Finite Group Databasebeta

← C83×#61721 #35289545

Subgroups — C83×#61721 #35289545

Order 21248.

1798 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3501C211yesmaximal, characteristic, complemented
2C2×#555×C83C211yesmaximal, characteristic
3C83×#3357C218yesmaximal, complemented
4C83×#3329C218yesmaximal, complemented
5C83×#3230C212yesmaximal, complemented
6C83×#3193C214yesmaximal, complemented
7C83×#3101C212yesmaximal, complemented
8C83×#3095C214yesmaximal, complemented
9C83×#2975C211yesmaximal, characteristic, complemented
10#555×C83—24no—
11#555×C83C2218yes—
12#554×C83—24no—
13#554×C83C2214yescomplemented
14D4×Q8×C83C2214yescomplemented
15#544×C83—21no—
16#543×C83—22no—
17#533×C83C2212yes—
18C2×C4⋊Q8×C83C2211yescharacteristic
19C2×#124×C83—22no—
20C2×C22⋊Q8×C83C2214yes—
21C2×#121×C83—21no—
22C2×#121×C83C2212yes—
23#518×C83—21no—
24C2×C4×D4×C83C2211yescharacteristic
25C2×C4×D4×C83C2211yescharacteristic
26C2×#117×C83—21no—
27C2×#117×C83C2212yes—
28#500×C83C2214yescomplemented
29#487×C83C2214yescomplemented
30#485×C83C22116yescomplemented
31#484×C83C2218yes—
32#483×C83C2214yes—
33#480×C83C2218yescomplemented
34#477×C83C2218yescomplemented
35#476×C83C2214yescomplemented
36#468×C83C22116yescomplemented
37#461×C83C2214yescomplemented
38#450×C83C2214yes—
39#448×C83C2218yescomplemented
40C4×SD16×C83C2218yescomplemented
41#436×C83C2218yescomplemented
42#431×C83C2212yes—
43#428×C83C2218yescomplemented
44#419×C83C2218yescomplemented
45#407×C83C2214yes—
46C22×Q8×C83—21no—
47C22×Q8×C83C2314yes—
48C22×D4×C83—22no—
49C22×D4×C83D411yescharacteristic
50C22×D4×C83C2311yescharacteristic
51C23×C4×C83—21no—
52C23×C4×C83C2312yes—
53C2×SD16×C83C2318yes—
54C2×M16×C83C2318yes—
55C4⋊Q8×C83—24no—
56C4⋊Q8×C83—22no—
57C4⋊Q8×C83C2312yes—
58C4⋊Q8×C83C2312yes—
59C4⋊Q8×C83C2314yes—
60#125×C83—24no—
61#125×C83—22no—
62#124×C83—216no—
63C22⋊Q8×C83—24no—
64C22⋊Q8×C83—24no—
65C22⋊Q8×C83—22no—
66C22⋊Q8×C83—28no—
67C22⋊Q8×C83—28no—
68C22⋊Q8×C83C2318yes—
69C22⋊Q8×C83C2314yes—
70#121×C83—28no—
71#121×C83—24no—
72#121×C83C2318yes—
73C4×Q8×C83—22no—
74C4×Q8×C83C2314yes—
75C4×D4×C83—24no—
76C4×D4×C83—24no—
77C4×D4×C83—22no—
78C4×D4×C83D414yes—
79C4×D4×C83C2314yes—
80C4×D4×C83C2314yes—
81#117×C83—24no—
82#117×C83—22no—
83#117×C83—28no—
84#117×C83C2318yes—
85C2×C4⋊C4×C83—21no—
86C2×C4⋊C4×C83—24no—
87C2×C4⋊C4×C83D411yescharacteristic
88C2×C4⋊C4×C83C2314yes—
89C2×C4⋊C4×C83C2311yescharacteristic
90C2×C4⋊C4×C83C2312yes—
91C2×C22⋊C4×C83—22no—
92C2×C22⋊C4×C83—24no—
93C2×C22⋊C4×C83—22no—
94C2×C22⋊C4×C83—21no—
95C2×C22⋊C4×C83D412yes—
96C2×C42×C83—22no—
97C2×C42×C83C2311yescharacteristic
98#107×C83C23116yes—
99#106×C83C2318yes—
100C4⋊C8×C83C2314yes—
101#103×C83C23116yes—
102#103×C83C2318yes—
103#102×C83C2318yes—
104C22⋊C8×C83C2318yes—
105C22⋊C8×C83C2314yes—
106C4×C8×C83C2314yes—
107C24×C83—21no—
108C24×C83C2×D412yes—
109C2×Q8×C83—22no—
110C2×Q8×C83—24no—
111C2×Q8×C83—24no—
112C2×Q8×C83—48no—
113C2×Q8×C83—82no—
114C2×Q8×C83C2414yes—
115C2×D4×C83—28no—
116C2×D4×C83—24no—
117C2×D4×C83—21no—
118C2×D4×C83—42no—
119C2×D4×C83—84no—
120C2×D4×C83C2×D414yes—
121C2×D4×C83C2412yes—
122C22×C4×C83—21no—
123C22×C4×C83—22no—
124C22×C4×C83—24no—
125C22×C4×C83—24no—
126C22×C4×C83—24no—
127C22×C4×C83—44no—
128C22×C4×C83—44no—
129C22×C4×C83C2×D412yes—
130C22×C4×C83C2×D411yescharacteristic
131C22×C4×C83C2×D414yes—
132C22×C4×C83C2×D411yescharacteristic
133C22×C4×C83C2411yescharacteristic
134C22×C4×C83C2414yes—
135SD16×C83—48no—
136M16×C83—48no—
137C2×C8×C83C2418yes—
138C4⋊C4×C83—24no—
139C4⋊C4×C83—24no—
140C4⋊C4×C83—28no—
141C4⋊C4×C83—21no—
142C4⋊C4×C83—22no—
143C4⋊C4×C83—24no—
144C4⋊C4×C83—22no—
145C4⋊C4×C83—22no—
146C4⋊C4×C83—22no—
147C4⋊C4×C83—24no—
148C4⋊C4×C83C2×D414yes—
149C4⋊C4×C83C2414yes—
150C4⋊C4×C83C2412yes—
151C4⋊C4×C83C2414yes—
152C22⋊C4×C83—24no—
153C22⋊C4×C83—28no—
154C22⋊C4×C83—28no—
155C22⋊C4×C83—28no—
156C22⋊C4×C83—24no—
157C22⋊C4×C83C2×D418yes—
158C42×C83—28no—
159C42×C83—24no—
160C42×C83C2×D412yes—
161C42×C83C2412yes—
162C23×C83—24no—
163C23×C83—22no—
164C23×C83—42no—
165C23×C83—42no—
166C23×C83—42no—
167C23×C83—42no—
168C23×C83—42no—
169C23×C83C22×D414yes—
170C23×C83C22×D411yescharacteristic
171Q8×C83—28no—
172Q8×C83—82no—
173Q8×C83—84no—
174Q8×C83—84no—
175D4×C83—28no—
176D4×C83—24no—
177D4×C83—42no—
178D4×C83—88no—
179C2×C4×C83—28no—
180C2×C4×C83—22no—
181C2×C4×C83—22no—
182C2×C4×C83—24no—
183C2×C4×C83—24no—
184C2×C4×C83—24no—
185C2×C4×C83—44no—
186C2×C4×C83—42no—
187C2×C4×C83—41no—
188C2×C4×C83—44no—
189C2×C4×C83—42no—
190C2×C4×C83—48no—
191C2×C4×C83—84no—
192C2×C4×C83—84no—
193C2×C4×C83#13714yes—
194C2×C4×C83C22×D414yes—
195C2×C4×C83C22×D411yescharacteristic
196C2×C4×C83C22×D412yes—
197C2×C4×C83C22×D412yes—
198C2×C4×C83C2511yescharacteristic
199C8×C83—28no—
200C22×C83—21no—
201C22×C83—21no—
202C22×C83—24no—
203C22×C83—24no—
204C22×C83—24no—
205C22×C83—42no—
206C22×C83—42no—
207C22×C83—42no—
208C22×C83—84no—
209C22×C83—84no—
210C22×C83C2×#13712yes—
211C22×C83C23×D411yescharacteristic
212C4×C83—22no—
213C4×C83—21no—
214C4×C83—24no—
215C4×C83—44no—
216C4×C83—44no—
217C4×C83—44no—
218C4×C83—42no—
219C4×C83C2×#13714yes—
220C4×C83#58512yes—
221#61721C8311yesmaximal, characteristic, direct product factor, Sylow
222C2×C83—22no—
223C2×C83—44no—
224C2×C83—82no—
225C2×C83#348511yescharacteristic
226C2×C83C22×#13711yescharacteristic
227C2×C83#358511yescharacteristic
228#3501C2×C8311yescharacteristic, complemented
229C2×#555C2×C8311yescharacteristic
230#3357C2×C8318yescomplemented
231#3329C2×C8318yescomplemented
232#3230C2×C8312yescomplemented
233#3193C2×C8314yescomplemented
234#3101C2×C8312yescomplemented
235#3095C2×C8314yescomplemented
236#2975C2×C8311yescharacteristic, complemented
237C83#6172111yescharacteristic, direct product factor, Sylow
238#555—24no—
239#555C22×C8318yes—
240#554—24no—
241#554C22×C8314yescomplemented
242D4×Q8C22×C8314yescomplemented
243#544—21no—
244#543—22no—
245#533C22×C8312yes—
246C2×C4⋊Q8C22×C8311yescharacteristic
247C2×#124—22no—
248C2×C22⋊Q8C22×C8314yes—
249C2×#121—21no—
250C2×#121C22×C8312yes—
251#518—21no—
252C2×C4×D4C22×C8311yescharacteristic
253C2×C4×D4C22×C8311yescharacteristic
254C2×#117—21no—
255C2×#117C22×C8312yes—
256#500C22×C8314yescomplemented
257#487C22×C8314yescomplemented
258#485C22×C83116yescomplemented
259#484C22×C8318yes—
260#483C22×C8314yes—
261#480C22×C8318yescomplemented
262#477C22×C8318yescomplemented
263#476C22×C8314yescomplemented
264#468C22×C83116yescomplemented
265#461C22×C8314yescomplemented
266#450C22×C8314yes—
267#448C22×C8318yescomplemented
268C4×SD16C22×C8318yescomplemented
269#436C22×C8318yescomplemented
270#431C22×C8312yes—
271#428C22×C8318yescomplemented
272#419C22×C8318yescomplemented
273#407C22×C8314yes—
274C22×Q8—21no—
275C22×Q8C23×C8314yes—
276C22×D4—22no—
277C22×D4D4×C8311yescharacteristic
278C22×D4C23×C8311yescharacteristic
279C23×C4—21no—
280C23×C4C23×C8312yes—
281C2×SD16C23×C8318yes—
282C2×M16C23×C8318yes—
283C4⋊Q8—24no—
284C4⋊Q8—22no—
285C4⋊Q8C23×C8312yes—
286C4⋊Q8C23×C8314yes—
287C4⋊Q8C23×C8312yes—
288#125—24no—
289#125—22no—
290#124—216no—
291C22⋊Q8—24no—
292C22⋊Q8—24no—
293C22⋊Q8—22no—
294C22⋊Q8—28no—
295C22⋊Q8—28no—
296C22⋊Q8C23×C8314yes—
297C22⋊Q8C23×C8318yes—
298#121—28no—
299#121—24no—
300#121C23×C8318yes—
301C4×Q8—22no—
302C4×Q8C23×C8314yes—
303C4×D4—24no—
304C4×D4—24no—
305C4×D4—22no—
306C4×D4D4×C8314yes—
307C4×D4C23×C8314yes—
308C4×D4C23×C8314yes—
309#117—24no—
310#117—22no—
311#117—28no—
312#117C23×C8318yes—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—24no—
315C2×C4⋊C4D4×C8311yescharacteristic
316C2×C4⋊C4C23×C8311yescharacteristic
317C2×C4⋊C4C23×C8312yes—
318C2×C4⋊C4C23×C8314yes—
319C2×C22⋊C4—22no—
320C2×C22⋊C4—24no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—21no—
323C2×C22⋊C4D4×C8312yes—
324C2×C42—22no—
325C2×C42C23×C8311yescharacteristic
326#107C23×C83116yes—
327#106C23×C8318yes—
328C4⋊C8C23×C8314yes—
329#103C23×C8318yes—
330#103C23×C83116yes—
331#102C23×C8318yes—
332C22⋊C8C23×C8314yes—
333C22⋊C8C23×C8318yes—
334C4×C8C23×C8314yes—
335C24—21no—
336C24C2×D4×C8312yes—
337C2×Q8—22no—
338C2×Q8—24no—
339C2×Q8—24no—
340C2×Q8—48no—
341C2×Q8—82no—
342C2×Q8C24×C8314yes—
343C2×D4—28no—
344C2×D4—24no—
345C2×D4—21no—
346C2×D4—42no—
347C2×D4—84no—
348C2×D4C2×D4×C8314yes—
349C2×D4C24×C8312yes—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—24no—
353C22×C4—24no—
354C22×C4—24no—
355C22×C4—44no—
356C22×C4—44no—
357C22×C4C2×D4×C8312yes—
358C22×C4C2×D4×C8311yescharacteristic
359C22×C4C2×D4×C8314yes—
360C22×C4C2×D4×C8311yescharacteristic
361C22×C4C24×C8311yescharacteristic
362C22×C4C24×C8314yes—
363SD16—48no—
364M16—48no—
365C2×C8C24×C8318yes—
366C4⋊C4—24no—
367C4⋊C4—24no—
368C4⋊C4—28no—
369C4⋊C4—21no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4C2×D4×C8314yes—
377C4⋊C4C24×C8314yes—
378C4⋊C4C24×C8312yes—
379C4⋊C4C24×C8314yes—
380C22⋊C4—24no—
381C22⋊C4—28no—
382C22⋊C4—28no—
383C22⋊C4—28no—
384C22⋊C4—24no—
385C22⋊C4C2×D4×C8318yes—
386C42—28no—
387C42—24no—
388C42C2×D4×C8312yes—
389C42C24×C8312yes—
390C23—24no—
391C23—22no—
392C23—42no—
393C23—42no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23C22×D4×C8314yes—
398C23C22×D4×C8311yescharacteristic
399Q8—28no—
400Q8—84no—
401Q8—84no—
402Q8—82no—
403D4—28no—
404D4—24no—
405D4—42no—
406D4—88no—
407C2×C4—28no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—48no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4#137×C8314yes—
422C2×C4C22×D4×C8314yes—
423C2×C4C22×D4×C8311yescharacteristic
424C2×C4C22×D4×C8312yes—
425C2×C4C22×D4×C8312yes—
426C2×C4C25×C8311yescharacteristic
427C8—28no—
428C22—21no—
429C22—21no—
430C22—24no—
431C22—24no—
432C22—24no—
433C22—42no—
434C22—42no—
435C22—42no—
436C22—84no—
437C22—84no—
438C22C2×#137×C8312yes—
439C22C23×D4×C8311yescharacteristic
440C4—22no—
441C4—21no—
442C4—24no—
443C4—42no—
444C4—44no—
445C4—44no—
446C4—44no—
447C4C2×#137×C8314yes—
448C4#585×C8312yes—
449C2—22no—
450C2—44no—
451C2—82no—
452C2C83×#348511yescharacteristic
453C2C22×#137×C8311yescharacteristic
454C2C83×#358511yescharacteristic
subgroupG/H
center, socleC22×C83C23×D4
commutator, Frattini, cosocleC2×C4C25×C83
prime radicalC2×#555×C83C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C4×D4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C83×#3501C2semidirect
3. C83×#3357C2semidirect
4. C83×#3329C2semidirect
5. C83×#3230C2semidirect
6. C83×#3193C2semidirect
7. C83×#3101C2semidirect
8. C83×#3095C2semidirect
9. C83×#2975C2semidirect
13. #554×C83C22semidirect
14. D4×Q8×C83C22semidirect
28. #500×C83C22semidirect
29. #487×C83C22semidirect
30. #485×C83C22semidirect
33. #480×C83C22semidirect
34. #477×C83C22semidirect
35. #476×C83C22semidirect
36. #468×C83C22semidirect
37. #461×C83C22semidirect
39. #448×C83C22semidirect
40. C4×SD16×C83C22semidirect
41. #436×C83C22semidirect
43. #428×C83C22semidirect
44. #419×C83C22semidirect
221. #61721C83direct (N × H)
228. #3501C2×C83semidirect
230. #3357C2×C83semidirect
231. #3329C2×C83semidirect
232. #3230C2×C83semidirect
233. #3193C2×C83semidirect
234. #3101C2×C83semidirect
235. #3095C2×C83semidirect
236. #2975C2×C83semidirect
237. C83#61721direct (N × H)
241. #554C22×C83semidirect
242. D4×Q8C22×C83semidirect
256. #500C22×C83semidirect
257. #487C22×C83semidirect
258. #485C22×C83semidirect
261. #480C22×C83semidirect
262. #477C22×C83semidirect
263. #476C22×C83semidirect
264. #468C22×C83semidirect
265. #461C22×C83semidirect
267. #448C22×C83semidirect
268. C4×SD16C22×C83semidirect
269. #436C22×C83semidirect
271. #428C22×C83semidirect
272. #419C22×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#35012
C2×#555×C832
C83×#33572
C83×#33292
C83×#32302
C83×#31932
C83×#31012
C83×#30952
C83×#29752
#6172183