Finite Group Databasebeta

← #35426432

Subgroups — #35426432

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
1#24910553C211yesmaximal, characteristic, complemented
2#24910373C211yesmaximal, characteristic
3#24908437C211yesmaximal, characteristic, complemented
4#24908233C211yesmaximal, characteristic, complemented
5#24908202C211yesmaximal, characteristic, complemented
6#24908197C211yesmaximal, characteristic, complemented
7#24908057C211yesmaximal, characteristic, complemented
8#24908036C211yesmaximal, characteristic, complemented
9#24907726C211yesmaximal, characteristic
10#24907712C211yesmaximal, characteristic
11#24907340C211yesmaximal, characteristic, complemented
12#24907290C211yesmaximal, characteristic
13#24907288C211yesmaximal, characteristic
14#102×Dic83C211yesmaximal, characteristic
15C2×#474×C83C211yesmaximal, characteristic
16C2×D4×Dic83C2211yescharacteristic
17C2×D4×Dic83C2211yescharacteristic
18C2×#14581726C2211yescharacteristic
19C2×#14581724C2211yescharacteristic
20C2×#14581713C2211yescharacteristic
21C2×#14581713C2211yescharacteristic
22C2×#14581703C2211yescharacteristic
23C2×#14581702C2211yescharacteristic
24C2×C8×Dic83C2211yescharacteristic
25#19127608—21no—
26#19127608—21no—
27#19127600—21no—
28#19127600C2211yescharacteristic
29#19127600C2211yescharacteristic
30#19127600C2211yescharacteristic
31#19127600C2211yescharacteristic
32#19127576—21no—
33#19127576—21no—
34#19127567C2211yescharacteristic
35#19127561—21no—
36C4⋊C4×Dic83—21no—
37C4⋊C4×Dic83C2211yescharacteristic
38C4⋊C4×Dic83C2211yescharacteristic
39C4×C4⋊Dic83C2211yescharacteristic
40C4×C4⋊Dic83C2211yescharacteristic
41C2×C83⋊#95—21no—
42#19127464C2211yescharacteristic
43#19127464C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127455C2211yescharacteristic
51#19127455C2211yescharacteristic
52C2×C4⋊Q8×C83C2211yescharacteristic
53C2×C4×D4×C83C2211yescharacteristic
54#474×C83C418yescomplemented
55C2×#107×C83C2211yescharacteristic
56C2×#107×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⋊Dic83D411yescharacteristic
73C2×C4⋊Dic83D411yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×C22⋊Dic83—21no—
79C2×C22⋊Dic83—21no—
80C2×C4×Dic83—21no—
81C2×C4×Dic83—21no—
82C2×C4×Dic83—21no—
83C2×C4×Dic83C2311yescharacteristic
84C2×C4×Dic83C2311yescharacteristic
85#14581726—41no—
86#14581724—41no—
87#14581713—22no—
88#14581713Q814yescomplemented
89#14581703—22no—
90#14581702—41no—
91C8×Dic83—41no—
92C83⋊#95—21no—
93C83⋊#95—21no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95D411yescharacteristic
102C83⋊#95D411yescharacteristic
103C22×Q8×C83—21no—
104C22×D4×C83C2311yescharacteristic
105C23×C4×C83—21no—
106C22×C8×C83C2311yescharacteristic
107C22×C8×C83C2311yescharacteristic
108C4⋊Q8×C83—22no—
109C4⋊Q8×C83C2×C414yescomplemented
110C4×D4×C83—22no—
111C4×D4×C83C2×C414yes—
112C2×C4⋊C4×C83—21no—
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C4⋊C4×C83C2311yescharacteristic
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C22⋊C4×C83—21no—
117C2×C42×C83C2311yescharacteristic
118#107×C83C2×C414yescomplemented
119#107×C83C2×C414yescomplemented
120C4⋊C8×C83C2×C414yescomplemented
121#102×C83C2×C414yes—
122#102×C83C2×C414yes—
123C22×Dic83—21no—
124C22×Dic83—21no—
125C22×Dic83—21no—
126C22×Dic83—21no—
127C22×Dic83—42no—
128C22×Dic83—42no—
129C22×Dic83—42no—
130C22×Dic83—42no—
131C22×Dic83—81no—
132C22×Dic83—81no—
133C22×Dic83C2×D411yescharacteristic
134C22×Dic83C2×D411yescharacteristic
135C22×Dic83C2×D411yescharacteristic
136C22×Dic83C2×D411yescharacteristic
137C2×C83⋊C8—21no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8C2×Q812yes—
143C4⋊Dic83—21no—
144C4⋊Dic83—22no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—41no—
152C4⋊Dic83—81no—
153C4⋊Dic83—81no—
154C4⋊Dic83—81no—
155C4⋊Dic83—81no—
156C4⋊Dic83C2×Q812yes—
157C22⋊Dic83—42no—
158C22⋊Dic83—42no—
159C4×Dic83—21no—
160C4×Dic83—21no—
161C4×Dic83—21no—
162C4×Dic83—21no—
163C4×Dic83—42no—
164C4×Dic83—81no—
165C4×Dic83—81no—
166C24×C83—21no—
167C2×Q8×C83—24no—
168C2×Q8×C83—81no—
169C2×Q8×C83—81no—
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—81no—
181C22×C4×C83—81no—
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—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83C22×C412yes—
196C4⋊C4×C83C22×C412yes—
197C4⋊C4×C83C22×C412yes—
198C22⋊C4×C83—24no—
199C42×C83C22×C412yes—
200C42×C83#4112yes—
201C23×C83—22no—
202C23×C83—22no—
203C23×C83—22no—
204C23×C83—41no—
205C23×C83—41no—
206C23×C83—41no—
207C23×C83—41no—
208C23×C83C22×D411yescharacteristic
209Q8×C83—84no—
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—41no—
222C2×C4×C83—41no—
223C2×C4×C83—41no—
224C2×C4×C83—41no—
225C2×C4×C83—41no—
226C2×C4×C83—41no—
227C2×C4×C83—84no—
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×Dic83#13712yes—
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×#13711yescharacteristic
277C22×C83#57511yescharacteristic
278C22×C83#57611yescharacteristic
279C22×C83#57611yescharacteristic
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#22979—831nomaximal, Sylow
298C2×C83—44no—
299C2×C83#236011yescharacteristic
300C2×C83C4×#13711yescharacteristic
301C2×C83#294811yescharacteristic
302C2×C83#331411yescharacteristic
303C2×C83#331911yescharacteristic
304C2×C83#338511yescharacteristic
305C2×C83#338911yescharacteristic
306C2×#474D8311yescharacteristic
307#2369—831no—
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×D4C2×D8311yescharacteristic
326#474Dic8318yescomplemented
327C2×#107—831no—
328C2×#107C2×D8311yescharacteristic
329C2×#107C2×D8311yescharacteristic
330C2×#106—831no—
331C2×C4⋊C8C2×D8311yescharacteristic
332C2×#102—831no—
333C2×#102—831no—
334C2×#102C2×D8311yescharacteristic
335C2×#102C2×D8311yescharacteristic
336C2×#97—831no—
337C2×#97—831no—
338C2×C4×C8—831no—
339#391—1661no—
340#391—1661no—
341#389—831no—
342#389—831no—
343#389—831no—
344#389—831no—
345#389—1661no—
346#385—1661no—
347#385—1661no—
348#382—831no—
349#380—1661no—
350C4×C4⋊C4—831no—
351C4×C4⋊C4—831no—
352C4×C4⋊C4—831no—
353C4×C4⋊C4—831no—
354C4×C4⋊C4—1661no—
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×D4C22×D8311yescharacteristic
368C23×C4—21no—
369C23×C4—1661no—
370C23×C4—1661no—
371C22×C8—831no—
372C22×C8—831no—
373C22×C8C22×D8311yescharacteristic
374C22×C8C22×D8311yescharacteristic
375C4⋊Q8—22no—
376C4⋊Q8C2×Dic8314yescomplemented
377C4×D4—22no—
378C4×D4—3321no—
379C4×D4—3321no—
380C4×D4—1662no—
381C4×D4—1662no—
382C4×D4C2×Dic8314yes—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—831no—
385C2×C4⋊C4—831no—
386C2×C4⋊C4—831no—
387C2×C4⋊C4—831no—
388C2×C4⋊C4—831no—
389C2×C4⋊C4—831no—
390C2×C4⋊C4—1661no—
391C2×C4⋊C4—1661no—
392C2×C4⋊C4—1661no—
393C2×C4⋊C4—1661no—
394C2×C4⋊C4C22×D8311yescharacteristic
395C2×C4⋊C4C22×D8311yescharacteristic
396C2×C4⋊C4C22×D8311yescharacteristic
397C2×C22⋊C4—21no—
398C2×C22⋊C4—1661no—
399C2×C22⋊C4—1661no—
400C2×C42—831no—
401C2×C42—831no—
402C2×C42—1661no—
403C2×C42—1661no—
404C2×C42—1661no—
405C2×C42C22×D8311yescharacteristic
406#107—3321no—
407#107C2×Dic8314yescomplemented
408#107C2×Dic8314yescomplemented
409#106—3321no—
410C4⋊C8C2×Dic8314yescomplemented
411#102—834no—
412#102—1662no—
413#102C2×Dic8314yes—
414#102C2×Dic8314yes—
415#97—3321no—
416#97—1662no—
417C4×C8—3321no—
418#95—831no—
419#95—831no—
420#95—1661no—
421#95—1661no—
422#95—1661no—
423#95—1661no—
424#95—1661no—
425#95—1661no—
426#95—1661no—
427#95—1661no—
428#95—1661no—
429C24—21no—
430C2×Q8—24no—
431C2×Q8—81no—
432C2×Q8—81no—
433C2×D4—21no—
434C2×D4—21no—
435C2×D4—21no—
436C2×D4C22×Dic8312yes—
437C2×D4Q8×D8312yes—
438C2×D4#1305059112yes—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—24no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—3322no—
446C22×C4—3322no—
447C22×C4—3322no—
448C22×C4—3322no—
449C22×C4—831no—
450C22×C4—831no—
451C22×C4—831no—
452C22×C4—831no—
453C22×C4—6641no—
454C22×C4—6641no—
455C22×C4—1661no—
456C22×C4—1661no—
457C22×C4—1661no—
458C22×C4—1661no—
459C22×C4D4×D8311yescharacteristic
460C22×C4D4×D8311yescharacteristic
461C22×C4C23×D8311yescharacteristic
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—832no—
467C2×C8—1661no—
468C2×C8—1661no—
469C2×C8—1661no—
470C2×C8—1661no—
471C2×C8—1661no—
472C2×C8C22×Dic8312yes—
473C2×C8C22×Dic8312yes—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—24no—
478C4⋊C4—3321no—
479C4⋊C4—832no—
480C4⋊C4—6641no—
481C4⋊C4—6641no—
482C4⋊C4—6641no—
483C4⋊C4—6641no—
484C4⋊C4—1662no—
485C4⋊C4—1661no—
486C4⋊C4—1661no—
487C4⋊C4—1661no—
488C4⋊C4—1661no—
489C4⋊C4—1661no—
490C4⋊C4—1661no—
491C4⋊C4—1661no—
492C4⋊C4C22×Dic8312yes—
493C4⋊C4C22×Dic8312yes—
494C4⋊C4C22×Dic8312yes—
495C22⋊C4—24no—
496C22⋊C4—3322no—
497C22⋊C4—3322no—
498C42—3322no—
499C42—6641no—
500C42—6641no—
501C42—1661no—
502C42—1661no—
503C42—1661no—
504C42—1661no—
505C42C22×Dic8312yes—
506C42#1305058912yes—
507C23—22no—
508C23—22no—
509C23—22no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23C2×D4×D8311yescharacteristic
515Q8—84no—
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—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—84no—
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
582C22#137×D8311yescharacteristic
583C22#1912850011yescharacteristic
584C22#1912850211yescharacteristic
585C22#1912850211yescharacteristic
586C4—22no—
587C4—22no—
588C4—42no—
589C4—42no—
590C4—42no—
591C4—3321no—
592C4—3321no—
593C4—3322no—
594C4—3321no—
595C4—6641no—
596C4—6641no—
597C4—6641no—
598C4—1662no—
599C4C2×Q8×Dic8311yescharacteristic
600C4#1912818411yescharacteristic
601C4#1912839311yescharacteristic
602C4#1912840911yescharacteristic
603C2—44no—
604C2#2491047211yescharacteristic
605C2#137×Dic8311yescharacteristic
606C2#2491631511yescharacteristic
607C2#2491954111yescharacteristic
608C2#2491961611yescharacteristic
609C2#2492036811yescharacteristic
610C2#2492041611yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#474×C83C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
cosocleC22×C4×C83C24
hypercenterC2×#474D83
hypocenterC83#22979
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. #24910553C2semidirect
3. #24908437C2semidirect
4. #24908233C2semidirect
5. #24908202C2semidirect
6. #24908197C2semidirect
7. #24908057C2semidirect
8. #24908036C2semidirect
11. #24907340C2semidirect
54. #474×C83C4semidirect
88. #14581713Q8semidirect
109. C4⋊Q8×C83C2×C4semidirect
118. #107×C83C2×C4semidirect
119. #107×C83C2×C4semidirect
120. C4⋊C8×C83C2×C4semidirect
321. C83#22979semidirect
326. #474Dic83semidirect
376. C4⋊Q8C2×Dic83semidirect
407. #107C2×Dic83semidirect
408. #107C2×Dic83semidirect
410. C4⋊C8C2×Dic83semidirect

Maximal subgroups

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

subgroupindex
#249105532
#249103732
#249084372
#249082332
#249082022
#249081972
#249080572
#249080362
#249077262
#249077122
#249073402
#249072902
#249072882
#102×Dic832
C2×#474×C832
#2297983