Finite Group Databasebeta

← #35409449

Subgroups — #35409449

Order 21248.

1258 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128253C211yesmaximal, characteristic
2C2×#19127751C211yesmaximal, characteristic
3C2×C4⋊C4×Dic83C211yesmaximal, characteristic
4#24908409C212yesmaximal, complemented
5#24908384C212yesmaximal, complemented
6#24907771C211yesmaximal, characteristic
7#24907770C211yesmaximal, characteristic
8#24907768C211yesmaximal, characteristic
9#103×Dic83C212yesmaximal, complemented
10#24907282C212yesmaximal, complemented
11C83×#1872C211yesmaximal, characteristic
12#19128253—24no—
13#19128253C418yescomplemented
14C2×#14581791C2211yescharacteristic
15C2×#14581790C2211yescharacteristic
16C2×#14581782—21no—
17C2×#14581771C2211yescharacteristic
18C2×#14581769—21no—
19C2×Q8×Dic83C2211yescharacteristic
20C2×#14581757—21no—
21C2×#14581757C2211yescharacteristic
22C2×#14581748—21no—
23C22×C4⋊Dic83C2211yescharacteristic
24C22×C4×Dic83—21no—
25C22×C4×Dic83C2211yescharacteristic
26#19127751—24no—
27C2×Dic83⋊C8C2212yes—
28C2×#14581717C2212yes—
29C2×#14581716C2212yes—
30C2×#14581715C2212yes—
31C2×C22⋊(C83⋊C8)C2211yescharacteristic
32C2×C8×Dic83C2212yes—
33C4⋊C4×Dic83—21no—
34C4⋊C4×Dic83—21no—
35C4⋊C4×Dic83C2212yes—
36C4⋊C4×Dic83C2212yes—
37C2×C83⋊#95—21no—
38#19127464C2212yes—
39#19127463C2212yes—
40#19127463C2212yes—
41C2×C22⋊Q8×C83C2211yescharacteristic
42C22×C4⋊C4×C83C2211yescharacteristic
43C2×#103×C83C2212yes—
44C2×C22⋊C8×C83C2211yescharacteristic
45#340×C83C2212yes—
46C22×Dic166—21no—
47C22×Dic166C2311yescharacteristic
48C23×Dic83—21no—
49C23×Dic83D411yescharacteristic
50C23×Dic83D411yescharacteristic
51C22×C83⋊C8C2312yes—
52#14581791—22no—
53#14581791C2×C414yescomplemented
54#14581790—22no—
55#14581790C2×C414yescomplemented
56#14581782—28no—
57#14581771—22no—
58#14581771C2×C414yes—
59#14581769—28no—
60Q8×Dic83—22no—
61Q8×Dic83C2×C414yescomplemented
62#14581757—28no—
63#14581757—22no—
64#14581757C2×C414yescomplemented
65#14581748—28no—
66C2×C4⋊Dic83—21no—
67C2×C4⋊Dic83—24no—
68C2×C4⋊Dic83—21no—
69C2×C4⋊Dic83C2311yescharacteristic
70C2×C4⋊Dic83C2312yes—
71C2×Dic83⋊C4—21no—
72C2×Dic83⋊C4—21no—
73C2×Dic83⋊C4—21no—
74C2×Dic83⋊C4—21no—
75C2×Dic83⋊C4—21no—
76C2×C22⋊Dic83—21no—
77C2×C4×Dic83—21no—
78C2×C4×Dic83—21no—
79C2×C4×Dic83—22no—
80C2×C4×Dic83—22no—
81C2×C4×Dic83—44no—
82C2×C4×Dic83C2×C414yes—
83C2×C4×Dic83D411yescharacteristic
84C2×C4×Dic83D411yescharacteristic
85C2×C4×Dic83D414yes—
86C2×C4×Dic83C2311yescharacteristic
87C2×C4×Dic83C2311yescharacteristic
88Dic83⋊C8—24no—
89#14581717—24no—
90#14581716—24no—
91#14581715—24no—
92C22⋊(C83⋊C8)—22no—
93C8×Dic83—24no—
94C83⋊#95—22no—
95C83⋊#95—22no—
96C22×Q8×C83C2311yescharacteristic
97C23×C4×C83—21no—
98C23×C4×C83C2311yescharacteristic
99C22×C8×C83C2312yes—
100C22⋊Q8×C83—22no—
101C22⋊Q8×C83C2×C414yes—
102C2×C4⋊C4×C83—21no—
103C2×C4⋊C4×C83—21no—
104C2×C4⋊C4×C83—24no—
105C2×C4⋊C4×C83C2311yescharacteristic
106C2×C4⋊C4×C83C2312yes—
107C2×C22⋊C4×C83—21no—
108#103×C83—24no—
109C22⋊C8×C83—22no—
110C2×Dic166—22no—
111C2×Dic166—22no—
112C2×Dic166—21no—
113C2×Dic166—24no—
114C2×Dic166—42no—
115C2×Dic166—42no—
116C2×Dic166C22×C412yes—
117C22×Dic83—24no—
118C22×Dic83—22no—
119C22×Dic83—21no—
120C22×Dic83—21no—
121C22×Dic83—41no—
122C22×Dic83—41no—
123C22×Dic83—42no—
124C22×Dic83—42no—
125C22×Dic83C22⋊C414yes—
126C22×Dic83C22⋊C414yes—
127C22×Dic83SD1614yes—
128C22×Dic83Q1614yes—
129C22×Dic83C2×D411yescharacteristic
130C22×Dic83C2×D411yescharacteristic
131C22×Dic83C2×D411yescharacteristic
132C22×Dic83C2×D411yescharacteristic
133C2×C83⋊C8—22no—
134C2×C83⋊C8—22no—
135C2×C83⋊C8—22no—
136C4⋊Dic83—24no—
137C4⋊Dic83—21no—
138C4⋊Dic83—22no—
139C4⋊Dic83—24no—
140C4⋊Dic83C22×C412yes—
141C4⋊Dic83#4114yes—
142Dic83⋊C4—24no—
143Dic83⋊C4—24no—
144Dic83⋊C4—24no—
145Dic83⋊C4—24no—
146Dic83⋊C4—24no—
147C22⋊Dic83—24no—
148C4×Dic83—24no—
149C4×Dic83—24no—
150C4×Dic83—44no—
151C4×Dic83—44no—
152C4×Dic83C22⋊C414yes—
153C4×Dic83C22⋊C414yes—
154C4×Dic83C22×C412yes—
155C4×Dic83C22×C412yes—
156C4×Dic83C2×D412yes—
157C4×Dic83C2×D412yes—
158C24×C83C2×D411yescharacteristic
159C2×Q8×C83—21no—
160C2×Q8×C83—22no—
161C2×Q8×C83—22no—
162C2×Q8×C83C22×C412yes—
163C22×C4×C83—22no—
164C22×C4×C83—22no—
165C22×C4×C83—24no—
166C22×C4×C83—21no—
167C22×C4×C83—21no—
168C22×C4×C83—22no—
169C22×C4×C83—42no—
170C22×C4×C83—42no—
171C22×C4×C83C22×C412yes—
172C22×C4×C83C2×D411yescharacteristic
173C22×C4×C83C2×D412yes—
174C22×C4×C83C2411yescharacteristic
175C2×C8×C83—22no—
176C2×C8×C83—22no—
177C2×C8×C83—22no—
178C4⋊C4×C83—24no—
179C4⋊C4×C83—24no—
180C4⋊C4×C83—21no—
181C4⋊C4×C83—22no—
182C4⋊C4×C83C22×C412yes—
183C4⋊C4×C83#4114yes—
184C22⋊C4×C83—24no—
185C23×C83—21no—
186C23×C83—21no—
187C23×C83—21no—
188C23×C83—21no—
189C23×C83C2×C22⋊C412yes—
190C23×C83C2×SD1612yes—
191C23×C83C2×Q1612yes—
192C23×C83C22×D411yescharacteristic
193Q8×C83—24no—
194Q8×C83—42no—
195C2×C4×C83—21no—
196C2×C4×C83—21no—
197C2×C4×C83—24no—
198C2×C4×C83—24no—
199C2×C4×C83—22no—
200C2×C4×C83—22no—
201C2×C4×C83—24no—
202C2×C4×C83—48no—
203C2×C4×C83—41no—
204C2×C4×C83—41no—
205C2×C4×C83—42no—
206C2×C4×C83—42no—
207C2×C4×C83—41no—
208C2×C4×C83—41no—
209C2×C4×C83C2×C22⋊C412yes—
210C2×C4×C83C23×C411yescharacteristic
211C2×C4×C83C22×D411yescharacteristic
212C2×C4×C83C2×#4112yes—
213C2×C4×C83C2×#4112yes—
214C2×Dic83—21no—
215C2×Dic83—21no—
216C2×Dic83—21no—
217C2×Dic83—21no—
218C2×Dic83—22no—
219C2×Dic83—22no—
220C2×Dic83—24no—
221C2×Dic83—44no—
222C2×Dic83—48no—
223C2×Dic83—41no—
224C2×Dic83—41no—
225C2×Dic83—42no—
226C2×Dic83—42no—
227C2×Dic83—41no—
228C2×Dic83—41no—
229C2×Dic83#103116yescomplemented
230C2×Dic83C2×C22⋊C412yes—
231C2×Dic83C2×C22⋊C412yes—
232C2×Dic83C2×C22⋊C412yes—
233C2×Dic83C2×C22⋊C412yes—
234C2×Dic83C2×SD1612yes—
235C2×Dic83C2×SD1612yes—
236C2×Dic83C2×Q1612yes—
237C2×Dic83C2×Q1612yes—
238Dic166—24no—
239Dic166—48no—
240Dic166—42no—
241C8×C83—42no—
242C83⋊C8—42no—
243C22×C83—22no—
244C22×C83—22no—
245C22×C83—22no—
246C22×C83—22no—
247C22×C83—22no—
248C22×C83—22no—
249C22×C83C2×#10314yes—
250C22×C83C22×C22⋊C411yescharacteristic
251C22×C83#52511yescharacteristic
252C22×C83#52511yescharacteristic
253C22×C83C22×SD1611yescharacteristic
254C22×C83C22×Q1611yescharacteristic
255C22×C83#57511yescharacteristic
256C22×C83#57511yescharacteristic
257C4×C83—22no—
258C4×C83—44no—
259C4×C83—42no—
260C4×C83—42no—
261C4×C83C4×#4112yes—
262C4×C83#53212yes—
263Dic83—22no—
264Dic83—22no—
265Dic83—44no—
266Dic83—42no—
267Dic83—42no—
268Dic83C2×#10314yes—
269Dic83C2×#10314yes—
270#21385—831nomaximal, Sylow
271C2×C83—24no—
272C2×C83#229711yescharacteristic
273C2×C83C22×#10311yescharacteristic
274C2×C83#289611yescharacteristic
275C2×C83#310111yescharacteristic
276C2×C83#310111yescharacteristic
277C2×C83#310411yescharacteristic
278C2×C83#310411yescharacteristic
279C2×#533—831no—
280C2×#431—831no—
281C2×C4×C4⋊C4—831no—
282#1987—832no—
283#1981—832no—
284#1872—831no—
285#1872—831no—
286#1872—831no—
287#1872D8311yescharacteristic
288C4×#103—832no—
289C4×#103—832no—
290C83#2138511yescharacteristic, complemented, Sylow
291#533—838no—
292#533—1664no—
293C2×C4⋊Q8—831no—
294C2×C4⋊Q8—831no—
295C2×#125—1661no—
296C2×C22⋊Q8—831no—
297C2×C22⋊Q8—1661no—
298C2×C22⋊Q8C2×D8311yescharacteristic
299C2×C4×Q8—831no—
300C2×C4×Q8—831no—
301C2×C4×Q8—1661no—
302C2×#117—1661no—
303C22×C4⋊C4—831no—
304C22×C4⋊C4C2×D8311yescharacteristic
305C22×C42—831no—
306C22×C42—1661no—
307#431—1664no—
308C2×C4⋊C8—832no—
309C2×#103—832no—
310C2×#103—832no—
311C2×#103—832no—
312C2×#103C2×D8312yes—
313C2×C22⋊C8—831no—
314C2×C22⋊C8C2×D8311yescharacteristic
315C2×C4×C8—832no—
316C4×C4⋊C4—832no—
317C4×C4⋊C4—832no—
318C4×C4⋊C4—1661no—
319C4×C4⋊C4—1661no—
320C2×#95—1661no—
321#340—832no—
322#340—832no—
323#340—832no—
324#340C2×D8312yes—
325C22×Q8—831no—
326C22×Q8—1661no—
327C22×Q8C22×D8311yescharacteristic
328C23×C4—21no—
329C23×C4—831no—
330C23×C4—831no—
331C23×C4—1661no—
332C23×C4C22×D8311yescharacteristic
333C22×C8—832no—
334C22×C8C22×D8312yes—
335C4⋊Q8—834no—
336C4⋊Q8—834no—
337C4⋊Q8—1662no—
338C4⋊Q8—1662no—
339#125—1668no—
340C22⋊Q8—22no—
341C22⋊Q8—834no—
342C22⋊Q8—1668no—
343C22⋊Q8—1662no—
344C22⋊Q8C4×D8314yes—
345C4×Q8—834no—
346C4×Q8—834no—
347C4×Q8—1662no—
348C4×Q8—1668no—
349C4×Q8—1662no—
350#117—1668no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—21no—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—832no—
355C2×C4⋊C4—831no—
356C2×C4⋊C4—1664no—
357C2×C4⋊C4—1661no—
358C2×C4⋊C4—1661no—
359C2×C4⋊C4—1661no—
360C2×C4⋊C4—1661no—
361C2×C4⋊C4—1661no—
362C2×C4⋊C4—1661no—
363C2×C4⋊C4—1661no—
364C2×C4⋊C4C22×D8311yescharacteristic
365C2×C4⋊C4C22×D8312yes—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—1661no—
368C2×C42—3324no—
369C2×C42—834no—
370C2×C42—831no—
371C2×C42—831no—
372C2×C42—834no—
373C2×C42—831no—
374C2×C42—831no—
375C2×C42—1662no—
376C2×C42—1662no—
377C2×C42—1661no—
378C2×C42—1661no—
379C4⋊C8—1664no—
380#103—24no—
381#103—1664no—
382#103—1664no—
383#103—1664no—
384C22⋊C8—22no—
385C22⋊C8—1662no—
386C4×C8—1664no—
387#95—1662no—
388#95—1662no—
389C24D4×D8311yescharacteristic
390C2×Q8—21no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—3322no—
394C2×Q8—3322no—
395C2×Q8—832no—
396C2×Q8—1664no—
397C2×Q8—1662no—
398C2×Q8—1662no—
399C2×Q8—1661no—
400C2×Q8C2×C4×D8312yes—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—22no—
405C22×C4—21no—
406C22×C4—21no—
407C22×C4—42no—
408C22×C4—42no—
409C22×C4—3322no—
410C22×C4—3322no—
411C22×C4—3321no—
412C22×C4—3321no—
413C22×C4—834no—
414C22×C4—834no—
415C22×C4—834no—
416C22×C4—831no—
417C22×C4—831no—
418C22×C4—834no—
419C22×C4—831no—
420C22×C4—831no—
421C22×C4—1664no—
422C22×C4—1662no—
423C22×C4—1661no—
424C22×C4—1661no—
425C22×C4C2×C4×D8312yes—
426C22×C4D4×D8312yes—
427C22×C4D4×D8311yescharacteristic
428C22×C4C23×D8311yescharacteristic
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—1662no—
433C2×C8—1662no—
434C2×C8—1662no—
435C4⋊C4—24no—
436C4⋊C4—21no—
437C4⋊C4—22no—
438C4⋊C4—24no—
439C4⋊C4—832no—
440C4⋊C4—834no—
441C4⋊C4—1664no—
442C4⋊C4—1664no—
443C4⋊C4—1664no—
444C4⋊C4—1664no—
445C4⋊C4—1664no—
446C4⋊C4—1664no—
447C4⋊C4—1661no—
448C4⋊C4—1662no—
449C4⋊C4—1664no—
450C4⋊C4C2×C4×D8312yes—
451C4⋊C4#1305058914yes—
452C22⋊C4—24no—
453C22⋊C4—1664no—
454C42—3324no—
455C42—3324no—
456C42—832no—
457C42—832no—
458C42—832no—
459C42—832no—
460C42—834no—
461C42—834no—
462C42—1664no—
463C42—1664no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23C22⋊C4×D8312yes—
469C23SD16×D8312yes—
470C23Q16×D8312yes—
471C23C2×D4×D8311yescharacteristic
472Q8—24no—
473Q8—42no—
474Q8—3328no—
475Q8—3322no—
476Q8—1664no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—48no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—3328no—
492C2×C4—3321no—
493C2×C4—3321no—
494C2×C4—3322no—
495C2×C4—3321no—
496C2×C4—3322no—
497C2×C4—3321no—
498C2×C4—3324no—
499C2×C4—832no—
500C2×C4—832no—
501C2×C4—8316no—
502C2×C4—832no—
503C2×C4—832no—
504C2×C4—832no—
505C2×C4—832no—
506C2×C4—832no—
507C2×C4—832no—
508C2×C4—1662no—
509C2×C4—1662no—
510C2×C4—1664no—
511C2×C4—1661no—
512C2×C4—1661no—
513C2×C4—1661no—
514C2×C4—1661no—
515C2×C4C22⋊C4×D8312yes—
516C2×C4C22×C4×D8311yescharacteristic
517C2×C4C2×D4×D8311yescharacteristic
518C2×C4#41×D8312yes—
519C2×C4C2×#1305058912yes—
520C8—42no—
521C8—3322no—
522C22—22no—
523C22—22no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22#103×D8314yes—
529C22C2×C22⋊C4×D8311yescharacteristic
530C22#1912820711yescharacteristic
531C22#1912820911yescharacteristic
532C22C2×SD16×D8311yescharacteristic
533C22C2×Q16×D8311yescharacteristic
534C22#1912849711yescharacteristic
535C22#1912850011yescharacteristic
536C4—22no—
537C4—42no—
538C4—42no—
539C4—44no—
540C4—3322no—
541C4—3322no—
542C4—3324no—
543C4—834no—
544C4—834no—
545C4—1662no—
546C4—1662no—
547C4C4×#1305058912yes—
548C4#1912824612yes—
549C2—24no—
550C2#2490994811yescharacteristic
551C2C2×#103×D8311yescharacteristic
552C2#2491597911yescharacteristic
553C2#2491746611yescharacteristic
554C2#2491746811yescharacteristic
555C2#2491749711yescharacteristic
556C2#2491749911yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#1872C2
socleC23×C83C22×D4
prime radicalC24×C83C2×D4
cosocleC22×C4×C83C24
hypercenter#1872D83
hypocenterC83#21385
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1872

Complements

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

NH ≅ G/Ntype
4. #24908409C2semidirect
5. #24908384C2semidirect
9. #103×Dic83C2semidirect
10. #24907282C2semidirect
13. #19128253C4semidirect
53. #14581791C2×C4semidirect
55. #14581790C2×C4semidirect
61. Q8×Dic83C2×C4semidirect
64. #14581757C2×C4semidirect
229. C2×Dic83#103semidirect
290. C83#21385semidirect

Maximal subgroups

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

subgroupindex
C2×#191282532
C2×#191277512
C2×C4⋊C4×Dic832
#249084092
#249083842
#249077712
#249077702
#249077682
#103×Dic832
#249072822
C83×#18722
#2138583