Finite Group Databasebeta

← #35462646

Subgroups — #35462646

Order 21248.

736 conjugacy classes of subgroups.

The subgroup lattice has 596 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24910410C211yesmaximal, characteristic
2#24908848C211yesmaximal, characteristic
3#24908848C211yesmaximal, characteristic
4#24908606C211yesmaximal, characteristic
5#24908596C211yesmaximal, characteristic
6#24908421C211yesmaximal, characteristic, complemented
7#24908412C211yesmaximal, characteristic, complemented
8#24908221C211yesmaximal, characteristic, complemented
9#24908187C211yesmaximal, characteristic, complemented
10#24908052C211yesmaximal, characteristic, complemented
11#24908031C211yesmaximal, characteristic, complemented
12#24907596C211yesmaximal, characteristic, complemented
13#24907470C211yesmaximal, characteristic
14#24907291C211yesmaximal, characteristic
15C83×#2550C211yesmaximal, characteristic, complemented
16C2×C4×D332C2211yescharacteristic
17C2×#14581750—21no—
18C2×#14581725C2211yescharacteristic
19C2×#14581725C2211yescharacteristic
20C2×C4⋊(C83⋊C8)C2211yescharacteristic
21C2×C4⋊(C83⋊C8)C2211yescharacteristic
22C2×C4⋊(C83⋊C8)C2211yescharacteristic
23C2×#14581712C2211yescharacteristic
24C2×#14581712C2211yescharacteristic
25C2×#14581712C2211yescharacteristic
26C2×#14581712C2211yescharacteristic
27C2×#14581703C2211yescharacteristic
28#19127603C2211yescharacteristic
29#19127603C2211yescharacteristic
30#19127593—21no—
31#19127593—21no—
32#19127571C2211yescharacteristic
33#19127558—21no—
34C4×C4⋊Dic83C2211yescharacteristic
35C4×C4⋊Dic83C2211yescharacteristic
36#19127463C2211yescharacteristic
37#19127463C2211yescharacteristic
38#19127463C2211yescharacteristic
39#19127463C2211yescharacteristic
40#19127463C2211yescharacteristic
41#19127463C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127456C2211yescharacteristic
47#19127456C2211yescharacteristic
48C2×C4⋊Q8×C83C2211yescharacteristic
49C2×#125×C83C2211yescharacteristic
50#398×C83—21no—
51#395×C83—21no—
52#391×C83—21no—
53#389×C83—21no—
54#389×C83C2211yescharacteristic
55#389×C83C2211yescharacteristic
56#382×C83C2211yescharacteristic
57C4×C4⋊C4×C83C2211yescharacteristic
58C4×C4⋊C4×C83C2211yescharacteristic
59C22×D332C2311yescharacteristic
60C22×C4×D83—21no—
61C22×C83⋊C8C2311yescharacteristic
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65C4×D332—22no—
66C4×D332Q814yescomplemented
67#14581750—44no—
68C2×C4⋊Dic83—21no—
69C2×C4⋊Dic83—21no—
70C2×C4⋊Dic83C2311yescharacteristic
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×(C2×D83)⋊C4—21no—
74C2×(C2×D83)⋊C4—21no—
75C2×(C2×D83)⋊C4—21no—
76C2×(C2×D83)⋊C4—21no—
77C2×(C2×D83)⋊C4—21no—
78C2×C4×Dic83—21no—
79C2×C4×Dic83—21no—
80#14581725—22no—
81#14581725—22no—
82C4⋊(C83⋊C8)—22no—
83C4⋊(C83⋊C8)—41no—
84C4⋊(C83⋊C8)—41no—
85#14581712—22no—
86#14581712—22no—
87#14581712—41no—
88#14581712—41no—
89#14581703—22no—
90C83⋊#95—21no—
91C83⋊#95—21no—
92C22×Q8×C83—21no—
93C4⋊Q8×C83—22no—
94C4⋊Q8×C83—41no—
95#125×C83—22no—
96#125×C83—41no—
97C2×C4⋊C4×C83—21no—
98C2×C4⋊C4×C83—21no—
99C2×C4⋊C4×C83—21no—
100C2×C4⋊C4×C83—21no—
101C2×C4⋊C4×C83—21no—
102C2×C4⋊C4×C83D411yescharacteristic
103C2×C4⋊C4×C83D411yescharacteristic
104C2×C4⋊C4×C83C2311yescharacteristic
105C2×C4⋊C4×C83C2311yescharacteristic
106C2×C4⋊C4×C83C2311yescharacteristic
107C2×C4⋊C4×C83C2311yescharacteristic
108C2×C42×C83—21no—
109C2×C42×C83C2311yescharacteristic
110C2×C42×C83C2311yescharacteristic
111C2×C42×C83C2311yescharacteristic
112#95×C83—21no—
113#95×C83—21no—
114#95×C83—21no—
115#95×C83—21no—
116#95×C83—21no—
117#95×C83D411yescharacteristic
118#95×C83D411yescharacteristic
119C23×D83—21no—
120C2×D332—21no—
121C2×D332—21no—
122C2×D332—21no—
123C2×D332—21no—
124C2×D332—21no—
125C2×D332C2×Q812yes—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C22×Dic83—21no—
129C2×C4×D83—24no—
130C2×C4×D83—42no—
131C2×C4×D83—42no—
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—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8—21no—
143C2×C83⋊C8—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—42no—
150C4⋊Dic83—81no—
151C4⋊Dic83C2×Q812yes—
152(C2×D83)⋊C4—24no—
153(C2×D83)⋊C4—42no—
154(C2×D83)⋊C4—42no—
155(C2×D83)⋊C4—81no—
156(C2×D83)⋊C4—81no—
157C4×Dic83—42no—
158C4×Dic83—42no—
159C2×Q8×C83—42no—
160C2×Q8×C83—42no—
161C2×Q8×C83—81no—
162C22×C4×C83—21no—
163C22×C4×C83—21no—
164C22×C4×C83—21no—
165C22×C4×C83—21no—
166C22×C4×C83C2×D411yescharacteristic
167C22×C4×C83C2×D411yescharacteristic
168C22×C4×C83C2×D411yescharacteristic
169C22×C4×C83C2×D411yescharacteristic
170C22×C4×C83C2×D411yescharacteristic
171C22×C4×C83C2×D411yescharacteristic
172C22×C4×C83C2411yescharacteristic
173C4⋊C4×C83—22no—
174C4⋊C4×C83—21no—
175C4⋊C4×C83—21no—
176C4⋊C4×C83—21no—
177C4⋊C4×C83—21no—
178C4⋊C4×C83—21no—
179C4⋊C4×C83—21no—
180C4⋊C4×C83—41no—
181C4⋊C4×C83—42no—
182C4⋊C4×C83—42no—
183C4⋊C4×C83—42no—
184C4⋊C4×C83—42no—
185C4⋊C4×C83—81no—
186C4⋊C4×C83#4112yes—
187C4⋊C4×C83#4112yes—
188C42×C83—21no—
189C42×C83—21no—
190C42×C83—21no—
191C42×C83—21no—
192C42×C83—81no—
193C42×C83C2×Q812yes—
194C42×C83#4112yes—
195C23×C83C22×D411yescharacteristic
196C22×D83—22no—
197C22×D83—41no—
198C22×D83—41no—
199C22×D83—41no—
200C22×D83—41no—
201C22×D83—41no—
202C22×D83—41no—
203Q8×C83—84no—
204C2×C4×C83—21no—
205C2×C4×C83—21no—
206C2×C4×C83—21no—
207C2×C4×C83—21no—
208C2×C4×C83—21no—
209C2×C4×C83—21no—
210C2×C4×C83—21no—
211C2×C4×C83—21no—
212C2×C4×C83—21no—
213C2×C4×C83—22no—
214C2×C4×C83—22no—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—21no—
218C2×C4×C83—21no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
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—41no—
228C2×C4×C83—41no—
229C2×C4×C83—41no—
230C2×C4×C83—41no—
231C2×C4×C83C22⋊Q812yes—
232C2×C4×C83C22⋊Q812yes—
233C2×C4×C83AGL(1,ℤ/8)12yes—
234C2×C4×C83C22×Q811yescharacteristic
235C2×C4×C83C2×#4111yescharacteristic
236C2×C4×C83C2×#4111yescharacteristic
237C2×C4×C83C2×#4111yescharacteristic
238C2×C4×C83H(5,𝔽2)11yescharacteristic
239C2×C4×C832−1+411yescharacteristic
240C2×Dic83—22no—
241C2×Dic83—41no—
242C2×Dic83—41no—
243C2×Dic83—41no—
244C2×Dic83—41no—
245C2×Dic83—41no—
246C2×Dic83—41no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249D332—22no—
250D332—22no—
251D332—41no—
252D332—41no—
253C4×D83—48no—
254C83⋊C8—41no—
255C83⋊C8—41no—
256C83⋊C8—41no—
257C83⋊C8—41no—
258C22×C83C2×C22⋊Q811yescharacteristic
259C22×C83#54611yescharacteristic
260C22×C83#54811yescharacteristic
261C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
262C22×C83#57511yescharacteristic
263C22×C83#57611yescharacteristic
264C22×C83#57811yescharacteristic
265C2×D83—42no—
266C2×D83—42no—
267C2×D83—42no—
268C2×D83—81no—
269C2×D83—81no—
270C2×D83—81no—
271C2×D83—81no—
272C4×C83—22no—
273C4×C83—22no—
274C4×C83—22no—
275C4×C83—42no—
276C4×C83—41no—
277C4×C83—41no—
278C4×C83—41no—
279C4×C83—42no—
280C4×C83—81no—
281C4×C83—81no—
282C4×C83#53311yescharacteristic
283C4×C83#54411yescharacteristic
284C4×C83#55511yescharacteristic
285C4×C83#55611yescharacteristic
286Dic83—42no—
287Dic83—81no—
288Dic83—81no—
289#26033—831nomaximal, Sylow
290C2×C83#243911yescharacteristic
291C2×C83#307711yescharacteristic
292C2×C83#308311yescharacteristic
293C2×C83#330811yescharacteristic
294C2×C83#331911yescharacteristic
295C2×C83#336611yescharacteristic
296C2×C83#337711yescharacteristic
297D83—82no—
298#2550D8311yescharacteristic, complemented
299#2354—831no—
300#2067—831no—
301#2067—831no—
302#2025—831no—
303#2023—831no—
304#1989—831no—
305#1987—831no—
306#1947—831no—
307#1940—831no—
308#1921—831no—
309#1918—831no—
310#1836—831no—
311#1804—831no—
312#1762—831no—
313C83#2603311yescharacteristic, complemented, Sylow
314C2×C4⋊Q8C2×D8311yescharacteristic
315C2×#125C2×D8311yescharacteristic
316C2×C4×D4—831no—
317C2×#117—1661no—
318C2×#107—831no—
319C2×#107—831no—
320C2×C4⋊C8—831no—
321C2×C4⋊C8—831no—
322C2×C4⋊C8—831no—
323C2×#102—831no—
324C2×#102—831no—
325C2×#102—831no—
326C2×#102—831no—
327C2×#97—831no—
328#398—21no—
329#395—21no—
330#391—21no—
331#390—831no—
332#390—831no—
333#389—21no—
334#389C2×D8311yescharacteristic
335#389C2×D8311yescharacteristic
336#388—1661no—
337#388—1661no—
338#384—831no—
339#382C2×D8311yescharacteristic
340#380—1661no—
341C4×C4⋊C4—831no—
342C4×C4⋊C4—831no—
343C4×C4⋊C4C2×D8311yescharacteristic
344C4×C4⋊C4C2×D8311yescharacteristic
345#340—831no—
346#340—831no—
347#340—831no—
348#340—831no—
349#340—831no—
350#340—831no—
351#340—831no—
352#340—831no—
353#340—831no—
354#340—831no—
355#336—831no—
356#336—831no—
357C22×Q8—21no—
358C22×D4—831no—
359C23×C4—1661no—
360C22×C8—831no—
361C22×C8—831no—
362C22×C8—831no—
363C22×C8—831no—
364C4⋊Q8—22no—
365C4⋊Q8—41no—
366#125—22no—
367#125—41no—
368C4×D4—834no—
369C4×D4—1662no—
370#117—3324no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C4⋊C4—21no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—831no—
377C2×C4⋊C4—831no—
378C2×C4⋊C4—831no—
379C2×C4⋊C4—1661no—
380C2×C4⋊C4—1661no—
381C2×C4⋊C4C22⋊D8311yescharacteristic
382C2×C4⋊C4C22⋊D8311yescharacteristic
383C2×C4⋊C4C22×D8311yescharacteristic
384C2×C4⋊C4C22×D8311yescharacteristic
385C2×C4⋊C4C22×D8311yescharacteristic
386C2×C4⋊C4C22×D8311yescharacteristic
387C2×C22⋊C4—1661no—
388C2×C22⋊C4—1661no—
389C2×C22⋊C4—1661no—
390C2×C22⋊C4—1661no—
391C2×C22⋊C4—1661no—
392C2×C42—21no—
393C2×C42—1661no—
394C2×C42—1661no—
395C2×C42C22×D8311yescharacteristic
396C2×C42C22×D8311yescharacteristic
397C2×C42C22×D8311yescharacteristic
398#107—1662no—
399#107—1662no—
400C4⋊C8—3321no—
401C4⋊C8—3321no—
402C4⋊C8—1662no—
403#102—3321no—
404#102—3321no—
405#102—1662no—
406#102—1662no—
407#97—1662no—
408#95—21no—
409#95—21no—
410#95—21no—
411#95—21no—
412#95—21no—
413#95—1661no—
414#95—1661no—
415#95C22⋊D8311yescharacteristic
416#95C22⋊D8311yescharacteristic
417C24—1661no—
418C2×Q8—42no—
419C2×Q8—42no—
420C2×Q8—81no—
421C2×D4—832no—
422C2×D4—1661no—
423C2×D4—1661no—
424C2×D4—1661no—
425C2×D4—1661no—
426C2×D4—1661no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—3322no—
432C22×C4—3322no—
433C22×C4—1661no—
434C22×C4—1661no—
435C22×C4—1661no—
436C22×C4—1664no—
437C22×C4C2×C22⋊D8311yescharacteristic
438C22×C4C2×C22⋊D8311yescharacteristic
439C22×C4C2×C22⋊D8311yescharacteristic
440C22×C4C2×C22⋊D8311yescharacteristic
441C22×C4C2×C22⋊D8311yescharacteristic
442C22×C4C2×C22⋊D8311yescharacteristic
443C22×C4C23×D8311yescharacteristic
444C2×C8—1661no—
445C2×C8—1661no—
446C2×C8—1661no—
447C2×C8—1661no—
448C2×C8—1661no—
449C2×C8—1661no—
450C2×C8—1661no—
451C2×C8—1661no—
452C2×C8—1661no—
453C2×C8—1661no—
454C2×C8—1661no—
455C2×C8—1661no—
456C4⋊C4—22no—
457C4⋊C4—21no—
458C4⋊C4—21no—
459C4⋊C4—21no—
460C4⋊C4—21no—
461C4⋊C4—21no—
462C4⋊C4—21no—
463C4⋊C4—41no—
464C4⋊C4—42no—
465C4⋊C4—42no—
466C4⋊C4—42no—
467C4⋊C4—42no—
468C4⋊C4—81no—
469C4⋊C4—3322no—
470C4⋊C4—832no—
471C4⋊C4—6641no—
472C4⋊C4—1661no—
473C4⋊C4—1661no—
474C4⋊C4—1661no—
475C4⋊C4—1661no—
476C4⋊C4—1661no—
477C4⋊C4#1305058912yes—
478C4⋊C4#1305059112yes—
479C22⋊C4—3322no—
480C22⋊C4—3322no—
481C22⋊C4—6641no—
482C22⋊C4—6641no—
483C22⋊C4—1664no—
484C42—21no—
485C42—21no—
486C42—21no—
487C42—21no—
488C42—81no—
489C42—3322no—
490C42—3322no—
491C42Q8×D8312yes—
492C42#1305059112yes—
493C23—3321no—
494C23—3321no—
495C23—3321no—
496C23—3321no—
497C23—3321no—
498C23—3321no—
499C23—1662no—
500C23C22×C22⋊D8311yescharacteristic
501Q8—84no—
502D4—3321no—
503D4—3321no—
504D4—1662no—
505D4—1662no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—3328no—
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—1662no—
543C2×C4#1458176812yes—
544C2×C4#1458176812yes—
545C2×C4#1458181712yes—
546C2×C4C2×Q8×D8311yescharacteristic
547C2×C4C2×#1305058911yescharacteristic
548C2×C4C2×#1305059111yescharacteristic
549C2×C4C2×#1305059111yescharacteristic
550C2×C4#1458183711yescharacteristic
551C2×C4#1458184011yescharacteristic
552C8—3321no—
553C8—3321no—
554C8—3321no—
555C8—3321no—
556C22—3322no—
557C22—3322no—
558C22—3322no—
559C22—6641no—
560C22—6641no—
561C22—6641no—
562C22—6641no—
563C22C2×#1458176811yescharacteristic
564C22#1912834511yescharacteristic
565C22#1912836211yescharacteristic
566C22C2×#1458181711yescharacteristic
567C22#1912849911yescharacteristic
568C22#1912850411yescharacteristic
569C22#1912851611yescharacteristic
570C4—22no—
571C4—22no—
572C4—22no—
573C4—41no—
574C4—41no—
575C4—42no—
576C4—42no—
577C4—41no—
578C4—81no—
579C4—81no—
580C4—3322no—
581C4—6641no—
582C4—6641no—
583C4#1912825611yescharacteristic
584C4#1912832811yescharacteristic
585C4#1912841711yescharacteristic
586C4#1912842511yescharacteristic
587C2—6642no—
588C2#2491115011yescharacteristic
589C2#2491725611yescharacteristic
590C2#2491731011yescharacteristic
591C2#2491945411yescharacteristic
592C2#2491961911yescharacteristic
593C2#2492017011yescharacteristic
594C2#2492030511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2550C2
socleC23×C83C22×D4
prime radicalC22×D332C23
hypercenter#2550D83
hypocenterC83#26033
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2550

Complements

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

NH ≅ G/Ntype
6. #24908421C2semidirect
7. #24908412C2semidirect
8. #24908221C2semidirect
9. #24908187C2semidirect
10. #24908052C2semidirect
11. #24908031C2semidirect
12. #24907596C2semidirect
15. C83×#2550C2semidirect
66. C4×D332Q8semidirect
298. #2550D83semidirect
313. C83#26033semidirect

Maximal subgroups

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

subgroupindex
#249104102
#249088482
#249086062
#249085962
#249084212
#249084122
#249082212
#249081872
#249080522
#249080312
#249075962
#249074702
#249072912
C83×#25502
#2603383