Finite Group Databasebeta

← C83×#9345 #35237169

Subgroups — C83×#9345 #35237169

Order 21248.

678 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3007C211yesmaximal, characteristic
2C83×#2880C211yesmaximal, characteristic
3C83×#1788C211yesmaximal, characteristic
4C83×#1520C211yesmaximal, characteristic, complemented
5C83×#1472C211yesmaximal, characteristic, complemented
6C83×#1305C211yesmaximal, characteristic, complemented
7C83×#1298C211yesmaximal, characteristic, complemented
8C22×#41×C83C2211yescharacteristic
9C2×#137×C83C411yescharacteristic, complemented
10C2×#137×C83C411yescharacteristic, complemented
11C2×#131×C83—21no—
12C2×C22⋊Q8×C83C2211yescharacteristic
13#452×C83—21no—
14#452×C83—21no—
15#451×C83—21no—
16#450×C83—21no—
17#417×C83C411yescharacteristic, complemented
18#417×C83C411yescharacteristic, complemented
19C2×#103×C83—21no—
20C2×#102×C83—21no—
21#408×C83—21no—
22#408×C83—22no—
23#407×C83—21no—
24#407×C83C2211yescharacteristic
25#407×C83C2211yescharacteristic
26C2×C22⋊C8×C83—21no—
27C2×C22⋊C8×C83C2211yescharacteristic
28C2×C22⋊C8×C83C2211yescharacteristic
29#380×C83C2211yescharacteristic
30#328×C83—21no—
31#328×C83—21no—
32#324×C83—22no—
33C2×#41×C83—21no—
34C2×#41×C83—21no—
35C2×#41×C83—42no—
36C2×#41×C83—42no—
37C2×#41×C83C2×C411yescharacteristic
38C2×#41×C83C2×C411yescharacteristic
39C2×#41×C83D411yescharacteristic
40C2×#41×C83D411yescharacteristic
41C22×Q8×C83C2×C411yescharacteristic
42C22×Q8×C83D411yescharacteristic
43C22×D4×C83—21no—
44C22×D4×C83D411yescharacteristic
45C23×C4×C83—21no—
46C23×C4×C83C2311yescharacteristic
47#137×C83—42no—
48#137×C83—42no—
49C2×Q16×C83—21no—
50C2×Q16×C83—21no—
51C2×SD16×C83—21no—
52C2×SD16×C83—21no—
53#131×C83—42no—
54#131×C83—42no—
55C2×M16×C83—22no—
56C2×M16×C83—21no—
57C2×M16×C83C2×C411yescharacteristic
58C2×M16×C83C2×C411yescharacteristic
59C22×C8×C83—22no—
60C22×C8×C83—21no—
61C22×C8×C83—21no—
62C22⋊Q8×C83—21no—
63C22⋊Q8×C83—21no—
64C22⋊Q8×C83—21no—
65C22⋊Q8×C83—21no—
66C2×C4⋊C4×C83—21no—
67C2×C4⋊C4×C83—21no—
68C2×C4⋊C4×C83C2×C411yescharacteristic
69C2×C22⋊C4×C83—21no—
70C2×C22⋊C4×C83—21no—
71#103×C83—21no—
72#103×C83—21no—
73#103×C83—42no—
74#102×C83—21no—
75#102×C83—21no—
76#102×C83—42no—
77C22⋊C8×C83—22no—
78C22⋊C8×C83—22no—
79C22⋊C8×C83—21no—
80C22⋊C8×C83—21no—
81C22⋊C8×C83—21no—
82C22⋊C8×C83—21no—
83C22⋊C8×C83—21no—
84C22⋊C8×C83—21no—
85C22⋊C8×C83D412yes—
86C22⋊C8×C83D411yescharacteristic
87C22⋊C8×C83D411yescharacteristic
88#95×C83—21no—
89C24×C83—21no—
90C24×C83#4111yescharacteristic
91#41×C83—22no—
92#41×C83—22no—
93#41×C83—22no—
94#41×C83—22no—
95#41×C83—41no—
96#41×C83—41no—
97#41×C83—41no—
98#41×C83—41no—
99#41×C83—44no—
100#41×C83—44no—
101C2×Q8×C83—21no—
102C2×Q8×C83—22no—
103C2×Q8×C83—21no—
104C2×Q8×C83—21no—
105C2×Q8×C83—41no—
106C2×Q8×C83—41no—
107C2×Q8×C83—41no—
108C2×Q8×C83C22⋊C411yescharacteristic
109C2×Q8×C83C22⋊C411yescharacteristic
110C2×D4×C83—21no—
111C2×D4×C83—22no—
112C2×D4×C83—21no—
113C2×D4×C83—21no—
114C2×D4×C83—21no—
115C2×D4×C83—21no—
116C2×D4×C83—41no—
117C2×D4×C83—44no—
118C2×D4×C83C22⋊C411yescharacteristic
119C2×D4×C83C22⋊C411yescharacteristic
120C22×C4×C83—21no—
121C22×C4×C83—21no—
122C22×C4×C83—21no—
123C22×C4×C83—21no—
124C22×C4×C83—21no—
125C22×C4×C83—21no—
126C22×C4×C83—21no—
127C22×C4×C83—41no—
128C22×C4×C83—41no—
129C22×C4×C83—44no—
130C22×C4×C83—41no—
131C22×C4×C83C22×C411yescharacteristic
132C22×C4×C83C2×D411yescharacteristic
133C22×C4×C83C2×D411yescharacteristic
134C22×C4×C83C2×D411yescharacteristic
135C22×C4×C83C2×D411yescharacteristic
136C22×C4×C83#4111yescharacteristic
137Q16×C83—42no—
138Q16×C83—42no—
139SD16×C83—42no—
140SD16×C83—42no—
141M16×C83—44no—
142M16×C83—41no—
143M16×C83—41no—
144M16×C83—41no—
145M16×C83—41no—
146C2×C8×C83—21no—
147C2×C8×C83—21no—
148C2×C8×C83—21no—
149C2×C8×C83—22no—
150C2×C8×C83—21no—
151C2×C8×C83—44no—
152C2×C8×C83—41no—
153C2×C8×C83—41no—
154C2×C8×C83—41no—
155C2×C8×C83—41no—
156C2×C8×C83—41no—
157C4⋊C4×C83—21no—
158C4⋊C4×C83—21no—
159C4⋊C4×C83—41no—
160C4⋊C4×C83—41no—
161C4⋊C4×C83—41no—
162C4⋊C4×C83—41no—
163C22⋊C4×C83—41no—
164C22⋊C4×C83—41no—
165C22⋊C4×C83—81no—
166C23×C83—21no—
167C23×C83—21no—
168C23×C83—21no—
169C23×C83—21no—
170C23×C83—41no—
171C23×C83—41no—
172C23×C83—42no—
173C23×C83—41no—
174C23×C83—41no—
175C23×C83—81no—
176C23×C83C4×D411yescharacteristic
177C23×C83#12111yescharacteristic
178C23×C83#12311yescharacteristic
179Q8×C83—21no—
180Q8×C83—21no—
181Q8×C83—21no—
182Q8×C83—42no—
183Q8×C83—41no—
184Q8×C83—41no—
185Q8×C83—81no—
186Q8×C83#9912yes—
187D4×C83—24no—
188D4×C83—21no—
189D4×C83—21no—
190D4×C83—21no—
191D4×C83—42no—
192D4×C83—42no—
193D4×C83—42no—
194D4×C83—42no—
195D4×C83#9912yes—
196C2×C4×C83—24no—
197C2×C4×C83—21no—
198C2×C4×C83—21no—
199C2×C4×C83—21no—
200C2×C4×C83—21no—
201C2×C4×C83—21no—
202C2×C4×C83—21no—
203C2×C4×C83—21no—
204C2×C4×C83—21no—
205C2×C4×C83—41no—
206C2×C4×C83—41no—
207C2×C4×C83—41no—
208C2×C4×C83—42no—
209C2×C4×C83—42no—
210C2×C4×C83—42no—
211C2×C4×C83—42no—
212C2×C4×C83—42no—
213C2×C4×C83—41no—
214C2×C4×C83—41no—
215C2×C4×C83—81no—
216C2×C4×C83—81no—
217C2×C4×C83—81no—
218C2×C4×C83—81no—
219C2×C4×C83C2×C22⋊C411yescharacteristic
220C2×C4×C83C4×D411yescharacteristic
221C2×C4×C83#12011yescharacteristic
222C2×C4×C83#12111yescharacteristic
223C8×C83—41no—
224C8×C83—42no—
225C8×C83—41no—
226C8×C83—41no—
227C8×C83—41no—
228C8×C83—81no—
229C22×C83—21no—
230C22×C83—21no—
231C22×C83—21no—
232C22×C83—21no—
233C22×C83—21no—
234C22×C83—21no—
235C22×C83—22no—
236C22×C83—21no—
237C22×C83—21no—
238C22×C83—41no—
239C22×C83—41no—
240C22×C83—41no—
241C22×C83—41no—
242C22×C83—41no—
243C22×C83—81no—
244C22×C83—81no—
245C22×C83—82no—
246C22×C83—82no—
247C22×C83#38611yescharacteristic
248C22×C83#47111yescharacteristic
249C22×C83#47311yescharacteristic
250C4×C83—21no—
251C4×C83—21no—
252C4×C83—22no—
253C4×C83—21no—
254C4×C83—41no—
255C4×C83—81no—
256C4×C83—81no—
257C4×C83—161no—
258C4×C83C2×#9911yescharacteristic
259C4×C83C2×#9911yescharacteristic
260#9345C8311yesmaximal, characteristic, direct product factor, Sylow
261C2×C83—21no—
262C2×C83—21no—
263C2×C83—41no—
264C2×C83—41no—
265C2×C83—42no—
266C2×C83—81no—
267C2×C83#188111yescharacteristic
268C2×C83#189811yescharacteristic
269C2×C83#190811yescharacteristic
270#3007C2×C8311yescharacteristic
271#2880C2×C8311yescharacteristic
272#1788C2×C8311yescharacteristic
273#1520C2×C8311yescharacteristic, complemented
274#1472C2×C8311yescharacteristic, complemented
275#1305C2×C8311yescharacteristic, complemented
276#1298C2×C8311yescharacteristic, complemented
277C83#934511yescharacteristic, direct product factor, Sylow
278C22×#41C22×C8311yescharacteristic
279C2×#137C4×C8311yescharacteristic, complemented
280C2×#137C4×C8311yescharacteristic, complemented
281C2×#131—21no—
282C2×C22⋊Q8C22×C8311yescharacteristic
283#452—21no—
284#452—21no—
285#451—21no—
286#450—21no—
287#417C4×C8311yescharacteristic, complemented
288#417C4×C8311yescharacteristic, complemented
289C2×#103—21no—
290C2×#102—21no—
291#408—21no—
292#408—22no—
293#407—21no—
294#407C22×C8311yescharacteristic
295#407C22×C8311yescharacteristic
296C2×C22⋊C8—21no—
297C2×C22⋊C8C22×C8311yescharacteristic
298C2×C22⋊C8C22×C8311yescharacteristic
299#380C22×C8311yescharacteristic
300#328—21no—
301#328—21no—
302#324—22no—
303C2×#41—21no—
304C2×#41—21no—
305C2×#41—42no—
306C2×#41—42no—
307C2×#41C2×C4×C8311yescharacteristic
308C2×#41C2×C4×C8311yescharacteristic
309C2×#41D4×C8311yescharacteristic
310C2×#41D4×C8311yescharacteristic
311C22×Q8C2×C4×C8311yescharacteristic
312C22×Q8D4×C8311yescharacteristic
313C22×D4—21no—
314C22×D4D4×C8311yescharacteristic
315C23×C4—21no—
316C23×C4C23×C8311yescharacteristic
317#137—42no—
318#137—42no—
319C2×Q16—21no—
320C2×Q16—21no—
321C2×SD16—21no—
322C2×SD16—21no—
323#131—42no—
324#131—42no—
325C2×M16—22no—
326C2×M16—21no—
327C2×M16C2×C4×C8311yescharacteristic
328C2×M16C2×C4×C8311yescharacteristic
329C22×C8—22no—
330C22×C8—21no—
331C22×C8—21no—
332C22⋊Q8—21no—
333C22⋊Q8—21no—
334C22⋊Q8—21no—
335C22⋊Q8—21no—
336C2×C4⋊C4—21no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4C2×C4×C8311yescharacteristic
339C2×C22⋊C4—21no—
340C2×C22⋊C4—21no—
341#103—21no—
342#103—21no—
343#103—42no—
344#102—21no—
345#102—21no—
346#102—42no—
347C22⋊C8—22no—
348C22⋊C8—22no—
349C22⋊C8—21no—
350C22⋊C8—21no—
351C22⋊C8—21no—
352C22⋊C8—21no—
353C22⋊C8—21no—
354C22⋊C8—21no—
355C22⋊C8D4×C8312yes—
356C22⋊C8D4×C8311yescharacteristic
357C22⋊C8D4×C8311yescharacteristic
358#95—21no—
359C24—21no—
360C24#41×C8311yescharacteristic
361#41—22no—
362#41—22no—
363#41—22no—
364#41—22no—
365#41—41no—
366#41—41no—
367#41—41no—
368#41—41no—
369#41—44no—
370#41—44no—
371C2×Q8—21no—
372C2×Q8—22no—
373C2×Q8—21no—
374C2×Q8—21no—
375C2×Q8—41no—
376C2×Q8—41no—
377C2×Q8—41no—
378C2×Q8C22⋊C4×C8311yescharacteristic
379C2×Q8C22⋊C4×C8311yescharacteristic
380C2×D4—21no—
381C2×D4—22no—
382C2×D4—21no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—44no—
388C2×D4C22⋊C4×C8311yescharacteristic
389C2×D4C22⋊C4×C8311yescharacteristic
390C22×C4—21no—
391C22×C4—21no—
392C22×C4—21no—
393C22×C4—21no—
394C22×C4—21no—
395C22×C4—21no—
396C22×C4—21no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—44no—
400C22×C4—41no—
401C22×C4C22×C4×C8311yescharacteristic
402C22×C4C2×D4×C8311yescharacteristic
403C22×C4C2×D4×C8311yescharacteristic
404C22×C4C2×D4×C8311yescharacteristic
405C22×C4C2×D4×C8311yescharacteristic
406C22×C4#41×C8311yescharacteristic
407Q16—42no—
408Q16—42no—
409SD16—42no—
410SD16—42no—
411M16—44no—
412M16—41no—
413M16—41no—
414M16—41no—
415M16—41no—
416C2×C8—21no—
417C2×C8—21no—
418C2×C8—21no—
419C2×C8—22no—
420C2×C8—21no—
421C2×C8—44no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—81no—
436C23—21no—
437C23—21no—
438C23—21no—
439C23—21no—
440C23—41no—
441C23—41no—
442C23—42no—
443C23—41no—
444C23—41no—
445C23—81no—
446C23C4×D4×C8311yescharacteristic
447C23#121×C8311yescharacteristic
448C23#123×C8311yescharacteristic
449Q8—21no—
450Q8—21no—
451Q8—21no—
452Q8—42no—
453Q8—41no—
454Q8—41no—
455Q8—81no—
456Q8#99×C8312yes—
457D4—24no—
458D4—21no—
459D4—21no—
460D4—21no—
461D4—42no—
462D4—42no—
463D4—42no—
464D4—42no—
465D4#99×C8312yes—
466C2×C4—24no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4C2×C22⋊C4×C8311yescharacteristic
490C2×C4C4×D4×C8311yescharacteristic
491C2×C4#120×C8311yescharacteristic
492C2×C4#121×C8311yescharacteristic
493C8—41no—
494C8—42no—
495C8—41no—
496C8—41no—
497C8—41no—
498C8—81no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—22no—
506C22—21no—
507C22—21no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—81no—
514C22—81no—
515C22—82no—
516C22—82no—
517C22#386×C8311yescharacteristic
518C22#471×C8311yescharacteristic
519C22#473×C8311yescharacteristic
520C4—21no—
521C4—21no—
522C4—22no—
523C4—21no—
524C4—41no—
525C4—81no—
526C4—81no—
527C4—161no—
528C4C2×#99×C8311yescharacteristic
529C4C2×#99×C8311yescharacteristic
530C2—21no—
531C2—21no—
532C2—41no—
533C2—41no—
534C2—42no—
535C2—81no—
536C2C83×#188111yescharacteristic
537C2C83×#189811yescharacteristic
538C2C83×#190811yescharacteristic
subgroupG/H
center, socleC22×C83#386
commutatorC22×C4C22×C4×C83
Frattini, cosocleC23×C4C23×C83
prime radicalC22×#41×C83C22
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
4. C83×#1520C2semidirect
5. C83×#1472C2semidirect
6. C83×#1305C2semidirect
7. C83×#1298C2semidirect
9. C2×#137×C83C4semidirect
10. C2×#137×C83C4semidirect
17. #417×C83C4semidirect
18. #417×C83C4semidirect
260. #9345C83direct (N × H)
273. #1520C2×C83semidirect
274. #1472C2×C83semidirect
275. #1305C2×C83semidirect
276. #1298C2×C83semidirect
277. C83#9345direct (N × H)
279. C2×#137C4×C83semidirect
280. C2×#137C4×C83semidirect
287. #417C4×C83semidirect
288. #417C4×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#30072
C83×#28802
C83×#17882
C83×#15202
C83×#14722
C83×#13052
C83×#12982
#934583