Finite Group Databasebeta

← C83×#19421 #35247245

Subgroups — C83×#19421 #35247245

Order 21248.

684 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3206C211yesmaximal, characteristic, complemented
2C2×#474×C83C211yesmaximal, characteristic, complemented
3C83×#3036C211yesmaximal, characteristic
4C83×#2975C211yesmaximal, characteristic, complemented
5C83×#1699C214yesmaximal, complemented
6C83×#1685C214yesmaximal, complemented
7C83×#1571C211yesmaximal, characteristic, complemented
8C83×#1549C211yesmaximal, characteristic, complemented
9C83×#1540C211yesmaximal, characteristic, complemented
10C2×AGL(1,ℤ/8)×C83—21no—
11#532×C83C2211yescharacteristic
12C2×C4⋊Q8×C83C2211yescharacteristic
13C2×C4×D4×C83C2211yescharacteristic
14#492×C83C2212yescomplemented
15#487×C83C2212yescomplemented
16#474×C83—22no—
17#474×C83C2214yes—
18#463×C83—22no—
19#461×C83—22no—
20#461×C83C2212yes—
21#459×C83C2212yes—
22#448×C83—22no—
23#436×C83—22no—
24#436×C83C2214yes—
25#431×C83C2211yescharacteristic
26#431×C83C2211yescharacteristic
27C2×#107×C83—21no—
28#423×C83C2211yescharacteristic
29C2×C4⋊C8×C83C2211yescharacteristic
30#417×C83—21no—
31C2×#102×C83—21no—
32#407×C83—22no—
33#335×C83C2214yescomplemented
34#332×C83C2212yescomplemented
35#332×C83C2214yescomplemented
36#331×C83C2212yescomplemented
37C2×#41×C83—21no—
38C22×Q8×C83—21no—
39C22×D4×C83D411yescharacteristic
40C23×C4×C83—21no—
41AGL(1,ℤ/8)×C83—44no—
42C2×SD16×C83—22no—
43C2×SD16×C83—24no—
44C2×D8×C83—22no—
45C2×M16×C83—21no—
46C2×M16×C83—24no—
47C22×C8×C83—21no—
48C4⋊Q8×C83—21no—
49C4⋊Q8×C83—21no—
50C4⋊Q8×C83—21no—
51C4⋊Q8×C83C2312yes—
52C4⋊Q8×C83C2311yescharacteristic
53#127×C83C2311yescharacteristic
54#124×C83—21no—
55#121×C83—21no—
56#121×C83—21no—
57C4×D4×C83—21no—
58C4×D4×C83—21no—
59C4×D4×C83—21no—
60C4×D4×C83—21no—
61C4×D4×C83D412yes—
62C4×D4×C83C2312yes—
63C2×C4⋊C4×C83—21no—
64C2×C4⋊C4×C83—21no—
65C2×C4⋊C4×C83D411yescharacteristic
66C2×C22⋊C4×C83—21no—
67C2×C42×C83C2311yescharacteristic
68#107×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—24no—
76#103×C83—22no—
77#102×C83—24no—
78#102×C83—22no—
79C22⋊C8×C83—21no—
80C22⋊C8×C83—24no—
81C22⋊C8×C83D412yes—
82C4×C8×C83—21no—
83C4×C8×C83C2312yes—
84C24×C83—21no—
85#41×C83—42no—
86#41×C83—42no—
87C2×Q8×C83—21no—
88C2×Q8×C83—22no—
89C2×Q8×C83—41no—
90C2×Q8×C83—82no—
91C2×D4×C83—21no—
92C2×D4×C83—22no—
93C2×D4×C83—22no—
94C2×D4×C83—21no—
95C2×D4×C83—21no—
96C2×D4×C83—21no—
97C2×D4×C83—41no—
98C2×D4×C83C2×D412yes—
99C22×C4×C83—21no—
100C22×C4×C83—22no—
101C22×C4×C83—22no—
102C22×C4×C83—41no—
103C22×C4×C83—41no—
104C22×C4×C83—82no—
105C22×C4×C83C2×D411yescharacteristic
106C22×C4×C83C2×D411yescharacteristic
107C22×C4×C83#4111yescharacteristic
108SD16×C83—44no—
109SD16×C83—84no—
110D8×C83—44no—
111M16×C83—42no—
112M16×C83—84no—
113C2×C8×C83—22no—
114C2×C8×C83—22no—
115C2×C8×C83—22no—
116C2×C8×C83—42no—
117C2×C8×C83C2×D414yes—
118C4⋊C4×C83—21no—
119C4⋊C4×C83—22no—
120C4⋊C4×C83—21no—
121C4⋊C4×C83—22no—
122C4⋊C4×C83—21no—
123C4⋊C4×C83—41no—
124C4⋊C4×C83—41no—
125C4⋊C4×C83C2×D412yes—
126C22⋊C4×C83—22no—
127C22⋊C4×C83—22no—
128C22⋊C4×C83—41no—
129C22⋊C4×C83—41no—
130C42×C83C2×D411yescharacteristic
131C42×C83C2×D411yescharacteristic
132C42×C83#4111yescharacteristic
133C42×C83C2411yescharacteristic
134C23×C83—22no—
135C23×C83—42no—
136C23×C83—41no—
137C23×C83—81no—
138C23×C83—81no—
139C23×C83#12111yescharacteristic
140Q8×C83—41no—
141Q8×C83—41no—
142Q8×C83—44no—
143Q8×C83—82no—
144D4×C83—22no—
145D4×C83—22no—
146D4×C83—41no—
147D4×C83—41no—
148D4×C83—42no—
149D4×C83—42no—
150D4×C83—81no—
151D4×C83—82no—
152C2×C4×C83—22no—
153C2×C4×C83—22no—
154C2×C4×C83—41no—
155C2×C4×C83—44no—
156C2×C4×C83—41no—
157C2×C4×C83—42no—
158C2×C4×C83—82no—
159C2×C4×C83—81no—
160C2×C4×C83—82no—
161C2×C4×C83—82no—
162C2×C4×C83#12111yescharacteristic
163C2×C4×C83#12111yescharacteristic
164C2×C4×C83#12111yescharacteristic
165C2×C4×C83AGL(1,ℤ/8)12yes—
166C2×C4×C83#13712yes—
167C2×C4×C83C22×D411yescharacteristic
168C2×C4×C83C22×D411yescharacteristic
169C2×C4×C83C2×#4111yescharacteristic
170C8×C83—24no—
171C8×C83—42no—
172C8×C83—42no—
173C8×C83—42no—
174C22×C83—21no—
175C22×C83—21no—
176C22×C83—42no—
177C22×C83—42no—
178C22×C83—42no—
179C22×C83—84no—
180C22×C83—81no—
181C22×C83—81no—
182C22×C83—81no—
183C22×C83#46812yes—
184C22×C83C2×#12111yescharacteristic
185C4×C83—21no—
186C4×C83—21no—
187C4×C83—21no—
188C4×C83—42no—
189C4×C83—82no—
190C4×C83—81no—
191C4×C83#46812yes—
192C4×C83C2×AGL(1,ℤ/8)11yescharacteristic
193C4×C83C2×#13711yescharacteristic
194C4×C83#57612yes—
195#19421C8311yesmaximal, characteristic, direct product factor, Sylow
196C2×C83—22no—
197C2×C83—82no—
198C2×C83—161no—
199C2×C83#304111yescharacteristic
200C2×C83C2×#46811yescharacteristic
201C2×C83#306911yescharacteristic
202#3206C2×C8311yescharacteristic, complemented
203C2×#474C2×C8311yescharacteristic, complemented
204#3036C2×C8311yescharacteristic
205#2975C2×C8311yescharacteristic, complemented
206#1699C2×C8314yescomplemented
207#1685C2×C8314yescomplemented
208#1571C2×C8311yescharacteristic, complemented
209#1549C2×C8311yescharacteristic, complemented
210#1540C2×C8311yescharacteristic, complemented
211C83#1942111yescharacteristic, direct product factor, Sylow
212C2×AGL(1,ℤ/8)—21no—
213#532C22×C8311yescharacteristic
214C2×C4⋊Q8C22×C8311yescharacteristic
215C2×C4×D4C22×C8311yescharacteristic
216#492C22×C8312yescomplemented
217#487C22×C8312yescomplemented
218#474—22no—
219#474C22×C8314yes—
220#463—22no—
221#461—22no—
222#461C22×C8312yes—
223#459C22×C8312yes—
224#448—22no—
225#436—22no—
226#436C22×C8314yes—
227#431C22×C8311yescharacteristic
228#431C22×C8311yescharacteristic
229C2×#107—21no—
230#423C22×C8311yescharacteristic
231C2×C4⋊C8C22×C8311yescharacteristic
232#417—21no—
233C2×#102—21no—
234#407—22no—
235#335C22×C8314yescomplemented
236#332C22×C8312yescomplemented
237#332C22×C8314yescomplemented
238#331C22×C8312yescomplemented
239C2×#41—21no—
240C22×Q8—21no—
241C22×D4D4×C8311yescharacteristic
242C23×C4—21no—
243AGL(1,ℤ/8)—44no—
244C2×SD16—22no—
245C2×SD16—24no—
246C2×D8—22no—
247C2×M16—21no—
248C2×M16—24no—
249C22×C8—21no—
250C4⋊Q8—21no—
251C4⋊Q8—21no—
252C4⋊Q8—21no—
253C4⋊Q8C23×C8312yes—
254C4⋊Q8C23×C8311yescharacteristic
255#127C23×C8311yescharacteristic
256#124—21no—
257#121—21no—
258#121—21no—
259C4×D4—21no—
260C4×D4—21no—
261C4×D4—21no—
262C4×D4—21no—
263C4×D4D4×C8312yes—
264C4×D4C23×C8312yes—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4D4×C8311yescharacteristic
268C2×C22⋊C4—21no—
269C2×C42C23×C8311yescharacteristic
270#107—24no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8D4×C8312yes—
274C4⋊C8C23×C8312yes—
275C4⋊C8C23×C8312yes—
276C4⋊C8C23×C8312yes—
277#103—24no—
278#103—22no—
279#102—24no—
280#102—22no—
281C22⋊C8—21no—
282C22⋊C8—24no—
283C22⋊C8D4×C8312yes—
284C4×C8—21no—
285C4×C8C23×C8312yes—
286C24—21no—
287#41—42no—
288#41—42no—
289C2×Q8—21no—
290C2×Q8—22no—
291C2×Q8—41no—
292C2×Q8—82no—
293C2×D4—21no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4—41no—
300C2×D4C2×D4×C8312yes—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—82no—
307C22×C4C2×D4×C8311yescharacteristic
308C22×C4C2×D4×C8311yescharacteristic
309C22×C4#41×C8311yescharacteristic
310SD16—44no—
311SD16—84no—
312D8—44no—
313M16—42no—
314M16—84no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—42no—
319C2×C8C2×D4×C8314yes—
320C4⋊C4—21no—
321C4⋊C4—22no—
322C4⋊C4—21no—
323C4⋊C4—22no—
324C4⋊C4—21no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4C2×D4×C8312yes—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C42C2×D4×C8311yescharacteristic
333C42C2×D4×C8311yescharacteristic
334C42#41×C8311yescharacteristic
335C42C24×C8311yescharacteristic
336C23—22no—
337C23—42no—
338C23—41no—
339C23—81no—
340C23—81no—
341C23#121×C8311yescharacteristic
342Q8—41no—
343Q8—41no—
344Q8—44no—
345Q8—82no—
346D4—22no—
347D4—22no—
348D4—42no—
349D4—41no—
350D4—41no—
351D4—42no—
352D4—81no—
353D4—82no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—41no—
357C2×C4—44no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—82no—
361C2×C4—81no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4#121×C8311yescharacteristic
365C2×C4#121×C8311yescharacteristic
366C2×C4#121×C8311yescharacteristic
367C2×C4AGL(1,ℤ/8)×C8312yes—
368C2×C4#137×C8312yes—
369C2×C4C22×D4×C8311yescharacteristic
370C2×C4C22×D4×C8311yescharacteristic
371C2×C4C2×#41×C8311yescharacteristic
372C8—24no—
373C8—42no—
374C8—42no—
375C8—42no—
376C22—21no—
377C22—21no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—84no—
382C22—81no—
383C22—81no—
384C22—81no—
385C22#468×C8312yes—
386C22C2×#121×C8311yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—82no—
392C4—81no—
393C4#468×C8312yes—
394C4C2×AGL(1,ℤ/8)×C8311yescharacteristic
395C4C2×#137×C8311yescharacteristic
396C4#576×C8312yes—
397C2—22no—
398C2—82no—
399C2—161no—
400C2C83×#304111yescharacteristic
401C2C2×#468×C8311yescharacteristic
402C2C83×#306911yescharacteristic
subgroupG/H
center, socleC22×C83C2×#121
commutator, Frattini, cosocleC42C24×C83
prime radicalC83×#3036C2
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×#3206C2semidirect
2. C2×#474×C83C2semidirect
4. C83×#2975C2semidirect
5. C83×#1699C2semidirect
6. C83×#1685C2semidirect
7. C83×#1571C2semidirect
8. C83×#1549C2semidirect
9. C83×#1540C2semidirect
14. #492×C83C22semidirect
15. #487×C83C22semidirect
33. #335×C83C22semidirect
34. #332×C83C22semidirect
35. #332×C83C22semidirect
36. #331×C83C22semidirect
195. #19421C83direct (N × H)
202. #3206C2×C83semidirect
203. C2×#474C2×C83semidirect
205. #2975C2×C83semidirect
206. #1699C2×C83semidirect
207. #1685C2×C83semidirect
208. #1571C2×C83semidirect
209. #1549C2×C83semidirect
210. #1540C2×C83semidirect
211. C83#19421direct (N × H)
216. #492C22×C83semidirect
217. #487C22×C83semidirect
235. #335C22×C83semidirect
236. #332C22×C83semidirect
237. #332C22×C83semidirect
238. #331C22×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#32062
C2×#474×C832
C83×#30362
C83×#29752
C83×#16992
C83×#16852
C83×#15712
C83×#15492
C83×#15402
#1942183