Finite Group Databasebeta

← #35467126

Subgroups — #35467126

Order 21248.

848 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127983C211yesmaximal, characteristic, complemented
2C2×#19127741C211yesmaximal, characteristic
3#24910488C211yesmaximal, characteristic
4#24909002C211yesmaximal, characteristic, complemented
5#24908954C211yesmaximal, characteristic, complemented
6#24908855C211yesmaximal, characteristic, complemented
7#24908788C211yesmaximal, characteristic, complemented
8#24908224C211yesmaximal, characteristic, complemented
9#24908045C211yesmaximal, characteristic, complemented
10#24907894C211yesmaximal, characteristic
11#24907770C211yesmaximal, characteristic
12#24907730C211yesmaximal, characteristic, complemented
13#24907660C211yesmaximal, characteristic
14#24907648C211yesmaximal, characteristic
15C83×#2440C211yesmaximal, characteristic
16C22×#13050574C2211yescharacteristic
17C2×#14581771C2211yescharacteristic
18C2×C4×Dic166C2211yescharacteristic
19C2×#14581751C2211yescharacteristic
20#19127983—24no—
21#19127741—24no—
22C2×#14581725C2211yescharacteristic
23C2×#14581725C2211yescharacteristic
24C2×#14581725C2211yescharacteristic
25C2×#14581723C2211yescharacteristic
26C2×#14581723C2211yescharacteristic
27C2×#14581723C2211yescharacteristic
28C2×C4⋊(C83⋊C8)C2211yescharacteristic
29C2×C4⋊(C83⋊C8)C2211yescharacteristic
30C2×#14581715C2211yescharacteristic
31C2×#14581715C2211yescharacteristic
32C2×#14581715C2211yescharacteristic
33C2×#14581715C2211yescharacteristic
34C2×C22⋊(C83⋊C8)C2211yescharacteristic
35#19127607C2211yescharacteristic
36#19127599C2211yescharacteristic
37#19127595—21no—
38#19127589—21no—
39#19127583—21no—
40C4×C4⋊Dic83C2211yescharacteristic
41#19127552—21no—
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×Dic166C2311yescharacteristic
62C2×#13050574—22no—
63C2×#13050574—22no—
64C2×#13050574—22no—
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67C22×C83⋊C8C2311yescharacteristic
68C22×C83⋊C8C2311yescharacteristic
69#14581771—22no—
70#14581771Q814yescomplemented
71C4×Dic166—22no—
72C4×Dic166—41no—
73#14581751—22no—
74#14581751—41no—
75C2×C4⋊Dic83C2311yescharacteristic
76C2×C4⋊Dic83C2311yescharacteristic
77C2×C4⋊Dic83C2311yescharacteristic
78C2×Dic83⋊C4—21no—
79C2×Dic83⋊C4—21no—
80C2×Dic83⋊C4—21no—
81C2×C22⋊Dic83—21no—
82C2×C22⋊Dic83—21no—
83C2×C4×Dic83—21no—
84C2×C4×Dic83—21no—
85C2×C4×Dic83—21no—
86#14581725—22no—
87#14581725—22no—
88#14581725—41no—
89#14581723—22no—
90#14581723—22no—
91#14581723D414yes—
92C4⋊(C83⋊C8)—22no—
93C4⋊(C83⋊C8)—22no—
94#14581715—22no—
95#14581715—22no—
96#14581715—22no—
97#14581715—41no—
98C22⋊(C83⋊C8)—22no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C22×Q8×C83—21no—
104C23×C4×C83—21no—
105C23×C4×C83C2311yescharacteristic
106C4⋊Q8×C83—22no—
107C4⋊Q8×C83—41no—
108C22⋊Q8×C83—44no—
109C2×C4⋊C4×C83—22no—
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—21no—
116C2×C4⋊C4×C83—41no—
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C4⋊C4×C83C2311yescharacteristic
119C2×C4⋊C4×C83C2311yescharacteristic
120C2×C4⋊C4×C83C2311yescharacteristic
121C2×C22⋊C4×C83—21no—
122C2×C22⋊C4×C83D411yescharacteristic
123C2×C22⋊C4×C83D411yescharacteristic
124C2×C42×C83C2311yescharacteristic
125C2×C42×C83C2311yescharacteristic
126#95×C83—21no—
127#95×C83—21no—
128#95×C83—21no—
129#95×C83D411yescharacteristic
130#95×C83D411yescharacteristic
131C2×Dic166—21no—
132C2×Dic166—21no—
133C2×Dic166—21no—
134C2×Dic166—21no—
135C2×Dic166—21no—
136C2×Dic166C2×Q812yes—
137C22×Dic83—21no—
138C22×Dic83—21no—
139C22×Dic83—21no—
140C22×Dic83—21no—
141#13050574—28no—
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⋊C8—21no—
149C2×C83⋊C8—21no—
150C2×C83⋊C8—21no—
151C2×C83⋊C8—21no—
152C2×C83⋊C8C2×D412yes—
153C2×C83⋊C8#4112yes—
154C4⋊Dic83—21no—
155C4⋊Dic83—21no—
156C4⋊Dic83—21no—
157C4⋊Dic83—21no—
158C4⋊Dic83C2×D412yes—
159C4⋊Dic83C2×Q812yes—
160Dic83⋊C4—24no—
161Dic83⋊C4—42no—
162Dic83⋊C4—42no—
163C22⋊Dic83—24no—
164C22⋊Dic83—42no—
165C4×Dic83—42no—
166C4×Dic83—42no—
167C4×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×C83#4112yes—
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—22no—
206C22⋊C4×C83—22no—
207C22⋊C4×C83—42no—
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×C83—21no—
217C23×C83—21no—
218C23×C83C22⋊Q812yes—
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—22no—
228C2×C4×C83—22no—
229C2×C4×C83—22no—
230C2×C4×C83—22no—
231C2×C4×C83—21no—
232C2×C4×C83—22no—
233C2×C4×C83—21no—
234C2×C4×C83—21no—
235C2×C4×C83—21no—
236C2×C4×C83—21no—
237C2×C4×C83—21no—
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—41no—
248C2×C4×C83—41no—
249C2×C4×C83—84no—
250C2×C4×C83C22⋊Q812yes—
251C2×C4×C83AGL(1,ℤ/8)12yes—
252C2×C4×C83#13712yes—
253C2×C4×C83C22×D411yescharacteristic
254C2×C4×C83C22×Q811yescharacteristic
255C2×C4×C83C2×#4111yescharacteristic
256C2×C4×C83C2×#4111yescharacteristic
257C2×C4×C83H(5,𝔽2)11yescharacteristic
258C2×C4×C832−1+411yescharacteristic
259C2×Dic83—22no—
260C2×Dic83—22no—
261C2×Dic83—22no—
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—
271Dic166—22no—
272Dic166—22no—
273Dic166—41no—
274Dic166—41no—
275C83⋊C8—22no—
276C83⋊C8—22no—
277C83⋊C8—41no—
278C83⋊C8—41no—
279C22×C83—22no—
280C22×C83—22no—
281C22×C83—22no—
282C22×C83—41no—
283C22×C83—41no—
284C22×C83—41no—
285C22×C83—41no—
286C22×C83C2×C22⋊Q811yescharacteristic
287C22×C83D4211yescharacteristic
288C22×C83#54711yescharacteristic
289C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
290C22×C83C2×#13711yescharacteristic
291C22×C83#57711yescharacteristic
292C22×C83#57811yescharacteristic
293C4×C83—22no—
294C4×C83—22no—
295C4×C83—22no—
296C4×C83—41no—
297C4×C83—42no—
298C4×C83—41no—
299C4×C83—81no—
300C4×C83—81no—
301C4×C83—81no—
302C4×C83#52511yescharacteristic
303C4×C83#53911yescharacteristic
304C4×C83D4×Q811yescharacteristic
305C4×C83#55411yescharacteristic
306Dic83—42no—
307Dic83—42no—
308Dic83—42no—
309Dic83—81no—
310#26353—831nomaximal, Sylow
311C2×C83—42no—
312C2×C83#244911yescharacteristic
313C2×C83#307911yescharacteristic
314C2×C83#308211yescharacteristic
315C2×C83#326511yescharacteristic
316C2×C83#327611yescharacteristic
317C2×C83#334411yescharacteristic
318C2×C83#335311yescharacteristic
319C2×#475—831no—
320C2×#428—831no—
321#2440D8311yescharacteristic
322#2362—831no—
323#2095—831no—
324#2086—831no—
325#2068—831no—
326#2057—831no—
327#1948—831no—
328#1920—831no—
329#1894—831no—
330#1872—831no—
331#1865—831no—
332#1854—831no—
333#1851—831no—
334C83#2635311yescharacteristic, complemented, Sylow
335C22×M16—831no—
336C2×C4⋊Q8C2×D8311yescharacteristic
337C2×C22⋊Q8—21no—
338C2×C22⋊Q8—831no—
339C2×C4×Q8—831no—
340C2×#117—831no—
341C22×C4⋊C4C2×D8311yescharacteristic
342#475—1664no—
343#428—1664no—
344C2×#107—831no—
345C2×#107—831no—
346C2×#107—831no—
347C2×#106—831no—
348C2×#106—831no—
349C2×#106—831no—
350C2×C4⋊C8—831no—
351C2×C4⋊C8—831no—
352C2×#103—831no—
353C2×#103—831no—
354C2×#103—831no—
355C2×#103—831no—
356C2×C22⋊C8—831no—
357#398—21no—
358#395—21no—
359#391—831no—
360#389—831no—
361#389C2×D8311yescharacteristic
362#389C2×D8311yescharacteristic
363#388—1661no—
364#388—1661no—
365#387—1661no—
366#385—21no—
367#384C2×D8311yescharacteristic
368#380C2×D8311yescharacteristic
369C4×C4⋊C4—831no—
370C4×C4⋊C4—1661no—
371C4×C22⋊C4C2×D8311yescharacteristic
372#340—831no—
373#340—831no—
374#340—831no—
375#340—831no—
376#340—831no—
377#340—831no—
378#336—831no—
379#336—831no—
380C22×Q8—21no—
381C22×Q8—831no—
382C23×C4—21no—
383C23×C4C22×D8311yescharacteristic
384C2×M16—1662no—
385C2×M16—1662no—
386C2×M16—1662no—
387C22×C8—831no—
388C22×C8—831no—
389C22×C8—831no—
390C22×C8—831no—
391C4⋊Q8—22no—
392C4⋊Q8—41no—
393C22⋊Q8—44no—
394C22⋊Q8—834no—
395C22⋊Q8—1662no—
396C4×Q8—3321no—
397C4×Q8—1662no—
398#117—3321no—
399#117—1662no—
400C2×C4⋊C4—22no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—21no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—21no—
405C2×C4⋊C4—21no—
406C2×C4⋊C4—21no—
407C2×C4⋊C4—41no—
408C2×C4⋊C4—831no—
409C2×C4⋊C4—831no—
410C2×C4⋊C4—831no—
411C2×C4⋊C4—1661no—
412C2×C4⋊C4—1661no—
413C2×C4⋊C4—1661no—
414C2×C4⋊C4C22×D8311yescharacteristic
415C2×C4⋊C4C22×D8311yescharacteristic
416C2×C4⋊C4C22×D8311yescharacteristic
417C2×C4⋊C4C22×D8311yescharacteristic
418C2×C22⋊C4—21no—
419C2×C22⋊C4—1661no—
420C2×C22⋊C4—1661no—
421C2×C22⋊C4C22⋊D8311yescharacteristic
422C2×C22⋊C4C22⋊D8311yescharacteristic
423C2×C42—1661no—
424C2×C42—1661no—
425C2×C42—1661no—
426C2×C42C22×D8311yescharacteristic
427C2×C42C22×D8311yescharacteristic
428#107—3321no—
429#107—1662no—
430#107—1662no—
431#106—834no—
432#106—1662no—
433#106—1662no—
434C4⋊C8—1662no—
435C4⋊C8—1662no—
436#103—3321no—
437#103—1662no—
438#103—1662no—
439#103—1662no—
440C22⋊C8—1662no—
441#95—21no—
442#95—21no—
443#95—21no—
444#95—1661no—
445#95—1661no—
446#95—1661no—
447#95—1661no—
448#95C22⋊D8311yescharacteristic
449#95C22⋊D8311yescharacteristic
450C24C2×C22⋊D8311yescharacteristic
451C2×Q8—42no—
452C2×Q8—42no—
453C2×Q8—81no—
454C2×Q8—832no—
455C2×Q8—1661no—
456C2×Q8—1661no—
457C2×Q8—1661no—
458C2×Q8—1661no—
459C2×Q8—1661no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—21no—
466C22×C4—21no—
467C22×C4—21no—
468C22×C4—42no—
469C22×C4—42no—
470C22×C4—81no—
471C22×C4—1661no—
472C22×C4—1661no—
473C22×C4—1661no—
474C22×C4—1661no—
475C22×C4C2×C22⋊D8311yescharacteristic
476C22×C4C2×C22⋊D8311yescharacteristic
477C22×C4C2×C22⋊D8311yescharacteristic
478C22×C4C2×C22⋊D8311yescharacteristic
479C22×C4C2×C22⋊D8311yescharacteristic
480C22×C4Q8×D8312yes—
481C22×C4#1305059112yes—
482C22×C4C23×D8311yescharacteristic
483M16—1668no—
484C2×C8—832no—
485C2×C8—832no—
486C2×C8—1661no—
487C2×C8—1661no—
488C2×C8—1661no—
489C2×C8—1661no—
490C2×C8—1661no—
491C2×C8—1661no—
492C2×C8—1661no—
493C2×C8—1661no—
494C2×C8—1661no—
495C2×C8—1661no—
496C4⋊C4—21no—
497C4⋊C4—21no—
498C4⋊C4—21no—
499C4⋊C4—21no—
500C4⋊C4—21no—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—42no—
504C4⋊C4—42no—
505C4⋊C4—42no—
506C4⋊C4—42no—
507C4⋊C4—42no—
508C4⋊C4—81no—
509C4⋊C4—3322no—
510C4⋊C4—3322no—
511C4⋊C4—832no—
512C4⋊C4—832no—
513C4⋊C4—1661no—
514C4⋊C4—1661no—
515C4⋊C4—1664no—
516C4⋊C4—1661no—
517C4⋊C4—1661no—
518C4⋊C4D4×D8312yes—
519C22⋊C4—22no—
520C22⋊C4—22no—
521C22⋊C4—42no—
522C22⋊C4—3322no—
523C22⋊C4—1664no—
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—3322no—
529C42—3322no—
530C42—3322no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23#1458176812yes—
538C23C22×C22⋊D8311yescharacteristic
539Q8—84no—
540Q8—3321no—
541Q8—3321no—
542Q8—1662no—
543Q8—1662no—
544C2×C4—21no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—22no—
554C2×C4—21no—
555C2×C4—22no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—84no—
573C2×C4—3321no—
574C2×C4—3321no—
575C2×C4—3321no—
576C2×C4—3321no—
577C2×C4—3321no—
578C2×C4—3321no—
579C2×C4—3321no—
580C2×C4—3321no—
581C2×C4—3321no—
582C2×C4—1662no—
583C2×C4—1662no—
584C2×C4—1662no—
585C2×C4#1458176812yes—
586C2×C4#1458181812yes—
587C2×C4#1458182312yes—
588C2×C4C2×D4×D8311yescharacteristic
589C2×C4C2×Q8×D8311yescharacteristic
590C2×C4#41×D8311yescharacteristic
591C2×C4C2×#1305059111yescharacteristic
592C2×C4#1458183711yescharacteristic
593C2×C4#1458183911yescharacteristic
594C8—3321no—
595C8—3321no—
596C8—1662no—
597C8—1662no—
598C22—22no—
599C22—22no—
600C22—22no—
601C22—41no—
602C22—41no—
603C22—41no—
604C22—41no—
605C22C2×#1458176811yescharacteristic
606C22D4×C22⋊D8311yescharacteristic
607C22#1912835011yescharacteristic
608C22C2×#1458181811yescharacteristic
609C22C2×#1458182311yescharacteristic
610C22#1912851211yescharacteristic
611C22#1912851611yescharacteristic
612C4—22no—
613C4—22no—
614C4—22no—
615C4—41no—
616C4—42no—
617C4—41no—
618C4—81no—
619C4—81no—
620C4—81no—
621C4—3322no—
622C4—3322no—
623C4—3322no—
624C4—6641no—
625C4#1912821411yescharacteristic
626C4#1912830011yescharacteristic
627C4#1912836411yescharacteristic
628C4#1912840511yescharacteristic
629C2—42no—
630C2#2491126811yescharacteristic
631C2#2491727211yescharacteristic
632C2#2491729911yescharacteristic
633C2#2491895411yescharacteristic
634C2#2491910911yescharacteristic
635C2#2491991211yescharacteristic
636C2#2492004711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2440C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
hypercenter#2440D83
hypocenterC83#26353
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×#19127983C2semidirect
4. #24909002C2semidirect
5. #24908954C2semidirect
6. #24908855C2semidirect
7. #24908788C2semidirect
8. #24908224C2semidirect
9. #24908045C2semidirect
12. #24907730C2semidirect
70. #14581771Q8semidirect
334. C83#26353semidirect

Maximal subgroups

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

subgroupindex
C2×#191279832
C2×#191277412
#249104882
#249090022
#249089542
#249088552
#249087882
#249082242
#249080452
#249078942
#249077702
#249077302
#249076602
#249076482
C83×#24402
#2635383