Finite Group Databasebeta

← C83×#19212 #35247036

Subgroups — C83×#19212 #35247036

Order 21248.

646 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3230C211yesmaximal, characteristic, complemented
2C83×#3208C211yesmaximal, characteristic
3C2×#486×C83C211yesmaximal, characteristic
4C83×#2960C211yesmaximal, characteristic, complemented
5C83×#1698C214yesmaximal, complemented
6C83×#1678C214yesmaximal, complemented
7C83×#1564C211yesmaximal, characteristic, complemented
8C83×#1548C211yesmaximal, characteristic, complemented
9C83×#1545C211yesmaximal, characteristic
10#533×C83C2211yescharacteristic
11C2×C4⋊Q8×C83C2211yescharacteristic
12C2×#127×C83C2211yescharacteristic
13#500×C83—21no—
14#492×C83—21no—
15#487×C83—21no—
16#486×C83—21no—
17#486×C83—21no—
18#486×C83—21no—
19#486×C83C2212yes—
20#486×C83C2212yes—
21#484×C83—22no—
22#477×C83C2214yescomplemented
23#460×C83C2214yes—
24#450×C83—22no—
25#446×C83—21no—
26#446×C83C2212yescomplemented
27C8×Q8×C83—21no—
28C8×Q8×C83C2212yescomplemented
29#431×C83C2211yescharacteristic
30#431×C83C2211yescharacteristic
31#424×C83—21no—
32#423×C83C2211yescharacteristic
33C2×#102×C83—21no—
34C2×#102×C83—21no—
35#405×C83—21no—
36C2×C4×C8×C83C2211yescharacteristic
37#335×C83C2212yescomplemented
38#334×C83C2212yescomplemented
39#333×C83C2214yescomplemented
40#331×C83C2214yes—
41C22×Q8×C83—21no—
42C22×D4×C83—21no—
43C22×D4×C83—21no—
44C22×D4×C83—21no—
45C2×SD16×C83—24no—
46C2×M16×C83—21no—
47C22×C8×C83D411yescharacteristic
48C22×C8×C83D411yescharacteristic
49C4⋊Q8×C83—21no—
50C4⋊Q8×C83—21no—
51C4⋊Q8×C83—21no—
52C4⋊Q8×C83D411yescharacteristic
53C4⋊Q8×C83D411yescharacteristic
54C4⋊Q8×C83C2312yes—
55#127×C83—21no—
56#127×C83—21no—
57#127×C83—21no—
58#127×C83C2312yes—
59#125×C83—21no—
60C22⋊Q8×C83—21no—
61C22⋊Q8×C83D412yes—
62C4×Q8×C83—21no—
63C4×Q8×C83C2312yes—
64#117×C83—21no—
65C2×C4⋊C4×C83—21no—
66C2×C4⋊C4×C83—21no—
67C2×C42×C83C2311yescharacteristic
68#107×C83—24no—
69C4⋊C8×C83—22no—
70C4⋊C8×C83—22no—
71C4⋊C8×C83—21no—
72C4⋊C8×C83C2312yes—
73C4⋊C8×C83C2312yes—
74C4⋊C8×C83C2312yes—
75#103×C83—24no—
76#102×C83—22no—
77#102×C83—22no—
78#102×C83—22no—
79#102×C83—22no—
80#102×C83—24no—
81C22⋊C8×C83—21no—
82C22⋊C8×C83—21no—
83#97×C83—22no—
84C4×C8×C83—22no—
85C4×C8×C83—21no—
86C4×C8×C83—21no—
87C4×C8×C83D411yescharacteristic
88C4×C8×C83D411yescharacteristic
89C4×C8×C83C2311yescharacteristic
90C4×C8×C83C2311yescharacteristic
91C24×C83—41no—
92C2×Q8×C83—21no—
93C2×Q8×C83—22no—
94C2×Q8×C83—41no—
95C2×Q8×C83—82no—
96C2×Q8×C83C2×D412yes—
97C2×D4×C83—22no—
98C2×D4×C83—22no—
99C2×D4×C83—22no—
100C2×D4×C83—41no—
101C2×D4×C83—41no—
102C2×D4×C83—41no—
103C2×D4×C83—42no—
104C2×D4×C83—42no—
105C2×D4×C83—82no—
106C2×D4×C83—82no—
107C22×C4×C83—41no—
108C22×C4×C83C2×D411yescharacteristic
109C22×C4×C83C2×D411yescharacteristic
110C22×C4×C83#4111yescharacteristic
111SD16×C83—84no—
112M16×C83—42no—
113C2×C8×C83—22no—
114C2×C8×C83—22no—
115C2×C8×C83—22no—
116C2×C8×C83—22no—
117C2×C8×C83—22no—
118C2×C8×C83C2×D411yescharacteristic
119C2×C8×C83C2×D411yescharacteristic
120C2×C8×C83C2×D411yescharacteristic
121C2×C8×C83C2×D411yescharacteristic
122C4⋊C4×C83—21no—
123C4⋊C4×C83—21no—
124C4⋊C4×C83—21no—
125C4⋊C4×C83—22no—
126C4⋊C4×C83—22no—
127C4⋊C4×C83—22no—
128C4⋊C4×C83—22no—
129C4⋊C4×C83—41no—
130C4⋊C4×C83—41no—
131C4⋊C4×C83C2×D412yes—
132C22⋊C4×C83—22no—
133C42×C83—22no—
134C42×C83C2×D411yescharacteristic
135C42×C83C2×D411yescharacteristic
136C42×C83#4111yescharacteristic
137C42×C83C2411yescharacteristic
138C23×C83—42no—
139C23×C83—82no—
140C23×C83—82no—
141C23×C83—82no—
142C23×C83#12111yescharacteristic
143Q8×C83—24no—
144Q8×C83—42no—
145Q8×C83—82no—
146Q8×C83—82no—
147D4×C83—42no—
148D4×C83—42no—
149D4×C83—44no—
150D4×C83—82no—
151D4×C83—82no—
152D4×C83—82no—
153D4×C83—82no—
154C2×C4×C83—22no—
155C2×C4×C83—22no—
156C2×C4×C83—22no—
157C2×C4×C83—22no—
158C2×C4×C83—42no—
159C2×C4×C83—82no—
160C2×C4×C83#12111yescharacteristic
161C2×C4×C83#12111yescharacteristic
162C2×C4×C83#12111yescharacteristic
163C2×C4×C83#13512yes—
164C2×C4×C83#13712yes—
165C2×C4×C83C22×D411yescharacteristic
166C2×C4×C83C22×D411yescharacteristic
167C2×C4×C83C2×#4111yescharacteristic
168C8×C83—21no—
169C8×C83—21no—
170C8×C83—21no—
171C8×C83—21no—
172C8×C83—42no—
173C8×C83—42no—
174C8×C83—42no—
175C22×C83—21no—
176C22×C83—21no—
177C22×C83—82no—
178C22×C83—82no—
179C22×C83—82no—
180C22×C83—164no—
181C22×C83#46412yes—
182C22×C83C2×#12111yescharacteristic
183C4×C83—21no—
184C4×C83—21no—
185C4×C83—42no—
186C4×C83—42no—
187C4×C83—82no—
188C4×C83#46412yes—
189C4×C83#47212yes—
190C4×C83C2×#13511yescharacteristic
191C4×C83C2×#13711yescharacteristic
192C4×C83#57712yes—
193#19212C8311yesmaximal, characteristic, direct product factor, Sylow
194C2×C83—22no—
195C2×C83—162no—
196C2×C83C2×#46411yescharacteristic
197C2×C83#305711yescharacteristic
198C2×C83C2×#47211yescharacteristic
199#3230C2×C8311yescharacteristic, complemented
200#3208C2×C8311yescharacteristic
201C2×#486C2×C8311yescharacteristic
202#2960C2×C8311yescharacteristic, complemented
203#1698C2×C8314yescomplemented
204#1678C2×C8314yescomplemented
205#1564C2×C8311yescharacteristic, complemented
206#1548C2×C8311yescharacteristic, complemented
207#1545C2×C8311yescharacteristic
208C83#1921211yescharacteristic, direct product factor, Sylow
209#533C22×C8311yescharacteristic
210C2×C4⋊Q8C22×C8311yescharacteristic
211C2×#127C22×C8311yescharacteristic
212#500—21no—
213#492—21no—
214#487—21no—
215#486—21no—
216#486—21no—
217#486—21no—
218#486C22×C8312yes—
219#486C22×C8312yes—
220#484—22no—
221#477C22×C8314yescomplemented
222#460C22×C8314yes—
223#450—22no—
224#446—21no—
225#446C22×C8312yescomplemented
226C8×Q8—21no—
227C8×Q8C22×C8312yescomplemented
228#431C22×C8311yescharacteristic
229#431C22×C8311yescharacteristic
230#424—21no—
231#423C22×C8311yescharacteristic
232C2×#102—21no—
233C2×#102—21no—
234#405—21no—
235C2×C4×C8C22×C8311yescharacteristic
236#335C22×C8312yescomplemented
237#334C22×C8312yescomplemented
238#333C22×C8314yescomplemented
239#331C22×C8314yes—
240C22×Q8—21no—
241C22×D4—21no—
242C22×D4—21no—
243C22×D4—21no—
244C2×SD16—24no—
245C2×M16—21no—
246C22×C8D4×C8311yescharacteristic
247C22×C8D4×C8311yescharacteristic
248C4⋊Q8—21no—
249C4⋊Q8—21no—
250C4⋊Q8—21no—
251C4⋊Q8D4×C8311yescharacteristic
252C4⋊Q8D4×C8311yescharacteristic
253C4⋊Q8C23×C8312yes—
254#127—21no—
255#127—21no—
256#127—21no—
257#127C23×C8312yes—
258#125—21no—
259C22⋊Q8—21no—
260C22⋊Q8D4×C8312yes—
261C4×Q8—21no—
262C4×Q8C23×C8312yes—
263#117—21no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C42C23×C8311yescharacteristic
267#107—24no—
268C4⋊C8—22no—
269C4⋊C8—22no—
270C4⋊C8—21no—
271C4⋊C8C23×C8312yes—
272C4⋊C8C23×C8312yes—
273C4⋊C8C23×C8312yes—
274#103—24no—
275#102—22no—
276#102—22no—
277#102—22no—
278#102—22no—
279#102—24no—
280C22⋊C8—21no—
281C22⋊C8—21no—
282#97—22no—
283C4×C8—22no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8D4×C8311yescharacteristic
287C4×C8D4×C8311yescharacteristic
288C4×C8C23×C8311yescharacteristic
289C4×C8C23×C8311yescharacteristic
290C24—41no—
291C2×Q8—21no—
292C2×Q8—22no—
293C2×Q8—41no—
294C2×Q8—82no—
295C2×Q8C2×D4×C8312yes—
296C2×D4—22no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—42no—
303C2×D4—42no—
304C2×D4—82no—
305C2×D4—82no—
306C22×C4—41no—
307C22×C4C2×D4×C8311yescharacteristic
308C22×C4C2×D4×C8311yescharacteristic
309C22×C4#41×C8311yescharacteristic
310SD16—84no—
311M16—42no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8C2×D4×C8311yescharacteristic
318C2×C8C2×D4×C8311yescharacteristic
319C2×C8C2×D4×C8311yescharacteristic
320C2×C8C2×D4×C8311yescharacteristic
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—22no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4C2×D4×C8312yes—
331C22⋊C4—22no—
332C42—22no—
333C42C2×D4×C8311yescharacteristic
334C42C2×D4×C8311yescharacteristic
335C42#41×C8311yescharacteristic
336C42C24×C8311yescharacteristic
337C23—42no—
338C23—82no—
339C23—82no—
340C23—82no—
341C23#121×C8311yescharacteristic
342Q8—24no—
343Q8—42no—
344Q8—82no—
345Q8—82no—
346D4—42no—
347D4—42no—
348D4—44no—
349D4—82no—
350D4—82no—
351D4—82no—
352D4—82no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4#121×C8311yescharacteristic
360C2×C4#121×C8311yescharacteristic
361C2×C4#121×C8311yescharacteristic
362C2×C4#135×C8312yes—
363C2×C4#137×C8312yes—
364C2×C4C22×D4×C8311yescharacteristic
365C2×C4C22×D4×C8311yescharacteristic
366C2×C4C2×#41×C8311yescharacteristic
367C8—21no—
368C8—21no—
369C8—21no—
370C8—21no—
371C8—42no—
372C8—42no—
373C8—42no—
374C22—21no—
375C22—21no—
376C22—82no—
377C22—82no—
378C22—82no—
379C22—164no—
380C22#464×C8312yes—
381C22C2×#121×C8311yescharacteristic
382C4—21no—
383C4—21no—
384C4—42no—
385C4—42no—
386C4—82no—
387C4#464×C8312yes—
388C4#472×C8312yes—
389C4C2×#135×C8311yescharacteristic
390C4C2×#137×C8311yescharacteristic
391C4#577×C8312yes—
392C2—22no—
393C2—162no—
394C2C2×#464×C8311yescharacteristic
395C2C83×#305711yescharacteristic
396C2C2×#472×C8311yescharacteristic
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
1. C83×#3230C2semidirect
4. C83×#2960C2semidirect
5. C83×#1698C2semidirect
6. C83×#1678C2semidirect
7. C83×#1564C2semidirect
8. C83×#1548C2semidirect
22. #477×C83C22semidirect
26. #446×C83C22semidirect
28. C8×Q8×C83C22semidirect
37. #335×C83C22semidirect
38. #334×C83C22semidirect
39. #333×C83C22semidirect
193. #19212C83direct (N × H)
199. #3230C2×C83semidirect
202. #2960C2×C83semidirect
203. #1698C2×C83semidirect
204. #1678C2×C83semidirect
205. #1564C2×C83semidirect
206. #1548C2×C83semidirect
208. C83#19212direct (N × H)
221. #477C22×C83semidirect
225. #446C22×C83semidirect
227. C8×Q8C22×C83semidirect
236. #335C22×C83semidirect
237. #334C22×C83semidirect
238. #333C22×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#32302
C83×#32082
C2×#486×C832
C83×#29602
C83×#16982
C83×#16782
C83×#15642
C83×#15482
C83×#15452
#1921283