Finite Group Databasebeta

← #35463075

Subgroups — #35463075

Order 21248.

894 conjugacy classes of subgroups.

The subgroup lattice has 634 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128104C211yesmaximal, characteristic, complemented
2C22×#14581725C211yesmaximal, characteristic
3#24911165C211yesmaximal, characteristic
4#24908935C211yesmaximal, characteristic, complemented
5#24908934C211yesmaximal, characteristic, complemented
6#24908791C211yesmaximal, characteristic, complemented
7#24908788C211yesmaximal, characteristic, complemented
8#24908193C211yesmaximal, characteristic, complemented
9#24908193C211yesmaximal, characteristic, complemented
10#24907651C211yesmaximal, characteristic
11#24907650C211yesmaximal, characteristic
12#24907622C211yesmaximal, characteristic, complemented
13#24907490C211yesmaximal, characteristic
14C22⋊C4×C83⋊C8C211yesmaximal, characteristic
15C83×#2440C211yesmaximal, characteristic
16C23×C83⋊C8C2211yescharacteristic
17C2×#14581792C2211yescharacteristic
18C2×#14581769—21no—
19C22×C4⋊Dic83C2211yescharacteristic
20#19128104—24no—
21C2×#14581725—24no—
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581725C2211yescharacteristic
25C2×#14581725C2211yescharacteristic
26C2×#14581725C2211yescharacteristic
27C2×#14581725C2211yescharacteristic
28C2×#14581725C2211yescharacteristic
29C2×#14581725C2211yescharacteristic
30C2×#14581725C2211yescharacteristic
31C2×#14581725C2211yescharacteristic
32C2×C22⋊(C83⋊C8)C2211yescharacteristic
33C2×C4×C83⋊C8C2211yescharacteristic
34C2×C4×C83⋊C8C2211yescharacteristic
35#19127633—21no—
36#19127622—21no—
37#19127599C2211yescharacteristic
38#19127599C2211yescharacteristic
39#19127576—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×C22⋊Q8×C83—21no—
52C22×C4⋊C4×C83C2211yescharacteristic
53#398×C83—21no—
54#395×C83—21no—
55#389×C83C2211yescharacteristic
56#389×C83C2211yescharacteristic
57#385×C83—21no—
58#384×C83C2211yescharacteristic
59#380×C83C2211yescharacteristic
60C4×C22⋊C4×C83C2211yescharacteristic
61C22×Dic166—21no—
62C23×Dic83—21no—
63C22×C83⋊C8—22no—
64C22×C83⋊C8—22no—
65C22×C83⋊C8Q814yescomplemented
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69C22×C83⋊C8C2311yescharacteristic
70#14581792—22no—
71#14581792—41no—
72#14581769—44no—
73C2×C4⋊Dic83—22no—
74C2×C4⋊Dic83—21no—
75C2×C4⋊Dic83—21no—
76C2×C4⋊Dic83—21no—
77C2×C4⋊Dic83—21no—
78C2×C4⋊Dic83—41no—
79C2×C4⋊Dic83C2311yescharacteristic
80C2×C4⋊Dic83C2311yescharacteristic
81C2×C4⋊Dic83C2311yescharacteristic
82C2×C4⋊Dic83C2311yescharacteristic
83C2×Dic83⋊C4—21no—
84C2×Dic83⋊C4—21no—
85C2×C22⋊Dic83—21no—
86#14581725—22no—
87#14581725—22no—
88#14581725—22no—
89#14581725—22no—
90#14581725—22no—
91#14581725—22no—
92#14581725—22no—
93#14581725—41no—
94#14581725—41no—
95#14581725D414yes—
96C22⋊(C83⋊C8)Q814yescomplemented
97C4×C83⋊C8—22no—
98C4×C83⋊C8—22no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C22×Q8×C83—21no—
103C23×C4×C83—21no—
104C23×C4×C83C2311yescharacteristic
105C4⋊Q8×C83—22no—
106C4⋊Q8×C83—41no—
107C22⋊Q8×C83—44no—
108C2×C4⋊C4×C83—22no—
109C2×C4⋊C4×C83—21no—
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83—21no—
112C2×C4⋊C4×C83—21no—
113C2×C4⋊C4×C83—21no—
114C2×C4⋊C4×C83—21no—
115C2×C4⋊C4×C83—41no—
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C4⋊C4×C83C2311yescharacteristic
119C2×C4⋊C4×C83C2311yescharacteristic
120C2×C22⋊C4×C83—21no—
121C2×C22⋊C4×C83D411yescharacteristic
122C2×C22⋊C4×C83D411yescharacteristic
123C2×C42×C83C2311yescharacteristic
124C2×C42×C83C2311yescharacteristic
125#95×C83—21no—
126#95×C83—21no—
127#95×C83—21no—
128#95×C83D411yescharacteristic
129#95×C83D411yescharacteristic
130C2×Dic166—42no—
131C2×Dic166—42no—
132C2×Dic166—81no—
133C22×Dic83—21no—
134C22×Dic83—21no—
135C22×Dic83—21no—
136C22×Dic83—21no—
137C22×Dic83—42no—
138C22×Dic83—42no—
139C22×Dic83—81no—
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—28no—
147C2×C83⋊C8C2×D412yes—
148C2×C83⋊C8C2×Q812yes—
149C2×C83⋊C8C2×Q812yes—
150C2×C83⋊C8C2×Q812yes—
151C2×C83⋊C8C2×Q812yes—
152C2×C83⋊C8#4112yes—
153C4⋊Dic83—21no—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83—21no—
159C4⋊Dic83—21no—
160C4⋊Dic83—42no—
161C4⋊Dic83—42no—
162C4⋊Dic83—42no—
163C4⋊Dic83—81no—
164C4⋊Dic83C2×D412yes—
165Dic83⋊C4—42no—
166Dic83⋊C4—42no—
167C22⋊Dic83—42no—
168C24×C83C2×D411yescharacteristic
169C2×Q8×C83—42no—
170C2×Q8×C83—42no—
171C2×Q8×C83—81no—
172C22×C4×C83—21no—
173C22×C4×C83—21no—
174C22×C4×C83—21no—
175C22×C4×C83—21no—
176C22×C4×C83—21no—
177C22×C4×C83—21no—
178C22×C4×C83—21no—
179C22×C4×C83—21no—
180C22×C4×C83—42no—
181C22×C4×C83—42no—
182C22×C4×C83—81no—
183C22×C4×C83C2×D411yescharacteristic
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83C2×D411yescharacteristic
188C22×C4×C83C2×Q812yes—
189C22×C4×C83C2×Q812yes—
190C22×C4×C83C2411yescharacteristic
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—21no—
195C4⋊C4×C83—21no—
196C4⋊C4×C83—21no—
197C4⋊C4×C83—21no—
198C4⋊C4×C83—42no—
199C4⋊C4×C83—42no—
200C4⋊C4×C83—42no—
201C4⋊C4×C83—42no—
202C4⋊C4×C83—42no—
203C4⋊C4×C83—81no—
204C4⋊C4×C83C2×D412yes—
205C22⋊C4×C83—42no—
206C22⋊C4×C83D814yes—
207C22⋊C4×C83Q1614yes—
208C42×C83—21no—
209C42×C83—21no—
210C42×C83—21no—
211C42×C83—21no—
212C23×C83—21no—
213C23×C83—21no—
214C23×C83—21no—
215C23×C83—21no—
216C23×C83C4⋊Q812yes—
217C23×C83C2×D812yes—
218C23×C83C2×Q1612yes—
219C23×C83C22×D411yescharacteristic
220Q8×C83—84no—
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—21no—
231C2×C4×C83—22no—
232C2×C4×C83—22no—
233C2×C4×C83—22no—
234C2×C4×C83—22no—
235C2×C4×C83—22no—
236C2×C4×C83—41no—
237C2×C4×C83—41no—
238C2×C4×C83—41no—
239C2×C4×C83—41no—
240C2×C4×C83—41no—
241C2×C4×C83—41no—
242C2×C4×C83—41no—
243C2×C4×C83—41no—
244C2×C4×C83—41no—
245C2×C4×C83—41no—
246C2×C4×C83—41no—
247C2×C4×C83—84no—
248C2×C4×C83C4⋊Q812yes—
249C2×C4×C83C2×D812yes—
250C2×C4×C83C2×D812yes—
251C2×C4×C83C2×Q1612yes—
252C2×C4×C83C2×Q1612yes—
253C2×C4×C83C22×D411yescharacteristic
254C2×C4×C83C22×Q811yescharacteristic
255C2×C4×C83C22×Q811yescharacteristic
256C2×C4×C83C2×#4111yescharacteristic
257C2×C4×C83H(5,𝔽2)11yescharacteristic
258C2×C4×C832−1+411yescharacteristic
259C2×Dic83—22no—
260C2×Dic83—41no—
261C2×Dic83—41no—
262C2×Dic83—41no—
263C2×Dic83—41no—
264C2×Dic83—41no—
265C2×Dic83—41no—
266C2×Dic83—41no—
267C2×Dic83—41no—
268C2×Dic83—41no—
269C2×Dic83—41no—
270C2×Dic83—41no—
271C2×Dic83—84no—
272Dic166—84no—
273C83⋊C8—22no—
274C83⋊C8—22no—
275C83⋊C8—22no—
276C83⋊C8C22⋊Q814yescomplemented
277C22×C83—22no—
278C22×C83—22no—
279C22×C83—22no—
280C22×C83—22no—
281C22×C83—22no—
282C22×C83—22no—
283C22×C83#50014yescomplemented
284C22×C83C2×C4⋊Q811yescharacteristic
285C22×C83D4211yescharacteristic
286C22×C83#54711yescharacteristic
287C22×C83C22×D811yescharacteristic
288C22×C83C22×Q1611yescharacteristic
289C22×C83#57611yescharacteristic
290C22×C83#57811yescharacteristic
291C4×C83—22no—
292C4×C83—22no—
293C4×C83—22no—
294C4×C83—22no—
295C4×C83—22no—
296C4×C83—42no—
297C4×C83—81no—
298C4×C83—81no—
299C4×C83—81no—
300C4×C83C2×C22⋊Q811yescharacteristic
301C4×C83#54411yescharacteristic
302C4×C83D4×Q811yescharacteristic
303C4×C83#55411yescharacteristic
304Dic83—42no—
305Dic83—81no—
306Dic83—81no—
307Dic83—81no—
308#26065—831nomaximal, Sylow
309C2×C83—24no—
310C2×C83#244011yescharacteristic
311C2×C83C2×#50011yescharacteristic
312C2×C83#316411yescharacteristic
313C2×C83D4×D811yescharacteristic
314C2×C83D4×Q1611yescharacteristic
315C2×C83#333411yescharacteristic
316C2×C83#333811yescharacteristic
317C2×#500—831no—
318C22×#107—831no—
319#2440—831no—
320#2440D8311yescharacteristic
321#2083—831no—
322#2083—831no—
323#2057—831no—
324#2057—831no—
325#1942—831no—
326#1942—831no—
327#1852—831no—
328#1852—831no—
329#1845—831no—
330#1808—831no—
331C8×C22⋊C4—831no—
332C83#2606511yescharacteristic, complemented, Sylow
333C23×C8—831no—
334C2×C4⋊Q8—831no—
335C2×C4⋊Q8C2×D8311yescharacteristic
336C2×C22⋊Q8—21no—
337C2×C22⋊Q8—1661no—
338C22×C4⋊C4—831no—
339C22×C4⋊C4C2×D8311yescharacteristic
340#500—1664no—
341C2×#107—831no—
342C2×#107—831no—
343C2×#107—831no—
344C2×#107—831no—
345C2×#107—831no—
346C2×#107—831no—
347C2×#107—831no—
348C2×#107—831no—
349C2×#107—831no—
350C2×#107—831no—
351C2×#107—1664no—
352C2×C22⋊C8—831no—
353C2×C4×C8—831no—
354C2×C4×C8—831no—
355#398—21no—
356#398—1661no—
357#395—21no—
358#395—1661no—
359#389—831no—
360#389—831no—
361#389C2×D8311yescharacteristic
362#389C2×D8311yescharacteristic
363#385—21no—
364#385—1661no—
365#384—831no—
366#384C2×D8311yescharacteristic
367#380—831no—
368#380C2×D8311yescharacteristic
369C4×C22⋊C4C2×D8311yescharacteristic
370#340—831no—
371#340—831no—
372#340—831no—
373#340—831no—
374#340—831no—
375#340—831no—
376#336—831no—
377#336—831no—
378C22×Q8—21no—
379C22×Q8—1661no—
380C23×C4—21no—
381C23×C4—1661no—
382C23×C4C22×D8311yescharacteristic
383C22×C8—831no—
384C22×C8—834no—
385C22×C8—831no—
386C22×C8—831no—
387C22×C8—831no—
388C22×C8—1662no—
389C22×C8—1662no—
390C4⋊Q8—22no—
391C4⋊Q8—41no—
392C4⋊Q8—3321no—
393C4⋊Q8—1662no—
394C22⋊Q8—44no—
395C22⋊Q8—3324no—
396C2×C4⋊C4—22no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—21no—
400C2×C4⋊C4—21no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—41no—
404C2×C4⋊C4—3321no—
405C2×C4⋊C4—831no—
406C2×C4⋊C4—831no—
407C2×C4⋊C4—831no—
408C2×C4⋊C4—831no—
409C2×C4⋊C4—1661no—
410C2×C4⋊C4—1661no—
411C2×C4⋊C4—1661no—
412C2×C4⋊C4—1661no—
413C2×C4⋊C4—1661no—
414C2×C4⋊C4—1662no—
415C2×C4⋊C4—1661no—
416C2×C4⋊C4C22×D8311yescharacteristic
417C2×C4⋊C4C22×D8311yescharacteristic
418C2×C4⋊C4C22×D8311yescharacteristic
419C2×C4⋊C4C22×D8311yescharacteristic
420C2×C22⋊C4—21no—
421C2×C22⋊C4—1661no—
422C2×C22⋊C4C22⋊D8311yescharacteristic
423C2×C22⋊C4C22⋊D8311yescharacteristic
424C2×C42C22×D8311yescharacteristic
425C2×C42C22×D8311yescharacteristic
426#107—3321no—
427#107—3321no—
428#107—834no—
429#107—1662no—
430#107—1662no—
431#107—1662no—
432#107—1662no—
433#107—1662no—
434#107—1662no—
435#107—1662no—
436C22⋊C8—834no—
437C4×C8—1662no—
438C4×C8—1662no—
439#95—21no—
440#95—21no—
441#95—21no—
442#95—1661no—
443#95—1661no—
444#95—1661no—
445#95C22⋊D8311yescharacteristic
446#95C22⋊D8311yescharacteristic
447C24C2×C22⋊D8311yescharacteristic
448C2×Q8—42no—
449C2×Q8—42no—
450C2×Q8—81no—
451C2×Q8—3322no—
452C2×Q8—3322no—
453C2×Q8—6641no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—42no—
463C22×C4—42no—
464C22×C4—81no—
465C22×C4—3322no—
466C22×C4—3322no—
467C22×C4—6641no—
468C22×C4—1661no—
469C22×C4—1661no—
470C22×C4—1661no—
471C22×C4—1661no—
472C22×C4C2×C22⋊D8311yescharacteristic
473C22×C4C2×C22⋊D8311yescharacteristic
474C22×C4C2×C22⋊D8311yescharacteristic
475C22×C4C2×C22⋊D8311yescharacteristic
476C22×C4C2×C22⋊D8311yescharacteristic
477C22×C4Q8×D8312yes—
478C22×C4Q8×D8312yes—
479C22×C4C23×D8311yescharacteristic
480C2×C8—832no—
481C2×C8—832no—
482C2×C8—832no—
483C2×C8—832no—
484C2×C8—832no—
485C2×C8—832no—
486C2×C8—1668no—
487C2×C8—1661no—
488C2×C8—1661no—
489C2×C8—1661no—
490C2×C8—1661no—
491C2×C8—1661no—
492C2×C8—1661no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4—21no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—42no—
501C4⋊C4—42no—
502C4⋊C4—42no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—81no—
506C4⋊C4—3322no—
507C4⋊C4—3322no—
508C4⋊C4—3322no—
509C4⋊C4—3322no—
510C4⋊C4—3322no—
511C4⋊C4—832no—
512C4⋊C4—6641no—
513C4⋊C4—1661no—
514C4⋊C4—1661no—
515C4⋊C4—1661no—
516C4⋊C4—1661no—
517C4⋊C4—1661no—
518C4⋊C4—1661no—
519C4⋊C4—1661no—
520C4⋊C4D4×D8312yes—
521C22⋊C4—42no—
522C22⋊C4—3322no—
523C22⋊C4#1305057514yes—
524C22⋊C4#1305058014yes—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23#1458179012yes—
534C23C2×#1305057512yes—
535C23C2×#1305058012yes—
536C23C22×C22⋊D8311yescharacteristic
537Q8—84no—
538Q8—6644no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—84no—
566C2×C4—3321no—
567C2×C4—3321no—
568C2×C4—3321no—
569C2×C4—3321no—
570C2×C4—3321no—
571C2×C4—3321no—
572C2×C4—3321no—
573C2×C4—3321no—
574C2×C4—3321no—
575C2×C4—3321no—
576C2×C4—3321no—
577C2×C4—6644no—
578C2×C4—1662no—
579C2×C4#1458179012yes—
580C2×C4C2×#1305057512yes—
581C2×C4C2×#1305057512yes—
582C2×C4C2×#1305058012yes—
583C2×C4C2×#1305058012yes—
584C2×C4C2×D4×D8311yescharacteristic
585C2×C4C2×Q8×D8311yescharacteristic
586C2×C4C2×Q8×D8311yescharacteristic
587C2×C4#41×D8311yescharacteristic
588C2×C4#1458183711yescharacteristic
589C2×C4#1458183911yescharacteristic
590C8—834no—
591C8—1662no—
592C8—1662no—
593C8—1662no—
594C22—22no—
595C22—22no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—22no—
600C22#1912810414yescomplemented
601C22C2×#1458179011yescharacteristic
602C22D4×C22⋊D8311yescharacteristic
603C22#1912835011yescharacteristic
604C22C22×#1305057511yescharacteristic
605C22C22×#1305058011yescharacteristic
606C22#1912850411yescharacteristic
607C22#1912851611yescharacteristic
608C4—22no—
609C4—22no—
610C4—22no—
611C4—22no—
612C4—22no—
613C4—42no—
614C4—81no—
615C4—81no—
616C4—81no—
617C4—3322no—
618C4—6641no—
619C4—6641no—
620C4—6641no—
621C4C22⋊Q8×D8311yescharacteristic
622C4#1912833011yescharacteristic
623C4#1912836411yescharacteristic
624C4#1912840511yescharacteristic
625C2—24no—
626C2#2491116211yescharacteristic
627C2C2×#1912810411yescharacteristic
628C2#2491799711yescharacteristic
629C2D4×#1305057511yescharacteristic
630C2D4×#1305058011yescharacteristic
631C2#2491980611yescharacteristic
632C2#2491985111yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2440C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#2440D83
hypocenterC83#26065
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2440

Complements

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

NH ≅ G/Ntype
1. C2×#19128104C2semidirect
4. #24908935C2semidirect
5. #24908934C2semidirect
6. #24908791C2semidirect
7. #24908788C2semidirect
8. #24908193C2semidirect
9. #24908193C2semidirect
12. #24907622C2semidirect
65. C22×C83⋊C8Q8semidirect
96. C22⋊(C83⋊C8)Q8semidirect
276. C83⋊C8C22⋊Q8semidirect
283. C22×C83#500semidirect
332. C83#26065semidirect
600. C22#19128104semidirect

Maximal subgroups

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

subgroupindex
C2×#191281042
C22×#145817252
#249111652
#249089352
#249089342
#249087912
#249087882
#249081932
#249076512
#249076502
#249076222
#249074902
C22⋊C4×C83⋊C82
C83×#24402
#2606583