Finite Group Databasebeta

← #35406584

Subgroups — #35406584

Order 21248.

974 conjugacy classes of subgroups.

The subgroup lattice has 624 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127740C211yesmaximal, characteristic
2#389×D83C211yesmaximal, characteristic
3C4×#14581750C211yesmaximal, characteristic
4#24908058C211yesmaximal, characteristic, complemented
5#24908054C211yesmaximal, characteristic, complemented
6#24908050C211yesmaximal, characteristic, complemented
7#24908046C211yesmaximal, characteristic, complemented
8#24908041C211yesmaximal, characteristic, complemented
9#24908035C211yesmaximal, characteristic, complemented
10#24908031C211yesmaximal, characteristic, complemented
11#24907181C211yesmaximal, characteristic
12#24907181C211yesmaximal, characteristic
13#24907181C211yesmaximal, characteristic
14#24907155C211yesmaximal, characteristic
15C83×#1919C211yesmaximal, characteristic, complemented
16C22×#13050573C2211yescharacteristic
17C22×#13050573C2211yescharacteristic
18C2×#14581750C2211yescharacteristic
19C2×#14581750C2211yescharacteristic
20C2×C4⋊C4×D83—21no—
21C2×C4⋊C4×D83C2211yescharacteristic
22C2×C4⋊C4×D83C2211yescharacteristic
23C2×C42×D83C2211yescharacteristic
24#19127740C418yescomplemented
25C2×#14581726C2211yescharacteristic
26C2×#14581725C2211yescharacteristic
27C2×#14581723C2211yescharacteristic
28#19127600C2211yescharacteristic
29#19127599C2211yescharacteristic
30#19127598—21no—
31#19127597C2211yescharacteristic
32#19127596—21no—
33C4⋊C4×Dic83C2211yescharacteristic
34C4⋊C4×Dic83C2211yescharacteristic
35C4×C4⋊Dic83C2211yescharacteristic
36C4×(C2×D83)⋊C4—21no—
37#19127548—21no—
38#19127546—21no—
39#95×D83—21no—
40C42×Dic83—21no—
41#19127464C2211yescharacteristic
42#19127464C2211yescharacteristic
43#19127464C2211yescharacteristic
44#19127463C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127456C2211yescharacteristic
51#19127455C2211yescharacteristic
52#19127455C2211yescharacteristic
53C2×#106×C83C2211yescharacteristic
54#389×C83C2211yescharacteristic
55C4×C4⋊C4×C83C2211yescharacteristic
56#340×C83C2211yescharacteristic
57#340×C83C2211yescharacteristic
58#340×C83C2211yescharacteristic
59#336×C83C2211yescharacteristic
60C22×C4×D83—21no—
61C22×C4×D83—21no—
62C22×C4×D83D411yescharacteristic
63C22×C4×D83D411yescharacteristic
64C22×C4×D83C2311yescharacteristic
65C2×#13050573—22no—
66C2×#13050573—22no—
67C2×#13050573—22no—
68C2×#13050573—22no—
69C2×#13050573—22no—
70C2×#13050573C2×C414yes—
71C22×C83⋊C8C2311yescharacteristic
72C22×C83⋊C8C2311yescharacteristic
73#14581750—22no—
74#14581750—22no—
75#14581750C2×C414yes—
76#14581750Q814yes—
77C2×C4⋊Dic83—21no—
78C2×C4⋊Dic83C2311yescharacteristic
79C2×C4⋊Dic83C2311yescharacteristic
80C2×C4⋊Dic83C2311yescharacteristic
81C2×C4⋊Dic83C2311yescharacteristic
82C2×Dic83⋊C4—21no—
83C2×Dic83⋊C4—21no—
84C2×Dic83⋊C4—21no—
85C2×Dic83⋊C4—21no—
86C4⋊C4×D83—22no—
87C4⋊C4×D83—22no—
88C4⋊C4×D83—22no—
89C4⋊C4×D83—44no—
90C4⋊C4×D83C2×C414yes—
91C2×(C2×D83)⋊C4—21no—
92C2×(C2×D83)⋊C4—21no—
93C2×C4×Dic83—21no—
94C2×C4×Dic83—21no—
95C2×C4×Dic83—21no—
96C2×C4×Dic83—21no—
97C2×C4×Dic83—21no—
98C2×C4×Dic83D411yescharacteristic
99C2×C4×Dic83D411yescharacteristic
100C2×C4×Dic83C2311yescharacteristic
101C42×D83—22no—
102C42×D83—22no—
103#14581726C2×C414yescomplemented
104#14581725C2×C414yescomplemented
105#14581723C2×C414yescomplemented
106C83⋊#95—21no—
107C83⋊#95—21no—
108C83⋊#95—21no—
109C83⋊#95—21no—
110C83⋊#95—21no—
111C83⋊#95—21no—
112C22×C8×C83C2311yescharacteristic
113C22×C8×C83C2311yescharacteristic
114C2×C4⋊C4×C83—21no—
115C2×C4⋊C4×C83C2311yescharacteristic
116C2×C4⋊C4×C83C2311yescharacteristic
117C2×C4⋊C4×C83C2311yescharacteristic
118C2×C4⋊C4×C83C2311yescharacteristic
119C2×C42×C83—21no—
120C2×C42×C83C2311yescharacteristic
121#106×C83C2×C414yescomplemented
122#95×C83—21no—
123#95×C83—21no—
124C23×D83C2×D411yescharacteristic
125C22×Dic83—21no—
126C22×Dic83—21no—
127C22×Dic83—21no—
128C22×Dic83—21no—
129C22×Dic83C2×D411yescharacteristic
130C22×Dic83C2×D411yescharacteristic
131C22×Dic83C2×D411yescharacteristic
132C2×C4×D83—24no—
133C2×C4×D83—22no—
134C2×C4×D83—22no—
135C2×C4×D83—22no—
136C2×C4×D83—22no—
137C2×C4×D83—41no—
138C2×C4×D83—41no—
139C2×C4×D83—42no—
140C2×C4×D83—42no—
141C2×C4×D83—42no—
142C2×C4×D83—42no—
143C2×C4×D83—42no—
144C2×C4×D83C22×C412yes—
145C2×C4×D83C2×Q812yes—
146C2×C4×D83#4112yes—
147C2×C4×D83#4112yes—
148C2×C4×D83#4112yes—
149C2×C4×D83#4112yes—
150#13050573—28no—
151#13050573—44no—
152C2×C83⋊C8—21no—
153C2×C83⋊C8—21no—
154C2×C83⋊C8—21no—
155C2×C83⋊C8—21no—
156C2×C83⋊C8—21no—
157C2×C83⋊C8C22×C412yes—
158C4⋊Dic83—21no—
159C4⋊Dic83—21no—
160C4⋊Dic83—21no—
161C4⋊Dic83—21no—
162C4⋊Dic83—21no—
163C4⋊Dic83—42no—
164C4⋊Dic83C22×C412yes—
165C4⋊Dic83C22×C412yes—
166C4⋊Dic83C2×Q812yes—
167Dic83⋊C4—24no—
168Dic83⋊C4—42no—
169Dic83⋊C4—42no—
170Dic83⋊C4—42no—
171(C2×D83)⋊C4—24no—
172(C2×D83)⋊C4—24no—
173C4×Dic83—21no—
174C4×Dic83—21no—
175C4×Dic83—24no—
176C4×Dic83—24no—
177C4×Dic83—22no—
178C4×Dic83—22no—
179C4×Dic83—42no—
180C4×Dic83—42no—
181C4×Dic83—42no—
182C22×C4×C83—21no—
183C22×C4×C83—21no—
184C22×C4×C83—21no—
185C22×C4×C83—21no—
186C22×C4×C83C2×D411yescharacteristic
187C22×C4×C83C2×D411yescharacteristic
188C22×C4×C83C2411yescharacteristic
189C2×C8×C83—21no—
190C2×C8×C83—21no—
191C2×C8×C83—21no—
192C2×C8×C83—21no—
193C2×C8×C83—21no—
194C2×C8×C83C22×C412yes—
195C4⋊C4×C83—21no—
196C4⋊C4×C83—21no—
197C4⋊C4×C83—21no—
198C4⋊C4×C83—21no—
199C4⋊C4×C83—21no—
200C4⋊C4×C83—42no—
201C4⋊C4×C83C22×C412yes—
202C4⋊C4×C83C22×C412yes—
203C4⋊C4×C83C2×Q812yes—
204C42×C83—21no—
205C42×C83—21no—
206C42×C83—42no—
207C23×C83C22×D411yescharacteristic
208C22×D83—21no—
209C22×D83—21no—
210C22×D83—21no—
211C22×D83—21no—
212C22×D83C4×D412yes—
213C22×D83C22⋊Q812yes—
214C22×D83#12312yes—
215C2×C4×C83—21no—
216C2×C4×C83—21no—
217C2×C4×C83—22no—
218C2×C4×C83—22no—
219C2×C4×C83—22no—
220C2×C4×C83—21no—
221C2×C4×C83—21no—
222C2×C4×C83—21no—
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×C83—41no—
232C2×C4×C83AGL(1,ℤ/8)12yes—
233C2×C4×C83C23×C411yescharacteristic
234C2×C4×C83C22×Q811yescharacteristic
235C2×C4×C83C2×#4111yescharacteristic
236C2×C4×C83C2×#4111yescharacteristic
237C2×C4×C83C2×#4111yescharacteristic
238C2×C4×C83C2×#4111yescharacteristic
239C2×Dic83—21no—
240C2×Dic83—21no—
241C2×Dic83—22no—
242C2×Dic83—22no—
243C2×Dic83—22no—
244C2×Dic83—21no—
245C2×Dic83—21no—
246C2×Dic83—21no—
247C2×Dic83—41no—
248C2×Dic83—41no—
249C2×Dic83—41no—
250C2×Dic83—41no—
251C2×Dic83—41no—
252C2×Dic83—41no—
253C2×Dic83—41no—
254C2×Dic83—41no—
255C2×Dic83—41no—
256C2×Dic83C4×D412yes—
257C2×Dic83C22⋊Q812yes—
258C2×Dic83#12312yes—
259C2×Dic83#13712yes—
260C4×D83—44no—
261C4×D83—44no—
262C4×D83—48no—
263C4×D83—84no—
264C4×D83#11714yes—
265C4×D83C4×Q814yes—
266C4×D83#12514yes—
267C4×D83#12614yes—
268C8×C83—22no—
269C8×C83—41no—
270C83⋊C8—22no—
271C83⋊C8—41no—
272C22×C83C2×C4×D411yescharacteristic
273C22×C83C2×C22⋊Q811yescharacteristic
274C22×C83C2×#12311yescharacteristic
275C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
276C22×C83C2×#13711yescharacteristic
277C22×C83#57511yescharacteristic
278C22×C83#57511yescharacteristic
279C2×D83—22no—
280C2×D83—22no—
281C2×D83—22no—
282C2×D83—22no—
283C2×D83—22no—
284C2×D83—22no—
285C2×D83#38714yes—
286C4×C83—22no—
287C4×C83—42no—
288C4×C83—41no—
289C4×C83—42no—
290C4×C83—42no—
291C4×C83—81no—
292C4×C83C2×#11711yescharacteristic
293C4×C83C2×C4×Q811yescharacteristic
294C4×C83C2×#12511yescharacteristic
295C4×C83C2×#12611yescharacteristic
296Dic83—22no—
297Dic83—42no—
298Dic83—41no—
299Dic83—42no—
300Dic83—42no—
301Dic83—81no—
302Dic83#38714yes—
303#21034—831nomaximal, Sylow
304C2×C83C2×#38711yescharacteristic
305C2×C83C4×AGL(1,ℤ/8)11yescharacteristic
306C2×C83C4×#13711yescharacteristic
307C2×C83#307911yescharacteristic
308C2×C83#308011yescharacteristic
309C2×C83#309411yescharacteristic
310C2×C83#309511yescharacteristic
311D83—24no—
312C2×#428—831no—
313C2×#389—831no—
314C4×#117—831no—
315#1921—831no—
316#1921—831no—
317#1920—831no—
318#1920—831no—
319#1919—831no—
320#1919D8311yescharacteristic, complemented
321#1918—831no—
322#1918—831no—
323#1736—831no—
324#1736—831no—
325#1736—831no—
326#1728—831no—
327C83#2103411yescharacteristic, complemented, Sylow
328C22×M16—831no—
329C22×M16—831no—
330C2×#117—831no—
331C2×#117—831no—
332C22×C4⋊C4—831no—
333C22×C4⋊C4—831no—
334C22×C4⋊C4—1661no—
335C22×C42—831no—
336#428—838no—
337C2×#107—831no—
338C2×#107—831no—
339C2×#106—831no—
340C2×#106C2×D8311yescharacteristic
341#389—831no—
342#389—831no—
343#389—831no—
344#389—1661no—
345#389—1661no—
346#389C2×D8311yescharacteristic
347C4×C4⋊C4—831no—
348C4×C4⋊C4—831no—
349C4×C4⋊C4—831no—
350C4×C4⋊C4C2×D8311yescharacteristic
351C4×C22⋊C4—1661no—
352C4×C22⋊C4—1661no—
353#376—1661no—
354C2×#95—1661no—
355C43—1661no—
356#340—831no—
357#340—831no—
358#340—831no—
359#340—831no—
360#340—831no—
361#340—831no—
362#340—831no—
363#340—831no—
364#340—831no—
365#340C2×D8311yescharacteristic
366#340C2×D8311yescharacteristic
367#340C2×D8311yescharacteristic
368#336—831no—
369#336—831no—
370#336—831no—
371#336C2×D8311yescharacteristic
372C23×C4—831no—
373C23×C4—831no—
374C23×C4—831no—
375C23×C4—1661no—
376C23×C4—1661no—
377C2×M16—834no—
378C2×M16—1662no—
379C2×M16—1662no—
380C2×M16—1662no—
381C2×M16—1662no—
382C2×M16—1662no—
383C22×C8—831no—
384C22×C8—831no—
385C22×C8C22×D8311yescharacteristic
386C22×C8C22×D8311yescharacteristic
387#117—834no—
388#117—834no—
389#117—1662no—
390#117—1662no—
391C2×C4⋊C4—21no—
392C2×C4⋊C4—3324no—
393C2×C4⋊C4—831no—
394C2×C4⋊C4—831no—
395C2×C4⋊C4—831no—
396C2×C4⋊C4—834no—
397C2×C4⋊C4—831no—
398C2×C4⋊C4—1662no—
399C2×C4⋊C4—1662no—
400C2×C4⋊C4—1662no—
401C2×C4⋊C4—1661no—
402C2×C4⋊C4—1661no—
403C2×C4⋊C4—1661no—
404C2×C4⋊C4—1661no—
405C2×C4⋊C4—1661no—
406C2×C4⋊C4C22×D8311yescharacteristic
407C2×C4⋊C4C22×D8311yescharacteristic
408C2×C4⋊C4C22×D8311yescharacteristic
409C2×C4⋊C4C22×D8311yescharacteristic
410C2×C22⋊C4—1661no—
411C2×C22⋊C4—1661no—
412C2×C42—21no—
413C2×C42—831no—
414C2×C42—831no—
415C2×C42—831no—
416C2×C42—1662no—
417C2×C42—1662no—
418C2×C42—1661no—
419C2×C42—1661no—
420C2×C42—1661no—
421C2×C42—1661no—
422C2×C42—1661no—
423C2×C42C22×D8311yescharacteristic
424#107—834no—
425#107—834no—
426#106—834no—
427#106C4×D8314yescomplemented
428#95—21no—
429#95—21no—
430#95—1661no—
431#95—1661no—
432#95—1661no—
433#95—1661no—
434#95—1661no—
435#95—1661no—
436C24—831no—
437C22×C4—21no—
438C22×C4—21no—
439C22×C4—21no—
440C22×C4—21no—
441C22×C4—3321no—
442C22×C4—3321no—
443C22×C4—3322no—
444C22×C4—3322no—
445C22×C4—3322no—
446C22×C4—3322no—
447C22×C4—3322no—
448C22×C4—831no—
449C22×C4—832no—
450C22×C4—832no—
451C22×C4—832no—
452C22×C4—832no—
453C22×C4—832no—
454C22×C4—832no—
455C22×C4—831no—
456C22×C4—831no—
457C22×C4—1662no—
458C22×C4—1662no—
459C22×C4—1662no—
460C22×C4—1664no—
461C22×C4—1661no—
462C22×C4—1661no—
463C22×C4—1661no—
464C22×C4—1661no—
465C22×C4—1662no—
466C22×C4D4×D8311yescharacteristic
467C22×C4D4×D8311yescharacteristic
468C22×C4C23×D8311yescharacteristic
469M16—3324no—
470M16—1668no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—832no—
477C2×C8—1661no—
478C2×C8—1661no—
479C2×C8—1661no—
480C2×C8—1661no—
481C2×C8—1661no—
482C2×C8C2×C4×D8312yes—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—42no—
489C4⋊C4—3322no—
490C4⋊C4—3322no—
491C4⋊C4—3322no—
492C4⋊C4—3322no—
493C4⋊C4—832no—
494C4⋊C4—832no—
495C4⋊C4—832no—
496C4⋊C4—1664no—
497C4⋊C4—1661no—
498C4⋊C4—1661no—
499C4⋊C4—1661no—
500C4⋊C4—1661no—
501C4⋊C4—1661no—
502C4⋊C4C2×C4×D8312yes—
503C4⋊C4C2×C4×D8312yes—
504C4⋊C4Q8×D8312yes—
505C22⋊C4—1664no—
506C22⋊C4—1664no—
507C42—21no—
508C42—21no—
509C42—42no—
510C42—3322no—
511C42—3322no—
512C42—3322no—
513C42—1662no—
514C42—1662no—
515C42—1664no—
516C42—1664no—
517C42—1661no—
518C42—1661no—
519C23—832no—
520C23—832no—
521C23—832no—
522C23—1661no—
523C23—1661no—
524C23—1661no—
525C23—1661no—
526C23C2×D4×D8311yescharacteristic
527C2×C4—21no—
528C2×C4—21no—
529C2×C4—22no—
530C2×C4—22no—
531C2×C4—22no—
532C2×C4—21no—
533C2×C4—21no—
534C2×C4—21no—
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—3324no—
545C2×C4—3328no—
546C2×C4—3324no—
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—832no—
557C2×C4—832no—
558C2×C4—832no—
559C2×C4—834no—
560C2×C4—834no—
561C2×C4—834no—
562C2×C4—834no—
563C2×C4—832no—
564C2×C4—6644no—
565C2×C4—1662no—
566C2×C4—1662no—
567C2×C4—1662no—
568C2×C4—1661no—
569C2×C4—1661no—
570C2×C4—1661no—
571C2×C4—1661no—
572C2×C4—1661no—
573C2×C4#1458181912yes—
574C2×C4C22×C4×D8311yescharacteristic
575C2×C4C2×Q8×D8311yescharacteristic
576C2×C4#41×D8311yescharacteristic
577C2×C4#41×D8311yescharacteristic
578C2×C4#41×D8311yescharacteristic
579C2×C4#41×D8311yescharacteristic
580C8—22no—
581C8—41no—
582C8—3321no—
583C8—1662no—
584C22—834no—
585C22—1662no—
586C22—1662no—
587C22—1662no—
588C22—1662no—
589C22—1662no—
590C22—1662no—
591C22C4×D4×D8311yescharacteristic
592C22C22⋊Q8×D8311yescharacteristic
593C22#123×D8311yescharacteristic
594C22C2×#1458181911yescharacteristic
595C22#137×D8311yescharacteristic
596C22#1912849711yescharacteristic
597C22#1912849811yescharacteristic
598C4—22no—
599C4—42no—
600C4—42no—
601C4—42no—
602C4—41no—
603C4—81no—
604C4—3322no—
605C4—3322no—
606C4—3322no—
607C4—3321no—
608C4—834no—
609C4—6641no—
610C4—1662no—
611C4#117×D8311yescharacteristic
612C4C4×Q8×D8311yescharacteristic
613C4#125×D8311yescharacteristic
614C4#126×D8311yescharacteristic
615C2—1664no—
616C2#387×D8311yescharacteristic
617C2C4×#1458181911yescharacteristic
618C2#2491628511yescharacteristic
619C2#2491727311yescharacteristic
620C2#2491728011yescharacteristic
621C2#2491740011yescharacteristic
622C2#2491741011yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1919C2
socleC23×C83C22×D4
prime radicalC23×D83C2×D4
cosocleC22×C4×C83C24
hypercenter#1919D83
hypocenterC83#21034
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1919

Complements

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

NH ≅ G/Ntype
4. #24908058C2semidirect
5. #24908054C2semidirect
6. #24908050C2semidirect
7. #24908046C2semidirect
8. #24908041C2semidirect
9. #24908035C2semidirect
10. #24908031C2semidirect
15. C83×#1919C2semidirect
24. #19127740C4semidirect
103. #14581726C2×C4semidirect
104. #14581725C2×C4semidirect
105. #14581723C2×C4semidirect
121. #106×C83C2×C4semidirect
320. #1919D83semidirect
327. C83#21034semidirect
427. #106C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×#191277402
#389×D832
C4×#145817502
#249080582
#249080542
#249080502
#249080462
#249080412
#249080352
#249080312
#249071812
#249071552
C83×#19192
#2103483