Finite Group Databasebeta

← C83×#61892 #35289716

Subgroups — C83×#61892 #35289716

Order 21248.

1770 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3502C211yesmaximal, characteristic, complemented
2C2×#559×C83C211yesmaximal, characteristic
3C83×#3400C218yesmaximal, complemented
4C83×#3375C218yesmaximal, complemented
5C83×#3215C214yesmaximal, complemented
6C83×#3208C212yesmaximal
7C83×#3110C214yesmaximal, complemented
8C83×#3104C212yesmaximal, complemented
9C83×#2983C211yesmaximal, characteristic, complemented
10#559×C83—24no—
11#559×C83C2218yes—
12Q82×C83C2214yescomplemented
13#557×C83—24no—
14#557×C83C2214yescomplemented
15#544×C83—21no—
16#544×C83—22no—
17#533×C83C2212yes—
18#533×C83C2214yes—
19C2×#127×C83C2211yescharacteristic
20C2×#124×C83—21no—
21C2×#124×C83C2212yes—
22C2×#123×C83—22no—
23#519×C83—21no—
24C2×C4×Q8×C83C2211yescharacteristic
25C2×C4×Q8×C83C2211yescharacteristic
26C2×C4×D4×C83—21no—
27C2×C4×D4×C83C2212yes—
28#497×C83C2218yescomplemented
29#492×C83C2214yes—
30#489×C83C2218yescomplemented
31#486×C83C2214yes—
32#484×C83C2214yes—
33#477×C83C2214yescomplemented
34#474×C83C2218yescomplemented
35#464×C83C22116yescomplemented
36#462×C83C2214yescomplemented
37#461×C83C2218yescomplemented
38#460×C83C2218yes—
39#451×C83C2214yes—
40#450×C83C2218yes—
41#446×C83C2218yescomplemented
42#440×C83C22116yescomplemented
43C4×Q16×C83C2218yescomplemented
44#432×C83C2214yes—
45#431×C83C2212yes—
46C22×Q8×C83D411yescharacteristic
47C22×Q8×C83C2311yescharacteristic
48C22×D4×C83—21no—
49C22×D4×C83—21no—
50C22×D4×C83C2312yes—
51C23×C4×C83—22no—
52C2×Q16×C83C2318yes—
53C2×SD16×C83C23116yes—
54C4⋊Q8×C83—24no—
55C4⋊Q8×C83—28no—
56C4⋊Q8×C83C2312yes—
57C4⋊Q8×C83C2314yes—
58C4⋊Q8×C83C2314yes—
59C4⋊Q8×C83C2312yes—
60#127×C83—22no—
61#127×C83C2314yes—
62#125×C83—24no—
63#125×C83—28no—
64#125×C83—24no—
65#125×C83—22no—
66#124×C83—28no—
67#124×C83—24no—
68#124×C83C2318yes—
69#123×C83—216no—
70C22⋊Q8×C83—24no—
71C22⋊Q8×C83—22no—
72C22⋊Q8×C83C2314yes—
73C4×Q8×C83—24no—
74C4×Q8×C83—24no—
75C4×Q8×C83—22no—
76C4×Q8×C83—24no—
77C4×Q8×C83—22no—
78C4×Q8×C83D414yes—
79C4×Q8×C83C2318yes—
80C4×Q8×C83C2314yes—
81C4×Q8×C83C2314yes—
82C4×Q8×C83C2314yes—
83C4×D4×C83—28no—
84C4×D4×C83—24no—
85C4×D4×C83C2318yes—
86#117×C83—24no—
87#117×C83—22no—
88C2×C4⋊C4×C83—21no—
89C2×C4⋊C4×C83D411yescharacteristic
90C2×C4⋊C4×C83D412yes—
91C2×C4⋊C4×C83C2311yescharacteristic
92C2×C22⋊C4×C83—22no—
93C2×C22⋊C4×C83—22no—
94C2×C22⋊C4×C83—22no—
95C2×C42×C83—21no—
96C2×C42×C83C2312yes—
97C2×C42×C83C2311yescharacteristic
98#107×C83C2318yes—
99C4⋊C8×C83C2318yes—
100C4⋊C8×C83C2314yes—
101#103×C83C2318yes—
102#103×C83C2318yes—
103#102×C83C23116yes—
104C22⋊C8×C83C2314yes—
105#97×C83C2318yes—
106C4×C8×C83C2314yes—
107C24×C83—22no—
108C2×Q8×C83—24no—
109C2×Q8×C83—21no—
110C2×Q8×C83—24no—
111C2×Q8×C83—22no—
112C2×Q8×C83—42no—
113C2×Q8×C83C2×D414yes—
114C2×Q8×C83C2412yes—
115C2×Q8×C83C2414yes—
116C2×D4×C83—22no—
117C2×D4×C83—24no—
118C2×D4×C83—24no—
119C2×D4×C83—44no—
120C2×D4×C83—44no—
121C2×D4×C83—82no—
122C2×D4×C83C2414yes—
123C22×C4×C83—21no—
124C22×C4×C83—21no—
125C22×C4×C83—28no—
126C22×C4×C83—84no—
127C22×C4×C83C2×D412yes—
128C22×C4×C83C2×D412yes—
129C22×C4×C83C2×D411yescharacteristic
130C22×C4×C83C2×D411yescharacteristic
131C22×C4×C83C2411yescharacteristic
132Q16×C83—48no—
133SD16×C83—416no—
134C2×C8×C83C2418yes—
135C4⋊C4×C83—24no—
136C4⋊C4×C83—21no—
137C4⋊C4×C83—24no—
138C4⋊C4×C83—22no—
139C4⋊C4×C83—24no—
140C4⋊C4×C83—22no—
141C4⋊C4×C83—24no—
142C4⋊C4×C83—24no—
143C4⋊C4×C83—22no—
144C4⋊C4×C83—28no—
145C4⋊C4×C83—24no—
146C4⋊C4×C83C2×D414yes—
147C4⋊C4×C83C2×D418yes—
148C4⋊C4×C83C2414yes—
149C4⋊C4×C83C2412yes—
150C22⋊C4×C83—24no—
151C22⋊C4×C83—28no—
152C22⋊C4×C83—28no—
153C22⋊C4×C83—28no—
154C42×C83—24no—
155C42×C83—28no—
156C42×C83—24no—
157C42×C83C2×D414yes—
158C42×C83C2×D412yes—
159C42×C83C2412yes—
160C42×C83C2414yes—
161C23×C83—24no—
162C23×C83—44no—
163C23×C83—44no—
164C23×C83—44no—
165C23×C83C22×D411yescharacteristic
166Q8×C83—28no—
167Q8×C83—28no—
168Q8×C83—24no—
169Q8×C83—42no—
170Q8×C83—84no—
171Q8×C83—82no—
172D4×C83—28no—
173D4×C83—44no—
174D4×C83—48no—
175D4×C83—84no—
176C2×C4×C83—24no—
177C2×C4×C83—22no—
178C2×C4×C83—24no—
179C2×C4×C83—24no—
180C2×C4×C83—22no—
181C2×C4×C83—22no—
182C2×C4×C83—42no—
183C2×C4×C83—41no—
184C2×C4×C83—42no—
185C2×C4×C83—42no—
186C2×C4×C83—88no—
187C2×C4×C83#13714yes—
188C2×C4×C83C22×D414yes—
189C2×C4×C83C22×D411yescharacteristic
190C2×C4×C83C22×D414yes—
191C2×C4×C83C22×D412yes—
192C2×C4×C83C22×D412yes—
193C2×C4×C83C2511yescharacteristic
194C8×C83—28no—
195C22×C83—21no—
196C22×C83—21no—
197C22×C83—44no—
198C22×C83—44no—
199C22×C83—44no—
200C22×C83—88no—
201C22×C83C2×#13712yes—
202C22×C83C23×D411yescharacteristic
203C4×C83—24no—
204C4×C83—22no—
205C4×C83—21no—
206C4×C83—24no—
207C4×C83—42no—
208C4×C83—44no—
209C4×C83—42no—
210C4×C83—44no—
211C4×C83C2×#13714yes—
212C4×C83#58512yes—
213#61892C8311yesmaximal, characteristic, direct product factor, Sylow
214C2×C83—22no—
215C2×C83—84no—
216C2×C83#348711yescharacteristic
217C2×C83C22×#13711yescharacteristic
218C2×C83#358511yescharacteristic
219#3502C2×C8311yescharacteristic, complemented
220C2×#559C2×C8311yescharacteristic
221#3400C2×C8318yescomplemented
222#3375C2×C8318yescomplemented
223#3215C2×C8314yescomplemented
224#3208C2×C8312yes—
225#3110C2×C8314yescomplemented
226#3104C2×C8312yescomplemented
227#2983C2×C8311yescharacteristic, complemented
228C83#6189211yescharacteristic, direct product factor, Sylow
229#559—24no—
230#559C22×C8318yes—
231Q82C22×C8314yescomplemented
232#557—24no—
233#557C22×C8314yescomplemented
234#544—21no—
235#544—22no—
236#533C22×C8312yes—
237#533C22×C8314yes—
238C2×#127C22×C8311yescharacteristic
239C2×#124—21no—
240C2×#124C22×C8312yes—
241C2×#123—22no—
242#519—21no—
243C2×C4×Q8C22×C8311yescharacteristic
244C2×C4×Q8C22×C8311yescharacteristic
245C2×C4×D4—21no—
246C2×C4×D4C22×C8312yes—
247#497C22×C8318yescomplemented
248#492C22×C8314yes—
249#489C22×C8318yescomplemented
250#486C22×C8314yes—
251#484C22×C8314yes—
252#477C22×C8314yescomplemented
253#474C22×C8318yescomplemented
254#464C22×C83116yescomplemented
255#462C22×C8314yescomplemented
256#461C22×C8318yescomplemented
257#460C22×C8318yes—
258#451C22×C8314yes—
259#450C22×C8318yes—
260#446C22×C8318yescomplemented
261#440C22×C83116yescomplemented
262C4×Q16C22×C8318yescomplemented
263#432C22×C8314yes—
264#431C22×C8312yes—
265C22×Q8D4×C8311yescharacteristic
266C22×Q8C23×C8311yescharacteristic
267C22×D4—21no—
268C22×D4—21no—
269C22×D4C23×C8312yes—
270C23×C4—22no—
271C2×Q16C23×C8318yes—
272C2×SD16C23×C83116yes—
273C4⋊Q8—24no—
274C4⋊Q8—28no—
275C4⋊Q8C23×C8314yes—
276C4⋊Q8C23×C8312yes—
277C4⋊Q8C23×C8314yes—
278C4⋊Q8C23×C8312yes—
279#127—22no—
280#127C23×C8314yes—
281#125—24no—
282#125—28no—
283#125—24no—
284#125—22no—
285#124—28no—
286#124—24no—
287#124C23×C8318yes—
288#123—216no—
289C22⋊Q8—24no—
290C22⋊Q8—22no—
291C22⋊Q8C23×C8314yes—
292C4×Q8—24no—
293C4×Q8—24no—
294C4×Q8—22no—
295C4×Q8—24no—
296C4×Q8—22no—
297C4×Q8D4×C8314yes—
298C4×Q8C23×C8314yes—
299C4×Q8C23×C8314yes—
300C4×Q8C23×C8318yes—
301C4×Q8C23×C8314yes—
302C4×D4—28no—
303C4×D4—24no—
304C4×D4C23×C8318yes—
305#117—24no—
306#117—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4D4×C8311yescharacteristic
309C2×C4⋊C4D4×C8312yes—
310C2×C4⋊C4C23×C8311yescharacteristic
311C2×C22⋊C4—22no—
312C2×C22⋊C4—22no—
313C2×C22⋊C4—22no—
314C2×C42—21no—
315C2×C42C23×C8312yes—
316C2×C42C23×C8311yescharacteristic
317#107C23×C8318yes—
318C4⋊C8C23×C8314yes—
319C4⋊C8C23×C8318yes—
320#103C23×C8318yes—
321#103C23×C8318yes—
322#102C23×C83116yes—
323C22⋊C8C23×C8314yes—
324#97C23×C8318yes—
325C4×C8C23×C8314yes—
326C24—22no—
327C2×Q8—24no—
328C2×Q8—21no—
329C2×Q8—24no—
330C2×Q8—22no—
331C2×Q8—42no—
332C2×Q8C2×D4×C8314yes—
333C2×Q8C24×C8312yes—
334C2×Q8C24×C8314yes—
335C2×D4—22no—
336C2×D4—24no—
337C2×D4—24no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—82no—
341C2×D4C24×C8314yes—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—28no—
345C22×C4—84no—
346C22×C4C2×D4×C8312yes—
347C22×C4C2×D4×C8312yes—
348C22×C4C2×D4×C8311yescharacteristic
349C22×C4C2×D4×C8311yescharacteristic
350C22×C4C24×C8311yescharacteristic
351Q16—48no—
352SD16—416no—
353C2×C8C24×C8318yes—
354C4⋊C4—24no—
355C4⋊C4—21no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—22no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—28no—
364C4⋊C4—24no—
365C4⋊C4C2×D4×C8314yes—
366C4⋊C4C2×D4×C8318yes—
367C4⋊C4C24×C8314yes—
368C4⋊C4C24×C8312yes—
369C22⋊C4—24no—
370C22⋊C4—28no—
371C22⋊C4—28no—
372C22⋊C4—28no—
373C42—24no—
374C42—28no—
375C42—24no—
376C42C2×D4×C8314yes—
377C42C2×D4×C8312yes—
378C42C24×C8312yes—
379C42C24×C8314yes—
380C23—24no—
381C23—44no—
382C23—44no—
383C23—44no—
384C23C22×D4×C8311yescharacteristic
385Q8—28no—
386Q8—28no—
387Q8—24no—
388Q8—42no—
389Q8—84no—
390Q8—82no—
391D4—28no—
392D4—44no—
393D4—48no—
394D4—84no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—88no—
406C2×C4#137×C8314yes—
407C2×C4C22×D4×C8314yes—
408C2×C4C22×D4×C8311yescharacteristic
409C2×C4C22×D4×C8314yes—
410C2×C4C22×D4×C8312yes—
411C2×C4C22×D4×C8312yes—
412C2×C4C25×C8311yescharacteristic
413C8—28no—
414C22—21no—
415C22—21no—
416C22—44no—
417C22—44no—
418C22—44no—
419C22—88no—
420C22C2×#137×C8312yes—
421C22C23×D4×C8311yescharacteristic
422C4—24no—
423C4—24no—
424C4—22no—
425C4—21no—
426C4—42no—
427C4—44no—
428C4—42no—
429C4—44no—
430C4C2×#137×C8314yes—
431C4#585×C8312yes—
432C2—22no—
433C2—84no—
434C2C83×#348711yescharacteristic
435C2C22×#137×C8311yescharacteristic
436C2C83×#358511yescharacteristic
subgroupG/H
center, socleC22×C83C23×D4
commutator, Frattini, cosocleC2×C4C25×C83
prime radicalC2×#127×C83C22
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C4×Q8×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C83×#3502C2semidirect
3. C83×#3400C2semidirect
4. C83×#3375C2semidirect
5. C83×#3215C2semidirect
7. C83×#3110C2semidirect
8. C83×#3104C2semidirect
9. C83×#2983C2semidirect
12. Q82×C83C22semidirect
14. #557×C83C22semidirect
28. #497×C83C22semidirect
30. #489×C83C22semidirect
33. #477×C83C22semidirect
34. #474×C83C22semidirect
35. #464×C83C22semidirect
36. #462×C83C22semidirect
37. #461×C83C22semidirect
41. #446×C83C22semidirect
42. #440×C83C22semidirect
43. C4×Q16×C83C22semidirect
213. #61892C83direct (N × H)
219. #3502C2×C83semidirect
221. #3400C2×C83semidirect
222. #3375C2×C83semidirect
223. #3215C2×C83semidirect
225. #3110C2×C83semidirect
226. #3104C2×C83semidirect
227. #2983C2×C83semidirect
228. C83#61892direct (N × H)
231. Q82C22×C83semidirect
233. #557C22×C83semidirect
247. #497C22×C83semidirect
249. #489C22×C83semidirect
252. #477C22×C83semidirect
253. #474C22×C83semidirect
254. #464C22×C83semidirect
255. #462C22×C83semidirect
256. #461C22×C83semidirect
260. #446C22×C83semidirect
261. #440C22×C83semidirect
262. C4×Q16C22×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#35022
C2×#559×C832
C83×#34002
C83×#33752
C83×#32152
C83×#32082
C83×#31102
C83×#31042
C83×#29832
#6189283