Finite Group Databasebeta

← C83×#19362 #35247186

Subgroups — C83×#19362 #35247186

Order 21248.

648 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3238C211yesmaximal, characteristic, complemented
2C83×#3217C211yesmaximal, characteristic
3C2×#486×C83C211yesmaximal, characteristic, complemented
4C83×#2966C211yesmaximal, characteristic, complemented
5C83×#1698C212yesmaximal, complemented
6C83×#1697C212yesmaximal, complemented
7C83×#1678C212yesmaximal, complemented
8C83×#1677C212yesmaximal, complemented
9C83×#1568C211yesmaximal, characteristic, complemented
10C83×#1558C211yesmaximal, characteristic, complemented
11C83×#1555C211yesmaximal, characteristic, complemented
12C2×C4⋊Q8×C83C2211yescharacteristic
13C2×#127×C83C2211yescharacteristic
14#529×C83C2211yescharacteristic
15#501×C83—21no—
16#496×C83—21no—
17#490×C83—21no—
18#489×C83—21no—
19#486×C83—21no—
20#486×C83—21no—
21#486×C83C2212yes—
22#486×C83C2212yes—
23#483×C83—21no—
24#480×C83—21no—
25#477×C83C2212yescomplemented
26#476×C83C2212yescomplemented
27#460×C83C2212yes—
28#459×C83C2212yes—
29#450×C83—21no—
30#447×C83—21no—
31#446×C83C2211yescharacteristic, complemented
32C8×Q8×C83C2211yescharacteristic, complemented
33#436×C83C2211yescharacteristic, complemented
34#435×C83—21no—
35C8×D4×C83—21no—
36C8×D4×C83C2211yescharacteristic, complemented
37#433×C83—21no—
38#432×C83C2211yescharacteristic
39#432×C83C2211yescharacteristic
40#432×C83C2211yescharacteristic
41C2×#102×C83—21no—
42C2×#102×C83—21no—
43#405×C83—21no—
44C2×C4×C8×C83C2211yescharacteristic
45#335×C83C2212yescomplemented
46#334×C83C2212yescomplemented
47#333×C83C2212yescomplemented
48#333×C83C2212yescomplemented
49#331×C83C2212yes—
50#331×C83C2212yes—
51C22×Q8×C83—21no—
52C22×D4×C83—21no—
53C22×D4×C83—21no—
54C22×D4×C83—21no—
55C2×SD16×C83—22no—
56C2×D8×C83—22no—
57C2×M16×C83—21no—
58C22×C8×C83—21no—
59C22×C8×C83D411yescharacteristic
60C22×C8×C83D411yescharacteristic
61C4⋊Q8×C83—21no—
62C4⋊Q8×C83—21no—
63C4⋊Q8×C83—21no—
64C4⋊Q8×C83C2312yes—
65#127×C83—21no—
66#127×C83—21no—
67#127×C83—21no—
68#127×C83C2312yes—
69#126×C83—21no—
70#125×C83D411yescharacteristic
71#124×C83D411yescharacteristic
72#123×C83—21no—
73C22⋊Q8×C83D411yescharacteristic
74#121×C83D411yescharacteristic
75C4×Q8×C83C2311yescharacteristic
76C4×D4×C83—21no—
77C4×D4×C83C2311yescharacteristic
78#117×C83—21no—
79C2×C4⋊C4×C83—21no—
80C2×C4⋊C4×C83—21no—
81C2×C42×C83C2311yescharacteristic
82#107×C83—22no—
83#106×C83—22no—
84C4⋊C8×C83—21no—
85C4⋊C8×C83—21no—
86C4⋊C8×C83C2312yes—
87C4⋊C8×C83C2311yescharacteristic
88C4⋊C8×C83C2311yescharacteristic
89C4⋊C8×C83C2312yes—
90#103×C83—22no—
91#102×C83—22no—
92#102×C83—22no—
93#102×C83—22no—
94#102×C83—22no—
95#102×C83—22no—
96#102×C83—22no—
97#102×C83—22no—
98C22⋊C8×C83—21no—
99C22⋊C8×C83—21no—
100C22⋊C8×C83—21no—
101C22⋊C8×C83—21no—
102C22⋊C8×C83—21no—
103#97×C83—21no—
104#97×C83—21no—
105#97×C83—21no—
106#97×C83—21no—
107C4×C8×C83—21no—
108C4×C8×C83D411yescharacteristic
109C4×C8×C83D411yescharacteristic
110C4×C8×C83C2311yescharacteristic
111C4×C8×C83C2311yescharacteristic
112C24×C83—41no—
113C2×Q8×C83—22no—
114C2×Q8×C83—41no—
115C2×Q8×C83—82no—
116C2×Q8×C83C2×D411yescharacteristic
117C2×D4×C83—21no—
118C2×D4×C83—22no—
119C2×D4×C83—22no—
120C2×D4×C83—22no—
121C2×D4×C83—42no—
122C2×D4×C83—42no—
123C2×D4×C83—41no—
124C2×D4×C83—41no—
125C2×D4×C83—41no—
126C2×D4×C83—82no—
127C2×D4×C83—82no—
128C2×D4×C83C2×D411yescharacteristic
129C22×C4×C83—21no—
130C22×C4×C83—41no—
131C22×C4×C83C2×D411yescharacteristic
132C22×C4×C83C2×D411yescharacteristic
133C22×C4×C83#4111yescharacteristic
134SD16×C83—82no—
135D8×C83—82no—
136M16×C83—42no—
137C2×C8×C83—21no—
138C2×C8×C83—21no—
139C2×C8×C83—22no—
140C2×C8×C83—22no—
141C2×C8×C83—41no—
142C2×C8×C83—41no—
143C2×C8×C83—42no—
144C2×C8×C83C2×D411yescharacteristic
145C2×C8×C83C2×D411yescharacteristic
146C2×C8×C83C2×D411yescharacteristic
147C2×C8×C83C2×D411yescharacteristic
148C4⋊C4×C83—21no—
149C4⋊C4×C83—21no—
150C4⋊C4×C83—21no—
151C4⋊C4×C83—21no—
152C4⋊C4×C83—22no—
153C4⋊C4×C83—22no—
154C4⋊C4×C83—41no—
155C4⋊C4×C83—41no—
156C4⋊C4×C83C2×D411yescharacteristic
157C4⋊C4×C83C2×D411yescharacteristic
158C22⋊C4×C83—21no—
159C22⋊C4×C83—21no—
160C22⋊C4×C83—21no—
161C22⋊C4×C83—21no—
162C22⋊C4×C83—21no—
163C42×C83—21no—
164C42×C83C2×D411yescharacteristic
165C42×C83C2×D411yescharacteristic
166C42×C83#4111yescharacteristic
167C42×C83C2411yescharacteristic
168C23×C83—21no—
169C23×C83—42no—
170C23×C83—82no—
171C23×C83—82no—
172C23×C83—82no—
173C23×C83#12111yescharacteristic
174Q8×C83—22no—
175Q8×C83—82no—
176Q8×C83—82no—
177D4×C83—22no—
178D4×C83—42no—
179D4×C83—42no—
180D4×C83—42no—
181D4×C83—42no—
182D4×C83—42no—
183D4×C83—82no—
184D4×C83—82no—
185D4×C83—82no—
186D4×C83—82no—
187C2×C4×C83—22no—
188C2×C4×C83—21no—
189C2×C4×C83—21no—
190C2×C4×C83—22no—
191C2×C4×C83—21no—
192C2×C4×C83—21no—
193C2×C4×C83—21no—
194C2×C4×C83—41no—
195C2×C4×C83—41no—
196C2×C4×C83—42no—
197C2×C4×C83—82no—
198C2×C4×C83#12111yescharacteristic
199C2×C4×C83#12111yescharacteristic
200C2×C4×C83#12111yescharacteristic
201C2×C4×C83C22×D411yescharacteristic
202C2×C4×C83C22×D411yescharacteristic
203C2×C4×C83C2×#4111yescharacteristic
204C8×C83—21no—
205C8×C83—21no—
206C8×C83—21no—
207C8×C83—21no—
208C8×C83—41no—
209C8×C83—41no—
210C8×C83—42no—
211C8×C83—42no—
212C22×C83—21no—
213C22×C83—21no—
214C22×C83—41no—
215C22×C83—41no—
216C22×C83—41no—
217C22×C83—82no—
218C22×C83—82no—
219C22×C83—82no—
220C22×C83—164no—
221C22×C83#47212yes—
222C22×C83C2×#12111yescharacteristic
223C4×C83—21no—
224C4×C83—21no—
225C4×C83—21no—
226C4×C83—21no—
227C4×C83—41no—
228C4×C83—41no—
229C4×C83—41no—
230C4×C83—82no—
231C4×C83#57511yescharacteristic
232C4×C83#57511yescharacteristic
233C4×C83#57712yes—
234#19362C8311yesmaximal, characteristic, direct product factor, Sylow
235C2×C83—22no—
236C2×C83—81no—
237C2×C83—162no—
238C2×C83#304011yescharacteristic
239C2×C83#305711yescharacteristic
240C2×C83C2×#47211yescharacteristic
241#3238C2×C8311yescharacteristic, complemented
242#3217C2×C8311yescharacteristic
243C2×#486C2×C8311yescharacteristic, complemented
244#2966C2×C8311yescharacteristic, complemented
245#1698C2×C8312yescomplemented
246#1697C2×C8312yescomplemented
247#1678C2×C8312yescomplemented
248#1677C2×C8312yescomplemented
249#1568C2×C8311yescharacteristic, complemented
250#1558C2×C8311yescharacteristic, complemented
251#1555C2×C8311yescharacteristic, complemented
252C83#1936211yescharacteristic, direct product factor, Sylow
253C2×C4⋊Q8C22×C8311yescharacteristic
254C2×#127C22×C8311yescharacteristic
255#529C22×C8311yescharacteristic
256#501—21no—
257#496—21no—
258#490—21no—
259#489—21no—
260#486—21no—
261#486—21no—
262#486C22×C8312yes—
263#486C22×C8312yes—
264#483—21no—
265#480—21no—
266#477C22×C8312yescomplemented
267#476C22×C8312yescomplemented
268#460C22×C8312yes—
269#459C22×C8312yes—
270#450—21no—
271#447—21no—
272#446C22×C8311yescharacteristic, complemented
273C8×Q8C22×C8311yescharacteristic, complemented
274#436C22×C8311yescharacteristic, complemented
275#435—21no—
276C8×D4—21no—
277C8×D4C22×C8311yescharacteristic, complemented
278#433—21no—
279#432C22×C8311yescharacteristic
280#432C22×C8311yescharacteristic
281#432C22×C8311yescharacteristic
282C2×#102—21no—
283C2×#102—21no—
284#405—21no—
285C2×C4×C8C22×C8311yescharacteristic
286#335C22×C8312yescomplemented
287#334C22×C8312yescomplemented
288#333C22×C8312yescomplemented
289#333C22×C8312yescomplemented
290#331C22×C8312yes—
291#331C22×C8312yes—
292C22×Q8—21no—
293C22×D4—21no—
294C22×D4—21no—
295C22×D4—21no—
296C2×SD16—22no—
297C2×D8—22no—
298C2×M16—21no—
299C22×C8—21no—
300C22×C8D4×C8311yescharacteristic
301C22×C8D4×C8311yescharacteristic
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8C23×C8312yes—
306#127—21no—
307#127—21no—
308#127—21no—
309#127C23×C8312yes—
310#126—21no—
311#125D4×C8311yescharacteristic
312#124D4×C8311yescharacteristic
313#123—21no—
314C22⋊Q8D4×C8311yescharacteristic
315#121D4×C8311yescharacteristic
316C4×Q8C23×C8311yescharacteristic
317C4×D4—21no—
318C4×D4C23×C8311yescharacteristic
319#117—21no—
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C42C23×C8311yescharacteristic
323#107—22no—
324#106—22no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8C23×C8311yescharacteristic
328C4⋊C8C23×C8311yescharacteristic
329C4⋊C8C23×C8312yes—
330C4⋊C8C23×C8312yes—
331#103—22no—
332#102—22no—
333#102—22no—
334#102—22no—
335#102—22no—
336#102—22no—
337#102—22no—
338#102—22no—
339C22⋊C8—21no—
340C22⋊C8—21no—
341C22⋊C8—21no—
342C22⋊C8—21no—
343C22⋊C8—21no—
344#97—21no—
345#97—21no—
346#97—21no—
347#97—21no—
348C4×C8—21no—
349C4×C8D4×C8311yescharacteristic
350C4×C8D4×C8311yescharacteristic
351C4×C8C23×C8311yescharacteristic
352C4×C8C23×C8311yescharacteristic
353C24—41no—
354C2×Q8—22no—
355C2×Q8—41no—
356C2×Q8—82no—
357C2×Q8C2×D4×C8311yescharacteristic
358C2×D4—21no—
359C2×D4—22no—
360C2×D4—22no—
361C2×D4—22no—
362C2×D4—42no—
363C2×D4—42no—
364C2×D4—41no—
365C2×D4—41no—
366C2×D4—41no—
367C2×D4—82no—
368C2×D4—82no—
369C2×D4C2×D4×C8311yescharacteristic
370C22×C4—21no—
371C22×C4—41no—
372C22×C4C2×D4×C8311yescharacteristic
373C22×C4C2×D4×C8311yescharacteristic
374C22×C4#41×C8311yescharacteristic
375SD16—82no—
376D8—82no—
377M16—42no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—42no—
385C2×C8C2×D4×C8311yescharacteristic
386C2×C8C2×D4×C8311yescharacteristic
387C2×C8C2×D4×C8311yescharacteristic
388C2×C8C2×D4×C8311yescharacteristic
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4C2×D4×C8311yescharacteristic
398C4⋊C4C2×D4×C8311yescharacteristic
399C22⋊C4—21no—
400C22⋊C4—21no—
401C22⋊C4—21no—
402C22⋊C4—21no—
403C22⋊C4—21no—
404C42—21no—
405C42C2×D4×C8311yescharacteristic
406C42C2×D4×C8311yescharacteristic
407C42#41×C8311yescharacteristic
408C42C24×C8311yescharacteristic
409C23—21no—
410C23—42no—
411C23—82no—
412C23—82no—
413C23—82no—
414C23#121×C8311yescharacteristic
415Q8—22no—
416Q8—82no—
417Q8—82no—
418D4—22no—
419D4—42no—
420D4—42no—
421D4—42no—
422D4—42no—
423D4—42no—
424D4—82no—
425D4—82no—
426D4—82no—
427D4—82no—
428C2×C4—22no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—82no—
439C2×C4#121×C8311yescharacteristic
440C2×C4#121×C8311yescharacteristic
441C2×C4#121×C8311yescharacteristic
442C2×C4C22×D4×C8311yescharacteristic
443C2×C4C22×D4×C8311yescharacteristic
444C2×C4C2×#41×C8311yescharacteristic
445C8—21no—
446C8—21no—
447C8—21no—
448C8—21no—
449C8—41no—
450C8—41no—
451C8—42no—
452C8—42no—
453C22—21no—
454C22—21no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—82no—
459C22—82no—
460C22—82no—
461C22—164no—
462C22#472×C8312yes—
463C22C2×#121×C8311yescharacteristic
464C4—21no—
465C4—21no—
466C4—21no—
467C4—21no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—82no—
472C4#575×C8311yescharacteristic
473C4#575×C8311yescharacteristic
474C4#577×C8312yes—
475C2—22no—
476C2—81no—
477C2—162no—
478C2C83×#304011yescharacteristic
479C2C83×#305711yescharacteristic
480C2C2×#472×C8311yescharacteristic
subgroupG/H
center, socleC22×C83C2×#121
commutator, Frattini, cosocleC42C24×C83
prime radicalC83×#3217C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C83×#3238C2semidirect
3. C2×#486×C83C2semidirect
4. C83×#2966C2semidirect
5. C83×#1698C2semidirect
6. C83×#1697C2semidirect
7. C83×#1678C2semidirect
8. C83×#1677C2semidirect
9. C83×#1568C2semidirect
10. C83×#1558C2semidirect
11. C83×#1555C2semidirect
25. #477×C83C22semidirect
26. #476×C83C22semidirect
31. #446×C83C22semidirect
32. C8×Q8×C83C22semidirect
33. #436×C83C22semidirect
36. C8×D4×C83C22semidirect
45. #335×C83C22semidirect
46. #334×C83C22semidirect
47. #333×C83C22semidirect
48. #333×C83C22semidirect
234. #19362C83direct (N × H)
241. #3238C2×C83semidirect
243. C2×#486C2×C83semidirect
244. #2966C2×C83semidirect
245. #1698C2×C83semidirect
246. #1697C2×C83semidirect
247. #1678C2×C83semidirect
248. #1677C2×C83semidirect
249. #1568C2×C83semidirect
250. #1558C2×C83semidirect
251. #1555C2×C83semidirect
252. C83#19362direct (N × H)
266. #477C22×C83semidirect
267. #476C22×C83semidirect
272. #446C22×C83semidirect
273. C8×Q8C22×C83semidirect
274. #436C22×C83semidirect
277. C8×D4C22×C83semidirect
286. #335C22×C83semidirect
287. #334C22×C83semidirect
288. #333C22×C83semidirect
289. #333C22×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#32382
C83×#32172
C2×#486×C832
C83×#29662
C83×#16982
C83×#16972
C83×#16782
C83×#16772
C83×#15682
C83×#15582
C83×#15552
#1936283