Finite Group Databasebeta

← C83×#9375 #35237199

Subgroups — C83×#9375 #35237199

Order 21248.

418 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3238C211yesmaximal, characteristic
2C83×#2992C211yesmaximal, characteristic
3C83×#1988C211yesmaximal, characteristic, complemented
4C83×#1558C211yesmaximal, characteristic, complemented
5C83×#1509C211yesmaximal, characteristic
6C83×#1290C211yesmaximal, characteristic, complemented
7C83×#1285C211yesmaximal, characteristic, complemented
8C2×C4⋊Q8×C83C2211yescharacteristic
9#529×C83C2211yescharacteristic
10#501×C83C412yescomplemented
11#489×C83C412yescomplemented
12#483×C83—21no—
13#480×C83—21no—
14#477×C83—21no—
15#476×C83—21no—
16#446×C83—21no—
17#435×C83—21no—
18#435×C83—21no—
19C8×D4×C83—21no—
20#433×C83—21no—
21#432×C83C2211yescharacteristic
22#432×C83C2211yescharacteristic
23#432×C83C2211yescharacteristic
24#424×C83—21no—
25C2×C4⋊C8×C83C2211yescharacteristic
26C4×C4⋊C4×C83C2211yescharacteristic
27#340×C83—21no—
28#340×C83—21no—
29#333×C83—21no—
30#333×C83—21no—
31#328×C83—21no—
32#327×C83—21no—
33C22×Q8×C83—21no—
34C2×M16×C83—21no—
35C22×C8×C83—21no—
36C22×C8×C83D411yescharacteristic
37C22×C8×C83D411yescharacteristic
38C4⋊Q8×C83—21no—
39C4⋊Q8×C83—21no—
40C4⋊Q8×C83—21no—
41C4⋊Q8×C83C2×C412yes—
42#126×C83—21no—
43#125×C83C2×C411yescharacteristic
44#124×C83C2×C411yescharacteristic
45#123×C83—21no—
46C22⋊Q8×C83D411yescharacteristic
47#121×C83D411yescharacteristic
48C4×Q8×C83D411yescharacteristic
49C4×D4×C83—21no—
50C4×D4×C83D411yescharacteristic
51#117×C83—21no—
52C2×C4⋊C4×C83—21no—
53C2×C4⋊C4×C83—21no—
54C2×C4⋊C4×C83—21no—
55C2×C42×C83—21no—
56C2×C42×C83C2311yescharacteristic
57#107×C83—22no—
58#106×C83—22no—
59C4⋊C8×C83—21no—
60C4⋊C8×C83—21no—
61C4⋊C8×C83—21no—
62C4⋊C8×C83—21no—
63C4⋊C8×C83—21no—
64C4⋊C8×C83—21no—
65C4⋊C8×C83D411yescharacteristic
66C4⋊C8×C83D411yescharacteristic
67#103×C83—22no—
68#102×C83—22no—
69C22⋊C8×C83—21no—
70C22⋊C8×C83—21no—
71C22⋊C8×C83—21no—
72C22⋊C8×C83—21no—
73C22⋊C8×C83—21no—
74#97×C83—21no—
75#97×C83—21no—
76#97×C83—21no—
77#97×C83C2×C412yes—
78C4×C8×C83—21no—
79#95×C83—21no—
80C2×Q8×C83—22no—
81C2×Q8×C83—41no—
82C2×Q8×C83—81no—
83C2×Q8×C83—81no—
84C2×Q8×C83C22⋊C411yescharacteristic
85C2×D4×C83—21no—
86C2×D4×C83C22⋊C411yescharacteristic
87C22×C4×C83—21no—
88C22×C4×C83—41no—
89C22×C4×C83—41no—
90C22×C4×C83C2×D411yescharacteristic
91C22×C4×C83C2×D411yescharacteristic
92C22×C4×C83#4111yescharacteristic
93M16×C83—81no—
94C2×C8×C83—22no—
95C2×C8×C83—21no—
96C2×C8×C83—21no—
97C2×C8×C83—21no—
98C2×C8×C83—21no—
99C2×C8×C83—41no—
100C2×C8×C83—41no—
101C2×C8×C83—41no—
102C2×C8×C83—81no—
103C4⋊C4×C83—21no—
104C4⋊C4×C83—21no—
105C4⋊C4×C83—21no—
106C4⋊C4×C83—21no—
107C4⋊C4×C83—22no—
108C4⋊C4×C83—22no—
109C4⋊C4×C83—22no—
110C4⋊C4×C83—41no—
111C4⋊C4×C83—41no—
112C4⋊C4×C83—41no—
113C4⋊C4×C83C22⋊C411yescharacteristic
114C4⋊C4×C83C22⋊C411yescharacteristic
115C22⋊C4×C83—21no—
116C22⋊C4×C83—21no—
117C22⋊C4×C83—21no—
118C22⋊C4×C83—21no—
119C22⋊C4×C83—21no—
120C42×C83—21no—
121C42×C83—81no—
122C42×C83C22×C411yescharacteristic
123C42×C83C2×D411yescharacteristic
124C42×C83C2×D411yescharacteristic
125C42×C83#4111yescharacteristic
126C23×C83—21no—
127C23×C83#12111yescharacteristic
128Q8×C83—41no—
129Q8×C83—82no—
130Q8×C83—82no—
131D4×C83—41no—
132D4×C83—41no—
133D4×C83—41no—
134C2×C4×C83—21no—
135C2×C4×C83—21no—
136C2×C4×C83—21no—
137C2×C4×C83—21no—
138C2×C4×C83—21no—
139C2×C4×C83—21no—
140C2×C4×C83—21no—
141C2×C4×C83—41no—
142C2×C4×C83—41no—
143C2×C4×C83—41no—
144C2×C4×C83—42no—
145C2×C4×C83—42no—
146C2×C4×C83—81no—
147C2×C4×C83—81no—
148C2×C4×C83—81no—
149C2×C4×C83—81no—
150C2×C4×C83C2×C22⋊C411yescharacteristic
151C2×C4×C83C4×D411yescharacteristic
152C2×C4×C83C4×D411yescharacteristic
153C2×C4×C83#12011yescharacteristic
154C2×C4×C83#12111yescharacteristic
155C2×C4×C83#12311yescharacteristic
156C8×C83—42no—
157C8×C83—41no—
158C8×C83—41no—
159C8×C83—41no—
160C8×C83—41no—
161C8×C83—81no—
162C22×C83—21no—
163C22×C83—21no—
164C22×C83—41no—
165C22×C83—41no—
166C22×C83—41no—
167C22×C83#38611yescharacteristic
168C22×C83#47111yescharacteristic
169C22×C83#47211yescharacteristic
170C4×C83—21no—
171C4×C83—21no—
172C4×C83—21no—
173C4×C83—21no—
174C4×C83—21no—
175C4×C83—41no—
176C4×C83—41no—
177C4×C83—41no—
178C4×C83—82no—
179C4×C83—82no—
180C4×C83#45511yescharacteristic
181C4×C83#45611yescharacteristic
182#9375C8311yesmaximal, characteristic, direct product factor, Sylow
183C2×C83—21no—
184C2×C83—21no—
185C2×C83—81no—
186C2×C83#188211yescharacteristic
187C2×C83#190711yescharacteristic
188C2×C83#190911yescharacteristic
189#3238C2×C8311yescharacteristic
190#2992C2×C8311yescharacteristic
191#1988C2×C8311yescharacteristic, complemented
192#1558C2×C8311yescharacteristic, complemented
193#1509C2×C8311yescharacteristic
194#1290C2×C8311yescharacteristic, complemented
195#1285C2×C8311yescharacteristic, complemented
196C83#937511yescharacteristic, direct product factor, Sylow
197C2×C4⋊Q8C22×C8311yescharacteristic
198#529C22×C8311yescharacteristic
199#501C4×C8312yescomplemented
200#489C4×C8312yescomplemented
201#483—21no—
202#480—21no—
203#477—21no—
204#476—21no—
205#446—21no—
206#435—21no—
207#435—21no—
208C8×D4—21no—
209#433—21no—
210#432C22×C8311yescharacteristic
211#432C22×C8311yescharacteristic
212#432C22×C8311yescharacteristic
213#424—21no—
214C2×C4⋊C8C22×C8311yescharacteristic
215C4×C4⋊C4C22×C8311yescharacteristic
216#340—21no—
217#340—21no—
218#333—21no—
219#333—21no—
220#328—21no—
221#327—21no—
222C22×Q8—21no—
223C2×M16—21no—
224C22×C8—21no—
225C22×C8D4×C8311yescharacteristic
226C22×C8D4×C8311yescharacteristic
227C4⋊Q8—21no—
228C4⋊Q8—21no—
229C4⋊Q8—21no—
230C4⋊Q8C2×C4×C8312yes—
231#126—21no—
232#125C2×C4×C8311yescharacteristic
233#124C2×C4×C8311yescharacteristic
234#123—21no—
235C22⋊Q8D4×C8311yescharacteristic
236#121D4×C8311yescharacteristic
237C4×Q8D4×C8311yescharacteristic
238C4×D4—21no—
239C4×D4D4×C8311yescharacteristic
240#117—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C42—21no—
245C2×C42C23×C8311yescharacteristic
246#107—22no—
247#106—22no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8D4×C8311yescharacteristic
255C4⋊C8D4×C8311yescharacteristic
256#103—22no—
257#102—22no—
258C22⋊C8—21no—
259C22⋊C8—21no—
260C22⋊C8—21no—
261C22⋊C8—21no—
262C22⋊C8—21no—
263#97—21no—
264#97—21no—
265#97—21no—
266#97C2×C4×C8312yes—
267C4×C8—21no—
268#95—21no—
269C2×Q8—22no—
270C2×Q8—41no—
271C2×Q8—81no—
272C2×Q8—81no—
273C2×Q8C22⋊C4×C8311yescharacteristic
274C2×D4—21no—
275C2×D4C22⋊C4×C8311yescharacteristic
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4C2×D4×C8311yescharacteristic
280C22×C4C2×D4×C8311yescharacteristic
281C22×C4#41×C8311yescharacteristic
282M16—81no—
283C2×C8—22no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—41no—
289C2×C8—41no—
290C2×C8—41no—
291C2×C8—81no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—21no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—41no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4C22⋊C4×C8311yescharacteristic
303C4⋊C4C22⋊C4×C8311yescharacteristic
304C22⋊C4—21no—
305C22⋊C4—21no—
306C22⋊C4—21no—
307C22⋊C4—21no—
308C22⋊C4—21no—
309C42—21no—
310C42—81no—
311C42C22×C4×C8311yescharacteristic
312C42C2×D4×C8311yescharacteristic
313C42C2×D4×C8311yescharacteristic
314C42#41×C8311yescharacteristic
315C23—21no—
316C23#121×C8311yescharacteristic
317Q8—41no—
318Q8—82no—
319Q8—82no—
320D4—41no—
321D4—41no—
322D4—41no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—81no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—81no—
339C2×C4C2×C22⋊C4×C8311yescharacteristic
340C2×C4C4×D4×C8311yescharacteristic
341C2×C4C4×D4×C8311yescharacteristic
342C2×C4#120×C8311yescharacteristic
343C2×C4#121×C8311yescharacteristic
344C2×C4#123×C8311yescharacteristic
345C8—42no—
346C8—41no—
347C8—41no—
348C8—41no—
349C8—41no—
350C8—81no—
351C22—21no—
352C22—21no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22#386×C8311yescharacteristic
357C22#471×C8311yescharacteristic
358C22#472×C8311yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—82no—
368C4—82no—
369C4#455×C8311yescharacteristic
370C4#456×C8311yescharacteristic
371C2—21no—
372C2—21no—
373C2—81no—
374C2C83×#188211yescharacteristic
375C2C83×#190711yescharacteristic
376C2C83×#190911yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC42C22×C4×C83
Frattini, cosocleC2×C42C23×C83
prime radical#121×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×#1988C2semidirect
4. C83×#1558C2semidirect
6. C83×#1290C2semidirect
7. C83×#1285C2semidirect
10. #501×C83C4semidirect
11. #489×C83C4semidirect
182. #9375C83direct (N × H)
191. #1988C2×C83semidirect
192. #1558C2×C83semidirect
194. #1290C2×C83semidirect
195. #1285C2×C83semidirect
196. C83#9375direct (N × H)
199. #501C4×C83semidirect
200. #489C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#32382
C83×#29922
C83×#19882
C83×#15582
C83×#15092
C83×#12902
C83×#12852
#937583