Finite Group Databasebeta

← C83×#61290 #35289114

Subgroups — C83×#61290 #35289114

Order 21248.

1774 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3512C212yesmaximal, complemented
2C83×#3355C214yesmaximal, complemented
3C83×#3352C214yesmaximal, complemented
4C83×#3337C218yesmaximal, complemented
5C2×#499×C83C211yesmaximal, characteristic, complemented
6C83×#3098C214yesmaximal, complemented
7C83×#3080C212yesmaximal, complemented
8C83×#3060C212yesmaximal, complemented
9C4×#135×C83C211yesmaximal, characteristic, complemented
10C83×#2931C211yesmaximal, characteristic, complemented
11C83×#2912C212yesmaximal, complemented
12C2×#135×C83C2211yescharacteristic
13C2×#131×C83C2212yes—
14#553×C83—28no—
15#553×C83C2218yes—
16#552×C83C2218yes—
17#541×C83—22no—
18#540×C83—22no—
19#537×C83C2214yes—
20#532×C83C2211yescharacteristic
21#529×C83—24no—
22C2×#125×C83C2212yes—
23#520×C83C2214yes—
24C4×#41×C83C2212yes—
25#499×C83C2214yes—
26#499×C83C2214yes—
27#485×C83C2218yes—
28#484×C83C2218yes—
29#482×C83C2218yes—
30#480×C83C2218yes—
31#479×C83C2218yes—
32#478×C83C2218yes—
33#470×C83C2214yes—
34#469×C83C2214yes—
35#465×C83C2218yes—
36C4×Q16×C83C2212yes—
37C4×SD16×C83C2214yes—
38C4×D8×C83C2212yes—
39#436×C83C2214yes—
40C8×D4×C83C2214yes—
41#428×C83C2214yes—
42#428×C83C2218yes—
43#427×C83C2214yes—
44#427×C83C2211yescharacteristic
45C2×#107×C83C2212yes—
46C2×#106×C83C2212yes—
47C2×#106×C83C2214yes—
48#423×C83C2211yescharacteristic
49#417×C83C2214yes—
50#416×C83C2212yes—
51#408×C83C2212yes—
52C2×C4×C8×C83C2211yescharacteristic
53C2×#41×C83C2312yes—
54C2×#41×C83C2312yes—
55#135×C83—24no—
56C2×Q16×C83C2311yescharacteristic
57C2×SD16×C83C2312yes—
58C2×D8×C83C2311yescharacteristic
59#131×C83—28no—
60C2×M16×C83C2314yes—
61C2×M16×C83C2312yes—
62C22×C8×C83C2314yes—
63C22×C8×C83C2311yescharacteristic
64C22×C8×C83C2311yescharacteristic
65C4⋊Q8×C83D411yescharacteristic
66#127×C83D411yescharacteristic
67#126×C83—28no—
68#126×C83—28no—
69#125×C83—28no—
70#125×C83—22no—
71#125×C83—22no—
72#125×C83C2318yes—
73#124×C83D412yes—
74#123×C83—28no—
75#123×C83—28no—
76C22⋊Q8×C83—24no—
77C22⋊Q8×C83C2318yes—
78#121×C83—24no—
79#121×C83—28no—
80#121×C83—28no—
81#121×C83D414yes—
82#121×C83C2318yes—
83C4×Q8×C83—24no—
84C4×Q8×C83C2314yes—
85C4×D4×C83—24no—
86C4×D4×C83—28no—
87C4×D4×C83—24no—
88C4×D4×C83—28no—
89C4×D4×C83C2314yes—
90C4×D4×C83C2314yes—
91#117×C83—28no—
92#117×C83—22no—
93#117×C83C2318yes—
94#117×C83C2312yes—
95C2×C4⋊C4×C83—24no—
96C2×C4⋊C4×C83C2318yes—
97C2×C4⋊C4×C83C2314yes—
98C2×C42×C83—22no—
99C2×C42×C83C2311yescharacteristic
100#107×C83C2318yes—
101#107×C83C2314yes—
102#107×C83C2314yes—
103#107×C83C2312yes—
104#106×C83C2314yes—
105#106×C83C2314yes—
106#106×C83C2318yes—
107#106×C83C2312yes—
108C4⋊C8×C83C2314yes—
109#103×C83C2318yes—
110#103×C83C2314yes—
111#102×C83C2318yes—
112#102×C83C2314yes—
113C22⋊C8×C83C2318yes—
114C4×C8×C83C2312yes—
115C4×C8×C83C2312yes—
116#41×C83—24no—
117#41×C83—24no—
118#41×C83—24no—
119#41×C83—24no—
120C2×Q8×C83C2×D412yes—
121C2×Q8×C83C2412yes—
122C2×D4×C83—22no—
123C2×D4×C83—28no—
124C2×D4×C83C2×D414yes—
125C2×D4×C83C2×D412yes—
126C2×D4×C83C2×D414yes—
127C2×D4×C83C2412yes—
128C22×C4×C83—24no—
129C22×C4×C83—28no—
130C22×C4×C83—22no—
131C22×C4×C83—24no—
132C22×C4×C83C2×D412yes—
133C22×C4×C83C2414yes—
134C22×C4×C83C2411yescharacteristic
135Q16×C83—22no—
136SD16×C83—24no—
137D8×C83—22no—
138M16×C83—24no—
139M16×C83—28no—
140C2×C8×C83—22no—
141C2×C8×C83—28no—
142C2×C8×C83#4114yes—
143C2×C8×C83C2412yes—
144C2×C8×C83C2414yes—
145C2×C8×C83C2411yescharacteristic
146C2×C8×C83C2411yescharacteristic
147C4⋊C4×C83—28no—
148C4⋊C4×C83—28no—
149C4⋊C4×C83—24no—
150C4⋊C4×C83—28no—
151C4⋊C4×C83—24no—
152C4⋊C4×C83—28no—
153C4⋊C4×C83C2×D414yes—
154C4⋊C4×C83C2414yes—
155C4⋊C4×C83C2418yes—
156C22⋊C4×C83—28no—
157C22⋊C4×C83—22no—
158C22⋊C4×C83—28no—
159C22⋊C4×C83—24no—
160C22⋊C4×C83—28no—
161C22⋊C4×C83C2×D418yes—
162C42×C83—24no—
163C42×C83—22no—
164C42×C83—28no—
165C42×C83C2×D411yescharacteristic
166C42×C83C2×D411yescharacteristic
167C42×C83C2412yes—
168C23×C83—22no—
169C23×C83C22×D414yes—
170C23×C83C22×D411yescharacteristic
171Q8×C83—24no—
172Q8×C83—22no—
173Q8×C83—22no—
174D4×C83—28no—
175D4×C83—24no—
176D4×C83—22no—
177D4×C83—22no—
178D4×C83—44no—
179D4×C83—44no—
180D4×C83—88no—
181C2×C4×C83—24no—
182C2×C4×C83—24no—
183C2×C4×C83—24no—
184C2×C4×C83—22no—
185C2×C4×C83—28no—
186C2×C4×C83—42no—
187C2×C4×C83—44no—
188C2×C4×C83—84no—
189C2×C4×C83—88no—
190C2×C4×C83—84no—
191C2×C4×C83C22×D412yes—
192C2×C4×C83C22×D414yes—
193C2×C4×C83C22×D412yes—
194C2×C4×C83C22×D411yescharacteristic
195C2×C4×C83C2×#4112yes—
196C2×C4×C832−1+412yes—
197C2×C4×C83C2511yescharacteristic
198C8×C83—24no—
199C8×C83—22no—
200C8×C83C2×#4112yes—
201C8×C83C2×#4112yes—
202C22×C83—22no—
203C22×C83—28no—
204C22×C83—24no—
205C22×C83—44no—
206C22×C83—42no—
207C22×C83#54712yes—
208C22×C83C23×D411yescharacteristic
209C4×C83—22no—
210C4×C83—22no—
211C4×C83—24no—
212C4×C83—44no—
213C4×C83—48no—
214C4×C83—42no—
215C4×C83#54712yes—
216C4×C83C22×#4111yescharacteristic
217C4×C83C2×2−1+411yescharacteristic
218#61290C8311yesmaximal, characteristic, direct product factor, Sylow
219C2×C83—22no—
220C2×C83—44no—
221C2×C83—82no—
222C2×C83C2×#54711yescharacteristic
223C2×C83#358412yes—
224#3512C2×C8312yescomplemented
225#3355C2×C8314yescomplemented
226#3352C2×C8314yescomplemented
227#3337C2×C8318yescomplemented
228C2×#499C2×C8311yescharacteristic, complemented
229#3098C2×C8314yescomplemented
230#3080C2×C8312yescomplemented
231#3060C2×C8312yescomplemented
232C4×#135C2×C8311yescharacteristic, complemented
233#2931C2×C8311yescharacteristic, complemented
234#2912C2×C8312yescomplemented
235C83#6129011yescharacteristic, direct product factor, Sylow
236C2×#135C22×C8311yescharacteristic
237C2×#131C22×C8312yes—
238#553—28no—
239#553C22×C8318yes—
240#552C22×C8318yes—
241#541—22no—
242#540—22no—
243#537C22×C8314yes—
244#532C22×C8311yescharacteristic
245#529—24no—
246C2×#125C22×C8312yes—
247#520C22×C8314yes—
248C4×#41C22×C8312yes—
249#499C22×C8314yes—
250#499C22×C8314yes—
251#485C22×C8318yes—
252#484C22×C8318yes—
253#482C22×C8318yes—
254#480C22×C8318yes—
255#479C22×C8318yes—
256#478C22×C8318yes—
257#470C22×C8314yes—
258#469C22×C8314yes—
259#465C22×C8318yes—
260C4×Q16C22×C8312yes—
261C4×SD16C22×C8314yes—
262C4×D8C22×C8312yes—
263#436C22×C8314yes—
264C8×D4C22×C8314yes—
265#428C22×C8314yes—
266#428C22×C8318yes—
267#427C22×C8314yes—
268#427C22×C8311yescharacteristic
269C2×#107C22×C8312yes—
270C2×#106C22×C8314yes—
271C2×#106C22×C8312yes—
272#423C22×C8311yescharacteristic
273#417C22×C8314yes—
274#416C22×C8312yes—
275#408C22×C8312yes—
276C2×C4×C8C22×C8311yescharacteristic
277C2×#41C23×C8312yes—
278C2×#41C23×C8312yes—
279#135—24no—
280C2×Q16C23×C8311yescharacteristic
281C2×SD16C23×C8312yes—
282C2×D8C23×C8311yescharacteristic
283#131—28no—
284C2×M16C23×C8312yes—
285C2×M16C23×C8314yes—
286C22×C8C23×C8314yes—
287C22×C8C23×C8311yescharacteristic
288C22×C8C23×C8311yescharacteristic
289C4⋊Q8D4×C8311yescharacteristic
290#127D4×C8311yescharacteristic
291#126—28no—
292#126—28no—
293#125—28no—
294#125—22no—
295#125—22no—
296#125C23×C8318yes—
297#124D4×C8312yes—
298#123—28no—
299#123—28no—
300C22⋊Q8—24no—
301C22⋊Q8C23×C8318yes—
302#121—24no—
303#121—28no—
304#121—28no—
305#121D4×C8314yes—
306#121C23×C8318yes—
307C4×Q8—24no—
308C4×Q8C23×C8314yes—
309C4×D4—24no—
310C4×D4—28no—
311C4×D4—24no—
312C4×D4—28no—
313C4×D4C23×C8314yes—
314C4×D4C23×C8314yes—
315#117—28no—
316#117—22no—
317#117C23×C8312yes—
318#117C23×C8318yes—
319C2×C4⋊C4—24no—
320C2×C4⋊C4C23×C8314yes—
321C2×C4⋊C4C23×C8318yes—
322C2×C42—22no—
323C2×C42C23×C8311yescharacteristic
324#107C23×C8314yes—
325#107C23×C8314yes—
326#107C23×C8312yes—
327#107C23×C8318yes—
328#106C23×C8312yes—
329#106C23×C8314yes—
330#106C23×C8314yes—
331#106C23×C8318yes—
332C4⋊C8C23×C8314yes—
333#103C23×C8314yes—
334#103C23×C8318yes—
335#102C23×C8314yes—
336#102C23×C8318yes—
337C22⋊C8C23×C8318yes—
338C4×C8C23×C8312yes—
339C4×C8C23×C8312yes—
340#41—24no—
341#41—24no—
342#41—24no—
343#41—24no—
344C2×Q8C2×D4×C8312yes—
345C2×Q8C24×C8312yes—
346C2×D4—22no—
347C2×D4—28no—
348C2×D4C2×D4×C8314yes—
349C2×D4C2×D4×C8312yes—
350C2×D4C2×D4×C8314yes—
351C2×D4C24×C8312yes—
352C22×C4—24no—
353C22×C4—28no—
354C22×C4—22no—
355C22×C4—24no—
356C22×C4C2×D4×C8312yes—
357C22×C4C24×C8314yes—
358C22×C4C24×C8311yescharacteristic
359Q16—22no—
360SD16—24no—
361D8—22no—
362M16—24no—
363M16—28no—
364C2×C8—22no—
365C2×C8—28no—
366C2×C8#41×C8314yes—
367C2×C8C24×C8312yes—
368C2×C8C24×C8314yes—
369C2×C8C24×C8311yescharacteristic
370C2×C8C24×C8311yescharacteristic
371C4⋊C4—28no—
372C4⋊C4—28no—
373C4⋊C4—24no—
374C4⋊C4—28no—
375C4⋊C4—24no—
376C4⋊C4—28no—
377C4⋊C4C2×D4×C8314yes—
378C4⋊C4C24×C8314yes—
379C4⋊C4C24×C8318yes—
380C22⋊C4—28no—
381C22⋊C4—22no—
382C22⋊C4—28no—
383C22⋊C4—24no—
384C22⋊C4—28no—
385C22⋊C4C2×D4×C8318yes—
386C42—24no—
387C42—22no—
388C42—28no—
389C42C2×D4×C8311yescharacteristic
390C42C2×D4×C8311yescharacteristic
391C42C24×C8312yes—
392C23—22no—
393C23C22×D4×C8314yes—
394C23C22×D4×C8311yescharacteristic
395Q8—24no—
396Q8—22no—
397Q8—22no—
398D4—28no—
399D4—24no—
400D4—22no—
401D4—22no—
402D4—44no—
403D4—44no—
404D4—88no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—22no—
409C2×C4—28no—
410C2×C4—42no—
411C2×C4—44no—
412C2×C4—84no—
413C2×C4—88no—
414C2×C4—84no—
415C2×C4C22×D4×C8312yes—
416C2×C4C22×D4×C8314yes—
417C2×C4C22×D4×C8312yes—
418C2×C4C22×D4×C8311yescharacteristic
419C2×C4C2×#41×C8312yes—
420C2×C42−1+4×C8312yes—
421C2×C4C25×C8311yescharacteristic
422C8—24no—
423C8—22no—
424C8C2×#41×C8312yes—
425C8C2×#41×C8312yes—
426C22—22no—
427C22—28no—
428C22—24no—
429C22—44no—
430C22—42no—
431C22#547×C8312yes—
432C22C23×D4×C8311yescharacteristic
433C4—22no—
434C4—22no—
435C4—24no—
436C4—44no—
437C4—48no—
438C4—42no—
439C4#547×C8312yes—
440C4C22×#41×C8311yescharacteristic
441C4C2×2−1+4×C8311yescharacteristic
442C2—22no—
443C2—44no—
444C2—82no—
445C2C2×#547×C8311yescharacteristic
446C2C83×#358412yes—
subgroupG/H
center, socleC22×C83C23×D4
commutator, Frattini, cosocleC2×C4C25×C83
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C83 ◁ #532×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C83×#3512C2semidirect
2. C83×#3355C2semidirect
3. C83×#3352C2semidirect
4. C83×#3337C2semidirect
5. C2×#499×C83C2semidirect
6. C83×#3098C2semidirect
7. C83×#3080C2semidirect
8. C83×#3060C2semidirect
9. C4×#135×C83C2semidirect
10. C83×#2931C2semidirect
11. C83×#2912C2semidirect
218. #61290C83direct (N × H)
224. #3512C2×C83semidirect
225. #3355C2×C83semidirect
226. #3352C2×C83semidirect
227. #3337C2×C83semidirect
228. C2×#499C2×C83semidirect
229. #3098C2×C83semidirect
230. #3080C2×C83semidirect
231. #3060C2×C83semidirect
232. C4×#135C2×C83semidirect
233. #2931C2×C83semidirect
234. #2912C2×C83semidirect
235. C83#61290direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#35122
C83×#33552
C83×#33522
C83×#33372
C2×#499×C832
C83×#30982
C83×#30802
C83×#30602
C4×#135×C832
C83×#29312
C83×#29122
#6129083