Finite Group Databasebeta

← #35407104

Subgroups — #35407104

Order 21248.

1122 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19127702C211yesmaximal, characteristic
2C2×#19127697C211yesmaximal, characteristic
3C2×#19127657C211yesmaximal, characteristic, complemented
4#24910079C211yesmaximal, characteristic, complemented
5#24908163C212yesmaximal, complemented
6#24908156C212yesmaximal, complemented
7#24908149C212yesmaximal, complemented
8#24908142C212yesmaximal, complemented
9#24907182C211yesmaximal, characteristic, complemented
10C2×#19127463C211yesmaximal, characteristic, complemented
11C83×#2311C211yesmaximal, characteristic
12C22×#13050574C2211yescharacteristic
13C23×C83⋊C8C2211yescharacteristic
14C2×#14581751C2211yescharacteristic
15C22×C4⋊Dic83C2211yescharacteristic
16#19127702C418yescomplemented
17#19127697—24no—
18C2×#14581716C2211yescharacteristic
19C2×#14581716C2211yescharacteristic
20C2×#14581716C2212yes—
21C2×#14581713C2211yescharacteristic
22C2×#14581713C2211yescharacteristic
23C2×#14581713C2212yes—
24#19127657—24no—
25C2×#14581703C2212yes—
26C2×C4×C83⋊C8C2212yes—
27#19127599C2212yes—
28#19127599C2212yes—
29#19127597—21no—
30#19127597—21no—
31#19127559—22no—
32#19127463C2212yes—
33#19127463C2212yes—
34#19127463C2212yes—
35#19127463C2212yes—
36C22×#41×C83C2211yescharacteristic
37C2×#117×C83C2211yescharacteristic
38C22×C4⋊C4×C83C2211yescharacteristic
39#391×C83C2212yes—
40#390×C83C2212yes—
41#386×C83—22no—
42#380×C83—21no—
43#380×C83—21no—
44C23×Dic83—21no—
45C2×#13050574—22no—
46C2×#13050574—22no—
47C2×#13050574C2×C414yes—
48C22×C83⋊C8—22no—
49C22×C83⋊C8—22no—
50C22×C83⋊C8D414yes—
51C22×C83⋊C8C2312yes—
52C22×C83⋊C8C2311yescharacteristic
53C22×C83⋊C8C2311yescharacteristic
54#14581751—22no—
55#14581751—22no—
56C2×C4⋊Dic83—21no—
57C2×C4⋊Dic83—22no—
58C2×C4⋊Dic83C2×C414yes—
59C2×C4⋊Dic83C2311yescharacteristic
60C2×C4⋊Dic83C2311yescharacteristic
61C2×C4⋊Dic83C2312yes—
62C2×Dic83⋊C4—22no—
63C2×Dic83⋊C4—22no—
64C2×C22⋊Dic83—21no—
65C2×C4×Dic83—21no—
66#14581716—22no—
67#14581716—22no—
68#14581716C2×C418yescomplemented
69#14581713—22no—
70#14581713—22no—
71#14581713C2×C418yescomplemented
72#14581703—24no—
73C4×C83⋊C8—24no—
74C83⋊#95—22no—
75C83⋊#95—22no—
76C2×#41×C83—22no—
77C2×#41×C83—22no—
78C2×#41×C83—22no—
79C2×#41×C83—22no—
80C2×#41×C83—22no—
81C2×#41×C83C2×C414yes—
82C22×Q8×C83C2311yescharacteristic
83C22×D4×C83—21no—
84C22×D4×C83C2311yescharacteristic
85C23×C4×C83—21no—
86C23×C4×C83—21no—
87C23×C4×C83C2311yescharacteristic
88#117×C83—22no—
89#117×C83—22no—
90C2×C4⋊C4×C83—22no—
91C2×C4⋊C4×C83—22no—
92C2×C4⋊C4×C83—21no—
93C2×C4⋊C4×C83D411yescharacteristic
94C2×C4⋊C4×C83D411yescharacteristic
95C2×C4⋊C4×C83C2312yes—
96C2×C22⋊C4×C83—22no—
97C2×C22⋊C4×C83—22no—
98C2×C22⋊C4×C83D412yes—
99C2×C42×C83C2312yes—
100#95×C83—22no—
101#95×C83—22no—
102C22×Dic83—22no—
103C22×Dic83—24no—
104C22×Dic83—21no—
105C22×Dic83—21no—
106C22×Dic83—81no—
107C22×Dic83—81no—
108#13050574—28no—
109C2×C83⋊C8—28no—
110C2×C83⋊C8—22no—
111C2×C83⋊C8—21no—
112C2×C83⋊C8—21no—
113C2×C83⋊C8—21no—
114C2×C83⋊C8—21no—
115C2×C83⋊C8—22no—
116C2×C83⋊C8C22×C414yes—
117C2×C83⋊C8C2×D412yes—
118C2×C83⋊C8C2×D412yes—
119C4⋊Dic83—24no—
120C4⋊Dic83—22no—
121C4⋊Dic83—21no—
122C4⋊Dic83—21no—
123C4⋊Dic83—21no—
124C4⋊Dic83—21no—
125C4⋊Dic83C22×C414yes—
126Dic83⋊C4—82no—
127Dic83⋊C4—82no—
128C22⋊Dic83—42no—
129C4×Dic83—42no—
130C24×C83—21no—
131C24×C83C2×D411yescharacteristic
132#41×C83—28no—
133#41×C83—28no—
134#41×C83—44no—
135#41×C83—44no—
136C2×Q8×C83—21no—
137C2×Q8×C83—21no—
138C2×Q8×C83—21no—
139C2×Q8×C83—21no—
140C2×Q8×C83—21no—
141C2×Q8×C83C22×C412yes—
142C2×D4×C83—24no—
143C2×D4×C83—21no—
144C2×D4×C83—21no—
145C2×D4×C83—21no—
146C2×D4×C83—21no—
147C2×D4×C83—21no—
148C2×D4×C83—42no—
149C2×D4×C83—42no—
150C2×D4×C83C22×C412yes—
151C22×C4×C83—21no—
152C22×C4×C83—24no—
153C22×C4×C83—22no—
154C22×C4×C83—42no—
155C22×C4×C83—42no—
156C22×C4×C83—42no—
157C22×C4×C83—81no—
158C22×C4×C83—81no—
159C22×C4×C83C22×C412yes—
160C22×C4×C83C2×D412yes—
161C22×C4×C83C2×D412yes—
162C22×C4×C83C2×D412yes—
163C22×C4×C83C2×D411yescharacteristic
164C22×C4×C83#4112yes—
165C22×C4×C83#4112yes—
166C22×C4×C83#4112yes—
167C22×C4×C83#4112yes—
168C22×C4×C83C2411yescharacteristic
169C4⋊C4×C83—22no—
170C4⋊C4×C83—22no—
171C4⋊C4×C83—22no—
172C4⋊C4×C83—22no—
173C4⋊C4×C83—42no—
174C22⋊C4×C83—24no—
175C22⋊C4×C83—82no—
176C22⋊C4×C83—82no—
177C42×C83—22no—
178C42×C83—22no—
179C23×C83—21no—
180C23×C83—21no—
181C23×C83—21no—
182C23×C83—21no—
183C23×C83—22no—
184C23×C83—41no—
185C23×C83—41no—
186C23×C83—41no—
187C23×C83—41no—
188C23×C83—41no—
189C23×C83—41no—
190C23×C83C4×D412yes—
191C23×C83#12112yes—
192C23×C83#12412yes—
193C23×C83C22×D411yescharacteristic
194Q8×C83—22no—
195Q8×C83—22no—
196Q8×C83—41no—
197Q8×C83—41no—
198D4×C83—22no—
199D4×C83—22no—
200D4×C83—48no—
201D4×C83—41no—
202D4×C83—41no—
203C2×C4×C83—22no—
204C2×C4×C83—22no—
205C2×C4×C83—22no—
206C2×C4×C83—22no—
207C2×C4×C83—22no—
208C2×C4×C83—22no—
209C2×C4×C83—22no—
210C2×C4×C83—41no—
211C2×C4×C83—41no—
212C2×C4×C83—48no—
213C2×C4×C83—42no—
214C2×C4×C83—42no—
215C2×C4×C83—42no—
216C2×C4×C83—84no—
217C2×C4×C83#11714yes—
218C2×C4×C83C4×D414yes—
219C2×C4×C83C4×D412yes—
220C2×C4×C83#12112yes—
221C2×C4×C83#12314yes—
222C2×C4×C83#12412yes—
223C2×C4×C83#12614yes—
224C2×C4×C83C23×C411yescharacteristic
225C2×C4×C83C22×D411yescharacteristic
226C2×C4×C83C2×#4111yescharacteristic
227C2×C4×C83C2×#4111yescharacteristic
228C2×C4×C83C2×#4111yescharacteristic
229C2×C4×C83C2×#4111yescharacteristic
230C2×Dic83—24no—
231C2×Dic83—42no—
232C2×Dic83—42no—
233C2×Dic83—41no—
234C2×Dic83—41no—
235C2×Dic83—41no—
236C2×Dic83—41no—
237C2×Dic83—41no—
238C2×Dic83—41no—
239C2×Dic83—84no—
240C83⋊C8—24no—
241C83⋊C8—22no—
242C83⋊C8—22no—
243C22×C83—22no—
244C22×C83—22no—
245C22×C83—22no—
246C22×C83—22no—
247C22×C83—22no—
248C22×C83—22no—
249C22×C83—42no—
250C22×C83—42no—
251C22×C83—42no—
252C22×C83—81no—
253C22×C83—81no—
254C22×C83—81no—
255C22×C83—81no—
256C22×C83#38814yes—
257C22×C83C2×C4×D411yescharacteristic
258C22×C83C2×#12111yescharacteristic
259C22×C83C2×#12411yescharacteristic
260C22×C83#57611yescharacteristic
261C22×C83#57711yescharacteristic
262C22×C83#57711yescharacteristic
263C22×C83#57811yescharacteristic
264C4×C83—42no—
265C4×C83—42no—
266C4×C83—42no—
267C4×C83—82no—
268C4×C83#38814yes—
269C4×C83C2×#11711yescharacteristic
270C4×C83C2×C4×D411yescharacteristic
271C4×C83C2×#12311yescharacteristic
272C4×C83C2×#12611yescharacteristic
273Dic83—44no—
274Dic83—81no—
275Dic83—81no—
276#21092—831nomaximal, Sylow
277C2×C83—24no—
278C2×C83—82no—
279C2×C83C2×#38811yescharacteristic
280C2×C83#294611yescharacteristic
281C2×C83#294711yescharacteristic
282C2×C83#306011yescharacteristic
283C2×C83#306111yescharacteristic
284C2×C83#313611yescharacteristic
285C2×C83#313711yescharacteristic
286C2×#417—831no—
287C2×#416—831no—
288C2×#405—831no—
289#2316—831no—
290#2311D8311yescharacteristic
291#1936—832no—
292#1935—832no—
293#1934—832no—
294#1933—832no—
295#1736—831no—
296C2×#340—831no—
297C83#2109211yescharacteristic, complemented, Sylow
298C22×#41C2×D8311yescharacteristic
299C22×M16—831no—
300C23×C8—831no—
301C2×#117—831no—
302C2×#117C2×D8311yescharacteristic
303C22×C4⋊C4—831no—
304C22×C4⋊C4C2×D8311yescharacteristic
305#417—838no—
306#416—1664no—
307C2×#103—832no—
308C2×#103—831no—
309C2×#103—831no—
310C2×#102—832no—
311C2×#102—831no—
312C2×#102—831no—
313#405—1664no—
314C2×#97—832no—
315C2×C4×C8—832no—
316#391C2×D8312yes—
317#390C2×D8312yes—
318#389—832no—
319#389—832no—
320#389—1661no—
321#389—1661no—
322#386—22no—
323#380—21no—
324#380—21no—
325#380—1662no—
326#340—832no—
327#340—832no—
328#340—832no—
329#340—832no—
330C2×#41—22no—
331C2×#41—22no—
332C2×#41—22no—
333C2×#41—22no—
334C2×#41—22no—
335C2×#41C4×D8314yes—
336C22×Q8C22×D8311yescharacteristic
337C22×D4—21no—
338C22×D4C22×D8311yescharacteristic
339C23×C4—21no—
340C23×C4—21no—
341C23×C4—1661no—
342C23×C4C22×D8311yescharacteristic
343C2×M16—834no—
344C2×M16—1662no—
345C2×M16—1662no—
346C22×C8—834no—
347C22×C8—831no—
348C22×C8—831no—
349C22×C8—832no—
350C22×C8—1662no—
351C22×C8—1662no—
352#117—22no—
353#117—22no—
354#117—1662no—
355#117—1662no—
356C2×C4⋊C4—22no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—834no—
360C2×C4⋊C4—832no—
361C2×C4⋊C4—831no—
362C2×C4⋊C4—831no—
363C2×C4⋊C4—1662no—
364C2×C4⋊C4—1662no—
365C2×C4⋊C4—1661no—
366C2×C4⋊C4—1662no—
367C2×C4⋊C4C22⋊D8311yescharacteristic
368C2×C4⋊C4C22⋊D8311yescharacteristic
369C2×C4⋊C4C22×D8312yes—
370C2×C22⋊C4—22no—
371C2×C22⋊C4—22no—
372C2×C22⋊C4—1661no—
373C2×C22⋊C4C22⋊D8312yes—
374C2×C42—1661no—
375C2×C42C22×D8312yes—
376#103—838no—
377#103—1662no—
378#103—1662no—
379#102—838no—
380#102—1662no—
381#102—1662no—
382#97—1664no—
383C4×C8—1664no—
384#95—22no—
385#95—22no—
386#95—1662no—
387#95—1662no—
388C24—21no—
389C24C2×C22⋊D8311yescharacteristic
390#41—28no—
391#41—28no—
392#41—44no—
393#41—44no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8C2×C4×D8312yes—
400C2×D4—24no—
401C2×D4—21no—
402C2×D4—21no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—42no—
407C2×D4—42no—
408C2×D4C2×C4×D8312yes—
409C22×C4—21no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—42no—
413C22×C4—42no—
414C22×C4—42no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—6641no—
418C22×C4—6641no—
419C22×C4—1661no—
420C22×C4—1661no—
421C22×C4—1662no—
422C22×C4—1664no—
423C22×C4C2×C4×D8312yes—
424C22×C4D4×D8312yes—
425C22×C4C2×C22⋊D8312yes—
426C22×C4C2×C22⋊D8312yes—
427C22×C4C2×C22⋊D8311yescharacteristic
428C22×C4#1305058912yes—
429C22×C4#1305058912yes—
430C22×C4#1305058912yes—
431C22×C4#1305059012yes—
432C22×C4C23×D8311yescharacteristic
433M16—1668no—
434C2×C8—834no—
435C2×C8—832no—
436C2×C8—832no—
437C2×C8—1668no—
438C2×C8—1662no—
439C2×C8—1661no—
440C2×C8—1661no—
441C2×C8—1662no—
442C2×C8—1661no—
443C2×C8—1661no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—42no—
449C4⋊C4—834no—
450C4⋊C4—6642no—
451C4⋊C4—6642no—
452C4⋊C4—1661no—
453C4⋊C4—1661no—
454C4⋊C4—1661no—
455C4⋊C4—1661no—
456C4⋊C4—1662no—
457C4⋊C4—1664no—
458C22⋊C4—24no—
459C22⋊C4—82no—
460C22⋊C4—82no—
461C22⋊C4—3322no—
462C42—22no—
463C42—22no—
464C42—3322no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—22no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23C4×C22⋊D8312yes—
477C23#1458176412yes—
478C23#1458177912yes—
479C23C22×C22⋊D8311yescharacteristic
480Q8—22no—
481Q8—22no—
482Q8—41no—
483Q8—41no—
484D4—22no—
485D4—22no—
486D4—48no—
487D4—41no—
488D4—41no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—48no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—84no—
503C2×C4—3322no—
504C2×C4—3322no—
505C2×C4—3321no—
506C2×C4—3321no—
507C2×C4—3321no—
508C2×C4—3321no—
509C2×C4—3321no—
510C2×C4—3321no—
511C2×C4—6644no—
512C2×C4—1664no—
513C2×C4#1458174814yes—
514C2×C4(C22⋊D83)⋊C414yes—
515C2×C4C4×C22⋊D8312yes—
516C2×C4#1458176412yes—
517C2×C4#1458177414yes—
518C2×C4#1458177912yes—
519C2×C4#1458178514yes—
520C2×C4C22×C4×D8311yescharacteristic
521C2×C4C2×D4×D8311yescharacteristic
522C2×C4C2×#1305058911yescharacteristic
523C2×C4C2×#1305058911yescharacteristic
524C2×C4C2×#1305058911yescharacteristic
525C2×C4C2×#1305059011yescharacteristic
526C8—1664no—
527C8—1662no—
528C8—1662no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—81no—
539C22—81no—
540C22—81no—
541C22—81no—
542C22#1912759514yes—
543C22C2×C4×C22⋊D8311yescharacteristic
544C22C2×#1458176411yescharacteristic
545C22C2×#1458177911yescharacteristic
546C22#1912850411yescharacteristic
547C22#1912850911yescharacteristic
548C22#1912851211yescharacteristic
549C22#1912851611yescharacteristic
550C4—42no—
551C4—42no—
552C4—42no—
553C4—82no—
554C4—3324no—
555C4—6641no—
556C4—6641no—
557C4#1912759514yes—
558C4C2×#1458174811yescharacteristic
559C4C2×(C22⋊D83)⋊C411yescharacteristic
560C4C2×#1458177411yescharacteristic
561C4C2×#1458178511yescharacteristic
562C2—24no—
563C2—82no—
564C2C2×#1912759511yescharacteristic
565C2#2491629111yescharacteristic
566C2#2491630211yescharacteristic
567C2#2491712211yescharacteristic
568C2#2491713011yescharacteristic
569C2#2491781711yescharacteristic
570C2#2491782811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2311C2
socleC23×C83C22×D4
prime radicalC22×#41×C83C22
cosocleC22×C4×C83C24
hypercenter#2311D83
hypocenterC83#21092
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×#117 ◁ #2311

Complements

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

NH ≅ G/Ntype
3. C2×#19127657C2semidirect
4. #24910079C2semidirect
5. #24908163C2semidirect
6. #24908156C2semidirect
7. #24908149C2semidirect
8. #24908142C2semidirect
9. #24907182C2semidirect
10. C2×#19127463C2semidirect
16. #19127702C4semidirect
68. #14581716C2×C4semidirect
71. #14581713C2×C4semidirect
297. C83#21092semidirect

Maximal subgroups

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

subgroupindex
C2×#191277022
C2×#191276972
C2×#191276572
#249100792
#249081632
#249081562
#249081492
#249081422
#249071822
C2×#191274632
C83×#23112
#2109283