Finite Group Databasebeta

← C83×#61888 #35289712

Subgroups — C83×#61888 #35289712

Order 21248.

1970 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3497C211yesmaximal, characteristic, complemented
2C2×#556×C83C211yesmaximal, characteristic, complemented
3C83×#3397C218yesmaximal, complemented
4C83×#3367C218yesmaximal, complemented
5C83×#3213C214yesmaximal, complemented
6C83×#3206C212yesmaximal, complemented
7C83×#3112C214yesmaximal, complemented
8C83×#3100C212yesmaximal, complemented
9C83×#2983C211yesmaximal, characteristic, complemented
10C2×2−1+4×C83—21no—
11#556×C83—24no—
12#556×C83C2218yes—
13#550×C83C2214yescomplemented
14D4×Q8×C83C2214yescomplemented
15#548×C83—24no—
16#537×C83—21no—
17#537×C83—22no—
18#532×C83C2212yes—
19#532×C83C2214yes—
20C2×C4⋊Q8×C83C2211yescharacteristic
21C2×#126×C83—22no—
22C2×#124×C83C2212yes—
23C2×C22⋊Q8×C83—21no—
24C2×C4×Q8×C83C2212yes—
25C2×C4×Q8×C83C2211yescharacteristic
26C2×C4×D4×C83C2211yescharacteristic
27C2×#117×C83—21no—
28#496×C83C2218yes—
29#492×C83C2214yescomplemented
30#489×C83C2218yes—
31#487×C83C2214yescomplemented
32#480×C83C2214yes—
33#477×C83C2218yescomplemented
34#474×C83C2214yes—
35#463×C83C22116yes—
36#461×C83C2218yescomplemented
37#460×C83C2218yescomplemented
38#459×C83C2214yes—
39#448×C83C2218yes—
40#447×C83C2214yes—
41#446×C83C2218yescomplemented
42#440×C83C22116yescomplemented
43C4×D8×C83C2218yes—
44#432×C83C2214yes—
45#431×C83C2212yes—
462−1+4×C83—48no—
47C2×#41×C83—24no—
48C2×#41×C83—24no—
49C2×#41×C83—22no—
50C22×Q8×C83—24no—
51C22×Q8×C83D411yescharacteristic
52C22×Q8×C83C2312yes—
53C22×D4×C83C2311yescharacteristic
54C23×C4×C83—21no—
55C2×SD16×C83C23116yes—
56C2×D8×C83C2318yes—
57C4⋊Q8×C83—22no—
58C4⋊Q8×C83C2314yes—
59C4⋊Q8×C83C2312yes—
60C4⋊Q8×C83C2314yes—
61#127×C83C2312yes—
62#127×C83C2314yes—
63#126×C83—216no—
64#124×C83—24no—
65#124×C83—22no—
66#124×C83—24no—
67#124×C83C2318yes—
68C22⋊Q8×C83—28no—
69C22⋊Q8×C83—24no—
70C22⋊Q8×C83—28no—
71#121×C83—24no—
72#121×C83—22no—
73#121×C83—24no—
74#121×C83—28no—
75#121×C83C2314yes—
76C4×Q8×C83—24no—
77C4×Q8×C83D414yes—
78C4×Q8×C83C2318yes—
79C4×Q8×C83C2314yes—
80C4×D4×C83—22no—
81C4×D4×C83—22no—
82C4×D4×C83—24no—
83C4×D4×C83C2318yes—
84C4×D4×C83C2314yes—
85C4×D4×C83C2314yes—
86#117×C83—28no—
87C2×C4⋊C4×C83—21no—
88C2×C4⋊C4×C83—22no—
89C2×C4⋊C4×C83—21no—
90C2×C4⋊C4×C83D411yescharacteristic
91C2×C4⋊C4×C83D412yes—
92C2×C4⋊C4×C83C2311yescharacteristic
93C2×C22⋊C4×C83—22no—
94C2×C22⋊C4×C83—21no—
95C2×C22⋊C4×C83—22no—
96C2×C42×C83—22no—
97C2×C42×C83C2311yescharacteristic
98C2×C42×C83C2312yes—
99#107×C83C2318yes—
100C4⋊C8×C83C2318yes—
101C4⋊C8×C83C2314yes—
102#103×C83C23116yes—
103#102×C83C2318yes—
104#102×C83C2318yes—
105C22⋊C8×C83C2314yes—
106#97×C83C2318yes—
107C4×C8×C83C2314yes—
108C24×C83—21no—
109#41×C83—416no—
110#41×C83—416no—
111#41×C83—48no—
112C2×Q8×C83—24no—
113C2×Q8×C83—22no—
114C2×Q8×C83—24no—
115C2×Q8×C83—22no—
116C2×Q8×C83—44no—
117C2×Q8×C83—416no—
118C2×Q8×C83C2×D414yes—
119C2×Q8×C83C2414yes—
120C2×D4×C83—24no—
121C2×D4×C83—21no—
122C2×D4×C83—24no—
123C2×D4×C83—22no—
124C2×D4×C83—24no—
125C2×D4×C83—22no—
126C2×D4×C83—42no—
127C2×D4×C83C2412yes—
128C2×D4×C83C2414yes—
129C22×C4×C83—21no—
130C22×C4×C83—22no—
131C22×C4×C83—24no—
132C22×C4×C83—28no—
133C22×C4×C83—24no—
134C22×C4×C83—44no—
135C22×C4×C83C2×D412yes—
136C22×C4×C83C2×D412yes—
137C22×C4×C83C2×D411yescharacteristic
138C22×C4×C83C2×D411yescharacteristic
139C22×C4×C83C2411yescharacteristic
140SD16×C83—416no—
141D8×C83—48no—
142C2×C8×C83C2418yes—
143C4⋊C4×C83—24no—
144C4⋊C4×C83—28no—
145C4⋊C4×C83—24no—
146C4⋊C4×C83—21no—
147C4⋊C4×C83—24no—
148C4⋊C4×C83—22no—
149C4⋊C4×C83C2×D414yes—
150C4⋊C4×C83C2×D418yes—
151C4⋊C4×C83C2412yes—
152C4⋊C4×C83C2414yes—
153C22⋊C4×C83—28no—
154C22⋊C4×C83—24no—
155C22⋊C4×C83—28no—
156C22⋊C4×C83—24no—
157C22⋊C4×C83—28no—
158C22⋊C4×C83—24no—
159C42×C83—28no—
160C42×C83C2×D412yes—
161C42×C83C2×D414yes—
162C42×C83C2412yes—
163C42×C83C2414yes—
164C23×C83—22no—
165C23×C83—24no—
166C23×C83—42no—
167C23×C83—42no—
168C23×C83—42no—
169C23×C83C22×D411yescharacteristic
170Q8×C83—28no—
171Q8×C83—28no—
172Q8×C83—44no—
173Q8×C83—48no—
174Q8×C83—44no—
175D4×C83—24no—
176D4×C83—28no—
177D4×C83—42no—
178D4×C83—48no—
179D4×C83—48no—
180D4×C83—44no—
181D4×C83—84no—
182D4×C83—82no—
183C2×C4×C83—24no—
184C2×C4×C83—22no—
185C2×C4×C83—22no—
186C2×C4×C83—24no—
187C2×C4×C83—24no—
188C2×C4×C83—42no—
189C2×C4×C83—41no—
190C2×C4×C83—416no—
191C2×C4×C83—44no—
192C2×C4×C83—42no—
193C2×C4×C83—48no—
194C2×C4×C83—48no—
195C2×C4×C83AGL(1,ℤ/8)14yes—
196C2×C4×C83C22×D414yes—
197C2×C4×C83C22×D411yescharacteristic
198C2×C4×C83C22×D412yes—
199C2×C4×C83C22×D414yes—
200C2×C4×C83C22×D412yes—
201C2×C4×C83C2511yescharacteristic
202C8×C83—28no—
203C22×C83—21no—
204C22×C83—21no—
205C22×C83—44no—
206C22×C83—42no—
207C22×C83—42no—
208C22×C83—44no—
209C22×C83—44no—
210C22×C83—42no—
211C22×C83—84no—
212C22×C83C2×AGL(1,ℤ/8)12yes—
213C22×C83C23×D411yescharacteristic
214C4×C83—24no—
215C4×C83—22no—
216C4×C83—21no—
217C4×C83—24no—
218C4×C83—44no—
219C4×C83—42no—
220C4×C83—44no—
221C4×C83C2×AGL(1,ℤ/8)14yes—
222C4×C83#58512yes—
223#61888C8311yesmaximal, characteristic, direct product factor, Sylow
224C2×C83—22no—
225C2×C83—84no—
226C2×C83—82no—
227C2×C83#348711yescharacteristic
228C2×C83C22×AGL(1,ℤ/8)11yescharacteristic
229C2×C83#358611yescharacteristic
230#3497C2×C8311yescharacteristic, complemented
231C2×#556C2×C8311yescharacteristic, complemented
232#3397C2×C8318yescomplemented
233#3367C2×C8318yescomplemented
234#3213C2×C8314yescomplemented
235#3206C2×C8312yescomplemented
236#3112C2×C8314yescomplemented
237#3100C2×C8312yescomplemented
238#2983C2×C8311yescharacteristic, complemented
239C83#6188811yescharacteristic, direct product factor, Sylow
240C2×2−1+4—21no—
241#556—24no—
242#556C22×C8318yes—
243#550C22×C8314yescomplemented
244D4×Q8C22×C8314yescomplemented
245#548—24no—
246#537—21no—
247#537—22no—
248#532C22×C8312yes—
249#532C22×C8314yes—
250C2×C4⋊Q8C22×C8311yescharacteristic
251C2×#126—22no—
252C2×#124C22×C8312yes—
253C2×C22⋊Q8—21no—
254C2×C4×Q8C22×C8312yes—
255C2×C4×Q8C22×C8311yescharacteristic
256C2×C4×D4C22×C8311yescharacteristic
257C2×#117—21no—
258#496C22×C8318yes—
259#492C22×C8314yescomplemented
260#489C22×C8318yes—
261#487C22×C8314yescomplemented
262#480C22×C8314yes—
263#477C22×C8318yescomplemented
264#474C22×C8314yes—
265#463C22×C83116yes—
266#461C22×C8318yescomplemented
267#460C22×C8318yescomplemented
268#459C22×C8314yes—
269#448C22×C8318yes—
270#447C22×C8314yes—
271#446C22×C8318yescomplemented
272#440C22×C83116yescomplemented
273C4×D8C22×C8318yes—
274#432C22×C8314yes—
275#431C22×C8312yes—
2762−1+4—48no—
277C2×#41—24no—
278C2×#41—24no—
279C2×#41—22no—
280C22×Q8—24no—
281C22×Q8D4×C8311yescharacteristic
282C22×Q8C23×C8312yes—
283C22×D4C23×C8311yescharacteristic
284C23×C4—21no—
285C2×SD16C23×C83116yes—
286C2×D8C23×C8318yes—
287C4⋊Q8—22no—
288C4⋊Q8C23×C8314yes—
289C4⋊Q8C23×C8312yes—
290C4⋊Q8C23×C8314yes—
291#127C23×C8312yes—
292#127C23×C8314yes—
293#126—216no—
294#124—24no—
295#124—22no—
296#124—24no—
297#124C23×C8318yes—
298C22⋊Q8—28no—
299C22⋊Q8—24no—
300C22⋊Q8—28no—
301#121—24no—
302#121—22no—
303#121—24no—
304#121—28no—
305#121C23×C8314yes—
306C4×Q8—24no—
307C4×Q8D4×C8314yes—
308C4×Q8C23×C8318yes—
309C4×Q8C23×C8314yes—
310C4×D4—22no—
311C4×D4—22no—
312C4×D4—24no—
313C4×D4C23×C8314yes—
314C4×D4C23×C8314yes—
315C4×D4C23×C8318yes—
316#117—28no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4D4×C8311yescharacteristic
321C2×C4⋊C4D4×C8312yes—
322C2×C4⋊C4C23×C8311yescharacteristic
323C2×C22⋊C4—22no—
324C2×C22⋊C4—21no—
325C2×C22⋊C4—22no—
326C2×C42—22no—
327C2×C42C23×C8311yescharacteristic
328C2×C42C23×C8312yes—
329#107C23×C8318yes—
330C4⋊C8C23×C8314yes—
331C4⋊C8C23×C8318yes—
332#103C23×C83116yes—
333#102C23×C8318yes—
334#102C23×C8318yes—
335C22⋊C8C23×C8314yes—
336#97C23×C8318yes—
337C4×C8C23×C8314yes—
338C24—21no—
339#41—416no—
340#41—416no—
341#41—48no—
342C2×Q8—24no—
343C2×Q8—22no—
344C2×Q8—24no—
345C2×Q8—22no—
346C2×Q8—44no—
347C2×Q8—416no—
348C2×Q8C2×D4×C8314yes—
349C2×Q8C24×C8314yes—
350C2×D4—24no—
351C2×D4—21no—
352C2×D4—24no—
353C2×D4—22no—
354C2×D4—24no—
355C2×D4—22no—
356C2×D4—42no—
357C2×D4C24×C8312yes—
358C2×D4C24×C8314yes—
359C22×C4—21no—
360C22×C4—22no—
361C22×C4—24no—
362C22×C4—28no—
363C22×C4—24no—
364C22×C4—44no—
365C22×C4C2×D4×C8312yes—
366C22×C4C2×D4×C8312yes—
367C22×C4C2×D4×C8311yescharacteristic
368C22×C4C2×D4×C8311yescharacteristic
369C22×C4C24×C8311yescharacteristic
370SD16—416no—
371D8—48no—
372C2×C8C24×C8318yes—
373C4⋊C4—24no—
374C4⋊C4—28no—
375C4⋊C4—24no—
376C4⋊C4—21no—
377C4⋊C4—24no—
378C4⋊C4—22no—
379C4⋊C4C2×D4×C8314yes—
380C4⋊C4C2×D4×C8318yes—
381C4⋊C4C24×C8312yes—
382C4⋊C4C24×C8314yes—
383C22⋊C4—28no—
384C22⋊C4—24no—
385C22⋊C4—28no—
386C22⋊C4—24no—
387C22⋊C4—28no—
388C22⋊C4—24no—
389C42—28no—
390C42C2×D4×C8312yes—
391C42C2×D4×C8314yes—
392C42C24×C8312yes—
393C42C24×C8314yes—
394C23—22no—
395C23—24no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23C22×D4×C8311yescharacteristic
400Q8—28no—
401Q8—28no—
402Q8—48no—
403Q8—44no—
404Q8—44no—
405D4—24no—
406D4—28no—
407D4—42no—
408D4—48no—
409D4—48no—
410D4—44no—
411D4—84no—
412D4—82no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—416no—
421C2×C4—44no—
422C2×C4—42no—
423C2×C4—48no—
424C2×C4—48no—
425C2×C4AGL(1,ℤ/8)×C8314yes—
426C2×C4C22×D4×C8314yes—
427C2×C4C22×D4×C8311yescharacteristic
428C2×C4C22×D4×C8312yes—
429C2×C4C22×D4×C8314yes—
430C2×C4C22×D4×C8312yes—
431C2×C4C25×C8311yescharacteristic
432C8—28no—
433C22—21no—
434C22—21no—
435C22—44no—
436C22—42no—
437C22—44no—
438C22—42no—
439C22—42no—
440C22—44no—
441C22—84no—
442C22C2×AGL(1,ℤ/8)×C8312yes—
443C22C23×D4×C8311yescharacteristic
444C4—24no—
445C4—24no—
446C4—22no—
447C4—21no—
448C4—44no—
449C4—42no—
450C4—44no—
451C4C2×AGL(1,ℤ/8)×C8314yes—
452C4#585×C8312yes—
453C2—22no—
454C2—82no—
455C2—84no—
456C2C83×#348711yescharacteristic
457C2C22×AGL(1,ℤ/8)×C8311yescharacteristic
458C2C83×#358611yescharacteristic
subgroupG/H
center, socleC22×C83C23×D4
commutator, Frattini, cosocleC2×C4C25×C83
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×#3497C2semidirect
2. C2×#556×C83C2semidirect
3. C83×#3397C2semidirect
4. C83×#3367C2semidirect
5. C83×#3213C2semidirect
6. C83×#3206C2semidirect
7. C83×#3112C2semidirect
8. C83×#3100C2semidirect
9. C83×#2983C2semidirect
13. #550×C83C22semidirect
14. D4×Q8×C83C22semidirect
29. #492×C83C22semidirect
31. #487×C83C22semidirect
33. #477×C83C22semidirect
36. #461×C83C22semidirect
37. #460×C83C22semidirect
41. #446×C83C22semidirect
42. #440×C83C22semidirect
223. #61888C83direct (N × H)
230. #3497C2×C83semidirect
231. C2×#556C2×C83semidirect
232. #3397C2×C83semidirect
233. #3367C2×C83semidirect
234. #3213C2×C83semidirect
235. #3206C2×C83semidirect
236. #3112C2×C83semidirect
237. #3100C2×C83semidirect
238. #2983C2×C83semidirect
239. C83#61888direct (N × H)
243. #550C22×C83semidirect
244. D4×Q8C22×C83semidirect
259. #492C22×C83semidirect
261. #487C22×C83semidirect
263. #477C22×C83semidirect
266. #461C22×C83semidirect
267. #460C22×C83semidirect
271. #446C22×C83semidirect
272. #440C22×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#34972
C2×#556×C832
C83×#33972
C83×#33672
C83×#32132
C83×#32062
C83×#31122
C83×#31002
C83×#29832
#6188883