Finite Group Databasebeta

← C83×#61889 #35289713

Subgroups — C83×#61889 #35289713

Order 21248.

1706 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3502C211yesmaximal, characteristic, complemented
2C2×#556×C83C211yesmaximal, characteristic
3C83×#3398C218yesmaximal, complemented
4C83×#3369C218yesmaximal, complemented
5C83×#3215C214yesmaximal
6C83×#3209C212yesmaximal, complemented
7C83×#3113C214yesmaximal, complemented
8C83×#3101C212yesmaximal
9C83×#2983C211yesmaximal, characteristic, complemented
10Q82×C83C2214yescomplemented
11#557×C83—24no—
12#557×C83C2214yescomplemented
13#556×C83—24no—
14#556×C83C2218yes—
15#544×C83—21no—
16#544×C83—22no—
17#533×C83C2212yes—
18#533×C83C2214yes—
19C2×C4⋊Q8×C83C2211yescharacteristic
20C2×#126×C83—22no—
21C2×#124×C83C2212yes—
22C2×C22⋊Q8×C83—21no—
23#519×C83—21no—
24C2×C4×Q8×C83C2212yes—
25C2×C4×Q8×C83C2211yescharacteristic
26C2×C4×D4×C83C2211yescharacteristic
27C2×#117×C83—21no—
28#497×C83C2218yes—
29#494×C83C2214yescomplemented
30#489×C83C2218yes—
31#487×C83C2214yescomplemented
32#483×C83C2214yes—
33#476×C83C2214yes—
34#475×C83C2218yescomplemented
35#464×C83C22116yes—
36#462×C83C2218yescomplemented
37#462×C83C2218yescomplemented
38#461×C83C2214yes—
39#451×C83C2218yes—
40#450×C83C2214yes—
41#446×C83C2218yescomplemented
42#441×C83C22116yescomplemented
43C4×SD16×C83C2218yes—
44#432×C83C2214yes—
45#431×C83C2212yes—
46C22×Q8×C83D411yescharacteristic
47C22×Q8×C83C2312yes—
48C22×D4×C83C2311yescharacteristic
49C23×C4×C83—21no—
50C2×Q16×C83C23116yes—
51C2×SD16×C83C2318yes—
52C4⋊Q8×C83—24no—
53C4⋊Q8×C83—28no—
54C4⋊Q8×C83—22no—
55C4⋊Q8×C83C2314yes—
56C4⋊Q8×C83C2312yes—
57C4⋊Q8×C83C2314yes—
58C4⋊Q8×C83C2314yes—
59C4⋊Q8×C83C2312yes—
60#126×C83—216no—
61#125×C83—24no—
62#125×C83—28no—
63#125×C83—22no—
64#125×C83—24no—
65#124×C83—24no—
66#124×C83C2318yes—
67C22⋊Q8×C83—28no—
68C22⋊Q8×C83—24no—
69C22⋊Q8×C83—22no—
70C22⋊Q8×C83C2314yes—
71C4×Q8×C83—24no—
72C4×Q8×C83—24no—
73C4×Q8×C83—24no—
74C4×Q8×C83—22no—
75C4×Q8×C83—24no—
76C4×Q8×C83D414yes—
77C4×Q8×C83C2318yes—
78C4×Q8×C83C2318yes—
79C4×Q8×C83C2314yes—
80C4×Q8×C83C2314yes—
81C4×D4×C83—22no—
82C4×D4×C83C2314yes—
83#117×C83—24no—
84#117×C83—22no—
85#117×C83—28no—
86C2×C4⋊C4×C83—21no—
87C2×C4⋊C4×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—21no—
94C2×C22⋊C4×C83—22no—
95C2×C42×C83—22no—
96C2×C42×C83C2311yescharacteristic
97C2×C42×C83C2312yes—
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—22no—
112C2×Q8×C83—44no—
113C2×Q8×C83C2×D414yes—
114C2×Q8×C83C2414yes—
115C2×Q8×C83C2414yes—
116C2×D4×C83—21no—
117C2×D4×C83—42no—
118C2×D4×C83C2412yes—
119C22×C4×C83—21no—
120C22×C4×C83—22no—
121C22×C4×C83—24no—
122C22×C4×C83—44no—
123C22×C4×C83C2×D412yes—
124C22×C4×C83C2×D412yes—
125C22×C4×C83C2×D411yescharacteristic
126C22×C4×C83C2×D411yescharacteristic
127C22×C4×C83C2411yescharacteristic
128Q16×C83—416no—
129SD16×C83—48no—
130C2×C8×C83C2418yes—
131C4⋊C4×C83—24no—
132C4⋊C4×C83—28no—
133C4⋊C4×C83—24no—
134C4⋊C4×C83—21no—
135C4⋊C4×C83—24no—
136C4⋊C4×C83—22no—
137C4⋊C4×C83—24no—
138C4⋊C4×C83—22no—
139C4⋊C4×C83—24no—
140C4⋊C4×C83—24no—
141C4⋊C4×C83—22no—
142C4⋊C4×C83—28no—
143C4⋊C4×C83—24no—
144C4⋊C4×C83C2×D414yes—
145C4⋊C4×C83C2×D418yes—
146C4⋊C4×C83C2414yes—
147C4⋊C4×C83C2412yes—
148C22⋊C4×C83—24no—
149C22⋊C4×C83—28no—
150C22⋊C4×C83—24no—
151C22⋊C4×C83—28no—
152C42×C83—28no—
153C42×C83—28no—
154C42×C83—24no—
155C42×C83C2×D412yes—
156C42×C83C2×D414yes—
157C42×C83C2412yes—
158C42×C83C2414yes—
159C23×C83—22no—
160C23×C83—42no—
161C23×C83—42no—
162C23×C83—42no—
163C23×C83C22×D411yescharacteristic
164Q8×C83—28no—
165Q8×C83—28no—
166Q8×C83—28no—
167Q8×C83—44no—
168Q8×C83—84no—
169Q8×C83—82no—
170D4×C83—24no—
171D4×C83—42no—
172C2×C4×C83—24no—
173C2×C4×C83—22no—
174C2×C4×C83—22no—
175C2×C4×C83—24no—
176C2×C4×C83—24no—
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#13714yes—
184C2×C4×C83C22×D414yes—
185C2×C4×C83C22×D414yes—
186C2×C4×C83C22×D411yescharacteristic
187C2×C4×C83C22×D412yes—
188C2×C4×C83C22×D412yes—
189C2×C4×C83C2511yescharacteristic
190C8×C83—28no—
191C22×C83—21no—
192C22×C83—21no—
193C22×C83—42no—
194C22×C83—42no—
195C22×C83—42no—
196C22×C83—84no—
197C22×C83C2×#13712yes—
198C22×C83C23×D411yescharacteristic
199C4×C83—24no—
200C4×C83—22no—
201C4×C83—21no—
202C4×C83—24no—
203C4×C83—44no—
204C4×C83—44no—
205C4×C83—44no—
206C4×C83—42no—
207C4×C83C2×#13714yes—
208C4×C83#58512yes—
209#61889C8311yesmaximal, characteristic, direct product factor, Sylow
210C2×C83—22no—
211C2×C83—82no—
212C2×C83#348711yescharacteristic
213C2×C83C22×#13711yescharacteristic
214C2×C83#358611yescharacteristic
215#3502C2×C8311yescharacteristic, complemented
216C2×#556C2×C8311yescharacteristic
217#3398C2×C8318yescomplemented
218#3369C2×C8318yescomplemented
219#3215C2×C8314yes—
220#3209C2×C8312yescomplemented
221#3113C2×C8314yescomplemented
222#3101C2×C8312yes—
223#2983C2×C8311yescharacteristic, complemented
224C83#6188911yescharacteristic, direct product factor, Sylow
225Q82C22×C8314yescomplemented
226#557—24no—
227#557C22×C8314yescomplemented
228#556—24no—
229#556C22×C8318yes—
230#544—21no—
231#544—22no—
232#533C22×C8312yes—
233#533C22×C8314yes—
234C2×C4⋊Q8C22×C8311yescharacteristic
235C2×#126—22no—
236C2×#124C22×C8312yes—
237C2×C22⋊Q8—21no—
238#519—21no—
239C2×C4×Q8C22×C8312yes—
240C2×C4×Q8C22×C8311yescharacteristic
241C2×C4×D4C22×C8311yescharacteristic
242C2×#117—21no—
243#497C22×C8318yes—
244#494C22×C8314yescomplemented
245#489C22×C8318yes—
246#487C22×C8314yescomplemented
247#483C22×C8314yes—
248#476C22×C8314yes—
249#475C22×C8318yescomplemented
250#464C22×C83116yes—
251#462C22×C8318yescomplemented
252#462C22×C8318yescomplemented
253#461C22×C8314yes—
254#451C22×C8318yes—
255#450C22×C8314yes—
256#446C22×C8318yescomplemented
257#441C22×C83116yescomplemented
258C4×SD16C22×C8318yes—
259#432C22×C8314yes—
260#431C22×C8312yes—
261C22×Q8D4×C8311yescharacteristic
262C22×Q8C23×C8312yes—
263C22×D4C23×C8311yescharacteristic
264C23×C4—21no—
265C2×Q16C23×C83116yes—
266C2×SD16C23×C8318yes—
267C4⋊Q8—24no—
268C4⋊Q8—28no—
269C4⋊Q8—22no—
270C4⋊Q8C23×C8314yes—
271C4⋊Q8C23×C8312yes—
272C4⋊Q8C23×C8314yes—
273C4⋊Q8C23×C8314yes—
274C4⋊Q8C23×C8312yes—
275#126—216no—
276#125—24no—
277#125—28no—
278#125—22no—
279#125—24no—
280#124—24no—
281#124C23×C8318yes—
282C22⋊Q8—28no—
283C22⋊Q8—24no—
284C22⋊Q8—22no—
285C22⋊Q8C23×C8314yes—
286C4×Q8—24no—
287C4×Q8—24no—
288C4×Q8—24no—
289C4×Q8—22no—
290C4×Q8—24no—
291C4×Q8D4×C8314yes—
292C4×Q8C23×C8318yes—
293C4×Q8C23×C8314yes—
294C4×Q8C23×C8318yes—
295C4×Q8C23×C8314yes—
296C4×D4—22no—
297C4×D4C23×C8314yes—
298#117—24no—
299#117—22no—
300#117—28no—
301C2×C4⋊C4—21no—
302C2×C4⋊C4—22no—
303C2×C4⋊C4—21no—
304C2×C4⋊C4D4×C8311yescharacteristic
305C2×C4⋊C4D4×C8312yes—
306C2×C4⋊C4C23×C8311yescharacteristic
307C2×C22⋊C4—22no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4—22no—
310C2×C42—22no—
311C2×C42C23×C8311yescharacteristic
312C2×C42C23×C8312yes—
313#106C23×C8318yes—
314C4⋊C8C23×C8314yes—
315C4⋊C8C23×C8318yes—
316#103C23×C8318yes—
317#103C23×C83116yes—
318#102C23×C8318yes—
319C22⋊C8C23×C8314yes—
320#97C23×C8318yes—
321C4×C8C23×C8314yes—
322C24—21no—
323C2×Q8—24no—
324C2×Q8—22no—
325C2×Q8—24no—
326C2×Q8—22no—
327C2×Q8—44no—
328C2×Q8C2×D4×C8314yes—
329C2×Q8C24×C8314yes—
330C2×Q8C24×C8314yes—
331C2×D4—21no—
332C2×D4—42no—
333C2×D4C24×C8312yes—
334C22×C4—21no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—44no—
338C22×C4C2×D4×C8312yes—
339C22×C4C2×D4×C8312yes—
340C22×C4C2×D4×C8311yescharacteristic
341C22×C4C2×D4×C8311yescharacteristic
342C22×C4C24×C8311yescharacteristic
343Q16—416no—
344SD16—48no—
345C2×C8C24×C8318yes—
346C4⋊C4—24no—
347C4⋊C4—28no—
348C4⋊C4—24no—
349C4⋊C4—21no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—22no—
357C4⋊C4—28no—
358C4⋊C4—24no—
359C4⋊C4C2×D4×C8314yes—
360C4⋊C4C2×D4×C8318yes—
361C4⋊C4C24×C8314yes—
362C4⋊C4C24×C8312yes—
363C22⋊C4—24no—
364C22⋊C4—28no—
365C22⋊C4—24no—
366C22⋊C4—28no—
367C42—28no—
368C42—28no—
369C42—24no—
370C42C2×D4×C8312yes—
371C42C2×D4×C8314yes—
372C42C24×C8312yes—
373C42C24×C8314yes—
374C23—22no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23C22×D4×C8311yescharacteristic
379Q8—28no—
380Q8—28no—
381Q8—28no—
382Q8—44no—
383Q8—84no—
384Q8—82no—
385D4—24no—
386D4—42no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—44no—
396C2×C4—42no—
397C2×C4—48no—
398C2×C4#137×C8314yes—
399C2×C4C22×D4×C8314yes—
400C2×C4C22×D4×C8314yes—
401C2×C4C22×D4×C8311yescharacteristic
402C2×C4C22×D4×C8312yes—
403C2×C4C22×D4×C8312yes—
404C2×C4C25×C8311yescharacteristic
405C8—28no—
406C22—21no—
407C22—21no—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—84no—
412C22C2×#137×C8312yes—
413C22C23×D4×C8311yescharacteristic
414C4—24no—
415C4—24no—
416C4—22no—
417C4—21no—
418C4—42no—
419C4—44no—
420C4—44no—
421C4—44no—
422C4C2×#137×C8314yes—
423C4#585×C8312yes—
424C2—22no—
425C2—82no—
426C2C83×#348711yescharacteristic
427C2C22×#137×C8311yescharacteristic
428C2C83×#358611yescharacteristic
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×#3398C2semidirect
4. C83×#3369C2semidirect
6. C83×#3209C2semidirect
7. C83×#3113C2semidirect
9. C83×#2983C2semidirect
10. Q82×C83C22semidirect
12. #557×C83C22semidirect
29. #494×C83C22semidirect
31. #487×C83C22semidirect
34. #475×C83C22semidirect
36. #462×C83C22semidirect
37. #462×C83C22semidirect
41. #446×C83C22semidirect
42. #441×C83C22semidirect
209. #61889C83direct (N × H)
215. #3502C2×C83semidirect
217. #3398C2×C83semidirect
218. #3369C2×C83semidirect
220. #3209C2×C83semidirect
221. #3113C2×C83semidirect
223. #2983C2×C83semidirect
224. C83#61889direct (N × H)
225. Q82C22×C83semidirect
227. #557C22×C83semidirect
244. #494C22×C83semidirect
246. #487C22×C83semidirect
249. #475C22×C83semidirect
251. #462C22×C83semidirect
252. #462C22×C83semidirect
256. #446C22×C83semidirect
257. #441C22×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#35022
C2×#556×C832
C83×#33982
C83×#33692
C83×#32152
C83×#32092
C83×#31132
C83×#31012
C83×#29832
#6188983