Finite Group Databasebeta

← C83×#18342 #35246166

Subgroups — C83×#18342 #35246166

Order 21248.

998 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3459C211yesmaximal, characteristic
2C83×#3238C212yesmaximal, complemented
3C83×#2547C211yesmaximal, characteristic, complemented
4C83×#1612C214yesmaximal, complemented
5C83×#1608C212yesmaximal, complemented
6C83×#1607C212yesmaximal, complemented
7C83×#1509C212yesmaximal, complemented
8C83×#1466C211yesmaximal, characteristic, complemented
9#543×C83C2211yescharacteristic
10#539×C83—22no—
11#539×C83—21no—
12#538×C83—22no—
13#538×C83C2212yes—
14#534×C83—21no—
15#534×C83C2211yescharacteristic
16C2×C4⋊Q8×C83C2211yescharacteristic, complemented
17#529×C83—22no—
18#529×C83C2212yes—
19C2×#124×C83—21no—
20#525×C83—21no—
21#525×C83—22no—
22C2×C4×D4×C83—21no—
23C2×C4×D4×C83C2211yescharacteristic
24#501×C83—22no—
25#489×C83—22no—
26#483×C83C2214yescomplemented
27#480×C83C2214yes—
28#477×C83—22no—
29#476×C83—22no—
30#432×C83C2212yescomplemented
31#397×C83C2212yes—
32#397×C83C2212yescomplemented
33#391×C83—21no—
34#380×C83—21no—
35#380×C83—21no—
36C4×C4⋊C4×C83C2211yescharacteristic, complemented
37#328×C83C2214yescomplemented
38#327×C83C2214yes—
39#323×C83C2212yes—
40#323×C83C2212yescomplemented
41C2×#41×C83—21no—
42C22×Q8×C83—21no—
43C22×D4×C83—21no—
44C22×D4×C83D411yescharacteristic
45C23×C4×C83—21no—
46C23×C4×C83D411yescharacteristic
47C23×C4×C83D411yescharacteristic
48C4⋊Q8×C83—22no—
49C4⋊Q8×C83—22no—
50#126×C83—22no—
51#126×C83—22no—
52#126×C83—22no—
53#125×C83—22no—
54#125×C83D412yes—
55#124×C83—22no—
56#124×C83—44no—
57#124×C83D412yes—
58#123×C83—22no—
59#123×C83—22no—
60#123×C83—22no—
61#123×C83—22no—
62#123×C83—22no—
63#123×C83—22no—
64C22⋊Q8×C83—21no—
65C22⋊Q8×C83—22no—
66C22⋊Q8×C83—22no—
67C22⋊Q8×C83—21no—
68C22⋊Q8×C83—22no—
69C22⋊Q8×C83C2312yes—
70#121×C83—22no—
71#121×C83—21no—
72#121×C83—22no—
73#121×C83—22no—
74#121×C83—21no—
75#121×C83C2312yes—
76#120×C83—22no—
77#120×C83—22no—
78#120×C83—22no—
79C4×Q8×C83D412yes—
80C4×D4×C83—22no—
81C4×D4×C83—22no—
82C4×D4×C83—22no—
83C4×D4×C83—22no—
84C4×D4×C83—22no—
85C4×D4×C83—22no—
86C4×D4×C83—22no—
87C4×D4×C83D412yes—
88#117×C83—21no—
89#117×C83—22no—
90C2×C4⋊C4×C83—21no—
91C2×C4⋊C4×C83—21no—
92C2×C4⋊C4×C83—21no—
93C2×C4⋊C4×C83D411yescharacteristic
94C2×C4⋊C4×C83C2312yes—
95C2×C22⋊C4×C83—21no—
96C2×C22⋊C4×C83—21no—
97C2×C22⋊C4×C83—22no—
98C2×C22⋊C4×C83—22no—
99C2×C22⋊C4×C83C2311yescharacteristic
100C2×C22⋊C4×C83C2311yescharacteristic
101C2×C42×C83—21no—
102C2×C42×C83—21no—
103C2×C42×C83C2311yescharacteristic
104#107×C83—24no—
105#106×C83—24no—
106C4⋊C8×C83—22no—
107#103×C83—24no—
108#102×C83—24no—
109C22⋊C8×C83C2314yes—
110#97×C83—22no—
111#95×C83—22no—
112#95×C83—22no—
113#95×C83C2312yes—
114C24×C83—21no—
115C24×C83C2×D411yescharacteristic
116C24×C83C2×D411yescharacteristic
117#41×C83—82no—
118C2×Q8×C83—21no—
119C2×Q8×C83—42no—
120C2×Q8×C83—42no—
121C2×Q8×C83—42no—
122C2×Q8×C83C2×D412yes—
123C2×D4×C83—22no—
124C2×D4×C83—22no—
125C2×D4×C83—22no—
126C2×D4×C83—22no—
127C2×D4×C83—21no—
128C2×D4×C83—22no—
129C2×D4×C83—22no—
130C2×D4×C83—41no—
131C2×D4×C83—42no—
132C2×D4×C83—42no—
133C2×D4×C83C2×D412yes—
134C22×C4×C83—21no—
135C22×C4×C83—21no—
136C22×C4×C83—21no—
137C22×C4×C83—22no—
138C22×C4×C83—21no—
139C22×C4×C83—21no—
140C22×C4×C83—22no—
141C22×C4×C83—21no—
142C22×C4×C83—21no—
143C22×C4×C83—22no—
144C22×C4×C83—22no—
145C22×C4×C83—42no—
146C22×C4×C83—42no—
147C22×C4×C83—82no—
148C22×C4×C83C2×D411yescharacteristic
149C22×C4×C83C2×D411yescharacteristic
150C22×C4×C83C2×D411yescharacteristic
151C22×C4×C83C2×D411yescharacteristic
152C22×C4×C83C2411yescharacteristic
153C2×C8×C83—24no—
154C4⋊C4×C83—22no—
155C4⋊C4×C83—21no—
156C4⋊C4×C83—21no—
157C4⋊C4×C83—22no—
158C4⋊C4×C83—22no—
159C4⋊C4×C83—22no—
160C4⋊C4×C83—22no—
161C4⋊C4×C83—22no—
162C4⋊C4×C83—22no—
163C4⋊C4×C83—24no—
164C4⋊C4×C83—42no—
165C4⋊C4×C83—41no—
166C4⋊C4×C83—41no—
167C4⋊C4×C83C2×D412yes—
168C4⋊C4×C83C2×D412yes—
169C22⋊C4×C83—22no—
170C22⋊C4×C83—22no—
171C22⋊C4×C83—22no—
172C22⋊C4×C83—22no—
173C22⋊C4×C83—22no—
174C22⋊C4×C83—22no—
175C22⋊C4×C83—22no—
176C22⋊C4×C83—22no—
177C22⋊C4×C83—22no—
178C22⋊C4×C83—22no—
179C22⋊C4×C83—22no—
180C22⋊C4×C83—22no—
181C22⋊C4×C83—44no—
182C22⋊C4×C83—44no—
183C42×C83—22no—
184C42×C83—22no—
185C42×C83—42no—
186C42×C83C2×D412yes—
187C42×C83C2×D412yes—
188C23×C83—21no—
189C23×C83—21no—
190C23×C83—21no—
191C23×C83—22no—
192C23×C83—21no—
193C23×C83—21no—
194C23×C83—41no—
195C23×C83—41no—
196C23×C83—41no—
197C23×C83—41no—
198C23×C83—42no—
199C23×C83—81no—
200C23×C83—81no—
201C23×C83AGL(1,ℤ/8)12yes—
202C23×C83C22×D411yescharacteristic
203Q8×C83—42no—
204Q8×C83—84no—
205Q8×C83—81no—
206D4×C83—44no—
207D4×C83—42no—
208D4×C83—44no—
209D4×C83—42no—
210D4×C83—42no—
211D4×C83—82no—
212D4×C83—81no—
213D4×C83—82no—
214C2×C4×C83—21no—
215C2×C4×C83—21no—
216C2×C4×C83—22no—
217C2×C4×C83—22no—
218C2×C4×C83—22no—
219C2×C4×C83—21no—
220C2×C4×C83—21no—
221C2×C4×C83—22no—
222C2×C4×C83—41no—
223C2×C4×C83—42no—
224C2×C4×C83—42no—
225C2×C4×C83—42no—
226C2×C4×C83—44no—
227C2×C4×C83—41no—
228C2×C4×C83—41no—
229C2×C4×C83—42no—
230C2×C4×C83—42no—
231C2×C4×C83—42no—
232C2×C4×C83—42no—
233C2×C4×C83—42no—
234C2×C4×C83—42no—
235C2×C4×C83—81no—
236C2×C4×C83—81no—
237C2×C4×C83—81no—
238C2×C4×C83—82no—
239C2×C4×C83—81no—
240C2×C4×C83#12012yes—
241C2×C4×C83#12011yescharacteristic
242C2×C4×C83#12011yescharacteristic
243C2×C4×C83C22×D412yes—
244C8×C83—44no—
245C22×C83—21no—
246C22×C83—21no—
247C22×C83—22no—
248C22×C83—42no—
249C22×C83—42no—
250C22×C83—41no—
251C22×C83—41no—
252C22×C83—41no—
253C22×C83—42no—
254C22×C83—41no—
255C22×C83—42no—
256C22×C83—42no—
257C22×C83—41no—
258C22×C83—81no—
259C22×C83—81no—
260C22×C83—84no—
261C22×C83C2×#12011yescharacteristic
262C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
263C22×C83#57511yescharacteristic
264C4×C83—22no—
265C4×C83—21no—
266C4×C83—22no—
267C4×C83—42no—
268C4×C83—41no—
269C4×C83—41no—
270C4×C83—42no—
271C4×C83—42no—
272C4×C83—82no—
273C4×C83—82no—
274C4×C83#45512yes—
275#18342C8311yesmaximal, characteristic, direct product factor, Sylow
276C2×C83—21no—
277C2×C83—21no—
278C2×C83—42no—
279C2×C83—81no—
280C2×C83—82no—
281C2×C83#300611yescharacteristic
282C2×C83C2×#45511yescharacteristic
283C2×C83#302811yescharacteristic
284#3459C2×C8311yescharacteristic
285#3238C2×C8312yescomplemented
286#2547C2×C8311yescharacteristic, complemented
287#1612C2×C8314yescomplemented
288#1608C2×C8312yescomplemented
289#1607C2×C8312yescomplemented
290#1509C2×C8312yescomplemented
291#1466C2×C8311yescharacteristic, complemented
292C83#1834211yescharacteristic, direct product factor, Sylow
293#543C22×C8311yescharacteristic
294#539—22no—
295#539—21no—
296#538—22no—
297#538C22×C8312yes—
298#534—21no—
299#534C22×C8311yescharacteristic
300C2×C4⋊Q8C22×C8311yescharacteristic, complemented
301#529—22no—
302#529C22×C8312yes—
303C2×#124—21no—
304#525—21no—
305#525—22no—
306C2×C4×D4—21no—
307C2×C4×D4C22×C8311yescharacteristic
308#501—22no—
309#489—22no—
310#483C22×C8314yescomplemented
311#480C22×C8314yes—
312#477—22no—
313#476—22no—
314#432C22×C8312yescomplemented
315#397C22×C8312yes—
316#397C22×C8312yescomplemented
317#391—21no—
318#380—21no—
319#380—21no—
320C4×C4⋊C4C22×C8311yescharacteristic, complemented
321#328C22×C8314yescomplemented
322#327C22×C8314yes—
323#323C22×C8312yes—
324#323C22×C8312yescomplemented
325C2×#41—21no—
326C22×Q8—21no—
327C22×D4—21no—
328C22×D4D4×C8311yescharacteristic
329C23×C4—21no—
330C23×C4D4×C8311yescharacteristic
331C23×C4D4×C8311yescharacteristic
332C4⋊Q8—22no—
333C4⋊Q8—22no—
334#126—22no—
335#126—22no—
336#126—22no—
337#125—22no—
338#125D4×C8312yes—
339#124—22no—
340#124—44no—
341#124D4×C8312yes—
342#123—22no—
343#123—22no—
344#123—22no—
345#123—22no—
346#123—22no—
347#123—22no—
348C22⋊Q8—21no—
349C22⋊Q8—22no—
350C22⋊Q8—22no—
351C22⋊Q8—21no—
352C22⋊Q8—22no—
353C22⋊Q8C23×C8312yes—
354#121—22no—
355#121—21no—
356#121—22no—
357#121—22no—
358#121—21no—
359#121C23×C8312yes—
360#120—22no—
361#120—22no—
362#120—22no—
363C4×Q8D4×C8312yes—
364C4×D4—22no—
365C4×D4—22no—
366C4×D4—22no—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—22no—
370C4×D4—22no—
371C4×D4D4×C8312yes—
372#117—21no—
373#117—22no—
374C2×C4⋊C4—21no—
375C2×C4⋊C4—21no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4D4×C8311yescharacteristic
378C2×C4⋊C4C23×C8312yes—
379C2×C22⋊C4—21no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—22no—
382C2×C22⋊C4—22no—
383C2×C22⋊C4C23×C8311yescharacteristic
384C2×C22⋊C4C23×C8311yescharacteristic
385C2×C42—21no—
386C2×C42—21no—
387C2×C42C23×C8311yescharacteristic
388#107—24no—
389#106—24no—
390C4⋊C8—22no—
391#103—24no—
392#102—24no—
393C22⋊C8C23×C8314yes—
394#97—22no—
395#95—22no—
396#95—22no—
397#95C23×C8312yes—
398C24—21no—
399C24C2×D4×C8311yescharacteristic
400C24C2×D4×C8311yescharacteristic
401#41—82no—
402C2×Q8—21no—
403C2×Q8—42no—
404C2×Q8—42no—
405C2×Q8—42no—
406C2×Q8C2×D4×C8312yes—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—21no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—41no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4C2×D4×C8312yes—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—22no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—82no—
432C22×C4C2×D4×C8311yescharacteristic
433C22×C4C2×D4×C8311yescharacteristic
434C22×C4C2×D4×C8311yescharacteristic
435C22×C4C2×D4×C8311yescharacteristic
436C22×C4C24×C8311yescharacteristic
437C2×C8—24no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—22no—
447C4⋊C4—24no—
448C4⋊C4—42no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4C2×D4×C8312yes—
452C4⋊C4C2×D4×C8312yes—
453C22⋊C4—22no—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—22no—
460C22⋊C4—22no—
461C22⋊C4—22no—
462C22⋊C4—22no—
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—44no—
466C22⋊C4—44no—
467C42—22no—
468C42—22no—
469C42—42no—
470C42C2×D4×C8312yes—
471C42C2×D4×C8312yes—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—22no—
476C23—21no—
477C23—21no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—81no—
484C23—81no—
485C23AGL(1,ℤ/8)×C8312yes—
486C23C22×D4×C8311yescharacteristic
487Q8—42no—
488Q8—84no—
489Q8—81no—
490D4—44no—
491D4—42no—
492D4—44no—
493D4—42no—
494D4—42no—
495D4—81no—
496D4—82no—
497D4—82no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—82no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4#120×C8312yes—
525C2×C4#120×C8311yescharacteristic
526C2×C4#120×C8311yescharacteristic
527C2×C4C22×D4×C8312yes—
528C8—44no—
529C22—21no—
530C22—21no—
531C22—22no—
532C22—42no—
533C22—42no—
534C22—41no—
535C22—41no—
536C22—42no—
537C22—41no—
538C22—41no—
539C22—42no—
540C22—41no—
541C22—42no—
542C22—81no—
543C22—81no—
544C22—84no—
545C22C2×#120×C8311yescharacteristic
546C22C2×AGL(1,ℤ/8)×C8311yescharacteristic
547C22#575×C8311yescharacteristic
548C4—22no—
549C4—22no—
550C4—21no—
551C4—42no—
552C4—42no—
553C4—41no—
554C4—41no—
555C4—42no—
556C4—82no—
557C4—82no—
558C4#455×C8312yes—
559C2—21no—
560C2—21no—
561C2—42no—
562C2—81no—
563C2—82no—
564C2C83×#300611yescharacteristic
565C2C2×#455×C8311yescharacteristic
566C2C83×#302811yescharacteristic
subgroupG/H
center, socleC22×C83C2×#120
commutator, Frattini, cosocleC22×C4C24×C83
prime radicalC83×#3459C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#3238C2semidirect
3. C83×#2547C2semidirect
4. C83×#1612C2semidirect
5. C83×#1608C2semidirect
6. C83×#1607C2semidirect
7. C83×#1509C2semidirect
8. C83×#1466C2semidirect
16. C2×C4⋊Q8×C83C22semidirect
26. #483×C83C22semidirect
30. #432×C83C22semidirect
32. #397×C83C22semidirect
36. C4×C4⋊C4×C83C22semidirect
37. #328×C83C22semidirect
40. #323×C83C22semidirect
275. #18342C83direct (N × H)
285. #3238C2×C83semidirect
286. #2547C2×C83semidirect
287. #1612C2×C83semidirect
288. #1608C2×C83semidirect
289. #1607C2×C83semidirect
290. #1509C2×C83semidirect
291. #1466C2×C83semidirect
292. C83#18342direct (N × H)
300. C2×C4⋊Q8C22×C83semidirect
310. #483C22×C83semidirect
314. #432C22×C83semidirect
316. #397C22×C83semidirect
320. C4×C4⋊C4C22×C83semidirect
321. #328C22×C83semidirect
324. #323C22×C83semidirect

Maximal subgroups

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

subgroupindex
C83×#34592
C83×#32382
C83×#25472
C83×#16122
C83×#16082
C83×#16072
C83×#15092
C83×#14662
#1834283