Finite Group Databasebeta

← C83×#61826 #35289650

Subgroups — C83×#61826 #35289650

Order 21248.

1726 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3502C211yesmaximal, characteristic, complemented
2C2×#554×C83C211yesmaximal, characteristic
3C83×#3385C218yesmaximal, complemented
4C83×#3372C218yesmaximal, complemented
5C83×#3236C214yesmaximal, complemented
6C83×#3230C212yesmaximal, complemented
7C83×#3110C214yesmaximal
8C83×#3101C212yesmaximal
9C83×#2983C211yesmaximal, characteristic, complemented
10Q82×C83C2214yescomplemented
11#557×C83—24no—
12#557×C83C2214yescomplemented
13#554×C83—24no—
14#554×C83C2218yes—
15#544×C83—21no—
16#544×C83—22no—
17#533×C83C2212yes—
18#533×C83C2214yes—
19C2×C4⋊Q8×C83C2211yescharacteristic
20C2×#125×C83C2212yes—
21C2×C22⋊Q8×C83—21no—
22C2×C22⋊Q8×C83—22no—
23#519×C83—21no—
24C2×C4×Q8×C83C2211yescharacteristic
25C2×C4×D4×C83C2211yescharacteristic
26C2×C4×D4×C83C2212yes—
27C22×C4⋊C4×C83—21no—
28#501×C83C2218yescomplemented
29#500×C83C2214yescomplemented
30#491×C83C2218yescomplemented
31#487×C83C2214yescomplemented
32#484×C83C2218yes—
33#483×C83C2214yes—
34#479×C83C22116yescomplemented
35#477×C83C2218yescomplemented
36#476×C83C2214yes—
37#474×C83C2218yes—
38#461×C83C2214yes—
39#461×C83C2218yes—
40#450×C83C2214yes—
41#446×C83C2218yescomplemented
42#440×C83C22116yes—
43C4×SD16×C83C2218yes—
44#432×C83C2214yes—
45#431×C83C2212yes—
46C22×Q8×C83—21no—
47C22×Q8×C83D411yescharacteristic
48C22×D4×C83C2311yescharacteristic
49C23×C4×C83—21no—
50C23×C4×C83—22no—
51C2×SD16×C83C2318yes—
52C4⋊Q8×C83—24no—
53C4⋊Q8×C83—28no—
54C4⋊Q8×C83—22no—
55C4⋊Q8×C83C2312yes—
56C4⋊Q8×C83C2314yes—
57C4⋊Q8×C83C2314yes—
58C4⋊Q8×C83C2312yes—
59C4⋊Q8×C83C2314yes—
60#125×C83—24no—
61#125×C83—28no—
62#125×C83—24no—
63#125×C83—24no—
64#125×C83—22no—
65#125×C83C2318yes—
66C22⋊Q8×C83—28no—
67C22⋊Q8×C83—22no—
68C22⋊Q8×C83—24no—
69C22⋊Q8×C83—216no—
70C22⋊Q8×C83C2314yes—
71C4×Q8×C83—24no—
72C4×Q8×C83—24no—
73C4×Q8×C83—24no—
74C4×Q8×C83—22no—
75C4×Q8×C83D414yes—
76C4×Q8×C83C2314yes—
77C4×Q8×C83C2318yes—
78C4×Q8×C83C2314yes—
79C4×D4×C83—24no—
80C4×D4×C83—22no—
81C4×D4×C83C2318yes—
82C4×D4×C83C2314yes—
83#117×C83—24no—
84#117×C83—22no—
85C2×C4⋊C4×C83—22no—
86C2×C4⋊C4×C83—21no—
87C2×C4⋊C4×C83—22no—
88C2×C4⋊C4×C83—28no—
89C2×C4⋊C4×C83D411yescharacteristic
90C2×C4⋊C4×C83D412yes—
91C2×C4⋊C4×C83C2311yescharacteristic
92C2×C4⋊C4×C83C2312yes—
93C2×C22⋊C4×C83—21no—
94C2×C22⋊C4×C83—22no—
95C2×C42×C83C2312yes—
96C2×C42×C83C2311yescharacteristic
97#107×C83C23116yes—
98#106×C83C2318yes—
99C4⋊C8×C83C2318yes—
100C4⋊C8×C83C2314yes—
101#103×C83C23116yes—
102#103×C83C2318yes—
103#102×C83C2318yes—
104C22⋊C8×C83C2314yes—
105#97×C83C2318yes—
106C4×C8×C83C2314yes—
107C24×C83—21no—
108C2×Q8×C83—24no—
109C2×Q8×C83—22no—
110C2×Q8×C83—24no—
111C2×Q8×C83—24no—
112C2×Q8×C83—82no—
113C2×Q8×C83C2×D414yes—
114C2×Q8×C83C2414yes—
115C2×D4×C83—24no—
116C2×D4×C83—21no—
117C2×D4×C83C2412yes—
118C22×C4×C83—21no—
119C22×C4×C83—22no—
120C22×C4×C83—24no—
121C22×C4×C83—28no—
122C22×C4×C83—44no—
123C22×C4×C83—84no—
124C22×C4×C83C2×D412yes—
125C22×C4×C83C2×D411yescharacteristic
126C22×C4×C83C2×D412yes—
127C22×C4×C83C2×D411yescharacteristic
128C22×C4×C83C2411yescharacteristic
129SD16×C83—216no—
130C2×C8×C83C2418yes—
131C4⋊C4×C83—28no—
132C4⋊C4×C83—24no—
133C4⋊C4×C83—28no—
134C4⋊C4×C83—21no—
135C4⋊C4×C83—22no—
136C4⋊C4×C83—24no—
137C4⋊C4×C83—22no—
138C4⋊C4×C83—24no—
139C4⋊C4×C83—22no—
140C4⋊C4×C83—24no—
141C4⋊C4×C83—22no—
142C4⋊C4×C83—24no—
143C4⋊C4×C83—24no—
144C4⋊C4×C83—28no—
145C4⋊C4×C83C2×D414yes—
146C4⋊C4×C83C2×D418yes—
147C4⋊C4×C83C2414yes—
148C4⋊C4×C83C2412yes—
149C4⋊C4×C83C2414yes—
150C22⋊C4×C83—24no—
151C22⋊C4×C83—24no—
152C22⋊C4×C83—28no—
153C42×C83—28no—
154C42×C83—24no—
155C42×C83C2×D414yes—
156C42×C83C2×D412yes—
157C42×C83C2412yes—
158C42×C83C2414yes—
159C23×C83—24no—
160C23×C83—22no—
161C23×C83—42no—
162C23×C83—42no—
163C23×C83C22×D411yescharacteristic
164Q8×C83—28no—
165Q8×C83—28no—
166Q8×C83—84no—
167Q8×C83—82no—
168Q8×C83—84no—
169D4×C83—24no—
170D4×C83—22no—
171D4×C832−1+414yes—
172C2×C4×C83—24no—
173C2×C4×C83—22no—
174C2×C4×C83—24no—
175C2×C4×C83—24no—
176C2×C4×C83—22no—
177C2×C4×C83—24no—
178C2×C4×C83—42no—
179C2×C4×C83—41no—
180C2×C4×C83—44no—
181C2×C4×C83—42no—
182C2×C4×C83—48no—
183C2×C4×C83—88no—
184C2×C4×C83#13714yes—
185C2×C4×C83C22×D414yes—
186C2×C4×C83C22×D411yescharacteristic
187C2×C4×C83C22×D412yes—
188C2×C4×C83C22×D414yes—
189C2×C4×C83C22×D412yes—
190C2×C4×C83C2511yescharacteristic
191C8×C83—28no—
192C22×C83—21no—
193C22×C83—21no—
194C22×C83—24no—
195C22×C83—48no—
196C22×C83—42no—
197C22×C83—42no—
198C22×C83C2×#13712yes—
199C22×C83C23×D411yescharacteristic
200C4×C83—24no—
201C4×C83—22no—
202C4×C83—21no—
203C4×C83—24no—
204C4×C83—44no—
205C4×C83—44no—
206C4×C83—44no—
207C4×C83—42no—
208C4×C83C2×#13714yes—
209C4×C83C2×2−1+412yes—
210#61826C8311yesmaximal, characteristic, direct product factor, Sylow
211C2×C83—22no—
212C2×C83—44no—
213C2×C83#348611yescharacteristic
214C2×C83C22×#13711yescharacteristic
215C2×C83#358511yescharacteristic
216#3502C2×C8311yescharacteristic, complemented
217C2×#554C2×C8311yescharacteristic
218#3385C2×C8318yescomplemented
219#3372C2×C8318yescomplemented
220#3236C2×C8314yescomplemented
221#3230C2×C8312yescomplemented
222#3110C2×C8314yes—
223#3101C2×C8312yes—
224#2983C2×C8311yescharacteristic, complemented
225C83#6182611yescharacteristic, direct product factor, Sylow
226Q82C22×C8314yescomplemented
227#557—24no—
228#557C22×C8314yescomplemented
229#554—24no—
230#554C22×C8318yes—
231#544—21no—
232#544—22no—
233#533C22×C8312yes—
234#533C22×C8314yes—
235C2×C4⋊Q8C22×C8311yescharacteristic
236C2×#125C22×C8312yes—
237C2×C22⋊Q8—21no—
238C2×C22⋊Q8—22no—
239#519—21no—
240C2×C4×Q8C22×C8311yescharacteristic
241C2×C4×D4C22×C8311yescharacteristic
242C2×C4×D4C22×C8312yes—
243C22×C4⋊C4—21no—
244#501C22×C8318yescomplemented
245#500C22×C8314yescomplemented
246#491C22×C8318yescomplemented
247#487C22×C8314yescomplemented
248#484C22×C8318yes—
249#483C22×C8314yes—
250#479C22×C83116yescomplemented
251#477C22×C8318yescomplemented
252#476C22×C8314yes—
253#474C22×C8318yes—
254#461C22×C8314yes—
255#461C22×C8318yes—
256#450C22×C8314yes—
257#446C22×C8318yescomplemented
258#440C22×C83116yes—
259C4×SD16C22×C8318yes—
260#432C22×C8314yes—
261#431C22×C8312yes—
262C22×Q8—21no—
263C22×Q8D4×C8311yescharacteristic
264C22×D4C23×C8311yescharacteristic
265C23×C4—21no—
266C23×C4—22no—
267C2×SD16C23×C8318yes—
268C4⋊Q8—24no—
269C4⋊Q8—28no—
270C4⋊Q8—22no—
271C4⋊Q8C23×C8312yes—
272C4⋊Q8C23×C8314yes—
273C4⋊Q8C23×C8314yes—
274C4⋊Q8C23×C8314yes—
275C4⋊Q8C23×C8312yes—
276#125—24no—
277#125—28no—
278#125—24no—
279#125—24no—
280#125—22no—
281#125C23×C8318yes—
282C22⋊Q8—28no—
283C22⋊Q8—22no—
284C22⋊Q8—24no—
285C22⋊Q8—216no—
286C22⋊Q8C23×C8314yes—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—24no—
290C4×Q8—22no—
291C4×Q8D4×C8314yes—
292C4×Q8C23×C8314yes—
293C4×Q8C23×C8318yes—
294C4×Q8C23×C8314yes—
295C4×D4—24no—
296C4×D4—22no—
297C4×D4C23×C8314yes—
298C4×D4C23×C8318yes—
299#117—24no—
300#117—22no—
301C2×C4⋊C4—22no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—22no—
304C2×C4⋊C4—28no—
305C2×C4⋊C4D4×C8311yescharacteristic
306C2×C4⋊C4D4×C8312yes—
307C2×C4⋊C4C23×C8311yescharacteristic
308C2×C4⋊C4C23×C8312yes—
309C2×C22⋊C4—21no—
310C2×C22⋊C4—22no—
311C2×C42C23×C8312yes—
312C2×C42C23×C8311yescharacteristic
313#107C23×C83116yes—
314#106C23×C8318yes—
315C4⋊C8C23×C8314yes—
316C4⋊C8C23×C8318yes—
317#103C23×C8318yes—
318#103C23×C83116yes—
319#102C23×C8318yes—
320C22⋊C8C23×C8314yes—
321#97C23×C8318yes—
322C4×C8C23×C8314yes—
323C24—21no—
324C2×Q8—24no—
325C2×Q8—22no—
326C2×Q8—24no—
327C2×Q8—24no—
328C2×Q8—82no—
329C2×Q8C2×D4×C8314yes—
330C2×Q8C24×C8314yes—
331C2×D4—24no—
332C2×D4—21no—
333C2×D4C24×C8312yes—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—28no—
338C22×C4—44no—
339C22×C4—84no—
340C22×C4C2×D4×C8312yes—
341C22×C4C2×D4×C8311yescharacteristic
342C22×C4C2×D4×C8312yes—
343C22×C4C2×D4×C8311yescharacteristic
344C22×C4C24×C8311yescharacteristic
345SD16—216no—
346C2×C8C24×C8318yes—
347C4⋊C4—28no—
348C4⋊C4—24no—
349C4⋊C4—28no—
350C4⋊C4—21no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—28no—
361C4⋊C4C2×D4×C8314yes—
362C4⋊C4C2×D4×C8318yes—
363C4⋊C4C24×C8314yes—
364C4⋊C4C24×C8312yes—
365C4⋊C4C24×C8314yes—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—28no—
369C42—28no—
370C42—24no—
371C42C2×D4×C8314yes—
372C42C2×D4×C8312yes—
373C42C24×C8312yes—
374C42C24×C8314yes—
375C23—24no—
376C23—22no—
377C23—42no—
378C23—42no—
379C23C22×D4×C8311yescharacteristic
380Q8—28no—
381Q8—28no—
382Q8—84no—
383Q8—84no—
384Q8—82no—
385D4—24no—
386D4—22no—
387D42−1+4×C8314yes—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—48no—
399C2×C4—88no—
400C2×C4#137×C8314yes—
401C2×C4C22×D4×C8314yes—
402C2×C4C22×D4×C8311yescharacteristic
403C2×C4C22×D4×C8312yes—
404C2×C4C22×D4×C8314yes—
405C2×C4C22×D4×C8312yes—
406C2×C4C25×C8311yescharacteristic
407C8—28no—
408C22—21no—
409C22—21no—
410C22—24no—
411C22—48no—
412C22—42no—
413C22—42no—
414C22C2×#137×C8312yes—
415C22C23×D4×C8311yescharacteristic
416C4—24no—
417C4—24no—
418C4—22no—
419C4—21no—
420C4—42no—
421C4—44no—
422C4—44no—
423C4—44no—
424C4C2×#137×C8314yes—
425C4C2×2−1+4×C8312yes—
426C2—22no—
427C2—44no—
428C2C83×#348611yescharacteristic
429C2C22×#137×C8311yescharacteristic
430C2C83×#358511yescharacteristic
subgroupG/H
center, socleC22×C83C23×D4
commutator, Frattini, cosocleC2×C4C25×C83
prime radicalC22×D4×C83C23
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×#3385C2semidirect
4. C83×#3372C2semidirect
5. C83×#3236C2semidirect
6. C83×#3230C2semidirect
9. C83×#2983C2semidirect
10. Q82×C83C22semidirect
12. #557×C83C22semidirect
28. #501×C83C22semidirect
29. #500×C83C22semidirect
30. #491×C83C22semidirect
31. #487×C83C22semidirect
34. #479×C83C22semidirect
35. #477×C83C22semidirect
41. #446×C83C22semidirect
210. #61826C83direct (N × H)
216. #3502C2×C83semidirect
218. #3385C2×C83semidirect
219. #3372C2×C83semidirect
220. #3236C2×C83semidirect
221. #3230C2×C83semidirect
224. #2983C2×C83semidirect
225. C83#61826direct (N × H)
226. Q82C22×C83semidirect
228. #557C22×C83semidirect
244. #501C22×C83semidirect
245. #500C22×C83semidirect
246. #491C22×C83semidirect
247. #487C22×C83semidirect
250. #479C22×C83semidirect
251. #477C22×C83semidirect
257. #446C22×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#35022
C2×#554×C832
C83×#33852
C83×#33722
C83×#32362
C83×#32302
C83×#31102
C83×#31012
C83×#29832
#6182683