Finite Group Databasebeta

← #35463138

Subgroups — #35463138

Order 21248.

774 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×#19127737C211yesmaximal, characteristic
3#24911176C211yesmaximal, characteristic
4#24908945C211yesmaximal, characteristic, complemented
5#24908939C211yesmaximal, characteristic, complemented
6#24908801C211yesmaximal, characteristic, complemented
7#24908793C211yesmaximal, characteristic, complemented
8#24908203C211yesmaximal, characteristic, complemented
9#24908198C211yesmaximal, characteristic, complemented
10#24907662C211yesmaximal, characteristic
11#24907660C211yesmaximal, characteristic
12#24907632C211yesmaximal, characteristic, complemented
13#24907486C211yesmaximal, characteristic
14#24907241C211yesmaximal, characteristic
15C83×#2439C211yesmaximal, characteristic
16C23×C83⋊C8C2211yescharacteristic
17C2×#14581784C2211yescharacteristic
18C2×#14581751C2211yescharacteristic
19#19128107—24no—
20#19127737—24no—
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581725C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×#14581723C2211yescharacteristic
27C2×#14581723C2211yescharacteristic
28C2×#14581723C2211yescharacteristic
29C2×#14581723C2211yescharacteristic
30C2×#14581723C2211yescharacteristic
31C2×C22⋊(C83⋊C8)C2211yescharacteristic
32C2×#14581703C2211yescharacteristic
33C2×#14581703C2211yescharacteristic
34#19127637—21no—
35#19127624—21no—
36#19127599C2211yescharacteristic
37#19127599C2211yescharacteristic
38#19127598—21no—
39#19127584—21no—
40#19127571C2211yescharacteristic
41#19127561C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×C4⋊Q8×C83C2211yescharacteristic
51C2×#117×C83C2211yescharacteristic
52#396×C83—21no—
53#393×C83—21no—
54#391×C83—21no—
55#389×C83C2211yescharacteristic
56#389×C83C2211yescharacteristic
57#387×C83—21no—
58#382×C83C2211yescharacteristic
59#380×C83C2211yescharacteristic
60C4×C22⋊C4×C83C2211yescharacteristic
61C22×C83⋊C8—22no—
62C22×C83⋊C8—22no—
63C22×C83⋊C8Q814yescomplemented
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68#14581784—22no—
69#14581784—41no—
70#14581751—22no—
71#14581751—41no—
72C2×C4⋊Dic83—21no—
73C2×C4⋊Dic83—21no—
74C2×C4⋊Dic83C2311yescharacteristic
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—21no—
80C2×Dic83⋊C4—21no—
81C2×Dic83⋊C4—21no—
82C2×C22⋊Dic83—21no—
83C2×C22⋊Dic83—21no—
84C2×C4×Dic83—21no—
85#14581725—22no—
86#14581725—22no—
87#14581725—41no—
88#14581725D414yes—
89#14581723—22no—
90#14581723—22no—
91#14581723—22no—
92#14581723—22no—
93#14581723—22no—
94#14581723—41no—
95C22⋊(C83⋊C8)Q814yescomplemented
96#14581703—22no—
97#14581703—22no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C22×Q8×C83—21no—
102C23×C4×C83C2311yescharacteristic
103C4⋊Q8×C83—22no—
104C4⋊Q8×C83—41no—
105#117×C83—22no—
106#117×C83—41no—
107C2×C4⋊C4×C83—21no—
108C2×C4⋊C4×C83—21no—
109C2×C4⋊C4×C83—21no—
110C2×C4⋊C4×C83C2311yescharacteristic
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C22⋊C4×C83—21no—
115C2×C22⋊C4×C83—21no—
116C2×C22⋊C4×C83D411yescharacteristic
117C2×C22⋊C4×C83D411yescharacteristic
118C2×C42×C83—21no—
119C2×C42×C83C2311yescharacteristic
120C2×C42×C83C2311yescharacteristic
121#95×C83—21no—
122#95×C83—21no—
123#95×C83—21no—
124#95×C83—21no—
125#95×C83—21no—
126#95×C83D411yescharacteristic
127#95×C83D411yescharacteristic
128C22×Dic83—21no—
129C22×Dic83—21no—
130C22×Dic83—21no—
131C22×Dic83—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—28no—
139C2×C83⋊C8C2×D412yes—
140C2×C83⋊C8C2×Q812yes—
141C2×C83⋊C8C2×Q812yes—
142C2×C83⋊C8C2×Q812yes—
143C2×C83⋊C8C2×Q812yes—
144C2×C83⋊C8#4112yes—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—42no—
153C4⋊Dic83—81no—
154C4⋊Dic83C2×D412yes—
155Dic83⋊C4—42no—
156Dic83⋊C4—42no—
157Dic83⋊C4—42no—
158Dic83⋊C4—81no—
159C22⋊Dic83—42no—
160C22⋊Dic83—81no—
161C4×Dic83—42no—
162C24×C83C2×D411yescharacteristic
163C2×Q8×C83—42no—
164C2×Q8×C83—42no—
165C2×Q8×C83—81no—
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83—21no—
169C22×C4×C83—21no—
170C22×C4×C83—21no—
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83—21no—
174C22×C4×C83C2×D411yescharacteristic
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83C2×D411yescharacteristic
177C22×C4×C83C2×D411yescharacteristic
178C22×C4×C83C2×D411yescharacteristic
179C22×C4×C83C2×Q812yes—
180C22×C4×C83C2×Q812yes—
181C22×C4×C83C2411yescharacteristic
182C4⋊C4×C83—21no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—21no—
186C4⋊C4×C83—21no—
187C4⋊C4×C83—21no—
188C4⋊C4×C83—21no—
189C4⋊C4×C83—42no—
190C4⋊C4×C83—42no—
191C4⋊C4×C83—81no—
192C4⋊C4×C83C2×D412yes—
193C22⋊C4×C83—41no—
194C22⋊C4×C83—41no—
195C22⋊C4×C83—42no—
196C22⋊C4×C83—81no—
197C42×C83—21no—
198C42×C83—21no—
199C42×C83—21no—
200C42×C83—21no—
201C42×C83—42no—
202C23×C83—21no—
203C23×C83—21no—
204C23×C83—21no—
205C23×C83—21no—
206C23×C83—21no—
207C23×C83—21no—
208C23×C83C4⋊Q812yes—
209C23×C83C22×D411yescharacteristic
210Q8×C83—84no—
211C2×C4×C83—21no—
212C2×C4×C83—21no—
213C2×C4×C83—22no—
214C2×C4×C83—22no—
215C2×C4×C83—22no—
216C2×C4×C83—22no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—22no—
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—21no—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
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—41no—
238C2×C4×C83—41no—
239C2×C4×C83—41no—
240C2×C4×C83—41no—
241C2×C4×C83C4⋊Q812yes—
242C2×C4×C83C22×D411yescharacteristic
243C2×C4×C83C22×Q811yescharacteristic
244C2×C4×C83C22×Q811yescharacteristic
245C2×C4×C83C2×#4111yescharacteristic
246C2×C4×C83H(5,𝔽2)11yescharacteristic
247C2×C4×C832−1+411yescharacteristic
248C2×Dic83—22no—
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—41no—
260C83⋊C8—22no—
261C83⋊C8—22no—
262C83⋊C8—22no—
263C83⋊C8C22⋊Q814yes—
264C22×C83—22no—
265C22×C83—22no—
266C22×C83—22no—
267C22×C83—41no—
268C22×C83—41no—
269C22×C83—41no—
270C22×C83—41no—
271C22×C83C2×C4⋊Q811yescharacteristic
272C22×C83D4211yescharacteristic
273C22×C83#54711yescharacteristic
274C22×C83#57511yescharacteristic
275C22×C83#57511yescharacteristic
276C22×C83#57711yescharacteristic
277C22×C83#57711yescharacteristic
278C4×C83—22no—
279C4×C83—41no—
280C4×C83—41no—
281C4×C83—41no—
282C4×C83—42no—
283C4×C83—41no—
284C4×C83—81no—
285C4×C83—81no—
286C4×C83—81no—
287C4×C83C2×C22⋊Q811yescharacteristic
288C4×C83#54411yescharacteristic
289C4×C83D4×Q811yescharacteristic
290C4×C83#55411yescharacteristic
291Dic83—42no—
292Dic83—81no—
293Dic83—81no—
294Dic83—81no—
295#26070—831nomaximal, Sylow
296C2×C83—42no—
297C2×C83#244011yescharacteristic
298C2×C83#316211yescharacteristic
299C2×C83#316311yescharacteristic
300C2×C83#326711yescharacteristic
301C2×C83#327111yescharacteristic
302C2×C83#334811yescharacteristic
303C2×C83#335011yescharacteristic
304C2×#501—831no—
305C2×#427—831no—
306#2441—831no—
307#2439D8311yescharacteristic
308#2085—831no—
309#2084—831no—
310#2059—831no—
311#2058—831no—
312#1944—831no—
313#1943—831no—
314#1854—831no—
315#1854—831no—
316#1847—831no—
317#1807—831no—
318#1752—831no—
319C83#2607011yescharacteristic, complemented, Sylow
320C23×C8—831no—
321C2×C4⋊Q8C2×D8311yescharacteristic
322C2×#125—831no—
323C2×#117—831no—
324C2×#117C2×D8311yescharacteristic
325#501—1664no—
326#427—1664no—
327C2×#107—831no—
328C2×#107—831no—
329C2×#107—831no—
330C2×#107—831no—
331C2×#106—831no—
332C2×#106—831no—
333C2×#106—831no—
334C2×#106—831no—
335C2×#106—831no—
336C2×#106—831no—
337C2×C22⋊C8—831no—
338C2×#97—831no—
339C2×#97—831no—
340#399—1661no—
341#396—21no—
342#396—1661no—
343#393—21no—
344#391—21no—
345#389—831no—
346#389—831no—
347#389—1661no—
348#389C2×D8311yescharacteristic
349#389C2×D8311yescharacteristic
350#387—21no—
351#387—1661no—
352#384—831no—
353#382C2×D8311yescharacteristic
354#380—831no—
355#380C2×D8311yescharacteristic
356C4×C22⋊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—
366C23×C4C22×D8311yescharacteristic
367C22×C8—831no—
368C22×C8—834no—
369C22×C8—831no—
370C22×C8—831no—
371C22×C8—831no—
372C22×C8—1662no—
373C22×C8—1662no—
374C4⋊Q8—22no—
375C4⋊Q8—41no—
376#125—3321no—
377#125—1662no—
378#117—22no—
379#117—41no—
380#117—3321no—
381#117—1662no—
382C2×C4⋊C4—21no—
383C2×C4⋊C4—21no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—831no—
386C2×C4⋊C4—831no—
387C2×C4⋊C4—831no—
388C2×C4⋊C4—831no—
389C2×C4⋊C4—1661no—
390C2×C4⋊C4—1661no—
391C2×C4⋊C4—1661no—
392C2×C4⋊C4—1661no—
393C2×C4⋊C4—1661no—
394C2×C4⋊C4—1661no—
395C2×C4⋊C4C22×D8311yescharacteristic
396C2×C4⋊C4C22×D8311yescharacteristic
397C2×C4⋊C4C22×D8311yescharacteristic
398C2×C4⋊C4C22×D8311yescharacteristic
399C2×C22⋊C4—21no—
400C2×C22⋊C4—21no—
401C2×C22⋊C4—1661no—
402C2×C22⋊C4—1661no—
403C2×C22⋊C4C22⋊D8311yescharacteristic
404C2×C22⋊C4C22⋊D8311yescharacteristic
405C2×C42—21no—
406C2×C42—1661no—
407C2×C42C22×D8311yescharacteristic
408C2×C42C22×D8311yescharacteristic
409#107—3321no—
410#107—834no—
411#107—1662no—
412#107—1662no—
413#106—3321no—
414#106—1662no—
415#106—1662no—
416#106—1662no—
417#106—1662no—
418#106—1662no—
419C22⋊C8—834no—
420#97—1662no—
421#97—1662no—
422#95—21no—
423#95—21no—
424#95—21no—
425#95—21no—
426#95—21no—
427#95—1661no—
428#95—1661no—
429#95—1661no—
430#95C22⋊D8311yescharacteristic
431#95C22⋊D8311yescharacteristic
432C24C2×C22⋊D8311yescharacteristic
433C2×Q8—42no—
434C2×Q8—42no—
435C2×Q8—81no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—21no—
442C22×C4—21no—
443C22×C4—21no—
444C22×C4—1661no—
445C22×C4—1661no—
446C22×C4—1661no—
447C22×C4—1661no—
448C22×C4C2×C22⋊D8311yescharacteristic
449C22×C4C2×C22⋊D8311yescharacteristic
450C22×C4C2×C22⋊D8311yescharacteristic
451C22×C4C2×C22⋊D8311yescharacteristic
452C22×C4C2×C22⋊D8311yescharacteristic
453C22×C4Q8×D8312yes—
454C22×C4Q8×D8312yes—
455C22×C4C23×D8311yescharacteristic
456C2×C8—832no—
457C2×C8—832no—
458C2×C8—832no—
459C2×C8—832no—
460C2×C8—832no—
461C2×C8—832no—
462C2×C8—1668no—
463C2×C8—1661no—
464C2×C8—1661no—
465C2×C8—1661no—
466C2×C8—1661no—
467C2×C8—1661no—
468C2×C8—1661no—
469C4⋊C4—21no—
470C4⋊C4—21no—
471C4⋊C4—21no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—42no—
477C4⋊C4—42no—
478C4⋊C4—81no—
479C4⋊C4—3322no—
480C4⋊C4—3322no—
481C4⋊C4—3322no—
482C4⋊C4—3322no—
483C4⋊C4—832no—
484C4⋊C4—6641no—
485C4⋊C4—6641no—
486C4⋊C4—1661no—
487C4⋊C4—1661no—
488C4⋊C4—1661no—
489C4⋊C4—1661no—
490C4⋊C4—1661no—
491C4⋊C4—1661no—
492C4⋊C4—1661no—
493C4⋊C4D4×D8312yes—
494C22⋊C4—41no—
495C22⋊C4—41no—
496C22⋊C4—42no—
497C22⋊C4—81no—
498C22⋊C4—3322no—
499C22⋊C4—6641no—
500C42—21no—
501C42—21no—
502C42—21no—
503C42—21no—
504C42—42no—
505C42—3322no—
506C23—21no—
507C23—21no—
508C23—21no—
509C23—21no—
510C23—21no—
511C23—21no—
512C23#1458179012yes—
513C23C22×C22⋊D8311yescharacteristic
514Q8—84no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—22no—
524C2×C4—21no—
525C2×C4—21no—
526C2×C4—21no—
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—21no—
530C2×C4—21no—
531C2×C4—21no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—41no—
535C2×C4—41no—
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—3321no—
555C2×C4—3321no—
556C2×C4—1662no—
557C2×C4#1458179012yes—
558C2×C4C2×D4×D8311yescharacteristic
559C2×C4C2×Q8×D8311yescharacteristic
560C2×C4C2×Q8×D8311yescharacteristic
561C2×C4#41×D8311yescharacteristic
562C2×C4#1458183711yescharacteristic
563C2×C4#1458183911yescharacteristic
564C8—834no—
565C8—1662no—
566C8—1662no—
567C8—1662no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22C2×#1458179011yescharacteristic
576C22D4×C22⋊D8311yescharacteristic
577C22#1912835011yescharacteristic
578C22#1912849911yescharacteristic
579C22#1912849911yescharacteristic
580C22#1912850911yescharacteristic
581C22#1912851211yescharacteristic
582C4—22no—
583C4—41no—
584C4—41no—
585C4—41no—
586C4—42no—
587C4—41no—
588C4—81no—
589C4—81no—
590C4—81no—
591C4—3322no—
592C4—6641no—
593C4—6641no—
594C4—6641no—
595C4C22⋊Q8×D8311yescharacteristic
596C4#1912833011yescharacteristic
597C4#1912836411yescharacteristic
598C4#1912840511yescharacteristic
599C2—42no—
600C2#2491116211yescharacteristic
601C2#2491798511yescharacteristic
602C2#2491798911yescharacteristic
603C2#2491898411yescharacteristic
604C2#2491904911yescharacteristic
605C2#2491997811yescharacteristic
606C2#2492000811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2439C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#2439D83
hypocenterC83#26070
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2439

Complements

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

NH ≅ G/Ntype
1. C2×#19128107C2semidirect
4. #24908945C2semidirect
5. #24908939C2semidirect
6. #24908801C2semidirect
7. #24908793C2semidirect
8. #24908203C2semidirect
9. #24908198C2semidirect
12. #24907632C2semidirect
63. C22×C83⋊C8Q8semidirect
95. C22⋊(C83⋊C8)Q8semidirect
319. C83#26070semidirect

Maximal subgroups

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

subgroupindex
C2×#191281072
C2×#191277372
#249111762
#249089452
#249089392
#249088012
#249087932
#249082032
#249081982
#249076622
#249076602
#249076322
#249074862
#249072412
C83×#24392
#2607083