Finite Group Databasebeta

← #35456122

Subgroups — #35456122

Order 21248.

792 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128107C211yesmaximal, characteristic, complemented
2C2×#19127988C211yesmaximal, characteristic
3#24912663C211yesmaximal, characteristic
4#24908944C211yesmaximal, characteristic, complemented
5#24908939C211yesmaximal, characteristic, complemented
6#24908848C211yesmaximal, characteristic
7#24908834C211yesmaximal, characteristic
8#24908420C211yesmaximal, characteristic, complemented
9#24908231C211yesmaximal, characteristic, complemented
10#24908226C211yesmaximal, characteristic, complemented
11#24908187C211yesmaximal, characteristic, complemented
12#24908090C211yesmaximal, characteristic
13#24908083C211yesmaximal, characteristic
14#24907291C211yesmaximal, characteristic
15C83×#2365C211yesmaximal, characteristic, complemented
16C2×#14581784C2211yescharacteristic
17C2×C4×D332C2211yescharacteristic
18#19128107—24no—
19#19127988—24no—
20C2×#14581725C2211yescharacteristic
21C2×#14581725C2211yescharacteristic
22C2×#14581723C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×C4⋊(C83⋊C8)C2211yescharacteristic
27C2×C4⋊(C83⋊C8)C2211yescharacteristic
28C2×C4⋊(C83⋊C8)C2211yescharacteristic
29C2×#14581712C2211yescharacteristic
30C2×#14581712C2211yescharacteristic
31C2×#14581712C2211yescharacteristic
32C2×#14581712C2211yescharacteristic
33C2×#14581703C2211yescharacteristic
34#19127623—21no—
35#19127623—21no—
36#19127603C2211yescharacteristic
37#19127603C2211yescharacteristic
38#19127575—21no—
39#19127575—21no—
40#19127571C2211yescharacteristic
41#19127567C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×C4⋊Q8×C83C2211yescharacteristic
51C2×C4×Q8×C83—21no—
52#391×C83—21no—
53#389×C83C2211yescharacteristic
54#389×C83C2211yescharacteristic
55#387×C83—21no—
56#387×C83—21no—
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C4⋊C4×C83C2211yescharacteristic
59C4×C4⋊C4×C83C2211yescharacteristic
60C4×C4⋊C4×C83C2211yescharacteristic
61C22×D332C2311yescharacteristic
62C22×C4×D83—21no—
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67#14581784—22no—
68#14581784—22no—
69C4×D332—22no—
70C4×D332—22no—
71C2×C4⋊Dic83—21no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×Dic83⋊C4—21no—
77C2×Dic83⋊C4—21no—
78C2×(C2×D83)⋊C4—21no—
79C2×(C2×D83)⋊C4—21no—
80C2×(C2×D83)⋊C4—21no—
81C2×(C2×D83)⋊C4—21no—
82C2×(C2×D83)⋊C4—21no—
83#14581725—22no—
84#14581725—22no—
85#14581723—22no—
86#14581723—22no—
87#14581723—22no—
88#14581723—22no—
89C4⋊(C83⋊C8)—22no—
90C4⋊(C83⋊C8)—22no—
91C4⋊(C83⋊C8)—22no—
92#14581712—22no—
93#14581712—22no—
94#14581712—22no—
95#14581712Q814yescomplemented
96#14581703—22no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C22×Q8×C83—21no—
100C4⋊Q8×C83—22no—
101C4⋊Q8×C83D414yes—
102C4×Q8×C83—44no—
103C2×C4⋊C4×C83—21no—
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83—21no—
106C2×C4⋊C4×C83D411yescharacteristic
107C2×C4⋊C4×C83D411yescharacteristic
108C2×C4⋊C4×C83C2311yescharacteristic
109C2×C4⋊C4×C83C2311yescharacteristic
110C2×C4⋊C4×C83C2311yescharacteristic
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C42×C83—21no—
113C2×C42×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×C83—21no—
121#95×C83—21no—
122#95×C83—21no—
123#95×C83D411yescharacteristic
124#95×C83D411yescharacteristic
125C23×D83—21no—
126C2×D332—21no—
127C2×D332—21no—
128C2×D332—21no—
129C2×D332—21no—
130C2×D332—21no—
131C2×D332C2×Q812yes—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83—21no—
135C2×C4×D83—42no—
136C2×C4×D83—81no—
137C2×C4×D83—81no—
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⋊C8—21no—
146C2×C83⋊C8—21no—
147C2×C83⋊C8—21no—
148C2×C83⋊C8C2×Q812yes—
149C2×C83⋊C8#4112yes—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—81no—
157C4⋊Dic83—81no—
158Dic83⋊C4—42no—
159Dic83⋊C4—42no—
160(C2×D83)⋊C4—42no—
161(C2×D83)⋊C4—42no—
162(C2×D83)⋊C4—42no—
163(C2×D83)⋊C4—81no—
164(C2×D83)⋊C4—81no—
165C2×Q8×C83—24no—
166C2×Q8×C83—42no—
167C2×Q8×C83—42no—
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—42no—
188C4⋊C4×C83—42no—
189C4⋊C4×C83C2×D412yes—
190C4⋊C4×C83C2×D412yes—
191C4⋊C4×C83C2×Q812yes—
192C42×C83—21no—
193C42×C83—21no—
194C42×C83—21no—
195C42×C83—21no—
196C42×C83—42no—
197C42×C83—42no—
198C42×C83—42no—
199C42×C83C2×D412yes—
200C42×C83#4112yes—
201C23×C83C22×D411yescharacteristic
202C22×D83—22no—
203C22×D83—41no—
204C22×D83—41no—
205C22×D83—41no—
206C22×D83—41no—
207C22×D83—41no—
208C22×D83—41no—
209Q8×C83—48no—
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×C83#12112yes—
238C2×C4×C83#12112yes—
239C2×C4×C83AGL(1,ℤ/8)12yes—
240C2×C4×C83C22×D411yescharacteristic
241C2×C4×C83C22×Q811yescharacteristic
242C2×C4×C83C2×#4111yescharacteristic
243C2×C4×C83C2×#4111yescharacteristic
244C2×C4×C832−1+411yescharacteristic
245C2×C4×C832−1+411yescharacteristic
246C2×Dic83—41no—
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—41no—
258D332—41no—
259C4×D83—84no—
260C83⋊C8—22no—
261C83⋊C8—22no—
262C83⋊C8—41no—
263C83⋊C8—41no—
264C22×C83C2×#12111yescharacteristic
265C22×C83#54711yescharacteristic
266C22×C83D4×Q811yescharacteristic
267C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
268C22×C83#57511yescharacteristic
269C22×C83#57711yescharacteristic
270C22×C83#57711yescharacteristic
271C2×D83—42no—
272C2×D83—42no—
273C2×D83—42no—
274C2×D83—81no—
275C2×D83—81no—
276C2×D83—81no—
277C2×D83—81no—
278C4×C83—22no—
279C4×C83—22no—
280C4×C83—22no—
281C4×C83—41no—
282C4×C83—41no—
283C4×C83—41no—
284C4×C83—42no—
285C4×C83—42no—
286C4×C83—42no—
287C4×C83—81no—
288C4×C83#53311yescharacteristic
289C4×C83#54111yescharacteristic
290C4×C83#54811yescharacteristic
291C4×C83D4×Q811yescharacteristic
292Dic83—81no—
293Dic83—81no—
294Dic83—81no—
295#25461—831nomaximal, Sylow
296C2×C83#242911yescharacteristic
297C2×C83#303611yescharacteristic
298C2×C83#304311yescharacteristic
299C2×C83#334511yescharacteristic
300C2×C83#335011yescharacteristic
301C2×C83#338411yescharacteristic
302C2×C83#338811yescharacteristic
303D83—82no—
304C2×#501—831no—
305C2×#476—831no—
306#2566—831no—
307#2365D8311yescharacteristic, complemented
308#2085—831no—
309#2084—831no—
310#2067—831no—
311#2065—831no—
312#1989—831no—
313#1949—831no—
314#1948—831no—
315#1940—831no—
316#1926—831no—
317#1925—831no—
318#1762—831no—
319C83#2546111yescharacteristic, complemented, Sylow
320C2×C4⋊Q8C2×D8311yescharacteristic
321C2×#125—831no—
322C2×C4×Q8—21no—
323C2×C4×D4—831no—
324#501—1664no—
325#476—1664no—
326C2×#107—831no—
327C2×#107—831no—
328C2×#106—831no—
329C2×#106—831no—
330C2×#106—831no—
331C2×#106—831no—
332C2×C4⋊C8—831no—
333C2×C4⋊C8—831no—
334C2×C4⋊C8—831no—
335C2×#102—831no—
336C2×#102—831no—
337C2×#102—831no—
338C2×#102—831no—
339C2×#97—831no—
340#396—1661no—
341#396—1661no—
342#391—21no—
343#390—831no—
344#390—831no—
345#389C2×D8311yescharacteristic
346#389C2×D8311yescharacteristic
347#387—21no—
348#387—21no—
349#385—1661no—
350#385—1661no—
351#384—831no—
352#382—831no—
353C4×C4⋊C4C2×D8311yescharacteristic
354C4×C4⋊C4C2×D8311yescharacteristic
355C4×C4⋊C4C2×D8311yescharacteristic
356C4×C4⋊C4C2×D8311yescharacteristic
357#340—831no—
358#340—831no—
359#340—831no—
360#340—831no—
361#340—831no—
362#340—831no—
363#336—831no—
364#336—831no—
365C22×Q8—21no—
366C22×D4—831no—
367C23×C4—1661no—
368C22×C8—831no—
369C22×C8—831no—
370C22×C8—831no—
371C22×C8—831no—
372C4⋊Q8—22no—
373C4⋊Q8D33214yes—
374#125—1662no—
375#125—1662no—
376C4×Q8—44no—
377C4×D4—1662no—
378C4×D4—1662no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—21no—
382C2×C4⋊C4—831no—
383C2×C4⋊C4—831no—
384C2×C4⋊C4—831no—
385C2×C4⋊C4—1661no—
386C2×C4⋊C4—1661no—
387C2×C4⋊C4—1661no—
388C2×C4⋊C4—1661no—
389C2×C4⋊C4C22⋊D8311yescharacteristic
390C2×C4⋊C4C22⋊D8311yescharacteristic
391C2×C4⋊C4C22×D8311yescharacteristic
392C2×C4⋊C4C22×D8311yescharacteristic
393C2×C4⋊C4C22×D8311yescharacteristic
394C2×C4⋊C4C22×D8311yescharacteristic
395C2×C22⋊C4—1661no—
396C2×C22⋊C4—1661no—
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—1662no—
407#107—1662no—
408#106—1662no—
409#106—1662no—
410#106—1662no—
411#106—1662no—
412C4⋊C8—1662no—
413C4⋊C8—1662no—
414C4⋊C8—1662no—
415#102—834no—
416#102—1662no—
417#102—1662no—
418#102—1662no—
419#97—1662no—
420#95—21no—
421#95—21no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—1661no—
426#95—1661no—
427#95C22⋊D8311yescharacteristic
428#95C22⋊D8311yescharacteristic
429C24—1661no—
430C2×Q8—24no—
431C2×Q8—42no—
432C2×Q8—42no—
433C2×D4—832no—
434C2×D4—1661no—
435C2×D4—1661no—
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—6641no—
445C22×C4—6641no—
446C22×C4—1661no—
447C22×C4—1661no—
448C22×C4—1661no—
449C22×C4C2×C22⋊D8311yescharacteristic
450C22×C4C2×C22⋊D8311yescharacteristic
451C22×C4C2×C22⋊D8311yescharacteristic
452C22×C4C2×C22⋊D8311yescharacteristic
453C22×C4C2×C22⋊D8311yescharacteristic
454C22×C4C2×C22⋊D8311yescharacteristic
455C22×C4C23×D8311yescharacteristic
456C2×C8—832no—
457C2×C8—832no—
458C2×C8—1661no—
459C2×C8—1661no—
460C2×C8—1661no—
461C2×C8—1661no—
462C2×C8—1661no—
463C2×C8—1661no—
464C2×C8—1661no—
465C2×C8—1661no—
466C2×C8—1661no—
467C2×C8—1661no—
468C4⋊C4—24no—
469C4⋊C4—22no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—41no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—3322no—
479C4⋊C4—3322no—
480C4⋊C4—6641no—
481C4⋊C4—6641no—
482C4⋊C4—1661no—
483C4⋊C4—1661no—
484C4⋊C4—1661no—
485C4⋊C4—1661no—
486C4⋊C4—1661no—
487C4⋊C4—1661no—
488C4⋊C4C2×D33212yes—
489C4⋊C4C2×D33212yes—
490C4⋊C4Q8×D8312yes—
491C22⋊C4—3322no—
492C22⋊C4—3322no—
493C22⋊C4—3322no—
494C22⋊C4—6641no—
495C22⋊C4—6641no—
496C42—21no—
497C42—21no—
498C42—21no—
499C42—21no—
500C42—42no—
501C42—42no—
502C42—42no—
503C42C2×D33212yes—
504C42#1305059012yes—
505C23—3321no—
506C23—3321no—
507C23—3321no—
508C23—3321no—
509C23—3321no—
510C23—3321no—
511C23—1662no—
512C23C22×C22⋊D8311yescharacteristic
513Q8—48no—
514D4—3321no—
515D4—3321no—
516D4—1662no—
517D4—1662no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—22no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
535C2×C4—21no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—41no—
544C2×C4—41no—
545C2×C4—3321no—
546C2×C4—3321no—
547C2×C4—3321no—
548C2×C4—3321no—
549C2×C4—3321no—
550C2×C4—3321no—
551C2×C4—3321no—
552C2×C4—3321no—
553C2×C4—3321no—
554C2×C4—6644no—
555C2×C4#1458176612yes—
556C2×C4#1458176612yes—
557C2×C4#1458181812yes—
558C2×C4C22×D33211yescharacteristic
559C2×C4C2×Q8×D8311yescharacteristic
560C2×C4#41×D8311yescharacteristic
561C2×C4C2×#1305059011yescharacteristic
562C2×C4#1458183911yescharacteristic
563C2×C4#1458184011yescharacteristic
564C8—3321no—
565C8—3321no—
566C8—1662no—
567C8—1662no—
568C22—3322no—
569C22—3322no—
570C22—3322no—
571C22—6641no—
572C22—6641no—
573C22—6641no—
574C22—6641no—
575C22C2×#1458176611yescharacteristic
576C22#1912835011yescharacteristic
577C22Q8×C22⋊D8311yescharacteristic
578C22C2×#1458181811yescharacteristic
579C22#1912849911yescharacteristic
580C22#1912850911yescharacteristic
581C22#1912850911yescharacteristic
582C4—22no—
583C4—22no—
584C4—22no—
585C4—42no—
586C4—41no—
587C4—42no—
588C4—42no—
589C4—41no—
590C4—41no—
591C4—81no—
592C4—6641no—
593C4—6641no—
594C4—6641no—
595C4#1912825511yescharacteristic
596C4#1912830911yescharacteristic
597C4#1912836311yescharacteristic
598C4Q8×D33211yescharacteristic
599C2—6642no—
600C2#2491105611yescharacteristic
601C2#2491692711yescharacteristic
602C2#2491698411yescharacteristic
603C2#2491993311yescharacteristic
604C2#2492000811yescharacteristic
605C2#2492035811yescharacteristic
606C2#2492040211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2365C2
socleC23×C83C22×D4
prime radicalC22×D332C23
hypercenter#2365D83
hypocenterC83#25461
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2365

Complements

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

NH ≅ G/Ntype
1. C2×#19128107C2semidirect
4. #24908944C2semidirect
5. #24908939C2semidirect
8. #24908420C2semidirect
9. #24908231C2semidirect
10. #24908226C2semidirect
11. #24908187C2semidirect
15. C83×#2365C2semidirect
95. #14581712Q8semidirect
307. #2365D83semidirect
319. C83#25461semidirect

Maximal subgroups

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

subgroupindex
C2×#191281072
C2×#191279882
#249126632
#249089442
#249089392
#249088482
#249088342
#249084202
#249082312
#249082262
#249081872
#249080902
#249080832
#249072912
C83×#23652
#2546183