Finite Group Databasebeta

← C83×#9385 #35237209

Subgroups — C83×#9385 #35237209

Order 21248.

548 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#2973C211yesmaximal, characteristic
2C83×#2564C211yesmaximal, characteristic
3C83×#1799C211yesmaximal, characteristic
4C83×#1520C212yesmaximal, complemented
5C83×#1294C212yesmaximal, complemented
6C2×C22⋊Q8×C83C2211yescharacteristic
7C2×C4×D4×C83C2211yescharacteristic
8#435×C83—24no—
9#432×C83—21no—
10C2×#99×C83—21no—
11#407×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
30C2×M16×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—
54#97×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—21no—
69C2×D4×C83—21no—
70C2×D4×C83—22no—
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—41no—
79C22×C4×C83—41no—
80C22×C4×C83—42no—
81C22×C4×C83—81no—
82C22×C4×C83—81no—
83C22×C4×C83—81no—
84C22×C4×C83C22⋊C411yescharacteristic
85C22×C4×C83C22⋊C411yescharacteristic
86C22×C4×C83C22⋊C411yescharacteristic
87C22×C4×C83C22×C411yescharacteristic
88C22×C4×C83C2×D412yes—
89C22×C4×C83C2×D411yescharacteristic
90C22×C4×C83#4112yes—
91M16×C83—82no—
92C2×C8×C83—24no—
93C2×C8×C83—82no—
94C4⋊C4×C83—22no—
95C4⋊C4×C83—22no—
96C4⋊C4×C83—42no—
97C4⋊C4×C83—41no—
98C4⋊C4×C83—41no—
99C4⋊C4×C83—81no—
100C4⋊C4×C83—82no—
101C22⋊C4×C83—22no—
102C22⋊C4×C83—22no—
103C22⋊C4×C83—42no—
104C22⋊C4×C83—42no—
105C22⋊C4×C83—42no—
106C22⋊C4×C83—42no—
107C22⋊C4×C83—81no—
108C42×C83—21no—
109C42×C83—21no—
110C23×C83—21no—
111C23×C83—21no—
112C23×C83—21no—
113C23×C83—41no—
114C23×C83—41no—
115C23×C83—41no—
116C23×C83—41no—
117C23×C83C2×C22⋊C411yescharacteristic
118C23×C83#12112yes—
119C23×C83AGL(1,ℤ/8)11yescharacteristic
120C23×C83#13711yescharacteristic
121C23×C83#13712yes—
122Q8×C83—82no—
123D4×C83—42no—
124D4×C83—42no—
125C2×C4×C83—22no—
126C2×C4×C83—21no—
127C2×C4×C83—21no—
128C2×C4×C83—21no—
129C2×C4×C83—21no—
130C2×C4×C83—21no—
131C2×C4×C83—42no—
132C2×C4×C83—42no—
133C2×C4×C83—41no—
134C2×C4×C83—41no—
135C2×C4×C83—42no—
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—41no—
148C2×C4×C83—41no—
149C2×C4×C83—81no—
150C2×C4×C83—82no—
151C2×C4×C83—82no—
152C2×C4×C83—82no—
153C2×C4×C83—82no—
154C2×C4×C83—82no—
155C2×C4×C83C4×D412yes—
156C2×C4×C83#12011yescharacteristic
157C2×C4×C83#12311yescharacteristic
158C8×C83—44no—
159C22×C83—21no—
160C22×C83—22no—
161C22×C83—21no—
162C22×C83—22no—
163C22×C83—41no—
164C22×C83—41no—
165C22×C83—42no—
166C22×C83—42no—
167C22×C83—41no—
168C22×C83—41no—
169C22×C83—41no—
170C22×C83—81no—
171C22×C83#38611yescharacteristic
172C22×C83#41711yescharacteristic
173C22×C83#41911yescharacteristic
174C22×C83#46812yes—
175C22×C83#47012yes—
176C4×C83—21no—
177C4×C83—21no—
178C4×C83—22no—
179C4×C83—42no—
180C4×C83—41no—
181C4×C83—41no—
182C4×C83—41no—
183C4×C83—81no—
184C4×C83—81no—
185C4×C83—81no—
186C4×C83—162no—
187#9385C8311yesmaximal, characteristic, direct product factor, Sylow
188C2×C83—21no—
189C2×C83—22no—
190C2×C83—21no—
191C2×C83—22no—
192C2×C83—81no—
193C2×C83#188411yescharacteristic
194C2×C83#189411yescharacteristic
195C2×C83#191311yescharacteristic
196#2973C2×C8311yescharacteristic
197#2564C2×C8311yescharacteristic
198#1799C2×C8311yescharacteristic
199#1520C2×C8312yescomplemented
200#1294C2×C8312yescomplemented
201C83#938511yescharacteristic, direct product factor, Sylow
202C2×C22⋊Q8C22×C8311yescharacteristic
203C2×C4×D4C22×C8311yescharacteristic
204#435—24no—
205#432—21no—
206C2×#99—21no—
207#407—21no—
208C2×C22⋊C8C22×C8312yes—
209#400C4×C8311yescharacteristic, complemented
210#399C4×C8311yescharacteristic, complemented
211#395C4×C8311yescharacteristic, complemented
212#393C4×C8311yescharacteristic, complemented
213#387—21no—
214#386—21no—
215#385—21no—
216#380—21no—
217#380C22×C8312yes—
218C2×#95C22×C8311yescharacteristic
219#342—21no—
220#328—24no—
221C22×Q8C2×C4×C8311yescharacteristic
222C22×D4D4×C8311yescharacteristic
223C23×C4—21no—
224C23×C4D4×C8311yescharacteristic
225C23×C4C23×C8311yescharacteristic
226C2×M16—22no—
227C22×C8—22no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C4×D4—22no—
231C4×D4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4C2×C4×C8311yescharacteristic
237C2×C4⋊C4C2×C4×C8311yescharacteristic
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4C2×C4×C8311yescharacteristic
243C2×C22⋊C4D4×C8311yescharacteristic
244C2×C42D4×C8311yescharacteristic
245C4⋊C8—22no—
246#99—42no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8D4×C8314yes—
250#97—22no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—22no—
257#95C2×C4×C8311yescharacteristic
258#95C2×C4×C8311yescharacteristic
259C24C22⋊C4×C8311yescharacteristic
260C24C2×D4×C8311yescharacteristic
261C2×Q8—22no—
262C2×Q8—81no—
263C2×D4—22no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—81no—
278C22×C4—81no—
279C22×C4—81no—
280C22×C4C22⋊C4×C8311yescharacteristic
281C22×C4C22⋊C4×C8311yescharacteristic
282C22×C4C22⋊C4×C8311yescharacteristic
283C22×C4C22×C4×C8311yescharacteristic
284C22×C4C2×D4×C8312yes—
285C22×C4C2×D4×C8311yescharacteristic
286C22×C4#41×C8312yes—
287M16—82no—
288C2×C8—24no—
289C2×C8—82no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—42no—
293C4⋊C4—41no—
294C4⋊C4—41no—
295C4⋊C4—81no—
296C4⋊C4—82no—
297C22⋊C4—22no—
298C22⋊C4—22no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—42no—
303C22⋊C4—81no—
304C42—21no—
305C42—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—41no—
310C23—41no—
311C23—41no—
312C23—41no—
313C23C2×C22⋊C4×C8311yescharacteristic
314C23#121×C8312yes—
315C23AGL(1,ℤ/8)×C8311yescharacteristic
316C23#137×C8312yes—
317C23#137×C8311yescharacteristic
318Q8—82no—
319D4—42no—
320D4—42no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
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—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4—82no—
349C2×C4—82no—
350C2×C4—82no—
351C2×C4C4×D4×C8312yes—
352C2×C4#120×C8311yescharacteristic
353C2×C4#123×C8311yescharacteristic
354C8—44no—
355C22—21no—
356C22—22no—
357C22—21no—
358C22—22no—
359C22—41no—
360C22—41no—
361C22—42no—
362C22—42no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—81no—
367C22#386×C8311yescharacteristic
368C22#417×C8311yescharacteristic
369C22#419×C8311yescharacteristic
370C22#468×C8312yes—
371C22#470×C8312yes—
372C4—21no—
373C4—21no—
374C4—22no—
375C4—42no—
376C4—41no—
377C4—41no—
378C4—41no—
379C4—81no—
380C4—81no—
381C4—81no—
382C4—162no—
383C2—21no—
384C2—22no—
385C2—21no—
386C2—22no—
387C2—81no—
388C2C83×#188411yescharacteristic
389C2C83×#189411yescharacteristic
390C2C83×#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
187. #9385C83direct (N × H)
199. #1520C2×C83semidirect
200. #1294C2×C83semidirect
201. C83#9385direct (N × H)
209. #400C4×C83semidirect
210. #399C4×C83semidirect
211. #395C4×C83semidirect
212. #393C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#29732
C83×#25642
C83×#17992
C83×#15202
C83×#12942
#938583