Finite Group Databasebeta

← C83×#61828 #35289652

Subgroups — C83×#61828 #35289652

Order 21248.

1914 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3497C211yesmaximal, characteristic
2C2×#557×C83C211yesmaximal, characteristic, complemented
3C83×#3386C218yesmaximal, complemented
4C83×#3368C218yesmaximal, complemented
5C83×#3234C214yesmaximal, complemented
6C83×#3226C212yesmaximal, complemented
7C83×#3112C214yesmaximal, complemented
8C83×#3103C212yesmaximal, complemented
9C83×#2983C211yesmaximal, characteristic, complemented
10C2×2−1+4×C83—21no—
11#557×C83—24no—
12#557×C83C2218yes—
13#550×C83C2214yes—
14D4×Q8×C83C2214yes—
15#548×C83—24no—
16#537×C83—21no—
17#537×C83—22no—
18#532×C83C2212yes—
19#532×C83C2214yes—
20C2×C4⋊Q8×C83—21no—
21C2×C4⋊Q8×C83C2211yescharacteristic
22C2×#125×C83—22no—
23C2×#125×C83C2212yes—
24C2×C4×Q8×C83—21no—
25C2×C4×Q8×C83C2211yescharacteristic
26C2×C4×Q8×C83C2212yes—
27C2×C4×Q8×C83C2211yescharacteristic
28#501×C83C2218yes—
29#500×C83C2214yes—
30#490×C83C2218yes—
31#486×C83C2214yes—
32#481×C83C2214yes—
33#480×C83C2218yes—
34#478×C83C22116yes—
35#477×C83C2218yes—
36#475×C83C2214yes—
37#474×C83C2218yes—
38#460×C83C2214yes—
39#460×C83C2218yes—
40#448×C83C2214yes—
41#446×C83C2218yes—
42#440×C83C22116yes—
43C4×SD16×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×C83—21no—
52C22×Q8×C83D411yescharacteristic
53C22×Q8×C83C2311yescharacteristic
54C2×SD16×C83C2318yes—
55C4⋊Q8×C83—28no—
56C4⋊Q8×C83—22no—
57C4⋊Q8×C83C2314yes—
58C4⋊Q8×C83C2312yes—
59C4⋊Q8×C83C2314yes—
60#127×C83C2312yes—
61#127×C83C2314yes—
62#125×C83—216no—
63#125×C83—24no—
64#125×C83C2318yes—
65#124×C83—24no—
66#124×C83—22no—
67C22⋊Q8×C83—24no—
68C22⋊Q8×C83—28no—
69#121×C83—24no—
70#121×C83—22no—
71#121×C83—24no—
72#121×C83—28no—
73#121×C83C2314yes—
74C4×Q8×C83—28no—
75C4×Q8×C83—24no—
76C4×Q8×C83—22no—
77C4×Q8×C83D414yes—
78C4×Q8×C83C2314yes—
79C4×Q8×C83C2318yes—
80C4×Q8×C83C2314yes—
81C4×D4×C83—24no—
82C4×D4×C83—22no—
83C4×D4×C83C2314yes—
84C4×D4×C83C2318yes—
85C2×C4⋊C4×C83—22no—
86C2×C4⋊C4×C83—21no—
87C2×C4⋊C4×C83—22no—
88C2×C4⋊C4×C83—21no—
89C2×C4⋊C4×C83—22no—
90C2×C4⋊C4×C83D411yescharacteristic
91C2×C4⋊C4×C83D412yes—
92C2×C4⋊C4×C83C2312yes—
93C2×C4⋊C4×C83C2311yescharacteristic
94C2×C42×C83—22no—
95C2×C42×C83—21no—
96C2×C42×C83C2312yes—
97C2×C42×C83C2311yescharacteristic
98#107×C83C23116yes—
99#106×C83C2318yes—
100C4⋊C8×C83C2314yes—
101C4⋊C8×C83C2318yes—
102#103×C83C2318yes—
103#102×C83C23116yes—
104#102×C83C2318yes—
105C22⋊C8×C83C2314yes—
106#97×C83C2318yes—
107C4×C8×C83C2314yes—
108#41×C83—416no—
109#41×C83—416no—
110#41×C83—48no—
111C2×Q8×C83—24no—
112C2×Q8×C83—24no—
113C2×Q8×C83—21no—
114C2×Q8×C83—24no—
115C2×Q8×C83—24no—
116C2×Q8×C83—22no—
117C2×Q8×C83—416no—
118C2×Q8×C83—44no—
119C2×Q8×C83C2×D414yes—
120C2×Q8×C83C2412yes—
121C2×D4×C83—24no—
122C2×D4×C83—24no—
123C2×D4×C83—22no—
124C2×D4×C83—24no—
125C2×D4×C83—22no—
126C2×D4×C83C2414yes—
127C22×C4×C83—21no—
128C22×C4×C83—21no—
129C22×C4×C83—22no—
130C22×C4×C83—28no—
131C22×C4×C83—24no—
132C22×C4×C83C2×D412yes—
133C22×C4×C83C2×D412yes—
134C22×C4×C83C2×D411yescharacteristic
135C22×C4×C83C2×D411yescharacteristic
136C22×C4×C83C2411yescharacteristic
137SD16×C83—216no—
138C2×C8×C83C2418yes—
139C4⋊C4×C83—28no—
140C4⋊C4×C83—24no—
141C4⋊C4×C83—28no—
142C4⋊C4×C83—24no—
143C4⋊C4×C83—28no—
144C4⋊C4×C83—21no—
145C4⋊C4×C83—22no—
146C4⋊C4×C83—24no—
147C4⋊C4×C83—22no—
148C4⋊C4×C83C2×D414yes—
149C4⋊C4×C83C2×D418yes—
150C4⋊C4×C83C2412yes—
151C4⋊C4×C83C2414yes—
152C4⋊C4×C83C2414yes—
153C22⋊C4×C83—24no—
154C22⋊C4×C83—28no—
155C22⋊C4×C83—24no—
156C42×C83—28no—
157C42×C83—24no—
158C42×C83C2×D414yes—
159C42×C83C2×D412yes—
160C42×C83C2412yes—
161C42×C83C2414yes—
162C23×C83—24no—
163C23×C83C22×D411yescharacteristic
164Q8×C83—24no—
165Q8×C83—28no—
166Q8×C83—22no—
167Q8×C83—48no—
168Q8×C83—48no—
169Q8×C83—44no—
170Q8×C832−1+414yescomplemented
171D4×C83—28no—
172D4×C83—48no—
173D4×C83—48no—
174D4×C83—44no—
175D4×C83—84no—
176D4×C83—82no—
177C2×C4×C83—24no—
178C2×C4×C83—24no—
179C2×C4×C83—22no—
180C2×C4×C83—22no—
181C2×C4×C83—22no—
182C2×C4×C83—24no—
183C2×C4×C83—24no—
184C2×C4×C83—42no—
185C2×C4×C83—41no—
186C2×C4×C83—416no—
187C2×C4×C83—42no—
188C2×C4×C83—44no—
189C2×C4×C83—48no—
190C2×C4×C83AGL(1,ℤ/8)14yes—
191C2×C4×C83C22×D414yes—
192C2×C4×C83C22×D411yescharacteristic
193C2×C4×C83C22×D412yes—
194C2×C4×C83C22×D414yes—
195C2×C4×C83C22×D412yes—
196C2×C4×C83C2511yescharacteristic
197C8×C83—28no—
198C22×C83—21no—
199C22×C83—21no—
200C22×C83—44no—
201C22×C83—44no—
202C22×C83—44no—
203C22×C83C2×AGL(1,ℤ/8)12yes—
204C22×C83C23×D411yescharacteristic
205C4×C83—24no—
206C4×C83—22no—
207C4×C83—24no—
208C4×C83—21no—
209C4×C83—24no—
210C4×C83—42no—
211C4×C83—44no—
212C4×C83—44no—
213C4×C83C2×AGL(1,ℤ/8)14yes—
214C4×C83C2×2−1+412yes—
215#61828C8311yesmaximal, characteristic, direct product factor, Sylow
216C2×C83—22no—
217C2×C83—84no—
218C2×C83#348611yescharacteristic
219C2×C83C22×AGL(1,ℤ/8)11yescharacteristic
220C2×C83#358611yescharacteristic
221#3497C2×C8311yescharacteristic
222C2×#557C2×C8311yescharacteristic, complemented
223#3386C2×C8318yescomplemented
224#3368C2×C8318yescomplemented
225#3234C2×C8314yescomplemented
226#3226C2×C8312yescomplemented
227#3112C2×C8314yescomplemented
228#3103C2×C8312yescomplemented
229#2983C2×C8311yescharacteristic, complemented
230C83#6182811yescharacteristic, direct product factor, Sylow
231C2×2−1+4—21no—
232#557—24no—
233#557C22×C8318yes—
234#550C22×C8314yes—
235D4×Q8C22×C8314yes—
236#548—24no—
237#537—21no—
238#537—22no—
239#532C22×C8312yes—
240#532C22×C8314yes—
241C2×C4⋊Q8—21no—
242C2×C4⋊Q8C22×C8311yescharacteristic
243C2×#125—22no—
244C2×#125C22×C8312yes—
245C2×C4×Q8—21no—
246C2×C4×Q8C22×C8311yescharacteristic
247C2×C4×Q8C22×C8312yes—
248C2×C4×Q8C22×C8311yescharacteristic
249#501C22×C8318yes—
250#500C22×C8314yes—
251#490C22×C8318yes—
252#486C22×C8314yes—
253#481C22×C8314yes—
254#480C22×C8318yes—
255#478C22×C83116yes—
256#477C22×C8318yes—
257#475C22×C8314yes—
258#474C22×C8318yes—
259#460C22×C8314yes—
260#460C22×C8318yes—
261#448C22×C8314yes—
262#446C22×C8318yes—
263#440C22×C83116yes—
264C4×SD16C22×C8318yes—
265#432C22×C8314yes—
266#431C22×C8312yes—
2672−1+4—48no—
268C2×#41—24no—
269C2×#41—24no—
270C2×#41—22no—
271C22×Q8—24no—
272C22×Q8—21no—
273C22×Q8D4×C8311yescharacteristic
274C22×Q8C23×C8311yescharacteristic
275C2×SD16C23×C8318yes—
276C4⋊Q8—28no—
277C4⋊Q8—22no—
278C4⋊Q8C23×C8314yes—
279C4⋊Q8C23×C8312yes—
280C4⋊Q8C23×C8314yes—
281#127C23×C8312yes—
282#127C23×C8314yes—
283#125—216no—
284#125—24no—
285#125C23×C8318yes—
286#124—24no—
287#124—22no—
288C22⋊Q8—24no—
289C22⋊Q8—28no—
290#121—24no—
291#121—22no—
292#121—24no—
293#121—28no—
294#121C23×C8314yes—
295C4×Q8—28no—
296C4×Q8—24no—
297C4×Q8—22no—
298C4×Q8D4×C8314yes—
299C4×Q8C23×C8314yes—
300C4×Q8C23×C8318yes—
301C4×Q8C23×C8314yes—
302C4×D4—24no—
303C4×D4—22no—
304C4×D4C23×C8314yes—
305C4×D4C23×C8318yes—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—21no—
308C2×C4⋊C4—22no—
309C2×C4⋊C4—21no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4D4×C8311yescharacteristic
312C2×C4⋊C4D4×C8312yes—
313C2×C4⋊C4C23×C8311yescharacteristic
314C2×C4⋊C4C23×C8312yes—
315C2×C42—22no—
316C2×C42—21no—
317C2×C42C23×C8312yes—
318C2×C42C23×C8311yescharacteristic
319#107C23×C83116yes—
320#106C23×C8318yes—
321C4⋊C8C23×C8314yes—
322C4⋊C8C23×C8318yes—
323#103C23×C8318yes—
324#102C23×C8318yes—
325#102C23×C83116yes—
326C22⋊C8C23×C8314yes—
327#97C23×C8318yes—
328C4×C8C23×C8314yes—
329#41—416no—
330#41—416no—
331#41—48no—
332C2×Q8—24no—
333C2×Q8—24no—
334C2×Q8—21no—
335C2×Q8—24no—
336C2×Q8—24no—
337C2×Q8—22no—
338C2×Q8—416no—
339C2×Q8—44no—
340C2×Q8C2×D4×C8314yes—
341C2×Q8C24×C8312yes—
342C2×D4—24no—
343C2×D4—24no—
344C2×D4—22no—
345C2×D4—24no—
346C2×D4—22no—
347C2×D4C24×C8314yes—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—28no—
352C22×C4—24no—
353C22×C4C2×D4×C8312yes—
354C22×C4C2×D4×C8312yes—
355C22×C4C2×D4×C8311yescharacteristic
356C22×C4C2×D4×C8311yescharacteristic
357C22×C4C24×C8311yescharacteristic
358SD16—216no—
359C2×C8C24×C8318yes—
360C4⋊C4—28no—
361C4⋊C4—24no—
362C4⋊C4—28no—
363C4⋊C4—24no—
364C4⋊C4—28no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—24no—
368C4⋊C4—22no—
369C4⋊C4C2×D4×C8314yes—
370C4⋊C4C2×D4×C8318yes—
371C4⋊C4C24×C8312yes—
372C4⋊C4C24×C8314yes—
373C4⋊C4C24×C8314yes—
374C22⋊C4—24no—
375C22⋊C4—28no—
376C22⋊C4—24no—
377C42—28no—
378C42—24no—
379C42C2×D4×C8314yes—
380C42C2×D4×C8312yes—
381C42C24×C8312yes—
382C42C24×C8314yes—
383C23—24no—
384C23C22×D4×C8311yescharacteristic
385Q8—24no—
386Q8—28no—
387Q8—22no—
388Q8—48no—
389Q8—44no—
390Q8—48no—
391Q82−1+4×C8314yescomplemented
392D4—28no—
393D4—48no—
394D4—48no—
395D4—44no—
396D4—84no—
397D4—82no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—416no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—48no—
411C2×C4AGL(1,ℤ/8)×C8314yes—
412C2×C4C22×D4×C8314yes—
413C2×C4C22×D4×C8311yescharacteristic
414C2×C4C22×D4×C8312yes—
415C2×C4C22×D4×C8314yes—
416C2×C4C22×D4×C8312yes—
417C2×C4C25×C8311yescharacteristic
418C8—28no—
419C22—21no—
420C22—21no—
421C22—44no—
422C22—44no—
423C22—44no—
424C22C2×AGL(1,ℤ/8)×C8312yes—
425C22C23×D4×C8311yescharacteristic
426C4—24no—
427C4—24no—
428C4—22no—
429C4—24no—
430C4—21no—
431C4—44no—
432C4—44no—
433C4—42no—
434C4C2×AGL(1,ℤ/8)×C8314yes—
435C4C2×2−1+4×C8312yes—
436C2—22no—
437C2—84no—
438C2C83×#348611yescharacteristic
439C2C22×AGL(1,ℤ/8)×C8311yescharacteristic
440C2C83×#358611yescharacteristic
subgroupG/H
center, socleC22×C83C23×D4
commutator, Frattini, cosocleC2×C4C25×C83
prime radicalC83×#3497C2
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
2. C2×#557×C83C2semidirect
3. C83×#3386C2semidirect
4. C83×#3368C2semidirect
5. C83×#3234C2semidirect
6. C83×#3226C2semidirect
7. C83×#3112C2semidirect
8. C83×#3103C2semidirect
9. C83×#2983C2semidirect
170. Q8×C832−1+4semidirect
215. #61828C83direct (N × H)
222. C2×#557C2×C83semidirect
223. #3386C2×C83semidirect
224. #3368C2×C83semidirect
225. #3234C2×C83semidirect
226. #3226C2×C83semidirect
227. #3112C2×C83semidirect
228. #3103C2×C83semidirect
229. #2983C2×C83semidirect
230. C83#61828direct (N × H)
391. Q82−1+4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#34972
C2×#557×C832
C83×#33862
C83×#33682
C83×#32342
C83×#32262
C83×#31122
C83×#31032
C83×#29832
#6182883