Finite Group Databasebeta

← C2×C83×#3523 #35290611

Subgroups — C2×C83×#3523 #35290611

Order 21248.

7022 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C83×#3523C2132yesmaximal, direct product factor
2C2×#557×C83C211yesmaximal, characteristic, complemented
3C2×#557×C83C213yesmaximal, complemented
4C2×#556×C83C213yesmaximal, complemented
5C2×#552×C83C216yesmaximal, complemented
6C2×D4×Q8×C83C212yesmaximal, complemented
7C2×D4×Q8×C83C211yesmaximal, characteristic, complemented
8C2×#541×C83C216yesmaximal, complemented
9C2×#533×C83C213yesmaximal, complemented
10C2×#529×C83C213yesmaximal
11C2×#519×C83C212yesmaximal, complemented
12C22×C4×Q8×C83C211yesmaximal, characteristic, complemented
13C23×Q8×C83C2211yescharacteristic
14C23×Q8×C83C2212yes—
15#557×C83C22116yescomplemented
16#557×C83C22148yescomplemented
17#556×C83C22148yescomplemented
18#552×C83C22196yescomplemented
19D4×Q8×C83C22132yescomplemented
20D4×Q8×C83C22116yescomplemented
21#541×C83C22196yescomplemented
22#533×C83C22148yescomplemented
23C2×C4⋊Q8×C83C2213yes—
24C2×C4⋊Q8×C83C2213yes—
25C2×C4⋊Q8×C83C2216yes—
26#529×C83C22148yes—
27C2×#126×C83C2216yes—
28C2×#126×C83C22112yes—
29C2×#125×C83C2213yes—
30C2×#125×C83C2216yes—
31C2×#125×C83C22112yes—
32C2×#125×C83C2216yes—
33C2×#124×C83C2213yes—
34C2×#123×C83C2216yes—
35C2×C22⋊Q8×C83C2213yes—
36C2×C22⋊Q8×C83C2216yes—
37C2×C22⋊Q8×C83C2216yes—
38C2×C22⋊Q8×C83C2216yes—
39C2×#121×C83C2211yescharacteristic
40#519×C83C22132yescomplemented
41C2×C4×Q8×C83C2211yescharacteristic
42C2×C4×Q8×C83C2213yes—
43C2×C4×Q8×C83C2211yescharacteristic
44C2×C4×Q8×C83C2213yes—
45C2×C4×Q8×C83C2216yes—
46C2×C4×Q8×C83C2212yes—
47C2×C4×Q8×C83C2216yes—
48C2×C4×Q8×C83C2213yes—
49C2×C4×Q8×C83C2216yes—
50C2×C4×Q8×C83C22116yescomplemented
51C2×C4×D4×C83C2216yes—
52C2×C4×D4×C83C2213yes—
53C2×#117×C83C2216yes—
54C2×#117×C83C2216yes—
55C2×#117×C83C2216yes—
56C22×C4⋊C4×C83C2213yes—
57C22×C42×C83C2213yes—
58C22×Q8×C83—28no—
59C22×Q8×C83—216no—
60C22×Q8×C83C2313yes—
61C22×Q8×C83C2313yes—
62C22×Q8×C83C2316yes—
63C22×Q8×C83C2318yes—
64C22×Q8×C83C23116yes—
65C22×Q8×C83C2311yescharacteristic
66C22×D4×C83C2312yes—
67C22×D4×C83C2311yescharacteristic
68C23×C4×C83C2311yescharacteristic
69C23×C4×C83C2313yes—
70C23×C4×C83C2313yes—
71C23×C4×C83C2316yes—
72C4⋊Q8×C83C23124yes—
73C4⋊Q8×C83C23124yes—
74C4⋊Q8×C83C23148yes—
75#126×C83C23148yes—
76#126×C83C23196yes—
77#125×C83C23124yes—
78#125×C83C23148yes—
79#125×C83C23196yes—
80#125×C83C23148yes—
81#124×C83C23124yes—
82#123×C83C23148yes—
83C22⋊Q8×C83C23124yes—
84C22⋊Q8×C83C23148yes—
85C22⋊Q8×C83C23148yes—
86C22⋊Q8×C83C23148yes—
87#121×C83C2318yes—
88C4×Q8×C83C2318yes—
89C4×Q8×C83C23124yes—
90C4×Q8×C83C2318yes—
91C4×Q8×C83C23124yes—
92C4×Q8×C83C23148yes—
93C4×Q8×C83C23116yes—
94C4×Q8×C83C23148yes—
95C4×Q8×C83C23124yes—
96C4×Q8×C83C23148yes—
97C4×D4×C83C23148yes—
98C4×D4×C83C23124yes—
99#117×C83C23148yes—
100#117×C83C23148yes—
101#117×C83C23148yes—
102C2×C4⋊C4×C83C2313yes—
103C2×C4⋊C4×C83C2313yes—
104C2×C4⋊C4×C83C2316yes—
105C2×C4⋊C4×C83C23124yes—
106C2×C4⋊C4×C83C2316yes—
107C2×C4⋊C4×C83C2316yes—
108C2×C4⋊C4×C83C2316yes—
109C2×C4⋊C4×C83C2316yes—
110C2×C4⋊C4×C83C2316yes—
111C2×C4⋊C4×C83C2313yes—
112C2×C4⋊C4×C83C2316yes—
113C2×C4⋊C4×C83C2311yescharacteristic
114C2×C4⋊C4×C83C23112yes—
115C2×C4⋊C4×C83C2316yes—
116C2×C22⋊C4×C83C2312yes—
117C2×C22⋊C4×C83C2316yes—
118C2×C22⋊C4×C83C2316yes—
119C2×C22⋊C4×C83C2316yes—
120C2×C22⋊C4×C83C2316yes—
121C2×C42×C83C2313yes—
122C2×C42×C83C2313yes—
123C2×C42×C83C2313yes—
124C2×C42×C83C2313yes—
125C2×C42×C83C23124yes—
126C2×C42×C83C23112yes—
127C2×C42×C83C2316yes—
128C24×C83C2412yes—
129C24×C83C2411yescharacteristic
130C2×Q8×C83—232no—
131C2×Q8×C83—264no—
132C2×Q8×C83—212no—
133C2×Q8×C83—224no—
134C2×Q8×C83—212no—
135C2×Q8×C83#4118yes—
136C2×Q8×C83C24112yes—
137C2×Q8×C83C24112yes—
138C2×Q8×C83C24124yes—
139C2×Q8×C83C2414yes—
140C2×D4×C83—28no—
141C2×D4×C83—24no—
142C2×D4×C83C2418yes—
143C2×D4×C83C2414yes—
144C22×C4×C83—212no—
145C22×C4×C83—212no—
146C22×C4×C83—224no—
147C22×C4×C83—24no—
148C22×C4×C83C2414yes—
149C22×C4×C83C24112yes—
150C22×C4×C83C24112yes—
151C22×C4×C83C24124yes—
152C22×C4×C83C2411yescharacteristic
153C22×C4×C83C2413yes—
154C22×C4×C83C2416yes—
155C22×C4×C83C2412yes—
156C22×C4×C83C2413yes—
157C22×C4×C83C2413yes—
158C22×C4×C83C2416yes—
159C22×C4×C83C2413yes—
160C22×C4×C83C2411yescharacteristic
161C4⋊C4×C83C24112yes—
162C4⋊C4×C83C24112yes—
163C4⋊C4×C83C24124yes—
164C4⋊C4×C83C24124yes—
165C4⋊C4×C83C24124yes—
166C4⋊C4×C83C24124yes—
167C4⋊C4×C83C24124yes—
168C4⋊C4×C83C24124yes—
169C4⋊C4×C83C24112yes—
170C4⋊C4×C83C24124yes—
171C4⋊C4×C83C2414yes—
172C4⋊C4×C83C24148yes—
173C4⋊C4×C83C24124yes—
174C22⋊C4×C83C2418yes—
175C22⋊C4×C83C24124yes—
176C22⋊C4×C83C24124yes—
177C22⋊C4×C83C24124yes—
178C22⋊C4×C83C24124yes—
179C42×C83C24112yes—
180C42×C83C24112yes—
181C42×C83C24112yes—
182C42×C83C24112yes—
183C42×C83C24148yes—
184C42×C83C24124yes—
185C23×C83—22no—
186C23×C83—22no—
187C23×C83—24no—
188C23×C83—24no—
189C23×C83—24no—
190C23×C832−1+414yes—
191C23×C83C2514yes—
192C23×C83C2512yes—
193C23×C83C2511yescharacteristic
194Q8×C83—224no—
195Q8×C83—248no—
196Q8×C83—224no—
197Q8×C83C2×#41116yes—
198D4×C83—216no—
199D4×C83—28no—
200C2×C4×C83—26no—
201C2×C4×C83—212no—
202C2×C4×C83—224no—
203C2×C4×C83—26no—
204C2×C4×C83—26no—
205C2×C4×C83—22no—
206C2×C4×C83—24no—
207C2×C4×C83—212no—
208C2×C4×C83—224no—
209C2×C4×C83—248no—
210C2×C4×C83—28no—
211C2×C4×C83C2×#41112yes—
212C2×C4×C832−1+414yes—
213C2×C4×C83C2512yes—
214C2×C4×C83C2516yes—
215C2×C4×C83C25112yes—
216C2×C4×C83C2514yes—
217C2×C4×C83C2516yes—
218C2×C4×C83C2516yes—
219C2×C4×C83C25112yes—
220C2×C4×C83C2516yes—
221C2×C4×C83C2512yes—
222C22×C83—22no—
223C22×C83—22no—
224C22×C83—28no—
225C22×C83—24no—
226C22×C83—24no—
227C22×C83—24no—
228C22×C83—48no—
229C22×C83C22×#4112yes—
230C22×C83C2×2−1+414yes—
231C22×C83C2×2−1+412yes—
232C22×C83C2×2−1+412yes—
233C22×C83C2611yescharacteristic
234C4×C83—26no—
235C4×C83—212no—
236C4×C83—26no—
237C4×C83—26no—
238C4×C83—22no—
239C4×C83—24no—
240C4×C83—212no—
241C4×C83C22×#41112yes—
242C4×C83C2×2−1+414yes—
243C2×#3523C8311yesmaximal, characteristic, direct product factor, Sylow
244C2×C83—24no—
245C2×C83—44no—
246C2×C83#352314yesdirect product factor
247C2×C83C23×#4111yescharacteristic
248C2×C83C22×2−1+411yescharacteristic
249C2×C83C22×2−1+411yescharacteristic
250#3523C2×C83132yesdirect product factor
251C2×#557C2×C8311yescharacteristic, complemented
252C2×#557C2×C8313yescomplemented
253C2×#556C2×C8313yescomplemented
254C2×#552C2×C8316yescomplemented
255C2×D4×Q8C2×C8312yescomplemented
256C2×D4×Q8C2×C8311yescharacteristic, complemented
257C2×#541C2×C8316yescomplemented
258C2×#533C2×C8313yescomplemented
259C2×#529C2×C8313yes—
260C2×#519C2×C8312yescomplemented
261C22×C4×Q8C2×C8311yescharacteristic, complemented
262C83C2×#352311yescharacteristic, direct product factor, Sylow
263C23×Q8C22×C8311yescharacteristic
264C23×Q8C22×C8312yes—
265#557C22×C83116yescomplemented
266#557C22×C83148yescomplemented
267#556C22×C83148yescomplemented
268#552C22×C83196yescomplemented
269D4×Q8C22×C83132yescomplemented
270D4×Q8C22×C83116yescomplemented
271#541C22×C83196yescomplemented
272#533C22×C83148yescomplemented
273C2×C4⋊Q8C22×C8313yes—
274C2×C4⋊Q8C22×C8313yes—
275C2×C4⋊Q8C22×C8316yes—
276#529C22×C83148yes—
277C2×#126C22×C8316yes—
278C2×#126C22×C83112yes—
279C2×#125C22×C8313yes—
280C2×#125C22×C8316yes—
281C2×#125C22×C83112yes—
282C2×#125C22×C8316yes—
283C2×#124C22×C8313yes—
284C2×#123C22×C8316yes—
285C2×C22⋊Q8C22×C8313yes—
286C2×C22⋊Q8C22×C8316yes—
287C2×C22⋊Q8C22×C8316yes—
288C2×C22⋊Q8C22×C8316yes—
289C2×#121C22×C8311yescharacteristic
290#519C22×C83132yescomplemented
291C2×C4×Q8C22×C8311yescharacteristic
292C2×C4×Q8C22×C8313yes—
293C2×C4×Q8C22×C8311yescharacteristic
294C2×C4×Q8C22×C8313yes—
295C2×C4×Q8C22×C8316yes—
296C2×C4×Q8C22×C8312yes—
297C2×C4×Q8C22×C8316yes—
298C2×C4×Q8C22×C8313yes—
299C2×C4×Q8C22×C8316yes—
300C2×C4×Q8C22×C83116yescomplemented
301C2×C4×D4C22×C8316yes—
302C2×C4×D4C22×C8313yes—
303C2×#117C22×C8316yes—
304C2×#117C22×C8316yes—
305C2×#117C22×C8316yes—
306C22×C4⋊C4C22×C8313yes—
307C22×C42C22×C8313yes—
308C22×Q8—28no—
309C22×Q8—216no—
310C22×Q8C23×C8313yes—
311C22×Q8C23×C8313yes—
312C22×Q8C23×C8316yes—
313C22×Q8C23×C8318yes—
314C22×Q8C23×C83116yes—
315C22×Q8C23×C8311yescharacteristic
316C22×D4C23×C8312yes—
317C22×D4C23×C8311yescharacteristic
318C23×C4C23×C8313yes—
319C23×C4C23×C8313yes—
320C23×C4C23×C8316yes—
321C23×C4C23×C8311yescharacteristic
322C4⋊Q8C23×C83124yes—
323C4⋊Q8C23×C83124yes—
324C4⋊Q8C23×C83148yes—
325#126C23×C83148yes—
326#126C23×C83196yes—
327#125C23×C83124yes—
328#125C23×C83148yes—
329#125C23×C83196yes—
330#125C23×C83148yes—
331#124C23×C83124yes—
332#123C23×C83148yes—
333C22⋊Q8C23×C83124yes—
334C22⋊Q8C23×C83148yes—
335C22⋊Q8C23×C83148yes—
336C22⋊Q8C23×C83148yes—
337#121C23×C8318yes—
338C4×Q8C23×C8318yes—
339C4×Q8C23×C83124yes—
340C4×Q8C23×C8318yes—
341C4×Q8C23×C83124yes—
342C4×Q8C23×C83148yes—
343C4×Q8C23×C83116yes—
344C4×Q8C23×C83148yes—
345C4×Q8C23×C83124yes—
346C4×Q8C23×C83148yes—
347C4×D4C23×C83148yes—
348C4×D4C23×C83124yes—
349#117C23×C83148yes—
350#117C23×C83148yes—
351#117C23×C83148yes—
352C2×C4⋊C4C23×C8313yes—
353C2×C4⋊C4C23×C8313yes—
354C2×C4⋊C4C23×C8316yes—
355C2×C4⋊C4C23×C83124yes—
356C2×C4⋊C4C23×C8316yes—
357C2×C4⋊C4C23×C8316yes—
358C2×C4⋊C4C23×C8316yes—
359C2×C4⋊C4C23×C8316yes—
360C2×C4⋊C4C23×C8316yes—
361C2×C4⋊C4C23×C8313yes—
362C2×C4⋊C4C23×C8316yes—
363C2×C4⋊C4C23×C8311yescharacteristic
364C2×C4⋊C4C23×C83112yes—
365C2×C4⋊C4C23×C8316yes—
366C2×C22⋊C4C23×C8312yes—
367C2×C22⋊C4C23×C8316yes—
368C2×C22⋊C4C23×C8316yes—
369C2×C22⋊C4C23×C8316yes—
370C2×C22⋊C4C23×C8316yes—
371C2×C42C23×C8313yes—
372C2×C42C23×C83112yes—
373C2×C42C23×C8316yes—
374C2×C42C23×C8313yes—
375C2×C42C23×C8313yes—
376C2×C42C23×C8313yes—
377C2×C42C23×C83124yes—
378C24C24×C8312yes—
379C24C24×C8311yescharacteristic
380C2×Q8—232no—
381C2×Q8—264no—
382C2×Q8—212no—
383C2×Q8—224no—
384C2×Q8—212no—
385C2×Q8#41×C8318yes—
386C2×Q8C24×C83112yes—
387C2×Q8C24×C83112yes—
388C2×Q8C24×C83124yes—
389C2×Q8C24×C8314yes—
390C2×D4—28no—
391C2×D4—24no—
392C2×D4C24×C8318yes—
393C2×D4C24×C8314yes—
394C22×C4—212no—
395C22×C4—212no—
396C22×C4—224no—
397C22×C4—24no—
398C22×C4C24×C8311yescharacteristic
399C22×C4C24×C8313yes—
400C22×C4C24×C8314yes—
401C22×C4C24×C8316yes—
402C22×C4C24×C8312yes—
403C22×C4C24×C8313yes—
404C22×C4C24×C8316yes—
405C22×C4C24×C8313yes—
406C22×C4C24×C83112yes—
407C22×C4C24×C83112yes—
408C22×C4C24×C83124yes—
409C22×C4C24×C8313yes—
410C22×C4C24×C8311yescharacteristic
411C4⋊C4C24×C83112yes—
412C4⋊C4C24×C83112yes—
413C4⋊C4C24×C83124yes—
414C4⋊C4C24×C83124yes—
415C4⋊C4C24×C83124yes—
416C4⋊C4C24×C83124yes—
417C4⋊C4C24×C83124yes—
418C4⋊C4C24×C83124yes—
419C4⋊C4C24×C83112yes—
420C4⋊C4C24×C83124yes—
421C4⋊C4C24×C8314yes—
422C4⋊C4C24×C83148yes—
423C4⋊C4C24×C83124yes—
424C22⋊C4C24×C8318yes—
425C22⋊C4C24×C83124yes—
426C22⋊C4C24×C83124yes—
427C22⋊C4C24×C83124yes—
428C22⋊C4C24×C83124yes—
429C42C24×C83112yes—
430C42C24×C83112yes—
431C42C24×C83112yes—
432C42C24×C83112yes—
433C42C24×C83148yes—
434C42C24×C83124yes—
435C23—22no—
436C23—22no—
437C23—24no—
438C23—24no—
439C23—24no—
440C232−1+4×C8314yes—
441C23C25×C8314yes—
442C23C25×C8312yes—
443C23C25×C8311yescharacteristic
444Q8—224no—
445Q8—248no—
446Q8—224no—
447Q8C2×#41×C83116yes—
448D4—216no—
449D4—28no—
450C2×C4—26no—
451C2×C4—212no—
452C2×C4—224no—
453C2×C4—26no—
454C2×C4—26no—
455C2×C4—22no—
456C2×C4—24no—
457C2×C4—212no—
458C2×C4—224no—
459C2×C4—248no—
460C2×C4—28no—
461C2×C4C2×#41×C83112yes—
462C2×C42−1+4×C8314yes—
463C2×C4C25×C8312yes—
464C2×C4C25×C8316yes—
465C2×C4C25×C83112yes—
466C2×C4C25×C8314yes—
467C2×C4C25×C8316yes—
468C2×C4C25×C8316yes—
469C2×C4C25×C83112yes—
470C2×C4C25×C8316yes—
471C2×C4C25×C8312yes—
472C22—22no—
473C22—22no—
474C22—28no—
475C22—24no—
476C22—24no—
477C22—24no—
478C22—48no—
479C22C22×#41×C8312yes—
480C22C2×2−1+4×C8314yes—
481C22C2×2−1+4×C8312yes—
482C22C2×2−1+4×C8312yes—
483C22C26×C8311yescharacteristic
484C4—26no—
485C4—212no—
486C4—26no—
487C4—26no—
488C4—22no—
489C4—24no—
490C4—212no—
491C4C22×#41×C83112yes—
492C4C2×2−1+4×C8314yes—
493C2—24no—
494C2—44no—
495C2C83×#352314yesdirect product factor
496C2C23×#41×C8311yescharacteristic
497C2C22×2−1+4×C8311yescharacteristic
498C2C22×2−1+4×C8311yescharacteristic
subgroupG/H
center, socleC23×C83C25
commutator, Frattini, cosocleC22C26×C83
prime radicalC2×#121×C83C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C83 ◁ G

Complements

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

NH ≅ G/Ntype
1. C83×#3523C2direct (N × H)
2. C2×#557×C83C2semidirect
3. C2×#557×C83C2semidirect
4. C2×#556×C83C2semidirect
5. C2×#552×C83C2semidirect
6. C2×D4×Q8×C83C2semidirect
7. C2×D4×Q8×C83C2semidirect
8. C2×#541×C83C2semidirect
9. C2×#533×C83C2semidirect
11. C2×#519×C83C2semidirect
12. C22×C4×Q8×C83C2semidirect
15. #557×C83C22semidirect
16. #557×C83C22semidirect
17. #556×C83C22semidirect
18. #552×C83C22semidirect
19. D4×Q8×C83C22semidirect
20. D4×Q8×C83C22semidirect
21. #541×C83C22semidirect
22. #533×C83C22semidirect
40. #519×C83C22semidirect
50. C2×C4×Q8×C83C22semidirect
243. C2×#3523C83direct (N × H)
246. C2×C83#3523direct (N × H)
250. #3523C2×C83direct (N × H)
251. C2×#557C2×C83semidirect
252. C2×#557C2×C83semidirect
253. C2×#556C2×C83semidirect
254. C2×#552C2×C83semidirect
255. C2×D4×Q8C2×C83semidirect
256. C2×D4×Q8C2×C83semidirect
257. C2×#541C2×C83semidirect
258. C2×#533C2×C83semidirect
260. C2×#519C2×C83semidirect
261. C22×C4×Q8C2×C83semidirect
262. C83C2×#3523direct (N × H)
265. #557C22×C83semidirect
266. #557C22×C83semidirect
267. #556C22×C83semidirect
268. #552C22×C83semidirect
269. D4×Q8C22×C83semidirect
270. D4×Q8C22×C83semidirect
271. #541C22×C83semidirect
272. #533C22×C83semidirect
290. #519C22×C83semidirect
300. C2×C4×Q8C22×C83semidirect
495. C2C83×#3523direct (N × H)

Maximal subgroups

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

subgroupindex
C83×#35232
C2×#557×C832
C2×#556×C832
C2×#552×C832
C2×D4×Q8×C832
C2×#541×C832
C2×#533×C832
C2×#529×C832
C2×#519×C832
C22×C4×Q8×C832
C2×#352383