Finite Group Databasebeta

← C83×#8324 #35236148

Subgroups — C83×#8324 #35236148

Order 21248.

1006 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#520×C83C211yesmaximal, characteristic
2C83×#1479C212yesmaximal, complemented
3C83×#1478C212yesmaximal, complemented
4C83×#1458C211yesmaximal, characteristic, complemented
5C83×#1458C211yesmaximal, characteristic, complemented
6C22×#41×C83C411yescharacteristic
7#520×C83—24no—
8#520×C83—24no—
9C2×C4×Q8×C83C411yescharacteristic
10C2×C4×Q8×C83C2211yescharacteristic
11C2×C4×D4×C83—21no—
12C2×C4×D4×C83—21no—
13C2×C4×D4×C83C411yescharacteristic
14C2×C4×D4×C83C2211yescharacteristic
15C2×#117×C83—21no—
16C2×#117×C83C2211yescharacteristic
17C22×C4⋊C4×C83—21no—
18C22×C4⋊C4×C83C411yescharacteristic
19#431×C83C2212yes—
20#431×C83C2212yes—
21Q8⋊C8×C83—24no—
22D4⋊C8×C83—24no—
23#324×C83—21no—
24#324×C83—21no—
25#323×C83—21no—
26#323×C83—21no—
27C2×#41×C83—22no—
28C2×#41×C83—22no—
29C2×#41×C83—22no—
30C2×#41×C83—22no—
31C2×#41×C83—22no—
32C2×#41×C83C814yescomplemented
33C22×Q8×C83C2×C411yescharacteristic
34C22×D4×C83—21no—
35C22×D4×C83C2×C411yescharacteristic
36C23×C4×C83—21no—
37C23×C4×C83—21no—
38C23×C4×C83—21no—
39C23×C4×C83—21no—
40C23×C4×C83C2×C411yescharacteristic
41C4×Q8×C83—22no—
42C4×Q8×C83—22no—
43C4×Q8×C83—22no—
44C4×Q8×C83D414yes—
45C4×D4×C83—28no—
46C4×D4×C83—28no—
47C4×D4×C83—22no—
48C4×D4×C83—22no—
49C4×D4×C83—22no—
50C4×D4×C83D414yes—
51#117×C83—28no—
52#117×C83—22no—
53#117×C83—22no—
54C2×C4⋊C4×C83—28no—
55C2×C4⋊C4×C83—21no—
56C2×C4⋊C4×C83—21no—
57C2×C4⋊C4×C83—21no—
58C2×C4⋊C4×C83—22no—
59C2×C4⋊C4×C83—22no—
60C2×C4⋊C4×C83C2×C411yescharacteristic
61C2×C4⋊C4×C83C2×C411yescharacteristic
62C2×C4⋊C4×C83D411yescharacteristic
63C2×C4⋊C4×C83D411yescharacteristic
64C2×C22⋊C4×C83—21no—
65C2×C22⋊C4×C83—21no—
66C2×C22⋊C4×C83D411yescharacteristic
67C2×C22⋊C4×C83D411yescharacteristic
68C2×C42×C83—21no—
69C2×C42×C83—21no—
70C2×C42×C83C2×C411yescharacteristic
71C2×C42×C83C2311yescharacteristic
72C4⋊C8×C83—22no—
73C4⋊C8×C83—22no—
74C22⋊C8×C83—22no—
75C22⋊C8×C83—22no—
76C4×C8×C83—22no—
77C4×C8×C83—22no—
78C24×C83—21no—
79C24×C83C22⋊C411yescharacteristic
80#41×C83—28no—
81#41×C83—28no—
82#41×C83—28no—
83#41×C83—28no—
84C2×Q8×C83—21no—
85C2×Q8×C83—21no—
86C2×Q8×C83—21no—
87C2×Q8×C83C22⋊C412yes—
88C2×Q8×C83C2×C812yes—
89C2×Q8×C83M1612yes—
90C2×D4×C83—24no—
91C2×D4×C83—24no—
92C2×D4×C83—24no—
93C2×D4×C83—21no—
94C2×D4×C83—21no—
95C2×D4×C83—21no—
96C2×D4×C83C22⋊C412yes—
97C2×D4×C83C2×C812yes—
98C2×D4×C83M1612yes—
99C22×C4×C83—24no—
100C22×C4×C83—24no—
101C22×C4×C83—21no—
102C22×C4×C83—24no—
103C22×C4×C83—21no—
104C22×C4×C83—21no—
105C22×C4×C83—21no—
106C22×C4×C83—21no—
107C22×C4×C83—21no—
108C22×C4×C83—21no—
109C22×C4×C83—24no—
110C22×C4×C83—24no—
111C22×C4×C83—24no—
112C22×C4×C83—82no—
113C22×C4×C83—82no—
114C22×C4×C83—82no—
115C22×C4×C83C22⋊C411yescharacteristic
116C22×C4×C83C22⋊C411yescharacteristic
117C22×C4×C83C22⋊C411yescharacteristic
118C22×C4×C83C2×C812yes—
119C22×C4×C83M1612yes—
120C22×C4×C83C22×C411yescharacteristic
121C22×C4×C83C2×D411yescharacteristic
122C22×C4×C83C2×D411yescharacteristic
123C2×C8×C83—42no—
124C2×C8×C83—42no—
125C4⋊C4×C83—24no—
126C4⋊C4×C83—24no—
127C4⋊C4×C83—21no—
128C4⋊C4×C83—21no—
129C4⋊C4×C83—22no—
130C4⋊C4×C83—24no—
131C4⋊C4×C83—22no—
132C4⋊C4×C83C22⋊C412yes—
133C4⋊C4×C83C22⋊C412yes—
134C22⋊C4×C83—24no—
135C22⋊C4×C83—24no—
136C22⋊C4×C83—22no—
137C22⋊C4×C83—22no—
138C42×C83—24no—
139C42×C83—21no—
140C42×C83—21no—
141C42×C83—24no—
142C42×C83C2×D412yes—
143C42×C83C2×D412yes—
144C23×C83—21no—
145C23×C83—22no—
146C23×C83—22no—
147C23×C83—21no—
148C23×C83—22no—
149C23×C83—41no—
150C23×C83—41no—
151C23×C83—42no—
152C23×C83—42no—
153C23×C83C22⋊C812yes—
154C23×C83C2×C22⋊C411yescharacteristic
155Q8×C83—22no—
156Q8×C83—22no—
157Q8×C83—22no—
158Q8×C83C22⋊C814yes—
159D4×C83—216no—
160D4×C83—22no—
161D4×C83—22no—
162D4×C83—22no—
163D4×C83C22⋊C814yes—
164C2×C4×C83—216no—
165C2×C4×C83—22no—
166C2×C4×C83—21no—
167C2×C4×C83—22no—
168C2×C4×C83—21no—
169C2×C4×C83—22no—
170C2×C4×C83—22no—
171C2×C4×C83—22no—
172C2×C4×C83—22no—
173C2×C4×C83—22no—
174C2×C4×C83—21no—
175C2×C4×C83—21no—
176C2×C4×C83—22no—
177C2×C4×C83—22no—
178C2×C4×C83—41no—
179C2×C4×C83—41no—
180C2×C4×C83—42no—
181C2×C4×C83—42no—
182C2×C4×C83—41no—
183C2×C4×C83—41no—
184C2×C4×C83—84no—
185C2×C4×C83—84no—
186C2×C4×C83—84no—
187C2×C4×C83C22⋊C812yes—
188C2×C4×C83C2×C22⋊C411yescharacteristic
189C2×C4×C83C2×C22⋊C411yescharacteristic
190C2×C4×C83#12012yes—
191C2×C4×C83#12012yes—
192C2×C4×C83C22×C811yescharacteristic
193C2×C4×C83C2×M1611yescharacteristic
194C2×C4×C83C2×M1611yescharacteristic
195C2×C4×C83C2×M1611yescharacteristic
196C8×C83—82no—
197C8×C83—82no—
198C22×C83—21no—
199C22×C83—21no—
200C22×C83—22no—
201C22×C83—24no—
202C22×C83—42no—
203C22×C83—44no—
204C22×C83—42no—
205C22×C83—44no—
206C22×C83—41no—
207C22×C83—42no—
208C22×C83—41no—
209C22×C83—42no—
210C22×C83#37911yescharacteristic
211C22×C83C2×C22⋊C811yescharacteristic
212C22×C83#40711yescharacteristic
213C4×C83—22no—
214C4×C83—22no—
215C4×C83—22no—
216C4×C83—24no—
217C4×C83—42no—
218C4×C83—42no—
219C4×C83—41no—
220C4×C83—41no—
221C4×C83C2×C22⋊C811yescharacteristic
222C4×C83C2×C22⋊C811yescharacteristic
223C4×C83#40711yescharacteristic
224C4×C83#40711yescharacteristic
225#8324C8311yesmaximal, characteristic, direct product factor, Sylow
226C2×C83—21no—
227C2×C83—21no—
228C2×C83—42no—
229C2×C83—44no—
230C2×C83#177911yescharacteristic
231C2×C83#179511yescharacteristic
232C2×C83#179611yescharacteristic
233C2×#520C2×C8311yescharacteristic
234#1479C2×C8312yescomplemented
235#1478C2×C8312yescomplemented
236#1458C2×C8311yescharacteristic, complemented
237#1458C2×C8311yescharacteristic, complemented
238C83#832411yescharacteristic, direct product factor, Sylow
239C22×#41C4×C8311yescharacteristic
240#520—24no—
241#520—24no—
242C2×C4×Q8C4×C8311yescharacteristic
243C2×C4×Q8C22×C8311yescharacteristic
244C2×C4×D4—21no—
245C2×C4×D4—21no—
246C2×C4×D4C4×C8311yescharacteristic
247C2×C4×D4C22×C8311yescharacteristic
248C2×#117—21no—
249C2×#117C22×C8311yescharacteristic
250C22×C4⋊C4—21no—
251C22×C4⋊C4C4×C8311yescharacteristic
252#431C22×C8312yes—
253#431C22×C8312yes—
254Q8⋊C8—24no—
255D4⋊C8—24no—
256#324—21no—
257#324—21no—
258#323—21no—
259#323—21no—
260C2×#41—22no—
261C2×#41—22no—
262C2×#41—22no—
263C2×#41—22no—
264C2×#41—22no—
265C2×#41C8×C8314yescomplemented
266C22×Q8C2×C4×C8311yescharacteristic
267C22×D4—21no—
268C22×D4C2×C4×C8311yescharacteristic
269C23×C4—21no—
270C23×C4—21no—
271C23×C4—21no—
272C23×C4—21no—
273C23×C4C2×C4×C8311yescharacteristic
274C4×Q8—22no—
275C4×Q8—22no—
276C4×Q8—22no—
277C4×Q8D4×C8314yes—
278C4×D4—28no—
279C4×D4—28no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4D4×C8314yes—
284#117—28no—
285#117—22no—
286#117—22no—
287C2×C4⋊C4—28no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—22no—
292C2×C4⋊C4—22no—
293C2×C4⋊C4C2×C4×C8311yescharacteristic
294C2×C4⋊C4C2×C4×C8311yescharacteristic
295C2×C4⋊C4D4×C8311yescharacteristic
296C2×C4⋊C4D4×C8311yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4D4×C8311yescharacteristic
300C2×C22⋊C4D4×C8311yescharacteristic
301C2×C42—21no—
302C2×C42—21no—
303C2×C42C2×C4×C8311yescharacteristic
304C2×C42C23×C8311yescharacteristic
305C4⋊C8—22no—
306C4⋊C8—22no—
307C22⋊C8—22no—
308C22⋊C8—22no—
309C4×C8—22no—
310C4×C8—22no—
311C24—21no—
312C24C22⋊C4×C8311yescharacteristic
313#41—28no—
314#41—28no—
315#41—28no—
316#41—28no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8C22⋊C4×C8312yes—
321C2×Q8C2×C8×C8312yes—
322C2×Q8M16×C8312yes—
323C2×D4—24no—
324C2×D4—24no—
325C2×D4—24no—
326C2×D4—21no—
327C2×D4—21no—
328C2×D4—21no—
329C2×D4C22⋊C4×C8312yes—
330C2×D4C2×C8×C8312yes—
331C2×D4M16×C8312yes—
332C22×C4—24no—
333C22×C4—24no—
334C22×C4—21no—
335C22×C4—24no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—24no—
343C22×C4—24no—
344C22×C4—24no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4C22⋊C4×C8311yescharacteristic
349C22×C4C22⋊C4×C8311yescharacteristic
350C22×C4C22⋊C4×C8311yescharacteristic
351C22×C4C2×C8×C8312yes—
352C22×C4M16×C8312yes—
353C22×C4C22×C4×C8311yescharacteristic
354C22×C4C2×D4×C8311yescharacteristic
355C22×C4C2×D4×C8311yescharacteristic
356C2×C8—42no—
357C2×C8—42no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—22no—
363C4⋊C4—24no—
364C4⋊C4—22no—
365C4⋊C4C22⋊C4×C8312yes—
366C4⋊C4C22⋊C4×C8312yes—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—22no—
370C22⋊C4—22no—
371C42—24no—
372C42—21no—
373C42—21no—
374C42—24no—
375C42C2×D4×C8312yes—
376C42C2×D4×C8312yes—
377C23—21no—
378C23—22no—
379C23—22no—
380C23—21no—
381C23—22no—
382C23—41no—
383C23—41no—
384C23—42no—
385C23—42no—
386C23C22⋊C8×C8312yes—
387C23C2×C22⋊C4×C8311yescharacteristic
388Q8—22no—
389Q8—22no—
390Q8—22no—
391Q8C22⋊C8×C8314yes—
392D4—216no—
393D4—22no—
394D4—22no—
395D4—22no—
396D4C22⋊C8×C8314yes—
397C2×C4—216no—
398C2×C4—22no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—21no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—84no—
418C2×C4—84no—
419C2×C4—84no—
420C2×C4C22⋊C8×C8312yes—
421C2×C4C2×C22⋊C4×C8311yescharacteristic
422C2×C4C2×C22⋊C4×C8311yescharacteristic
423C2×C4#120×C8312yes—
424C2×C4#120×C8312yes—
425C2×C4C22×C8×C8311yescharacteristic
426C2×C4C2×M16×C8311yescharacteristic
427C2×C4C2×M16×C8311yescharacteristic
428C2×C4C2×M16×C8311yescharacteristic
429C8—82no—
430C8—82no—
431C22—21no—
432C22—21no—
433C22—22no—
434C22—24no—
435C22—42no—
436C22—44no—
437C22—44no—
438C22—42no—
439C22—42no—
440C22—41no—
441C22—42no—
442C22—41no—
443C22#379×C8311yescharacteristic
444C22C2×C22⋊C8×C8311yescharacteristic
445C22#407×C8311yescharacteristic
446C4—22no—
447C4—24no—
448C4—22no—
449C4—22no—
450C4—42no—
451C4—42no—
452C4—41no—
453C4—41no—
454C4C2×C22⋊C8×C8311yescharacteristic
455C4C2×C22⋊C8×C8311yescharacteristic
456C4#407×C8311yescharacteristic
457C4#407×C8311yescharacteristic
458C2—21no—
459C2—21no—
460C2—44no—
461C2—42no—
462C2C83×#177911yescharacteristic
463C2C83×#179511yescharacteristic
464C2C83×#179611yescharacteristic
subgroupG/H
center, socleC22×C83#379
commutatorC2×C4C22×C8×C83
Frattini, cosocleC2×C42C23×C83
prime radicalC22×#41×C83C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C83 ◁ C2×C42×C83 ◁ G

Complements

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

NH ≅ G/Ntype
2. C83×#1479C2semidirect
3. C83×#1478C2semidirect
4. C83×#1458C2semidirect
5. C83×#1458C2semidirect
32. C2×#41×C83C8semidirect
225. #8324C83direct (N × H)
234. #1479C2×C83semidirect
235. #1478C2×C83semidirect
236. #1458C2×C83semidirect
237. #1458C2×C83semidirect
238. C83#8324direct (N × H)
265. C2×#41C8×C83semidirect

Maximal subgroups

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

subgroupindex
C2×#520×C832
C83×#14792
C83×#14782
C83×#14582
#832483