Finite Group Databasebeta

← #35462897

Subgroups — #35462897

Order 21248.

864 conjugacy classes of subgroups.

The subgroup lattice has 628 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128001C211yesmaximal, characteristic
2C2×#19127981C211yesmaximal, characteristic
3#24912484C211yesmaximal, characteristic, complemented
4#24908964C211yesmaximal, characteristic, complemented
5#24908959C211yesmaximal, characteristic, complemented
6#24908801C211yesmaximal, characteristic, complemented
7#24908791C211yesmaximal, characteristic, complemented
8#24908219C211yesmaximal, characteristic, complemented
9#24908187C211yesmaximal, characteristic, complemented
10#24907724C211yesmaximal, characteristic
11#24907710C211yesmaximal, characteristic
12#24907629C211yesmaximal, characteristic, complemented
13#24907473C211yesmaximal, characteristic
14#24907288C211yesmaximal, characteristic
15D4×C4⋊C4×C83C211yesmaximal, characteristic
16C2×#14581792C2211yescharacteristic
17C2×#14581784C2211yescharacteristic
18#19128001—24no—
19#19127981—24no—
20C2×#14581725C2211yescharacteristic
21C2×#14581725C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581723C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×C4⋊(C83⋊C8)C2211yescharacteristic
27C2×C4⋊(C83⋊C8)C2211yescharacteristic
28C2×C4⋊(C83⋊C8)C2211yescharacteristic
29C2×#14581713C2211yescharacteristic
30C2×#14581713C2211yescharacteristic
31C2×#14581713C2211yescharacteristic
32C2×#14581713C2211yescharacteristic
33C2×#14581703C2211yescharacteristic
34#19127634—21no—
35#19127621—21no—
36#19127605—21no—
37#19127599C2211yescharacteristic
38#19127599C2211yescharacteristic
39#19127598—21no—
40#19127567C2211yescharacteristic
41C4×C4⋊Dic83C2211yescharacteristic
42#19127463C2211yescharacteristic
43#19127463C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127456C2211yescharacteristic
49#19127456C2211yescharacteristic
50C2×C4×D4×C83C2211yescharacteristic
51C2×C4×D4×C83C2211yescharacteristic
52C2×C4×D4×C83C2211yescharacteristic
53C22×C4⋊C4×C83—21no—
54#389×C83C2211yescharacteristic
55#389×C83C2211yescharacteristic
56#385×C83—21no—
57#385×C83—21no—
58#384×C83C2211yescharacteristic
59#380×C83—21no—
60C4×C4⋊C4×C83C2211yescharacteristic
61C22×Dic166—21no—
62C22×C83⋊C8C2311yescharacteristic
63C22×C83⋊C8C2311yescharacteristic
64C22×C83⋊C8C2311yescharacteristic
65C22×C83⋊C8C2311yescharacteristic
66#14581792—22no—
67#14581792—41no—
68#14581784—22no—
69#14581784—41no—
70C2×C4⋊Dic83—21no—
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×Dic83⋊C4—21no—
76C2×Dic83⋊C4—21no—
77C2×Dic83⋊C4—21no—
78C2×Dic83⋊C4—21no—
79C2×C4×Dic83—21no—
80#14581725—22no—
81#14581725—22no—
82#14581725—22no—
83#14581725—41no—
84#14581723—22no—
85#14581723—41no—
86C4⋊(C83⋊C8)—22no—
87C4⋊(C83⋊C8)—22no—
88C4⋊(C83⋊C8)—22no—
89#14581713—22no—
90#14581713—22no—
91#14581713D414yes—
92#14581713Q814yescomplemented
93#14581703—22no—
94C83⋊#95—21no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C22×D4×C83C2311yescharacteristic
100C23×C4×C83—21no—
101C23×C4×C83—21no—
102C23×C4×C83—21no—
103C4×D4×C83—22no—
104C4×D4×C83—22no—
105C4×D4×C83—22no—
106C4×D4×C83—41no—
107C4×D4×C83—41no—
108C4×D4×C83Q814yescomplemented
109C2×C4⋊C4×C83—21no—
110C2×C4⋊C4×C83—21no—
111C2×C4⋊C4×C83—21no—
112C2×C4⋊C4×C83—21no—
113C2×C4⋊C4×C83—44no—
114C2×C4⋊C4×C83D411yescharacteristic
115C2×C4⋊C4×C83D411yescharacteristic
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C4⋊C4×C83C2311yescharacteristic
119C2×C22⋊C4×C83—21no—
120C2×C22⋊C4×C83—21no—
121C2×C22⋊C4×C83—21no—
122C2×C42×C83C2311yescharacteristic
123C2×C42×C83C2311yescharacteristic
124C2×C42×C83C2311yescharacteristic
125#95×C83—21no—
126#95×C83—21no—
127#95×C83D411yescharacteristic
128#95×C83D411yescharacteristic
129C2×Dic166—42no—
130C2×Dic166—42no—
131C2×Dic166—81no—
132C22×Dic83—21no—
133C22×Dic83—21no—
134C22×Dic83—21no—
135C22×Dic83—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—
144C2×C83⋊C8—21no—
145C2×C83⋊C8—21no—
146C2×C83⋊C8C2×D412yes—
147C2×C83⋊C8C2×Q812yes—
148C4⋊Dic83—21no—
149C4⋊Dic83—21no—
150C4⋊Dic83—21no—
151C4⋊Dic83—21no—
152C4⋊Dic83—21no—
153C4⋊Dic83—21no—
154C4⋊Dic83—42no—
155C4⋊Dic83C2×D412yes—
156C4⋊Dic83C2×Q812yes—
157Dic83⋊C4—42no—
158Dic83⋊C4—42no—
159Dic83⋊C4—42no—
160Dic83⋊C4—81no—
161C4×Dic83—81no—
162C24×C83—21no—
163C2×D4×C83—21no—
164C2×D4×C83—21no—
165C2×D4×C83—21no—
166C2×D4×C83C2×D412yes—
167C2×D4×C83C2×Q812yes—
168C2×D4×C83C2×Q812yes—
169C22×C4×C83—21no—
170C22×C4×C83—24no—
171C22×C4×C83—21no—
172C22×C4×C83—21no—
173C22×C4×C83—42no—
174C22×C4×C83—42no—
175C22×C4×C83—42no—
176C22×C4×C83—42no—
177C22×C4×C83—42no—
178C22×C4×C83—42no—
179C22×C4×C83—81no—
180C22×C4×C83—81no—
181C22×C4×C83C2×D411yescharacteristic
182C22×C4×C83C2×D411yescharacteristic
183C22×C4×C83C2×D411yescharacteristic
184C22×C4×C83C2×D411yescharacteristic
185C22×C4×C83C2×D411yescharacteristic
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83C2411yescharacteristic
188C4⋊C4×C83—22no—
189C4⋊C4×C83—21no—
190C4⋊C4×C83—21no—
191C4⋊C4×C83—21no—
192C4⋊C4×C83—21no—
193C4⋊C4×C83—21no—
194C4⋊C4×C83—41no—
195C4⋊C4×C83—42no—
196C4⋊C4×C83—42no—
197C4⋊C4×C83—42no—
198C4⋊C4×C83—81no—
199C4⋊C4×C83C2×Q812yes—
200C22⋊C4×C83—24no—
201C22⋊C4×C83—42no—
202C22⋊C4×C83—42no—
203C42×C83—21no—
204C42×C83—21no—
205C42×C83—21no—
206C42×C83—21no—
207C42×C83C2×Q812yes—
208C42×C83#4112yes—
209C23×C83—22no—
210C23×C83—22no—
211C23×C83—22no—
212C23×C83—41no—
213C23×C83—41no—
214C23×C83—41no—
215C23×C83—41no—
216C23×C83C22×D411yescharacteristic
217D4×C83—22no—
218D4×C83—22no—
219D4×C83—22no—
220D4×C83C4⋊Q814yes—
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—22no—
228C2×C4×C83—21no—
229C2×C4×C83—21no—
230C2×C4×C83—21no—
231C2×C4×C83—21no—
232C2×C4×C83—21no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—48no—
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—84no—
247C2×C4×C83—84no—
248C2×C4×C83C22⋊Q812yes—
249C2×C4×C83C22⋊Q812yes—
250C2×C4×C83AGL(1,ℤ/8)12yes—
251C2×C4×C83C22×D411yescharacteristic
252C2×C4×C83C22×Q811yescharacteristic
253C2×C4×C83C22×Q811yescharacteristic
254C2×C4×C83C2×#4111yescharacteristic
255C2×C4×C83H(5,𝔽2)11yescharacteristic
256C2×C4×C832−1+411yescharacteristic
257C2×Dic83—22no—
258C2×Dic83—22no—
259C2×Dic83—41no—
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—
269Dic166—84no—
270C83⋊C8—22no—
271C83⋊C8—22no—
272C83⋊C8—41no—
273C83⋊C8—41no—
274C22×C83—24no—
275C22×C83—42no—
276C22×C83—42no—
277C22×C83—42no—
278C22×C83—42no—
279C22×C83—42no—
280C22×C83—42no—
281C22×C83C2×C22⋊Q811yescharacteristic
282C22×C83D4211yescharacteristic
283C22×C83D4×Q811yescharacteristic
284C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
285C22×C83#57511yescharacteristic
286C22×C83#57611yescharacteristic
287C22×C83#57811yescharacteristic
288C4×C83—22no—
289C4×C83—22no—
290C4×C83—22no—
291C4×C83—42no—
292C4×C83—41no—
293C4×C83—41no—
294C4×C83—41no—
295C4×C83—81no—
296C4×C83—81no—
297C4×C83C2×C4⋊Q811yescharacteristic
298C4×C83#54411yescharacteristic
299C4×C83#54711yescharacteristic
300C4×C83#55411yescharacteristic
301Dic83—42no—
302Dic83—42no—
303Dic83—81no—
304Dic83—81no—
305#26052—831nomaximal, Sylow
306C2×C83—44no—
307C2×C83#244011yescharacteristic
308C2×C83#307711yescharacteristic
309C2×C83#308311yescharacteristic
310C2×C83#326411yescharacteristic
311C2×C83#326911yescharacteristic
312C2×C83#338611yescharacteristic
313C2×C83#338911yescharacteristic
314C2×#478—831no—
315C2×#474—831no—
316#2550—831no—
317D4×C4⋊C4D8311yescharacteristic
318#2088—831no—
319#2087—831no—
320#2059—831no—
321#2057—831no—
322#1947—831no—
323#1940—831no—
324#1864—831no—
325#1862—831no—
326#1847—831no—
327#1804—831no—
328#1762—831no—
329C83#2605211yescharacteristic, complemented, Sylow
330C2×C4⋊Q8—831no—
331C2×#125—831no—
332C2×C4×D4C2×D8311yescharacteristic
333C2×C4×D4C2×D8311yescharacteristic
334C2×C4×D4C2×D8311yescharacteristic
335C22×C4⋊C4—21no—
336#478—1664no—
337#474—1664no—
338C2×#107—831no—
339C2×#107—831no—
340C2×#107—831no—
341C2×#107—831no—
342C2×#106—831no—
343C2×#106—831no—
344C2×C4⋊C8—831no—
345C2×C4⋊C8—831no—
346C2×C4⋊C8—831no—
347C2×#102—831no—
348C2×#102—831no—
349C2×#102—831no—
350C2×#102—831no—
351C2×#97—831no—
352#398—1661no—
353#395—1661no—
354#391—1661no—
355#389—831no—
356#389—831no—
357#389—1661no—
358#389C2×D8311yescharacteristic
359#389C2×D8311yescharacteristic
360#385—21no—
361#385—21no—
362#384C2×D8311yescharacteristic
363#382—831no—
364#380—21no—
365C4×C4⋊C4—831no—
366C4×C4⋊C4C2×D8311yescharacteristic
367#340—831no—
368#340—831no—
369#340—831no—
370#340—831no—
371#340—831no—
372#340—831no—
373#336—831no—
374#336—831no—
375C22×Q8—1661no—
376C22×D4C22×D8311yescharacteristic
377C23×C4—21no—
378C23×C4—21no—
379C23×C4—21no—
380C22×C8—831no—
381C22×C8—831no—
382C22×C8—831no—
383C22×C8—831no—
384C4⋊Q8—3321no—
385C4⋊Q8—1662no—
386#125—3321no—
387#125—1662no—
388C4×D4—22no—
389C4×D4—22no—
390C4×D4—22no—
391C4×D4—41no—
392C4×D4—41no—
393C4×D4Dic16614yescomplemented
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—44no—
399C2×C4⋊C4—831no—
400C2×C4⋊C4—831no—
401C2×C4⋊C4—831no—
402C2×C4⋊C4—831no—
403C2×C4⋊C4—1661no—
404C2×C4⋊C4—1661no—
405C2×C4⋊C4—1661no—
406C2×C4⋊C4—1661no—
407C2×C4⋊C4—1661no—
408C2×C4⋊C4C22⋊D8311yescharacteristic
409C2×C4⋊C4C22⋊D8311yescharacteristic
410C2×C4⋊C4C22×D8311yescharacteristic
411C2×C4⋊C4C22×D8311yescharacteristic
412C2×C4⋊C4C22×D8311yescharacteristic
413C2×C22⋊C4—21no—
414C2×C22⋊C4—21no—
415C2×C22⋊C4—21no—
416C2×C42—1661no—
417C2×C42C22×D8311yescharacteristic
418C2×C42C22×D8311yescharacteristic
419C2×C42C22×D8311yescharacteristic
420#107—3321no—
421#107—1662no—
422#107—1662no—
423#107—1662no—
424#106—3321no—
425#106—1662no—
426C4⋊C8—1662no—
427C4⋊C8—1662no—
428C4⋊C8—1662no—
429#102—834no—
430#102—834no—
431#102—1662no—
432#102—1662no—
433#97—1662no—
434#95—21no—
435#95—21no—
436#95—1661no—
437#95—1661no—
438#95—1661no—
439#95—1661no—
440#95—1661no—
441#95C22⋊D8311yescharacteristic
442#95C22⋊D8311yescharacteristic
443C24—21no—
444C2×Q8—3322no—
445C2×Q8—3322no—
446C2×Q8—6641no—
447C2×D4—21no—
448C2×D4—21no—
449C2×D4—21no—
450C2×D4D4×D8312yes—
451C2×D4Q8×D8312yes—
452C2×D4C2×Dic16612yes—
453C22×C4—21no—
454C22×C4—24no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—42no—
458C22×C4—42no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—42no—
462C22×C4—42no—
463C22×C4—81no—
464C22×C4—81no—
465C22×C4—1661no—
466C22×C4—1661no—
467C22×C4—1661no—
468C22×C4—1661no—
469C22×C4C2×C22⋊D8311yescharacteristic
470C22×C4C2×C22⋊D8311yescharacteristic
471C22×C4C2×C22⋊D8311yescharacteristic
472C22×C4C2×C22⋊D8311yescharacteristic
473C22×C4C2×C22⋊D8311yescharacteristic
474C22×C4C2×C22⋊D8311yescharacteristic
475C22×C4C23×D8311yescharacteristic
476C2×C8—832no—
477C2×C8—832no—
478C2×C8—1661no—
479C2×C8—1661no—
480C2×C8—1661no—
481C2×C8—1661no—
482C2×C8—1661no—
483C2×C8—1661no—
484C2×C8—1661no—
485C2×C8—1661no—
486C2×C8—1661no—
487C2×C8—1661no—
488C4⋊C4—22no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—41no—
495C4⋊C4—42no—
496C4⋊C4—42no—
497C4⋊C4—42no—
498C4⋊C4—81no—
499C4⋊C4—3322no—
500C4⋊C4—3322no—
501C4⋊C4—3322no—
502C4⋊C4—3322no—
503C4⋊C4—832no—
504C4⋊C4—832no—
505C4⋊C4—6641no—
506C4⋊C4—1661no—
507C4⋊C4—1661no—
508C4⋊C4—1661no—
509C4⋊C4—1661no—
510C4⋊C4—1661no—
511C4⋊C4—1661no—
512C4⋊C4C2×Dic16612yes—
513C22⋊C4—24no—
514C22⋊C4—42no—
515C22⋊C4—42no—
516C42—21no—
517C42—21no—
518C42—21no—
519C42—21no—
520C42—6641no—
521C42C2×Dic16612yes—
522C42#1305059012yes—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—41no—
527C23—41no—
528C23—41no—
529C23—41no—
530C23C22×C22⋊D8311yescharacteristic
531Q8—6644no—
532D4—22no—
533D4—22no—
534D4—22no—
535D4#1458179114yes—
536C2×C4—21no—
537C2×C4—21no—
538C2×C4—21no—
539C2×C4—21no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—21no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—48no—
553C2×C4—41no—
554C2×C4—41no—
555C2×C4—41no—
556C2×C4—41no—
557C2×C4—41no—
558C2×C4—41no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—84no—
562C2×C4—84no—
563C2×C4—3321no—
564C2×C4—3321no—
565C2×C4—3321no—
566C2×C4—3321no—
567C2×C4—3321no—
568C2×C4—3321no—
569C2×C4—3321no—
570C2×C4—3321no—
571C2×C4—3321no—
572C2×C4—3321no—
573C2×C4—1662no—
574C2×C4—1662no—
575C2×C4#1458177112yes—
576C2×C4#1458177112yes—
577C2×C4#1458181812yes—
578C2×C4C2×D4×D8311yescharacteristic
579C2×C4C2×Q8×D8311yescharacteristic
580C2×C4C22×Dic16611yescharacteristic
581C2×C4C2×#1305059011yescharacteristic
582C2×C4#1458183711yescharacteristic
583C2×C4#1458184011yescharacteristic
584C8—3321no—
585C8—3321no—
586C8—1662no—
587C8—1662no—
588C22—24no—
589C22—42no—
590C22—42no—
591C22—42no—
592C22—42no—
593C22—42no—
594C22—42no—
595C22C2×#1458177111yescharacteristic
596C22D4×C22⋊D8311yescharacteristic
597C22Q8×C22⋊D8311yescharacteristic
598C22C2×#1458181811yescharacteristic
599C22#1912849911yescharacteristic
600C22#1912850411yescharacteristic
601C22#1912851611yescharacteristic
602C4—22no—
603C4—22no—
604C4—22no—
605C4—42no—
606C4—41no—
607C4—41no—
608C4—41no—
609C4—81no—
610C4—81no—
611C4—3322no—
612C4—3322no—
613C4—6641no—
614C4—6641no—
615C4C2×#1458179111yescharacteristic
616C4#1912833211yescharacteristic
617C4#1912835211yescharacteristic
618C4#1912840611yescharacteristic
619C2—44no—
620C2#2491115911yescharacteristic
621C2#2491726011yescharacteristic
622C2#2491731411yescharacteristic
623C2#2491893911yescharacteristic
624C2#2491901411yescharacteristic
625C2#2492038411yescharacteristic
626C2#2492041711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingD4×C4⋊C4×C83C2
socleC23×C83C22×D4
prime radicalC22×D4×C83C23
hypercenterD4×C4⋊C4D83
hypocenterC83#26052
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ D4×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. #24912484C2semidirect
4. #24908964C2semidirect
5. #24908959C2semidirect
6. #24908801C2semidirect
7. #24908791C2semidirect
8. #24908219C2semidirect
9. #24908187C2semidirect
12. #24907629C2semidirect
92. #14581713Q8semidirect
108. C4×D4×C83Q8semidirect
329. C83#26052semidirect
393. C4×D4Dic166semidirect

Maximal subgroups

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

subgroupindex
C2×#191280012
C2×#191279812
#249124842
#249089642
#249089592
#249088012
#249087912
#249082192
#249081872
#249077242
#249077102
#249076292
#249074732
#249072882
D4×C4⋊C4×C832
#2605283