Finite Group Databasebeta

← C83×#61844 #35289668

Subgroups — C83×#61844 #35289668

Order 21248.

1756 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3522C211yesmaximal, characteristic
2C2×Q82×C83C211yesmaximal, characteristic, complemented
3C83×#3385C216yesmaximal, complemented
4Q8×Q16×C83C212yesmaximal, complemented
5C83×#3375C216yesmaximal, complemented
6C83×#3362C212yesmaximal, complemented
7C83×#3238C213yesmaximal, complemented
8C83×#3226C213yesmaximal, complemented
9C83×#3112C213yesmaximal, complemented
10C83×#3108C213yesmaximal, complemented
11C83×#2984C211yesmaximal, characteristic, complemented
12#559×C83C2213yes—
13Q82×C83—21no—
14Q82×C83—23no—
15Q82×C83C2212yes—
16Q82×C83C2216yes—
17Q82×C83C2211yescharacteristic
18#554×C83C2213yes—
19#550×C83C2211yescharacteristic
20#548×C83—21no—
21#547×C83—23no—
22#542×C83—23no—
23#532×C83C2213yes—
24C2×C4⋊Q8×C83—23no—
25C2×C4⋊Q8×C83C2213yes—
26#529×C83C2213yes—
27#520×C83—21no—
28C2×C4×Q8×C83—21no—
29C2×C4×Q8×C83C2213yes—
30C2×C4×Q8×C83C2211yescharacteristic
31#501×C83C2216yes—
32#500×C83C2216yes—
33#489×C83C2216yes—
34#486×C83C2216yes—
35#483×C83C2213yes—
36#480×C83C2213yes—
37#480×C83C2216yes—
38#479×C83C2216yes—
39#477×C83C2216yes—
40#477×C83C2216yes—
41#476×C83C2216yes—
42#474×C83C22112yes—
43#464×C83C2216yes—
44#460×C83C2216yes—
45#451×C83C2211yescharacteristic
46#448×C83C2213yes—
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×C83—23no—
57C22×Q8×C83D411yescharacteristic
58C22×Q8×C83C2311yescharacteristic
59C2×Q16×C83C2312yes—
60C2×SD16×C83C2316yes—
61C4⋊Q8×C83—26no—
62C4⋊Q8×C83—26no—
63C4⋊Q8×C83—212no—
64C4⋊Q8×C83—23no—
65C4⋊Q8×C83—23no—
66C4⋊Q8×C83—23no—
67C4⋊Q8×C83C2316yes—
68C4⋊Q8×C83C2316yes—
69C4⋊Q8×C83C2313yes—
70#127×C83C2313yes—
71#126×C83—23no—
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×C83C2313yes—
93C4×Q8×C83C2316yes—
94C4×Q8×C83C2311yescharacteristic
95C4×Q8×C83C2313yes—
96C4×D4×C83—23no—
97C4×D4×C83—23no—
98C4×D4×C83—23no—
99C4×D4×C83C2316yes—
100C4×D4×C83C2313yes—
101#117×C83—23no—
102C2×C4⋊C4×C83—23no—
103C2×C4⋊C4×C83—23no—
104C2×C4⋊C4×C83—23no—
105C2×C4⋊C4×C83D413yes—
106C2×C4⋊C4×C83C2313yes—
107C2×C42×C83—23no—
108C2×C42×C83C2313yes—
109#107×C83C2316yes—
110#107×C83C23112yes—
111#106×C83C2316yes—
112C4⋊C8×C83C2316yes—
113C4⋊C8×C83C2316yes—
114#103×C83C2316yes—
115#103×C83C2316yes—
116#103×C83C2312yes—
117#102×C83C2316yes—
118#102×C83C23112yes—
119C22⋊C8×C83C2311yescharacteristic
120C22⋊C8×C83C2313yes—
121#97×C83C2316yes—
122C4×C8×C83C2316yes—
123#41×C83—44no—
124#41×C83—86no—
125C2×Q8×C83—24no—
126C2×Q8×C83—24no—
127C2×Q8×C83—21no—
128C2×Q8×C83—23no—
129C2×Q8×C83—26no—
130C2×Q8×C83—26no—
131C2×Q8×C83—86no—
132C2×Q8×C83C2×D413yes—
133C2×Q8×C83C2×D411yescharacteristic
134C2×Q8×C83C2411yescharacteristic
135C2×Q8×C83C2412yes—
136C2×D4×C83—23no—
137C2×D4×C83—23no—
138C2×D4×C83—23no—
139C2×D4×C83C2413yes—
140C22×C4×C83—21no—
141C22×C4×C83—23no—
142C22×C4×C83—23no—
143C22×C4×C83—26no—
144C22×C4×C83C2×D413yes—
145C22×C4×C83C2×D413yes—
146C22×C4×C83C2411yescharacteristic
147Q16×C83—24no—
148SD16×C83—212no—
149C2×C8×C83C2412yes—
150C2×C8×C83C2416yes—
151C4⋊C4×C83—23no—
152C4⋊C4×C83—23no—
153C4⋊C4×C83—26no—
154C4⋊C4×C83—26no—
155C4⋊C4×C83—26no—
156C4⋊C4×C83—26no—
157C4⋊C4×C83—23no—
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×C83C2416yes—
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—24no—
183Q8×C83—26no—
184Q8×C83—22no—
185Q8×C83—22no—
186Q8×C83—86no—
187Q8×C83—86no—
188Q8×C83—83no—
189Q8×C83#13714yes—
190Q8×C832−1+414yes—
191D4×C83—26no—
192D4×C83—46no—
193D4×C83—46no—
194D4×C83—83no—
195C2×C4×C83—24no—
196C2×C4×C83—26no—
197C2×C4×C83—22no—
198C2×C4×C83—26no—
199C2×C4×C83—22no—
200C2×C4×C83—21no—
201C2×C4×C83—26no—
202C2×C4×C83—23no—
203C2×C4×C83—21no—
204C2×C4×C83—46no—
205C2×C4×C83—46no—
206C2×C4×C83—86no—
207C2×C4×C83C22×D413yes—
208C2×C4×C83C22×D411yescharacteristic
209C2×C4×C83C22×D413yes—
210C2×C4×C83C22×D413yes—
211C2×C4×C83C22×D413yes—
212C2×C4×C83C2511yescharacteristic
213C8×C83—26no—
214C8×C83—22no—
215C22×C83—21no—
216C22×C83—21no—
217C22×C83—43no—
218C22×C83—43no—
219C22×C83—43no—
220C22×C83#57612yes—
221C22×C83C23×D411yescharacteristic
222C4×C83—24no—
223C4×C83—21no—
224C4×C83—23no—
225C4×C83—23no—
226C4×C83—23no—
227C4×C83—41no—
228C4×C83—41no—
229C4×C83—46no—
230C4×C83—43no—
231C4×C83C2×#13716yes—
232C4×C83C2×2−1+412yes—
233#61844C8311yesmaximal, characteristic, direct product factor, Sylow
234C2×C83—22no—
235C2×C83—83no—
236C2×C83#348611yescharacteristic
237C2×C83C22×#13711yescharacteristic
238C2×C83C2×#57611yescharacteristic
239#3522C2×C8311yescharacteristic
240C2×Q82C2×C8311yescharacteristic, complemented
241#3385C2×C8316yescomplemented
242Q8×Q16C2×C8312yescomplemented
243#3375C2×C8316yescomplemented
244#3362C2×C8312yescomplemented
245#3238C2×C8313yescomplemented
246#3226C2×C8313yescomplemented
247#3112C2×C8313yescomplemented
248#3108C2×C8313yescomplemented
249#2984C2×C8311yescharacteristic, complemented
250C83#6184411yescharacteristic, direct product factor, Sylow
251#559C22×C8313yes—
252Q82—21no—
253Q82—23no—
254Q82C22×C8312yes—
255Q82C22×C8316yes—
256Q82C22×C8311yescharacteristic
257#554C22×C8313yes—
258#550C22×C8311yescharacteristic
259#548—21no—
260#547—23no—
261#542—23no—
262#532C22×C8313yes—
263C2×C4⋊Q8—23no—
264C2×C4⋊Q8C22×C8313yes—
265#529C22×C8313yes—
266#520—21no—
267C2×C4×Q8—21no—
268C2×C4×Q8C22×C8313yes—
269C2×C4×Q8C22×C8311yescharacteristic
270#501C22×C8316yes—
271#500C22×C8316yes—
272#489C22×C8316yes—
273#486C22×C8316yes—
274#483C22×C8313yes—
275#480C22×C8313yes—
276#480C22×C8316yes—
277#479C22×C8316yes—
278#477C22×C8316yes—
279#477C22×C8316yes—
280#476C22×C8316yes—
281#474C22×C83112yes—
282#464C22×C8316yes—
283#460C22×C8316yes—
284#451C22×C8311yescharacteristic
285#448C22×C8313yes—
286#446C22×C8316yes—
287C8×Q8C22×C8312yes—
288#440C22×C83112yes—
289C4×Q16C22×C8316yes—
290C4×SD16C22×C8316yes—
291#432C22×C8313yes—
292#431C22×C8313yes—
293C2×#41—21no—
294C2×#41—23no—
295C22×Q8—23no—
296C22×Q8D4×C8311yescharacteristic
297C22×Q8C23×C8311yescharacteristic
298C2×Q16C23×C8312yes—
299C2×SD16C23×C8316yes—
300C4⋊Q8—26no—
301C4⋊Q8—26no—
302C4⋊Q8—212no—
303C4⋊Q8—23no—
304C4⋊Q8—23no—
305C4⋊Q8—23no—
306C4⋊Q8C23×C8316yes—
307C4⋊Q8C23×C8316yes—
308C4⋊Q8C23×C8313yes—
309#127C23×C8313yes—
310#126—23no—
311#125C23×C8313yes—
312#124—23no—
313#124—23no—
314#124C23×C8313yes—
315#123—23no—
316#123—26no—
317C22⋊Q8—23no—
318C22⋊Q8—26no—
319C22⋊Q8C23×C8311yescharacteristic
320#121—23no—
321#121—23no—
322#121C23×C8313yes—
323C4×Q8—26no—
324C4×Q8—22no—
325C4×Q8—21no—
326C4×Q8—23no—
327C4×Q8—23no—
328C4×Q8D4×C8311yescharacteristic
329C4×Q8D4×C8313yes—
330C4×Q8C23×C8316yes—
331C4×Q8C23×C8316yes—
332C4×Q8C23×C8313yes—
333C4×Q8C23×C8311yescharacteristic
334C4×Q8C23×C8313yes—
335C4×D4—23no—
336C4×D4—23no—
337C4×D4—23no—
338C4×D4C23×C8313yes—
339C4×D4C23×C8316yes—
340#117—23no—
341C2×C4⋊C4—23no—
342C2×C4⋊C4—23no—
343C2×C4⋊C4—23no—
344C2×C4⋊C4D4×C8313yes—
345C2×C4⋊C4C23×C8313yes—
346C2×C42—23no—
347C2×C42C23×C8313yes—
348#107C23×C8316yes—
349#107C23×C83112yes—
350#106C23×C8316yes—
351C4⋊C8C23×C8316yes—
352C4⋊C8C23×C8316yes—
353#103C23×C8312yes—
354#103C23×C8316yes—
355#103C23×C8316yes—
356#102C23×C8316yes—
357#102C23×C83112yes—
358C22⋊C8C23×C8311yescharacteristic
359C22⋊C8C23×C8313yes—
360#97C23×C8316yes—
361C4×C8C23×C8316yes—
362#41—44no—
363#41—86no—
364C2×Q8—24no—
365C2×Q8—24no—
366C2×Q8—21no—
367C2×Q8—23no—
368C2×Q8—26no—
369C2×Q8—26no—
370C2×Q8—86no—
371C2×Q8C2×D4×C8313yes—
372C2×Q8C2×D4×C8311yescharacteristic
373C2×Q8C24×C8311yescharacteristic
374C2×Q8C24×C8312yes—
375C2×D4—23no—
376C2×D4—23no—
377C2×D4—23no—
378C2×D4C24×C8313yes—
379C22×C4—21no—
380C22×C4—23no—
381C22×C4—23no—
382C22×C4—26no—
383C22×C4C2×D4×C8313yes—
384C22×C4C2×D4×C8313yes—
385C22×C4C24×C8311yescharacteristic
386Q16—24no—
387SD16—212no—
388C2×C8C24×C8312yes—
389C2×C8C24×C8316yes—
390C4⋊C4—23no—
391C4⋊C4—23no—
392C4⋊C4—26no—
393C4⋊C4—26no—
394C4⋊C4—26no—
395C4⋊C4—26no—
396C4⋊C4—23no—
397C4⋊C4—21no—
398C4⋊C4—23no—
399C4⋊C4—23no—
400C4⋊C4C2×D4×C8313yes—
401C4⋊C4C2×D4×C8313yes—
402C4⋊C4C2×D4×C8316yes—
403C4⋊C4C24×C8316yes—
404C4⋊C4C24×C8311yescharacteristic
405C4⋊C4C24×C8313yes—
406C22⋊C4—21no—
407C22⋊C4—23no—
408C22⋊C4—23no—
409C22⋊C4—23no—
410C22⋊C4—23no—
411C22⋊C4—26no—
412C42—26no—
413C42—26no—
414C42—23no—
415C42C2×D4×C8313yes—
416C42C2×D4×C8313yes—
417C42C24×C8313yes—
418C42C24×C8313yes—
419C23—23no—
420C23C22×D4×C8311yescharacteristic
421Q8—24no—
422Q8—26no—
423Q8—22no—
424Q8—22no—
425Q8—86no—
426Q8—86no—
427Q8—83no—
428Q8#137×C8314yes—
429Q82−1+4×C8314yes—
430D4—26no—
431D4—46no—
432D4—46no—
433D4—83no—
434C2×C4—24no—
435C2×C4—26no—
436C2×C4—22no—
437C2×C4—26no—
438C2×C4—22no—
439C2×C4—21no—
440C2×C4—26no—
441C2×C4—23no—
442C2×C4—21no—
443C2×C4—46no—
444C2×C4—46no—
445C2×C4—86no—
446C2×C4C22×D4×C8313yes—
447C2×C4C22×D4×C8311yescharacteristic
448C2×C4C22×D4×C8313yes—
449C2×C4C22×D4×C8313yes—
450C2×C4C22×D4×C8313yes—
451C2×C4C25×C8311yescharacteristic
452C8—26no—
453C8—22no—
454C22—21no—
455C22—21no—
456C22—43no—
457C22—43no—
458C22—43no—
459C22#576×C8312yes—
460C22C23×D4×C8311yescharacteristic
461C4—23no—
462C4—23no—
463C4—23no—
464C4—24no—
465C4—21no—
466C4—41no—
467C4—46no—
468C4—43no—
469C4—41no—
470C4C2×#137×C8316yes—
471C4C2×2−1+4×C8312yes—
472C2—22no—
473C2—83no—
474C2C83×#348611yescharacteristic
475C2C22×#137×C8311yescharacteristic
476C2C2×#576×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×Q82×C83C2semidirect
3. C83×#3385C2semidirect
4. Q8×Q16×C83C2semidirect
5. C83×#3375C2semidirect
6. C83×#3362C2semidirect
7. C83×#3238C2semidirect
8. C83×#3226C2semidirect
9. C83×#3112C2semidirect
10. C83×#3108C2semidirect
11. C83×#2984C2semidirect
233. #61844C83direct (N × H)
240. C2×Q82C2×C83semidirect
241. #3385C2×C83semidirect
242. Q8×Q16C2×C83semidirect
243. #3375C2×C83semidirect
244. #3362C2×C83semidirect
245. #3238C2×C83semidirect
246. #3226C2×C83semidirect
247. #3112C2×C83semidirect
248. #3108C2×C83semidirect
249. #2984C2×C83semidirect
250. C83#61844direct (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×Q82×C832
C83×#33852
Q8×Q16×C832
C83×#33752
C83×#33622
C83×#32382
C83×#32262
C83×#31122
C83×#31082
C83×#29842
#6184483