Finite Group Databasebeta

← C83×#9323 #35237147

Subgroups — C83×#9323 #35237147

Order 21248.

612 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3110C211yesmaximal, characteristic
2C83×#2275C211yesmaximal, characteristic
3C83×#1986C211yesmaximal, characteristic, complemented
4C83×#1506C211yesmaximal, characteristic, complemented
5C83×#1490C211yesmaximal, characteristic
6C83×#1300C211yesmaximal, characteristic, complemented
7C83×#1285C211yesmaximal, characteristic, complemented
8#533×C83C2211yescharacteristic
9C2×C4×D4×C83—21no—
10C2×C4×D4×C83C2211yescharacteristic
11#484×C83—21no—
12#474×C83C412yescomplemented
13#461×C83C412yescomplemented
14#450×C83—21no—
15#440×C83—22no—
16#432×C83C2211yescharacteristic
17#432×C83C2211yescharacteristic
18C4×M16×C83C2211yescharacteristic
19C4×C4⋊C4×C83—21no—
20C4×C4⋊C4×C83C2211yescharacteristic
21C4×C22⋊C4×C83—22no—
22C4×C22⋊C4×C83—21no—
23#376×C83C2211yescharacteristic
24C2×#95×C83—21no—
25#339×C83—22no—
26#328×C83—21no—
27#328×C83—21no—
28D4⋊C8×C83—22no—
29C22×D4×C83D411yescharacteristic
30C23×C4×C83—22no—
31C23×C4×C83—21no—
32C2×SD16×C83—22no—
33C2×M16×C83—21no—
34C4⋊Q8×C83C2×C411yescharacteristic, complemented
35C4⋊Q8×C83C2×C411yescharacteristic, complemented
36#125×C83—21no—
37C22⋊Q8×C83—21no—
38C22⋊Q8×C83—21no—
39C4×Q8×C83—21no—
40C4×Q8×C83—21no—
41C4×D4×C83—28no—
42C4×D4×C83—21no—
43C4×D4×C83—21no—
44C4×D4×C83C2×C412yes—
45C4×D4×C83D412yes—
46#117×C83—21no—
47C2×C4⋊C4×C83—21no—
48C2×C4⋊C4×C83D411yescharacteristic
49C2×C22⋊C4×C83—22no—
50C2×C22⋊C4×C83—21no—
51C2×C42×C83—21no—
52C2×C42×C83—22no—
53C2×C42×C83D412yes—
54C2×C42×C83C2311yescharacteristic
55#107×C83—22no—
56C4⋊C8×C83—21no—
57C4⋊C8×C83C2×C412yescomplemented
58#103×C83—22no—
59#102×C83—22no—
60C22⋊C8×C83—21no—
61C22⋊C8×C83—21no—
62#97×C83—21no—
63#97×C83—21no—
64#97×C83—21no—
65C4×C8×C83—21no—
66#95×C83—22no—
67#95×C83—21no—
68#95×C83—21no—
69#95×C83D411yescharacteristic
70#95×C83D411yescharacteristic
71C24×C83—21no—
72C2×Q8×C83—21no—
73C2×Q8×C83—21no—
74C2×D4×C83—24no—
75C2×D4×C83—21no—
76C2×D4×C83C22⋊C412yes—
77C22×C4×C83—21no—
78C22×C4×C83—21no—
79C22×C4×C83—22no—
80C22×C4×C83—21no—
81C22×C4×C83—28no—
82C22×C4×C83—22no—
83C22×C4×C83—22no—
84C22×C4×C83—82no—
85C22×C4×C83—82no—
86C22×C4×C83—82no—
87C22×C4×C83C2×D411yescharacteristic
88C22×C4×C83C2×D411yescharacteristic
89C22×C4×C83C2×D411yescharacteristic
90SD16×C83—44no—
91M16×C83—22no—
92M16×C83—41no—
93C2×C8×C83—22no—
94C2×C8×C83—41no—
95C2×C8×C83—41no—
96C4⋊C4×C83—24no—
97C4⋊C4×C83—21no—
98C4⋊C4×C83—21no—
99C4⋊C4×C83—21no—
100C4⋊C4×C83—21no—
101C4⋊C4×C83—21no—
102C4⋊C4×C83—41no—
103C4⋊C4×C83—41no—
104C4⋊C4×C83C22⋊C412yes—
105C22⋊C4×C83—28no—
106C22⋊C4×C83—22no—
107C22⋊C4×C83—22no—
108C22⋊C4×C83—41no—
109C42×C83—24no—
110C42×C83—42no—
111C42×C83—41no—
112C42×C83—82no—
113C42×C83C22×C411yescharacteristic
114C42×C83C2×D411yescharacteristic
115C42×C83#4111yescharacteristic
116C42×C83#4111yescharacteristic
117C23×C83—24no—
118C23×C83—22no—
119C23×C83—42no—
120C23×C83—41no—
121C23×C83—41no—
122C23×C83#12011yescharacteristic
123Q8×C83—42no—
124Q8×C83—81no—
125D4×C83—24no—
126D4×C83—22no—
127D4×C83#10414yes—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—22no—
131C2×C4×C83—22no—
132C2×C4×C83—22no—
133C2×C4×C83—24no—
134C2×C4×C83—22no—
135C2×C4×C83—41no—
136C2×C4×C83—41no—
137C2×C4×C83—42no—
138C2×C4×C83—42no—
139C2×C4×C83—41no—
140C2×C4×C83—42no—
141C2×C4×C83—41no—
142C2×C4×C83—41no—
143C2×C4×C83—41no—
144C2×C4×C83—41no—
145C2×C4×C83—82no—
146C2×C4×C83—82no—
147C2×C4×C83—88no—
148C2×C4×C83C2×C22⋊C411yescharacteristic
149C2×C4×C83C4×D411yescharacteristic
150C2×C4×C83C4×D411yescharacteristic
151C2×C4×C83#12111yescharacteristic
152C2×C4×C83#12111yescharacteristic
153C2×C4×C83#12311yescharacteristic
154C8×C83—42no—
155C8×C83—42no—
156C8×C83—81no—
157C22×C83—21no—
158C22×C83—21no—
159C22×C83—24no—
160C22×C83—44no—
161C22×C83—44no—
162C22×C83—42no—
163C22×C83—42no—
164C22×C83#38611yescharacteristic
165C22×C83#45511yescharacteristic
166C22×C83#45611yescharacteristic
167C4×C83—21no—
168C4×C83—21no—
169C4×C83—21no—
170C4×C83—21no—
171C4×C83—21no—
172C4×C83—42no—
173C4×C83—44no—
174C4×C83—42no—
175C4×C83—42no—
176C4×C83—81no—
177C4×C83—81no—
178C4×C83C2×#10412yes—
179#9323C8311yesmaximal, characteristic, direct product factor, Sylow
180C2×C83—21no—
181C2×C83—21no—
182C2×C83—44no—
183C2×C83#188211yescharacteristic
184C2×C83#189611yescharacteristic
185C2×C83#190411yescharacteristic
186#3110C2×C8311yescharacteristic
187#2275C2×C8311yescharacteristic
188#1986C2×C8311yescharacteristic, complemented
189#1506C2×C8311yescharacteristic, complemented
190#1490C2×C8311yescharacteristic
191#1300C2×C8311yescharacteristic, complemented
192#1285C2×C8311yescharacteristic, complemented
193C83#932311yescharacteristic, direct product factor, Sylow
194#533C22×C8311yescharacteristic
195C2×C4×D4—21no—
196C2×C4×D4C22×C8311yescharacteristic
197#484—21no—
198#474C4×C8312yescomplemented
199#461C4×C8312yescomplemented
200#450—21no—
201#440—22no—
202#432C22×C8311yescharacteristic
203#432C22×C8311yescharacteristic
204C4×M16C22×C8311yescharacteristic
205C4×C4⋊C4—21no—
206C4×C4⋊C4C22×C8311yescharacteristic
207C4×C22⋊C4—22no—
208C4×C22⋊C4—21no—
209#376C22×C8311yescharacteristic
210C2×#95—21no—
211#339—22no—
212#328—21no—
213#328—21no—
214D4⋊C8—22no—
215C22×D4D4×C8311yescharacteristic
216C23×C4—22no—
217C23×C4—21no—
218C2×SD16—22no—
219C2×M16—21no—
220C4⋊Q8C2×C4×C8311yescharacteristic, complemented
221C4⋊Q8C2×C4×C8311yescharacteristic, complemented
222#125—21no—
223C22⋊Q8—21no—
224C22⋊Q8—21no—
225C4×Q8—21no—
226C4×Q8—21no—
227C4×D4—28no—
228C4×D4—21no—
229C4×D4—21no—
230C4×D4C2×C4×C8312yes—
231C4×D4D4×C8312yes—
232#117—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4D4×C8311yescharacteristic
235C2×C22⋊C4—22no—
236C2×C22⋊C4—21no—
237C2×C42—21no—
238C2×C42—22no—
239C2×C42D4×C8312yes—
240C2×C42C23×C8311yescharacteristic
241#107—22no—
242C4⋊C8—21no—
243C4⋊C8C2×C4×C8312yescomplemented
244#103—22no—
245#102—22no—
246C22⋊C8—21no—
247C22⋊C8—21no—
248#97—21no—
249#97—21no—
250#97—21no—
251C4×C8—21no—
252#95—22no—
253#95—21no—
254#95—21no—
255#95D4×C8311yescharacteristic
256#95D4×C8311yescharacteristic
257C24—21no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×D4—24no—
261C2×D4—21no—
262C2×D4C22⋊C4×C8312yes—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—28no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—82no—
271C22×C4—82no—
272C22×C4—82no—
273C22×C4C2×D4×C8311yescharacteristic
274C22×C4C2×D4×C8311yescharacteristic
275C22×C4C2×D4×C8311yescharacteristic
276SD16—44no—
277M16—22no—
278M16—41no—
279C2×C8—22no—
280C2×C8—41no—
281C2×C8—41no—
282C4⋊C4—24no—
283C4⋊C4—21no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4C22⋊C4×C8312yes—
291C22⋊C4—28no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—41no—
295C42—24no—
296C42—42no—
297C42—41no—
298C42—82no—
299C42C22×C4×C8311yescharacteristic
300C42C2×D4×C8311yescharacteristic
301C42#41×C8311yescharacteristic
302C42#41×C8311yescharacteristic
303C23—24no—
304C23—22no—
305C23—42no—
306C23—41no—
307C23—41no—
308C23#120×C8311yescharacteristic
309Q8—42no—
310Q8—81no—
311D4—24no—
312D4—22no—
313D4#104×C8314yes—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—88no—
334C2×C4C2×C22⋊C4×C8311yescharacteristic
335C2×C4C4×D4×C8311yescharacteristic
336C2×C4C4×D4×C8311yescharacteristic
337C2×C4#121×C8311yescharacteristic
338C2×C4#121×C8311yescharacteristic
339C2×C4#123×C8311yescharacteristic
340C8—42no—
341C8—42no—
342C8—81no—
343C22—21no—
344C22—21no—
345C22—24no—
346C22—44no—
347C22—44no—
348C22—42no—
349C22—42no—
350C22#386×C8311yescharacteristic
351C22#455×C8311yescharacteristic
352C22#456×C8311yescharacteristic
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—42no—
359C4—44no—
360C4—42no—
361C4—42no—
362C4—81no—
363C4—81no—
364C4C2×#104×C8312yes—
365C2—21no—
366C2—21no—
367C2—44no—
368C2C83×#188211yescharacteristic
369C2C83×#189611yescharacteristic
370C2C83×#190411yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC42C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radicalC22×D4×C83D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
3. C83×#1986C2semidirect
4. C83×#1506C2semidirect
6. C83×#1300C2semidirect
7. C83×#1285C2semidirect
12. #474×C83C4semidirect
13. #461×C83C4semidirect
34. C4⋊Q8×C83C2×C4semidirect
35. C4⋊Q8×C83C2×C4semidirect
57. C4⋊C8×C83C2×C4semidirect
179. #9323C83direct (N × H)
188. #1986C2×C83semidirect
189. #1506C2×C83semidirect
191. #1300C2×C83semidirect
192. #1285C2×C83semidirect
193. C83#9323direct (N × H)
198. #474C4×C83semidirect
199. #461C4×C83semidirect
220. C4⋊Q8C2×C4×C83semidirect
221. C4⋊Q8C2×C4×C83semidirect
243. C4⋊C8C2×C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#31102
C83×#22752
C83×#19862
C83×#15062
C83×#14902
C83×#13002
C83×#12852
#932383