Finite Group Databasebeta

← C83×#61906 #35289730

Subgroups — C83×#61906 #35289730

Order 21248.

1748 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3522C211yesmaximal, characteristic
2C2×#557×C83C211yesmaximal, characteristic, complemented
3C83×#3400C216yesmaximal, complemented
4C83×#3392C212yesmaximal, complemented
5C83×#3372C216yesmaximal, complemented
6C83×#3363C212yesmaximal, complemented
7C83×#3218C213yesmaximal, complemented
8C83×#3206C213yesmaximal, complemented
9C83×#3112C213yesmaximal, complemented
10C83×#3108C213yesmaximal, complemented
11C83×#2984C211yesmaximal, characteristic, complemented
12#559×C83C2213yes—
13Q82×C83C2211yescharacteristic
14#557×C83—21no—
15#557×C83—23no—
16#557×C83C2212yes—
17#557×C83C2216yes—
18#554×C83C2213yes—
19#550×C83C2211yescharacteristic
20#548×C83—21no—
21#547×C83—23no—
22#542×C83—23no—
23#532×C83C2213yes—
24C2×C4⋊Q8×C83C2213yes—
25#529×C83C2213yes—
26C2×#125×C83—23no—
27#520×C83—21no—
28C2×C4×Q8×C83—21no—
29C2×C4×Q8×C83C2213yes—
30C2×C4×Q8×C83C2211yescharacteristic
31#497×C83C2216yes—
32#492×C83C2216yes—
33#491×C83C2216yes—
34#487×C83C2216yes—
35#483×C83C2211yescharacteristic
36#480×C83C2213yes—
37#479×C83C2216yes—
38#477×C83C2216yes—
39#464×C83C2216yes—
40#462×C83C2216yes—
41#461×C83C2216yes—
42#461×C83C22112yes—
43#460×C83C2216yes—
44#451×C83C2213yes—
45#448×C83C2213yes—
46#448×C83C2216yes—
47#446×C83C2216yes—
48C8×Q8×C83C2212yes—
49#440×C83C22112yes—
50C4×Q16×C83C2216yes—
51C4×SD16×C83C2216yes—
52#432×C83C2213yes—
53#431×C83C2213yes—
54C2×#41×C83—21no—
55C2×#41×C83—23no—
56C22×Q8×C83D411yescharacteristic
57C22×Q8×C83C2313yes—
58C2×Q16×C83C2316yes—
59C2×SD16×C83C2316yes—
60C2×SD16×C83C23112yes—
61C4⋊Q8×C83—23no—
62C4⋊Q8×C83—23no—
63C4⋊Q8×C83—23no—
64C4⋊Q8×C83C2316yes—
65C4⋊Q8×C83C2316yes—
66C4⋊Q8×C83C2313yes—
67#127×C83C2313yes—
68#126×C83—23no—
69#125×C83—26no—
70#125×C83—26no—
71#125×C83—212no—
72#125×C83C2313yes—
73#124×C83—23no—
74#124×C83—23no—
75#124×C83C2313yes—
76#123×C83—23no—
77#123×C83—26no—
78C22⋊Q8×C83—23no—
79C22⋊Q8×C83—26no—
80C22⋊Q8×C83C2311yescharacteristic
81#121×C83—23no—
82#121×C83—23no—
83#121×C83C2313yes—
84C4×Q8×C83—26no—
85C4×Q8×C83—22no—
86C4×Q8×C83—21no—
87C4×Q8×C83—23no—
88C4×Q8×C83—23no—
89C4×Q8×C83D411yescharacteristic
90C4×Q8×C83D413yes—
91C4×Q8×C83C2316yes—
92C4×Q8×C83C2316yes—
93C4×Q8×C83C2313yes—
94C4×Q8×C83C2311yescharacteristic
95C4×Q8×C83C2313yes—
96C4×D4×C83—23no—
97C4×D4×C83—23no—
98C4×D4×C83—23no—
99C4×D4×C83C2313yes—
100C4×D4×C83C2316yes—
101#117×C83—23no—
102C2×C4⋊C4×C83—23no—
103C2×C4⋊C4×C83—23no—
104C2×C4⋊C4×C83—23no—
105C2×C4⋊C4×C83—23no—
106C2×C4⋊C4×C83D413yes—
107C2×C4⋊C4×C83C2311yescharacteristic
108C2×C42×C83—23no—
109C2×C42×C83C2313yes—
110#107×C83C2316yes—
111#106×C83C2312yes—
112C4⋊C8×C83C2316yes—
113C4⋊C8×C83C2316yes—
114#103×C83C2312yes—
115#103×C83C23112yes—
116#103×C83C2316yes—
117#103×C83C2316yes—
118#102×C83C2316yes—
119C22⋊C8×C83C2313yes—
120C22⋊C8×C83C2311yescharacteristic
121#97×C83C2316yes—
122C4×C8×C83C2316yes—
123#41×C83—44no—
124#41×C83—86no—
125C2×Q8×C83—24no—
126C2×Q8×C83—23no—
127C2×Q8×C83—23no—
128C2×Q8×C83—46no—
129C2×Q8×C83C2×D413yes—
130C2×Q8×C83C2×D411yescharacteristic
131C2×Q8×C83C2416yes—
132C2×Q8×C83C2411yescharacteristic
133C2×D4×C83—23no—
134C2×D4×C83—23no—
135C2×D4×C83—23no—
136C2×D4×C83C2413yes—
137C22×C4×C83—21no—
138C22×C4×C83—23no—
139C22×C4×C83—23no—
140C22×C4×C83—26no—
141C22×C4×C83C2×D413yes—
142C22×C4×C83C2×D413yes—
143C22×C4×C83C2411yescharacteristic
144Q16×C83—46no—
145SD16×C83—46no—
146SD16×C83—412no—
147C2×C8×C83C2412yes—
148C2×C8×C83C2416yes—
149C4⋊C4×C83—23no—
150C4⋊C4×C83—23no—
151C4⋊C4×C83—26no—
152C4⋊C4×C83—26no—
153C4⋊C4×C83—26no—
154C4⋊C4×C83—26no—
155C4⋊C4×C83—26no—
156C4⋊C4×C83—26no—
157C4⋊C4×C83—21no—
158C4⋊C4×C83—21no—
159C4⋊C4×C83—23no—
160C4⋊C4×C83—23no—
161C4⋊C4×C83C2×D413yes—
162C4⋊C4×C83C2×D413yes—
163C4⋊C4×C83C2×D416yes—
164C4⋊C4×C83C2412yes—
165C4⋊C4×C83C2411yescharacteristic
166C4⋊C4×C83C2413yes—
167C22⋊C4×C83—21no—
168C22⋊C4×C83—23no—
169C22⋊C4×C83—23no—
170C22⋊C4×C83—23no—
171C22⋊C4×C83—23no—
172C22⋊C4×C83—26no—
173C42×C83—26no—
174C42×C83—26no—
175C42×C83—23no—
176C42×C83C2×D413yes—
177C42×C83C2×D413yes—
178C42×C83C2413yes—
179C42×C83C2413yes—
180C23×C83—23no—
181C23×C83C22×D411yescharacteristic
182Q8×C83—26no—
183Q8×C83—26no—
184Q8×C83—26no—
185Q8×C83—22no—
186Q8×C83—46no—
187Q8×C83—83no—
188Q8×C83#13714yes—
189D4×C83—26no—
190D4×C83—46no—
191D4×C83—46no—
192D4×C83—83no—
193C2×C4×C83—26no—
194C2×C4×C83—26no—
195C2×C4×C83—22no—
196C2×C4×C83—22no—
197C2×C4×C83—21no—
198C2×C4×C83—26no—
199C2×C4×C83—23no—
200C2×C4×C83—21no—
201C2×C4×C83—46no—
202C2×C4×C83—42no—
203C2×C4×C83—46no—
204C2×C4×C83—86no—
205C2×C4×C83C22×D413yes—
206C2×C4×C83C22×D411yescharacteristic
207C2×C4×C83C22×D413yes—
208C2×C4×C83C22×D413yes—
209C2×C4×C83C22×D413yes—
210C2×C4×C83C2511yescharacteristic
211C8×C83—22no—
212C8×C83—26no—
213C22×C83—21no—
214C22×C83—21no—
215C22×C83—43no—
216C22×C83—43no—
217C22×C83—43no—
218C22×C83#57712yes—
219C22×C83C23×D411yescharacteristic
220C4×C83—21no—
221C4×C83—23no—
222C4×C83—23no—
223C4×C83—23no—
224C4×C83—46no—
225C4×C83—41no—
226C4×C83—41no—
227C4×C83—42no—
228C4×C83—43no—
229C4×C83C2×#13716yes—
230C4×C83#58512yes—
231#61906C8311yesmaximal, characteristic, direct product factor, Sylow
232C2×C83—22no—
233C2×C83—83no—
234C2×C83#348711yescharacteristic
235C2×C83C22×#13711yescharacteristic
236C2×C83C2×#57711yescharacteristic
237#3522C2×C8311yescharacteristic
238C2×#557C2×C8311yescharacteristic, complemented
239#3400C2×C8316yescomplemented
240#3392C2×C8312yescomplemented
241#3372C2×C8316yescomplemented
242#3363C2×C8312yescomplemented
243#3218C2×C8313yescomplemented
244#3206C2×C8313yescomplemented
245#3112C2×C8313yescomplemented
246#3108C2×C8313yescomplemented
247#2984C2×C8311yescharacteristic, complemented
248C83#6190611yescharacteristic, direct product factor, Sylow
249#559C22×C8313yes—
250Q82C22×C8311yescharacteristic
251#557—21no—
252#557—23no—
253#557C22×C8312yes—
254#557C22×C8316yes—
255#554C22×C8313yes—
256#550C22×C8311yescharacteristic
257#548—21no—
258#547—23no—
259#542—23no—
260#532C22×C8313yes—
261C2×C4⋊Q8C22×C8313yes—
262#529C22×C8313yes—
263C2×#125—23no—
264#520—21no—
265C2×C4×Q8—21no—
266C2×C4×Q8C22×C8313yes—
267C2×C4×Q8C22×C8311yescharacteristic
268#497C22×C8316yes—
269#492C22×C8316yes—
270#491C22×C8316yes—
271#487C22×C8316yes—
272#483C22×C8311yescharacteristic
273#480C22×C8313yes—
274#479C22×C8316yes—
275#477C22×C8316yes—
276#464C22×C8316yes—
277#462C22×C8316yes—
278#461C22×C8316yes—
279#461C22×C83112yes—
280#460C22×C8316yes—
281#451C22×C8313yes—
282#448C22×C8313yes—
283#448C22×C8316yes—
284#446C22×C8316yes—
285C8×Q8C22×C8312yes—
286#440C22×C83112yes—
287C4×Q16C22×C8316yes—
288C4×SD16C22×C8316yes—
289#432C22×C8313yes—
290#431C22×C8313yes—
291C2×#41—21no—
292C2×#41—23no—
293C22×Q8D4×C8311yescharacteristic
294C22×Q8C23×C8313yes—
295C2×Q16C23×C8316yes—
296C2×SD16C23×C8316yes—
297C2×SD16C23×C83112yes—
298C4⋊Q8—23no—
299C4⋊Q8—23no—
300C4⋊Q8—23no—
301C4⋊Q8C23×C8316yes—
302C4⋊Q8C23×C8316yes—
303C4⋊Q8C23×C8313yes—
304#127C23×C8313yes—
305#126—23no—
306#125—26no—
307#125—26no—
308#125—212no—
309#125C23×C8313yes—
310#124—23no—
311#124—23no—
312#124C23×C8313yes—
313#123—23no—
314#123—26no—
315C22⋊Q8—23no—
316C22⋊Q8—26no—
317C22⋊Q8C23×C8311yescharacteristic
318#121—23no—
319#121—23no—
320#121C23×C8313yes—
321C4×Q8—26no—
322C4×Q8—22no—
323C4×Q8—21no—
324C4×Q8—23no—
325C4×Q8—23no—
326C4×Q8D4×C8311yescharacteristic
327C4×Q8D4×C8313yes—
328C4×Q8C23×C8316yes—
329C4×Q8C23×C8316yes—
330C4×Q8C23×C8313yes—
331C4×Q8C23×C8311yescharacteristic
332C4×Q8C23×C8313yes—
333C4×D4—23no—
334C4×D4—23no—
335C4×D4—23no—
336C4×D4C23×C8313yes—
337C4×D4C23×C8316yes—
338#117—23no—
339C2×C4⋊C4—23no—
340C2×C4⋊C4—23no—
341C2×C4⋊C4—23no—
342C2×C4⋊C4—23no—
343C2×C4⋊C4D4×C8313yes—
344C2×C4⋊C4C23×C8311yescharacteristic
345C2×C42—23no—
346C2×C42C23×C8313yes—
347#107C23×C8316yes—
348#106C23×C8312yes—
349C4⋊C8C23×C8316yes—
350C4⋊C8C23×C8316yes—
351#103C23×C8316yes—
352#103C23×C8312yes—
353#103C23×C8316yes—
354#103C23×C83112yes—
355#102C23×C8316yes—
356C22⋊C8C23×C8311yescharacteristic
357C22⋊C8C23×C8313yes—
358#97C23×C8316yes—
359C4×C8C23×C8316yes—
360#41—44no—
361#41—86no—
362C2×Q8—24no—
363C2×Q8—23no—
364C2×Q8—23no—
365C2×Q8—46no—
366C2×Q8C2×D4×C8313yes—
367C2×Q8C2×D4×C8311yescharacteristic
368C2×Q8C24×C8316yes—
369C2×Q8C24×C8311yescharacteristic
370C2×D4—23no—
371C2×D4—23no—
372C2×D4—23no—
373C2×D4C24×C8313yes—
374C22×C4—21no—
375C22×C4—23no—
376C22×C4—23no—
377C22×C4—26no—
378C22×C4C2×D4×C8313yes—
379C22×C4C2×D4×C8313yes—
380C22×C4C24×C8311yescharacteristic
381Q16—46no—
382SD16—46no—
383SD16—412no—
384C2×C8C24×C8312yes—
385C2×C8C24×C8316yes—
386C4⋊C4—23no—
387C4⋊C4—23no—
388C4⋊C4—26no—
389C4⋊C4—26no—
390C4⋊C4—26no—
391C4⋊C4—26no—
392C4⋊C4—26no—
393C4⋊C4—26no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—23no—
397C4⋊C4—23no—
398C4⋊C4C2×D4×C8313yes—
399C4⋊C4C2×D4×C8313yes—
400C4⋊C4C2×D4×C8316yes—
401C4⋊C4C24×C8312yes—
402C4⋊C4C24×C8311yescharacteristic
403C4⋊C4C24×C8313yes—
404C22⋊C4—21no—
405C22⋊C4—23no—
406C22⋊C4—23no—
407C22⋊C4—23no—
408C22⋊C4—23no—
409C22⋊C4—26no—
410C42—26no—
411C42—26no—
412C42—23no—
413C42C2×D4×C8313yes—
414C42C2×D4×C8313yes—
415C42C24×C8313yes—
416C42C24×C8313yes—
417C23—23no—
418C23C22×D4×C8311yescharacteristic
419Q8—26no—
420Q8—26no—
421Q8—26no—
422Q8—22no—
423Q8—46no—
424Q8—83no—
425Q8#137×C8314yes—
426D4—26no—
427D4—46no—
428D4—46no—
429D4—83no—
430C2×C4—26no—
431C2×C4—26no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—21no—
435C2×C4—26no—
436C2×C4—23no—
437C2×C4—21no—
438C2×C4—46no—
439C2×C4—42no—
440C2×C4—46no—
441C2×C4—86no—
442C2×C4C22×D4×C8313yes—
443C2×C4C22×D4×C8311yescharacteristic
444C2×C4C22×D4×C8313yes—
445C2×C4C22×D4×C8313yes—
446C2×C4C22×D4×C8313yes—
447C2×C4C25×C8311yescharacteristic
448C8—22no—
449C8—26no—
450C22—21no—
451C22—21no—
452C22—43no—
453C22—43no—
454C22—43no—
455C22#577×C8312yes—
456C22C23×D4×C8311yescharacteristic
457C4—23no—
458C4—23no—
459C4—23no—
460C4—21no—
461C4—41no—
462C4—42no—
463C4—43no—
464C4—46no—
465C4—41no—
466C4C2×#137×C8316yes—
467C4#585×C8312yes—
468C2—22no—
469C2—83no—
470C2C83×#348711yescharacteristic
471C2C22×#137×C8311yescharacteristic
472C2C2×#577×C8311yescharacteristic
subgroupG/H
center, socleC22×C83C23×D4
commutator, Frattini, cosocleC2×C4C25×C83
prime radicalC83×#3522C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C4×Q8×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×#557×C83C2semidirect
3. C83×#3400C2semidirect
4. C83×#3392C2semidirect
5. C83×#3372C2semidirect
6. C83×#3363C2semidirect
7. C83×#3218C2semidirect
8. C83×#3206C2semidirect
9. C83×#3112C2semidirect
10. C83×#3108C2semidirect
11. C83×#2984C2semidirect
231. #61906C83direct (N × H)
238. C2×#557C2×C83semidirect
239. #3400C2×C83semidirect
240. #3392C2×C83semidirect
241. #3372C2×C83semidirect
242. #3363C2×C83semidirect
243. #3218C2×C83semidirect
244. #3206C2×C83semidirect
245. #3112C2×C83semidirect
246. #3108C2×C83semidirect
247. #2984C2×C83semidirect
248. C83#61906direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#35222
C2×#557×C832
C83×#34002
C83×#33922
C83×#33722
C83×#33632
C83×#32182
C83×#32062
C83×#31122
C83×#31082
C83×#29842
#6190683