Finite Group Databasebeta

← C83×#60884 #35288708

Subgroups — C83×#60884 #35288708

Order 21248.

1622 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3509C211yesmaximal, characteristic
2C83×#3504C211yesmaximal, characteristic
3C83×#3242C218yesmaximal, complemented
4C83×#3237C214yesmaximal, complemented
5C83×#3236C214yesmaximal, complemented
6C83×#3166C211yesmaximal, characteristic, complemented
7C83×#3162C211yesmaximal, characteristic, complemented
8C83×#3139C211yesmaximal, characteristic, complemented
9C2×#491×C83C211yesmaximal, characteristic, complemented
10C83×#3097C214yesmaximal
11C83×#3082C214yesmaximal, complemented
12C83×#2923C211yesmaximal, characteristic
13#557×C83—22no—
14#557×C83—22no—
15#556×C83—22no—
16#554×C83—24no—
17#554×C83—22no—
18#552×C83—24no—
19#544×C83C2212yes—
20#544×C83C2214yes—
21#541×C83C2214yes—
22#536×C83C2212yes—
23#533×C83C2214yes—
24#533×C83C2211yescharacteristic
25#533×C83C2211yescharacteristic
26#533×C83C2212yes—
27C2×C4⋊Q8×C83C2211yescharacteristic
28C2×#125×C83C2211yescharacteristic
29C2×#125×C83C2212yes—
30#520×C83C2212yes—
31#519×C83C2212yes—
32C4×#41×C83C2211yescharacteristic
33#501×C83C2212yes—
34#501×C83C2212yes—
35#501×C83C2214yes—
36#501×C83C2214yes—
37#499×C83C2214yes—
38#491×C83C2214yes—
39#491×C83C2214yes—
40#491×C83C2212yes—
41#491×C83C2212yes—
42#487×C83C2214yes—
43#485×C83C2218yes—
44#485×C83C2218yes—
45#484×C83C2218yes—
46#483×C83C2218yes—
47#479×C83C2218yes—
48#479×C83C2218yes—
49#477×C83C2218yes—
50#475×C83C2218yes—
51#433×C83C2214yes—
52#432×C83C2214yes—
53#428×C83C2214yes—
54#427×C83C2214yes—
55#424×C83C2212yes—
56#419×C83C2214yes—
57C2×#103×C83C2214yes—
58#408×C83C2212yes—
59C4×M16×C83C2211yescharacteristic
60C2×#97×C83C2211yescharacteristic
61C2×#41×C83C2311yescharacteristic
62C22×Q8×C83C2312yes—
63C2×M16×C83C2312yes—
64C22×C8×C83C2312yes—
65C4⋊Q8×C83—22no—
66C4⋊Q8×C83—22no—
67C4⋊Q8×C83—24no—
68C4⋊Q8×C83—22no—
69C4⋊Q8×C83C2312yes—
70C4⋊Q8×C83C2312yes—
71C4⋊Q8×C83C2314yes—
72C4⋊Q8×C83C2312yes—
73C4⋊Q8×C83C2312yes—
74#126×C83—28no—
75#125×C83—28no—
76#125×C83—22no—
77#125×C83—24no—
78#125×C83—24no—
79#125×C83—22no—
80#125×C83—24no—
81#125×C83C2314yes—
82#125×C83C2312yes—
83#125×C83C2312yes—
84#125×C83C2311yescharacteristic
85#125×C83C2311yescharacteristic
86#125×C83C2312yes—
87#124×C83—22no—
88#123×C83—24no—
89C22⋊Q8×C83—22no—
90C22⋊Q8×C83—24no—
91C22⋊Q8×C83—22no—
92C22⋊Q8×C83—24no—
93C22⋊Q8×C83—24no—
94C22⋊Q8×C83—24no—
95C22⋊Q8×C83—24no—
96C22⋊Q8×C83C2318yes—
97C4×Q8×C83—22no—
98C4×Q8×C83—24no—
99C4×Q8×C83—24no—
100C4×Q8×C83—24no—
101C4×Q8×C83—22no—
102C4×Q8×C83—22no—
103C4×Q8×C83D412yes—
104C4×Q8×C83C2318yes—
105C4×D4×C83—24no—
106C4×D4×C83—22no—
107C4×D4×C83D414yes—
108C4×D4×C83D412yes—
109#117×C83—24no—
110#117×C83—24no—
111#117×C83C2314yes—
112#117×C83C2312yes—
113#117×C83C2312yes—
114#117×C83C2311yescharacteristic
115#117×C83C2312yes—
116C2×C4⋊C4×C83—22no—
117C2×C4⋊C4×C83—24no—
118C2×C4⋊C4×C83—22no—
119C2×C4⋊C4×C83C2314yes—
120C2×C4⋊C4×C83C2312yes—
121C2×C42×C83C2311yescharacteristic
122C2×C42×C83C2312yes—
123#107×C83C2314yes—
124#107×C83C2318yes—
125#107×C83C2314yes—
126#106×C83C2314yes—
127#106×C83C2318yes—
128#106×C83C2314yes—
129C4⋊C8×C83C2318yes—
130#103×C83C2318yes—
131#103×C83C2318yes—
132#103×C83C2318yes—
133#103×C83C2318yes—
134C22⋊C8×C83C2318yes—
135#97×C83C2312yes—
136#97×C83C2312yes—
137#97×C83C2312yes—
138C4×C8×C83C2312yes—
139#41×C83—24no—
140C2×Q8×C83—22no—
141C2×Q8×C83—22no—
142C2×Q8×C83—24no—
143C2×Q8×C83—22no—
144C2×Q8×C83—28no—
145C2×Q8×C83C2×D411yescharacteristic
146C2×Q8×C83C2414yes—
147C2×D4×C83C2×D412yes—
148C2×D4×C83C2×D411yescharacteristic
149C22×C4×C83—22no—
150C22×C4×C83—24no—
151C22×C4×C83—22no—
152C22×C4×C83C2×D412yes—
153C22×C4×C83C2×D414yes—
154C22×C4×C83C2412yes—
155C22×C4×C83C2411yescharacteristic
156M16×C83—24no—
157C2×C8×C83—24no—
158C2×C8×C83C2414yes—
159C2×C8×C83C2414yes—
160C4⋊C4×C83—24no—
161C4⋊C4×C83—22no—
162C4⋊C4×C83—24no—
163C4⋊C4×C83—24no—
164C4⋊C4×C83—24no—
165C4⋊C4×C83—24no—
166C4⋊C4×C83—24no—
167C4⋊C4×C83—24no—
168C4⋊C4×C83—22no—
169C4⋊C4×C83—22no—
170C4⋊C4×C83—28no—
171C4⋊C4×C83—22no—
172C4⋊C4×C83—24no—
173C4⋊C4×C83—22no—
174C4⋊C4×C83—24no—
175C4⋊C4×C83—28no—
176C4⋊C4×C83C2×D412yes—
177C4⋊C4×C83C2×D414yes—
178C4⋊C4×C83C2414yes—
179C4⋊C4×C83C2412yes—
180C4⋊C4×C83C2412yes—
181C4⋊C4×C83C2414yes—
182C22⋊C4×C83—22no—
183C22⋊C4×C83—28no—
184C22⋊C4×C83—24no—
185C22⋊C4×C83—22no—
186C22⋊C4×C83C2×D412yes—
187C22⋊C4×C83C2×D414yes—
188C42×C83—28no—
189C42×C83—24no—
190C42×C83—22no—
191C42×C83—22no—
192C42×C83C2×D414yes—
193C42×C83C2×D412yes—
194C42×C83C2411yescharacteristic
195C42×C83C2411yescharacteristic
196C42×C83C2412yes—
197C23×C83C22×D412yes—
198C23×C83C22×D411yescharacteristic
199Q8×C83—22no—
200Q8×C83—28no—
201Q8×C83—44no—
202Q8×C83—44no—
203Q8×C83—84no—
204Q8×C83—82no—
205D4×C83—24no—
206D4×C83—22no—
207C2×C4×C83—24no—
208C2×C4×C83—24no—
209C2×C4×C83—22no—
210C2×C4×C83—24no—
211C2×C4×C83—24no—
212C2×C4×C83—22no—
213C2×C4×C83—42no—
214C2×C4×C83—44no—
215C2×C4×C83—44no—
216C2×C4×C83—82no—
217C2×C4×C83—84no—
218C2×C4×C83C22×D412yes—
219C2×C4×C83C22×D414yes—
220C2×C4×C83C22×D412yes—
221C2×C4×C83C22×D412yes—
222C2×C4×C83C22×D411yescharacteristic
223C2×C4×C832−1+414yes—
224C2×C4×C83C2511yescharacteristic
225C8×C83—24no—
226C8×C83—24no—
227C22×C83—22no—
228C22×C83—24no—
229C22×C83—22no—
230C22×C83#53612yes—
231C22×C83C23×D411yescharacteristic
232C4×C83—22no—
233C4×C83—24no—
234C4×C83—22no—
235C4×C83—22no—
236C4×C83—44no—
237C4×C83—42no—
238C4×C83—44no—
239C4×C83—42no—
240C4×C83—44no—
241C4×C83#53612yes—
242C4×C83C2×2−1+412yes—
243#60884C8311yesmaximal, characteristic, direct product factor, Sylow
244C2×C83—22no—
245C2×C83—42no—
246C2×C83C2×#53611yescharacteristic
247C2×C83#358612yes—
248#3509C2×C8311yescharacteristic
249#3504C2×C8311yescharacteristic
250#3242C2×C8318yescomplemented
251#3237C2×C8314yescomplemented
252#3236C2×C8314yescomplemented
253#3166C2×C8311yescharacteristic, complemented
254#3162C2×C8311yescharacteristic, complemented
255#3139C2×C8311yescharacteristic, complemented
256C2×#491C2×C8311yescharacteristic, complemented
257#3097C2×C8314yes—
258#3082C2×C8314yescomplemented
259#2923C2×C8311yescharacteristic
260C83#6088411yescharacteristic, direct product factor, Sylow
261#557—22no—
262#557—22no—
263#556—22no—
264#554—24no—
265#554—22no—
266#552—24no—
267#544C22×C8312yes—
268#544C22×C8314yes—
269#541C22×C8314yes—
270#536C22×C8312yes—
271#533C22×C8314yes—
272#533C22×C8311yescharacteristic
273#533C22×C8311yescharacteristic
274#533C22×C8312yes—
275C2×C4⋊Q8C22×C8311yescharacteristic
276C2×#125C22×C8311yescharacteristic
277C2×#125C22×C8312yes—
278#520C22×C8312yes—
279#519C22×C8312yes—
280C4×#41C22×C8311yescharacteristic
281#501C22×C8312yes—
282#501C22×C8312yes—
283#501C22×C8314yes—
284#501C22×C8314yes—
285#499C22×C8314yes—
286#491C22×C8314yes—
287#491C22×C8314yes—
288#491C22×C8312yes—
289#491C22×C8312yes—
290#487C22×C8314yes—
291#485C22×C8318yes—
292#485C22×C8318yes—
293#484C22×C8318yes—
294#483C22×C8318yes—
295#479C22×C8318yes—
296#479C22×C8318yes—
297#477C22×C8318yes—
298#475C22×C8318yes—
299#433C22×C8314yes—
300#432C22×C8314yes—
301#428C22×C8314yes—
302#427C22×C8314yes—
303#424C22×C8312yes—
304#419C22×C8314yes—
305C2×#103C22×C8314yes—
306#408C22×C8312yes—
307C4×M16C22×C8311yescharacteristic
308C2×#97C22×C8311yescharacteristic
309C2×#41C23×C8311yescharacteristic
310C22×Q8C23×C8312yes—
311C2×M16C23×C8312yes—
312C22×C8C23×C8312yes—
313C4⋊Q8—22no—
314C4⋊Q8—22no—
315C4⋊Q8—24no—
316C4⋊Q8—22no—
317C4⋊Q8C23×C8312yes—
318C4⋊Q8C23×C8312yes—
319C4⋊Q8C23×C8312yes—
320C4⋊Q8C23×C8314yes—
321C4⋊Q8C23×C8312yes—
322#126—28no—
323#125—28no—
324#125—22no—
325#125—24no—
326#125—24no—
327#125—22no—
328#125—24no—
329#125C23×C8312yes—
330#125C23×C8311yescharacteristic
331#125C23×C8314yes—
332#125C23×C8312yes—
333#125C23×C8311yescharacteristic
334#125C23×C8312yes—
335#124—22no—
336#123—24no—
337C22⋊Q8—22no—
338C22⋊Q8—24no—
339C22⋊Q8—22no—
340C22⋊Q8—24no—
341C22⋊Q8—24no—
342C22⋊Q8—24no—
343C22⋊Q8—24no—
344C22⋊Q8C23×C8318yes—
345C4×Q8—22no—
346C4×Q8—24no—
347C4×Q8—24no—
348C4×Q8—24no—
349C4×Q8—22no—
350C4×Q8—22no—
351C4×Q8D4×C8312yes—
352C4×Q8C23×C8318yes—
353C4×D4—24no—
354C4×D4—22no—
355C4×D4D4×C8314yes—
356C4×D4D4×C8312yes—
357#117—24no—
358#117—24no—
359#117C23×C8312yes—
360#117C23×C8312yes—
361#117C23×C8314yes—
362#117C23×C8311yescharacteristic
363#117C23×C8312yes—
364C2×C4⋊C4—22no—
365C2×C4⋊C4—24no—
366C2×C4⋊C4—22no—
367C2×C4⋊C4C23×C8312yes—
368C2×C4⋊C4C23×C8314yes—
369C2×C42C23×C8311yescharacteristic
370C2×C42C23×C8312yes—
371#107C23×C8314yes—
372#107C23×C8314yes—
373#107C23×C8318yes—
374#106C23×C8314yes—
375#106C23×C8314yes—
376#106C23×C8318yes—
377C4⋊C8C23×C8318yes—
378#103C23×C8318yes—
379#103C23×C8318yes—
380#103C23×C8318yes—
381#103C23×C8318yes—
382C22⋊C8C23×C8318yes—
383#97C23×C8312yes—
384#97C23×C8312yes—
385#97C23×C8312yes—
386C4×C8C23×C8312yes—
387#41—24no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—24no—
391C2×Q8—22no—
392C2×Q8—28no—
393C2×Q8C2×D4×C8311yescharacteristic
394C2×Q8C24×C8314yes—
395C2×D4C2×D4×C8312yes—
396C2×D4C2×D4×C8311yescharacteristic
397C22×C4—22no—
398C22×C4—24no—
399C22×C4—22no—
400C22×C4C2×D4×C8312yes—
401C22×C4C2×D4×C8314yes—
402C22×C4C24×C8312yes—
403C22×C4C24×C8311yescharacteristic
404M16—24no—
405C2×C8—24no—
406C2×C8C24×C8314yes—
407C2×C8C24×C8314yes—
408C4⋊C4—24no—
409C4⋊C4—22no—
410C4⋊C4—24no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—24no—
414C4⋊C4—24no—
415C4⋊C4—24no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—28no—
419C4⋊C4—22no—
420C4⋊C4—24no—
421C4⋊C4—22no—
422C4⋊C4—24no—
423C4⋊C4—28no—
424C4⋊C4C2×D4×C8312yes—
425C4⋊C4C2×D4×C8314yes—
426C4⋊C4C24×C8314yes—
427C4⋊C4C24×C8312yes—
428C4⋊C4C24×C8312yes—
429C4⋊C4C24×C8314yes—
430C22⋊C4—22no—
431C22⋊C4—28no—
432C22⋊C4—24no—
433C22⋊C4—22no—
434C22⋊C4C2×D4×C8312yes—
435C22⋊C4C2×D4×C8314yes—
436C42—28no—
437C42—24no—
438C42—22no—
439C42—22no—
440C42C2×D4×C8314yes—
441C42C2×D4×C8312yes—
442C42C24×C8311yescharacteristic
443C42C24×C8311yescharacteristic
444C42C24×C8312yes—
445C23C22×D4×C8312yes—
446C23C22×D4×C8311yescharacteristic
447Q8—22no—
448Q8—28no—
449Q8—44no—
450Q8—44no—
451Q8—84no—
452Q8—82no—
453D4—24no—
454D4—22no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—22no—
461C2×C4—42no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—82no—
465C2×C4—84no—
466C2×C4C22×D4×C8312yes—
467C2×C4C22×D4×C8314yes—
468C2×C4C22×D4×C8312yes—
469C2×C4C22×D4×C8312yes—
470C2×C4C22×D4×C8311yescharacteristic
471C2×C42−1+4×C8314yes—
472C2×C4C25×C8311yescharacteristic
473C8—24no—
474C8—24no—
475C22—22no—
476C22—24no—
477C22—22no—
478C22#536×C8312yes—
479C22C23×D4×C8311yescharacteristic
480C4—22no—
481C4—24no—
482C4—22no—
483C4—22no—
484C4—44no—
485C4—42no—
486C4—44no—
487C4—42no—
488C4—44no—
489C4#536×C8312yes—
490C4C2×2−1+4×C8312yes—
491C2—22no—
492C2—42no—
493C2C2×#536×C8311yescharacteristic
494C2C83×#358612yes—
subgroupG/H
center, socleC22×C83C23×D4
commutator, Frattini, cosocleC2×C4C25×C83
prime radicalC2×#41×C83C23
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C83 ◁ C4×#41×C83 ◁ G

Complements

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

NH ≅ G/Ntype
3. C83×#3242C2semidirect
4. C83×#3237C2semidirect
5. C83×#3236C2semidirect
6. C83×#3166C2semidirect
7. C83×#3162C2semidirect
8. C83×#3139C2semidirect
9. C2×#491×C83C2semidirect
11. C83×#3082C2semidirect
243. #60884C83direct (N × H)
250. #3242C2×C83semidirect
251. #3237C2×C83semidirect
252. #3236C2×C83semidirect
253. #3166C2×C83semidirect
254. #3162C2×C83semidirect
255. #3139C2×C83semidirect
256. C2×#491C2×C83semidirect
258. #3082C2×C83semidirect
260. C83#60884direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#35092
C83×#35042
C83×#32422
C83×#32372
C83×#32362
C83×#31662
C83×#31622
C83×#31392
C2×#491×C832
C83×#30972
C83×#30822
C83×#29232
#6088483