Finite Group Databasebeta

← #35426417

Subgroups — #35426417

Order 21248.

870 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910563C211yesmaximal, characteristic, complemented
2D4×C4⋊Dic83C211yesmaximal, characteristic
3#24908433C211yesmaximal, characteristic, complemented
4#24908227C211yesmaximal, characteristic, complemented
5#24908202C211yesmaximal, characteristic, complemented
6#24908192C211yesmaximal, characteristic, complemented
7#24908050C211yesmaximal, characteristic, complemented
8#24908036C211yesmaximal, characteristic, complemented
9#24907726C211yesmaximal, characteristic
10#24907726C211yesmaximal, characteristic
11#24907340C211yesmaximal, characteristic, complemented
12#24907290C211yesmaximal, characteristic
13#24907288C211yesmaximal, characteristic
14#102×Dic83C211yesmaximal, characteristic
15C2×#478×C83C211yesmaximal, characteristic
16C2×D4×Dic83C2211yescharacteristic
17C2×D4×Dic83C2211yescharacteristic
18C22×C4⋊Dic83—21no—
19C2×#14581724C2211yescharacteristic
20C2×#14581724C2211yescharacteristic
21C2×#14581713C2211yescharacteristic
22C2×#14581713C2211yescharacteristic
23C2×#14581703C2211yescharacteristic
24C2×#14581702C2211yescharacteristic
25C2×C8×Dic83C2211yescharacteristic
26#19127600—21no—
27#19127600C2211yescharacteristic
28#19127600C2211yescharacteristic
29#19127600C2211yescharacteristic
30#19127600C2211yescharacteristic
31#19127587—21no—
32#19127587—21no—
33#19127576—21no—
34#19127576—21no—
35#19127571C2211yescharacteristic
36#19127561—21no—
37C4⋊C4×Dic83—21no—
38C4⋊C4×Dic83C2211yescharacteristic
39C4⋊C4×Dic83C2211yescharacteristic
40C4×C4⋊Dic83C2211yescharacteristic
41C4×C4⋊Dic83C2211yescharacteristic
42#19127464C2211yescharacteristic
43#19127464C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127455C2211yescharacteristic
51#19127455C2211yescharacteristic
52C2×#125×C83C2211yescharacteristic
53C2×C4×D4×C83C2211yescharacteristic
54#478×C83C418yescomplemented
55C2×#107×C83C2211yescharacteristic
56C2×#106×C83C2211yescharacteristic
57C2×C4⋊C8×C83C2211yescharacteristic
58C2×#102×C83C2211yescharacteristic
59C2×#102×C83C2211yescharacteristic
60C23×Dic83—21no—
61C23×Dic83—21no—
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64D4×Dic83—22no—
65D4×Dic83—22no—
66D4×Dic83—41no—
67D4×Dic83—41no—
68C2×C4⋊Dic83—21no—
69C2×C4⋊Dic83—21no—
70C2×C4⋊Dic83—21no—
71C2×C4⋊Dic83—21no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83—21no—
74C2×C4⋊Dic83—44no—
75C2×C4⋊Dic83D411yescharacteristic
76C2×C4⋊Dic83D411yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×C4⋊Dic83C2311yescharacteristic
79C2×C4⋊Dic83C2311yescharacteristic
80C2×C4⋊Dic83C2311yescharacteristic
81C2×C22⋊Dic83—21no—
82C2×C22⋊Dic83—21no—
83C2×C4×Dic83—21no—
84C2×C4×Dic83—21no—
85C2×C4×Dic83—21no—
86C2×C4×Dic83C2311yescharacteristic
87C2×C4×Dic83C2311yescharacteristic
88#14581724—41no—
89#14581724—41no—
90#14581713—22no—
91#14581713Q814yes—
92#14581703—22no—
93#14581702—41no—
94C8×Dic83—41no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95D411yescharacteristic
103C83⋊#95D411yescharacteristic
104C22×D4×C83C2311yescharacteristic
105C23×C4×C83—21no—
106C22×C8×C83C2311yescharacteristic
107C22×C8×C83C2311yescharacteristic
108#125×C83—22no—
109#125×C83C2×C414yescomplemented
110C4×D4×C83—22no—
111C4×D4×C83C2×C414yes—
112C2×C4⋊C4×C83—21no—
113C2×C4⋊C4×C83—21no—
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C22⋊C4×C83—21no—
118C2×C42×C83C2311yescharacteristic
119#107×C83C2×C414yescomplemented
120#106×C83C2×C414yescomplemented
121C4⋊C8×C83C2×C414yescomplemented
122#102×C83C2×C414yes—
123#102×C83C2×C414yes—
124C22×Dic83—21no—
125C22×Dic83—21no—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C22×Dic83—42no—
129C22×Dic83—42no—
130C22×Dic83—42no—
131C22×Dic83—42no—
132C22×Dic83—81no—
133C22×Dic83—81no—
134C22×Dic83C2×D411yescharacteristic
135C22×Dic83C2×D411yescharacteristic
136C22×Dic83C2×D411yescharacteristic
137C22×Dic83C2×D411yescharacteristic
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8—21no—
143C2×C83⋊C8C2×Q812yes—
144C4⋊Dic83—21no—
145C4⋊Dic83—22no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—41no—
153C4⋊Dic83—42no—
154C4⋊Dic83—42no—
155C4⋊Dic83—42no—
156C4⋊Dic83—42no—
157C4⋊Dic83—81no—
158C4⋊Dic83—81no—
159C4⋊Dic83C2×Q812yes—
160C22⋊Dic83—42no—
161C22⋊Dic83—42no—
162C4×Dic83—21no—
163C4×Dic83—21no—
164C4×Dic83—21no—
165C4×Dic83—21no—
166C4×Dic83—42no—
167C4×Dic83—81no—
168C4×Dic83—81no—
169C24×C83—21no—
170C2×D4×C83—21no—
171C2×D4×C83—21no—
172C2×D4×C83—21no—
173C2×D4×C83C22×C412yes—
174C2×D4×C83C2×Q812yes—
175C2×D4×C83#4112yes—
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83—21no—
179C22×C4×C83—24no—
180C22×C4×C83—42no—
181C22×C4×C83—42no—
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2×D411yescharacteristic
184C22×C4×C83C2411yescharacteristic
185C2×C8×C83—21no—
186C2×C8×C83—21no—
187C2×C8×C83—21no—
188C2×C8×C83—21no—
189C2×C8×C83C22×C412yes—
190C2×C8×C83C22×C412yes—
191C4⋊C4×C83—24no—
192C4⋊C4×C83—24no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83C22×C412yes—
197C4⋊C4×C83C22×C412yes—
198C4⋊C4×C83C22×C412yes—
199C22⋊C4×C83—24no—
200C42×C83C22×C412yes—
201C42×C83#4112yes—
202C23×C83—22no—
203C23×C83—22no—
204C23×C83—22no—
205C23×C83—41no—
206C23×C83—41no—
207C23×C83—41no—
208C23×C83—41no—
209C23×C83C22×D411yescharacteristic
210D4×C83—22no—
211D4×C83—22no—
212D4×C83—22no—
213D4×C83C4×Q814yes—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—22no—
219C2×C4×C83—22no—
220C2×C4×C83—22no—
221C2×C4×C83—48no—
222C2×C4×C83—41no—
223C2×C4×C83—41no—
224C2×C4×C83—41no—
225C2×C4×C83—41no—
226C2×C4×C83—41no—
227C2×C4×C83—41no—
228C2×C4×C83C4×D412yes—
229C2×C4×C83C4×D412yes—
230C2×C4×C83C23×C411yescharacteristic
231C2×C4×C83C22×Q811yescharacteristic
232C2×C4×C83C2×#4111yescharacteristic
233C2×C4×C83C2×#4111yescharacteristic
234C2×C4×C83H(5,𝔽2)11yescharacteristic
235C2×C4×C832−1+411yescharacteristic
236C2×Dic83—21no—
237C2×Dic83—21no—
238C2×Dic83—21no—
239C2×Dic83—21no—
240C2×Dic83—21no—
241C2×Dic83—22no—
242C2×Dic83—21no—
243C2×Dic83—21no—
244C2×Dic83—21no—
245C2×Dic83—21no—
246C2×Dic83—21no—
247C2×Dic83—21no—
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—
260C2×Dic83—84no—
261C2×Dic83AGL(1,ℤ/8)12yes—
262C8×C83—22no—
263C8×C83—22no—
264C83⋊C8—22no—
265C83⋊C8—41no—
266C22×C83—24no—
267C22×C83—42no—
268C22×C83—42no—
269C22×C83—42no—
270C22×C83—42no—
271C22×C83—42no—
272C22×C83—42no—
273C22×C83C2×C4×D411yescharacteristic
274C22×C83#54611yescharacteristic
275C22×C83D4×Q811yescharacteristic
276C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
277C22×C83#57511yescharacteristic
278C22×C83#57711yescharacteristic
279C22×C83#57711yescharacteristic
280C4×C83—22no—
281C4×C83—22no—
282C4×C83—42no—
283C4×C83—42no—
284C4×C83—42no—
285C4×C83C2×C4×Q811yescharacteristic
286C4×C83#52011yescharacteristic
287C4×C83#55211yescharacteristic
288C4×C83#55411yescharacteristic
289Dic83—22no—
290Dic83—42no—
291Dic83—41no—
292Dic83—41no—
293Dic83—41no—
294Dic83—81no—
295Dic83—81no—
296Dic83—81no—
297#22978—831nomaximal, Sylow
298C2×C83—44no—
299C2×C83#236011yescharacteristic
300C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
301C2×C83#295211yescharacteristic
302C2×C83#330711yescharacteristic
303C2×C83#331011yescharacteristic
304C2×C83#338411yescharacteristic
305C2×C83#338811yescharacteristic
306C2×#478D8311yescharacteristic
307#2370—831no—
308D4×C4⋊C4—831no—
309#1992—831no—
310#1948—831no—
311#1943—831no—
312#1941—831no—
313#1920—831no—
314#1918—831no—
315#1864—831no—
316#1864—831no—
317#1771—831no—
318#1762—831no—
319#1762—831no—
320C4×#102—831no—
321C83#2297811yescharacteristic, complemented, Sylow
322C2×#125C2×D8311yescharacteristic
323C2×C4×D4—831no—
324C2×C4×D4—831no—
325C2×C4×D4C2×D8311yescharacteristic
326C22×C4⋊C4—1661no—
327#478Dic8318yescomplemented
328C2×#107C2×D8311yescharacteristic
329C2×#106—831no—
330C2×#106—831no—
331C2×#106C2×D8311yescharacteristic
332C2×C4⋊C8C2×D8311yescharacteristic
333C2×#102—831no—
334C2×#102—831no—
335C2×#102C2×D8311yescharacteristic
336C2×#102C2×D8311yescharacteristic
337C2×#97—831no—
338C2×#97—831no—
339C2×C4×C8—831no—
340#389—831no—
341#389—831no—
342#389—831no—
343#389—831no—
344#389—1661no—
345#387—1661no—
346#387—1661no—
347#385—1661no—
348#385—1661no—
349#384—831no—
350#380—1661no—
351C4×C4⋊C4—831no—
352C4×C4⋊C4—831no—
353C4×C4⋊C4—831no—
354C4×C4⋊C4—831no—
355C4×C4⋊C4—1661no—
356#340—831no—
357#340—831no—
358#340—831no—
359#340—831no—
360#340—831no—
361#340—831no—
362#340—831no—
363#340—831no—
364#336—831no—
365#336—831no—
366C22×D4C22×D8311yescharacteristic
367C23×C4—21no—
368C23×C4—1661no—
369C23×C4—1661no—
370C22×C8—831no—
371C22×C8—831no—
372C22×C8C22×D8311yescharacteristic
373C22×C8C22×D8311yescharacteristic
374#125—22no—
375#125C2×Dic8314yescomplemented
376C4×D4—22no—
377C4×D4—3321no—
378C4×D4—3321no—
379C4×D4—1662no—
380C4×D4—1662no—
381C4×D4C2×Dic8314yes—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—3324no—
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—1661no—
397C2×C4⋊C4C22×D8311yescharacteristic
398C2×C4⋊C4C22×D8311yescharacteristic
399C2×C4⋊C4C22×D8311yescharacteristic
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1661no—
402C2×C22⋊C4—1661no—
403C2×C42—831no—
404C2×C42—831no—
405C2×C42—1661no—
406C2×C42—1661no—
407C2×C42—1661no—
408C2×C42C22×D8311yescharacteristic
409#107C2×Dic8314yescomplemented
410#106—3321no—
411#106—3321no—
412#106C2×Dic8314yescomplemented
413C4⋊C8C2×Dic8314yescomplemented
414#102—834no—
415#102—1662no—
416#102C2×Dic8314yes—
417#102C2×Dic8314yes—
418#97—3321no—
419#97—1662no—
420C4×C8—3321no—
421#95—831no—
422#95—831no—
423#95—1661no—
424#95—1661no—
425#95—1661no—
426#95—1661no—
427#95—1661no—
428#95—1661no—
429#95—1661no—
430C24—21no—
431C2×D4—21no—
432C2×D4—21no—
433C2×D4—21no—
434C2×D4C22×Dic8312yes—
435C2×D4Q8×D8312yes—
436C2×D4#1305059112yes—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—24no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—3322no—
444C22×C4—3322no—
445C22×C4—3322no—
446C22×C4—3322no—
447C22×C4—831no—
448C22×C4—831no—
449C22×C4—831no—
450C22×C4—831no—
451C22×C4—6641no—
452C22×C4—6641no—
453C22×C4—1661no—
454C22×C4—1661no—
455C22×C4—1661no—
456C22×C4—1661no—
457C22×C4D4×D8311yescharacteristic
458C22×C4D4×D8311yescharacteristic
459C22×C4C23×D8311yescharacteristic
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—832no—
465C2×C8—1661no—
466C2×C8—1661no—
467C2×C8—1661no—
468C2×C8—1661no—
469C2×C8—1661no—
470C2×C8C22×Dic8312yes—
471C2×C8C22×Dic8312yes—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—24no—
476C4⋊C4—24no—
477C4⋊C4—3322no—
478C4⋊C4—3321no—
479C4⋊C4—3322no—
480C4⋊C4—3322no—
481C4⋊C4—3322no—
482C4⋊C4—832no—
483C4⋊C4—6641no—
484C4⋊C4—6641no—
485C4⋊C4—1662no—
486C4⋊C4—1661no—
487C4⋊C4—1661no—
488C4⋊C4—1661no—
489C4⋊C4—1661no—
490C4⋊C4—1661no—
491C4⋊C4—1661no—
492C4⋊C4—1661no—
493C4⋊C4C22×Dic8312yes—
494C4⋊C4C22×Dic8312yes—
495C4⋊C4C22×Dic8312yes—
496C22⋊C4—24no—
497C22⋊C4—3322no—
498C22⋊C4—3322no—
499C42—3322no—
500C42—6641no—
501C42—6641no—
502C42—1661no—
503C42—1661no—
504C42—1661no—
505C42—1661no—
506C42C22×Dic8312yes—
507C42#1305058912yes—
508C23—22no—
509C23—22no—
510C23—22no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23C2×D4×D8311yescharacteristic
516D4—22no—
517D4—22no—
518D4—22no—
519D4Q8×Dic8314yes—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—48no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
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—832no—
546C2×C4—6644no—
547C2×C4—6644no—
548C2×C4—1661no—
549C2×C4—1661no—
550C2×C4—1661no—
551C2×C4—1662no—
552C2×C4—1661no—
553C2×C4—1661no—
554C2×C4—1661no—
555C2×C4—1661no—
556C2×C4—1661no—
557C2×C4—1661no—
558C2×C4—1661no—
559C2×C4—1661no—
560C2×C4D4×Dic8312yes—
561C2×C4D4×Dic8312yes—
562C2×C4C23×Dic8311yescharacteristic
563C2×C4C2×Q8×D8311yescharacteristic
564C2×C4C2×#1305058911yescharacteristic
565C2×C4C2×#1305059111yescharacteristic
566C2×C4#1458183811yescharacteristic
567C2×C4#1458183911yescharacteristic
568C8—22no—
569C8—22no—
570C8—3321no—
571C8—1662no—
572C22—24no—
573C22—42no—
574C22—42no—
575C22—42no—
576C22—42no—
577C22—42no—
578C22—42no—
579C22C2×D4×Dic8311yescharacteristic
580C22#1912834211yescharacteristic
581C22#1912836411yescharacteristic
582C22AGL(1,ℤ/8)×D8311yescharacteristic
583C22#1912849711yescharacteristic
584C22#1912850811yescharacteristic
585C22#1912850811yescharacteristic
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—3322no—
592C4—3321no—
593C4—3321no—
594C4—3321no—
595C4—6641no—
596C4—6641no—
597C4—6641no—
598C4—1662no—
599C4C2×Q8×Dic8311yescharacteristic
600C4#1912818411yescharacteristic
601C4#1912839311yescharacteristic
602C4#1912840911yescharacteristic
603C2—44no—
604C2#2491047211yescharacteristic
605C2AGL(1,ℤ/8)×Dic8311yescharacteristic
606C2#2491634711yescharacteristic
607C2#2491943611yescharacteristic
608C2#2491948111yescharacteristic
609C2#2492036511yescharacteristic
610C2#2492040511yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#478×C83C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
cosocleC22×C4×C83C24
hypercenterC2×#478D83
hypocenterC83#22978
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#478

Complements

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

NH ≅ G/Ntype
1. #24910563C2semidirect
3. #24908433C2semidirect
4. #24908227C2semidirect
5. #24908202C2semidirect
6. #24908192C2semidirect
7. #24908050C2semidirect
8. #24908036C2semidirect
11. #24907340C2semidirect
54. #478×C83C4semidirect
109. #125×C83C2×C4semidirect
119. #107×C83C2×C4semidirect
120. #106×C83C2×C4semidirect
121. C4⋊C8×C83C2×C4semidirect
321. C83#22978semidirect
327. #478Dic83semidirect
375. #125C2×Dic83semidirect
409. #107C2×Dic83semidirect
412. #106C2×Dic83semidirect
413. C4⋊C8C2×Dic83semidirect

Maximal subgroups

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

subgroupindex
#249105632
D4×C4⋊Dic832
#249084332
#249082272
#249082022
#249081922
#249080502
#249080362
#249077262
#249073402
#249072902
#249072882
#102×Dic832
C2×#478×C832
#2297883