Finite Group Databasebeta

← #35422164

Subgroups — #35422164

Order 21248.

782 conjugacy classes of subgroups.

The subgroup lattice has 586 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127740C211yesmaximal, characteristic
2#24910355C211yesmaximal, characteristic
3#24910339C211yesmaximal, characteristic
4#24908207C211yesmaximal, characteristic, complemented
5#24908052C211yesmaximal, characteristic, complemented
6#24908051C211yesmaximal, characteristic, complemented
7#24908038C211yesmaximal, characteristic, complemented
8#24908037C211yesmaximal, characteristic, complemented
9#24907557C211yesmaximal, characteristic
10#24907548C211yesmaximal, characteristic
11#24907493C211yesmaximal, characteristic
12#24907363C211yesmaximal, characteristic, complemented
13C4×#14581726C211yesmaximal, characteristic, complemented
14#24907246C211yesmaximal, characteristic
15C83×#1941C211yesmaximal, characteristic, complemented
16C22×#13050573C2211yescharacteristic
17C2×#14581750C2211yescharacteristic
18C2×C4⋊C4×D83C2211yescharacteristic
19#19127740C418yescomplemented
20C2×#14581726C2211yescharacteristic
21C2×#14581726C2211yescharacteristic
22C2×#14581725C2211yescharacteristic
23C2×#14581723C2211yescharacteristic
24C2×(C2×D83)⋊C8C2211yescharacteristic
25C2×#14581702C2211yescharacteristic
26#19127600C2211yescharacteristic
27#19127600C2211yescharacteristic
28#19127593—21no—
29#19127591—21no—
30#19127588—21no—
31#19127582—21no—
32#19127568—21no—
33#19127558—21no—
34#19127558C2211yescharacteristic
35#19127558C2211yescharacteristic
36C4⋊C4×Dic83—21no—
37C4⋊C4×Dic83C2211yescharacteristic
38C4⋊C4×Dic83C2211yescharacteristic
39C4×C4⋊Dic83C2211yescharacteristic
40C4×C4⋊Dic83C2211yescharacteristic
41C4×(C2×D83)⋊C4C2211yescharacteristic
42#19127548—21no—
43#19127464C2211yescharacteristic
44#19127464C2211yescharacteristic
45#19127463C2211yescharacteristic
46#19127463C2211yescharacteristic
47#19127463C2211yescharacteristic
48#19127463C2211yescharacteristic
49#19127463C2211yescharacteristic
50#19127463C2211yescharacteristic
51#19127456C2211yescharacteristic
52#19127455C2211yescharacteristic
53C2×#106×C83C2211yescharacteristic
54C2×#106×C83C2211yescharacteristic
55C2×C4×C8×C83C2211yescharacteristic
56#389×C83C2211yescharacteristic
57#389×C83C2211yescharacteristic
58#340×C83C2211yescharacteristic
59#340×C83C2211yescharacteristic
60C22×C4×D83—21no—
61C22×C4×D83C2311yescharacteristic
62C2×#13050573—22no—
63C2×#13050573—22no—
64C2×#13050573C2×C414yes—
65C22×C83⋊C8C2311yescharacteristic
66C22×C83⋊C8C2311yescharacteristic
67#14581750—22no—
68#14581750C2×C414yes—
69C2×C4⋊Dic83—21no—
70C2×C4⋊Dic83—21no—
71C2×C4⋊Dic83C2311yescharacteristic
72C2×C4⋊Dic83C2311yescharacteristic
73C2×C4⋊Dic83C2311yescharacteristic
74C2×C4⋊Dic83C2311yescharacteristic
75C2×Dic83⋊C4—21no—
76C4⋊C4×D83—22no—
77C4⋊C4×D83C2×C414yes—
78C2×(C2×D83)⋊C4—21no—
79C2×(C2×D83)⋊C4—21no—
80C2×(C2×D83)⋊C4—21no—
81C2×(C2×D83)⋊C4D411yescharacteristic
82C2×(C2×D83)⋊C4D411yescharacteristic
83C2×C4×Dic83—21no—
84C2×C4×Dic83—21no—
85C2×C4×Dic83—21no—
86C2×C4×Dic83—21no—
87C2×C4×Dic83—21no—
88C2×C4×Dic83C2311yescharacteristic
89#14581726—22no—
90#14581726C2×C414yes—
91#14581725C2×C414yes—
92#14581723C2×C414yes—
93(C2×D83)⋊C8—22no—
94#14581702—22no—
95C83⋊#95—21no—
96C83⋊#95—21no—
97C83⋊#95—21no—
98C83⋊#95—21no—
99C83⋊#95—21no—
100C83⋊#95—21no—
101C83⋊#95—21no—
102C83⋊#95—21no—
103C83⋊#95D411yescharacteristic
104C83⋊#95D411yescharacteristic
105C22×C8×C83C2311yescharacteristic
106C22×C8×C83C2311yescharacteristic
107C2×C4⋊C4×C83—21no—
108C2×C4⋊C4×C83—21no—
109C2×C4⋊C4×C83C2311yescharacteristic
110C2×C4⋊C4×C83C2311yescharacteristic
111C2×C4⋊C4×C83C2311yescharacteristic
112C2×C4⋊C4×C83C2311yescharacteristic
113C2×C42×C83C2311yescharacteristic
114#106×C83—22no—
115#106×C83C2×C414yes—
116C4×C8×C83—22no—
117#95×C83—21no—
118#95×C83—21no—
119C23×D83C2×D411yescharacteristic
120C22×Dic83—21no—
121C22×Dic83—21no—
122C22×Dic83—21no—
123C22×Dic83—21no—
124C22×Dic83C2×D411yescharacteristic
125C22×Dic83C2×D411yescharacteristic
126C22×Dic83C2×D411yescharacteristic
127C2×C4×D83—21no—
128C2×C4×D83—21no—
129C2×C4×D83—21no—
130C2×C4×D83—21no—
131C2×C4×D83—24no—
132C2×C4×D83—81no—
133C2×C4×D83—81no—
134C2×C4×D83C22×C412yes—
135C2×C4×D83#4112yes—
136#13050573—28no—
137C2×C83⋊C8—21no—
138C2×C83⋊C8—21no—
139C2×C83⋊C8—21no—
140C2×C83⋊C8—21no—
141C2×C83⋊C8—21no—
142C2×C83⋊C8C22×C412yes—
143C4⋊Dic83—21no—
144C4⋊Dic83—21no—
145C4⋊Dic83—21no—
146C4⋊Dic83—21no—
147C4⋊Dic83—21no—
148C4⋊Dic83—21no—
149C4⋊Dic83—42no—
150C4⋊Dic83—81no—
151C4⋊Dic83C22×C412yes—
152C4⋊Dic83C22×C412yes—
153Dic83⋊C4—24no—
154(C2×D83)⋊C4—24no—
155(C2×D83)⋊C4—22no—
156(C2×D83)⋊C4—41no—
157(C2×D83)⋊C4—42no—
158(C2×D83)⋊C4—81no—
159C4×Dic83—21no—
160C4×Dic83—21no—
161C4×Dic83—24no—
162C4×Dic83—81no—
163C4×Dic83—81no—
164C4×Dic83—81no—
165C4×Dic83—81no—
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83—21no—
169C22×C4×C83—21no—
170C22×C4×C83C2×D411yescharacteristic
171C22×C4×C83C2×D411yescharacteristic
172C22×C4×C83C2411yescharacteristic
173C2×C8×C83—21no—
174C2×C8×C83—21no—
175C2×C8×C83—21no—
176C2×C8×C83C22×C412yes—
177C2×C8×C83#4112yes—
178C2×C8×C83#4112yes—
179C4⋊C4×C83—21no—
180C4⋊C4×C83—21no—
181C4⋊C4×C83—21no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—21no—
184C4⋊C4×C83—21no—
185C4⋊C4×C83—42no—
186C4⋊C4×C83—81no—
187C4⋊C4×C83C22×C412yes—
188C4⋊C4×C83C22×C412yes—
189C42×C83—21no—
190C42×C83—21no—
191C23×C83C22×D411yescharacteristic
192C22×D83—21no—
193C22×D83—21no—
194C22×D83—21no—
195C22×D83—21no—
196C22×D83—21no—
197C22×D83—21no—
198C22×D83C4×D412yes—
199C2×C4×C83—21no—
200C2×C4×C83—21no—
201C2×C4×C83—22no—
202C2×C4×C83—22no—
203C2×C4×C83—21no—
204C2×C4×C83—21no—
205C2×C4×C83—21no—
206C2×C4×C83—41no—
207C2×C4×C83—41no—
208C2×C4×C83—41no—
209C2×C4×C83—41no—
210C2×C4×C83—41no—
211C2×C4×C83—41no—
212C2×C4×C83—41no—
213C2×C4×C83—41no—
214C2×C4×C83—41no—
215C2×C4×C83—41no—
216C2×C4×C83#13712yes—
217C2×C4×C83C23×C411yescharacteristic
218C2×C4×C83C2×#4111yescharacteristic
219C2×C4×C83C2×#4111yescharacteristic
220C2×C4×C83C2×#4111yescharacteristic
221C2×C4×C832−1+411yescharacteristic
222C2×C4×C832−1+411yescharacteristic
223C2×Dic83—21no—
224C2×Dic83—21no—
225C2×Dic83—22no—
226C2×Dic83—22no—
227C2×Dic83—21no—
228C2×Dic83—21no—
229C2×Dic83—21no—
230C2×Dic83—21no—
231C2×Dic83—21no—
232C2×Dic83—21no—
233C2×Dic83—41no—
234C2×Dic83—41no—
235C2×Dic83—41no—
236C2×Dic83—41no—
237C2×Dic83—41no—
238C2×Dic83—41no—
239C2×Dic83—41no—
240C2×Dic83—41no—
241C2×Dic83—41no—
242C2×Dic83—41no—
243C2×Dic83C4×D412yes—
244C4×D83—22no—
245C4×D83—22no—
246C4×D83—22no—
247C4×D83—22no—
248C4×D83—84no—
249C8×C83—41no—
250C8×C83#11714yes—
251C83⋊C8—22no—
252C83⋊C8—41no—
253C22×C83C2×C4×D411yescharacteristic
254C22×C83#54711yescharacteristic
255C22×C83#54711yescharacteristic
256C22×C83C2×#13711yescharacteristic
257C22×C83#57511yescharacteristic
258C22×C83#57611yescharacteristic
259C22×C83#57711yescharacteristic
260C2×D83—22no—
261C2×D83—22no—
262C2×D83—22no—
263C2×D83—41no—
264C2×D83—41no—
265C2×D83—41no—
266C2×D83—41no—
267C4×C83—22no—
268C4×C83—41no—
269C4×C83—42no—
270C4×C83—42no—
271C4×C83—81no—
272C4×C83—81no—
273C4×C83C2×#11711yescharacteristic
274C4×C83#51911yescharacteristic
275C4×C83#55211yescharacteristic
276C4×C83#55211yescharacteristic
277Dic83—22no—
278Dic83—41no—
279Dic83—41no—
280Dic83—42no—
281Dic83—42no—
282Dic83—81no—
283Dic83—81no—
284#22623—831nomaximal, Sylow
285C2×C83#234411yescharacteristic
286C2×C83C4×#13711yescharacteristic
287C2×C83#294611yescharacteristic
288C2×C83#334611yescharacteristic
289C2×C83#335011yescharacteristic
290C2×C83#335511yescharacteristic
291C2×C83#335611yescharacteristic
292D83—42no—
293C2×#428—831no—
294#2347—831no—
295#2345—831no—
296#1944—831no—
297#1941D8311yescharacteristic, complemented
298#1921—831no—
299#1920—831no—
300#1919—831no—
301#1918—831no—
302#1825—831no—
303#1823—831no—
304#1809—831no—
305#1776—831no—
306C4×#107—831no—
307#1753—831no—
308C83#2262311yescharacteristic, complemented, Sylow
309C22×M16—831no—
310C2×#117—831no—
311C22×C4⋊C4—831no—
312#428—838no—
313C2×#107—831no—
314C2×#107—831no—
315C2×#107—831no—
316C2×#106—831no—
317C2×#106C2×D8311yescharacteristic
318C2×#106C2×D8311yescharacteristic
319C2×C22⋊C8—831no—
320C2×#97—831no—
321C2×C4×C8C2×D8311yescharacteristic
322#389—831no—
323#389—831no—
324#389C2×D8311yescharacteristic
325#389C2×D8311yescharacteristic
326#388—1661no—
327#388—1661no—
328#387—1661no—
329#387—1661no—
330#383—1661no—
331#380—831no—
332#380—831no—
333#380—1661no—
334C4×C4⋊C4—831no—
335C4×C4⋊C4—831no—
336C4×C4⋊C4—831no—
337C4×C4⋊C4—831no—
338C4×C4⋊C4—1661no—
339C4×C22⋊C4—831no—
340C4×C22⋊C4—1661no—
341#340—831no—
342#340—831no—
343#340—831no—
344#340—831no—
345#340—831no—
346#340—831no—
347#340—831no—
348#340—831no—
349#340C2×D8311yescharacteristic
350#340C2×D8311yescharacteristic
351#336—831no—
352#336—831no—
353C23×C4—831no—
354C23×C4—1661no—
355C2×M16—834no—
356C2×M16—1662no—
357C2×M16—1662no—
358C22×C8—831no—
359C22×C8—831no—
360C22×C8C22×D8311yescharacteristic
361C22×C8C22×D8311yescharacteristic
362#117—834no—
363#117—1662no—
364C2×C4⋊C4—21no—
365C2×C4⋊C4—21no—
366C2×C4⋊C4—831no—
367C2×C4⋊C4—831no—
368C2×C4⋊C4—834no—
369C2×C4⋊C4—831no—
370C2×C4⋊C4—831no—
371C2×C4⋊C4—1662no—
372C2×C4⋊C4—1661no—
373C2×C4⋊C4—1661no—
374C2×C4⋊C4—1661no—
375C2×C4⋊C4C22×D8311yescharacteristic
376C2×C4⋊C4C22×D8311yescharacteristic
377C2×C4⋊C4C22×D8311yescharacteristic
378C2×C4⋊C4C22×D8311yescharacteristic
379C2×C22⋊C4—831no—
380C2×C22⋊C4—831no—
381C2×C22⋊C4—1661no—
382C2×C22⋊C4—1661no—
383C2×C22⋊C4—1661no—
384C2×C42—831no—
385C2×C42—1661no—
386C2×C42—1661no—
387C2×C42—1661no—
388C2×C42—1661no—
389C2×C42—1661no—
390C2×C42C22×D8311yescharacteristic
391#107—834no—
392#107—834no—
393#107—1662no—
394#106—22no—
395#106—834no—
396#106C4×D8314yes—
397C22⋊C8—1662no—
398#97—1662no—
399C4×C8—22no—
400#95—21no—
401#95—21no—
402#95—831no—
403#95—831no—
404#95—1661no—
405#95—1661no—
406#95—1661no—
407#95—1661no—
408#95—1661no—
409#95—1661no—
410#95—1661no—
411#95—1661no—
412C24—831no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—831no—
418C22×C4—832no—
419C22×C4—832no—
420C22×C4—831no—
421C22×C4—831no—
422C22×C4—6641no—
423C22×C4—6641no—
424C22×C4—1664no—
425C22×C4—1661no—
426C22×C4—1661no—
427C22×C4—1661no—
428C22×C4—1661no—
429C22×C4—1661no—
430C22×C4—1661no—
431C22×C4—1661no—
432C22×C4—1661no—
433C22×C4D4×D8311yescharacteristic
434C22×C4D4×D8311yescharacteristic
435C22×C4C23×D8311yescharacteristic
436M16—1668no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—21no—
440C2×C8—832no—
441C2×C8—1661no—
442C2×C8—1661no—
443C2×C8—1661no—
444C2×C8—1661no—
445C2×C8—1661no—
446C2×C8C2×C4×D8312yes—
447C2×C8#1305058912yes—
448C2×C8#1305059112yes—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—42no—
456C4⋊C4—81no—
457C4⋊C4—3322no—
458C4⋊C4—832no—
459C4⋊C4—832no—
460C4⋊C4—6641no—
461C4⋊C4—1664no—
462C4⋊C4—1661no—
463C4⋊C4—1661no—
464C4⋊C4—1661no—
465C4⋊C4—1661no—
466C4⋊C4—1661no—
467C4⋊C4—1661no—
468C4⋊C4C2×C4×D8312yes—
469C4⋊C4C2×C4×D8312yes—
470C22⋊C4—3321no—
471C22⋊C4—3322no—
472C22⋊C4—6641no—
473C22⋊C4—1664no—
474C22⋊C4—1662no—
475C42—21no—
476C42—21no—
477C42—6641no—
478C42—6641no—
479C42—6641no—
480C42—6641no—
481C42—1664no—
482C42—1661no—
483C42—1661no—
484C23—832no—
485C23—1661no—
486C23—1661no—
487C23—1661no—
488C23—1661no—
489C23—1661no—
490C23—1661no—
491C23C2×D4×D8311yescharacteristic
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—3321no—
510C2×C4—3321no—
511C2×C4—3321no—
512C2×C4—3321no—
513C2×C4—3321no—
514C2×C4—3321no—
515C2×C4—3321no—
516C2×C4—3321no—
517C2×C4—3321no—
518C2×C4—3321no—
519C2×C4—832no—
520C2×C4—6644no—
521C2×C4—1661no—
522C2×C4—1661no—
523C2×C4—1662no—
524C2×C4—1662no—
525C2×C4—1661no—
526C2×C4—1661no—
527C2×C4—1661no—
528C2×C4—1661no—
529C2×C4—1661no—
530C2×C4—1661no—
531C2×C4—1662no—
532C2×C4—1662no—
533C2×C4—1662no—
534C2×C4—1662no—
535C2×C4#1458182112yes—
536C2×C4C22×C4×D8311yescharacteristic
537C2×C4#41×D8311yescharacteristic
538C2×C4C2×#1305058911yescharacteristic
539C2×C4C2×#1305059111yescharacteristic
540C2×C4#1458183911yescharacteristic
541C2×C4#1458184011yescharacteristic
542C8—41no—
543C8—3321no—
544C8—1662no—
545C8#1458175014yes—
546C22—3321no—
547C22—3321no—
548C22—3321no—
549C22—3321no—
550C22—1662no—
551C22—1662no—
552C22—1662no—
553C22C4×D4×D8311yescharacteristic
554C22#1912835111yescharacteristic
555C22#1912835411yescharacteristic
556C22C2×#1458182111yescharacteristic
557C22#1912849811yescharacteristic
558C22#1912850311yescharacteristic
559C22#1912850711yescharacteristic
560C4—22no—
561C4—42no—
562C4—42no—
563C4—41no—
564C4—81no—
565C4—81no—
566C4—3321no—
567C4—3321no—
568C4—3322no—
569C4—3322no—
570C4—6641no—
571C4—6641no—
572C4—1662no—
573C4C2×#1458175011yescharacteristic
574C4#1912817711yescharacteristic
575C4#1912838511yescharacteristic
576C4#1912838911yescharacteristic
577C2—3322no—
578C2#2491033111yescharacteristic
579C2C4×#1458182111yescharacteristic
580C2#2491629011yescharacteristic
581C2#2491995411yescharacteristic
582C2#2492001411yescharacteristic
583C2#2492006711yescharacteristic
584C2#2492007711yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1941C2
socleC23×C83C22×D4
prime radicalC23×D83C2×D4
cosocleC22×C4×C83C24
hypercenter#1941D83
hypocenterC83#22623
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1941

Complements

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

NH ≅ G/Ntype
4. #24908207C2semidirect
5. #24908052C2semidirect
6. #24908051C2semidirect
7. #24908038C2semidirect
8. #24908037C2semidirect
12. #24907363C2semidirect
13. C4×#14581726C2semidirect
15. C83×#1941C2semidirect
19. #19127740C4semidirect
297. #1941D83semidirect
308. C83#22623semidirect

Maximal subgroups

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

subgroupindex
C2×#191277402
#249103552
#249103392
#249082072
#249080522
#249080512
#249080382
#249080372
#249075572
#249075482
#249074932
#249073632
C4×#145817262
#249072462
C83×#19412
#2262383