Finite Group Databasebeta

← C83×#8517 #35236341

Subgroups — C83×#8517 #35236341

Order 21248.

1040 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#537×C83C211yesmaximal, characteristic
2C83×#1785C211yesmaximal, characteristic, complemented
3C83×#1782C211yesmaximal, characteristic, complemented
4C83×#1511C212yesmaximal, complemented
5C83×#1510C212yesmaximal, complemented
6C22×#41×C83C411yescharacteristic, complemented
7C22×#41×C83C2211yescharacteristic
8#537×C83—24no—
9#537×C83—24no—
10C2×C22⋊Q8×C83—21no—
11C2×C22⋊Q8×C83C411yescharacteristic, complemented
12C2×C22⋊Q8×C83C2211yescharacteristic
13C2×#121×C83—21no—
14C2×#121×C83—21no—
15C2×#121×C83—21no—
16C2×#121×C83C411yescharacteristic, complemented
17C2×#121×C83C2211yescharacteristic
18C22×C4⋊C4×C83C411yescharacteristic, complemented
19C2×#101×C83—21no—
20C2×#100×C83—21no—
21C2×#99×C83—21no—
22C2×#99×C83—21no—
23#407×C83C2212yes—
24#381×C83C2212yes—
25#328×C83—24no—
26#327×C83—24no—
27C2×#41×C83—22no—
28C2×#41×C83—22no—
29C2×#41×C83—22no—
30C2×#41×C83—22no—
31C2×#41×C83—42no—
32C2×#41×C83—42no—
33C2×#41×C83—42no—
34C2×#41×C83—42no—
35C22×Q8×C83C2×C411yescharacteristic
36C22×Q8×C83D411yescharacteristic
37C22×D4×C83—21no—
38C22×D4×C83—21no—
39C22×D4×C83—21no—
40C22×D4×C83C2×C411yescharacteristic
41C22×D4×C83D411yescharacteristic
42C22×D4×C83D411yescharacteristic
43C22×D4×C83D411yescharacteristic
44C23×C4×C83—21no—
45C23×C4×C83—21no—
46C23×C4×C83C2×C411yescharacteristic
47C23×C4×C83C2×C411yescharacteristic
48C23×C4×C83C2311yescharacteristic
49C2×M16×C83—22no—
50C22⋊Q8×C83—28no—
51C22⋊Q8×C83—22no—
52C22⋊Q8×C83—22no—
53C22⋊Q8×C83—22no—
54C22⋊Q8×C83D414yes—
55#121×C83—28no—
56#121×C83—28no—
57#121×C83—28no—
58#121×C83—22no—
59#121×C83—22no—
60#121×C83—22no—
61#121×C83D414yes—
62C2×C4⋊C4×C83—21no—
63C2×C4⋊C4×C83—21no—
64C2×C4⋊C4×C83—21no—
65C2×C4⋊C4×C83—22no—
66C2×C4⋊C4×C83—22no—
67C2×C4⋊C4×C83C2×C411yescharacteristic
68C2×C4⋊C4×C83C2×C411yescharacteristic
69C2×C22⋊C4×C83—21no—
70C2×C22⋊C4×C83—21no—
71C2×C22⋊C4×C83—21no—
72C2×C22⋊C4×C83—21no—
73C2×C22⋊C4×C83—22no—
74#101×C83—42no—
75#100×C83—42no—
76#99×C83—42no—
77#99×C83—42no—
78C22⋊C8×C83—22no—
79C22⋊C8×C83—22no—
80#95×C83—22no—
81#95×C83—22no—
82C24×C83—21no—
83C24×C83C22⋊C411yescharacteristic
84C24×C83C22⋊C411yescharacteristic
85C24×C83C2×D411yescharacteristic
86#41×C83—48no—
87#41×C83—48no—
88#41×C83—48no—
89#41×C83—48no—
90C2×Q8×C83—22no—
91C2×Q8×C83—21no—
92C2×Q8×C83—22no—
93C2×Q8×C83—41no—
94C2×Q8×C83—41no—
95C2×Q8×C83—41no—
96C2×Q8×C83C22⋊C412yes—
97C2×D4×C83—24no—
98C2×D4×C83—22no—
99C2×D4×C83—22no—
100C2×D4×C83—22no—
101C2×D4×C83—22no—
102C2×D4×C83—24no—
103C2×D4×C83—24no—
104C2×D4×C83—22no—
105C2×D4×C83—21no—
106C2×D4×C83—22no—
107C2×D4×C83—41no—
108C2×D4×C83—41no—
109C2×D4×C83—41no—
110C2×D4×C83—41no—
111C2×D4×C83—41no—
112C2×D4×C83—42no—
113C2×D4×C83—42no—
114C2×D4×C83—82no—
115C2×D4×C83—82no—
116C2×D4×C83C22⋊C412yes—
117C22×C4×C83—22no—
118C22×C4×C83—22no—
119C22×C4×C83—21no—
120C22×C4×C83—21no—
121C22×C4×C83—24no—
122C22×C4×C83—21no—
123C22×C4×C83—21no—
124C22×C4×C83—21no—
125C22×C4×C83—24no—
126C22×C4×C83—21no—
127C22×C4×C83—21no—
128C22×C4×C83—41no—
129C22×C4×C83—41no—
130C22×C4×C83—41no—
131C22×C4×C83—41no—
132C22×C4×C83—42no—
133C22×C4×C83—42no—
134C22×C4×C83—42no—
135C22×C4×C83—82no—
136C22×C4×C83C22⋊C411yescharacteristic
137C22×C4×C83C22⋊C411yescharacteristic
138C22×C4×C83C22×C411yescharacteristic
139C22×C4×C83C2×D411yescharacteristic
140C22×C4×C83C2×D412yes—
141C22×C4×C83C2×D412yes—
142M16×C83—42no—
143M16×C83—42no—
144C2×C8×C83—42no—
145C4⋊C4×C83—24no—
146C4⋊C4×C83—24no—
147C4⋊C4×C83—21no—
148C4⋊C4×C83—21no—
149C4⋊C4×C83—24no—
150C4⋊C4×C83C22⋊C412yes—
151C4⋊C4×C83C22⋊C412yes—
152C22⋊C4×C83—24no—
153C22⋊C4×C83—24no—
154C22⋊C4×C83—24no—
155C22⋊C4×C83—24no—
156C22⋊C4×C83—42no—
157C22⋊C4×C83—42no—
158C23×C83—21no—
159C23×C83—21no—
160C23×C83—21no—
161C23×C83—21no—
162C23×C83—21no—
163C23×C83—22no—
164C23×C83—21no—
165C23×C83—21no—
166C23×C83—41no—
167C23×C83—41no—
168C23×C83—41no—
169C23×C83—41no—
170C23×C83—42no—
171C23×C83—42no—
172C23×C83—41no—
173C23×C83—41no—
174C23×C83—41no—
175C23×C83—41no—
176C23×C83#9912yes—
177C23×C83C2×C22⋊C411yescharacteristic
178C23×C83#12012yes—
179Q8×C83—44no—
180Q8×C83—42no—
181Q8×C83—42no—
182D4×C83—44no—
183D4×C83—44no—
184D4×C83—42no—
185D4×C83—42no—
186D4×C83—44no—
187D4×C83—48no—
188D4×C83—84no—
189D4×C83—84no—
190C2×C4×C83—24no—
191C2×C4×C83—24no—
192C2×C4×C83—21no—
193C2×C4×C83—21no—
194C2×C4×C83—21no—
195C2×C4×C83—21no—
196C2×C4×C83—21no—
197C2×C4×C83—21no—
198C2×C4×C83—22no—
199C2×C4×C83—22no—
200C2×C4×C83—22no—
201C2×C4×C83—21no—
202C2×C4×C83—21no—
203C2×C4×C83—21no—
204C2×C4×C83—42no—
205C2×C4×C83—42no—
206C2×C4×C83—48no—
207C2×C4×C83—42no—
208C2×C4×C83—42no—
209C2×C4×C83—42no—
210C2×C4×C83—42no—
211C2×C4×C83—41no—
212C2×C4×C83—41no—
213C2×C4×C83—84no—
214C2×C4×C83—82no—
215C2×C4×C83—82no—
216C2×C4×C83—82no—
217C2×C4×C83#9912yes—
218C2×C4×C83C2×C22⋊C411yescharacteristic
219C2×C4×C83C2×C22⋊C411yescharacteristic
220C2×C4×C83#12012yes—
221C8×C83—82no—
222C22×C83—21no—
223C22×C83—21no—
224C22×C83—22no—
225C22×C83—22no—
226C22×C83—22no—
227C22×C83—22no—
228C22×C83—42no—
229C22×C83—42no—
230C22×C83—41no—
231C22×C83—41no—
232C22×C83—41no—
233C22×C83—42no—
234C22×C83—41no—
235C22×C83—42no—
236C22×C83—41no—
237C22×C83—42no—
238C22×C83—81no—
239C22×C83—81no—
240C22×C83—81no—
241C22×C83—81no—
242C22×C83—82no—
243C22×C83—82no—
244C22×C83#37911yescharacteristic
245C22×C83C2×#9911yescharacteristic
246C22×C83C2×#9911yescharacteristic
247C4×C83—22no—
248C4×C83—22no—
249C4×C83—21no—
250C4×C83—21no—
251C4×C83—42no—
252C4×C83—42no—
253C4×C83—41no—
254C4×C83—42no—
255C4×C83—162no—
256#8517C8311yesmaximal, characteristic, direct product factor, Sylow
257C2×C83—21no—
258C2×C83—21no—
259C2×C83—42no—
260C2×C83—42no—
261C2×C83—81no—
262C2×C83—82no—
263C2×C83#178111yescharacteristic
264C2×C83#179511yescharacteristic
265C2×C83#179611yescharacteristic
266C2×#537C2×C8311yescharacteristic
267#1785C2×C8311yescharacteristic, complemented
268#1782C2×C8311yescharacteristic, complemented
269#1511C2×C8312yescomplemented
270#1510C2×C8312yescomplemented
271C83#851711yescharacteristic, direct product factor, Sylow
272C22×#41C4×C8311yescharacteristic, complemented
273C22×#41C22×C8311yescharacteristic
274#537—24no—
275#537—24no—
276C2×C22⋊Q8—21no—
277C2×C22⋊Q8C4×C8311yescharacteristic, complemented
278C2×C22⋊Q8C22×C8311yescharacteristic
279C2×#121—21no—
280C2×#121—21no—
281C2×#121—21no—
282C2×#121C4×C8311yescharacteristic, complemented
283C2×#121C22×C8311yescharacteristic
284C22×C4⋊C4C4×C8311yescharacteristic, complemented
285C2×#101—21no—
286C2×#100—21no—
287C2×#99—21no—
288C2×#99—21no—
289#407C22×C8312yes—
290#381C22×C8312yes—
291#328—24no—
292#327—24no—
293C2×#41—22no—
294C2×#41—22no—
295C2×#41—22no—
296C2×#41—22no—
297C2×#41—42no—
298C2×#41—42no—
299C2×#41—42no—
300C2×#41—42no—
301C22×Q8C2×C4×C8311yescharacteristic
302C22×Q8D4×C8311yescharacteristic
303C22×D4—21no—
304C22×D4—21no—
305C22×D4—21no—
306C22×D4C2×C4×C8311yescharacteristic
307C22×D4D4×C8311yescharacteristic
308C22×D4D4×C8311yescharacteristic
309C22×D4D4×C8311yescharacteristic
310C23×C4—21no—
311C23×C4—21no—
312C23×C4C2×C4×C8311yescharacteristic
313C23×C4C2×C4×C8311yescharacteristic
314C23×C4C23×C8311yescharacteristic
315C2×M16—22no—
316C22⋊Q8—28no—
317C22⋊Q8—22no—
318C22⋊Q8—22no—
319C22⋊Q8—22no—
320C22⋊Q8D4×C8314yes—
321#121—28no—
322#121—28no—
323#121—28no—
324#121—22no—
325#121—22no—
326#121—22no—
327#121D4×C8314yes—
328C2×C4⋊C4—21no—
329C2×C4⋊C4—21no—
330C2×C4⋊C4—21no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4C2×C4×C8311yescharacteristic
334C2×C4⋊C4C2×C4×C8311yescharacteristic
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—21no—
338C2×C22⋊C4—21no—
339C2×C22⋊C4—22no—
340#101—42no—
341#100—42no—
342#99—42no—
343#99—42no—
344C22⋊C8—22no—
345C22⋊C8—22no—
346#95—22no—
347#95—22no—
348C24—21no—
349C24C22⋊C4×C8311yescharacteristic
350C24C22⋊C4×C8311yescharacteristic
351C24C2×D4×C8311yescharacteristic
352#41—48no—
353#41—48no—
354#41—48no—
355#41—48no—
356C2×Q8—22no—
357C2×Q8—21no—
358C2×Q8—22no—
359C2×Q8—41no—
360C2×Q8—41no—
361C2×Q8—41no—
362C2×Q8C22⋊C4×C8312yes—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—22no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—24no—
369C2×D4—24no—
370C2×D4—22no—
371C2×D4—21no—
372C2×D4—22no—
373C2×D4—41no—
374C2×D4—41no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—42no—
379C2×D4—42no—
380C2×D4—82no—
381C2×D4—82no—
382C2×D4C22⋊C4×C8312yes—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—21no—
386C22×C4—21no—
387C22×C4—24no—
388C22×C4—21no—
389C22×C4—21no—
390C22×C4—21no—
391C22×C4—24no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—41no—
395C22×C4—41no—
396C22×C4—41no—
397C22×C4—41no—
398C22×C4—42no—
399C22×C4—42no—
400C22×C4—42no—
401C22×C4—82no—
402C22×C4C22⋊C4×C8311yescharacteristic
403C22×C4C22⋊C4×C8311yescharacteristic
404C22×C4C22×C4×C8311yescharacteristic
405C22×C4C2×D4×C8311yescharacteristic
406C22×C4C2×D4×C8312yes—
407C22×C4C2×D4×C8312yes—
408M16—42no—
409M16—42no—
410C2×C8—42no—
411C4⋊C4—24no—
412C4⋊C4—24no—
413C4⋊C4—21no—
414C4⋊C4—21no—
415C4⋊C4—24no—
416C4⋊C4C22⋊C4×C8312yes—
417C4⋊C4C22⋊C4×C8312yes—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—24no—
421C22⋊C4—24no—
422C22⋊C4—42no—
423C22⋊C4—42no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—22no—
430C23—21no—
431C23—21no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23—42no—
437C23—42no—
438C23—41no—
439C23—41no—
440C23—41no—
441C23—41no—
442C23#99×C8312yes—
443C23C2×C22⋊C4×C8311yescharacteristic
444C23#120×C8312yes—
445Q8—44no—
446Q8—42no—
447Q8—42no—
448D4—44no—
449D4—44no—
450D4—42no—
451D4—42no—
452D4—44no—
453D4—48no—
454D4—84no—
455D4—84no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—48no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—84no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4#99×C8312yes—
484C2×C4C2×C22⋊C4×C8311yescharacteristic
485C2×C4C2×C22⋊C4×C8311yescharacteristic
486C2×C4#120×C8312yes—
487C8—82no—
488C22—21no—
489C22—21no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—42no—
495C22—42no—
496C22—41no—
497C22—41no—
498C22—41no—
499C22—42no—
500C22—42no—
501C22—41no—
502C22—42no—
503C22—41no—
504C22—81no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—82no—
509C22—82no—
510C22#379×C8311yescharacteristic
511C22C2×#99×C8311yescharacteristic
512C22C2×#99×C8311yescharacteristic
513C4—22no—
514C4—22no—
515C4—21no—
516C4—21no—
517C4—41no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—162no—
522C2—21no—
523C2—21no—
524C2—42no—
525C2—42no—
526C2—82no—
527C2—81no—
528C2C83×#178111yescharacteristic
529C2C83×#179511yescharacteristic
530C2C83×#179611yescharacteristic
subgroupG/H
center, socleC22×C83#379
commutatorC22×C4C22×C4×C83
Frattini, cosocleC23×C4C23×C83
prime radicalC2×#537×C83C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C23×C4×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#1785C2semidirect
3. C83×#1782C2semidirect
4. C83×#1511C2semidirect
5. C83×#1510C2semidirect
6. C22×#41×C83C4semidirect
11. C2×C22⋊Q8×C83C4semidirect
16. C2×#121×C83C4semidirect
18. C22×C4⋊C4×C83C4semidirect
256. #8517C83direct (N × H)
267. #1785C2×C83semidirect
268. #1782C2×C83semidirect
269. #1511C2×C83semidirect
270. #1510C2×C83semidirect
271. C83#8517direct (N × H)
272. C22×#41C4×C83semidirect
277. C2×C22⋊Q8C4×C83semidirect
282. C2×#121C4×C83semidirect
284. C22×C4⋊C4C4×C83semidirect

Maximal subgroups

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

subgroupindex
C2×#537×C832
C83×#17852
C83×#17822
C83×#15112
C83×#15102
#851783