Finite Group Databasebeta

← #2084×D83 #35433376

Subgroups — #2084×D83 #35433376

Order 21248.

1086 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#106×D83C212yesmaximal
2#389×D83C211yesmaximal, characteristic
3#384×D83C211yesmaximal, characteristic
4#24908943C212yesmaximal, complemented
5#24908942C212yesmaximal, complemented
6#24908941C211yesmaximal, characteristic, complemented
7#24908940C211yesmaximal, characteristic, complemented
8#24908939C211yesmaximal, characteristic, complemented
9#340×D83C212yesmaximal
10#336×D83C211yesmaximal, characteristic
11C83×#2084C211yesmaximal, characteristic, complemented
12C22×C8×D83C2212yes—
13C2×C4⋊C4×D83—21no—
14C2×C4⋊C4×D83—21no—
15C2×C4⋊C4×D83—21no—
16C2×C4⋊C4×D83C2212yes—
17C2×C4⋊C4×D83C2211yescharacteristic
18C2×C4⋊C4×D83C2211yescharacteristic
19C2×C42×D83C2211yescharacteristic
20C2×#14581724C2212yes—
21C2×#14581723C2212yes—
22C2×#14581723C2212yes—
23#106×D83—28no—
24#19127600C2211yescharacteristic
25#19127599C2211yescharacteristic
26#19127598—21no—
27#19127597C2211yescharacteristic
28#19127596—21no—
29#19127571C2211yescharacteristic
30#19127570—21no—
31#19127570—21no—
32#19127570C2212yes—
33#95×D83—21no—
34#19127464C2212yes—
35#19127463C2212yes—
36#19127463C2212yes—
37#19127456C2211yescharacteristic
38#19127455C2212yes—
39C2×#106×C83C2212yes—
40#389×C83C2211yescharacteristic
41#384×C83C2211yescharacteristic
42#340×C83C2212yes—
43#336×C83C2211yescharacteristic
44C22×C4×D83—21no—
45C22×C4×D83—22no—
46C22×C4×D83—21no—
47C22×C4×D83D411yescharacteristic
48C22×C4×D83D411yescharacteristic
49C22×C4×D83C2311yescharacteristic
50C22×C83⋊C8C2312yes—
51C2×C8×D83—24no—
52C2×C8×D83—24no—
53C2×C8×D83Q818yes—
54C2×C4⋊Dic83—21no—
55C2×C4⋊Dic83—21no—
56C2×C4⋊Dic83—21no—
57C2×C4⋊Dic83C2312yes—
58C2×C4⋊Dic83C2311yescharacteristic
59C2×C4⋊Dic83C2311yescharacteristic
60C2×Dic83⋊C4—21no—
61C2×Dic83⋊C4—22no—
62C2×Dic83⋊C4—22no—
63C2×Dic83⋊C4—22no—
64C2×Dic83⋊C4—21no—
65C2×Dic83⋊C4—21no—
66C2×Dic83⋊C4—21no—
67C4⋊C4×D83—22no—
68C4⋊C4×D83—24no—
69C4⋊C4×D83—24no—
70C4⋊C4×D83—24no—
71C4⋊C4×D83—44no—
72C4⋊C4×D83—44no—
73C4⋊C4×D83—44no—
74C4⋊C4×D83D414yes—
75C2×C4×Dic83D411yescharacteristic
76C2×C4×Dic83D411yescharacteristic
77C2×C4×Dic83C2311yescharacteristic
78C42×D83—24no—
79#14581724—24no—
80#14581723—24no—
81#14581723—24no—
82C83⋊#95—21no—
83C83⋊#95—21no—
84C83⋊#95—21no—
85C22×C8×C83C2312yes—
86C2×C4⋊C4×C83—21no—
87C2×C4⋊C4×C83—21no—
88C2×C4⋊C4×C83—21no—
89C2×C4⋊C4×C83C2312yes—
90C2×C4⋊C4×C83C2311yescharacteristic
91C2×C4⋊C4×C83C2311yescharacteristic
92C2×C42×C83C2311yescharacteristic
93#106×C83—24no—
94#95×C83—21no—
95C23×D83C2×D411yescharacteristic
96C22×Dic83—21no—
97C22×Dic83—21no—
98C22×Dic83—22no—
99C22×Dic83C2×D411yescharacteristic
100C22×Dic83C2×D411yescharacteristic
101C22×Dic83C2×D411yescharacteristic
102C2×C4×D83—22no—
103C2×C4×D83—22no—
104C2×C4×D83—22no—
105C2×C4×D83—22no—
106C2×C4×D83—24no—
107C2×C4×D83—42no—
108C2×C4×D83—42no—
109C2×C4×D83—44no—
110C2×C4×D83—42no—
111C2×C4×D83—44no—
112C2×C4×D83—42no—
113C2×C4×D83—42no—
114C2×C4×D83—44no—
115C2×C4×D83SD1614yes—
116C2×C4×D83SD1614yes—
117C2×C4×D83C2×D412yes—
118C2×C4×D83C2×Q814yes—
119C2×C4×D83#4114yes—
120C2×C4×D83#4112yes—
121C2×C83⋊C8—22no—
122C2×C83⋊C8—22no—
123C2×C83⋊C8C2×Q814yes—
124C8×D83—216no—
125C4⋊Dic83—21no—
126C4⋊Dic83—22no—
127C4⋊Dic83—22no—
128C4⋊Dic83—22no—
129C4⋊Dic83—42no—
130C4⋊Dic83—42no—
131C4⋊Dic83—42no—
132C4⋊Dic83C2×D412yes—
133Dic83⋊C4—24no—
134Dic83⋊C4—44no—
135Dic83⋊C4—42no—
136Dic83⋊C4—44no—
137Dic83⋊C4—42no—
138Dic83⋊C4—42no—
139Dic83⋊C4—44no—
140C4×Dic83—22no—
141C4×Dic83—22no—
142C4×Dic83—22no—
143C22×C4×C83—21no—
144C22×C4×C83—21no—
145C22×C4×C83—22no—
146C22×C4×C83C2×D411yescharacteristic
147C22×C4×C83C2×D411yescharacteristic
148C22×C4×C83C2411yescharacteristic
149C2×C8×C83—22no—
150C2×C8×C83—22no—
151C2×C8×C83C2×Q814yes—
152C4⋊C4×C83—21no—
153C4⋊C4×C83—22no—
154C4⋊C4×C83—22no—
155C4⋊C4×C83—22no—
156C4⋊C4×C83—42no—
157C4⋊C4×C83—42no—
158C4⋊C4×C83—42no—
159C4⋊C4×C83C2×D412yes—
160C42×C83—22no—
161C23×C83C22×D411yescharacteristic
162C22×D83#12112yes—
163C22×D83#12312yes—
164C22×D83C4⋊Q812yes—
165C22×D83C2×SD1612yes—
166C22×D83C2×SD1612yes—
167C22×D83AGL(1,ℤ/8)12yes—
168C22×D83#13712yes—
169C2×C4×C83—21no—
170C2×C4×C83—21no—
171C2×C4×C83—21no—
172C2×C4×C83—21no—
173C2×C4×C83—22no—
174C2×C4×C83—41no—
175C2×C4×C83—41no—
176C2×C4×C83—42no—
177C2×C4×C83—41no—
178C2×C4×C83—41no—
179C2×C4×C83—42no—
180C2×C4×C83—42no—
181C2×C4×C83—41no—
182C2×C4×C83C2×SD1612yes—
183C2×C4×C83C2×SD1612yes—
184C2×C4×C83C22×D411yescharacteristic
185C2×C4×C83C22×Q812yes—
186C2×C4×C83C2×#4112yes—
187C2×C4×C83C2×#4111yescharacteristic
188C2×Dic83—21no—
189C2×Dic83—21no—
190C2×Dic83—21no—
191C2×Dic83—21no—
192C2×Dic83—22no—
193C2×Dic83—41no—
194C2×Dic83—41no—
195C2×Dic83—42no—
196C2×Dic83—41no—
197C2×Dic83—41no—
198C2×Dic83—42no—
199C2×Dic83—42no—
200C2×Dic83—41no—
201C2×Dic83#12112yes—
202C2×Dic83#12312yes—
203C2×Dic83C4⋊Q812yes—
204C2×Dic83C2×SD1612yes—
205C2×Dic83C2×SD1612yes—
206C4×D83—28no—
207C4×D83—28no—
208C4×D83—48no—
209C4×D83—88no—
210C4×D83—84no—
211C4×D83C22⋊Q818yes—
212C4×D83#12518yes—
213C8×C83—24no—
214C83⋊C8—24no—
215C22×C83C2×#12111yescharacteristic
216C22×C83C2×#12311yescharacteristic
217C22×C83C2×C4⋊Q811yescharacteristic
218C22×C83C22×SD1611yescharacteristic
219C22×C83C22×SD1611yescharacteristic
220C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
221C22×C83C2×#13711yescharacteristic
222C2×D83#39814yes—
223C2×D83#46014yes—
224C2×D83#46114yes—
225C2×D83#48114yes—
226C2×D83#48314yes—
227C2×D83#49814yes—
228C2×D83#50114yes—
229C4×C83—22no—
230C4×C83—22no—
231C4×C83—42no—
232C4×C83—81no—
233C4×C83—82no—
234C4×C83C2×C22⋊Q812yes—
235C4×C83C2×#12512yes—
236Dic83—22no—
237Dic83—22no—
238Dic83—42no—
239Dic83—81no—
240Dic83—82no—
241Dic83#39814yes—
242C2×#2084—831nomaximal, Sylow
243C2×C83C2×#39811yescharacteristic
244C2×C83C2×#46111yescharacteristic
245C2×C83C2×#46011yescharacteristic
246C2×C83C2×#48311yescharacteristic
247C2×C83C2×#48111yescharacteristic
248C2×C83C2×#49811yescharacteristic
249C2×C83C2×#50111yescharacteristic
250D83#208418yesdirect product factor
251C22×#106—832no—
252C2×#389—831no—
253C2×#384—831no—
254#2084—832no—
255#2084—831no—
256#2084—831no—
257#2084—832no—
258#2084—831no—
259#2084D8311yescharacteristic, direct product factor
260C2×#340—832no—
261C2×#336—831no—
262C83C2×#208411yescharacteristic, complemented, Sylow
263C23×C8—832no—
264C22×C4⋊C4—831no—
265C22×C4⋊C4—832no—
266C22×C4⋊C4—831no—
267C22×C4⋊C4—1661no—
268C22×C4⋊C4—1661no—
269C22×C4⋊C4—1661no—
270C22×C42—831no—
271C2×#106—832no—
272C2×#106—832no—
273C2×#106—832no—
274C2×#106—1668no—
275C2×#106C2×D8312yes—
276#389—831no—
277#389—831no—
278#389—831no—
279#389—1661no—
280#389—1661no—
281#389C2×D8311yescharacteristic
282#384—832no—
283#384—831no—
284#384—1661no—
285#384—1661no—
286#384C2×D8311yescharacteristic
287C2×#95—1661no—
288#340—832no—
289#340—832no—
290#340—832no—
291#340C2×D8312yes—
292#336—832no—
293#336—831no—
294#336C2×D8311yescharacteristic
295C23×C4—831no—
296C23×C4—831no—
297C23×C4—831no—
298C23×C4—1662no—
299C23×C4—1661no—
300C23×C4—1661no—
301C22×C8—832no—
302C22×C8—838no—
303C22×C8—1664no—
304C22×C8—1664no—
305C22×C8C22×D8312yes—
306C2×C4⋊C4—21no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—21no—
309C2×C4⋊C4—3324no—
310C2×C4⋊C4—3324no—
311C2×C4⋊C4—3324no—
312C2×C4⋊C4—831no—
313C2×C4⋊C4—834no—
314C2×C4⋊C4—831no—
315C2×C4⋊C4—832no—
316C2×C4⋊C4—1664no—
317C2×C4⋊C4—1664no—
318C2×C4⋊C4—1664no—
319C2×C4⋊C4—1662no—
320C2×C4⋊C4—1662no—
321C2×C4⋊C4—1661no—
322C2×C4⋊C4—1662no—
323C2×C4⋊C4—1662no—
324C2×C4⋊C4—1661no—
325C2×C4⋊C4—1661no—
326C2×C4⋊C4—1661no—
327C2×C4⋊C4—1661no—
328C2×C4⋊C4—1661no—
329C2×C4⋊C4—1661no—
330C2×C4⋊C4C22×D8311yescharacteristic
331C2×C4⋊C4C22×D8312yes—
332C2×C4⋊C4C22×D8311yescharacteristic
333C2×C42—831no—
334C2×C42—831no—
335C2×C42—831no—
336C2×C42—1664no—
337C2×C42C22×D8311yescharacteristic
338#106—24no—
339#106—1664no—
340#106—1664no—
341#106—1664no—
342#95—21no—
343#95—1661no—
344#95—1661no—
345#95—1661no—
346C24—831no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—3322no—
351C22×C4—3322no—
352C22×C4—3324no—
353C22×C4—3322no—
354C22×C4—3324no—
355C22×C4—3324no—
356C22×C4—3322no—
357C22×C4—3322no—
358C22×C4—831no—
359C22×C4—832no—
360C22×C4—834no—
361C22×C4—834no—
362C22×C4—832no—
363C22×C4—831no—
364C22×C4—831no—
365C22×C4—834no—
366C22×C4—834no—
367C22×C4—1664no—
368C22×C4—1662no—
369C22×C4—1662no—
370C22×C4—1662no—
371C22×C4—1662no—
372C22×C4—1661no—
373C22×C4—1662no—
374C22×C4—1661no—
375C22×C4D4×D8311yescharacteristic
376C22×C4D4×D8311yescharacteristic
377C22×C4C23×D8311yescharacteristic
378C2×C8—22no—
379C2×C8—22no—
380C2×C8—834no—
381C2×C8—16616no—
382C2×C8—1662no—
383C2×C8—1662no—
384C2×C8Q8×D8314yes—
385C4⋊C4—21no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—3322no—
393C4⋊C4—3322no—
394C4⋊C4—3324no—
395C4⋊C4—3322no—
396C4⋊C4—3324no—
397C4⋊C4—3322no—
398C4⋊C4—3324no—
399C4⋊C4—3322no—
400C4⋊C4—3322no—
401C4⋊C4—832no—
402C4⋊C4—1664no—
403C4⋊C4—1662no—
404C4⋊C4—1662no—
405C4⋊C4—1662no—
406C4⋊C4—1661no—
407C4⋊C4D4×D8312yes—
408C42—22no—
409C42—1662no—
410C42—1662no—
411C42—1662no—
412C23—832no—
413C23—832no—
414C23—832no—
415C23—832no—
416C23—832no—
417C23—832no—
418C23—832no—
419C23C2×D4×D8311yescharacteristic
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—42no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—3328no—
434C2×C4—3321no—
435C2×C4—3321no—
436C2×C4—3322no—
437C2×C4—3321no—
438C2×C4—3321no—
439C2×C4—3322no—
440C2×C4—3322no—
441C2×C4—3321no—
442C2×C4—832no—
443C2×C4—832no—
444C2×C4—832no—
445C2×C4—838no—
446C2×C4—838no—
447C2×C4—832no—
448C2×C4—832no—
449C2×C4—6644no—
450C2×C4—6648no—
451C2×C4—1668no—
452C2×C4—1668no—
453C2×C4—1662no—
454C2×C4—1661no—
455C2×C4—1661no—
456C2×C4—1661no—
457C2×C4—1661no—
458C2×C4SD16×D8312yes—
459C2×C4SD16×D8312yes—
460C2×C4C2×D4×D8311yescharacteristic
461C2×C4C2×Q8×D8312yes—
462C2×C4#41×D8312yes—
463C2×C4#41×D8311yescharacteristic
464C8—24no—
465C8—1664no—
466C22—834no—
467C22—834no—
468C22—834no—
469C22—834no—
470C22—834no—
471C22—834no—
472C22—834no—
473C22#121×D8311yescharacteristic
474C22#123×D8311yescharacteristic
475C22C4⋊Q8×D8311yescharacteristic
476C22C2×SD16×D8311yescharacteristic
477C22C2×SD16×D8311yescharacteristic
478C22AGL(1,ℤ/8)×D8311yescharacteristic
479C22#137×D8311yescharacteristic
480C4—22no—
481C4—22no—
482C4—42no—
483C4—81no—
484C4—82no—
485C4—3322no—
486C4—834no—
487C4—6641no—
488C4—6642no—
489C4—1662no—
490C4—1662no—
491C4C22⋊Q8×D8312yes—
492C4#125×D8312yes—
493C2—838no—
494C2#398×D8311yescharacteristic
495C2#461×D8311yescharacteristic
496C2#460×D8311yescharacteristic
497C2#483×D8311yescharacteristic
498C2#481×D8311yescharacteristic
499C2#498×D8311yescharacteristic
500C2#501×D8311yescharacteristic
subgroupG/H
centerC23C2×D4×D83
commutator, cosocleC22×C4×C83C24
FrattiniC22×C4C23×D83
FittingC83×#2084C2
socleC23×C83C22×D4
prime radicalC23×D83C2×D4
hypercenter#2084D83
hypocenterC83C2×#2084
LCSG ▷ C22×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2084

Complements

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

NH ≅ G/Ntype
4. #24908943C2semidirect
5. #24908942C2semidirect
6. #24908941C2semidirect
7. #24908940C2semidirect
8. #24908939C2semidirect
11. C83×#2084C2semidirect
250. D83#2084direct (N × H)
259. #2084D83direct (N × H)
262. C83C2×#2084semidirect

Maximal subgroups

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

subgroupindex
C2×#106×D832
#389×D832
#384×D832
#249089432
#249089422
#249089412
#249089402
#249089392
#340×D832
#336×D832
C83×#20842
C2×#208483