Finite Group Databasebeta

← C2×#398×D83 #35570614

Subgroups — C2×#398×D83 #35570614

Order 21248.

4158 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊C4×D83C211yesmaximal, characteristic
2C22×C4⋊C4×D83C211yesmaximal, characteristic
3C22×C4⋊C4×D83C212yesmaximal
4C2×#19127635C212yesmaximal, complemented
5C2×#19127634C212yesmaximal, complemented
6C2×#19127633C211yesmaximal, characteristic, complemented
7C2×#19127632C211yesmaximal, characteristic, complemented
8C2×#19127631C211yesmaximal, characteristic, complemented
9#398×D83C2116yesmaximal, direct product factor
10C2×#95×D83C211yesmaximal, characteristic
11C2×#95×D83C212yesmaximal
12C2×#398×C83C211yesmaximal, characteristic, complemented
13C23×C4×D83C2212yes—
14C23×C4×D83C2211yescharacteristic
15C23×C4×D83C2211yescharacteristic
16C23×C4×D83C2211yescharacteristic
17C23×C4×D83C2212yes—
18C22×C4⋊Dic83C2211yescharacteristic
19C22×C4⋊Dic83C2212yes—
20C22×C4⋊Dic83C2211yescharacteristic
21C22×Dic83⋊C4C2212yes—
22C22×Dic83⋊C4C2211yescharacteristic
23C22×Dic83⋊C4C2212yes—
24C22×Dic83⋊C4C2212yes—
25C22×Dic83⋊C4C2211yescharacteristic
26C2×C4⋊C4×D83—28no—
27C2×C4⋊C4×D83—216no—
28C2×C4⋊C4×D83—28no—
29C2×C4⋊C4×D83C2218yes—
30C2×C4⋊C4×D83C22116yes—
31C2×C4⋊C4×D83C2218yes—
32#19127635C22116yescomplemented
33#19127634C22116yescomplemented
34#19127633C2218yescomplemented
35#19127632C2218yescomplemented
36#19127631C2218yescomplemented
37C2×C83⋊#95C2211yescharacteristic
38C2×C83⋊#95C2212yes—
39C2×C83⋊#95C2212yes—
40C2×C83⋊#95C2212yes—
41C2×C83⋊#95C2212yes—
42#95×D83C22116yes—
43#95×D83C2218yes—
44C22×C4⋊C4×C83C2211yescharacteristic
45C22×C4⋊C4×C83C2211yescharacteristic
46C22×C4⋊C4×C83C2212yes—
47#398×C83C2218yescomplemented
48C2×#95×C83C2211yescharacteristic
49C2×#95×C83C2212yes—
50C24×D83C2311yescharacteristic
51C23×Dic83C2312yes—
52C23×Dic83C2311yescharacteristic
53C23×Dic83C2311yescharacteristic
54C23×Dic83C2311yescharacteristic
55C23×Dic83C2312yes—
56C22×C4×D83—24no—
57C22×C4×D83—28no—
58C22×C4×D83—24no—
59C22×C4×D83—28no—
60C22×C4×D83—28no—
61C22×C4×D83—28no—
62C22×C4×D83—28no—
63C22×C4×D83—24no—
64C22×C4×D83—28no—
65C22×C4×D83—28no—
66C22×C4×D83D418yes—
67C22×C4×D83D418yes—
68C22×C4×D83Q8116yes—
69C22×C4×D83C2318yes—
70C22×C4×D83C2314yes—
71C22×C4×D83C2314yes—
72C22×C4×D83C2314yes—
73C22×C4×D83C2318yes—
74C2×C4⋊Dic83—24no—
75C2×C4⋊Dic83—28no—
76C2×C4⋊Dic83—24no—
77C2×C4⋊Dic83C2314yes—
78C2×C4⋊Dic83C2318yes—
79C2×C4⋊Dic83C2314yes—
80C2×Dic83⋊C4—24no—
81C2×Dic83⋊C4—28no—
82C2×Dic83⋊C4—28no—
83C2×Dic83⋊C4—28no—
84C2×Dic83⋊C4—24no—
85C2×Dic83⋊C4C2318yes—
86C2×Dic83⋊C4C2314yes—
87C2×Dic83⋊C4C2318yes—
88C2×Dic83⋊C4C2318yes—
89C2×Dic83⋊C4C2314yes—
90C4⋊C4×D83—264no—
91C4⋊C4×D83—232no—
92C4⋊C4×D83—232no—
93C83⋊#95C2314yes—
94C83⋊#95C2318yes—
95C83⋊#95C2318yes—
96C83⋊#95C2318yes—
97C83⋊#95C2318yes—
98C23×C4×C83C2312yes—
99C23×C4×C83C2311yescharacteristic
100C23×C4×C83C2311yescharacteristic
101C23×C4×C83C2311yescharacteristic
102C23×C4×C83C2312yes—
103C2×C4⋊C4×C83—28no—
104C2×C4⋊C4×C83—24no—
105C2×C4⋊C4×C83—24no—
106C2×C4⋊C4×C83C2314yes—
107C2×C4⋊C4×C83C2314yes—
108C2×C4⋊C4×C83C2318yes—
109#95×C83C2314yes—
110#95×C83C2318yes—
111C23×D83C2×D414yes—
112C23×D83C2×D414yes—
113C23×D83C2×Q818yes—
114C23×D83#4114yes—
115C23×D83#4118yes—
116C23×D83C2412yes—
117C22×Dic83—22no—
118C22×Dic83—24no—
119C22×Dic83—22no—
120C22×Dic83—24no—
121C22×Dic83—24no—
122C22×Dic83—24no—
123C22×Dic83—24no—
124C22×Dic83—22no—
125C22×Dic83—24no—
126C22×Dic83—24no—
127C22×Dic83C2×D414yes—
128C22×Dic83C2×D414yes—
129C22×Dic83C2×Q818yes—
130C22×Dic83C2412yes—
131C22×Dic83C2414yes—
132C22×Dic83C2412yes—
133C22×Dic83C2412yes—
134C22×Dic83C2414yes—
135C2×C4×D83—232no—
136C2×C4×D83—264no—
137C2×C4×D83—232no—
138C2×C4×D83—28no—
139C2×C4×D83—216no—
140C2×C4×D83—28no—
141C2×C4×D83—216no—
142C2×C4×D83—216no—
143C2×C4×D83—216no—
144C2×C4×D83—216no—
145C2×C4×D83—28no—
146C2×C4×D83—216no—
147C2×C4×D83—216no—
148C2×C4×D83—432no—
149C2×C4×D83—416no—
150C2×C4×D83C2×D4116yes—
151C2×C4×D83C2×D4116yes—
152C2×C4×D83C2×Q8132yes—
153C4⋊Dic83—28no—
154C4⋊Dic83—28no—
155C4⋊Dic83—216no—
156Dic83⋊C4—216no—
157Dic83⋊C4—28no—
158Dic83⋊C4—28no—
159Dic83⋊C4—216no—
160Dic83⋊C4—216no—
161C24×C83C2411yescharacteristic
162C22×C4×C83—22no—
163C22×C4×C83—24no—
164C22×C4×C83—22no—
165C22×C4×C83—24no—
166C22×C4×C83—24no—
167C22×C4×C83—24no—
168C22×C4×C83—24no—
169C22×C4×C83—22no—
170C22×C4×C83—24no—
171C22×C4×C83—24no—
172C22×C4×C83C2×D414yes—
173C22×C4×C83C2×D414yes—
174C22×C4×C83C2×Q818yes—
175C22×C4×C83C2412yes—
176C22×C4×C83C2414yes—
177C22×C4×C83C2412yes—
178C22×C4×C83C2412yes—
179C22×C4×C83C2414yes—
180C4⋊C4×C83—216no—
181C4⋊C4×C83—28no—
182C4⋊C4×C83—28no—
183C23×C83C22×D412yes—
184C23×C83C22×D412yes—
185C23×C83C22×Q814yes—
186C23×C83C2×#4112yes—
187C23×C83C2×#4114yes—
188C23×C83C2511yescharacteristic
189C22×D83#121116yes—
190C22×D83C22⋊Q8132yes—
191C22×D83#123116yes—
192C22×D83#125132yes—
193C22×D83C4⋊Q8116yes—
194C22×D83C22×D414yes—
195C22×D83C22×D414yes—
196C22×D83C22×Q818yes—
197C22×D83C2×#4114yes—
198C22×D83C2×#4118yes—
199C2×C4×C83—28no—
200C2×C4×C83—216no—
201C2×C4×C83—28no—
202C2×C4×C83—22no—
203C2×C4×C83—24no—
204C2×C4×C83—22no—
205C2×C4×C83—24no—
206C2×C4×C83—24no—
207C2×C4×C83—24no—
208C2×C4×C83—24no—
209C2×C4×C83—22no—
210C2×C4×C83—24no—
211C2×C4×C83—24no—
212C2×C4×C83—48no—
213C2×C4×C83—44no—
214C2×C4×C83C22×D414yes—
215C2×C4×C83C22×D414yes—
216C2×C4×C83C22×Q818yes—
217C2×Dic83—28no—
218C2×Dic83—216no—
219C2×Dic83—28no—
220C2×Dic83—22no—
221C2×Dic83—24no—
222C2×Dic83—22no—
223C2×Dic83—24no—
224C2×Dic83—24no—
225C2×Dic83—24no—
226C2×Dic83—24no—
227C2×Dic83—22no—
228C2×Dic83—24no—
229C2×Dic83—24no—
230C2×Dic83—48no—
231C2×Dic83—44no—
232C2×Dic83C22×D414yes—
233C2×Dic83C22×D414yes—
234C2×Dic83C22×Q818yes—
235C4×D83—232no—
236C4×D83—264no—
237C4×D83—232no—
238C4×D83—432no—
239C4×D83—416no—
240C22×C83C2×#12114yes—
241C22×C83C2×C22⋊Q818yes—
242C22×C83C2×#12314yes—
243C22×C83C2×#12518yes—
244C22×C83C2×C4⋊Q814yes—
245C22×C83C23×D411yescharacteristic
246C22×C83C23×D411yescharacteristic
247C22×C83C23×Q812yes—
248C22×C83C22×#4111yescharacteristic
249C22×C83C22×#4112yes—
250C2×D83#398164yesdirect product factor
251C2×D83C2×#12118yes—
252C2×D83C2×C22⋊Q8116yes—
253C2×D83C2×#12318yes—
254C2×D83C2×#125116yes—
255C2×D83C2×C4⋊Q818yes—
256C4×C83—24no—
257C4×C83—28no—
258C4×C83—24no—
259C4×C83—44no—
260C4×C83—42no—
261Dic83—24no—
262Dic83—28no—
263Dic83—24no—
264Dic83—44no—
265Dic83—42no—
266C22×#398—831nomaximal, Sylow
267C2×C83C2×#39818yescomplemented
268C2×C83C22×#12111yescharacteristic
269C2×C83C22×C22⋊Q812yes—
270C2×C83C22×#12311yescharacteristic
271C2×C83C22×#12512yes—
272C2×C83C22×C4⋊Q811yescharacteristic
273D83C2×#398116yesdirect product factor
274C23×C4⋊C4—831no—
275C23×C4⋊C4—832no—
276C23×C4⋊C4—831no—
277C2×#398—8316no—
278C2×#398—831no—
279C2×#398—832no—
280C2×#398—831no—
281C2×#398—831no—
282C2×#398—832no—
283C2×#398D8311yescharacteristic, direct product factor
284C22×#95—831no—
285C22×#95—832no—
286C83C22×#39811yescharacteristic, complemented, Sylow
287C24×C4—831no—
288C24×C4—831no—
289C24×C4—832no—
290C24×C4—832no—
291C24×C4—831no—
292C22×C4⋊C4—831no—
293C22×C4⋊C4—832no—
294C22×C4⋊C4—832no—
295C22×C4⋊C4—831no—
296C22×C4⋊C4—832no—
297C22×C4⋊C4—832no—
298C22×C4⋊C4—831no—
299C22×C4⋊C4—831no—
300C22×C4⋊C4—838no—
301C22×C4⋊C4—8316no—
302C22×C4⋊C4—838no—
303C22×C4⋊C4—1668no—
304C22×C4⋊C4—16616no—
305C22×C4⋊C4—1668no—
306C22×C4⋊C4C2×D8311yescharacteristic
307C22×C4⋊C4C2×D8312yes—
308C22×C4⋊C4C2×D8311yescharacteristic
309#398—838no—
310#398—838no—
311#398—8316no—
312#398—838no—
313#398—8316no—
314#398C2×D8318yesdirect product factor
315C2×#95—832no—
316C2×#95—831no—
317C2×#95—832no—
318C2×#95—832no—
319C2×#95—832no—
320C2×#95—838no—
321C2×#95—8316no—
322C2×#95C2×D8312yes—
323C2×#95C2×D8311yescharacteristic
324C25—831no—
325C23×C4—831no—
326C23×C4—831no—
327C23×C4—832no—
328C23×C4—838no—
329C23×C4—838no—
330C23×C4—8316no—
331C23×C4—834no—
332C23×C4—834no—
333C23×C4—838no—
334C23×C4—832no—
335C23×C4—831no—
336C23×C4—838no—
337C23×C4—834no—
338C23×C4—1668no—
339C23×C4—1668no—
340C23×C4—1664no—
341C23×C4—1668no—
342C23×C4—1668no—
343C23×C4—1664no—
344C23×C4—1668no—
345C23×C4—1664no—
346C23×C4—1668no—
347C23×C4—1668no—
348C23×C4C22×D8312yes—
349C23×C4C22×D8311yescharacteristic
350C23×C4C22×D8311yescharacteristic
351C23×C4C22×D8311yescharacteristic
352C23×C4C22×D8312yes—
353C2×C4⋊C4—24no—
354C2×C4⋊C4—28no—
355C2×C4⋊C4—24no—
356C2×C4⋊C4—838no—
357C2×C4⋊C4—834no—
358C2×C4⋊C4—838no—
359C2×C4⋊C4—838no—
360C2×C4⋊C4—834no—
361C2×C4⋊C4—834no—
362C2×C4⋊C4—838no—
363C2×C4⋊C4—834no—
364C2×C4⋊C4—16632no—
365C2×C4⋊C4—16664no—
366C2×C4⋊C4—16632no—
367C2×C4⋊C4—1668no—
368C2×C4⋊C4—1668no—
369C2×C4⋊C4—1664no—
370C2×C4⋊C4—1664no—
371C2×C4⋊C4—1668no—
372C2×C4⋊C4—1668no—
373C2×C4⋊C4—1664no—
374C2×C4⋊C4—1664no—
375C2×C4⋊C4C22×D8314yes—
376C2×C4⋊C4C22×D8314yes—
377C2×C4⋊C4C22×D8318yes—
378#95—834no—
379#95—838no—
380#95—838no—
381#95—838no—
382#95—838no—
383#95C22×D8318yes—
384#95C22×D8314yes—
385C24—834no—
386C24—834no—
387C24—838no—
388C24—834no—
389C24—838no—
390C24—832no—
391C24C23×D8311yescharacteristic
392C22×C4—22no—
393C22×C4—24no—
394C22×C4—22no—
395C22×C4—24no—
396C22×C4—24no—
397C22×C4—24no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—24no—
402C22×C4—33232no—
403C22×C4—33216no—
404C22×C4—834no—
405C22×C4—834no—
406C22×C4—838no—
407C22×C4—834no—
408C22×C4—832no—
409C22×C4—832no—
410C22×C4—8316no—
411C22×C4—8316no—
412C22×C4—8332no—
413C22×C4—832no—
414C22×C4—834no—
415C22×C4—16616no—
416C22×C4—16616no—
417C22×C4—1668no—
418C22×C4—16616no—
419C22×C4—16616no—
420C22×C4—1668no—
421C22×C4—16616no—
422C22×C4—16632no—
423C22×C4—16664no—
424C22×C4—16632no—
425C22×C4—1668no—
426C22×C4—16616no—
427C22×C4—16616no—
428C22×C4—1664no—
429C22×C4—1664no—
430C22×C4—1662no—
431C22×C4—1664no—
432C22×C4—1664no—
433C22×C4—1662no—
434C22×C4—1664no—
435C22×C4—1662no—
436C22×C4—1664no—
437C22×C4—1664no—
438C22×C4D4×D8314yes—
439C22×C4D4×D8314yes—
440C22×C4Q8×D8318yes—
441C22×C4C23×D8312yes—
442C22×C4C23×D8314yes—
443C22×C4C23×D8312yes—
444C22×C4C23×D8312yes—
445C22×C4C23×D8314yes—
446C4⋊C4—28no—
447C4⋊C4—216no—
448C4⋊C4—28no—
449C4⋊C4—16616no—
450C4⋊C4—16616no—
451C4⋊C4—1668no—
452C4⋊C4—1668no—
453C4⋊C4—16616no—
454C4⋊C4—16616no—
455C4⋊C4—1668no—
456C4⋊C4—1668no—
457C23—8316no—
458C23—8332no—
459C23—8316no—
460C23—8332no—
461C23—8316no—
462C23—834no—
463C23—834no—
464C23—838no—
465C23—834no—
466C23—838no—
467C23C2×D4×D8312yes—
468C23C2×D4×D8312yes—
469C23C2×Q8×D8314yes—
470C23#41×D8312yes—
471C23#41×D8314yes—
472C23C24×D8311yescharacteristic
473C2×C4—28no—
474C2×C4—216no—
475C2×C4—28no—
476C2×C4—22no—
477C2×C4—24no—
478C2×C4—22no—
479C2×C4—24no—
480C2×C4—24no—
481C2×C4—24no—
482C2×C4—24no—
483C2×C4—22no—
484C2×C4—24no—
485C2×C4—24no—
486C2×C4—48no—
487C2×C4—44no—
488C2×C4—33232no—
489C2×C4—33216no—
490C2×C4—3328no—
491C2×C4—3324no—
492C2×C4—834no—
493C2×C4—834no—
494C2×C4—838no—
495C2×C4—16632no—
496C2×C4—16664no—
497C2×C4—16632no—
498C2×C4—1664no—
499C2×C4—1664no—
500C2×C4—1662no—
501C2×C4—1664no—
502C2×C4—1664no—
503C2×C4—1668no—
504C2×C4—16616no—
505C2×C4—1668no—
506C2×C4—1662no—
507C2×C4—1664no—
508C2×C4—1662no—
509C2×C4—1664no—
510C2×C4—1664no—
511C2×C4C2×D4×D8314yes—
512C2×C4C2×D4×D8314yes—
513C2×C4C2×Q8×D8318yes—
514C22—8364no—
515C22—838no—
516C22—8316no—
517C22—838no—
518C22—8316no—
519C22—838no—
520C22#121×D8314yes—
521C22C22⋊Q8×D8318yes—
522C22#123×D8314yes—
523C22#125×D8318yes—
524C22C4⋊Q8×D8314yes—
525C22C22×D4×D8311yescharacteristic
526C22C22×D4×D8311yescharacteristic
527C22C22×Q8×D8312yes—
528C22C2×#41×D8311yescharacteristic
529C22C2×#41×D8312yes—
530C4—24no—
531C4—28no—
532C4—24no—
533C4—44no—
534C4—42no—
535C4—3324no—
536C4—3322no—
537C4—1664no—
538C4—1668no—
539C4—1664no—
540C2—8316no—
541C2#398×D8318yesdirect product factor
542C2C2×#121×D8311yescharacteristic
543C2C2×C22⋊Q8×D8312yes—
544C2C2×#123×D8311yescharacteristic
545C2C2×#125×D8312yes—
546C2C2×C4⋊Q8×D8311yescharacteristic
subgroupG/H
centerC24C23×D83
commutator, cosocleC23×C83C25
FrattiniC23C24×D83
FittingC2×#398×C83C2
socleC24×C83C24
prime radicalC24×D83C23
hypercenterC2×#398D83
hypocenterC83C22×#398
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C24 ◁ C2×#398

Complements

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

NH ≅ G/Ntype
4. C2×#19127635C2semidirect
5. C2×#19127634C2semidirect
6. C2×#19127633C2semidirect
7. C2×#19127632C2semidirect
8. C2×#19127631C2semidirect
9. #398×D83C2direct (N × H)
12. C2×#398×C83C2semidirect
32. #19127635C22semidirect
33. #19127634C22semidirect
34. #19127633C22semidirect
35. #19127632C22semidirect
36. #19127631C22semidirect
47. #398×C83C22semidirect
250. C2×D83#398direct (N × H)
267. C2×C83C2×#398semidirect
273. D83C2×#398direct (N × H)
283. C2×#398D83direct (N × H)
286. C83C22×#398semidirect
314. #398C2×D83direct (N × H)
541. C2#398×D83direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊C4×D832
C2×#191276352
C2×#191276342
C2×#191276332
C2×#191276322
C2×#191276312
#398×D832
C2×#95×D832
C2×#398×C832
C22×#39883