescharacteristic
462C22×C4C2×C22⋊D8311yescharacteristic
463C22×C4C2×C22⋊D8311yescharacteristic
464C22×C4C2×C22⋊D8311yescharacteristic
465C22×C4C2×C22⋊D8311yescharacteristic
466C22×C4C23×D8311yescharacteristic
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—
476C2×C8—1661no—
477C2×C8—1661no—
478C2×C8—1661no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—42no—
484C4⋊C4—81no—
485C4⋊C4—81no—
486C4⋊C4—3322no—
487C4⋊C4—3322no—
488C4⋊C4—3322no—
489C4⋊C4—6641no—
490C4⋊C4—6641no—
491C4⋊C4—6641no—
492C4⋊C4—6641no—
493C4⋊C4—1661no—
494C4⋊C4—1661no—
495C4⋊C4—1661no—
496C4⋊C4—1661no—
497C4⋊C4—1661no—
498C4⋊C4—1661no—
499C4⋊C4—1661no—
500C4⋊C4—1661no—
501C22⋊C4—24no—
502C22⋊C4—24no—
503C22⋊C4—42no—
504C22⋊C4—81no—
505C22⋊C4—81no—
506C42—21no—
507C42—21no—
508C42—42no—
509C42C2×Dic16612yes—
510C42#1305058912yes—
511C42#1305058912yes—
512C42#1305059012yes—
513C23—22no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23C22×C22⋊D8311yescharacteristic
521Q8—22no—
522Q8—22no—
523Q8—41no—
524Q8—41no—
525Q8—6644no—
526D4—22no—
527D4—22no—
528D4—41no—
529D4—41no—
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—22no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
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—84no—
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—3321no—
564C2×C4—3321no—
565C2×C4—3321no—
566C2×C4#1458177112yes—
567C2×C4#1458177112yes—
568C2×C4#1458177912yes—
569C2×C4#1458177912yes—
570C2×C4C22×Dic16611yescharacteristic
571C2×C4C2×#1305058911yescharacteristic
572C2×C4C2×#1305058911yescharacteristic
573C2×C4C2×#1305059011yescharacteristic
574C2×C4#1458183811yescharacteristic
575C2×C4#1458183811yescharacteristic
576C8—3321no—
577C8—3321no—
578C8—3321no—
579C8—3321no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—81no—
584C22—81no—
585C22—81no—
586C22—81no—
587C22C2×#1458177111yescharacteristic
588C22C2×#1458177911yescharacteristic
589C22#1912826511yescharacteristic
590C22#1912850411yescharacteristic
591C22#1912850911yescharacteristic
592C22#1912851211yescharacteristic
593C22#1912851611yescharacteristic
594C4—22no—
595C4—22no—
596C4—22no—
597C4—22no—
598C4—41no—
599C4—41no—
600C4—42no—
601C4—81no—
602C4—81no—
603C4—6641no—
604C4—6641no—
605C4—6641no—
606C4—6641no—
607C4#1912840811yescharacteristic
608C4#1912840811yescharacteristic
609C4#1912841611yescharacteristic
610C4#1912841611yescharacteristic
611C2—82no—
612C2#2491263611yescharacteristic
613C2#2491731411yescharacteristic
614C2#2491732511yescharacteristic
615C2#2491781711yescharacteristic
616C2#2491782811yescharacteristic
617C2#2491855111yescharacteristic
618C2#2491856611yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2326C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
hypercenter#2326D83
hypocenterC83#29151
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ #376 ◁ #2326

Complements

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

NH ≅ G/Ntype
1. #24913582C2semidirect
2. #24908614C2semidirect
4. #24908604C2semidirect
8. #24907598C2semidirect
9. #24907590C2semidirect
10. #24907478C2semidirect
12. #24907468C2semidirect
14. #24907320C2semidirect
99. #124×C83Q8semidirect
325. C83#29151semidirect
382. #124Dic166semidirect

Maximal subgroups

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

subgroupindex
#249135822
#249086142
#249086092
#249086042
#249085992
#249082802
#249082752
#249075982
#249075902
#249074782
#249074732
#249074682
#249074632
#249073202
C83×#23262
#2915183