Finite Group Databasebeta

← C83×#9383 #35237207

Subgroups — C83×#9383 #35237207

Order 21248.

570 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#2972C211yesmaximal, characteristic
2C83×#2564C211yesmaximal, characteristic
3C83×#1799C211yesmaximal, characteristic
4C83×#1520C212yesmaximal, complemented
5C83×#1294C212yesmaximal, complemented
6C2×C22⋊Q8×C83C2211yescharacteristic
7C2×C4×D4×C83C2211yescharacteristic
8C8×D4×C83—24no—
9#431×C83—21no—
10C2×#99×C83—21no—
11C2×C22⋊C8×C83—21no—
12C2×C22⋊C8×C83C2212yes—
13#400×C83C411yescharacteristic, complemented
14#399×C83C411yescharacteristic, complemented
15#395×C83C411yescharacteristic, complemented
16#393×C83C411yescharacteristic, complemented
17#387×C83—21no—
18#386×C83—21no—
19#385×C83—21no—
20#380×C83—21no—
21#380×C83C2212yes—
22C2×#95×C83C2211yescharacteristic
23#342×C83—21no—
24#328×C83—24no—
25C22×Q8×C83C2×C411yescharacteristic
26C22×D4×C83D411yescharacteristic
27C23×C4×C83—21no—
28C23×C4×C83D411yescharacteristic
29C23×C4×C83C2311yescharacteristic
30C22×C8×C83—22no—
31C22×C8×C83—22no—
32C22⋊Q8×C83—22no—
33C22⋊Q8×C83—22no—
34C4×D4×C83—22no—
35C4×D4×C83—22no—
36C2×C4⋊C4×C83—21no—
37C2×C4⋊C4×C83—21no—
38C2×C4⋊C4×C83—21no—
39C2×C4⋊C4×C83—22no—
40C2×C4⋊C4×C83C2×C411yescharacteristic
41C2×C4⋊C4×C83C2×C411yescharacteristic
42C2×C22⋊C4×C83—21no—
43C2×C22⋊C4×C83—21no—
44C2×C22⋊C4×C83—22no—
45C2×C22⋊C4×C83—21no—
46C2×C22⋊C4×C83C2×C411yescharacteristic
47C2×C22⋊C4×C83D411yescharacteristic
48C2×C42×C83D411yescharacteristic
49C4⋊C8×C83—22no—
50#99×C83—42no—
51C22⋊C8×C83—22no—
52C22⋊C8×C83—22no—
53C22⋊C8×C83D414yes—
54C4×C8×C83—22no—
55#95×C83—21no—
56#95×C83—21no—
57#95×C83—21no—
58#95×C83—21no—
59#95×C83—21no—
60#95×C83—22no—
61#95×C83C2×C411yescharacteristic
62#95×C83C2×C411yescharacteristic
63C24×C83C22⋊C411yescharacteristic
64C24×C83C2×D411yescharacteristic
65C2×Q8×C83—22no—
66C2×Q8×C83—81no—
67C2×D4×C83—22no—
68C2×D4×C83—22no—
69C2×D4×C83SD1612yes—
70C2×D4×C83Q1612yes—
71C22×C4×C83—21no—
72C22×C4×C83—21no—
73C22×C4×C83—21no—
74C22×C4×C83—21no—
75C22×C4×C83—21no—
76C22×C4×C83—21no—
77C22×C4×C83—42no—
78C22×C4×C83—42no—
79C22×C4×C83—42no—
80C22×C4×C83—81no—
81C22×C4×C83—81no—
82C22×C4×C83—81no—
83C22×C4×C83C22⋊C411yescharacteristic
84C22×C4×C83C22⋊C411yescharacteristic
85C22×C4×C83C22⋊C411yescharacteristic
86C22×C4×C83C22×C411yescharacteristic
87C22×C4×C83C2×D412yes—
88C22×C4×C83C2×D411yescharacteristic
89C22×C4×C83#4112yes—
90C2×C8×C83—24no—
91C2×C8×C83—82no—
92C2×C8×C83—82no—
93C4⋊C4×C83—22no—
94C4⋊C4×C83—22no—
95C4⋊C4×C83—42no—
96C4⋊C4×C83—41no—
97C4⋊C4×C83—41no—
98C4⋊C4×C83—81no—
99C4⋊C4×C83—82no—
100C22⋊C4×C83—22no—
101C22⋊C4×C83—22no—
102C22⋊C4×C83—42no—
103C22⋊C4×C83—42no—
104C22⋊C4×C83—42no—
105C22⋊C4×C83—42no—
106C22⋊C4×C83—81no—
107C42×C83—21no—
108C42×C83—21no—
109C23×C83—21no—
110C23×C83—21no—
111C23×C83—41no—
112C23×C83—41no—
113C23×C83—41no—
114C23×C83—41no—
115C23×C83#10312yes—
116C23×C83C2×C22⋊C411yescharacteristic
117C23×C83#12112yes—
118C23×C83C2×SD1611yescharacteristic
119C23×C83C2×Q1611yescharacteristic
120C23×C83#13512yes—
121Q8×C83—82no—
122D4×C83—24no—
123D4×C83—24no—
124C2×C4×C83—21no—
125C2×C4×C83—21no—
126C2×C4×C83—22no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—42no—
130C2×C4×C83—42no—
131C2×C4×C83—41no—
132C2×C4×C83—41no—
133C2×C4×C83—42no—
134C2×C4×C83—41no—
135C2×C4×C83—41no—
136C2×C4×C83—41no—
137C2×C4×C83—41no—
138C2×C4×C83—41no—
139C2×C4×C83—41no—
140C2×C4×C83—41no—
141C2×C4×C83—41no—
142C2×C4×C83—41no—
143C2×C4×C83—41no—
144C2×C4×C83—41no—
145C2×C4×C83—41no—
146C2×C4×C83—41no—
147C2×C4×C83—81no—
148C2×C4×C83—84no—
149C2×C4×C83—82no—
150C2×C4×C83—82no—
151C2×C4×C83—82no—
152C2×C4×C83#10312yes—
153C2×C4×C83C4×D412yes—
154C2×C4×C83#12011yescharacteristic
155C2×C4×C83#12311yescharacteristic
156C8×C83—44no—
157C22×C83—21no—
158C22×C83—22no—
159C22×C83—21no—
160C22×C83—22no—
161C22×C83—22no—
162C22×C83—42no—
163C22×C83—41no—
164C22×C83—41no—
165C22×C83—42no—
166C22×C83—41no—
167C22×C83—41no—
168C22×C83—81no—
169C22×C83#38611yescharacteristic
170C22×C83C2×#10311yescharacteristic
171C22×C83#41611yescharacteristic
172C22×C83#46512yes—
173C22×C83#46712yes—
174C4×C83—22no—
175C4×C83—21no—
176C4×C83—21no—
177C4×C83—22no—
178C4×C83—42no—
179C4×C83—41no—
180C4×C83—41no—
181C4×C83—81no—
182C4×C83—81no—
183C4×C83—81no—
184C4×C83—162no—
185#9383C8311yesmaximal, characteristic, direct product factor, Sylow
186C2×C83—21no—
187C2×C83—22no—
188C2×C83—21no—
189C2×C83—22no—
190C2×C83—42no—
191C2×C83#188411yescharacteristic
192C2×C83#189211yescharacteristic
193C2×C83#191311yescharacteristic
194#2972C2×C8311yescharacteristic
195#2564C2×C8311yescharacteristic
196#1799C2×C8311yescharacteristic
197#1520C2×C8312yescomplemented
198#1294C2×C8312yescomplemented
199C83#938311yescharacteristic, direct product factor, Sylow
200C2×C22⋊Q8C22×C8311yescharacteristic
201C2×C4×D4C22×C8311yescharacteristic
202C8×D4—24no—
203#431—21no—
204C2×#99—21no—
205C2×C22⋊C8—21no—
206C2×C22⋊C8C22×C8312yes—
207#400C4×C8311yescharacteristic, complemented
208#399C4×C8311yescharacteristic, complemented
209#395C4×C8311yescharacteristic, complemented
210#393C4×C8311yescharacteristic, complemented
211#387—21no—
212#386—21no—
213#385—21no—
214#380—21no—
215#380C22×C8312yes—
216C2×#95C22×C8311yescharacteristic
217#342—21no—
218#328—24no—
219C22×Q8C2×C4×C8311yescharacteristic
220C22×D4D4×C8311yescharacteristic
221C23×C4—21no—
222C23×C4D4×C8311yescharacteristic
223C23×C4C23×C8311yescharacteristic
224C22×C8—22no—
225C22×C8—22no—
226C22⋊Q8—22no—
227C22⋊Q8—22no—
228C4×D4—22no—
229C4×D4—22no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4C2×C4×C8311yescharacteristic
235C2×C4⋊C4C2×C4×C8311yescharacteristic
236C2×C22⋊C4—21no—
237C2×C22⋊C4—21no—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4C2×C4×C8311yescharacteristic
241C2×C22⋊C4D4×C8311yescharacteristic
242C2×C42D4×C8311yescharacteristic
243C4⋊C8—22no—
244#99—42no—
245C22⋊C8—22no—
246C22⋊C8—22no—
247C22⋊C8D4×C8314yes—
248C4×C8—22no—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—22no—
255#95C2×C4×C8311yescharacteristic
256#95C2×C4×C8311yescharacteristic
257C24C22⋊C4×C8311yescharacteristic
258C24C2×D4×C8311yescharacteristic
259C2×Q8—22no—
260C2×Q8—81no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4SD16×C8312yes—
264C2×D4Q16×C8312yes—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—81no—
275C22×C4—81no—
276C22×C4—81no—
277C22×C4C22⋊C4×C8311yescharacteristic
278C22×C4C22⋊C4×C8311yescharacteristic
279C22×C4C22⋊C4×C8311yescharacteristic
280C22×C4C22×C4×C8311yescharacteristic
281C22×C4C2×D4×C8312yes—
282C22×C4C2×D4×C8311yescharacteristic
283C22×C4#41×C8312yes—
284C2×C8—24no—
285C2×C8—82no—
286C2×C8—82no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—42no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—81no—
293C4⋊C4—82no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—42no—
297C22⋊C4—42no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—81no—
301C42—21no—
302C42—21no—
303C23—21no—
304C23—21no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23—41no—
309C23#103×C8312yes—
310C23C2×C22⋊C4×C8311yescharacteristic
311C23#121×C8312yes—
312C23C2×SD16×C8311yescharacteristic
313C23C2×Q16×C8311yescharacteristic
314C23#135×C8312yes—
315Q8—82no—
316D4—24no—
317D4—24no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—22no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—81no—
342C2×C4—84no—
343C2×C4—82no—
344C2×C4—82no—
345C2×C4—82no—
346C2×C4#103×C8312yes—
347C2×C4C4×D4×C8312yes—
348C2×C4#120×C8311yescharacteristic
349C2×C4#123×C8311yescharacteristic
350C8—44no—
351C22—21no—
352C22—22no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—42no—
357C22—41no—
358C22—41no—
359C22—42no—
360C22—41no—
361C22—41no—
362C22—81no—
363C22#386×C8311yescharacteristic
364C22C2×#103×C8311yescharacteristic
365C22#416×C8311yescharacteristic
366C22#465×C8312yes—
367C22#467×C8312yes—
368C4—22no—
369C4—21no—
370C4—21no—
371C4—22no—
372C4—42no—
373C4—41no—
374C4—41no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—162no—
379C2—21no—
380C2—22no—
381C2—21no—
382C2—22no—
383C2—42no—
384C2C83×#188411yescharacteristic
385C2C83×#189211yescharacteristic
386C2C83×#191311yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC22×C4C22×C4×C83
Frattini, cosocleC23×C4C23×C83
prime radicalC22×D4×C83D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C23×C4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
4. C83×#1520C2semidirect
5. C83×#1294C2semidirect
13. #400×C83C4semidirect
14. #399×C83C4semidirect
15. #395×C83C4semidirect
16. #393×C83C4semidirect
185. #9383C83direct (N × H)
197. #1520C2×C83semidirect
198. #1294C2×C83semidirect
199. C83#9383direct (N × H)
207. #400C4×C83semidirect
208. #399C4×C83semidirect
209. #395C4×C83semidirect
210. #393C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#29722
C83×#25642
C83×#17992
C83×#15202
C83×#12942
#938383