Finite Group Databasebeta

← C83×#18412 #35246236

Subgroups — C83×#18412 #35246236

Order 21248.

994 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3506C211yesmaximal, characteristic
2C83×#3229C211yesmaximal, characteristic, complemented
3C83×#3226C211yesmaximal, characteristic, complemented
4C83×#2547C211yesmaximal, characteristic, complemented
5C83×#1612C218yesmaximal, complemented
6C83×#1503C211yesmaximal, characteristic, complemented
7C83×#1502C211yesmaximal, characteristic, complemented
8C83×#1460C211yesmaximal, characteristic, complemented
9#559×C83—22no—
10#556×C83—21no—
11#555×C83—21no—
12#553×C83—22no—
13#550×C83—21no—
14#547×C83—21no—
15#538×C83C2214yes—
16#535×C83—21no—
17#533×C83C2211yescharacteristic
18#532×C83C2211yescharacteristic
19C2×C4⋊Q8×C83C2211yescharacteristic
20C2×#124×C83—21no—
21#525×C83—22no—
22#520×C83—21no—
23C2×C4×D4×C83C2211yescharacteristic
24#500×C83—21no—
25#498×C83—21no—
26#487×C83—21no—
27#486×C83—21no—
28#483×C83C2214yes—
29#480×C83C2214yes—
30#475×C83—22no—
31#474×C83—22no—
32#431×C83C2211yescharacteristic
33#431×C83C2211yescharacteristic
34#397×C83C2214yes—
35#391×C83—21no—
36#380×C83—22no—
37C4×C4⋊C4×C83C2211yescharacteristic
38#328×C83C2214yes—
39#327×C83C2214yes—
40#323×C83C2214yes—
41C2×#41×C83—22no—
42C2×#41×C83—21no—
43C22×Q8×C83—21no—
44C22×D4×C83D411yescharacteristic
45C23×C4×C83D412yes—
46C4⋊Q8×C83—21no—
47C4⋊Q8×C83—21no—
48C4⋊Q8×C83—21no—
49C4⋊Q8×C83—21no—
50C4⋊Q8×C83D411yescharacteristic
51C4⋊Q8×C83D411yescharacteristic
52C4⋊Q8×C83D411yescharacteristic
53#127×C83—22no—
54#127×C83D411yescharacteristic
55#126×C83—24no—
56#125×C83—21no—
57#124×C83—22no—
58#124×C83—24no—
59#124×C83—21no—
60#124×C83—44no—
61#123×C83—24no—
62#123×C83—24no—
63C22⋊Q8×C83—22no—
64C22⋊Q8×C83—21no—
65C22⋊Q8×C83C2312yes—
66#121×C83—22no—
67#121×C83—22no—
68#121×C83—22no—
69#121×C83—24no—
70#121×C83—21no—
71#121×C83C2312yes—
72#120×C83—24no—
73C4×Q8×C83—22no—
74C4×Q8×C83—21no—
75C4×Q8×C83D412yes—
76C4×D4×C83—22no—
77C4×D4×C83—24no—
78C4×D4×C83—24no—
79C4×D4×C83—21no—
80C4×D4×C83—21no—
81C4×D4×C83—21no—
82C4×D4×C83—21no—
83C4×D4×C83—21no—
84C4×D4×C83D412yes—
85#117×C83—22no—
86#117×C83—22no—
87#117×C83—21no—
88C2×C4⋊C4×C83—22no—
89C2×C4⋊C4×C83—21no—
90C2×C4⋊C4×C83—21no—
91C2×C4⋊C4×C83D411yescharacteristic
92C2×C4⋊C4×C83C2312yes—
93C2×C22⋊C4×C83—24no—
94C2×C22⋊C4×C83C2312yes—
95C2×C42×C83—21no—
96C2×C42×C83C2311yescharacteristic
97#107×C83—24no—
98#106×C83—24no—
99C4⋊C8×C83—21no—
100C4⋊C8×C83—21no—
101#103×C83—24no—
102#102×C83—24no—
103C22⋊C8×C83C2312yes—
104C22⋊C8×C83C2312yes—
105C4×C8×C83—21no—
106C4×C8×C83—21no—
107#95×C83—24no—
108#95×C83C2312yes—
109C24×C83C2×D412yes—
110#41×C83—42no—
111#41×C83—42no—
112#41×C83—84no—
113C2×Q8×C83—21no—
114C2×Q8×C83—21no—
115C2×Q8×C83—41no—
116C2×Q8×C83—41no—
117C2×Q8×C83—42no—
118C2×Q8×C83—42no—
119C2×Q8×C83C2×D412yes—
120C2×D4×C83—22no—
121C2×D4×C83—24no—
122C2×D4×C83—24no—
123C2×D4×C83—21no—
124C2×D4×C83—21no—
125C2×D4×C83—21no—
126C2×D4×C83—21no—
127C2×D4×C83—22no—
128C2×D4×C83—22no—
129C2×D4×C83C2×D412yes—
130C22×C4×C83—24no—
131C22×C4×C83—22no—
132C22×C4×C83—22no—
133C22×C4×C83—22no—
134C22×C4×C83—22no—
135C22×C4×C83—22no—
136C22×C4×C83—44no—
137C22×C4×C83C2×D411yescharacteristic
138C22×C4×C83C2×D412yes—
139C22×C4×C83C2×D411yescharacteristic
140C22×C4×C83C2411yescharacteristic
141C2×C8×C83—22no—
142C2×C8×C83—22no—
143C4⋊C4×C83—24no—
144C4⋊C4×C83—21no—
145C4⋊C4×C83—21no—
146C4⋊C4×C83—21no—
147C4⋊C4×C83—21no—
148C4⋊C4×C83—21no—
149C4⋊C4×C83—21no—
150C4⋊C4×C83—22no—
151C4⋊C4×C83—22no—
152C4⋊C4×C83—22no—
153C4⋊C4×C83—22no—
154C4⋊C4×C83—42no—
155C4⋊C4×C83—41no—
156C4⋊C4×C83—41no—
157C4⋊C4×C83C2×D412yes—
158C4⋊C4×C83C2×D412yes—
159C22⋊C4×C83—24no—
160C22⋊C4×C83—24no—
161C22⋊C4×C83—24no—
162C22⋊C4×C83—22no—
163C22⋊C4×C83—22no—
164C22⋊C4×C83—22no—
165C22⋊C4×C83—22no—
166C22⋊C4×C83—22no—
167C22⋊C4×C83—48no—
168C42×C83—24no—
169C42×C83—22no—
170C42×C83—42no—
171C42×C83C2×D411yescharacteristic
172C42×C83C2×D411yescharacteristic
173C42×C83C2×D411yescharacteristic
174C42×C83C2×D411yescharacteristic
175C23×C83—22no—
176C23×C83—22no—
177C23×C83—22no—
178C23×C83—22no—
179C23×C83—42no—
180C23×C83—42no—
181C23×C83C22×D411yescharacteristic
182Q8×C83—42no—
183Q8×C83—42no—
184Q8×C83—42no—
185Q8×C83—82no—
186Q8×C83—82no—
187D4×C83—42no—
188D4×C83—42no—
189D4×C83—42no—
190D4×C83—42no—
191D4×C83—42no—
192D4×C83—42no—
193D4×C83—42no—
194D4×C83—84no—
195D4×C83—84no—
196C2×C4×C83—22no—
197C2×C4×C83—22no—
198C2×C4×C83—22no—
199C2×C4×C83—22no—
200C2×C4×C83—22no—
201C2×C4×C83—41no—
202C2×C4×C83—44no—
203C2×C4×C83—44no—
204C2×C4×C83—42no—
205C2×C4×C83—42no—
206C2×C4×C83—42no—
207C2×C4×C83—42no—
208C2×C4×C83—42no—
209C2×C4×C83—42no—
210C2×C4×C83—42no—
211C2×C4×C83—82no—
212C2×C4×C83—84no—
213C2×C4×C83#12011yescharacteristic
214C2×C4×C83#12011yescharacteristic
215C2×C4×C83#12011yescharacteristic
216C2×C4×C83#12011yescharacteristic
217C2×C4×C83#13512yes—
218C2×C4×C83AGL(1,ℤ/8)12yes—
219C2×C4×C83C22×D411yescharacteristic
220C2×C4×C83C22×D411yescharacteristic
221C8×C83—42no—
222C8×C83—42no—
223C22×C83—21no—
224C22×C83—21no—
225C22×C83—44no—
226C22×C83—42no—
227C22×C83—42no—
228C22×C83—42no—
229C22×C83—42no—
230C22×C83—42no—
231C22×C83—42no—
232C22×C83—82no—
233C22×C83C2×#12011yescharacteristic
234C22×C83C2×#13511yescharacteristic
235C22×C83C2×AGL(1,ℤ/8)11yescharacteristic
236C4×C83—21no—
237C4×C83—21no—
238C4×C83—21no—
239C4×C83—42no—
240C4×C83—42no—
241C4×C83—42no—
242C4×C83—42no—
243C4×C83—82no—
244C4×C83—82no—
245C4×C83#44912yes—
246C4×C83#44912yes—
247C4×C83#45512yes—
248#18412C8311yesmaximal, characteristic, direct product factor, Sylow
249C2×C83—21no—
250C2×C83—21no—
251C2×C83—82no—
252C2×C83—82no—
253C2×C83C2×#44911yescharacteristic
254C2×C83C2×#45511yescharacteristic
255C2×C83#302811yescharacteristic
256#3506C2×C8311yescharacteristic
257#3229C2×C8311yescharacteristic, complemented
258#3226C2×C8311yescharacteristic, complemented
259#2547C2×C8311yescharacteristic, complemented
260#1612C2×C8318yescomplemented
261#1503C2×C8311yescharacteristic, complemented
262#1502C2×C8311yescharacteristic, complemented
263#1460C2×C8311yescharacteristic, complemented
264C83#1841211yescharacteristic, direct product factor, Sylow
265#559—22no—
266#556—21no—
267#555—21no—
268#553—22no—
269#550—21no—
270#547—21no—
271#538C22×C8314yes—
272#535—21no—
273#533C22×C8311yescharacteristic
274#532C22×C8311yescharacteristic
275C2×C4⋊Q8C22×C8311yescharacteristic
276C2×#124—21no—
277#525—22no—
278#520—21no—
279C2×C4×D4C22×C8311yescharacteristic
280#500—21no—
281#498—21no—
282#487—21no—
283#486—21no—
284#483C22×C8314yes—
285#480C22×C8314yes—
286#475—22no—
287#474—22no—
288#431C22×C8311yescharacteristic
289#431C22×C8311yescharacteristic
290#397C22×C8314yes—
291#391—21no—
292#380—22no—
293C4×C4⋊C4C22×C8311yescharacteristic
294#328C22×C8314yes—
295#327C22×C8314yes—
296#323C22×C8314yes—
297C2×#41—22no—
298C2×#41—21no—
299C22×Q8—21no—
300C22×D4D4×C8311yescharacteristic
301C23×C4D4×C8312yes—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8—21no—
306C4⋊Q8D4×C8311yescharacteristic
307C4⋊Q8D4×C8311yescharacteristic
308C4⋊Q8D4×C8311yescharacteristic
309#127—22no—
310#127D4×C8311yescharacteristic
311#126—24no—
312#125—21no—
313#124—22no—
314#124—24no—
315#124—21no—
316#124—44no—
317#123—24no—
318#123—24no—
319C22⋊Q8—22no—
320C22⋊Q8—21no—
321C22⋊Q8C23×C8312yes—
322#121—22no—
323#121—22no—
324#121—22no—
325#121—24no—
326#121—21no—
327#121C23×C8312yes—
328#120—24no—
329C4×Q8—22no—
330C4×Q8—21no—
331C4×Q8D4×C8312yes—
332C4×D4—22no—
333C4×D4—24no—
334C4×D4—24no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—21no—
338C4×D4—21no—
339C4×D4—21no—
340C4×D4D4×C8312yes—
341#117—22no—
342#117—22no—
343#117—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—21no—
347C2×C4⋊C4D4×C8311yescharacteristic
348C2×C4⋊C4C23×C8312yes—
349C2×C22⋊C4—24no—
350C2×C22⋊C4C23×C8312yes—
351C2×C42—21no—
352C2×C42C23×C8311yescharacteristic
353#107—24no—
354#106—24no—
355C4⋊C8—21no—
356C4⋊C8—21no—
357#103—24no—
358#102—24no—
359C22⋊C8C23×C8312yes—
360C22⋊C8C23×C8312yes—
361C4×C8—21no—
362C4×C8—21no—
363#95—24no—
364#95C23×C8312yes—
365C24C2×D4×C8312yes—
366#41—42no—
367#41—42no—
368#41—84no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—41no—
372C2×Q8—41no—
373C2×Q8—42no—
374C2×Q8—42no—
375C2×Q8C2×D4×C8312yes—
376C2×D4—22no—
377C2×D4—24no—
378C2×D4—24no—
379C2×D4—21no—
380C2×D4—21no—
381C2×D4—21no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—22no—
385C2×D4C2×D4×C8312yes—
386C22×C4—24no—
387C22×C4—22no—
388C22×C4—22no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—44no—
393C22×C4C2×D4×C8311yescharacteristic
394C22×C4C2×D4×C8312yes—
395C22×C4C2×D4×C8311yescharacteristic
396C22×C4C24×C8311yescharacteristic
397C2×C8—22no—
398C2×C8—22no—
399C4⋊C4—24no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—21no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—22no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—22no—
410C4⋊C4—42no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4C2×D4×C8312yes—
414C4⋊C4C2×D4×C8312yes—
415C22⋊C4—24no—
416C22⋊C4—24no—
417C22⋊C4—24no—
418C22⋊C4—22no—
419C22⋊C4—22no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—48no—
424C42—24no—
425C42—22no—
426C42—42no—
427C42C2×D4×C8311yescharacteristic
428C42C2×D4×C8311yescharacteristic
429C42C2×D4×C8311yescharacteristic
430C42C2×D4×C8311yescharacteristic
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—42no—
436C23—42no—
437C23C22×D4×C8311yescharacteristic
438Q8—42no—
439Q8—42no—
440Q8—42no—
441Q8—82no—
442Q8—82no—
443D4—42no—
444D4—42no—
445D4—42no—
446D4—42no—
447D4—42no—
448D4—42no—
449D4—42no—
450D4—84no—
451D4—84no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—41no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—82no—
468C2×C4—84no—
469C2×C4#120×C8311yescharacteristic
470C2×C4#120×C8311yescharacteristic
471C2×C4#120×C8311yescharacteristic
472C2×C4#120×C8311yescharacteristic
473C2×C4#135×C8312yes—
474C2×C4AGL(1,ℤ/8)×C8312yes—
475C2×C4C22×D4×C8311yescharacteristic
476C2×C4C22×D4×C8311yescharacteristic
477C8—42no—
478C8—42no—
479C22—21no—
480C22—21no—
481C22—44no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—82no—
489C22C2×#120×C8311yescharacteristic
490C22C2×#135×C8311yescharacteristic
491C22C2×AGL(1,ℤ/8)×C8311yescharacteristic
492C4—21no—
493C4—21no—
494C4—21no—
495C4—42no—
496C4—42no—
497C4—42no—
498C4—42no—
499C4—82no—
500C4—82no—
501C4#449×C8312yes—
502C4#449×C8312yes—
503C4#455×C8312yes—
504C2—21no—
505C2—21no—
506C2—82no—
507C2—82no—
508C2C2×#449×C8311yescharacteristic
509C2C2×#455×C8311yescharacteristic
510C2C83×#302811yescharacteristic
subgroupG/H
center, socleC22×C83C2×#120
commutator, Frattini, cosocleC22×C4C24×C83
prime radicalC83×#3506C2
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×#3229C2semidirect
3. C83×#3226C2semidirect
4. C83×#2547C2semidirect
5. C83×#1612C2semidirect
6. C83×#1503C2semidirect
7. C83×#1502C2semidirect
8. C83×#1460C2semidirect
248. #18412C83direct (N × H)
257. #3229C2×C83semidirect
258. #3226C2×C83semidirect
259. #2547C2×C83semidirect
260. #1612C2×C83semidirect
261. #1503C2×C83semidirect
262. #1502C2×C83semidirect
263. #1460C2×C83semidirect
264. C83#18412direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#35062
C83×#32292
C83×#32262
C83×#25472
C83×#16122
C83×#15032
C83×#15022
C83×#14602
#1841283