Finite Group Databasebeta

← #35426422

Subgroups — #35426422

Order 21248.

850 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
1C2×#19127978C211yesmaximal, characteristic
2#24910374C211yesmaximal, characteristic
3#24908437C211yesmaximal, characteristic, complemented
4#24908231C211yesmaximal, characteristic, complemented
5#24908198C211yesmaximal, characteristic, complemented
6#24908193C211yesmaximal, characteristic, complemented
7#24908055C211yesmaximal, characteristic, complemented
8#24908034C211yesmaximal, characteristic, complemented
9#24907723C211yesmaximal, characteristic
10#24907709C211yesmaximal, characteristic
11#24907338C211yesmaximal, characteristic, complemented
12#24907291C211yesmaximal, characteristic
13#24907291C211yesmaximal, characteristic
14C4×#14581712C211yesmaximal, characteristic
15C83×#2369C211yesmaximal, characteristic, complemented
16C2×C4×D332C2211yescharacteristic
17C2×C4×D332C2211yescharacteristic
18C2×C4×D332C2211yescharacteristic
19#19127978C418yescomplemented
20C2×#14581725C2211yescharacteristic
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×C4⋊(C83⋊C8)C2211yescharacteristic
25C2×#14581712C2211yescharacteristic
26C2×#14581712C2211yescharacteristic
27C2×#14581712C2211yescharacteristic
28C2×#14581712C2211yescharacteristic
29C2×#14581703C2211yescharacteristic
30C2×#14581703C2211yescharacteristic
31C2×C4×C83⋊C8C2211yescharacteristic
32#19127599C2211yescharacteristic
33#19127599C2211yescharacteristic
34#19127572—21no—
35#19127572—21no—
36#19127567C2211yescharacteristic
37#19127560—21no—
38#95×D83—21no—
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127456C2211yescharacteristic
48#19127456C2211yescharacteristic
49C2×C4⋊Q8×C83C2211yescharacteristic
50#391×C83—21no—
51#391×C83—21no—
52#389×C83—21no—
53#389×C83C2211yescharacteristic
54#389×C83C2211yescharacteristic
55C4×C4⋊C4×C83—21no—
56C4×C4⋊C4×C83C2211yescharacteristic
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C4⋊C4×C83C2211yescharacteristic
59C4×C4⋊C4×C83C2211yescharacteristic
60C22×D332C2311yescharacteristic
61C22×C4×D83—21no—
62C22×C4×D83—21no—
63C22×C4×D83—21no—
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C4×D332—22no—
69C4×D332—22no—
70C4×D332—22no—
71C4×D332—41no—
72C4×D332—41no—
73C4×D332C2×C414yes—
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×Dic83⋊C4—21no—
78C2×Dic83⋊C4—21no—
79C2×(C2×D83)⋊C4—21no—
80C2×(C2×D83)⋊C4—21no—
81C2×(C2×D83)⋊C4—21no—
82#14581725—41no—
83#14581725C2×C414yescomplemented
84#14581725C2×C414yescomplemented
85#14581723—41no—
86C4⋊(C83⋊C8)C2×C414yescomplemented
87#14581712—22no—
88#14581712C2×C414yes—
89#14581712C2×C414yes—
90#14581712Q814yescomplemented
91#14581703—22no—
92#14581703—41no—
93C4×C83⋊C8—41no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C22×Q8×C83—21no—
99C4⋊Q8×C83—22no—
100C4⋊Q8×C83C2×C414yescomplemented
101C2×C4⋊C4×C83—21no—
102C2×C4⋊C4×C83—21no—
103C2×C4⋊C4×C83—21no—
104C2×C4⋊C4×C83D411yescharacteristic
105C2×C4⋊C4×C83D411yescharacteristic
106C2×C4⋊C4×C83C2311yescharacteristic
107C2×C4⋊C4×C83C2311yescharacteristic
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C4⋊C4×C83C2311yescharacteristic
110C2×C42×C83—21no—
111C2×C42×C83—21no—
112C2×C42×C83—21no—
113C2×C42×C83C2311yescharacteristic
114C2×C42×C83C2311yescharacteristic
115C2×C42×C83C2311yescharacteristic
116#95×C83—21no—
117#95×C83—21no—
118#95×C83—21no—
119#95×C83—21no—
120#95×C83—21no—
121#95×C83D411yescharacteristic
122#95×C83D411yescharacteristic
123C23×D83—21no—
124C2×D332—21no—
125C2×D332—21no—
126C2×D332—21no—
127C2×D332C22×C412yes—
128C2×D332C2×Q812yes—
129C2×D332#4112yes—
130C22×Dic83—21no—
131C22×Dic83—21no—
132C22×Dic83—21no—
133C2×C4×D83—24no—
134C2×C4×D83—42no—
135C2×C4×D83—42no—
136C2×C4×D83—42no—
137C2×C4×D83—42no—
138C2×C4×D83—81no—
139C2×C4×D83—81no—
140C2×C4×D83—81no—
141C2×C4×D83—81no—
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⋊C8—21no—
149C2×C83⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8C22×C412yes—
152C2×C83⋊C8C22×C412yes—
153C2×C83⋊C8C2×Q812yes—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—21no—
159C4⋊Dic83C22×C412yes—
160Dic83⋊C4—81no—
161Dic83⋊C4—81no—
162(C2×D83)⋊C4—24no—
163(C2×D83)⋊C4—42no—
164(C2×D83)⋊C4—42no—
165C2×Q8×C83—24no—
166C2×Q8×C83—81no—
167C2×Q8×C83—81no—
168C22×C4×C83—21no—
169C22×C4×C83—21no—
170C22×C4×C83—21no—
171C22×C4×C83—21no—
172C22×C4×C83C2×D411yescharacteristic
173C22×C4×C83C2×D411yescharacteristic
174C22×C4×C83C2×D411yescharacteristic
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83C2×D411yescharacteristic
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2411yescharacteristic
179C4⋊C4×C83—24no—
180C4⋊C4×C83—22no—
181C4⋊C4×C83—21no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—41no—
187C4⋊C4×C83—81no—
188C4⋊C4×C83—81no—
189C4⋊C4×C83C22×C412yes—
190C4⋊C4×C83C22×C412yes—
191C4⋊C4×C83C2×Q812yes—
192C42×C83—21no—
193C42×C83—21no—
194C42×C83—21no—
195C42×C83—21no—
196C42×C83—42no—
197C42×C83—81no—
198C42×C83—81no—
199C42×C83C22×C412yes—
200C42×C83#4112yes—
201C23×C83C22×D411yescharacteristic
202C22×D83—22no—
203C22×D83—22no—
204C22×D83—22no—
205C22×D83—41no—
206C22×D83—41no—
207C22×D83—41no—
208C22×D83—41no—
209Q8×C83—84no—
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—21no—
213C2×C4×C83—21no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—22no—
217C2×C4×C83—22no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—22no—
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×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83—41no—
232C2×C4×C83—41no—
233C2×C4×C83—41no—
234C2×C4×C83—41no—
235C2×C4×C83—41no—
236C2×C4×C83—41no—
237C2×C4×C83C4×D412yes—
238C2×C4×C83C4×D412yes—
239C2×C4×C83#13712yes—
240C2×C4×C83C23×C411yescharacteristic
241C2×C4×C83C22×Q811yescharacteristic
242C2×C4×C83C2×#4111yescharacteristic
243C2×C4×C83C2×#4111yescharacteristic
244C2×C4×C83H(5,𝔽2)11yescharacteristic
245C2×C4×C832−1+411yescharacteristic
246C2×Dic83—22no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255D332—22no—
256D332—22no—
257D332—22no—
258D332C4×Q814yes—
259C4×D83—84no—
260C4×D83—84no—
261C4×D83—84no—
262C83⋊C8—22no—
263C83⋊C8—22no—
264C83⋊C8—22no—
265C83⋊C8—41no—
266C22×C83C2×C4×D411yescharacteristic
267C22×C83#54611yescharacteristic
268C22×C83D4×Q811yescharacteristic
269C22×C83C2×#13711yescharacteristic
270C22×C83#57511yescharacteristic
271C22×C83#57611yescharacteristic
272C22×C83#57611yescharacteristic
273C2×D83—24no—
274C2×D83—42no—
275C2×D83—42no—
276C2×D83—42no—
277C2×D83—42no—
278C2×D83—42no—
279C2×D83—42no—
280C4×C83—22no—
281C4×C83—22no—
282C4×C83—22no—
283C4×C83—42no—
284C4×C83—42no—
285C4×C83—42no—
286C4×C83—41no—
287C4×C83—41no—
288C4×C83—41no—
289C4×C83—81no—
290C4×C83C2×C4×Q811yescharacteristic
291C4×C83#52011yescharacteristic
292C4×C83#55211yescharacteristic
293C4×C83#55411yescharacteristic
294Dic83—42no—
295Dic83—81no—
296Dic83—81no—
297#22979—831nomaximal, Sylow
298C2×C83#236011yescharacteristic
299C2×C83C4×#13711yescharacteristic
300C2×C83#294811yescharacteristic
301C2×C83#331411yescharacteristic
302C2×C83#331911yescharacteristic
303C2×C83#338511yescharacteristic
304C2×C83#338911yescharacteristic
305D83—44no—
306C2×#474—831no—
307#2369D8311yescharacteristic, complemented
308#2349—831no—
309#1993—831no—
310#1949—831no—
311#1943—831no—
312#1942—831no—
313#1921—831no—
314#1918—831no—
315#1864—831no—
316#1862—831no—
317#1771—831no—
318#1762—831no—
319#1762—831no—
320C4×#102—831no—
321C83#2297911yescharacteristic, complemented, Sylow
322C2×C4⋊Q8C2×D8311yescharacteristic
323C2×C4×D4—831no—
324C2×C4×D4—831no—
325C2×C4×D4—831no—
326#474—838no—
327C2×#107—831no—
328C2×#107—831no—
329C2×#107—831no—
330C2×#106—831no—
331C2×C4⋊C8—831no—
332C2×#102—831no—
333C2×#102—831no—
334C2×#102—831no—
335C2×#102—831no—
336C2×#97—831no—
337C2×#97—831no—
338C2×C4×C8—831no—
339#391—21no—
340#391—21no—
341#389—21no—
342#389—831no—
343#389—831no—
344#389C2×D8311yescharacteristic
345#389C2×D8311yescharacteristic
346#385—1661no—
347#385—1661no—
348#382—831no—
349#380—1661no—
350C4×C4⋊C4—21no—
351C4×C4⋊C4C2×D8311yescharacteristic
352C4×C4⋊C4C2×D8311yescharacteristic
353C4×C4⋊C4C2×D8311yescharacteristic
354C4×C4⋊C4C2×D8311yescharacteristic
355C2×#95—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×Q8—21no—
367C22×D4—831no—
368C23×C4—1661no—
369C23×C4—1661no—
370C23×C4—1661no—
371C22×C8—831no—
372C22×C8—831no—
373C22×C8—831no—
374C22×C8—831no—
375C4⋊Q8—22no—
376C4⋊Q8C4×D8314yescomplemented
377C4×D4—3321no—
378C4×D4—3321no—
379C4×D4—834no—
380C4×D4—1662no—
381C4×D4—1662no—
382C4×D4—1662no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4—831no—
387C2×C4⋊C4—831no—
388C2×C4⋊C4—831no—
389C2×C4⋊C4—1661no—
390C2×C4⋊C4—1661no—
391C2×C4⋊C4C22⋊D8311yescharacteristic
392C2×C4⋊C4C22⋊D8311yescharacteristic
393C2×C4⋊C4C22×D8311yescharacteristic
394C2×C4⋊C4C22×D8311yescharacteristic
395C2×C4⋊C4C22×D8311yescharacteristic
396C2×C4⋊C4C22×D8311yescharacteristic
397C2×C22⋊C4—1661no—
398C2×C22⋊C4—1661no—
399C2×C22⋊C4—1661no—
400C2×C42—21no—
401C2×C42—21no—
402C2×C42—21no—
403C2×C42C22×D8311yescharacteristic
404C2×C42C22×D8311yescharacteristic
405C2×C42C22×D8311yescharacteristic
406#107—3321no—
407#107—834no—
408#107—834no—
409#106—3321no—
410C4⋊C8—834no—
411#102—834no—
412#102—834no—
413#102—834no—
414#102—1662no—
415#97—3321no—
416#97—1662no—
417C4×C8—3321no—
418#95—21no—
419#95—21no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—1661no—
424#95—1661no—
425#95—1661no—
426#95—1661no—
427#95C22⋊D8311yescharacteristic
428#95C22⋊D8311yescharacteristic
429C24—1661no—
430C2×Q8—24no—
431C2×Q8—81no—
432C2×Q8—81no—
433C2×D4—832no—
434C2×D4—832no—
435C2×D4—832no—
436C2×D4—1661no—
437C2×D4—1661no—
438C2×D4—1661no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—3322no—
444C22×C4—3322no—
445C22×C4—3322no—
446C22×C4—3322no—
447C22×C4—6641no—
448C22×C4—6641no—
449C22×C4—6641no—
450C22×C4—6641no—
451C22×C4—1661no—
452C22×C4—1661no—
453C22×C4—1661no—
454C22×C4—1664no—
455C22×C4C2×C22⋊D8311yescharacteristic
456C22×C4C2×C22⋊D8311yescharacteristic
457C22×C4C2×C22⋊D8311yescharacteristic
458C22×C4C2×C22⋊D8311yescharacteristic
459C22×C4C2×C22⋊D8311yescharacteristic
460C22×C4C2×C22⋊D8311yescharacteristic
461C22×C4C23×D8311yescharacteristic
462C2×C8—832no—
463C2×C8—832no—
464C2×C8—832no—
465C2×C8—1661no—
466C2×C8—1661no—
467C2×C8—1661no—
468C2×C8—1661no—
469C2×C8—1661no—
470C2×C8—1661no—
471C2×C8—1661no—
472C2×C8—1661no—
473C2×C8—1661no—
474C4⋊C4—22no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—24no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—41no—
482C4⋊C4—81no—
483C4⋊C4—81no—
484C4⋊C4—832no—
485C4⋊C4—6641no—
486C4⋊C4—6641no—
487C4⋊C4—1661no—
488C4⋊C4—1661no—
489C4⋊C4—1661no—
490C4⋊C4—1661no—
491C4⋊C4—1661no—
492C4⋊C4C2×C4×D8312yes—
493C4⋊C4C2×C4×D8312yes—
494C4⋊C4Q8×D8312yes—
495C22⋊C4—3322no—
496C22⋊C4—3322no—
497C22⋊C4—1664no—
498C42—21no—
499C42—21no—
500C42—21no—
501C42—21no—
502C42—42no—
503C42—81no—
504C42—81no—
505C42C2×C4×D8312yes—
506C42#1305059012yes—
507C23—3321no—
508C23—3321no—
509C23—3321no—
510C23—3321no—
511C23—1662no—
512C23—1662no—
513C23—1662no—
514C23C22×C22⋊D8311yescharacteristic
515Q8—84no—
516D4—834no—
517D4—1662no—
518D4—1662no—
519D4—1662no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
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—6644no—
556C2×C4—6644no—
557C2×C4—6644no—
558C2×C4—1662no—
559C2×C4C4×C22⋊D8312yes—
560C2×C4C4×C22⋊D8312yes—
561C2×C4#1458182312yes—
562C2×C4C22×C4×D8311yescharacteristic
563C2×C4C2×Q8×D8311yescharacteristic
564C2×C4#41×D8311yescharacteristic
565C2×C4C2×#1305059011yescharacteristic
566C2×C4#1458183811yescharacteristic
567C2×C4#1458184011yescharacteristic
568C8—3321no—
569C8—1662no—
570C8—1662no—
571C8—1662no—
572C22—3322no—
573C22—3322no—
574C22—3322no—
575C22—3322no—
576C22—3322no—
577C22—3322no—
578C22—1664no—
579C22C2×C4×C22⋊D8311yescharacteristic
580C22#1912833911yescharacteristic
581C22Q8×C22⋊D8311yescharacteristic
582C22C2×#1458182311yescharacteristic
583C22#1912849911yescharacteristic
584C22#1912850411yescharacteristic
585C22#1912850411yescharacteristic
586C4—22no—
587C4—22no—
588C4—22no—
589C4—41no—
590C4—41no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—41no—
595C4—81no—
596C4—3322no—
597C4—6641no—
598C4—6641no—
599C4C4×Q8×D8311yescharacteristic
600C4#1912818311yescharacteristic
601C4#1912839211yescharacteristic
602C4#1912841111yescharacteristic
603C2—3324no—
604C2#2491046411yescharacteristic
605C2C4×#1458182311yescharacteristic
606C2#2491631311yescharacteristic
607C2#2491953811yescharacteristic
608C2#2491961311yescharacteristic
609C2#2492036911yescharacteristic
610C2#2492041711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2369C2
socleC23×C83C22×D4
prime radicalC22×D332C23
cosocleC22×C4×C83C24
hypercenter#2369D83
hypocenterC83#22979
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2369

Complements

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

NH ≅ G/Ntype
3. #24908437C2semidirect
4. #24908231C2semidirect
5. #24908198C2semidirect
6. #24908193C2semidirect
7. #24908055C2semidirect
8. #24908034C2semidirect
11. #24907338C2semidirect
15. C83×#2369C2semidirect
19. #19127978C4semidirect
83. #14581725C2×C4semidirect
84. #14581725C2×C4semidirect
86. C4⋊(C83⋊C8)C2×C4semidirect
90. #14581712Q8semidirect
100. C4⋊Q8×C83C2×C4semidirect
307. #2369D83semidirect
321. C83#22979semidirect
376. C4⋊Q8C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191279782
#249103742
#249084372
#249082312
#249081982
#249081932
#249080552
#249080342
#249077232
#249077092
#249073382
#249072912
C4×#145817122
C83×#23692
#2297983