Finite Group Databasebeta

← #35414083

Subgroups — #35414083

Order 21248.

1222 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19128253C211yesmaximal, characteristic
2C2×#19127751C211yesmaximal, characteristic
3C2×#19127570C211yesmaximal, characteristic
4#24908422C212yesmaximal, complemented
5#24908263C212yesmaximal, complemented
6#24907771C211yesmaximal, characteristic
7#24907770C211yesmaximal, characteristic
8#24907768C211yesmaximal, characteristic
9#24907467C212yesmaximal, complemented
10#24907466C212yesmaximal, 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×Dic83⋊C4—21no—
25C22×Dic83⋊C4—21no—
26C22×C4×Dic83C2211yescharacteristic
27#19127751—24no—
28C2×Dic83⋊C8C2212yes—
29C2×#14581717C2212yes—
30C2×#14581716C2212yes—
31C2×#14581715C2212yes—
32C2×C22⋊(C83⋊C8)C2211yescharacteristic
33C2×C8×Dic83C2212yes—
34#19127570—21no—
35#19127570—21no—
36#19127570C2212yes—
37#19127570C2212yes—
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—21no—
68C2×C4⋊Dic83—42no—
69C2×C4⋊Dic83C2312yes—
70C2×C4⋊Dic83C2311yescharacteristic
71C2×Dic83⋊C4—21no—
72C2×Dic83⋊C4—21no—
73C2×Dic83⋊C4—21no—
74C2×Dic83⋊C4—21no—
75C2×Dic83⋊C4—22no—
76C2×Dic83⋊C4—22no—
77C2×Dic83⋊C4—22no—
78C2×Dic83⋊C4—22no—
79C2×Dic83⋊C4—21no—
80C2×Dic83⋊C4—44no—
81C2×Dic83⋊C4—44no—
82C2×C22⋊Dic83—21no—
83C2×C4×Dic83—21no—
84C2×C4×Dic83—21no—
85C2×C4×Dic83C2×C414yes—
86C2×C4×Dic83D411yescharacteristic
87C2×C4×Dic83D411yescharacteristic
88C2×C4×Dic83D414yes—
89C2×C4×Dic83C2311yescharacteristic
90C2×C4×Dic83C2311yescharacteristic
91Dic83⋊C8—24no—
92#14581717—42no—
93#14581716—42no—
94#14581715—42no—
95C22⋊(C83⋊C8)—22no—
96C8×Dic83—24no—
97C22×Q8×C83C2311yescharacteristic
98C23×C4×C83—21no—
99C23×C4×C83C2311yescharacteristic
100C22×C8×C83C2312yes—
101C22⋊Q8×C83—22no—
102C22⋊Q8×C83C2×C414yes—
103C2×C4⋊C4×C83—21no—
104C2×C4⋊C4×C83—21no—
105C2×C4⋊C4×C83—42no—
106C2×C4⋊C4×C83C2312yes—
107C2×C4⋊C4×C83C2311yescharacteristic
108C2×C22⋊C4×C83—21no—
109#103×C83—42no—
110C22⋊C8×C83—22no—
111C2×Dic166—22no—
112C2×Dic166—22no—
113C2×Dic166—21no—
114C2×Dic166—24no—
115C2×Dic166—42no—
116C2×Dic166—42no—
117C2×Dic166C22×C412yes—
118C22×Dic83—22no—
119C22×Dic83—22no—
120C22×Dic83—21no—
121C22×Dic83—21no—
122C22×Dic83—22no—
123C22×Dic83—42no—
124C22×Dic83—42no—
125C22×Dic83—42no—
126C22×Dic83C22⋊C414yes—
127C22×Dic83C22⋊C414yes—
128C22×Dic83SD1614yes—
129C22×Dic83Q1614yes—
130C22×Dic83C2×D411yescharacteristic
131C22×Dic83C2×D411yescharacteristic
132C22×Dic83C2×D411yescharacteristic
133C22×Dic83C2×D411yescharacteristic
134C2×C83⋊C8—22no—
135C2×C83⋊C8—22no—
136C2×C83⋊C8—22no—
137C4⋊Dic83—21no—
138C4⋊Dic83—24no—
139C4⋊Dic83—22no—
140C4⋊Dic83—22no—
141C4⋊Dic83—42no—
142C4⋊Dic83C22×C412yes—
143Dic83⋊C4—24no—
144Dic83⋊C4—24no—
145Dic83⋊C4—24no—
146Dic83⋊C4—24no—
147Dic83⋊C4—24no—
148Dic83⋊C4—44no—
149Dic83⋊C4—44no—
150Dic83⋊C4—44no—
151Dic83⋊C4—44no—
152C22⋊Dic83—24no—
153C4×Dic83—24no—
154C4×Dic83—24no—
155C4×Dic83C22⋊C414yes—
156C4×Dic83C22⋊C414yes—
157C4×Dic83C22×C412yes—
158C4×Dic83C22×C412yes—
159C4×Dic83C2×D412yes—
160C4×Dic83C2×D412yes—
161C24×C83C2×D411yescharacteristic
162C2×Q8×C83—21no—
163C2×Q8×C83—22no—
164C2×Q8×C83—22no—
165C2×Q8×C83C22×C412yes—
166C22×C4×C83—22no—
167C22×C4×C83—22no—
168C22×C4×C83—21no—
169C22×C4×C83—21no—
170C22×C4×C83—22no—
171C22×C4×C83—42no—
172C22×C4×C83—42no—
173C22×C4×C83—42no—
174C22×C4×C83C22×C412yes—
175C22×C4×C83C2×D411yescharacteristic
176C22×C4×C83C2×D412yes—
177C22×C4×C83C2411yescharacteristic
178C2×C8×C83—22no—
179C2×C8×C83—22no—
180C2×C8×C83—22no—
181C4⋊C4×C83—24no—
182C4⋊C4×C83—21no—
183C4⋊C4×C83—22no—
184C4⋊C4×C83—22no—
185C4⋊C4×C83—42no—
186C4⋊C4×C83C22×C412yes—
187C22⋊C4×C83—24no—
188C23×C83—21no—
189C23×C83—21no—
190C23×C83—21no—
191C23×C83—21no—
192C23×C83C2×C22⋊C412yes—
193C23×C83C2×SD1612yes—
194C23×C83C2×Q1612yes—
195C23×C83C22×D411yescharacteristic
196Q8×C83—24no—
197Q8×C83—42no—
198C2×C4×C83—21no—
199C2×C4×C83—21no—
200C2×C4×C83—24no—
201C2×C4×C83—24no—
202C2×C4×C83—22no—
203C2×C4×C83—22no—
204C2×C4×C83—41no—
205C2×C4×C83—41no—
206C2×C4×C83—41no—
207C2×C4×C83—41no—
208C2×C4×C83—42no—
209C2×C4×C83—42no—
210C2×C4×C83—42no—
211C2×C4×C83—84no—
212C2×C4×C83C2×C22⋊C412yes—
213C2×C4×C83C23×C411yescharacteristic
214C2×C4×C83C22×D411yescharacteristic
215C2×C4×C83H(5,𝔽2)12yes—
216C2×C4×C832−1+412yes—
217C2×Dic83—21no—
218C2×Dic83—21no—
219C2×Dic83—28no—
220C2×Dic83—22no—
221C2×Dic83—22no—
222C2×Dic83—41no—
223C2×Dic83—41no—
224C2×Dic83—41no—
225C2×Dic83—41no—
226C2×Dic83—42no—
227C2×Dic83—42no—
228C2×Dic83—42no—
229C2×Dic83—84no—
230C2×Dic83#103116yescomplemented
231C2×Dic83C2×C22⋊C412yes—
232C2×Dic83C2×C22⋊C412yes—
233C2×Dic83C2×C22⋊C412yes—
234C2×Dic83C2×C22⋊C412yes—
235C2×Dic83C2×SD1612yes—
236C2×Dic83C2×SD1612yes—
237C2×Dic83C2×Q1612yes—
238C2×Dic83C2×Q1612yes—
239C2×Dic83AGL(1,ℤ/8)12yes—
240C2×Dic83#13712yes—
241Dic166—24no—
242Dic166—42no—
243Dic166—48no—
244C8×C83—42no—
245C83⋊C8—42no—
246C22×C83—22no—
247C22×C83—22no—
248C22×C83—22no—
249C22×C83—22no—
250C22×C83—22no—
251C22×C83—22no—
252C22×C83C2×#10314yes—
253C22×C83C22×C22⋊C411yescharacteristic
254C22×C83#53511yescharacteristic
255C22×C83#53611yescharacteristic
256C22×C83C22×SD1611yescharacteristic
257C22×C83C22×Q1611yescharacteristic
258C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
259C22×C83C2×#13711yescharacteristic
260C4×C83—22no—
261C4×C83—42no—
262C4×C83—42no—
263C4×C83—82no—
264C4×C83#52012yes—
265C4×C83#53712yes—
266Dic83—22no—
267Dic83—42no—
268Dic83—42no—
269Dic83—82no—
270Dic83C2×#10314yes—
271Dic83C2×#10314yes—
272Dic83#41714yes—
273#21860—831nomaximal, Sylow
274C2×C83—24no—
275C2×C83#231111yescharacteristic
276C2×C83C22×#10311yescharacteristic
277C2×C83C2×#41711yescharacteristic
278C2×C83#320811yescharacteristic
279C2×C83#320911yescharacteristic
280C2×C83#322911yescharacteristic
281C2×C83#323011yescharacteristic
282C2×#533—831no—
283C2×#431—831no—
284C2×#384—831no—
285#1990—832no—
286#1956—832no—
287#1872—831no—
288#1872—831no—
289#1872—831no—
290#1872D8311yescharacteristic
291#1803—832no—
292#1803—832no—
293C83#2186011yescharacteristic, complemented, Sylow
294#533—838no—
295#533—1664no—
296C2×C4⋊Q8—831no—
297C2×C4⋊Q8—831no—
298C2×#125—1661no—
299C2×C22⋊Q8—831no—
300C2×C22⋊Q8—1661no—
301C2×C22⋊Q8C2×D8311yescharacteristic
302C2×C4×Q8—831no—
303C2×C4×Q8—831no—
304C2×C4×Q8—1661no—
305C2×#117—1661no—
306C22×C4⋊C4—831no—
307C22×C4⋊C4—1661no—
308C22×C4⋊C4—1661no—
309C22×C4⋊C4C2×D8311yescharacteristic
310C22×C42—831no—
311#431—1664no—
312C2×C4⋊C8—832no—
313C2×#103—832no—
314C2×#103—832no—
315C2×#103—832no—
316C2×#103C2×D8312yes—
317C2×C22⋊C8—831no—
318C2×C22⋊C8C2×D8311yescharacteristic
319C2×C4×C8—832no—
320#384—832no—
321#384—832no—
322#384—1661no—
323#384—1661no—
324#340—832no—
325#340—832no—
326#340—832no—
327#340C2×D8312yes—
328C22×Q8—831no—
329C22×Q8—1661no—
330C22×Q8C22×D8311yescharacteristic
331C23×C4—21no—
332C23×C4—831no—
333C23×C4—831no—
334C23×C4—1661no—
335C23×C4C22×D8311yescharacteristic
336C22×C8—832no—
337C22×C8C22×D8312yes—
338C4⋊Q8—834no—
339C4⋊Q8—834no—
340C4⋊Q8—1662no—
341C4⋊Q8—1662no—
342#125—1668no—
343C22⋊Q8—22no—
344C22⋊Q8—834no—
345C22⋊Q8—1668no—
346C22⋊Q8—1662no—
347C22⋊Q8C4×D8314yes—
348C4×Q8—834no—
349C4×Q8—834no—
350C4×Q8—1662no—
351C4×Q8—1668no—
352C4×Q8—1662no—
353#117—1668no—
354C2×C4⋊C4—21no—
355C2×C4⋊C4—21no—
356C2×C4⋊C4—42no—
357C2×C4⋊C4—3324no—
358C2×C4⋊C4—3324no—
359C2×C4⋊C4—3322no—
360C2×C4⋊C4—831no—
361C2×C4⋊C4—832no—
362C2×C4⋊C4—1662no—
363C2×C4⋊C4—1662no—
364C2×C4⋊C4—1662no—
365C2×C4⋊C4—1662no—
366C2×C4⋊C4—1661no—
367C2×C4⋊C4—1661no—
368C2×C4⋊C4—1661no—
369C2×C4⋊C4—1661no—
370C2×C4⋊C4—1661no—
371C2×C4⋊C4—1661no—
372C2×C4⋊C4—1661no—
373C2×C4⋊C4C22×D8312yes—
374C2×C4⋊C4C22×D8311yescharacteristic
375C2×C22⋊C4—21no—
376C2×C22⋊C4—1661no—
377C2×C42—834no—
378C2×C42—831no—
379C2×C42—831no—
380C2×C42—834no—
381C2×C42—831no—
382C2×C42—831no—
383C2×C42—1661no—
384C2×C42—1661no—
385C4⋊C8—1664no—
386#103—42no—
387#103—3322no—
388#103—3322no—
389#103—3322no—
390C22⋊C8—22no—
391C22⋊C8—1662no—
392C4×C8—1664no—
393C24D4×D8311yescharacteristic
394C2×Q8—21no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—3322no—
398C2×Q8—3322no—
399C2×Q8—832no—
400C2×Q8—1664no—
401C2×Q8—1662no—
402C2×Q8—1662no—
403C2×Q8—1661no—
404C2×Q8C2×C4×D8312yes—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—22no—
410C22×C4—42no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—3322no—
414C22×C4—3322no—
415C22×C4—3322no—
416C22×C4—834no—
417C22×C4—834no—
418C22×C4—831no—
419C22×C4—831no—
420C22×C4—831no—
421C22×C4—834no—
422C22×C4—831no—
423C22×C4—834no—
424C22×C4—1662no—
425C22×C4—1661no—
426C22×C4—1661no—
427C22×C4—1662no—
428C22×C4—1662no—
429C22×C4C2×C4×D8312yes—
430C22×C4D4×D8312yes—
431C22×C4D4×D8311yescharacteristic
432C22×C4C23×D8311yescharacteristic
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—1662no—
437C2×C8—1662no—
438C2×C8—1662no—
439C4⋊C4—21no—
440C4⋊C4—24no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—3322no—
445C4⋊C4—3324no—
446C4⋊C4—3324no—
447C4⋊C4—3324no—
448C4⋊C4—3324no—
449C4⋊C4—832no—
450C4⋊C4—1664no—
451C4⋊C4—1664no—
452C4⋊C4—1664no—
453C4⋊C4—1664no—
454C4⋊C4—1664no—
455C4⋊C4—1662no—
456C4⋊C4—1662no—
457C4⋊C4—1661no—
458C4⋊C4—1664no—
459C4⋊C4C2×C4×D8312yes—
460C22⋊C4—24no—
461C22⋊C4—1664no—
462C42—832no—
463C42—832no—
464C42—832no—
465C42—832no—
466C42—834no—
467C42—834no—
468C42—1664no—
469C42—1664no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—21no—
474C23C22⋊C4×D8312yes—
475C23SD16×D8312yes—
476C23Q16×D8312yes—
477C23C2×D4×D8311yescharacteristic
478Q8—24no—
479Q8—42no—
480Q8—3328no—
481Q8—3322no—
482Q8—1664no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—24no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—41no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—84no—
497C2×C4—3321no—
498C2×C4—3321no—
499C2×C4—3321no—
500C2×C4—3321no—
501C2×C4—3322no—
502C2×C4—3322no—
503C2×C4—3322no—
504C2×C4—832no—
505C2×C4—832no—
506C2×C4—832no—
507C2×C4—8316no—
508C2×C4—832no—
509C2×C4—832no—
510C2×C4—832no—
511C2×C4—832no—
512C2×C4—832no—
513C2×C4—832no—
514C2×C4—832no—
515C2×C4—6644no—
516C2×C4—1662no—
517C2×C4—1662no—
518C2×C4—1668no—
519C2×C4—1661no—
520C2×C4—1661no—
521C2×C4C22⋊C4×D8312yes—
522C2×C4C22×C4×D8311yescharacteristic
523C2×C4C2×D4×D8311yescharacteristic
524C2×C4#1458183712yes—
525C2×C4#1458183912yes—
526C8—42no—
527C8—3322no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22#103×D8314yes—
535C22C2×C22⋊C4×D8311yescharacteristic
536C22#1912826611yescharacteristic
537C22#1912827311yescharacteristic
538C22C2×SD16×D8311yescharacteristic
539C22C2×Q16×D8311yescharacteristic
540C22AGL(1,ℤ/8)×D8311yescharacteristic
541C22#137×D8311yescharacteristic
542C4—22no—
543C4—42no—
544C4—42no—
545C4—82no—
546C4—3322no—
547C4—3322no—
548C4—834no—
549C4—834no—
550C4—834no—
551C4—6642no—
552C4—1662no—
553C4#1912818012yes—
554C4#1912827812yes—
555C2—24no—
556C2#2491005011yescharacteristic
557C2C2×#103×D8311yescharacteristic
558C2#417×D8311yescharacteristic
559C2#2491840211yescharacteristic
560C2#2491840911yescharacteristic
561C2#2491860511yescharacteristic
562C2#2491861211yescharacteristic
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#21860
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. #24908422C2semidirect
5. #24908263C2semidirect
9. #24907467C2semidirect
10. #24907466C2semidirect
13. #19128253C4semidirect
53. #14581791C2×C4semidirect
55. #14581790C2×C4semidirect
61. Q8×Dic83C2×C4semidirect
64. #14581757C2×C4semidirect
230. C2×Dic83#103semidirect
293. C83#21860semidirect

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×#191275702
#249084222
#249082632
#249077712
#249077702
#249077682
#249074672
#249074662
C83×#18722
#2186083