Finite Group Databasebeta

← C83×#19396 #35247220

Subgroups — C83×#19396 #35247220

Order 21248.

686 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×#3208C211yesmaximal, characteristic
2C83×#3078C211yesmaximal, characteristic, complemented
3C2×#460×C83C211yesmaximal, characteristic
4C83×#2982C211yesmaximal, characteristic, complemented
5C83×#1678C214yesmaximal, complemented
6C83×#1666C214yesmaximal, complemented
7C83×#1570C211yesmaximal, characteristic, complemented
8C83×#1545C211yesmaximal, characteristic
9C83×#1542C211yesmaximal, characteristic, complemented
10C22×SD16×C83—21no—
11#533×C83C2211yescharacteristic
12C2×#127×C83C2211yescharacteristic
13C2×C4×Q8×C83C2211yescharacteristic
14#492×C83C2212yes—
15#486×C83C2212yes—
16#479×C83—22no—
17#477×C83C2212yescomplemented
18#475×C83C2212yescomplemented
19#460×C83—22no—
20#460×C83—22no—
21#460×C83C2214yes—
22#450×C83—22no—
23C8×Q8×C83—22no—
24C8×Q8×C83C2214yescomplemented
25#431×C83—22no—
26#431×C83C2211yescharacteristic
27#431×C83C2211yescharacteristic
28#428×C83—21no—
29#423×C83C2211yescharacteristic
30C2×C4⋊C8×C83C2211yescharacteristic
31#419×C83—21no—
32C2×#102×C83—21no—
33#334×C83C2214yescomplemented
34#333×C83C2212yescomplemented
35#332×C83C2212yescomplemented
36#331×C83C2214yes—
37C22×Q8×C83D411yescharacteristic
38C22×D4×C83—21no—
39C22×D4×C83—21no—
40C22×D4×C83—21no—
41C2×SD16×C83—24no—
42C2×SD16×C83—24no—
43C2×SD16×C83—44no—
44C2×M16×C83—21no—
45C22×C8×C83—21no—
46C4⋊Q8×C83C2311yescharacteristic
47C4⋊Q8×C83C2311yescharacteristic
48#127×C83—21no—
49#127×C83—21no—
50#127×C83—21no—
51#127×C83C2312yes—
52#125×C83—21no—
53C22⋊Q8×C83—21no—
54C22⋊Q8×C83—21no—
55C4×Q8×C83—21no—
56C4×Q8×C83—21no—
57C4×Q8×C83—21no—
58C4×Q8×C83—21no—
59C4×Q8×C83D412yes—
60C4×Q8×C83C2312yes—
61#117×C83—21no—
62C2×C4⋊C4×C83—21no—
63C2×C4⋊C4×C83D411yescharacteristic
64C2×C42×C83—21no—
65C2×C42×C83C2311yescharacteristic
66#107×C83—22no—
67#106×C83—22no—
68C4⋊C8×C83—24no—
69C4⋊C8×C83—21no—
70C4⋊C8×C83—21no—
71C4⋊C8×C83D412yes—
72C4⋊C8×C83C2312yes—
73C4⋊C8×C83C2312yes—
74C4⋊C8×C83C2312yes—
75#103×C83—22no—
76#103×C83—22no—
77#102×C83—24no—
78#102×C83—24no—
79C22⋊C8×C83—21no—
80C22⋊C8×C83D412yes—
81C4×C8×C83—24no—
82C4×C8×C83—21no—
83C4×C8×C83C2312yes—
84C24×C83—41no—
85C2×Q8×C83—21no—
86C2×Q8×C83—22no—
87C2×Q8×C83—21no—
88C2×Q8×C83—21no—
89C2×Q8×C83SD1614yescomplemented
90C2×Q8×C83C2×D412yes—
91C2×D4×C83—22no—
92C2×D4×C83—22no—
93C2×D4×C83—22no—
94C2×D4×C83—41no—
95C2×D4×C83—41no—
96C2×D4×C83—41no—
97C2×D4×C83—42no—
98C2×D4×C83—42no—
99C2×D4×C83—82no—
100C2×D4×C83—82no—
101C22×C4×C83—21no—
102C22×C4×C83—21no—
103C22×C4×C83C2×D411yescharacteristic
104C22×C4×C83C2×D411yescharacteristic
105C22×C4×C83#4111yescharacteristic
106SD16×C83—48no—
107SD16×C83—84no—
108M16×C83—42no—
109C2×C8×C83—22no—
110C2×C8×C83—22no—
111C2×C8×C83—22no—
112C2×C8×C83—42no—
113C2×C8×C83C2×D414yes—
114C4⋊C4×C83—22no—
115C4⋊C4×C83—21no—
116C4⋊C4×C83—22no—
117C4⋊C4×C83—21no—
118C4⋊C4×C83—21no—
119C4⋊C4×C83—21no—
120C4⋊C4×C83—21no—
121C4⋊C4×C83—41no—
122C4⋊C4×C83—41no—
123C4⋊C4×C83C2×D412yes—
124C22⋊C4×C83—41no—
125C42×C83—22no—
126C42×C83—22no—
127C42×C83—41no—
128C42×C83C2×D411yescharacteristic
129C42×C83C2×D411yescharacteristic
130C42×C83#4111yescharacteristic
131C42×C83C2411yescharacteristic
132C23×C83—42no—
133C23×C83—82no—
134C23×C83—82no—
135C23×C83—82no—
136C23×C83#12111yescharacteristic
137Q8×C83—22no—
138Q8×C83—24no—
139Q8×C83—42no—
140Q8×C83—81no—
141Q8×C83C2×SD1614yes—
142D4×C83—42no—
143D4×C83—42no—
144D4×C83—44no—
145D4×C83—82no—
146D4×C83—82no—
147D4×C83—82no—
148D4×C83—82no—
149C2×C4×C83—22no—
150C2×C4×C83—24no—
151C2×C4×C83—22no—
152C2×C4×C83—22no—
153C2×C4×C83—41no—
154C2×C4×C83—41no—
155C2×C4×C83—41no—
156C2×C4×C83—81no—
157C2×C4×C83#12111yescharacteristic
158C2×C4×C83#12111yescharacteristic
159C2×C4×C83#12111yescharacteristic
160C2×C4×C83C2×SD1612yes—
161C2×C4×C83#13512yes—
162C2×C4×C83C22×D411yescharacteristic
163C2×C4×C83C22×D411yescharacteristic
164C2×C4×C83C2×#4111yescharacteristic
165C8×C83—24no—
166C8×C83—42no—
167C8×C83—42no—
168C8×C83—42no—
169C22×C83—21no—
170C22×C83—21no—
171C22×C83—82no—
172C22×C83—82no—
173C22×C83—82no—
174C22×C83—164no—
175C22×C83#46512yes—
176C22×C83C2×#12111yescharacteristic
177C4×C83—21no—
178C4×C83—24no—
179C4×C83—21no—
180C4×C83—21no—
181C4×C83—42no—
182C4×C83—81no—
183C4×C83—81no—
184C4×C83#46512yes—
185C4×C83C22×SD1611yescharacteristic
186C4×C83C2×#13511yescharacteristic
187C4×C83#57712yes—
188#19396C8311yesmaximal, characteristic, direct product factor, Sylow
189C2×C83—22no—
190C2×C83—162no—
191C2×C83#304311yescharacteristic
192C2×C83C2×#46511yescharacteristic
193C2×C83#306811yescharacteristic
194#3208C2×C8311yescharacteristic
195#3078C2×C8311yescharacteristic, complemented
196C2×#460C2×C8311yescharacteristic
197#2982C2×C8311yescharacteristic, complemented
198#1678C2×C8314yescomplemented
199#1666C2×C8314yescomplemented
200#1570C2×C8311yescharacteristic, complemented
201#1545C2×C8311yescharacteristic
202#1542C2×C8311yescharacteristic, complemented
203C83#1939611yescharacteristic, direct product factor, Sylow
204C22×SD16—21no—
205#533C22×C8311yescharacteristic
206C2×#127C22×C8311yescharacteristic
207C2×C4×Q8C22×C8311yescharacteristic
208#492C22×C8312yes—
209#486C22×C8312yes—
210#479—22no—
211#477C22×C8312yescomplemented
212#475C22×C8312yescomplemented
213#460—22no—
214#460—22no—
215#460C22×C8314yes—
216#450—22no—
217C8×Q8—22no—
218C8×Q8C22×C8314yescomplemented
219#431—22no—
220#431C22×C8311yescharacteristic
221#431C22×C8311yescharacteristic
222#428—21no—
223#423C22×C8311yescharacteristic
224C2×C4⋊C8C22×C8311yescharacteristic
225#419—21no—
226C2×#102—21no—
227#334C22×C8314yescomplemented
228#333C22×C8312yescomplemented
229#332C22×C8312yescomplemented
230#331C22×C8314yes—
231C22×Q8D4×C8311yescharacteristic
232C22×D4—21no—
233C22×D4—21no—
234C22×D4—21no—
235C2×SD16—24no—
236C2×SD16—24no—
237C2×SD16—44no—
238C2×M16—21no—
239C22×C8—21no—
240C4⋊Q8C23×C8311yescharacteristic
241C4⋊Q8C23×C8311yescharacteristic
242#127—21no—
243#127—21no—
244#127—21no—
245#127C23×C8312yes—
246#125—21no—
247C22⋊Q8—21no—
248C22⋊Q8—21no—
249C4×Q8—21no—
250C4×Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8D4×C8312yes—
254C4×Q8C23×C8312yes—
255#117—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4D4×C8311yescharacteristic
258C2×C42—21no—
259C2×C42C23×C8311yescharacteristic
260#107—22no—
261#106—22no—
262C4⋊C8—24no—
263C4⋊C8—21no—
264C4⋊C8—21no—
265C4⋊C8D4×C8312yes—
266C4⋊C8C23×C8312yes—
267C4⋊C8C23×C8312yes—
268C4⋊C8C23×C8312yes—
269#103—22no—
270#103—22no—
271#102—24no—
272#102—24no—
273C22⋊C8—21no—
274C22⋊C8D4×C8312yes—
275C4×C8—24no—
276C4×C8—21no—
277C4×C8C23×C8312yes—
278C24—41no—
279C2×Q8—21no—
280C2×Q8—22no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8SD16×C8314yescomplemented
284C2×Q8C2×D4×C8312yes—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—82no—
294C2×D4—82no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4C2×D4×C8311yescharacteristic
298C22×C4C2×D4×C8311yescharacteristic
299C22×C4#41×C8311yescharacteristic
300SD16—48no—
301SD16—84no—
302M16—42no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—42no—
307C2×C8C2×D4×C8314yes—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4C2×D4×C8312yes—
318C22⋊C4—41no—
319C42—22no—
320C42—22no—
321C42—41no—
322C42C2×D4×C8311yescharacteristic
323C42C2×D4×C8311yescharacteristic
324C42#41×C8311yescharacteristic
325C42C24×C8311yescharacteristic
326C23—42no—
327C23—82no—
328C23—82no—
329C23—82no—
330C23#121×C8311yescharacteristic
331Q8—24no—
332Q8—22no—
333Q8—42no—
334Q8—81no—
335Q8C2×SD16×C8314yes—
336D4—42no—
337D4—42no—
338D4—44no—
339D4—82no—
340D4—82no—
341D4—82no—
342D4—82no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—81no—
351C2×C4#121×C8311yescharacteristic
352C2×C4#121×C8311yescharacteristic
353C2×C4#121×C8311yescharacteristic
354C2×C4C2×SD16×C8312yes—
355C2×C4#135×C8312yes—
356C2×C4C22×D4×C8311yescharacteristic
357C2×C4C22×D4×C8311yescharacteristic
358C2×C4C2×#41×C8311yescharacteristic
359C8—24no—
360C8—42no—
361C8—42no—
362C8—42no—
363C22—21no—
364C22—21no—
365C22—82no—
366C22—82no—
367C22—82no—
368C22—164no—
369C22#465×C8312yes—
370C22C2×#121×C8311yescharacteristic
371C4—21no—
372C4—24no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—81no—
377C4—81no—
378C4#465×C8312yes—
379C4C22×SD16×C8311yescharacteristic
380C4C2×#135×C8311yescharacteristic
381C4#577×C8312yes—
382C2—22no—
383C2—162no—
384C2C83×#304311yescharacteristic
385C2C2×#465×C8311yescharacteristic
386C2C83×#306811yescharacteristic
subgroupG/H
center, socleC22×C83C2×#121
commutator, Frattini, cosocleC42C24×C83
prime radicalC2×#127×C83C22
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
2. C83×#3078C2semidirect
4. C83×#2982C2semidirect
5. C83×#1678C2semidirect
6. C83×#1666C2semidirect
7. C83×#1570C2semidirect
9. C83×#1542C2semidirect
17. #477×C83C22semidirect
18. #475×C83C22semidirect
24. C8×Q8×C83C22semidirect
33. #334×C83C22semidirect
34. #333×C83C22semidirect
35. #332×C83C22semidirect
89. C2×Q8×C83SD16semidirect
188. #19396C83direct (N × H)
195. #3078C2×C83semidirect
197. #2982C2×C83semidirect
198. #1678C2×C83semidirect
199. #1666C2×C83semidirect
200. #1570C2×C83semidirect
202. #1542C2×C83semidirect
203. C83#19396direct (N × H)
211. #477C22×C83semidirect
212. #475C22×C83semidirect
218. C8×Q8C22×C83semidirect
227. #334C22×C83semidirect
228. #333C22×C83semidirect
229. #332C22×C83semidirect
283. C2×Q8SD16×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#32082
C83×#30782
C2×#460×C832
C83×#29822
C83×#16782
C83×#16662
C83×#15702
C83×#15452
C83×#15422
#1939683