Finite Group Databasebeta

← #2646×D83 #35570845

Subgroups — #2646×D83 #35570845

Order 21248.

2374 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24913436C212yesmaximal, complemented
2#24913435C214yesmaximal, complemented
3#24913434C214yesmaximal, complemented
4#24913433C211yesmaximal, characteristic, complemented
5#24913432C214yesmaximal, complemented
6#400×D83C214yesmaximal, complemented
7#398×D83C214yesmaximal, complemented
8#385×D83C214yesmaximal
9#380×D83C212yesmaximal
10C2×#95×D83C211yesmaximal, characteristic
11C83×#2646C211yesmaximal, characteristic, complemented
12C23×C4×D83C2212yes—
13C23×C4×D83C2211yescharacteristic
14C2×C4⋊C4×D83C2212yes—
15C2×C4⋊C4×D83C2212yes—
16C2×C4⋊C4×D83C2214yes—
17C2×C4⋊C4×D83C2214yes—
18C2×C22⋊C4×D83C2214yes—
19#19127643C2214yescomplemented
20#19127643C2214yescomplemented
21#19127642C2214yescomplemented
22#19127642C2214yescomplemented
23#19127641C2214yescomplemented
24#19127640C2214yescomplemented
25#19127639C2214yescomplemented
26#19127635C2214yescomplemented
27#19127635C2214yescomplemented
28#19127634C2214yescomplemented
29#19127634C2214yescomplemented
30#19127633C2214yescomplemented
31#19127632C2214yescomplemented
32#19127631C2214yescomplemented
33#19127576C2214yes—
34#19127575C2214yes—
35#19127575C2214yes—
36#19127574C2214yes—
37#19127573C2214yes—
38#19127572C2214yes—
39#19127572C2214yes—
40#19127561C2212yes—
41#19127560C2214yes—
42#19127559C2214yes—
43#19127558C2212yes—
44#19127558C2212yes—
45C2×C83⋊#95C2212yes—
46C2×C83⋊#95C2211yescharacteristic
47#95×D83C2214yes—
48#95×D83C2214yes—
49#95×D83C2214yes—
50#95×D83C2214yes—
51#95×D83C2214yes—
52#400×C83C2214yescomplemented
53#398×C83C2214yescomplemented
54#385×C83C2214yes—
55#380×C83C2212yes—
56C2×#95×C83C2211yescharacteristic
57C24×D83C2311yescharacteristic
58C23×Dic83C2312yes—
59C23×Dic83C2311yescharacteristic
60C22×C4×D83—28no—
61C22×C4×D83—44no—
62C22×C4×D83—44no—
63C22×C4×D83—44no—
64C22×C4×D83Q818yes—
65C22×C4×D83C2314yes—
66C22×C4×D83C2314yes—
67C22×C4×D83C2314yes—
68C22×C4×D83C2312yes—
69C22×C4×D83C2314yes—
70C22×C4×D83C2312yes—
71C2×C4⋊Dic83C2312yes—
72C2×C4⋊Dic83C2312yes—
73C2×C4⋊Dic83C2314yes—
74C2×C4⋊Dic83C2314yes—
75C2×Dic83⋊C4C2314yes—
76C2×Dic83⋊C4C2314yes—
77C2×Dic83⋊C4C2314yes—
78C2×Dic83⋊C4C2314yes—
79C2×Dic83⋊C4C2314yes—
80C2×Dic83⋊C4C2314yes—
81C4⋊C4×D83—44no—
82C4⋊C4×D83—44no—
83C4⋊C4×D83—48no—
84C4⋊C4×D83—48no—
85C2×C22⋊Dic83C2314yes—
86C2×(C2×D83)⋊C4C2314yes—
87C2×(C2×D83)⋊C4C2314yes—
88C22⋊C4×D83—216no—
89C83⋊#95C2314yes—
90C83⋊#95C2314yes—
91C83⋊#95C2314yes—
92C83⋊#95C2314yes—
93C83⋊#95C2314yes—
94C83⋊#95C2314yes—
95C83⋊#95C2314yes—
96C83⋊#95C2314yes—
97C83⋊#95C2314yes—
98C83⋊#95C2314yes—
99C83⋊#95C2314yes—
100C83⋊#95C2314yes—
101C23×C4×C83C2312yes—
102C23×C4×C83C2311yescharacteristic
103C2×C4⋊C4×C83C2312yes—
104C2×C4⋊C4×C83C2312yes—
105C2×C4⋊C4×C83C2314yes—
106C2×C4⋊C4×C83C2314yes—
107C2×C22⋊C4×C83C2314yes—
108#95×C83C2314yes—
109#95×C83C2314yes—
110#95×C83C2314yes—
111#95×C83C2314yes—
112C23×D83—22no—
113C23×D83—24no—
114C23×D83—22no—
115C23×D83C2×Q814yes—
116C23×D83#4118yes—
117C23×D83C2411yescharacteristic
118C23×D83C2411yescharacteristic
119C22×Dic83—24no—
120C22×Dic83—42no—
121C22×Dic83—42no—
122C22×Dic83—42no—
123C22×Dic83C2×Q814yes—
124C22×Dic83C2414yes—
125C22×Dic83C2414yes—
126C22×Dic83C2414yes—
127C22×Dic83C2412yes—
128C2×C4×D83—24no—
129C2×C4×D83—24no—
130C2×C4×D83—28no—
131C2×C4×D83—28no—
132C2×C4×D83—28no—
133C2×C4×D83—28no—
134C2×C4×D83—28no—
135C2×C4×D83—28no—
136C2×C4×D83—28no—
137C2×C4×D83—28no—
138C2×C4×D83—28no—
139C2×C4×D83—28no—
140C2×C4×D83—416no—
141C2×C4×D83—42no—
142C2×C4×D83—42no—
143C2×C4×D83—44no—
144C2×C4×D83—44no—
145C2×C4×D83—432no—
146C2×C4×D83C2×Q818yes—
147C2×C4×D83C2×Q818yes—
148C4⋊Dic83—42no—
149C4⋊Dic83—44no—
150C4⋊Dic83—42no—
151C4⋊Dic83—44no—
152Dic83⋊C4—44no—
153Dic83⋊C4—44no—
154Dic83⋊C4—44no—
155Dic83⋊C4—44no—
156Dic83⋊C4—44no—
157Dic83⋊C4—44no—
158C22⋊Dic83—28no—
159(C2×D83)⋊C4—28no—
160(C2×D83)⋊C4—28no—
161C24×C83C2411yescharacteristic
162C22×C4×C83—24no—
163C22×C4×C83—42no—
164C22×C4×C83—42no—
165C22×C4×C83—42no—
166C22×C4×C83C2×Q814yes—
167C22×C4×C83C2414yes—
168C22×C4×C83C2414yes—
169C22×C4×C83C2414yes—
170C22×C4×C83C2412yes—
171C4⋊C4×C83—42no—
172C4⋊C4×C83—42no—
173C4⋊C4×C83—44no—
174C4⋊C4×C83—44no—
175C22⋊C4×C83—28no—
176C23×C83—21no—
177C23×C83—22no—
178C23×C83—21no—
179C23×C83C22×Q812yes—
180C23×C83C2×#4114yes—
181C23×C83C2511yescharacteristic
182C22×D83—21no—
183C22×D83—22no—
184C22×D83—21no—
185C22×D83—216no—
186C22×D83—216no—
187C22×D83—28no—
188C22×D83—28no—
189C22×D83#125116yes—
190C22×D83C22×Q812yes—
191C22×D83C22×Q812yes—
192C22×D83C2×#4114yes—
193C22×D83C2×#4114yes—
194C22×D83H(5,𝔽2)12yes—
195C22×D83H(5,𝔽2)14yes—
196C22×D832−1+412yes—
197C2×C4×C83—22no—
198C2×C4×C83—22no—
199C2×C4×C83—24no—
200C2×C4×C83—24no—
201C2×C4×C83—24no—
202C2×C4×C83—24no—
203C2×C4×C83—24no—
204C2×C4×C83—24no—
205C2×C4×C83—24no—
206C2×C4×C83—24no—
207C2×C4×C83—24no—
208C2×C4×C83—44no—
209C2×C4×C83—48no—
210C2×C4×C83C22×Q814yes—
211C2×Dic83—22no—
212C2×Dic83—22no—
213C2×Dic83—24no—
214C2×Dic83—24no—
215C2×Dic83—24no—
216C2×Dic83—24no—
217C2×Dic83—24no—
218C2×Dic83—24no—
219C2×Dic83—24no—
220C2×Dic83—24no—
221C2×Dic83—24no—
222C2×Dic83—44no—
223C2×Dic83—48no—
224C2×Dic83C22×Q814yes—
225C4×D83—216no—
226C4×D83—48no—
227C4×D83—416no—
228C4×D83—416no—
229C4×D83—416no—
230C4×D83—416no—
231C22×C83—24no—
232C22×C83—24no—
233C22×C83—22no—
234C22×C83—22no—
235C22×C83C2×#12514yes—
236C22×C83C23×Q811yescharacteristic
237C22×C83C22×#4112yes—
238C22×C83C2×H(5,𝔽2)11yescharacteristic
239C22×C83C2×H(5,𝔽2)12yes—
240C22×C83C2×2−1+411yescharacteristic
241C2×D83—24no—
242C2×D83—24no—
243C2×D83—22no—
244C2×D83—22no—
245C2×D83—232no—
246C2×D83C2×#12514yes—
247C2×D83C2×#12514yes—
248C2×D83#53818yes—
249C2×D83#53918yes—
250C2×D83#54314yes—
251C2×D83#54414yes—
252C4×C83—24no—
253C4×C83—44no—
254C4×C83—44no—
255C4×C83—44no—
256Dic83—24no—
257Dic83—44no—
258Dic83—44no—
259Dic83—44no—
260C2×#2646—831nomaximal, Sylow
261C2×C83—24no—
262C2×C83C22×#12511yescharacteristic
263C2×C83C2×#53812yes—
264C2×C83C2×#53912yes—
265C2×C83C2×#54311yescharacteristic
266C2×C83C2×#54411yescharacteristic
267D83—24no—
268D83#264618yesdirect product factor
269#2646—834no—
270#2646—834no—
271#2646—834no—
272#2646—832no—
273#2646—831no—
274#2646D8311yescharacteristic, direct product factor
275C2×#400—834no—
276C2×#398—834no—
277C2×#385—834no—
278C2×#380—832no—
279C22×#95—831no—
280C83C2×#264611yescharacteristic, complemented, Sylow
281C24×C4—831no—
282C24×C4—832no—
283C22×C4⋊C4—832no—
284C22×C4⋊C4—834no—
285C22×C4⋊C4—834no—
286C22×C4⋊C4—832no—
287C22×C22⋊C4—834no—
288#400—834no—
289#400—834no—
290#400—834no—
291#400—834no—
292#400—834no—
293#400—834no—
294#400—834no—
295#400C2×D8314yescomplemented
296#398—834no—
297#398—834no—
298#398—834no—
299#398—834no—
300#398—834no—
301#398—834no—
302#398—834no—
303#398C2×D8314yescomplemented
304#385—834no—
305#385—834no—
306#385—834no—
307#385—834no—
308#385—834no—
309#385—834no—
310#385—834no—
311#385C2×D8314yes—
312#380—832no—
313#380—832no—
314#380—832no—
315#380—834no—
316#380—834no—
317#380C2×D8312yes—
318C2×#95—832no—
319C2×#95—831no—
320C2×#95—834no—
321C2×#95—834no—
322C2×#95—834no—
323C2×#95—834no—
324C2×#95—834no—
325C2×#95C2×D8311yescharacteristic
326C25—831no—
327C23×C4—3324no—
328C23×C4—3324no—
329C23×C4—3324no—
330C23×C4—831no—
331C23×C4—838no—
332C23×C4—832no—
333C23×C4—832no—
334C23×C4—834no—
335C23×C4—832no—
336C23×C4—834no—
337C23×C4—834no—
338C23×C4—834no—
339C23×C4—1668no—
340C23×C4C22×D8312yes—
341C23×C4C22×D8311yescharacteristic
342C2×C4⋊C4—3328no—
343C2×C4⋊C4—3328no—
344C2×C4⋊C4—3324no—
345C2×C4⋊C4—3324no—
346C2×C4⋊C4—834no—
347C2×C4⋊C4—832no—
348C2×C4⋊C4—834no—
349C2×C4⋊C4—832no—
350C2×C4⋊C4—834no—
351C2×C4⋊C4—834no—
352C2×C4⋊C4—834no—
353C2×C4⋊C4—834no—
354C2×C4⋊C4—834no—
355C2×C4⋊C4—834no—
356C2×C4⋊C4C22×D8312yes—
357C2×C4⋊C4C22×D8312yes—
358C2×C4⋊C4C22×D8314yes—
359C2×C4⋊C4C22×D8314yes—
360C2×C22⋊C4—834no—
361C2×C22⋊C4—834no—
362C2×C22⋊C4—834no—
363C2×C22⋊C4—16616no—
364C2×C22⋊C4C22×D8314yes—
365#95—834no—
366#95—834no—
367#95—834no—
368#95—834no—
369#95—834no—
370#95—834no—
371#95—834no—
372#95—834no—
373#95—834no—
374#95—834no—
375#95—834no—
376#95—834no—
377#95C22×D8314yes—
378#95C22×D8314yes—
379#95C22×D8314yes—
380#95C22×D8314yes—
381C24—831no—
382C24—834no—
383C24—838no—
384C24—831no—
385C24—1664no—
386C24—1662no—
387C24—1662no—
388C24C23×D8311yescharacteristic
389C22×C4—24no—
390C22×C4—42no—
391C22×C4—42no—
392C22×C4—42no—
393C22×C4—3322no—
394C22×C4—3322no—
395C22×C4—3324no—
396C22×C4—3324no—
397C22×C4—33232no—
398C22×C4—33216no—
399C22×C4—3322no—
400C22×C4—3322no—
401C22×C4—3322no—
402C22×C4—834no—
403C22×C4—838no—
404C22×C4—832no—
405C22×C4—838no—
406C22×C4—834no—
407C22×C4—834no—
408C22×C4—834no—
409C22×C4—1668no—
410C22×C4—1668no—
411C22×C4—1668no—
412C22×C4—1668no—
413C22×C4—1668no—
414C22×C4—1668no—
415C22×C4—1668no—
416C22×C4—1668no—
417C22×C4—1668no—
418C22×C4—1668no—
419C22×C4—1664no—
420C22×C4—1664no—
421C22×C4—1664no—
422C22×C4Q8×D8314yes—
423C22×C4C23×D8314yes—
424C22×C4C23×D8314yes—
425C22×C4C23×D8314yes—
426C22×C4C23×D8312yes—
427C4⋊C4—42no—
428C4⋊C4—42no—
429C4⋊C4—44no—
430C4⋊C4—44no—
431C4⋊C4—3324no—
432C4⋊C4—3324no—
433C4⋊C4—3324no—
434C4⋊C4—3324no—
435C4⋊C4—3324no—
436C4⋊C4—3324no—
437C4⋊C4—3322no—
438C4⋊C4—3322no—
439C4⋊C4—3324no—
440C4⋊C4—3324no—
441C22⋊C4—28no—
442C22⋊C4—1668no—
443C22⋊C4—1668no—
444C22⋊C4—1668no—
445C23—22no—
446C23—21no—
447C23—21no—
448C23—832no—
449C23—834no—
450C23—834no—
451C23—832no—
452C23—832no—
453C23—8316no—
454C23—832no—
455C23—834no—
456C23—1668no—
457C23—16616no—
458C23—1668no—
459C23—16616no—
460C23—1662no—
461C23—1661no—
462C23—1661no—
463C23C2×Q8×D8312yes—
464C23#41×D8314yes—
465C23C24×D8311yescharacteristic
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—24no—
470C2×C4—24no—
471C2×C4—24no—
472C2×C4—24no—
473C2×C4—24no—
474C2×C4—24no—
475C2×C4—24no—
476C2×C4—24no—
477C2×C4—44no—
478C2×C4—48no—
479C2×C4—3328no—
480C2×C4—33216no—
481C2×C4—33216no—
482C2×C4—33216no—
483C2×C4—33216no—
484C2×C4—3328no—
485C2×C4—3324no—
486C2×C4—834no—
487C2×C4—16616no—
488C2×C4—1664no—
489C2×C4—1664no—
490C2×C4—1664no—
491C2×C4—1664no—
492C2×C4—1664no—
493C2×C4—1664no—
494C2×C4—1664no—
495C2×C4—1664no—
496C2×C4—1664no—
497C2×C4—1662no—
498C2×C4—1662no—
499C2×C4C2×Q8×D8314yes—
500C22—22no—
501C22—22no—
502C22—24no—
503C22—24no—
504C22—834no—
505C22—834no—
506C22—834no—
507C22—838no—
508C22—838no—
509C22—834no—
510C22—16632no—
511C22—1662no—
512C22—1662no—
513C22—1664no—
514C22—1664no—
515C22#125×D8314yes—
516C22C22×Q8×D8311yescharacteristic
517C22C2×#41×D8312yes—
518C22H(5,𝔽2)×D8312yes—
519C22H(5,𝔽2)×D8311yescharacteristic
520C222−1+4×D8311yescharacteristic
521C4—24no—
522C4—44no—
523C4—44no—
524C4—44no—
525C4—3324no—
526C4—3324no—
527C4—3324no—
528C4—1664no—
529C2—24no—
530C2—838no—
531C2—1664no—
532C2C2×#125×D8311yescharacteristic
533C2#538×D8312yes—
534C2#539×D8312yes—
535C2#543×D8311yescharacteristic
536C2#544×D8311yescharacteristic
subgroupG/H
center, FrattiniC23C24×D83
commutator, socle, cosocleC23×C83C25
FittingC83×#2646C2
prime radicalC24×D83C23
hypercenter#2646D83
hypocenterC83C2×#2646
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #2646

Complements

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

NH ≅ G/Ntype
1. #24913436C2semidirect
2. #24913435C2semidirect
3. #24913434C2semidirect
4. #24913433C2semidirect
5. #24913432C2semidirect
6. #400×D83C2semidirect
7. #398×D83C2semidirect
11. C83×#2646C2semidirect
19. #19127643C22semidirect
20. #19127643C22semidirect
21. #19127642C22semidirect
22. #19127642C22semidirect
23. #19127641C22semidirect
24. #19127640C22semidirect
25. #19127639C22semidirect
26. #19127635C22semidirect
27. #19127635C22semidirect
28. #19127634C22semidirect
29. #19127634C22semidirect
30. #19127633C22semidirect
31. #19127632C22semidirect
32. #19127631C22semidirect
52. #400×C83C22semidirect
53. #398×C83C22semidirect
268. D83#2646direct (N × H)
274. #2646D83direct (N × H)
280. C83C2×#2646semidirect
295. #400C2×D83semidirect
303. #398C2×D83semidirect

Maximal subgroups

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

subgroupindex
#249134362
#249134352
#249134342
#249134332
#249134322
#400×D832
#398×D832
#385×D832
#380×D832
C2×#95×D832
C83×#26462
C2×#264683