Finite Group Databasebeta

← #35407101

Subgroups — #35407101

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×#19127700C211yesmaximal, characteristic
2C2×#19127698C211yesmaximal, characteristic
3C2×#19127656C211yesmaximal, characteristic, complemented
4#24910081C211yesmaximal, characteristic, complemented
5#24910056C211yesmaximal, characteristic
6#24908161C212yesmaximal, complemented
7#24908154C212yesmaximal, complemented
8#24908147C212yesmaximal, complemented
9#24908140C212yesmaximal, complemented
10#24907182C211yesmaximal, characteristic, complemented
11C2×#340×C83C211yesmaximal, characteristic, complemented
12C22×#13050590C2211yescharacteristic
13C22×#13050574C2211yescharacteristic
14C2×#14581751C2211yescharacteristic
15C2×#14581751C2211yescharacteristic
16#19127700C418yescomplemented
17#19127698—24no—
18C2×#14581717C2211yescharacteristic
19C2×#14581717C2211yescharacteristic
20C2×#14581715C2212yes—
21C2×#14581714C2211yescharacteristic
22C2×#14581714C2211yescharacteristic
23C2×#14581712C2212yes—
24#19127656—24no—
25C2×#14581702C2212yes—
26C2×C8×Dic83C2212yes—
27#19127606C2212yes—
28#19127602C2212yes—
29#19127600—21no—
30#19127600—21no—
31#19127600C2212yes—
32#19127600C2212yes—
33#19127580—22no—
34#19127561—22no—
35#19127558—21no—
36#19127558—21no—
37#19127463C2212yes—
38#19127463C2212yes—
39C23×C8×C83C2211yescharacteristic
40C22×C4⋊C4×C83C2211yescharacteristic
41C22×C4⋊C4×C83C2211yescharacteristic
42#340×C83C2212yes—
43#340×C83C2212yes—
44C2×#13050590—22no—
45C2×#13050590—22no—
46C2×#13050590—22no—
47C2×#13050590—22no—
48C2×#13050590—22no—
49C2×#13050590C2×C414yes—
50C22×Dic166C2311yescharacteristic
51C22×D332C2311yescharacteristic
52C22×C22⋊D83—21no—
53C22×C4×D83—21no—
54C2×#13050574—22no—
55C2×#13050574—22no—
56C2×#13050574C2×C414yes—
57C22×C83⋊C8C2312yes—
58#14581751—22no—
59#14581751—22no—
60#14581751—22no—
61#14581751—22no—
62C2×C4⋊Dic83—22no—
63C2×C4⋊Dic83—22no—
64C2×C4⋊Dic83D411yescharacteristic
65C2×C4⋊Dic83D411yescharacteristic
66C2×C4⋊Dic83C2311yescharacteristic
67C2×C4⋊Dic83C2311yescharacteristic
68C2×C22⋊Dic83—21no—
69C2×C22⋊Dic83D412yes—
70C2×(C2×D83)⋊C4—22no—
71C2×(C2×D83)⋊C4—22no—
72C2×C4×Dic83—21no—
73C2×C4×Dic83C2312yes—
74#14581717—22no—
75#14581717—22no—
76#14581715C2×C418yescomplemented
77#14581714—22no—
78#14581714—22no—
79#14581712C2×C418yescomplemented
80#14581702—24no—
81C8×Dic83—24no—
82C83⋊#95—22no—
83C83⋊#95—22no—
84C83⋊#95—22no—
85C83⋊#95—22no—
86C23×C4×C83—21no—
87C23×C4×C83—21no—
88C23×C4×C83C2311yescharacteristic
89C22×C8×C83—22no—
90C22×C8×C83—22no—
91C22×C8×C83D414yes—
92C22×C8×C83C2311yescharacteristic
93C22×C8×C83C2311yescharacteristic
94C2×C4⋊C4×C83—21no—
95C2×C4⋊C4×C83—22no—
96C2×C4⋊C4×C83—22no—
97C2×C4⋊C4×C83—22no—
98C2×C4⋊C4×C83—21no—
99C2×C4⋊C4×C83C2×C414yes—
100C2×C4⋊C4×C83C2312yes—
101C2×C4⋊C4×C83C2312yes—
102C23×D83—21no—
103#13050590—28no—
104#13050590—28no—
105#13050590—44no—
106#13050590—44no—
107C2×Dic166—21no—
108C2×Dic166—21no—
109C2×Dic166—21no—
110C2×Dic166—21no—
111C2×Dic166—21no—
112C2×Dic166C22×C412yes—
113C2×D332—21no—
114C2×D332—21no—
115C2×D332—21no—
116C2×D332—21no—
117C2×D332—21no—
118C2×D332C22×C412yes—
119C2×C22⋊D83—24no—
120C2×C22⋊D83—42no—
121C2×C22⋊D83—42no—
122C22×Dic83—21no—
123C22×Dic83—21no—
124C22×Dic83—21no—
125C22×Dic83C2×D412yes—
126C22×Dic83C2×D412yes—
127C2×C4×D83—24no—
128C2×C4×D83—42no—
129C2×C4×D83—42no—
130#13050574—28no—
131C2×C83⋊C8—22no—
132C2×C83⋊C8—22no—
133C2×C83⋊C8C22×C414yes—
134C4⋊Dic83—22no—
135C4⋊Dic83—22no—
136C4⋊Dic83—21no—
137C4⋊Dic83—21no—
138C4⋊Dic83—21no—
139C4⋊Dic83—21no—
140C4⋊Dic83—82no—
141C4⋊Dic83—82no—
142C22⋊Dic83—24no—
143C22⋊Dic83—42no—
144(C2×D83)⋊C4—82no—
145(C2×D83)⋊C4—82no—
146C4×Dic83—22no—
147C4×Dic83—22no—
148C4×Dic83—42no—
149C24×C83C2×D411yescharacteristic
150C22×C4×C83—22no—
151C22×C4×C83—24no—
152C22×C4×C83—22no—
153C22×C4×C83—42no—
154C22×C4×C83—81no—
155C22×C4×C83—81no—
156C22×C4×C83—81no—
157C22×C4×C83—81no—
158C22×C4×C83C22×C412yes—
159C22×C4×C83C2×D411yescharacteristic
160C22×C4×C83C2×D412yes—
161C22×C4×C83#4112yes—
162C22×C4×C83#4112yes—
163C22×C4×C83#4112yes—
164C22×C4×C83#4112yes—
165C22×C4×C83C2411yescharacteristic
166C2×C8×C83—28no—
167C2×C8×C83—21no—
168C2×C8×C83—21no—
169C2×C8×C83—21no—
170C2×C8×C83—21no—
171C2×C8×C83C2×D412yes—
172C2×C8×C83C2×D412yes—
173C4⋊C4×C83—24no—
174C4⋊C4×C83—22no—
175C4⋊C4×C83—22no—
176C4⋊C4×C83—22no—
177C4⋊C4×C83—42no—
178C4⋊C4×C83C22×C414yes—
179C23×C83—21no—
180C23×C83—21no—
181C23×C83—21no—
182C23×C83—21no—
183C23×C83C4×D412yes—
184C23×C83#12112yes—
185C23×C83#12412yes—
186C23×C83C22×D411yescharacteristic
187C22×D83—22no—
188C22×D83—41no—
189C22×D83—41no—
190C22×D83—41no—
191C22×D83—41no—
192C22×D83—41no—
193C22×D83—41no—
194C2×C4×C83—24no—
195C2×C4×C83—42no—
196C2×C4×C83—42no—
197C2×C4×C83—42no—
198C2×C4×C83—42no—
199C2×C4×C83—42no—
200C2×C4×C83—84no—
201C2×C4×C83—84no—
202C2×C4×C83#11714yes—
203C2×C4×C83C4×D412yes—
204C2×C4×C83C4×D414yes—
205C2×C4×C83#12112yes—
206C2×C4×C83#12314yes—
207C2×C4×C83#12412yes—
208C2×C4×C83#12614yes—
209C2×C4×C83C23×C411yescharacteristic
210C2×C4×C83C22×D411yescharacteristic
211C2×C4×C83C2×#4111yescharacteristic
212C2×C4×C83C2×#4111yescharacteristic
213C2×C4×C83C2×#4111yescharacteristic
214C2×C4×C83C2×#4111yescharacteristic
215C22⋊D83—48no—
216C2×Dic83—22no—
217C2×Dic83—22no—
218C2×Dic83—22no—
219C2×Dic83—22no—
220C2×Dic83—22no—
221C2×Dic83—22no—
222C2×Dic83—22no—
223C2×Dic83—41no—
224C2×Dic83—41no—
225C2×Dic83—41no—
226C2×Dic83—41no—
227C2×Dic83—41no—
228C2×Dic83—41no—
229C2×Dic83—41no—
230C2×Dic83—41no—
231D332—22no—
232D332—22no—
233D332—41no—
234D332—41no—
235C4×D83—48no—
236Dic166—22no—
237Dic166—22no—
238Dic166—41no—
239Dic166—41no—
240C8×C83—22no—
241C8×C83—22no—
242C83⋊C8—24no—
243C22×C83—22no—
244C22×C83—22no—
245C22×C83—22no—
246C22×C83—22no—
247C22×C83—22no—
248C22×C83—22no—
249C22×C83#38814yes—
250C22×C83C2×C4×D411yescharacteristic
251C22×C83C2×#12111yescharacteristic
252C22×C83C2×#12411yescharacteristic
253C22×C83#57611yescharacteristic
254C22×C83#57711yescharacteristic
255C22×C83#57711yescharacteristic
256C22×C83#57811yescharacteristic
257C2×D83—42no—
258C2×D83—42no—
259C2×D83—42no—
260C2×D83—81no—
261C2×D83—81no—
262C2×D83—81no—
263C2×D83—81no—
264C4×C83—44no—
265C4×C83—82no—
266C4×C83#38814yes—
267C4×C83C2×#11711yescharacteristic
268C4×C83C2×C4×D411yescharacteristic
269C4×C83C2×#12311yescharacteristic
270C4×C83C2×#12611yescharacteristic
271Dic83—42no—
272Dic83—42no—
273Dic83—42no—
274Dic83—81no—
275Dic83—81no—
276#21092—831nomaximal, Sylow
277C2×C83—24no—
278C2×C83C2×#38811yescharacteristic
279C2×C83#294611yescharacteristic
280C2×C83#294711yescharacteristic
281C2×C83#306011yescharacteristic
282C2×C83#306111yescharacteristic
283C2×C83#313611yescharacteristic
284C2×C83#313711yescharacteristic
285D83—82no—
286C2×#417—831no—
287C2×#416—831no—
288C2×#405—831no—
289#2316—831no—
290#2311—831no—
291#1936—832no—
292#1935—832no—
293#1934—832no—
294#1933—832no—
295#1736—831no—
296C2×#340D8311yescharacteristic, complemented
297C83#2109211yescharacteristic, complemented, Sylow
298C22×#41—831no—
299C22×M16—831no—
300C23×C8C2×D8311yescharacteristic
301C2×#117—831no—
302C2×#117—831no—
303C22×C4⋊C4C2×D8311yescharacteristic
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#391—832no—
317#390—832no—
318#389—832no—
319#389—832no—
320#389—1661no—
321#389—1661no—
322#386—1662no—
323#380—1662no—
324#380—1661no—
325#380—1661no—
326#340—832no—
327#340—832no—
328#340C2×D8312yes—
329#340C2×D8312yes—
330C2×#41—834no—
331C2×#41—1662no—
332C2×#41—1662no—
333C2×#41—1662no—
334C2×#41—1662no—
335C2×#41—1662no—
336C22×Q8—831no—
337C22×D4—831no—
338C22×D4—1661no—
339C23×C4—21no—
340C23×C4—21no—
341C23×C4—1661no—
342C23×C4C22×D8311yescharacteristic
343C2×M16—834no—
344C2×M16—1662no—
345C2×M16—1662no—
346C22×C8—22no—
347C22×C8—22no—
348C22×C8—832no—
349C22×C8D33214yes—
350C22×C8C22×D8311yescharacteristic
351C22×C8C22×D8311yescharacteristic
352#117—1662no—
353#117—1662no—
354#117—1662no—
355#117—1662no—
356C2×C4⋊C4—21no—
357C2×C4⋊C4—22no—
358C2×C4⋊C4—22no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4—831no—
362C2×C4⋊C4—831no—
363C2×C4⋊C4—831no—
364C2×C4⋊C4—831no—
365C2×C4⋊C4—1662no—
366C2×C4⋊C4—1662no—
367C2×C4⋊C4C4×D8314yes—
368C2×C4⋊C4C22×D8312yes—
369C2×C4⋊C4C22×D8312yes—
370C2×C22⋊C4—832no—
371C2×C22⋊C4—1661no—
372C2×C22⋊C4—1662no—
373C2×C22⋊C4—1662no—
374C2×C42—832no—
375C2×C42—1661no—
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—1662no—
385#95—1662no—
386#95—1662no—
387#95—1662no—
388C24—1661no—
389C24D4×D8311yescharacteristic
390#41—3324no—
391#41—3324no—
392#41—1668no—
393#41—1668no—
394C2×Q8—832no—
395C2×Q8—1661no—
396C2×Q8—1661no—
397C2×Q8—1661no—
398C2×Q8—1661no—
399C2×Q8—1661no—
400C2×D4—3322no—
401C2×D4—3322no—
402C2×D4—832no—
403C2×D4—1664no—
404C2×D4—1661no—
405C2×D4—1661no—
406C2×D4—1661no—
407C2×D4—1661no—
408C2×D4—1661no—
409C22×C4—22no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—42no—
413C22×C4—81no—
414C22×C4—81no—
415C22×C4—81no—
416C22×C4—81no—
417C22×C4—3322no—
418C22×C4—3322no—
419C22×C4—832no—
420C22×C4—832no—
421C22×C4—1661no—
422C22×C4—1661no—
423C22×C4—1661no—
424C22×C4—1664no—
425C22×C4C2×C4×D8312yes—
426C22×C4D4×D8311yescharacteristic
427C22×C4C2×D33212yes—
428C22×C4#1305058912yes—
429C22×C4#1305058912yes—
430C22×C4#1305059012yes—
431C22×C4#1305059112yes—
432C22×C4C23×D8311yescharacteristic
433M16—1668no—
434C2×C8—28no—
435C2×C8—21no—
436C2×C8—21no—
437C2×C8—21no—
438C2×C8—21no—
439C2×C8—834no—
440C2×C8—1662no—
441C2×C8—1662no—
442C2×C8C2×D33212yes—
443C2×C8C2×D33212yes—
444C4⋊C4—22no—
445C4⋊C4—24no—
446C4⋊C4—22no—
447C4⋊C4—22no—
448C4⋊C4—42no—
449C4⋊C4—6642no—
450C4⋊C4—6642no—
451C4⋊C4—1661no—
452C4⋊C4—1661no—
453C4⋊C4—1661no—
454C4⋊C4—1661no—
455C4⋊C4—1662no—
456C4⋊C4—1662no—
457C4⋊C4C2×C4×D8314yes—
458C22⋊C4—3322no—
459C22⋊C4—6642no—
460C22⋊C4—6642no—
461C22⋊C4—1664no—
462C42—3322no—
463C42—1662no—
464C42—1662no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—3321no—
470C23—3321no—
471C23—3321no—
472C23—3321no—
473C23—3321no—
474C23—3321no—
475C23—1662no—
476C23(C22⋊D83)⋊C412yes—
477C23#1458176712yes—
478C23#1458177812yes—
479C23C2×D4×D8311yescharacteristic
480Q8—3321no—
481Q8—3321no—
482Q8—1662no—
483Q8—1662no—
484D4—3328no—
485D4—3321no—
486D4—3321no—
487D4—1662no—
488D4—1662no—
489C2×C4—24no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—84no—
496C2×C4—84no—
497C2×C4—3321no—
498C2×C4—3321no—
499C2×C4—3328no—
500C2×C4—3321no—
501C2×C4—3321no—
502C2×C4—3321no—
503C2×C4—3321no—
504C2×C4—3321no—
505C2×C4—3321no—
506C2×C4—1662no—
507C2×C4—1662no—
508C2×C4—1662no—
509C2×C4—1662no—
510C2×C4—1662no—
511C2×C4—1662no—
512C2×C4—1662no—
513C2×C4#1458175014yes—
514C2×C4(C22⋊D83)⋊C412yes—
515C2×C4C4×D33214yes—
516C2×C4#1458176712yes—
517C2×C4#1458177614yes—
518C2×C4#1458177812yes—
519C2×C4#1458178714yes—
520C2×C4C22×C4×D8311yescharacteristic
521C2×C4C22×D33211yescharacteristic
522C2×C4C2×#1305058911yescharacteristic
523C2×C4C2×#1305058911yescharacteristic
524C2×C4C2×#1305059011yescharacteristic
525C2×C4C2×#1305059111yescharacteristic
526C8—22no—
527C8—22no—
528C8—1664no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—22no—
535C22—3322no—
536C22—3322no—
537C22—3322no—
538C22—6641no—
539C22—6641no—
540C22—6641no—
541C22—6641no—
542C22#1912759314yes—
543C22C2×(C22⋊D83)⋊C411yescharacteristic
544C22C2×#1458176711yescharacteristic
545C22C2×#1458177811yescharacteristic
546C22#1912850211yescharacteristic
547C22#1912850711yescharacteristic
548C22#1912850711yescharacteristic
549C22#1912851411yescharacteristic
550C4—44no—
551C4—82no—
552C4—3322no—
553C4—3322no—
554C4—3322no—
555C4—6641no—
556C4—6641no—
557C4#1912759314yes—
558C4C2×#1458175011yescharacteristic
559C4C2×C4×D33211yescharacteristic
560C4C2×#1458177611yescharacteristic
561C4C2×#1458178711yescharacteristic
562C2—24no—
563C2—6642no—
564C2C2×#1912759311yescharacteristic
565C2#2491628911yescharacteristic
566C2#2491630011yescharacteristic
567C2#2491712611yescharacteristic
568C2#2491713511yescharacteristic
569C2#2491781511yescharacteristic
570C2#2491782611yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC2×#340×C83C2
socleC23×C83C22×D4
prime radicalC22×#13050590C22
cosocleC22×C4×C83C24
hypercenterC2×#340D83
hypocenterC83#21092
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. C2×#19127656C2semidirect
4. #24910081C2semidirect
6. #24908161C2semidirect
7. #24908154C2semidirect
8. #24908147C2semidirect
9. #24908140C2semidirect
10. #24907182C2semidirect
11. C2×#340×C83C2semidirect
16. #19127700C4semidirect
76. #14581715C2×C4semidirect
79. #14581712C2×C4semidirect
296. C2×#340D83semidirect
297. C83#21092semidirect

Maximal subgroups

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

subgroupindex
C2×#191277002
C2×#191276982
C2×#191276562
#249100812
#249100562
#249081612
#249081542
#249081472
#249081402
#249071822
C2×#340×C832
#2109283